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Bioengineering Analysis of Orthodontic Mechanics

Analysis of Orthodontic
Bloengi neering
Analysis of Orthodontic
Professor, Department of Orthodontics
St. Louis University Medical Center
St. Louis, Missouri
600 South Washington Square
Philadelphia, PA 191064198
Library of Congress Cataloging in Publication Data
Nikolai, Robert J.
Bioengineering analysis of orthodontic mechanics
Bibliography: p.
Includes index.
1. Orthodontic appliances. 2. Force and energy.
3. Bioengineering. I. Title. [DNLM: 1. Biomedical
engineering. 2. Orthodontic appliances. 3. Orthodontics.
WU 400 N693b]
RK527.N54 1984
ISBN 0-8121-0922-8
Copyright © 1985 by Lea & Febiger. Copyright under the Inter-
national Copyright Union. All Rights Reserved. This book is
protected by copyright. No part of it may be reproduced in any
manner or by any means without written permission of the
Prioted in the United States of America
Print Number:
To My Parents
The advancement of orthodontics and the quality of orthodontic treat-
ment are dependent upon a thorough knowledge of basic science. Although there are many orthodontic texts that cover the biologic basic
sciences, such as growth and development, neuromuscular physiology,
genetics, and anatomy, considerably less attention has been paid to bioengineering. Certainly no basic science could be more relevant to the
orthodontic researcher and to the practicing orthodontist than bioengineering. Dr. Nikolai has written the first comprehensive text that can
serve as a keystone to the understanding of bioengineering principles in
general and their application to clinical orthodontics.
This text develops the fundamental concepts in an orderly way
which will allow the orthodontist to understand and use his orthodontic
appliances. Since most dentists in their predoctoral training receive little
information relative to bioengineering, the development of basic concepts and their application is most important and necessary before moving on to more complicated clinical situations. The sequential presentation of material from the elementary principles to the more complicated
force systems of multibanded appliances will allow both student and
practitioner to gain a better understanding of how orthodontic appliances work and how to improve clinical therapy based on this understanding.
Bioengineering has a broad application to orthodontics and, hence,
the scope of the text is very wide. It includes a discussion of the force
systems produced from orthodontic appliances, the behavior of the materials that are used, and correlates the relationship between the torce
systems and the biologic changes that occur in the dentition and periodontium. The excellent use of clinical problems and examples through-
out the text insures that this is not just another engineering text, but
truly a book on orthodontic bioengineering, which can serve both the
student and the experienced practitioner.
Professor and Head
Department of Orthodontics
School of Dental Medicine
The University of Connecticut
As viewed by the observant bioengineer, the orthodontic appliance is a
unique assembly of small, structural elements. When activated, the appliance exerts a system of forces upon the human dentition. These forces
cause, or at least serve to catalyze, biologic remodeling within the dentofacial complex toward achievement of finite, permanent displacements of teeth.
From both structural and clinical perspectives, the principal component of the fixed, intraoral appliance is the arch wire. This wire is currently available to the clinician in at least five materials (metal alloys),
round and rectangular cross-sections in various sizes, single- and mul-
tistrand formats, and in straight lengths or precontoured into arch
forms. Prior to engagement and activation the practitioner may bend or
twist the wire, using special tools to place or modify the arch form and
detail the wire geometry to the specific patient. The appliance contacts
the dentition through crown-affixed brackets and tubes, and the structural assembly may also include various kinds of metallic springs and
polymeric elements to exert forces. Moreover, extraoral adjuncts may
contribute to the total appliance.
The complexity of the orthodontic appliance as a structure is extended through its dynamic behavior and that of its foundation, the
dentition; over time the appliance deactivates slowly in concert with
movements within the biologic complex to which it is attached. The
overall analysis requires an integrated examination of the appliance and
the force system it delivers to the dentition. The characteristics of that
delivered force system influence the format of the potential displacements of the teeth.
The interested bioengineer now envisions a multifaceted analysis
problem. The experienced practitioner, as well as the clinician-in-training, perhaps viewing structural analysis as "foreign territory," should
gain some familiarity with the engineering of the orthodontic appliance,
and this text has been prepared to assist in that education. Herein the
subjects of force and structural analysis are discussed within the framework of practical, orthodontic treatment and at a level appropriate to the
intended audience. The presentation is seemingly theoretical for the
most part, although much of the theory has been demonstrated to be
well founded through bench and clinical experimentation. The topic
coverage is largely a composite of analytic and graphical mechanics,
materials science, and mechanics of deformable bodies (subject titles
familiar to the engineering student), but presented with minimal mathematical involvement and with direct application to orthodontics. Substantial attention is given to the interactions between the activated appliance and the biologic systems and processes affected by the transmitted
This text is an outgrowth of class notes prepared and carefully revised, over a period in excess of ten years, for a bioengineering course
presented to first-year students in an orthodontics graduate program.
Not only intended for students, the book should serve the interested,
experienced practitioner as well, whether or not previously exposed to
formalized graduate or continuing-education coursework in orthodontic biomechanics. Because examples are used throughout, the text as-
sumes at least superficial familiarity with orthodontic therapy that
would be gained, for example, through an introductory or survey exposure within the professional dental curriculum.
Principles and procedures accumulate from the start in the book, but
no preparation in mathematics and science, beyond traditional predental and dental curricula, are required. The text begins with a discussion
of needed mathematical "tools." The subsequent three chapters introduce concepts of engineering mechanics in the orthodontic setting. Con-
sidered next is the response to applied force of the dentition and its
supporting tissues. Structural analysis procedures are then introduced
and applied to simple appliance elements. Chapters are devoted, individually, to bending and torsional analyses of arch wires and to extraoral
appliances. Finally, force and structural analyses of entire representative
orthodontic mechanics are undertaken.
Orthodontic bioengineering is becoming widely recognized as an
important area of formal study for the aspiring clinician. The design and
preparation of appliances have been traditionally learned from preceptors, and refined through actual clinical experience. Recent advances in
appliances and techniques, however, have resulted from the joint contributions of practitioner expertise and engineering science. Although
the latter has demonstrated its potential impact toward overall advance-
ment of orthodontic treatment, the organized application of bioengineering to orthodontics remains relatively obscure, apparently, to the
majority of practitioners. Accordingly, the objective in preparing this
text was to provide a means whereby a bioengineering approach to the
analysis and design of orthodontic appliances and procedures might be
outlined to the orthodontic community.
Several individuals have intangibly, although importantly, contributed to the completion of this work. Vasil Vasileff, now retired from the
School of Dental Medicine at Southern Illinois University, and formerly
a faculty member at the St. Louis University Medical Center, presented
the author with his first opportunity to interact with graduate orthodontics students. Kenneth Marshall, former Chairman of the Depart-
ment of Orthodontics, hired and charged the author, trained in engi-
neering rather than in dentistry, to develop the bioengineering course
within the graduate curriculum. Lysle Johnston, present Chairman of
the Department, through his urging and encouragement, was instrumental in making this text, begun in earnest during a sabbatical leave, a
The author would be remiss without acknowledging the influence of
the students in the graduate orthodontics program at St. Louis University, particularly those whose thesis research projects he has supervised;
their comments, criticisms, and suggestions were of substantial aid in
this effort. Finally, to Charles Burstone, eminent researcher, clinician,
and educator, who kindly consented to contribute the foreword to this
text, the author expresses his sincere gratitude.
St. Louis, Missouri
1. Mathematical Topics
Constants, variables, and functions 1
Frames of reference 3
Displacements of particles and solid bodies
An introduction to vector algebra 16
Dimensions and units 20
Measurements, computations, and numerical accuracy
2. Introduction to Analysis of Orthodontic Force
Mechanics 24
Force 25
Vector addition and decomposition of concentrated forces 29
The moment of a concentrated force 36
The couple 40
Resultants of force systems exerted on rigid bodies 46
Distributed forces and their resultants 50
Concepts of mechanical equilibrium and their applicability to
orthodontic mechanics 56
3. Material Behavior of the Orthodontic Appliance
Internal structure of a solid material 73
Load-deformation behavior of a structural member 78
Mechanical stress
Mechanical and structural properties: Standardized testing
Chemical and thermal influences 103
Selection of materials 110
4. Energy Analyses in Orthodontics
Concepts leading to the process laws
Displacement 114
Mechanical work 115
Energy 117
Heat transfer and thermal energy 121
The conservation-of-energy law 122
Available energy 124
Activation and deactivation processes 125
Strain hardening and heat treatment of metals and alloys
A work-energy analysis of the preparation of an arch wire
Synopsis 144
5. Response of Dentition and Periodontium to Force
Mechanical response of the individual tooth to applied
Transverse crown force systems and tooth displacements
Extrusion, intrusion, and long-axis rotation 163
Response of the periodontium to force 169
Displacements related to magnitude and duration of force
Controlling the force-time pattern 178
Physiologically proper orthodontic forces 182
Dentofacial Orthopedics 189
Synopsis 191
6. Introduction to Structural Analysis of the Orthodontic
orthodontic appliance: A structure or a machine? 196
Attachment of the orthodontic appliance to the dentition 197
A continuous-arch-wire appliance model 205
An overview of the structural analysis of an orthodontic
appliance 208
The activation and deactivation characteristics of representative
appliance elements 211
The action and response of tip-back bends: An illustration of the
four-step procedure in orthodontic structural analysis 224
"Control" of the orthodontic appliance 228
Synopsis 231
7. Behavior of Orthodontic Wire in Bending
deformations, strains, and stresses 235
Force systems within the beam 241
Beam stiffnesses 248
Additional topics in elastic bending 252
Application of elastic beam theory to orthodontic arch-wire
activation in bending 257
Inelastic behavior in bending 261
Orthodontic wire loops 264
8. Delivery of Torque by the Orthodontic Appliance
Structural theory for the straight, circular shaft
Extension of theory to shafts having rectangular
cross-sections 285
Application of shaft theory to the orthodontic arch wire 288
Isolation of the activating torque 288
Responses of the appliance and the dentition to torsional
activation 289
Structural influences on active and responsive force
systems 295
Anterior-segment torquing mechanics 299
Rectangular-wire torquing 299
Torquing spurs in the appliance 305
Inelastic behavior in third-order mechanics 310
Wire loops, springs, and torsion 314
Synopsis 318
9. Extraoral Appliances
The cervical-pull, face-bow appliance
Occiusal-plane analysis 326
Buccal-view analysis of the cervical appliance
A coronal-plane view and comments on the asymmetric
problem 343
Canine retraction with headgear 344
Extraoral force delivered to an anterior segment or an entire
Delivery of extraoral force to the mandibular arch 358
Dual-force headgear 361
The chin-cap assembly 362
Reverse-pull appliances 365
Synopsis 369
10. Force and Structural Analyses of Representative
Orthodontic Mechanics
Individual tooth malalignments
Leveling displacements 374
Rotational corrections
Bilateral action 385
Interarch mechanics 388
Intra-arch vertical positioning 392
Intra-arch retraction mechanics 402
Prepared, posterior-segment anchorage 412
Class III mechanics 417
Class II mechanics 424
Synopsis 435
Appendix I Glossary of Terms
Appendix II List of Symbols
Orthodontic bioengineering has emerged as an important subfield of
orthodontics, not only for research but for formal study by the practitioner-in-training as well as the experienced clinician. Bioengineering is
appropriately termed an "applied science" and, as such, uses numerous
mathematical concepts and procedures. This chapter introduces and
explores the mathematical topics required in this text. These developments are not unduly rigorous and most will not be totally new to the
reader. Familiarization with the necessary mathematics is undertaken
initially in this text, rather than to introduce each topic individually as
the need arises. As necessary, the reader may refer back to this chapter
for a review of any specific concept. Moreover, a foundation is laid for
uninterrupted and continuous study of the principles and applications
of bioengineering to orthodontics.
Constants, Variables, and Functions
Any object having physical attributes or characteristics may be described
qualitatively or quantitatively. For example, a specific orthodontic wire
may be characterized as being shiny, made of stainless steel, round, and
having a cross-sectional diameter equal to .018 in. The first three de-
scriptive words or phrases refer to qualitative attributes; the last is a
quantitative characteristic. Descriptions, whether qualitative or quantitative, are helpful only insofar as their references or bases are familiar
and understood. In describing the orthodontic wire, the adjectives were
ordered from the least to the most definitive: "shiny" implies a comparison with some herein undefined luster or smoothness of surface; several
types of stainless steel, differing from one another metallurgically, are
used in fabricating orthodontic appliances; and "round" is meant to
describe the shape of the wire cross-section (but, alternatively, it might
refer to a curvature placed in the wire). Finally, the reference to the
"diameter" of the cross-section with the measurement given in inches is
Bioengineering Analysis of Orthodontic Mechanics
the most definite because the dimension is well understood and the inch
is an established unit of length. This example suggests that descriptions
in quantifiable terms are generally clearer because of familiarity with the
bases of those terms. A particular stainless steel may be described in
terms of the constituent materials of the alloy, given in percent by
weight or by volume, or implicitly, for example, by the American Iron
and Steel Institute grade-and-type designation. (Given the AISI code for
the alloy, its composition may be learned by consulting an appropriate
metals handbook.)
Because many objective descriptions undergo some form of change
in the most general discussion, in practical considerations some limits or
bounds on the extent of change are often defined or implied. The magnitude of a force, exerted by an appliance on a tooth, may lessen over a
period of days as the tooth moves and the appliance partially or totally
deactivates. In a structural analysis undertaken to be valid for only a
short time, however, the force may be considered as unchanging. The
temperature in the oral cavity may vary in a complex fashion and may be
affected temporarily by ingesting hot or cold food or beverages. Often,
however, it is sufficient to consider oral temperature as unchanging, but
elevated some 30°F relative to normal room temperature. (In a study of
the characteristics of a metallic arch-wire material, for example, even the
fact that intraoral temperature is elevated with respect to the ambient air
may be inconsequential.) A constant, then, is defined as a quantity that
retains a fixed, unchanging value or level within the bounds of a particular discussion or investigation.
A quantity that cannot be regarded as constant while under study,
either because its variation is the subject of the study or the contribution
of its changing is not negligible in overall considerations, may experience change associated with movement from one location to another, or
directly or indirectly as time passes. Location or position may be related
to time. The force in a helically-coiled spring may be altered by changing
the position of one end of the spring with respect to the other. The
position or orientation of a malposed tooth under the influence of an
activated orthodontic appliance may change with time. A variable, is, in
general description, a definable characteristic or parameter under study
that takes on more than one quantifiable value during the course of a
discussion or investigation. Variables may be categorized as independent
or dependent. Although this distinction may often depend on viewpoint,
a few variables (e.g., time and temperature) are generally independent,
and other parameters assume their values in some manner associated
with the magnitude(s) of the independent variable(s). In the analysis of
the helical spring, the force in the spring may be said to be dependent on
the amount of deformation referenced from the passive (zero-appliedforce) configuration; alternatively, the extent of the deformation may be
viewed as dependent on the size of the force applied to the spring. In
the example of tooth location versus time, however, the parameter
"time" can never be realistically considered as dependent on the orientation or position of a malposed tooth.
Mathematical Topics
Whenever two or more variables are involved in a common discussion or investigation, and whenever these variables are interrelated such
that one or more unique values of one variable are assumed upon the
assignment of specific values to the remaining variables, the first is dependent on, or said to be a function of, the other (independent) variables.
A functional relationship is usually expressed explicitly in the depend-
ent variable. For example, if the symbol, y represents a quantity that
takes on a particular value upon the assignment of a value to x, the
general relationship is written as y = function (x), or simply y = f(x). If
the "data" relating y to x originate from an experiment, the function f(x)
is subsequently sought toward the facility of determining all values of y
within some range or bounds of the independent variable x.
The variation in a parametric value might be expressed mathematically in a smooth, continuous manner, and an analytic relationship, an
equation, may be written to represent it. This representation can always
be set down in a graphical or tabular form, but there are times when a
variable takes on individual, discrete, seemingly unrelated values and
an equation cannot be derived easily, if at all. The graph or table is then
necessary to visualize a pattern in and the range of the variable values.
In this text, the typical equation is simple in form and often derivable
from the graphical representation of the relationship among the variables.
Frames of Reference
When expressing graphically the change in value of a variable, the values and change may be associated with the length of, or distance along,
a reference line or axis. Many variables have the possibility of assuming
the value of zero, and this or another convenient reference point or
origin is located on the axis. To associate length or distance with the
variable values, a scale is needed. In Figure 1-1, an axis is established for
the parameter "time." The origin of the plot coincides with a relative
time of zero (hours). The chosen scale assigned one hour to a specific
interval, and twelve adjacent intervals "equals" one-half day. In this
particular representation, amounts of time elapsed or time change are of
interest and negative time readings have no real meaning. Hence, since
time is not an algebraic quantity, there is no need to extend the reference
1 hr
FIGuRE 1-1. A one-dimensional plot relating time as a variable, in hours, to distance
along an axis.
Bloengineering Analysis of Orthodontic Mechanics
line in both directions from the origin, which has, therefore, been placed
at the left end of the axis with time increasing to the right.
The graphical relationship of one dependent variable to another, or
possibly to two other independent variables, is usually established
within a two-dimensional, rectangular-coordinates framework. A pair of
reference axes are defined as the basis for the graph; the axes are perpendicular to one another with the origin of each the intersection point,
the now-defined origin of the framework as a whole. Ordinarily, one
axis is positioned horizontally and the other vertically. When the variables may only assume positive values, the origin is placed at the lowerleft so that variable values associated with the horizontal axis will increase from left to right and values associated with the vertical axis will
increase upward. As with the one-dimensional "plot" of Figure 1-1, a
variable name (e.g., force) and units of measurement (e.g., ounces) are
appended to each axis. A scale is implied by the distance associated with
an individual unit division along each axis.
When preparing a graph of the relationship between two variables,
symbolized, for example, by x and y (without specifying the units of
measurement of either), points would be individually plotted in the x-y
plane and each point would be designated by a number pair, or coordinates, (x,y). When plotting a two-dimensional relationship with respect
to a set of horizontal and vertical axes, the x-coordinate of a point is
often termed the abscissa and the y-coordinate the ordinate. When all
points have been plotted, a curve is sketched as a "best fit" through all
of the plotted points. An attempt may then be made to obtain an analytic
expression that relates x and y at or near to every point on the drawn
curve. Figure 1-2 shows a straight-line relationship that is defined as to
position and orientation by just two (x,y) coordinate pairs. A nonlinear
relationship is also displayed that may necessitate the plotting of many
points prior to sketching the curve to obtain an accurate representation.
The scales chosen for the axes need not be identical, regardless of
whether or not similar units of measurement are appropriate to the variables symbolized by x and y. The straight line in Figure 1-2 was drawn
after the coordinates (1,2) and (2,4) were plotted. If the lower bound for
x and y is zero, the plot appropriately begins at the origin of this x-y
frame (0,0). The equation of the line is easily determined to be y = 2x.
The curve sketched with respect to the same pair of coordinate axes
suggests a functional relationship between x and y that is quite different
from that of the straight line. Although the numerical relationship of x to
y can be readily obtained for any specific point on the curve, the equation relating the two variables at every point on the curve is not easily
derived from the plotted points.
The two plots of Figure 1-2 may be unrelated to one another or may
be "members" of a family of curves. If the latter is the case, a third
variable is involved and, for each value of interest of that third variable,
an individual relationship exists between x and y. Plotting a family of
curves in the plane is a method of visually describing the relationship
among three variables within a two-dimensional reference frame. Gen-
URE 1-2. Relationships between pairs of variables displayed, referenced to an
coordinate framework.
when plotting one curve or a family of curves with respect to a
me framework consisting of horizontal and vertical axes, the dependI variable, expressed graphically as a function of one or two independ-
I variables, is scaled on the vertical axis.
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 1-3. Relationships of an individual dependent variable to pairs of independent
variables, sketched within an x-y-z-coordinate framework.
From a two-dimensional plot, in addition to enabling the determinalion of one or more values of the dependent variable for a given value of
the independent variable(s), the slope of the curve at one or more specific
points is often useful. This slope represents the rate of change of the
dependent variable with respect to a unit increase in the scaled dependent variable and is, geometrically, the inclination of the tangent to the
curve at the specified point. The slope at a point may be positive or
negative, depending upon whether the dependent variable is increasing
or decreasing in value from that point. If the plot of the relationship
between two variables is a straight line, the slope is the same at all
points. Along the nonlinear curve in Figure 1-2, the slope decreases to
x = 1, then increases with increasing values of x.
The rectangular-coordinate framework in two dimensions may be
extended to three with the addition of a third axis, mutually perpendicu-
lar to both axes in the plane and intersecting the origin of the plane
frame. In the coordinate system in space, a functional relationship
among three or possibly four variables may be displayed. Figure 1-3
shows a curve and a surface referenced to an x-y-z framework. Both the
curve and the surface might represent functions in which z symbolizes
the dependent variable and x and y the independent variables. The surface may serve to display a function for all values of x and y within some
bounds; the curve, drawn in the surface, may graphically provide the
functional relationship for only those values of the independent variables that are of particular interest, A function of three independent variables may be displayed in the spatial framework as a family of curves or
Mathematical Topics
The concept of slope can be extended to three dimensions.
Along a surface the rate of change of z, for example, the dependent
variable, may be considered at a point of the surface independently with
respect to unit changes in x and y or with respect to a coordinate in the
direction of a curve in the surface. A unique tangent plane is associated
with each point on a surface, and a slope may be computed for any
specified direction in that plane.
The interpretation of a functional relationship displayed in a threedimensional reference framework is difficult, particularly when
sketched in two dimensions. Although the pictorial representation of
analytic functions in three dimensions is generally beyond the needs of
this text, coordinate systems have other uses, and an important one is
the designation of reference directions in a plane or in space. In the
study of dental anatomy, a coordinate frame is often placed in an individual tooth; one axis coincides with the long axis of the dental unit.
Mutually perpendicular directions associated with this framework are
occlusogingival (or occlusoapical), faciolingual (or labiolingual or buccolingual), and mesiodistal. Such a reference frame is shown in Figure 1-4.
Note that these axes extend on both sides of the origin of the framework;
rather than adopting an algebraic scheme, a sense is specified (and the
direction is implied) in discussions. For example, as indicated in Figure
1-4, "distal" indicates a specific sense in the mesiodistal direction. In a
FIGURE 1-4. Mutually perpendicular coordinate directions used in dentistry.
Bioengineering Analysis of Orthodontic Mechanics
framework associated with the entire dental arch or dentition, the refer-
ence directions are generally labeled anteroposterior, lateral or transverse, and vertical.
A plane of reference is generally associated directly with the coordinate perpendicular to the plane, often with the additional indication of
the "side" of the plane being viewed. For example, the occiusal plane is
perpendicular to the occlusogingival direction with the view toward the
tooth crowns. A buccal plane is perpendicular to a buccolingual direction; an identical lingual plane exists, but the implied view is toward the
lingual surface(s). The faciolingual and mesiodistal planes are perpendicular to one another, references associated with an individual dental
unit, and the line common to both planes is coincident with the long axis
of the tooth. Note that the common terminology for the planes of reference for the dentition, or each arch, as a whole deviates from this pattern; the coronal, the occlusal (or transverse), and the sagittal planes are
the mutually perpendicular references, and two of the associated, perpendicular axes have designations unlike their counterpart planes.
Displacements of Particles and Solid Bodies
The distance between two points in space (e.g., A and B) is measured in
length units along a straight line connecting the two points. The displacement "path" of a particle moving from point A to point B, however, need not be along that line. The actual distance traveled by the
particle in moving from A to B is measured along the path followed, and
may be sizable compared to the straight-line measurement. The displacement of a particle from point A is in magnitude the distance from A
to a specific point on the path, and is dependent upon the elapsed time
since its coincidence with point A. Displacements of orthodontic interest
are generally small in absolute measurement, but are comparable to
characteristic intraoral dimensions. Such displacements may be difficult
to obtain accurately because (1) often few, if any, potential references
with the dentofacial complex remain stationary over time, (2) position
measurements cannot be monitored continuously, and (3) total move-
ments may be an inseparable combination of orthodontic and orthopedic displacements and growth.
Although full coverage of this subject is properly relegated to the
study of cephalometrics, several portions of the topic fit into the discussions of this chapter. First, any displacement to be quantified, must be
measured with respect to a reference frame. The displacement path of a
particle must often be estimated by connecting straight-line distances
between successive positions of the particle; the larger the number of
position measurements, the more accurate the estimation. The displacement of a solid body as a whole is fundamentally described in terms of
the displacement of one particle of the body and the angular movement
of a line within the body passing through this particle.
Mathematical Topics
arc D"D'
FIGURE 1-5. Successive positions of a tooth undergoing planar orthodontic
displacement, including a change in angular orientation.
Figure 1-5 depicts the displacement of a tooth as well as the displace-
ments of two particles (or points) of the tooth; in this example, the
points and movements are planar. The displacement of point D may be
expressed in terms of the displacement of point C and the relative movement of D with respect to C. The angular displacement of the tooth is the
difference in orientations, 02 — 01, of the line through points C and D. In
fact, all lines of the tooth in the plane of the sketch, or parallel to it,
experience the same angular displacement. Because the distance between points C and D in the tooth does not change during the movement, the displacement path of the particle at D with respect to, or as
viewed from, point C is a circular arc. The radius of that circular arc is
the line segment in the tooth from C to D, the change in angular position
of the segment CD represents the angular displacement of the tooth as a
whole, and the relative curvilinear displacement of the particle at D with
respect to point C is along a circular path, the arc D'D', and equals the
product of the length of the segment CD and the angular change, 02
01. In
this computation the angular change must be expressed in radi-
ans, where 22/7 radians is comparable to 180°. The units of the length CD
and the relative displacement DD' are then identical.
The completion of the determination of the "absolute" displacement
of the particle coinciding with point D must be deferred until later in this
Bioerigineering Analysis of Orthodontic Mechanics
chapter because this displacement is, mathematically, a vector quantity.
This section has, however, introduced or reviewed concepts associated
with particle and angular whole-body displacements and recalled the
relationship among the size of the angle, the radius, and the length of
the circular arc for a circular segment.
As noted in the preceding section, an angle is the measurement of the
orientation of one line with respect to another in the plane containing
the lines. The actual quantification implies a circle centered at the point
of intersection of the lines; the curvilinear length of the circular arc between the lines divided by the radius of the circle yields the size of the
angle in radians. The example of Figure 1-6 shows an angle of 22/21
radians, or 600. (Although the dimensionless units of degrees are more
familiar than radians, many computations to obtain an angle as the result yield the answer in radians.) Angles are often categorized by size:
acute angles are smaller than 90°, right angles are exactly 90°, obtuse
angles are larger than 90° but less than 1800; and straight angles are
exactly 1800. Although circles are described as encompassing 360°, the
smaller interior angle between two radii never exceeds 180°. The sum of
acute and adjacent obtuse angles within the intersection of two lines is
180°; these angles are termed "supplementary" to one another.
A triangle is a plane, three-sided, closed figure made up of three line
segments. The intersection of each pair of line segments defines an
QA = OB =
AB = s
600 = 22/21 radians
s = (22/21)r
FIGURE 1-6. Angular measurement.
Mathematical Topics
angle. Each angle is measured interior to the figure and the three angular measurements always sum to 1800 or 22/7 ("pi") radians. Also, for
every triangle, the ratio of the length of a side to a function of the angle
opposite that side is a constant. That function is defined as the "sine" of
the angle; it is a dimensionless quantity that has bounds for angles between 0° and 180° of 0 and +1.
Triangles are categorized according to the size of the largest interior
angle. A right triangle contains, opposite to the longest side termed the
"hypotenuse," a right angle. The other two angles, necessarily acute,
must sum to 90°; such angles are said to be "complementary" to one
another. A right triangle is pictured in Figure 1-7. The following trigonometric functions are defined for an acute angle within a right triangle:
sine A =
cosine A =
tangent A =
"Opposite" and "adjacent" refer to the positions of sides of the triangle
with respect to angle A and the ratios are of side lengths. Similar relationships may be written explicitly for angle B. The defined trigonometric functions are generally abbreviated "sin," "cos," and "tan." Table 1-1
contains trigonometric functional values and conversions between degrees and radians for angles between 0° and 90°.
Three additional trigonometric functions—the cosecant (csc), the
FIGURE 1-7. A right triangle with angles and sides labeled symbolically.
Bioengineering Analysis of Orthodontic Mechanics
TABLE 1-1. Degrees-radians equivalents and trigonometric functions
0 848
Mathematical Topics
(sec), and the cotangent (cot or ctn)—are defined as the reciprocals of the sine, cosine, and tangent, respectively. The lengths of the
sides of a right triangle are interrelated through the Pythagorean theorem;
for the triangle of Figure 1-7 the relationship is
= a2 + b2
or, in words, the length of the hypotenuse equals the square root of the
sum of the squares of the other two sides. Also, from the foregoing
definitions and because A and B are mutually complementary:
sin A = cos
cot A = tan
tan A = cot
csc A = sec
cos B = sin A
sec A = csc B
An oblique triangle, incorporating no right angle, is sketched in Figure 1-8. The law of sines for this triangle is
Another relationship, valid for the oblique triangle, also relating sides
and angles, is the law of cosines:
d2 =
e2 =f2+
2ef(cos D)
—2fd(cos E)
2de(cos F)
f2 =
+ f2
Although the oblique triangle of Figure 1-8 is specifically an obtuse triangle, these two laws may also be used in analyzing an acute triangle.
FIGURE 1-8. An oblique triangle with angles and sides labeled symbolically.
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 1-9. A line segment referred to an x-y-coordinate framework.
For the right triangle, the sine and cosine laws are replaced by the defini-
tions of the sine and cosine functions and the Pythagorean theorem.
Figure 1-9 shows a line drawn in the plane of the page with the origin
of an x-y-coordinate framework coinciding with one end of the line segment. Because the reference frame is rectangular, the line makes complementary angles with the x- and y-axes. These angles, labeled and
are called the direction angles of the line OL. The "shadow" or projection of the line segment on the x-axis is the apparent length of the line
seen from a vantage point far out on the y-axis; that length equals OL cos
Similarly, the projection of OL on the y-axis is equal to OL cos The
cosines of the direction angles are termed direction cosines. Enclosing the
line OL in a rectangle, having adjacent sides coincident with the x- and
y-axes and making the segment OL a diagonal of the rectangle, and
noting the two equal right triangles formed (which share a common
hypotenuse), the following equations may be written:
(x-projection)2 + (p-projection)2 = 0L2
= 1
The development just completed using Figure 1-9 may be extended
to three dimensions. In Figure 1-10 the coordinate framework is again
appended to one end of the line segment (although, in general, the
segment may be remote with respect to the frame); the end points of the
segment are (0,0,0) and (XN,YN,ZN). The direction angles are defined as in
Figure 1-9, except there are three; for instance, in Figure 1-10 the angle 0,
Mathematical Topics
FIGURE 1-10. A line segment in space referred to an x-y-z-coordinate framework.
measured between the x-axis and the line segment and that measurement is made in the plane containing ON and the x-axis. The projections
of the segment ON on the coordinate axes are
y-projection = QN(cos
x-projection = ON(cos
z-projection = ON(cos
respectively, and the three projections are also related to the coordinates
of end points of the segment and its direction cosines:
y-projection =
z-projection = ZN — 0
An alternative two-step process may be used to obtain the projections.
First, for example, project the line on the y-axis and into the x-z plane.
The angle 4, measured between the segment ON and the x-z plane, is
complementary to
and the projection in the plane equals ON cos q5.
Second, decompose the x-z plane projection, OQ, into x- and z-axis projections using the complementary angles a and y, measured in the x-z
plane. The three projections may then be expressed as follows:
Bioengineering Analysis of Orthodontic Mechanics
x-projection = ON(cos 4)cos a
y-projection = ON(cos Os,,)
z-projection = ON(cos q5)cos y
The length of the line segment is related to the coordinate-axis projections through the three-dimensional form of the Pythagorean theorem:
(x-projection)2 + (y-projection)2 + (z-projection)2
Dividing through by the square of the segment length and noting the
resulting forms defining the direction cosines, the expression becomes
The comparable relationship in the two-dimensional situation, previously examined, is the complementary nature of the two direction angles. (No simple relationship among the three direction angles exists
Before closing this section, it is important to note that a three-dimensional geometry problem may always be decomposed into two or three
two-dimensional problems. The original problem is, seemingly, expanded from one into two or three, but the gain is in terms of each of the
two or three "projections" being in its own plane setting.
An Introduction to Vector Algebra
physical quantities associated with orthodontic bioengineering
possess characteristics in addition to magnitude. A most important entity, the point or concentrated force must be known in magnitude, in
direction (and sense), and in point of application to be fully described. A
multifaceted description is similarly associated with the displacement of
a particle or a point of a solid body. Both the point force and displacement obey the laws of vector mathematics.
A vector is a mathematical quantity possessing both magnitude and
direction. The equations describing vectors are unique, and vector quantities combine with scalars and with other vectors according to specific
mathematical laws and procedures. Vector quantities may be expressed
in both graphic and analytic formats. The vector is pictured as a directed
line segment. When drawn to scale, the length of the segment is proportional to the magnitude of the vector. The inclination or orientation of the
segment, the angle between it and a specified reference line or axis,
indicates the direction of the vector. The sense of the vector is denoted by
an arrowhead affixed to one end of the segment. Given all other characteristics of a vector, with only two possibilities for sense, it may be specified algebraically. Sense is often linked with direction; two vectors having identical angles with a reference axis, but differing in sense, are said
Mathematical Topics
be opposite in direction to one another. Without being specific as to
the physical quantity represented, a vector is shown in Figure 1-11 with
an x-y-coordinate frame included in the sketch that serves as a reference
for direction and sense. The displayed vector, indicated as having a
magnitude of 100 units with the directed line segment subdivided into 5
equal length units, implies a linear scale factor of 100 to 5 or 20 to 1.
Figure 1-11 is similar to Figure 1-9 except for the absence of the arrowhead in the latter sketch and the coordinate framework not attached
to an end of the line segment in the former. The angles labeled in Figure
1-11 are those which the line of the vector, if extended, would make with
the x- and y-axes; therefore, these angles are properly termed the direction angles of the vector. Individual multiplications of the vector length
by the direction cosines gives the x- and y-projections of the vector. Converting these (scalar) projections to vector quantities by including the
senses and multiplying each by the scale factor for the sketch yields the
x-, and y-components of the given vector. The lengths of the x- and
y-projections are indicated as 4 and 3 units; with a 20-to-i scale factor,
the x- and y-component magnitudes are 80 and 60, respectively. The
rectangular components of a vector are, by definition, mutually perpendicular and individually parallel to a coordinate axis. Hence, the sum of
the squares of the component magnitudes (or the lengths of their line-
segment representations) equals the square of the magnitude (or the
length) of the original vector.
The analytic expression for a vector is typically written in terms of its
component magnitudes (projections) and unit vectors (dimensionless
and of magnitude one). Vector quantities appear in boldface type in this
80 units
FIGURE 1-11. A vector positioned in the x-y plane.
Bioengineering Analysis of Orthodontic Mechanics
text to distinguish them from scalars. The unit vectors i and j, defined as
associated in direction and sense with the x- and y-directions, are shown
in Figure 1-11; note the positive senses corresponding to increasing coordinate values. The equation for the vector displayed may be written as
V = 80i + 60j
Shown in Figure 1-12 is a vector in space. The ends of the vector are
coincident with points located with respect to the chosen x-y-z-coordinate framework; the "tail" of the vector coincides with point D(3, 1,2)
and the vector symbol extends to point E(5,5,6). In this example, the
vector represents a 60-g force. A parallelepiped, oriented to the coordinate frame, "surrounds" the vector, which lies on a main diagonal. The
length DE of the force vector is found through the three-dimensional
form of the Pythagorean theorem:
DE = [(5
+ (5
1)2 + (6
2)21112 =
The scale factor for the sketch is, then, 60 g to 6 length units or 10 g to 1
length unit. The lengths of the sides DA (2 units), DB (4 units), and DC
(also 4 units) of the parallelepiped correspond to the projections of the
given vector, which are parallel to the x-, y-, and z-axes, respectively.
The direction angles
and of the force vector are the angles EDA,
EDB, and EDC, respectively. Similarities in and differences between Figures 1-10 and 1-12 should now be apparent. Defining k as the unit vector
FIGURE 1-12. A vector referenced to a three-dimensional framework.
unit = log
Mathematical Topics
parallel to the z-axis and with sense corresponding to increasing z, the
vector equation for the 60-g force may now be written:
The terms of Equation 1-14 are the x-, y-, and z-components of the force
vector. By way of application, Figure 1-12 might be displaying a force
exerted by a stretched elastic, pulling from a point (3 cm, 1 cm, 2 cm) in
a lateral-anteroposterior-vertical reference frame. (Continuing, the point
(5 cm, 5 cm, 6 cm) is on the line of the force, but the "head" (arrowhead) of the vector located at that point in the figure is merely a result of
the particular choice of scale factor for the sketch.) Equation 1-14 charac-
terizes the force as to magnitude, direction, and sense at a particular
time, but does not indicate the point of application of the force.
To complete the displacement discussion begun in an earlier section
of this chapter, a vector addition must be carried out. The displacement
of point D, the root apex in Figure 1-5, equals the sum of the displacement vector of point C in the crown and the relative displacement, D
with respect to C. In Figure 1-11 the vector components having magnitudes of 80 and 60 units "add" to give the resultant vector V with magnitude of 100 units. (The sum of two vectors depends on both the individ-
ual magnitudes and their directions and senses.) In that example, the
(rectangular) components were perpendicular to one another; the components of the apex displacement vector are not 900 apart. Note, however, that the parallelogram law of vector addition, when employed graphi-
cally, is easily undertaken regardless of the relative orientations of the
vectors to be summed.
The vector addition of the two displacement components is shown in
Figure 1-13. The displacement of point C, in reality obtained, perhaps,
from successive cephalograms, is drawn to a scale; in this example, the
scale magnifies the actual displacement size. The magnitude of the relative displacement is the scalar product of the change in long-axis angulation (in radians) and the length of the axis segment CD. The directions
and senses are from C to C' and D" to D', respectively, from Figure 1-5.
To accomplish the summation, the two vectors are placed in series, with
the tail of one meeting the arrowhead of the other, and they form adjaC
(or D)
(or D")
FIGURE 1-13. Vector addition. The relationship among the displacement vectors of
points in Figure 1-5.
Bioengineering Analysis of Orthodontic Mechanics
cent sides of a parallelogram. The third and fourth sides of the parallelo-
gram are added (dashed) to complete the figure and the desired vector,
the absolute displacement of point D, is coincident with a diagonal. The
length of the resultant vector is measured and multiplied by the scale
factor to obtain the magnitude of the displacement of point D.
Because the two vectors summed in Figure 1-13 are nearly parallel
and have comparable senses, the result in this example is almost as if the
component magnitudes were added. Parallel components, however, are
the exception and not the rule; the 80-unit and 60-unit components in
Figure 1-11 sum to give a resultant magnitude of 100 units, not 140.
Additional examples of vector addition are presented in Chapter 2.
Dimensions and Units
The quantifiable parameters of orthodontic bioengineering are generally
dimensional and their magnitudes depend upon the units in which the
parameters are expressed. Four dimensions are basic or fundamental:
length, time, mass, and temperature. All other dimensions of interest may
be derived from these four; the most important is the dimension of
force. The result of a dimensional analysis of an equation of classical
physics yields force as the product of mass, length, and the inverse of
time squared [mass(length)/(time2)}. Dimensions help to describe a
quantity, and also to distinguish among apparently similar parameters.
For example, pressure is dimensionally mass/length!(time2) or force!
(length2) and, therefore, is not identical to, or a type of, force, although
the two are related. One quantity already discussed, which is dimensionless (or nondimensional), is angular measurement; another, which
will be introduced in Chapter 3, is strain.
The numerical value assigned to a dimensional quantity (and some
nondimensional quantities as well) is meaningful only when units are
appended to that value. Each fundamental or derived dimension generally has several units or sets of units that may be associated with it. The
numbers 1, 3, and 36 are different from each other, but when used as
expressions of length in yards, feet, and inches, respectively, they are
Because of biologic and materials-science influences on their field,
dental practitioners are familiar with two systems of units. Metric units
are common to the life sciences; length is measured in meters and mass
in grams or fractions!multiples thereof (e.g., millimeters [mm] and kilograms [kg]). In this country the physical sciences have long employed
the American engineering system of units; typically, lengths are in feet
and forces in pounds-force. In both unit systems the basic unit of time is
the second. The derived, composite unit of force in the metric system is
the "newton;" the gram-force is more common now in the United
States, but is a distortion of the metric mass unit. In the American engi-
Mathematical Topics
neering scheme, force is considered a fundamental dimension and the
"slug" is the derived unit of mass (although pound-mass is also used in
a manner comparable to gram-force). Conversions from one metric-units
set to another is straightforward; "kilo" indicates 1000 and "centi"
means "divide by 100," for example. Such conversions in the American
engineering system are more difficult (e.g., 5280 ft/mile and 16 oz/lb);
this is probably a principal reason why the scheme has never been
widely accepted outside of the United States.
Orthodontists, until now purchasing arch wires having cross-seclions measured in inches, but themselves measuring arch lengths and
discrepancies in millimeters, should be pleased about the pressures
from abroad toward a worldwide move to the Système International
d'Unités (French), or the SI units system. A modified metric scheme, the
fundamental SI units of length, time, mass, and temperature are the
meter, the second, the kilogram, and the degree Kelvin, respectively.
Examples of SI units associated with derived dimensions include the
newton (force), the pascal (stress or pressure; one newton per square
meter), and the joule (energy; one newton-meter). Currently, the conversion to one international system of units remains incomplete. Hence,
the clinician must possess some proficiency in converting from one unit
or set of units to another in the same system and in transposing or
converting between the metric and American engineering schemes.
Table 1-2 is provided to assist the reader with a partial listing of conversion factors within and between unit systems.
ants, Computations, and Numerical Accuracy
Although not professing to be mathematicians, orthodontists are con-
fronted with, and make use of, a substantial amount of quantitative
information. Treatment planning is influenced by computations performed after measurements are taken from casts and cephalograms. Today's vendors of wires, elastics, and the whole array of appliance ele-
ments are oriented in the materials and engineering sciences, and
clinicians must be able to communicate appropriately with them. To
understand the mechanical and structural functions expected of their
appliances, and to match sets of appliances properly with the treatment
plan, the practitioner must be able to undertake at least a cursory analysis of the force systems created by activation of those appliances. Physiologically-proper forces, to be exerted on the dentition, must be approximated.
To follow the development and examples in this text, and more importantly to make clinical use of the knowledge to be gained in the study
of the bioengineering topics to follow, the mathematical tools discussed
in this chapter become indispensable. When possible, emphases will be
placed on the appropriate graphic techniques to carry out an analysis, in
particular to help visualize relationships among variables and in the
Bioengineering Analysis of Orthodontic Mechanics
TABLE 1-2. Conversions within and between units systems
Length (L)
Area (L2)
Force (F)
to Obtain
Energy (F-L)
Stress (FIL2)
0. 07 04
Temperature (T)
degrees F—32
degrees C
degrees C
degrees F—32
Time (t)
1. Degrees F = °Fahrenheit
Degrees C =
2. Angular measurement is nondimen-
sional but is quantified in degrees
or radians.
3. Joules (J) and pascals (Pa) are
derived units in the SI system.
4. Absolute temperature is measured
in °Kelvin = °Celsius + 273 or
kine = ¶ahrenheit + 460
manipulations of force and displacement parameters. When necessary,
analytic procedures will be used and dimensional and units analyses
included as a partial check on the correctness of terms and equations.
In any analysis requiring mathematical manipulations, after an overview of the entire undertaking and coming to a decision regarding the
appropriate procedure to be followed, the input or given data must be
gathered or examined. The techniques of obtaining linear and angular
measurements, and the accuracy of these data, are expected to be those
ordinarily achieved in acceptable cephalometric exercises. Determinations of force magnitudes should generally be to the nearest ½ oz or
15 g. When input data for a problem to be solved originate from a previous statistical analysis, the accuracy of that data is to be determined; if
mean values of parameters are to be used, their standard deviations
should be sought.
Mathematical Topics
Analyses within this text are to be undertaken and solutions pursued
through principles of force and structural mechanics interfaced with
knowledge obtained from clinical experience and experimental research.
Quantifiable results will be sought by using mathematical procedures
introduced in this chapter. After any solution has been obtained it
should be carefully checked. Two possible means of checking results are
(1) to proceed to the solution by an alternate method or approach, if
available, or to change the order of steps in the solution procedure followed, or at least to substitute the numerical answers obtained into the
mathematical relationships initially used to ensure satisfaction; and (2)
to make sure that no mistakes exist in dimensions or units in the procedural steps. A dimensional analysis of an algebraic equation, for example, must show identical, net dimensions and units for each term.
The solution to a problem involving mathematical operations is only
as accurate as the data employed and the accuracy of the computations
performed. Deficiencies in either can render a solution inadequate, even
if the procedures are all correct. An accurate set of calculations cannot
make up for inaccuracies in initial data. Hand-held, electronic calculators and digital computers will yield numerical solutions to six or more
figures, but no more digits should appear in the answer than can be
justified. The evaluation of the accuracy of mathematical procedures is a
complex matter and will not be investigated here. It is perhaps sufficient
to indicate that most input data in orthodontic bioengineering problems
are accurate to no more than three significant figures and, therefore, no
more than three significant figures should be recorded in the solutions
Davis, H.F., and Snider, A.D.: Introduction to Vector Analysis. 4th Ed. Boston,
Allyn & Bacon, 1979, Chapter 1.
Drooyan, I., Hadel, W., and Carico, C.C.: Trigonometry. 3rd Ed. New York,
Macmillan, 1979, Chapters 1, 4, 5, and Appendix A.
Greener, E.H., Harcourt, J.K., and Lautenschlager, E.P.: Materials Science in
Dentistry. Baltimore, Williams & Wilkins, 1972, Chapter 1.
Spiegel, M.R.: Mathematical Handbook of Formulas and Tables. Schaum's Outline Series. New York, McGraw-Hill, 1968.
Spiegel, M.R.: Theory and Problems of Vector Analysis. New York, Schaum,
1959, Chapter 1.
Thurow, R.C.: Atlas of Orthodontic Principles. St. Louis, C.V. Mosby, 1970,
Chapter 5.
Wilson, W.A., and Tracey, J.I.: Analytic Geometry. Boston, D.C. Heath, 1949,
Chapters 1—3, 13.
Introduction to Analysis
of Orthodontic Force
The activation of an orthodontic appliance creates a system of forces that
are transmitted through the members of that appliance to the dentition,
there providing the potential to produce displacements of dental units.
Relationships exist between force systems and potential displacements
of bodies upon which the forces act. Hence, an understanding of the
fundamental principles and procedures of force analysis is indispensable to the practitioner who fabricates, places, and activates the appliance
toward the desired end: the controlled movement of teeth.
A large portion of the analysis of forces to be discussed has its foundations in the subfield of classical physics known as particle mechanics.
Several of the basic concepts may be familiar to the reader from an undergraduate survey course in general physics. Nevertheless, the discussion here commences with no assumptions of previous knowledge of
the subject.
as an area of study within the physical sciences, is concerned
with the state of rest or motion of bodies subjected to forces. Mechanics
is subdivided by the physicist according to the sizes of the bodies studied (from the smallest subatomic parts of matter—quantum mechanics—
to the planets and galaxies—astronomy) and with respect to the degree
of motion (from no movement whatsoever to bodies moving at rates of
the order of the speed of light). Although evidence exists that the early
Greek scientists understood and used some aspects of mechanics several
hundred years nc., the fundamental principles of particle mechanics, as
understood today, were formulated by Sir Isaac Newton in the seven-
teenth century. Newtonian mechanics pertain to bodies of moderate
size, if moving then not at the excessive speeds where mass and velocity
may become relatively, mutually indistinguishable, and is separable
from relativity theory and celestial mechanics.
Introduction to Analysis of Orthodontic Force
Newtonian mechanics is divided into branches known as statics,
which treats bodies at rest and under the action of forces, and dynamics,
which deals with moving bodies. Dynamics is further subdivided into
kinematics, the study of motion itself, and kinetics, in which relationships between the force systems and the characteristics of body motion
are explored. The subareas of study within mechanics also may be categorized according to interest in whole-body motions and displacements
of solids, the deformations that solid bodies experience under loading,
and the flow of fluids. The principles of Newtonian mechanics, together
with their interactions with the biologic systems and processes involved,
govern the study of orthodontic forces. The movements of individual
teeth are examples of whole-body displacements. Because the activation
of the orthodontic appliance requires deformation of one or more elements of the appliance, applicable principles of mechanics of deformable
bodies are discussed in subsequent chapters.
This chapter first introduces the principles of Newtonian partide
mechanics. The extensions of these principles, enabling the treatment of
bodies that are large in comparison to the particle, are then discussed.
Some modeling (idealizing from the actual) and approximations will be
found necessary or beneficial and, in each instance, the validity and
limitations of the model or approximation must be assured and understood. Although introduced individually, note that in large measure the
concepts and principles "fit" together or accumulate to ultimately form a
packaged procedure to be employed in analyses wherein total descrip•tions are required of force systems induced in the orthodontic appliance
and in the dentition to which the appliance is attached.
is defined as an act upon a body that changes or tends to change
the state of rest or the motion of that body. The categories of forces are
numerous, and forces are described in many different ways. Two bodies
are associated with every force; one body exerts the force and the other
receives the force. A first manner of categorizing forces is according to the
proximity of the two involved bodies with respect to each other. Contact
forces arise due to actual physical contact between the bodies; body
forces exist between bodies that are some distance from one another.
The most common, sizable body force is the attraction of the Earth for a
body on or near the Earth's surface: the weight of the body. Newton's
gravitational law enables the determination of the weight of a body,
which is directly proportional to the product of the masses of the Earth
and the body and inversely proportional to the square of the distance
between the centers of the Earth and the body. Other body forces include those existing in the presence of electric or magnetic fields, but
these do not occur routinely in the study of orthodontic force systems
and thus will receive little attention. Notably, the weights of teeth and
Bioengineering Analysis of Orthodontic Mechanics
appliance parts are small and may be neglected in comparison to the
contact forces present within the activated appliance.
In analysis procedures, forces are associated with the areas or volumes of bodies with which they directly interact, and contact forces are
associated with the surface areas over which the contact exists. If the
contact area is small in comparison with the total surface area of the
body, the force is modeled as exerted on a surface point or particle of
the body and is termed a concentrated or point force. On the other hand,
if the contact area is relatively large, the pattern of the force intensity over
the area may be an important consideration and, therefore, the force is
explicitly termed distributed. For example, the force exerted by an elastic
stretched against a hook, soldered to a bracket or to an arch wire, is
reasonably concentrated, but the force arising between the periodontal
ligament and the root, in response to appliance activation transmitted to
the tooth, must be analyzed with regard to its distribution over the sizable surface area of contact. By definition, forces exerted on an individual
particle are point forces that may actually or potentially only pull or
push the particle.
As previously mentioned, two bodies are necessary to the existence
of every force. Moreover, whenever one force is created or present, a
counterpart exists as well; when one body exerts a force on another, the
second body also exerts a force on the first. Between the two bodies
there is an action and a reaction, and which force is which depends solely
on point of view. In analysis procedures, however, just one of the two
bodies will be examined at a time and that one body will be studied with
respect to the forces exerted on it (not those forces exerted by it). Reactive
forces must not be confused with responsive forces; reactions exist as
counterparts to both active and responsive forces. In the study of a body
subjected to forces, the active forces are those purposely created or those
that "load" the body; the responsive forces are also exerted on the body,
but are a direct result of the creation of active forces. The responsive
forces are often undesired, but being present they must be included in
the analysis. As a final categorization of forces for present purposes,
forces are said to be external to a body under study when they are exerted by other bodies not part of the specific analysis. Internal forces
arise within a body as a result of loading or activating the body under
study. These internal forces "carry" or transmit the action through the
body to the responsive locations, for example, to connections with other
In addition to the gravitational law, Sir Isaac Newton formulated a
set of three laws governing the mechanics of a particle. Newton's First
Law states that a particle subjected to a balanced system of concentrated
forces will remain at rest, if originally at rest, or will move with constant
speed in a straight line, if originally in motion. ("Balanced" force system: the net force is zero.) The Second Law indicates that, if the particle
is subjected to an unbalanced system of forces, the particle will be accelerated in the direction of the net force exerted; in equation form, the net
Introduction to Analysis of Orthodontic Force
force and the product of the particle mass and its acceleration, with the
three quantities expressed in consistent, related units, are identical in
magnitude, direction, and sense. (Acceleration is the time rate of change
of velocity and, like the concentrated force, is mathematically a vector
quantity.) The Third Law states that paired active and reactive forces are
equal in magnitude, but are directly opposed to one another and are
exerted on adjacent particles. A corollary to the Third Law is the existence of internal forces, as previously defined, in canceling pairs; the
ramification of this corollary is demonstrated later in this chapter.
Newton's First Law might be considered a special case of the Second
Law, and in orthodontic force analysis a strong argument may be made
to the point that the inertia of an appliance element or a tooth is negligibly small, in particular after the almost instantaneous response, in the
form of soft-tissue deformations, immediately following the activation of
an appliance. If the inertia, the product of mass and acceleration for a
particle, may be neglected in comparison with the individual components of the force system created through activation, the orthodontic
system may be said to exist in a kind of quasi-static state; the displacements that take place are small and occur over a relatively long period of
time and, at any instant, a force analysis may be carried out without
erring appreciably, as if the system was at rest. Clearly, this approximation has profound consequences; for the purpose of most analyses involving the entire force system, a dynamic body or group of bodies is
effectively replaced by a static body or assembly.
In Chapter 1 the concentrated force was reported to fulfill the mathematical requisites of a vector quantity. In the complete description of a
point force, then, its characteristics as a vector must be stated: magni-
tude, direction, and sense. Moreover, from a physical standpoint the
location of the force on or in the body must be given. Figure 2-1 shows a
point force exerted on a molar crown. From the sketch, the force is seen
to have a magnitude of 200 g; in this two-dimensional idealization, the
direction of the force (and that of the dashed line of action of the force) is
700 from the long axis of the tooth. The sense of the force may be said to
be up and to the left with respect to the sketch as a whole, and the point
of application is point B with the force pushing on that point. (If desired,
an equation in vector form may be written for this or any point force,
following the establishment of a reference frame, and all of the necessary characteristics to describe the force, except point of application,
would be contained in that equation.) Figure 2-1, then, provides a complete visual description of the 200-g force.
The potential or actual effects of a force exerted on a body, whether
active or responsive, are displacement and/or deformation. Although both
effects are of interest in orthodontic structural analysis, only the association of displacement with force is discussed in this chapter. Disregard-
ing deformation resulting from force application is tantamount to as-
suming that the body, on which the force or force system acts, is
perfectly rigid. Although this is a reasonable assumption for some bod-
Bioengineering Analysis of Orthodontic Mechanics
200 g
FIGURE 2-1. A concentrated force exerted on a tooth crown.
ies, including individual teeth, it is not a valid assumption for others, for
example, elastics. In considering effects of force application, however,
bodies that are clearly nonrigid may be reasonably analyzed only within
a framework that recognizes differences in passive and active configurations.
The rigid body, then, is a model of Newtonian mechanics used in
analysis procedures when only completion of the description of the total
force system or the whole-body displacement is sought. (In Chapter 1,
the displacement of point D of the tooth was obtained within the assumption that the tooth was rigid; otherwise, the relative displacement,
point D with respect to point C in Figure 1-5, could not have been described as it was.) An important feature of rigid-body mechanics allows
the "sliding" of a point force to any location on its line of action found
most convenient for analysis purposes. This principle of transmissibility
states that the mechanical, whole-body effect of a concentrated force,
acting at a specific point, is unchanged by replacing the given force by
another force having the same magnitude, direction, and sense as the
original force, but acting at some different point on the line of action.
Figure 2-2 shows an 80-g force in two different locations on its line of
action (dashed); the potential for displacement of the tooth as a whole is
the same, whether the force is pulling on the lingual side or pushing on
the labial crown surface.
Before proceeding further, the general dependence of orthodontic
force upon the independent variable "time" must be mentioned. The
Introduction to Analysis of Orthodontic Force
80 g
FIGURE 2-2. Two concentrated torces having identical displacement potentials. The
principle of transmissibility.
concentrated orthodontic force often changes with time and that change
is typically reflected in the magnitude characteristic, although alteration
of direction may also occur with time. Magnitudes of force generated by
appliance activation decrease during the between-appointment periods
and an accompanying change in displacement potential generally oc-
curs. The changes with time may be divided into short-term occurrences, primarily due to soft-tissue deformations immediately upon acti-
vation, and long-term changes associated with tissue remodeling.
Magnitude increases and, perhaps, directional alterations take place
when the orthodontist reactivates the appliance. Detailed discussions of
these phenomena are undertaken later in this text; in the present chapter, force systems are analyzed as if all involved entities are "frozen" at
a particular instant in time.
ition and Decomposition of Concentrated Forces
fundamental analysis manipulations with point forces are dis-
cussed in this section. First, toward enabling the prediction of the form
and direction of a potential rigid-body displacement associated with a
system of active forces, the forces are combined in such a way that the
mechanical effect of the original system remains unchanged while the
resultant is as simple as possible. Second, often in the process of obtaining the resultant of a force system, the replacement of a force by a set of
force components with each component having a designated direction is
Beginning the first procedure with the fundamental operation, a pair
of concentrated forces having intersecting lines of action has, as its resultant, a single force. The two original forces define a plane; the resultant lies in that plane and its line of action intersects the point where the
Bioengineering Analysis of Orthodontic Mechanics
lines of action of the original forces come together. The means of reduc-
ing the two concurrent forces to a single force, mechanically equivalent to
the combination of the original forces, is known as the parallelogram law
of vector addition.
To illustrate, Figure 2-3 shows a pair of point forces: one originating
from an interarch elastic (EL) and the other from an extraoral appliance
(HG); both are exerted on a maxillary molar bracket or buccal tube. The
view is from the buccal on the right side. To obtain the resultant graphi-
cally, the directions and senses are sketched exactly as they exist in a
two-dimensional picture of the appliance action. The force magnitudes
are represented in the lengths of the vectors according to the given scale;
because it is twice the magnitude, the length of the headgear force vector is twice that of the elastic force. The given forces intersect at point 0,
the center of the bracket in the buccal view, and their vector representations form adjacent sides of a parallelogram. The dashed lines are added
to complete the geometric figure. The vector representing the resultant
is drawn as the diagonal of the parallelogram; the "tail" of the resultant
vector coincides with the point where the "tails" of the given vectors
meet. Its sense is then known, the direction of the resultant may be
determined with respect to a chosen reference line with a protractor,
and the length of the vector representation is converted to the force
magnitude through the scale as indicated. Note that if one of the forces
is seemingly pushing on the point and the other is pulling on the point
of intersection of their lines of action, one or the other vector must be
moved along its line of action until either both "tails" or both "arrowheads" or "tips" of the vectors meet at the point. Then the parallelogram
may be completed and the sense of the resultant is determined; either all
three "tails" or all three "tips" meet at the common point.
EL = 200 g
HG = 400 g
+ 02
Scale: 1 in. = bOg
lengths = 2.8 in.
H = 280 g
= 28°
FIGURE 2-3. Graphical determination of the resultant of frio concurrent forces. The
parallelogram law.
Introduction to Analysis of Orthodontic Force
With the aid of a rough sketch, the resultant of two concurrent point
forces may be obtained analytically. Seen in the parallelogram construclion of Figure 2-3 is a pair of congruent triangles with a shared side, a
diagonal, coinciding with the resultant vector. Two sides and the included angle of each triangle are known through the characteristics of
the given forces. The laws of cosines and sines may then be used to
obtain the magnitude of the resultant force and the orientation of its line
of action with that of a given force or some other known reference.
Either the vector lengths or the actual magnitudes of the given forces,
because length and force size are proportional, may be used in the computations which would proceed as follows:
+ HG2 — 2(EL)HG(cos 0)
[2002 + 4002 2(200)400(cos 400)]h/2
= 278 g or approximately 10 oz
The rough sketch is necessary, accompanying the analytic solution, to
determine and illustrate the sense of the resultant force and location of
its line of action.
If the two forces to be "summed" are perpendicular to one another,
the graphical procedure remains essentially unchanged, but the parallelogram becomes a rectangle. The pair of congruent triangles become a set
of adjacent right triangles, and the analytic solution is simplified somewhat with the cosine and sine laws replaced by the Pythagorean theorem and the definition of one trigonometric function. An example solulion is given in Figure 2-4.
The vector summations of Figures 2-3 and 2-4 may be handled alternatively, using a corollary of the parallelogram law: the triangle law.
= 6.8 oz
P = 3 oz
tan 6 =
Q = 6 oz
0 = 27°
FIGURE 2-4. The resultant of two concurrent, mutually-perpendicular forces.
= 0.5
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 2-5. Resultants obtained for the given forces of Figure 2-3 (top) and Figure 2-4
(bottom) using the triangle law.
Graphical solutions for the examples of Figures 2-3 and 2-4, using the
triangle law, are shown in Figure 2-5. Recall that the "tails" or "tips" of
the given force vectors must coincide when using the parallelogram law;
the solution by the triangle law requires that the "tip" of one given force
and the "tail" of the other be coincident. The resultant is drawn from the
"tail" of one given force to the "lip" of the other so that the three forces
form the sides of a triangle. Note that in summing two concurrent forces
by means of the triangle law, one given force may have to be displaced
to an artificial location in order that the resultant emerges in the construction along its proper line of action; alternatively, keeping both
given forces on their original lines of action in the graphical approach,
the resultant obtained is correct in all characteristics except for its actual
point of application. Hence, care must be taken in using the triangle law
so that the proper line of action of the resultant is ultimately determined
and designated correctly.
Introduction to Analysis of Orthodontic Force
The resultant of any number of point forces having lines of action all
intersecting at one common point is a single, concentrated force; the line
of action of the resultant passes through that same point. To determine
the resultant of a system of three or more concurrent forces, the parallel-
ogram law or triangle law may be used repeatedly. Two of the given
forces may be combined into one, that one combined with a third given
force, and so forth until all of the given forces have been included in the
process; the resultant emerges from the final combination involving the
last given force. Because any two of the concurrent forces define a
unique plane in which their resultant will lie, the two procedures may be
used in a three-dimensional problem. If all of the concurrent forces act in
the same plane, the triangle law may be extended to a polygonal law.
Graphically, the given forces are drawn to scale with the "tip" of one
touching the "tail" of the next until all have been included, in any order
desired. The magnitude, direction, and sense of the resultant are determined by closing the polygon with the vector resultant having its "tip"
touching that of the last given force drawn (and its "tail" touching that
of the force with which the sketch was begun). The number of sides of
the polygon is one more than the number of given forces. In this construction all but two of the given forces must be moved from their actual
lines of action. The line of action of the resultant is known, though, to
pass through the point common to the lines of action of the given forces.
An example of the procedure using the polygonal law is illustrated with
three given forces in Figure 2-6. In this example the forces are "added"
in the order indicated by the equation in the figure. The dashed lines
suggest the graphical procedure by two successive applications of the
parallelogram law.
In a force-system analysis, the decomposition of given forces into com-
ponents is often desirable. The procedure is the reverse of that of the
parallelogram or triangle law of vector addition. In Figure 2-3 the 200and 400-g forces could be considered components of the 278-g force. In
analyses of orthodontic force systems, however, the component directions often coincide with the mutually perpendicular, dental-coordinate
directions. In Figure 2-4, with proper orientation, the 3- and 6-oz forces
become occlusogingival and mesiodistal components of the 6.8-oz force.
The number of mutually perpendicular, nonzero components equals, at
most, the number of coordinate dimensions in a given problem: up to
two in the plane problem and three in the spatial problem.
Although the determination of vector components perpendicular to
one another was outlined in Chapter 1, by way of review consider the
example of Figure 2-7 in which the occlusal and mesial components of
the 300-g force against the molar are obtained. Graphically, the parallelogram (actually a rectangle because perpendicular components are
desired) is constructed around the 300-g force as its diagonal, drawn
to scale. The components are adjacent sides of the rectangle; their
"tails" touch that of the given vector. The component magnitudes are
obtained from their lengths and the scale factor for the sketch. Analyti-
Bioengineering Analysis of Orthodontic Mechanics
A = 4 oz
B = 6 oz
R = 7.5 oz
C = 5 oz
FIGURE 2-6. Three concurrent, coplanar forces and their resultant obtained through the
polygonal law.
cally, each rectangular-component magnitude is the product of its direction cosine and the magnitude of the given force as indicated in the
The three-dimensional problem of decomposition into mutually per-
pendicular components cannot be easily handled graphically. The analytic solution, however, can be a direct extension of that used in the
plane problem. A rough sketch, particularly helpful toward determining
component senses, should be used. In proceeding, first establish the
coordinate directions and, perhaps, sketch the corresponding rectangular parallelepiped around the given force. Second, obtain a set of angles,
either (1) the direction angles (angles between the given force and the
coordinate axes), or (2) the angles to project the given vector into one of
the coordinate planes and onto the axis perpendicular to that plane and,
subsequently, to project the component in the plane onto the coordinate
axes defining the plane. Again, the procedure illustrated by Figure 2-8 is
a review of the spatial relationship between the vector and its mutually
Introduction to Analysis of Orthodontic Force
300 0
280 g
F0 = F cos 70° = 300(0.34) = 1000
Fm = F cos 20° = F sin 700 = 300(0.94) = 280 g
FIGURE 2-7. Decomposition of a concentrated force into two rectangular components.
The parallelogram law in reverse.
perpendicular components previously discussed in Chapter 1 (see Fig.
1-12). Note that only two of the three direction angles need to be meas-
ured or estimated; the third angle can then be determined from the
geometric relationship requiring the sum of the squares of the directions
cosines to be unity.
a: apical
d: distal
0: occlusal
Fd = 12 cos 30° = 10.4 oz
F0 = 12 cos 75° = 3.1 oz
F, = 12 cos 65° = 5.1 oz
(cos 300)2 + (cos 750)2 + (cos 650)2 = 1
FIGURE 2-8. Three mutually perpendicular components of a concentrated force not
located in any of the three coordinate planes.
Bioengineering Analysis of Orthodontic Mechanics
The Moment of a Concentrated Force
action of a concentrated force upon a particle tends to produce,
simply, a push or pull displacement; the particle is small and the force
seemingly covers the particle as a whole. The magnitude, direction, and
sense of the force, together with characteristics of the particle and its
supporting structure, are collectively a measurement of the potential of
the force to displace the particle. A force, exerted on a small portion of
the total surface area of a relatively large body, tends to produce a displacement of the body which is generally rotational. The moment of a
force about, or with respect to, a specified point or line is a measure of
the potential of that force to rotate the body, upon which the force acts,
about the particular point or line.
Consider a rigid body that is hanging from a cord attached to just one
point of the body. Label that point 0. A force F is exerted on the body so
that its line of action does not pass through point 0. Define the line OB
so that point B is on the line of action of the concentrated force F and the
two lines intersect at right angles to one another. As depicted in Figure
2-9, the line OB and the line of action of force F define a plane. If the
distance from point 0 measured along OB is d to the line of action of F,
the magnitude of the moment of force F about point 0 is the product of
the force magnitude F and the distance d. Given the manner of support
of the body in the figure, the application of force F would displace the
body from the position shown and cause an angular change in the initially vertical line through point G (the center of gravity, through which
the weight resultant acts).
The moment of a force about a point is a vector quantity; the direction of the moment vector is perpendicular to the plane defined by the
force vector and the point. The sense of the moment vector is determined by a rule associated with the apparent rotational tendency, clockwise or counterclockwise as viewed from above the plane of the force
and the point about which the moment is determined: out of the plane if
counterclockwise and into the plane if clockwise. The moment vector is
located such that its line of action passes through the point about which
the moment is computed, and may be specifically positioned wherever
convenient along its line of action. The moment vector of magnitude Fd
is shown in Figure 2-9. Force and moment vectors may not be directly
combined in a force-system analysis and, because the two types of vectors may appear in the same sketch, the short slash is incorporated near
the arrowhead in the symbol for the moment vector.
To obtain the moment of a force about a point, three quantities must
be completely known: (1) the force itself; (2) the point about which the
moment is to be found or moment center (point 0 in Fig. 2-9); and (3) the
length of the moment arm extending from the moment center and traversing the shortest distance to the line of action of the force (line OB in Fig.
2-9). From the determination of the moment of a force about a point, the
Introduction to Malysis of Orthodontic Force
Mzz = F(d)
= F(d)(cos 0)
FIGURE 2-9. A concentrated torce and moment vectors with respect to lines (axes)
through point 0.
Bioengineering Analysis of Orthodontic Mechanics
potential of the force to produce rotation about a particular line through
the point may be obtained. The rotational potential with respect to a
line, a moment axis, has the greater application in orthodontic mechanics.
In Figure 2-9, the moment of F about point 0 is also the moment of the
given force with respect to the line Z-Z. To obtain the moment about
another line through point 0, line L-L in Figure 2-9 for example, the
magnitude of the moment about line Z-Z is multiplied by the cosine of
the angle between L-L and Z-Z. The vector component is the directed
projection of the moment about Z-Z on line L-L.
Apparent now is the general, three-dimensional nature of the moment representation and computation. The moment axis typically does
not lie in the plane of the force and the moment arm. In fact, if the
moment axis is in the plane of the force and moment arm, either the axis
intersects the line of action of the force and, therefore, the length of the
moment arm is zero, or the axis is parallel to the force; in both cases the
tendency for rotation about the specified axis is nonexistent, and, correspondingly, the moment with respect to the specific axis is zero. In the
general problem, the force for which the rotational potential is sought
likely will not be perpendicular to the moment axis, and one of two
approaches may be used: (1) the procedure followed in Figure 2-9 in
which the moment vector with respect to point 0 was projected onto
line L-L; or (2) the process of Figure 2-10. In the latter approach, the force
R is decomposed into components parallel (P) and perpendicular (Q) to
the moment axis S-S. Now, the moment of the resultant of several con-
current forces must be equal to the vector sum of the moments of the
individual forces with respect to the same axis (a theorem established by
the French mathematician Varignon in the late 1600s, before the advent
of vector algebra). Since the moment of P about axis S-S is zero, the
magnitude of the moment of R equals that of Q: Qb; the sense is as
shown in the figure according to the rule previously established.
In problems in which the forces of interest all exist in the same plane,
the moment concept may be put into a two-dimensional format. Rotational potentials are obtained with respect to axes that are perpendicular
to the plane of the force. In sketches these axes will appear in end views,
or as apparent points in the plane. Moment vectors with respect to these
moment centers will all have a common direction: perpendicular to the
plane of the forces. They may then be treated as algebraic quantities
with the distinction between moments having clockwise or counterclockwise rotational potentials made merely by a sign convention.
To determine the net rotational tendency with respect to a particular
moment axis, owing to the action of several forces, either of two procedures may be followed. The moment vectors may be obtained for each
individual force with respect to a particular point on the moment axis,
these concurrent vectors may be combined into one by methods of vector addition already discussed, and the resultant moment vector may
then be projected onto the moment axis to obtain the magnitude and
sense of the rotational potential of the force system. Alternatively, the
resultant of the system of forces may be sought and the moment of the
resultant then found with respect to the specified moment axis.
Introduction to Analysis of Orthodontic Force
= Q(b)
FIGURE 2-10. A force in space, decomposed into components parallel and
perpendicular to a given line, and its moment vector with respect to that line.
The former procedure is employed in Figure 2-11 to obtain the net
moment of a pair of concurrent forces with respect to the z-axis of the
reference framework. In this instance a two-dimensional solution may
be carried out in the plane of the given forces; the z-axis would be perpendicular to the plane of the forces and in the sketch it is, in effect,
point 0. The curved-arrow notation indicates the senses of the moments
of the 120- and 90-g forces and of the net moment. Two moment computations are displayed with magnitudes and senses obtained with respect
to moment axes perpendicular to the plane of the forces through points
O and E. The net moment being equal to the moment of the resultant of
the two forces enables the determinations of the actual moment arms of
the 150-g force with respect to points 0 and E; the moment-arm lengths
are symbolized by o and e, respectively. Viewing the 120- and 90-g forces
as x- and y-components of the l50-g force, whether or not to decompose
a force into rectangular components to obtain its moment will depend
upon the ease of the actual decomposition process and the establishment of the moment arms of the components versus the determination
of the actual moment arm of the given force. The latter can often be a
Bioengineering Analysis of Orthodontic Mechanics
M0 = 90(3)1 + 120(2)
30 g-mm
= 420 g-mm J
o = -h-- =
= 0.2 mm
= 2.8 mm
FIGURE 2-11. Determination of the moment of a force about a point in two dimensions
by using rectangular components of that force in the plane.
seemingly difficult geometry program while the moment arms of the
components become clear after the coordinates of the moment center
and one point on the line of action of the force are known.
The more complex problems of determining the net rotational effect
of a system of forces, which may be two-dimensional but nonconcurrent
or three-dimensional in nature, have been outlined as to general approach. They are more easily discussed in detail, however, following the
introduction in the next section, "The Couple."
The Couple
A pair of point forces having equal magnitudes and identical directions,
opposite in sense to one another and having noncoincident lines of
action, when considered as one mechanical entity is termed a couple.
Because the forces have the same magnitude, but are oppositely directed, the net potential of this special force system to translate the body
upon which it acts is nil. The lines of action of the two forces are parallel,
but are not coincident; as a result, this force system tends to rotate the
body acted upon. Also, since the forces are nonconcurrent, they cannot
be combined and, therefore, the couple cannot be simplified further.
The couple, occupying a specific plane as defined by the two forces,
is illustrated in Figure 2-12. The rotational potential of the couple with
respect to point 0, or about a moment axis perpendicular to the plane
through 0, is the sum of the moments of the individual forces with
respect to the point. Using the procedure and notation of Figure 2-11,
Introduction to Analysis of Orthodontic Force
M0 = 300(1)
300 g
+ 300(3)
= 600 g-mm)
= 300(2)
= 600 g-mm)
= 300(1)) + 300(1))
M0 = 300(3)) + 300(1)
= 600 g-mm)
300 g
FIGURE 2-12. A couple and computations of its inherent moment.
noting that with respect to point 0 one moment is clockwise and the
other is counterclockwise, the net moment is found to be 600 g-mm
counterclockwise. Using the points B, C, and D as individual moment
centers, the same moment is obtained in the three computations—the
same as that with respect to point 0. In fact, for all moment centers in or
moment axes perpendicular to the plane of the couple, the same result
emerges. A couple has, then, an inherent moment, and the magnitude of
the couple itself is that of the inherent moment, the product of the size
of one of the forces and the perpendicular distance between the two
forces. Because the moments of the individual forces are vector quantities, so also is that of the couple and it follows that the couple itself may
be represented mathematically as a vector. The direction of the couple is
associated with the plane of the pair of forces and the sense with its
rotational potential, either clockwise or counterclockwise, as viewed
looking into the plane of the couple.
Figure 2-13 displays a couple and its vector representation (and that
of its moment). The direction of the vector C is perpendicular to the
plane of the couple (and the "moment arm" b between the forces) and
the sense is determined according to the convention noted earlier for
moment vectors: out of the plane if viewed as counterclockwise, into the
plane if clockwise. Note the slash through the symbol, again to differentiate between force and moment vectors.
A couple is completely described when specified are its magnitude,
the plane of the two forces, its sense, and, if the body is nonrigid, the
actual points of application of the two forces. The value of the couple as
Bloengineering Analysis of Orthodontic Mechanics
C = Pb
FIGURE 2-13. A couple in its plane and its vector representation perpendicular to that
an analysis tool, however, is in the determination of whole-body displacement potential and, when so used, the specific points of application are unimportant. Although its vector representation in Figure 2-13
is shown midway between the two forces, because the moment of the
couple is the same with respect to every moment axis perpendicular to
the plane of the forces, the vector has no particular line-of-action location and may be drawn through any point of the plane of the couple.
This "freedom" associated with the couple vector has far-reaching implications with respect to certain force-analysis procedures.
To review, a force acting on a rigid body has the potential, generally,
to both translate and rotate the body. Sliding the force along its line of
action will not change this potential. The tendency to rotate the body
about a specific axis is associated with the vector characteristics of the
force and also with the location of the line of action of that force because
the axis and the line define a moment arm. Moving the line of action of
the force with respect to the moment axis alters the force direction or the
moment arm and, therefore, changes the translational or rotational potential of the force.
The couple, however, has been shown to possess an inherent rotational potential and no tendency to produce translation. Apparently,
then, as long as the characteristics of the couple are unchanged, the
forces making up the couple may be altered without affecting wholebody response; it follows, though, that an alteration of one force of the
pair necessitates a comparable change in the other. Indeed, the use of
properties of and operations on forces already discussed, including the
Introduction to Analysis of Orthodontic Force
parallelogram law, demonstrates that the following alterations may be
made without affecting the mechanical effect of the couple on the body
as a whole: (1) the forces may be moved ("slid") along their individual
lines of action; (2) the forces may be rotated in the plane of the couple,
keeping their lines of action mutually parallel and the distance between
them unchanged; (3) the couple may be moved as a whole from the
given plane to another parallel plane; and (4) the magnitudes of the
forces may be altered, while preserving the equality of force magnitudes
and the moment of the couple, the latter by making a corresponding
change in the distance between the lines of action of the forces.
Figure 2-14 shows four mechanically-equivalent couples, one being a
curved-arrow representation which is convenient in two-dimensional
analyses. These permissible movements of the couple, and the allowable
correlated changes in the force magnitudes and the distances between
their lines of action, are embodied in the vector representation of the
couple. The couple acting on a rigid body is, mathematically, a "free"
vector, having no specific line-of-action location as noted previously.
Frce and moment vectors may not be directly combined, and the
slash through the moment-vector symbol is a reminder of this fact. Perhaps unfortunately, however, no symbolic distinction is made that
would indicate the difference between the vector representing the moment of a single point force and that depicting a couple with its inherent
800 g
200 g
.75 mm
600 9-mm
800 g
FIGURE 2-14. Four mechanically-equivalent couples.
300 9
Rioengineering Analysis of Orthodontic Mechanics
moment. Note that the moment vector, representing the rotational po-
tential of the concentrated force with respect to a specific moment center
or axis, is not a "free" vector. On the other hand, because a couple is a
free vector and through its two-force representation, a couple may be
combined with a third force. The form of the resultant from this combination is particularly dependent on the direction of the third force with
respect to the couple vector.
Figure 2-15 shows the relative position of the force and couple vectors, enabling the placement of the two forces of the couple in a plane
containing the third force, which leads to a simple resultant. Figure 2-16
illustrates the combination procedure. With the three forces in the same
plane, the couple is represented as two forces having magnitudes equal
to that of the single force; the distance between the forces making up the
couple is then determined. Next, the couple is translated and rotated in
the plane so that one force of the couple shares the line of action of
single force and is opposed (in sense) to it. The now directly opposite
forces cancel one another and one concentrated force from the original
three remains, having the same vector characteristics as the original single force, but with a line of action some distance from it.
The procedure of combining a force and a couple may also be carried
out in reverse. Given a single force having specific characteristics, including location of line of action, a force and a couple may replace it
without altering the whole-body displacement potential. The transla-
800 g-mm
FIGURE 2-15. Mutually-perpendicular force and couple vectors referenced to the plane
of the force system.
Introduction to Analysis of Orthodontic Force
200 g
200 g
200 g
FIGURE 2-16. Reducing the force and couple of Figure 2-15 to a single force.
tional potential remains unchanged if the force portion of the new sys-
tem has the same vector characteristics as the given force. To leave the
rotational potential unaltered, the magnitude, direction, and sense of
the couple must correlate with the position of the line of action of the
force of the substituted system with respect to that of the given force. In
Figure 2-17, in occlusal view the force exerted on the edgewise orthodontic bracket is "moved" by means of the addition of the couple to a
parallel position through the long-axis of the tooth. Although seemingly
adding to the complexity of a force system, the procedure provides a
means to "free" the force vector from its specified location and permit
lateral movement of its line of action.
A couple is a special two-force system. In orthodontic application,
active couples are often created upon bracket engagement of arch wires
C = Qe
FIGURE 2-17. Moving the line of action of a force 0 through introduction of a couple C.
The reverse of the process illustrated in Figure 2-16.
Bioengineering Analysis of Orthodontic Mechanics
into which permanent bends or twists have been placed. The dimension
of the couple magnitude is the product of force and length. The couple
tends to cause only rotation of the body on which it acts. In addition to
its importance as a displacement-producing system, the couple is useful
in analysis because it alone enables the transfer of a concentrated force,
exerted on a rigid body, from its given location to a parallel, alternative
line of action associated with the same rigid body.
Resultants of Force Systems Exerted on Rigid Bodies
definition of force, presented earlier in this chapter, was stated in
terms of its potential to influence the motion of the body acted upon. In
discussions of forces, moments, and couples thus far, references to
translational and rotational displacements have been made. The word
"resultant" was used previously in discussing the replacement of two
forces by one through use of the parallelogram or triangle law. In the
analysis of a class of particle- and rigid-body-mechanics problems, including certain applications to orthodontics, relationships are sought
between an active system of forces and a displacement pattern to be
achieved by, or to result from, force application. Because the force systems often are somewhat complicated in their actual formats, the replacement of the given system by another, which is mechanically equivalent but more easily related to the potential or actual displacement of
the body, is usually desirable, if not necessary. Generally, this new or
replacement system is simpler, and is termed a resultant of the original
force system. It is possible to reduce any force system acting on a body,
when relationships between the given system and whole-body displacement potentials are sought, to one force and one couple, and in some
instances to just a single force or just one couple. Clearly, these are the
most simple resultants.
In the process of determining the displacement format of a body
subjected to a set of forces, the resultant of the entire, active force system
is sought. (Recall that the active forces "load" the body and are distinguished from responsive forces, which come into being because of contact between the body and whatever supports it. Generally, the respon-
sive force system is present in some form, however abbreviated, even
before the loading is applied to the body, simply because of the member
weights and the physical contact of the body with its supporting structure. Loading the body, then, usually alters the responsive force system
in some manner.) Although a more simple resultant of the active force
system may exist, and often does, the resultant sought is usually that
referenced to a point in the body known as its center of resistance. The
location of the center of resistance depends upon the size and shape of
the body, the distribution of the weight of the body, and the manner in
which the body is supported. In general, the location of the center of
resistance of a body must be found experimentally or approximately by
Introduction to Analysis of Orthodontic Force
appropriate analytic analysis; simple force systems are applied and
displacement responses are observed or determined and specific displacement formats are sought.
Upon the application of loading to a body, which results in an activeplus-responsive force system, and if the entire force system exerted on
the body is not completely self-balancing, one of three forms of displacement occurs. Translation is that displacement within which no line in the
body experiences a change in angular position (orientation) with respect
to a fixed reference line. If the body is rigid, during translation all points
in the body follow parallel paths; although these paths may be straight
lines, they may also be curvilinear without violating the requisites for
translation. In order for a translational displacement to result from the
application of force, the resultant of the applied, active force system reduce to a single concentrated force, and the line of action of that
force must pass through the center of resistance of the body.
When the resultant of the active force system is a couple, the resulting displacement is termed pure rotation. All points in or on the body
move in circular paths; the centers of those circles are all on one line or
axis that pierces the center of resistance of the body. The vector representing the resultant couple is parallel to the axis of rotation. The most
common form of displacement is a combination of translation and pure
rotation and is known as generalized rotation. The resultant of the active
force system is usually expressed as one force and one couple. The displacement may be difficult to describe except to note that the angulation
of the body has changed. If the applied force system is two-dimensional
and the resultant of the active force system reduces to a single force, its
line of action does not pierce the center of resistance of the body, and for
small displacements or if the resultant remains constant in all characteristics with respect to the body acted upon, the motion is circular about
an axis perpendicular to the plane of the forces but not passing through
the center of resistance.
Except when the active force system consists entirely of couples, at
least a portion of the resultant problem is the simplification of a concurrent force system. Although the parallelogram or triangle law may be
used to reduce the concurrent system to a single force, an alternative
procedure may be more simple if numerous forces are involved; its advantage in an analytic approach is organization. Useful in both two- and
three-dimensional problems, the process follows a sequence of four
steps: (1) a coordinate framework is established; (2) each given force is
decomposed into its coordinate components; (3) the algebraic sum of
each set of components is determined, thereby obtaining the resultant
components; and (4) the resultant components may be combined into
one single force, using the parallelogram law or the three-dimensional
extension of it. A re-examination of a previous example serves to illustrate this procedure (see Fig. 2-3).
Figure 2-18 shows the elastic and headgear forces referenced to a
chosen framework in a buccal-plane view, the given forces decomposed
into components along the reference axes, the resultant components,
Bloengineering Analysis of Orthodontic Mechanics
365 g
55 9
200 g
400 9
163 g
Rd = 400 sin 66° — 200 sin 74°
= 400 cos 66°+ 200 cos 74°
= '11732 + 2182 = 278 g
FIGURE 2-18. The resultant of several concurrent forces obtained through the use of
force components. A two-dimensional example.
and the single-force resultant together with the computations providing
the force characteristics sought. The resultant components as computed
suggest that the displacement should have a strong, occlusally directed
aspect, but some potential for distal movement also exists. The resultant
force does not pass through the center of resistance of the tooth in this
two-dimensional analysis.
The procedure for determining the resultant of a general force system, consisting of both point forces and couples exerted on a rigid body,
is as follows. First, when the entire active force system is under investigation, a reference point is chosen that often will be the center of resistance of the body if its location is known. A coordinate framework is
established, centered at the reference point, with coordinate axes o*
ented according to the force-system layout. Second, each concentrated
force is transferred from its given line of action to a parallel line passing
through the reference point. Displacing the force in this manner requires
the addition of a couple to the force system. Following the approach
shown in Figure 2-17, the magnitude of the couple added equals the
product of the force magnitude and the perpendicular distance between
the new and given lines of action of the force. The sense of the couple is
that of the moment of the original force about the reference point. In a
three-dimensional problem, the couple will be represented in vector
fashion with its direction perpendicular to the plane of the given and
new force vectors. The given force system has now been transformed
into a set of concurrent point forces at the chosen reference point and a
set of couples (including those given and those created in "moving"
Introduction to Analysis of Orthodontic Force
Third, the forces are combined and the set reduced to a single concentrated force, using the parallelogram or triangle law or following the
process outline with Figure 2-18 in which each force is decomposed into
coordinate components. The couples, properly represented by free vectors, can all be gathered at any reference point and reduced to one couple. The choice of reduction procedures is the same as that for reducing
a system of concurrent force vectors. The given force system now has
been replaced by an equivalent system consisting of one point force, or
its two or three rectangular coordinate components, at the reference
point and a single couple (or its two or three components). Fourth and
finally, to the extent possible or desirable, the force and couple may be
combined; in this part of the process, the direct association with the
ref eMnce point is lost (unless, as determined, no couple portion of the
resultant exists as associated with the point). In the special circumstance
when the force and couple vectors are perpendicular to one another, this
pair may be reduced further to a single force. In the plane problem, if
both force and couple, as found with respect to the chosen reference
point, are nonzero, these vectors are always mutually perpendicular
and, therefore, may be combined to yield the single-force resultant.
A two-dimensional resultant problem, using the procedure just outlined, is presented in Figure 2-19. The rectangular plate is viewed from
above, initially resting on a frictionless surface. The three forces and the
couple are then applied, and the resultant is sought toward description
of the displacement produced by the loading. If the plate is homogeneous (its weight uniformly distributed throughout its volume), point 0 at
the geometric center is the center of resistance. By the choice of the
300 g
= 300(4)) + 100(0)
= 300— 100 = 200 g
+ 200(4)) + 960)
= —200 g 01 200 g
°x =
== 2mm
FIGURE 2-19. The resultant of a general two-dimensional force system acting on a solid
Bioengineering Analysis of Orthodontic Mechanics
orientation of the x-y-coordinate framework, no decomposition of any of
the given forces is necessary in this example. Each force is then moved
to point 0; the line of action of one of the given forces passes through 0,
so the creation of the concurrent force system there requires the introduction of two additional couples. The resultant force components are
determined, then the force portion of the resultant at point 0. The three
couples may be combined in algebraic fashion, since they occupy a common plane. Note that the couple portion of the resultant has characteristics identical to those of the sum of the moments of the given forces with
respect to point 0; this is always the case, whether the problem is twoor three-dimensional (with vector addition of moments necessary in the
spatial situation). The motion resulting from the application of the force
system may now be described: point 0 moves in the direction and sense
of the 280-g force with accompanying counterclockwise rotation of the
plate. The final sketch shows the single-force resultant following combination of the force and couple portions of the resultant (see the procedure shown in Fig. 2-16); the line of action of the single force is fixed and
is totally unrelated to the initial choice of point 0 as the reference point
for the solution.
The typical resultant problems in orthodontic mechanics involve a
single tooth or a dental segment. Because the active force system is
ordinarily applied to the labial or lingual crown surface(s), the problem is
generally three-dimensional. Although seemingly a somewhat indirect
and lengthy process, from the standpoint of ease of viewing, the elimination of the need to express couples in the vector format, and the definitions of orthodontic displacements, it is usually convenient to decom-
pose the actual spatial situation into two or three two-dimensional
portions. Employing as the reference the center of resistance as seen in
each dental coordinate plane needed, in each plane the force-system
components are reduced to force and couple resultant components. If
then desired, as a final step in the solution, the components may be
combined and the resultant expressed in the three-dimensional format.
Distributed Forces and Their Resultants
Earlier, contact forces were noted as being associated with surface areas
of the bodies acted upon. If such forces directly affect only a small portion of the total surface area of the body, in modeling, that small area is
collapsed around a surface point and a concentrated force is exerted
there. On the other hand, when a force system is spread over a relatively
sizable portion of the surface area in a continuous manner, the system is
termed a distributed force. A resultant, a point force and/or a couple, may
be determined to replace the distributed force system for purposes of
examination of whole-body displacement tendencies. When considering
associated deformations, however, it is necessary to differentiate among
forces according to the sizes of contact areas involved.
Introduction to Analysis of Orthodontic Force
Two familiar examples of distributed force are that exerted by a load
of grain or coal against the floor and sides of a railroad car and the force
of water pushing against a dam. In the analysis of any distributed force,
the pattern of distribution must be established and, to this end, the total
contact area is partitioned into small portions. The amount of the total
force associated with each small part of the subdivided area is determined; for each partial area the fraction of the total force divided by that
partial area is the intensity of force there (measured, for example, in
pounds per square inch). Generally, the intensity of force varies from
one location to another over the total area covered by the distribution. In
the railroad-car example, the distributed force arises from the weight of
the contents; the depth of the grain at any specific location measured
vertically from the car floor is proportional to the local intensity of contact force against the floor. Water pressure increases directly with depth,
and the pressure at a point or level beneath the water surface is the
intensity of force there. If the car floor and the dam-water interface are
plane surfaces, the forces against each partial area are parallel to one
another. In problems involving curved surfaces, the distributions may
be decomposed into components yielding two or three systems of parallel, distributed forces. The number of parallel forces in a specific force
distribution equals the number of parts into which the contact area is
divided. The larger the number of partial areas, the better the approximation by a set of discrete forces of a continuous distribution. If the
intensity of force at any location can be expressed mathematically in
terms of the coordinates of the location within the contact area, an exact
representation of the distribution may be available.
Two examples of distributed force occurring with the activation of
an orthodontic appliance are the "loading" of the periodontal ligament
by a tooth root and the force system generated within the adhesive
when an engaged arch wire transfers contact force through a bonded
bracket pad (or when a masticatory force is unintentionally exerted on
the bracket). Distributed forces can be external or internal to a body
under study; in an analysis of the railroad-car floor, the force of the grain
or coal is a distributed, external load, and every activated arch wire
"contains" distributed, internal force. The dimensions of force intensity
are force divided by length squared and the intensity of internal force is
known as stress. Another vector quantity, stress is discussed in detail in
Chapter 3.
The systems of distributed force of orthodontic interest may be represented or modeled by one or more sets of parallel forces, with all forces
in a set having the same sense. The resultant of a set of such forces is a
single force having the direction and sense of the set, with its line of
action passing through the center of the distribution (generally not the
center of the associated contact area). Although the problem is inherently three-dimensional, many force distributions vary in only one direction parallel to the contact area. In such problems, the two-dimensional sketch of interest wifi contain directions referencing the
magnitude of the force intensity from point to point and the associated
Bioengineering Analysis of Orthodontic Mechanics
coordinate locating points of the contact area; the area itself will appear
only in edge view.
In Figure 2-20, two particular distributions for which the determination of resultants is straightforward are displayed with their resultants.
The distributions are assumed uniform in the direction perpendicular to
the page and, for simplicity, the areas are assumed to be rectangular
with the dimension not seen (perpendicular to the sketch) symbolized
by w. The independent coordinate x, extending from zero to L, locates
points along the edge of the area and, generally, the intensity (at any
point represented by the height of the diagram) may be expressed mathematically as a function of the coordinate x. For any distribution of parallel forces, all having the same sense, the resultant magnitude equals the
average intensity multiplied by the area covered by the distribution. For
the uniformly distributed loading on the left in the figure, the intensity,
which is the same everywhere, is clearly the average intensity. For the
linearly increasing, triangular distribution, the average intensity is onehalf of the sum of the minimum (zero) and maximum intensities. Note
that the resultant force may be computed as the area, within the boundary of the distribution seen in the figure, multiplied by the width w of
the rectangular area covered by the distributed force system. In a dimensional analysis, force/(length2) multiplied by length gives force/length
for the "area" of the sketched diagram, and this multiplied by another
— — — — ——1
P = h0Lw
FIGURE 2-20. The resultants of uniform and linearly-varying, parallel, distributed forces.
Introduction to Analysis of Orthodontic Force
length (perpendicular to the plane of the sketch) yields force. Also note
the locations of the resultant forces; their lines of action pierce the cen-
ters of the rectangular and triangular "areas" shown in the figure.
Again, this development is restricted to distributions over rectangular
areas where the intensifies vary only in one direction along one edge of
an area. Do not confuse the two areas that have been mentioned: the
rectangular contact area over which the force is distributed and the area
within the distribution-of-force sketch.
Two materials touching one another share a contact area. The resistance
to movement tangent to this area, of one material relative to the other, is
known as friction. Friction may be generated between dry solid surfaces;
the resistance to displacement may be lessened if a lubricant is placed
between the contacting surfaces. Friction may exist between two solid
surfaces, at a solid-fluid interface, or between fluid layers. Friction be-
tween solids is termed rolling or sliding, depending upon the form of
relative movement attempted; rolling resistance is highly dependent on
the amount of localized deformation where contact occurs. The resistance that precludes actual motion is termed static friction; that which
exists during motion is called dynamic friction. Both the static and dynamics forms of sliding friction are of orthodontic interest. Sliding friction is
generated between arch wire and bracket when the wire "guides" the
bracket during mesiodistal movement of an individual tooth or when
the arch wire is slipped through posterior crown attachments in, for
example, the retraction of an anterior dental segment.
A classical model of frictional analysis is depicted in Figure 2-21.
Before the active force P is applied, to produce intended horizontal
movement to the left, the fixed surface on which the block rests responds only to the weight of the block with an upward force, perpendicular to the plane contact area. This force, often symbolized by N (representing "normal") is actually the resultant of a distributed force system
against the bottom, rectangular face of the block and, if the block is
homogeneous, initially the distribution is uniform over the contact area.
With the application of the force P at a relatively low magnitude, the
responsive distribution of the force against the bottom surface of the
block becomes nonuniform and the magnitude, direction, and point of
application of the resultant of the distribution are altered. With P sufficiently small, however, the block remains stationary and the changes in
characteristics of the responsive force from the horizontal surface take
place because the contacting surfaces are not perfectly smooth. (If absolutely no friction existed, the application of P, however small in magni-
tude, would result in motion with no change from the static (P = 0)
configuration in the responsive surface force.) With the application of
the force
the now-inclined, responsive, resultant force R may be de-
Bioengineering Analysis of Orthodontic Mechanics
p ,-p 777'?
FIGURE 2-21. Active and responsive forces exerted on a block supported by a rough
horizontal surface.
composed into components perpendicular (normal) and parallel (tan-
gential) to the contact area. Since no potential for vertical motion exists,
the normal component is N and the tangential component, in the direction of but opposite in sense to the intended displacement, is the resistive force of friction. Increasing the magnitude of P results in a conesponding increase in the magnitude of the frictional force f, but motion
does not occur until the force of friction reaches a maximum value corre-
sponding to a critical magnitude of the applied force P. A slight, additional increase in P beyond this critical size results in motion, and
the force of friction generally decreases slightly from the "maximum
1' as the static situation becomes dynamic. A plot of the frictional force
versus the applied force P for the model described is displayed in
Figure 2-22.
Controlled experiments in which frictional forces between plane surfaces were measured have determined that the maximum f and the level
of the frictional force following the initiation of motion are highly dependent on the relative roughness of the contacting surfaces and thtmagnitude of the normal force component N. In the classical model, fmax and
the dynamic frictional force are individually assumed proportional to N;
the proportionality constants are the static and dynamic coefficients of
friction, quantified empirically for pairs of contacting surfaces. Note that
in the static model only the maximum frictional-force magnitude is equal
to the product of the static frictional coefficient and the normal force
component N and the model does not consider the manner of distribution
of contact force between the surfaces.
The classical block-on-plane analysis gives a good fundamental basis
for examining the problem of friction between arch wire and bracket
slot. The block and plane are analogous to the wire and slot, but while
Introduction to Analysis of Orthodontic Force
(Pent, fmax)
Activating Force P
FIGURE 2-22. Relationship between the magnitudes of active and frictional forces, P
and f, for the block and surface of Figure 2-2 1.
only one contact area exists in the foregoing example (Fig. 2-21), the
orthodontic problem is three-dimensional in that the wire may contact
the bracket slot and the accompanying ligation at as many as four locations simultaneously. The direction of the frictional force is bounded by
the angulations of wire and bracket slot; the sense opposes the relative
motion and depends upon whether force is exerted on the bracket or the
wire (action-reaction). If the bracket is to move distally on the wire, the
frictional force against the bracket is generally toward the mesial (and its
action-reaction counterpart against the wire is directed distally). If the
wire is to slide toward the posterior through a stationary bracket, the
frictional force from the wire and against the bracket acts toward the
distal (since the force of friction against the wire is directed mesially,
opposing the intended motion).
Oscillation between static and dynamic forms of friction occurs between the wire and slot in an activated retraction appliance as the relative movement takes place in small "steps." As viewed buccally, one or
more of four separate forms of contact between bracket and arch wire
may be observed during canine retraction. These four forms of contact
are shown in Figure 2-23 and a pair of components, normal and frictional forces, would generally exist at each individual contact area. Influencing the normal component of contact force, and therefore the level of
friction, are the angulation between arch wire and bracket and the tightness of the ligation. The wire material and that of the ligature, for a
given bracket, also influence the frictional resistance. Beyond the classical model, the difference in contact areas that exists with round versus
rectangular wire likely affects the magnitude of frictional resistance. For
a more complete discussion of friction between bracket and wire, refer to
Frank and Nikolai (1980).
Bioengineering Analysis of Orthodontic Mechanics
Zero angulation,
One-edge contact,
Two-edge contact,
zero clearance
Snug wire/slot fit,
zero clearance
FIGURE 2-23. Bucca! views of possible contact modes between bracket and orthodontic
arch wire.
Concepts of Mechanical Equilibrium and Their Applicability to
Orthodontic Mechanics
The definition of "mechanical equilibrium" of a solid body represents an
extension of Newton's First Law for a particle. In effect, the practical
definition says that the net tendencies for translation and rotation of the
body are nil: if the magnitudes of the force and couple portions of
the entire force system exerted on the body are both
the body is in mechanical equilibrium. If the body was stationary before
this "balanced" force system was applied, the system adds zero net
potential for whole-body motion. The balanced force system creates no
inertia, where inertia is proportional to the products of mass and the
acceleration components of the body. When the total force system is
unbalanced, one or the other or both portions of the resultant of that
system are nonzero, an extension of Newton's Second Law governs,
Introduction to Analysis of Orthodontic Force
and the body undergoes accelerated motion and has been given some
An important notion in this discussion is that of the "entire force
system": all of the external forces and couples, both active and responsive,
exerted on the body under study. The definition of mechanical equilibrium may be extended to apply to groups of bodies or assemblies. Inter-
nal forces, by definition being within the body or assembly, exist in
cancelling pairs and, therefore, collectively contribute nothing to a resultant computation. Internal forces, however, may become external, for
example, when an assembly is disassembled for purposes of analysis
and one body or member is studied individually. Pairs of forces associated with the connection(s) of this body to others in the assembly are
internal to the assembly, but one of each pair of forces is external to the
one member.
Under the influence of an activated orthodontic appliance, the dentilion should not be static. Over a finite period of time the teeth are to be
moyed and, correspondingly, within the appliance itself, relative movements and changes in as-activated shapes may occur. The concept of the
resultant of a system of forces has been discussed toward establishing a
procedure by which forms of displacement caused by active forces might
be predicted. Also created with the activation of the appliance, however,
are responsive force systems in need of description: the force system
transmitted through the tooth and exerted by the root on the periodontal ligament, and that delivered by the appliance to the "anchorage," for
Force analyses of the entire tooth, segment, and the appliance and its
elements or parts are to be carried out, but exact results need not be
obtained. Note, then, that although appliance activation generally results in the creation of some translational and rotational inertia, in the
dentition as well as in the appliance, these levels of inertia are not sizable. Both parametric parts of the inertia, the mass and the time rate of
change of velocity (the acceleration), are small. Although the entire complex is more dynamic immediately following activation, during the short
lime period of soft-tissue deformations, than during the much longer
period of tissue remodeling, tooth displacement, and appliance deactivation, the contributions of inertia to the analysis may reasonably be
neglected, even in comparison to the smallest of active forces. The pair
of governing vector equations of the orthodontic dynamics problem
may, then, be written as
Vector sum of all forces
inertia =
(Vector sum of all moments +jot&ti
liriertia)mass center =
With the inertia terms deleted, the relationships are those associated
with the definition of mechanical equilibrium. The equations are necessary and sufficient conditions for a mechanically static system. Thus, at
Bioengineering Analysis of Orthodontic Mechanics
any instant following activation, the dentition and appliance are said to
exist in a quasi-static state and the equations of mechanical equilibrium
may be employed to obtain a close approximation to the relative magni-
tudes of the components of the active and responsive force systems.
The rationale for the consideration of the mechanical equilibrium
problem and its analysis is, therefore, established. The general problem
may be stated as follows: Given a body or group of interconnected bodies subjected to a system of external forces and couples, such that the
state produced by the system is mechanically static, determine the relationships that must hold among the characteristics of the external force
system. Often the problem is one in which several of the characteristics
of the force system are quantified and those characteristics that are not
given explicitly are to be obtained.
Nearly indispensable in proceeding toward the solution of all but the
simplest of problems is the use of the free-body diagram, and several of
these diagrams may be needed to solve one problem. In the solution
procedure, a body or assembly of bodies is earmarked for study and, as
the name suggests, is "freed" from its surroundings (its supporting
structure and all of its connections to other bodies); this is depicted in an
isolated sketch. All of the external forces and couples exerted, both active and responsive, are shown in the sketch. As has been the convention in previous force diagrams, forces and couples exerted on (not by)
the body are displayed. Not only the completed known portions of the
force system, but also those forces and couples with unknown characteristics, must be properly depicted. Moreover, to aid in moment computa-
tions, distances and dimensions must be included. Omitting existing
parts of the force system, including nonexistent forces or couples, and
incorrectly identifying support structures of force-transmission mechanisms (connections) are typical mistakes made in these problems. Once
the free-body diagram has been correctly completed, however, the interrelationships among the known and unknown force-system characteristics may be expressed in a rather straightforward manner. For each freebody diagram drawn, a set of scalar equations may be written, obtained
from the pair of governing vector relationships written previously. In
words, these equations say:
All of the forces exerted on the body or group of bodies, pictured and
under study, must balance, and the moments of all of the forces and coupies, with respect to any chosen moment center or axis, must also balance.
(This is an alternative way of stating that both the force and couple
portions of the resultant of the entire force system must have zero mag-
nitudes.) The number of independent, scalar, algebraic relationships
that may be written from each free-body diagram will depend upon the
type of force system and whether the problem is one-, two-, or threedimensional.
In the quasi-static orthodontic problem, because body forces are
nearly always negligible in comparison with contact forces, understand-
Introduction to Analysis of Orthodontic Force
ing the manner of transfer of force through a connection is fundamen-
tally important. Connections of specific interest include those between
the tooth crown and appliance element(s) and between individual parts
of the orthodontic appliance. Before examining connections within the
appliance, however, the displaying of responsive forces and couples
having unknown characteristics and several general types of connections are considered.
The complete description of a concentrated support or connection
force generally requires the definition of several scalar quantities. The
magnItude of a force or couple is the most common unknown in the
equilibrium problem. To define the direction of the line of action of a
force, one scalar quantity, the size of an angle relative to a reference line,
is needed in the two-dimensional problem; two angles are required in
the spatial problem. Although nearly always known in practical problems of orthodontic interest, the point of application of a concentrated
force is described through the dimensions of the body or in terms of the
coordinates of the point in relation to a specified reference framework.
In approaching an equilibrium problem in which the vector characteristics of a force are sought, with the point of application of the force
known the unknowns may be expressed as the magnitude and one or
two angles or as the two or three components of the force referenced to
an established set of coordinates. Figure 2-24 illustrates the two methods
of displaying an unknown force at a point in two dimensions. On the
left, the force is completely described by the magnitude, the one angle,
and a convention to define sense of the vector; that convention could be
an understanding that the force would appear to be pulling on the point
-__,,_,, ,
Scalar unknowns: F, 6
Scalar unknowns:
FIGURE 2-24. Two methods of illustrating a concentrated force in two dimensions,
unknown except for point of application.
Bioengineering Analysis of Orthodontic Mechanics
and the angle would be measured counterclockwise from the positive
extent of the x-axis, the angle having a value between 0 and 360°. On the
right, the force is completely described in terms of its x- and y-components; two algebraic quantities provide the magnitude and sense of each
component. Because the sign convention leading to the determination of
sense is more straightforward in the component description of a force,
expressing the unknowns as the components themselves is, seemingly,
the more convenient procedure in an analytic approach to the equilibrium problem.
To clarify, then, although sense often is noted as an individual force
characteristic, sense is never a separate unknown in the equilibrium problem. A force at a point is totally described in terms of its components. In
picturing an unknown force in a free-body diagram, an assumption of
sense may be made for each component; in solving an algebraic equation
for a component magnitude, the sign accompanying the answer will
indicate whether the assumed sense was correct (plus sign) or incorrect
(minus sign).
Shown in Table 2-1 are several types of connections between bodies,
the associated force systems exerted on each body (action and reaction),
and the number of scalar quantities involved, exclusive of location of the
connection. The first is, effectively, one-dimensional (although it may
exist within a two- or three-dimensional problem), the next three are
two-dimensional, and the final two are three-dimensional. The first two
entries in the table indicate push-contact between two bodies that share
a contact area. In the general two-dimensional analysis, the resultant of
the distributed force system over the contact area would be displayed as
two force components, one perpendicular to and the other tangent to
the area at the center of the distribution. When it is reasonable to assume
that friction between the bodies is negligible, the force of one body upon
the other is perpendicular to the contact surface and, possibly, only one
characteristic is an unknown: the force magnitude. The tangential component is the frictional force discussed previously, and it generally adds
a second unknown at the contact site (except when proportional to the
normal component, and then the sense of the frictional force must be
correctly indicated).
The third entry in the table, depicting a frictionless hinge connection
(with the pin through the two members perpendicular to the plane of
the sketch), generally transmits a force having unknown magnitude and
direction. The two scalar quantities may be represented as the magnitudes of the components of the force; the components must be mutually
perpendicular, but the pair orientation as a whole may be arbitrarily
selected (often most convenient with one component coinciding in direction with the long dimension of one of the interconnected members).
Shown as the fourth entry is the fixed support. Whereas the hinged
connection possesses no inherent resistance to the angular movement of
one body with respect to the other in the plane of the sketch and in the
absence of friction, the fixed support or connection provides that resist-
Introduction to Analysis of Orthodontic Force
TABLE 2-1. Connections between members and supports and associated force systems
Force system
Description of contact or connection
Bar against
Reactive composaNs of bar
Force components exerted
by surface
Bar against
Action and reaction
counterpart forces perpendicular to plane of
contact. (One componeat magnitude is Unknown.)
Force of
bar against
Notmal force
\ \ surtace
Ft eaction
are unknown.)
by pin
Bar restrained
by frictiortless
pin connection
/ p3
Force components
from bar
Direction of resultant
contact force generalty
not longitudinal. (Two
component magnitudes
are unknown
Force system exerted
by support on bar
Couple represents rasistance to rotation inherent in support
(Action and reaction
interchanged compared
to toregoing examples.
(Three component
magnitudes are unknown.)
Action of bar against
Fixed support of
bar in plane
Frictronless support of shaft in
Fixed support
of bar
Indicated is potential
for bar to slide to the
right. Friction opposes
this tendency. (Two
component magnitudes
No resistance to longitudinat shaft displacement assumed. (Four
component magnifudea
are unknown)
Complete resistance to
linear and angular dinplacements in all directions. (Six component
magnitudes are unknown.)
ance, represented by the couple. In the plane problem only one unknown, its magnitude, is associated with such a couple; the sense of the
couple (clockwise or counterclockwise) is obtained through a sign convention in the manner of that of an unknown force component.
A number of three-dimensional connections are relatively common
within structures and machines, encompassing from one to six potential
unknowns, but just two of interest are displayed in Table 2-1. For the
bearing-shaft connection, negligible resistance to shaft translation
through and rotation within the bearing is assumed. This leaves a force
vector and a couple vector, both perpendicular to the shaft axis, generally transmitted through the connection. Expressing each vector in com-
Bioengineering Analysis of Orthodontic Mechanics
ponent form, a total of four potential unknowns is present. The force is
associated with the positional resistance of one• body relative to the
other, and often equilibrium can exist in the absence of the couple; in
such instances the couple is regarded as a secondary effect in primarily
giving the connection added stability. The final entry in the table is the
totally fixed, rigid connection. This connection, and a weld is a good
example, prohibits both translation and rotation of one body with respect to the other in any direction. The force and couple are expressed as
sets of three components each, giving a total of six potential unknowns.
Before proceeding to a discussion of connections associated with the
orthodontic appliance, a straightforward and applicable statics problem
is examined. Consider an elastic that is stretched between two connection points in the orthodontic complex. The elastic at either end might be
activated against a hook attached to a bracket, a hook soldered or
welded to the arch wire, or a loop bent into the wire, but assume that the
elastic is stretched between just two connection locations. A reasonable
assumption, based on the flexibility of the elastic, is that a single force is
transmitted through the elastic and, therefore, through each connectioti
Shown in Figure 2-25 are seemingly possible free-body diagrams of
the elastic, the first with both forces of apparently unequal magnitudes
drawn in arbitrary directions. If, however, the elastic is in mechanical
equilibrium, and its weight is comparatively negligible, the moments of
the two forces must balance. Choose first a moment center at point A;
with the line of action of P passing through A its moment arm, and
therefore its moment, is zero. Hence, the moment of Q with respect to A
FIGURE 2-25. Preparation of a free-body diagram for a static elastic stretched between
two fixed points.
Introduction to Analysis of Orthodontic Force
also be zero, and thus the line of action of Q must also pass
through point A. The direction of Q is then determined as shown in the
second diagram of Figure 2-25. Now, making a similar argument using
point B as the moment center, the direction of the line of action of I? is
determined; the two forces, accordingly, must share a common line of
action as shown in the third sketch. Finally, because the entire system of
forces must balance in any static situation, the forces P and Q must be
equal in magnitude and opposite in sense. Their common line of action
is determined by the positions of the two connection points, which both
lie on the line. Hence, for any fwo-force member in equilibrium, the two
forces must be equal in magnitude, opposite in sense, and share one line
of action. Generally, the forces may be either pulling away from or pushmust
.ing toward one another, but an elastic has no resistance to pushing
forces and, therefore, when activated it must be "in tension."
Now, with the background of Table 2-1 and the force analysis of a
stretched elastic, a number of connections associated with orthodontic
mechanics are analyzed in Table 2-2. Although all are shown in a twodimensional format, with several a third dimension may be necessary to
represent the connection force system completely. In the first entry just
one potential unknown is present, the magnitude of P1. provided the
orientation (angulation) of the stretched elastic has been or can be measured. Because the bracket in the second example has finite mesiodistal
width, the couple, due to angulation of the wire within the bracket, as
well as the force may be generated. This is somewhat analogous to the
fixed support in two dimensions shown in Table 2-1. Friction, not included in this sketch but potentially present, would add a third scalar
quantity. Examining this connection in an occlusal-plane view, assuming the presence of ligation, another force-couple pair must be present,
and with friction neglected and round wire engaged, this connection is
analogous to that of the bearing-and-shaft connection of Table 2-1.
The third entry in Table 2-2 assumes that the rectangular wire will
not rotate within the bracket slot; the couple shown is torsional in nature. Also generally present is an accompanying force transverse to the
wire, and a total of three potential scalar unknowns in this connection
not including friction. The fourth entry presents a somewhat simplified
representation with friction included; analogous to the second entry in
Table 2-1, this is just one of the possible configurations, in a view from
the buccal perspective, already displayed. (See Fig. 2-23.) The fifth entry
in Table 2-2 shows the resistance to rotation and translation provided by
a buccal tube. The component H5 may arise from a step bend in the
engaged arch wire, a stop, and/or from friction. (If the arch wire was tied
back, H5 would have the opposite sense.) Two additional components, a
buccolingual force and another antirotation couple, would be seen in an
occlusal view, reflecting a total of five potential unknowns associated
with this connection. Finally, the connection of a J-hook to an arch wire
has little; if any, inherent resistance to rotation. This is the three-dimensional extension of the pin connection with three associated potential
unknown force components, two of them existing in the buccal-plane
Bioengineering Analysis of Orthodontic Mechanics
TABLE 2-2. Orthodontic appliance interconnections and contact force systems exerted
on the appliance members
Force system
I-look affixed to
buccal tube and
Force trom elastic
against hook
Hook force against
Direct tension Lines of
action along the
stretched elastic.
Force system exerted
Bracket and arch
Bracket force system
against wire
Second-order angulation and contact
creates couple
Bracket and
Force system from
Force system from
Third-order angulation
and contact creates tingual root torque.
Bracket and arch
Force components
exerted by
force components
Friction opposes relative displacement of
wire and bracket, No
second-order contact
Stopped arch wire
and buccal tube
Generated by stopped
arch wire
Reaction of tube
against wire
Slop contact with mesial extent of tube produces horizontal force
J-hook and hook
Representative of highpull-headgear force
delivery to maxillary
anterior segment of
arch wire
view. Note that the most general and complex connection, analogous to
the final entry in Table 2-1, can be generated between the ligated bracket
and a rectangular arch wire: all three force components (the mésiodistal
component from a stop or friction) and all three couple components may
exist for a total of six potential scalar unknown quantities.
The two vector equations governing the equilibrium problem, which
indicated that both the force and couple portions of the resultant of the
entire force system must be zero, suggest that a maximum of six scalar
relationships may be written among the characteristics of the force system properly displayed in the free-body diagram. Although the majority
of the quasi-static problems to be examined are three-dimensional,
Introduction to Analysis of Orthodontic Force
many of these problems may be modeled into two dimensions. Also,
those problems that must be treated in three dimensions may be decom-
posed into several interrelated two-dimensional analyses and approached in parts, each in its own plane. As noted earlier, a relationship
exists between the maximum number of independent scalar equations
available, relating characteristics of the force system, and the dimensional level and format of that system. Several examples illustrating this
relationship and the solution procedures follow.
In Figure 2-3, two forces were shown acting on a bracket. Returning
to that problem: the equilibrating force exerted on the bracket through
its attachment to the band or its bond to the facial crown surface is to be
determined. To be pictured in a free-body diagram, then, is the bracket
itself and the forces exerted on it. The three existing forces are: the elastic force, the headgear force, and the unknown force. Ignoring the possibility that the two known forces may be slightly out of plane, relative to
each other, and neglecting the bracket thickness, the problem then becomes two-dimensional. Moreover, the force system is concurrent with
the lines of action of all three forces passing through the bracket center.
The free-body diagram is shown in Figure 2-26. All characteristics of
the elastic and headgear forces are known. The point of application of
EL = 200 g
= 400 g
Rm =
Rg = B8
97:O=_147O3O0S1fl 66° = 0
400 COS 66° — 200 COS 74° = 0
B8 = 163 = 55 = 218 g
+ 2182
278 g
FIGURE 2-26. Free-body diagram of an orthodontic bracket subjected to a twodimensional set of concurrent forces.
Bioengineeririg Analysis of Orthodontic Mechanics
the third force is known, but its magnitude and direction are not. There-
fore, two unknown scalar quantities are unknown and are pictured as
mesially- and gingivally-directed components in the facial view. In the
problem which is two-dimensional with the forces concurrent, only two
independent scalar relationships may be written: the algebraic sum of
force components in two mutually perpendicular directions in the plane
must be zero to ensure a force balance. The solutions for the components of the bracket force and, subsequently, the resultant of those components are carried out in the figure. The solution yielding algebraically
positive results indicates correct initial assumptions of the senses of the
components. No moment-balance equation is written. (Clearly, a moment equation using the bracket center as the moment center is identically satisfied since the moment arms of all three forces are zero; hence,
no useful information is obtained from the moment-balance relationship. Another moment center might be chosen, and a moment balance
written with respect to it, but the equation obtained would only be useful in replacing one of the two scalar force relationships.) The force just
obtained is noted to be identical in all characteristics, except sense, to the
resultant of the elastic and headgear forces found in Figure 2-3; in all
equilibrium problems involving a concurrent force system, the resultant
of the given forces must be balanced by the force(s) that involve the
The lever is a second example that has applicability to orthodontic
appliance analysis. Figure 2-27 shows a straight member that could represent an arch-wire segment balanced on a "fulcrum" by the forces P1
and P2 at the ends of the member. The fulcrum force and the relationship between the forces P1 and P2 to ensure equilibrium are sought. It is
= P0 — Pi — P2 = 0
P0 =
P2L2 = P1L1
FIGURE 2-27. A lever subjected to a two-dimensional parallel force system.
Introduction to Analysis of Orthodontic Force
assumed that the applied forces exist in a common plane and are both
perpendicular to the long dimension of the member. Hence, the problem is two-dimensional and the free-body diagram reflects a parallel
force system because neither of the applied forces has a component in
the direction of the long dimension of the member. The responsive (fulcrum) force likewise can have no long-dimension component and reflect
a balanced system. The magnitude of the fulcrum force is obtained by
balancing the forces, algebraically summing the forces in the one direction of all three lines of action. The relationship between P1 and P2 is
obtained through a moment balance using any convenient moment center; in this example, point C, which is immediately above the fulcrum, is
In general, for a two-dimensional parallel force system, two independent scalar equations of equilibrium may be written. One force and
one moment equation were used to analyze the lever example. The general vector moment-balance equation previously described indicates that
the center of mass of the body or assembly under study is to be used as
the moment center. With the inertia effects absent or disregarded, however, it may be demonstrated that a force balance and a moment balance
about the mass center demand that moments balance with respect to
any selected moment center. It follows that the two scalar equilibrium
equations for the problem of Figure 2-27 could have been two moment
equations, and those equations would be independent of one another so
long as the line through the two moment centers is not parallel to the
direction common to the three forces. Figure 2-28 shows a straight
"beam" subjected to a transverse concentrated "load" of 300 g. The
problem of determining the responses at the supports is an inversion of
that of Figure 2-27; sketching the free-body diagram and using the equa-
300 g
FIGURE 2-28. A structural member supported at its ends and subjected to a transverse,
concentrated load. The correct free-body diagram shows a parallel force system in one
Bioengineering Analysis of Orthodontic Mechanics
lions of Figure 2-27 with P0 = 300 g, the magnitudes of the left and right
support may be shown to equal 100 g and 200 g, respectively.
In the final example of this section, a cantilevered straight member is
subjected to a loading consisting of forces at its right end, parallel and
perpendicular to its long dimension, and a couple, all in a common
plane. The left end of the member is engaged in a fixed support, and the
components of the support force system are sought. The free-body diagram of the member includes the support resistances to any movement
of its left end in the plane of the sketch. This is a general two-dimensional equilibrium problem and a total of three independent scalar equations may be written to relate characteristics of the total force system. An
analysis is undertaken in Figure 2-29. A choice of sets of equations is
available: two force and one moment equations, one force and two
100 g
2100 g-mm
80 g
2100 g-mm
— 100 = 0
= 100 g
M8 —
+ 2100
+ CA
CA = 2100 —2800 = —700 g-mm
700 g-mm
FIGURE 2-29. A cantilevered structural member loaded at its "free" end. A twodimensional system of active and responsive forces and couples.
Introduction to Analysis of Orthodontic Force
moment equations (but the line connecting the two moment centers may
not be perpendicular to the direction of the force-component balance),
or three moment equations (but the three moment centers may not lie on
one straight line).
The free-body diagram in Figure 2-29 contains three unknown forcesystem characteristics: the magnitudes of the two force components and
that of the couple. The reference frame adjacent to the free-body diagram defines the chosen directions and assumed positive senses for the
force-balance relationships written below the sketches. Note that the
assumed senses associated with the unknown magnitudes need not coincide with the sign conventions for the equations. In the solution, note
that a negative result was obtained for the support couple; this indicates
that the assumed sense designated in the free-body diagram was incorrect. Also, recalling that the moment of a couple is inherent, note that its
contribution to a moment balance in a two-dimensional analysis is the
same, regardless of the moment-center location. The couple portion of
the load on the member was included in the moment-balance relationship written, even though the moment center chosen at the right end of
the member eliminated contributions from the other applied forces (because their moment arms with respect to point B are zef o) and the couple and moment center appear to be coincident. A partial check of the
solution obtained in the figure may be pursued by writing a moment-
balance equation using point A at the left end of the member as the
moment center.
This chapter introduced the concept of force. Various force manipula-
tions were discussed, including the determinations of the moment of a
force and the resultant of a system of forces. The couple, a special force
system, was defined and its use as a tool in force analysis, as well as its
occurrence as a load and a mechanical response, was considered. Modeling of actual force systems and the bodies acted upon, toward simplification of analyses, were discussed. The resultant of an active force system is sought in order to describe the whole-body displacement that
may be produced by the system. A rationale was developed for analyzing the quasi-static problem, applicable to the orthodontic appliance,
slowly deactivating between appointments, using the governing equadons of mechanical equilibrium.
Force systems produced by activated orthodontic appliances provide
the catalyst for the processes that result in the displacements of teeth.
Forces are also generated in orthopedic actions toward alterations of
growth patterns and to cause movements of dentofacial bones. Hence, a
first step in evaluating the potential of a given appliance, with regard to
its tendency to produce desired displacements in a controlled manner,
Bioengineering Analysis of Orthodontic Mechanics
in the design of a new appliance, is an examination of the forces
created upon activation that are associated with the actual or proposed
appliance. This, however, is only the beginning of the total structural
analysis. Thus far, deformations have not been considered, nor has the
interface between the biologic and mechanical systems. To initiate the
action of the orthodontic mechanics, the appliance must be activated;
one or more appliance elements must be deformed from their passive
configurations, thereby "loading" the appliance and inducing forces
throughout it. Because only a part of the total appliance force system can
be quantified through direct measurement and the use of equilibrium
equations, relationships between force and deformation must be
sought. Such relationships involve the material characteristics of the
appliance members as well as their sizes and shapes and will be discussed in the next chapter.
Frank, C.A., and Nikolai, R.J.: A comparative study of frictional resistances be-
tween orthodontic bracket and arch wire. Am. J. Orthod., 78:593-609, 1980.
Suggested Readings
F,P., and Johnston, E.R., Jr.: Vector Mechanics for Engineers. 3rd Ed. New
York, McGraw-Hill, 1977, Chapters 1 to 4, 6, and 8.
Jarabak, J.R., and Fizzell, J.A.: Technique and Treatment with Light-wire Edgewise Appliances. 2nd Ed. St. Louis, C.V. Mosby, 1972, Chapter 1.
McLean, W.G., and Nelson, E.W.: Engineering Mechanics: Statics and Dynamics. New York, Schaum Publishing, 1962, Chapters 2, 3, 5, and 8.
Mulligan, T.F.: Common Sense Mechanics. Phoenix, CSM, 1982, Chapters 2
and 3.
Smith, R.J., and Burstone, C.J.: Mechanics of tooth movement. Am. J. Orthod.,
85:294—307, 1984.
Thurow, R.C.: Atlas of Orthodontic Principles. St. Louis, C.V. Mosby, 1970,
Chapter 4.
Thurow, R.C.: Edgewise Orthodontics. 4th Ed. St. Louis, CV. Mosby, 1982,
Chapter 2.
Material Behavior of
the Orthodontic
In planning the treatment of a malocclusion, the orthodontist selects a
series of appliances for sequential use in the correction. Each appliance
is a kind of structure. Activated through a deformation from its passive,
as-prepared configuration, the appliance attempts to return to that passive state, but is prevented, temporarily, by the resistance of the dentilion to which it is affixed. The appliance responds to the activation by
exerting a system of forces against those teeth and portions of the dentofacial complex that hold it in an activated state. Creation of contact
forces between the appliance and dentition is accompanied by force sys-
tems within the individual members of the appliance, transmitted
through connections from one member to another. The responses of the
appliance members to orthodontic "loading" are fundamentally dependent on the materials of those members.
The orthodontist uses a variety of materials: metallic-alloy arch
wires, bands, brackets, and headgear components, polymeric "elastics"
(bands, modules, 0-rings, chains, threads), adhesives, alginates, acrylics, plasters, among others. It is important, then, that the clinician be
acquainted with the basic concepts of materials science as applicable to
the specialty. When designing and preparing each arch wire, auxiliary,
or retainer, the orthodontist must be able to predict the response of each
appliance member to "in-service" conditions. The materials are subjected not only to mechanical actions associated with the induced force
systems, but the environment in which the materials must exist also
influences their behavior.
The basis for explaining or predicting the response of a structural or
machine member under specified conditions of use is found in its properties. A property is, in a general sense, a descriptive quantity that gives a
specific characteristic to the member. Properties may be categorized in
many ways, but initially they should be divided into those associated
directly with the material or materials of the member and those termed
Bioengineering Analysis of Orthodontic Mechanics
properties, which are descriptive of its size, shape, and appearance. Material properties are further subdivided into characteristics that
are independent of external influences, simply termed "material" properties, and those that are associated in some way with the conditions of
use or the use environment: mechanical, chemical, thermal, and magnetic, for example. Consider, for instance, the decision of selecting a
spring to produce a particular orthodontic displacement. Properties that
enter into this design decision include the cross-sectional wire size, the
coil diameter, and the overall passive length of the spring (physical characteristics), the stiffness and strength of the wire material (mechanical)
and the spring itself (structural), the resistance of the material to corrosion (chemical), and the effects of a somewhat elevated and varying
temperature (thermal).
The materials of the members of the orthodontic appliance are the
focus of this chapter. Behavior of ancillary materials, such as plasters,
will not be discussed here. The responses of biologic materials during
orthodontic treatment will not be overlooked, but are relegated to a subphysical
sequent chapter. Physical, mechanical, chemical, and thermal are the property categories of primary interest to the orthodontist; optical, electrical,
and magnetic properties, although not to be generally disregarded, are
of little importance within the scope of this text and will not be discussed
in this chapter.
Properties may be interdependent and, as previously mentioned, are
often influenced by conditions of use of the material and in-service environment. The characteristics of a stationary body, directly associated
with its response to applied or induced forces, which include both material and physical contributions, are known as structural properties. The
response to activation of an orthodontic wire loop will depend upon
the wire material, the wire cross-section, the loop geometry, and the
amounts and directions of the bends made in forming the loop. The
behavior of an elastic module used intraorally will differ in some respects from the results of an evaluation of its mechanical characteristics
on the bench in the laboratory under ordinary conditions of temperature
and humidity or its response as part of an extraoral appliance.
Material-property values are generally obtained in controlled experimentation. In-service conditions, as necessary, are created in the laboratory and the required measurements are taken. When widespread use of
the results is expected, and to ensure reliability of comparisons, testing
procedures must be standardized. Much of the standardization in the
United States in testing, particularly in the determination of mechanical
properties, has been and is established by the American Society for Testing Materials. Property quantification and certification of testing procedures in the dental fields is carried out under the auspices of the American Dental Association.
In the sections to follow, the nature and structure of solid materials
are surveyed and the interaction between load and deformation is examined and pertinent structural characteristics are described. The concepts
of stress and strain are introduced, mechanical and structural properties
Material Behavior of the Orthodontic Appliance
obtained through standardized testing are defined and discussed as
needed for developments in subsequent chapters, and chemical and
thermal characteristics of interest are considered. The chapter concludes
with a set of guidelines to be used when selecting orthodontic-appliance
Internal Structure of a Solid Material
Any fluid or solid material is a conglomerate of particles. Forces, generally attractive, exist between pairs of particles, and their magnitudes are
determined by particle densities and distances between particle centers.
The forces of attraction in a fluid are relatively small and, as a result, the
material readily conforms to its container. The particles in a solid are
dense and relatively closely packed; the associated cohesive forces are
large enough that, generally, the solid retains shape in the absence of
any containment.
The smallest subdivided unit of matter, which is representative of a
pure material (chemical element), is the atom. Submicroscopic in size,
the atom is envisioned as a nucleus surrounded by electrons in a kind of
planetary arrangement. The nucleus contains protons and neutrons;
these subparticles have approximately identical masses, the proton possesses a small, positive, electrical charge, and the neutron is electrically
neutral. The mass of an electron is approximately 1/2000th that of the
proton, yet it possesses a negative charge equal in magnitude, but opposite in sense (sign) to that of the proton. The atomic number of an element
equals the number of protons in its nucleus. The atomic weight is approxi-
mately that of the nucleus, neglecting the small contribution from the
electrons, a figure that numerically is, in essence, the sum of the protons
and neutrons and varies with element from slightly more than 1 to in
excess of 250. In the element model, the electrons exist in concentric
orbits, or "shells," with the nucleus at the center.
Heavier atoms may possess more electron shells than light atoms,
but independent of the material element each shell—its position designated radially with respect to the nucleus—has a specific electron capacity. Except for the outside shell, which may be incomplete, all shells are
totally "filled." The outer shell contains, at most, 8 electrons; elements
with like numbers of electrons in their outer shells have similar chemical
behaviors and the number of outer-shell electrons is particularly significant in connection with mechanical, electrical, and magnetic properties.
Atoms hold together because of the intra-atomic attraction between elecIrons and protons. Overall, an atom is said to be electrically neutral if the
numbers of protons and electrons are equal, regardless of the number of
electron shells and the number of electrons in the outer shell.
Although possessing a net charge of zero with numbers of protons
and electrons equal, the stability of an atom with regard to its potential
to combine with other atoms is largely tied to the outer shell and the
Bioengineering Analysis of Orthodonfic Mechanics
number of "valence" electrons in it. Certain materials are inert or chemi-
cally inactive because the pure chemical elements exhibit filled outer
shells together with having no net electrical charge. Many elements do
not occur in their pure forms in nature because their atomic instability
has resulted in chemical reaction or combination with other elements. In
short, the most stable form is that achieved in a configuration with the
outer shell of electrons completely filled. With the exception of the noble
gases, to attain this stabilized state an element must either effectively
gain or lose from one to four electrons. Filling the outer shell results in
the atom assuming a net negative charge, and the atom that loses its
valence electrons assumes a positive charge; in either instance the result
is a transformation from atom to ion.
Ionization yields the potential for the formation of primary, interatomic bonds, of which three forms exist. Ionic bonds are formed when the
valence electrons shed by one group of atoms are transferred to, and
complete the outer shells of, other atoms. Table salt is the result of sodium losing its one valence electron to chlorine, electrically neutral with
seven electrons in its outer shell. Covalent bonding is the sharing of
electrons. The simplest example is the combination of two hydrogen
atoms, another is the sharing among four hydrogen atoms and a carbon
atom to form methane, and a third is the formation of diamond solely
from carbon atoms. Metallic bonds may be formed by atoms of elements
having just a few (generally one or two) valence electrons which are
shed easily; formed is a structure of positive ions and "free" electrons or
an "electron cloud" and between them the bonding occurs. The type of
primary bond is usually indicated by the number of valence electrons in
one of the elements; in a few materials, more than one form of bonding
can exist. Ionic and covalent bonding results in combinations of atoms
known as molecules. Although the molecule is electrically neutral, or
nearly so, due to the nonuniform distribution of electrical charges, intermolecular bonds (also termed secondary or van der Waals bonds) may form.
Several types of secondary bonds are defined, but for all forms their
strengths are substantially less, often an order of magnitude lower than
that of the typical primary bond.
Interatomic and intermolecular forms of attraction pull the particles
toward one another, but spaces also exist within the particle structures,
primarily because mutually repulsive forces between their electron
shells become dominant when two atoms become too close to each
other. Figure 3-1 depicts the forces of attraction and repulsion, both
shown in dashed plots, versus the distance between the centers of a pair
of atoms. The solid curve, the net force versus interatomic spacing,
crosses the horizontal axis at the point of equal attractive and repulsive
forces—the so-called "equilibrium distance." This equilibrium spacing
or distance between atoms varies with the material elements because of
atomic size and the number of adjacent atoms in the molecular structure. Also, temperature is an important, controlling parameter. As the
temperature is raised (lowered) the mean spacing between atoms increases (decreases), accounting for the thermal expansion (contraction)
of materials.
Material Behavior of the Orthodontic Appliance
Force of attraction
Atomic Spacing
/ Force of repulsion
ed: Equilibrium distance
FIGURE 3-1. Plots of interatomic force versus distance between atoms.
Several material properties are related to the bonding characteristics
of atoms and molecules. Density is determined by atomic weight and
spacing. Overall strength of a material depends upon bond strengths.
Material stiffness is related jointly to magnitudes of bond forces and
particle spacings. Thermal and electrical conductivities are low in ionicand covalent-bonded materials and high in materials that are metalli-
cally bonded wherein the free electrons serve as carriers of energy. Prop-
erties are also dependent upon the arrangement of atoms within the
material, and atomic arrangements are generally classified as molecular,
crystalline, or amorphous in structure.
The chemical composition of a limited number of atoms forms a molecule, having a net electrical charge of zero due to primary bonding. From
the characteristics of the constituent atoms, the numbers of bonds, the
bond lengths, and possible relative positions of the atoms (molecular
geometries) may be determined. Often, for a molecular composition,
more than one atomic arrangement is possible; such structural variations
are termed isomers. Molecular materials, solids in particular, are held
together through secondary bonding. Such solids have low melting
points, indicative of the relative weak intermolecular bonding, but the
larger molecules have greater strengths because of the increased num-
Bioengineering Analysis of Orthodontic Mechanics
bers of such bonds. Basic to the understanding of molecular structures is
the study of hydrocarbons. In a so-called "unsaturated" hydrocarbon,
multiple carbon-to-carbon bonds exist that enable the formation of a
large molecule from several smaller molecules in a chemical reaction
known as polymerization. One form of this reaction, termed "addition
polymerization," beginning with like monomers of ethylene, is illustrated
in Figure 3-2. The polymeric molecule, or polymer, obtained is made up
of many repeating units, called mers. A number of polymeric materials
are familiar to the orthodontist: waxes, alginates (impression materials),
acrylics (for retainers), adhesives (for direct or indirect bracket bonding),
and the elastomeric (rubber or rubberlike) bands, modules, "chains,"
and "threads."
Although polymerization can develop lengthy molecular "chains,"
and the formation of van der Waals bonds, or "cross-linking" (vulcanization of rubber, for example) can create two- and three-dimensional
molecular networks having some strength, the majority of engineering
materials are made up of thrde-dimensional, repeating patterns of atoms
called crystals. The primary bonding is typically metallic, but secondary
bonding can result in similar highly ordered patterns such as in the
molecular crystal of table salt. The pictorial conception of a crystal is a
spatial structure of straight lines with the atomic nuclei (in table salt, the
sodium-ion centers) at their intersections. The three-dimensional, repeating element of the crystalline structure is known as the unit cell.
Some 14 different unit-cell geometries are defined and the most common
cubic crystal representations are depicted in Figure 3-3. Various material
properties may be correlated with crystalline structure, and often the
unit cell is indicative of the material class of the solid. Some materials,
however, can exist as a solid in more than one crystal structure (polymorphism or allotropism, analogous to isomerism in polymers). A prime
example is iron, which may exhibit a body- or face-centered-cubic crystal
structure, depending on its temperature or the presence of alloying elements.
Materials that do not display the regular, repetitive patterns found in
crystalline solids and many polymers, but instead reflect a disordered
H: Hydrogen
C: Carbon
FIGURE 3-2. The ethylene monomer and a portion of the long-chain molecule resulting
from addition polymerization.
Material B.ehavior of the Orthodontic Appliance
Simple cubic
Body-centered cubic
Face-centered cubic
FIGURE 3-3. Representations of simple cubic, body-centered cubic, and face-centered
cubic unit cells.
molecular structure, are termed amorphous. Gases and liquids are amor-
phous ("without form") as are a number of oxides including glasses.
Although such materials are not generally of specific interest to the orthodontist, some aggregates of crystalline or molecular and amorphous
materials may be. Plastics are often reinforced with various "fillers." The
divisions between crystals (grain boundaries) contain amorphous matter. Polycrystalline materials are, by definition, aggregates of small crys-
tals. Those polymers having highly disordered molecular structures
might be categorized as aggregate materials.
The internal, submicroscopic structures of many materials are highly
ordered, but imperfections exist. Although often cumulatively occupying
a small portion or volume of the material, certain imperfections, particularly in crystalline solids, cannot be ignored owing to their substantial
impact upon properties. Within the crystals themselves atoms may be
missing from individual cells, foreign atoms may have replaced some of
those of the pure material, atoms may have become displaced to "in-between" (interstitial) locations within the cells, or foreign atoms may
have entered the structure and assumed interstitial positions. An extra
line or plane of atoms, termed a "dislocation," may be present within
the crystal. Imperfections inherently exist at external surfaces where the
exposure of surface atoms makes them more vulnerable to reactions
with matter within the adjacent environment. Within crystalline materials at internal or grain boundaries, amorphous matter up to ten or more
atomic distances in thickness is present which generally possesses at
least some relevant property values inferior to those of the individual
Certain imperfections are created to enhance property values. For
example, brass is a solid solution of copper and zinc, termed "substitutional" because zinc atoms are substituted for some copper atoms in the
crystal structure to produce the alloy. Steel formation involves elevating
the temperature of iron to a level such that the relatively small carbon
atoms can fit into the middle of the face-centered-cubic cell structure of
the iron. As indicated previously, some material properties are highly
sensitive to imperfections. Although the weight fraction of carbon in
steel is often less than 5%, the presence of the carbon greatly improves
its strength properties over those of pure iron. On the other hand, den-
Bloengineering Analysis of Orthodontic Mechanics
sity and material stiffness are termed "structure insensitive" properties
because they are affected only in proportion to the imperfection level or
numbers. (For example, the density of stainless steel is equal to the sum
of the products of the fractional amounts, by mass, and the densities of
the constituent elements.)
Atoms, molecules, and crystals within a solid are retained in particular positions because of the existence of an overall, internal, force balance. The potential for relative displacement is often present, however,
either in the form of some external effect or simply because the particles
are being maintained in a strained configuration. Actually, atoms are
continually in motion, vibrating about their equilibrium positions with
amplitudes dependent upon temperature; apparently, completely static
states occur only at absolute zero. If temperatures become sufficiently
elevated and the particles are initially in strained positions, the vibra-
tions can result in displacements to less constrained configurations.
Mechanical action can cause particle displacements, often to more constrained configurations, as can an electric or magnetic field if the particle
characteristics are such as to respond to the field. A combination of
mechanical and thermal inputs can cause a displacement that cannot be
achieved individually at the same level as the combined action.
The movements of atoms within a crystal often involve point imperfections; with relatively little mechanical thermal input, an atom can be
displaced into a neighboring vacancy or can move a smaller foreign atom
interstitially and take its position. Displacements of molecules usually
require less input than average atomic movements because secondary
bonds are weaker than primary bonds; however, bond density is sometimes an overriding factor. The displacement of entire crystals typically
occurs with the grains sliding with respect to one another; secondary
bonds with grain-boundary matter are broken and then re-established
when the resistance to further movement exceeds what remains of the
initial input.
A somewhat superficial discussion of the internal structure of a solid
material is contained in the preceding paragraphs. Much should be familiar from previous dental-materials coursework, and a more complete
treatment of these topics may be found in the materials-science texts in
the "Suggested Readings" at the end of the chapter. The objective of this
section has been to examine internal material structure to the point of
recognizing the existence of relationships with pertinent properties of
orthodontic appliance materials. Those relationships will be recalled as
the associated properties and material behavior are considered in subsequent sections of this chapter.
Load-Deformation Behavior of a Structural Member
generation of forces within the members of an orthodontic appliance, in addition to those associated with interatomic and intermolecuThe
Material Behavior of the Orthodontic Appliance
bonding, and the transferral of induced forces to and through the
dentition and into the periodontium, results in the remodeling of the
periodontal support system and the displacements of teeth toward corlar
rection of the malocclusion. Although the rigid-body model was used in
the discussion of forces in Chapter 2, the rationale was a simplification
in force analyses when deformations could reasonably be neglected in
comparison to the dimensions of the body or member under study. In
fact, for any stationary, supported body, loading and deformation cannot take place exclusive of one another. To the typical engineering struc-
ture, a "load" (an external force system) is applied which results in a
pattern of deformation; although perhaps unwanted, that deformation
must be included in design or analysis procedures. On the other hand,
the clinician produces a deformation, in activating an orthodontic appliance, to induce the desired force system. Although the cause-and-effect
roles seemingly are reversed in loading the appliance compared to the
typical structure, the parts of the orthodontic appliance may reasonably
be termed "structural members" because they carry and deliver force.
Accordingly, certain appliance-member characteristics defined in this
section will be referred to as structural properties.
The practitioner generally produces and observes deformation before, during, and after activation of the orthodontic appliance. Each deformation, defined as any localized or overall change in shape of a mem-
ber, results in an alteration of the force system within the appliance.
Most familiar are the deformations necessary to activate the appliance,
producing the initial levels of force to be delivered to the dentition. Over
a period of time, with movements of teeth, the initial appliance deformations are lessened with corresponding changes in force magnitudes. At a
succeeding appointment, if the appliance has not totally deactivated
(lost all force) in the meantime, the practitioner may unload the appliance with configuration changes accompanying disengagement. An
appliance member may be said to be structurally passive when it is subjected only to gravitational attraction and any forces responsive only to
its small weight. Prior to appliance engagement and activation, the clinician may affect loading and unloading deformations that take an appliance element, typically a length of arch wire, from one passive configu-
ration to another. In the formation of arches or auxiliaries from
as-received wire, the orthodontist must apply (often with pliers), and
then release, force systems in order to produce the desired shape
Several types of deformation are to be recognized. First, consider two
particles within a structural member. A change in the distance between
particles reflects a normal deformation. If the distance is increased, the
deformation is termed tensile or extensional; if the distance is decreased,
the deformation is compressive. A second fundamental deformation is
associated with a pair of lines of particles of the material. If the angle
between the two lines changes, a shearing deformation has occurred.
Normal deformations are depicted in Figure 3-4; note the changes in
relative positions of particles at right angles to the normal deformation,
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 3-4. Normal deformations, extension and compression, illustrated on an atomic
with bonding forces and distances, the so-called "Poisson
effect." The schematic of a shearing deformation appears in Figure 3-5.
Both normal and shearing deformations can occur simultaneously
within a member—indeed, even at the same location. These fundamental forms of deformation may or may not be visually recognizable, depending upon their sizes and the volumes of material affected; an obvious deformation is the extension associated with the activation of an
Two additional forms of deformation include both of the fundamental forms. Bending is identified externally by a longitudinal shape change
in the member, often from a straight to a curved configuration, as depicted in Figure 3-6. In the analysis of bending, the member is conceptualized as an assembly of longitudinal fibers, all having the same shape.
FIGURE 3-5. An atomic representation of shear deformation.
Material Behavior of the Orthodontic Appliance
Passive bar
Bar deformed in bending
FIGURE 3-6. An illustration of bending detormation, together with two measures of this
form of deformation.
all fibers are attached to one another in a solid member, bending results in the stretching of some fibers and the compressing of others. In Figure 3-6, if external forces hold the member in the deformed
configuration the lower fibers are maintained in compression and the
upper fibers in tension. Two external measures of bending deformation
are defined; as shown in the figure, a transverse deflection 6 denotes the
displacement of a cross-section of the member perpendicular to the longitudinal dimension and an angular coordinate U indicates the rotation of
a cross-section.
The final type of deformation to be introduced is twisting. Figure 3-7
illustrates that the external measure of a twisting deformation is an angular coordinate 4) which denotes the change in orientation of a transverse
reference line in the cross-section. In a twisted member, because the
amounts of rotation generally vary from one cross-section to another,
and since cross-sections are interconnected, the characteristic internal
deformation is a shearing deformation. (Shearing action often also exists
within a member subjected to bending activation, but the accompanying
deformation is negligible except, possibly, when the longitudinal dimension of the member is small.)
Beginning with a structural member in a passive but supported configuration, the application of external force produces an associated deformation. To maintain the deformed configuration induced by the force
system, the force system itself must be maintained and the member is
said to be constrained. Releasing the load results in a second configuration change, and the member again becomes passive. By definition, the
application of force causes an elastic deformation of the member if the
passive configurations, before loading and following unloading, are
geometrically identical. If, upon unloading, the member does not return
to its initial, passive configuration, the loading produces inelastic
Bioengineeririg Analysis of Orthodontic Mechanics
Passive bar
Bar deformed
in torsion
FIGURE 3-7. A twist deformation and an angle 4' as a measure of it.
As a first example to study, totally from an external standpoint, the
load-deformation behavior of a structural member, the bending of an
arch-wire segment is examined. Consider an initially straight piece of
wire between two fixed supports. The span, the distance between the
supports, is sizable in comparison to the cross-sectional dimensions of
the wire. The wire segment has been cut from a longer piece of as-
Material Behavior of the Orthodontic Appliance
----Deflection ô
FIGURE 3-8. An arch-wire segment subjected to bending deformation produced by a
transverse, midspan force.
received wire. A point force is applied midway between the supports to
generate a bending deformation; specifically, the external deformation
of interest is the transverse deflection of the cross-section directly under
the applied load. Shown schematically in Figure 3-8 are the undeformed
shape (dashed) and a deformed configuration (solid), the latter generated by the applied force. The load-deformation curve, given in Figure
3-9, is the result of the plotting of corresponding values of applied force
and deflected position of the midspan cross-section relative to its passive
location. The load is slowly increased in small increments and the accu-
mulated deflection grows until the simple structure collapses (is no
S: Stiffness
p1: Proportional limit
el: Elastic limit
FIGURE 3-9. A p'ot of applied force versus midspan deflection for the wire segment of
Figure 3-8.
Bloengineering Analysis of Orthodontic Mechanics
longer able to sustain the load). The overall shape of the curve is charac-
teristic of crystalline materials, regardless of the member geometry or
the type of deformation induced. The initial portion of the plot is linear
with the slope then decreasing as the loading (plotted as the ordinate on
the graph) becomes relatively substantial. The end point of the linear
portion of the plot is appropriately termed the proportional limit. The
coordinates of the final point of the curve denote the largest values of
deformation and force before failure.
The application of a relatively light force to the member of Figure 3-8
results in a small deflection; the two values are the coordinates of a point
on the curve of Figure 3-9, most likely below the proportional limit. If
the load is held constant and the temperature is not elevated substantially, the deformation of the metallic member will generally remain unchanged, virtually independent of time. Upon release of the load the
deformation will disappear and, in general, the original straight-line
shape will be regained, provided the elastic limit has not been exceeded.
The elastic limit is the extreme point on the force-deformation plot from
which unloading will occur with no permanent deformation resulting
from the load-unload cycle.
Points 1 and 2 in Figure 3-9 are below and above the elastic limit,
respectively. An unloading plot from point 1 traces the loading plot
between the origin and point 1. The dashed unloading curve from point
2 is notably linear and parallel to the linear portion of the loading plot;
the deformation coordinate of the point of intersection of this unloading
curve with the abscissa (deformation axis) denotes the permanent deformation of the segment and a new passive configuration of the now bent
wire. Internally, the loading to point 1 stretches some interatomic bonds
and pushes other atoms toward one another, but no bonds are broken
and, as the load is released, the atoms return to their prior equilibrium
positions. Loading to point 2, however, severs some bonds and results
in movements of crystals relative to one another; new bonds are established that are reflected upon unloading in a new, altered, passive configuration. The elastic limit for a crystalline member is found experimentally to be only slightly beyond the proportional limit and, to simplify
computations, the elastic limit is often "placed" at the end of the linear
portion of the plot, coincident with the proportional limit.
The elastic limit divides the structural character of the wire segment
and the load-deformation plot into regions of totally elastic behavior and
inelastic response. The slope of the linear part of the curve is the bending
stiffness at the "midspan" location of the arch-wire segment. In an orthodontic activation, the loading of a wire is intended to be to a point not
beyond the proportional limit so that the totally deactivated configuration is known: generally identical to the preload, passive geometry.
Hence, in an elastic activation, the stiffness when quantified enables the
determination of (1) the magnitude of the load corresponding to the
activating deformation—the load is the product of stiffness and - the
amount of activation—and (2) the reduction in the as-activated level of
Material Behavior of the Orthodontic Appliance
loading during deactivation—the product of stiffness and quantified
deactivation increment.
The coordinates of the elastic-limit point on the plot are known as the
elastic range and the elastic strength. In this example, the elastic range is
the largest transverse bending deformation of the midspan cross-section
that the wire segment can experience and subsequently deactivate
totally or be unloaded without sustaining a permanent bend. The elastic
strength is the maximum magnitude of applied, midspan, bending force
without inelastic material behavior occurring somewhere in the segment. A fourth notable elastic structural property is termed resilience.
An energy-related quantity, and as such to be discussed in more detail
in Chapter 4, resilience is, quantified, the area under the load-deformation diagram up to the elastic limit; for crystalline materials that area is
triangular. Dependent collectively upon both elastic-limit coordinates,
which themselves are intradependent with stiffness, a member exhibiting low stiffness but a high elastic range may possess a resilience similar
to that of a member having greater stiffness but a lesser elastic range.
Although the activation of an arch wire in bending is not intended to
take the member beyond its elastic limit, the nonlinear portion of the
load-deformation diagram is also important because inelastic bending is
necessary by the practitioner to place permanent bends. Taking the example wire segment to point 2 on the load-deformation curve results in
the appearance of a "gable bend" midway between the supports. How
severe a permanent bend the wire can sustain without fracture depends
upon the ductility of the wire in bending. Two measurements of ductility
are the deformation to fracture, the horizontal coordinate in Figure 3-9 of
the final point of the plot, compared to the elastic range, and the toughness, the total area under the load-deformation diagram, compared to
the resilience. Both of these quantities just defined, compared to their
elastic counterparts, are substantial for the very ductile ligature wire, for
example. In contrast, both comparisons are 1:1 for a perfectly brittle
member; the fracture point and elastic limit coincide on the load-deformation diagram.
A second, pertinent, structural example is that of a stretched elastomeric module. Schematics showing the passive and an activated configuration are presented in Figure 3-10. One end of the module is fixed and
the member is elongated by the application of a concentrated force at the
other end. The overall deformation (e) is the extended length (L) minus
the passive length (L0). As with the arch wire in bending, the larger the
load, the greater the deformation, but the module exhibits a two- or
three-part load-deformation diagram, quite different from that of the
arch wire. An elastomer is a rubber or rubberlike, often synthetic, polymeric material. The nature of the bonding of atoms and molecules, and
the imperfections, make the material inherently more flexible than a
crystalline solid of the same size and shape. A typical two-part diagram
is presented in Figure 3-11; the solid curve represents loading to failure
by rupture (separation). Along the lengthy, linear portion of the plot,
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 3-10. Activation of an elastomeric module in direct tension.
the elongation under load is reflected principally in the uncoiling of long
molecular chains. (A typical three-part curve suggests, in the initial nonlinear portion, that secondary bonds must be broken before the major
uncoiling of chain molecules can begin; that first part of the curve exhibits decreasing slope with increasing applied force.) As the uncoiling is
completed, material stiffness increases as primary bonds begin to be
stretched; with continued increasing of the load the slope of the curve
grows and rupture eventually occurs. Accurate determination of the
coordinates of the elastic limit of the module is difficult because of the
time-dependent nature of the mechanical behavior of the material (to be
discussed subsequently). Most elastomers, however, are relatively brittle; their elastic limits are "high" on the load-deformation curves, beyond the linear initial or middle portions of these plots. Inelastic action,
therefore, is never expected in clinical use, as the name "elastomer"
The dashed curve in Figure 3-11 is representative of rather rapid
unloading of the module. Unloaded from a point below its elastic limit,
some deformation remains immediately after removal of the activating
force, but over time, full recovery of the preload, passive configuration
takes place as secondary bonds are re-established (dashed line). Strktly
speaking, orthodontic deactivation cannot be shown in Figure 3-11 be-
cause the force in the module depends upon the time elapsed while
Material Behavior of the Orthodontic Appliance
Extensional Deformation e
FIGURE 3-11. The force-deformation diagram for the module of Figure 3-10.
under load, in addition to the amount of elongating activation. Hence,
stiffness, as the slope of the load-deformation curve, can only be approximated because it, too, is changing with time. As previously mentioned,
time-dependent behavior is considered further in a subsequent section
of this chapter.
Mechanical Stress
concept of internal force was introduced in Chapter 2. The interatomic and intermolecular bonding forces discussed previously in this
chapter are fundamentally part of any internal force system. Superimposed on the bonding forces, however, are internal force distributions
arising from the external loading of the structure through the application
of mechanical force or through the imposition of an activating deformation and maintenance of a constrained configuration. The loading of one
member creates internal forces in the whole of the structure; these forces
seemingly travel through the loaded member, across the connections
and into adjacent members, and ultimately into the supports or foundation of the structure. These internal forces, both from bonding and from
external loading, not only exist throughout the lengths of the members,
but are also distributed over the entirety of the member cross-sections.
Bioengineering Analysis of Orthodontic Mechanics
In general, the intensity of internal force can vary in all directions within
a member. (In structural analyses, although the internal force system
arising from external loading is superimposed on the bonding force system and, likely, also a residual internal-force distribution remaining
from manufacturing or preactivation fabrication processes, the analyses
often ignore all but the first on the basis of magnitude comparisons.)
As a first example in this section, consider the activated elastomeric
module of Figure 3-10. Held in this constrained state, although
stretched, the module is in mechanical equilibrium with its supports.
The internal force system can be exposed in part by making an imaginary cut through the module and arguing that the resulting two parts
are each in equilibrium as is the whole of the module. The specific loca-
tion of the "cut" along the module length is arbitrary in this example
and is simply made perpendicular to the longitudinal direction. The two
portions of the module are shown in Figure 3-12 together with the activating and responsive forces on the entire module. The external forces at
the module ends are equal and opposite from a force balance; the exposed internal forces are equal and opposite because they are action-reaction counterparts. Assumed is a one-material, homogeneous module
and, with the line of action of the external forces coinciding with the
longitudinal axis of geometric symmetry, the distribution of internal
force, arising from the loading, is essentially uniform over the cross-section as illustrated. Also, with the resultant of the internal force system
equal and opposite to the external force (from a longitudinal force balance on either module portion), the internal distribution is wholly pulling against the cross-section of the "cut." Finally, because the external
loading is only at the ends of the module, and with its weight negligible,
the pattern of internal forces on all cross-sections having identical orientations is the same from one end of the module to the other, regardless
of the longitudinal location of the "cut." The average intensity of the
internal force
in Fig. 3-12) is the resultant, internal-force magnitude
(F) divided by the cross-sectional area (As) exposed by the "cut." In the
determination of the initiation of inelastic material behavior or failure,
intensity of internal force is the critical parameter. In short, but sometimes oversimplified, the larger the cross-section of a structural member,
the greater is the load that it can carry.
Stress is defined as the intensity of internal force, per unit area, at a
point within a machine or structural member. It is a vector quantity,
obtained by dividing a vector (force) by a scalar (area). The dimensions
of stress are force divided by length-squared (FIL2). A "cut" exposes an
area internal to the member, and that area is divided into a number of
subareas with a portion of the internal-force resultant exerted on each
subarea. As the number of subareas increases, the size of each subarea
becomes smaller and the pattern of internal-force intensity (stress) becomes more detailed. In the module example, the stress everywhere is
normal to the area and is the same on every subarea of the exposed
cross-section, but this is a special situation; in general, the stress may be
oblique to the area and may vary over the whole of a cross-sectional area
Material Behavior of the Orthodontic Appliance
FIGURE 3-12. Activation and response at the ends of an elastomeric module, Internal
force distribution over a cross-section of the module.
Bioengineering Analysis of Orthodontic Mechanics
Side view
FIGURE 3-13. A general stress vector decomposed into normal- and shear-stress
of a member. Shown in Figure 3-13, within the total cross-section, is a
typical subarea (As) (enlarged) with the stress s associated with it. Nearly
always convenient is the decomposition of the stress vector into components normal (perpendicular) to and tangential to the area; these components are known as the normal and shear stresses, symbolized by a or s,,
and r or s5, respectively. When the normal stress is apparently pulling
against the area, as in Figure 3-13, it is called tensile stress. A normal
stress that pushes against the area is a compressive stress, and is directly
analogous to pressure in a fluid. Note the correspondence between the
two categories of normal stress and the forms of deformation previously
As an example illustrating the generation of shear stresses, consider
an orthodontic bracket pad bonded to a facial crown surface and the
bracket subjected to an external force parallel to the facial surface. Figure
3-14, on the left in perhaps a mesial or distal view, shows schematically
the enamel crown surface, an exaggerated sketch of the deformed adhesive, and the loaded bracket. On the right, following a "cut" through the
adhesive, which is made parallel to the load Q, a free-body diagram has
been drawn of the bracket, pad, and a portion of the adhesive as a
whole. Neglecting weight and the faciolingual thickness, the distributed
internal force system exposed by the "cut" is entirely shear. A force
balance indicates that the resultant internal force must equal Q. The
average shear stress in the adhesive equals the resultant of the distribu-
tion divided by the area in shear k.
In the examples of the module in tension and the adhesive in shear,
the resultant internal force equals the product of average stress and the
corresponding area. The computation in arch-wire bending, however, is
somewhat more complex. Figure 3-15 depicts a wire segment with the
"right-section cut" (perpendicular to the longitudinal dimension) exposing an internal cross-sectional area (seen only in edge view in the figure)
and a typical internal force system including both forms of normal stress
as well as shear stress, with none of the three uniformly distributed. In
this segment, no inelastic behavior is present and the resultant of the
tensile- and compressive-stress distributions is a pair of equal, opposite,
and parallel forces—a "bending couple." The resultant of the shear-
Material Behavior of the Orthodontic Appliance
Average shear stress
where A,.
total area exposed
by the cut'
FIGURE 3-14. Shear loading of a bonded bracket and shear-stress distribution within
the bonding adhesive.
distribution is an internal force tangent to the area exposed by the
"cut" and parallel to the direction of external force causing the bending deformation. A complete discussion of bending is undertaken in
Chapter 7.
Mechanical and Structural Properties: Standardized Testing
response of a structural member to external loading depends
upon the size and shape, or geometry, of the member, the relationship
of the loading pattern or characteristics to that geometry, and the mechanical properties of the material of the member. Problems of analysis
Bioengineering Analysis of Orthodontic Mechanics
Wire segment
Position and
direction ot
shear stresses
ot shear stress
FIGURE 3-15. Typical stress patterns internal to a structural member subjected to inplane bending.
or design of structures are often approached by taking experimentally
determined mechanical-property values, substituting them into formulas or equations that express the governing principles of mechanics and
include the contributions of geometric and loading parameters, and
thereby obtain or check what might be termed structural potential or
capacity. The differences, if any, in bending characteristics of two, .016-
in. -diameter, arch wires, for example, are founded in the mechanical
properties of the wire materials. Hence, a familiarity with these material
properties is important to the practitioner.
Many of the fundamental mechanical properties of a solid material
are commonly quantified through the performance of a static tension test.
Each test specimen is carefully prepared to minimize experimental varia-
tion or "error" associated with the testing itself. The test volume (the
middle portion, longitudinally) of the specimen is uniform in cross-sectional shape and size; the cross-section often exhibits at least one axis of
geometric symmetry. The straight specimen is to be held at its ends,
some distance away from the test volume such that the manner of attachment of the loading system to the specimen will not influence the
test volume and the data gathered.
An example test specimen is shown in the "grips" of the testing
apparatus in Figure 3-16. The grips transfer the loading, measured by
the testing machine, to the specimen in the resultant form of a longitudinal force having a line of action that passes through the centers of the
cross-sections of the test volume. With the weight of the specimen ne-
glected (compared to the load), the specimen is a two-force member,
similar to the module of Figure 3-12 and as described in Chapter 2. The
grips are attached to the "heads" of the testing machine; one head is
movable and the other is stationary. The testing machine has the capability to apply a slowly increasing load (thus the term "static" test) and
Load P
FIGURE 3-16. A tensile-test specimen in the grips of a standard test apparatus.
Bioengineering Analysis of Orthodontic Mechanics
to monitor continuously the magnitude of that load. A longitudinal or
axial dimension within the test volume is known as the "gauge length."
Its initial value, before external loading is begun, is shown in Figure 3-16
as 4?,,. An "extensometer" is attached to the specimen at the extremes of
the gauge length, or an alternative procedure is employed, to enable
instantaneous measurements of the gauge length which increases during the tension test. (The extensometer weight must also be negligible
compared to the specimen load or counterbalanced so as not to create
notable testing error or destroy the mechanical symmetry.) In the test
itself, the loading is begun from zero and gradually increased until rupture of the specimen occurs. Readings of applied load (which equals the
internal axial tensile force) and the corresponding extensional deformation (or extended length) of the gauge length are taken, either manually
within equal loading increments or, by means of transducers and a stripchart recorder, recorded continuously by the testing machine and plotted as the test is proceeding.
The extension of the gauge length during the tension test will depend upon its initial length; therefore, this extension or elongation is
"normalized" with respect to the initial length. Axial deformation per
unit length is a form of engineering normal strain:
The test under discussion produces longitudinal tensile strain; if the loading sense is reversed, the gauge length is lessened under pushing force
and longitudinal compressive strain is induced. Corresponding to the normal strains are normal stresses against cross-sectional areas exposed by
"cutting" the specimen perpendicular to its longitudinal axis. The uniformly distributed internal force system, as in the module example, has
an intensity that depends on the size of the exposed area. The engineering tensile stress equals the resultant, internal force divided by the pas-
sive, cross-sectional area. Note that by converting force to stress and
deformation to strain, the net effect is the dividing out of the volume of
material in the test section and, thereby, eliminating the influences of
specimen size from the test results. (As the test specimen elongates
under load, the cross-sectional area generally reduces; recall the Poisson
effect illustrated in Figure 3-4. Engineering stress and strain are obtaine&
by dividing out the undefonned, as-prepared area and length of the test
volume. True stress and strain are obtained by dividing the instantaneous,
deformed area and length into the corresponding internal force and elon-
gallon, respectively. All stresses and strains discussed in this chapter
and throughout the text are engineering stresses and strains.)
From the paired readings taken during the tension test, converting
force to stress and elongation or stretched length to strain, a tensile
stress-strain diagram is prepared (either by hand or plotted by the testing
machine). With modifications of coordinates of data only through division by constants, the stress-strain diagram in tension for a specific ma-
Material Behavior of the Orthodontic Appliance
x: tracture point
Tensile Strain
Tensile Strain
FIGURE 3-17. Stress-strain diagrams in tension, obtained from static testing, for mild,
structural steel (left) and for a stainless steel (right).
terial has the same shape as the tensile, load-versus-deformation plot for
a structural member of the same material. Shown in Figure 3-17 are
tensile stress-strain diagrams depicting the behavior of mild, structural
steel, which exhibits a yielding phenomenon, and a stainless steel,
which does not notably yield. Yielding is the straining of the material at
virtually a constant stress well below the level at which rupture occurs.
Illustrated in Figure 3-18 are diagrams for a ceramic material and twoand three-part diagrams for typical polymeric materials. The mechanical
properties of interest are obtained from the stress-strain diagram for the
material and from measurements taken directly from the specimen, and
are analogous to corresponding structural properties, many of which
have been defined previously.
A crystalline material obeys Hooke's Law, which states that the stress-
strain ratio is constant up to the proportional limit; the constant in this
linear stress-strain relationship is the modulus of elasticity E (Young's
modulus) in tension, the elastic stiffness of the material and the slope of
the initial portion of the diagram. The ultimate strength of the material is
the value of the maximum ordinate of the curve; as indicated in Figure
3-17 this may or may not be the stress level at fracture (but if the high
point of the cprve is reached in loading, although data may be taken at
greater strains, the imminent fracture generally cannot be prevented).
The coordinates of the fracture/rupture point are ordinarily termed the
fracture strain and fracture stress or strength. The proportional and elastic
limits are close together on the curve; the coordinates of the elastic limit
are termed the elastic-limit strain and elastic strength of the material. Accordingly, the simple relationship a = Ec, which is the expression of
Bioengineering Analysis of Orthodontic Mechanics
A ceramic
x: fracture
Polymeric materials
Tensile Strain
FIGURE 3-18. Static stress-strain diagrams in tension for ceramic and polymeric
Hooke's Law in direct tension, holds virtually up to the elastic limit. For
those crystalline materials that yield, the yield strength is the stress level
at which this phenomenon is exhibited; because yielding immediately
follows the initiation of inelastic action in tension, the values of elastic
strength and yield strength are, for practical purposes, equal.
In the discussion of mechanical properties, the word "modulus"
implies "per unit volume;" accordingly, the modulus of resilience of the
material is quantitatively the area under the stress-strain diagram up to
the elastic limit and the modulus of toughness is the total area under the
diagram to the point of fracture. A material is relatively ductile or brittle,
depending on the extent of the stress-strain diagram beyond the elastic
limit. Clearly, the ceramic material of Figure 3-18 reflects no ductility at
all; the plot being linear to fracture indicates the nonexistence of an
elastic limit and the inability of the material to take a permanent strain.
Typical measures of ductility are, analogous to the content of an earlier
discussion of structural characteristics, the fracture strain referenced to
the elastic-limit strain and the modulus of toughness relative to the modulus of resilience. A third measure is obtained by taking the two por-
tions of the fracture specimen from the testing machine, mating the
fractured surfaces, measuring the as-fractured gauge length, determining the corresponding strain value, and finally multiplying by 100 to
obtain the percent elongation at fracture. A fourth measure of ductility,
Material Behavior of the Orthodontic Appliance
similar to the third, is obtained from the pretest cross-sectional area and
the cross-sectional area of the fractured specimen at the fracture location: the percent reduction in area.
Noncrystalline materials, such as the polymers of Figure 3-18, seldom exhibit any useful level of ductility and often do not obey Hooke's
Law. As a result, there is little distinction between elastic strength and
ultimate strength. Although not exactly correct, the elastic limit may be
taken as the point where the slope of the stress-strain curve begins to
increase substantially toward rupture. The stiffness per unit volume is
not a constant in those ranges where the curve is nonlinear; hence, the
simple relation between stiffness modulus and the coordinates of the
elastic limit, (cr/e)ej = E for crystalline materials, is not valid for those
materials exhibiting a nonlinear first portion of their stress-strain diagrams. To determine the stress level, for a given activation in tension of
an elastic for example, it is best to directly measure the force and divide
that magnitude by the cross-sectional area.
Mechanical properties of materials are obtained from simple experiments, tension tests for example, which generate the fundamental forms
of deformation cited previously. Because their values often depend
upon the form of deformation generated in the test, these properties
when quantified must be presented as "in tension," "in compression,"
or "in shear." In addition, the rate of loading during the test will influence mechanical-property values to some extent; for example, distinctions are made between static and impact tests. Furthermore, environmental conditions, particularly temperature in the test locality, may
affect values, as may specimen-storage time and conditions prior to testing. Finally, the as-received internal material configuration will partially
determine the values of some mechanical properties, in particular for
a metallic material the coordinates of the elastic limit and, therefore,
its modulus of resilience. (The influence on properties of permanent
deformations and accompanying "residual stresses" is considered in
Chapter 4.)
Material behavior in static compression is similar to that exhibited in
a tension test, but some differences in test specimens and procedures,
and in obtained mechanical-property values, do exist. In an axial compression test, with the sense of the loading reversed from that of the
tension test, the as-prepared specimen often has a uniform cross-sectional geometry from one end to the other, and is short and thick to
ensure symmetric loading throughout the test and to avoid the possible
occurrence of lateral deflection or buckling. Load and deformation data
are taken, converted to compressive stress and strain, and the diagram
is constructed. For a crystalline material, the modulus of elasticity is
generally the same in tension and compression, although the extension
of the linear portion of the curve to a greater elastic-strength magnitude
in compression than in tension is not uncommon. Noncrystalline materials may exhibit a substantially altered stress-strain diagram in compression compared to the tension plot. With molecular materials not
extensively cross-linked, the chain molecules that uncoil at low to mod-
Bioengineering Analysis of Orthodontic Mechanics
loads in tension clearly would not do so to any great extent in
compression. Although some tensile strain is developed laterally from
the Poisson effect accompanying the axial loading, with compressive
loading, interatomic bond deformations begin almost immediately. Ultierate
mate strength in compression is defined and computed in the same
manner as in tension, but the failure mode must be substantially different from the tensile rupture. The load-carrying capacity of a member in
direct compression is lost when one of the following occurs: extensive
lateral deformation (buckling); longitudinal splitting (also known as internal buckling); or a "slipping fracture," on a plane inclined with respect to the loading axis, due to the exceeding of the ultimate strength of
the material in shear. (Although absent on right cross-sections in members undergoing direct tensile or compressive loading, shear stresses
exist on all other planes exposed by "cuts" through the members and
are largest on planes at 45° with the loading axis.) The nonlinear portion
of the stress-strain curve in compression, seemingly related to ductility,
is often reduced comparatively in materials that can sustain at least moderate amounts of inelastic action in tension; the opposite may be true
with apparently brittle materials because right-section rupture cannot
occur under compressive loading.
Examples of actual or potential shear deformation, associated with
orthodontic appliances, were cited previously; a schematic was shown
in Figure 3-5 and the shear strain, generally symbolized by y or
defined as the tangent of the angle developed as two intersecting line
segments at a point within the material, passively perpendicular, are
inclined to one another with the member under load. In a test to determine mechanical properties in shear, the specimen and loading are typically similar to the bonded bracket assembly of Figure 3-14 except that
the specimen is quite small, is of one material, and in particular the
moment distance of the applied force with respect to the specimen support is minimized. Readings of load and angular deformation obtained
are converted to shear stress and strain, and the reduced data are plotted to give the stress-strain diagram. Two example diagrams are presented in Figure 3-19.
The form of the plot for the crystalline material is, by now, familiar;
mechanical properties as derived from the diagram are defined exactly
as in the tension or compression test except each name ends with the
phrase "in shear." The slope of the linear part of the shear-test curve,
however, is generally often termed the modulus of rigidity G. From stiffness magnitudes, many metallic materials seem to be more flexible in
shear; for arch-wire alloys, the ratio of elastic moduli in identical units,
shear to axial tension, is about 0.4. (Although measured differently,
both normal and shear strains are nondimensional, so these stiffnesses
are dimensionally alike.) Hooke's Law for pure shear loading is written
T = G-y and is valid up to the proportional limit in shear. For ductile
materials in general, the elastic and ultimate strengths in shear are about
one-half of the comparable values in tension; for brittle materials, the
ratio of ultimate strength in shear to that in tension approaches unity.
Material Behavior of the Orthodontic Appliance
A crystalline
Shear Strain
FIGURE 3-19. Static stress-strain diagrams in shear for crystalline and polymeric
The forms of the stress-strain diagram in shear and in compression are
similar for most molecular and amorphous materials; the initial nonlinear portion of the three-part curves in tension of some noncrystalline
materials, as shown in Figure 3-18, is rarely seen under compressive or
shear loading.
Previously depending upon mechanical-property data generated
elsewhere, in 1977 the Council on Dental Materials and Devices of the
American Dental Association announced the ADA Specification Number 32 for orthodontic wires not containing precious metals. The Specification includes a detailed description of the preparation and testing of
arch-wire specimens toward the determination of the modulus of elasticity and "yield strength" for the arch-wire material in flexure (bending).
A schematic of the cantilevered-beam test is given in Figure 3-20. A
bending couple is applied to one end of the specimen where only rota-
tion is permitted; at the other end of the test span the wire is held
against a fixed, knife-edge stop. The angular deformation measured is
the rotation of the shaft, which is also the rotation of the end cross-section of the specimen engaging the shaft with respect to the passive orientation.
A typical plot of the couple versus the rotational deformation, for a
specimen of metallic material not exhibiting a well-defined yield point, is
Bioengineering Analysis of Orthodontic Mechanics
Load couple C
FIGURE 3-20. An illustration of the fiexure-test arrangement dictated by ADA
Specification No. 32.
presented in Figure 3-21. To obtain an equivalent yield point, approximating the elastic limit, the dashed line is drawn parallel to the initial,
linear portion of the plot, intersecting the deformation axis at the specified offset (approximately 2.9° according to Specification No. 32), and
the yield strength of the material is determined from the couple magnitude at the point. (The actual computations to quantify the elastic modu-
lus and yield strength in flexure are outlined in Chapter 7.) A third
material property of orthodontic interest may then be computed; the
ratio of yield strength to modulus of elasticity is termed the springback of
the material and is a close approximation to the elastic-limit strain. Table
3-1 contains approximate values of these three properties in bending for
five arch-wire materials.
In addition to the static flexure test, an experimental procedure that
may be terminated after the data confirms exceeding of the elastic limit,
another is required which, in part, provides an indication of the ductility
of the wire material. In this second portion of the Specification, a 90°
bend is placed in the wire, then removed, and then replaced at the same
cross-section and in the same direction. The radius of the bend is specified. The 90-to-0-to-90° bending cycle is continued at a specific rate until
fracture occurs. The number of cycles to failure is then compared with
the Specification requirements, which vary somewhat with wire size.
Before concluding this section, several additional comments associated with testing and mechanical properties are in order. Indicating that
a material is relatively "hard" or "soft" does not pertain, strictly speaking, to any previously discussed property. Hardness is defined as the
resistance of a material to localized, permanent, compressive deformation (indentation); it is, therefore, largely a surface phenomenon. Hardness numbers are determined experimentally by subjecting a material
specimen to an indenter of a specific geometry exerting a designated
load. Indices of hardness familiar to engineers are the Brinell and Rockwell hardness scales. Although hardness is very nearly, directly proportional to ultimate tensile strength for a number of metallic materials, this
property must not be confused with material stiffness or resilience.
Mechanical properties of some materials are difficult to quantify by
testing because their stress-strain characteristics change with time under
Material Behavior of the Orthodontic Appliance
yield point
Angular Deformation 0
FIGURE 3-21. Couple versus angular deformation; typical plot of flexure-test data
obtained according to ADA Specification No. 32.
load. Although not a factor at room or oral temperature with most metals, often it is with molecular materials, particularly certain polymers.
Subjecting a typical, orthodontic, elastic module, thread, or "chain" to a
moderate tensile load, and holding that load constant while monitoring
deformation, a continuing elongation will be observed over a period of
hours. The material is said to creep and, from data taken, a strain-versustime curve may be generated similar to the plot of Figure 3-22. If the
temperature or time period is sufficiently high or long, the straight-line
portion of the curve will give way to a nonlinear increase in strain fol-
lowed by rupture, even though the stress level may be substantially
below the ultimate strength of the material as gained from a static, tension test at room temperature. A material that creeps will also "relax."
Relaxation is the decrease with time of load carried under conditions of
constant strain; a relaxation plot is presented in Chapter 4.
Bioengineering Analysis of Orthodontic Mechanics
TABLE 3-1. Mechanical pmperties in bending for selected, orthodontic, arch-wire
Modulus of
(x106 lb/in.2)
(x103 lb/in.2)
(x102 in/in.)
A gold alloy
Type 302
stainless steel
(Cr-Co alloy)
(Ni-Ti alloy)
Beta titanium
(Ti-Mo alloy)
upon unloading from the (equivalent) yield point: the yieldstrength-to-elastic-modulus ratio. Yield strength and springback values will vary with cold working or heat
*The recovered, unit
The final topic considered in this section is fatigue, the progressive
failure of a material undergoing loading that changes with time. Usually
such loading is cyclic, exhibiting a repeating pattern; the maximum
stress generated may be substantially below the ultimate strength of the
material obtained from a static test, and still result in an eventual fracture. Examples of cyclic loading pertinent to orthodontics include masti-
Stage 3
FIGURE 3-22. A strain-versus-time plot illustrating the three stages of creep.
Material Behavior of the Orthodontic Appliance
Ultimate strength
Endurance limit
Number of Cycles to Fracture
FIGURE 3-23. A typical plot of the results of tests to fracture of a metallic member
subjected to cyclic loading in bending.
(relatively high-frequency cycling) and the daily activation,
continuous-force, deactivation pattern associated with an extraoral
appliance and interrupted loading (low-frequency cycling). A common
fatigue failure is the progressive fracture of a bent-wire segment emerging from the acrylic which is part of a retainer; the removal and replacement of the appliance represents one cycle. Figure 3-23 shows the re-
suits of a series of fatigue tests in bending. The curve intersects the
stress axis at the ultimate-strength level. The inclined portion of the plot
is obtained from fractures occurring at maximum stress levels below the
ultimate-strength value; the lower the stress, the greater the number of
cycles to failure. Below a particular stress level, termed the endurance
limit, fracture does not occur regardless of the number of load cycles.
With the retainer in which the wire eventually fractures, the maximum
stress in the activated wire ioop is less than the ultimate strength but
greater than the endurance limit of the wire material. Besides indicating
level of ductility, the previously described bending test to fracture
within ADA Specification No. 32, owing to the cyclic-loading pattern
required, also indirectly provides an approximate, relative measure of
arch-wire resistance to fatigue.
Chemical and Thermal Influences
With orthodontic application in mind, a discussion of the chemical
and thermal behaviors of solids is undertaken. The chemical and thermal material properties themselves are not the principal focus of atten-
Bioengineering Analysis of Orthodontic Mechanics
lion, but rather the potential for interactions that may result in deleterious effects upon the mechanical or physical characteristics of
orthodontic materials. Some, and often all, of the members of the ortho-
dontic appliance must exist and "work" for substantial periods of time
in the oral cavity. Materials used in the mouth must be biologically admissible and the potential for interference with acceptability, because of
chemical reactions that may occur over time, either before or during
intraoral engagement, must be realized and understood.
The oral-cavity environment is inherently corrosive. The oral fluids
are strong, potential reactants toward oxidation of metals. Saliva is
known to contain salts and acids are often liberated during mastication;
ingested food and drink vary widely in their levels of acidity or alkalinity
(pH). Food debris may become lodged and remain for relatively long
periods of time and provide the catalyst for initiation of corrosion; although generally a dental-care matter, the tendency for this occurrence
is increased by the presence of appliances. Both metallic materials (e.g.,
restorations, wires, bands) and molecular solids (e.g., "elastics," cements, adhesives, acrylics) may be increasingly vulnerable to chemical
degradation in the moist oral environment. The effects of chemistry and
temperature upon the materials themselves, as well as the products of
chemical reactions, must be of concern.
The extraoral environment may also influence the structural capabilities of orthodontic appliance materials. The time period between manufacture and actual use can be substantial. Storage may occur with the
manufacturer, with the vendor, and with the practitioner. Molecular
materials, in particular, often have a finite "shelf-life," which may be
reduced by high temperature and humidity. Parts of extraoral appliances come into contact with skin and hair, and the oils as well as perspiration, often together with particles suspended in the air, can and do
have a degrading effect.
Corrosion is defined as a deterioration because of a chemical reaction
that results in apparent disappearance of the material attacked. Corrosion may occur by a chemical solution contacting the material; ionization
ensues and the material is dissolved in the fluid. For example, silver will
be corroded by a solution containing sulfides with silver sulfide formed
in the solvent. A more common form of corrosion is known as electrochemical oxidation. Characteristic of metallic materials, the process begins
with a reaction initiated by a fluid that removes electrons, thereby form-
ing positive material ions. In turn, chemical combination of these ions
with electrically negative ions in the fluid may take place. Rusting starts
with the stripping of three electrons from the iron element, forming the
ferric ion. The degree to which electrons are bound to the atoms in
metals varies, and so also does their tendency to form compounds in the
presence of nonmetals having incomplete outer electron shells. Gold
and platinum are relatively inactive and, in mining, are often found in
the pure form; iron and aluminum, for example, as active metals, are
continually oxidized and must be purified (reduced) chemically before
metallurgical processing.
Material Behavior of the Orthodontic Appliance
Generally, corrosion occurs as a combination of the two forms described. The facility for the ionization of a metal is dependent upon the
nature of the fluid-solvent and the inherent activity level of the material;
the latter is often termed the electrode potential of the metal and is a direct
indicator of its corroding tendency. The determination of the ranking of
metals by electrode potential is accomplished through use of the "standard cell" shown in Figure 3-24. Immersed in the solution is the test-metal
electrode on the left and a hydrogen electrode on the right, the two
connected externally through a potentiometer. Ionization begins at both
electrodes; however, depending upon the electrode potential of the
metal with respect to hydrogen, electrons will flow one way or the
other, registering a potential difference on the meter. The experiment is
TM ±
Positive ion of test metal or alloy
H2: Hydrogen gas
Hydrogen ion
FIGURE 3-24. A standard electrochemical cell.
Bioengineering Analysis of O,thodontic Mechanics
repeated, changing only the test metal. With all potential-difference val-
ues recorded, a ranking such as that given in Table 3-2 is obtained with
hydrogen as the reference ion.
Consider a cell such as that shown in Figure 3-25 with iron as the
electrode on the left and water as the solution. From Table 3-2, iron has a
higher electrode potential than tin and, upon ionizing, electrons removed from the iron flow through the connector to the hydrogen electrode. The hydrogen gas is also ionized and water molecules are decom-
posed into hydrogen and hydroxyl (OH) ions. The electrons coming
from the iron unite with hydrogen ions and hydrogen gas is liberated.
3-2. Electrode-potential rankings, with respect to hydrogen, of selected metals
and alloys
The ranking to follow is in the order of most anodic, at the top of the list, to most
cathodic at the bottom of the list.
Aluminum, active
Aluminum, passivated
Stainless steel, active
Nickel, active
Lead-tin solder
nconel*, active
Nickel, passivated
lnconel*, passivated
Stainless steel, passivated
lnconel is a nickel-chromium alloy.
tMonel is a copper-nickel alloy.
Material Behavior of the Orthodontic Appliance
Electron flow
Fe3t Iron ions
OH—: Hydroxyl ion
Fe(OH)3: Rust
FIGURE 3-25. A galvanic cell illustrating the electrochemical corrosion of iron.
ferric ions react with the water and oxygen therein to form ferric
hydroxide: rust. The rusting of iron will occur in such a galvanic cell, in
fact, whenever the iron is connected externally to an electrode having a
lower potential (occupying a lower position in Table 3-2) and the elecThe
trodes are immersed in a solution (the electrolyte) wherein hydroxyl
ions become available with ionization. In general, the electrode supplying the electrons is called the anode and the receiver of electrons is the
cathode. The anode experiences this galvanic corrosion and the location of
the corrosion products will depend upon the relative ease with which
the ions, forming those products, are able to diffuse through the electrolyte. In a galvanic cell containing iron and a less active metal, because
ferric ions are smaller than hydroxyl ions and because the product requires three OH ions for every ferric ion, rust will usually be found at or
near the cathode.
The formation of a galvanic cell with the physical contact of an orthodontic-appliance member and an amalgam restoration, in the presence
of saliva as the electrolyte, results in momentary ionization and electron
flow and the dental "galvanic shock." This is an example of a composition
cell established between two dissimilar metals. A tin roof, placed on a
Bloengineering Analysis of Orthodontic Mechanics
building having steel siding, becomes cathodic and corrosion begins in
the presence of moisture and direct contact between roof and siding.
Important in addition are the more subtle forms of electrochemical
cells. A stress cell may form between two different locations on the same
material, because of an imbalance in electrode potential. A vivid example of stress corrosion is the etching or corroding of grain boundaries; the
atoms at the boundaries are more loosely bound and, therefore, are
potentially anodic with respect to those within the crystals. It follows,
then, that a fine-grained metal is potentially more vulnerable to corrosive attack than is a coarse-grained metal. An imbalance in electrode
potential at different surface locations of the same material results from
localized inelastic action. The region of a permanent bend in an arch
wire is anodic in comparison to the passively-straight portions of the
wire. A concentration cell may occur involving one material (or two) when
the electrolyte is nonhomogeneous. The surface area exposed to an electrolyte is potentially more active where the concentration is lower. In a
one-material cell, the region exposed to a diluted electrolyte is anodic
with respect to a region in contact with a concentrated electrolyte. The
significance here is the anodic nature of cracks, crevices, and areas
under accumulations of debris or other surface contaminations. Moreover, often the deposits of corrosion products aggravate the situation
and increase the rate of degradation; localized pitting of a material is a
typical manifestation.
Electrochemical corrosion requires the presence of two electrodes,
physical contact between them for conduction of electrons, and an ionizing, ion-carrying fluid, the electrolyte. Retardation or elimination of
corrosion tendencies necessitates minimization of the electrode potential
or a break in the "circuit." In two-material systems, corrosion may be
prevented by insulating one material from the other or one or both materials from the electrolyte. With one material, isolate it from the electrolyte, keep its surface smooth, clean, and free of sharp edges, and reduce
its inherent electrode potential. Steel coated with zinc is "galvanized" to
separate the steel from moisture. Because zinc is more active than steel,
it is the potential anode should a crack or scratch expose the steel. Orthodontic stainless steel generally resists corrosion well because its surface is protected by passivation with chromium oxide, the nickel in the
alloy allows a reduction in carbon content (and a lessening, then of the
potential for chromium-carbide formation while retaining strength and
ductility properties), and the result is a position in Table 3-2 just above
The foregoing discussion has dealt primarily with "wet" corrosion in
which a liquid electrolyte or at least moisture is necessary to the reaction. Dry corrosion is the result of a chemical reaction of the affected
material with a gas, often air. With metals the specific gaseous reactant
is oxygen and an oxide scale is formed, initially on the metal surface. In
general, the rate of increase of the film thickness depends upon the
existing thickness, the diffusion coefficients for the reactants, and the
temperature. Continuation of the reaction is dependent upon the pene-
Material Behavior of the Orthodontic Appliance
trabiity of the scale film by either the gas or the metal ions. An example
of dry oxidation is the passivation of stainless steel; the chromium oxide
film is very thin and highly impenetrable.
Dry corrosion of molecular materials also occurs; of particular interest is the aging of elastomers. In this oxidation reaction the initial result is
an increase in cross-linking (secondary bonding), which causes harden-
ing, stiffening, and embrittlement. Heat, light, stress, and ozone concentration influence aging, which takes place because of the existence of
free-end molecular chains. Antioxidants may be incorporated during
polymerization to combine in a monofunctional manner with the chain
ends to provide some resistance to aging. The practitioner must be
aware of aging and its effects, particularly with regard to the time period
and location of storage of polymeric, orthodontic-appliance elements. In
a moist environment polymeric materials experience a combination of
solution and aging corrosion. "Elastics" that have been stored may contain aging cracks that are catalysts for further and accelerated degradation following activation; intraoral placement of these elements tends to
quicken deterioration compared to extraoral application.
Another type of degradation occurring in the orthodontic setting involves extraoral appliances, in particular the neck pads, head caps, and
other nonmetallic elements that come into contact with the skin and
hair. A form of wet corrosion occurs as skin and hair oils, and perspiration attack these molecular materials over time. Although causing some
aging, of equal or perhaps more concern is the chemical reaction initi-
ated by the hydrocarbons that ultimately breaks down the material
structure into small molecules, thereby drastically reducing mechanical
strength. The progress of this deterioration is relatively slow, and material discolorations indicate the start of the process. Frequent cleansings
of washable parts of the extraoral appliance, for example, retards this
form of corrosion.
The effects of temperature on a material are, to a great extent, manifested through influences upon properties already discussed. The fundamental material response to change in temperature is on the level of
atomic activity which, in turn, affects volume. A typical unconstrained
solid will expand upon experiencing an increase in temperature; such a
material will contract as its temperature is lowered. (Some polymeric
materials, however, react differently with temperature changes due to
accompanying bonding alterations.) A structural member that is constrained against undergoing volumetric change will be subjected to induced thermal stresses if its temperature is altered; these stresses are
superimposed upon those that exist due to mechanical loading.
The increased vibrational motions of atoms and molecules, accompanying increased temperatures, generally affect mechanical behavior in
that ductility characteristics are enhanced but strength properties are
reduced. The rates of, or tendencies toward, creep and relaxation grow
as the temperature rises. The greater the atomic motions, the more easily materials may be ionized; hence, increased temperatures generally
raise the rates of corrosion and oxidation.
Bioengineering Analysis of Orthodonfic Mechanics
In general, the effect of an incremental change in temperature is
greater upon nonmetals than upon metals. Pertinent to orthodontic application, the influence of the difference between room and oral temperature, or between that of ingested ice cream and coffee, is inconsequential upon arch-wire materials. The effects of such temperature
differentials, however, are discernible in many of the elastomeric materials. Although the metallic appliance components do not experience
large temperature ranges during actual orthodontic therapy, they may
in soldering, welding, or heat-treatment procedures prior to engagement and activation. Note that when the temperature is raised to about
900°F and above, the carbon atoms in stainless steels tend to migrate
toward the grain boundaries; they combine with chromium ions to form
chromium carbide, and the passivating potential of the alloy is lessened.
Hence, care must be taken in soldering or welding this material, and
other metals and alloys, to concentrate the heat and hold it for only a
very short time.
Selection of Materials
In concluding this chapter, comments pertaining to the selection of ma-
terials are appropriate. Although materials and appliance members associated with orthodontic therapy are of specific interest, the same criteria are reviewed and examined in the intelligent selection of materials for
any structural or machine application.
The relationship of the relevant characteristics or properties of the
material to its intended use must be the first and overriding considera-
tion in material selection. The orthodontic appliance is a structural
mechanism and, with appliance members available in varieties of physical sizes, attention is focused primarily on mechanical properties. Archwire materials must be ductile if arch fabrication, including the place-
ments of permanent bends and twists, is necessary. Moreover, arch
wires must be moderately flexible while possessing substantial elastic
strength to provide desired activation and deactivation responses. All
parts of the appliance must exhibit strength characteristics to sustain
without failure the stresses induced in appliance activation, both at the
moment of engagement and over time periods ranging from a few days
to, perhaps, several years. Possible chemical reactions, which may affect
mechanical behavior or result in unwanted product compounds, must
be a factor of serious consideration, particularly in the selection of intra-
oral appliance materials. Ideally, the mechanical characteristics of a
structural material should not be influenced by time or by temperature
fluctuations that may occur in the service environment; however, with
real materials, one or both of these parameters may exert significant
impact on mechanical behavior.
Material Behavior of the Orthodontic Appliance
Other factors to be evaluated are generally subordinate to those directly associated with in-service use and are given significant weight in
the selection decision only when, from principal considerations, two or
more materials seem equally suitable or when a compromise is warranted and can be tolerated. The ranking of these additional factors will
vary, depending on the specific application and, often, on financial influences. Costs will clearly impact the selection, and they include the
initial investment, expenses associated with subsequent fabrication, and
costs related to storage of the material. A compromise directly related to
initial cost has led to the virtual disappearance of gold alloys as archwire materials. Availability is another factor that may influence initial
cost and may, at least temporarily, impact the selection procedure. In
any business involving a product, inventories must be kept, and orthodontic practices are no exception. Accordingly, the effects of storage on
the material and the needs for handling and working with the material, in
transforming it from the condition "as received" to the state of readiness
for immediate use, come into consideration. Chemical degradation limits the "shelf life" of a number of materials used by the orthodontist.
Moreover, many cements, adhesives, alginates, and acrylics, for example, require storage in controlled environments or careful mixing to produce chemical reactions resulting in optimum material characteristics.
Not to be ignored, but often rightfully near or at the end of the list of
influencing factors, is appearance. Although advertising may attempt to
inflate the importance of external attractiveness, the potential user must
place this factor in its proper position in the list of selection criteria.
Although often an apparently difficult decision, a sacrifice within the
primary selection factor, that of relation of material properties to inservice use, should rarely be made solely to accommodate appearance.
Suggested Readings
Burstone, C.J.: Application of bioengineering to clinical orthodontics. In Current
Orthodontic Concepts and Techniques. 2nd Ed. Edited by T. M. Graber and
B. F. Swain. Philadelphia, W.B. Saunders, 1975, Chapter 3.
Burstone, C.J., and Goldberg, A.J.: Beta titanium: A new orthodontic alloy. Am.
J. Orthod., 77:121—132, 1980.
Greener, E.H., Harcourt, J.K., and Lautenschlager, E.P.: Materials Science in
Dentistry. Baltimore, Williams & Wilkins, 1972, Chapters 3—6, 8, 11, and 13.
Hayden, H.W., Moffatt, W.G., and Wuiff, H.C.: The Structure and Properties of
Materials. Vol. III. New York, John Wiley & Sons, 1965, Chapters 1—3, 6, 7,
and 10.
Jarabak, JR., and Fizzell, J.A.: Technique and Treatment with Light-wire Edgewise Appliances. 2nd Ed. St. Louis, C.V. Mosby, 1972, Chapter 3.
Bioengineering Analysis of Orthodontic Mechanics
Kusy, R.P., and Greenberg, A.R.: Effects of composition and cross-section on
the elastic properties of orthodontic arch wires. Angle Orthod., 51:325—341,
Richards, C.W.: Engineering Materials Science. Belmont, CA, Wadsworth Publishing, 1961, Chapters 1—4, 8, 9, 12, and 13.
Phillips, R.W.: Skinner's Science of Dental Materials. 7th Ed. Philadelphia, W.B.
Saunders, 1973, Chapters 2, 3, 11, 15—19, and 32.
Thurow, R.C.: Edgewise Orthodontics. 4th Ed. St. Louis, C.V. Mosby, 1982,
Chapters 3, 7, and 8.
Van Vlack, L.H.: Elements of Materials Science. 2nd Ed. Reading, Massachusetts, Addison-Wesley Publishing, 1964, Chapters 1—3, 6, 7, and 12.
Energy Analyses in
The theoretical developments within classical mechanics have provided
two analysis formulations. The four laws offered by Isaac Newton are
the foundation for one approach toward the solution of problems in
mechanics, and two mathematicians—Jean Bernoulli, a Swiss, and J. L.
Lagrange, a Frenchmen—are generally given credit for the more con-
temporary hypotheses and methods of work and energy.
In Chapter 2 the procedures of Newtonian mechanics were introduced as based upon a set of equations of equilibrium or motion, depending upon the state of rest or movement of the mechanical system.
The free-body diagram was introduced and employed as a principal tool
in the solution of problems. Generally, the objective was the altering,
simplifying, or completing of the description of the set of forces and
couples exerted at a particular instant of time upon or within the body or
the elements of a structure or machine. In the analysis procedure employing work and energy, the mechanical system undergoes an actual or
contrived process that may be of short or long duration and results in
changes in one or more of the characteristics of the system.
To prepare for an analysis through use of the principles of work and
energy, the body or set of elements or bodies included within the mechanical system under study must be defined. The initial state of the
system is described at a particular time; physical, geometric, kinematic,
thermal, chemical, and mechanical properties—any characteristics that
may experience change during the ensuing process—are noted. The
process that is to take place results from an interaction between the system
and its surroundings. Therefore, the surroundings, which will influence
the changes in the mechanical system, must also be characterized. Of
particular importance are (1) the set of forces and couples to be exerted
by the surroundings upon the mechanical system under study and (2)
any heat sources or heat "sinks" in the surroundings that may interact
with the system. With the system and its surroundings adequately described to define the initial state, an examination of the process is undertaken. A process is the interaction between a defined system and its surroundings that results in a change of state (alteration of character, changes
Bloengineering Analysis of Orthodontic Mechanics
in property values) of the system. The final state of the mechanical system is described in terms of the properties of the system following com-
pletion of the process. In summary, the analysis is, in general, toward
the solution of the following problem: Given the initial state of a mechanical system to be studied and the description of the interaction be-
tween the system and its surroundings during the defined process,
characterize the final state of the system.
Several processes associated with orthodontic treatment may be analyzed more directly through energy methods than by Newtonian mechanics. Included, and to be discussed in this chapter, are the processes
of activation and deactivation of the orthodontic appliance, the placements of permanent bends and twists in an arch wire prior to intraoral
engagement, the various forms of heat treatment to which arch wire and
some auxiliaries may be subjected, and the actual process of manufacturing arch wire. Because each process to be analyzed generally involves
some displacement or motion of the system, and the interaction may
include the transfer of heat toward affecting the thermal as well as the
mechanical characteristics of the system, it is appropriate that the body
or group of bodies or system elements to be studied be termed a "thermodynamic system."
Concepts Leading to the Process Laws
In this section appropriate concepts are introduced or reviewed toward
stating and understanding the process laws that govern the work-andenergy approach to mechanics problems.
As described in Chapter 1, the displacement of a particle is the change in
its position with respect to a reference framework. A straight-line, particle displacement is shown in Figure 4-1. The initial position of the parti-
cle is defined by the vector r1, drawn from point 0 to point P. The
particle then moves to point Q. Labeling the coordinate s along the displacement path, the distance covered may be symbolized as
prefix represents "change in" the parameter. Because the
the delta
displacement is along a straight line in this example, the distance traveled by the particle equals the magnitude of the displacement vector.
The displacement vector may be expressed more generally as the difference between the final and initial position vectors, r2 — r1.
The displacement of just one point or particle of the body cannot
indicate whether or not the orientation or angulation of the body was
concurrently changed. If the body is rigid and is moved, but without an
angular change, the displacement of the body is a translation and the
displacements of all particles of the body have identical vector characteristics. Alternatively, if the orientation of the reference line in the rigid
Energy Analyses in Orthodontics
Q .—
FIGURE 4-1. The displacement of a particle along a straight-line path.
body is changed in the displacement, some form of rotation has occurred. If the body is nonrigid, linear displacements of particles or angular displacements of lines of the body may be the result of widespread or
localized deformations. Displacements accompanying deformations,
both visible external changes in shape and alterations internal to the
body, are of substantial interest in this chapter. Orthodontic displacements to be studied subsequently include that of one end of an "elastic"
with respect to the other in an extensional activation, the displacements
that occur in the elongation or compression from the passive state of a
helical-coiled spring, the occiusogingival or faciolingual bending deactivation of an arch wire accompanying tooth movement, and the placement of a tip-back or toe-in bend in an arch wire.
Mechanical Work
Displacements produced mechanically, whether or not the body experiences deformation, are associated with forces or force systems. Both
active and responsive forces may cause displacements; on the other
hand, displacements can result in the creation of forces. Figure 4-2
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 4-2. A concentrated force exerted upon, and during its displacement doing
work on, a particle.
shows a force exerted on the particle of Figure 4-1. As the particle under-
goes a displacement with the force acting on it, regardless of the relationship between the force and the displacement, the force is said to be
doing mechanical work on the particle if the force has a nonzero component in the direction of the displacement. Although, in general, the
magnitude or direction of the force may change during the particle displacement, if the vector characteristics of the force shown in Figure 4-2
are all constant, the work done by the force F on the particle as it moves
from point P to point Q is equal to F(cos
Although computed
from two vectors, mechanical work is a scalar quantity. The component
of the force involved is that having the direction of the displacement
vector. If the force component and displacement vector have the same
sense, the angle 4 is less than 900 and the work is algebraically positive;
if the senses are opposite (q5 is between 90° and 180°), the work done is
negative. If the force is continually perpendicular to the displacement
vector, no work is done on the particle by the force.
The force of Figure 4-2 would properly be termed an external force if
exerted on an isolated particle or a particle on the outside surface of a
body; if the particle is internal to the body, the contact force would also
be internal. Internal forces, occurring in canceling pairs, do zero net
work on a rigid body; however, within a nonrigid body the net work of
a pair of equal and opposite internal forces may be nonzero because the
Energy Analyses in Orthodontics
particles on which they act may undergo unlike displacements. In the
absence of friction, the net work done by the force systems connecting
one body to another is zero if the parts of the connection are rigid. A
couple exerted on a body does work if the body experiences an associated rotation; the work is positive if, in the pertinent plane view, the
couple and angular displacement have the same sense. Friction may also
do work; if the force of friction is resistive to the displacement, the frictional force and displacement vectors will have opposite senses and the
work of friction will be negative.
Work is done by the orthodontist in the placement of permanent
bends or twists in arch wires; the forces are transmitted by the hands of
the practitioner to and through the pliers and to the wires. In the placement of a tip-back bend, for example, the force system exerted by the
pliers on the end of the wire, causing the rotational displacement beyond the elastic limit of the wire resulting in permanent deformation, is
a couple. The clinician does work on an elastic in stretching it, on a
spring in compressing it, and on an arch wire in actively engaging it.
Appliance elements do work on teeth as their force systems produce
desired or unwanted displacements of dental units. In the typical bend
or twist placement, activation, or deactivation, the force doing the work
changes in magnitude with the displacement, requiring an averaging or
a graphical procedure to quantify the work done; examples presented
later in the chapter illustrate the methods of computing work in these
The typical process in which mechanical work is done involves a
change of state that is reflected partially in the displacements of particles
of the mechanical system under study. A cause-and-effect relationship
exists between force and displacement in the process. The external
forces that may do work on the system are exerted by elements of the
surroundings of the mechanical system, indicating that this form of
work is an interaction between system and surroundings. On the other
hand, the work of internal forces occurs entirely within the system being
analyzed. In the overall analysis of a process, these two categories of
work are often separated because of this difference and its ramifications.
Newton's Second Law for a particle of mass in under the action of a
single, constant force F suggests that this force produces a proportional
increase in the speed v of the particle with time t. For a one-dimensional
motion, the relationship may be written as
where Av and the displacement coordinate s have the same sense, identical to that of the force, as shown in Figure 4-3. The work done by F
during the displacement of the particle from point P to point Q may be
Bioengineering Analysis of Orthodontic Mechanics
4-3. Changes in kinematic characteristics of a particle as work is done on it.
determined. Multiplying both sides of Equation 4-1 by As, noting that
speed is the time rate of displacement, and with appropriate mathematical operations, the following formula for mechanical work is obtained:
Work = F(As)
+ Au)2
= + m[(v
The quantity ½mv2 is recognized as the kinetic energy of the particle;
Equation 4-2 indicates that the work done on the particle results in a
change in the kinetic energy of the particle. This relationship may be
expanded to yield the law of kinetic energy: the work of all of the forces,
internal as well as external, acting on a mechanical system, equals the
change in kinetic energy of the system. The speeds used in the formula
must be referenced to a stationary coordinate framework or one moving
in a straight line with constant speed; such reference frames are termed
Newtonian. Kinetic energy is energy of motion and the kinetic-energy
law is a form of, and in one sense the basis for, the more general process
law toward which the discussion of this section is directed.
A rationale for partitioning the forms of work done in a process by
the external and internal forces was suggested earlier. A parallel division
separates the work done that is dependent on the displacement path(s)
from that which is dependent only on the initial and final states of the
mechanical system. An example of path-independent work is that which
is performed by the force of gravity (weight). For processes during
which the center of gravity of the system undergoes a change in elevation that is small compared to the radius of the Earth, the magnitude of
the work done is the product of the weight and the elevation change.
Because the force of gravity acts directly downward, gravitational work
is negative when the elevation increases. A second example of path-independent work is that which is done on or by a helical-coiled spring in
activating or deactivating it in direct tension or compression. In this
example the magnitude of the force changes with the amount of spring
deformation. Note that the force activating the spring and the resulting
displacement always have the same sense; therefore, the work of activation on the spring is positive and by the spring (and on the attached
body) is negative.
Energy Analyses in Orthodontics
Passive state 0
Initial state 1
Work12 =
Activated state A
Displaced state 2
Work done by gravity (left) and within a spring as it is stretched (right).
Figure 4-4 shows the forces and displacements associated with the
work done by an external weight force (left) and that by the internal
forces in a spring (right). Because the amounts of work done are dependent only on the initial and final configurations of the system, and with
the form of the law of kinetic energy, such path-independent work is
defined as energy change. Specifically, the work done by the weight
force is, in magnitude, set equal to the change in gravitational potential
energy, and the work done by the internal spring forces is, in absolute
value, set equal to the change in strain energy of the spring. Moreover, in
the examples of Figure 4-4, increases in energy levels of the bodies, increased elevation of the center of gravity and increased deformation of
the spring, are seen to correspond algebraically to negative amounts of
work done on the bodies.
To this point in the discussion, two forms of external energy, kinetic
and gravitational potential, and one form of internal energy, strain energy, have been introduced. Energy is defined as the capacity of a body
or system to produce an effect; thus far that effect has been, exclusively,
mechanical work. Energy is a scalar quantity and generally is referenced
so as to be non-negative. Although kinetic energy has an apparently
natural reference (zero speed), gravitational potential energy has no
such convenient reference level. Although initially it might seem reasonable to set strain energy equal to zero for the passive state of the spring,
this form of internal energy is added to the spring in the manufacturing
process when it is given its coiled shape. In actual process analyses, only
changes in energy levels are determined. Zero-energy references are
unnecessary, but for each individual process such references may be
chosen in a reasonable manner, for each energy form, if so desired.
Bioengineering Analysis of Orthodontic Mechanics
A work-energy analysis of standard testing procedures may be undertaken to determine certain mechanical-property values. Such tests
were discussed initially in the previous chapter. The general process of
interest is the loading of the test specimen. Because the loading is slow,
the kinetic-energy change for the process is negligible. Thus, the law of
kinetic energy reduces to "The total work of all forces, when individual
contributions are algebraically summed, equals zero." With the work of
the weight force, the change in gravitational potential energy, also negligible, the process equation becomes the sum of two work contributions,
that done by the external force system in loading the specimen (a positive quantity) and the work of the internal, responsive forces (a negative
quantity), which must be zero. Substituting for the work of the internal
forces, the final form of the process says that the work done in loading
the specimen equals the increase in strain energy of the specimen.
Figures 4-5 and 4-6 show the stress-strain and load-deflection diagrams associated with the standard uniaxial tension and ADA bending
tests, respectively, which were described in Chapter 3. (In obtaining the
diagram of Figure 4-5, the initial volume of the gauge-length portion of
the specimen has been divided out, converting force and deformed
length into stress and strain.) With the loading force system increasing
in magnitude with deformation, and the resulting need to sum quantities of the form F[AsI (in the tension test cr[A€J) to obtain the external
work done, the total work in loading the specimen from its passive state
to some deformed configuration is the area under the diagram up to that
activated state. The process law says this area also equals the increase in
strain energy. Therefore, in Figure 4-5, the process that takes the specimen from point 0 to point el, the elastic limit, requires positive, external
area OABO: modulus of resilience
area OCDO: modulus of toughness
FIGURE 4-5. A stress-strain diagram generated from the data of a standard tensile test.
Energy Analyses in Orthodontics
area OMNO: resilience
area OPQO: toughness
Angular Deformation
FIGURE 4-6. A load-deformation diagram plotted from the data of a bending test of a
cantilevered beam.
work per unit volume, and increases the strain energy per unit volume
an amount given by the triangular area under the stress-strain plot up to
the dashed line. This area is also the magnitude of the mechanical property known as the modulus of resilience in tension ("modulus of" denoting
per-unit-volume). The amount of strain energy per unit volume that
may be added to the specimen before fracture occurs is equal to the
entire area under the stress-strain diagram, and is known as the modulus
of toughness in tension. Analogous areas under the diagram of Figure 4-6
give the resilience and toughness of the arch-wire specimen in bending.
Note that both the resilience and toughness magnitudes are dependent upon the "stress history" of the specimen. In their initial states the
specimens contain unknown levels of internal energy; these levels are
dependent upon the mechanical and/or thermal processes that occurred
prior to the tests (including manufacturing operations). Just as a maximum load exists, which the specimen can sustain without fracture, the
specimen also possesses a finite capacity for strain energy, and fracture
will occur when this maximum level is exceeded.
Heat Transfer and Thermal Energy
In general, a process may be initiated or accompanied by the transfer of
heat between the system and the surroundings. When heat is transferred to the system, a heat source must exist in the surroundings. If heat
leaves the system, it is absorbed by a heat sink in the surroundings. In the
absence of perfect insulation between system and surroundings, and
Bioengineering Analysis of Orthodontic Mechanics
with a temperature differential between them, heat will move from one
to the other. Amounts of heat transfer are determinable; heat transfer,
like mechanical work, is a scalar quantity, but it has sense, always moving from a region of higher temperature to one of lower temperature.
When heat flows to or from a system of finite size and capacity, the
temperature of the system changes; the temperature increases when
heat is transferred into the system and decreases when heat flows away
from the system. Three forms of heat transfer are distinguishable: (1)
conduction, by physical contact between system and surroundings; (2)
convection, the transporting of heat between system and surroundings
by a third medium, often a fluid; and (3) radiation, the emission and
absorption of rays traveling through space between system and surroundings.
If the temperature is uniform within a system at a particular lime,
then temperature is a property of the system and is state-related. A
change in temperature of a body is reflected in the level of activity of the
elemental particles that make up the body, sometimes termed the "particle kinetic energy" of the body. Appropriate, then, is the introduction of
the concept of thermal energy, a form of internal energy dependent on the
temperature of the body or system. Like other forms of energy, thermal
energy is inherently a scalar, non-negative quantity. Its zero level might
be said to coincide with a system temperature of absolute zero, but
because only changes in thermal-energy levels occur in process analyses, as with other forms of energy, a fixed reference level is
Two types of internal energy now have been mentioned: strain and
thermal energies. Although individually they directly depend upon
apparently separate, more fundamental properties, deformation and
temperature, strain and thermal energies are not mutually exclusive. For
example, the results of the tension tests of two specimens to fracture,
initially identical in every characteristic except for a finite difference in
the temperatures of the test environments, will generally yield significant differences in resilience and toughness. The lower the test-environ-
ment temperature, the more brittle the response of the specimen.
Hence, although the strain and thermal forms of energy may be differentiated from each other, the more important consideration is their contribution to the total internal energy of the system under discussion.
Existing experimental evidence suggests that the total internal energy of
a body may depend upon other parameters besides deformation and
temperature, but only significant changes in the strain and thermal
forms of internal energy occur in processes of orthodontic interest.
The Conservation-of-Energy Law
A process occurs when an interaction between the system, defined for
study, and its surroundings takes place. Generally, but not always, a
process results in a change of state of the system, usually reflected in an
alteration in the total energy level of the system. The law of conservation
Energy Analyses in Orthodontics
of energy says, in simple terms, that energy can neither be created nor
eliminated within the system. If, in a process, the total energy of the
system increases, energy has been transferred from the surroundings
into the system. If the total energy of the system decreases, an energy
transfer to the surroundings from the system has taken place. If no net
energy change of the system occurred during a process (but perhaps
levels of individual forms of energy have changed), the net energy trans-
fer to or from the system is zero.
Two forms of energy transfer have been defined: mechanical work of
the external forces exerted on the system and heat. Several types of
energy, possessed by or contained within the system have been described: the two external forms of note are gravitational potential and
kinetic energy and the two significant forms of internal energy are strain
and thermal energy. In the general law to be stated, energy transfers are
positive when proceeding from the surroundings to the system; energy
changes of the system are positive when the final levels exceed the initial
levels. The work of the weight of the system equals the decrease in
gravitational potential energy; one or the other, but not both, is included
in the formulation. In orthodontic applications, changes in gravitational
potential energy as well as electric and magnetic energy contributions
rarely occur. With the foregoing definitions, terminology, assumptions,
and sign conventions, a form of the law of conservation of energy, sufficiently general for orthodontic applications, may be narrated as follows:
The mechanical work of external, contact forces and couples performed on
the system plus the heat transfer into the system during a process equals
the increase in internal energy of the system during that process.
This "equation" is also known as the first law of thermodynamics. It is
somewhat similar in form, but is actually independent of the law of
kinetic energy in that the external work term may be eliminated between
the two formulas, yielding an explicit expression for the work of the
internal forces. A dimensional analysis of the first-law equation indicates that energy, energy transfer, and work are dimensionally identical.
The dimension of work is the product of force and length, so the same is
true for heat and energy change (although a variety of units are employed: pound-feet, newton-meters (joules), gram-millimeters, ounceinches, and British thermal units (Btus, for example).
The law of conservation of energy is altered somewhat for application to orthodontic processes. First, changes in gravitational potential
and kinetic energies may be reasonably neglected. Second, a third form
of energy transfer may be defined, to be termed losses, energy escaping
from or released by the system in an uncontrolled and usually undesired
manner. Because the sense of losses is always from the system to the
surroundings, with the sign convention previously assigned to energy
transfers, losses are inherently negative quantities. With these alterations, the law of conservation of energy, modified for strict application
to orthodontic processes, may be rewritten as follows:
Bioengineering Analysis of Orthodontic Mechanics
The mechanical work of the external, contact forces on the system plus the
heat flow into the system minus the losses equals the net increase in the
strain and thermal energies of the system.
A schematic of the orthodontic process law is given in Figure 4-7; therein
SE, TE, and En represent strain, thermal, and total energy, respectively.
Available Energy
A process generally begins with the system in total (mechanical, ther-
mal, and chemical) equilibrium. The final state as well is usually an
equilibrium configuration. Any equilibrium state, however, may be constate (1)
En, = SE, + TE,
Losses L
Heat H
state (1)
Enf = SEf + TEf
FIGURE 4-7. A schematic depicting an orthodontically related process and the workenergy law that governs it.
Energy Analyses in Orthodontics
strained such that, if released from its constraints, without any catalyst
an energy transfer to the surroundings occurs as the system proceeds
toward a minimum-energy configuration. Orthodontic examples of
these constraints are the maintenance of an elastic in a stretched state
and an arch wire at an elevated temperature in a furnace. The available
energy of a system is that portion of the internal energy that is released
by removing all constraints and permitting the system to come to a completely passive equilibrium state. The total internal energy of a system
may be expressed as the sum of its available and unavailable ("locked
in") parts. The concept of available energy becomes more meaningful in
the study of orthodontic processes within the following sections of this
Activation and Deactivation Processes
When activating an orthodontic appliance, the practitioner applles exter-
nal force and does mechanical work on one or more elements of that
appliance. Because negligible heat transfer occurs in such processes, this
quantity of work equals the increase in internal energy; since little if any
change in temperature of the system takes place, the internal energy
increase is reflected in the strain-energy form. The appliance is taken in
the process from a passive state to a constrained, activated state. The
visible evidence of the process and the added strain energy are the deformations of activation. The activation process occurs over a short time
interval and negligible losses accompany it, but losses may occur over
the relatively long time period to follow when the appliance is "working" on the dentition. If no losses, as defined previously, occur subsequent to activation, and the activation does not take the material(s) be-
yond their elastic limits anywhere in the appliance,
of the
strain-energy "input" is available to produce displacements of teeth or
other movements within the dentofacial complex.
As a first example, consider the activation of a linear, helical-coiled
spring in compression. (This spring might be wound around an arch
wire and positioned against the mesial extent of a canine bracket, for
example; when activated the spring provides the distal driving force to
do the work of retraction.) The design of the spring is such that, even
when the compressive force pushes all of the coils into full contact with
one another, the elastic limit of the spring-wire material is not exceeded.
The laws of kinetic energy and conservation of energy say that the net
contribution from the external and internal forces is zero and the work
done by the external forces equals the increase in strain energy of the
spring. The spring, compressed from the right end is shown in Figure
4-8; the initial configuration is drawn dashed, and the final, activated
state is superimposed. The activated spring is "cut" to expose the internal force system and a free-body diagram of the right-hand portion is
included in the figure. Note that the responsive force against the left end
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 4-8. The process of compressing an open-coil, helical spring.
of the spring does no work because no displacement occurs there. The
relationship between the external force that compresses the spring and
the spring deformation is illustrated in Figure 4-9. The spring is termed
"linear" when the plot of this relationship is a straight line. The force is
seen to increase with the spring deformation and the work done to produce the 8-mm compression of the spring from its passive configuration
is the cross-hatched area under the plot:
Work =
(160)8 = 640
The process law indicates that, with negligible losses during activation,
640 g-mm of strain energy have been gained by the spring.
Figure 4-9 shows the loading or activation plot of the linear spring.
Because the elastic limit of the spring has not been exceeded, and if the
spring was formed from a hard-metal alloy such as stainless steel, no
losses of energy will occur over time as an activated configuration of the
spring is maintained. As a result, the unloading or deactivation plot will
trace the loading curve, indicating that all of the added strain energy of
activation is available energy. A second process may now be analyzed in
which the spring is an energy source in the surroundings and under
study is the system on which the spring will do work (e.g., a tooth). The
work capacity of the spring in this second process, during which the
spring is deactivated to some extent, equals the 640 g-mm of strain energy added to it in the activation process.
Energy Analyses in Orthodontics
Compressive Deformation (millimeters)
FIGURE 4-9. The force-deformation plot for the linear spring of Figure 4-8.
The analyses of the activation and deactivation processes for an
"elastic" are similar to those just discussed for the spring, but notable
differences exist in the response of the elastic. Figure 4-10 shows the
activation plot for an elastic band or module, fixed at one end and
stretched by an external activating force at the other end. As with the
spring, the process laws indicate the equality of magnitudes of work
done by the external and internal forces and the increase in internal
a (inches)
Passive state
FIGURE 4-10. The activation ot an elastic module and the associated torce-elongation
As noted in Chapter 3, because of the generally amorphous
nature of elastic materials, the initial portion of the force-deformation
plot is often nonlinear. To simplify the computation of the area under
the loading curve, a straight line is sketched that enables equal, partial
areas to be cut above and below the actual plot. The obtained triangular
area is the mechanical work of the activating, external force,
(18)1 =
and is also the strain energy added to the elastic in the short-term activation process that stretches it, in this example, 1 in.
Although springs and activated arch wires may be expected to deactivate partially, if not totally, during between-appointments periods, the
activated elastic module often maintains much of its initial stretch for a
substantial length of time. Because of the relatively weak, secondary
bonding within certain amorphous materials, when maintained in a
stretched configuration many of these materials lose tension over
time. The force-versus-time plot for the example polymeric element
under constant strain conditions is illustrated in Figure. 4-11. The phenomenon exhibited, defined in Chapter 3, is "relaxation"—loss of force
with time and without deactivation. An energy-analysis schematic of
this process is provided in Figure 4-12.
Like the spring, the activation of the elastic module provides it with
available energy to do work on another system; however, although the
activating process should not take the material of the module beyond its
elastic limit, the amount of available energy may decrease with time due
Energy Analyses in Orthodontics
Time (hours)
FIGURE 4-1 1. A relaxation plot characteristic of some polymeric materials.
to relaxation. In the relatively long-term deactivation process, it is gener-
ally impossible to quantitatively separate the energy transferred from
the elastic module into external-work and loss portions. To illustrate the
effect, however, consider maintaining (constant extensional strain) the
activation depicted graphically in Figure 4-10 for several hours. Figure
4-13 shows the force-deformation relationship upon subsequent total
unloading of the module. Note that this plot is also nonlinear and the
force at the initiation of unloading is somewhat reduced, due to relaxation, from the level immediately following the activation process. Again,
the straight line is sketched in order to aid in the computation of the
= 6.5 oz-in. = strain energy released
The difference between the initially induced strain-energy increment
and the strain energy released are the losses of relaxation and the energy
no longer available to do work. In other words, the 6.5 oz-in. was the
available energy in the elastic at the time of total deactivation. In general, the magnitude of such losses depends on the material of the elastic,
the magnitude of the initial force relative to the geometry of the elastic,
and the length of time the activation is maintained. Because the oral
environment is potentially more harmful to the polymeric materials of
"elastic" elements than the normal air environment, intraoral "elastics"
Bioengineering Analysis of Orthodontic Mechanics
20 oz
Initial state
to + 1
15 oz
Subsequent state
FIGURE 4-12. An energy-analysis schematic of the relaxation process.
should be expected to experience somewhat more dramatic relaxation
and, therefore, greater losses than "elastics" that activate extraoral
Consider now the process of activating an arch-wire segment in
bending. As shown dashed in Figure 4-14, the segment, supported at its
ends, is initially (passively) straight. The clinician applies an occlusogingival force between the supports to engage the segment in a bracket. As
with the spring and elastic, the force during the process increases from
zero to a maximum value at the point of bracket engagement. Once
more the process law says that the mechanical work of the external
alyses in Orthodontics
(e1, F2)
e (inches)
The unloading diagram for the elastio module of Figure 4-10.
equals the increase in internal (strain) energy of the segider study. (It is assumed that no displacements occur at the
s during short-term activation, so no work is done by the reForces
forces (Qi'
and couples (C1,
there.) The force-deforma-
t for an activation to a state or configuration below the elastic
hown in Figure 4-15. Negligible losses occur during and follow'ation. Hence, the work done in this activation of the segment is
as in the previous examples:
+ (2)1.5 = 1.5 oz-mm
oz-mm, also the strain energy added, is the energy available to
during deactivation. In the orthodontic deactivation process, in
system under study remains the wire segment, the possibility
at, in addition to the desired tooth movement, one or both of the
s might be displaced. If so, in total deactivation, the desired
Bioengineering Analysis of Orthodontic Mechanics
t: left
FIGURE 4-14. The activation of an arch-wire segment in bending.
energy transfer to the tooth would be less than the 1.5 oz-mm by the
amount of work done by the responsive forces in displacing the supports. (Similarly, in actual situations, the spring or elastic module could
deactivate with both ends of the element undergoing movement.)
If the tooth is somewhat farther out of alignment than was suggested
in the foregoing discussion, the activation process could result in localized inelastic behavior of the segment. Point 1 on the loading curve in
Figure 4-16 is the termination of the previous elastic activation; point 2
represents the final state of the present activation process. Once again
using straight-line aids, dashed in the figure, the work done in activating the segment is quantified as the sum of three parts making up the
total area under the plot:
+ (2)1.5 + 2(1) + + (1)1 = 4.0 oz-mm
The 4.0 oz-mm of work equals the strain energy input to the wire and,
although no substantial losses as defined are expected if the wire material is relatively ductile (and the deformation at fracture is substantially
in excess of 3 mm), the inelastic activation of the wire limits the energy
available to do work. The deactivation process from point 2, depicting
maximum movement of the tooth and assuming no work done at the
supports of the wire segment, is shown as the solid line in Figure 4-17.
Energy Analyses in Orthodontics
Deflection 8 (millimeters)
FIGURE 4-15. An elastic activation plot for the wire segment of Figure 4-14.
The cross-hatched area under the deactivation plot is
— (2.8)2.1 = 2.9 oz-mm
and is the energy available to do work on the tooth. The unavailable
portion of the internal-energy input during activation is
4.0 — 2.9 = 1.1 oz-mm
7eering Analysis of Orthodontic Mechanics
4-16. Loading of the wire segment of Figure 4-14 such that the elastic limit has
been "locked in" the wire at the activation site. This unavailable
differs conceptually and actually from the losses sustained by the
due to relaxation; although both phenomena result, in effect, in a
of potentially available energy to do work, energy was transother than through external work, from the elastic, but not from
re, between the end of the activation process and the conclusion
ctivation. (Relaxation of metals generally occurs only when they
Energy Analyses in Orthodontics
1.1 oz-mm
(2.5, 2.8)
FIGURE 4-17. The deactivation of, and the available and unavailable energies for, the
wire segment activated beyond its elastic limit.
have been strained almost to their fracture point or when they are main-
tained under load at substantially elevated temperatures.)
Relaxation and inelastic action have been noted in this section as
counterproductive toward maximizing the available energy of the activation process. Friction is a third phenomenon that "detours" a portion of
the strain energy available to a deactivation process. Recalling the linear
spring analyzed earlier in this section, now consider a canine as the
system with the linear spring activated against it at the initiation of a
process. A free-body diagram of the tooth during the process is shown
Bioengineering Analysis of Orthodontic Mechanics
Holding couple C
Spring force P
FIGuRE 4-18. A buccal-view, free-body diagram of a canine during its retraction into a
first-promo/ar extraction site.
in Figure 4-18. Ideally in the process no localized deformations should
occur and, therefore, no change in the internal energy level of the tooth.
In the absence of friction and tipping, the process law would say that the
work done by the spring force (equal, in total deactivation, to the strain
energy added in activating the spring) plus the work done by the periodontal force (sense of force to the right, sense of displacement to the left:
negative work) equals zero.
Considering bodily movement wherein no work is done against the
couples, a portion of the energy transferred from the compressed spring
during the displacement will be carried into the stationary arch wire by
friction. If the displacement is indeed bodily movement, the division of
available energy in the spring between the periodontium and the arch
wire is in the same ratio as the periodontal and frictional forces, inasmuch as a mesiodistal force balance must exist throughout the process
and the displacements of the particles, on which the three forces act, are
all identical excepting locations. If tipping occurs, negative work is done
Energy Analyses in Orthodontics
by the couples which further lessens the desired energy transfer from
the tooth to the periodontium. In this example, typical of processes in
the presence of unwanted friction, considering the arch wire as the system under study for the moment, the energy transferred into the wire by
the work of the frictional force is initially reflected in an increase in the
thermal form of internal energy. If the arch wire is stationary (the frictional force having done work during the displacement of the bracket
with respect to the wire), with the momentary, localized increase in wire
temperature, all of that energy increment will be transferred immediately into the surroundings with an accompanying return of the wire to
the ambient temperature. Hence, in such instances, in a very real sense,
energy transfer in the form of the work of friction is a loss.
Strain Hardening and Heat Treatment of Metals and Alloys
Mentioned earlier was the existence of a relationship between the level
of internal energy, with the system under study in an unconstrained
state, and the mechanical properties of the materials of the system, the
metals and alloys in particular. In addition, discussed previously was a
sequence of processes in which energy was transferred into a material
followed by restoration of the initial equilibrium state with its surround-
ings; the processes as a whole resulted in a net change in the level of
internal energy because of inelasticity during loading. Apparent, then,
is the ability to design processes which have as their objective to alter the
internal-energy level and, thereby, change the mechanical properties of
the system materials. Also indicated in prior discussion was an upper
limit to the internal-energy capacity, per unit volume, of a material; failure hypotheses known as the maximum-strain and maximum-internalenergy theories are often employed by practicing engineers in designing
load-carrying members and structures. High levels of internal energy,
principally in the form of strain energy, are accompanied by residual or
"locked in" stresses. Superposition of loading on such stress states may
lead to unexpected fractures; hence, a process, which will at least partially release residual stresses, may be carried out to advantage prior to
the application of the load. Although the level of internal energy cannot
be readily quantified, good indicators of high levels are substantial hardness and lack of ductility.
Two categories of processes, carried out on metals or alloys before
in-service use, alter the unconstrained-state internal energy in the material. Strain hardening is a process in which energy in the form of mechani-
cal work is transferred into the material, taking the material beyond its
elastic limit. When strained elastically, the interatomic bonds are only
stretched; upon release of the straining action, the deformation disappears. The loading of a material to a state beyond its elastic limit causes
the breaking of interatomic bonds and the establishment of new bonds;
permanent, relative displacements between grains occur in crystalline
Bioengineering Analysis of Orthodontic Mechanics
materials with the result, upon unloading, of an altered external shape
and, generally, a different stored-energy level compared to that of the
originally passive state.
The strain-hardening process is undertaken toward an increase in
internal energy; however, the process can, particularly in a high-temperature environment, result in a reduction in internal energy. When undertaken at normal room temperature or below, the process is called cold
working. Strain hardening may be carried out at an elevated temperature
to enhance the interaction between the strain and thermal forms of internal energy; this process is known as hot working. The strain-hardening
process, when initiated from a relatively low internal-energy state with a
moderate energy transfer, results in increases in the elastic strength and
range, the resilience, and, perhaps, the ultimate strength; however, the
process reduces the ductility. Because for each material an upper bound
exists to the amount of internal energy per unit volume it can retain
without undergoing drastic structural change (including, perhaps, fracture), the degree to which the material can be cold- or hot-worked is
limited. In general, when the point of maximum strain-energy capacity
is approached, control of the process is lost and, subsequently, a sizable
(and, perhaps, explosive) energy loss occurs accompanying fracture.
Heat treatment is the general name given to those processes in which
the catalyst toward changing the internal-energy level in the metallic
material is the transfer of energy into the system entirely in the form of
heat. The process is carried out in three steps:
1. The system temperature is elevated by placing it in a high-temperature environment (e.g., a furnace or a hot salt bath) or by electricresistance/induction heating; heat transfer is through convection in
using the furnace or bath and by conduction with electrical procedures.
2. Upon reaching the desired temperature, the system is maintained
there for a specific period of time. Because no losses are occurring
with the system at the temperature of the furnace or bath, while
energy is continually flowing into the surroundings with the direct
electrical methods, the potential for temperature control is greater
with the system enveloped in the high-temperature environment.
3. The system is returned to its initial-state temperature.
Although the lime taken to elevate the temperature is not particularly
critical, the step-3 time is if an objective of the heat treatment is a specific
alteration of the atomic arrangement within the material. Note that sol-
dering and welding procedures also use heat to initiate processes. Although not discussed in detail in this text, a by-product of either process
may be a heat treatment. Accordingly, these procedures must be carried
out carefully with the realization of the potential effects upon localized
internal-energy levels and mechanical properties.
The difference between strain-hardening and heat-treatment processes is clear with regard to the forms of energy transfer initiating these
Energy Analyses in Orthodontics
Losses (uncontrolled energy transfers from the system) are
substantial in heat-treatment processes; losses are generally negligible
with cold working as long as the energy input does not result in fracture.
Ariother distinction between the process categories is in the internalenergy change achieved: typically an increase from strain hardening,
but heat treatment may result in either raising or lowering the internalenergy level. Energy schematics showing strain-hardening and heattreatment processes are presented in Figure 4-19.
Three types of heat-treatment processes should be familiar to the
orthodontist. Following the placement of bends and/or twists in an arch
wire, a cold-working procedure, a stress-relief heat treatment may be
desirable, particularly if the bends are sharp and when complex loops
have been fabricated. The purpose of a stress-relief heat treatment is the
release of at least a portion of the residual stresses "locked in" the material, and recovery of some of the ductility lost through strain hardening.
Depending on the material, the dimensions of the member, and the
degree of strain hardening, the stress relief may also raise or lower the
elastic limit and resilience and/or the ultimate strength. Step 2 in the
stress-relief process in a furnace of stainless-steel orthodontic wire is
generally carried out at 700 to 900°F for 3 to 5 mm; the time necessary at
lower temperatures is excessive, from a practical standpoint, and higher
temperatures will likely produce carbide precipitation and loss of corrosion resistance.
An annealing heat treatment is carried out at a temperature substan1
Initial state
Work W
Losses L
Strain Hardening
> En1
Heat Treatment
FIGURE 4-19. Energy-analysis schematics of general strain-hardening (left) and heattreatment processes (right).
Bioengineering Analysis of Orthodontic Mechanics
tially above that to stress-relieve. The temperature and resulting particle
kinetic energy are high enough to dissolve at least a portion of the inter-
nal bonding and permit atomic reorganization. An anneal produces a
substantial loss of internal energy and an accompanying reduction of all
strength properties and resilience if the material was previously strain
hardened. The process also increases ductility, making the material
more malleable. The annealing of stainless-steel orthodontic wire requires only a few minutes at temperatures of 1800 to 2000°F. The mate-
rial is fully softened and any carbides are brought into solution. The
cooling (third) step in the process is carried out rapidly by quenching in
water or oil to prevent reformation of chromium carbides. The time at
temperature is generally kept short to minimize the potential for grain
growth and, therefore, maximize ductility and corrosion resistance.
A hardening heat treatment is usually a long-term process with a step-
2 time of several hours, carried out at a temperature somewhat below
that necessary to anneal and with the material often cooled rapidly by
quenching. Hardening generally will improve strength properties, particularly with respect to their levels as annealed. Although the particular
stainless steels from which orthodontic wire is manufactured will not
take a hardening heat treatment, the strength and resilience of gold and
chrome-cobalt-alloy wires are raised upon hardening (but at the "expense" of some reduction in ductility).
With the effects on properties of strain hardening and heat treatments now outlined, an overview of the processes employed in the
manufacture of orthodontic stainless-steel wire may now be undertaken. Initially a relatively large ingot of the material is heated to a temperature near that necessary to anneal, in order to facilitate the working
operations to follow. At the elevated temperature the material is "hot
rolled" (hot worked), which results in a reduction in cross-section and
corresponding lengthening of the member. Now the ingot has been
transformed into a straight bar having a round cross-section. To reduce
the bar to the final cross-sectional size and shape, the bar is drawn
through a series of dies. Each successive drawing is a strain-hardening
process that decreases the ductility of the bar. Accordingly, heat treatments must be interspersed between drawings to keep the internal-energy level from becoming excessive. The formation of rectangular wire is
accomplished either by drawing the initially round bar through a succession of dies that gradually change the cross-section from circular to rectangular or by performing a rolling operation on the round wire follow-
ing the drawing processes.
At some point in the overall wire-formation procedure an annealing
heat treatment must take place to ensure that the wire, as received by
the practitioner, has the metallurgical format characterized by the carbon atoms in solid solution. The anneal should occur as near to the end
of manufacturing operations as permissible, given the strength and resilience requirements of the final product. All of the orthodontic stainless-steel wires currently marketed contain the same principal constituents and virtually identical percentages of iron, chromium, nickel, and
Energy Malyses in Orthodontics
carbon. High- and moderate-resilience wires, and "standard" wires pos-
sess different strength, resilience, and ductility characteristics, notable
in the hands of the clinician, because production detailing varies somewhat among wire manufacturers and according to desired mechanical
properties. The as-received, high-resilience wires exhibit more strain
hardening and, therefore, a higher internal-energy level; ligature wire,
having a low elastic limit and substantial ductility, is fully annealed as
With the marketing of the "beta titanium" (titanium-molybdenum
alloy) wire, practitioners now have their choice among five arch-wire
materials. Each material has specific time and temperature formulations
for the different heat-treatment processes, and wires of the same material are often available with several different internal-energy levels. Accordingly, clinicians must be attentive to the vendor's instructional literature when placing permanent bends or twists or when considering or
undertaking soldering, welding, or heat-treatment processes.
Work-Energy Analysis of the Preparation of an Arch Wire
Orthodontists must be aware of the possibility of a wire fracturing in the
mouth during a between-appointments period. Care must be taken not
to induce excessive force or energy into a wire during fabrication and
activation. Mastication may load the appliance beyond that of activation
and add to the internal energy present in the engaged wire. Because the
amount of internal energy in a passive wire, even prior to bend or twist
placements, cannot be quantified and may be substantial, the danger of
raising the internal-energy level excessively is more subtle than that of
overloading the wire with respect to force magnitude.
Consider the preparation of an arch from a straight segment of asreceived, stainless-steel wire. The experienced clinician can, by the
"feel" of the wire, gauge its relative stiffness in bending, but the level of
ductility and the toughness cannot readily be estimated except from
specifications often known only by the wire manufacturer. The wire
segment is cut to an appropriate initial length and the arch form and
offset bends are placed. The amount of additional internal energy localized at any cross-section is related directly to the size of the permanent
deformation placed there. The strain energy induced in preparation of
the overall arch form is distributed primarily over that which will be the
anterior portion of the wire; the stored-energy increment per unit volume is nominal. The localized increases in internal energy will be greater
at the sites of the offset bends. If, for example, this arch wire is to provide guidance and, with the buccal segments, anchorage for intraoral
canine-retraction mechanics, second-order bends are placed including
tip-back bends at the terminal molars; toe-in bends are also placed at the
molar sites.
Substantial amounts of additional internal, strain energy are induced
Bioengineering Analysis of Orthodontic Mechanics
at the locations of the permanent bends, particularly just mesial to the
buccal tubes where the lip-back and toe-in bends are superimposed on
one another. Moreover, wire engagement likely activates the bends, and
the work of the clinician in placing the appliance inputs additional strain
energy at the molar sites. At no other cross-sections do both permanent
deformations and induced orthodontic loading exist in this example, if
fracture would occur, the most likely location is one of the "critical"
cross-sections just mesial to a terminal molar. If the wire is ductile and
possesses substantial toughness as received, there need be little fear of
fracture subsequent to a moderate amount of inelastic bending followed
by activation. The more highly resilient wires are generally more brittle,
however, and the clinician may wish to increase the "effective toughness" of the wire as a safety measure before placement and activation of
the appliance. This may be accomplished without the loss of resilience
only through a stress-relief heat treatment, and only if the process is
conducted properly so as to result in a net decrease in the internal energy of the passive wire (in particular, localized energy decreases at the
critical cross-sections).
Figures 4-20 and 4-21 depict the effects of bend placements, of activation, and the internal-energy capacities for a pair of otherwise identical
stainless-steel wires, without and with a stress-relief heat treatment
prior to activation. The figures indicate conditions at the critical crosssections for the example under discussion. In Figure 4-20 curves 1 and 2
FIGURE 4-20. The force-deformation diagram for an example arch wire, activated in
bending tollowing permanent-bend placement.
Energy Analyses in Orthodontics
FIGURE 4-21. The force-deformation diagram for an arch wire, activated in bending
subsequent to permanent-bend placement and a stress-relief heat treatment
represent the loading and unloading of the wire during the placement of
the tip-back and toe-in bends; note the permanent set and the new positions of the elastic limit following bend placements. Curve 3, coincident
with part of curve 2, illustrates an elastic activation process as the wire is
engaged in the brackets and buccal tubes. The triangular area under
curve 3 represents the available strain energy of activation of the wire at
the molar, revised through bend placements (strain hardening) from its
as-received value. Curve 4 (dashed) is the remaining portion of the plot
to fracture and the cross-hatched area represents the internal energy per
unit volume that may be added to the as-activated level before fracture
will occur. Mastication, which causes superposition of loading increments upon the activated state, would make the dot in the figure move
along curve 4 and, perhaps, take the wire material at the critical section
beyond the "new" elastic limit. The cyclic loading and unloading pattern of mastication may result in fatigue of the wire (defined in Chapter 3), which will effectively reduce the toughness, even when the elastic
limit is not exceeded.
As in Figure 4-20, curves 1 and 2 of Figure 4-21 indicate the place-
ment of permanent bends in the wire at the molars. To free at least a
portion of the "locked in" (residual) stresses at the critical cross-sections,
Bioengineering Analysis of Orthodontic Mechanics
stress-relief heat treatment is undertaken. This process has little effect
on the as-bent geometry and no mechanical work is involved, so it may
be represented simply as the dot marked "HT" on the plot of Figure
4-21. With some ductility remaining in the passive wire following bend
placement, the stress relief results in a further raising of the elastic limit
and an accompanying increase in the resilience (beyond the levels
achieved through bend placement only). Curve 3, here as in Figure 4-20,
represents the engagement of wire in the buccal tubes and brackets,
activating the tip-back and toe-in bends. Because the heat treatment also
raises the ultimate strength of the wire with negligible effect on the
deformation to fracture, both the maximum tolerable load and the internal-energy capacity are increased. As in Figure 4-20, curve 4 (dashed)
and the cross-hatched area of Figure 4-21 are the remainder of the potential loading curve to fracture and the revised toughness, respectively.
With identical bend placements in and activating deformations of the
two wires, the gains in both load and additional internal-energy capacilies attributable to the stress-relief heat treatment are apparent from a
visual comparison of the two figures.
first four chapters contain the concepts of mechanics and materials science needed by the clinician to embark on an intensive, but not
exceedingly complex, study of the orthodontic appliance in the manner
of a structural bioengineer. To complement the use of Newtonian mechanics in appliance analysis and design, in this chapter the approach of
work and energy has been discussed toward examining the mechanical
and thermodynamic behavior of the appliance elements over a finite
time interval. The concepts of mechanical work, heat transfer, external
and internal energies, and system with its surroundings were introduced, followed by the statements of the law of kinetic energy and the
principle of conservation of energy. Examples of orthodontic processes
discussed were the activation and deactivation of a coiled spring, an
elastic module, and an arch-wire segment. Strain hardening and heat
treatments have been considered with emphasis on their applications to
Prior to the pursuit of direct analyses of orthodontic-appliance members and assemblies, an examination of the kinematics and kinetics of
tooth movement is undertaken in the upcoming chapter. Also appropriate is a discussion of the interface of the physiology and the mechanics,
and the response of the periodontium and the dentofacial complex at
large to the application of force. These and related topics are the subject
of Chapter 5 and, with the tools of analytical mechanics, complete the
basis from which a bioengineering study of orthodontic therapy and
appliances may be approached.
Energy Analyses in Orthodontics
Suggested Readings
Allegheny Ludlum Steel Corporation: Stainless Steel Handbook. Pittsburgh, Al-
legheny Ludlum Steel, 1959.
Beer, F.P., and Johnston, E.R.: Vector Mechanics for Engineers-Statics. 3rd Ed.
New York, McGraw-Hill, 1977, Chapter 10.
Fillmore, G.M., and Tomlinson, J.L.: Heat treatment of cobalt-chromium alloys
of various tempers. Angle Orthod., 49:126—130, 1979.
Fillmore, G.M., and Tomlinson, J.L.: Heat treatment of cobalt-chromium alloy
wire. Angle Orthod., 46:187—195, 1976.
Greener, E.H., Harcourt, J.K., and Lautenschlager, E.P.: Materials Science in
Dentistry. Baltimore, Williams & Wilkins, 1972, Chapter 7.
Hayden, H.W., Moffatt, W.G., and Wulff, J.: The Structure and Properties of
Materials. Vol. III. New York, John Wiley & Sons, 1965, Chapter 8.
Howe, G.L., Greener, E.H., and Crimmins, P.S.: Mechanical properties and
stress relief of stainless steel orthodontic wire. Angle Orthod., 38:244—249,
Kohl, R.W.: Metallurgy in orthodontics. Angle Orthod., 24:37—52, 1964.
Marcotte, M.R.: Optimum time and temperature for stress relief heat treatment
of stainless steel wire. J. Dent. Res., 52:1171—1175, 1973.
McLean, W.G., and Nelson, E.W.: Engineering Mechanics, New York, Schaum
Publishing, 1962, Chapter 18.
Phillips, R.W.: Skinner's Science of Dental Materials. 7th Ed. Philadelphia, W.B.
Saunders, 1973, Chapters 16 and 34.
Thurow, R.C.: Edgewise Orthodontics. 4th Ed. St. Louis, C.V. Mosby, 1982,
Chapters 1, 7, and 8.
Zemansky, M.W., and Van Ness, H.D.: Basic Engineering Thermodynamics.
New York, McGraw-Hill, 1966, Chapters 3 and 4.
Response of Dentition
and Periodontium to
activation of an appliance by the orthodontist creates a complex
system of forces that is transmitted generally through the dentition and
into the dentofacial complex at large. The system of forces is developed
initially within the appliance due to the mechanical work of activation in
which one or more appliance members are deformed from the passive
configuration. The force system is carried throughout the appliance,
transmitted from one member to another through their interconnections, and to individual tooth crowns. Force systems delivered to
crowns are carried longitudinally through each tooth and are transmitted by the roots to and through the periodontal ligament into alveolar
bone and to the maxilla or mandible and beyond.
The alveolar process and the bones of the face respond to several
categories of force. Physiologic forces exist continuously, but perhaps are
most prominent during the eruption of teeth. Functional forces of mastication and deglutition periodically produce force against the dentition,
as may the tongue and lips. Orthodontic forces are exerted through me-
chanical connections of activated appliance members to the tooth
crowns and induce movements of individual teeth or segments. Forces
delivered to the dentofacial complex that result in movements of the
maxilla or mandible, or relative displacements of portions of these hard
tissues (e.g., palate splitting), are generally termed orthopedic.
An individual tooth receives crown force principally from the connection of arch wire to orthodontic bracket. Through incorporation of
stops, loops, ligation, and use of round and rectangular wire, forces and
couples may be transmitted in faciolingual, mesiodistal, and occlusogingival directions between arch wire
crown through the "universal"
bracket. Moreover, contact forces may reach the crown through other
than the bracket connections. Eyelets, staples, or buttons may be
welded to the orthodontic band or bonded directly to the crown surface
where desired and space permits, and appliance members (e.g., elastics,
Response of Dentition and Periodontium to Force
ligatures) activated against them. Adjacent teeth within the same arch
may push against one another through direct crown contact; they also
may "pull" on each other, interconnected through the transseptal-fiber
system. Finally, teeth in the opposing arches may exert force upon one
another through occlusal surface contact. The individual tooth is highly
resistant to localized deformation and is positioned in a flexible supporting structure.
The application of crown force displaces the tooth as a whole. The
support response to short-term forces, such as those of mastication, is
essentially elastic. Tooth displacements arising from forces existing over
extended periods of time—orthodontic forces, for example—must b97'
analyzed in two parts: the initial displacement associated with the in*
mediate response of the nonrigid alveolar process and the
displacement occurring with the biologic remodeling of the tissues.
The deformation of the periodontal ligament, accompanying initial
tooth displacement, produces a responsive force system against the
root. This system is the action-reaction counterpart of that delivered by
the root to the ligament, and arises due to the resistance to deformation
inherent in the ligament and the bone behind it. Because the stiffness of
the ligament itself is relatively low, the deformation of it occurs through-
out the entire contact area between ligament and root; hence, the mechanical response to tooth displacement is a force system that is distributed over virtually the entire root-surface area. Initially, that response
may reflect highly varying intensity from one location to another over
the root-ligament interface; if the applied force remains for a period of
time, remodeling of the alveolar process will alter that intensity pattern
toward a more uniform distribution. The ligament is less than 0.5 mm in
thickness everywhere, its weight and inertia are negligible, and the alveolar bone-ligament and root-ligament interface areas are virtually equal.
Accordingly, the force distributions transmitted from root to ligament
and from ligament to bone are nearly identical in all characteristics. Al-
though mature bone is much stiffer than the periodontal ligament,
forces against the bone can result in slight deformations, particularly
from pressure against the facial or lingual alveolar crestal regions. Upon
reaching the alveolar bone, ligament force against it resulting from
crown loading travels into and through the cortical plate and into the
bone proper. This force does not disappear; instead, it is distributed
farther throughout the bone volume, decreasing in intensity with distance inward from the bone-ligament interface as the effective area of
distribution grows.
Several reasonable idealizations are proposed at the outset in this
bioengineering analysis. The tooth is assumed to be a rigid body; it is
displaced, but not deformed, under the action of crown loading and the
accompanying response from the periodontium. The weight of the tooth
is small compared to the contact forces that may be exerted on it; the
weight of each tooth, as well as that of the periodontal ligament, may be
neglected. Furthermore, even in the short period of initial displacement
following creation of crown force, the inertias of all components of the
Bioengineering Analysis of Orthodontic Mechanics
structural and biologic systems are never of significant magnitude to
warrant inclusion in the analysis. In structurally characterizing the penodontium, its response to loading is time-dependent and its biomatenial
components must be separated due to differences in mechanical properties. As mentioned previously, the overall stiffness of the periodontal
ligament is low compared to that of bone, and although the response
mechanisms are different from one another, the stiffnesses of the periodontal ligament in tension and compression generally will be assumed
equal in the light of present knowledge. The comparable stiffnesses of
alveolar bone are at least one order of magnitude higher than those of
the ligament and the greater hardness and stiffnesses of the cortical
plate compared to the spongy bone beneath it may be a factor in portions of the analysis. Mechanical strengths of the teeth and periodontal
components need not be of concern; inherent (anatomic) controls
against structural failures exist in response to most forms of possible
crown loading. In general, replacing strength criteria in ordinary structural (engineering) analyses are pathologic considerations in the study
of the periodontium and its influence on the appropriate characteristics
of induced forces and the design of appliances.
The total analysis or design problem associated with orthodontic
therapy is in two distinct and interesting parts. The "engineering" of
appliances will be discussed in subsequent chapters. The biomechanics
of the living system to which those appliances are attached is the subject
of this chapter.
Mechanical Response of the Individual Tooth to Applied Force
Under the action of an applied force system the single tooth experiences
a whole-body displacement due to the nonrigid nature of its supporting
structure. To adequately describe such a displacement, a reference frame
is established with respect to the initial position and normal orientation
of the aligned tooth. This framework includes three mutually perpendicular axes. One axis is coincident with the long axis of the aligned tooth,
the second is directed faciolingually and passes through the center of the
proper bracket location, and the third is oriented mesiodistally. The cen-
ter of the framework, the point common to all three axes, is located
approximately at the center of the crown and is labeled point CC. This
reference system, described previously in Chapter 1, is shown within a
typical tooth in Figure 5-1. Another useful reference point is the center
of resistance of the tooth; the long axis passes through it and the point is
designated the cre. Four additional reference points will be introduced
subsequently; however, because the tooth is a rigid body, the displacements of these four points and the cre may be readily obtained from the
absolute displacement of the long axis, a line perpendicular to the long
axis through CC, and point CC itself. (Recall the two-dimensional displacement example in Chapter 1 and the associated Figure 1-5.)
Response of Dentition and Periodontium to Force
FIGuRE 5-1. A localized reference frame fixed within the tooth and having its origin at
the crown-center point CC.
Defined in Chapter 2 were three general forms of rigid-body displacement: pure translation from an active force system mechanically
equivalent to a single force with its line of action passing through the
center of resistance; pure rotation from an active force system having a
single couple as its resultant; and a combination of translation and pure
rotation termed generalized rotation resulting from action equivalent to a
force and couple referenced to the center of resistance. Orthodontic displacements as defined have clinical origins, but each may be categorized
under one of the three general forms just mentioned. Although multiple-rooted teeth do not possess the near-axial symmetry of the singlerooted teeth, no special differentiation in overall displacement designations is necessary; point-wise descriptions of root-apex movements,
however, may be different for specific, whole-body displacements. The
partitioning of the response to crown loading into immediate and longterm components was previously mentioned; nevertheless, a description of displacement ordinarily defines the position and orientation of
the tooth, at a specified instant of time after loading, referenced to the
preload configuration.
The description of an orthodontic (individual-tooth) displacement is
often a complex undertaking because the load is principally applied
away from the long axis and through the bracket. For this reason, the
displacement is decomposed into what is seen individually in the occlusal plane and in two mutually perpendicular vertical planes. Three
forms of orthodontic translation are recognized: extrusion, intrusion, and
bodily movement. The first two may be viewed from a vertical or transverse perspective, but ordinarily they are not discernible in an occlusal
view because particle displacement vectors are parallel to the passive,
Bioengineering Analysis of Orthodontic Mechanics
long-axis orientation. In bodily movement, particle displacement vectors
are generally parallel to the occlusal plane or perpendicular to the reference, long-axis angulation.
Rotational movements, whether they are "pure" or "generalized,"
are typically described in terms of the initial and displaced orientations
of the faciolingual axis or long axis and the intersection of the axis positions in the sequential configurations: the center of rotation. The two
positions of the reference axis define a plane; the center of rotation is in
that plane and is often labeled cr0, and the rotation is about an axis
perpendicular to the plane and pierces the cro. These reference lines and
the cro are exemplified for a "transverse rotation" in Figure 5-2; note that
the cro may be located within or outside of the confines of the tooth
itself. Transverse rotations are those tooth displacements during which
the long-axis orientation (angulation) changes. During pure long-axis rotation, the angulation of the long axis is not altered. Transverse rotations
are known to the clinician as "tipping" or "torquing" displacements,
depending on whether the center of rotation is apical or occlusal of the
center of resistance. Simple tipping is a transverse rotational displacement
produced by a single active force directed along the faciolingual or mesi-
odistal axis through point CC. Transverse rotations are called crown
movements when the center of rotation is at or near the root apex and root
movements when the cro is located within the crown, more specifically
between bracket level and the occlusal surface (or incisal edge).
During orthodontic therapy, the force system exerted on the crown is
transmitted from the activated appliance, primarily through the bracket.
The connections of arch wire, "elastics," and auxiliaries to brackets carry
a force system to the crown having the general resultant consisting of a
concentrated force and a couple. Individually, the force and couple vectors may have any direction; for analysis purposes each vector is decomposed into components in the occlusogingival, faciolingual, and mesio-
FIGURE 5-2. Several transverse rotations viewed from a facial perspective and, in each
sketch, showing the location of the center of rotation, cro.
Response of Dentition and Periodontiurn to Force
distal directions. As previously mentioned, the crown may also be
subjected to additional forces during the period of activation; included
are mesiodistal contact from an adjacent tooth, force transmitted
through an eyelet, staple, or button, the force of occlusion, and transseptal-fiber pull. With the exception of the fiber force, which is exerted
at approximately the cementoenamel-j unction level, the applied force
system is generally located at the occlusogingival position of point CC on
the long axis, and any movement of the line of action of a force (to the
cre, for example), requires the addition of a couple according to the pro-
cedure outlined in Chapter 2.
To describe the responsive force system exerted on a tooth by the
periodontal ligament surrounding the root, the composition of the ligament must be examined. Completely enveloping the root from cementoenamel junction to apex (apices), the ligament is a composite of fibers,
cells, nerves, and blood vessels embedded in a soft-tissue matrix often
termed the "ground substance." From a structural standpoint, two components, in essence, respond to crown loading: the fibers (in bundles) in
tension and, in compression, principally the ground substance. The
apparent shear resistance of the ligament is generated through redirection of the fibers from their primarily radial configurations with respect
to the long axis. Accordingly, the response of the individual ligament is
generally a distribution of tensile, compressive, and shear forces over
the entire root-surface area while the action on the crown is a system of
concentrated forces. The initial intensity of ligament force against a
point of the root, produced by appliance activation, varies directly as the
displacement of that point from the passive, preactivation state.
In analyses of bodily movement and transverse rotations, a twodimensional model is used; for the other orthodontic displacements,
two-dimensional modeling is unnecessary, but advantage is taken of the
near-axial symmetry of the single-rooted tooth. Certain potential errors
are recognized in the modeling of the distributed force transfer at the
root-ligament interface. First, a limit exists to the amount of compressive
ligament deformation; upon reaching this limit, the stiffness of the ped-
odontium increases substantially. Second, although most fibers are
embedded in bone, some terminate in free gingiva; the latter result in a
more flexible support structure occlusal to the alveolar crest than apical
to it. Third, the remodeling of alveolar bone results in a redistribution of
the responsive ligament force; hence, the distributed force changes, not
only in resultant magnitude with gradual deactivation, but also in pattern with the long-term, biologic remodeling of the periodontium. The
significance of each of these three modeling faults will be noted as they
arise within discussions in the sections to follow.
Transverse Crown Force Systems and Tooth Displacements
Attention is now restricted to the force-displacement analysis of a single
tooth subjected to an active force system that is equivalent to a force and
Bioengineering Analysis of Orthodontic Mechanics
couple at point CC; the vector representations of this action have no
occlusogingival components. Therefore, excluded from the discussion of
this section are extrusion, intrusion, and long-axis rotation. Moreover,
the active resultant at point CC and the problem itself may be decom-
posed, when necessary, into two plane-view (occlusogingival and
faciolingual) analyses. Finally, changes in long-axis orientation are generally sufficiently small that the convenient configuration for instantaneous analysis can be the tooth position immediately following activation,
and changes in angles between forces and the long axis, and in moment
arms, with respect to the passive (preactivation) state, ordinarily, may
reasonably be neglected.
The orthodontic displacement defined as bodily movement is examined
first to enable the introduction of terminology and procedures that will
be used throughout this section. The resultant of the crown force system
to achieve this translational displacement is mechanically equivalent to a
single point force having a line of action passing through the center of
resistance; hence, at point CC the equivalent active system is a force and
a related couple. Two typical sets of orthodontic mechanics that may
seek to produce bodily movement are canine retraction, with the displacement directed distally, and incisal-segment retraction with the displacement directed lingually. If the active force system is transmitted to
the crown at a bracket on the facial surface, the position of the line of
action of the "driving force" may or may not be such as to exhibit a
tendency only for a transverse displacement.
In anterior-retraction mechanics, the line of action of the driving
force, with correct positioning of the bracket, passes through the long
axis, and therefore, this force has no moment about the long axis (and
no potential to cause long-axis rotation). On the other hand, the driving
force in canine retraction, when transmitted to the crown by a labial or
lingual bracket, does have a moment arm with respect to the long axis of
slightly more than one-half the faciolingual crown width: an accompa-
nying tendency for rotation about the long axis is thus exhibited. The
two actions are shown in Figure 5-3, and throughout this section the
existence of an occlusal-plane couple is assumed, when necessary, to
eliminate long-axis rotational potential effectively from any analysis and
permit plane-view study. (This couple counters the long-axis moment of
the driving force. The effects of the eccentricity of the bracket with re-
spect to the long axis are discussed in Chapter 6.)
Shown on the left in Figure 5-4 is the two-dimensional model of
transverse crown loading referred to point CC; the plane view contains
the long axis and either the faciolingual or the mesiodistal axis of the
localized reference frame established previously. To "move" the loading
apically the distance d to the center of resistance, a couple must be introduced that compensates for the change in rotational potential associated
with the change in the line of action of F. This compensating couple C',
with F now located at the cre, and the couple C from the original loading,
are shown in the center sketch of Figure 5-4; because the force F was
moved a distance d perpendicular to its direction, the magnitude of C' is
Response of Dentition and Periodontium to Force
FIGURE 5-3. Transverse crown forces applied faciolingually (left) and mesiodistally
(right) through the bracket. (When the line of action does not pierce the long axis, the
transverse force must be accompanied by an occlusal-plane couple of appropriate
magnitude if long-axis rotation is to be prevented.)
the product of F and d. The couple C, having a whole-body, rotational-
displacement potential independent of its location on the tooth, now
may simply be placed at the cre and, in order to yield a force-only resultant there, must be identical to C' in all characteristics except sense. Finally, on the right in Figure 5-4 is shown in most simple form the active
force system referenced to the cre. Established completely, then, is the
in-plane relationship between the force and couple portions of the resultant action at point CC; for bodily movement the cou pie-force ratio
must equal the CC-to-cre distance d, measured along the long axis (more
specifically, perpendicular to the lines of action of the applied force
through CC and the cre).
In general, the applied force system is first combined into a resultant
at point CC to obtain the couple-force ratio, and that resultant is then
transferred to the cre to obtain at least a qualitative picture of the tooth
displacement to be anticipated from the given loading. The actual location of the center of resistance, generally dependent upon the size and
(C = Fd)
FIGURE 5-4. The crown loading in bodily movement: the force and couple at point CC
(left); the force moved to the cre with the compensating couple introduced (center); and
the force system at the ore reduced to the single-force resultant (right).
Bioengineering Analysis of Orthodontic Mechanics
and the manner of support of the body, must be obtained for the
tooth by varying, in an experimental format, the couple-force ratio and,
for each value, determining the displacement pattern of points on the
root surface. Such studies have been carried out using both theoretical
modeling (Nikolai, 1974) and laboratory experimentation (Burstone et
al., 1982); the cre has been located close to midway between cementoenamel junction and root apex, making d approximately equal to one-half
the total tooth length (if the bracket is positioned at midcrown, occlusoshape
Although a substantial simplification, discussion of the distributed
responsive force system resulting from transverse crown loading within
a two-dimensional model does encompass the essence of that response
and, in particular, clearly differentiates among the individual forms of
associated, orthodontic displacements. Neglecting the inertia of the
tooth, the resultant of the periodontal-ligament response to deformation
by the root, referenced to the cre, must be equal and opposite to the
equivalent crown loading at the point. For bodily movement that response is a single force F' as shown on the left in Figure 5-5, identical in
all characteristics to the driving force F in Figure 5-4 except sense. Because the tooth experiences no change in angulation in this displacement, F' may be expressed as the sum of a tensile component F exerted
on the trailing surface (in the plane view appearing as a curve) and a
compressive component against the leading surface as shown in the
center sketch of Figure 5-5. If the two halves of the total root surface as
divided were identical reflections of one another, and the stiffnesses of
the ligament were equal in tension and compression, the two compo-
nents would be equal and, herein, are modeled as such. (The slight
angulations of the root surfaces and the deviations of some fiber directions from the normal with respect to the long axis are reasonably neglected in the modeling.)
A sketch of the in-plane distributed force of the ligament upon the
root is shown on the right in Figure 5-5. Although in bodily movement
the displacements of all tooth points are identical, the tensile and compressive distributions are shown nonuniform to properly recognize the
5-5. The plane view of the responsive force of the ligament against the root
during bodily movement: the resultant force at the cre (left); the tensile and compressive
components (center); and the modeled tensile and compressive distributed forces
Response of Dentition and Periodontium to Force
tapering of the root. Accordingly, the intensities are highest at the apex
where the root circumference is smallest and lowest where the root
circumference is maximum at the cementoenamel junction. The difference in tensile stiffnesses of intra-alveolar and extra-alveolar fiber bundles is ignored as modeled here, based on the relatively small portion
of the total root area in which the latter fibers are embedded. (The twodimensional modeling of the stress distribution, and use of the projected
root areas, are tantamount to replacing the approximately round or oval
root cross-section of the single-rooted tooth by a square or rectangular
cross-section. Shown in Figure 5-6 are the as-modeled and typical, anatomically correct, root "slices" and the distributed force systems exerted
on each periphery.) From the discussion of stress in Chapter 3, the maximum ligament tensile and compressive intensities, as suggested in the
model, are dependent on the magnitude of the driving force and the
surface area and anatomy of the root.
The transverse displacement known as simple tipping derives its name
from the simple active force system—a single point force P with line of
action perpendicular to the long axis and piercing point CC, which produces an alteration in long-axis angulation. The active force may have
faciolingual and/or mesiodistal components; one component is shown
on the left in Figure 5-7. The couple-force ratio for this loading is zero; no
applied crown couple exists in a plane containing the long axis. The
transferral of the active force to the cre, however, requires the addition of
a couple
equal in magnitude to the product of P and d (where, again,
d is the long-axis distance from the bracket to the cre). The active force
system referenced to the center of resistance is shown in the center
sketch of Figure 5-7; the two components indicate that the potential
displacement is a translation of the cre in the direction and sense of P
and a rotation or tipping of the long axis with sense corresponding to
that of C1,. A typical displaced configuration with respect to an initiallyinclined long axis is shown on the right in Figure 5-7. The intersection of
the two long-axis positions is the center of rotation for the simple-tipping displacement, slightly apical of the cre. Note that simple tipping
includes displacements of both point CC and the root apex (apices).
The resultant of the periodontal-ligament response to simple-tipping
a likeness of the
actual root cross-section
the modeled
root cross-section
FIGURE 5-6. 0cc/usa! views of root cross-sections and distributed force systems
exerted thereon: the anatomically correct (left) and as-modeled (right) root slices for a
single-rooted tooth.
Bioengineering Analysis of Orthodontic Mechanics
= Pd
FIGURE 5-7. A plane view of the simple-tipping displacement: the active crown force at
point CC (left); the equivalent loading referred to the cre (center); and a typical
displacement showing the crc location (right).
action, referred to the cre, is equal and opposite to the action referred
there (Fig. 5-7, center), and is shown on the left in Figure 5-8. Equivalent
to this resultant of P' and is the set of four forces shown in the middle
sketch of Figure 5-8; the pair located more apically are individually
smaller in magnitude than those positioned closer to the cementoenamel
junction. Each of these four forces is the resultant of a quadrant of the
total distributed ligament response sketched on the right in Figure 5-8;
each quadrant is not exactly triangular due to the tapering of the root.
Both compression and tension exist on either "side" of the root. Proceeding occiusally from the apex, the stress (intensity) levels in the
model decrease from relative maximum values at the apex, go to zero
and change signs at approximately the cro level, and then increase toward relative maximum values at the cementoenamel junction. The absolute maximum stresses occur at the root apex, due to the minimum
root circumference there. Throughout this section in the modeling of the
FIGURE 5-8. The responsive force system against the root during simple tipping: the
resultant referred to the cre (left); an equivalent set of two tensile and two compressive
forces (center); and the modeled distributed force system in four parts (right).
Response of Den tition and Periodontium to Force
distributed response, the tensile and compressive ligament-material
stiffnesses are assumed equal and unchanging over the entire root
length as noted previously.
Considered next is the response of a single-rooted tooth to the application of a second- or third-order crown couple. This example might be
termed "academic" because, in the presence of friction and some existing faciolingual arch-wire resistance, the crown couple will always be
accompanied by some amount of transverse force. Nevertheless, pure
transverse rotation is worthy of brief discussion. Shown from left to right
in Figure 5-9 are the activating couple referred to point CC, the couple
moved to the cre, and the displacement format associated with the couple-only loading. The couple-force ratio is mathematically infinite and,
with pure rotation occurring, the cre remains stationary and the cro coin-
cides with it. The resultant of the periodontal-ligament response,
this couple expressed as four discrete (in this instance, equal) forces, and
the distribution of the response modeled in two dimensions, are depicted from left to right in Figure 5-10. Figure 5-10 is analogous to Figure
5-8; again, the nonlinearity and the maximum intensity at the apex are
due to the root taper, but the stress changes sign between tensile and
compressive farther from the apex than in simple tipping (associated
with the difference in cro locations).
Root movement is the term given to the orthodontic tipping-torquing
displacement in which the crown position remains essentially unchanged. To accomplish the displacement, a second- or third-order cou-
ple is applied to the crown in the presence of a transverse "holding"
force. A typical root movement is the uprighting displacement, to
change tooth angulation with the vertical toward proper orientation of
the long axis with respect to the occlusal plane. The principal part and
5-9. A plane view of pure, transverse rotation: the couple loading at point CC
(left); the couple moved to the cre (center); and the displacement format showing the
coincidence of the cro and the cre (right).
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 5-10. The response to transverse couple loading: the resultant at the cre (left);
an equivalent set of four forces (center); and the modeled distributed force system of
the ligament against the root (right).
motive action of the applied force system is the couple; the role of the
force is prevention of crown displacement which occurs when the couple alone is exerted on the crown (Fig. 5-9, right). The force H and the
couple Ch are shown on the left in Figure 5-11 at the crown center point
CC. In the center sketch of Figure 5-11 the displacement format is illustrated with the cro at bracket level; note the direction of cre movement
and the sense of the change in long-axis angulation. Correspondingly,
shown on the right in Figure 5-11 are the force and couple referred to the
cre, which are needed to produce the displacement. The magnitude of
this couple is C,, — Hd, where the product Hd arises from the movement
of the force H from point CC to the cre. Accordingly, the couple-force
ratio necessary to produce this displacement is greater than d that required for bodily movement.
C,, - Hd
FIGURE 5-11. A plane view of root movement: the components of the active force
system referred to point CC (left); the displacement and cro location (center); and the
equivalent loading at the cre (right).
Response of Dentition and Periodontium to Force
In root movement the apex experiences the largest displacement of
all points of the tooth; the maximum intensity of the ligament responses
is located at the root apex. Shown on the left in Figure 5-12 is the resultant of the distributed force system exerted by the periodontal ligament,
referred to the cre. The force H' is equal and opposite to H and the
couple Ch is in magnitude Ch Hd. This force and couple may be combined into a single force located somewhat apical of the cre, and then it
may be divided into tensile and compressive components H; and
the trailing and leading root surfaces, respectively, and plane-view modeled as illustrated in the center sketch of Figure 5-12. These forces are the
resultants of the tensile and compressive portions of the distributed re-
sponse shown on the right in Figure 5-12. If the center of rotation is
occlusal of the cementoenamel junction, nowhere along either the trailing or leading surface of the root does the intensity of the distribution go
to zero and change sense. In this way the distributed response is similar
to that for bodily movement; however, during root movement the difference in maximum and minimum intensifies is substantially greater
than during bodily movement, given the same magnitude of transverse
force for the two displacements (Nikolai, 1975). The clinical objective of
root movement is to locate the cro within the tooth crown; to achieve the
desired result, the mechanics must be capable of developing a holding
force of sufficient magnitude to obtain the needed couple-force ratio.
The last of the recognized transverse orthodontic displacements, and
perhaps the most difficult to achieve clinically, is crown movement. The
objective is a transverse displacement of the tooth crown without moving the root apex; this is, as ideally performed, a tipping displacement
with the center of rotation at the apex. The displacement is shown on
the left in Figure 5-13. From the relationship between the point-wise
displacement of the center of resistance and the sense of the long-axis
change in angulation, the applied force system referred to the cre must
be as shown in the center sketch of Figure 5-13, including both a force
and a couple. Furthermore, with the cro at the root apex, the response of
FIGURE 5-12. The periodontal response during root movement; the resultant located at
the cre (left); an equivalent system of two forces (center); and the modeled tensile and
compressive distributions of force (right).
Bioengineering Analysis of Orthodontic Mechanics
Cd - Cq
FIGURE 5-13. A plane view of crown movement: the displacement and the crc location
(left); the applied force and couple referenced to the cre (center); and the crown force
system at point CC (right).
the periodontal ligament against the root as modeled will divide into a
compressive distribution on the half-root surface corresponding to the
sense of the active force Q and tension on the remaining surface area.
The resulting implication is in combining Q and Cq into a single force;
that force must be located between the cre and the cementoenamel junction. It then follows that the active crown force system referenced to
point CC consists of a force and a couple as shown on the right in Figure
5-13, the magnitude of the couple is less than the product Qd as indicated, and the couple-force ratio is less than d. Figure 5-14 depicts the
resultant components Q' and Cq' of the periodontal-ligament response
referred to the cre on the left (components equal and opposite to those of
FIGURE 5-14. The ligament response during crown movement: the components of the
resultant of the response referenced to the cre (left); an equivalent, single-force
resultant divided into tensile and compressive parts (center); and the modeled
distributed response against the root (right).
Response of Dentition and Periodontium to Force
the active force system in the center in Figure 5-13), the resultant liga-
ment response as a single force divided into compressive and tensile
and Qj, in the center, and the compressive and tensile
distributed responses on the right. Crown movement is the only transverse orthodontic displacement for which the maximum intensities of
responsive force occur at the cementoenamel junction.
To summarize, all transverse orthodontic tooth movements as defined are produced by a combination of a force approximately perpendicular to the long axis (parallel to the occiusal plane) and a couple in a
plane containing the long axis. Looking into that plane, when both components have nonzero magnitudes, the sense of the moment of the force
with respect to the cre is opposite to that of the couple. The couple-force
ratio, which controls the position of the cro, is given simply as a ratio of
magnitudes and equals the occlusogingival distance from point CC to
the location of an equivalent, single-force load. (Although the center of
resistance is a fixed point in a tooth—its location is independent of the
crown loading—the center of rotation is highly dependent on the characteristics of the crown force system.) In the model used in this section,
the cro is always located on the long axis or its extension; in reality this
may be somewhat (but not substantially) in error in root movement produced by a third-order couple and a sufficiently stiff holding-force potential that will place the axis of rotation coincident, or nearly so, with
the arch wire and, thus, the cro within the bracket slot as viewed
mesially or distally. In the analyses of these movements, when necessary the loading may be decomposed into components contained in
faciolingual and mesiodistal planes through the long axis; often the
transverse action is totally in one or the other of these planes. In the
faciolingual plane, adjacent teeth exerting pushing forces through
crown contact can control the cro location; in the absence of a stiff palatal
bar, no such strong control can exist exclusive of ordinary orthodontic
mechanics in the mesiodistal plane.
Shown in Figure 5-15 is a plot, theoretically derived, of occlusogingival center-of-rotation location versus couple-force ratio for the labiolingual, transverse displacements of an average-size, maxillary central incisor (Nikolai, 1974). The partitioning of the plot into two curves results
from the couple-force ratio becoming infinitely large as the bodily-movement displacement is approached. Notable are the substantial effects on
cro location as the ratio nears d from above or below. Given in Table 5-1
are results associated with five mesiodistal displacements of a typical
canine. The root length and overall long-axis length of the tooth were
17 mm and 22 mm, respectively. The analysis was theoretical assuming linear relationships between intensities and displacements of rootligament interface points, equal ligament stiffnesses in tension and
compression, and accounting for root taper. Notable are the substantial differences in extremum (maximum, minimum) interface force intensities among the four tooth displacements generated with the same
magnitude of transverse force.
Bioengineering Analysis of Orthodontic Mechanics
to 0.9
Distance to cro occliisally
from apex, normalized
by root length
to 0.45
to 0.45
ratio (cm)
to 0.9
FIGURE 5-15. Lab/olin qua! displacements of a typical, maxillary, central incisor: a plot of
cro location along the long axis and its extension versus the ratio of couple to
transverse force as the loading components at point CC.
Finally, the remarks of this section need not be restricted to singlerooted teeth. Although the analysis may divide the overall root-surface
area into multiple portions for individual teeth having two or three
roots, this is not a significant, complicating factor; neither is the fact of
root apices away from the long axis when restricting attention only to
transverse displacements. From recent experimentation, the center of
resistance of a multiple-rooted (molar) tooth is apparently slightly apical
of the midpoint of the overall apices-to-cementoenamel-j unction length
(Burstone et al., 1981). Hence, center-of-rotation locations for transverse
displacements, in relation to total root length, may be insignificantly
influenced by the number of roots.
Response of Dentition and Periodontium to Force
TABLE 5-1. Couple-force ratios, center-of-rotation locations, and maximum stress values tar five
transverse canine displacements
Simple tipping
Bodily movement
Crown movement
Root movement
Center of
Maximum stresses
Pure torquing
The location ot the center of rotafion is given from the root apex and as a fraction of the root length.
The model canine has a projected area (mesiodistal perspective) of 0.9 cm2, a root length of 17 mm, and a
length of
22 mm.
fModitied from Nikolai, R.J.: On optimum orthodontic force theory as applied to canine retraction. Am. J. Orthod., 68:290—
302, 1975.)
Extrusion, Intrusion, and Long-Axis Rotation
remaining three distinct orthodontic displacements are characterized by several common features. The active force system includes no
net force component perpendicular to the long axis of the tooth; therefore, no potential exists to alter the long-axis position or orientation. The
resistance to any of these three displacements, as provided by the periodontal ligament, appears to be principally shear in nature. Because the
inherent shear resistance of the matrix material of the ligament is low,
however, the essence of the response originates with the fiber bundles
and, as one or a combination of these displacements begins, fiber orientations are altered toward positions tangential to the root surface. Twodimensional modeling in analyses of these displacements is unnecessary
and unwarranted. With single-rooted teeth the anatomy and force system approach a configuration of geometric and mechanical symmetry
with respect to the long axis, and the modeling more appropriately takes
advantage of this near-symmetry. The analysis is somewhat more involved, however, particularly with long-axis rotation, when the tooth
under study is multirooted and the near-symmetry is absent.
In pure extrusion the activating system is a concentrated force acting
along the long axis and referred to point CC. In the clinical setting,
however, the extrusive force is exerted at the crown surface. Moving
that force first to point CC and then to the cre, as shown in the leftto-right sequence in Figure 5-16, the extrusive force originating on the
facial crown surface is seen to be accompanied by a third-order couple
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 5-16. An applied crown force with the combined potential for extrusion and
lingual crown tipping: at the facial surface (left); an equivalent system at point CC
(center); and the equivalent system at the cre (right).
and, with it, the potential to also tip the tooth; the crown would move
lingually and the root apex (apices) labially with the cro in the position
indicated in Figure 5-9. Accordingly, the occlusally directed crown force
must be augmented if the tipping potential is to be countered. (As mentioned earlier, the eccentricity of the typical appliance is discussed generally in Chapter 6.)
In response to a purely extrusive potential, the periodontal ligament
exerts a distributed force system having a resultant directed apically and
positioned along the long axis. The distributed response arises almost
exclusively from reorientation and stretching of the ligament fibers although, depending on actual anatomy, there may be some compressive
resistance over a small area near the cementoenamel junction. The intraalveolar fibers will provide the majority of the total resistance because of
their numbers and higher tensile stiffnesses compared to the supra alveolar fibers; otherwise, the distribution is apparently nearly uniform both
circumferentially and longitudinally, and is modeled as such, for the
single-rooted tooth. Other than to note the increased root-surface area
inherent in the multirooted tooth, no substantial difference in extrusion
analysis is necessary for multirooted versus single-rooted teeth. Of particular note, however, is the relatively weak periodontal resistance to
extrusive displacements with little, if any, ligament reinforcement from
bone compared to other tooth movements; control in this regard must be
provided in the decay characteristic of the activating force, perhaps
through the arch wire, and by occlusion.
Intrusion and extrusion are similar orthodontic displacements in that
both activating loads are, ideally, axial forces, both pure movements are
translational, and both resistances to displacement are associated with
stretched ligament fibers. Differences between the movements include
the sense of the load and of displacement potential and the creation of
Response of Dentition and Periodontium to Force
apical pressure as a portion of the force system responsive to an intru-
sive loading. If the intrusive force is applied through the bracket at the
facial surface, transferring it to point CC will introduce a third-order
couple, opposite in sense to its counterpart in Figure 5-16, indicating
that a tendency for labial crown tipping may accompany the intrusion.
Shown in Figure 5-17 are the action and response associated with
pure intrusive displacement potential. Of particular note is the presence
of pressure at the root apex associated with localized ligament compression. With a multirooted tooth this pressure exists not only at each root
apex, but also in the bifurcation region. In the absence of substantial
fiber stiffness, the pressures in the compression areas, which are small
in comparison to the overall root-surface area, would be high even with
a light, intrusive load. In their passive configurations, many of the intraalveolar fiber bundles are obliquely oriented with respect to the rootligament interface. This "hammock effect" protects the root apex or apices from excessive pressures resulting from masticatory, occlusal, and
intrusive orthodontic forces. If the intrusive force is substantial or is
present continuously for a long time period, the pressure zones of the
root-ligament interface grow and can envelope the entire root-surface
area. Hence, because of the tapering of the root toward the apex, under
intrusive force both fiber tensions and pressures from the ligament can
exist at common points on the root surface; these pressures away from
C = F,e
FIGURE 5-17. The force system associated with pure intrusion: the active and
responsive resultants (left); and the active force referred to the ore together with the
responsive distribution of tension and pressure (right).
Bioengineering Analysis of Orthodontic Mechanics
the apex, relatively low compared to compressive intensifies at the apex,
are not shown in Figure 5-17 (and such pressures do not exist at all along
the root in response to extrusive loading).
A notable difficulty occurs in the attempt to quantify extrusive and
intrusive movements in comparison with similar analyses of other orthodontic displacements. The problem arises when tipping accompanies
extrusion or intrusion, which occurs often in clinical practice, thereby
altering the perpendicular-to-occlusal-plane distance between occlusal
surface (incisal edge) and root apex. Shown in Figure 5-18 are two sets of
initial (dashed) and final (solid) positions of a maxillary central incisor as
viewed mesially or distally. The two initial positions are identical. The
amounts of extrusive or intrusive displacement must be determined in
terms of two reference locations, one fixed within the tooth (e.g., the
incisal edge) and a second outside the tooth (perhaps fixed within the
maxilla or mandible, although "landmarks" within hard tissues may
also move during the period of orthodontic treatment).
The orthodontic displacement known as long-axis rotation is characterized by a change in the faciolingual- and mesiodistal-axis angulations
as viewed from an occlusal perspective. To produce this displacement
the active force system must exhibit a moment about the long axis, and
such a moment is associated with a force, having a transverse component, applied tangentially to a facial, lingual, mesial, or distal crown
surface. In pure, long-axis rotation no movement of the long axis occurs;
hence, the center of rotation is on the long axis as shown in Figure 5-19.
To obtain the pure displacement a first-order couple is required as illus-
Cant of the palatal plane
Dashed: initial position
Sohd: Displaced position
FIGURE 5-18. Combined orthodontic displacements of a maxillary central incisor as
viewed distally: lingual crown tipping and intrusion (left); and labial root torquing and
apparent extrusion (right).
Response of Dentition and Periodontium to Force
FIGURE 5-19. An occiusal view of an applied first-order couple to generate long-axis
rotation (left), and the initial (dashed) and displaced (solid) configurations together with
cr0 location (right).
trated, and that couple might be made up of a mesiodistal force applied
through the bracket on the facial surface and a parallel force, but with
opposite sense, exerted through crown contact with an adjacent tooth. If
no provision is made or no potential exists for creation of the latter force,
the moment with respect to the long axis is present but the expected
movement as seen in an occiusal view is a "rolling" displacement with
the cro located away from (lingual of, in this example) the long axis.
Figure 5-20 shows the sequence with the active force depicted on the
left, the equivalent system at the cre (which coincides with point CC in
an occiusal view) in the center sketch, and the displaced position (solid)
of the crown on the right.
The resultant of the periodontal-ligament force system, in response
to a first-order crown couple, is a couple identical in all characteristics to
the load except for sense. In the pure displacement, points on the surface of the root move circumferentially with respect to their counterparts
on the alveolar-bone surface, lengthening and reorienting the fiber bundles toward configurations tangential to the root and perpendicular to
the long axis. Fiber strains and stresses vary directly with the displacements of root-surface points, and the amounts of these displacements
are directly proportional to the radial distances from the long axis to the
points. Accordingly, the intensity of distributed responsive force is minimum at the root apex (of a single-rooted tooth) and increases in magnitude occiusally to a maximum near the alveolar crest. By reason of their
attachment in the gingiva, the radial extra-alveolar fibers are more flexi-
Bioengineering Analysis of Orthodontic Mechanics
• cro
FIGURE 5-20. A rolling displacement in 0cc/usa! view: the applied force at the facial
surface (left); the equivalent force system at point CC (center); and the displaced
configuration and cr0 location (right).
ble and, as a result, the fiber tensions are less than those in neighboring,
intra-alveolar fibers. A schematic of the load and the distributed periodontal response is given in Figure 5-21; on the right is a qualitative plot
of fiber stress against longitudinal position along the root surface. In the
model, the single-rooted tooth is assumed conical in shape; therefore, at
a specific occiusogingival level the fiber stresses do not vary in magnitude circumferentially. In reality, the variability in root-surface contour
Fiber Stress
FIGURE 5-21. Pure, long-axis rotation: a transverse view of the crown loading and the
periodontal response (left); and the variation in fiber stress with longitudinal distance
from the gingiva! margin to the root apex for a single-rooted tooth (right).
Response of Dentition and Periodontium to Force
FIGURE 5-22. Responsive periodontal distribution against the roots of a maxillary first
molar undergoing long-axis rotation: apical view (left) and transverse view (right).
results in some localized compression of the ligament, but the percentage of the root surface experiencing pressure is relatively small.
Of all the distinct orthodontic displacements, the multirooted tooth
undergoing long-axis rotation presents the most substantial difference
in response, compared to the single-rooted tooth. Apical of the bifurcation location, an assumption of axial symmetry is totally incorrect. The
"long axis" of each root deviates from the long axis of the tooth as a
whole with, generally, substantial distances between the long axis and
the apices. With the cro on the long axis of the tooth, in this displacement the individual roots experience bodily movement in a curvilinear
fashion, with pressures against the leading surfaces and fiber tensions
exerted on the trailing surfaces catalyzing the bone remodeling necessary to achieve the movement. Apical and mesial views of the distributed periodontal responses to long-axis rotation of a maxillary first
molar are depicted in Figure 5-22.
Response of the Periodontium to Force
applied to the tooth crown, originating in mastication, the activated orthodontic appliance, or from the lips, tongue, or facial tissues,
results in the creation of a response within the supporting structure of
the tooth. The individual tooth is positioned locally within a socket or
alveolus, lined with a soft tissue—the periodontal ligament—which separates the tooth root from the alveolar bone. The basal bone interconnects
the units of the dentition; the alveolar processes of the upper arch blend
Bioengineering Analysis of Orthodontic Mechanics
into the maxilla and those of the lower arch into the mandible. The
periodontal ligament completely surrounds the tooth root from cementoenamel junction to apex (apices). The ligament is composed of collagenous fibers in bundles, blood vessels, nerves, and cells embedded in an
amorphous, viscous matrix (often termed "ground substance"). The ligament is a composite material capable of sustaining and transmitting
tensile loading primarily through the fibers; pressure is transmitted
principally by the matrix material. The ligament passively exhibits nearuniform thickness of approximately 0.3 mm and low stiffness, at least an
order of magnitude less than that of calcified bone. The primary function
of the ligament is to cushion the bone beneath it from the action of
impulsive, masticatory loading of the dental units.
The alveolar bone, also, is a heterogeneous material. Although the
underlying portions of the bone are somewhat spongy, the layer adjacent to the periodontal ligament is dense and exhibits rather high surface hardness. This layer, the cortical plate (lamina dura), and the cementum anchor the radial fibers that run into and through the ligament.
The alveolar bone is relatively stiff as a whole, although some deflection
apparently occurs in the thin, crestal regions under the action of relatively high forces applied to the tooth crowns. Forces exerted on the
dentofacial complex may be designated as orthodontic if their intent is the
movement of teeth through remodeling of the alveolar process; during
such remodeling, no significant alteration of basal bone occurs. On the
other hand, forces may be exerted with the intention of displacing basal
bone without producing relative displacements of teeth within the alveolus; such forces are rightfully termed orthopedic.
The internal force system in the periodontium, arising from the application of a crown force, possesses an involved format, made so by the
geometric and mechanical as well as the biologic complexities of the
responding tissues. The problem, however, can be simplified somewhat
through modeling, with a reasonable analysis of the distribution in the
alveolus of this internal force pattern including three individual consid-
erations. First, because the periodontal ligament is thin compared to
bone dimensions, the distributed force system carried to it by the tooth
root is insignificantly altered while transmitted through it. Specifically,
the tension-pressure system at the ligament and cortical-plate interface
may be considered identical to that between the root and ligament.
Admittedly, a small reduction in intensities exists due to the slightly
larger ligament surface area adjacent to the plate compared to the area
against the root, but the difference is reasonably ignored.
Second, the distribution of normal stress into alveolar bone, in response to loading at the plate surface by the ligament, may be envisioned through the aid of Figure 5-23. Shown on the left is a flat surface
(in edge view) of a solid subjected to a compressive force and representations of resulting pressures at various distances from the point of application of the load. In a homogeneous material, such pressures have
been shown to decrease in magnitude directly with the radial distance
from the location of the load. (Changing the sense of the load, an ap-
Response of Dentition and Periodontium to Force
p: compressive
stress (pressure)
FIGURE 5-23. A simplified representation of the alveolar-bone response to pressure
loading by the periodontal ligament: the variation of compressive-stress magnitude with
distance from the load (left); and the accompanying induced normal and shear stresses
plied tensile force induces corresponding tensile stresses directed radially.) On the right in Figure 5-23 are shown stress-at-a-point sketches for
points A and B. Note the presence of circumferential tensile stresses at
right angles to the radially directed pressures at points off the force line.
(In general, for either sense of the normal surface load, the two mutually
perpendicular normal stresses at each point will exhibit opposite senses.
The shear stresses, present at all points like B and C not on the line of
action of the concentrated load, indicate a tendency to distort the small
elements of bone material surrounding the points.) The two normal
stresses being of the same order of magnitude, the circumferential
stresses also decrease in size with increasing distance from the applica-
tion-location of the load. In the problem of analyzing the transfer of
force from the root through the ligament and into alveolar bone, the root
typically transfers a nonuniformly distributed load rather than a concentrated force. Figure 5-23 depicts, then, a substantial simplification of the
response at the bone-ligament interface and within the alveolar bone to
transverse crown loading; however, the model does reflect the essential
features of that response.
Third, a portion of the load applied to one tooth crown can be trans-
mitted directly to adjacent teeth through either crown contact or the
transseptal fibers (to be discussed subsequently). Distributions of force
by these means depend upon the characteristics of the load, the relative
positions of the teeth, and the continuity of the fiber network. The effect
of this distribution is twofold: a reduction from expected value of the
force transmitted into the periodontal ligament surrounding the loaded
tooth and the displacement(s) of adjacent teeth that were not subjected
Bioengineering Analysis of Orthodontic Mechanics
to external loading. The phenomenon is most apparent when the crown
loading is mesiodistal in direction, although the effects of the transseptal-fiber system are also present in other orthodontic displacements.
The preceding discussion provides a simplistic, qualitative description of the principal effects of the force system within the periodontium
at any particular instant in response to crown loading. No general discussion of periodontal response is properly undertaken, however, without considering the influence of time upon the applied and responsive
forces. An examination of the response to masticatory action serves well
to introduce time as an analysis variable. Chewing forces are primarily
intrusive and exist only infrequently during the average day. Moreover,
when present, forces of mastication exhibit a periodic pattern, oscillating
with jaw movement between zero and some finite value. The responsive
forces are induced in the same pattern with the periodontal ligament
tending to isolate the impact of mastication from the bone beyond it. The
fiber bundles form a hammock-like structure around the root, stretching
and relaxing in primarily an elastic manner in response to the occlusal
impulses. Although the loading is transmitted into the bone, it is shortterm and "dulled" by the flexibility of the ligament; normally no lasting
effects are produced in the ordinary, passive configuration of the dentition and its supporting system. Note that the individual tooth is inherently well protected against the effects of all short-term loading with the
possible exception of extrusive force (although occlusion tends to prevent abnormal displacement). The overall response of the periodontium
tends toward that of a viscoelastic material, with a rather substantial
time period under uninterrupted force required to produce lasting
The response of the periodontium to the internal force system created by a sustained load, such as that produced by an orthodontic appliance, is fundamentally in two sequential parts. Immediately upon appli-
cation of the crown force system, the tooth undergoes an initial
displacement owing to the flexibility of the periodontal ligament. As
discussed previously, the form of the displacement depends upon the
characteristics of the applied force system. With the exception of extrusion and, possibly, long-axis rotation, the extent of this initial displacement is limited by the complete compression of the ligament between
root and alveolar bone. Transversely and intrusively, with allowance for
possible small crestal deflections, the narrow periodontal ligament restricts an initial displacement to a maximum of approximately 0.5 mm
reflected at the root apex (apices) or the alveolar crest. Any displacement
is indicative of the appliance doing mechanical work upon the dentition
which reduces the strain energy of activation and the magnitudes of the
components of the force system exerted on the dentition. The initial
displacement ordinarily will not deactivate the appliance totally and the
result is a "stand-off' between the appliance-teeth and periodontal systems, both in constrained configurations with respect to their previously
passive states. If left in these constrained states, in a period of days the
Response of Dentition and Periodontium to Force
long-term portion of the periodontal response would become evident
with the biologic remodeling of the periodontium allowing further dental-unit displacement accompanied by additional deactivation of the
appliance. If allowed to continue indefinitely, the process might end
only upon complete deactivation of the appliance.
The long-term response of the periodontal ligament to continued
deformation involves two forms of biologic remodeling: the partial replacement of fibrous matter carrying tensile stress by new, stress-relieved material and the resorption of alveolar bone to relieve and redis-
tribute the pressure. In the portion of the ligament in which fiber
stretching is developed and sustained, due to either direct tension or
what would otherwise be shear, the tensile strains apparently heighten
the activity of fibroblasts, builder cells within the ligament that initiate the
formation of new collagenous fibers. The level of activity of these cells is
assumed to be at least partially dependent on the amount of localized
pressure developed between the stretched fibers as well as the magnitude of tensile strain.
Notable are the categories of fibers within the periodontal ligament.
The intra-alveolar or principal fiber bundles radiate from the root and proceed into and through the ligament and into alveolar bone. These fibers
are relatively short and differentiation among groups is generally made
according to direction as noted in a facial view of the tooth and alveolus.
Proceeding apically, included are the crestal, the horizontal, the oblique,
the interradicular (attached to multirooted teeth), and the apical fibers. All
but the interradicular fibers are depicted passively in Figure 5-24. The
extra-alveolar (supra-alveolar) or gin gival fibers are passively longer and are
embedded in the cementum and in the gingiva occlusal to the alveolar
crest. In this group the cementoepithelial fibers radiate from the cementum
near the cementoenamel junction, the circumferential bundles are totally
within the gingiva and surround each tooth, and the transseptal fibers
weave their way mesiodistally from tooth to tooth, linking the dental
units to one another. Given the displacement of a root point, the amount
of accompanying fiber strain depends upon the passive length of the
fiber. Of the radial groups, the cementoepithelial fibers are the longest
and require the greatest length of time to remodel; to hasten readaptation and overall remodeling, and, in particular, toward retention of
long-axis rotational displacements, these fibers are sometimes severed
with subsequent reattachment occurring naturally in the displaced configuration.
With nonextrusive tooth displacements the proximity of the root to
the alveolar-bone surface protects the ligament against failure due to
excessive tensile strain. The real concern is sustained compressive deformation of the periodontal ligament. Where pressures exist substantially
in excess of normal blood pressure (25 to 35 g/cm2), occlusion of the
vessels occurs and with it an interference in the supply of nutrition to
the tissues. Pressures of sufficient magnitude not only interfere with
normal blood flow, but drive the blood supply from the area of high
Bioengineering Analysis of Orthodontic Mechanics
_— Crestal fibers
Horizontal fibers
Oblique fibers
— Apical fibers
FIGURE 5-24. The groups of intra-alveolar fiber bundles between root and cortical plate
as seen in a transverse view of a single-rooted tooth.
Response of Dentition and Periodontium to Force
pressure, reduce or prevent cellular activity, and even force the displace-
ment of the ligament matrix material to regions of lower stress. Sustained high pressure is accompanied by potential for regressive biologic
alterations of the ligament (hyalinization, necrosis) and prevents remodeling of the alveolar bone at the cortical plate-ligament interface. Accordingly, not only to lessen the pain associated with tissue deformation, but
also to guard against the possibility of ligament pathology associated
with sustained high pressures, the magnitudes of the force-system com-
ponents transmitted by appliances to tooth crowns must be limited.
The fibrous attachment of the periodontal ligament to the cortical
plate enables the transmission of tensile as well as compressive distributed loading into the alveolar bone. A number of cells, in addition to the
aforementioned fibroblasts, exist in the periodontal space and contribute
to the remodeling process in the absence of excessive pressure. In regions of the bone-ligament interface where continuing fiber tension is
present, the fibroblastic activity is accompanied by that of the osteoblasts.
These cells play a role in the formation of osteoid, which develops along
the outer surface of the cortical plate, adhering to both the surface and
the stretched fibers emerging from the plate. In time this new material
mineralizes to form bundle bone and eventually calcifies. Each small but
distinct movement of the tooth root causes an incremental change in
interface stress magnitudes and in osteoblastic action; as a result, the
new bone appears in layers (lamellae). The overall effect on osteoblastic
and fibroblastic activity, attributable to the initial magnitude of the fiber
tension, is not well understood. Notable, however, is the apparent fact
that removal of the fiber tension has little effect on the continuation of
bone maturation, once the osteoid has been laid down. Moreover, a
reversal of the interface loading to create compression of the ligament
against bone does not result in the immediate removal of osteoid; this
new tissue exhibits an inherent resistance, significantly more so than
mature bone, to such remodeling.
Although sustained crown loading results in fiber remodeling and
bone apposition in periodontal regions where the radial component (di-
rected out from the long axis) of induced stress is tensile, where the
radial stress component is compressive, another form of biologic action
occurs. If transverse movement of teeth is to take place, resulting from
the application of orthodontic force, alveolar bone must be removed to
accommodate the leading root surfaces while new bone is formed in the
vacated space(s). The bone-resorption process is stimulated by yet another cell form, and the location(s) of the initiation of the process and the
time required to complete it are apparently dependent upon the stress
pattern within the periodontium which, in turn, depends upon the form
and magnitude of the applied crown force system. Osteoclasts in the
periodontium seemingly are activated, beyond their normal levels, by
moderate amounts of stress in certain configurations to break down the
protein matrix of bone. When the bone resorption occurs at the bone
surface adjacent to the periodontal ligament, the process is termed direct
or frontal resorption. Where the bone-ligament interface stresses are
Bioeng!neering Analysis of Orthodontic Mechanics
high, the process begins at some depth within the alveolar bone; this
internal bone remodeling is known as undermining or rear resorption.
(Recall that the radial and circumferential normal stresses decrease with
distance from a normal surface loading; see Fig. 5-23).
Two hypotheses are offered in regard to the mechanical catalyst associated with the remodeling of alveolar bone. On the one hand, there is a
strong tendency to relate the remodeling format to the radial stress—
bone apposition to tension or the tensile stress component normal to the
bone-ligament interface and resorption to the radial compressive stress
or pressure. Alternatively, transverse remodeling of long bones is said
to be keyed to the sense of the longitudinal stress—apposition by pressure and resorption by tension. Once again recall the discussion of Figure 5-23, particularly the fact that, at a point within the bone, the normal
stresses at right angles are opposite in sense to one another. If the analogy is drawn between the longitudinal and transverse directions associated with the long bone and the circumferential and radial directions,
respectively, within the alveolar bone, the so-called "piezoelectric effect" theory espoused by the orthopedist does not inherently conflict
with the pressure theory of the orthodontist.
Much remains to be learned and understood regarding the biologic
response of the periodontium to mechanical stress. Moreover, the nahire of the orthodontic appliance and of therapy adds to the complexity
of the matter. The effect of biologic remodeling "in isolation" would
undoubtedly be a modification of the initially varying stress pattern toward a uniform distribution throughout the ligament and bone, as well
as a reduction in the average pressure within the periodontium. Although the crown loading is lessened in overall magnitude as incremental displacements due to remodeling occur, it generally continues to
exist and maintains a nonuniform pattern of stress against and within
the alveolar bone. This matter is further complicated when several components are included in the crown force system and each is dependent
for its magnitude on individual stiffnesses of appliance elements; the
couple-force ratio, for example, may change as the overall displacement
progresses. Finally, the clinician may interrupt the ongoing process and
start another by first unloading, then modifying, and subsequently reactivating the appliance.
In concluding this section, a comment on the resistance of cementum
to resorption and the activity of cementoblasts is appropriate. These cells
potentially sustain and repair the hard outer surface of the root. Both the
cortical plate and the cementum are subjected essentially to the same
pressures created through the application of crown force; however, a
physiologic difference between the seemingly similar tissues is reflected
in a greater resistance to resorption inherent in the cementum. The
cementoblast resides in the periodontal space and its routine activity is
undoubtedly affected by sustained high pressure. Other factors have a
role, but apparently the combination of conditions that sometimes result
in root resorption during orthodontic treatment is not now well understood.
Response of Dentition and Periodontium to Force
Displacements Related to Magnitude and Duration of Force
force system applied to the tooth crown, toward the creation of a
displacement, originates in the straining of elements of an appliance in
the attachment of it to the dentition. The crown force system may consist of one or several components; they are individually concentrated
forces or couples, each having a direction, sense, and an initial magniThe
tude dependent on the manner of appliance-to-crown contact, the
amount of activation deformation, and the localized stiffness of the associated appliance element. The components of the applied force system
generally lessen in magnitude as time proceeds from the instant of activation, and the significant portions of the force-versus-time pattern for
the fixed appliance are (1) the initial magnitude, (2) the rapid decrease in
magnitude associated with the deformation of the periodontal ligament,
(3) the long-term decay resulting from remodeling of the alveolus (to be
examined subsequently in more detail), and (4) any reactivation by the
practitioner to begin a new cycle. Notable also is the possible creation of
additions to the crown force system resulting from tooth displacement
caused by the principal activation. For example, an arch wire may be
initially passive within a bracket, but mesiodistal crown tipping can
eliminate the original second-order, wire-to-bracket clearance and gradually result in activation of a counter-tipping couple. In such a situation,
although the original crown force system is decaying with time, a portion of the load is inactive initially, then grows from zero to some maximum magnitude dependent in a complex manner upon several parameters, and also, subsequently, decays in the absence of interference with
the original activation.
The intensity of internal force, or stress, at a particular instant and at
a specific point within the periodontium, is directly related to the characteristics of the crown force system at that instant and the specific perio-
dontal location. The change in stress at a point with time follows the
same temporal pattern as that of the crown load. The maximum stresses
exist in the periodontal ligament and against root and the cortical plate
at alveolar-crest or root-apex locations, depending upon crown loading,
as discussed previously. Ideally, any tooth movement should be accomplished in a direct and reasonably rapid fashion while minimizing patient discomfort. Largely because of the differences in biologic responses
to stimuli found in virtually any sample of man, however, there currently exists no wide acceptance of one "best" force-time pattern to accomplish even the simplest single-tooth displacements. Nevertheless,
the practitioner must be cognizant of several basic guidelines, whichever
treatment philosophy is followed:
1. Continuous interruption of the blood supply to the periodontal ligament must be limited in time.
2. The amount of time under load, required to resorb sufficient alveolar
Bioengineering Analysis of Orthodontic Mechanics
to permit a measurable tooth displacement, is apparently dependent upon the magnitude of pressure against the periodontal ligament.
3. Relatively light forces can generally produce the desired orthodontic
Controlling the Force-Time Pattern
The clinician, with the cooperation of the patient, can and must control
the magnitude-versus-time pattern of applied orthodontic force. Two
displacements, together with appliance-element stiffnesses, are the key
items in that control. The combination of appliance-member displacement(s) (activation) and stiffness(es) determine the initial level of applied crown force. The immediate, soft-tissue deformation and accompanying tooth displacement, and the same stiffness(es) determine the
state at which biologic remodeling must begin and to what extent, if at
all, the periodontal ligament has been hyalinized. In an over-simplified
manner, orthodontic forces are sometimes categorized as "light" or
"heavy," according to the initial magnitude of force generated within
the appliance. In reality, the important consideration is not whether the
initial appliance-element force is 50 g, 150 g, or 450 g, but rather (1) the
maximum value and distribution of pressure in the individual periodontal ligament following soft-tissue deformation and (2) the stiffness of the
appliance element generating the force. Accordingly, the classification
of appliances as generally "light-wire" or "heavy-wire" devices, with
reference to stiffness is somewhat more appropriate.
When periodontal-ligament pressures are sufficient to occlude blood
vessels throughout a substantial portion of the tissue at the initiation of
crown loading, as suggested in the aforementioned guidelines, the existence of a "rest and recovery" portion of the activation-deactivation cycle
is imperative. Prior to the end of each cycle and reactivation, the maximum pressures must have dropped (as a result of remodeling and tooth
displacement) to levels that permit the restoration of circulation and the
resumption of nutritional activity within the ligament. This requires
almost complete deactivation of the appliance or the unloading of it.
There need be little concern of losing the displacement achieved since
whatever osteoid has been laid down is highly resistant to resorption
compared to calcified bone.
The intermittent orthodontic force exhibits a cyclic pattern with a period equal to the time between successive appointments. Applied by a
heavy-wire appliance, the initial activation distance should be approximately twice, and generally never more than three times the expected,
corresponding soft-tissue deformation. (Greater activations may produce some inelastic straining of the appliance as well as, perhaps, excessive force magnitudes.) Although the initial force may be rather high, so
also is the appliance stiffness and immediately with the ligament deformation this force decays substantially. Bone remodeling then begins
with undermining resorption likely in the higher pressure zones until
Response of Dentition and Periodontium to Force
applied force drops off further with tooth movement. Again, this
force is termed intermittent because the combination of moderate activalion and high appliance stiffness results initially in ligament necrosis,
but ultimately in pressure reduction to low levels long before the next
appointment to ensure the resumption of blood circulation within the
soft tissue.
A second type of acceptable orthodontic force is that typically applied by an extraoral or removable appliance. The interrupted force has a
characteristic magnitude-lime pattern created by the patient who periodically unloads the appliance, thereby totally relieving all force for a
period of time. A typical interrupted, orthodontic force follows a 24hour cyclic pattern with the appliance in place and active for, perhaps,
10 to 14 hours a day. The force may be heavy and decay little; hence the
inactive period each day must be sufficient to keep the periodontal liga-
ment healthy over the total period of use of the appliance.
Continuous orthodontic forces do not, by their definition, decay to
zero during the between-appointments period. Exerted by rather highly
flexible appliance elements, activations must be to relatively low force
levels so as to occlude no more than a small percentage of the vessels
within the periodontal ligament (and not substantially interfere with
nutritional activity). Bone remodeling in pressure zones is expected to
occur primarily as direct resorption of the cortical-plate surface. Because
the continuous-force cycle includes no "rest period," little interference
with normal biologic functioning within the soft tissues can be tolerated.
Figure 5-25 provides a qualitative comparison between the magnitudetime patterns of acceptable light and continuous, and relatively heavy,
but intermittent, orthodontic forces.
To produce the total desired displacement by a particular appliance
configuration or set of mechanics often requires several months. A new
activation-deactivation cycle begins each time the clinician reactivates
the appliance, and this reactivation may occur four or more times in
the overall process, for example, in distally driving canines into firstpremolar extraction sites. If the teeth to be moved are receptive to ortho-
dontic force, a reactivation is necessary only when the appliance has
exhausted all or most of the mechanical-work capability provided by a
prior activation or reactivation. In the absence of some appliance failure,
the total displacement is accomplished incrementally, and an examination of one activation-deactivation cycle provides the substance of the
entire tooth-movement process, even though perhaps only one-fourth
or less of that total displacement is depicted.
Figure 5-26 shows two time-versus-displacement plots for one acti-
vation-deactivation cycle and a between-appointments period of 3
weeks. The displacements are simple-tipping of canine teeth on opposite sides of the same arch. The solid plot represents an initial force
magnitude of 60 g; the dashed plot was obtained following activation to
150 g. Each crown displacement takes place in 3 stages. Stage 1 is the
almost immediate movement reflecting the deformation of the periodontal ligament by the tooth root; little difference in this first portion of the
overall displacement is noted for the 2 teeth as, apparently, 60 g was
Bioengineering Analysis of Orthodontic Mechanics
soft-tissue deformation
undermining resorption
Heavy, intermittent force
Light, continuous
FIGURE 5-25. A one-cycle comparison of the magnitude-versus-time patterns of
intermittent and continuous orthodontic force.
sufficient to completely compress the ligament at the root apex and alve-
olar-crest levels. During stage 2 the remodeling of fibers and bone begins and further displacement is virtually nonexistent. The length of this
stage may vary from several days to several weeks or longer, depending
on the level of pressure against the cortical plate. From the figure, stage
2 for the tooth initially receiving 60 g of load was completed in about 4
days; for the canine subjected to 150 g this second stage extended to
nearly 2 weeks. As bone is resorbed in the pressure zones and the effect
of resorption reaches the bone-ligament interface where complete softtissue compression had occurred upon loading, additional tooth movement signals the beginning of stage 3. When direct resorption takes
place, stage 3 displacement is more gradual than when substantial undermining resorption is necessary before any further tooth movement
can occur. This is reflected in Figure 5-26 in that, after stage 3 began for
the tooth receiving 150% more initial force, the displacement progressed
at a more rapid rate such that, when the patient appeared on day 22, the
practitioner measured nearly the same accumulated canine crown displacement on both sides of the arch (Gianelly and Goldman, 1971).
The overall results depicted in Figure 5-26 are indicative of the preference of some clinicians for heavy-force mechanics and others for lightforce appliances and techniques. Conceivably, however, had a third,
similar canine been subjected to an initial force of several hundred
grams, stage 2 might not have been completed in 21 days; hence, no rest
period for the ligament would have occurred between appointments, a
)flSO of Dentition and Periodontium to Force
Stage 3
Stage 2
Stage 1
Time (days)
E 5-26. The three-stage pattern of tooth movement. Plots of displacement versus
Or a pair of canine teeth, one subjected to an initial distal driving force of 60 g
and the other to 150 g (dashed).
Bioengineering Analysis of Orthodontic Mechanics
reactivation of the appliance might endanger ligament vitality, and the
result of excessive initial force applied 3 weeks earlier is likely an exten-
sion of the total time to complete the full tooth movement.
The graphics of Figure 5-26, in illustrating the three-stage nature
of orthodontic tooth movement, are indicative of the first, betweenappointments period associated with a long-term displacement; the
1.5 mm is perhaps one-fifth of the total movement desired. A reactivation will produce another displacement-time pattern, similar to the first
cycle if (1) nearly total deactivation of the appliance had taken place, (2)
the periodontal ligament had been fully revitalized if hyalinization oc-
curred early in the cycle, and (3) the reactivation displacement of the
appliance was the same as that at the start of the first cycle. In general,
the lighter the force, the less significant may be stage 1 and the shorter is
stage 2. Assuming that measurable movement does not take place during the first cycle, subsequent reactivations can be to somewhat larger
initial loads since cellular activity has been ongoing and the tooth or
teeth are loosened with slightly wider periodontal space than before the
orthodontic force was first applied. Also, the apparently insignificant
difference in one-cycle displacements shown in Figure 5-26 may be the
long-term result, indicating that an initial magnitude of force might be
selected from a range of acceptable values. It is clear, however, that the
range is bounded. If the force is too light, the rate of stage 3 displacement will be low and the between-appointments displacement small.
With the load magnitude too large, little more than stage 1 displacements will occur and, with subsequent reactivations and no relief for the
tissues from the high pressures, accumulated damage to the periodontal
ligament is likely.
Physiologically Proper Orthodontic Forces
An orthodontic force or force system may be said to be "physiologically
proper" when the force-time pattern produces a reasonable rate of tooth
movement without causing extensive interference with the normal biologic activity within the periodontium and, therefore, no irreversible
damage to the tissues. Although very light forces fulfill this prescription,
the efficient management of treatment demands that displacements be
performed as rapidly as possible. Some discomfort is often associated
with activation or reactivation of an appliance due to the relationship
between the pressures developed and the pain threshold of the patient;
again, efficient management to the end of gaining and maintaining good
patient cooperation suggests minimization of this discomfort.The orthodontist, then, should strive to apply force that is controlled both with
respect to initial value and, through astute selection of appliance elements, in terms of its decay pattern over time.
From the foregoing discussion, the principal indicator of appropriate
force is the pressure generated in the periodontium, more specifically, in
the periodontal ligament. Because this pressure is actually compressive
stress, and stress level is generally related to applied force through the
area over which the stress is distributed, a first consideration is the num-
Response of Dentition and Periodontium to Force
TABLE 5-2. Root-surface areas in centimeters squared for average normal permanent
Central Incisor
Lateral Incisor
First Premolar
Second Premolar
First Molar
Second Molar
Third Molar
1 .50
1 .30
ber of teeth directly affected by the applied force system and the size of
the roots of those teeth. Table 5-2 provides the root surface areas of
individual teeth of average size. Note that these areas vary from the
smallest to the largest by a factor of 3.3. Given the value of an applied
force and the tooth or segment to receive that force, the average stress in
the periodontal ligament may be obtained by dividing the force magnitude by the affected root-surface area. Using this approach, then, nearly
twice the force magnitude to displace a mandibular central incisor
should be applied to move a maxillary canine. Again, however, the key
parameter is specifically compressive stress and three, additional, important factors must be considered in this analysis: (1) the direction of the
applied load, (2) the couple-force ratio (or type of displacement desired),
and (3) the load-time pattern.
Refining the procedure of selecting an initial orthodontic loading
magnitude to emphasize pressure and direction of load as criteria suggests replacing the overall root-surface area with projected areas. Table 5-3
gives these projected root-surface areas from three perspectives: occluTABLE 5-3. Projected root-surface areas in centimeters squared for average normal
permanent teeth presented from three perspectives
Mandibular arch
Maxillary arch
Central Incisor
Lateral Incisor
First Premolar
Second Premolar
First Molar
Second Molar
Third Molar
FL: Faciolingual perspective
MD: Mesiodistal perspective
OG: Occiusogirgival perspective
Bioengineering Analysis of Orthodontic Mechanics
sogingival, faciolingual, and mesiodistal. The only orthodontic displace-
ments I or which pressure does not cover such an area are extrusion and
long-axis rotation, although only a partial (occlusogingival) area may be
involved in an intrusive displacement. Recalling the differences in stress
distributions produced in the various transverse tooth movements (Figs.
5-5, 5-8, 5-10, 5-12, and 5-14), Table 5-1 presented sets of theoretically
determined couple-force ratios, center-of-rotation locations, and maxi-
mum stress values for five such displacements of an average canine.
Noting particularly that the distal driving force is identical in four of the
displacements, because of the sizable differences in maximum stress
values between simple tipping and bodily movement, for example, a
substantially larger crown load may be used in attempting the latter
The influence of root shape should also be brought into the determina-
tion of appropriate crown loading when warranted. Perhaps the best
example of root-shape impact is in the comparison between long-axis
rotation of the single- and multirooted tooth. The specific factor is the
substantially greater amount of bone resorption required to bodily move
the bifurcated-root apices that follow circular paths around the long axis.
In other instances, unusual root curvatures or cross-sectional shapes
may provide significant input in determining load magnitude. A final,
important factor is individual biology. Resistance to or ease of remodeling
is affected by alveolar bone density and to an extent by cortical-plate and
periodontal-ligament thicknesses. A reasonable manner of including
this parameter is to estimate by clinical observation and patient input, a
nearly "optimum" orthodontic force magnitude for a specific tooth displacement early in treatment. Subsequently, by proportions according
to projected root-surface area and type of movement primarily, appropriate load values may be established for succeeding displacements to be
undertaken in the same dentition and within the same treatment philosophy.
With all principal influencing factors now mentioned, a hypothetical
procedure may be outlined which leads to the determination of the suggested, initial level of physiologically proper force with which to begin a
specific orthodontic displacement in the average patient. The approach
to be outlined arises from the impressions of clinicians, based on their
experiences, together with theoretical and bench-experimental analyses.
The fundamental bases are (1) the knowledge of normal blood pressure
and, therefore, a reasonable approximation of the ligament pressure
necessary to interrupt blood flow at a point within that tissue, (2) the
assumption that vessel occlusion over a sizable volume within the periodontal ligament for a period of several weeks is necessary to produce
irreversible pathology, and (3) the understanding that, whenever direct
pressure exists within a portion of the ligament (and at the interfaces),
elsewhere in that tissue fiber tension is the direct stress that should not
interfere substantially with nutritional activity in those areas.
The procedure begins with the following hypothesis: an intermittent,
simple-tipping force should have an initial value such that the cone-
Response of Dentition and Periodonfium to Force
sponding average stress within the affected periodontal ligament is approximately 30 g/cm2. In other words, the initial magnitude of the trans-
verse-tipping force should be the product of 60 g/cm2 and the
mesiodistal or faciolingual projection of the total root area transferring
responsive tension and pressure. (In transverse movements, two projected areas are under normal stress—mesial and distal, for example.
Hence, 60 g/cm2 is multiplied by one of these areas or 30 g/cm2 by the
sum of the two areas.) If the applied force is to be continuous and no
"rest period" will occur over a period of four or more weeks, the initial
value should be approximately 0.4 of the corresponding intermittent
magnitude (based on the assumption that such a force will decay, beyond stage 1, about 40% during one between-appointments period).
For bodily movement, the stress distribution along the root length is
substantially more nearly uniform than the response in simple tipping
(see Figs. 5-5 and 5-8). Also, the ratio of maximum stresses to applied
force is much lower in bodily movement (see Table 5-1). Accordingly, a
larger initial force can be tolerated in bodily movement, and the suggested value is based on the average of the alveolar-crest and root-apex
stresses taken from Table 5-1. The resulting formulas for the initial val-
ues of the force and couple portions of the intermittent, transverse
crown load to produce bodily movement are
Transverse force = 3.5(simple-tipping force)
Couple = 0.5(transverse force)(total tooth length)
(Assumed is the crown force system effectively applied at point CC on
the long axis, midway between incisal edge [occlusal surface] and the
cementoenamel junction.) Again, examining Table 5-1 and Figure 5-11,
the couple-force ratio for root movement is understandably somewhat
greater than that for bodily movement. Once more employing as a portion of the basis the average of alveolar-crest and root-apex stresses
equal for all transverse movements, the formulas to follow are offered
for the torquing of roots through intermittent loading:
Transverse (holding) force = 0.8(comparable bodilymovement force)
Couple-force ratio = 0.6(total tooth length)
When several teeth are to undergo similar displacements as a single
entity, for example, in the retraction of four incisors in either arch, the
computations may be made for the segment as a whole using an average
tooth length and the sum of the projected root areas of the individual
teeth. Proceeding as above, for transverse crown movement the following formulas are suggested:
Transverse force = 1. 7(comparable bodily-movement force)
Couple-force ratio = 0.4(total tooth length)
Bioengineering Analysis of Orthodontic Mechanics
Although not occurring often in actual clinical practice, but appropriate
to include for completeness in the set of formulas for proposed, physiologically proper transverse force systems is this equation for pure torquing:
Couple = (comparable simple-tipping force)(root length)
The couple-force ratio is undefined because no separate crown force
The three nontransverse orthodontic displacements—intrusion, extrusion, and long-axis rotation—might be characterized collectively by
the dominance of fiber tensions within the responsive force system. Vir—
tually no pressure occurs anywhere between root and periodontium in
response to purely extrusive action; in long-axis rotation, the sizes and
locations of pressure zones depend entirely on root anatomy. To complete the set of formulas for the force-system components, then, for the
remainder of the eight, previously defined displacements, in the absence of pressure as a principal influence, collective judgments of clinicians, based on their treatment observations, are the principal bases for
the following suggestions:
Extrusion: Force =
projection of root-surface
area in cm2)
Intrusion: Force =
projection of root-surface
area in cm2)
Long-Axis Rotation: Couple =
projection of root-surface
area in cm2)(average
crown width)
Several addenda to these formulas are pertinent:
If the tooth is against the cortical plate during an attempted intrusive
displacement, the applied force computed from the formula should
be doubled.
If a multirooted tooth is intruded, the magnitude of force obtained
from the formula should be increased 50%.
If long-axis rotation of a multirooted tooth is envisioned, the couple
value calculated from the formula is multiplied by 1.5 if the tooth has
two roots or by 2.5 if the tooth has three roots.
It is worth repeating that the foregoing formulas are all presented for
computations of initial, intermittent crown loadings. If the crown force
system is continuous with only a moderate decrease in magnitude during the between-appointments period, beyond that associated with the
rapid, soft-tissue deformation, the initial magnitudes should be 50 to
60% less, depending on the amount of long-term decay. On the other
Response of Dentition and Periodontium to Force
hand, the interrupted force that is cycled daily may have a magnitude
larger than that of the corresponding intermittent force due to (1) the
increased frequency of rest periods and (2) the generally longer accumu-
lated lime of total absence of the interrupted force between appointments. A reasonable procedure is to make the magnitude of interrupted,
orthodontic force the product of the corresponding, intermittent force
and the ratio of 24 to the number of hours the interrupted force is active
during the day.
Implicit in the considerations of responsive forces arising from the
application of crown loading is the existence at any specific time of the
quasi-static state defined in Chapter 2. Hence, the resultant of the distributed force system exerted on the root of a tooth under orthodontic
load is identical, for practical purposes of analysis, in all characteristics
except sense to the resultant of the force system applied by the appliance
to the crown. The appliance as well as the dentition exhibit this quasistatic state. The reaction against the appliance accompanying the action
of the appliance is transmitted to the anchorage location(s). Focusing on
the appliance momentarily, the force system exerted by the anchorage is
equal in magnitude but opposite in sense to that exerted by the crown(s)
at the displacement site(s). In general, "anchorage" refers to stability,
and in orthodontics the anchorage supports the appliance, receiving the
responsive force system transmitted through the appliance to it. Although in orthodontic therapy the entire force system developed by the
appliance is sometimes intended to produce displacements, more often
a portion of the appliance is attached to anchorage which, by definition,
is to be displaced very little or not at all. The rationale for the use of the
extraoral appliance is the location of anchorage external to the oral cavity, against the back of the neck or the cranium. Intraoral anchorage is
often necessary or desirable, and the knowledge of the influences of the
various force parameters on orthodontic displacement have led to the
differential force concept.
The actual meaning of
differential force,
not unrelated to the
hypothesis of a physiologically proper force, is best explained by an
example. Consider the distal movement of a mandibular canine into the
space created by the extraction of the formerly adjacent first premolar.
Using intraoral mechanics exclusively, the distal driving force is created
by extending a helical-coiled spring or an elastic module between the
canine and the first molar. The forces against the canine and molar,
holding the activating appliance member in a stretched configuration,
are equal in magnitude (from a quasi-static analysis of that member) and
tend to move both teeth. The force against the molar, however, may be
distributed also to the second molar (if "tied" to the first molar) or the
second premolar (through crown contact). If the anchorage includes all
three posterior teeth mentioned, from the projected areas given in Table
5-3, the anchorage area is approximately 250% larger than that of the
canine. Accordingly, although the forces at the anchorage and displace-
ment sites are equal in magnitude, the average periodontal-ligament
stress is 3.5 times greater at the canine. The size of the active force (upon
Bioengineering Analysis of Orfhodontic Mechanics
canine crown) should be just large enough to produce primarily
direct resorption of bone and a reasonable rate of canine movement. The
same force magnitude, then, should not be sufficiently great to create
adequate pressure to produce appreciable displacement of the anchorage during the between-appointments period.
Note that if the active force is relatively large, so also is the responsive force, and both units may move comparable distances over time.
Furthermore, extreme force magnitudes can result in displacement of
the anchorage while, in effect, producing ankylosis of the canine. Refer-
ring once more to Table 5-1, noting that the distal forces for four
movements were equal, changes in the couple-force ratio are seen to
alter the maximum-stress values and the overall stress pattern with respect to a uniform distribution as a reference. Apparently, then, a uniform stress distribution is ideal for the anchorage, although it is difficult
to obtain in the clinical setting. More realistically, the creation of a crown
couple with the appliance to counter in sense the tipping potential of a
force against anchorage is a step toward reinforcing that anchorage, beyond that achieved by maximizing the root area within the anchorage.
The point is made, then, that the phrase differential force is a misnomer;
the differentials are actually in the average stresses and/or the stress-distribution patterns within the ligament at the displacement and anchorage
The parameters that influence the determination of the initial magnitude of a physiologically proper orthodontic force have been indicated in
the foregoing discussion, and a set of formulas was given. Through the
literature, additional terms have been introduced to describe or categorize forces exerted on the tooth crown. The lightest of these actions have
been termed threshold forces, just large enough to result in some remodeling which permits measurable movement over a lengthy, perhaps unrestricted, period of time. Differentiation between "light" and "heavy"
orthodontic forces was undertaken previously. Upper-bound or maximum orthodontic forces produce ligament necrosis for a time, are likely
to be initially painful to the patient, but result in tooth movement with
no significant, long-term pathology. Forces are termed excessive if they
crush the periodontal ligament over a substantial portion of the area
under pressure, probably result in little or no tooth movement during a
between-appointments period, and if continuous beyond several weeks
may cause irreversible tissue damage.
During the 1950s and 1960s research was undertaken toward determination of optimum-force magnitudes associated with specific, orthodontic displacements. Values were sought that would produce desired
movements most rapidly and cause only minimal interference with normal biologic activity and patient discomfort. A principal motivator in
these investigations was the solidification, through in vivo experimentation, of the validity of the differential-force concept. In the late 1960s,
however, papers began to appear in the literature exhibiting research
results that tended to discount an optimum-force theory. The influence
of the magnitude-time pattern on displacement achieved has been
Response of Dentition and Periodontium to Force
Light force,,,1!'
\HeavY force
\ Excessive
As-Activated Force Magnitude
FIGURE 5-27. Qualitative plot of expected, be!ween-appointments displacement versus
the initial level of orthodontic force.
shown to be a significant factor in addition to parameters rightfully con-
sidered in the determination of the initial force values. Collectively, from
these efforts, the acceptance of the hypothesis of an existing range of
orthodontic force magnitudes has emerged for each specific displacement, which produces no long-term physiologic conflict. The plot of
initial force magnitude versus displacement rate of Figure 5-27 provides
a qualitative illustration of the present understanding of expected results
with a typical patient.
Dento facial Orthopedics
Although the typical displacements performed during orthodontic treatment are of individual teeth or dental-unit segments, the total therapy in
a specific case may, in part, call for dentofacial orthopedics. Examples of
orthopedic forces are those induced by an appliance designed to widen the
maxillary arch through palate splitting, to tip the maxilla (altering the
angulation as seen in a buccal view), and to exert influence on the pat-
tern of growth of the mandible. The appliance may be intraoral or
extraoral, but the applied force must always be distributed over a substantial area and, correspondingly, the magnitude of the total load is
large. The load may or may not be transmitted through the dentition,
but the biologic remodeling is not intended to be within alveolar bone; in
general, relative displacements of adjacent teeth in the same arch are
In palate splitting, the intention is to induce transverse tension along
Bioengineering Analysis of Orthodontic Mechanics
the median suture. The appliance used is totally intraoral, producing
bilateral action with force transmitted at least partially through the posterior teeth and often, in part, by direct appliance contact with the hard
palate. That portion of the load imparted through the lingual crown
surfaces must originate from an appliance element sufficiently rigid such
that the individual teeth are not displaced relative to their alveoli. Because remodeling is to occur in tensile zones, the magnitude of physiologically proper force is not directly associated with the suture-bone in-
terface area, but with the force-time pattern and the faciolingual
projected areas of the tooth roots through which the force is to be transmitted. Because patient discomfort is often used as an indicator of sufficient or excessive activation-reactivation of the appliance, the appliance
should be designed so that the pressure generated in those periodontal
ligaments, through which the active force system is transmitted, approaches a uniform distribution.
In orthopedic action designed to tip the maxilla, at least a portion of
the anchorage may be extraoral; the mandibular arch may also be used
in the anchorage. Depending on whether vertical displacements are desired only anteriorly, only posteriorly, or divided between anterior and
posterior locations, the vertical component(s) of the load are applied
anteriorly or posteriorly as individual forces, as equal and opposite
forces at both locations to form a couple, or as unequal and opposite
forces at both locations to generate a force-and-couple resultant. Heavy,
continuous arch wires are in place to prevent individual tooth displacements. The desired remodeling is primarily along the zygomatic process; the stress distribution at that interface is nonuniform and the center
of rotation for the maxilla as a whole will be located, anteroposteriorly,
approximately directly below the point in a buccal view where the stress
distribution, or its extension, goes to zero (or changes sense from tension to compression or vice versa). Figure 5-28 illustrates two forms of
HG: Headgear
Ill: Class Ill
00: Occlusion
Class II
FIGURE 5-28. Tipping the maxilla to open the bite anteriorly (left) and to close an open
bite primarily through posterior vertical action (right).
Response of Dentition and Periodontium to Force
Solid: Appliance activated
Dashed: Appliance disengaged
Time (hours)
FIGURE 5-29. The typical interrupted, force-time pattern of an extraoral appliance.
maxilla tipping. Note that horizontal (posteriorly directed) components
of force nearly always accompany the wanted vertical components,
owing to the nature of the appliances. The magnitude of force applied
should be gauged by imagining the maxilla as a single tooth with total
length equal to the anteroposterior dimension. The tipping load is a
force, a couple, or a force and couple, and the couple-force ratio for the
desired displacement format may be estimated. Also note that the max-
illa is supported, in essence, only superiorly; as modeled as a large,
individual tooth, then, it has only approximately one-half of the equiva-
lent periodontal support of a "tooth" with comparable support (root)
To control mandibular growth downward and forward, orthopedic
force may be applied by an extraoral appliance. The force is most often
applied by a chin cap rather than through the mandibular dentition. The
maximum levels of force used are governed by the acceptable pressures
induced against the temporomandibular-joint members. In order that
large forces may be employed, the force-time relationship is the typical
interrupted pattern of extraoral appliances as depicted in Figure 5-29.
This appliance is discussed in more detail in Chapter 9.
This chapter has endeavored to examine the biomechanical response of
the dentofacial complex, with emphasis on the dentition and its sup-
Bioengineering Analysis of Orthodontic Mechanics
porting structures, to applied force during orthodontic treatment. The
delivered force system, most often exerted through the facial crown surfaces, originating in the activation of an appliance, follows a discernible
path into and through the alveolar process and beyond. The presence of
force within the tissues, augmenting the normal physiologic activity,
catalyzes biologic processes that result in tissue remodeling, thus per-
mitting measurable displacements of teeth—individually or in segments—or of an entire arch.
Eight distinct forms of orthodontic (individual-tooth) displacement
have been defined. For each, the characteristics of the crown force system, the distributed responsive pattern of tension and pressure within
the periodontal ligament and against the root, and the displacement
format, were discussed. Without becoming deeply involved in the physiology, the remodeling of the periodontal structures has been considered with emphasis on the role of induced internal-force intensity, in
particular that of compressive stress (pressure). The movement of teeth
by means of orthodontic therapy is known to be controlled by the forcetime pattern of the loading as well as the initial (as-activated) load characteristics; this pattern, in turn, is dependent on the structural properties of certain appliance elements and by the actual displacements that
occur. The seemingly closed loop notwithstanding, advancements
through research are leading to the refinements of procedures to determine and apply physiologically proper ranges of orthodontic force. A
"cookbook approach" to the quantification of initial levels of crown
loading has been offered.
The concepts of differential and optimum force were introduced and
discussed briefly. Through the former, the term anchorage was defined.
Some attention was given to the biomechamcal analysis of dentofacial
orthopedic therapy.
This chapter has provided descriptions of and guidelines for desirable characteristics of force systems delivered to the dentition in orthodontic treatment. These force systems are developed within and transmitted to the crowns by the orthodontic appliances. Beginning in the
subsequent chapter and continuing throughout the remainder of this
text, structural analyses of appliance elements are undertaken, individually and in relation to their roles in the appliance as a whole. The ultimate objective is the refinement and improvement of appliance designs
toward devices that provide ideal magnitude-time patterns of orthodontic force at those locations where displacements are desired, while concurrently minimizing unwanted, secondary effects elsewhere.
Burstone, C.J., Pryputniewicz, R.J., Bullock, C., and Hubert, M.: Centers of
rotation of the human maxillary central incisor. J. Dent. Res., 61 IADR Abstract 1095, 1982.
Response of Dentition and Periodontium to Force
Burstone, C.J., Pryputniewicz, R.J., and Weeks, R.: Center of resistance of the
human mandibular molar. J. Dent. Res., 60 IADR Abstract 822, 1981.
Gianelly, A.A., and Goldman, H.M.: Biologic Basis of Orthodontics. Philadelphia, Lea & Febiger, 1971, Chapter 4.
Nikolai, R.J.: On optimum orthodontic force theory as applied to canine retraclion. Am. J. Orthod., 68:290—302, 1975.
Periodontal ligament reaction and displacement of a maxillary central incisor subjected to transverse crown loading. J. Biomech., 7:93—99, 1974.
C.H., and Johnston, L.E.: A clinical investigation of the concepts of
differential and optimal force in canine retraction. Angle Orthod., 44:113—
119, 1974.
Burstone, C.J.: Application of bioengineering to clinical orthodontics. in Current
Orthodontic Concepts and Techniques. 2nd Ed. Edited by T.M. Graber and
B.F. Swain. Philadelphia, W.B. Saunders, 1975, Chapter 3.
Burstone, C.J.: The biomechanics of tooth movement. in Vistas in Orthodontics. Edited by B.S. Kraus and R.A. Riedel. Philadelphia, Lea & Febiger, 1962,
Chapter 5.
Burstone, C.J., Baldwin, J.J., and Lawless, D.T.: Application of continuous
forces to orthodontics. Angle Orthod., 31:1—14,
Burstone, C.J., and Pryputniewicz, R.J.: Holographic determination of centers of
rotation produced by orthodontic forces. Am. J. Orthod., 77:396—409, 1980.
Graber, T.M.: Orthodontics, Principles and Practice, 3rd Ed. Philadelphia, W.B.
Saunders, 1972, Chapter 10.
Jarabak, J.R., and Fizzell, J.A.: Techniques and Treatment with Light-wire Edgewise Appliances, 2nd Ed. St. Louis, C.V. Mosby, 1972, Chapter 7.
Reitan, K.: Some factors determining the evaluation of forces in orthodontics.
Am. J. Orthod., 43:32—45, 1957.
Reitan, K.: Bone formation and resorption during reversed tooth movement, in
Vistas in Orthodontics. Edited by B.S. Kraus and R.A. Riedel. Philadelphia,
Lea & Febiger, 1962, Chapter 3.
Ricketts, R.M., et al.: Bioprogressive Therapy, Book 1. Denver, Rocky Mountain
Orthodontics, 1979, Section 1, Part 6.
Smith, R.J., and Burstone, C.J.: Mechanics of tooth movement. Am. J. Orthod.,
85:294—307, 1984.
Thurow, R.C.: Edgewise Orthodontics. 4th Ed. St. Louis, C.V. Mosby, 1982,
Chapters 10—12.
Thurow, R.C.: Atlas of Orthodontic Principles. 2nd Ed. St. Louis, C.V. Mosby,
1977, Chapters 2, 14.
Introduction to
Structural Analysis of
the Orthodontic
Beginning in this chapter and continuing throughout the remainder of
this text, force and structural analyses of orthodontic-appliance elements or members, and assemblies thereof, are undertaken. In the following discussions, the terminology and procedures used in structural
analysis are introduced, and the orthodontic appliance is compared to
the typical engineering structure. The manner of attachment of the appliance to the dentition is examined in detail, highlighting the associated
force system and devoting particular attention to orthodontic brackets.
The continuous-arch-wire appliance is then modeled to simplify a geo-
metrically and mechanically complex member for analysis purposes
without altering its essential characteristics. A four-step analysis Focedure is introduced. The activation and deactivation behavior of several
of the more common orthodontic-appliance elements is discussed. This
chapter concludes with a section that examines the "control" of treatment progress and the roles therein of the appliance, the clinician, and
the patient.
A structure is an interconnected assembly of individual members; the
assembly is designed to withstand without failure a specific loading and
to transmit the force system induced by that loading into some foundation or supporting substructure. The load may be "dead" weight, some
other static action, or it may be dynamic, varying characteristically with
time. A "machine" may be distinguished from a structure. The typical
structure is designed primarily to sustain a load and its principal components are usually stationary; a machine is designed primarily to transmit a
force system from one site within the assembly—the input—to another—
the output; moreover, the machine typically has movable parts. Buildings and bridges are familiar structures and a pair of pliers is an example
Introduction to Structural Analysis of the Orthodontic Appliance
of a machine. Because both structures and machines include, when "in
service," members within which internal force systems exist, structuralanalysis procedures are as appropriate to machines as they are to structures.
Each member of the assembly contributes toward the performance of
the desired function of the machine or structure. The loading of the
structure or the input to the machine results in subjecting each element
or member to a force system. The force system delivered to a specific
member may be transmitted entirely through its connections to other
members, or, it may result, in part, directly from external activation.
Each member must be analyzed initially to determine the type of force
system exerted on it and then the relationship of each component of that
force system to the external load or input. After the external force system has been quantified, the internal force distribution throughout the
volume of the member must be obtained toward consideration of possible failure of the member.
Failure of a structural or machine member occurs when either stresses
or deformations exceed their allowable values. Excessive stresses may
result in outright fracture of the member or in drastic reduction of its
ductility, for example. Excessive deformations may eliminate clearances
that must exist or may result in structural instability or collapse of the
structure. Stresses generally vary from point to point within the mem-
ber, and deformations ordinarily vary from one cross-section of the
member to another; hence, the analysis is undertaken to determine not
only the maximum values of stress and deformation, but also where
within or along the member they occur. From the discussions within
Chapter 3, the material and geometry (size and shape) of a structural or
machine member are known to influence, in addition to the applied
force system, the induced stresses and deformations.
The typical structural analysis proceeds according to one of two formats:
Given the design or actual structure or machine and the failure criteria, to be obtained is the maximum allowable loading or input force
2. Given the design, the loading or input, and the possible modes of
failure, computations are made to determine whether or not a failure
might occur in the actual structure or machine.
The analysis procedure follows a well-defined series of steps, regardless
of the format. First, a force analysis is undertaken, for the structure or
machine as a whole and for each member of the assembly, to obtain in
terms of the loading or input the components of the force system exerted
on the entire assembly and on each member at the connections. Newton's laws (Chap. 2) and the relationships governing continuity of displacements may be used to carry out this first step. Second, the failure
criteria are established (fracture, excessive inelastic behavior, formation
or extension of a crack, or deformation beyond established bounds) and
Bioengineering Analysis of Orthodontic Mechanics
quantified for each member and for the structure as a whole. Each maxi-
mum permissible stress or deformation is modified in value by a "factor
of safety" to account for inadequacies in the theory or model used in the
analysis. Third, "critical" cross-sections and locations of maximum
stress therein are pinpointed within each member of the structure or
machine. Fourth and finally, the stresses and deformations at the critical
locations are obtained in terms of the force system exerted, and are
compared to the failure criteria as quantified to establish the structural
integrity, or lack of it, of the assembly.
A structural analysis of an existing structure or machine would be
undertaken when a change in the loading or input force system is contemplated. Such an analysis would also be carried out as a primary facet
in the preparation of a new design. The overriding design objective is
always the creation of an assembly that is capable of sustaining, without
failure, the force system to be induced in it. Design constraints with
regard to space, materials, and perhaps aesthetics may also require satisfaction. Several alternative designs may meet the principal design criteria, and one may have to be selected through secondary considerations
or judgmental factors. Often, after choosing one design from among
those considered, further modification of it occurs that will not result in
infringement on the failure criteria; excess material may be removed
from regions of low stress, for example. In short, the preceding comments are intended to suggest that design and analysis are closely related.
The Orthodontic Appliance: A Structure or a Machine?
The function of the orthodontic appliance is the application of specific
force-system components to selected intraoral sites to produce desired
displacements of dental units. The appliance is similar to the typical
structure or machine; forces exist at the connections of its elements and
within its members when "in service," and it, therefore, may appropriately be subjected to structural analysis. The principal "loading" of the
orthodontic appliance, however, is unlike that of a structure or machine;
its load source is not an external force system (although the forces of
occlusion or mastication may apply secondary, external loading). The
"input" to the appliance is the deforming of it from its passive configuration to induce internal forces that are to be transmitted to the displace-
ment sites. The "output" is the force system delivered to the tooth
crowns to produce selected movements of portions of the "foundation"
to which the appliance is attached.
In a strict sense, the action of the appliance is dynamic; most components of the internal force system decrease in magnitude with the passage of time in the activated state. This change is so gradual, however,
and the masses of the appliance elements are so small, that analyses of
the primary force system in the activated appliance may be undertaken
Introduction to Structural Analysis of the Orthodontic Appliance
using static relationships with negligible error. Restricting attention only
to the generally quasi-static appliance system, failures might be assumed to occur immediately upon activation or not at all during the
between-appointments period. Forces of occlusion, however, do provide real dynamic loading, superimposed on the appliance activation.
Although this source of "load" is comparatively minor by itself, it can
result in loss of structural integrity and failure with, for example, a frac-
ture within the bond between bracket pad and enamel surface. The
"loading" of the removable appliance is dynamic when examined in the
context of a time frame of days or weeks, and progressive failure—
resulting in the fracture of a wire component of a removable retainer—is
not uncommon. Apparently required, then, is at least the direct incorporation in a static-appliance analysis of a factor of safety to account for the
dynamic loads that may be intermittently applied to it.
The structural analysis of the ideal orthodontic appliance may be
undertaken through a hypothetical design of the appliance. Principal
design parameters include specific stiffnesses and elastic ranges of the
members of the appliance to be activated. Generally, the activation is
such that inelastic material behavior does not occur anywhere in the
appliance; hence, maximum stresses may not exceed elastic-strength
values. Also important are the appropriate member stiffnesses at anchorage locations, a somewhat subtle consideration to be taken up in
later discussions of specific mechanics. The appliance design must con-
sider the possible paths of the force transmitted from the activation
site(s) to the anchorage location(s), and the distribution of anchorage
force. Pertinent in addition are the shape changes in the appliance and
resulting alterations in the force system as deactivation due to tooth
displacements takes place; not all of these changes in appliance configurations are mirror images of the activation deformations. And not to be
overlooked, the materials of members must be chosen with full regard
for the biologic environment in which they will be placed.
:achment of the Orthodontic Appliance to the Dentition
intraoral orthodontic appliance is supported by the tooth crowns
and its overall configuration is maintained through its connections to
them. These connections also provide the final measure of continuity
that permits the transfer of active force from the appliance to the teeth to
generate the wanted displacements, regardless of the type of appliance.
Although these connections may be made in a variety of ways, considering all phases, types, and philosophies of treatment, the most common
is through the orthodontic bracket affixed to the facial or lingual surface of
the tooth crown. The bracket as attached is meant to be integral with the
crown; hence, it should exhibit high overall stiffness, although strength
is more often the principal concern. Failures in bracket-to-crown attach-
ment may occur by fracture, either at or near the solid joint of the
Bioengineering Analysis of Orthodontic Mechanics
bracket to its pad or to the band, or in the bond of the pad or band to the
crown enamel. The latter is more common.
Attachment of the bracket assembly to the crown may be through
cementation of the metal (stainless steel) band that encircles the crown
or by bonding of the bracket pad to the facial or lingual surface. The
merits or disadvantages of one scheme versus the other may be argued
from several standpoints; however, of interest here is the structural
comparison. The force system transferred from bracket to crown must
travel through the cement or adhesive and the strength of the attachment depends on the size and shape of the attachment area, the characteristics of the force system transferred through it, and the inherent unit
strength of the cement or adhesive itself (internally, between it and the
band or pad, and its attachment to the enamel). The cement or adhesive
may be stressed in many directions because of the various force-system
components that may be present—individually and in combinations. In
addition, tensile and sheat strengths differ for a specific product and
between products. A detailed comparison would, therefore, be lengthy
and inappropriate here; however, two general comments are pertinent.
First, the bonding area of the band is at least double that of the typical
bracket pad; hence, the inherent strengths of the bonding adhesives
must be substantially greater than those of band cements. Second, the
more nearly uniform the stress distribution within the adhesive, the
stronger the bond; accordingly, failures of bonds of pads to enamel surfaces are more likely under loadings that tend to produce "peeling"
(from first- or third-order couples or from occlusogingival or mesiodistal
forces that have a moment arm with respect to the adhesive) or rotation
(from second-order couples) than when the force system is direct tension, shear, or compression.
The bracket-band and bracket-pad assemblies are compared pictorially in Figure 6-1 as to stress generation within the bond resulting from
the application of a facially directed force. Although the uniform stress
distribution is desirable, the average intensity is less within the band
cement because of the greater area of attachment. Note also that a failure
within the band cement will not result in complete disruption of the
force-transmission path for the particular loading shown. Figure 6-2 illustrates several other simple force systems against the bonded bracketpad assembly and the resulting stress distributions within the adhesive.
tensile stress
FIGURE 6-1. Applied and responsive force systems exerted on bracket-pad (left) and
bracket-band (right) assemblies.
Introduction to Structural Analysis of the Orthodontic Appliance
FIGURE 6-2. Force systems exerted on the bonded-bracket-pad assembly as loaded by
lingually directed compression (left), shear-peeling (center), and a pure couple (right).
Although brackets transmit the bulk of orthodontic force to tooth
crowns, other means are also in common use. Buccal tubes are often
attached, in place of brackets, to terminal molar crowns. Buttons, eyelets,
and staples are often affixed either to the band or bonded directly to the
crown surface. They may be located either mesial to distal with respect
to the bracket on the facial surface, on the lingual crown surface, or, if
space permits, on the mesial or distal crown surfaces. Most commonly, a
ligature tie or an elastic is activated against the button or eyelet to create
a moment about the long axis of the tooth as shown in Figure 6-3. Other
orthodontic appliance elements may transfer force to the lingual crown
surface. A transpalatal bar may be used to deliver buccolingual force or
tipping (torquing) couples bilaterally, for example. Labiolingual treatment therapy divides the transmission of force from the appliance to the
dentition between the facial and lingual crown surfaces in order that
mesiodistal displacements might be accomplished without the tendency
for long-axis rotation. Removable-appliance therapy may make no use of
brackets at all. Lateral force generation must then be accomplished by
Arch wire
= P(e)
FIGURE 6-3. An elastic extended between a button and the arch wire to generate a
moment about the long axis of the tooth (occiusal views).
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 6-4. Occlusal views of a distal driving force exerted on the canine bracket (left)
and the equivalent force system referred to point CC (right).
pushing, and both facial and lingual surfaces are contacted by elements
of the appliance.
Accessibility contributes heavily to the rationale for placing the orthodontic bracket on the facial surface. Often resulting, however, is the
eccentric transfer of at least a portion of the appliance force system, as
noted in Chapter 5. Because the center of resistance is unreachable directly, the next most favorable reference point is on the long axis of the
tooth and occiusoapically at bracket level (point CC in the local reference
frame introduced in Chap. 5). Although through the bracket, labiolingual force and any couple may be effectively applied at point CC, mesiodistal and occiusogingival forces may not. These two components of
force, commonly existing in the activated appliance, because their lines
of action are in excess of one-half of the faciolingual crown width from
point CC, embody sizable rotational potentials often undesired.
Figure 6-4 (similar to Figure 2-17) presents occlusal views of a distal
driving force delivered to the canine bracket and the equivalent force
system at point CC. The couple (right) embodies the rotational tendency
of the force Q applied to the bracket; if the rotation is unwanted, an
equal but opposite (in sense) first-order couple must be generated
within the arch-wire-to-bracket force system or by means of some other
appliance-to-crown contact (through an eyelet- or button-to-arch-wire
ligation, for example). On the other hand, distolingual rotation of the
canine can be accomplished by using the distal driving force and a
mesially directed force applied through point CC, the latter created
through crown contact with the adjacent premolar; this force system is
presented in Figure 6-5. An occlusogingival force at the bracket tends to
FIGURE 6-5. Occlusal view of a pair of forces acting on the canine crown as a couple
capable of producing distolingual rotation.
Introduction to Structural Analysis of the Orthodontic Appliance
extrude and tip the crown lingually or intrude and produce facial crown
Upping, depending on the sense of the force. Countering an unwanted
rotational potential is often attempted through placement of an active,
third-order twist in an engaged, rectangular wire. Figure 6-6 illustrates
the problem in attempting pure extrusion and the ideal elimination of
the accompanying Upping potential. (Alternatively, if lingual crown Upping without extrusion is desired, it may be obtained through the sole
application of a third-order torque having the appropriate sense or with
a lingually directed force applied through point CC. The slight difference
in the two displacements, in terms of the locations of the centers of
rotation, was discussed in Chapter 5.)
The force system carried to the orthodontic bracket from the appliance activation site(s) is transmitted primarily by the arch wire. The
bracket is designed principally to accommodate the arch wire and the
ligation completes the engaged confinement of that wire. Figure 6-7 (left)
presents a view of the ordinary edgewise bracket from a mesial or distal
perspective. The faciolingual and occlusogingival dimensions of the
bracket slot admit a range of cross-sectional sizes of round and rectangular wire. The "wings" occlusal and gingival of the slot provide support
C3 = Fje)
Initial position: dashed
FIGURE 6-6. Extrusive displacements: with tipping, achieved with a force located oft of
the long axis (left) and with a force and couple having their resultant along the long
axis of the incisor (right).
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 6-7. The typical edgewise bracket: mesiodistal view (left) and facial view (right).
for the ligature tie or elastic "0-ring" which, when in place, closes the
slot. (Pins, clasps, cams, and "locks" provide resistance to arch-wire
disengagement within other types of bracket assemblies.) The faciolingual dimension from the lingual surface of the slot to the lingual extent
of the bracket itself is as small as strength and stiffness demands will
allow to minimize the distance from arch wire to the long axis (although
this "concept" is exchanged in straight-wire-therapy appliances for offsets built into the brackets). Figure 6-7 (right) illustrates a facial view of
the typical bracket. Edgewise brackets are classified by occiusogingival
slot dimension, by mesiodistal width, and by the absence or presence of
a vertical slot ("single" versus "twin"). Irrespective of slot size, the four
common bracket designations are: narrow-single, wide-single, narrowtwin, and wide-twin (in order of increasing mesiodistal bracket width).
Bracket materials must be strong, stiff, and, true of all intraoral appliance elements, chemically enert in the oral environment.
The so-called "universal" bracket design enables the transfer from
the arch wire of all six possible components of the force system carried
by the wire to it. The occlusogingival and faciolingual components of
concentrated force are transferred by direct contact of wire and slot or
ligation as viewed from the mesial or distal. Note that the facially directed force from the wire pushes against the ligation, which may be less
stiff and have less strength than the bracket slot itself. Ordinarily, the
arch wire may be able to slide mesiodistally through the bracket slot,
depending on the wire-to-slot clearance and the type and tightness of
ligation. The transfer of the mesiodistal force component generally does
not rely upon friction; instead, it depends more positively on a stop
affixed to the arch wire or a ioop placed in the wire. The stop or loop
may make mesiodistal contact with a bracket, creating an action-reaction
pair of "push" forces, or force may be transferred between the loop and
the bracket by a simple auxiliary element capable of carrying tension (a
fled length of ligature wire, for example). Typical transfers of the three
force components between arch wire and bracket are illustrated in Figure 6-8.
The capability of the arch wire to effectively transfer first- and sec-
ond-order couples through the bracket and into the tooth crown is related directly to the mesiodistal bracket width. Each couple consists of a
pair of forces generated through wire-to-bracket contact at the mesial
Introduction to Structural Analysis of the Orthodontic Appliance
FIGURE 6-8. Transfers of concentrated force components from arch wire to bracket:
occiusogingival (left), faciolingual and lingually directed (center), and mesiodistal (right).
and distal extents of the bracket. As illustrated in Figure 6-9, and re-
called from discussions in Chapter 2, the couple magnitude depends on
force size and the shortest distance between the lines of action of the
forces. Accordingly, the potential couple size increases with the mesiodistal bracket width. The clearance between arch wire and slot is also a
factor, which is discussed further in Chapter 7. In the generation of a
first-order couple, one of the forces arises from contact between wire
and ligation; hence, if the ligation has low stiffness, the capacity to transfer this couple through the connection of arch wire to bracket is substantially impaired. Accordingly, several alternative means of applying a
first-order couple or moment to the crown have been developed and
three are examined in Chapter 10.
A third-order couple may be transferred by a straight, round wire to
the bracket only by friction, but two more positive mechanisms are available to apply the third-order couple to the tooth crown. A rectangular
arch wire, having a diagonal cross-sectional dimension greater than the
occiusogingival width of the bracket slot, may transfer this couple
through two-edge contact within the slot. (The bracket slot must be stiff
enough not to "open" under this loading to the extent that the wire may
spin within it.) The distance between the two forces forming the couple,
then, is smaller than the diagonal dimension of the wire cross-section.
Arch wire
Bracket pad
FIGURE 6-9. Bending couples transferred from arch wire to bracket-ligation system:
first-order couple (left), and second-order couple (right).
Bioengineering Analysis of Orthodontic Mechanics
An alternative procedure is to loop the round arch wire passing through
the bracket such that, as activated, two-point contact is made against (1)
the ligation closing the bracket slot and (2) the crown occiusal or gingival
to the slot (depending on the desired sense of the couple). The perpendicular distance between the forces is substantially greater than that of
the rectangular-wire procedure, but one of the forces pushes against the
ligation and, in the absence of an accompanying lingual driving or holding force, demands at least a moderately stiff and strong closure of the
bracket slot. Figure 6-10 depicts the transfer of a third-order couple from
arch wire to crown using the two mechanisms just described. More detailed discussions of arch-wire torsion are contained in Chapter 8.
Orthodontic force may be transmitted to the bracket other than from
the arch wire. Typically, coiled springs or "elastics" are used to produce
mesiodistal driving forces within one arch against individual teeth. Because of their small cross-sectional sizes and relatively high flexibilities,
"elastics" are most commonly used to transfer interarch force. Various
auxiliaries may be engaged and activated against individual brackets;
generally, such devices are anchored in some fashion adjacent to the
tooth that is to receive the active force. These auxiliaries are discussed
later in this chapter and in Chapter 10.
The partial connection of the appliance to the dentition, and the
transfer of some orthodontic force, may be other than through the orthodontic bracket. Although similar to the bracket and, like it, transferring
force directly to a crown through a bonded base or band, the buccal tube
FIGURE 6-10. Transfers of third-order couples from arch wire to crown: by a rectangular
wire to the bracket (left) and by a ligated looped round wire (right).
Introduction to Structural Analysis of the Orthodontic Appliance
special characteristics. Generally affixed to the terminal molars in one or both arches, the buccal tube is open only at its mesial and
distal ends and, therefore, transmits buccally directed force from a wire
just as well as lingually directed or occlusogingival force. Its mesiodistal
dimension is greater than that of the widest brackets; hence, the tube
may more effectively transfer first- and second-order couples. Buccal
tubes are also made to accommodate the "inner arms" of the face bow, a
principal element of a class of extraoral appliances (discussed in Chap.
9). Connections of appliance elements may avoid the bracket or buccal
tube entirely and transfer force to the crown directly. Examples already
mentioned include the button, eyelet, and staple against which elastics
or ligature-wire ties may be activated, and the attachments for lingually
affixed appliance elements. Forces may also be delivered first to the arch
wire, and then carried by it to the brackets, buccal tubes, and to the
tooth crowns. Because such forces nearly always have mesiodistal com-
ponents, delivery demands rigid attachment of spurs or hooks to, or
loops bent into, the arch wire. Typical carriers of such force include the
interarch elastic and the J-hook, the latter an extraoral-appliance element.
Continuous-Arch-Wire Appliance Model
continuous arch wire is a curved structural member, typically extending from terminal molar to terminal molar, and as engaged and
activated it generally contacts through brackets and buccal tubes six or
more—often at least ten—tooth crowns. At each primary connection
site, where arch wire and bracket or tube make physical contact, as
many as six independent components of a transmitted force system may
exist. Although an essentially static analysis may be undertaken, there M
no externally applied loading and all components of force are generated
through deformations of appliance members occurring in the engagement-activation process. In short, the analysis of a highly indeterminate,
geometrically complex structure is required if the force system and the
structural characteristics of the wire are to be examined in detail. The
task is formidable for the highly skilled structural analyst having sophisticated computer assistance, and is not a problem to be undertaken by
the orthodontist. The clinician should be capable, however, of pursuing
an approximate analysis that includes the principal features of the activated arch wire or appliance. To this end a model of the continuous arch
wire is now proposed and developed.
The ideal arch is geometrically symmetric with respect to the midsagittal plane. Often, this symmetry exists in the arch exhibiting a malocclusion, and thereby demands bilateral treatment mechanics. The concur-
rent presence of symmetry in both the dentition and the appliance as
activated allows an analysis involving just one-half (the left or right half)
of the orthodontic mechanism. Such an analysis requires a figurative
Bioengineering Analysis of Orthodontic Mechanics
Arch wire—
FIGURE 6-1 1. Components of the force system in the arch wire transferred across the
midsagittal plane in the presence of symmetry: occkisal view (left) and coronal view
(right). Either sense of each component can exist.
"cut" through all portions of the appliance that span the midsagittal
plane. In many instances only the arch wire must be "cut," and the
symmetry demands that, of the six force-system components that could,
in general, be transmitted across the plane, only the three shown in the
sketches of Figure 6-11 may possibly be nonzero. Any analysis model
should take advantage of midsagittal-plane symmetry when the geometry and force system permit it.
Various phases of orthodontic therapy often treat groups of teeth as
if each group is an individual entity, either in the application of force to
produce wanted displacement (in the retraction of an anterior segment,
for instance) or in arranging anchorage (in intraoral, canine-retraction
mechanics, for example). In portions of orthodontic treatment, the adjacent central and lateral incisors may be considered as a single unit, and
so also may the second premolar, the first molar, and possibly the first
premolar (if not extracted) and the second molar (if fully erupted and
aligned). Such segmentation can substantially simplify analyses, particularly in the presence of midsagittal-plane symmetry.
Two distinct curvatures generally exist in the dental arch—that of the
anterior portion as observed from an occlusal perspective and the anteroposterior curvature as seen in a buccal view, the latter often termed the
"curve of Spee." In the analyses of many orthodontic procedures, the
effects of the curvature of the anterior segment may be suppressed without significantly violating the overall structural integrity of the appliance; such suppression serygs to simplify the geometric aspects of the
problem. With the exception of analyses of anteroposterior leveling mechanics, the effects of the curve of Spee often may also be neglected.
Furthermore, localized first- or second-order bends, although their ef-
Introduction to Structural Analysis of the Orthodontic Appliance
Actual arch form: dashed
FIGURE 6-12. 0cc/usa! and bucca/ views of the continuous arch-wire model.
if active or stabilizing may be prominent and, therefore, must generally be included, need not be shown explicitly in sketches of the arch
wire to be analyzed; the same may be said of third-order twists and of
the details of arch-wire loops. Often, then, the overall shape of the contirtuous arch wire may be modeled as shown in Figure 6-12, consisting of
three solidly connected, straight-section segments. The central and lateral incisors would always engage the middle (anterior) segment. First
premolars (if present) and dental units distal to them would contact the
posterior segments of the arch-wire model. The canines would generally
be located at the anterior extremities of the posterior segments, but,
more appropriately, in nonextraction therapy, they may be part of the
anterior or middle segment of the model.
Although each arch-wire or appliance analysis is, in reality, threedimensional in general character, the best approach is often through the
use of several plane views that are commensurate with the component
forces and couples already mentioned. Perhaps the most useful view,
particularly with midsagittal-plane symmetry, is that from a buccal perspective. Explicit in this plane are anteroposterior (labiolingual in the
anterior segment, mesiodistal in the posterior segments) and occlusogingival components of force, third-order couples in the anterior segment, and second-order couples in the posterior segments. Practically
any analysis of mechanics intending to produce anteroposterior displacements will necessitate a buccal-view evaluation. Anteroposterior
movement may also be examined in an occlusal-plane view, but the
particular value in this plane analysis is in evaluating mechanics designed to produce mesiodistal displacements in the anterior segment
and buccolingual movements in the posterior segments.
The force-system components between arch wire and crowns apand faciolingual
pearing in the occiusal view include
components of force and first-order
Anteroposterior or lateral
differences in the action of an asymmetric
left side versus right side, may be viewed from the occlusal perspective.
An analysis in the coronal plane, from an anterior or posterior view, is
Bioengineering Analysis of Orthodontic Mechanics
used less frequently. The coronal-plane perspective is helpful, however,
in studying anterior-segment consolidation, third-order torque in the
buccal segments, and the transpalatal bar, for example, or in examining
occlusogingival actions, left versus right side, in asymmetric situations.
Occlusogingival forces, mesiodistal forces, and second-order couples in
the anterior segment, and buccolingual forces and third-order couples in
the posterior segment, appear in the transverse-plane view.
An Overview of the Structural Analysis of an Orthodontic Appliance
The complexity of an orthodontic-appliance analysis increases with the
number of teeth to which the appliance is attached, when those teeth are
directly involved in attempting the desired displacement pattern. The
movements of an individual tooth, with the appliance activated to extrude or intrude it, tip or torque it, or correct a rotation, all may make
use of practically an entire arch for anchorage and often are not strongly
dependent on the overall geometry of the continuous arch wire and
dentition. These analyses are not initiated here, but will be discussed
subsequently. The more immediate objective is to introduce and outline
the structural-analysis procedure for those mechanics that involve the
whole of the appliance attached to a dental arch or make use of appliances in both arches andlor an extraoral appliance. Indeed, the problem
requires a careful examination of the actions, responses, and displacement potentials throughout the dentition as a whole and of the structure
affixed to it.
To begin the analysis, a peripheral examination of the active configu-
ration versus the passive shape of the appliance as a whole must be
made. Note specifically the numbers and positions of the individual
teeth to which the appliance is engaged. At each bracket or buccal tube
where contact exists, the active or passive nature of the connection, with
respect to each of the six individual components of the most general
force system, is examined. Often, several components are zero at a contact location and in many instances only one component is nonzero, at
least until a displacement occurs that may activate one or more additional components. In all but the simplest analyses, some modeling is
generally warranted in accordance with the guidelines of the previous
section. If the individual activation displacements and force systems associated with adjacent teeth are similar, the dental segment so identified
may be treated in the analysis as a single entity. If one or more of the
arch-wire curvatures are not relevant to the analysis, they may be neglected in formulating the model. Note that after the modeling to simplify the analysis procedures has been undertaken, the model must be
carefully scrutinized regarding its propriety, not only at the instant of
activation, but also as to its continued validity throughout the days or
weeks during which displacements of teeth and corresponding changes
in the activated-appliance configuration are occurring. After the model
Introduction to Structural Analysis of the Orthodontic Appliance
formulated and found acceptable, the next steps are to check for the
presence or absence of midsagittal-plane symmetry and to determine
the number of necessary plane views together with the specific views to
be employed.
The entire appliance assembly, in general, is a combination of the
intraoral appliances contacting the individual (maxillary and mandibular) dental arches, any interarch attachments, and, if used, the extraoral
appliance. Each of the existing subassemblies, considering prior modeling, is separated from the others and the dentition, and is isolated; using
the necessary plane views, the force system exerted on each subassembly is sketched. Included are all components arising from appliancecrown contacts, intraoral forces associated with interarch interconneclions, and force systems created through the activation of "headgear."
The directions of concentrated forces and dimensions or distances that
may be needed as moment arms are also shown in these sketches. Next,
the transmission of force through the appliance members is qualitatively
traced from the location(s) of principal activation(s) to (1) the sites of
wanted displacements and (2), if existing, the appliance-crown conneclions and any other contact locations intended to experience little or no
movement during the between-appointments period. If no anchorage
exists, the activation sites and the force system against the appliance will
be divided (halved) on either side of a plane of symmetry; in the majority of such instances the activation is bilateral with symmetry with respect to the midsagittal plane. The transmission of force is then followed
from the activation locations on either side of the plane to the plane of
symmetry itself, where action-reaction matching of the force-system
components is carried out. The total number of force and couple components, exerted on the complete appliance assembly, may now be deter-
mined. From discussions in Chapter 2, in each plane view a total of
three, scalar, force- and moment-balance relationships are available
from Newton's laws and the concept of the quasi-static state of the appliance. The number of force and couple components to be quantified will
exceed the number of quasi-static, balance equations by the number of
needed "load" versus deformation relationships associated with the activation of the appliance.
The complete analysis will likely require a disassembly of the appliances "on paper," separating the auxiliaries from the arch wires, and an
examination of each appliance member in a manner similar to that previously described for the principal appliance subassemblies. This detailing
will "expose" the force systems that exist at the sites of interconnections
between appliance members, and these systems are interrelated in pairs
through Newton's law of action and reaction. Each member of the appliance assembly may be placed in one of two categories: (1) those members that are physically deformed from their passive states to achieve the
activation of the assembly as a whole; and (2) those members in which
force systems arise in response to the activation. For the members in the
first category, if the entire force system external to and within the appli-
ance assembly is to be quantified, the amounts of activating displace-
Bioengineering Analysis of Orthodontic Mechanics
ment-deformation must be measurable and the load-deformation rela-
tionships must be known.
The structural analysis of an orthodontic appliante is accomplished
in two principal, sequential parts. First, all of the forces and couples
acting on the appliance as a whole and on each appliance member at the
interconnections are determined. This procedure, described in a general
manner previously, requires the use of force- and moment-balance
equations and load-deformation relationships. If the purpose is only to
obtain the force systems delivered to anchorage sites, to decide whether
or not anchorage reinforcement is advisable, for example, then the analysis may be terminated following the completion of the first portion. The
second part of the analysis examines each appliance member internally.
For each, the critical sections and points thereon are located and, as
potential sites for initiation of structural failure, are checked for stress
levels and amounts of deformation. These stresses and deformations are
determined from geometry and material characteristics of the individual
member and the force system exerted on it, and may then be related
back to the parameters of the activation itself. This second portion of the
analysis may be undertaken for one of several reasons:
1. A failure may have occurred and the need exists to make some modification of the appliance or in the activation of it.
2. Under consideration is the possibility of replacing one or more mem-
bers of the appliance or altering the character of the activation to
modify the response, and the structural integrity of the appliance
under the new conditions is to be examined.
3. An entirely new design is contemplated, including the shapes and
sizes of the appliance members and their materials, with selections to
be based in large part on allowable stresses and deformations as related to those that will be generated in the new appliance.
The complete structural analysis of the orthodontic appliance incorporating a continuous arch wire can be a complex undertaking as previously noted. The foregoing, general approach to the process as carefully
outlined partially attests to the magnitude of the task. With appropriate
modeling, however, and with the knowledge to be gained in the remainder of this chapter and those to follow, the clinician will be able to incorporate the principal features of an appliance in a reasonably straightforward analysis procedure. To this end, the foregoing discussion might be
condensed into the organized, four-step process suggested here:
1. Knowing the specific displacement(s) desired, both in form and extent, the individual components of the total force system acting on
the dentition, which should produce those displacements, are identi-
fied together with any components present to control the movements.
2. One or more free-body diagrams of the orthodontic appliance as a
whole, and subassemblies and individual elements thereof as
Introduction to Structural Analysis of the Orthodontic Appliance
are sketched. These diagrams include the action-reaction
counterparts of the components identified in step 1. The entire force
system exhibited in each diagram is investigated through Newton's
laws (action-reaction and the force and moment balances).
3. With the aid of the results of step 2 and, once more, using the actionreaction law, the force system(s) transmitted to the responsive locations (anchorage) are evaluated to determine what, if any, unwanted
displacements might occur. Stress and deformation computations for
individual appliance members may also be undertaken if a possible,
structural failure is of concern.
4. As warranted from the results of step 3, appropriate precautions or
alterations should be considered and undertaken to minimize any
potential, undesired "side effects" of the appliance objective. Such
action could involve, for example, reductions in induced force magni-
tudes, making member substitutions in or additions to the existing
appliance, or altering parameters intended to control the orthodontic
The Activation and Deactivation Characteristics of Representative
Appliance Elements
The completion of the first portion of the structural analysis of an ortho-
dontic appliance initially requires the quantification of the forces and
couples induced in those auxiliary elements or at the individual, specific
locations along the arch wire where mechanical work is done by the
clinician on the appliance to achieve activation. Examples of this work
are the compressing of a spring, the stretching of an elastic, and the
deflecting of an arch wire to gain bracket engagement. The work of
activation is usually concentrated at only a few locations, properly
termed activation sites, and at each site knowledge of the amount of localized deformation and the associated element or wire stiffness enable the
determination of the induced force system. A subsequent force-system
analysis, assuming that all of the dentition to which the appliance is
affixed behaves like a rigid foundation or support-base, is sufficient to
approximate the initial magnitudes of the responsive forces throughout
the dentition. Although somewhat idealized, such an analysis of the
appliance is appropriate because a state of maximum, overall internal
loading and deformation exists under the assumption of a rigid dentition and periodontium.
The soft-tissue deformations, which in reality permit some immediate tooth movement following activation, serve to partially "relieve" the
appliance, releasing a portion of the strain energy induced by the mechanical work of activation and, thereby, reducing the stresses and deformations from their maximum values. As time proceeds following acti-
Bioengineering Analysis of Orthodontic Mechanics
vation and stage 1 of tooth movement (defined in Chap. 5), additional
deactivation will likely occur during the between-appointments period
with stage 3 displacements. Although some of these movements may
serve to generate force systems at locations that were passive when the
appliance was initially engaged (the guiding arch wire attempting to
keep a canine upright during retraction, for example), this work is accompanied by a transfer of strain energy from one location to another
within the appliance. Overall, stage 3 displacements result in a net, continuing decrease in strain-energy level, stresses, and deformations in
and of the appliance with respect to the as-initially-activated state.
The orthodontist may disengage the appliance before complete deactivation occurs as a result of tooth movement or possible relaxation of
some appliance elements. If the unloaded configuration of the appliance
is not identical to the original passive configuration, signaled are energy
losses, energy transfers other than to the dentition in producing displacements or to the clinician in the unloading process, or increases in
unavailable, stored energy in the appliance resulting from exceeding the
elastic limit somewhere during activation. Inelastic behavior may be reflected in the metallic appliance members and relaxation in the nonmetallic members. Although the configuration change following a load-unload cycle is more often the rule than the exception, of at least equal and
probably of more importance is the fact that the induced increment of
strain energy of activation generally cannot be completely controlled
with respect to the subsequent work done by the appliance in moving
With the influence of the clinician removed as soon as activation is
complete, the appliance will seek to take on a minimum-energy configuration. The added strain energy, transferred to the appliance at the activa-
tion site(s), is immediately distributed throughout the appliance such
that the assembly as a whole assumes the deformed configuration characteristic of the lowest energy level possible, consistent with the constraints to which it is subjected. Moreover, with every small tooth dis-
placement that takes place, the appliance adjusts itself to a new,
minimum-energy configuration. The ideal appliance, then, is one that
most nearly permits the orthodontist complete control over the manner
in which the strain-energy input of activation will be distributed and
dissipated over time.
The first individual orthodontic element to be discussed is not part of
an appliance at all, and is used before brackets are affixed and only
when and where interproximal space is to be created to accommodate
the placement of bands. These "spacers" are small polymeric rings or
discs, placed interproximally adjacent to gingival margins, and are
"loaded" or activated, in their placements, in direct compression. Figure
6-13 shows a free-body diagram of the activated spacer and the actionreaction, counterpart forces exerted upon the adjacent teeth. Compliance with Newton's laws requires that all four forces shown be equal in
magnitude. This magnitude at the time of activation is determined by
the amount of deformation of the spacer required to place it and the
Introduction to Structural Analysis of the Orthodontic Appliance
FIGURE 6-13. The spacer unloaded and activated in direct compression (center) and
the adjacent teeth subjected to spacer action (facial views).
load-deformation relationship for the element. Typically, the average
stiffness of a spacer in compression is moderate with the cross-section in
the mesiodistal plane (as activated) sizable compared to its dimension in
the direction of loading. The width of the ordinary orthodontic band is,
however, only a fraction of the periodontal-ligament width at the gingival margin and, as a result, the ligament need not be fully compressed
by this loading to create the needed space. Because the forces exerted by
the spacer on the two teeth will be of equal magnitude throughout the
active period, the absolute displacements of the individual dental units
will be controlled by the resistance of each tooth to movement, either
inherent (projected root surface area, crown contacts along the dental
arch) or created artificially through an anchoring of one tooth or the
other in an attempt to limit the movement to just one unit.
Helical-coiled springs may be used as auxiliary elements of both
intraoral and extraoral orthodontic appliances. Typically activated by a
force applied along the axis of the spring at one end and supported at
the other, the member may be loaded in either direct tension or direct
compression. Regardless of the sense of the applied force, neglecting its
weight compared to contact forces generally makes the spring, like the
spacer, a two-force member. Helical springs, to be activated in tension,
are usually "wound" (fabricated) to assume a closed-coil, passive configuration while compression springs must exhibit an open-coil (space-between-coils), passive state. Tension springs may be activated (stretched)
almost indefinitely and are inherently stable when loaded; however, an
elastic limit does exist as the activation actually produces a rather uni-
form bending deformation of the coiled wire. On the other hand, the
compression spring can be deformed only until all coils contact one another; this spring is ordinarily designed such that the closed-coil state is
achieved with a force producing maximum stresses less than the elasticlimit stress of the spring material. Of some concern must be the stability
of an activated compression spring in the absence of lateral support
potential and, to guard against the possibility of lateral buckling (uncon-
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 6-14. Helical tension and compression springs (left and center); typical loaddeformation diagram for the linear spring (right).
trolled, large deformation), this spring as an appliance element is usu-
ally wound around the arch wire.
The majority of metallic, helical springs are termed "linear" in that
they exhibit straight-line load-deformation behavior beneath the elastic
limit. The structural properties of a helical spring are primarily dependent on the spring material, the wire cross-section, the coil diameter, and
the passive length. Helical springs in tension and compression are
shown schematically in Figure 6-14. A typical load-versus-deformation
plot for a linear spring is also presented (similar to Fig. 4-9). Given the
plot for a specific spring, a measurement of its axial deformation from
the passive state provides the magnitude of force at its ends. Figure 6-15
shows a free-body diagram of a compression spring wound around an
FIGURE 6-15. A helical compression spring making lateral contact with the arch wire
about which it is wound, midway between the spring ends.
Introduction to Structural Analysis of the Orthodontic Appliance
arch wire, as activated and making lateral contact with the wire midway
between the ends of the spring. A force balance indicates that the end
forces have vertical components that must sum to the force of the arch
wire against the spring coils. The lateral forces arise if the spring tends to
buckle laterally or if the section of arch wire, over which the spring is
wound, is curved. (An available spring assembly of interest is activated
by a pulling force against its "jacket" and an axial wire transferring force
from an end coil; the axial wire passes through the spring body. This
element exhibits favorable characteristics of both standard tension and
compression springs; it is loaded in tension, but activation actually compresses the spring. Accordingly, the assembly incorporates the closedcoil "guard" against exceeding the elastic limit and the axial wire and
jacket prevent buckling.)
To activate a pulling force in a specific direction within the orthodontic appliance, or to transmit a tensile force from one site to another, an
elastic may be used in place of the tension spring. The popularity of the
elastic, generally descriptive of the array of bands, 0-rings, modules,
chains, and threads of rubber or rubberlike polymeric materials, is associated with its favorable characteristics of high flexibility, small crosssectional dimensions, and low cost. Free-body diagrams of the elastic
band (or ring or the looped and tied thread) and the module, loaded in
direct tension, are shown in Figure 6-16. A typical load-deformation
plot, similar to a portion of Figure 3-11, is also presented. An undesirable feature of the elastic, associated with many polymeric materials, is
the tendency toward relaxation. Defined in Chapter 4, the result of reP
FIGURE 6-16. An elastic band or looped thread (left) and an elastic module or chain
(center) subjected to two-point tension, and an example load-deformation diagram
Bioengineering Analysis of Orthodontic Mechanics
laxation is the loss of force magnitude with lime under condilions of
constant stretch (strain).
Figure 6-17 is a four-part plot for an elaslic showing (1) inilial activalion, (2) maintenance of constant strain for a finite lime period, (3) rather
rapid unloading to a stress-free state, and (4) recovery in the absence of
reacivalion. Notable in parlicular is the fourth segment of the plot
which indicates that, for some polymers and given sufficient lime, the
original passive configuration may be nearly, if not totally, regained if
the elaslic limit of the material was not exceeded during activation. (As
noted in Chapter 4, these elements are termed elastics, partly because of
their sizable elaslic ranges. Strictly speaking, though, the materials are
quite brittle with measurable elongations to failure by rupture not
sizably beyond the elastic-limit, extensional deformalions.) Although
several comparable characteristics already menlioned suggest preference for the elaslic over the tension spring, a noteworthy advantage of
the metallic spring is the virtual absence of lime-dependence in its structural properties.
Convenient to transmit tensile force, when moderately high, extensional sliffness is desired, is the looped and lied (twisted, "pig—tailed")
length of ligature wire. The ductility of this wire is substantial and the
elaslic strength of the "dead soft" material is quite low; this combinalion
permits the
procedure. Figure 6-18 qualitalively compares
the tensile stress-strain behavior of the annealed (as-received), stainlesssteel ligature wire with the same wire material having average resilience.
FIGURE 6-17. A four-part hysteresis diagram for an 'elastic" appliance element
Introduction to Structural Analysis of the Orthodontic Appliance
Strain-hardened wire
Annealed wire
Tensile Strain
FIGURE 6-18. Tensile stress-strain diagrams tor annealed and resilient stainless-steel
wires having identical cross-sections.
Note that the tying process inelastically deforms the wire, adding to the
strain energy stored locally in the wire; each twist reduces the ductility
and the effective deformation to fracture. If the number of twists are
excessive and this cold-working process itself does not cause fracture, a
small additional load superimposed upon the activated ligation, in overall appliance activation or later from mastication, for example, might
cause the ligature to rupture. The longitudinal flexibility of the ligature
tie is practically nonexistent, despite its small cross-section; the elastic
modulus of its material, generally stainless steel, is high and the relatively low stiffness site is that of the twist-tie.
In summary, ligature wire should be expected to carry responsive
force; it will not perform as an activating element like the elastic module
or the coiled spring. In its intended use (to close the bracket slot) and to
generate secondary activation (holding force), this wire serves well as
long as the localized embrittlement at the twist-tie is not so extensive
that fracture occurs with the superposition of masticatory force upon
appliance activation.
The rubber-band elastic may be stretched between two points, creating the action-response force pair shown in Figure 6-16, or it may be
activated in three- or four-point tension. In the "up-and-down elastic"
application (between the maxillary and mandibular arches) the element
may be looped around three or perhaps four brackets or spurs (affixed to
arch wire). Unless constrained locally at several points, the activated
Bioengineering Analysis of Orthodontic Mechanics
elastic will assume a configuration with the longitudinal strain the same
throughout its stretched length. If the load-deformation behavior of an
elastic band has been or can be obtained in two-point tension, an analysis of the three- or four-point, equalized-tension problem can be accomplished. First, the magnitude of the tensile force existing throughout the
band is obtained from its effective elongation, which is one-half the difference between the passive and activated circumferential lengths, and
the load-deformation plot. Second, with the force magnitude the same
on either side of the bracket or spur (because the strains are the same),
as is shown in Figure 6-19, the direction of the delivered force (to the
bracket) bisects the angle formed by the elastic-band segments. The
magnitude of the delivered force is obtained from a force-balance on
— — — - I. — — -
0 = 2(P cos
FIGURE 6-19. A free-body diagram of the portion of an elastic band contacting a
bracket; the analysis yields the direction of the force delivered to the bracket.
Introduction to Structural Analysis of the Orthodontic Appliance
the segment free-body diagram shown, with the equation given in the
Although elastics are always activated in direct tension and helical
springs are generally loaded uniaxially, the arch-wire segment spanning
several brackets may be subjected to one loading pattern or several patterns concurrently. The most common arch-wire activation is in bending, and when activated in this manner, the wire behaves structurally as
a beam. Detailed discussions of arch-wire bending are contained in
Chapter 7, but some general comments are appropriate here. First, a
distinction is drawn between occlusogingival and faciolingual bending,
simply in terms of the direction of activation and the plane containing
the external force system. Second, the anchorage may be totally mesial
or distal of the activation site, or it may be divided between locations
mesial and distal to the activated section of the wire. For example, a
symmetric arch might use the posterior segments for anchorage in mechanics designed to intrude or extrude the incisors; the model of such an
arch is shown in buccal view on the left in Figure 6-20 and resembles the
cantilevered beam of Figure 2-29. A lingually malposed second premolar
might be tipped into alignment by a facially directed force, activated by
tying the adjacent arch-wire section into the premolar bracket. In this
example, the anchorage is split between the dental units mesial and
distal to the premolar, and an occlusal-view, free-body diagram of the
arch-wire segment, showing the concentrated faciolingual forces, appears on the right in Figure 6-20. Third, innumerable load-deformation
plots exist for a single arch wire in bending. Although each plot is approximately a straight line for metallic beams, if the deformations are not
large and the elastic limit is not exceeded, the slope (stiffness) is dependent not only on the material and the size and shape of the cross-section,
but also upon the characteristics of the loading, including its position
with respect to the anchorage and, for rectangular cross-sections, the
direction of the loading with respect to the cross-sectional geometry.
When activating an arch wire to produce a crown force, the displacement must be measurable so that the wire "works" during the betweenappointments period with the force induced having a physiologically
proper magnitude. Although such activations may readily be obtained
through displacements perpendicular (normal, transverse) to the long
FIGURE 6-20. Buccal and occiusal views of arch wires activated in bending by
transverse forces: the wire as a cantilevered beam (left) and with divided anchorage
Bioengineering Analysis of Orthodontic Mechanics
dimension of the wire, no matter how small the cross-sectional area the
longitudinal (mesiodistal) wire stiffness is much too great. Two procedures are used to effectively reduce the mesiodistal stiffness of the arch
wire: (1) More flexible elements are incorporated, nearly always bilaterally, in series with the arch wire, generally stretched between the ends
of the wire and posterior anchorage. An example of this arrangement is
shown in the buccal view on the left in Figure 6-21. Often this is done in
anterior-retraction mechanics incorporating stops, springs, and, perhaps, ligature wire or elastic thread. (2) A loop is bent into the arch wire
such as shown in the facial view on the right in Figure 6-21. The more
wire incorporated in the loop, the lower the resulting mesiodistal stiffness. Because they are formed through inelastic bending procedures,
and most often activated in bending, loops are examined in some detail
in Chapter 7. Among other uses, loops are used in space-closing procedures and to "unravel" crowded incisors.
Couples, as well as concentrated forces, may be transmitted between
arch wire and bracket or buccal tube. Two of the three couples or couple
components load the arch wire in bending; the third is associated with
activation in torsion. To produce an active, second-order bending couple, in the passive configuration an angular malalignment must exist, as
viewed from a facial perspective, between bracket slot or buccal tube
and the section of arch wire to be engaged. Accordingly, clinicians place
second-order bends in the wire or use preangulated brackets and tubes.
The mesiodistal bracket width can greatly influence the magnitude of
the second-order couple. Besides defining the distance between two
contact forces (generated between wire and diagonally opposed edges of
the bracket slot or buccal tube) forming the couple, for a given arch wire
increased width (1) reduces any second-order, angular clearance between wire and bracket slot or buccal tube, (2) decreases the adjacent
interbracket distances, and (3) tends to increase the as-activated, con-
tact-force magnitudes. In short, the wider the bracket or tube, the
greater the potential, second-order bending stiffness.
Many characteristics of the first-order couple are similar to those of
its second-order counterpart. Located in an occlusal plane, the first-
Buccal tube
Arch wire
FIGURE 6-21. Methods of increasing the longitudinal flexibility of the arch wire: a spring
stretched between hooks affixed to the wire and a terminal-molar buccal tube (left) and
a loop bent into the arch wire (right).
Introduction to Structural Analysis of the Orthodontic Appliance
order couple arises from a difference in passive angulation between the
arch-wire section to be engaged and the axis of buccal tube or the lingual
surface of the bracket slot. Again, the action is bending, and stiffness is
substantially affected by an alteration in the mesiodistal bracket or tube
width. Notable with the first-order couple generated specifically between bracket and arch wire, however, is the dependence of stiffness on
ligation, inasmuch as one-half of the force pair exists due to contact
between wire and the ligation. Moreover, the ligation may affect the
bending stiffness from two aspects: (1) through its own stiffness, and (2)
in determining the effective mesiodistal width of the bracket. Because
the ligation can weaken the connection of arch-wire to bracket and
lessen efficiency in transferring first-order couples, other means are
available to produce long-axis rotations (eyelet-to-arch-wire ties,
wedges, and levers, for example); procedures are discussed in Chapter
6-22 shows a typical second-order couple transfer on the left
and first-order couple transfers suggesting conventional ligature ties on
the right. As with the transfer of a transverse force, anchorage for an
active first- or second-order couple may be totally mesial or completely
distal to the activation site, or the anchorage may be split (divided) with
it partly mesial and partly distal to the activation location.
Appropriately noted at this juncture is a comparison of similar parameters appearing in a load-deformation plot in bending for an arch
wire, depending on the form of activation. If the wire is activated by a
transverse force, the magnitudes of the force and the transverse bending
deflection at the point of activation along the wire are plotted. At least
the initial portion of the plot is linear and the slope—the stiffness—has
dimensions of force divided by length. If the bending activation is by a
first- or second-order couple, the deformation is angular; the stiffness in
this mode has dimensions of the product of force and length per unit of
rotation. Example plots for small-deformation, elastic behavior, together
with sample units for stiffness, are given in Figure 6-23. An area under
any load-deformation plot represents energy or energy transferred, always having dimensions of force times length [FL].
Couples created in planes containing the longitudinal "axis" of the
arch wire induce bending; the couple applied to the wire such that its
10. Figure
plane is perpendicular to the mesiodistal direction creates a twisting
FIGURE 6-22. Arch wires activated in bending by couples transferred to them by
brackets: an active second-order couple exerted on a wire and transmitted to divided
anchorage (left) and bilateral, first-order couples delivered by a wire segment to correct
central incisor rotations (right).
Bioengineering Analysis of Orthodontic Mechanics
Deflection (mm)
Rotation (deg)
FIGURE 6-23. Arch-wire load-deformation diagrams in bending; the active force system
is a transverse force (left) and a couple (right).
deformation. The twisting or torsional couple can arise from a rectangu-
lar wire and an edgewise bracket, the wire and bracket slot passively
angulated, one with respect to the other. The rectangular wire may experience third-order activation following permanent twist placements
by the clinician or through the use of "pretorqued" brackets. Torquing
spurs bent into round wire are also used to generate third-order action.
As with other forms of arch-wire "loading," third-order activation may
be sustained by anchorage entirely on one side or the other of the activation site, or the anchorage may be divided between locations mesial and
distal to the location of the activation. Free-body diagrams of an archwire segment subjected to a third-order couple with the anchorage split,
and of a cantilevered wire loaded by a third-order couple, are shown on
the left and right, respectively, in Figure 6-24. Note that the bracket
exerts less influence on the third- than on the first- or second-order
stiffness of the arch wire. The mesiodistal bracket width is a factor in
C action
FIGURE 6-24. Free-body diagrams of an arch-wire segment loaded in torsion; the
segment on the left reflects divided anchorage, but only one anchorage site exists for
the segment on the right.
Introduction to Structural Analysis of the Orthodontic Appliance
torsional wire stiffness only insofar as it affects interbracket distances.
(As noted previously, a third-order couple in a rectangular wire will be
transmitted to a tooth crown through a bracket only if the occlusogingival slot width is less than the diagonal dimension of the wire cross-section.) The form of the load-deformation plot for third-order arch-wire
behavior is the same as that for first- or second-order-couple loading; for
twisting as for bending by couples the loads, deformations, and stiffnesses individually have identical dimensions. In short, the load-deformation plot on the right in Figure 6-23 could qualitatively be representative of the arch wire subjected to active torsion or bending. Chapter 8 is
entirely devoted to the third-order behavior of arch wires and associated
For all of the fundamental activations previously discussed, uniaxial
tension and compression, the cantilevered or divided-anchorage archwire beam subjected to a concentrated transverse force or couple, and
the wire activated by a twisting (third-order) couple, force- and momentbalance analyses made of the member at any specific time following
activation results in several quasi-static relationships that are less than
the number of unknown characteristics necessary to complete the description of the entire force system. With the exception of the arch wire
subjected to a couple loading and having divided anchorage, one additional relationship is needed; for the exception noted, two are needed.
For each of these problems, the load-deformation plot provides the
equivalent of one equation; with the deformation quantified with respect to the passive state, the plot yields the magnitude of the induced
load. In those problems of couple loading and partitioned anchorage,
one additional relationship is obtained, dependent on the relative distances between the load and the anchorage sites. Thus, the first portion
of the structural analysis is handled, completing the description in all
characteristics of the external force system exerted on the arch-wire segment. Detailed discussions of such analyses are taken up in subsequent
The load-deformation behavior for a helical spring in tension or compression and for the arch wire in bending or torsion is virtually independent of time, at least within the elastic limit, because of its material
properties. Moreover, provided no inelastic action has occurred, the
deactivation or unloading plot will retrace the activation curve. Furthermore, should the elastic limit be exceeded in activation of a metallic
appliance element, the deactivation (unloading) plot will be nearly linear
with the same slope as that of the linear portion of the activation curve.
(This was first discussed in Chapter 3.) Accordingly, for metallic appliance members, the relationship between load and deformation during
deactivation is, in essence, known from the plot of these parameters
obtained from activation data. Such an extrapolation, however, cannot
be made for appliance members of polymeric materials because, generally, their structural behavior is nonlinear and time-dependent. A typical activation-deactivation pattern for a polymer was described previously in this section (see Fig. 6-17).
Bioengineering Analysis of Orthodontic Mechanics
The Action and Response of Tip-Back Bends: An Illustration of the
Four-Step Procedure in Orthodontic Structural Analysis
Until in-depth studies of bending and torsional behavior of individual
members are undertaken, structural evaluations of orthodontic appliances cannot be completed. Nevertheless, in view of the concepts and
procedures introduced in this chapter and the rather general discussions
herein, with particular consideration of the analysis initiated, a look at
an arch-wire example is appropriate.
Chosen for examination is a continuous arch wire that fully engages
the mandibular dentition from first molar to first molar. As often occurs,
this example appliance is geometrically and mechanically symmetric
with respect to the midsagittal plane. All permanent teeth are fully
erupted mesial to the first molars and have been aligned occlusogingivally and faciolingually. Because of the symmetry and alignments,
together with the nature of the activations, the analysis is undertaken
through a right-side buccal view of one-half of the arch wire and the
involved dental units. (The additional potential for faciolingual tooth
movements is recognized, owing to the development of force systems
not within the "plane" containing the long axes, but actions and responses appearing in coronal-plane views are not investigated here.)
The intended activations in this example are those produced by bilateral tip-back bends in the wire at the terminal molars; the angular, passive malalignments between wire segments and buccal-tube slots give
rise, upon full engagement, to second-order couples exerted by the wire
on the molar crowns (through the buccal tubes with the action-reaction
counterpart couples against the wire). Three variations in the overall
"loading" are to be examined, associated with the bend geometry and
the occlusogingival position of the buccal tubes with respect to the "line
of bracket slots": (1) the couple alone exerted on the molar; (2) the couple accompanied by an extrusive force; and (3) an intrusive force as well
as the couple exerted on the molar crown.
Before beginning the force analysis, some modeling is appropriate,
as mentioned earlier in this chapter. Proposed is the consideration of the
right-side half-arch as "L"-shaped as viewed from an occlusal perspective; the corner of the "L" marks the intersection of anterior and posterior portions of the arch model. The central and lateral incisors engage
the anterior section and the canine, premolars, and first molar the posterior section of the wire. Because the intended action is against the first
molar, it is separated in the model from the middle segment consisting
of the canine and premolars.
Shown on the left in Figure 6-25 is the right-side, first molar sub-
jected to the couple created by the tip-back bend. The displacement
tendency is that of distal crown tipping with a center of rotation coinciding with the center of resistance in the approximate location indicated.
On the right in Figure 6-25 is a free-body diagram of the half arch wire.
Introduction to Structural Analysis of the Orthodontic Appliance
FIGURE 6-25. Buccal views of a terminal molar subjected to tip-back-bend action in the
absence of net occlusogingival force (left), and a free-body diagram of one-half of the
arch-wire model showing fhe responsive, contact force system exerted by the dental
units (right).
Newton's law of action and reaction yields the couple at the first-molar
site. Because the half-arch is quasi-static, like the dentition, the entire
force system acting on it must be balanced in this view. In the absence of
substantial second-order bends mesial to the tip-back and, then, assuming second-order clearance between premolar and canine brackets and
the wire, together with third-order clearance in the anterior section (or
round wire engaged and containing no torquing spurs), the response to
the couple loading must be in the form of occlusogingival forces mesial
to the activation site. The pair of equal, parallel vertical forces shown
form another couple, opposite in sense to the couple against the lip-back
bend; the product of one of the forces and the distance between the pair
of forces must equal the magnitude of the tip-back-bend couple in order
to yield a balanced force system exerted on the half arch wire.
The specific locations of the vertical responsive forces depend on the
bending stiffness and the passive geometry of the arch wire relative to
the bracket placements and their orientations, but most likely both will
act within the middle segment—perhaps against the second premolar
and canine. The net effect of the response tends to rotate the line of
middle-segment brackets clockwise, as viewed in Figure 6-25. If the arch
wire has high bending stiffness, the response to the couple loading may
travel to the incisal segment with a tendency there toward intrusion.
The sketch on the left in Figure 6-26 shows the combined effect of the
lip-back bend and an occlusogingival malalignment of buccal tube and
wire, the latter creating a tendency for extrusion. The action-reaction
counterparts of the force and couple against the molar are shown against
the posterior end of the half arch wire on the right in the figure. Again,
assuming ample second-order clearance throughout the middle segment
and no third-order activity in the incisal region, the response mesial to
the molar is again in the form of occlusogingival force. In fact, the total
force system exerted on the half arch wire by the engaged dentition may
be balanced, with the given action at the terminal molar, by a single,
resultant gingivally directed force as shown. The location of this balancing force is probably at the first-premolar site if the wire exhibits moder-
Bioengineering Analysis of Orthodontic Mechanics
0 (= P6)
FIGURE 6-26. Buccal views of a terminal molar subjected to tip-back-bend action in
combination with a net extrusive force (left) and a free-body diagram of one-half of the
arch-wire model showing the response of the dentition (right).
ate bending
stiffness, but all three teeth in the middle segment are likely
subjected to intrusive action. If the wire is very flexible, the second premolar will receive the greatest portion of the distributed, intrusive potential.
The third variation is the most common, found in mechanics designed to establish posterior anchorage; the action against the terminal
molars is a combination of distal crown tipping and intrusion. Shown on
the left in Figure 6-27 is the right-side molar subjected to the counterclockwise couple and the gingivally directed force, The action-reaction
counterparts act against the posterior extent of the half arch wire on the
right in the figure. For a third time, assume that the response mesial to
the terminal molar is void of couples. Note that, although the addition of
one responsive, occlusogingival force could provide, in the free-body
diagram of the half arch wire, a vertical force balance, a moment balance
would be impossible. Accordingly, two vertical forces of mutually opposing senses are necessary to enable both force and moment balances.
Moreover, because the intermediate vertical force must have a magnitude equal to the sum of the other two forces and a sizable counterclockwise moment is necessary for the overall moment balance, the more
FIGURE 6-27. Buccal views of a terminal molar subjected to tip-back-bend action
together with a net intrusive force (left) and a free-body diagram of one-half of the
arch-wire model showing the response of the dentition (right).
Introduction to Structural Analysis of the Orthodontic Appliance
mesial of the two parts of the vertical, responsive, force system is likely
located in the incisal region as shown. Hence, the intermediate, vertical
force is exerted on the middle segment.
For each of the three active force systems against the terminal molars, three of the four steps in the previously outlined analysis procedure
have been discussed. Figures 6-25 through 6-27 show the desired (or
obtained) actions at the activation site, the terminal molar (step 1), and
the action-reaction transition to the appliance (the arch wire—with symmetry, the half arch) and the subsequent balancing of the force system
on the wire (step 2). The effects of the response elsewhere in the dental
arch, gained from a second action-reaction transition, were then noted;
the three variations in response in the order considered are pictured
from left to right in Figure 6-28 (step 3). The displacements that occur in
these mechanics are to be restricted to the terminal molar. Undesired
side effects, then, are movements within the middle and incisal segments produced by the responsive force systems. The ideal may not be
realized, however, inasmuch as the molar demands a sizable active force
system commensurate with its root-surface area, and this area is larger
than that of any of the teeth mesial to the molar.
Beyond keeping the active force system against the molar as small as
reasonably possible, considering the smaller root-surface areas of teeth
receiving responsive force, little further discussion of step 4 is reasona-
ble here. The distribution of responsive force is associated with the
structural behavior of the appliance, and in this instance knowledge of
arch-wire bending (flexure) is required—the topic of Chapter 7. It is
notably necessary, in particular when the tip-back couple is accompanied by the intrusive force (see Fig. 6-27), to unitize the middle segment
rather than permit an individual tooth to take a majority of the vertical
response. Of concern, in addition, must be the intrusive responsive
force against the incisal segment seen in the third variation (Fig. 6-27
and Fig. 6-28, right). The potential for intrusive displacement is not only
strong because of the relatively small root-surface area of the half incisal
segment, but the line of action misses the center of resistance by some
distance if these teeth have typical angulations and, as a result, the
tendency for flaring is also present. One suggestion, assuming the intrusion and flaring are unwanted, is to place up-and-down elastics to transfer a portion of the gingivally directed force to the maxillary arch. Alter-
natively, the patient could wear, for a portion of each day, an
FIGURE 6-28. The transfers of responsive force systems to the dentition resulting from
the terminal-molar activations of Figures 6-25 (left), 6-26 (center), and 6-27 (right).
Bioengineering Analysis of Orthodontic Mechanics
appropriate extraoral appliance engaging the incisal segment of the
mandibular arch wire. Clearly, these "remedies" draw on concepts and
discussions yet ahead in this text and, as already mentioned, the distribution of the responsive force system is partially dependent on bending
characteristics of the arch wire that receive in-depth attention in the
chapter to follow. In short, the example discussed here is typical in
format of the whole of Chapter 10; as noted at the beginning of this
section, the purpose was to provide a glimpse of a portion of the structural analysis procedure, the foundation for which has been laid in the
foregoing sections of the present chapter.
"Control" of the Orthodontic Apptiance
of its relation to the structural behavior of the mechanism over
time, before closing this chapter several comments pertaining to the
"control of the appliance," to use the clinician's terminology, are in
order. A difficult concept to define directly, "control" may seem to make
reference to displacements of teeth, but a broader interpretation should
be understood. In anticipation of activation or reactivation within the
oral cavity, the appliance is prepared with regard to the existing configu-
ration of the dentition and the realignment desired, expecting certain
tooth movements to occur in the interval between appointments. Because the displacements of dental units are accompanied by geometric
and mechanical alterations of the appliance from the as-activated state,
and the orthodontist has no "hands on" influence during the betweenappointments period, control must be built into the appliance design
and the activation of that appliance.
After the patient has left the office or clinic, three factors govern: (1)
the biologic response of the tissues to the induced force system; (2) the
minimum-energy principle; and (3) the actions of the patient. The first
factor was discussed in Chapter 5. The second factor has been mentioned, but to repeat the principle indicates that, given the locations and
formats of connections of the appliance to the dentition and anchorage,
the appliance will assume a configuration, within the constraints imposed by those connections and the dentition, which makes the total
strain-energy increment associated with activation quantitatively as
small as possible. This strain-energy increment, directly related to the
appliance deformation from its original passive state and the accompanying induced forces, is distributed throughout the mechanism. Any
crown displacement causes a change in the appliance configuration (due
to the connection of mechanism to dentition), a reduction in the strain
energy (due to an energy transfer in the form of work done by the
appliance in moving the tooth or teeth), and a redistribution of what
remains of the increment to define a new, minimum-energy state. The
third factor is well understood as to its potential adverse impact. If any
Introduction to Structural Analysis of the Orthodontic Appliance
portion of the appliance is removable (elastics, an extraoral appliance, or
a retainer, for example), the patient may infringe upon the control estab-
lished by the clinician. Patient cooperation and control go together; a
patient who is 100% cooperative leaves the maximum amount of appliance control to the orthodontist. The ideal orthodontic appliance would
permit the operator to predict, in extent to within the limits imposed by
the incompletely known biologic response factor, the displacements of
all points of the appliance where force is transmitted to the dentition
and, in effect, to completely but remotely be in control of the deactivation process at all times. Of obvious importance is control of displacements at the anchorage sites. In this regard the extraoral appliance presents a paradox: the rationale for development and use of headgear is the
location of the anchorage away from the dentition, and yet, with this
appliance, the patient may interfere with operator control to a substantial and, perhaps, critical extent.
The process of activating and engaging the orthodontic appliance
produces a system of forces within the mechanism that reaches the dentition through the appliance-to-dentition connection sites. Although
permitting some initial deactivation with primarily soft-tissue displacements, the resistance of the periodontium that prevents large initial
tooth movements generally keeps the appliance under load (activated)
for an extended period of lime. Accordingly, during active treatment
two sets of forces exist as action-reaction counterparts at the connection
sites. One set is primarily and continually related to the deformed configuration of the appliance with respect to its passive state; the other
determines in large measure the course of dental-unit displacements
over the finite time period. Control of the overall force system, then,
must account for this system's influence on both the appliance and the
dentition, and all characteristics of each component of the system (force
or couple location, magnitude, and direction and sense), together with
variations in these characteristics with time, must be considered.
With respect to the dentition, force and couple levels should be large
enough initially so that, after the soft-tissue movements, sufficient magnitudes remain to produce measurable orthodontic displacements during the between-appointments period. On the other hand, force magni-
tudes cannot be so great as to inflict undue pain and/or result in
ankylosis. Concurrently, the force system delivered to intraoral anchorage sites, if any, must be small enough (through distribution as well as
by limiting the size of the resultant) not to cause unwanted movements
there. Furthermore, the initial force levels chosen will be affected by
their expected decay patterns over lime (discussed in Chapter 5). Displacement paths of the dental units will be directly influenced by directional characteristics of the force system; displacements early in the be-
tween-appointments period may alter initial force directions with
respect to teeth and, thereby, affect subsequent movements within the
same period. Particularly difficult to control are couple-force ratios over
time, inasmuch as the individual decay patterns of couple and force are
Bioengineering Analysis of Orthodontic Mechanics
related to separate stiffness and deformation parameters. The biologic
system reacts to the local force system as a whole, and not to the individual components.
Ideally, force control relative to the appliance should be built into the
design of the mechanism. The force system created with the activation
of the appliance must not permit any type of failure at the instant of
activation when the stresses and localized appliance displacements are
generally greatest. To the point mentioned previously, the allowable
"load" on the appliance must account for the expectation of superposition of masticatory forces. Particularly with regard to the possibility of
fatigue failure, special precautions must be taken with appliances or
appliance elements that are removable and, as a result, are subjected to
interrupted force systems.
Displacement control pertains primarily to the orthodontic tooth
movements, but not to the complete exclusion of the appliance. Allow-
ances must be made, and again largely in the design, such that the
mechanism, as it deactivates, will not make unwanted contact with the
tissues, thereby giving rise to the potential for trauma or spurious displacements. Displacements, like forces, are vector quantities having direction as well as magnitude. As discussed in Chapter 5, the directions
of tooth movements are influenced by the characteristics of the delivered
force systems and, to some extent, by alveolar-bone biology. The
bounds on dental displacement magnitudes are directly related to the
extent of appliance activation and the between-appointments time period; even in unlimited time, orthodontic tooth movements will not continue beyond total deactivation of the appliance.
With metallic appliance elements the orthodontist uses the passive
configuration of the appliance and the elastic limits of appliance elements
as guides in activation. Ideally, if the activation does not exceed the
elastic limit, the appliance will deactivate to its former, passive state.
This does not mean, however, that the movements will be, at their
greatest extents, exactly as "read" from the activation, because this
"reading" assumes immovable anchorage. Although the majority of
metallic, activated appliance elements are expected to produce intermittent force as defined in Chapter 5, displacement control of elements
producing continuous force depends upon unloading at the appropriate
time by the clinician or the patient, or on intraoral abutments. Because
interrupted forces are continuous during their "on" periods, the same is
true for this third class of loading. When the continuous force is produced by a polymeric element, the deterioration-aging and relaxation of
the material may hinder the maximizing of controlled displacements,
but in an uncontrolled situation (the patient missing one or more appointments, for example), a drop in force magnitude or discontinuance
of the force due to failure of the element may be quite helpful.
Note that the orthodontist can use the clearances that may exist between arch wire and bracket or buccal tube, particularly with regard to
controlling rotations. The wire may be used as an active appliance element, it may be a guide to control the direction of a displacement with a
Introduction to Structural Analysis of the Orthodontic Appliance
driving force produced by a separate appliance element, or it may be the
"trigger" of an abutment system to limit the extent of a specific displace-
ment component. In canine retraction on a continuous arch wire, some
distal crown tipping may be permitted; however, when second-order
clearance disappears, a couple countering the tipping moment of the
driving force arises with bracket-wire contact. In preparing for Class IT
mechanics, posterior-segment crowns may be tipped distally; secondorder bends are placed in the posterior portion of the mandibular arch
wire of the Class II appliance, but may become active only when force
against the segments begins to upright these teeth and "burn" (eliminate) the anchorage. These two sets of mechanics, and others, are examined in detail in Chapter 10, and means of control of forces and displacements are discussed concurrently.
objective of this chapter was to provide the reader with a foundation, based in part on the concepts and procedures of structural analysis,
for the understanding of the topics of arch-wire bending and torsion,
extraoral-appliance force systems, and a variety of orthodontic treatment mechanics, discussed from a bioengineering standpoint in the
final four chapters of this text. The uniqueness of the orthodontic appliance, neither a structure nor a machine by their engineering definitions,
has been noted. The sequence of procedures in a general structural analysis was outlined, the entire process was seen as divisible into two main
portions (related to the external and internal force systems associated
with individual appliance members), and a four-step approach to the
analysis of the orthodontic appliance was introduced.
The complexity of the appliance was noted as to its geometry, material characteristics, and the force systems transmitted between appliance
and dentition. When evaluating the continuous arch wire or an appliance as a whole, the primary actions and responses may be examined in
a less complicated environment through reasonable modeling procedures; preparation of an arch-wire model has been undertaken with the
importance of validity evaluation emphasized. Substantial attention was
given to the connection sites of appliance to dentition, in particular to
the orthodontic bracket and its characteristics with influences on both
the properties of the transmitted force system and the structural behavior of the arch wire.
The orthodontic appliance was discussed as a quasi-static and indeterminate structure, and in this light the force-deformation characteristics of a number of the more basic auxiliary elements were examined.
Mention was also made of the various formats and components of arch-
wire activation, the many stiffnesses that may enter into an analysis,
and the associated anchorage configurations; however, it is only an
overview with the two chapters to follow devoted to these considera-
Bioengineering Analysis of Orthodontic Mechanics
tions. An examination of the effects of activated, tip-back bends served
as a first experience involving an arch wire with the four-step analysis
Finally, the clinician's concept of "control" was considered with note
of its applicability both to the desired displacements within the dentition
and the structural behavior of the orthodontic appliance.
This text is not intended to convert orthodontic students or practicing clinicians into structural analysts. The practitioner does, however,
contribute significantly to the advancements in design of orthodontic
appliances and, as important, is potentially in the best position to knowledgeably critique existing designs from the standpoint of their "in-service" performances. Accordingly, the orthodontist needs at minimum a
superficial understanding of the contributions of the structural parameters of the appliance elements to the total behavior of the mechanism to
properly employ it in therapy and to evaluate the results achieved with
respect to treatment objectives.
Suggested Readings
Hsieh, Y.Y.: Elementary Theory of Structures. Englewood Cliffs, N.J., Prentice-
Hall, 1970, Chapters 1, 2.
Jarabak, JR., and Fizzell, J.A.: Technique and Treatment with Light-wire Edgewise Appliances. 2nd Ed. St. Louis, C.V. Mosby, 1972, Chapters 3, 8.
Laursen, H.I.: Structural Analysis. 2nd Ed. New York, McGraw-Hill, 1978,
Chapters 1, 2.
Behavior of Orthodontic
Wire in Bending
The orthodontic arch wire, more than any other form of structural mem-
ber, resembles a beam. The beam is characterized by its geometry, its
loading, and its response to the force systems exerted upon it. The longitudinal dimension of this structural member is large in comparison with
its cross-sectional measurements; the interbracket distance, a characteristic length for the arch-wire segment, is ordinarily at least ten times the
typical cross-sectional dimension. Beam loading, as well as its responsive force system, generally consists of concentrated and distributed
forces directed perpendicular to the longitudinal dimension; couples
may be included in the force system. The applied forces and responses
may exist in more than one plane; the force system exerted on an archwire beam may be directed occlusogingivally and/or faciolingually. The
activation of a beam is reflected in characteristic displacements with respect to a passive configuration. The primary reference for beam deformations is the passive, longitudinal beam axis that pierces the centers of
all of the beam cross-sections. Also associated with beam deformations
are the right cross-sections which, in the passive state, by definition, are
everywhere perpendicular to the beam axis. The two forms of beam
deformation of principal importance are the transverse deflection, the displacement of a beam-axis point perpendicular to the passive orientation
of the axis at the point, and the cross-sectional rotation, not a twisting but
rather a tipping, angular displacement of the right cross-section.
In keeping with the model established in Chapter 6, the typical archwire segment, to be activated in bending, is envisioned as passively flat
and straight. Depending on the plane(s) in which the bending occurs,
facial or occlusal views of the arch-wire beam are used in analysis. Most
often, the bending force systems are exerted by ligated brackets and
buccal tubes, although forces may be transmitted to the arch wire
through hooks or loops—by elastics stretched against them, for example. When an auxiliary force acts on a wire in a direction oblique to it,
that force is decomposed into longitudinal and transverse components;
only the transverse component contributes directly to bending action.
Occlusogingival and faciolingual bracket and buccal-tube force systems
Bioengineering Analysis of Orthodontic Mechanics
(but not mesiodistal forces from, for example, stops, ties to anchorage,
or friction) are associated with the arch-wire bending problem. First- and
second-order couples, originating from angulations and two-point contacts between wire and bracket or buccal tube, also enter into the bending analysis. The arch-wire segment, activated in bending through en-
gagement in brackets and buccal tubes, may test the analyst to
distinguish between the "load" and the responsive portion of the force
system; moreover, the arch-wire beam problem is generally quasi-statically indeterminate, necessitating the use of force-deformation relationships in the solution in addition to force and moment balances. The
responsive force system, exerted by the "support(s)" of the arch-wire
beam, is the action-reaction counterpart of the force system against the
anchorage. Clearances between bracket or tube and wire often play a
role in the bending problem. Occlusogingival, dimensional differences
may be important to the determination of a transverse deflection.
Changes in shape from the passive beam (arch) configuration are generally consequences of bending activation; depending on the amount of
second-order clearance, for example, a faciolingual-plane couple may or
may not be developed at a support location as part of the bending force
The many facets of the arch-wire bending problem already mentioned suggest a complex analysis procedure. Reasonable simplifications, however, beginning with the straight-segmented, passive,
arch-wire model, enable a manageable examination of the significant
contributions to the analysis. A first bending example appears to be an
inversion of the lever of Figure 2-27. The member of Figure 7-1, however, is subjected to a transverse load located between two anchorage
sites. Characterizing the beam by its longitudinal axis, it is shown in the
Midspan Deflection
FIGURE 7-1. A simple example of a split-anchorage beam: passive (dashed) and
activated beam-axis configurations (upper left); a free-body diagram of the beam (lower
left); and the force-deflection pattern of activation at midspan (right).
Behavior of Orthodontic Wire in Bending
sketch in the upper left in the figure, passively as a dashed, straight line
and having a solid, curvilinear shape following activation. Such an arch-
wire beam might be maintained in its deformed configuration by the
relative occiusogingival positions of three brackets. Assuming that the
initial position of the intermediate bracket represents a malalignment to
be corrected, the brackets at either end of the wire beam are affixed to
teeth that are not to be displaced; the anchorage might be termed "divided" or "split." To make this first example straightforward, distances
between adjacent brackets are equal and sufficient second-order clearances are assumed to exist such that no couples arise with activation.
The free-body diagram of the activated beam, shown in Figure 7-1
(lower left), is identical to that for a lever having the fulcrum at its midpoint. Force and moment balances indicate that the support forces Qi
and QT are equal in magnitude and each is half that of the force P. but
none of the three can be quantified by a quasi-static analysis alone. To
approach the solution from this point, the configuration of the active
state, with respect to the passive beam-axis shape, must be examined.
Consider the wire as first engaged in only the support brackets, with
activation then achieved by depressing the midsection of the wire to
reach the slot of the intermediate bracket. The greater the vertical malalignment, the larger the value of P upon full engagement. From foregoing discussions of material behavior (Chap. 3), the expectation is a
linear force-deflection relationship for the metallic wire as shown on the
right in Figure 7-1. The slope of the plot is a bending stiffness, the quantity needed to complete the solution; the product of this stiffness and the
activating deflection yields the magnitude of P.
With the foregoing as an introduction, this chapter develops with a
discussion in the following four sections of the geometric and mechanical parameters involved in elastic beam analyses. Next, direct application to activation and deactivation of arch wires in bending is under-
taken. Subsequently considered is the inelastic bending of metallic
beams toward an understanding of the influence of the placement of
permanent bends in an arch wire which is then to be activated. The
chapter closes with a structural analysis of orthodontic wire loops—their
characteristics and the influences of loop geometry and wire properties
upon activation and deactivation in bending.
am Deformations, Strains, and Stresses
reference frame established for determining bending deformations
is affixed to the longitudinal axis of the passive, straight beam. The beam
axes for arch wires are easily located due to the symmetry of circular and
rectangular cross-sections. The general beam problem is three-dimensional with two coordinates of a rectangular framework locating a point
in the cross-section with respect to its geometric center and the third
coordinate locating the cross-section. More specifically in the develop-
Bioengineering Analysis of Orthodontic Mechanics
ment to follow, the origin of the frame coincides with the geometric
center of the reference cross-section of the passive beam. The longitudinal coordinate x is measured along the undeformed (straight) beam axis.
The y and z axes, completing the framework, are coincident with perpendicular axes of symmetry within the reference, right cross-section of
the beam before loading. Although the force systems associated with the
activation of arch-wire beam may be located in either the x-z or y-z plane
or in both, when the force system is three-dimensional in character,
separate "in-plane" analyses may be undertaken and the results subsequently superimposed. Accordingly, a plane of bending is defined as that
plane containing both the undeformed and the deformed longitudinal
axes of a beam, as well as the external load and responsive force system.
Arch-wire bending may be reasonably analyzed in parts, individually in
the faciolingual and occlusogingival planes as necessary, followed if de-
sired by superposition of the results. In discussions to follow the x-z
plane is the plane of bending and an angle 6 is introduced as a cross-sectional, rotational coordinate.
As many as five displacement parameters may be associated with
in-plane bending. They are described here in their usual order of importance. First is the linear displacement of a beam-axis point perpendicular
to the passive beam axis, quantified by a z-coordinate measurement.
Second is the rotation of a cross-section, symbolically indicated by the
angle 6. These are the principal displacement parameters and both may
be expressed mathematically as functions of the longitudinal coordinate
x. Shown in the upper sketch of Figure 7-2 is a cantilevered beam sub-
jected to a concentrated load at its free end; in a separate sketch, the
undeformed (dashed) and deformed configurations of the beam axis are
depicted. A reasonable location for the origin of the x-z frame is the
center of the fixed-end cross-section, and indicated in the figure are the
transverse-deflection and cross-sectional-rotation parameters for an arbitrary value of the longitudinal coordinate and for the "free" end of the
beam. The collection of transverse displacements of all beam-axis points
yields the deformed beam axis. Because, for all values of x associated
with the activated beam, continuity demands that the longitudinal axis
and rotated right cross-section be perpendicular to one another, 6 represents the slope of the beam axis, generally varying from one cross-section to another.
A third displacement parameter designates the longitudinal shortening of the beam. With axial (longitudinal) force systems excluded in the
bending analysis, the curvilinear length of the beam axis is unchanged
by activation, but the x-coordinate distance between the ends of the
loaded beam is less than the length. Nearly always in the elastic bending
of engaged, activated arch wires, this form of displacement may be neglected in its influence on other beam parameters, as is implied in the
symbols z(L) and 6(L) in the example of Figure 7-2. (In leveling procedures within orthodontic treatment, however, because of high mesiodistal wire stiffness and, often, the real presence of longitudinal loading,
this shortening can have a sizable influence on displacement potential,
Behavior of Orthodontic Wire in Bending
FIGURE 7-2. Sketches of a cantllevered beam serving as an example to
diagrammatically define the bending displacement z, the transverse deflection, and 0,
the rotational displacement of a cross-section.
particularly of incisors.) The fourth and fifth displacement parameters
pertain to deformations of the cross-section itself. In the activation of
metallic beams, the induced transverse force systems will rarely alter the
passive, in-plane, depth dimension of the cross-section; this potential
deformation, which might be significant with soft materials or very short
beams, is routinely neglected here. Finally, experimental research has
shown that, under bending action, the right cross-sections of some
beams will warp, but no warping occurs when the end view of the plane
of loading coincides with an axis of symmetry of the beam cross-section.
In summary, in all beam theories the z and 0 deformations are recognized as sizable. Small-deformation theories of bending neglect all forms
of deformation except these two. When transverse and angular displacements of cross-sections are substantial, however, the effects of longitudinal shortening must be considered; such analyses are highly involved
and, because of their complexity, they are not discussed theoretically in
this text.
Bioengineering Analysis of Orthodontic Mechanics
In beginning an examination of structural bending, the solid beam is
considered to be composed of a large number of longitudinal fibers, each
having a very small cross-sectional area in comparison with that of the
beam as a whole. One fiber coincides with the longitudinal beam axis.
All fibers of the beam as modeled are passively straight and, in the plane
of bending, all deformed fibers have identical curvatures. Because these
fibers are interconnected, however, reflecting the structural integrity of
the beam as a unit, the majority of these fibers experience longitudinal
strain upon activation. Depending on the location of the fiber, this
"bending strain" may be tensile, compressive, or zero.
Figure 7-3 shows an exaggerated, activated shape of a beam segment
in in-plane bending. With the outline of the passive segment superimposed (dashed) and the segment subdivided to simulate the fibrous idealization, the fibers on the side of the beam axis nearer to its center of
curvature are seen to have been contracted; the fibers on the other side
of the axis are stretched. Note also that the amount of fiber (bending)
strain varies with the distance from the beam axis. As shown in Figure
7-3, a local coordinate u is used to measure this distance with the posi-
Neutral surface
FIGURE 7-3. A beam segment in passive (dashed) and activated configurations.
Illustrated are the neutral surface, bending strains of the beam fibers," and the
coordinate u that locates specific fibers with respect to the neutral surface.
Behavior of Orthodontic Wire in Bending
five sense taken toward the stretched fibers. An in-plane bending activa-
tion results in all fibers having the same u-coordinate value experiencing
equal bending strains and, considering only circular and rectangular
cross-sections, the group of fibers undergoing no longitudinal strain coincides with the cross-sectional axis of symmetry perpendicular to the
plane of loading. In Figure 7-3 the apparent beam-axis fiber is actually an
edge view of the collection of unstrained fibers known as the neutral
surface of the beam. Because the cross-sections of arch-wire beams ordinarily do not warp upon activation of the member, the fiber-strain pattern is associated solely with cross-sectional rotational deformation and
the strain values are, therefore, proportional to the u-coordinate locations of the fibers.
The overall beam shape in plane bending is represented by the curvature of the beam axis, within and coincident with the edge view of the
neutral surface. In general, each point on the bending axis is associated
with a beam cross-section and has its own center and radius of curva-
ture. If all bending-axis points in a beam segment possess the same
center and radius of curvature, that segment is said to be experiencing
"pure bending," terminology that will be further explained in a subsequent section of this chapter. Figure 7-4 shows a very short segment of a
beam with a typical, relative rotation of one neighboring cross-section
with respect to another. The portion of beam axis within the segment
takes on, approximately, the shape of a circular arc with activation, and
the center of curvature of that arc is point C. The associated radius of
curvature is r and its inverse will be symbolized in this chapter by k. The
beam-axis segment is of curvilinear length 3x, where 5 generally indicates "small amount of' or "small change in" the variable that follows it.
The angle 4 denotes the relative cross-sectional rotation. Because & is a
very short arc length and approaches in magnitude, and because the
longitudinal (bending) strain is the deformation per unit length of a
fiber having a passive length 8x, the relationship
is obtained. The coordinate u locates the beam fibers experiencing bending strain e,, and k is the average curvature of the short beam-axis segment. On any specific beam cross-section located by the coordinate x,
then, the rate of increase of bending strain from the neutral surface, a
constant for the cross-sectional location, is directly dependent on only
the beam-axis curvature at that location.
On the typical cross-section, in-plane bending deformation of an initially straight beam produces compressive strains in those beam fibers
on one side of the neutral surface and tensile strains in the remaining
fibers. The magnitude and sense of any fiber strain at a point in the
beam depend on location, defined by the two coordinates x and u for
in-plane bending, the geometry and material of the beam, and the force
system producing the bending deformation. On any beam cross-section,
Bioengineering Analysis of Orthodontic Mechanics
5x =
= k(8x)
= e0(ôx)
FIGURE 7-4. The geometry from which may be derived, for an activated beam, the
relationship between the intensity of bending strains on a specific cross-section and the
beam-axis curvature there.
Behavior of Orthodontic Wire in Bending
FIGURE 7-5. Fiber-strain and elastic fiber-stress patterns on a typical circular or
rectangular cross-section of a beam subjected to in-plane bending.
the fiber strains are proportional to their distances from the neutral sur-
faces and, for beams with circular or rectangular cross-sections, the max-
imum tensile and compressive strains are equal in magnitude.
A typical fiber-strain pattern is shown on the left in Figure 7-5 with,
for cross-sections of interest here, the neutral surface (seen in edge view)
midway between the pictured longitudinal boundaries of the beam. The
corresponding bending-stress pattern is obtained from the stress-strain
law for the beam material and the established strains. Because archwire-beam materials are reasonably homogeneous, the stress-strain relationship for any one fiber is that for the material as a whole. Since such
materials are metaffic, they behave within their elastic limits according to
Hooke's law. The theoretical development and results to follow reflect
equal material stiffnesses in tension and compression, a reasonable as-
sumption (verified by experimental research) well beyond the elastic
limit for beam material of interest. Shown on the right in Figure 7-5,
then, is the elastic, fiber- (bending-) stress distribution associated with
the adjacent strain pattern. At any specific point of the cross-section, for
totally elastic material behavior, the ratio of stress to strain equals
Young's modulus for the beam material. Increasing the beam loading to
produce inelastic action will not destroy the antisymmetry seen in the
figure, but it will change the relationship between the bending (fiber)
stress s,. and u from the linear pattern shown to a nonlinear pattern,
corresponding to the general plot of stress versus strain for a typical,
ductile, crystalline material (see Fig. 3-17, right).
orce Systems Within the Beam
The beam of Figure 7-1 may be described as having a "span" of magnihide 2L, simply supported (because the responses at the beam ends are
"simply" concentrated forces, and activated by a transverse, point load
Bioengineering Analysis of Orthodontic Mechanics
"midspan." The responsive forces obtained earlier seem to suggest
that the load is divided and its halves are transmitted left and right from
the midspan site to the beam ends where they are transferred from the
member to the supports. This is an oversimplification, but correctly implied is the generation of a force system within the beam, created with
application of the load. The internal force system may be "exposed" by
figuratively cutting the beam perpendicular to the beam axis at a desired
longitudinal location, revealing two adjacent right cross-sections. One
or the other beam portion is isolated for study; either may be chosen
since the force system exerted on one cross-section is the action-reaction
counterpart of that exerted on the other exposed cross-section.
As an example, the beam of Figure 7-1 has been "cut" half way be-
tween the left support and the load, and a free-body diagram of the
shorter, left-hand portion is shown in Figure 7-6. (Even though activation produces deformation, for purposes of force analysis the passiveconfiguration sketch may be used when longitudinal shortening is minimal.) In the activated state, any part of the beam is static (or quasi-static)
as is the entire member. Although the force system exerted by one portion of the beam upon the other at the "cut" site is distributed throughout the cross-section, resultant components perpendicular and tangent
to the cross-section are initally considered. First, the presence of the
support response at the end of the segment and the need for a transverse force balance suggest the existence of a tangential or shear component V. (The symbol V is used because the majority of beams are ori-
ented with the passive longitudinal axis horizontal; the shear force,
then, is directed vertically.) Second, because no axial loading is present,
a horizontal force balance indicates that the net longitudinal force com-
ponent at the "cut" is zero. (When present in the practical structural
problem, axial forces and their effects may be handled separately and
superimposed upon those associated with bending.) Third, since the
7-6. A free-body diagram of the left quarter segment of the beam of Figure 7-1,
exposing components of the resultant of the internal force system at a specific crosssection and permitting the determination ot their characteristics.
Behavior of Orthodontic Wire in Bending
support response and the shear component in the example are not col-
linear, a moment balance cannot exist without the presence of a couple
M at the "cut" site. (The symbol M is used because the couple is known
as the bending "moment" to the structural analyst.) The magnitudes
and correct senses of V and M from quasi-static analysis are given in the
figure for the example, simply-supported, split-anchorage beam. In obtaining these components a force analysis of the entire beam is necessary
prior to making the "cut" in order that V and M may be expressed in
terms of the load and longitudinal dimensions.
In the beam problems of interest here, an in-plane bending analysis
will involve external loadings consisting of concentrated forces and/or
couples located in the x-z plane. The internal components V and M will
be partially dependent on the load and will generally vary with the
longitudinal coordinate x from one cross-section to another along the
beam length. Using the example analysis as a guide, V may be seen to be
unchanged along a beam segment upon which no external force is ap-
plied while M, computed from force and distance, varies in a linear
fashion (increasing or decreasing with distance from the force to the
"cut" site). Proceeding from one cross-section to another longitudinally,
an encounter with an applied, concentrated force makes V "jump" to a
new value; M is also affected, but to a less dramatic extent. In the same
analysis procedure, encountering an applied, concentrated couple
makes the internal bending moment "jump," but the couple has no
effect on V.
In detailed beam analyses, plots of V and M versus the longitudinal
coordinate x are often sketched. Such diagrams, showing typical varia-
tional patterns of these resultant components of the internal force system, are presented for the example problem below the free-body diagram in Figure 7-7. The determination of the critical cross-section, the
location of the largest M value is important in beam analysis; the figure
provides that information for the example problem. Although not exhibited in the example, a segment of finite length may exist in a beam in
which the bending couple M remains constant. Correspondingly, the
shear force V must be zero throughout the segment and the segment is
said to be in pure bending. Within the segment the deformed beam axis
has the shape of a circular arc and, therefore, every cross-section within
the segment has the same center and radius of curvature and, as mentioned previously, the same bending strain pattern.
The foregoing comments suggest the existence of relationships between the couple M and the bending stress and deformation patterns for
the general beam. Appropriate first, however, is a look at the influence
of the shear component V upon bending. Although V is related to M
(reflected in Figure 7-7 in that the slope of the M-diagram is everywhere
equal to the corresponding shear force), its direct effect on important
bending parameters is minimal. Experimental results have shown that
transverse deformations attributable to V are generally very small. The
component V is the resultant of the transverse shearing-stress pattern,
and the average and maximum shearing stresses are typically less than
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 7-7. Shear and bending-moment diagrams below a free-body diagram of the
beam of Figure 7-1.
10% of their counterparts in bending. Only when the beam is extremely
short are the shear effects significant, and such beams are nonexistent in
the orthodontic appliance. Hence, the remarks to follow focus solely on
one component of the internal-force system in bending.
Behavior of Orthodontic Wire in Bending
Noted previously was the presence of a fiber-stress pattern perpendicular to a right cross-section within the activated beam (Fig. 7-5). For
the two cross-sectional shapes of interest and considering in-plane
bending, that pattern is divided in half by the neutral surface. Directly
viewing the cross-section, the edge of the neutral surface seen is termed
the neutral axis and it coincides with the y-coordinate axis defined earlier.
Bending stresses are zero on the neutral axis that divides the distribution
into tensile and compressive parts; the resultants of the two parts of the
bending-stress pattern are equal, parallel forces, and their resultant is
the bending couple M. In general, then, like M, the magnitudes of these
forces and the sr-to-u ratio may vary from one cross-section to another,
mathematically dependent on the longitudinal coordinate x. A diagrammatic representation of the relationship among the bending stresses, the
pair of force resultants, and M is sketched in Figure 7-8. The bending(fiber-) stress value at a point depends upon the coordinates x and u to
specify the location within the beam, upon the loading that enables the
determination of M, and finally upon the cross-sectional geometry of
the beam. To fill several needs an analytic expression in these parameters is now developed.
Figure 7-9 depicts a typical beam cross-section in edge view, and a
line of fibers parallel to the y-axis and the associated internal force upon
which attention is focused. The desired, explicit expression for M is
obtained by summing the moments of all such "fiber-line" forces about
the neutral axis, also seen only in edge view in the figure. Because each
fiber has a very small cross-section and is activated in direct tension (or
compression), the 0-notation is again used and the fiber-line force equals
the product of stress (unchanging in the y-direction) and the fiber-line
OP =
F, =
FIGURE 7-8. The internal bending couple as the resultant of the fiber-stress distribution.
Bioengineering Analysis of Orthodontic Mechanics
Neutral axis
3M = (SP)u
a "tiber"
Outline ot tensile
tiber-stress distribution
FIGURE 7-9. The contribution of the force in one beam fiber to the internal bending
couple M.
The moment arm for each fiber-line force 8P is the coordinate value u,
and the sum of the moments is M:
M = sum of [u(sn)6A1
Again referring to Figure 7-5 and recalling the beam-material model, if
the in-plane bending does not take any fiber beyond its elastic limit, the
value of is proportional to the distance between the neutral axis and
the fiber-line (which is, again, the coordinate value u). Using this proportion, Equation 7-3 may be rewritten:
M = —[sum of (u 2SAfl
The quantity in brackets is a geometric parameter of the beam cross-section; it is termed a second momenf of the cross-sectional area and is often
symbolized by the letter I. The ratio of Ito c, where c is one-half of the
beam depth measured in the plane of bending, is known as the secf ion
modulus for the cross-section. Values for the second moment of area and
the section modulus are given in Figure 7-10 for circular and rectangular
cross-sections; note that, with the loading plane vertical with respect to
the sketches, the beam depths 2c are the diameter and the dimension h
for the circular and rectangular cross-sections, respectively. Returning to
Equation 7-3, the ratio of bending stress to coordinate u may be written as
Sn: u
(sn)max : c
Behavior of Orthodontic Wire in Bending
For the circular cross-section:
For the rectangular cross-section:
where c = —
FIGURE 7-10. Second moments of area I arid section moduli I/c for circular and
rectangular cross-sections.
desired equation may now be expressed in either of two forms:
= (Sn)max1
This equation relates M, the bending stress in a particular line of fibers
(perpendicular to the plane of bending), and a geometric parameter of
the cross-section (generally a constant throughout the beam length).
The arch-wire-beam activation ordinarily should be fully elastic in
order that the totally deactivated and pre-load passive configurations are
identical. Accordingly, the maximum magnitude of loading to be exerted on the beam, without anywhere exceeding the elastic limit of the
beam material, is of interest. This quantity is known as the elastic strength
in bending. From the form of Equation 7-6 and Figure 7-8, the maximum
fiber stresses on a cross-section are located at the points farthest from
the neutral axis, and these maximum stresses are proportional to the
value of M. Considering the entire beam, the elastic strength depends
on the maximum value of M, the internal bending couple at the "critical
cross-section," which is dependent on the loading and manner of support of the beam and its longitudinal geometry. Accordingly, the elastic
strength may be expressed analytically by setting the maximum bending
stress in Equation 7-6 equal to the elastic limit value for the beam material (in tension or compression), and specifying the critical cross-section:
Bioengineering Analysis of Orthodontic Mechanics
= (Sn)erL
The critical bending couple is, therefore, computed from mechanical and
geometric property values for the beam. The relationship between
and the loading parameters is obtained from a free-body analysis of one
of the two beam segments isolated by "cutting" the whole member at its
critical section. Finally, then, the elastic strength of a given beam subjected to a particular loading configuration may be determined as maximum or limiting values of parameters associated with the active force
Beam Stiffnesses
The theory of elastic bending has its origins in experimental observation
and research proceeding to an empirical development. The initially
straight member takes on a curved form when activated or loaded, with
the associated deformations largely expressible in terms of the shape of
the longitudinal beam axis. Passively plane right cross-sections have
been found to remain plane with activation for problems of interest,
even when the deformations are substantial, leading to a simple, linear
fiber-strain pattern (see Fig. 7-5). Hooke's law then provides the bending-stress distribution on a cross-section; the resultant is the internal
bending couple. This couple is related to the external force system exerted on the beam through force and moment balances. To this point in
the discussion, three governing relationships have been mentioned
which contain variables dependent on the longitudinal coordinate x: (1)
the curvature of the beam axis and the u-coordinate provide the fiber
strains; (2) fiber stresses are everywhere proportional to fiber strains;
and (3) the internal bending couple is obtained from the bending-stress
distribution and the cross-sectional geometry.
An explicit expression, or a set of expressions, for the deformations
of the activated beam, in terms of the beam material, geometry, and
loading, is now required. An intermediate result is obtained from eliminating and u from the aforementioned three relationships;
In essence, this equation interrelates deformation (k), force (M), and
stiffness (El) parameters; the first two are generally dependent on x, but
the third is a constant and is known as the unit bending stiffness. The next
step in the derivation eliminates k in favor of its differential expression in
terms of z and x and, although the subsequent mathematical operations
are beyond the scope of this discussion, several associated comments
are appropriate. First, relatively simple equations are obtained only
when the slope of the beam axis is, for all x values, small compared to
Behavior of Orthodontic Wire in Bending
unity (0 less than 45°), and when longitudinal-dimension changes with
activation are negligible; therefore, in invoking these restrictions, the
result is a small-deformation theory. Second, the mathematical procedures that must be undertaken yield equations for both the cross-section-rotation parameter 0 and the transverse deflection z. In these derivations use is made of physical constraints to the actual deformations
that exist along the beam; such constraints are integral in the supports of
the beam and, for the arch-wire beam, exist where the wire engages the
brackets or buccal tubes. Third, a number of assumptions involving geometry and material have been made and it is well to recall the list here:
cross-sectional symmetry; in-plane bending; elastic activation; Hookean
material; and values of the elastic modulus F equal in tension and compression.
The split-anchorage, arch-wire beam of Figure 7-1 again serves well
as an illustrative example. The span is 2L and the bracket-wire clearances are assumed sufficiently large that transverse forces alone are developed upon activation. With midspan loading, the responsive forces
at the anchorage locations have magnitudes that are each one-half that
of the active point load. Given is the magnitude of the midspan deflection necessary to activate the member; in perspective, the occlusogingival or faciolingual crown malalignment of a canine or premolar has been
measured. If the midspan, transverse stiffness of the wire beam can be
determined, and the activation is totally elastic, the product of this stiff-
ness and the activating displacement is the magnitude of the midspan
With the external force analysis completed for the entire beam in
terms of the applied force F, the next step is the determination of the
bending couple on an arbitrary cross-section as a function of P and x. An
x-z-coordinate frame is established with its origin taken, for example, at
the left-support position with the x-axis coincident with the passive con-
figuration of the beam axis. The undeformed (dashed) and deformed
(solid) beam configurations and the reference frame are shown in Figure
7-11. Because of the geometric and mechanical symmetry in the example
problem with respect to the midspan cross-section, the M-diagram (Fig.
7-7) and the transverse-deflection pattern exhibit the same symmetry.
FIGURE 7-11. An x-z-coordinate frame superimposed upon the passive (dashed) and
activated (solid) configurations of the longitudinal beam axis for the Figure 7-1 example.
Bloengineering Analysis of Orthodontic Mechanics
With the slope of the beam axis horizontal at midspan and the symme-
try, in effect the solution may be pursued from this point using only
one-half of the beam. The equation for the couple M is written from
free-body analyses of segments of the beam, or obtained from the M-diagram, and substituted in Equation 7-8. The curvature k is then expressed
implicitly in terms of z and x as noted previously, and subsequent mathematical operations yield equations for 0 and z. In the example given,
the equation of interest is
obtained by substituting x = L into the more general expression. A linear relationship between P and z is noted and the ratio of P to z is the
midspan, transverse-stiffness equation desired. This stiffness is seen to
be directly proportional to the unit bending stiffness El and inversely
proportional to a power of the characteristic, longitudinal dimension.
With these quantities determined for the wire beam selected and the
malalignment (z at x = L) measured, Equation 7-9 provides the value of
P and the forces transmitted to anchorage also become known.
From the results of this example problem the parametric influences
on bending stiffnesses may be discussed in general. A first effect is that
of beam material with stiffness proportional to Young's modulus E. A
second influence is that of the size and shape of the cross-section. Because the parameter I is a fourth-power function of the cross-sectional
dimensions, small changes in cross-section can substantially affect the
bending stiffness. For instance, due to the differences in values of I,
replacement of a round wire by a square wire having an edge length
equal to the round-wire diameter increases the bending stiffness by 70%.
Third, a characteristic beam length or span has sizable impact on bending stiffness. In arch-wire bending, this influence is largely that of inter-
bracket distance. In the point-load example, the third power of the
length appears in the transverse-stiffness formula; in general, as will be
seen, this L-influence will be determined by the type of loading and
which stiffness (transverse or rotational) is being computed.
Two other considerations related to the example problem are of interest. First is the influence of the position of the load between the supports
upon the transverse stiffness at the beam-axis point under that load. For
purposes of comparison, shown in Figure 7-12 is the example beam with
the load moved from midspan, left to the one-third-span location. The
free-body diagram indicates that the mechanical symmetry is no longer
present; the desired stiffness is given in the sketch. A comparison of the
expression with Equation 7-9 shows that a lateral shift in the load position a distance equal to 17% of the span (214 results in an increase of
about 27% in the stiffness. Second is the influence of the presence or
absence of second-order clearances at anchorage in occlusogingival
bending (or first-order clearances in faciolingual bending).
Figure 7-13 depicts the original example problem once more, except
Behavior of Orthodontic Wire in Bending
At x =
= 7.6k
FIGURE 7-12. The split-anchorage example beam with the applied, transverse force
moved to the one-third-span location: a free-body diagram of the entire beam and the
expression for the transverse stiffness at the load site (above, right).
At x = L:
FIGURE 7-13. The split-anchorage example beam with no rotational deformation
permitted at its ends: a free-body diagram of the entire beam and the expression for
the transverse stiffness at the load site (above, right).
Bioengineering Analysis of Orthodontic Mechanics
that the supports permit no beam rotations at those locations (indicative
of zero second-order clearance if the figure shows a facial view). The
free-body diagram reflects the presence of couples as part of the support
responses; in effect, these couples oppose the rotational tendencies.
With the additional constraints on the shape of the deformed beam axis—
its slope forced to be zero at the supports—the stiffness should be expected to rise compared to that expressed in Equation 7-8. Indeed, the
transverse, midspan stiffness given on the right in Figure 7-13 for the
beam having "fixed ends" is four times that for the same beam material,
length, and cross-section, but with ample clearance permitting end rotations.
Additional Topics in Elastic Bending
With elastic strength and stiffness defined, a third structural parameter
of interest in bending is elastic range. A counterpart to elastic strength,
range is the maximum measure of a flexural deformation which the
beam can experience without an inelastic response occurring somewhere in the member. Hence, elastic strength is directly associated with
load and elastic range with the corresponding deformation. In the splitanchorage problem of Figure 7-1, the range of interest is the maximum,
elastic, transverse deflection at midspan. The M-diagram indicates the
largest bending stresses in the beam to be at midspan, the location of the
is equal to PLI2 (Fig. 7-7). The elastic
critical cross-section, and
range is obtained by first substituting
for PLI2 in Equation 7-9 and
then eliminating Mcrjt from the resulting expression using Equation 7-7.
The largest, elastic midspan deflection may then be expressed as
In general, elastic range is influenced by the characteristic length
dimension, the beam depth in the plane of bending, and elastic-limit
strain for the beam material. The power to which the length dimension
is raised in the range expression is always one less than that appearing
in the corresponding stiffness equation. Instead of the elastic-limit
strain, the ratio of elastic-limit stress to modulus of elasticity is often
explicit in the range equation; these two mechanical properties are typically more readily available, from reference materials or experimental
data, than is the strain value. The existence or amount of second-order
clearance at the supports affects the range as well as the stiffness. If the
beam is fixed or "built-in" at both ends such that no second-order clearance is present at either site, the value of
is lessened to PL/4 and the
Behavior of Orthodontic Wire in Bending
elastic range becomes
For the same reason that the stiffness was increased, the elastic range is
reduced when rotational beam deformation is constrained. In this example, requiring the beam-axis curvature to be zero and exhibit horizontal
slope at its ends results in a 50% reduction in range.
Elastic range and strength are significant parameters in beam analysis and design because they signify the onset of inelastic material behavior. In the discussion of orthodontic bending, because the actual loading
is often induced by a transverse or rotational deformation, range is perhaps the more important of the two. However, because arch-wire activation should normally not produce inelastic bending, the pertinent of the
two parameters (depending upon the loading) should be known or
quantified before engagement of the wire. Because of the interrelationship among range, stiffness, and strength, with second- (or first-) order
clearance already seen as influencing the first two, it affects strength as
well. Just noted was the halving of
in the example problem, as it
relates to the load on the beam, by eliminating that clearance at both
supports. Returning to Figure 7-7 and recalling Equation 7-7, making
no changes in the example problem other than eliminating clearances
at the supports results in a twofold increase in the critical value of the
load P (the elastic strength). Typically, alterations in bending parameters that raise (lower) beam stiffness will also raise (lower) the corresponding elastic strength and concurrently lower (raise) the elastic
Thus far the discussion of bending behavior has concentrated on the
beam having split anchorage and subjected to a transverse, point load
applied between the supports. Two additional problems having orthodontic application are the split-anchorage beam under in-plane, couple
loading and the cantilevered beam activated by either a transverse, concentrated load or a couple at its "free" end. As with the example problem, of interest are the anchorage responses, the locations of the critical
cross-sections and the associated, internal bending couples, the stiffnesses, ranges, and strengths, and appropriate comparisons among results. Shown in the upper sketch in Figure 7-14 is the free-body diagram
of a beam of span 2L activated by an external bending couple of magnitude M0. (The loading may either be an applied couple, or the couple
induced as a result of a clockwise, midspan, rotational deformation On.)
Rotational clearances are ample at the supports and the antisymmetry of
the deformed member helps in determining the responsive forces at the
ends of the beam. The internal bending couple is zero at each beam end,
increases linearly toward the middle cross-section, and at midspan it
"jumps" an amount equal to M0; the value of
is M012. The stiffness
of interest is the ratio of the applied-induced couple to the cross-sec-
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 7-14. Free-body diagrams of split-anchorage beams subjected to couple
loading at mids pan. No rotational resistance at the anchorage sites exists for the beam
in the upper sketch; the anchorage prevents rotation of the ends of the beam in the
lower sketch.
tional rotation at midspan and the relationship is
The cross-sectional, rotational range for this beam is
and the elastic strength may be expressed as
M0 =
Behavior of Orthodontic Wire in Bending
Shown in the lower sketch in Figure 7-14 is the same beam except
that no cross-sectional, rotational clearance exists at either end support,
and as a result the anchorage response includes a couple in addition to
the transverse force at each end of the beam. Computations of the values
of the responsive components require the use of indeterminate-beamanalysis methods and the constraints of zero slope of the beam axis at its
ends. The relationship between couple and cross-sectional rotation at
midspan for this beam is
and the rotational range is
Pertinent here is a comparison of stiffnesses and ranges for the beams
that are identical except for clearances at the anchorage sites; constraining the cross-sectional rotations at the ends of the member results in a
33% increase in stiffness and the same percentage decrease in elastic
range. Also of particular note is that the elimination of rotational clearances at the beam ends has no effect on the critical bending couple and,
therefore, the elastic strengths are equal for the two beams.
Free-body diagrams of cantilevered beams are shown in Figure 7-15.
Because of the existence of just one anchorage site (support), prevention
of both rotational and translational displacements must be present
there. Although some small amount of cross-sectional, rotational clearance could exist in reality, the equations to follow assume a complete
constraint and, therefore, zero slope of the beam axis at its left end. The
support response against the beam on the left in the figure includes the
expected, transverse force and couple, and the internal bending couple
= PL
7-15. Free-body diagrams of beams cantilevered from their left ends. The
free-end loads are a transverse force exerted on the beam at left and an in-plane
couple on the beam at right.
Bioengineering Analysis of Orthodontic Mechanics
varies with x in a linear manner from zero at the right end of the member
to the maximum,
value of PL at the built-in, left-end support. On
the other hand, the same beam on the right is loaded by a concentrated
couple at its free end and, with no net potential in the load to translate
the member, the response at the support is just an opposing couple. The
internal bending couple is unchanged from one cross-section to another
along the beam length (pure bending throughout); hence, even if the
applied couple M0 loading one beam was equal in magnitude to the
product PL for the other beam, the differences in M-diagrams indicate
unlike responses of the two, perhaps otherwise identical, members to
their loadings.
For the beam on the left in Figure 7-15, the stiffness of interest is the
ratio of the load to the transverse deflection of the free end and the
relationship is
The cross-sectional rotation of the free end of the same beam is
therefore, at least this amount of rotational clearance must be present
within whatever maintains the activation (e.g., a bracket) if the loading
is not to also include a couple. The range of interest in this example is
the largest, transverse, free-end deflection that will induce maximum
fiber (bending) stresses at the left end not exceeding the elastic limit of
the beam material, and that range is
With Mcrit equal to PL at the built-in end of the cantilever, the elastic
strength of the beam is
P = (sn)ei4
For the cantilevered beam subjected to pure bending on the right in
Figure 7-15, the rotational stiffness of its free end is
and the corresponding transverse deflection is
Behavior of Orthodontic Wire in Bending
the transverse position of the right end of the beam is main-
tained by some partial support that would exert a vertical force upward
and, by its presence, change the problem entirely, a transverse deflection downward is induced by the given loading.) The range for crosssectional rotations of the free end of this beam is
and, with
equal to M0, its elastic strength is given by
= (Sn)e11
of Elastic Beam Theory to Orthodontic Arch-Wire
in Bending
The orthodontic appliance is activated by imposing on a passive configu-
ration the deformation of one or more elements of that appliance. The
deformations of metallic elements are characteristically bending or twisting; the former predominates. Examples of bending activations are those
of arch wires in occlusogingival and faciolingual leveling procedures,
utility arches used to intrude or extrude incisal segments, and cantilevered arch-wire segments used to displace individual teeth. Responsive
bending deformations also occur in, for example, a face bow in the activation of a cervical-pull headgear and in the buccal-segment portions of
an arch wire into which anterior torque has been induced. The majority
of orthodontic wire loops are activated in bending. Note that under consideration here is the deformation imposed following whatever forming
of the as-received member is undertaken by the practitioner. Although
such forming procedures take the material beyond its elastic limit and
result in geometric alterations from the as-received configuration, the
activating deformation is intended to be elastic in nature. Inelastic bending is discussed in a subsequent section of this chapter.
The question arises regarding the validity of applying an engineering
bending theory to orthodontic arch wires, inasmuch as these members
have very small cross-sectional dimensions compared to the typical
structural member. Note that, within the theory, the format of the elastic
bending-stress distribution, exhibiting a linear variation from the neutral
surface, is as simple as the resultant, internal-force analysis and the
continuity of strains will permit. To be sure, the imposition of residual
stresses in the manufacturing process and in the placements of bends
and twists, prior to appliance engagement, will have some effect on the
subsequent structural behavior of the activated wire, but such influences
of substance are usually localized and
reflected directly in the mechanical properties of the wire material. Arch-wire-bending experimen-
Bioengineering Analysis of Orthodontic Mechanics
tation with straight segments has resulted in reasonable agreement between theoretical predictions and actual measurements, provided the
deformations are small and assuming Young's modulus in bending is
equal to the magnitude of the material stiffness derived from a tensile
The ADA specification for orthodontic wire, discussed briefly in
Chapter 3, includes an elastic-bending (flexure) evaluation and the theory wherein the pattern of couple loading versus cross-sectional rotation
is examined and structural properties are determined. Although the theory has apparently been accepted by the orthodontic community, care
must be taken to avoid exceeding the limits of its applicability. The theory will yield erroneous results when the activating deformations are
substantial; its accuracy is suspect, for example, when a transverse deflection of a continuous arch wire exceeds in magnitude that of an adjacent interbracket distance or one-half of the distance between supports
in a split-anchorage arrangement. In addition, the theory as discussed
here applies, in a strict sense, only to passively straight beams. To this
point, the arch-form curvatures of continuous wires across the incisal
segment and throughout the posterior segments are not locally severe
and are present only in the occlusal plane. The arch-wire model introduced in Chapter 6 suggests that only in the vicinity of the canines is the
curvature sufficient to void practical use of the theory. The presence
within the beam span of a permanent bend however, may affect the
accuracy of a theoretical analysis in the plane of that bend and, in particular, at the specific location of the bend. In an overall analysis, the inf usion of error originates not so much with the residual stresses, which are
usually highly localized, but rather with the change in passive geometry
associated with the bend placement.
The selection of an arch wire for use in an appliance, for which the Th
activation will result in bending deformation of that wire, must be based
primarily upon its structural characteristics in bending. The most important parameter is stiffness, which is not only the key to the solution of
the indeterminate problem but also provides the elastic activation and
deactivation rates (load per unit of deformation) in bending. Elastic
range is significant because prevention of inelasticity during activation is
desirable. If the load to be applied is measurable, the strength parameter
provides an upper bound for an elastic activation. Appearing in the
bending formulas of this chapter are the modulus of elasticity E and the
stress and strain values at the elastic limit. Young's modulus is affected
only slightly, but the elastic-limit coordinates may be substantially
changed, by strain-hardening or heat treatment of the arch wire prior to
in-service use. The recent additions of nickel-titanium and titanium-molybdenum alloys to the list of available orthodontic-wire materials has
broadened the spectrum of values of all three cited mechanical properties. The three structural characteristics are also dependent on wire geometry, more specifically on the cross-sectional dimensions, the shape
of the cross-section, and length (span) dimensions. The dependence of
stiffness, range, and strength in elastic bending upon arch-wire geomet-
Behavior of Orthodontic Wire in Bending
TABLE 7-1. Dependence of stiffness, range, and strength in bending upon material and
geometric properties of round and rectangular wires
*tr; Deformaton is transverse deflection.
tro: Deformation is cross-section rotation.
nc and material properties, for both transverse-deflection and cross-sec-
tional-rotational formats, but restricted to solid (single-strand) wires, is
given in Table 7-1.
Additional comments regarding cross-sectional geometry are in
order here. Note was previously made of the influence of shape and the
fact that, because of the additional material present, the bending stiffness of a square wire is potentially greater than that of the round wire,
the two cross-sections having equal depth dimensions. Rectangular
wires also exhibit greater strength; however, since only the beam depth
is the cross-sectional influence in the range formulas, shape does not
impact elastic range. In addition, the differences in the two dimensions
of a rectangular (nonsquare) wire produce, for a given orientation of the
wire in the brackets, differences in the structural bending properties for
occlusogingival versus faciolingual bending. Still another influencing
factor is the physical configuration of the wire, be it solid, layered, or
stranded. Because in layered or stranded wire the longitudinal elements
may slide somewhat with respect to one another upon activation (and
deactivation), for a particular overall cross-sectional size and shape,
these wires exhibit less strength, greater range, and, most significantly,
lower stiffness than their solid counterparts. Braided stainless-steel wire
is available in both round and rectangular cross-sections and will take
some permanent bending, exhibiting ductility, without substantially
compromising the braiding.
Without the use of loops placed in the wire, the practitioner is generally unable to vary the longitudinal dimension(s) of the arch-wire beam
to take advantage of the influence of this geometric parameter upon the
structural properties under consideration. Except when using a specialized cantilever or when leaving a particular tooth or segment out of the
activation, the mesiodistal geometry of the dentition controls the archwire beam spans. From a practical standpoint, then, a quantitative ex-
Bioengineering Analysis of Orthodontic Mechanics
TABLE 7-2. Effect of material of a solid arch wire, activated in bending by a transverse
force, upon stiffness, range, and strength
1 .86
A gold
Type 302
'Given figures are not actual values, but, in each column, are compared to a typical value for the stainless
amination of the effects of material and cross-sectional size and shape
upon stiffness, range, strength in bending is appropriate. Table 7-2 presents the comparative influences of five wire materials, considering wire
beams having all other characteristics identical and taking stainless steel
as the reference material (setting its three property values each equal to
unity). Table 7-3 examines the impact of cross-sectional geometry comparing solid round and rectangular beams having all other characteristics identical. The smaller cross-sectional dimension of each nonsquare,
rectangular wire is assumed oriented occiusogingivally. Values of the
parameters are, in each column, normalized with respect to a noted
reference wire and, where pertinent, the plane of bending. The fact that
the second moment of area (I) varies with the fourth power of the crosssectional dimensions causes the substantial differences in stiffness and
strength between the smallest and largest wires listed, even though
their depth dimensions differ by a factor not greater than two.
One additional influencing factor, referred to earlier in the stiffness
discussion, is the location of the load between specific, split-anchorage
positions. An illustrative orthodontic example is associated with the distal displacement of a canine into a first-premolar extraction site, using an
arch wire for guidance. The wire beam is supported anteriorly by the
incisors and also on each side by a posterior segment. Considering the
half arch (one side), the primary load on the wire is a second-order
couple exerted by the canine bracket, created as the tooth tends to tip
under the action of the distal driving force. With no anchorage loss, the
Behavior of Orthodontic Wire in Bending
TABLE 7-3. Influence of cross-sectional dimensions of the arch wire upon stiffness,
range, and strength in occiusogingiva! and faciolingual bending by a transverse force
.016 x .016
.016 x .022 ogt
.016 x .022
.017 x .025 og
.017 x .025 fI
.019 x .026 og
.019 x .026 fI
Wire size
.0215 x .0275 og
.0215 x .0275 Ii
*Given figures are not actual values, but, in each column, are compared to the value for the wire indicated
by the asterisk (*)
tog: Occiusogingival bending.
4f1: Faciolingual bending.
tooth, and with it the couple, move an average distance of 7 mm in the
retraction process, assuming approximately 20 mm (mesiodistally) between lateral-incisor and second-premolar bracket centers. Although a
detailed consideration of such a moving-load problem is too complex to
include here, the closer the load is to either anchorage site, the higher
are the relevant bending stiffness and strength and the lower is the
range. In the canine retraction example, none of the three parameters
changes with canine position as much as 30%; therefore, a reasonable,
approximate analysis may be undertaken with the canine located mid-
way through its intended displacement: at the midspan location between the anchorage sites.
elastic Behavior in Bending
fundamental, characteristic bending deformation is the rotation of
one beam cross-section relative to a neighboring cross-section. The beam
fibers that are farthest from the neutral surface of the member experience the largest bending strains and, if the relative rotation becomes
sufficiently great, the extreme fibers under tensile stress can rupture and
a progressive failure of the entire member by fracture may be initiated. If
Bioengineering Analysis of Orthodonfic Mechanics
the beam material is metallic and ductile, however, as is true of many
orthodontic, arch-wire materials, the stretched fibers can experience
substantial strain beyond their elastic limit without rupture, and thereby
enable the placement of permanent bends.
In plane bending of a straight beam the fiber strains have been noted
to increase linearly with distance from the neutral surface (see Fig. 7-5).
If the beam material obeys Hooke's law, then the fiber-stress pattern is
the same as that for the bending strains with the limit of Hookean be-
havior approximately coinciding with the generation of elastic-limit
stresses in the extreme fibers. Whenever the beam loading exceeds the
elastic-strength level, however, some extreme fibers are strained beyond
their elastic limit and, upon unloading, a portion of the relative rotation
remains and the member exhibits permanent deformation (with respect
to the previous, passive shape). The beam has been permanently bent.
In the fabrication by the practitioner of an arch wire from an as-received,
straight piece of wire, permanent bending deformation is widely distrib-
uted over at least the anterior portion of the wire as the arch form is
prepared. At any cross-section in the anterior region, the permanent
bend may be far from severe. On the other hand, the placement of a
loop in that wire requires localized bends that take the wire material
substantially beyond its elastic limit and, therefore, demands considerable ductility.
Experimental investigations of the bending behavior of structural
members have shown that often beam cross-sections remain plane (do
not warp) under load, even when the load produces a moderate amount
of inelastic action. In other words, the strain pattern remains linear well
beyond the elastic limit of the beam material and through substantial
cross-sectional rotations. For metals, however, the slope of the typical
stress-strain diagram decreases after the elastic limit is exceeded: hence,
taking a metallic beam beyond its elastic limit results in a nonlinear
bending-stress pattern such as that shown on the left in Figure 7-16.
Because of the linear relationship between fiber strain and the coordinate u, inelastic behavior and stress nonlinearity begin in the extreme
fibers and proceed toward the neutral surface as the load is increased
beyond the elastic strength. The portion of the cross-section incorporating the fibers not strained to the elastic limit is known as the elastic core,
as indicated in the figure. Note that even with inelastic behavior the
antisymmetry of the bending-stress pattern remains for the round and
rectangular beam cross-sections.
The sketches on the left in Figure 7-16 depict the rotational deformation and bending-stress patterns of the loaded (activated) beam. As verified by bench experimentation, deactivation from this state is tanta-
mount to superimposing a reversed, totally elastic loading on the
activated configuration. In other words, the resulting bending-stress
pattern upon partial unloading is obtained by, in effect, subtracting the
stresses of Figure 7-5 from those shown on the left in Figure 7-16. If
unloading is continued to its completion, the extreme fibers reach the
zero stress state before the load is totally withdrawn and subsequently
Behavior of Orthodontic Wire in Bending
FIGURE 7-16. Bending deformation and stress patterns associated with inelastic
bending: cross-sectional rotation and bending-stress pattern of the activated state (left)
and permanent rotation and remaining residual bending stresses following total
deactivation (right).
undergo a reversal of stress. A residual-stress pattern of the form shown
on the right in Figure 7-16 ultimately accompanies the permanent bend
induced. The pattern indicates a reversal in sense of the stresses (as
load-induced) in fibers distant to the neutral surface and the incomplete
unloading of fibers within the elastic core near the neutral surface. The
resultant of this residual-stress distribution may be expressed as four
longitudinal forces (two couples) that must exactly cancel one another in
the final reduction to yield a net internal bending couple M of zero
Typically in orthodontic therapy, the placement of first- and secondorder bends in the arch wire is followed by engagement of that wire. An
activation, then, intended to be elastic, is superimposed on prior inelastic action. Because the bend placements produce the residual-bending-
stress patterns just discussed and result in additional strain energy
"locked" into the wire, some alterations in structural properties have
occurred and the response to activation of the bent wire will be changed
somewhat in comparison to that of an otherwise identical, straight wire.
In general, the described inelastic action, upon unloading, results in
increased elastic range and strength and reduced ductility. The severity
of the permanent deformation, how localized or widely distributed it is,
and the longitudinal position and extent of that deformation with respect to the activating load and anchorage, are significant. Of substantial
importance also is the directional nature of the effects of bend placements and, therefore, the direction-sense relationships of the bends to
the activation. For example, if the permanent bend is second-order, the
responses during activation will differ, depending on whether the activating, superimposed bending action is in the same direction and sense,
in the same direction but opposite sense, or in another (e.g., the first-
Bioengineering Analysis of Orthodontic Mechanics
order) direction. Quantification of these property changes appears to be
virtually impossible, but further consideration of the phenomena is contained in the section that follows.
Orthodontic Wire Loops
Although the orthodontic arch wire is often activated in bending to
create faciolingual or occiusogingival tooth movement, routine mesiodistal displacements through arch-wire deformation cannot be accomplished directly because of the extremely high longitudinal stiffness of
wire. To reduce this stiffness at a specific location, a wire loop may be
incorporated in the arch, effectively providing a spring of manageable
flexibility between segments of the arch wire.
In general, the orthodontic wire loop may be bent directly into the
continuous arch wire, or a loop or spring may be fabricated from a piece
of wire and activated between a pair of neighboring teeth. The "legs" of
a wire loop extend occiusogingivally from the arch-form plane into the
"body" of the loop, the shape of which serves to primarily characterize
the loop. The loop having the simplest geometry is shown in Figure
7-17. Generally, this ioop is wholly contained in one plane, its body
gingival to the line of bracket slots, and its position usually within the
interbracket space between adjacent teeth. The loop shape is attained by
the placement of localized bends in a straight piece of wire; hence, the
wire material must exhibit substantial ductility to sustain such inelastic
strains without fracture. Again, the basic rationale for incorporating
loops in appliances is to gain mesiodistal flexibility, although some loops
simply serve as stops or as hooks along the arch wire.
A typical wire ioop is characterized by the size, shape, and material
7-17. A simple wire loop: passive (left) and activated by mesiodistal pulling
Behavior of Orthodontic Wire in Bending
FIGURE 7-18. Orthodontic wire loops: 'teardrop," '7," "L, ' and "box," from left to
from which it is fabricated, its overall occiusogingival dimension, its
as-formed geometry, and whether it "opens" or "closes" when activated mesiodistally. Several loop geometries are shown in Figure 7-18,
and the association of names such as "teardrop," "T," "L," and "box" is
usually made by viewing the individual sketches. The loop in Figure
7-17 and three of the four loops in Figure 7-18 are termed "opening
loops;" the area within the loop form tends to enlarge and many (or all)
of the bends tend toward reductions in curvature as the loop is activated
mesiodistally by pulling action. On the other hand, the same activation
of the box loop in Figure 7-18 results in a reduction of the area within the
rectangular geometry and two of the four bends tend toward increased
curvature; this loop is described as a "closing loop."
Several properties influence the mesiodistal stiffness of a wire loop.
Figure 7-19 serves to define this stiffness in terms of load and deformation parameters and, as indicated by the diagram, below the elastic limit
Activation a
FIGURE 7-19. The mesiodistal-pulling activation of a loop (left) and the associated
force-deformation pattern including the graphic definition of the pertinent stiffness
Bioengineering Analysis of Orthodontic Mechanics
the loop reflects the Hookean behavior of its material. Because the load-
ing of the ioop produces bending deformation, this stiffness is affected
by the wire material (through its modulus of elasticity E) and by the size
and shape of the cross-section (through the parameter I). The curvilinear length of wire within the ioop is also highly influential: the more
wire present in the ioop, the lower the stiffness. Also impacting stiffness
are the number and severity of bend placements, the procedures in
forming the bends and the overall loop geometry, and the relationships
between direction and sense of activation and those of the bend placements in fabrication. Strictly speaking, two mesiodistal stiffnesses must
be noted, inasmuch as mesiodistal activation may be pulling or pushing,
the latter depicted on the left in Figure 7-20; these stiffnesses are generally unequal due to the presence of residual-sfress patterns left by fabrication of the loop. Unfortunately, the influences on stiffness are too
complex to permit an analytic determination; desired quantifications
must be obtained experimentally.
Although the foregoing discussion pertains to mesiodistal loop stiffnesses, noteworthy in addition is the potential, multifaceted nature of
loop activation and the possible involvement of as many stiffnesses as
there are modes of activation. As can be seen on the right in Figure 7-17,
upon mesiodistal pulling activation the permanent bends at the base of
the loop tend to remain generally as bent (because the internal bending
couples there from mesiodistal activation are small) and a second-order
phenomenon appears. If appropriate compensation is not provided in
loop formation, second-order couples arise in the presence of inadequate wire-bracket clearances. The magnitudes of these tipping couples
depend heavily on the passive angulations of the loop legs with respect
to one another, the associated stiffness, the amount of mesiodistal activation, and the second-order clearances. Although not inherent in the
typical loop, first-order angulation can be incorporated in the leg fabrication to enable the generation of first-order-couple action upon engagement.
FIGURE 7-20. Loop activations other than by mesiodistal pulling: the opening of a
simple reverse-closing loop by mesiodistal pushing (left) and an occiusogingival
activation of a stepped "L' loop (right).
Behavior of Orthodontic Wire in Bending
Another mode of activation is introduced by fabricating a passive
"step," either occiusogingivally as shown on the right in Figure 7-20 or
faciolingually. The magnitude of the occiusogingival forces shown as
part of the activation force systems will depend upon the size of the
step, the related stiffness, and the amount of occlusogingival activation.
The loop containing a faciolingual step, or the loop formed in rectangular wire and passively exhibiting a third-order, relative angulation between leg ends, may undergo torsional as well as bending deformations
upon activation, and will be discussed further in Chapter 9. In summary, all of the individual stiffnesses are generally influenced by the
wire and loop properties already mentioned.
The previous section of this chapter dealt with inelastic bending and
the residual-stress pattern induced with the placement of a permanent
bend was noted. The typical orthodontic loop is a composite of straight
and bent portions of an initially straight segment of wire. An activation
of a ioop may tend to increase or decrease the curvature of a specific
permanent bend, and the sense of that activation apparently has an
influence on the associated stiffness related to the residual-stress pattern
left by fabrication (see Fig. 7-16). In fact, in identical mesiodistal activa-
tions, the closing-loop counterpart of the opening loop of Figure 7-17
has been shown to exhibit a greater mesiodistal pulling stiffness, even
though the closing ioop contains slightly more wire (Lane and Nikolai,
1980). Furthermore, the severity of the permanent bend, in terms of the
as-bent curvature of the beam axis at the bend, interacting with the
beam depth, may also impact stiffnesses, including those akin to firstand second-order effects.
Upon activation, much of the deformation seems to occur within the
straight sections of the loop while the permanent bends tend to retain
their bent shapes. To enhance flexibility, at locations of bends through
angles of 90° or greater within the loop, helices may be incorporated. In
the typical loop the localized mesiodistal and second-order stiffnesses
may be reduced by the presence of helices; apparently most influential is
the increase in curvilinear length of wire necessarily accompanying a
helix. Helices placed where the ioop legs blend into the arch form can
reduce the potential of a loop to tip teeth adjacent to it. Furthermore,
depending on loop geometry, the presence of a helix may affect other
stiffnesses as well. For example, the loop of Figure 7-17 depends somewhat for its activation on the opening of the angle between its legs.
Comparing it with the loop of Figure 7-21, the incorporation of the helix
in the loop body increases the mesiodistal flexibility while the helices at
the base of the loop in particular reduce second-order stiffnesses and the
tipping potential just previously noted.
The mesiodistal elastic ranges of a ioop are influenced, like the associated stiffnesses, by properties of the wire and the loop geometry; analogous to the straight-wire bending discussion, the impact upon a range
of the wire depth dimension in the plane of bending is not as great as it
is upon stiffness. A particularly important influence on loQp range is the
severity of the bend(s) in the body of the ioop. The free-body diagram of
Figure 7-22 shows the location and relative size of the critical, internal
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 7-21. An opening loop containing he/ices: passive (left) and activated by
mesiodistal pulling (right).
bending couple within a teardrop ioop. The maximum bending strains
induced on activation are found within a permanent bend where residual bending stresses also exist. Of concern is not only an activation possibly taking the wire material within this bend beyond the elastic limit,
FIGURE 7-22. A free-body diagram of one-ha/f of the loop of Figure 7-17 showing the
location of and an expression for the critical internal bending couple.
Behavior of Orthodontic Wire in Bending
TABLE 7-4. Typical values of stiffness and elastic range for stainless-steel wire loops
subjected to mesiodistal-pulling activation*
in loop
17 x 25
21 x 25
17 x 25
21 x 25
17 x 25
21 x 25
17 x 25
21 x 25
with Helix
From Engel, EL. (1977).
The occiusogingival height of all loops was approximately 6.5 mm.
None of the loops was stress-relieved following fabrication.
The above stiffness and range magnitudes are averages involving several vendors' wires.
but also that the bend placement has locally embrittled the wire mate-
rial, substantially reducing ductility within the bend. Accordingly, care
should be taken both to limit the severity of bends within the body of the
loop and to activate the loop to a state somewhat below its predetermined elastic range. Table 7-4 contains approximate magnitudes of stiffness and range associated with mesiodistal-pulling activation for several
common loops, but it is important to remember that both properties may
be influenced rather significantly by fabrication procedures and, therefore, variations in obtained property values for seemingly identical loops
prepared by different practitioners should be expected.
The discussions of this chapter have examined the concepts of structural
bending behavior and the application of the theory to orthodontic appliance members—the arch wire in particular. Although the bending problem for the actual, continuous arch wire is almost hopelessly complex,
reasonable modeling can result in a manageable problem and analysis.
Substantial care was taken in the development of the theory, but with
some voids necessary because of mathematical procedures beyond the
scope of the text; the thrust was toward understanding the parametric
influences on this most common form of appliance activation occurring
in orthodontic therapy.
The structural response of an arch-wire beam is dependent on sev-
eral geometric and mechanical properties, most of which may be selected and controlled by the practitioner. The ke.y parameter is stiffness
and, following its definition with respect to the specific bending prob-
Bioengineering Analysis of Orthodontic Mechanics
under analysis, the orthodontist must make a decision as to wire
material, cross-sectional size and shape, and a solid versus a twisted,
layered, or braided longitudinal format. Material ductility is an additional consideration if permanent bends are to be placed prior to activation and, in this regard, a superficial examination of inelastic bending
was undertaken. Consequences of superposition of elastic activation on
prior inelastic behavior received attention and led to a discussion of the
functions, characteristics, and behavior in bending of orthodontic wire
It is important to note once again that an American Dental Association specification standardizes the evaluation of the orthodontic arch
wire in both elastic and inelastic bending. Property values obtained from
the results of structural tests are quantified through use of engineering
bending theory. Perhaps this fact gives support to the developments
undertaken in Chapter 7 that are not only pertinent but also necessary to
the structural considerations of the example orthodontic mechanics discussed in Chapter 10.
E.L.: A force-activation comparison of retraction loops used with the
"018" and "022" bracket assemblies. Master's thesis, Saint Louis University,
Lane, D.F., and Nikolai, R.J.: Effects of stress relief on the mechanical properties
of orthodontic wire loops. Angle Orthod., 50:139—145, 1980.
Suggested Readings
American Dental Association Specificaton No. 32 for orthodontic wires not con-
taining precious metals. J. Am. Dent. Assoc., 95:1169—1171, 1977.
Andreasen, G.F., and Barrett, RD.: An evaluation of cobalt-substituted nitinol
wire in orthodontics. Am. J. Orthod., 63:462—470, 1973.
Andreasen, G.F., and Hilleman, T.B.: An evaluation of 55 cobalt substituted
nitinol wire for use in orthodontics. J. Am. Dent. Assoc., 82:1373—1375, 1971.
Andreasen, G.F., and Morrow, R.E.: Laboratory and clinical analysis of nitinol
wire. Am. J. Orthod., 73:142—151, 1978.
Brantley, W.A.: Comments on stiffness measurements for orthodontic wires. J.
Dent. Res., 55:705, 1976.
Brantley, W.A., Augat, W.S., Myers, C.L., and Winders, R.V.: Bending deformation studies of orthodontic wires. J. Dent. Res., 57:609—615, 1978.
Brantley, W.A., and Myers, C.L.: Measurement of bending deformation for
small diameter orthodontic wires. J. Dent. Res., 58:1696—1700, 1979.
Behavior of Orthodontic Wire in Bending
Burstone, C.J.: Variable modulus orthodontics. Am. J. Orthod., 80:1—16, 1981.
Burstone, C.J.: Application of bioengineering to clinical orthodontics. in Current
Orthodontic Concepts and Techniques. 2nd Ed. Edited by T.M. Graber and
B.F. Swain. Philadelphia, W.B. Saunders, 1975, Chapter 3.
Burstone, C.J., and Goldberg, A.J.: Maximum forces and deflections from orthodontic appliances. Am. J. Orthod., 84:95—103, 1983.
Burstone, C.J., and Goldberg, A.J.: Beta titanium: a new orthodontic alloy. Am.
J. Orthod., 77:121—132, 1980.
Creekmore, T.D.: The importance of interbracket width in orthodontic tooth
movement. J. Clin. Orthod., 10:530—534, 1976.
Goldberg, 1.' and Burstone, C.J.: An evaluation of beta-titanium alloys for use in
orthodontic appliances. J. Dent. Res., 58:593—599, 1979.
Goldberg, A.J., Vanderby, R., and Burstone, C.J.: Reduction in the modulus of
elasticity in orthodontic wires. J. Dent. Res., 56:1227—1231, 1977.
Jarabak, J.R., and Fizzell, J.A.: Technique and Treatment with Light-wire
Edgewise Appliances. 2nd Ed. St. Louis, C. V. Mosby, 1972, Chapters 2, 3.
Kusy, R.P.: On the use of nomograms to determine the elastic property ratios of
orthodontic arch wires. Am. J. Orthod., 83:374—381, 1983.
Kusy, R.P.: Comparison of nickel-titanium and beta titanium wire sizes to conventional orthodontic arch wire materials. Am. J. Orthod., 79:625—629, 1981.
Kusy, R.P., and Greenberg, A.R.: Effects of composition and cross-section on
the elastic properties of orthodontic wires. Angle Orthod., 51 :325—342, 1981.
Kusy, R.P., and Greenberg, AR.: Comparison of elastic properties of nickel-titanium and beta titanium arch wires. Am. J. Orthod., 82:199—205, 1982.
Lopez, I., Goldberg, J., and Burstone, C.J.: Bending characteristics of nitinol
wire. Am. J. Orthod., 75:569—575, 1979.
Nikolai, R.J.: On optimum orthodontic force theory as applied to canine retraction. Am. J. Orthod., 68:290—302, 1975.
Popov, E.P.: Introduction to Mechanics of Solids. Englewood Cliffs, N.J., Prentice-Hall, 1968, Chapters 6, 11.
Smith, JO., and Sidebottom, 0. M.: Elementary Mechanics of Deformable Bodies. London, Macmillan, 1969, Chapters 7, 8.
Thurow, R.C.: Edgewise Orthodontics. 4th Ed. St. Louis, C. V. Mosby, 1982,
Chapter 4, 5, 9.
Yoshikawa, D.K., Burstone, C.J., Goldberg, A.J., and Morton, J.: Flexure modulus of orthodontic stainless steel wires. J. Dent. Res.: 60:139—145, 1981.
Waters, N.E., Houston, W.J.B., and Stephen, C.D.: The characterization of arch
wires for the initial alignment of irregular teeth. Am. J. Orthod., 79:373—389,
Delivery of Torque by
the Orthodontic
force system transmitted by and through a structural or machine
member, capable of producing pure rotational displacement about a lon-
gitudinal axis, is known as torque. The resultant of a torsional force
system is a couple; therefore, the terms "torque" and "couple" are
sometimes used interchangeably. In a strict sense, and in the typical
engineering setting, however, torque is the more correct term to describe a distributed force system carried by a "shaft" from one location
to another within a structure or machine.
In orthodontic terminology, torque is often associated with the angulations of long axes of teeth and pertains to the positioning of root apices
with respect to crowns. Second-order torque was described in Chapter 5
as a force system used to alter a long-axis orientation in a faciolingual
plane that results in mesiodistal displacement of the root tip. Activated
uprighting springs exert a second-order torque, and these auxiliaries are
examined later in this chapter. Receiving principal attention herein,
however, are the structural considerations of third-order torque which,
with respect to the dentition, pertain to faciolingual root movement and
control. Although third-order torque may be delivered to an individual
dental unit to produce a wanted displacement, perhaps the most common occurrence is in the mechanics of retraction of the maxillary anterior or incisal segment. To reduce overjet, to upright procumbent anteriors, and to angulate incisors with respect to vertical requires that the
appliance deliver third-order torque. Third-order torque is often part of
the force system exerted on buccal segments to alter posterior arch
width, to cite another example.
In typical edgewise orthodontic mechanics, third-order torque is
generated when, in order to achieve bracket engagement, a rectangular
wire must be twisted about its longitudinal axis during the activation
process. The third-order, passive "mismatch" between wire and
bracket, as seen in a mesiodistal view, is created or possibly augmented
by a permanent deformation of the wire, by the use of pretorqued brack272
Delivery of Torque by the Orthodontic Appliance
or by a combination of the two. To generate third-order action
within a round-wire appliance, torquing spurs or loops are bent into or
affixed to the round arch wire and, as seen in a mesial or distal view,
they are passively angulated with respect to the long axes of the teeth to
be displaced. A round-wire torquing auxiliary, overlaying the main arch
wire from canine to canine, is used when appropriate in the incisal-segment retraction mechanics of Begg therapy (Begg and Kesling, 1977).
Such torquing spurs may be incorporated in a continuous arch wire as
The generation of torque at an activation site results in the delivery of
a torsional force system to the dentition. The action-reaction counterpart
of the desired torque is that exerted by the dentition on the appliance,
and the wire must carry it to the anchorage site(s). The quantification of
torque delivered, then, involves an analysis of the wire and, as with the
activated arch-wire beam, presents the analyst with an indeterminate
problem. Regardless of the number of anchorage locations, the number
of unknown torque resultants (couples) exceeds the number of available, moment-balance relationships of quasi-statics. The analysis must
make use of the known or measurable activating twist(s) that are necessarily performed to gain bracket engagement. Torque as the force system
generated and twist as the form of activating deformation are interrelated in the structural response of the loaded member. The complete
description of the third-order force system, paralleling the solution of
the bending problem, requires the augmentation of the equations of
quasi-statics by relationships incorporating torsional stiffness. Torsional
action, specifically in the anterior segment, may be generated by an
extraoral appliance as well as through intraoral mechanics. Intraoral torsional mechanics are covered in this chapter, but all extraoral-appliance
discussions are deferred to Chapter 9.
In the sections that follow, torsional theory is developed for the
straight, circular shaft and the extensions and modifications of that theory to accommodate the rectangular cross-section. Next, the theory is
discussed with respect to its applicability to the orthodontic arch wire.
Orthodontic mechanics of anterior-segment retraction are then detailed
with emphasis on the role of third-order behavior; both the edgewise
and round-wire approaches are examined. Because permanent bends or
twists are often placed prior to third-order activation, inelastic behavior
of metallic materials in torsion is explored and, subsequently, the superposition of an elastic activation on the residual-stress patterns of clinical
fabrication. Finally, loops and springs used in torquing mechanics as
well as other auxiliaries having torsional aspects to their activation-deactivation behavior are considered.
Structural Theory for the Straight, Circular Shaft
In engineering terminology a "shaft" is a slender member providing
mechanical means by which, through rotational motion about an axis,
Bloengineering Analysis of Orthodontic Mechanics
power may be transmitted from one location to another. Power is work
done per unit time and work requires a force system acting through an
accompanying displacement (see Chap. 4). The shaft displacement is a
rotation about its longitudinal axis and the resultant of the force system
producing or reacting to the displacement is a couple. Power is transferred to, carried along, and taken from the shaft; the principal force
system existing within the member is known as torque. Transient responses occur during start-up and shut-down of the mechanical system,
but while the shaft is turning at constant rotational speed, the input and
output torques (couples) balance one another and the shaft experiences
an unchanging, overall twist deformation. Early designs of shafts were
based on observations within simple experimentation. The typical shaft
had a circular cross-section associated with the relative ease of fabrication, the facility of attachment of pulleys and gears to the member, and
the character of the shaft motion.
A theoretical basis for the design of the circular shaft has been firmly
established through research. Force systems may be interrelated
through Newton's laws. Torque is the resultant of an internal distribution of shearing stresses on a shaft cross-section and is transmitted longitudinally within the member, unchanged between locations of forcesystem input or output. The torque-twist relationship is associated with
the stress-strain behavior of the shaft material in shear and with the
shaft dimensions, and is independent of the shaft speed under steadystate (constant rotational motion) conditions. Torque may also be transmitted to, from, and through a stationary shaft, and the arch wire may
perform the mechanical function of such a shaft. Although the character
of shaft loading differs substantially from that of a beam, analogies may
be drawn between bending and torsional responses. Influential in their
structural responses are cross-sectional size and shape, longitudinal
dimensions, and the mechanical properties of the beam and shaft materials. Accordingly, of interest and principal subjects of the developments
to follow are torsional stiffness, range, and strength.
The essential features of the torsional behavior of a shaft may be
obtained from a rather simple example. Figure 8-1 illustrates a solid,
circular member of radius R and deformable length L. The member is
fixed against all movement at its left end and any bending deformations
FIGURE 8-1. Side view (left) and end view (right) of a straight, solid, circular shaft.
Delivery of Torque by the Orthodontic Appliance
and stresses (caused by shaft weight, for example) are assumed negligi-
ble (or such effects can be superimposed on the torsion analysis). The
line MN is on the outer surface and, in the passive state, is parallel to the
axis of circular symmetry of the member; point 0 is on the axis of geometric symmetry. The end cross-section is a right cross-section; the
angle MNO is a right (900) angle.
Figure 8-2 depicts the deformation of the shaft, externally visable,
following the application of torsional loading to its "free" end. The resultant of the load is a couple T, which represents torque and distinguishes this force system from an applied bending couple. Because the
shaft is not rigid and only its fixed end is restrained from movement, a
rotational deformation is distributed along the length of the member as
characterized externally by (1) the distorted shape of the generator (labeled MN' as activated); (2) the unaltered, straight-line configuration of
the shaft axis; and (3) the rotation of the reference, free-end radius to the
as-activated position ON'. The radii remaining straight lines and the
shaft axis undeformed by the loading are a combined result of the characteristics of the load and the symmetry of the member; as the magnitude of the applied couple is increased, indefinitely until fracture begins,
the pattern of deformation remains that shown in the figure. The
amount of "twist" of the free end (with respect to the fixed end) of the
shaft, the angle NON' in Figure 8-2, is, however, directly related to the
magnitude of the couple loading. The evidence of longitudinal shear
resulting from torsional activation is obtained by imagining the unwrapping of the outer surface, "skin," of the deformed shaft. The flattened
skin is rectangular, having dimensions L by the shaft circumference, and
is pictured in Figure 8-3. Corresponding to the change in inclination of
the generator, any small area of the skin that is passively rectangular
deforms into the shape of a parallelogram, as shown, upon activation of
FIGURE 8-2. The shaft of Figure 8-1 subjected to a torsional couple applied at its
Bioengineering Analysis of Orthodontic Mechanics
Shaft circumfererice
FIGURE 8-3. The outer surface of the shaft of Figure 8-2; the shaft "skin" has been
unwrapped and flattened into a plane, but not otherwise further deformed.
the shaft; the demise of the right angles of the rectangle is the indication
of shearing strain.
In the example under discussion the load is the torque input at the
right end of the shaft; the torque is transmitted the length of the member
and into the left-end support. A free-body diagram of the entire shaft is
shown on the left in Figure 8-4; only the single couple loads the member
and a moment balance for the static shaft yields a resultant, responsive
couple of the same magnitude T at the fixed end. An analysis of any
FIGURE 8-4. Free-body diagrams of the shaft of Figure 8-2; the entire shaft of length L
(left) and a portion of the shaft of arbitrary length t (right) measured from the free end.
Delivery of Torque by the Orthodontic Appliance
portion of the shaft, such as that shown on the right in Figure 8-4,
indicates that the resultant of the internal force system on every right
section is a torsional couple equal to T. Further indication that the internal torque is the same on all right cross-sections between the shaft ends
is the constant slope of the deformed generator MN' in Figure 8-3. (The
internal bending couple in a beam changes from one right cross-section
to another, when the loading includes transverse forces, due to the
moment-arm changes; the shaft, however, is directly analogous to the
beam in pure bending.) The deformed generator has an apparent curvilinear shape as viewed in Figure 8-2, but, in the general torsion problem,
it is made up of a series of straight lines in the "plane skin" view with
sites of slope changes corresponding to locations of external torsional
In between torque input-output locations, relative rotational displacements of cross-sections exist; the amount of relative twist is directly
proportional to the longitudinal distance between cross-sections. In the
example shaft, the absolute rotation of the left-end cross-section is zero
as constrained; the relative rotations are simply additive along the shaft
length (because no intermediate external couples are present) and sum
to the rotation of the free-end cross-section. This free-end displacement,
indicated by the angle NON' in Figure 8-2, is the twist produced by the
input couple T; it is symbolized by the Greek letter 4 (phi) and is analogous to the bending-deformation angle 0. Note that torsional loading of
the straight, circular shaft warps no cross-section of the member.
The character of unit deformation (strain) on the surface of the shaft
is determined by comparing the active and passive configurations of a
skin element. Consider a small rectangle, with its sides parallel and per-
pendicular to the shaft axis, as the passive element. Torsional loading
displaces the element circumferentially and distorts it with negligible
changes in the side lengths. As deformed, the element takes the shape
of a parallelogram, indicative of the presence of shearing strain. The
passive and activated elements are shown in Figure 8-3 and, in an enlarged view, in Figure 8-5. As noted in Figure 8-5, the angular distortion
of the element (the angle between the passive and active configurations
of the generator at a specific, cross-sectional location) is the localized,
surface shearing strain. Shearing strain was initially defined in Chapter
3; Figures 8-5 and 3-14 are similar, although exhibiting rotational and
straight (linear) shear, respectively. For a shaft experiencing only torsional loading, all deformed generators are similar, indicating axial symmetry in the surface shearing-strain pattern. For the example problem
under study, the surface shearing strains are everywhere equal and are
by the magnitude (in radians) of the angle NMN' in Figures 8-3
and 8-5.
For any shaft cross-section, the axial symmetry results in internal
shearing strains varying only with the radial coordinate. The circumferential shearing strains for any specific radial coordinate r are all equal
and the maximum shearing strains exist at the shaft surface (where
Bioengineering Analysis of Orthodontic Mechanics
strain (es)max
FIGURE 8-5. Small element of the outer surface of a straight, circular shaft shown
enlarged (from Figure 8-3) and exaggerated in passive (dashed) and activated (solid)
positions and shapes.
r = R). Because torsional loading does not deform the shaft axis, where
r = 0 no shearing strains exist. Moreover, for the circular shaft, since
radii from the shaft axis remain straight with activation (ON becomes
ON' in Figure 8-2), no radial shearing strains exist and the magnitude of
circumferential shearing strain at a point is directly proportional to the
coordinate r.
Shearing strains at a point generally vary with the passive orientation
(edge directions) of the rectangular element there. Shearing strains are
always accompanied by shearing stresses. Shearing stresses of equal
magnitude exist at a point at right angles to one another in pairs, and the
stresses of interest in the arch-wire torsion problem are longitudinal and
circumferential; note the element orientation in Figure 8-5. For a crystalline material, paired shearing stress and strain are proportional to one
another up to the elastic limit in shear. A stress-strain diagram in shear
for a ductile, metallic material is depicted in Figure 8-6; this is a repetition of one plot in Figure 3-19. The material stiffness in shear, also
known as the modulus of rigidity, is the slope of the linear portion of the
curve and is usually symbolized by G. The stiffness G and the modulus
of elasticity in tension or compression are interrelated; for metals and
alloys G is typically about O.4E. Also for ductile metallics, the elasticlimit stress in shear is approximately one-half of its counterpart in tension.
Delivery of Torque by the Orthodontic Appliance
Elastic limit
in shear
Shearing Strain
FIGURE 8-6. Characteristic plot of stress versus strain in shear for a metallic material.
The sketch in Figure 8-7 is of one side of a three-dimensional, internal, material element of the shaft showing shearing stresses acting on
areas appearing in edge view from this perspective, which is radial and
the same as that of Figure 8-5. (The reference line shown is parallel to the
shaft axis.) Because no normal stresses exist on the element, it is said to
be subjected to pure shear. (Tensile and compressive stresses do exist in
the shaft, but on elements having orientations differing from that of
Figure 8-7.) When the torsional loading of the shaft nowhere causes
stresses exceeding the elastic limit, the shearing strains and correspond-
ing shearing stresses are proportional to one another at every point
within the shaft.
Linear shearing action may be compared to that of a pair of scissors.
The shearing strength of a shaft tends to resist the attempt of torsional
loading to rotate each cross-section with respect to its neighbor. Another
example of rotational shearing-stress development is within the adhesive attaching the bonded bracket-pad to the tooth when a second-order
couple is transmitted from arch wire to the bracket and into the crown.
Figure 8-8 is a view along the shaft axis into an internal cross-section
through which a torque T is transmitted. Circumferential shearing
stresses are shown on the exposed areas of typical, diametrically opposed shaft elements. (A corresponding elemental area appears in edge
view in Figure 8-7; the edge labeled a-a is positioned perpendicular to
Bioengineering Analysis of Orthodontic Mechanics
to the
FIGURE 8-7. Equal circumferential and longitudinal shearing stresses exerted on an
internal element of the activated shaft.
the shaft axis.) Each elemental area may be termed
representing a
very small portion of the total cross-sectional area A. The resultant
shearing force on the small area M has the direction and sense of the
shearing stress and is equal to the product (M); this elemental force
may be symbolized as 6F5. The elemental forces exerted on the two elemental areas shown are of the same magnitude and direction, but are of
opposite sense, due to area location with respect to point 0 (the end
view of the shaft axis). These two forces form a couple of magnitude
The sum of all such couples over the entire cross-section is the
resultant internal torque T.
Deli ve,y of Torque by the Orthodontic Appliance
FIGURE 8-8. An end view into an internal cross-section of the shaft of Figure 8-2
showing circumferential shearing stresses on two elemental areas.
The torque transmitted through the right cross-section of Figure 8-8
may be initially expressed as
sum of
Refinement of this formula requires knowledge of the manner in which
the shearing stress varies through the area A. Figure 8-6 shows a singlevalued relationship between stress and strain and, because this shearing
strain varies only with radial location on a specific cross-section, shearing stresses are likewise independent of circumferential position of the
elemental area. For totally elastic behavior, stress and strain are proportional to one another and, therefore, the variation with r of the stress as
well as the strain exhibits the pattern of Figure 8-9. Mathematically expressed, the pattern indicates
= (ss)max:R
Bioengineering Analysis of Orthodontic Mechanics
FIGuRE 8-9. Linear relationship between radial position and magnitude of
oircumferential shearing stress on the cross-seofion of Figure 8-8 for totally elastic
Substituting Equation 8-2 into Equation 8-1 yields
[sum of (r2&A)]
resulting in the summation now associated solely with the cross-sectional area. The sum is the second moment of area with respect to the
shaft axis and, for the solid circular cross-section, is equal to
This geometric parameter is known as the polar second moment of area,
is often symbolized by the letter I, and is analogous to the second mo-
ment of area I, which appears in the bending analyses of Chapter 7.
21.) In terms of
the maximum shearing stress, occurring at the outside surface of the
(Referring to Figure 7-10, for circular cross-sections, J =
shaft, and for totally elastic material behavior, Equation 8-1 can now be
written as
Delivery of Torque by the Orthodontic Appliance
An analogous expression, explicit in the internal bending couple M, was
generated in Chapter 7; the two equations are similar in form.
The structural response of a shaft in torsion is characterized by plots
of internal torque versus the relative rotations of cross-sections at the
ends of shaft segments through which torque is transmitted unchanged.
For the simple shaft example of Figure 8-2 the internal torque is constant
throughout the entire length L (and equals the applied torque) and the
left-end cross-section cannot move or deform. Accordingly, the appropriate plot, given in Figure 8-10, depicts the applied torque in terms of
the free-end twist 4). The slope of the linear portion of the graph is a
torsional stiffness and a formula for this parameter may be readily derived. Needed are the stress-strain relationship for the shaft material in
shear below its elastic limit from Figure 8-6
= G(e5)
and Equations 8-2 and 8-5 previously developed. Referring to Figures
8-2, 8-3, and 8-5, and noting that the arc = length NN' is small in comparison to L, NN' can be equated not only to R4), but with negligible
error also to the product of L and the angle NMN; this yields the relationship
R4) = L =
Angle of Twist 4'
FIGURE 8-10. Plot of activating couple versus free-end twist angle for the shaft of Figure
Bioengineering Analysis of Orthodontic Mechanics
The formula sought, the ratio of torque to twist, obtained through elimi-
nation of shearing stress and strain from the equations noted, is
Torsional stiffness is seen to depend directly upon the shaft radius and
the modulus of rigidity of the material and inversely upon a characteristic length. Again, note the similarity between this formula and its counterpart in the bending developments of the previous chapter.
The coordinates of the elastic-limit point of the plot in Figure 8-10 are
critical twist angle and the critical torque. In general, the critical crosssection of an activated shaft is that where the internal torque is a maximum and, typically, a critical shaft segment exists. In the example shaft
of Figure 8-2 all cross-sections are critical and the critical segment is the
entire shaft. If a continuous arch wire is delivering third-order action
(torque) to the incisors, the critical "shaft" segments are generally between the canines and the lateral incisors. Under critical conditions, the
maximum shearing strains in the shaft are the elastic-limit values for the
shaft material. The critical twist as described, then, is the elastic range
for the shaft and, from Equations 8-6 and 8-7, its formula may be
The maximum elastic twist of the critical segment is seen to depend on
mechanical properties of the shaft material, the shaft radius, and the
length of the critical segment. Once again, the analogy may be drawn
between the bending and torsion analyses; similar parameters of the
structural member exert like influences on the elastic range.
The critical torque is that magnitude of internal torque corresponding
to the generation of elastic-limit stresses in shear at the shaft surface of
the critical segment. Through a moment balance, this level of internal
torque may be related to the system of torsional couples applied to the
shaft. The critical torque may be directly expressed, then, by substituting the elastic-limit value for the maximum shearing stress in Equation
In the example problem, because the entire shaft is the critical segment
and the internal torque is everywhere equal to the applied couple at the
free end of the shaft, the value obtained from Equation 8-10 is the elastic
strength of the shaft in torsion.
Delivery of Torque by the Orthodontic Appliance
Extension of Theory to Shafts Having Rectangular Cross-Sections
Although perhaps not of direct interest to many edgewise practitioners,
the foregoing development of torsional theory for circular shafts was
undertaken for three reasons:
1. The axial symmetry, not only in geometry but also in loading and
structural response, leads to the least complex formulas for torsional
stiffness, range, and strength.
2. This theory provides a basis from which to generate corresponding
relationships for the shaft having a rectangular cross-section.
3. Third-order force systems cannot be directly transmitted between
round wire and bracket, but can be delivered from round wire to
crown by means of torquing spurs and, therefore, circular-shaft theory has application to specific orthodontic appliances.
In extending the theory to admit rectangular cross-sections, contributions from the principal influencing parameters—shaft material, length
dimension, and cross-sectional size—might be expected to be similar to
those exhibited in the circular-shaft formulas. Material properties (elastic-limit strain and stress and modulus of rigidity), when appearing in
the equations, each did so to the first power (direct or indirect proportionality); the shaft radius or diameter, however, contributed in a nonlinear manner in two of the three formulas. In this chapter the dimensions of the rectangular cross-section are symbolized by b and c where,
by definition, c is less than or at most equal to b. With bending deformation, the response of the rectangular wire depends on the direction of
activation; in torsion no such interdependence exists, but one of the two
cross-sectional dimensions of the rectangular shaft will be seen to be a
more dominant influence than the other.
The first analyses of rectangular shafts in torsion were carried out
experimentally and by analogy, and were deemed reasonably accurate
only when the ratio of b to c was large. In time an exact mathematical
solution was obtained, general in terms of b-to-c ratio but highly complex in form. A third approach was that of empirical adaptation, be-
ginning with the form taken from the circular-shaft derivations and
using experimentation to complete the formulas. The results of the ex-
perimental studies enabled the clarification of how the individual
dimensions b and c should appear in the equations and the quantifications of dimensionless coefficients included in the formulas to generalize them.
The empirically determined formulas for stiffness, range, and
strength for the straight shaft having a rectangular cross-section, analogous to Equations 8-8 through 8-10, are as follows:
Bioengineering Analysis of Orthodontic Mechanics
The coefficients alpha (a) and beta (/3) have values that are dependent on
the ratio of b to c; the relationships are presented graphically in Figure
8-11 and representative coefficient magnitudes are given in Table 8-LBoth coefficients are seen to increase monotonically, and, as the b-to-c
ratio becomes large, they individually approach as an asymptote the
value of one-third. A cursory examination of Equations 8-11 suggests
that square (b = c) shafts are stronger and exhibit higher torsional stiffness than "ribbon" shafts (b >>c) having the same cross-sectional area.
The cross-sectional stiffness parameter, analogous to I for the circular
shaft, is seen to be proportional to b(c3). In general, the smaller of the
two cross-sectional dimensions, c, seems to be the more influential, but
the contribution of the b-to-c ratio also has some impact. The materialand length-parameter influences are assumed, by the chosen forms of
the empirical expressions, to be the same as for the circular shaft; the
determination that these parameters do not impact the values of a or /3,
for a specific ratio of b to c, ensures the validity of that assumption. Also
note that the individual influences of the two cross-sectional dimensions
depend upon the plane of bending when the member is subjected to
transverse (flexural) loading; in torsion, the only load "direction" is longitudinal and, therefore, the cross-sectional dimensions of the shaft affect a specific dependent parameter (e.g., elastic range) in a singular
Ratio of b to c
FIGURE 8-11. Plots of the coefficients and /3 as functions of the ratio of crosssectional dimensions of the rectangular shaft.
Delivery of Torque by the Orthodontic Appliance
TABLE 8-1. Coefficient values for specific ratios of cross-sectional dimensions for the
rectangular shaft
The complete mathematical solution to the rectangular shaft problem
provides, in addition to expressions for stiffness, range, and strength
within the elastic limit of the shaft material, the strain and stress patterns throughout the member. Several points of interest arise from this
theoretical development and from experimental observations. Associated with the absence of axial symmetry in the geometry are the appear-
ance of warping of the passively plane cross-section under torsional
loading and a complex shearing-strain pattern on the noncircular crosssection. A portion of the strain pattern is exhibited in Figure 8-12; for
linear elastic behavior the shearing-stress pattern is identical to
FIGURE 8-12. Typical cross-section of a rectangular shaft. Characteristic distributions of
shearing strains along major and minor axes of geometric symmetry and a diagonal
associated with torsional loading are antis ymmetric.
Bioengineering Analysis of Orthodontic Mechanics
Strains are seen to be zero at the center of the cross-section and at each
of the four corners; the corners remain right angles when the shaft is
loaded. The shearing strains change in magnitude nonlinearly from the
center (axis) point outward; the patterns are antisymmetric along any
line passing through the center point and the cross-section boundary.
Shown in the figure are the strain patterns along the axes of symmetry in
the plane of the cross-section and a diagonal through a corner. The
maximum values of shearing strain on the cross-section occur at the
outer-surface points that are along one axis of symmetry; these are the
closest surface points to the shaft axis.
App'ication of Shaft Theory to the Orthodontic Arch Wire
analysis of the behavior and effects of an orthodontic arch wire in
the presence of third-order activation can be a formidable bioengineering problem. To the foregoing theoretical developments and applicable
quasi-static relationships must often be added complications imposed by
the existence of several input and output torques and the curvature of
the arch-wire "shaft." When the entire, continuous arch wire is involved
in third-order mechanics, use of the model developed in Chapter 6 can
be both reasonable and prudent. Combining the dental units into segments reduces the number of force-system components to the principal
contributions. In general, the pure torque input to a curved shaft, as it is
transmitted longitudinally through the member, is modified gradually
and a bending component appears. Considering the block-U-shaped
wire model, a third-order activation in the anterior section, for example,
provokes a bending response in the posterior sections, arising from the
900 change in longitudinal orientation between anterior and posterior
portions of the model arch.
With the assistance of the arch-wire model, the following problems
of interest may be examined: (1) the torquing of one tooth with a
straight-shaft transmission of load to anchorage; (2) unilateral or bilat-
eral third-order activation of the terminal molar(s) or posterior segment(s) with, possibly, some response located in the anterior segment;
and (3) perhaps of greatest interest, the torquing of four incisors or six
anterior teeth as a segment, with midsagittal-plane symmetry present
and responses in the posterior segments.
Isolation of the Activating Torque
A third-order couple is almost never exerted on a tooth in the total absence of other force-system components. This torque can be effectively
transmitted to the crown by a facial or lingual arch or, in some instances,
by a transpalatal bar. Pure third-order loading produces a center of rotation coincident with the center of resistance and, therefore, a crown-displacement potential toward the facial or lingual, depending on the sense
Delivery of Torque by the Orthodontic Appliance
of the torque. If a faciolingual driving force is not part of the initial
activation, then the combination of appliance-crown contact and bending stiffness of the appliance will result in the appearance of a faciolingual force tending to restrain the crown. Furthermore, an occlusogingival force may also be present, resulting from a "secondary response" to
be described in the following subsection. For purposes of analysis, however, it is both possible and practical to examine the effects of thirdorder, active torque upon the arch wire and any existing anchorage, in
the absence of other contributions to the force system at the activation
site(s), and subsequently superimpose the influences of the remaining
components of the total force system.
The couple transmitted from the appliance, reflected in two-point
crown contact and a resulting pair of forces in the mesiodistal plane of
the tooth, is associated with structural characteristics of the appliance
and the amount of third-order activation with respect to passive configu-
rations. In conventional edgewise mechanics, permanent localized
twists of the rectangular wire, placed prior to activation, create the potential for third-order action. Recently, pretorqued brackets have become widely available. Third-order, anterior torque may also be activated extraorally, as noted in Chapter 9. Interbracket twists or brackets
pretorqued at different angles may be used to influence the distribution
of torque within a dental segment. As noted earlier, Begg therapy uses a
somewhat different torquing format to be contrasted, subsequently,
with that of the twisted, rectangular wire; once again, however, the
couple loading is produced through an angular deformation. To the
point, regardless of the approach or treatment philosophy, the creation
of third-order torque occurs as a result of a particular form or component
of activating deformation; hence, it is possible, and also contributory to
the relative ease of overall analysis, to isolate torsional action and its
effects from the total appliance load.
Responses of the Appliance and the Dentition to Torsional Activation
third-order activation of an orthodontic appliance results in
action-reaction counterpart couples exerted on the dentition and appliance at the site(s) of the activation. The load on the appliance is transmitted through it to the support (anchorage) location(s). The anchorage
responses are described in format through application of the equations
of quasi-statics to the appliance. A single moment balance is initially
employed to relate load and response, but individual forces as well as
couples may appear with couple loading, due to wire curvature, and
necessitate the added use of force balances. In this regard, separate considerations of primary and secondary responses are undertaken here, with
primary responses examined first.
An active, third-order couple may be generated against an individual
tooth within the dental arch. The reaction is a single torsional couple
exerted on the appliance, and this torque is transmitted to anchorage,
ordinarily through an arch wire. If the activation site is a terminal molar,
Bioengineering Analysis of Orthodontic Mechanics
1torque T
FIGURE 8-13. A straight shaft restrained against rotation at both ends and subjected to
torsional loading at an intermediate cross-section.
or if a segmented arch wire is engaged, a single anchorage site is possible (although several adjacent teeth may contribute). The basic, quasistatic analysis through a moment balance simply requires that the net
activation and anchorage torques be equal in magnitude but of opposite
sense (assuming negligible wire curvature). Figure 8-2 is applicable here;
the fixed end represents the anchorage. Alternatively, the anchorage
may be split or divided with the torsional load partitioned and carried by
the wire to support sites on either side of the activation location. Such a
straight shaft is depicted in Figure 8-13 and the given equation is the
result of the moment balance. If the active torque is located midway
between the anchorage sites, the responsive torques are equal if no variation in geometric or material characteristics exists from one end of the
shaft to the other. Otherwise, twist-deformation equations must be used
to interrelate the responsive torques.
The primary response to an incisal or anterior segment activated in
torsion is divided between the posterior segments when a continuous
arch wire carries the reactive load distally. Often the entire force system
as well as the geometry exhibits symmetry with respect to the midsagittal plane. The sketch on the left in Figure 8-14 shows the right half of the
activated arch-wire model in buccal view and the essence of the force
F3 = F5 + F6
FIGURE 8-14. Free-body diagrams of the buccal view of an arch-wire model subjected
to anterior, torsional loading. Posterior-segment response shown in its resultant form
(left) and as distributed among dental units (right).
Deliver,' of Torque by the Orthodontic Appliance
system exerted on it: a third-order couple against the half anterior seg-
ment and an equal, but opposite, second-order couple from the posterior
segment. On the right in Figure 8-14, the posterior response is shown in
some detail, including occlusogingival forces, which must obey an oc-
clusogingival force balance, and diminishing second-order responses
from individual dental units within the posterior segment. The distribution of torque within the anterior segment is considered subsequently.
Another example of third-order mechanics exhibiting midsagittalplane symmetry is that of bilateral torquing of the terminal molars or the
buccal segments. Figure 8-15 illustrates transverse views of two activated appliances from a posterior perspective. On the left is the archwire model, sketched as if the anterior teeth are not engaged. The bilateral torsional activation results in bending deformation of the anterior
portion of the arch, exaggerated in the absence of anchorage response
there. On the right is shown a transpalatal bar engaging the molars,
contacting the lingual crown surfaces. If the bar is very stiff, under bilateral torque no response of consequence will exist between it and the
palatal vault. In both sketches the moment balance is inherent in the
activation; the couples on the left and right sides of the arch are equal in
magnitude, but opposite in sense to one another.
As noted in Chapter 7, when the responsive force system in bending
is distributed over several individual dental units, that distribution is
nonuniform with the tooth or teeth nearest to the activation site providing the largest portion(s) of the response. The same is true in torsion.
For example, if the first molar on the left side is torqued, the second
premolar and, if present, the second molar are the principal anchorage
units. If the buccal segment on the left side is alone subject to third-order
activation, and a continuous wire engages all three segments, the response is distributed principally in the anterior segment, and the intensity of the distribution decreases with curvilinear distance along the wire
from the activation site. The free-body diagram of the model arch wire
from a posterior perspective is shown in Figure 8-16. Torque in the buc-
FIGURE 8-15. Bilateral torsional activation of the buccal segments posteriorly viewed;
reactions against an arch wire (left) and a transpalatal bar (right).
Bloengineering Analysis of Orthodontic Mechanics
FIGURE 8-16. Free-body diagram from a posterior perspective of a continuous arch wire
subjected to unilateral torquing associated with activation of the left buccal segment.
segment is transformed into bending potential in the anterior region,
resulting in occlusogingival force responses there and, in the absence of
substantial bracket-wire clearances, second-order couples. If any internal force is carried beyond the anterior segment, the remaining bending
potential becomes torsional, with further change in longitudinal direction, and is delivered to the right-side posterior segment.
On the left in Figure 8-17 is a free-body diagram of the transpalatal
bar with right-side, unilateral activation. With this sense of the applied
torque, the response may be partly adjacent to the activation site and
exerted by the palate in opposing potential rotation of the bar. On the
other hand, because only pressure (no tension but, perhaps, some small
amount of frictional shearing force) can be generated between bar and
FIGURE 8-17. Unilateral, right-side couple loading of a transpalatal bar and the
response as viewed from the posterior: clockwise activation (left) and counterclockwise
activation (right).
Delivery of Torque by the Orthodontic Appliance
palate, reversing the sense of the activating couple results in a different
response format, shown on the right in Figure 847.
Two forms of secondary response exist as the result of third-order
activation of an incisal or anterior segment. From a buccal perspective
the arch-wire model resembles a beam, cantilevered from the posterior
segments. Figure 8-18 (left) depicts the activated configuration of such a
beam, loaded by a couple at its "free end." The sketch represents the
torqued arch wire well except the deformed shape suggests that no constraint is present to prevent occlusogingival displacement of the anterior
segment. Such constraint is, in fact, provided by the anterior brackets.
Accordingly, consider the arch wire engaged in the posterior segments
and, before undergoing third-order activation, the anterior segment is
aligned occlusogingivally with the bracket slots. Accompanying the
torque from the twist to gain bracket engagement is an anterior intrusive
potential. The complete free-body diagram from a buccal perspective is
shown in the center of Figure 8-18, together with the action-reaction
counterpart force system against the maxillary anterior crowns in the
application of lingual root torque on the right. Note that changing the
sense of the anterior torque reverses all aspects of the response shown in
the figure.
Activation of the anterior torque provokes another secondary response which must be viewed from an occlusal perspective. Suggested
on the left in Figure 8-19, resulting from the combination of loading and
wire curvature, is a tendency for the posterior sections of the arch wire
to assume buccally displaced positions. The associated, free-body diagram from an occlusal perspective is shown on the right in the figure;
the action-reaction counterparts of the transverse forces induced tend to
widen the dental arch in the posterior in the absence of arch-wire constriction before activation. The same arch-widening potential is present
for either sense of the third-order action in the anterior portion of the
Secondary responses exist in another example previously examined,
that of bilateral activation of torsional couples in the posterior segments,
if the arch wire also engages the anterior segment. The twist activations
produce bending potential in the anterior portion of the wire as noted
previously in Figure 8-15, left. Resistance to immediate achievement of
that anterior deformation is provided by bracket-wire engagement. In
this example, two activation sites must exist and the internal force sys-
segment support
FIGURE 8-18. Buccal-view diagrams of the arch wire and anterior teeth with third-order,
anterior activation. Vertical displacement potential associated with the torquing action
(left) and a free-body diagram (center) together with crown loading (right).
Bloengineering Analysis of Orthodontic Mechanics
FIGURE 8-19. Occlusal-view diagrams of the arch wire associated with anterior,
torsional loading: the posterior, buccal-displacement potential (left) and the free-body
diagram assuming no initial, transverse, arch constriction (right).
tems resulting from third-order loading on both sides of the arch are
carried mesiafly by the arch wire toward the midline. A free-body diagram of the wire, viewed from the posterior, is illustrated on the left in
Figure 8-20. The canines in this example are assumed to be within the
posterior segments. Occiusogingival, responsive force systems are exerted by the incisal pairs and, to balance the system, forces must also be
induced in the buccal segments. The force system exerted against one
maxillary incisal pair is shown on the right in Figure 8-20. Note that
changing the senses of the posterior-segment torques reverses the pattern of the bending potential in the anterior section of the wire and the
senses of the secondary responses.
FIGURE 8-20. Posterior views of bilateral, buccal-segment torquing with anteriorsegment appliance engagement: free-body diagram of the arch wire (left) and the force
system transmitted to an incisor pair (right).
Deliveiy of Torque by the Orthodontic Appliance
Structural influences on Active and Responsive Force Systems
Shaft stiffness is dependent on material, cross-sectional shape and
size, and longitudinal dimension. Examined first in this subsection are
cantilevered wire "shafts" within the context of the previously introduced arch-wire model. Figure 8-21 shows schematically a rectangularwire segment subjected to an activating torque near one end, anchored
at the other end, and having an effective length L. In conventional,
edgewise mechanics, the potential for torsional action is achieved by
placing permanent, third-order twists adjacent to the anchorage sites
and/or through use of "pretorqued" brackets (having slots angulated
with respect to the passive wire) at the activation sites. Superficially,
perhaps the simplest problem is the torquing of a terminal molar with
anchorage provided principally by the remainder of the posterior segment. The initial value of the molar couple is the product of the activating twist (within the elastic range) and the torsional stiffness of the wire
segment. In estimating this stiffness, the nonrigidity of the anchorage
unit must be considered; in effect, the real characteristic length L of the
straight segment is greater than the interbracket distance between the
molar and the adjacent tooth. Note in addition that distribution of the
responsive couple becomes an important consideration when the active
and resultant responsive torques are equal in magnitude, and the tooth
adjacent to the molar is smaller in root-surface area.
To approach a uniform distribution of the response among the sev-
eral, available, posterior teeth (in the absence of discrete, permanent
twists between pairs), the potential for the wire between the individual
FIGURE 8-21. The cantilevered, rectangular-wire shaft modeL
Bioengineering Analysis of Orthodontic Mechanics
units of the anchorage to twist must be minimized. Therefore, within
the anchoring segment the torsional stiffness of the engaged segment of
wire must be as large as possible. Furthermore, an attempt to isolate or
contain the anchorage, such as not to involve the anterior teeth, requires
that a portion of wire between the posterior and anterior teeth exhibit
high torsional and bending flexibility. (Termination of the activated wire
segment between the anterior and posterior dental segments is one
obvious way of isolating the anterior teeth from any effects of molar
In the presence of midsagittal-plane symmetry, the analysis of incisal- or anterior-segment torquing may be accomplished by considering
either side of the arch. When the wire is "cut" at the midline, it becomes, in effect, a pair of mirror-image cantilevers. Assuming for this
discussion that the four incisors are torqued as a unit, each cantilever is
supported by a posterior segment with a canine at the mesial extent of
the anchorage. The permanent twist in the cantilevered wire segment
(in the absence of pretorqued brackets) is placed immediately mesial to
the canine bracket. The product of torsional stiffness of the cantilever
and the activating twist yields the third-order couple delivered initially
to the incisal pair. In essence, the torque is transmitted mesially from
anchorage; hence, a portion of the torque is first taken by the lateral
incisor and the remainder is delivered to the central incisor. Unless an
appropriate twist is placed in the wire between the two incisors, or an
equivalent arrangement is attained through use of pretorqued brackets,
the lateral incisor will generally receive the larger portion of the resultant
couple at activation. (Likewise, in the distribution of the responsive
force system, the canine will receive the greatest portion of that resultant
at activation.) In fact, because of the flexibility of the periodontal ligament, the first-phase movement of the lateral incisor prevents its acceptance of the entire activation and permits some third-order action to
reach the central incisor. As further third-order displacement of the lateral incisor occurs with time, the apportionment of the resultant couple
within the pair changes and the central incisor receives an increasing
percentage of the total torque. Because the root-surface area of the canine is generally greater than that of the lateral incisor, the distribution
of the responsive force system within the posterior segment does not
raise the level of concern indicated in the previous example. As a result,
the decision as to choices of wire and arch configuration may focus more
directly on the requirements at the activation site.
Although providing an interesting analysis problem, the effective
application of third-order action to a posterior segment is clinically feasible only under special conditions. A first difficulty is that of properly
distributing the active torque over a segment including three to five
teeth, which demands locally high appliance stiffness or differential
twists between the teeth of the segment. Second, transmission of the
resultant couple through a continuous arch wire results in the anterior
segment becoming the first line of support (anchorage). This couple
Delivery of Torque by the Orthodontic Appliance
must have a sizable magnitude if wanted displacements are to occur,
and the root-surface area of the anterior segment will be substantially
deficient to prevent generally undesired movement there. If bilateral
action is wanted, only the dental units to undergo displacement should
engage the appliance; no anchorage potential is needed and, ideally, the
anterior segment should not be involved. If unilateral action is needed,
then progressive torquing (one tooth at a time) should be the approach
to keep the resultant-couple magnitude small throughout the mechanics
and some, if not all, of the responsive force system should be transmitted other than to the anterior segment (e.g., to the palatal vault if in the
maxillary arch or by some means directly to the other posterior segment
in the same arch).
In the preceding discussion involving cantilevered shafts, the potential problem of insufficient anchorage was noted. The availability of support, both mesial and distal to the activation site, lessens the tendency
for undesired anchorage displacement. (Two response couples, their
magnitudes summing to that of the activation-site couple, are ordinarily
associated with a larger, total root-surface area than that of the single
anchorage site.) Consider, for example, the torquing of a first molar
using a rectangular-wire segment engaging the first and second premo-
lars and the first and second molars. To accomplish the task, in the
absence of pretorqued brackets, localized permanent twists are placed
immediately mesial to the second molar and distal to the second premolar brackets. These twists are generally made equal in magnitude,
such that the wire engaging only the divided anchorage sites would be
passive (Fig. 8-22, above). If the twists are unequal, engagement only into
anchorage produces equal but opposite torques against the second
molar and the premolar pair (Fig. 8-22, below). Activation at the molar,
through elastic twisting to permit bracket engagement there, superimposes a couple load on the molar and response couples at the anchorage
If a uniform, straight, wire segment is the principal appliance element, the partitioning of the molar load for transmission to the two
anchorage sites depends on the characteristic, shaft-segment lengths.
Because of the division of the mesial response between the premolars,
resulting from the flexibility of the periodontal ligament, the effective
distance from activation site to mesial support is greater than that to the
distal support. Increasing the characteristic length lowers the stiffness;
hence, the second molar absorbs more of the activating couple than does
the premolar pair. (Since the second premolar, being adjacent to the first
molar and having a smaller root surface area compared to it, receives
more than half of the total mesial response torque, the second molar
taking the larger portion of the total response may be advantageous.)
The division of torque between mesial and distal anchorage sites for this
example problem is depicted on the left in Figure 8-23 and corresponds
to the pretwisting of Figure 8-22, above. To illustrate the potential influence available to the clinician, as to the partitioning of torque in a di-
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 8-22. Permanent twists placed in a rectangular-wire segment prior to third-order
activation in the presence of split anchorage: equal twists (above) and unequal twists
vided-anchorage arrangement, the superposition shown on the right in
Figure 8-23 reflects the pretwist geometry of Figure 8-22, below, and the
activation itself. The facility is present to effectively modify the split of
the responsive torque (as apparently dictated by interbracket distances)
through pretwisting to take advantage of the relative sizes of the anchorage units that must absorb that torque. Pretorqued brackets that are
appropriately chosen can accomplish the same end.
Dellveiy of Torque by the Orthodontic Appliance
FIGURE 8-23. The partitioning of anchorage in the example of the torqued first molar
with equal (left) and unequal (right) permanent pretwists.
Anterior-Segment Torquing Mechanics
Rectangular- Wire Torquing
The conventional, edgewise practitioner uses a rectangular arch wire
in anterior torquing or retraction mechanics. Third-order action potential is achieved through the presence, in the passive state, of an angular
differential between the wire cross-section and the bracket slot at each
anterior location where torque transfer is desired. The longitudinal dimension of the wire shaft seems to be controlled by the mesiodistal,
dental-arch dimensions in the anterior segment. The wire must "fill" the
brackets; that is, the diagonal dimension of the wire cross-section must
exceed the occlusogingival width of the bracket slot in order to maintain
a twist activation. The wire material must be reasonably resilient, exhibiting a substantial elastic range in torsion, to accommodate sizable twist
activations. Generally, one of two orthodontic displacements is attempted in these mechanics: bodily movement or root movement.
In the absence of pretorqued brackets, the clinician typically prepares
the arch wire for third-order activation by placing permanent twists
immediately mesial to the intra-arch anchorage sites. The often-bilateral
twist placements may be quantified as shown schematically in Figure
8-24: the angle between the arch-form plane and the axis of symmetry of
the pliers. The amount of torsional activation achieved upon complete
Bioengineering Analysis of Orthodontic Mechanics
30° of
incisal torque
FIGURE 8-24. A buccal view of a rectangular arch wire held at midarch with pliers,
indicating that 30° of torque has been placed.
appliance engagement is, however, actually less than this permanent-
twist angle; the principal portion of the difference is the third-order
clearance between wire and bracket slot. This clearance may be computed from the actual cross-sectional dimensions of the arch wire and
the occiusogingival bracket-slot width. The clearance is depicted on the
left in Figure 8-25 and calculation of the clearance for a .019- by .026-in.
wire in a .022-in, slot is presented on the right in the figure. Given in
Table 8-2 are the third-order clearances for four combinations of wire
and slot sizes; nominal dimensions were employed in the computations.
True third-order clearances depend upon the actual wire and slot dimensions and, in effect, upon a stiffness of the bracket slot. In response to
Third-order clearance =
Angle VWX — Angle VWY
For a .019- by 0.26-in, wire
in a 0.22-in. (slot) bracket:
Tangent of LVWY
LVWY = 36.2°
Sine of LVWX =
V'192 + 262
LVWX = 43.1°
43.1 — 36.2
= 6.9°
FIGURE 8-25. Third-order clearance between rectangular wire and edgewise bracket:
schematic with geometry (left) and an example quantification (right).
Deliveiy of Torque by the Orthodontic Appliance
TABLE 8-2. Third-order clearances between rectangular arch wires and bracket slots
Wire size
.016 x
.017 x
.019 x
.021 x
Slot size
the third-order activation, the slot will "open" slightly as the wire attempts to deactivate, torsionally, within it. Although stainless-steel
brackets apparently exhibit adequate "slot stiffnesses," vendors have
had to reinforce the slots of some plastic brackets and they continue to
seek reduced slot flexibility through improved designs and materials.
Many clinicians conveniently refer to "torque," quantitatively, in
terms of the size of the permanent-twist placement(s) in degrees.
Pretorqued brackets are sized, in part, according to the third-order angulations of their slots with respect to a faciolingual reference. Just as the
units of orthodontic force are, properly, grams or ounces, however, ac-
tual torque is determined in gram-millimeters or ounce-inches, for example. To quantify torque correctly as a couple, the combination of shaft
stiffness and twist activation are necessary. Earlier in this chapter primary influences were seen to be the cross-sectional dimensions of a
solid, rectangular wire (shaft) in determining structural parameters in
torsion. Table 8-3 presents relative theoretical values of elastic stiffness,
range, and strength in torsion, per unit shaft length and a common
material, for four rectangular wires: typical working- and stabilizingwire sizes associated with the .018- and the .022-in, bracket systems. The
torsional stiffness of the largest wire is seen to be nearly five times that
of the smallest wire in the table. Also noteworthy is the inverse relationship between size and range. (Nominal dimensions, Equations 8-11, and
Figure 8-11 were used in the preparation of Table 8-3.) Additional influ-
ences on all three dependent, structural parameters are the wire material, longitudinal dimension, and the physical character of the wire
shaft. Pertinent to physical character, rectangular wire is now marketed
8-3. Comparisons of structural properties in torsion of rectangular arch wires
influenced solely by cross-sectional dimensions
Wire size
.016x .016
fl17 x .025
.019 x .026
.021 x .027
*Given figures not actual values, but, in each column, compared to value for wire adiacent to asterisk (*)
Bioengineering Analysis of Orthodontic Mechanics
in several multistranded configurations. Because the strands may move
with respect to one another in activation or deactivation processes, the
stiffness in torsion of stranded wire is less than that of the solid wire of
the same overall cross-section and material. Moreover, the stiffness is
dependent on the number and size of the individual strands and the
"braiding" pattern within the wire as a whole. Elastic range and
strength are similarly affected.
Torque is generated in the appliance as the operator produces thirdorder deformation of the rectangular wire to achieve engagement of the
anterior brackets; the larger the amount of necessary activating twist,
the greater the initial level of torque in the wire. Figure 8-26 illustrates a
torque-versus-twist plot for a full-size (for .022-in, bracket slots), stainless-steel, maxillary arch wire. The "torque" is actually the sum of the
four couples transmitted to the four maxillary incisors; the twist is that
measured at the midsagittal plane, relative to the before-engagement
orientation there. The nonlinear portion near the origin of the plot is
associated with bracket-wire clearance, bracket-slot flexibility, and manner of ligation; beyond this portion the relationship is, essentially, linear
to the elastic limit, reflecting the overall pattern of Figure 8-10. The thirdorder range of this arch is beyond 35° and the slope of the linear portion
of the curve, the "working stiffness," was determined to be approximately 217 g-mm/deg. A relatively simple relationship among the relevant parameters for the entire segment is the following:
Whole-segment torque = (Working stiffness) )<
x (Induced twist — Clearance)
The nominal bracket-wire clearance for the arch of Figure 8-26 is approximately 3°; the horizontal-axis intercept (dotted extension of the linear
portion of the plot) is slightly greater than 3°, reflecting the flexibiities of
the bracket-slot and ligation.
If the elastic range is sufficiently great to disallow inelastic behavior
during activation, and not minimizing the importance of bracket-wire
clearance, the dominant, indispensable parameter to convert the activating twist into torque induced is the third-order stiffness. Shaft theory,
examined earlier in this chapter, indicates that, in addition to material,
stiffness of a solid, rectangular arch wire depends on cross-sectional size
and longitudinal dimension. In the bilateral delivery of torque to the
incisal segment, "longitudinal dimension" refers, in essence, to the
amount of wire between the canine and the lateral incisor. Table 8-4
presents third-order stiffnesses obtained through bench experimentation (Wagner, 1981) for 16 maxillary incisal-segment torquing arches.
The material was moderately resilient stainless steel, and the occlusalview arch form coincided with the Bonwill-Hawley diagram. Particularly
notable are the decreases in stiffness related to the localized, inelastic
wire deformation: the permanent-twist placements and the "V" bends.
The values in Table 8-4 should be looked upon as indicating trends because of differences in arch lengths among patients; further reductions
Torque by the Orthodontic Appliance
Angle of Twist (degrees)
6. A typical third-order activation plot for a full-size, rectangular-wire, incisalrquing arch.
ss may
be attained through use of additional wire in loops more
than the simple opening loops used in the experimentation.
torquing mechanics generally involves one of two
attempted orthodontic displacement: bodily or root movement.
is the more difficult; separate activations of the torque and
ng force are necessary, and the two must be maintained over
specific couple-to-force ratio. If the angulations (in a sagittal
Bioengineering Analysis of Orthodontic Mechanics
TABLE 8-4. Torsional stiffnesses for representative, stainless-steel, rectangular-wire,
incisal-segment torquing arches (in gram-millimeters per degree)*
Bend, twist
None (flat arch)
V-bends, no twists
V-bends and twists
Teardrop loops
and twists
'From Wagner, WA. (1981).
view) are not to change during the displacement, the initial torque mag-
nitude will remain constant (assuming no relaxation of the wire). Accordingly, the driving force should also experience no decrease in size
with time and segment movement. In reality, this is an impossible set of
mandates to the appliance and, therefore, some compromise must be
accepted and undertaken in actual therapy. Alternatively, the torque is
the principal activation in root movement. The accompanying faciolingual force is primarily responsive; as a secondary activation it arises
against the tendency of the torque to move the crowns as well as the
roots. The objective is to produce a displacement characterized by a
center of rotation located occlusoapically near the bracket level; the ligation and anchorage must be sufficiently strong and stiff to permit rapid
development of a holding force and adequate to prevent sizable dis-
placements of crown points while the torque deactivates with the
wanted movements of the roots.
The discussions of transverse, orthodontic displacements in Chapter
5 revealed that the couple-to-force ratios for bodily and root movement
are quantitatively similar. Using the physiologically proper force theory
proposed in Chapter 5, for average-sized, maxillary, incisal segments
the total intermittent torque requirement is between 3000 and 3500
g-mm at activation. The accompanying faciolingual force is 200 to 300 g
and is split between the two sides of the arch. All four rectangular wires
of Table 8-4 can generate in excess of 3500 g-mm of torque within their
elastic ranges. For a chosen rectangular arch-wire size and configuration, together with bracket selection to enable determination of clearance, substitution of the torque requirement and the working stiffness
into Equation 8-12 enables estimation of the appropriate amount of twist
The combination of arch curvature and anterior torsional activation
produces several secondary effects that were mentioned previously. Of
these effects, perhaps the one of most interest and concern pertains to
control of the vertical position of the segment while faciolingual bodily
or root movement is progressing between appointments. Any changes
in the level of third-order activation in the anterior segment is accompanied by a tendency for occlusogingival movement of the anterior section
of the wire. Consider, for example, the anterior portion of the wire at
Delivery of Torque by the Orthodontic Appliance
bracket level with posterior engagement complete, but immediately be-
fore torsional activation and engagement into the anterior brackets. Figure 8-18 shows the result following activation of lingual root-torque in
the maxillary anterior segment. In the absence of vertical constraint, the
twist activation causes downward movement of the anterior portion of
the arch wire; in the clinical situation, the existence of localized (bending) stiffness results in the development of an extrusive force exerted by
the wire upon the anterior brackets. If the torque deactivates somewhat,
subsequently, the accompanying tendency for extrusion is also reduced.
Depending on the specific arch, each 10 to 15° change in elastic twist
may be accompanied by as much as a millimeter of vertical movement of
the anterior section of the arch wire if unrestrained. This deflection,
together with measured vertical stiffnesses between 100 and 200 g/mm
for the arches of Table 8-4 (Wagner, 1981), exemplify the very real potential for extrusive or intrusive movements of the anterior teeth during
torquing mechanics.
Compensation for the vertical-displacement tendency can be created;
the effect can also be used to produce vertical movement in concert with
the torquing mechanics. Necessary at the outset is consideration of the
desired vertical positions of the anterior teeth with respect to their locations prior to the torquing mechanics. Also note that both activation and
deactivation of the twist affect the vertical-displacement potential. As an
example, consider the attempt at bodily-movement, anterior retraction
while holding the vertical positions of the maxillary anterior teeth. As
indicated previously, change in long-axis angulations and reduction in
torque magnitude as activated are cause and effect. To approach maintenance of the initial couple-to-force ratio over time and bodily retraction,
the driving force must be generated by a low-stiffness appliance element. Ideally, the lingual root torque should not lessen with displacement and, correspondingly, the vertical-displacement potential should
also remain unchanged with time. Accordingly, the position of the anterior section of the arch wire, prior to third-order activation, should be
somewhat apical of the line of brackets; the amount of this vertical "malalignment" is to be proportional to the magnitude of the forthcoming
twist activation. Again, the rule of thumb is 1 mm for every 10 to 15° of
actual twist activation. If intrusion is wanted with these torquing mechanics, the passive, vertical "malalignment" of the wire and brackets
must be exaggerated.
Torquing Spurs in the Appliance
an alternative to the use of rectangular wire, torquirig spurs
formed in round wire may be used to deliver third-order couples (with
or without accompanying driving or holding forces) to the anterior teeth
(or, for that matter, to any tooth). The wire must be ductile to accept the
permanent bends of spur fabrication without fracture and, concurrently,
resilient to sustain third-order activations approaching (and often exceeding) 90° without becoming inelastic. The individual spur may be
Bloengineering Analysis of Orthodontic Mechanics
FIGURE 8-27. Facial views of torquing spurs: a single, narrow spur engaging an
individual crown (left) and a T-loop contacting a pair of teeth (right).
affixed to an arch wire or an integral part of the wire. The spurs may be
attached to or part of a continuous arch wire or incorporated in an auxiliary (as in the Stage III appliance of pure Begg therapy). A single spur
may contact an individual crown as shown on the left in Figure 8-27,
engaged between the wings of a siamese or in a Begg bracket, or one
spur may act on several teeth as suggested by the T-loop on the right in
Figure 8-27.
The typical lingual root torque generated by a torquing spur is de-
picted in the series of sketches in Figure 8-28. On the left, the spur is
shown passively; it is often fabricated to lie near to or in the arch-form
plane. In a view from the mesial or distal perspective, the center sketch
shows the spur (dashed) following activation, an incisor with a bracket
on the facial surface, and the forces exerted by the spur tip and by the
arch wire on the bracket. On the right, the lingual holding- or drivingforce component, generally present and exerted by the arch wire, has
been added. A vertical (occlusogingival) force component may also
exist, as described previously. The labially and lingually directed forces
Arch wire
FIGURE 8-28. Sagittal views of a torquir,g spur and its effects: typical passive
angulation (left), tip and in-bracket force pair acting on the crown (center), and with the
holding or driving force from the arch wire added (right).
Delivery of Torque by the Orthodontic Appliance
at the bracket could be combined into a single labiolingual force; how-
ever, the labially directed force in the figure is the counterpart of the
spur-tip force, the two forming the couple, and combining the forces at
the bracket results in loss of identity of the couple-and-force loading.
Of interest is a comparison of the third-order couples generated by
rectangular wire and a torquing spur. The differences in the directions of
the pairs of forces and the moment arms are shown in Figure 8-29. Those
forces exerted by the rectangular wire are directed nearly occlusogingivally while the force pair from the spur acts labiolingually. Because the
net translational effect of the force pair is zero (with the individual forces
parallel and equal in magnitude), this difference in directions is meaningless with respecf to the action delivered to the dentition. The inherent moment arms for the rectangular-wire and spur couples are the labi-
olingual dimension of the wire cross-section and the occlusogingival
height of the spur, respectively; they differ by approximately one order
of magnitude (a factor of ten). Hence, if the size of each force generated
by the spur is about one-tenth that of each force of the pair generated
within the bracket, the results are approximately equal third-order couples applied by the two appliances. Two other comparisons are noteworthy. First, one force of the pair exerted by the spur, if the wire is
engaged in an edgewise bracket, acts against the ligation (which tends to
be the weakest, most flexible part of the bracket assembly); alternatively,
the Begg bracket incorporates a vertical slot. Second, although a torque
having either sense (clockwise or counterclockwise as viewed from a
mesial or distal perspective) may easily be generated by a rectangular
wire, the spur tip can only push against the facial surface; hence, to
apply a labial-root (lingual-crown) torque with a labial-appliance spur,
the tip must contact the crown occlusally to the bracket.
Torquing the maxillary incisal segment with spurs is conventional in
arm for
arm for
FIGURE 8-29. Sagittal views of maxillary incisors loaded by rectangular-wire torque (left)
and spur torque (right) showing differences in force directions and inherent moment
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 8-30. The typical passive geometry of a Begg torquing auxiliary.
III of Begg therapy. Figure 8-30 shows a passive Begg torquing
auxiliary in an occlusal view. Spurs, approximately 3 mm in height, are
bent into the wire; one spur contacts each incisor facial surface with the
auxiliary also engaging the canines. The auxiliary is usually formed in
.012-, .014-, or .016-in, wire and is typically employed in conjunction
with a .020-in, main arch wire. One purpose of the main arch wire is to
help to counter the extrusive potential associated with the activation of
the auxiliary. Third-order stiffnesses of Begg torquing auxiliaries are
substantially less than those of the edgewise wires; correspondingly, the
activations of the spurs are greater with the passive spur configurations
generally in or near to the arch-form plane. Torsional- and vertical-stiffness data for Stage III appliances are presented in Table 8-5 (Leaver and
TABLE 8-5. Torsional and vertical incisal-segment stiffnesses for six Begg, Stage lii
maxillary-arch appliancest
Wire size
Spur angle
Overlaying a .020-in. main arch wire.
Leaver, SR., and Nikolai, R.J. (1978).
Delivery of Torque by the Orthodontic Appliance
Nikolai, 1978); the passive angle of the spurs with the arch-form plane
has, apparently, little influence on either stiffness.
Figure 8-31 illustrates a torquing arch incorporating spurs; however,
this design is quite different from the Begg auxiliary. First, it is a continuous, maxillary arch wire engaging all teeth through the terminal molars. Second, spurs will contact only the central incisors; the torque is
transferred bilaterally along the wire from posterior anchorage directly
to the central incisors, with third-order action then apparently carried to
the lateral incisors by the transseptal-fiber system within the alveolus.
The passive geometry of the arch reflects the attempt to negate unwanted secondary effects; a curve of Spee is placed to counter the potential for extrusion that accompanies third-order activation and the posterior sections are curved toward the lingual to offset the tendency for
torsional activation to enlarge the posterior arch width. The appliance
may be formed in highly resilient stainless-steel wire with the passive
configuration of the spurs in the arch-form plane. The spur height is a
clinical variable with its upper bound determined by the distance from
bracket line to gingival margin and, for heights of 5 to 6 mm, the typical
torsional stiffness of the arch fabricated in .016-in, stainless steel wire is
approximately 45 g-mm/deg (Mellion and Nikolai, 1982). Comparing
this value to the Begg counterpart in Table 8-5, the difference is principally due to the greater "shaft" length (canine to central incisor) in the
arch under discussion. Note that the torquing spur itself is, in effect, a
cantilever beam; the greater height reduces the bending stiffness of the
spur, but it also enlarges the moment arm of the torsional couple. Be-
8-31. An occiusal view of the passive configuration of a hybrid, torquing arch
Bioengineering Analysis of Orthodontic Mechanics
the overall activation of the anterior section of the arch is primarily
twisting, with only a small contribution from spur bending, for otherwise identical, round-wire torquing arches, the actual torque generated
(in g-mm, for instance) is virtually proportional to the height of the
To summarize, regardless of the type of anterior-segment torquing
appliance used, the key load parameter is the third-order couple to be
transferred from the wire to the tooth crown. This couple is determined
principally by a third-order stiffness and the amount of rotational activation. Rectangular-wire stiffnesses are relatively high and the amounts of
third-order activation should be correspondingly small; torsional stiffnesses of spur-incorporated round wires are less, thus creating the need
for greater angular activations.
Inelastic Behavior in Third-Order Mechanics
Prior to twist activation of an arch wire the clinician may alter its config-
uration, through bending and/or torsional deformation, taking the wire
beyond its elastic limit to achieve a desired, pre-engagement shape.
Such a process changes the external geometry of the wire and induces
residual stresses; consequently, mechanical and structural parameters
are affected. Substantial ductility is required of the wire so that the
working of it, which strains the material beyond its elastic limit, will not
cause immediate—or, while the appliance remains activated, progressive—fracture.
Inelastic twisting is examined first in this section. Recall that thirdorder loading of a straight shaft produces a circumferential strain pattern
on a cross-section characterized by zero magnitude at its geometric center (through which the longitudinal shaft axis passes) with, generally,
the strain increasing with radial distance from the center. On a circular
cross-section, the circumferential strain at a point is directly proportional
to the radial coordinate of the point. On the rectangular cross-section,
these strains vary radially in a nonlinear manner. The maximum strains
occur, then, at points on the external boundary of the cross-section. For
circular shafts with their axisymmetric circumferential strain patterns, all
points on the outer perimeter of the cross-section are subjected to maxi-
mum values. The largest circumferential strains on the rectangular
cross-section occur at the two outer-surface points nearest the center
(and are on one of the axes of symmetry). The patterns of shearing
strains on the two cross-sections of interest are maintained, with increasing load, throughout the elastic range and well into the inelastic
region of material behavior. Below the elastic limit, stress and strain are
proportional; the patterns of both circumferential stress and strain are
shown in Figure 8-9 for the circular shaft and in Figure 8-12 for the
rectangular shaft.
The formula for elastic range of a shaft indicates direct dependence of
Delivery of Torque by the Orthodontic Appliance
critical twist angle upon the characteristic longitudinal dimension (Equa-
tions 8-9 and 8-11). The practitioner typically localizes the permanenttwist placement. This establishes a sizable twist-set within a relatively
small portion of shaft length and, as a result, a concentration of residual
shearing stress is created. Shown on the left in Figure 8-32 is the loadunload plot for the short section of shaft (wire) in which a permanent
twist is placed. On the right in the figure is a curve describing in graphic
fashion the circumferential stress-strain relation during the twist placement for the element on the shaft outer surface experiencing the greatest
shearing strain during the process; the final point of the plot reflects the
residual shearing strain and stress. For a round wire, a direct analogy
can be drawn between inelastic bending and twisting when discussing
stress patterns. In bending, the variation in normal stress with distance
from the neutral axis was considered and, in torsion, the shearing-stress
dependence on radial distance from the shaft center is examined.
The sketch on the left in Figure 8-33 shows the pattern of increasing
circumferential shearing stress with the radial coordinate r for the critical
section of the circular shaft that has undergone torsional loading beyond
the elastic range for the member as a whole. Within the elastic core of
radius re., the relationship between the stress and r is linear; beyond the
core, shaft elements have been inelastically strained and the relationship
is nonlinear. The center sketch depicts the change-in-stress pattern assodated with the unloading of the shaft section. As noted previously in
Chapters 3 and 7, unloading of a structural member always has elastic
characteristics, thus the linear relationship between stress-change and
the coordinate r for the metallic shaft material. Although the resultants
of the load and unload stress distributions shown must be equal in magnitude (because the couple returns to zero upon unloading), the differences in distribution yield, upon superposition of one upon the other,
Shear Strain e0
Angle of Twist 0
(stress reversal)
FIGURE 8-32. Load-deformation plot for a shaft section in placement of a permanent
twist (left) and a stress-strain plot for an element in the section at the shaft outer
surface (right).
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 8-33. The load (left), unload (center), and net residual (right), circumferential,
shearing-stress patterns for a metallic shaft strained beyond its elastic limit.
net residual-stress pattern shown on the right in Figure 8-33. The
difference between the load and unload third-order deformations is the
permanent twist (Fig. 8-32, left). Upon overall unloading, elements near
the shaft axis do not completely lose the stress induced with loading. At
a specific radial coordinate beyond the elastic core, the loading stress
and unloading stress-change exactly cancel one another, and near the
outer surface of the shaft the elements experience a stress-reversal upon
unloading. For the circular shaft this residual-stress pattern is the same
along all radial-coordinate lines, commensurate with the axial symmetry. In the rectangular shaft the patterns are similar to that on the right in
Figure 8-33, but they are more nonlinear and relative maximum values
vary with radial-coordinate direction. Some changes in pattern occur as
the diagonals of the rectangular cross-section are approached; however,
the stress magnitudes are not critical in these regions.
If the wire material can adequately take a permanent twist without
failure, the subsequent concern is the superimposition of an elastic activation. From a practical standpoint, considerations are restricted to rectangular wires. (Round wires that easily fit into brackets will spin within
them under torsional activation, with or without placement of permanent twists; torquing spurs incorporated in round wire to enable transfer
of third-order actions are bent into the wire.) Two points of practical note
are to be made. First, the activating (elastic) twist to gain bracket engagement and create the torque is undertaken in a sense opposite to that of the
preceding, permanent-set twist. Thus, the largest shearing stresses of
the activation itself, occurring near and at the outer surface of the wire,
will be, at the sites of permanent-twist placement, superimposed on the
largest residual stresses having the same sense. Accordingly, the maximum net shearing stresses will be at points on the outer surface of the
cross-section and will be greater than if no permanent twists had been
placed. Distant from the outer surface the activating and residual
stresses are of opposite senses and will tend to cancel one another. In
short, placements of permanent twists followed by opposite-sense,
third-order activation results in concentrations of sizable shearing
stresses at outer-surface locations of the shaft. Localized inelastic action
Deliveiy of Torque by the Orthodontic Appliance
may even occur, which will tend to reduce the concentration of stresses
and distribute them inwardly. A wire material with substantial ductility
is, therefore, absolutely necessary to avoid approach of the fracture
point with activation. Second, bench-experimentation has shown that
permanent-twist placements tend to lower the third-order stiffness of
the arch wire (Table 8-4). The greater the amount of the permanent set,
the larger the decrease in the stiffness. For a given magnitude of twist
placement, the percentage reduction in stiffness seems to be essentially
independent of cross-sectional size.
In appliances incorporating torquing spurs, third-order elastic action
is superimposed upon a residual-stress pattern remaining from the formation of those spurs. The analysis is complicated from several standpoints, and only a superficial examination is warranted here. Whether
formed directly in the arch wire or prepared separately, the spurs are
bent into an initially straight segment of wire. Localized bends are
placed to form the spur tip and where the spur joins the arch form. If the
spur is formed directly in the arch wire, the three bends are located in a
common plane; if the spur is attached to the wire, the attachment is
partially accomplished through inelastic bending that wraps the legs of
the spur around the arch wire, and this bending is in a plane that is
perpendicular to that of the spur-tip preparation. The activation of the
torquing arch of Figure 8-31, for example, is in two parts: the wire segments between the central incisors and the canines experience the bulk
of the elastic deformation as they are loaded in torsion while the spurs
experience a small amount of bending deformation. Each spur, behaving much like a cantilevered beam, undergoes its maximum bending
strains at the junctions with the arch form and experiences virtually zero
strain of activation at the tip (apex). Whatever bending activation does
occur at the junctions is in a plane at right angles to that of the permanent bends. Torsional couples are transformed into bending couples at
these junctions and, although the stress analysis is extremely difficult
there, it is likely that these are critical locations (of maximum total
stress). If the spurs are formed separately, then wound around the main
arch wire, the bending activation is directly superimposed upon the
residual-stress patterns left by the winding. From the discussion of inelastic bending in Chapter 7, it is preferable to orient the windings such
that the activation of the spurs tends to further wind, rather than partially unwind, the bent wire at the junction. The maximum net bending
stress will be less with the former procedure.
As noted earlier, third-order activation in the anterior portion of the
arch wire produces second-order response in the posterior regions due
to the curvature of the arch form. In edgewise, anterior-retraction mechanics, the clinician may place loops just mesial to the lead teeth of the
posterior segments. These loops, incorporated in the arch wire, are
formed by inelastic bending in the faciolingual plane. Potential wire fail-
ure associated with the superimposition of the activating and residual
bending-stress patterns within the loops need not be of great concern if
the wire material is ductile, but the influence of these loops on stiff-
Bioengineering Analysis of Orthodontic Mechanics
is notable. First, the presence of the loops effectively increases
the longitudinal dimension of the shaft portion of the arch wire and,
thereby, lowers the third-order stiffness as indicated in Table 8-4. Second, viewing the arch wire from a buccal perspective, the anterior portion is cantilevered from the posterior segments, the loops also effectively increase the beam length and, as a result, the vertical stiffness is
reduced in the anterior section to possibly lessen the impact of the generally unwanted, secondary effect discussed previously. The more wire
incorporated in these loops, the greater are the third-order and vertical
flexibilities in the anterior sections of the arch. Note, however, that the
loops oriented gingivally must be passively open to be effective if the
third-order activation is to produce lingual-root torque. Loops may also
have a more direct association with torsion, either in terms of their own
internal force systems or their effects upon the teeth; such auxiliaries are
discussed in the following section.
Wire Loops, Springs, and Torsion
Structural aspects of orthodontic wire loops were first considered in
Chapter 7, and that discussion was restricted to loops that provide mesiodistal flexibility to the continuous arch wire and were activated in flexure within the plane of their bends. Appropriate here in this chapter is
the extension of that discussion to consider loops bent directly into the
wire or auxiliaries that either experience internal torsional loading when
activated or are intended to transmit torquing couples to individual
teeth in order to impart root-movement potential or control. These loops
and auxiliaries are exemplified by the "half-box" loop and the uprighting spring.
The usual objective in activating a half-box loop is to exert a faciolingual force on an individual tooth. The loop, as prepared from a straight
segment of wire through placement of four, right-angle bends, is shown
on the left in Figure 8-34. The vertical legs are typically 5 to 6 mm in
length and the horizontal section spans one or more interbracket dis-
Passive, buccal view
Occiusal view
FIGURE 8-34. A half-box loop shown passively in a faciolirigual view (left), and as
activated and subjected to its typical force system from an occiusal perspective (right).
Delivery of Torque by the Orthodontic Appliance
depending on the specific application. The loop is activated in
the direction perpendicular to the plane of the loop as formed. The
sketch on the right in Figure 8-34 shows an occlusal view of the ioop as
ordinarily activated. The active and responsive force systems are included in the figure; capability must be present for generating the firstorder couple(s) to disallow contact between the loop and the gingiva.
The faciolingual forces at the activation and anchorage sites are equal
and opposite to one another, and the resultant couple is equal in magnihide to that formed by the two forces. (Additional actions may be imparted at the activation site, provided the equilibrating responses can be
developed. The loop could also actively transmit a first-order couple, an
occlusogingival force, a second-order couple, and, if the ioop is bent in
rectangular wire, a third-order couple to an appropriate malaligned
In Figure 8-35, the three sections of the half-box loop are separated,
making the force systems at the "corners" external. Following activation, the reaction of the tooth crown at the activation site loads one leg in
bending; the mesiodistal view of the vertical leg on the left shows the
force system of a cantilever (plus torsion if C1 is present). The bending
couple in the left leg becomes a twisting couple (Qv) in the midsection
of the loop with the right-angle turn in the wire. The horizontal section,
as shown from an occlusal perspective in the center sketch of Figure
8-35, is subjected to a combination of twisting and bending; action-reaclion analysis gives the force system at its left end, and force and moment
balances would yield the responsive components at the right end of the
section. A mesiodistal view of the right leg of the loop is shown on the
right in Figure 8-35. The 90° turn converts the twisting and bending
couples at the right end of the midsection into bending and twisting
couples, respectively, at the gingival end of the right leg of the half-box
FIGURE 8-35. Free-body diagrams of the three straight-segment sections of the
activated halt-box loop of Figure 8-34.
Bioengineering Analysis of Orthodontic Mechanics
ioop, and this is the critical section of the ioop (where the internal bend-
ing and twisting moments are concurrently maximum). The right leg of
the loop, then, is also subjected to combined bending and twisting.
(With additional components from activation at the occiusal end of the
left vertical leg, the analysis becomes more complex. The numerous variations will not be examined here, but in most instances the critical section wifi be located at the bend diagonally opposite to the activation
Of the various structural properties of the loop, interest centers on
the faciolingual stiffness since this characteristic, together with the activating displacement, yields the initial crown load. Generally, this stiffness is dependent on wire material, cross-sectional size and shape, and
curvilinear length (the amount of wire in the loop). From the foregoing
bending and torsional analyses, the mesiodistal wire dimension and the
occlusogingival lengths of the vertical legs will be the most influential
geometric parameters. All other properties held constant, the greater the
separation between activation and anchorage sites, the greater the
length of the horizontal section and the larger the overall flexibility of
the loop. Should the action of the loop include an occlusogingival loading component, the midsection length in particular will influence the
occlusogingival stiffness.
The uprighting spring is an auxiliary designed to exert a secondorder couple on a tooth crown, adding to an appliance the capability of
controlling root-tip position (long-axis angulation) in the faciolingual
plane. The spring is formed, usually in round wire, by placing several
permanent bends to obtain the desired shape. One leg of the spring
engages the bracket slot (which may be either horizontal or vertical) at
the activation site and the other leg is generally hooked over the main
arch wire. The passive (dashed) and active (solid) configurations of the
spring are shown in Figure 8-36. As indicated, a helix is often incorporated in the spring at its apex, both to lessen stiffness and to increase
elastic range.
Shown on the left in Figure 8-37 is the free-body diagram in a
faciolingual plane of an activated spring. The second-order couple is the
action-reaction counterpart of the desired component of the force system, but the pair of forces—one at the bracket and the other exerted on
the hook by the contacting arch wire—arises in response to balance the
system. The analysis parallels that for the half-box loop with C1 absent
except for differences in activation planes and in transposition of active
and responsive force components. For the spring, the size of the couple
is determined by the associated stiffness and the magnitude of the displacement necessary to engage the hook with the main arch wire. The
action-reaction counterparts exerted by the legs of the spring on the
tooth and the arch wire are illustrated on the right in Figure 8-37. Note
that the arch wire, through its own bending stiffness in the faciolingual
plane, can sustain at least a portion of the usually unwanted responsive
force that tends to extrude the tooth. The force exerted by the leg
hooked over the arch wire is divided and transmitted by the wire to the
Delivery of Torque by the Orthodontic Appliance
Line of bracket slots
FIGURE 8-36. Passive (dashed) and activated (solid) configurations of an uprighting
spring in a facial view.
teeth on either side of the interbracket contact. Shown in Figure 8-38 is a
pair of free-body diagrams of segments of the spring, formed by a "cut"
immediately to the right of the helix. The left-hand portion of the spring
is subjected to a combination of bending and compression. The moment
balance shows that the couple at the "cut" is slightly smaller in magnitude than the couple delivered to the bracket. The right-hand portion of
the spring behaves as a cantilevered beam. The maximum internal bend-
ing couple exists, then, continuously throughout that portion of the
spring between the bracket and the helix.
FIGURE 8-37. Free-body diagram of the uprighting spring (left) and the force system
exerted by the spring (right).
Bioengineering Analysis of Orthodontic Mechanics
- b)
FIGURE 8-38. Free-body diagrams of two portions of the uprighting spring exposing an
internal bending couple adjacent to the helix.
extrusion is a portion of the desired tooth movement, the
spring should be designed to generate an adequate couple at the bracket
while suppressing the vertical forces that respond to the activation. Examining the moment balance with the sketch on the left in Figure 8-37,
the proper procedure is to make the distance between the bracket and
the "hook-on" site (a moment arm) as large as possible. Alterations in
spring-wire material, cross-section, occlusogingival height, or number
of helices all influence the second-order stiffness of the spring and, as a
result, the magnitude of the as-activated couple for a given, passive
geometry. These adjustments also affect the range and strength of the
spring, but will not impact the ratio of second-order couple to extrusive
force at the bracket. On the other hand, changing the characteristic mesiodistal dimension of the spring alters both the ratio of couple to force
and the relevant stiffness.
and the previous chapter have dealt with the force-system and
structural analyses of the orthodontic arch wire, subjected through clini-
cal activation to bending and torsional loadings. Engineering theory,
experimental research, and clinical observations have been integrated
toward presentation of the current development of the subject matter
and the means to undertake evaluations of existing intraoral appliances
Delivery of Torque by the Orthodontic Appliance
and, perhaps, to proceed toward new designs. Study of the fundamen-
tal bending problem, with its activations perpendicular to the longitudinal wire dimension, has proved to be more straightforward than that of
the third-order problem. The clinician, however, must become familiar
with actual torque and its relationship to torsional deformation and the
third-order, structural characteristics of the arch wire and bracket systems if an integrated understanding of the appliance, in all of its aspects
and capabilities, is to be achieved.
Helpful to the reader in following the developments of this chapter is
the parallelism between bending and torsional theoretical concepts.
Stiffness links activating deformation to induced force system. Stiffness,
range, and strength have the same general definitions, and are influenced by the same wire parameters and to similar extents, whether considering bending or torsion. The curvature of the arch form provides
difficulties in the clinical arena as it causes transformation of torsional
couples into bending couples, and vice versa, but the continuous-archwire model aids in understanding this phenomenon and the associated
secondary effects that have been described.
The emphasis in this chapter, not explicitly stated previously, has
focused on the instantaneous, structural aspects of the third-order problem. Little attention is given to the time-influencing, deactivation process because the present level of knowledge has apparently been gained
primarily from clinical experimentation; the actual results have often
been less satisfactory than those envisioned. The mechanics of anterior
retraction, for example, are in need of substantial, controlled study.
Often in actual therapy, the achieved angulations are not those desired
and an additional step in treatment becomes necessary, demanding revised and lengthened third-order mechanics.
The foregoing discussions have attempted to enlighten the orthodontic student and practitioner on a perplexing subject and a difficult
portion of therapy. Hopefully, some gain has been achieved in the direc-
tion of further understanding of torsion and third-order mechanics.
Begg, P.R., and Kesling, P.C.: Begg Orthodontic Theory and Technique. 3rd Ed.
Philadelphia, W.B. Saunders, 1977, Chapters 6 and 11.
Leaver, S.R., and Nikolai, R.J.: Mechanical analysis of the Begg torquing auxiliary. Aust. Orthod. J., 5:133—141, 1978.
Mellion, N.P., and Nikolai, R.J.: Torquing maxillary incisors with a continuous,
round-wire appliance. J. Dent. Res. (Special Ed.), Abst. 1107, March, 1982.
Wagner, J.A.: Mechanical Analysis of Anterior Torque in Rectangular Orthodontic Wire. Master's thesis, Saint Louis University, 1981.
Bioengineering Analysis of Orthodontic Mechanics
Suggested Readings
Andreasen, G.F.: Selection of the square and rectangular wires in clinical prac-
lice. Angle Orthod., 42:81—84, 1972.
Arbuckle, G.R., and Sondhi, A.: Canine root movement: an evaluation of root
springs. Am. J. Orthod., 77:626—635, 1977.
Blodgett, G.B., and Andreasen, CF.: Comparison of two methods of applying
lingual root torque to maxillary incisors. Angle Orthod., 38:216—224, 1968.
Boman, V.R.: A radiographic study of response to torquing spring action. Angle
Orthod., 32:54—58, 1962.
Burstone, C.J.: The biomechanics of tooth movement. In Vistas in Orthodontics.
Edited by B.S. Kraus and R.A. Riedel. Philadelphia, Lea & Febiger, 1962,
Chapter 5.
Gianelly, A.A.: Bodily retraction of maxillary incisors with round wires. Am. J.
Orthod., 66:1—8, 1974.
Holdaway, R.A.: Changes in relationship of points A and B. Am. J. Orthod.,
42:176—193, 1956.
Hurd, J.J., and Nikolai, R.J.: Maxillary control in Class II, Division 1 Begg treatment. Am, J. Orthod., 72:641—652, 1977.
Jarabak, J.R., and Fizzell, J.A.: Technique and Treatment with Light-wire Edgewise Appliances. 2nd Ed. St. Louis, C.V. Mosby, 1972, Chapters 2, 3.
Kusy, R.P.: Comparison of nickel-titanium and beta titanium wire sizes to conventional orthodontic wire materials. Am J. Orthod., 79:625—629, 1981.
Kusy, R.P.: On the use of nomograms to determine the elastic property ratios of
orthodontic arch wires. Am. J. Orthod., 83:374—381, 1983.
Kusy, R.P., and Greenberg, A.R.: Effects of composition and cross-section on
the elastic properties of orthodontic wires. Angle Orthod,. 51 :325—342, 1981.
Kusy, R.P., and Greenberg, A.R.: Comparison of elastic properties of nickel-titanium and beta titanium arch wires. Am, J. Orthod., 82:199—205, 1982.
Mitchell, D.L., and Kinder, J.D.: A comparison of two torquing techniques on
the maxillary central incisors. Am. 1. Orthod., 63:407—413, 1973.
Neuger, R.L.: The measurement and analysis of moments applied by a lightwire torquing auxiliary and how these moments change magnitude with respect to various changes in configuration and application. Am. J. Orthod.,
53:492—513, 1967.
Newman, G.V.: A biomechanical analysis of the Begg light arch wire technique.
Am. J. Orthod., 49:721—739, 1963.
Popov, E.P.: Introduction to Mechanics of Solids. Englewood Cliffs, NJ, Prentice-Hall, 1968, Chapter 5.
Delivery of Torque by the Orthodontic Appliance
Rausch, ED.: Torque
45:817—830, 1959.
and its application to orthodontics. Am. 1. Orthod.,
Schrody, D.W.: A mechanical evaluation of buccal segment reaction to edgewise
torque. Angle Orthod., 44:120-126, 1974.
Smith, J. 0., and Sidebottom, 0. M.: Elementary Mechanics of Deformable Bodies. London, Macmillan, 1969, Chapter 9.
Steyn, C.L.: Measurement of edgewise torque in vitro. Am. J. Orthod., 71 :565—
573, 1977.
Thurow, R.C.: Edgewise Orthodontics. 4th Ed. St. Louis, C.V. Mosby, 1982,
Chapters 6 and 9.
Ext rao ral
A class of appliances familiar to the practitioner is characterized by the
extraoral positions of the activating elements and supporting structure.
These appliances, which individually or collectively are often termed
"headgear," receive widespread use, primarily because they have the
advantage of remotely located responsive force (anchorage). The task of
force-displacement analysis is eased because the effects of the responsive force system, broadly distributed, need be of little concern; therefore, the practitioner may focus attention on the active force system and
its potential to produce or restrain orthodontic or orthopedic displacements. The principal components of the typical extraoral orthodontic
appliance are:
1. A face bow or a pair of "J-hooks" that delivers the active force to
intraoral locations where, either directly or indirectly, the force is
transferred to the arch wire or through brackets or buccal tubes to the
tooth crowns.
2. The active force generated bilaterally by one or a pair of traction
bands or by one or more pairs of elastics or springs.
3. The responsive force distributed to cervical, occipital, or parietal anchorage areas by a neck pad or head cap.
The headgear apparently most popular with practitioners may be
described as those that provide active force primarily directed toward
the posterior, have intraoral mechanical connections to the maxillary
arch wire or to teeth of the maxillary arch, and possess geometric and
mechanical symmetry with respect to the midsagittal plane. Recent innovations, however, have improved the potential for controlled, unilateral displacement; delivery of extraoral force to the mandibular arch
receiving increased acceptance, and extraoral appliances from
which anteriorly directed forces are supported have been designed and
Extraoral appliances may be classified from several standpoints. Distinctions between types may be drawn according to the origin or direc322
Extracral Appliances
tion of the active force. This force may be generated against a neck pad
or head cap. Clinicians often refer to headgear as cervical-, straight-, or
high-pull, indicating the buccal-view angulation and line-of-action position of the active force. The appliance having posterior anchorage and
the active force on one or both sides divided into two separate, mechani-
cal actions is termed a "dual-force" headgear. Those appliances that
generate anteriorly directed active force are becoming known as "reverse-pull" mechanisms. Headgear may also be categorized according to
the intraoral locations of active-force delivery: the maxillary and/or mandibular arch, the first or terminal molars, the canines, or the incisal or
anterior segment(s). The chin-cap assembly might be placed in a separate category because it delivers active force to an extraoral location: the
chin. Although it is often employed to produce orthopedic action or
restraint, because of similarities to other types of headgear (in terms of
anchorage and activating elements) the chin-cap assembly is included in
the discussions to follow.
The development of this chapter proceeds with the mechanical eval-
uation of headgear, generally subdivided according to the location of
active-force delivery as seen from an occlusal or buccal view. Discussions are primarily focused on the relationships among active force systems, alone or in concert with intraoral mechanics, and the corresponding, potential, orthodontic and/or orthopedic displacements. There is
little attention to structural considerations of the extraoral appliance itself, beyond that which follows in the introductory section of this chapter. The analysis problem is three-dimensional; the individual, active
forces generally have three nonzero components with respect to a sagittal-coronal-transverse reference framework. The spatial problem is examined first in an occlusal view in which anteroposterior and faciolingual aspects may be considered, and then from a buccal view where
vertical and occlusogingival aspects, together with the anteroposterior
contributions, may be analyzed. All of the needed geometric parameters
will be assumed known or the facility to take required anatomic and
other measurements is available. The characteristics of the activating
force system (from traction band[s], elastics, or springs) are also assumed given or measurable. Sought in the analyses are descriptions of
potential displacements that are desired or, if the clinical objective is to
negate unwanted but likely movements arising from intraoral mechanics, the degree to which the objective can be met.
Before beginning the individual analyses, several overall, pertinent
considerations should be kept in mind. First, the analysis of the force
system must be undertaken with the extraoral appliance in its activated
configuration. Strictly speaking, this is always the rule with any structure; however, often the difference between passive and activated formats are small enough to produce negligible variance in the directions or
moment arms of forces. These differences are generally not substantially
influential, for example, in the bending or torsional analyses of arch
wires. The point is well made here, however, because the deformations
associated with the activations of face bows and J-hooks may be sizable.
Bioengineering Analysis of Orthodontic Mechanics
In fact, with the face bow in particular, several adjustments in the sagit-
tal and transverse planes made "at the chair" are undertaken in anticipation of the deformations to accompany activation. Second, with the exception of some mixed-dentition treatment, the headgear rarely acts in
the absence of intraoral-appliance interaction. When present, the arch
wire is nearly always interconnected with the delivering element of
extraoral force. Hence, the tracing of the transmission of force is always
necessary, even though originating extraorally, throughout whatever
intraoral appliance is engaged, whether activated independently or not.
Third, a substantial portion of the prediction of displacement or displacement restraint originates from a buccal-view analysis. In effect, a
complete division of the entire dentofacial complex and appliance(s) is
made at the midsagittal plane, and the analysis is carried out on the right
half separately. With complete symmetry with respect to the midsagittal
plane, the force system transmitted from one side to the other (e.g.,
through the center of a face bow) has no mechanical influence in the
buccal view. With an asymmetric situation associated with appliance
geometry and/or loading, however, such is not the case and appropriate
notations are made when applicable. Fourth, the size of the force to be
generated in the activating element and transferred to a bow or J-hook
end is determined by the factors discussed in Chapter 5 and an estima-
tion of the amount of generated, active force to be, in effect, lost in
transit to the intraoral site of force application. This force drop is small
with face-bow transmission, but friction and, perhaps, a portion of the
force not delivered in the desired direction may result in sizable losses in
or diversions of force carried by J-hooks. Also noteworthy is the pattern
of active force versus time: continuous when the headgear is in place
and activated, but overall often a cyclic, interrupted function, as illustrated in Figure 5-29. Often when used in permanent-dentition treatment, the extraoral appliance is worn for just 10 to 14 hr each day. This
interrupted force pattern must influence the choice of magnitude of the
extraoral traction when the headgear complements intraoral mechanics;
the sizes of intraoral and extraoral force must be selected to account for
their interaction.
Fifth, details of structural analysis are suppressed in this chapter.
The emphasis here is not with designs of face bows or head caps or
potential structural failures. In short, the components of the extraoral
appliance must be capable of generating, carrying, and transmitting
forces without excessive deformation, fracture, or loss of stability. Because face-bow and J-hook materials are ductile, exceeding an elastic
limit, whether to place a permanent bend or upon activation, generally
should not be of concern. The practitioner must remember, however,
that elastics and traction bands relax (the former more quickly than the
latter) and, therefore, must be replaced periodically. Routinely, activation should be accompanied by the actual movements of forces generated in and by the traction bands, elastics, and springs, and activating
elements should not be chosen solely by the distances over which they
must be elongated.
Extraoral Appliances
The Cervical-Pull, Face-Bow Appliance
cervical-pull appliance consists of a face bow, a neck pad, and the
element(s) that, when engaged, induce the activating force. The face
bow incorporates an inner bow, which fits intraorally, and the outer
bow. The two parts of the face bow are interconnected, either solidly
(e.g., welded) or by means of a hinge. When the face bow is in place,
that connection is to be slightly anterior of and between the lips and,
when geometric symmetry is present, intersecting the midsagittal plane.
The inner-bow arm ends engage buccal tubes that are affixed to the facial
crown surfaces of the maxillary first or second molars. The active force is
delivered bilaterally to the outer-bow arm ends, transmitted anteriorly
through the outer-bow arms, then through the bows connection into the
inner bow, posteriorly through the inner-bow arms and through stops
into the buccal tubes. The active force is generated by the elongation of
an elastic traction band, one or more pairs of elastics, or a pair of
springs, attached to the outer-bow arm ends. The accompanying responsive force(s) are transmitted to the neck pad which, in turn, distributes that force over the back of the neck.
The forces delivered to the outer-bow arm ends generally have com-
ponents in all three dentofacial-coordinate directions. Although the
principal components (left and right sides) are directed posteriorly,
small lateral components usually exist due to the transverse distance
between the outer-bow arm ends that exceeds the neck width. Vertical
components are often present, due, in part, to the anatomic positions of
the maxillary molar crowns with respect to the cervical region. The elastic(s) or spring, stretched between an outer-bow arm end and a fastener
on the neck pad, is a two-force element under axial load. Figure 9-1
shows a free-body diagram of an elastic traction band and the neck pad
as a unit. The responsive force of the neck against the pad is primarily
pressure distributed over a curved, rectangular area. Depending on the
actual direction of pull, a minor component of the distributed response
may exist tangential to the neck pad and perpendicular to the plane
view. Initially and, perhaps, irregularly over time, a frictional distribution may also exist. The essential features, however, are given in the
two-dimensional sketch. The plane of the diagram generally makes an
angle that is between 15 and 200 with the occlusal plane. The pressure
distribution is apparently somewhat nonuniform, but not to the point of
The tensile strain in the elastic band is essentially constant from one
end to the other (if not immediately upon activation, then shortly there-
after when movements of the head have occurred and the band has
sought a minimum-energy configuration); accordingly, the magnitudes
of the contact forces with the outer-bow arm ends are equal. The resultant of the pressure distribution, then, must dissect the angle between
the lines of action of the forces exerted by the outer-bow arm ends. The
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 9-1. Responsive force system exerted against cervical anchorage.
maximum pressure will be along the line of action of the resultant of
those two forces which, in the symmetric situation, will be in the midsagittal plane as shown in the sketch. Note that the substitution for the
traction band of elastics or springs, fastened directly to the neck pad,
results in no substantial change in Figure 9-1 if the individual activating
elements generate forces of the same magnitude. If one elastic or spring
is stiffer than its counterpart on the other side of the head, the neck pad
will tend to migrate toward the side of the stiffer element and, although
the forces to the outer-bow arms may be unequal as activated, with time
and the neck-pad migration, these forces will approach one another in
0cc/usa/-Plane Analysis
Figure 9-2 depicts a free-body diagram in an occlusal- or transverseplane view of an activated face bow that is geometrically and mechanically symmetric with respect to the midsagittal plane. The symmetry
requires equal lateral force magnitudes, each P0 sin
offsetting one
another. An anteroposterior force balance, with the symmetry, yields
= P0 cos 00, existing due to buccal-tube contact against the stopped,
inner-bow arm end. Also generally appearing in the occlusal view, as
Extraoral Appliances
FIGURE 9-2. A free-body diagram in the occiusal plane of an activated, symmetric face
part of the responsive force systems of the buccal tubes against the
inner-bow arm ends, are first-order couples and lateral forces. Because
they are not directly related to the forces activating the appliance or to
each other, midsagittal-plane symmetry and equilibrium conditions
demand that, separately, the couples (C0) and lateral forces (L0) be equal
in magnitude but opposite in sense to one another.
An occiusal view of the force systems exerted on the molar crowns is
presented in Figure 9-3; the posteriorly and lingually directed force components and couples are action-reaction with their counterparts in Fig-
ure 9-2. If the face bow produces molar movement, with the driving
forces at the buccal surfaces and not through the centers of resistance,
the occlusal-view displacements tend to be distolingual-rolling. The
length of the buccal tube and the bending stiffness of the inner-bow arm
together can produce first-order couples, however. With the senses
shown in the figure(s), the couples C0 may exist immediately upon acti-
vation, due to placements of toe-out bends in the inner-bow arms, or
they may arise in time with initial molar movements, to counter the
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 9-3. 0cc/usa! view of the force system transmitted to molar crowns by the face
bow of Figure 9-2.
distolingual-rotational component of displacement. (The generation of
each couple is through two-point buccolingual contact of an inner-bow
arm end within the buccal tube. The difference in magnitudes of the two
contact forces, if any, is the lateral-force component L0. The action-reaction within the buccal tube is shown in Figure 9-4. The potential size of
the couple is enhanced by the length of the buccal tube, which is also the
magnitude of the inherent moment arm.) The pair of lateral force components, with the senses indicated, reflect a tendency to decrease the
posterior width of the maxillary arch. They would be created upon activation when the passive distance between the inner-bow arm ends in
less than that between the buccal tubes. Another more subtle source is
in response to the potential for labial tipping of the molar crowns, present in occlusally directed components of force delivered by the innerbow arm ends. This latter phenomenon is more easily seen in a view
from a posterior perspective to be presented subsequently.
Generally, as noted, the lateral forces and first-order couples may
exist in response to bends placed in the inner-bow arms prior to activation, or in response to initial molar displacements. Their senses may be
those of Figures 9-2 and 9-3, or the senses of the couples and/or those of
the lateral forces may be opposite to those shown. For example, opposite senses for the couples would result from placements of toe-in bends.
Lateral forces created in the activation of an appliance exhibiting a passive distance between inner-bow arm ends greater than the posterior
arch width measured between the buccal tubes would be opposite in
sense to those shown. The lateral components may also exist in response to a potential in the force system delivered to the molars to tip
the crowns lingually. In Figure 9-3 the anteriorly directed forces F0 repre-
Extraoral Appliances
nner-bow arm
these forces
form C0
FIGuRE 9-4. Two-point contact in an occiusal view, between inner-bow arm end and
buccal tube, which generates C0.
sent buccal contact between the molars and intraoral mechanics (a tiedback, continuous arch wire or activated Class III elastics, for example).
These forces are included in the sketch because they, and the intraoral
mechanics, may exist in conjunction with the headgear. If the function
of the extraoral appliance is to transfer forces, responsive to intraoralmechanics activation, from the molars to the cervical region, the poten-
Bioengineering Analysis of Orthodontic Mechanics
tial displacements associated with the forces F0 are to be negated by the
First-order couples, opposite in sense to those shown and
generated by toe-in bends in the inner-bow arms (and/or in, if present,
the continuous arch wire), would help to stabilize the molars.
The cervical-pull appliance may be modified to produce asymmetric
action against the maxillary terminal molars; desired is the creation of
either a greater distal movement of one molar than the other or the
displacement of one molar while maintaining the position of the other.
Several face-bow designs have been offered, and mechanical asymmetry
must be present to achieve the wanted displacement pattern. Figure 9-5
shows a free-body diagram of a face bow possessing geometric, midsag-
+ C) — P2c2}
+ C)— P1c1}
FIGURE 9-5. A free-body diagram in 0cc/usa! view of a face bow activated by unequal
elastic or spring forces.
Extraoral Appliances
ittal-plane symmetry in the passive state and activated by concentrated
loads against the outer-bow arm ends, which differ in magnitude. As
mentioned previously, such a differential in delivered forces cannot be
maintained by a stretched traction band or by unlike elastics or springs
activated against a neck pad. Recommended instead is the activation of a
pair of elastics or springs, with one of the pair differing from the other in
stiffness and/or amount of activating deformation, against a well-fitting
head cap. Below the figure are the equations for the magnitudes of the
distal driving forces, action-reaction counterparts of Qi and Q2. The formulas are obtained from an anteroposterior balance of forces exerted on
the face bow and a moment balance about a convenient point such as B.
Note that P1 and P2 are occlusal-plane components of the total elastic or
spring forces. If the distances c1 and c2 are small compared to the lateral
dimension c between the buccal-tube slots, with the outer-bow arms as
activated close to but not touching the cheeks, a reasonable approximation is
and Q2 individually approaching P1 and
A disadvantage in the asymmetric design just described is the dependence upon stability of the head cap. Another procedure to obtain
unequal driving forces, which will accommodate neck-pad anchorage, is
to engage an asymmetric face bow. These face bows are available with
the inner and outer bows hinged or solidly connected to one another.
The hinged face bow, with separate free-body diagrams of the inner and
outer bows, is sketched in occlusal views in Figure 9-6. The axis of the
hinge is vertical and any frictional couple that might be present at the
hinge has been neglected. To maximize the potential differential in sizes
between Qi and Q2, the hinge is located as distant from the midsagittal
plane as is functionally permissible. Because the magnitudes of the
forces delivered to the outer-bow arm ends (from a traction band) are
equal (P0) and given the hinge location, to prevent the outer bow from
rotating about the hinge (from a moment imbalance, clockwise with the
hinge to the left of the midsagittal plane as shown in the figure), resulting in one outer-bow arm contacting the cheek and interfering with the
wanted force system, one outer-bow arm must be longer than the other.
The longer arm is on the same side of the midsagittal plane as the hinge;
the objective is a balance of moments of the two traction-band forces
about the hinge axis, point H. Since these two forces are of equal magnitude, the direction of the force between bows, transmitted through the
hinge, may be obtained through the graphical construction shown in
Figure 9-6; the line of action of Ph bisects the angle between the lines of
action of the forces exerted on the outer-bow arm ends, and the magnitude Ph is somewhat less than twice one outer-bow force as indicated.
The hinge force delivered to the inner bow is noted to include a lateral
component, a generally unwanted side effect of the asymmetric face
bow. Because of the hinge position and since the angle between Ph and
Pa is not large, the intersection of the line of action of Ph and the transverse line between the buccal-tube centers, point E, is some distance
away from point B. The larger this distance, the greater the mechanical
asymmetry. Force and moment balances from the free-body diagram of
4— b1
where b = b1 + b2
FIGURE 9-6. 0cc/usa/-view, free-body diagrams of inner- and outer-bow portions of an
asymmetric, hinged face bow.
Extraoral Appliances
the inner bow yield the formulas given in Figure 9-6 for Qi and Q2; in an
approximate analysis it is often reasonable to equate cos 0d to unity (one)
since this angle is usually small.
Although the hinged face bow should be able to deliver a substantial
differential in the distal driving forces delivered to the molars, inherent
with the hinge connection between bows is a lack of stability; the outer
bow may rotate rather easily and an arm contact the cheek. A patient,
wearing the appliance at night, for example, will amend the delivered
force system when sleeping with either side of the face against the
Several asymmetric face-bow designs exist that exhibit a solid connection between inner and outer bows. One such design has the connection between bows off the midsagittal plane and the passive posterior
extents of the outer-bow arms the same. This face bow, with a traction
band and neck pad has been shown to produce only a small differential
in the forces against the molars (Drenker, 1959); the mechanical asym-
metry is only slight and, in essence, is due to the bending-flexibility
differential in outer-bow arms associated with their unequal lengths.
A larger differential has been obtained with a design displaying the
rigid inner-to-outer-bow connection, on or off the midsagittal plane, and
outer-bow-arm ends exhibiting different posterior extents. This type of
face bow is shown in Figure 9-7; again, the activation is delivered by a
01 =
where b = b1 + b2
FIGURE 9-7. 0cc/usa/-view force diagrams for an asymmetric face bow exhibiting a
solid, midline connection between inner and outer bows.
Bioengineering Analysis of Orthodontic Mechanics
elastic band. The force diagram on the left suggests the
achievement of unequal actions delivered to the molars; the outer-bow
arm lengths and the patient anatomy must result in the line of action of
the resultant of the traction forces (each of magnitude P0), passing to
one side of point B. The free-body diagram on the right in Figure 9-7
shows only the anteroposterior force components; because the lateral
force components are colinear, their analysis may be undertaken independent of the anteroposterior force system. A moment balance indicates that here also, as with the hinged face bow, the magnitudes of the
forces Qi and Q2 are in the same ratio as the distances b2 and b1.
The size of the differential in delivered distal driving forces, using
the asymmetric face bow and traction-band activation, depends on the
eccentricity with respect to point B of the lines of action of the resultant
active force. If this eccentricity is greater than one-half of the intermolar
width measured between the buccal-tube centers, the mesiodistal forces
to the molars will be opposite in sense to one another; although improbable in practice, "on-paper" analysis indicates this is a possibility. If the
intent of the differential is to move just one molar, the achievement of a
near-zero force against one and a sizable distal driving force against the
other molar requires careful adjustments by the practitioner, particularly
with respect to outer-bow arm lengths and as-activated configurations.
Some assistance can be gained through concurrent use of intraoral mechanics delivering a balancing, mesial force to the molar receiving the
smaller distal headgear force.
As previously indicated, all asymmetric face bows exhibiting unequal
distal driving forces also exert lateral action. The net active lateral force
has been symbolized in Figures 9-6 and 9-7 as Pi. but its division between the molars cannot be quantified by force and moment balances
alone. In an overview of the problem, it is first necessary to realize that a
posteriorly directed force transmitted to the inner bow, whether positioned in the midsagittal plane or not, tends to widen the passive distance between the arm ends as indicated on the left in Figure 9-8 in the
FIGURE 9-6. Occlusal-view diagrams displaying lateral-force action delivered by an
inner bow to the molars.
Extraoral Appliances
presence of lingually directed forces against the arm ends (action-reac-
lion counterparts of the buccally directed forces on the molars). To be
superimposed on this effect at the molars is the response of P1. which
produces forces at the molars that are identical in sense (although perhaps not equal in magnitude) as is indicated in the center sketch of the
figure. The superposition, arguing only on the basis of senses of forces,
results in an excess of lateral force on the short-arm side and a relatively
small, net lateral load on the molar adjacent to the long arm of the outer
bow. Assuming that the lateral forces from the posterior component of
action are somewhat larger than those associated with the asymmetry,
and that P1 is approximately halved between the molars, the net lateral
response at the molars will generally be as shown in the diagram on the
right in Figure 9-8.
A third contributor to the lateral action exists if the inner-bow arm
ends do not fit passively into the buccal tubes, due for example, to expansion or constriction of the inner bow with the intention of altering
the intermolar width. Activation of this effect alone creates forces
against the molars that are equal in magnitude, but opposite in sense to
one another. This facet of the problem is also quasi-statically indeterminate and a complete analysis of the total problem is beyond the level of
the present discussion. If the face bow reflects geometric and tractionforce symmetry with respect to the midsagittal plane, this symmetry will
extend to the delivered lateral force components; with asymmetric geometry and action, the lateral forces transmitted to the molars will likely
differ substantially from one another in magnitude, and may be of the
same or opposite sense depending on the relative influences of the three
possible contributions as discussed.
In concluding the occlusal-view analysis of asymmetric face bows,
note that these appliance elements are "reversible." The longer outerbow arm can be positioned on either side of the face by simply flip-flopping the face bow. Two configurations of the bow then are available, and
the geometric and mechanical configurations individually are mirror
images of one another relative to the
Buccal-View Analysis of the Cervical Appliance
An analysis from a buccal perspective yields substantive component
relationships between the active forces delivered by the extraoral appliance and potential intraoral displacements. If the molars alone receive
the action of a face-bow appliance, should tipping, bodily, or torquing
displacements be expected and in which direction(s)? If a stiff arch wire
carries the loading anteriorly from the maxillary molars, might the maxila be rotated and, if so, might the anterior bite be opened or deepened?
One-half of the face bow is seen in the typical buccal view, and an
analysis involving just that half bow and the interacting half-arch would
be convenient. In general, "cutting" the face bow at the midsagittal
plane exposes an internal force system there with resultant force and
couple vectors having, individually, any conceivable directions in space.
Bioengineering Analysis of Orthodontic Mechanics
In the presence of midsagittal-plane symmetry, however, the internal
force system transferred across the plane may be shown to include only
a mesiodistal force and a first-order couple, and neither of these components appears in a buccal view. Midsagittal-plane symmetry is often
present in extraoral mechanics, and the buccal-view analyses to follow
will assume that symmetry unless otherwise noted.
Figure 9-9 shows a free-body diagram of the right half of a symmetric
face bow as viewed from a buccal perspective. One outer-bow arm and
one inner-bow arm appear in the sketch. The weight of the half face bow
is neglected and contact force systems exist at three locations (although
anteriorly where the left portion of the face bow connects to the right, no
force-system components are present in this view as noted previously).
The elastic, spring, or traction band contacts the outer-bow arm end and
the force Pb is the buccal-plane component of the total force transmitted
through this connection. (From a practical standpoint, the total activating force generally makes a small angle with the associated sagittal plane
and, accordingly, in magnitude, Pb often differs negligibly from it.) The
force system exerted by the buccal tube on the inner-bow arm end in this
view is shown in its components; Q0, was seen earlier in Figure 9-2,
perpendicular to the occiusal plane, and Cb is a second-order couple.
Measurement in this plane of the magnitude and direction of the force
delivered to the outer-bow arm end, the anteroposterior distances i and
o, and the angle 0b between the bow arms in this view, with force and
moment balances, enable the determinations of the magnitudes of the
three components at the inner-bow arm end.
Although the figure shows the half face bow in its activated state,
FIGURE 9-9. A free-body diagram in a bucca! view of one-half of a symmetric face bow.
Extraoral Appliances
note that the sense of the couple Cb is directly related to the vertical
movement of the anterior tip of the face bow during activation. When
activated, the face bow in Figure 9-9 rotated clockwise (from a buccal
perspective); its anterior tip moved down, generating the counterclockwise, responsive couple Cb. The sense of the couple corresponds to the
location of the line of action of Pb with respect to the inner-bow arm end.
Had that line of action been positioned above the inner-bow arm end,
the sense of the couple would have been clockwise and the displacements of the face bow generally and its anterior tip would have been
counterclockwise and up, respectively, during activation. This knowledge is useful in the preparation of the face bow before engagement; as
activated, the anterior tip must be positioned in the embrasure between
the lips.
Figure 9-10 shows a free-body diagram in the buccal view of the right
maxillary molar; the buccal tube (which receives the force, originating
extraorally, from the inner-bow arm end) is in effect, integral with the
tooth. The three components of the force system exerted on the crown
through the tube are action-reaction counterparts of the components of
Figure 9-9. For now, any effects of intraoral mechanics upon the molar
are not included. The distributed periodontal response to the molar
crown loading, as it would appear in this view, is shown as three resultant components referenced to the center of resistance of the molar. Estimating the long-axis distance a between the tube and the center of resist-
ance as one-half the overall tooth length, with the analysis of the
free-body diagram of the half face bow, (Fig. 9-9), force and moment
R, =
= Cb + Q0(a)
FIGURE 9-10. A free-body diagram in a buccal view of a right maxillai'y molar subjected
to face-bow action.
Bioengineering Analysis of Orthodontic Mechanics
and C, as indicated. Since
permit the determinations of
this force system is action-reaction with the resultant components of the
tooth action on the periodontium, the cervical-pull headgear exernplitied displays the following orthodontic-displacement tendencies associated with the buccal view: overall distal movement, distal crown tipping, and extrusion.
A more direct procedure to determine the displacement potential in
the buccal view would avoid the buccal-tube force analysis and proceed
directly to the center of resistance. Such a process can be validly undertaken provided that the molar alone receives the head-gear force and is
not in contact with any other appliance elements. Shown first in Figure
9-11 is a sketch showing the half face bow of Figure 9-8 and the molar of
Figure 9-10. The force-system transmission through the inner-bow-armend and buccal-tube contact is now internal. Needed here, besides the
magnitude and direction of Ph, is a line parallel to Pb through the center
of resistance of the molar; the dimension e and the direction of Pb could
be taken directly from a headplate radiograph, for example. The force Pb
is then "moved" to the new location at the center of resistance, decomposed into mesiodistal and occlusogingival components, and the couple
inserted to compensate for the change in lines of action of Ph (see Fig.
2-17). The transformed force system, now at the center of resistance of
the molar, is displayed in Figure 9-12 and is action-reaction with the
system of Figure 9-10; each component reflects an orthodontic-displacement tendency as previously described.
The graphical force analyses of Figures 9-11 and 9-12, together with
the associated displacement-potential interpretation, indicate that the
direction of Ph and the proximity of its line of action to the center of
resistance of the molar form the basis for the displacement-format prediction in the sagittal plane. The direction of the activating force is that
FIGURE 9-11. A buccal view of the molar, the half face bow, and the activating force on
the right side.
Extraoral Appliances
= Pb(e)
FIGURE 9-12. A force system at the center of resistance of the molar, mechanically
equivalent to Pb of Figure 9-10.
of the traction band, elastic, or spring as it connects to the end of the
outer-bow arm. Restricting attention to neck-pad anchorage, in the buccal view the line of action of Pb must pass through points at the outerbow end and at the back of the neck. Figure 9-13 shows an activated,
cervical-pull face bow oriented such that the line of action of Pb passes
through the center of resistance of the molar. Although no potential for
second-order tipping results from this configuration, a sizable extrusive
component of force is present with Pb directed toward the cervical region. The method of eliminating the extrusive potential, generally present in the cervical-pull appliance, is shown in Figure 9-14; the outer-bow
arm bent down allows Pb to approach a direction parallel to the occiusal
plane, but also produced is a large distal-crown-tipping couple. The
physical parameters available in this extraoral appliance with which to
FIGURE 9-13. A buccal view of the face-bow configuration to minimize the potential for
second-order tipping with the appliance subjected to cervical-pull action.
Bioengineering Analysis of Orthodontic Mechanics
C0 = Pb(e)
FIGURE 9-14. Outer-bow arms bend down to eliminate the potential for extrusion in
cervical-pull action.
vary the location and direction of the line of action of Pb are the length of
the outer-bow arm and the angulation of the outer-bow arm, as activated, with respect to the occlusal plane. Specific combinations of arm
length and angulation can result in coincident lines of action of
example, referring to Figure 9-13, an identical action can be obtained
with a shorter outer-bow arm and a larger angle 6h or a longer arm and a
smaller angle. It is possible to create a mesial-crown-tipping couple,
opposite in sense to the couples in Figures 9-12 and 9-14, but a long,
outer-bow arm is needed that must be bent severely upward with respect to the occlusal plane. A line of action is obtained that passes superior to the center of resistance of the molar, but a sizable potential for
extrusion is also created (again, because the line of action of
through the cervical region).
The buccal-view analyses discussed thus far have assumed that the
maxillary molars are the sole recipients of the extraorally generated force
system. Although primarily providing distal-driving potential, and
often also producing a tendency toward extrusion, the face-bow cervical-pull headgear may as well cause second-order rotations. If an arch
wire is engaged into and stopped and tied against the buccal tubes, the
force system delivered to the buccal tubes may be transmitted anteriorly
and, therefore, distributed throughout the dentition and into the maxilla
at large. To distribute all components of the delivered force in a nearuniform fashion to the entire arch—particularly the occlusogingival
(vertical) and second-order rotational components—the arch wire must
possess high bending stiffness. With such a wire engaged, the arch approaches a rigid entity and the procedure of Figures 9-11 and 9-12 may
be used to predict displacement of the half arch as a unit, but a needed
reference is the center of resistance for the half arch. Although not
Extraoral Appliances
known precisely, for an arch with all spaces closed the anteroposterior
coordinate of the center of resistance will approximately divide the total
root-surface area of the teeth engaged in half; this places the reference
point slightly anterior of the second premolar in a nonextraction case
with engagement terminating with the first molars. With the known
geometry of the maxilla, the vertical coordinate may be reasonably
placed near the apex level of the premolars. (Note that reducing the
bending stiffness of the arch wire will effectively move the center of
resistance posteriorly.)
Figure 9-15 depicts an appliance assembly in which the cervical-pull
force is, in effect, delivered to the entire half arch. (The magnitude of Pb
here must be much larger than that delivered to and absorbed by the
molar alone; assuming similar displacement formats, the delivered force
to the half arch should be magnified by a factor equal to the ratio of
root-surface areas affected.) With the angle of the activated outer bow
approximately the same as the inclination of the applied extraoral force,
in addition to the distal action that would tend to reduce an overjet, the
rotational potential is clockwise with the line of action of Pb occlusal of
the center of resistance. The resulting tendency is to close down the bite
in the anterior region. To create the opposite potential, toward opening
the bite anteriorly, the line of action of must be located superior to the
center of resistance. Hence, the outer-bow arms must be long and must
be bent upward. Moreover, because of the sizable force required, the
face bow must be very stiff since the application of force to its outer-bow
arm ends will tend to deflect them toward the occlusal plane. Also seen
in this arrangement, sketched in Figure 9-16, is a strong tendency for
extrusion; this must be countered by posterior occlusion if opening of
the bite (in the anterior region) is to be part of the overall result.
A somewhat unusual application of the cervical-pull face-bow headgear is examined in closing this subsection. The observed, vertical move-
FIGURE 9-15. Cervical-pull, face-bow delivery of extraoral force to the entire maxillary
arch to reduce overjet and an anterior open bite.
Bioengineering Analysis of Orthodontic Mechanics
= Pb(e)
FIGURE 9-16. Face-bow geometry necessary for reduction of an overbite.
ment of the anterior tip of the face bow during activation has suggested
a means of intruding maxillary incisors or up to six anterior teeth. Spurs
are affixed to the face bow to contact the incisal edges. Upon activation,
the force applied to the outer-bow arm ends, in the conventional appliance transmitted through the outer and into and through the inner bow
in an uninterrupted manner to the molars, is transferred instead in part
to the anterior teeth by means of the spurs. The molars provide intraoral
anchorage and a fulcrum. A free-body diagram in the buccal view is
shown in Figure 9-17. Again, symmetry with respect to the midsagittal
plane is assumed and one half of the face bow is sketched. To obtain the
desired action, the force must be located so as to produce a counter-
clockwise moment about the end of the inner-bow arm. Force- and
moment-balance relationships are written in the figure. The three components at the inner-bow arm end plus the force Pa against one half of
the involved anterior teeth yield a total of four unknowns, one more
00 =
0, = F8 +
= F8e) —
FIGURE 9-17. Modification of the face bow to gain the potential for anterior intrusion.
Extraoral Appliances
than the number of available quasi-static equations. To complete the
solution, a reading of 2Fa would have to be taken with a force gauge
following activation of the appliance.
Coronal-Plane View and Comments on the Asymmetric Problem
The delivery of force by the symmetric face-bow, cervical-pull headgear, viewed from the posterior into a coronal plane, is shown in Figure
9-18. This third plane view of the activated appliance indicates the potential, noted earlier, for faciolingual displacement of the molars arising
from the vertical components
transferred from the inner-bow arm
ends and the locations of their lines of action with respect to the centers
of resistance. Because the bow arms have circular cross-sections, with
the delivery of force to the molars at the buccal surfaces and not through
the centers of resistance, and if the senses of the vertical forces are as
FIGURE 9-18. Posterior views of the face bow (above) and the molars (below) showing
buccolingual responses to vertical force components.
Bioengineering Analysis of Orthodontic Mechanics
shown, a substantial tendency exists for lingual crown tipping. If the
inner bow is initially passive buccolingually, the initiation of lingual
crown tipping will generate the lateral forces (L) as indicated in the figure. Due to their characteristic lengths, however, the inner-bow arms
are rather
therefore, the magnitudes of these lateral components will build slowly and sizable lingual displacements could occur
before the forces would become significantly resistive. Alternatively, the
inner bow may be activated buccolingually such that the lateral forces
exist immediately upon engagement of the appliance, either with the
senses shown or their opposites. In the latter configuration the lateral
forces would enhance the lingual-tipping potential unless the vertical
have opposite senses to those shown, and as such provide the
potential for facial crown tipping. In summary, without the tendency for
development of third-order resistive couples, the existing buccolingual
stiffness of the inner bow is alone the available inherent deterrent to the
tipping tendency associated with the vertical force components delivered to the molars.
Throughout this section, in viewing force diagrams from a buccal
perspective, the half face bow has been used and, with midsagittalplane symmetry present, the force system transmitted between the left
and right halves of the face bow has no components in the midsagittal
plane. In the asymmetric situation when, for example, the component in
the buccal plane of force to the right-side outer-bow arm end is larger
than its counterpart on the left side, perhaps all three midsagittal-plane
components (anteroposterior force, vertical force, and third-order couple) will exist in the right-side buccal view. These components are not
large compared to the individual, corresponding components transmitted from the inner-bow arm end to the buccal tube, but they render the
problem quasi-statically indeterminate. Even the sense of the midsagittal-plane components may be difficult to determine; however, if the face
bow is geometrically symmetric in the passive state, the observation of
the displacement of the joint of the inner bow to outer bow upon activation can provide the sense(s) of one or more of the components. The full
analysis of the asymmetric headgear, to extend that portion that was
undertaken earlier (see Figs. 9-5 through 9-7), is beyond the general
level of presentation of this text and, perhaps, is of questionable clinical
Canine Retraction with Headgear
movement of the maxillary or mandibular canine teeth into spaces
left by the extractions of first premolars requires the controlled action of
distally directed force. That force may be activated against posterior-segment anchorage, but to avoid the potential of unwanted mesial movement of those segments, associated with the response to the activation,
extraoral anchorage may be used. Driving forces may be created through
Extraoral Appliances
the extensional activation of a traction band, elastics, or springs, and
transmitted to the dentition through J-hooks. The elastics or springs are
also attached to a head cap or neck pad; the traction band transmits the
responsive force through a neck pad to the cervical region. The extraoral
ends of the J-hooks connect to the activating element(s) in a hinged
fashion such that only a force—no couple—may be transmitted through
the connection. Hence, the ends of the 1-hook tend to line up with the
direction of the activating force. The end of the curved portion of the
J-hook is formed around the arch wire and makes contact with the mesial extent of the canine bracket (in the absence of the need for an interconnecting, sliding jig); again, only a force can be transmitted through
the connection. The principal intent is to deliver a distally directed force,
without mechanical interruption, to the canine. The arch-wire system
provides overall guidance—attempting, initially, to keep the canine in
faciolingual alignment as the movement begins, and thereafter to influence the displacement format.
Whether these mechanics are symmetric or not with respect to the
midsagittal plane, sufficient in a transverse perspective is an occlusal
view and an analysis of just one side of the appliance because the Jhooks do not contact one another. The noteworthy features of Figure
9-19 are the occlusal-plane components of the right-side force of activation P0. and two responsive force components. If the elastic or spring is
fastened to a head cap, P0 is posteriorly directed; with a cervical-pull
strap, as indicated previously, the force has a small lingual component.
The J-hook connects to other elements only at its ends. Similar to a
two-force member, the J-hook attempts to align its ends on the line of
action of P0 upon activation. Because one end is intraoral and the other is
extraoral, the alignment cannot be accomplished and L0 is created, exerted by whatever guides the J-hook. The third force on the J-hook, K3,
is the collective response to the activation in this view of the bracket and
arch wire. These three forces must balance as indicated by the closed
force triangle. The action-reaction counterpart of R0 is shown decomposed into distal and lingual components in the occlusal view of the
canine that completes the figure. The distal component actually acts
against the bracket and is the driving force: two comments concerning
that component are pertinent:
1. Because its line of action does not pierce the long axis of the canine,
antirotation mechanics are necessary (dashed in the figure) if distolingual rotation is to be prevented.
2. A portion of this component may not reach the canine, but due to
friction it is taken into the arch wire and carried to the posterior segment.
The lingual component is generally undesired; acting through the arch
wire, its tendency, at least during the initial portion of the canine movement, is to constrict the intercanine arch width. As the displacement
proceeds, though, relative to an unchanging direction of R0, the distal
Bioengineering Analysis of Orthodontic Mechanics
— '(Counterrotation
FIGURE 9-19. 0cc/usa/-view diagrams of the force systems exerted on the J-hook,
canine crown, and arch wire during retraction of the right maxillary canine.
component increases and the lingual component decreases in magni-
tude due to arch curvature.
Figure 9-20 shows a right-side buccal perspective of the maxillary
canine together with the guiding arch wire, the J-hook, a sliding jig, and
the active force component Ph depicted as posteriorly directed, characteristic of a straight-pull headgear. With the canine receiving the entire
force in the absence of friction, the displacement analysis may be carried
Extraoral Appliances
Arch wire
FIGURE 9-20. A right-side buccal view of a maxillary canine under retraction through
use of a straight-pull headgear.
out in somewhat the manner of Figures 9-11 and 9-12. Noting the posi-
tion of the center of resistance of the canine in this view relative to Pb,
without the arch wire engaged the distal displacement would be largely
uncontrolled crown tipping. To the extent that this tipping is unwanted,
the arch wire through its bending stiffness is the available source of a
counter-tipping couple, either present immediately upon arch-wire engagement through an active, second-order bend, or allowed to initiate
and grow (in magnitude) following initial canine tipping that eliminates
any second-order clearance and establishes two-point contact between
bracket slot and arch wire. If the applied force
includes a vertical
component, that component will be transferred from the J-hook to the
arch wire at their contact location. In the absence of the sliding jig, that
vertical component will be largely carried to the canine but, depending
on the bending stiffness of the arch wire, it may be distributed anteriorly
and posteriorly from the contact location. With cervical anchorage, an
occlusally directed component will be part of and add the tendency to
extrude the canine and enhance anteroposterior, whole-arch curvature.
On the other hand, activation of high-pull action from a head cap produces an opposite, vertical, displacement potential.
Figure 9-21 shows the activation of a high-pull headgear to the mandibular canine in a buccal view. The location of the line of action of
passing occlusal to the center of resistance of the tooth indicates a tendency for distal crown tipping which the arch wire may resist. The direction of reflects the existence, in addition to the distal driving component, of a vertical component with potential to extrude the canine and,
with an engaged arch wire possessing moderate stiffness, to reduce toward reversal of any curve of Spee. Generally, the delivery of extraoral
force to mandibular locations demands the consideration of several fac-
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 9-21. A right-side buccal view of mandibular canine retraction with a high-pull
tors beyond the direction of applied force with respect to the arch-form
plane and the line-of-action location relative to a center of resistance.
This discussion is undertaken in a separate, subsequent section of this
Extraoral Force Delivered to an Anterior Segment or an Entire Arch
section considers the bilateral delivery of force, originating
extraorally, to various locations along the arch wire with the force carried into the oral cavity by J-hooks. The extraoral forces are generated by
elastics or springs fastened posteriorly to a head cap or a neck pad, or
possibly by a traction band transferring responsive force through a neck
pad to cervical anchorage. The activating element(s) are connected to the
J-hooks that carry the force to the arch wire. Contact with the arch wire
on each side is made between the central and lateral incisors, distal to
the lateral incisor, or distal to the canines. Through incorporation of
sliding jigs, with the J-hooks contacting their mesial extents, the
extraoral force may be transmitted directly to the buccal segments.
Generally, force-transfer points on the arch wire are chosen accord-
Extraoral Appliances
ing to the intended distribution of the force delivered from the extraoral
source. The extent of that distribution depends largely on characteristics
of the arch wire. Again, the connections allow transfer of force in any
direction, but 1-hooks will not transfer couples. If the incisal segment
alone is to receive the force, the transfer points are between the central
and lateral incisors; if the six anterior teeth are to be displaced as a unit,
the 1-hooks will likely deliver the force on each side between the lateral
incisor and the canine. In these two instances, either the arch wire is not
stopped posteriorly or loops in the wire are activated to aid in the displacement. If the intent is to affect the entire arch (and produce orthope-
dic movement), the transfer points are distal to the canines and the
buccal segments are appropriately stopped and ligated together to ensure both anterior and posterior distribution of the active force.
Shown in Figures 9-22 and 9-23 are occiusal-plane views of applied
and responsive force systems with delivery (action-reaction) to the maxillary incisal segment and to the maxilla as a whole, respectively. Both
free-body diagrams reflect geometric and mechanical symmetry relative
to the midsagittal plane. In Figure 9-22 the distribution is shown uni-
FIGURE 9-22. A free-body diagram in an occlusa( view depicting extraoral-force delivery
to the maxillary incisal segment.
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 9-23. A free-body diagram in an 0cc/usa! view depicting extraoral-force delivery
to the maxillary arch as a unit.
form over the four teeth, a reasonable expectation in view of the contactpoint locations and the fact that each force essentially reaches just two
dental units. In the whole-arch distribution of Figure 9-23, the occiusalplane component of the contact force on one side divides with one por-
tion transmitted toward the anterior and the remainder carried first to
the terminal molar and then, by means of ligation, anteriorly throughout
the buccal segment. To attempt to quantify the distributions of force is to
undertake the solution of a highly quasi-statically indeterminate problem, particularly in the latter instance. With a stabilizing arch wire engaged, however, those distributions probably appear somewhat as they
are sketched, and a rough approximation for one side is obtained by
dividing P0 by the number of teeth affected in the half arch.
Several additional points of consideration are pertinent to this occlusal-view discussion. Again, due to the direction of P(, and the location of
Extraoral Appliances
its line of action with respect to the anterior end of the J-hook, the lateral
forces appear as first noted in the canine-retraction procedure with
headgear. The magnitudes P0. the design of the head cap, and the softtissue facial anatomy of the patient relative to the intraoral J-hook connection points all influence the size of the lateral force F0. For example, if
cervical pull is appropriate, the narrower width of the neck results in
relatively small lateral forces when elastics or springs are activated
against a neck pad. In contrast to the canine-retraction process, because
in the present mechanics the force from the J-hook is transmitted directly to the arch wire, the entire force delivered is effective. Continuing
in the occlusal view, the intent is to transfer—wherever the delivery
points along the arch wire are—posteriorly directed forces; the applied
forces have that direction. Friction may, however, divert some of the
force generated in the activating element. Frictional resistance will accompany the presence of the lateral force and essentially in direct proportion to the size of that force. The frictional force is transmitted by
direct contact from the J-hook to its guide, into the anterior extent of the
head cap, and to the side of the face or head. Also, in the arrangement of
Figure 9-22, friction between arch wire and brackets may take a portion
of the delivered force, intended exclusively for the incisal segment, posteriorly into the buccal segments. Precautions taken to keep the delivered force in the anterior segment include reducing the cross-sectional
wire size in the buccal segments and using light ligations—generally
avoiding bracket-wire binding and angulations. A last point to be considered is midsagittal-plane symmetry. Although with the face-bow and
canine-retraction headgear, asymmetric or unilateral action is now and
then required, not so with the extraoral appliance presently being discussed. A symmetric activation is a possibility, using unlike elastics or
springs, but with force delivery to the arch wire the inability to sufficiently control the distribution of that force seemingly renders the attempt useless.
As in previous analyses within this chapter, the view of the extraoral
appliance from a buccal perspective leads to the substance of the potential displacement format. Of particular importance are the direction and
location of the line of action of the activating force with respect to the
dental units to be affected. As in earlier discussions of symmetric headgear, attention is focused on one elastic or spring or traction-band force
and the right half of the arch (to which the J-hook is attached). In the
buccal view the J-hook will align with the direction of the force generated in the activating element, so both the angulation of the force and its
line-of-action location are determined by two points: the connection of
the activating element to the head cap or neck pad and the point of
contact of the anterior end of the J-hook with the arch wire (or spur
affixed to it). The relative magnitude and the sense of the vertical component of the active force are controlled largely by the anchorage arrangement (high-, straight-, or cervical-pull). The direction of the buccalview component of active force may range from 45° or more above the
occlusal plane to, perhaps, 15° below it. This angulation is also affected
Bioengineering Analysis of Orthodontic Mechanics
somewhat by the point of delivery to the arch wire; for example, for a
given head cap the high-pull angulation is greater with delivery distal to
the canine than when the J-hook attachment is between the central and
lateral incisors.
Although these headgear are often employed to reinforce buccal anchorage intraorally, they may be used to produce wanted displacements
and the function of the arch wire is solely to control the distribution of
the extraoral force. The analysis procedure, first illustrated in Figures
9-11 and 9-12, may be used. (This procedure may also be used in sepa-
rate analyses of the effects on the same portion of the dentition of
extraoral-force application and those of intraoral mechanics, the two to
be superimposed.) The process requires initial identification of the portion of the half arch to receive the headgear force. Next, the location of
the center of resistance of the segment is approximated in the manner
discussed previously (influenced by the distribution of root-surface area
and by the distribution of force determined by the extent and the bending stiffness of the arch wire). The characteristics of the delivered force
are then evaluated in terms of the dental segment receiving it and the
force is "moved" to the center of resistance. Figures 9-24 and 9-25 depict
buccal views of straight- and high-pull-headgear action to the entire
maxillary arch. To distribute the force as uniformly as possible, the attachment is to a location near the center of the half arch; a stabilizing
arch wire having a large cross-section should be used. The extraoral
force is transferred to the center of resistance of the half arch, the necessary couple is added in association with the line-of-action displacement
in Figure 9-24, and in Figure 9-25 the force is decomposed into anteroposterior and vertical components. The proper size of extraoral force is
determined by the procedures discussed in Chapter 5; the magnitude
FIGURE 9-24. A buccal view of straight-pull, extraora! force delivered to the maxillary
Extraoral Appliances
FIGURE 9-25. A buccal view of high-pull extraoral force delivered to the maxillaiy arch.
must be relatively large to produce an orthopedic displacement of the
In Figure 9-25 the potential whole-arch displacement components are
intrusive and toward the posterior; if the line of action passes through
the center of resistance, bodily movement (translation—no rotation)
should occur. Any overjet should be reduced and any intrusion of a
previously leveled arch would diminish an existing anterior deep bite.
With the straight-pull action of Figure 9-24, the potential for reducing
overjet is again present, but vertical displacements are associated with
the overall rotational potential (clockwise) suggested by the couple. Although no whole-arch vertical movement should take place, clockwise
rotation of the maxilla, as viewed from the right side, results in closing
or deepening of the bite anteriorly. If the mechanics are changed only by
using a flexible rather than a stiff arch wire, the center of rotation will be
more anteriorly located and the vertical action would be more concentrated in the locality of the connection of the J-hook to the arch wire. The
high-pull headgear would tend to intrude the middle portion of the half
arch, reducing any curve of Spee toward reversing that curvature. Addi-
tional effects of using the more flexible arch wire would be a minor
shifting of the force distribution posteriorly and possibly reducing the
stability of the mechanics. (In Figure 9-24 this reduction in stability
would become a concern only after some rotation had occurred, creating
an angle between the occlusal-plane and J-hook directions. Action
would then, in effect, be transferred toward the anterior, resulting in
more pronounced anterior bite closure and a reversed curve of Spee.)
Extraoral force is delivered to the anterior or incisal segment to pro-
Bioengineering Analysis of Orthodontic Mechanics
duce localized displacements there or to offset potential, unwanted dis-
placements associated with intraoral mechanics. To concentrate action to
the incisal segment, the engaged arch wire may be flexible and left rela-
lively unrestricted in the posterior segments to slide through brackets
and buccal tubes. Figure 9-26 shows the action on an incisal segment
from a straight-pull headgear. Replacing the I-hook force with an equivalent force system at the center of resistance of the segment reveals, in
addition to the potential for lingual driving, a rotational tendency toward increasing the long-axis angulation with respect to the occlusal
plane. In the absence of crown contacts with canines and without resistance from the arch wire from the posterior, a simple-tipping displacement is produced. Substitution of cervical-pull force has little effect on
the moment arm (and the potential for rotation) but adds the tendency
for extrusion. Replacing the straight-pull with high-pull action provides
an intrusive potential and reduces the length of the moment arm.
The delivery of extraoral force to the incisal or anterior segment may
also generate a third-order torque in the arch wire. Figure 9-27 shows
the delivery of high-pull extraoral force to a hook, affixed to the arch
wire and extending gingivally from it, conceivably in an effort to direct
the line of action of the delivered force through the center of resistance
of the segment. Because the active force has a moment arm with respect
to the arch wire, a force and couple are transmitted into the wire. For the
objective (bodily movement of the segment) to be achieved, the couple
(torque) must be transferred to the anterior teeth; hence, the arch wire
must be rectangular or have torquing loops incorporated into it.
Figure 9-28 shows a free-body diagram of an anterior section of arch
wire with the hook attached, indicating the force and couple delivered to
= Pb(e)
FIGURE 9-26. A buccal view of straight-pull-headgear force to the maxillary incisal
Extraoral Appliances
C re
Arch wire
FIGURE 9-27. A buccal view of high-pull-headgear force to the maxilla,y incisal
the wire; also shown is the action-reaction transfer of the force system
from the wire to the segment. Similar force systems may also be produced by a straight- or cervical-pull appliance, but differing from that in
Figure 9-28 in the vertical component. The anchorage assembly is chosen according to the desired direction of the applied force; controlling
the magnitude of the torque delivered are the magnitudes of the applied
F0 =
Ta = F0(e)
FIGURE 9-28. Force diagrams in the buccal view showing headgear-generated torque
in an anterior section of a rectangular arch wire (left) and the transfer of that torque to
the incisal pair (right).
Bioengineering Analysis of Orthodontic Mechanics
and its moment arm with respect to the center of the wire crosssection. Also, if the hook extends occlusally from the wire, the sense of
the torque is reversed from that of Figure 9-28. Note that the couple
generated in the arch wire differs conceptually and, therefore, in the
manner of interpretation with regard to displacement potential, from
the couple obtained in moving the line of action of the extraoral force
directly to the center of resistance of the segment. For example, in Figure
9-27 the extraoral force delivers a torque to the arch wire, but the mechanically equivalent force system at the center of resistance includes no
couple. Furthermore, the displacement potential of an extraoral force is
highly influenced by any labiolingual action or resistance from the arch
wire; again using the example of Figure 9-27, with anteroposterior holding force generated by intraoral mechanics exerted against the anterior
crowns (in addition to the headgear action), the anticipated displacement is lingual root-torque rather than bodily movement.
To obtain extraoral force transmission to the buccal segments using
J-hooks, because the J-hooks cannot reach these segments directly, either the force must be transmitted through the arch wire or sliding jigs
must be used to deliver the force to the premolars (or to the retracted
canines) and carried distally by crown contact. The former procedure
has been discussed previously; the advantage of the latter is the facility
to keep the extraoral force, in the absence of friction, from affecting the
anterior teeth. The headgear assembly including the sliding jigs may be
used to drive the buccal segments distally, with the arch wire as a guide,
or to reinforce the anchorage provided by those segments in response to
intraoral mechanics. Generally, a straight-pull appliance is used in this
particular application; any vertical components of delivered force will be
transferred to the arch wire at the points where the J-hooks contact the
mesial extents of the sliding jigs. (To transmit action having a vertical
component to the posterior segments from extraoral activation, a face
bow must be used.)
Although the examples cited in this section may seem to restrict the
delivery of extraoral force through J-hooks to the maxillary arch, headgear of this type may also interact with the mandibular dentition. Figures 9-29 and 9-30 illustrate force transmission to produce incisal-segforce
ment uprighting and to reinforce posterior anchorage, respectively.
Note that, although in Figure 9-29 the high-pull force cannot pass
through the center of resistance, a cervical- or straight-pull headgear to a
pair of hooks affixed gingivally to a rectangular arch wire might substantially reduce the potential for lingual crown-tipping and, with the arch
wire providing a holding force, third-order rotation opposite in sense to
that suggested by the figure can be achieved. Although posteriorly di-
rected active-force components exist with delivery to either arch, the
contrasts in effects (in one arch versus the other) lie in part in the vertical
components of force and the line-of-action locations with respect to centers of resistance. General considerations of headgear to the mandibular
arch are discussed in the following section.
Extraoral Appliances
FIGURE 9-29. A buccal view of high-pull action to the mandibular incisal segment.
FIGURE 9-30. A buccal view of straight-pull action to reinforce anchorage in the
mandibular arch.
Bioengineering Analysis of Orthodontic Mechanics
Delivery of Extraoral Force to the Mandibular Arch
Although the extraoral appliance most often delivers force to the maxil-
law arch wire and dentition—rightly given principal attention in this
chapter—extraoral force also may be directed to the mandibular arch.
Many of the applications are counterparts to extraoral actions to the
maxillary dentition: retraction of canines, anterior retraction, and reinforcement of posterior anchorage, for example. In several of the preceding sections, some contrasts between maxillary- and mandibular-arch
delivery of extraoral force have been mentioned. In the view into the
occlusal plane the differences in the force diagrams are not highly notable. The comparable lateral widths are smaller in the mandibular arch,
which results in slightly larger fractions of activating force projected into
the coronal plane compared to their maxillary-arch counterparts, but
this differential is unlikely to be clinically significant. Comparisons as
viewed from a buccal perspective, however, suggest somewhat more
substantial differences.
Generally, using the cervical-pull force to the maxillary arch as a
reference, the angulation of the line of action of the headgear force is
steeper to the mandibular arch than to comparable locations in the maxillary arch, even with the teeth in occlusion. With the mouth opened,
not only the angulation, but also the magnitude of the headgear force
may be increased, depending on the location and orientation of the activating force. Differences in line-of-action locations with respect to centers of resistance are also noteworthy. To approach the center of resistance of the maxilla, superior to the line of brackets, a high-pull headgear
must deliver force to the arch wire anterior to the center of resistance
(see Fig. 9-25). To align the active force with the center of resistance of
the mandibular arch, gingival of the bracket line, the high-pull appliance
must attach to the arch wire posterior to the center of resistance. When a
vertical component of headgear force exists, for example with high-pull
action, an intrusive displacement potential is created with force delivery
to the maxillary arch, but an extrusive tendency is developed with activation of this headgear to the mandibular arch.
The analysis approach to determining the displacement potential of a
given appliance or choosing the particular headgear to produce a desired displacement is the same in format, whether the appliance engages
the maxillary or mandibular arch. Of particular interest, however, is a
comparison of internal forces in the dentofacial complex between the
dentition and the head cap or neck pad. Activation of an extraoral appliance to the maxillary arch creates an internal force system within the
bones of the skull. For example, the high-pull headgear generates internal compression, shown in simplistic form in the schematic in Figure
9-31. In effect, because both the active and responsive portions of the
headgear force system are against the same structure (the head), the
force transmitted into the maxilla travels posteriorly and internally
Extraoral Appliances
F posterior
FIGURE 9-31. A buccal view depicting the internal torces generated in the skull by a
high-pull headgear to the maxillanj arch.
through continuous tissue structure to the anchorage location. The consequences of the internal compression developed are not substantial due
to the sizable volume of bone through which the force is distributed.
The extraoral force delivered to the mandibular arch is carried
through the dentition and into the mandible, and then it must also be
transmitted to the head cap or neck pad. One or a combination of three
paths must be taken by this force: through the dentition (only possible
when the teeth are occluded); through the masseter, temporal, and pterygoid muscles, having skull and mandible connections and resistance
only to tensile loadings (similar to an elastic); and through the temporomandibular joints (TMJ). Figure 9-32 shows a free-body diagram in the
buccal view of the half mandible (midsagittal-plane symmetry assumed), with a straight-pull headgear delivering force. The position of
the buccal-plane component of the headgear force Pb, with respect to the
TMJ, tends to rotate the mandible open. In this configuration there is
little or no transfer of the headgear force through the dentition; the force
of occlusion would be vertically downward and, therefore, could not
In essence, as shown in the figure, the responsive force system with
forms a pair of opposing couples in the buccal view: the
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 9-32. A free-body diagram in the buccal view of the mandible with the
mandibular arch subjected to straight-pull extraoral action.
anteroposterior component of the TMJ force with Ph and the vertical
component of the TMJ force with the resultant of the tensions generated
in the musculature. Hence, the resultant force against the TMJ is larger
this headgear pushes the condyle posteriorly and superiorly,
compressing the articular disc against the postglenoid process.
A similar free-body diagram in the buccal view of the mandible subjected to high-pull extraoral force is displayed in Figure 9-33. With this
headgear arrangement, the contribution to the responsive force system
of the resistance of the musculature is apparently small and a portion of
the response is through the occlusion, wherever it exists. The senses of
the TMJ-force components are the same as in Figure 9-32, but the total
TMJ-force magnitude is less than that of Pb. The dashed rectangle at the
condyle suggests the relative magnitudes of the vertical and anteriorly
directed responsive components there; the force triangle (upper right)
shows the relationship among the three balanced forces and suggests
the effect upon the two responsive forces of a change in angulation of
Note that an increase in the angle of with the occlusal plane creates an
Extraoral Appliances
FIGURE 9-33. A free-body diagram in the buccal view of the mandible with the
mandibular arch subjected to high-pull-headgear force.
increase in the magnitude of
and a decrease in the size of Ftmj. If
the angulation of Pb is lessened such that its line of action passes below
the TMJ, a jaw-opening moment is present,
disappears, the
muscle forces appear to counteract the potential jaw opening, and the
force diagram begins to resemble that of Figure 9-32.
In summary, the use of headgear to the mandibular arch may be
effective in producing wanted dental-unit segment displacements.
Force-displacement analyses would proceed along lines identical to
those for headgear to the maxillary arch. Of some concern in delivering
extraoral force to the mandibular arch, however, may be the effects of
jaw position on the activating forces and the responsive-force return
paths, particularly through the TMJ.
Dual-Force Headgear
Assuming clinical acceptability of extraoral action to either arch, consid-
eration may be given to the delivery of headgear force to both arches
simultaneously. Potential applications include buccal anchorage reinforcement for concurrent interarch mechanics, simultaneous retraction
of canines in both arches, and concurrent retraction of incisal-anterior
segments. The activating forces, two to each side of the arch, result,
generally, in an increase in the resultant magnitude of responsive force
at anchorage compared to that generated by headgear to the single arch.
Bioengineering Analysis of Orthodontic Mechanics
One head cap or a head cap to support the maxillary-arch activation and
a neck pad to support the force system to the mandibular arch might be
employed. The larger, total responsive force suggests that the area over
which the anchorage force is distributed should be increased; this is
particularly important when a dual-force headgear assembly is supported by just a head cap. If complementing displacements are to be
produced in the two arches, the magnitudes of force to the individual
dentitions must be coordinated to avoid interarch interferences. Practical limitations exist with regard to the directions and locations of the
lines of actions of forces to the two arches. For example, arranging a
high-pull activation to the mandibular arch and a cervical- or straightpull force system to the maxillary arch would require the lines of action
of activating forces on each side to cross one another.
The overall analysis of a dual-force headgear is accomplished, if one
pair of forces is transmitted to each arch, by separate force-displacement
analyses of the actions to the individual arches. In this approach, the
procedures discussed in preceding sections of this chapter are employed. Conceptually, the application of dual-force action to a single
arch may be considered. To be determined initially is whether the actions of the two forces (per side) are (1) separable in their effects on
different portions of the dentition or (2) complementary toward a more
uniform distribution of force through a segment or the arch. If the
former is the case, two individual analyses are undertaken; if the latter is
the situation, the resultant of the two active forces (on each side) is
obtained and a single analysis is carried out within the procedures already discussed.
The Chin-Cap Assembly
Although ordinarily categorized as an orthopedic appliance, discussion
of the chin-cap assembly is warranted here because of its geometric and
mechanical similarities to orthodontic headgear and its possible inclusion in the reverse-pull (extraoral) appliance examined in the succeeding
section. The components of the assembly include a molded or fitted cap
that extends superior to the point of the chin without encroaching on the
lower lip and also posterior to distribute force (pressure) frontally and
beneath the chin. Because sizable magnitudes are generally employed,
the force is generated most often by a pair of traction bands; springs or
heavy elastics may also be used. The activating elements are connected
to a head cap that transmits the responsive force to the cranium. Deferring, as mentioned, consideration of its potential role in reverse-pull
extraoral mechanics, the chin-cap assembly has one of two functions: as
an orthopedic device to attempt to retard or restrict the growth of the
mandible in individuals exhibiting Class III tendencies, or as an orthodontic appliance to produce or reinforce posterior forces of occlusion
toward intrusion of the buccal segments and closure of the anterior open
Extraoral Appliances
Chin cap
FIGURE 9-34. A free-body diagram of the chin cap in an 0cc/usa! view.
Figure 9-34 shows an occiusal-plane view of the force system exerted
on the chin cap. Little of note appears in this sketch except for the midsagittal-plane symmetry. The force F0 is the action-reaction counterpart
of the resultant of the posteriorly directed component of force distribu-
tion of the chin cap against the chin; its magnitude depends on the
amount of traction and the direction of the activating force with respect
to the occiusal plane.
The right-side buccal view of Figure 9-35 shows one-half of the chin
Chin cap
FIGURE 9-35. A buccal view of the chin cap subjected to high-pull-headgear force.
Bioengineering Analysis of Orthodontic Mechanics
subjected to active and responsive sagittal-plane force components.
High-pull action gives rise to both anteroposterior and superior force
distributions between the cap and chin. The design of the assembly
should be such that, in this view, the line of action of the active force
passes through the center of resistance of the mandible; this establishes
the need for a vertical force component. Because the activating element
contacts the chin cap posterior to the point of the chin, the fit must be
good and the material of the cap must extend substantially to the posterior beneath the chin so as to counteract any tendency for cap rotation
(clockwise in this view) and dislodgement. Figure 9-36 shows verticalpull action intended to produce occlusion and, by action and reaction,
intrusion of the molars and, possibly, the premolars. The action is meant
to be perpendicular to the occlusal plane and the force is directed
through the center of resistance of the set of dental units to be intruded.
To this end the force of the chin cap upon the chin is to be relatively
concentrated beneath the posterior portion of the mandible and the ma-
terial of the chin cap must, therefore, extend posterior of the line of
action of the active force.
As with the extraoral force delivered to the mandibular arch, the
activated chin-cap assembly creates internal, responsive force systems
that travel between the locations of active-force delivery and the anchorage. In the appliance shown in Figure 9-36 the responsive force is intended to be transmitted from the mandible entirely through the buccal
segments; no force is meant to be carried through the TMJ. With the
orthopedic appliance activated by high-pull action to oppose growth,
Chin cap
FIGURE 9-36. A buccal view of the chin cap subjected to vertical-pull action.
Extraoral Appliances
however, the location and direction of the line of action of the active
force is similar to that of Figure 9-33; hence, activation
produces notable responsive forces in the TMJ. As long as the lines of
action of the active forces to the chin
pass superior to the condyles,
tend to rotate the lower jaw open and, conse-
the appliance does
quently, the inuscl
responding to it.
not generally assume any role in
Reverse-Pull Appliances
With conventional extraoral anchorage at cranial or cervical locations,
the principal action in the direction of force delivery with headgear is
toward the posterior. The attraction of extraoral anchorage combined
with the clinical need at times for anteriorly directed active force has led
to the development of "reverse-pull" headgear. Because the anteropostenor action of the reverse-pull appliance is opposite in sense to that of
other headgear, and since the transfer of responsive force to extraoral
anchorage must be through pressure, the principal anchorage for the
reverse-pull device must be at the front of the head. Specifically, three
locations are available over which the responsive force may be distributed: the forehead, the region between the nose and the upper lip and
encroaching into the cheek regions, and the front (protuberance) of the
Several reverse-pull designs have been marketed, and all use some
form of head cap. A view into the transverse plane highlights the essentials of the reverse-pull appliance. Figure 9-37 schematically illustrates
2F0 + 2P0
FIGURE 9-37. A free-body diagram in the occiusal view showing action and response of
the reverse-pull headgear.
Bioengineering Analysis of Orthodontic Mechanics
the typical occiusal-view components of the force system exerted on the
usually symmetric headgear. The two pairs of forces posteriorly directed
are the forces F0 exerted by the activating elements and the forces P0
directed from the head cap against the anchorage element. The force Q0
from pressure contact between the anchorage element and the
delivery region of the responsive force. Neglecting any minor lateral
components of the force system, in this view an anteroposterior force
balance and action-reaction indicates that the anchorage force is greater
than the activating force by the magnitude of force transmitted from the
head cap. The head cap exists only to provide overall stability to the
device, but its presence places an additional burden of force on the anarises
Head band
Distribution of
responsive force
from forehead
Cheek pad
FIGURE 9-38. A free-body diagram in the buccal view of a "cantilevered," reverse-pull
extraoral appliance.
Extraoral Appliances
Three basic designs of reverse-pull appliances are currently marketed. Not mentioned here in any particular order, one is a device having a plastic bar cantilevered from a stiff head cap and, perhaps, partially
supported by cheek pads. Elastics are activated from the "free" end of
the bar and attached intraorally. A free-body diagram in the buccal view
of one-half of the (symmetric) appliance (halved at the midsagittal plane)
is presented in Figure 9-38. To keep the force R against the cheek minimal, a sizable couple Cb must be developed as shown; actually, together
Pb and Cb form the resultant of a nonuniform pressure distribution as
illustrated, demanding substantial vertical width and overall stiffness
and stability in the head band. Figure 9-39 provides a free-body diagram
F9, = F, h +
+ a fraction of F9
Chin cap
FIGURE 9-39. A free-body diagram in the buccal view of a "split-anchorage," reversepull, extraor& appliance.
Bioengineering Analysis of Orthodontic Mechanics
in the buccal view of a design that divides the frontal anchorage between
the chin and the forehead (or encircles the face much like a baseball
catcher's mask). Although perhaps providing the greatest area over
which to distribute the responsive force, considering the three designs,
allowance should be incorporated for the normal relative movement of
the mandible. The wire bridge between the head band and chin cap
should, therefore, be free to slide vertically with respect to the head
band. The inclined strap (shown dashed) must be relatively flexible, but
must carry sufficient force F9 to hold the appliance in place. A static
analysis of the horizontal forces is given within the figure; the responsive force magnitudes
depend on both the elastic and strap
Figure 9-40 shows the force systems in buccal view exerted on a
reverse-pull appliance that uses a chin cap for anchorage. The force system on the left exists before the elastics are in place; after the elastics
have been activated (F,,) the force system is that shown on the right in
the figure. Both sketches are provided to show the differences primarily
in the magnitude and location of Q,,, resulting from placement of the
elastics. The connection between the wire bridge and the chin cap must
be rigid since F,, tends to rotate the bridge about that location. This
potential for rotation is then transferred into and through the chin cap to
the chin; hence, the design of the chin cap, the location of the connection of the high-pull fraction band to the chin cap, and the band traction
must all be coordinated to prevent the rotation. (A cervical strap attached to the chin cap can help offset the rotational potential.) Here
again, the head cap primarily provides stability and the resultant pressure delivered to the anchorage is the superposition of the contributions
from the elastics and the head cap (through the fraction bands). In designs where the chin is used partially or totally as anchorage, the potential exists for interference with normal growth of the mandible; also,
internal forces are created within the TMJ that would not otherwise be
present (see Fig. 9-33).
The force transmitted from the wire bridge to intraoral locations is
obviously restricted to an extent in direction. Vertical components are
virtually nonexistent. If the elastics are not to cross the occlusion, lateral
Chin cap
FIGURE 9-40. Free-body diagrams in the buccal view of a reverse-pull appliance
incorporating a chin cap before (left) and after (right) activating the elastic(s) anchored
to the bridge.
Extraoral Appliances
components may be present, largely depending on the points of
delivery of the force. Without crossing the occiusal surfaces and deliver-
ing force to the lingual crown surfaces, the buccal segments are not
directly accessible; however, force could be carried to posterior locations
by the arch wire or by sliding jigs. Beyond these apparent limitations,
the reverse-pull appliance may be used in a variety of treatment protocols
calling for anteriorly directed force. Force may be transmitted to either
arch (although of some concern must be the transmission of force to the
maxillary dentition from the chin cap in that jaw movements alter characteristics of that force). Incisors crown-tipped lingually or canines distally inclined may be uprighted. Torque may be induced in the anterior
portion of a rectangular arch wire by delivering the force occlusally or
gingivally (with respect to the arch wire) to affixed spurs or hooks (see
Figs. 9-27 and 9-28), and labiolingual root movement may be accomplished with the incorporation of a posteriorly directed holding force
generated intraorally. Although requiring a sizable magnitude of force,
the reverse-pull headgear may be used to advance the maxilla and, with
the third of the discussed designs above, simultaneous, potential resistance to growth or protrusion of the mandible is inherently incorporated.
Although usually two elastics, equidistant from the midsagittal plane,
would be used to deliver anteriorly directed force, thereby maintaining
the symmetry of the appliance, if not so large as to impair the stability of
the device one force could be generated in an elastic attached to the
bridge and delivered toward alignment of a single, malposed tooth.
This chapter investigated the force systems created in the activation of
the extraoral appliance. Discussion has centered on the displacement
potential of each type of headgear examined. Although references have
been made to intraoral holding forces and anchorage reinforcement, detailed evaluations of orthodontic-appliance assemblies incorporating
both intraoral and extraoral components were purposely excluded. Examples of the various classifications of headgear received attention, but
not all combinations of activating elements, delivery mechanisms, and
anchorage units were discussed. Not specifically undertaken, for example, was an examination of the delivery of high-pull action to a face bow.
The force-displacement analysis of this particular headgear, however,
should not be difficult for the reader who understands the procedures
and results expected from the individual applications of cervical-pull
face-bow and high-pulliJ-hook appliances.
Although perhaps differing significantly in appearance and composition, several headgear designs may fit a specific clinical need. No attempt is made here to critically evaluate designs that appear to be mechanically similar; however, for a particular application, the reader
should now be prepared to carry out such a comparison, and to impart
Bioengineering Analysis of Orthodontic Mechanics
the facility to do so is a principal objective and thrust of this chapter.
With the possible exception of the rather recently developed reversepull devices, the basic headgear discussed herein are firmly entrenched
in the accepted array of orthodontic appliances. It is necessary to comprehend their potential and shortcomings with regard to use alone, with
an arch wire merely to distribute the delivered force, or together with
active intraoral mechanics. In the chapter to follow, a variety of orthodontic procedures discussed in a force-and-structural analysis setting
and the interaction between intraoral and extraoral appliances in simultaneous use are explained.
Drenker, E.W.: Unilateral cervical traction with a Kloehn extraoral mechanism.
Angle Orthod., 29:201—205, 1959.
Suggested Readings
Alba, J.A., Chaconas, S.)., and Caputo, A.A.: Orthopedic effect of the
extraoral chin cup appliance on the mandible. Am. J. Orthod., 69:29—41,
de Alba, J.A., Chaconas, S.J., and Emison, W.: Stress distribution under highpull extraoral chin cup traction. Angle Orthod., 52:69—78, 1982.
Armstrong, M.M.: Controlling the magnitude, direction, and duration of
extraoral force. Am. J. Orthod., 59:217—242, 1971.
Badell, M.C.: An evaluation of extraoral combined high-pull traction and cervical traction to the maxilla. Am. J. Orthod., 69:431—466, 1976.
Baldini, C.: Unilateral headgear: Lateral forces as unavoidable side effects. Am.
J. Orthod., 77:333, 1980.
Baldini, G., Haack, D.C., and Weinstein, S.: Bilateral buccolingual forces produced by extraoral traction. Angle Orthod., 51:301—318, 1981.
Baldridge, J.P.: Unilateral action with headcap. Angle Orthod., 31:63—68, 1961.
Barton, J.J.: High-pull headgear versus cervical fraction: A cephalometric comparison. Am. J. Orthod., 62:517—539, 1972.
Block, A.J.: An analysis of midline and off-center extraoral force. Angle Orthod.,
32:19—26, 1962.
Fischer, T.J.: The cervical facebow and mandibular rotation. Angle Orthod.,
50:54—62, 1980.
Greenspan, R.A.: Reference charts for controlled extraoral force application to
maxillary molars. Am. J. Orthod., 58:486—491, 1970.
Extraoral Appliances
Haack, D.C., and Weinstein, S.: The mechanics of centric and eccentric cervical
traction. Am. J. Orthod., 44:345—357, 1958.
Hershey, H.G., Houghton, C.W., and Burstone, C.J.: Unilateral facebows: A
theoretical and laboratory analysis. Am. I. Orthod., 79:229—249, 1981.
Jacobson, A.: A key to the understanding of extraoral forces. Am. J. Orthod.,
75:361—386, 1979.
Jarabak, J.R., and Fizzell, J.A.: Technique and Treatment with Light-wire Edgewise Appliances. 2nd Ed. St. Louis, C.V. Mosby, 1972, Chapter 7.
Kloehn, S.J.: An appraisal of the results of treatment of Class II malocciusions
with extraoral forces. In Vistas in Orthodontics. Edited by B.S. Kraus and
R.A. Riedel. Philadelphia, Lea & Febiger, 1962, pp. 227—258.
Kuhn, R.J.: Control of anterior vertical dimension and proper selection of
extraoral anchorage. Angle Orthod., 38:340—349, 1968.
Merrifield, L.L., and Cross, J.J.: Directional forces. Am. J. Orthod., 57:435—465,
Oosthuizen, L., Dijkman, J.F.P., and Evans, W.G.: A mechanical appraisal of the
Kloehn extraoral assembly. Angle Orthod., 43:221—232, 1973.
Osvaldik-Traph, M., and Droschl, H.: Upper headgear versus lower headgear,
yokes, and Class II elastics. Angle Orthod., 49:57—61, 1979.
Perez, C.A., de Alba, J.A., Caputo, A.A., and Chaconas, S.J.: Canine retraction
with J hook headgear. Am. J. Orthod., 78:538—547, 1980.
Ringenberg, Q.J., and Butts, W.C.: A controlled cephalometric evaluation of
single-arch cervical traction therapy. Am. J. Orthod. 57:179—185, 1970.
Tabash, J.W., Sandrik, J.L., Bowman, D., Lang, R.L., and Klapper, L.: Force
measurement and design of a torquing high-pull headgear. Am. J. Orthod.,
86:74—78, 1984.
Thurow, R.C.: Edgewise Orthodontics. 4th Ed. St. Louis, C.V. Mosby, 1982,
Chapter 19.
Thurow, R.C.: Atlas of Orthodontic Principles. 2nd Ed. St. Louis, C.V. Mosby,
1977, Chapter 16.
Watson, W.G.: A computerized appraisal of the high-pull face-bow. Am. J. Orthod., 62:561—579, 1972.
Worms, W.W., Isaacson, R.J., and Speidel, T.M.: A concept and classification of
centers of rotation and extraoral force systems. Angle Orthod., 43:384—401,
Force and Structural
Analyses of
Orthodontic Mechanics
The preceding chapters introduced the concepts and procedures neces-
sary to undertake force and structural analyses of orthodontic appliance
elements. Although all preparations were discussed with the intended
application in mind, that application has not been examined in its totality to this juncture. It is appropriate to indicate initially that the analyses
that follow are exemplary; although specific analyses may be more interesting and meaningful to some than to others, the intended emphasis is
on the analysis procedure and on the logical, orderly succession of steps
within it. Discussions are carried out with pertinent, accompanying diagrams. Of the sample mechanics examined, the overall procession is
from the relatively straightforward to the more complex, wherein the
level of complexity is associated with that of the involved dentition, that
of the appliance and the force system exerted upon it, and the sophisti-
cation of the modeling that must be undertaken to ensure an understandable and self-validating solution.
At the outset, it is helpful to recollect several of the more prominent
concepts previously introduced. The tooth crowns are subjected to mechanical force systems exerted by the appliance; the characteristics of
these crown force systems determine the potential displacement patterns of the dental units (Chap. 5). The appliance, as activated, exists in
a state of quasi-equilibrium which implies an almost exact balance of
force and moments within the total force system (Chap. 2). The forceanalysis problem is inherently indeterminate and, therefore, active configurations relative to passive constraints and stiffnesses (Chap. 3) are
keys to the completion of a solution. Differential force and anchorage are
to be clearly understood, inasmuch as the engaged orthodontic structure
is attached to a nonrigid "foundation," the dentition. Finally, the assem372
Force and Structural Analyses of Representative Orthodontic Mechanics
bly under study is, strictly speaking, a dynamic one, which means that
time is a principal independent parameter and deactivation is an ongoing, between-appointments process that must not be overlooked
while carrying out instantaneous analyses.
Each individual analysis follows a common procedural outline. An
overview of the as-activated force system is undertaken in the context of
the intended objectives of the appliance. As necessary and warranted,
the actual appliance and the involved dentition or segment thereof are
then modeled with care; the intent is the subsequent accomplishment of
an analysis in the absence of undue involvement and in the presence of
the principal aspects of the problem. The analysis itself is pursued,
using the four-step approach outlined in Chapter 6. Following the discussion of active, reactive, and responsive force systems exerted on and
existing within the appliance, exerted on the dentition, and existing
within the periodontium, the significant structural characteristics of the
appliance may be considered and, possibly, a rationale for improving
the design established. Toward quantification of the components of the
force systems, needed measurements at activation must be indicated
before the computations of force magnitudes are begun.
Although each example pertains to only a portion of the total therapy
plan for the patient, the individual tooth and segment displacements
considered must be examined within the framework of overall treatment
objectives. The functional requisites are to achieve, or at least approach
to the extent possible, ideal tooth positions and orientations as well as
ideal arch-form geometry. Proper interdigitation and relationships of
dentition to basal bone are sought. The positions and orientations of the
maxilla and mandible, with treatment completed, are expected to reflect
facial harmony and skeletal balance. Operational relationships—
occlusion and TMJ function—must be proper at the conclusion of active
treatment. Finally, correctional procedures are undertaken to promote
retention of the as-treated, realigned, dental-unit configurations and
supporting-tissue positions.
The implicit, biomechanical objectives of this final chapter are two in
each example. First, given the force system generated at the activation
site(s) by the appliance or a portion of it, an overall force analysis involving the appliance structure and the supporting dentition is undertaken
toward examination of all potential dental-unit displacements. Both the
displacements desired and intended and those unwanted—generally
associated with the responsive force system—are scrutinized. Second, a
structural evaluation of the appliance and its elements may be pursued
with a view toward potential failures and, perhaps, ways in which the
design might be positively modified.
Although any one of the examples to be considered might easily be
subjected to discussion to the point of devoting the equivalent of a complete chapter to its structural analysis, including numerous alternative
designs, clearly some limitations must be envoked. In general, detailed
numerical involvements are avoided. Most studies are through instantaneous, quasi-static analyses, although references may be made to rele-
Bioengineering Analysis of Orthodontic Mechanics
vant processes and long-term expectations. Little detailed attention is
given to periodontal response and remodeling beyond that discussed in
Chapter 5. All examples pertain to fixed appliance therapies and associated auxiliaries; removable or functional appliances, retainers, and orthopedic devices (such as palate-splitting mechanisms) are excluded (although the analysis procedures developed in this text may be applied to
Numerous appliances and auxiliaries are in common clinical use to
accomplish a specific displacement objective; examples are taken from
several treatment philosophies and there is no intention to rate or rank
the individual appliances or approaches to the mechanics or to favor or
support one treatment technique over another. As previously indicated,
the examples are intended to illustrate the analysis procedures and,
hopefully, to enable the individual practitioner to analyze existing or
proposed appliances and mechanics toward judging their suitability to
accomplish specific treatment objectives.
Individual Tooth Malalignments
This section of the chapter focuses on sample mechanics used to correct
individual, malposed teeth. The objective is the delivery of a specific
force system to the tooth crown by the appliance to produce the displacement necessary to bring the malaligned unit into proper intra-arch
position, correcting crown location, long-axis angulation, and first-order
orientation as warranted. Assumed is the existence of space in the arch
to accommodate each desired displacement. Activation to the extent of
producing inelastic behavior, anywhere within the appliance, is generally considered for purposes of analysis to be a structural failure. In
addition to maintaining the integrity of the appliance in its interconnected members and the attachments to teeth, awareness must exist of
the distribution of responsive, anchorage force and displacement control
during the desired tooth movement.
Leveling Displacements
To begin, a premolar requires a facially or lingually directed, simpletipping displacement to achieve proper alignment. A round arch wire is
engaged into the facially placed brackets of the premolar and, minimally, the teeth immediately mesial and distal to the premolar. The dental units ordinarily adjacent to the premolar are assumed present. Given
a choice of brackets and reasonable first-order orientations of all involved teeth, a wire is assumed available which, when fully engaged
and activated, will not have been inelastically bent as a consequence of
that activation; in short, the faciolingual malalignment is not excessively
Force and Structural Analyses of Representative Orthodontic Mechanics
Suppressing the arch-wire curvature in the vicinity of the premolar,
the structural analysis is that of a passively-straight wire beam, supported on either side of the malalignment site and elastically deflected in
the activation process. Because its faciolingual stiffness at the premolar
is nonzero, the deflection to engage produces the wanted, simple-tipping force. The direction (and sense) of that force corresponds to that of
the deflection, the location is that of the engagement of wire into premolar bracket, and the initial magnitude of the force depends on the
amount of the malalignment and the localized bending stiffness (which
is, in turn, dependent on a host of parameters discussed in Chapter 7).
A reasonable, additional assumption is that of similar mesiodistal
bracket widths, and neglected are any small differences between interbracket distances on either side of the premolar long axis.
The top sketch of Figure 10-1 shows a lingually malposed premolar,
the adjacent teeth, and the engaged arch wire. The center sketch presents the force system exerted on the three teeth by the activated wire.
Symmetry associated with the assumption of equal interbracket distances yields mirror-image responsive forces (Q) and couples (C) against
the crowns neighboring the premolar. The magnitudes Q are individ-
ually not more than one-half that of P (see Fig. 7-1); teeth mesial and
distal to the anchorage units shown may also share the responsive-force
burden, but those units adjacent to the activation site receive the largest
portions of the response. The couples arise from the curvature induced
in the wire by the activating deflection; they exist only upon elimination
of any first-order clearance between wire and ligated brackets. The freebody diagram of the involved wire segment is shown in the lower sketch
of Figure 10-1. The force system is the action-reaction counterpart of that
in the middle sketch except that a distribution of the responsive force
system to more than two crowns is suggested. The first-order curvature
effects decay more rapidly than those associated directly with the activating deflection; accordingly, only the two crowns adjacent to the premolar are shown subjected to responsive couples. The quasi-static analysis is straightforward, aided by the symmetry; the force balance
requires the total response on one side of the activation site to equal
one-half of P and the moment balance demands that the responsive
couples be equal in magnitude.
Three of the four steps in the analysis procedure introduced in Chapter 6 are covered in the foregoing narrative. The potential side effects of
the process of aligning the premolar are lingually directed simple tip-
ping and first-order angular displacements of the adjacent teeth. The
former can apparently be handled by making the initial magnitude of P
as small as practicable and by ensuring a broad distribution of the responsive force. Note, however, that the larger malpositions require high
elastic range and bending flexibility. Continuous wires exhibiting low
stiffness will concentrate the bending deformations at the activation site
and, thus, cause the two teeth neighboring the premolar to carry nearly
all of the responsive force. Moreover, the more flexible the wire, the
Bioengineering Analysis of Orthodontic Mechanics
j- (P)
FIGURE 10-1. 0cc/usa! views of a lingually ma/aligned premolar, adjacent teeth, and an
engaged leveling wire (top), the force system exerted on the premo/ar and split
anchorage by the arch wire (center), and the free-body diagram of the wire segment
Force and Structural Analyses ot Representative Orthodontic Mechanics
greater the tendency for occurrences of first-order couples that can pro-
duce rotational displacements.
A second example of this section is that of moving a "high" canine
into proper occlusogingival alignment. To provide adequate space and
reduce arch-length discrepancy, the adjacent first premolar has been
extracted. A continuous arch wire is placed and, again, its curvature in
the arch-form plane is ignored in the analysis. Because required movements of five or more millimeters are not uncommon, the extrusive force
is generated by stretching a relatively flexible, "elastic" element between
the canine bracket (affixed to the facial crown surface) and the arch wire.
Activation of the elastic induces the extrusive force against the canine
shown in views from the facial and mesial perspectives within Figure
10-2. The latter reflects the eccentricity of the force and suggests that a
lingual-crown-tipping displacement might accompany the extrusive
movement (see Fig. 5-16). A free-body diagram of the elastic is shown in
the upper right of the figure; from action-reaction and quasi-static analyses the magnitudes of the three forces encountered thus far, all approximately parallel to the long axis of the canine, are equal at any time. The
initial magnitude of Fe is best measured directly with an appropriate
force gauge. A free-body diagram of the arch wire is shown in the lower
portion of Figure 10-2. The symmetry of the previous example is absent
here because of the extracted premolar, but the anchorage is again split
and a force balance requires the sum of the two, resultant, responsiveforce magnitudes to equal that of the active force. From a moment balance with respect to the attachment location of the elastic, the resultant
responses are inversely proportional to their individual distances from
the line of action of the elastic force (in the absence of second-order
couples as part of the response).
Note that the force delivered to the anterior anchorage is greater
than—approximately twice the magnitude of—that delivered to the
posterior anchorage (see Fig. 2-28). The prudent arch-wire choice in this
example is that exhibiting the highest bending stiffness possible, given
the bracket-slot size. The arch wire need not be deflected in the activa-
tion of the appliance. The greater the wire stiffness, the more nearly
uniform is the distribution of force within the anchorage unit. The anterior segment in this example must sustain a responsive resultant approximately two-thirds the magnitude of the force applied to the canine
bracket. The lateral-incisor root is smaller than that of the canine and, if
the arch wire is flexible, an active force of sufficient size to produce the
desired canine displacement would undoubtedly move (intrude, tip?)
the incisor as well. Use of a full size rectangular wire has the added
benefit that, when the canine has been displaced occlusally to the extent
that the elastic may be discarded and canine-bracket engagement is possible (without excessive force or inelastic material behavior), a torsional
couple may be delivered to the canine crown to correct any third-order
malpositioning caused by the eccentricity of the active, extrusive force.
Moving a terminal molar into proper alignment presents a somewhat
different problem in that, unlike the two previous examples, restricting
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 10-2. Mesiodistal (upper left) and facial (upper center) views of a high"
(impacted) canine subjected to an elastic force, a free-body diagram of the stretched
elastic (upper right), and a free-body diagram of the arch-wire segment that transmits
the responsive elastic force to split, intraoral anchorage (bottom).
procedures to intra-arch mechanics demands that anchorage be entirely
mesial to the molar. (A palatal bar might be used depending on the
nature of the malalignment, but this option is not examined here.) The
desired displacement may be attempted by cantilevering an arch-wire
segment (or an appropriate auxiliary) from the adjacent buccal-segment
teeth, unitized to distribute the responsive force system. The cantilever
Force and Structural Analyses of Representative Orthodontic Mechanics
might be positioned on the facial or lingual aspect or the mechanics
divided between both sides of the segment; only minor differences in
the three analyses exist, and facial-surface mechanics are assumed in
this discussion.
The top sketch in Figure 10-3 shows the malaligned terminal molar,
with its root structure tipped mesially, in need of assistance to erupt
completely. Also indicated schematically is the anchorage together with
an arch wire and a cantilever. Although the continuous arch wire might
extend distally to the malposed molar, in these example mechanics that
wire terminates just distal to the bracket affixed to the crown of the tooth
immediately mesial to the molar. In the left center of the figure the molar
is shown subjected to crown loading (exerted by the cantilever) including an extrusive force and a second-order couple to correct the long-axis
angulation. Not seen in this buccal view is a third-order couple, possibly
present to maintain or correct the long-axis alignment in the mesiodistal
perspective in light of the eccentricity of with respect to the cre of the
molar. The bottom sketch of Figure 10-3 presents a free-body diagram of
the cantilever in the buccal perspective. The action-reaction counterpart
of the active force system exerted on the cantilever is carried through it
mesially and transferred to the anchorage unit. The force balance is
straightforward; the moment balance leads to the second relationship
given next to the diagram. The sketch in the right center of the figure
shows the responsive force system delivered to the anchorage.
The analysis thus far indicates that the force systems against the
molar crown and anchorage are similar; the forces are equal in magnitude but opposite in sense and, therefore, in displacement tendency.
The second-order couple exerted on the anchorage is, however, larger
than the active couple. Because the tooth mesial to the terminal molar
has a comparable (if a first or second molar) or smaller (if a premolar)
root-surface area, the anchorage unit must include at least two teeth
and, if available, three. The continuous arch wire should exhibit a high
bending stiffness to distribute the responsive force more widely. As
noted earlier, the cantilever could simply be an extension of the continuous wire if the malalignment is minor. If the occlusogingival malposition
is substantial, however, necessitating a sizable activation, the bending
flexibility and elastic range must be high. Concurrently, sizable localized
second- and third-order stiffnesses may be warranted for displacement
control. This combination of characteristics is perhaps best obtained by
maximizing the mesiodistal length (C) of the cantilever while choosing a
"working" (medium-size, rectangular) wire cross-section. Another consideration is delivery of the responsive force system at the mesial end of
the cantilever near the middle of the anchorage unit to more uniformly
distribute that response.
Rotational Corrections
Changing views, a common individual-unit malalignment is the rotated tooth. To attempt the correction, the active force system must pro-
Bioengineering Analysis of Orthodontic Mechanics
Fb =
Cb =
FIGURE 10-3. Buccal views of a terminal molar to be aligned and intraoral anchorage
(top), the crown force system exerted by a cantilever on the molar (center left) and on
the anchorage (center right), and a free-body diagram of the cantilever (bottom).
Force and Structural Analyses of Representative Orthodontic Mechanics
a moment about the desired axis (the cro in the occiusal view)
having the sense (clockwise or counterclockwise) to oppose the malposilion. If the cro is to be on the long axis, the resultant of the active force
system should ideally be a couple in an occlusogingival plane. Such a
force system might nearly be produced by an arch wire engaging a
bracket slot, but angulated with respect to the slot. If a distolingual
rotation is desired, for example, the wire would push against the distal
extent of the bracket slot and against the ligation at the mesial extent of
the slot. Two less than efficient aspects of this approach are the dependence, in part, on the ligation to maintain the activation and the small
moment arm equal to the mesiodistal width of the bracket slot. An increase in the size of the moment arm is the improvement demonstrated
in the mechanics illustrated in Figure 10-4. If a small interproximal space
exists, an elastic may be stretched between an eyelet or button (affixed to
the band or directly bonded to the crown surface in the lingual position
shown) and the arch wire. The bracket attached to the facial surface is
assumed aligned occlusogingivally with the arch wire, but perhaps not
initially engaging the wire. On the upper left in the figure is an occlusal
view showing the tooth, bracket, lingual button, and the elastic. In the
upper right view the active force system against the crown is just the
Here, Fe = F,,,
the forces
form a couple, C0.
FIGURE 10-4. Occlusa/ views of a rotated tooth and the use of an elastic and arch wire
to attempt the correction.
Bioengineering Analysis of Orthodontic Mechanics
force in the absence of bracket-wire contact. The displacement
tendency includes clockwise rotation (from the moment of the force
about the long axis) and facial crown tipping (from the force itself). The
lower left view shows the added responsive force from the arch wire
following tipping and bracket-wire contact. As time proceeds and movegrows. If
ment occurs, F, diminishes and
arises as a secondary
activation as indicated, its magnitude will never exceed that of Fe. The
sketch on the lower right in the figure depicts the ideal situation (if pure,
long-axis rotation is desired) with the two forces equal and forming the
couple C0. In comparison with the first-order couple formed in the
bracket and mentioned earlier, these mechanics exhibit an activating
element with substantial elastic range (the elastic) and a moment arm
approximately equal to one-half the mesiodistal width of the tooth. The
arch wire used should have sizable bending stiffness to prevent tipping
toward the labial beyond that to engage the bracket and wire.
When a rotational correction is desired with little or no change in the
faciolingual, long-axis position, and the full couple cannot be instanta-
neously generated, a sizable moment with a small, active force is
needed. This combination requires a large moment arm, and the auxiliary of Figure 10-5 has the wanted characteristic. As suggested by the
top sketch in the figure, the "rotation lever" might be positioned on the
lingual and one end rigidly affixed to the crown of the rotated tooth. The
lever is activated with a ligature tie or an elastic to the facially placed
arch wire, several teeth mesiodistal from the malposed unit. The activating deformation is partially in the lever; its passive configuration is
shown dashed. A free-body diagram of the lever is given in the center of
the figure; the action-reaction counterpart of the couple is the wanted
force system, but it cannot exist in the absence of the two forces of
magnitude f as indicated by a quasi-static analysis. For f to be light despite a sizable activation, the lever must exhibit high bending flexibility
and elastic range; not only should D be large, but the lever cross-section
must be small. The lever is better tied to the arch wire with an elastic
than with a segment of ligature wire because of the difference in their
extensional stiffnesses. The force system delivered to the rotated tooth
crown is shown in the lower portion of the figure together with the
expression for the net active moment with respect to the long axis. The
responsive force
does not exist in the absence of crown contact with
the arch wire.
Another means of correcting a rotated tooth uses a "V-spring"
("wedge"). The tooth, arch wire, spring, and tie are depicted in the top
sketch in Figure 10-6. The spring is activated by elastically reducing the
passive angle between its "legs," wedging the auxiliary between the
arch wire and tooth crown, and tying the spring to the bracket. To maintain the activation, the arch wire must have substantial bending stiffness
and must be secured in the bracket affixed to the crown. A free-body
diagram of the spring and the force system delivered to the tooth crown
are also shown in the figure. The desired wedge force is
exerted by
one leg of the spring on the crown. This force, together with a compo-
Force and Structural Analyses of Representative Orthodontic Mechanics
Passive lever
= f(D)
Effective couple
equals F(D — d)
FIGURE 10-5. 0cc/usa! views of a lingual-lever system to correct a rotation. Shown are
a composite sketch of the mechanics (top), a free-body diagram of the lever (center),
and the force system transmitted to the tooth crown (bottom).
of the tie force, forms the couple meant to correct the
rotation. Also shown in sketches are forces from the ligature and friction. The size of the frictional force between arch wire and bracket will
be the principal influence on the location of the axis of rotation relative
nent (parallel to
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 10-6. 0cc/usa! views of a 'wedge" used to correct a rotation. Shown are a
schematic of the mechanics (top), a free-body diagram of the wedge (lower left), and
the force system delivered to the tooth crown (lower right).
to the long axis. If this force is sufficiently great, the rotation could be, in
the absence of crown contact (on the side opposite to that of the spring
position), about an occlusogingival axis through the bracket. The effectiveness of the spring depends on the combined flexibilities of the tie
and spring in the presence of adequate elastic range. Increments of
movement following activation will be small if the tie is a segment of
ligature wire and the spring stiffness is high.
The last rotational-correction scheme to be examined is similar to the
first in this series in that the activation is again tangential to the crown.
The activating force here, however, is directed mesiodistally. The upper
portion of Figure 10-7 depicts the presence of crown contact from the
adjacent tooth; the driving force at the facial surface is accompanied by
the crown-contact response. These two forces form a couple and, in the
absence of significant friction, the first-order rotation induced will be
counterclockwise about the long axis of the malposed unit. In the lower
portion of the figure there is no crown contact and a rolling displacement should be expected with the center of resistance moving in the
direction of the force and the center of rotation located lingual to the
long axis. In both instances here the arch-wire influence is suppressed
and it is assumed that the tooth is not ligated to the wire. In all of these
examples of rotational correction, if the couple is to be formed, the "second" force arises from a secondary activation. When "in-place," first-
order rotations are wanted, it is important to note the location and
source of the secondary force together with gauging the appropriate
stiffness of the element producing that force, so that the format of the
expected, actual movement can knowledgeably be assessed.
Force and Structural Analyses of Representative Orthodontic Mechanics
Potential displacement
is long-axis rotation.
No adjacentcrown contact
Potential displacement
is a generalized,
rolling rotation.
FIGURE 10-7. Occlusal views of an attempted rotational correction, activated by a
mesiodistal force against a bracket with (top) and without (bottom) crown contact
between the rotated tooth and the adjacent dental unit.
Bilateral Action
Another class of displacements appropriate to this section encompasses the intra-arch, bilateral movements of individual teeth in a symmetric manner with respect to the midsagittal plane. As an initial example, consider the closing of space between the maxillary central incisors.
The appliance consists of an arch-wire segment, gabled between two
brackets if warranted, and an elastic encircling the pair of incisor brackets. Appropriate sketches are given in Figure 10-8. If the action is truly
bilateral, then two activation sites and no anchorage exist; action-response is effectively replaced by "action-action." The forces at either
end of the elastic drive the central incisors toward each other and the
ligated arch-wire segment is expected to produce first- and/or secondorder couples to control or rectify the angulations of the teeth during the
movements. The free-body diagrams in the figure are balanced by
paired, mirror-image force-system components. If no tipping is to occur
during the space closure, the "guiding" wire segment must possess
high bending stiffness. To prevent mesiolingual "rolling" the ligation
must be snug and the ties stiff, but concurrently not induce sizable frictional resistance.
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 10-8. The closing of space between maxillary central incisors: the teeth and the
mechanics (left), tree-body diagrams of the elastic and the wire segment (center), and
the delivered crown force systems (right), all in facial views.
A second bilateral-action example involves the terminal molars of
either arch, and the mechanics were part of the general analysis of facebow therapy in Chapter 9. The innerbow could be fabricated prior to
insertion to constrict or to expand the arch width at the molars and/or to
induce first-order couples (through placement of toe-in or toe-out
bends) as depicted in occlusal perspective in Figure 10-9. If the inner
bow is, in effect, a separate auxiliary and engages only the molars, as in
the previous example no anchorage exists and only action at both molars. The buccal tube in its mesiodistal length provides a relatively large
inherent moment arm for the couple and improved first-order control
compared to any bracket, thus allowing the operator latitude in choice of
bow-arm or wire size as it affects bending stiffness. Figure 10-10 shows
posterior views of active, bilateral, buccolingual forces and third-order
couples. The couples necessitate the use of rectangular wire (and tube
slots) or torquing spurs. If an arch wire is continuous and engages all
brackets anterior to the molars, with bilateral action present so also is
bilateral anchorage. A proper analysis in this instance would involve
either half arch; although ideally every wire-bracket connection mesial
to the molars could be mechanically passive, from a practical standpoint
contact (and, therefore, force transfer) will exist at most, if not all, such
Another means of generating mirror-image, bilateral, force systems
to posterior teeth in the maxillary dentition is the transpalatal bar. Spanning the palatal vault, this appliance typically interconnects canines or
molars through attachments on their lingual crown surfaces. Buccolingual forces, third-order couples, and, perhaps, also first-order couples
may be activated on both sides of the maxillary arch. A free-body diagram of the transpalatal bar and, through action-reaction analysis, typical, delivered force systems to the crowns are given in a posterior view
in Figure 10-11. The transpalatal bar is fabricated from a member having
a substantial cross-section in order that it maintains its shape and stabil-
Force and Structural Analyses of Representative Orthodontic Mechanics
Outer bow
FIGURE 10-9. lntra-arch bilateral action to the terminal molars from an 'inner bow':
force systems against the arm ends (top) and their action-reaction counterparts exerted
on the molar crowns (bottom) in 0cc/usa! views.
ity when activated. The bar is usually formed to give the tongue ample
room, but without exerting undue pressure on the hard palate. The
stiffnesses at the crown-contact locations are likewise substantial; hence,
activating displacements must be relatively small. With its bulk the
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 10-10. Posterior views of bilateral action against the terminal molars: actionreaction force systems exerted on the appliance (top) and the molars (bottom).
transpalatal bar is more often used to maintain arch width and/or long-
axis orientations as an auxiliary to other mechanics than to directly generate tooth movements.
lnterarch Mechanics
Individual teeth may experience displacements induced by interarch
mechanics. Perhaps the simplest example is in a finishing step in therapy using the "up and down" elastic to effect proper interdigitation. The
mechanics are similar to those illustrated earlier to extrude the "high"
canine, but the anchorage here is provided by teeth in the opposing
arch. Figure 10-12 shows a maxillary premolar subjected to an occlusally
directed force from an elastic stretched and placed around the bracket.
The elastic also encircles brackets affixed to the facial surfaces of the
opposing teeth of the mandibular arch. A free-body diagram of the elastic and an illustration of the responsive forces delivered to the involved
mandibular teeth complete the figure. Extrusion of all three teeth may
Force and Structural Analyses of Representative Orthodontic Mechanics
FIGURE 10-11. A free-body diagram of a transpalatal bar, as viewed from the posterior
(top), and the force systems delivered bilaterally to first maxillary molars (bottom).
depending on the magnitudes of the elastic forces and the individual bending stiffnesses of the engaged arch wires. If extrusion of only
the maxillary premolar is desired, the maxillary arch wire, if present,
should exhibit occiusogingival bending flexibility to permit the desired
displacement. In addition, because of the eccentricity of the active force,
a labial crown torque from the wire would be helpful. The mandibular
arch wire should be stiff to provide restraint against occiusally directed
displacement and tipping and to distribute the responsive force system
beyond the two mandibular teeth shown.
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 10-12. Mechanics to complete interdigitation of a maxil!asy premolar (left), a
free-body diagram of the involved elastic (center), and the forces exerted on the crowns
through elastic-bracket and adjacent crown contacts (right).
Interarch mechanics may also be used to correct a unilateral crossbite. As an example, consider a mandibular premolar that is improperly
tipped lingually and must be uprighted. If other mandibular teeth are
not involved, a button or eyelet might be affixed to the lingual crown
surface of the premolar and an elastic stretched over the occlusal surface
and attached to the maxillary arch wire. Appropriate sketches are given
in Figure 10-13. Because the elastic force has a vertical component, in the
absence of a controlling, mandibular arch wire, some extrusion should
be expected to accompany the wanted labial crown tipping. The maxil-
lary arch wire should be stiff in bending to distribute the responsive
force, equal in magnitude to that of the active force, to at least several of
the maxillary teeth neighboring the elastic-wire connection.
Two additional concerns are noteworthy in this example. First, the
relative positions of the arches influence the size of the elastic force here
(and in the preceding example); e.g., the mean force is expected to be
greater in a mouth-breathing patient. Second, with the elastic running
across an occlusal surface, any occlusion or mastication tends to wear or
cut the elastic, reducing its life. With their low cost and relaxation tendencies, however, such elastics should be routinely replaced every day, if
not more often.
In these last two examples of this section a pertinent question is
"Why not use intra-arch mechanics instead?" With the "up and down"
elastics the intent may well be to produce some extrusion in both arches.
Force and Structural Analyses of Representative Orthodontic Mechanics
arch wire
Responstve force
delivered to the
maxillary arch
Angle less than 900 yields
potential for extrusion
A portion of
the elastic
FIGURE 10-13. Mechanics to correct a mandibular premolar in cross-bite: mesiodistal
views of force systems exerted on the elastic, the premolar crown, and the maxillary
arch wire.
In the cross-bite example, a larger activation is possible resulting in more
displacement potential between appointments and appliance adjustments. Moreover, the anchorage is more unifiable in the maxillary arch.
In summary, there are advantages and disadvantages to consider with
either approach.
Bioengineering Analysis of Orthodontic Mechanics
lntra-arch Vertical Positioning
This section undertakes analyses of mechanics in one arch designed to
produce displacements that are primarily occlusogingival in direction
and are intended to move two or more teeth concurrently. Examples
include correcting the anterior open or deep bite and modifying the
curve of Spee. In all instances, geometric and mechanical symmetry
with respect to the midsagittal plane is assumed. Views of the teeth,
appliance, and force systems are from the buccal perspective. Analyses
may, then, involve just one-half of the arch and appliance in each example. One-half of the U-shaped arch model introduced in Chapter 6 is
used and within the L-shaped half arch the dentition is considered as
two or three segments, depending on the mechanics.
Examined first is the "utility" arch, formed in conventional wire to
engage brackets affixed to the facial crown surfaces of the four or six
anterior teeth, bypassing the teeth of the buccal segment mesial to the
terminal molars, and engaging buccal tubes attached to the molar
crowns. As appropriate or warranted, each buccal segment may be unified by means of a separate wire segment and ligation. Shown schematically in the top sketch of Figure 10-14 is a mandibular utility arch. Two
segments of the half dental arch are involved: the half anterior segment
including the central and lateral incisor and, perhaps, the canine, all
represented by the incisor drawn, and the buccal segment including
available premolar(s), molar(s), and, possibly, the canine. In this initial
example the intent is the intrusion of the anterior teeth; the passive
configuration of the arch, with the ends of the wire engaging the buccal
tubes, is shown dashed. A free-body diagram of the as-modeled, half
utility arch is given in the center of the figure. Engagement into the
anterior brackets creates intrusive force as the arch attempts to restore its
passive configuration. If the utility arch is formed in round wire and no
torquing spurs are included or attached, and no anteroposterior activation is present, the half anterior segment exerts only the vertical of mag-
nitude Fa. The resultant force system from the buccal tube, the only
other site of contact with the half arch, is obtained through a quasi-static
analysis; required for force and moment balances are a vertical force and
a second-order couple in this view. The action-reaction law and the balanced free-body diagram yield the overall force system exerted by half of
the appliance on half of the mandibular dentition as indicated in the
lower sketches of Figure 10-14; also given are the quasi-static relationships involving the parts of the force system and the anteroposterior
distance from anterior brackets to buccal tube. (Recall a previous, similar
analysis given in Figure 2-29.)
The utility arch in Figure 10-14 is activated by elastic bending, displacing the anterior section of the arch occlusally to enable bracket engagement. The amount of force induced (Fa) depends on a localized,
Force and Structural Analyses of Representative Orthodontic Mechanics
Cp = Fa(e)
FIGuRE 10-14. A utility arch wire in the mandibular arch: buccal views of the passive
(dashed) and activated (solid) mechanics (top); a free-body diagram of the half arch
wire (center); and the force system delivered to the terminal molar (lower left) and half
anterior segment (lower right).
Bioengineering Analysis of Orthodontic Mechanics
vertical stiffness of the arch wire and the amount of vertical displace-
ment necessary to engage the anterior segment. The stiffness is dependent on the wire from which the arch is formed and the distance 1?. The
passive position of the anterior section of the arch is independent of the
stiffness and is controllable by the clinician in, for example, the size of
the tip-back bend placed mesial to the buccal tube. The displacement(s)
following activation depend on the magnitudes of F,,, E, and the involved root-surface areas. If intrusion of the incisors is desired, the canines are not engaged and, if F,, is sufficiently light, the terminal molars
alone may provide enough anchorage; a conservative operator would,
however, reinforce the anchorage with an additional molar or premolar
on each side.
In the lower right sketch of Figure 10-14, note the position of the line
of action of the intrusive force with respect to the center of resistance;
apparently, the intrusion is likely to be accompanied by labial crown
tipping. One means of counteracting this additional (assumed unwanted) displacement component is illustrated in the maxillary, utility-
arch analysis of Figure 10-15. As suggested in the top sketch in the
dashed, passive configuration, the intent here (as in the previous example) is to open the anterior bite—in this instance through intrusion of
the maxillary anterior teeth. The analysis procedure through the
sketches of Figure 10-15 follows that of the foregoing example. Note in
the center sketch that the anteroposterior forces must be equal in magnitude; they are essentially independent of other components of the force
system and their size is determined by a longitudinal stiffness, a function largely of the characteristics of "tie-back" element and the "omega
loop" bent into the arch, and the amount of anteroposterior activating
deformation. In the representations of the anterior and posterior segments and the exerted force systems, the moments about the centers of
resistance of the forces P,, and oppose those of F,, and C,,, respectively.
Note also, because the roots of the maxillary central incisors are substantially larger than their counterparts in the mandibular arch, two-tooth
posterior segments are necessary, minimally, to provide anchorage for
intrusion of four incisors and are insufficient if the involved anterior
segment also includes the canines.
Returning to the mandibular arch, the treatment of an anterior open
bite may call for extrusion of the mandibular incisors and a suitable
utility arch may be engaged. The vertical forces and second-order couple
of Figure 10-16 are reverse in sense from their counterparts in Figure
10-14, but added in the new figure is a third-order anterior couple, generated, perhaps, through use of rectangular wire and appropriate activation. The extrusive force tends to produce lingual crown tipping because
of its position in relation to the cre. If the labiolingual long-axis orientation is to be maintained, or if labial crown tipping is to accompany the
extrusion, the torsional couple (or a labially directed force component)
must be present. Directing attention to the posterior response, usually
particularly important when creating anterior extrusion (in either arch)
is the use of more than sufficient anchorage or previous anchorage prep-
Force and Structural Analyses of Representative Orthodontic Mechanics
= F,,(L)
= Fa
FIGURE 10-15. A tied-back, maxillary utility arch: a buccal view with the passive anterior
position dashed (top); a free-body diagram of the half arch wire (center); and the force
systems exerted on the posterior (lower left) and half anterior (lower right) segments.
aration; the sense of the second-order couple is toward mesial crown
tipping and the reduction of arch length. Noting, however, from the
free-body diagram in the center of Figure 10-16 the moment-balance
equation for the half arch, although the solution is indeterminate, the
relationship suggests that the presence of Ca serves to reduce the magni-
Bioengineering Analysis of Orthodontic Mechanics
Fa =
C8 =
FIGURE 10-16. A mandibular utility arch incorporating third-order, anterior-segment
control: passive (dashed) and activated (solid) configurations (top); free-body diagram
of the arch (center), and the force systems delivered to the posterior (lower left) and
half anterior (lower right) segments.
Force and Structural Analyses of Representative Orthodontic Mechanics
tude of the posterior couple. In selecting a rectangular wire from which
to prepare the utility arch, if the torquing couple is to be secondarily
activated in holding action, seemingly a thin, wide, arch wire is appropriate toward maximizing the ratio of third-order stiffness to vertical
stiffness in the anterior segment. A small occlusogingival dimension
enhances the bending flexibility while a substantial faciolingual dimension adds to the potential for torsional stiffness.
The examples of utility-arch use to this point suggest that a variety of
anterior-segment displacements are possible. The relative ease of analysis, in the presence of the assumed symmetry, should now be apparent.
The validity of the modeling is not at question; the involved segments
are well defined and, practically speaking, unitized if individually leveled prior to engagement of the utility arch.
In a final example utilizing the appliance, the roles and locations of
action and anchorage are reversed. Noting the relative magnitudes of
root-surface areas, anterior segment versus terminal molars, under spe-
cific conditions it is reasonable to use the utility arch to extrude depressed molars with the six anterior teeth providing support. Figure
10-17 is in the format of the preceding several figures. The free-body
diagram of the half arch shows the necessary, third-order component of
the anterior force system (provided through use of rectangular wire, for
example). As is also suggested in Figure 10-14, the portion of the anterior vertical resultant against the incisors reflects a tendency toward flar-
ing these teeth; the torsional component provides a counter-tipping
moment, thereby giving added stability potential to the anchorage. Even
though the combined root area of three anterior teeth exceeds that of the
terminal molar, however, the differential is small enough that care must
be taken in choosing the wire size and amount of activating deflection to
minimize the tendency for reciprocal actions. Because of the differences
in root areas of the two sets of central incisors, the potential anchorage is
"stronger" in the maxillary than in the mandibular arch. The appliance
is more attractive in this application if some intrusion of the anterior
segment can be tolerated or is desired together with the molar displacement.
The second category of tooth movements noted in the opening paragraph of this section is restricted to occlusogingival displacements, but
the appliance contacts all dental units in one arch. The specific mechanics intend to produce, enhance, reduce, or eliminate a curve of Spee, the
pattern of collective occlusal-surf ace heights as viewed from the buccal
perspective. For this analysis, in the presence of midsagittal-plane symmetry, the half arch is divided into three segments: an anterior half-segment including central and lateral incisors and, possibly, the canine, a
middle segment including one or both premolars and, if not part of the
anterior half-segment, the canine, and a posterior segment including the
erupted molars and, occasionally, the second premolar. If a tooth has
been extracted and site closure has not yet occurred, the modeling will
generally place the extraction space between two segments. After prepa-
Bioengineering Analysis of Orthodontic Mechanics
F8 =
Ca = F8(€')
FIGURE 10-17. A utility arch to extrude mandibular terminal molars: buccal views of the
activation and anchorage sites and the arch (top); a free-body diagram of the half arch
wire (center); and the force systems exerted on the terminal molar (lower left) and half
anterior segment (lower right).
Force and Structural Analyses of Representative Orthodontic Mechanics
ration of the continuous arch, its passive shape should be superimposed
over the line of brackets in the buccal view. In an attempt to gauge just
how the activated arch will be configured occiusogingivally after working for a time, the relative positions (wire to brackets) should be such as
to distribute the localized activations approximately evenly over the half
arch; the engaged wire will attempt to assume a minimum-energy (of
deformation) orientation (Chap. 4). Figure 10-18 exemplifies such a superimposition; the objective is to reduce or eliminate a distinct curve of
In a first example, the intent is to reduce the curvature in the mandib-
ular arch through placement of a flat arch wire. The relevant sketches
are those of Figures 10-18 and 10-19. Assumed is the arch wire fabricated
in round wire of a size relative to the bracket slots such that no second-
order couples are developed upon activation. The free-body diagram
within Figure 10-19 shows a resultant force for each segment; the quasistatic analysis indicates that Fmjd is in magnitude the sum of
and Fa.
Typically (with second molars erupted), the posterior-segment, root-surface area exceeds that of the anterior segment (and, perhaps, also that of
the middle segment). Aware, in addition, that extrusion is, generally,
more easily accomplished than intrusion (with moderate-size forces),
the likely displacement of substance will occur in the middle segment (if
not blocked by occlusal contacts). Light forces generated by a wire of
small diameter will enhance the possibility of anterior-segment intrusion.
An examination of similar mechanics in the maxillary arch is of interest. The sketches in Figure 10-20 are comparable to those in Figure 10-19;
the magnitude relationship among the three forces is the same, but directions (senses) of potential displacements are reversed. In the maxillary arch, in the absence of posterior occlusion, extrusion of the molars is
possible (particularly if the second molars do not engage the wire). To
deter a similar displacement in the anterior segment, unitizing the six
anterior teeth through appropriate ligation is warranted. If the group-
Buccal tube
Passive arch wire
FIGURE 10-18. Buccal view of the superimposition of a mandibular leveling arch wire
and the line of brackets prior to engagement. The purpose of the superimposition is to
anticipate the occlusogingival displacements to be generated by the arch wire over a
working time period.
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 10-19. A free-body diagram in the buccal view of the activated arch wire of
Figure 10-18 (top), and the occiusogingival forces delivered to the posterior, middle,
and half anterior segments of the mandibular dentition.
ings of dental units are such as to define the middle segment as the two
adjacent premolars, from the relative size of Fmid some intrusion of those
teeth is inevitable also.
Within the initial assumptions made, the displacements achieved in
mechanics intended to modify the curve of Spee depend on the relative
sizes of the root-surface areas of the segments, occiusal contacts with
opposing teeth, the magnitudes of the forces generated in activation,
and, to some extent, the type of displacement (intrusion versus extrusion). The sizes of forces produced are governed by the characteristics of
the arch wire that determine bending stiffnesses and the difference in
Force and Structural Analyses of Representative Orthodontic Mechanics
FIGURE 10-20. Reducing the curve of Spee in the maxillary arch with a round arch
wire, as activated anteroposteriorly passive (but stopped against posteriorly directed,
relative displacement through the buccal tubes): free-body diagram of the half arch wire
(top) and occiusogingival forces delivered to the segments of the half arch (center)
immediately after activation; and antero posterior forces generated secondarily against
the terminal molar (lower left) and half anterior segment (lower right) with on going
leveling. Action-reaction counterparts appear dashed in the free-body diagram (top).
severity of curves exemplified in Figure 10-18. The clinician may influence all pertinent parameters and must consider each in designing the
appliance and segmenting the dentition.
Bioengineering Analysis of Orthodontic Mechanics
In addition to the possible appearance of couples in the force system,
due to absence of second-order clearances, another secondary effect
must be mentioned before concluding this section. In the process of
reducing the severity of a curve of Spee, the deactivation of the appliance results in an increase in its effective anteroposterior length. Ligations in the anterior segment and, if present, stops or friction at the
terminal molars will induce anteroposterior force in the manner illustrated (dashed) in the free-body diagram in Figure 10-20. As the deactivation continues the forces grow; a horizontal force balance yields equal
magnitudes of force delivered to the anterior and combined posterior
segments. These forces (for one-half of the arch) are shown in the lower
sketches of the figure. Because the anteroposterior stiffness of the wire is
substantial, the horizontal force components can become sizable with
only a small curvature reduction. A consequence of this development is
the flaring of the incisors possibly accompanied by distal crown tipping
of the terminal molars. To counteract this generally undesirable side
effect, the clinician must either ensure relative freedom of the arch wire
to slide through the molar tubes as the curve is flattened, or initially
activate the appliance horizontally in opposition to the expected effect of
curvature reduction by tying back posterior ends of the arch wire with
flexible auxiliaries. With the latter approach, initial horizontal forces are
developed upon activation that are opposite (in sense) to those shown in
Figure 10-20; as the appliance deactivates, these forces decay along with
the vertical components. Careful control of the magnitude of this horizontal action is necessary if interproximal spaces exist at this time.
Intra-arch Retraction Mechanics
Anteroposterior therapy procedures to close spaces, correct procum-
bency, reduce overjet, and eliminate extraction sites often may be gener-
ally categorized as "retraction mechanics." Such therapy procedures
may involve interarch action and response (subsequently examined in
this chapter), extraoral appliance delivery (discussed in Chapter 9) or—
the present focus of attention—treatment within the single arch. Either
a continuous- or segmented-arch approach might be undertaken. The
usual common characteristics of the various procedures include posteriorly directed driving forces as the primary actions, posterior-segment
anchorage, and, often, bilateral action in the presence of midsagittalplane (geometric and mechanical) symmetry. Two example displacements are analyzed in this section: the retraction of a canine into a space
created by the extraction of a first premolar, and the retraction of an
incisal segment.
In conventional, continuous-arch, edgewise therapy the retraction of
a canine is attempted with the tooth subjected to a distal-driving force
induced by a stretched elastic or helical spring and the dental unit experiencing movement with respect to a stationary, guiding arch wire. With
Force and Structural Analyses of Representative Orthodontic Mechanics
the driving-force element attached at midcrown level, occiusal to the cre
of the canine, the potential tipping displacement can be transformed by
the arch wire, generating a second-order, counter-tipping couple, to a
movement characterized by little, if any, change in long-axis angulation.
If the line of action of the driving force is buccal or lingual of the long
axis (i.e., the distally directed force system is not split faciolingually and
two driving elements used), the bracket-wire connection must also provide first-order control (unless long-axis rotation is a desired part of the
overall displacement).
Shown in Figure 10-21 are sketches associated with the analysis of a
process on a guiding arch wire. Providing the distaldriving action is a compressed, linear, helical-coiled spring, stabilized
through its winding about the arch wire and maintained in the activated
state by a segment of ligature wire tied between the mesial end of the
spring and the buccal tube affixed to the terminal molar. Note first in the
figure a composite illustration of one-half of the arch wire, the involved
dental units, the spring, and the piece of ligature wire mentioned. A
free-body diagram of the compressed spring is also shown. The magnitude F, of the spring force is the product of the modulus of the spring
and the deformation from the passive state (Chap. 4). The ligature-wire
loop, as indicated in its free-body diagram, like the spring, is a two-force
member; mesiodistal force balances in the two diagrams, with the
action-reaction law, indicate that all four force magnitudes are equal.
Action-reaction between the ligature loop and the buccal tube yields the
force shown, directed mesially, against the molar crown. Through
crown contact, that responsive force is distributed with a portion of it
transferred to the adjacent second premolar. (If the second molar was
erupted and aligned, through a ligature tie to it the responsive force
could be further distributed.) The second-order couple
should be
present, resulting from a tip-back bend placed in the arch wire, to reinforce the anchorage by amending a tipping potential toward a bodilymovement format.
A sketch of the canine shows the contact force from the distal end of
the spring against the mesial extent of the canine bracket; action-reaction
indicates that the distal-driving force also equals F, in size. The countertipping couple
is exerted by the arch wire. Although if total bodily
movement is desired,
must be activated simultaneously with the
spring (and in the proper couple-force ratio), and consideration must be
given to the potential impedance to movement of a placed second-order
bend along the "path" to be taken by the canine bracket. The alternative
is to allow C, to be a secondary activation, growing in magnitude from
zero following the disappearance of second-order clearance between
canine bracket and arch wire. Some tipping will then occur, the amount
depending on the spring characterisEics, the initial bracket-wire clear-
ance, and the bending stiffness of the arch wire; the couple will approach an upper bound equal to the concurrent moment of the driving
force about the cre of the canine, but the inherent deterrent of a permanent bend in the wire is not present.
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 10-21. Buccal-view sketches associated with the force analysis of a
canine-retraction procedure on a guiding arch wire. The distal-driving force is
provided by a compressed, coiled spring.
Force and Structural Analyses of Representative Orthodontic Mechanics
A free-body diagram of one-half of the arch wire is also given in
Figure 10-21. Through the action-reaction law once again, the couples
exerted by the canine and the posterior segment are obtained. Because
neither the spring nor the ligature loop have direct contact with the
wire, those forces do not appear in the diagram. The placement of the
lip-back bend, however, may produce in the arch wire in the molar
region, upon activation, an occlusogingival force between wire and buc-
cal tube. That force on the wire and its action-reaction counterpart
against the molar tube are shown dashed in the two sketches. An occlusogingival force balance in the free-body diagram of the half arch wire
requires the existence of another vertical force, also shown dashed. If
these vertical forces exist as secondary effects, a moment balance in the
wire free-body diagram results in the two couples, although opposite in
sense, being of different magnitudes. Note also that, depending on the
character of the tip-back bend, the vertical forces may be as drawn or all
have senses opposite to those shown.
These mechanics exemplify the differential-force concept. Whatever
the level of activation of the spring, the mesiodistal forces against the
canine and anchorage crowns are equal in magnitude and bodily-movement potentials are established, or nearly so, at both activation and support sites. If only intraoral anchorage is used (although it could well be
reinforced with headgear), the differential must be in the root-surface
areas. Accordingly, inclusion of as many teeth as are available for support is prudent. If desired, through appropriate placements of stops—
the arch wire against the posterior attachments—distribution of the responsive force can be directed into the arch wire, transmitted anteriorly
through it, and the incisal segment can be added to the anchorage.
Before leaving this example, two additional noteworthy facets are
illustrated in the occlusal-view sketches of Figure 10-22. Indicated on the
left is the division of the distal-driving force from the spring into that
part actually reaching the crown of the canine and the frictional component that enters the arch wire. Although present in the actual, clinical
situation, this friction—arising from the canine-bracket and ligation contact with the arch wire—is ignored in Figure 10-21 and the accompanying discussion. To quantify the division of the spring force is virtually
impossible, but a reasonable assumption if employing light to moderate
force magnitude is to expect that approximately one-half of the spring
action actually reaches the canine. The presence of the friction tends to
displace the arch wire toward the posterior with the canine. The incisal
segment is, then, vulnerable to displacement if the wire is not properly
stopped against the posterior, anchorage teeth. (Refer to the Frank and
Nikolai (1980) paper, not only for the discussion of bracket-wire friction,
but also for the comments regarding the selection of the guiding arch
The sketches on the right in Figure 10-22 exhibit the substance of the
first-order effects of the buccal location of the driving-force element and
the arch wire. Recalling the immediately preceding comments about friction, perhaps a counter-rotation tie mesial to the canine is preferable to
Bioengineering Analysis of Orthodontic Mechanics
delivered to canine
Portion of
Carried posteriorly
by friction
Arch wire
FIGURE 1 0-22. 0cc/usa/-view sketches associated with the force analysis of a
canine-retraction procedure on a guiding arch wire: frictional-force analysis (left) and
first-order effects (right).
dependence on the ligation to generate the moment to oppose the po-
tential for long-axis rotation. Tracing the force analysis through the arch
wire and into the supporting dentition, the responsive first-order couple
is best generated in the posterior segment—with the aid of a long buccal
tube—and kept away from the lateral incisor.
A somewhat different approach to intraoral canine retraction replaces the distal-driving element and the guiding arch wire by a single
auxiliary activated between the canine and the buccal segment. Although two, or perhaps four, canines may be moved concurrently into
adjacent extraction spaces, these mechanics in any one quadrant are
virtually independent of ongoing therapy elsewhere in the oral cavity.
Two or three teeth of a posterior segment are ligated together and the
desired canine displacement is initiated within the hypothesis of differential-force treatment.
An example auxiliary, fabricated from a straight segment of ortho-
dontic wire, is shown from a buccal perspective in the top sketch in
Figure 10-23. If bodily movement is wanted, as is typical, the design of
the auxiliary is intended to yield mesiodistal flexibility (to enable generation of a relatively light driving force in an activation of several millime-
ters), a corresponding, second-order stiffness level (in an attempt to
maintain the as-activated, couple-force ratio nearly constant over time as
movement takes place), and, assuming buccal placement of the auxiliary, to deter distolingual rotation of the canine, a high first-order stiff-
Force and Structural Analyses of Representative Orthodontic Mechanics
10-23. Buccal views of segmented-arch, canine-retraction mechanics: the
fabricated retraction spring (top); a free-body diagram of the activated spring (center);
and the crown force systems exerted on the canine and the posterior-segment
anchorage (bottom).
A free-body diagram of the "retraction spring" is shown in the
center of the figure. A horizontal force balance requires and to be
equal in magnitude. As noted in the previous example, to generate a
second-order couple without some accompanying occlusogingival
action is difficult; such secondary forces, shown dashed in Figure 10-23,
must be mutually opposite in direction against the auxiliary (from a ver-
tical force balance), but may be as sketched or with both senses re-
Bioengineering Analysis of Orthodontic Mechanics
The moment balance, in the presence of the vertical forces,
yields unequal
C magnitudes. If the opposing molars are in
occlusion, perhaps the better alternative in vertical-component senses is
that shown in the figure; accordingly, if some distal crown tipping occurs in the retraction process, a deterrent to extrusion is present.
The action-reaction law permits the proper illustrating of the crown
force systems exerted by the auxiliary on the involved dentition in the
buccal-view sketches of the lower portion of Figure 10-23. Unlike the
previous example, here the distal-driving force and counter-tipping couple against the canine crown are individually activated. Maintaining the
proper couple-force ratio concurrent with a translational displacement
is, strictly speaking, impossible; the magnitude will decrease as the
canine crown moves toward the posterior while
diminishes only if
distal root movement occurs. Accordingly, mesiodistal flexibility of the
auxiliary must be high, as noted previously, to minimize the rate of
decay with movement, and the initial couple-force ratio should be less
than that required for bodily movement (one-half of the total canine
length; see Chap. 5), observing that the ratio will grow as the displacement progresses.
The sketch of the terminal molar in Figure 10-23 indicates that the
moments, with respect to the cre, of the mesially directed and secondorder responsive components are opposite to one another as desired;
the couple works against the mesial-crown-tipping potential of the
force. These components, together with the vertical responsive force,
represent the resultants against the posterior-segment anchorage. If
arch-length preservation is critical, as much root-surface area as is avail-
able should be used in distributing the responsive force system. Note
that, in the absence of the continuous arch wire, the incisors cannot be
made part of the anchorage. Note also that, although the friction of the
continuous-arch retraction mechanics is absent in this approach, perhaps more attention needs to be focused on displacement control. For
the auxiliary, a rectangular arch wire is suggested; the faciolingual dimension should be sizable to enhance occiusogingival-plane bending
stiffness and first-order control while the occlusogingival dimension
should be small to minimize the necessary complexity of the buccal-view
geometry of the retraction spring.
Intra-arch therapy designed to retract the anterior or incisal segment
exhibits features similar to those of canine-retraction
larly in buccal-view evaluations, but both obvious and subtle differences
exist as well. For the arch as a whole, an analysis begins by consideration of three segments: an anterior segment and two posterior segments.
The canines belong to the posterior segments if just the tour incisors are
to be retracted; if six teeth are to be moved, the canines are part of the
anterior segment. Each of the three segments is unitized;
of crowded incisors, if necessary, and all leveling have been previously
accomplished. Although the responsive force system is bilateral and the
retraction of the anterior teeth is intended to be undertaken in unison,
midsagittal-plane symmetry presents the opportunity for a one-half-
Force and Structural Analyses of Representative Orthodontic Mechanics
analysis. Involved in the discussion from this point, then, are two
or three anterior teeth and one associated posterior segment. As with
canine retraction, both continuous and segmented-arch-wire procedures
are used, and both are examined subsequently. Of substantial concern,
particularly in the simultaneous retraction of incisors and canines, is the
stability of the posterior-segment anchorage.
Anterior retraction encompasses all displacements of the four or six
teeth characterized by some lingual movement of the crowns and/or root
apices. Depending on the initial positions and angulations of anterior
teeth (as viewed from a buccal perspective), one of several possible orthodontic displacements may be required. Using the U-shaped archwire model—halved at the midline—the displacements are categorically
transverse and first-order control need not provoke the degree of concern inherent in canine-retraction mechanics, particularly when only the
incisors are to undergo displacement.
Shown schematically in the upper portion of Figure 10-24 are the
incisors and posterior teeth engaging one-half of a continuous arch wire.
The intent of the mechanics is to close space between the lateral incisors
and the canines and, in doing so, to complete the displacement with the
desired labiolingual, long-axis angulations of the maxillary incisors, The
free-body diagram of the half arch wire and the force systems delivered
to the half-incisal and posterior segments are shown in the center and
lower portions of the figure, respectively. The anteroposterior, active
and responsive forces are ordinarily generated in one of two ways. A
hook is affixed to or an "omega loop" may be bent into the arch wire
mesial to the molar buccal tube (assuming facial-surface mechanics) or
an appropriate opening or closing loop is incorporated in the arch distal
to the lateral incisor. The arch is "tied back" either with flexible elastic
elements or, in the presence of the loops (providing the mesiodistal
flexibility), with segments of ligature wire. With the former method,
illustrated in the figure, the posterior ends of the arch wire are left "free"
to slide within the attachments to the buccal segments. The third-order
couple Ca may be initially activated, secondarily activated for control
only, or not present at all, depending on the character of the retraction
displacement desired. Correspondingly, a responsive couple against the
posterior segment may or may not exist. Vertical force components may
also be present, as noted previously in the canine-mechanics discussion,
but for simplicity they are not included here.
Figure 10-25 presents appropriate illustrations for mechanics using a
retraction spring to activate an anterior-segment force system. (An analogy is drawn here with Figure 10-23 and the discussion accompanying
it.) Overall, in this approach, arch-wire segments individually engage
the incisal and posterior dental segments. To create incisal-segment
torque (a moment about the bracket line), the retraction spring is activated against a hook affixed occlusally or gingivally of the wire-segment
level, as discussed in Chapter 9 (see Fig. 9-28). The posterior end of the
spring is attached to the buccal-tube assembly affixed to the terminal
Bioengineering Analysis of Orthodontic Mechanics
FIGURE 10-24. Buccal views of continuous-arch mechanics to retract an incisal
segment: representations of the buccal and half -incisal segments and the appliance
(top); free-body diagram of the half arch wire (center); and active and responsive force
systems delivered to the dental segments (bottom).
Force and Structural Analyses of Representative Orthodontic Mechanics
FIGURE 10-25. Buccal views of segmented-arch mechanics to retract an incisal
segment: the buccal and half-incisal segments and the appliance (top); free-body
diagram of the retraction spring (center); and representations of force delivery to the
haif-incisal and buccal segments (bottom).
Because of the greater root-surface area, compared to that of the canine, a larger driving force is appropriate against the haif-incisal segment. Accordingly, the inherent potential for the responsive force system to displace the anchorage mesially is increased. Moreover, the
Bioengineering Analysis of Orthodontic Mechanics
counter-tipping couple cannot be generated in the posterior segment
with the mechanics illustrated in Figure 10-25. If all six anterior teeth are
to be retracted together, first premolars have been extracted, and third
molars are unavailable for inclusion in the anchorage segments, unless
mesial migration of the posterior teeth is a desired adjunct to the retraction process, an appropriate extraoral appliance (e.g., a face-bow headgear) is almost a necessity to transfer away at least a portion of the
responsive force system. Alternatively, at minimum, posterior anchorage must be established and precede the retraction procedure. Posterioranchorage preparation is discussed in the section that follows.
Prepared, Posterior-Segment Anchorage
In one or more phases of typical treatment mechanics the posterior segments ordinarily are recipients of responsive force. Moreover, these seg-
ments are expected to sustain such force systems without experiencing
significant displacements. During canine or anterior retraction, for example, mesial movements of the buccal segments are often unwanted
and may be intolerable. As mentioned in the preceding section,, headgear may be used to reinforce posterior anchorage, but the orthodontist
cannot be assured of complete patient cooperation during the betweenappointments period. Accordingly, prior to the activation of mechanics
against the posterior teeth, the clinician may "prepare" the buccal segments to withstand the responsive forces to come by making that anchorage as strong and resistive as possible.
This section does not fully examine the process of posterior-anchorage preparation; rather, it focuses exclusively on the posterior segment.
The procedures are nearly always bilateral and the preparation may be
undertaken in either arch. Several biomechanics concepts form the basis
for this process: (1) the strongest resistance is that which, upon action
against it, responds with force carried to its supporting structure that
approaches a uniform distribution; (2) the resorption of newly apposed
bone is more difficult than the removal of mature bone; and (3) the most
easily performed orthodontic displacement is that of extrusion. The objective of posterior-segment anchorage preparation is the distal crown
tipping, without loss of arch length, of at least the terminal molar and,
preferably, all of the teeth in the segment that contribute to the anchorage. The force to be resisted is likely delivered to the
molar (and
then distributed throughout the segment); it often originates with an
interarch "elastic," and must not be directed with respect to the long
axis of the molar such that the potential for extrusion is present. Anticipating that some loss of anchorage may occur in almost all instances of
force applied against it, the amount or degree of anchorage preparation
must be dependent on the magnitude of the responsive forCe system,
the length of time the responsive force is expected to be present, and the
format of anchorage displacement expected, should it be realized.
Force and Structural Analyses of Representative Orthodontic Mechanics
buccal-segment anchorage is most typically prepared for
subsequent activation against it of inclined, interarch (Class III or Class
II) elastics. Such an'elastic may be stretched between one hook affixed to
an engaged arch wire (the hook located between lateral-incisor and canine brackets) and another hook, part of the buccal-tube assembly, affixed to the terminal molar on the same side but in the opposing arch.
With the mandible closed, typical distances between ends of the
stretched elastic, parallel and perpendicular to the occlusal plane, are 22
to 30 mm and 8 to 9 mm, respectively. The angle of the elastic with the
occlusal plane is, then, in the range of 15 to 20° with the teeth occluded.
As shown in Figure 10-26, if the angle of interest is 18° and the force in
the stretched elastic is 4 oz, the horizontal and vertical components are
108 g and 35 g respectively. Moreover, the 35 g is a sizable extrusive
force against the molar if its long axis is vertical. In addition, as the
mandible opens, the elastic is further elongated and the angle between
the stretched elastic and the long axis is diminished, producing a compound increase in extrusion potential.
In partial preparation for interarch elastic force, the terminal molar,
together with the adjacent teeth in the buccal segment, are displaced in
distal crown tipping to establish angulation, relative to the anticipated
stretched-elastic position and direction, similar to that of a tent peg. The
angle between the occlusal extent of the long axis (of the "peg") and the
elastic should be made greater than 90° as indicated in Figure 10-27 to
eliminate the potential for extrusion. Subsequently, when the elastic is
actually activated, creating the potential to tip the molar crown mesially
and roll the tooth mesiolingually, a ligated, stabilizing arch wire is engaged to produce both counter-tipping (second-order) and counter-ro-
108 g
4 oz
4 oz
FIGURE 10-26. Horizontal and vertical components of force delivered by a typical, 4 oz,
Class II elastic force.
Bioengineering Analysis of Orthodontic Mechanics
90° pIus
FIGURE 10-27. A buccal view of a mandibular, terminal molar with crown tipped distally
so that the Class II elastic force does not include an extrusive component.
tating (first-order) couples and reinforce the anchorage. As noted previ-
ously, this provides as strong a resistance to movement as possible,
considering the available root-surface area.
To prepare the anchorage without losing arch length in the process,
the proper displacement is crown movement as defined in Chapter 5.
The desired angulation is to be achieved without mesial migration of the
root apex and with minimum displacement of the crown. Accordingly,
the cm of the tooth should be positioned at the level of the root apex.
From a buccal perspective, a distal driving action is required together
with a second-order couple to "hold" the apex. In the clinical setting,
because achievement and maintenance of the necessary couple-force
ratio is virtually impossible, perhaps an appropriate compromise incorporates an initial simple-tipping displacement followed immediately by
bodily movement. The distal driving force may be created in several
Force and Structural Analyses of Representative Orthodontic Mechanics
ways, but it is a "pushing" action delivered by the arch wire to one of
the teeth in the buccal segment through a stop. This force is distributed
within the segment through crown contact or interligation of the brackets and tube.
Within the buccal portion of the arch wire, second-order bends are
fabricated, to be passive upon wire engagement, thus permitting some
initial tipping without interference as shown on the left in Figure 10-28.
In the center sketch of the figure the tipping has eliminated the secondorder clearance, but some mesial displacement of apices has occurred.
With the distal driving force still active, further potential tipping is inhibited by the secondary activation of second-order couples (sketch on
the right in Figure 10-28) and, ideally, some bodily movement follows.
Continued distal crown displacement is accompanied by distal movement of apices as well, the latter toward restoration of the initial apical
positions (a procedure inherently difficult because of the need to resorb
new bone at the apices). Note the intent to displace the posterior teeth
not as a unit, but as individual "dominos," particularly during the tipping portion of the movement.
An alternative approach to posterior anchorage preparation is again
in a sequence of two displacements, but in reverse order to that just
described. Active second-order bends initially produce pure couple
loading (in the idealized absence of friction); the crowns move distally
and the apices mesially. Subsequently, the distal driving force is added
toward alteration of the displacement to bodily movement. This procedure has its shortcomings compared to that described formerly with (1)
the centers of rotation in pure, second-order torquing further from the
Arch wire
FIGURE 10-28. Distal crown movement in steps to prepare posterior anchorage: initial,
second-order clearance between wire and bracket as the displacement is begun by a
distal driving force (left); the second-order clearance eliminated by tipping (center); and
subsequent activation by the wire of a counter-tipping couple (right).
Bioengineeririg Analysis of Orthodontic Mechanics
than in simple tipping, (2) friction likely inhibiting the crown
movement along the wire, (3) the addition of the distal driving force
after the initial torquing displacement as a separate, not a secondary,
activation, and (4) the control of the couple-force ratio toward produchon of bodily movement more difficult when the transverse force follows the couple rather than vice versa.
Clinicians may describe the amount of anchorage preparation in a
qualitative manner as "mild," "moderate," or "severe." Involved are
the time interval over which these mechanics are continued and the
initial force level together with the sizes of the second-order bends to
achieve a particular set of long-axis angulations with respect to vertical.
The degree of preparation is jointly dependent on the expected responsive force system against the anchorage in magnitude, displacement-format tendency and time interval of existence of that response, the size of
the anchorage (in terms of root-surface area) relative to the action, and
the amount of "slipping" of anchorage that can be tolerated. At the risk
of being repetitive, an additional, important consideration is that new,
uncalcified bone is more difficult to resorb than mature bone; hence, the
elimination ("burning") of anchorage generally requires more time than
the establishment of it—an inherent asset not to be overlooked.
Following preparation of anchorage it is put to use. Typically this
anchorage in the maxillary arch must resist Class III elastics and, in the
mandibular arch, Class II elastics. To prevent the responsive forces from
the elastics from tipping the prepared posterior teeth, stabilizing, second-order bends are placed in the wire segments or buccal portions of
the continuous arch wire engaging these teeth. Moreover, to further
protect the terminal molars in particular (to which the elastics are attached) against possible extrusion, arch wires are fabricated such that,
upon engagement, intrusive forces are induced when possible in addition to the second-order force system from the tip-back bends. As time
proceeds following anchorage preparation, the newly laid-down bone in
the mesial, crestal regions adjacent to the posterior teeth matures and
becomes more susceptible to resorption. Concurrently, the time during
which the anchorage will continue to be "in use" is growing shorter.
As the wanted displacements proceed toward their desired extents,
the active force can be increased in magnitude. The accompanying responsive force will likewise grow, heightening the potential for the
"slipping" of anchorage. Because the anchorage is not to remain as prepared following treatment—rather, the buccal teeth will need to be uprighted—the beginning of the elimination of the long-axis inclinations in
the closing stages of Class III or Class II mechanics, for example, is
appropriate. Care must be taken, however, to ensure that the anchorage
"burning" procedure approaches the exact reversal in format of the anchorage-preparation process. If a portion of the preparation remains following discontinuation of the mechanics against the anchorage, a com-
bination of the "memory" of the periodontal-ligament fibers and
ongoing physiologic processes will help to restore the upright orientations of the posterior teeth. "Up and down" elastics can be used subsequently, as necessary, to achieve the desired intercuspation.
Force and Structural Analyses of Representative Orthodontic Mechanics
Class III Mechanics
For potentially good occlusion and dentofacial harmony, the maxillary
and mandibular arches should be properly positioned, anteroposteriorly, with respect to each other. Without attempting a strict definition or
detailed description, a Class III tendency or malposition generally suggests that the location of the mandibular dentition is improperly anterior
with respect to the maxilla. The mechanical objective of Class III mechanics is often to exert posteriorly directed force against the mandibular
arch. Alternatively, this phase of therapy may intend to impart action to
advance the maxillary dentition. Class III mechanics, however, typically
focus on the mandibular arch and may be designed, for example, to
produce posterior displacement of the incisors, the six anterior teeth, or
the entire arch to establish posterior anchorage (as described in the previous section), or to exert holding force against (or attempt to reverse)
undesired, excessive, forward growth of the mandible.
Class III mechanics generally require attachment of appliances to
dental units in both arches, interarch force transfer is ordinarily present,
and headgear may be used. The action and response are bilateral; geometric and mechanical symmetry relative to the midsagittal plane is typi-
cal and is so assumed in the analyses to follow. The U-shaped arch
model is appropriate, and usually necessary is the division of each half
arch from a buccal perspective into three segments: an incisal or anterior
(half) segment, a middle segment, and a posterior segment. The teeth
typically included in each of the three segments have been noted in a
previous section of this chapter.
The interarch elastics, which are the principal activating elements of
intraoral Class III mechanics, are each stretched between a gingival hook
affixed to the buccal tube integral to the crown of a maxillary terminal
molar and a hook attached to the mandibular arch wire just distal to the
lateral-incisor position. A free-body diagram of the right-side elastic is
shown in buccal view in Figure 10-29. The magnitude of tensile force in
the activated elastic is symbolized by Fe and the forces exerted on the
ends of the auxiliary have been decomposed into horizontal (occlusal-
plane) and vertical components. The second molar is assumed to be
included in the maxillary buccal segment; the angulation of the elastic is
= 3.7 (V,,,)
(with the elastic angulated
15° trom the occiusa] plane)
FIGURE 10-29. A free-body diagram in the buccal view of a right-side? Class Ill elastic.
Bioengineering Analysis of Orthodontic Mechanics
approximately 15° with the occlusal plane. With this orientation the hori-
zontal components are about 3.7 times as large as the vertical components and, for practical purposes, are equal to the elastic force itself.
Ordinarily engaging the maxillary dentition is a continuous, stabiliz-
ing arch wire for the purpose of distributing the force of the elastic
against the terminal molar. Although no direct connection between the
elastic and wire exists, the elastic force can be transferred to the arch
wire by the buccal-tube assembly. If the maxillary arch, in particular the
maxillary buccal segment, is to provide anchorage in these mechanics,
stabilizing second-order bends are placed in the posterior portion of the
arch wire including tip-back bends, perhaps active, at the terminalmolar positions. The second-order couples are intended to prevent mesial crown tipping of the posterior teeth and, often, the tip-back bend is
sufficiently severe to also create an intrusive force against the terminal
molar. In addition, an extraoral appliance to the maxillary arch, perhaps
including a face bow with direct attachments to the terminal molars,
may be used. Again, if anchorage reinforcement is the intention, the
headgear principally transfers away the horizontal component of force
originating with the elastic.
Although dependent on the objective of the mechanics to some ex-
tent, typically a full-size arch wire is placed in the mandibular arch.
From the hooks affixed to the wire, through which the elastic force is
carried to the mandibular dentition, the horizontal component may be
directed anteriorly, posteriorly, or both, depending on clearances and
stops. If the objective is to establish posterior mandibular anchorage, the
horizontal component is transmitted directly to the posterior segment—
perhaps first to the terminal molar, then distributed from that site
throughout the segment. In this instance the vertical component is unwanted, so neutralization is attempted through concurrent activation of
the arch wire itself (to be detailed subsequently) together with distribuHon of the component mesially and distally from the site of the hook to
involve as much root-surface area as possible.
The force and structural analyses of Class III mechanics is a formidable undertaking, but feasible after study of the previous chapters and, in
particular, with understanding of the examples cited thus far that consider component parts of the mechanics and their analyses. The model
has been proposed, several reasonable assumptions have been made,
and the appliance elements have been discussed. The specific application to be examined in some detail is that of anchorage preparation in
the mandibular arch. Referring again to Figure 10-29, of the four components present only one, the horizontal component delivered to the mandibular arch, is wanted. A significant portion of the mechanics is de-
voted to the suppression of the potential effects of the other three
components. The initial magnitude of the elastic force, as chosen, must
take into account the number of posterior teeth to undergo distal crown
tipping, some expected relaxation of the elastic, the angulation of the
elastic, the presence of friction, and the possible additional "loss" of
force to the anterior teeth.
Force and Structural Analyses of Representative Orthodontic Mechanics
The maxillary force system is illustrated in the buccal views in Figure
10-30. To be noted first is the tendency of the elastic force to extrude the
terminal molar and to tip the crown mesially. The maxillary arch wire
and, perhaps, molar occlusion are available to resist extrusion, in addition to the periodontal support. In developing the force analysis of the
maxillary half arch, three separate sketches are presented to highlight
the three-part activation of the wire. (None is a complete free-body diagram.) From top to bottom, the first shows the secondary action which is
a direct outcome of activating the elastic; with a stop mesial to and contacting the buccal tube, the elastic force tends to displace the posterior
extent of the wire and the molar downward and forward. The second
sketch illustrates the action-reaction counterpart to the primary activation of the tip-back bend, producing resistance to both mesial crown
tipping and extrusion. The third sketch depicts the primary activation of
a straight-pull headgear.
The composite free-body diagram of the maxillary half arch wire is
shown in the upper portion of Figure 10-31. Force-system components,
exerted by the dentition, have been added to those activating the half
arch to balance the free-body diagram. The headgear force is intended to
offset the horizontal component of the elastic force transferred to the
wire through the buccal attachment to the molar and the stop, although
some of the elastic action may find its way to the incisors (H1), partially
because of the high mesiodistal stiffness of the wire. Although Vmid
could alone provide the balance of vertical forces, the combination of the
From the Class IN elastic
From the tip-back bend
From the straight-pull
FtGURE 10-30. Right-side buccal views of the maxillary-arch force systems associated
with Class Ill mechanics intended to prepare posterior anchorage in the mandibular
Bloengineering Analysis of Orthodontic Mechanics
FIGURE 10-31. A right-side, buccal-view, free-body diagram of the maxillaiy half arch
wire in Class Ill mechanics (top). The crown force systems delivered to the posterior,
midarch, and haif-incisal segments (bottom).
force and couple associated with the tip-back bend necessitates the pres-
ence of a vertical-force component in the anterior region to obtain a
moment balance and complete the quasi-static analysis.
The action-reaction law now enables an examination of the force system induced against the maxillary dentition (lower portion of Fig. 10-31).
Looking once more at the terminal molar, which also represents the
posterior segment, the horizontal component of the elastic force is carried first into the arch wire and then is transferred to extraoral anchorage
by means of the headgear. The second-order couple opposes the mesial
tipping potential of the elastic force, particularly important when the
headgear is absent. The tendency of the elastic to extrude the molar
must be countered by the arch wire and/or occlusion. The combination
of the tip-back bend and the bending stiffness of the wire must be capa-
Force and Structural Analyses of Representative Orthodontic Mechanics
of providing the principal resistance to extrusion in the absence of
posterior occlusion. The elastics tend to promote occlusion; however,
note that posterior extrusion is potentially manifested in anterior biteopening that may or may not be a desirable adjunct to the mechanics.
Depending on the relative magnitudes of the elastic and headgear
forces, the horizontal force delivered anteriorly, if any, can have either
sense. The sketches in Figure 10-31 assume that the headgear force dominates, which yields the senses of the horizontal incisal components
shown. The vertical elastic-force component combined with the tip-back
action induces the vertical components exerted on the middle and halfincisal segments. In the sketch representing the half-incisal segment the
moments of the two force components about the cre oppose one another;
in the absence of the headgear force and with the wire stopped as previously described, there is a tendency to flare the incisors. An interincisal
force from the mandibular arch would add to the flaring potential if
exerted on the lingual surfaces. The middle segment of the half arch
wire may function as a fulcrum; the elastic action tends to rotate the arch
wire and the maxilla counterclockwise as viewed from the right side.
The anteroposterior position of the middle-segment force depends
largely on the characteristics of the posterior-segment force system; the
senses of the couple and vertical forces of the posterior segment suggest
that the resultant response in the middle segment is positioned more
posterior than anterior in that segment.
The appliances under discussion are facially affixed; hence, in addition to the tendencies seen in the buccal views of Figure 10-31, the elastic
action has the potential to also rotate and lingually tip the molars. The
arch wire must provide the resistance to these displacement tendencies,
too, and toe-in bends may be placed to generate countering, first-order
couples. Moreover, the arch wire should be rectangular, not only to
enhance bending stiffness potential relative to cross-sectional shape, but
also to enable creation of stabilizing, third-order couples to counter any
labiolingual, crown-tipping potential.
Before proceeding to an analysis of the mandibular-arch mechanics,
a review of the action and response associated with tip-back bends may
be worthwhile; this force system may exist in both arches. A free-body
diagram of one half of a mandibular arch wire was illustrated in Figure
6-27. The active tip-back bend was accompanied by an intrusive force;
this is the combination of force and couple shown in Figures 10-30 and
10-31. Recall that, although a responsive force from the middle segment
of the dentition would provide a vertical force balance, these two forces
would form a couple with the same sense (counterclockwise as viewed)
and, therefore, a moment balance is impossible. Thus, a third vertical
force must be present, anterior to the middle-segment force, likely small
because of its distance from the terminal molar, but creating a substantial, clockwise moment with respect to the buccal tube. Note further that
the activation of a prepared second-order bend, depending on the mesiodistal position of the bend with respect to the bracket or tube and the
overall occlusogingival passive position of the wire relative to the ortho-
Bioengineering Analysis of Orthodontic Mechanics
dontic attachment, may produce a vertical force of either sense accompa-
flying the couple. Shown in Figure 6-26 was the second-order couple in
combination with an extrusive force. This force system can be balanced
by just the addition of one responsive force as long as the couple-force
ratio is not extreme. The location of the responsive force is anterior of
the buccal-tube force system by an amount equal to the couple-force
ratio. The couple formed by the two vertical forces is clockwise and
opposite in sense to that of the tip-back couple. In both sketches the
specific location of the middle-segment response depends on the relationship between the magnitudes of the force and couple exerted by the
buccal tube.
The analysis of the right half of the mandibular working arch wire is
illustrated in Figure 10-32. Initially, on the left, separate principal actions
against the wire—from the elastic and second-order effects—are shown.
These are then combined and the free-body diagram of the half arch
wire is presented. The desired, horizontal component of the elastic force
is meant to be wholly transferred, by the arch wire, distally to the posterior segment to become the crown-movement driving force. A portion of
that horizontal component can reach the incisors, however, if not ini-
tially then after some tipping of the posterior teeth has occurred. The
location of the connection of the elastic to the arch wire, together with
the elastic angulation and the small root-surface areas of the teeth,
creates a strong tendency to extrude the incisors. Accordingly, a reverse
From the elastic
From the second-order bends
Posterior segment
Incisal half-segment
FIGURE 10-32. Right-side buccal views of the mandibular-arch force systems
associated with Class Ill mechanics designed to prepare posterior anchorage in the
mandibular arch.
Force and Structural Analyses of Representative Orthodontic Mechanics
curve of Spee in the wire is necessary to induce intrusive force against
the incisors and counter the vertical-component action of the elastic
force. The potential to extrude the middle segment is created simultaneously; occlusion is needed to effectively counter this tendency. The secmay or may not be active initially, and it may
ond-order couple,
appear with the opposite sense after some tipping has occurred, as
noted in the previous section. With appropriate changes in some magnitudes and, perhaps, some slight alteration in the position of the middlesegment force, both vertical-force-component and moment balances can
be achieved with or without the second-order effects present.
Once again, action-reaction enables transfer of attention from the
appliance to the dentition. The distal driving force is transmitted with
the aid of a stop from the arch wire to the buccal tube and molar, then
distributed to other teeth within the posterior segment. The source of
the intrusive force is the curvature placed in the wire; the tip-back couple, perhaps initially active, is also shown and further represents second-order contact within the segment. Occiusal force from the maxillary
arch may also be present and is indicated. Already noted is the tendency
for the vertical force, from the passive curvature of the wire, to extrude
the middle segment. Responsive components against the haif-incisal
segment include the opposing vertical forces, one arising with the elastic
action and the other from the wire curvature, and the horizontal component that tends to produce lingual crown tipping. (Unlike the horizontal
force against the maxillary incisors, this component can have only one
sense.) If the horizontal component is unwanted, labiolingual clearance
must exist initially between wire and incisor brackets; as that clearance
disappears with the posterior displacements, the arch wire must be adjusted or replaced to recreate the clearance. An alternative approach is to
deliver the elastic force not by the arch wire, but, instead, to a sliding jig
(see Chap. 9) that by-passes the middle segment and directly transfers
the horizontal component to the terminal molar.
As mentioned previously, these mechanics are generally characterized by the presence of Class III elastics, and in the example just detailed
only one of the four components (see Fig. 10-29) provides the desired
action. It follows, then, that various additional applications of Class III
mechanics can be categorized with reference, at least partially, to the
forces transmitted by the activated elastics. Considering another example in which the action is in the mandibular arch, the vertical component
of the elastic force might be used to help reduce anteroposterior arch
curvature through extrusion of the middle segments. (A light, mandibular arch wire would be engaged to concentrate the elastic action.) The
incisal-anterior segment might be retracted with the horizontal component providing the driving action, but the accompanying potential of the
activated elastics to extrude those teeth must be recognized and appropriately dealt with, depending on the need or desire to close the bite
through this displacement.
Reversing the action-response roles of the "ends" of the elastics,
movements in the maxillary arch are also possible with anchorage pro-
Bioengineering Analysis of Orthodontic Mechanics
Unified dentition
FIGURE 10-33. An orthopedic force system, in a right-side buccal view, associated with
maxillary-arch mechanics intended to reduce an anterior deep bite by rotating the
maxilla as a unit.
vided in large measure by the mandibular dentition. With the mandibular posterior teeth providing some stability against undue extrusion of
their interarch counterparts, previously prepared maxillary, posterior
anchorage may be eliminated. In fact, advancement of the posterior segments anteriorly in the maxillary arch may be undertaken. The limit to
whole arch forward movement is in the sufficiency of the mandibular-
arch anchorage and, although headgear may be used to augment the
anchorage potential, extraoral-appliance application to the mandibular
arch demands extraordinary considerations, in comparison to maxillaryarch attachment, which have been discussed in Chapter 9.
Another important application of Class III mechanics in deep-bite
cases is the coupling of the elastic force in the posterior to that from an
anterior-positioned, high-pull headgear to rotate the entire maxilla. A
heavy, continuous arch wire must be used to provide vertical unification
of the maxillary dentition through as large a bending stiffness as possible. The active force system is shown in Figure 10-33. The two forces
appearing in the buccal view should be parallel and equal in magnitude
if the rotation is to be about the cra (center of rotation of the maxilla),
partitioning the displacement effectively between the anterior and the
posterior segments. Note, however, that the existence of posterior occlusion can hinder the downward movement of the posterior portion of
the maxilla and, in effect, move the cra posteriorly.
Class II Mechanics
The intraoral appliance assembly used in typical, edgewise Class II mechanics appears, excepting the headgear, to be the mirror image in the
buccal view of the Class III assembly discussed in the previous section.
Force and Structural Analyses of Representative Orthodontic Mechanics
The appliances include arch wires, generally both continuous, individu-
ally engaging the maxillary and mandibular dentitions; the desirable
structural characteristics and detailing of the wires depend on the clinical objectives. In Class II mechanics, force is transferred between arches
by bilateral elastics that are individually stretched between hooks at
mandibular-posterior and maxillary anterior sites. These mechanics, like
those of the preceding section, receive their name from the type of malocclusion they often help to correct: the maxillary dentition is forward of
the mandibular teeth, with respect to the ideal dentofacial-complex configuration, and the visible manifestation is known as "overjet."
In Class II mechanics the stretched elastics may provide one or more
active components of the desired force system. If the intent is to retract
the incisors or the six anterior teeth or, perhaps, the entire maxillary
dentition, the elastics exert at least part of the posteriorly directed driving force. Torque against the anterior teeth, to hold or alter the labiolingual, long-axis angulations, may be a part of the active force system. An
extraoral appliance may be used concurrently to provide additional driving action and, possibly, third-order action as well. In particular, to offset the often unwanted extrusive potential of the vertical component of
the elastic force against the maxillary arch, a high-pull headgear may
augment the intraoral mechanics. On the other hand, if the malocclu-
sion includes an anterior open bite and that correction has not been
undertaken earlier, the vertical component of the Class II elastic force is
welcome if the initial position of the maxillary-anterior teeth relative to
the soft tissue is such that extrusion can be tolerated as a corrective
The action of the appliance may be intended to be at the mandibular
posterior attachment sites, alternatively, with anchorage provided by
the maxillary arch. The clinical objective may be to upright the mandibu-
lar molars or the "burning" of prepared anchorage that is no longer
needed. Protraction of the mandibular dentition to aid or augment
growth of a potentially deficient mandible may be attempted at the appropriate time. Another possible objective might be that of posterior bite
closure through extrusion and, perhaps, accompanying mesial crown
tipping of the mandibular posterior teeth. As noted in the discussion of
Class III mechanics, the interarch elastic is a two-force member with
each force considered the resultant of components parallel and perpendicular to the occlusal plane; maybe just one, or perhaps all four of the
components of the forces transmitted by the inclined elastic may be desired as active initiators of displacement in a specific situation. The detailed analysis undertaken here, however, is that of the Class II mechanics used to produce controlled, posteriorly directed, bodily movement,
displacement potential delivered to the maxillary incisal segment.
In anticipation of the forthcoming Class II mechanics, posterior-anchorage preparation may have been completed in the mandibular arch;
the degree of preparation is dependent on the available root-surface
area, the expected magnitude and duration of the elastic forces, and the
tolerable amount of anchorage "slipping" concurrent with the Class II
Bioengineering Analysis of Orthodonfic Mechanics
mechanics. A full-size stabilizing wire is ordinarily engaged in the man-
dibular arch for two principal reasons: (1) to unify on each side the
posterior teeth toward distribution of the long-axis-parallel component
of the elastic force tending to extrude the terminal molar (if and when
the occlusal direction is less than perpendicular to the stretched elastic),
and (2) to permit creation of sizable second-order couples with little wire
deformation and, therefore, inhibit mesial crown tipping of the posterior
teeth in favor of establishing bodily-movement potential. Second-order
bends may be placed in the posterior sections of the wire to create immediately active, counter-tipping couples along with intrusive action from
the arch wire, particularly at the terminal-molar sites. Should the elastics
begin to produce mesial crown tipping of the terminal molars, crown
contact will distribute that effect anteriorly and sequentially involve the
teeth as far forward as the canines in the resistance. If some flaring of
upright, mandibular incisors is a desired displacement adjunct, a portion of the anteriorly directed effect of the elastic may be diverted into
the arch wire by a judiciously located stop mesial to the buccal tube and
carried within the wire to the incisal segment.
The activated, Class II elastic pulls on a gingivally directed hook affixed to the maxillary arch wire (or against a loop bent into that wire)
located in the interbracket space distal to the lateral-incisor position. (If
the six-tooth anterior segment or the entire maxillary arch is to be retracted, the elastic attachment may be distal to the canine site.) This wire
is rectangular, to permit activation of third-order couples, but is a working wire having sufficient flexibilily to enable moderate, activating deflections and twists without creating force systems having excessive initial magnitudes.
Two maxillary-arch-wire designs are rather common: one with gingivally oriented, simple, closing loops and the other without loops. If
the wire contains no loops, then elastics (perhaps together with headgear) generate the wanted, posteriorly directed force components that
are carried by the wire to the incisors. To permit retracting displacements, the arch wire must be permitted to sllde through the posterior
brackets and buccal tubes. To enhance clearances in an effort to minimize frictional resistance, the posterior sections of the maxillary arch
wire might be reduced in an acid bath prior to engagement. Furthermore, for the same reason, permanent bends are not placed in the posterior sections of the wire (which means that no stabilizing couples are
possible without eliminating the clearance and inducing resistance to
the retraction effort).
The placement of loops in the maxillary arch, alternatively, changes
the picture substantially if those loops are to be activated, either initially
or secondarily (and serve other than just an elastic attachment). Bilateral
stops are affixed to the wire mesial to the buccal tubes. Depending on
the positions and stability of the maxillary posterior segments, and perhaps other, more subtle considerations, the arch wire may be tied back
to open and activate the loops somewhat and generate the initial posteri-
orly directed force against the incisors. Subsequent activation of the
Force and Structural Analyses of Representative Orthodontic Mechanics
II elastics augments the wanted force initially. As retraction occurs, the ioop force deactivates more rapidly because of its greater mesiodistal stiffness compared to the extensional stiffness of the elastic and,
in time, the loop force may change sense and oppose the horizontal
component of the elastic force. The initial mesial-movement potential
against the maxillary posterior teeth is changed then to a distalization
tendency. On the other hand, if the loops are passively open and the
stops are initially up against the buccal tubes, activating the elastics
tends to close the loops, secondarily activating them. A division of the
horizontal component of the elastic force occurs; a portion of it is carried
to the incisors and the remainder to the posterior segment, and both
responses to the elastic force are posteriorly directed from the start. The
loop attempts to return to its passive configuration and, as time proceeds, the ratio of force magnitudes delivered to the segments, incisal
compared to posterior, diminishes. Accordingly, this last arrangement
would be most appropriate for whole-arch retraction.
Figure 10-34 schematically compares the changes in forces to the seg-
ments over time, resulting from immediate, opening activation versus
Passive, partially closed
FIGURE 10-34. Active and delivered mesiodistal forces in the maxillary arch arising fmm
the combined use of Class II elastics and simple, vertical loops. Right-side, buccalview, free-body diagrams of the half arch wires, and changes in the force magnitudes
with time (top to bottom), comparing mechanics with the loop initially opened elastically
(left) and initially passive (right).
Bioengineering Analysis of Qrfhodontic Mechanics
secondary closing activation. Not to be overlooked in this discussion is
the effect of the loops on the localized flexibilities (other than the mesio-
distal influence) of the anterior portion of the arch wire relative to the
posterior portions. Depending on ioop geometry, a substantially increased tendency for the vertical components of the elastic forces to
extrude the incisors may or may not be created. The third-order flexibility of the anterior arch-wire section, however, is invariably enhanced by
the presence of any loops (see Chap. 8).
Considering further the vertical-component influence on the maxillaw incisors, three counter-extrusive procedures, when warranted, are
commonly used, either individually or together. First, if the tied-back,
looped arch is used, the legs of each loop may be gabled with respect to
one another. The result is the activation of an intrusive-force distribution
against the incisors with engagement of the wire and prior to placement
of the interarch elastics. Second, active second-order bends in the postenor segment together with intrusive action against the terminal molars
concurrently induces, in response, intrusive force against the incisal segment. An accompanying feature of this force system, specifically in the
created second-order couples, is the strengthening of the posterior-segment anchorage in the maxillary arch. The couples tend to counter the
tipping potential of the mesiodistal force attempting to restore the passive state of the opened loop. Third, an auxiliary-appliance deterrent to
incisor extrusion is present in the inclusion in the mechanics of a highpull headgear. This extraoral appliance, delivering force directed upward and rearward, not only helps with the posteriorly directed action,
but also opposes the vertically downward component of the Class II
elastic action. As indicated in Chapter 9, the J-hooks delivering the
extraoral force should transfer their action to the arch wire at locations
between the lateral and central incisors. The line of action of the headgear force may be directed centrally through the arch wire or occlusal or
gingival to it, depending on the hook configuration. The potential then
exists to create, with the extraoral force, third-order, incisal action in
either direction or not at all.
The force and structural analysis of Class II mechanics, used to bodily retract the maxillary-incisal segment, is initiated under the assumption of complete symmetry with respect to the midsagittal plane. The
maxillary and mandibular dentitions on one side of the midline are each
partitioned into three segments with the first maxillary premolars assumed in this example to have been extracted to accommodate the anticipated reduction in overjet. The incisal half-segment consists of a central
and adjacent lateral incisor in each arch. Crowded incisors in the mandibular arch led in part to the decision to extract first premolars there
also. Retraction of four canines was previously completed and each mid-
dle segment, then, includes a canine and a second premolar. With the
second molars erupted and appropriately positioned in all quadrants,
the posterior segments in the modeling become the molar pairs; buccal
tubes are affixed to the second (terminal) molar crowns. Before beginning this example analysis, it is worthwhile to recollect the intra-arch,
anterior-retraction mechanics discussed previously.
Force and Structural Analyses ot Representative Orthodontic Mechanics
The Class II elastic, like the elastic that identifies Class III mechanics,
is recognized as exerting four significant components of force. All four
appear in a buccal view; two buccolingual components are very small
and, therefore, neglected. A free-body diagram of the right-side, Class II
elastic would look much like that in Figure 10-29 except for the angulahon and, perhaps, slight differences in the relative magnitudes of the
components parallel and perpendicular to the occlusal plane. In addition, if anchorage "slipping" in the mandibular arch and extrusion of the
maxillary incisors are unwanted, only one action-reaction component of
the elastic forces developed—the posteriorly directed component delivered to the maxillary arch wire—is desired. Accordingly, the mechanics
must incorporate provisions to suppress the displacement potentials of
the other three components.
The Class II elastic is stretched on one end against a hook that is part
of the buccal-tube attachment to the mandibular second-molar crown;
tipping and extrusive displacement potentials are thereby created. Figure 10-35 includes right-side, buccal views of the free-body diagram of
the Class II elastic and the force system from the elastic against the
terminal-molar crown. The sketch of the second molar also contains
force-system components arising from contact with the arch wire, the
adjacent first molar, and the occluding maxillary molar. The mandibular
arch wire is continuous and full size (stabilizing). In this figure the force
system exerted by the posterior segment on the wire is first shown divided into that associated with the elastic and the components arising
from second-order bends. The two portions are then combined in the
adjacent sketch and the entire force system exerted on the half arch wire
is completed through force and moment balances.
The horizontal component from the activated elastic is distributed
throughout the posterior and middle segments through crown contact
(if not from the instant of activation, then following sequential tipping
From the Class II &astic
From the tip-back bend
FGURE 10-35. Free-body diagram ot a right-side Class II elastic and right-side buccal
views of the mandibular-arch force systems associated with Class II mechanics.
Bioengineering Analysis of Orthodontic Mechanics
displacements until the contacts are established) and, via the arch wire
(through friction and, possibly, a posterior stop) to the incisors as well.
The vertical components against the posterior teeth, together with the
second-order couples, generate the vertical responses against the middle and incisal segments. Of some concern, because of the size of the
force relative to the root-surface area of the as-modeled segment, is the
extrusion potential against the middle segment. Accordingly, careful
adjustment of the arch wire in the buccal region may be prudent to
"transfer" the first molar from the posterior to the middle segment. In
addition, some displacement control should be expected from occlusion
of the middle-segment teeth.
Action-reaction involving the elastic, the posterior segment (the second molar), and the half arch wire must be used carefully in completing the force analysis. Force and moment balances and action-reaction
permit the description of the force system delivered to the incisal seg-
ment. The second-order effects between the wire and second molar
result in a small, intrusive force divided between the central and lateral
incisors. If the posterior- and middle-segment anchorage is insufficient
to contain the mesial-driving potential of the elastic force, some labial
crown tipping of the incisors is possible. Note in the sketch of the in-
cisal segment that the moments of the two force components about
the segment crc have the same sense; thus, a compound tendency for
incisor flaring is possible. One partial remedy is not to place the posterior stops in the arch wire at all, thereby allowing transmission of
the horizontal response forward beyond the canines only by means
of friction.
Four separate bilateral actions may exist in these mechanics against
the maxillary working arch wire. First, the elastics exert force on the wire
through hooks or loops distal to the lateral-incisor locations. Second,
torquing couples are delivered to the incisors. Third, a headgear force
system may be transmitted to the incisal segment. Fourth, if the arch
wire contains active opening or closing loops, the posterior segments
(minimally the second molars) exert force on the wire. In the specific
example mechanics under analysis here, assumed is the potential existence of bracket-wire clearances distal from the canines, and no stops
affixed to the wire to transfer horizontal force between arch wire and
buccal-segment crowns. This eliminates the fourth source of force in the
above list from consideration in this example.
On the left in Figure 10-36 are shown, separately, the actions on half
of the maxillary arch wire of the Class II elastic and J-hook from the
high-pull headgear, and the response to torsional activation in the incisal segment, all from the right-side buccal perspective. Note that the
J-hook force is shown with its line of action piercing the incisal section of
the arch wire; actually, the line of action is the dashed line. The headgear force has been "moved" and the compensating couple placed is
largely, if not totally, that shown in the lower left sketch. In the lower
right in the figure, the three contributions have been joined, together
with balancing components, to complete a free-body diagram of the
Force and Structural Analyses of Representative Orthodontic Mechanics
From the Class II elastic
V, = V11 —
From the high-pull headgear
From the lingual root torque
FIGURE 10-36. Right-side buccal views of the maxillary-arch force systems associated
with Class II mechanics intended to bodily retract the maxillaly incise! segment.
right half of the continuous arch wire. The horizontal components (parallel to the occlusal plane) from the elastic and headgear add and gener-
ate the distal driving force against the incisors. The driving force is
smaller than that sum by only the frictional resistance to the sliding of
the wire through the buccal attachments. Assuming the third-order stabilizing couple is produced entirely by the eccentrically (gingivally) attached J-hooks, and no activating twist is placed in the wire prior to its
engagement, the net vertical force against the incisal segment has a
sense dependent on the individual magnitudes and angulations of the
headgear and elastic forces. Responsive vertical forces against the buccal
segments are negligible due to the proximity of the headgear and elastic
attachments to the incisal segments. By action-reaction, the crown force
systems against the segments are obtained. No forces of any consequence, neglecting friction, exist against the molars or middle segments,
so they have not been pictured. The sketch representing the incisal pair
(Fig. 10-36, upper right) shows the resultant lingual driving force, the
counter-tipping, third-order couple, and a resultant of the vertical components of the elastic and headgear forces which often ideally should be
The example mechanics under discussion are designed to bodily
move the 4 maxillary incisors lingually to reduce an overjet. Assuming a
continuous force system, reasonable light-force figures are 155 to 165 g
and 1700 to 1800 g-mm for the driving force and counter-tipping torque,
respectively, for the segment. (The couple-force ratio must be approximately one-half of the average tooth length; see Chap. 5.) Achieving the
Bioengineering Analysis of Orthodontic Mechanics
driving force of nearly 3 oz per side may seem straightforward; however, the expected relaxation of elastics over time and the division of the
force between the Class II elastics, likely worn continuously, and the
headgear, probably engaged intermittently, must be considered.
Example calculations with sketches are presented in Figure 10-37,
suggesting that, as a basis, the product of the vertical force component
and time worn be equated for the elastic and headgear action on each
side. Now note once again that, in this example, the torque is delivered
through a pair of eccentrically attached J-hooks. The third-order couple
induced is the product of the total headgear force and the perpendicular
distance from the line of action to the center of the wire (e in Fig. 10-36).
With the headgear force calculated in Figure 10-37, the torque is deter-
mined. Two difficulties are now apparent: (1) the dependence of the
torque on the driving force; and (2) the inability to achieve sufficient
torque magnitude (without intolerably long hooks attached interincisally). The only answer, then, is the establishment of at least a portion
of the total torque requirement through third-order activation of the
rectangular arch wire, which brings with it the secondary considerations
discussed in Chapter 8. The overall problem is complex and perplexing,
and to date it has been approached clinically by retracting the incisors in
the presence of some torque, expecting that the long-axis angulations
Projected root-surface area
= 2(0.5 + 04) = 1.8 cm2
Force = 3.5(Area) (60)12.5
= 155 g against 4 incisors
Couple-torce ratio = 0.5 (average
total tooth length) = 105 cm
For each side of arch:
Distal driving force = 80 g = 2.8 or
It elastics worn continuously and
headgear for 12 hr each day.
Vh2 = 2(V11) (vertical balance)
It the Class Il elastic makes a 20°
angle with the ocolusal plane,
as does the headgear force,
H11 = 40 g,
= 15 g, F11 = 43 g,
To accompany the 155 g against
= 30 g, Fhg = 85 g
the incisal segment, needed
Total headgear force (both sides)
is a couple = 155(1 0.5)
To generate this couple with the headgear force, e must be 10 mm.
FIGURE 10-37. The force system delivered to the maxillary incisal segment from the
combined action of elastics, headgear, and arch wire, designed to bodily move the
incisors toward the posterior without vertical displacement.
Force and Structural Analyses of Representative Orthodontic Mechanics
will have to be corrected after the desired crown positions have been
With alterations, the mechanics under discussion may be used to
produce posteriorly directed, en masse movement of the maxillary dentition. First, assuming that any mesiodistal spaces have been previously
closed, the entire dentition should be unitized through appropriate in-
terbracket ligations. Stops should probably be placed mesial to the
brackets or tubes affixed to the first molars so that the arch wire, in
addition to crown contact, is used to help distribute the driving force.
Second, the driving force, if partially delivered by Class II elastics,
should probably be transferred to the arch wire just distal to the canines.
This is somewhat anterior to the center of resistance of the maxilla and is
a compromise, noting that the activating forces should be as nearly hori-
zontal as possible. On the other hand, consideration must be given to
the position of the line of action of the resultant of the headgear and
elastic forces on each side; the length of the moment arm of the resultant, with respect to the cra, must be minimized to suppress the potential
for rotation of the maxIlla if bodily movement is wanted. Third, the
magnitude of the resultant driving force must be increased to perhaps in
excess of five times the figure determined for the incisal segment (depending on the root-surface area present), to be divided between the
two sides of the arch. Again, if Class II elastics are used, responsive
forces carried to the mandibular arch are enlarged, and more severe
anchorage preparation than that for incisal-segment retraction
A worthwhile, last example of this section is the use Class II mechanics in protraction of the mandibular dentition, perhaps in an attempt to
aid growth at the proper time when an otherwise retrognathic mandible
is expected. In these mechanics the active and responsive ends of the
elastics are reversed. The action is in the mandibular arch; the anteriorly
directed elastic component delivered to the mandibular terminal molar
is the desired one of the four components generated. A series of diagrams in Figure 10-38 illustrate the essentials of the force system. The
driving-force component must be distributed anteriorly to the incisal
segment by the arch wire; crown contact is relied upon in the buccal
segments. A unitized, maxillary dentition is the first line of anchorage,
but reinforcement with extraoral assistance is warranted. To prevent
unwanted anterior bite closure a high-pull headgear is attached to the
maxillary arch wire at the same anteroposterior positions as the elastic
attachments. Because the resultant force against the maxillary arch tends
to move it posteriorly, this force must be distributed as uniformly as
possible throughout the entire maxillary dentition; moreover, the elastic
and headgear forces must be relatively light to minimize the potential for
reciprocal displacements in the two arches. To provide some differential
in the root-surface areas, second molars should be left out of the mandibular strap-up (and moved mesially at a later time). A suitable reverse-
pull headgear, one not involving a chin cap, could help reinforce the
maxillary-arch anchorage.
Bioengineenng Analysis of Orthodontic Mechanics
Mesial to
the terminal
FIGURE 1 0-38. Right-side buccal views of the force systems intended to produce
protraction mechanics in the mandibular arch.
Force and Structural Analyses of Representative Orthodontic Mechanics
Beginning with displacements of individual teeth and proceeding to
controlled movements of dental segments, starting with an individual
arch wire and advancing to complex appliances incorporating wires in
both arches, interarch elements, and extraoral auxiliaries, the intent of
this chapter was to gain familiarity with orthodontic-mechanics analysis
from a structural standpoint. Example analyses were purposely chosen
from existing treatment procedures so that, typically, the reader would
begin with some clinical understanding of the mechanics examined. The
appliances were disassembled and the active and responsive force systems, exerted on both the appliance members and the dentition, were
completed and scrutinized. The analyses were more qualitative than
quantitative in studying these examples, and a principal objective was
the identification of the force-system components created throughout
the appliance resulting from its activation. All forces and couples delivered to the dentition have the potential to move teeth and a good appliance design enables selective, controlled displacements where they are
desired while impeding or suppressing unwanted tooth movements
Not discussed were a substantial number of procedures and devices,
and brief comments about several are appropriate before concluding this
chapter. Positioners and retainers may be expected to perform minor
tooth movements. They use the palate, the mucosa, and the gingiva as
anchorage sites, providing broad distributions of responsive forces and
strict displacement control. The emphasis in the set of example mechanics examined was on labially affixed appliances. Lingual mechanics in
contrast are characterized by sizable and distinct wire curvatures in the
arch-form plane, potentially reduced interbracket distances (although
this depends on mesiodistal bracket widths, and narrower brackets are
often used to accommodate lingual tooth anatomy), and pretorqued
brackets because of the third-order inclinations of lingual surfaces of the
anterior teeth in particular. No fundamental differences, however, are
inherently present in force- and structural-analysis procedures, lingual
versus labial appliances.
Functional appliances, on the other hand, are unique because they
are removable, labiolingual in their contacts with the dentition, and activated in part against the oral and facial musculature. These appliances
can be complex in their designs, and the developments of this text are
only indirectly applicable to their analyses. Palate-splitting appliances
are symmetric, employ reciprocal-force (bilateral) action, and are truly
orthopedic devices. They attempt to transmit heavy, distributed force
into the halves of the hard palate to open the median suture (to catalyze
bone apposition there) without tipping the posterior teeth buccally. The
overall force analysis is straightforward; the difficulties in the design
pertain to the path and distribution of the active force and the stiffness
of pertinent elements of the device with respect to the step-wise ad justments of the appliance following incremental displacements.
Bioengineering Analysis of Orthodontic Mechanics
Although the mechanics just mentioned and others did not receive
detailed attention in this chapter, the patterns of analysis in the examples presented—the modeling, the force-displacement correlations, the
free-body diagrams, the force and moment balances, the differentialforce concept, the four-step analysis procedure, the emphasis on the
role of stiffness, the consideration of modes of failure, and the other
"tools" from the previous nine chapters—should enable the conscientious reader and student or practitioner of orthodontics, regardless of
the amount of clinical experience, to begin to undertake force and structural analyses of most mechanics of interest. The subsequent step is the
consideration of appliance-design modifications, as warranted, toward
improving the quality of orthodontic care.
This text was prepared primarily toward closure of an apparent gap
in the orthodontic educational literature—discussion of the appliance as
a structural mechanism from a bioengineering viewpoint. Comments on
the textual material of a constructive nature, submitted in writing to the
author or publisher, are encouraged.
Frank, C.A., and Nikolai, R.J.: A comparative study of frictional resistances be-
tween orthodontic bracket and arch wire. Am. J. Orthod., 78:593—609, 1980.
Glossary of Terms
activation—The process of deforming an appliance member from its passive state (e.g., the stretching of an elastic) and completing its engagement to produce an intraoral force system transmitted by the
appliance to the dentition.
activation site—The intraoral location of the activating process, often
where the orthodontic force system is to be transmitted to the dentition.
aging—A long-term chemical process in which molecular (secondary)
bonds are broken concurrent with oxidation, resulting in the appearance of cracks and embrittlement that proceed with time from
the material surface inward.
alveolar bone—The hard tissue that locally provides support for the in-
dividual teeth.
alveolus—The composite support, including alveolar bone and periodontal ligament, that envelops the root of the tooth.
amorphous—Descriptive of an irregular material microstructure, usually
with the molecule as the basic structural unit.
angle of twist—The measure of torsional (third-order) deformation of a
shaft or an orthodontic wire.
anneal—A heat-treatment process employing a relatively high temperature that results in recrystallization of microstructure and produces
marked changes in mechanical properties, including substantially
reduced resilience, of metallic materials.
anode—The electrode of a galvanic cell having the larger negative potential and from which electrons flow, leaving positive ions, and
dissolving (corroding) in the long-term oxidation process.
Bioengineering Analysis of Orthodontic Mechanics
apical fibers—The fibers of the periodontal ligament extending into the
cementum at the root apex.
apposition—The cellular process resulting in the addition of new alveolar bone at the interface of the periodontal ligament and the tooth
atom—The basic unit of microstructure exhibiting the nature of the
chemical element.
available energy—Energy retained in a body owing to existing deformational or thermal constraints that, upon release of the constraints,
would be transferred from the body and could be controlled to produce a desired effect.
axisymmetric—descriptive of symmetry in two dimensions relative to a
line (axis) perpendicular to the plane.
basal bone—The hard tissue of the maxilla or mandible that blends into
the alveoli and supports the dentition as a whole.
beam—A structural member characterized by a large longitudinal dimension (usually oriented horizontally) and small cross-sectional
dimensions, and loaded (activated) perpendicular to the longitudinal dimension.
beam (bending) axis—A reference line interconnecting the geometric
centers of all of the right cross-sections of a beam.
beam deflection—A characteristic, linear deformation of a beam measured from and perpendicular to the beam axis at a specific cross-section, ordinarily with reference to the passive configuration of the
structural member.
beam fiber—A characteristic element of a solid modeled beam, parallel
to the beam axis, having the same length as the beam and a very
small cross-section compared to that of the entire member.
beam rotation—A characteristic, angular beam deformation: the change
in orientation of a specific cross-section of the beam during activation as measured in the plane of bending.
bending couple—A couple in the plane of bending, either internal or
external to a beam, associated with its deformation relative to the
passive shape.
bending strain—The normal strain of a beam fiber at a particular location
on a specific right cross-section of the structural member.
Glossary of Terms
bending stress—The normal stress induced in a beam fiber at a particu-
lar location along its length.
bioengineering—Engineering principles and practice applied to living
bodily movement—Translational displacement of a tooth or dental segment, ordinarily understood to be perpendicular to the long axis.
braided—Descriptive of a multistrand orthodontic wire suggesting nonparallel, interwoven strands.
buckling—A structural failure of a long, slender member characterized
by sudden, transverse deformation.
cathode—The electrode of a galvanic cell having the algebraically
greater potential and the receiver of electron flow from the anode.
cementoblasts—The cells that specifically contribute to sustaining and
rebuilding cementum.
cementoepithelial fibers—The fibers of the periodontal ligament that extend into the cementum and over the alveolar crest into the free and
attached gingiva.
center of resistance—A singular point in a body through which the line
of action of a resultant force must pass to produce translation of that
center of rotation—The point about which a body rotates in a nontranslational plane (two-dimensional) displacement.
cervical-pull—Descriptive of an extraoral appliance having the responsive force delivered to and distributed over the back of the neck.
chin cap—An element of an extraoral appliance that transfers force to
the chin in dentofacial orthodontic or orthopedic therapy.
circumferential fibers—Periodontal-ligament fibers located in the attached gingiva that encircle the tooth without extending beyond the
ligament itself.
Class II elastic—An appliance element engaged intraorally and activated
between anterior maxillary and posterior mandibular locations on
one side of the dentition in Class II mechanics therapy.
Class II mechanics—A phase of orthodontic treatment using appliances
ordinarily designed to retract maxillary anterior teeth and/or
protract mandibular posterior teeth.
Bioengineering Analysis of Orthodontic Mechanics
Class HI elastic—An appliance element engaged intraorally and acti-
vated between anterior mandibular and posterior maxillary localions on one side of the dentition in Class III mechanics therapy.
Class III mechanics—A phase of orthodontic treatment using appliances
ordinarily designed to retract mandibular anterior teeth and/or
protract maxillary posterior teeth.
closing loop—An auxiliary fabricated of orthodontic wire that, upon
mesiodistal-pulling activation, exhibits a reduction in the plane area
enclosed within its geometry.
cold working—Inelastic (plastic) deformation of a metallic member at
normal (room) temperature to alter the member geometry and/or
change mechanical properties related to the amount and distribution of energy stored in the member.
collinear forces—Concentrated forces that share the same line of action
(but not necessarily the same point of application).
component (of a vector)—That portion of the vector associated with a
reference line or axis angulated with respect to the line of action of
the given vector.
compressive deformation—The shortening of a characteristic dimension
of a body associated with pushing force.
compressive strain—Internal compressive deformation per unit length.
compressive stress—Internal pushing force per unit area, the intensity
normth (perpendicular) to the associated area.
concentrated force—An external contact force with the area of contact
very small compared to the total surface area of the body subjected
to the force.
concurrent forces—Several concentrated forces exerted at a common
constant—A quantity that does not change within the context of its involvement in a problem or discussion.
continuous orthodontic force—Action of an appliance against the dentilion that decreases little in magnitude during the between-appointments period.
corrosion—The dissolving of a metallic material in a chemical, oxidation
Glossary of Terms
couple—A pair of concentrated forces having equal magnitudes, the
same direction but opposite senses, and noncollinear (parallel) lines
of action.
couple-force ratio—The ratio of magnitudes of the crown couple to the
crown force, having net units of length (e.g., mm), in the two-dimensional analysis of transverse tooth movement.
covalent bond—The bond between two or more atoms achieved by a
sharing of electrons.
creep—Continuing deformation over time of a body subjected to a constant load.
crestal fibers—Periodontal-ligament fibers that emerge from the cementum and extend into the gingiva directly adjacent to the alveolar crest.
critical cross-section—The right cross-section of a beam, shaft, or wire
where the internal bending or torsional couple has its maximum
crown movement—The transverse tooth displacement for which the cen-
ter of rotation coincides with the root apex (or the level of root
apices for a multirooted tooth).
crystalline—Descriptive of the organized, latticed microstructure of a
granular material.
decomposition of a (concentrated) force—Replacing the given force by a
mechanically equivalent set of concurrent force components, each
of which has a line of action coinciding with an axis of a reference
frame originating at the point of application of the given force.
deformation—Any change in the geometry (size and/or shape) of a body
produced by the application of force.
dental segment—Two or more adjacent teeth in the same arch.
differential-force—Pertaining to the distributions of two related forces,
equal in magnitude (and ordinarily in the same direction, but of
opposite sense), over areas of differing sizes, thus yielding unequal
average force intensities.
dimensions—The fundamental descriptions of measurement (mass,
length, time, force, and temperature).
direct resorption—The removal of alveolar bone at the bone-ligament
interface catalyzed by the application of compressive force.
Bioengineering Analysis ot Orthodontic Mechanics
direction angle—The angle between a line or a vector and a reference
displacement—Any movement of a particle or a body as a whole.
distributed force—A force that involves a substantial portion of the surface area or volume of the body acted upon.
dual-force appliance—An extraoral appliance characterized by the bilateral delivery of force to two separate intraoral locations on each side
of the midsagittal plane.
eccentric activation—An orthodontic-appliance activation that creates a
resultant force against the facial or lingual surface of a crown with a
line of action not piercing the long axis of the tooth.
elastic—A flexible appliance element, ordinarily activated in two-point
extension, that exhibits substantial elastic range (noun); descriptive
of material behavior such that, upon unloading from a deformed
state, recovery is totally to the configuration prior to loading (adjective).
elastic core—The inner portion of the cross-section of a beam, shaft, or
wire that remains elastic (while the outer portion is strained beyond
the elastic limit) when the member is subjected to substantial loading (activation) in bending or torsion.
elastic deformation—A deformation not sufficiently severe to take the
most strained element of a body beyond the elastic limit of the material.
elastic limit—The limit of load, stress, deformation, or strain beyond
which the loaded (activated) body will exhibit permanent deformation (a new passive shape) upon complete unloading (deactivation).
elastic range—The deformation or strain coordinate of the elastic limit.
elastic strength—The load or stress coordinate of the elastic limit.
electrode potential—The relative ease or difficulty with which a metal or
alloy will give up its valence (outer-shell) electrons in, for example,
a galvanic cell.
electron—A negatively charged particle of an atom that is in continual
motion around the nucleus.
endurance limit—In a dynamic, cyclic loading pattern, the greatest amplitude of load, stress, deformation, or strain that does not cause a
Glossary of Terms
failure within the expected fatigue life of the structural member; the
horizontal asymptote of a fatigue-life plot.
energy—The capacity of a body to produce an effect.
engineering strain—Strain computed with reference to the undeformed
(passive) geometry of the strained body.
engineering stress—Stress calculated using undeformed (passive) geometry of the stressed body.
excessive force—Force delivered by an orthodontic appliance of such
magnitude that it effectively ankyloses a tooth and absolutely no
desired movement is detectable following a between-appointments
extraorat appliance—An orthodontic or orthopedic device activated
against the dentofacial complex that induces its responsive force
system outside of the oral cavity.
extrusion—A translational form of tooth displacement with movement
occiusally directed and parallel to the long axis.
face bow—An appliance member that transmits force, delivered
extraorally to it on each side of the face, to the maxillary first or
second molar crowns.
failure—The development of excessive stress or deformation or a breakage (fracture, rupture) that interferes with or prevents the functioning of a structure or machine.
fatigue—The deterioration or gradual loss of desirable structural characteristics, possibly culminating in failure, of a member subjected to
fluctuating (cyclic) loading.
fiber—An actual or modeled (e.g., periodontal-ligament or beam) element, characterized by a substantial length compared to the dimensions of its cross-section, that transmits longitudinal force, often
tensile force.
fiber strain—Unit longitudinal deformation of a fiber.
fiber stress—Longitudinal fiber force divided by the cross-sectional area
of the fiber.
fibroblasts—The cells that specifically contribute to the rebuilding and
remodeling (lengthening) of the periodontal-ligament fiber.
Bioengineering Analysis of Orthodontic Mechanics
first-order—Pertaining to a rotational tooth movement or displacement
potential visible in an occlusal view.
force—A mechanical action of one body on another that tends to displace and/or deform the body receiving it.
fracture—A failure of a structural or machine member through interruption or loss of a force transmission path.
free end—One end of a beam, shaft, or wire not constrained in any way
from displacement resulting from an applied force system.
tree-body diagram—The figurative removal from a structure or machine
of a member or a portion of a member, depicting in an isolated
sketch the geometry and force system exerted, preparatory to a
force and/or structural analysis of that part of the assembly.
friction—A resistance to the relative displacement of contacting bodies in
a direction tangent to the plane of contact, owing principally to
surface roughnesses and contact pressure.
frontal resorption—See direct resorption.
function—A mathematical quantity that assumes a value upon assignment of magnitudes to one or more variable in the problem or discussion; the interaction of two or more bodies toward a collectively
produced effect.
functional force—Force arising from relative displacements of interconnected bodies producing contact(s) between surfaces during routine
tasks (e.g., chewing/masticatory force).
generalized rotation—A whole body, nontranslational displacement
characterized by a center of rotation not coincident with the center
of resistance of the body.
gingival fibers—The extra- or supra-alveolar group of longer, periodontal-ligament fibers that are embedded in the gingival tissue and extend into the cementum.
ground substance—The matrix of the composite-material, periodontal
ligament that provides the principal resistance to compressive force
(pressure) exerted on the ligament.
hardening heat treatment—A heat-treatment process designed to increase the surface hardness and, perhaps, to suitably affect other
properties of a material.
Glossary of Terms
hardness—The resistance of a material surface to indentation.
head cap—The member of an extraoral appliance that transfers responsive force to the cranium.
headgear—A synonym for "extraoral orthodontic appliance(s)."
heat—A form of energy transfer between two bodies or systems owing
to a difference in temperatures, one from the other, in the absence
of a perfect insulator between them.
heat sink—A receptor of energy transfer in the form of heat from a
thermodynamic system.
heat source—A provider of energy transfer in the form of heat to a
thermodynamic system.
heat treatment—A process characterized by the transfer of energy in the
form of heat to a metallic material, and subsequently from it, to alter
its mechanical and/or thermal properties.
heavy force—Orthodontic force induced by appliance members exhibiting relatively high characteristic stiffness(es).
high-pull—Descriptive of the activating bilateral force, directed posteriorly and superiorly, of an extraoral orthodontic or orthopedic appliance.
hinged face bow—A face bow having the outer and inner bows interconnected by a hinge that permits rotation of one relative to the other
about an axis perpendicular to the plane of the bows.
Hookean material—A crystalline material exhibiting a linear relationship
between induced normal or shear stress and corresponding strain
when subjected to relatively small levels of activation.
horizontal fibers—The principal, periodontal-ligament fibers extending
from cementum to alveolar bone that are radially directed relative to
the passive orientation of the long axis of the tooth.
hot working—Inelastically deforming a metallic member in an elevated-
temperature environment toward desired changes in geometry
and/or mechanical characteristics of the material.
incisal pair—Adjacent central and lateral incisors.
inelastic deformation—Deformation of a member sufficiently substantial
to exceed the elastic limit of the material and, upon total unloading
Bioengineering Analysis of Orthodontic Mechanics
(deactivation), resulting in a "permanent set" (a change in geome-
try/shape from that of the passive state prior to the deformation).
inner bow—The intraoral portion of the engaged face bow.
intermittent force—An active, orthodontic force that decays to zero magnitude, or nearly so, prior to the end of a between-appointments
interrupted force—An orthodontic or orthopedic force that is inactive for
intervals of time during the between-appointments period, often
exhibiting a cyclic, long-term, magnitude-time pattern (e.g., force
exerted by an extraoral appliance worn only at night).
intrusion—A translational form of tooth movement directed apically and
parallel to the long axis.
ionic bond—The interatomic bond involving unlike elements and cre-
ated by the transfer of valence (outer-shell) electrons from one atom
to another.
J-hook—The element of an extraoral appliance in the shape of the letter
"J" that transfers force between extraoral and intraoral locations.
labial root (lingual crown) torque—The third-order couple of a transverse
force system applied to the crown(s) of a tooth or dental segment
having the potential for rotational displacement resulting in labial
movement of the root(s) and/or lingual movement of the crown(s).
leading root surface—That portion of the tooth root under compression
(pressure) during a transverse tooth movement.
leveling—That phase of orthodontic therapy in which occiusogingival
and/or faciolingual malalignments are eliminated toward ideal arch
form (in the occlusal plane) and the desired shape of the curve of
leveling wire—Orthodontic wire, ordinarily circular (in cross-section)
and exhibiting low flexural stiffness, used in the leveling phase of
light force—Orthodontic force produced by an appliance element having
relatively low characteristic stiffness(es).
line of action—A line having the same direction as the associated concentrated-force vector and passing through the point of application
of the force.
Glossary of Terms
lingual root (labial crown) torque—The third-order couple of a transverse
force system applied to the crown(s) of a tooth or dental segment
having the potential for rotational displacement resulting in lingual
movement of the root(s) and/or labial movement of the crown(s).
long-axis rotation—A nontranslational tooth movement characterized by
the coincidence of the long axis and the axis of rotation.
longitudinal axis—The line composed of the centers of all of the right
cross-sections of a beam, shaft, or wire.
machine—A mechanism that produces a desired output when receiving
a necessary input.
maximum force—An orthodontic force having the greatest magnitude
within a range of force values that will produce tooth movement at
a clinically acceptable rate.
mechanical work—A form of energy transfer characterized by force acting through a distance.
mechanics—The branch of physics that focuses on force systems and
their effects on stationary and moving bodies; in orthodontics, descriptive of the procedures and/or appliance(s) used in a specific
phase of therapy.
mer—The characteristic molecule of a polymer.
metallic bond—The interatomic bond characterized by the release of valence (outer-shell) electrons, yielding an organized array of positive
ions and an electron cloud ("gas") with electrostatic forces maintaining the shape of the material.
midsagittal plane—The imaginary, vertical plane that separates the left
and right sides of the dentofacial complex.
modulus of elasticity—The ratio of change in normal stress to accompa-
nying change in normal strain between states below the proportional limit within a Hookean material; the slope of the normalstress versus normal-strain diagram below the proportional limit.
modulus of resilience—The amount of work done per unit volume on a
material in the energy-transfer process that takes the material from
a passive state to its elastic-limit configuration; the area under the
stress-strain diagram up to the elastic limit.
modulus of rigidity—The ratio of change in shear stress to accompanying
change in shear strain between states below the proportional limit
Bioengineering Analysis of Orthodontic Mechanics
within a 1-lookean material; the slope of the shear-stress versus
shear-strain diagram below the proportional limit.
modulus of toughness—The amount of work done per unit volume on a
material in the energy-transfer process that takes the material from
a passive state to fracture; the total area under the stress-strain diagram.
moment (of a concentrated force)—The measure of the rotational potential of a force with respect to a specific line (axis) or point; in magni-
tude the product of the size of the force and the distance (measured
perpendicular to the line of action) from the force to the moment
axis or moment center.
moment of inertia (second moment of area)—A property of a body de-
pending on mass, mass distribution, size, and shape that is characteristic of the resistance of the body to motion; also a property of the
cross-sectional area of a beam or wire, referenced to a line (axis) in
the plane of the cross-section, that depends on the shape and size of
the cross-section and the specific reference line.
neck pad—The element of the cervical-pull, extraoral appliance that
transfers the distributed responsive force to the back of the neck.
neutral axis—The intersection of the neutral surface with the right crosssection of a beam or wire activated in flexure.
neutral surface—The collection of beam fibers that are unstrained in
plane bending (flexural) activation.
neutron—A basic, uncharged particle within the nucleus of an atom.
normal deformation—The deformation of a body accompanying the application to it of pulling or pushing force; tensile or compressive
deformation perpendicular (normal) to a reference area of a body.
normal strain—Normal deformation per unit length of an element of a
loaded (activated) body.
normal stress—The intensity of internal force (force per unit area) per-
pendicular to a reference area within a loaded (activated) body.
oblique fibers—The principal fibers of the periodontal ligament that are
embedded in the cementum and the alveolar bone and are inclined
toward the root apices, providing hammock-like response and protection for the root apex (apices) of a tooth subjected to intrusively
directed (e.g., masticatory) force.
Glossary of Terms
opening loop—An auxiliary fabricated of orthodontic wire that, upon
mesiodistal-pulling activation, exhibits an enlargement of the plane
area enclosed within its geometry.
origin—The intersection of the coordinate axes of a reference frame.
orthodontic attachment—The element affixed to the tooth crown that
transmits force from the arch wire or auxiliary to the dentition (e.g.,
a bracket).
orthodontic displacement—Tooth movement achieved through biomechanical remodeling of the periodontal ligament and/or the alveolar
orthodontic force—Force generated by the orthodontic appliance that
contributes to the correction of a malocclusion.
orthopedic force—Force generated to assist or retard bone growth, to
displace teeth other than through remodeling of alveoli, or to move
bones of the dentofacial complex to catalyze sutural remodeling.
osteoblasts—The cells that specifically contribute to the apposition of
bone in regions of the ligament-bone interface under tension.
osteoclasts—The cells that specifically contribute to the resorption of
bone in regions of moderate compression (pressure).
outer bow—The extraoral portion of the engaged face bow.
oxidation—A chemical reaction with oxygen contained in the reactants
and yielding a metallic or nonmetallic oxide or hydroxide as a
parallelogram law—The fundamental, graphical procedure used to replace two concurrent, concentrated forces by a single, mechanically
equivalent point force; the basic rule of vector addition.
passivation—The process by which a material surface is protected
against corrosion.
passive—The configuration of a body unconstrained externally and not
subjected to mechanical or thermal loading (activation).
ligament—The soft tissue surrounding the root of a tooth
and overlaying the alveolar bone.
periodontal-ligament fibers—The material components of the composite
ligament that give it resistance to failure when subjected to moderate tensile (stretching) action.
Bloengineering Analysis of Orthodontic Mechanics
physiologic force—Force inherent in a biologic system (e.g., the pressure
of the blood against the vessel walls; the force working in the tootheruption process).
plane of bending—The plane defined by the passive and activated configurations of the beam axis in two-dimensional bending (flexure).
polar moment of inertia (polar second moment of area)—A property of a
body depending on mass, mass distribution, size, and shape that is
characteristic of the resistance of the body to motion; also a property
of the cross-sectional area of a shaft or wire, referenced to a line
(axis) perpendicular to the plane of the cross-section, that depends
on the shape and size of the cross-section and the specific reference
polygonal law—An extension of the triangle law, a corollary of the paral-
lelogram law, through which a system of more than two concurrent, concentrated forces may be replaced by a single point force
having the same mechanical effect as that of the system.
polymer—An amorphous material consisting of long chains and networks of repeating molecules.
preangulated bracket—An orthodontic bracket having its slot inclined to
the ordinary mesiodistal direction, permitting the generation of second-order action from a passively straight wire.
pretorqued bracket—An orthodontic bracket having its slot rotated with
respect to the ordinary faciolingual direction, permitting the generation of third-order action from a passively straight (untwisted) rectangular wire of suitable cross-sectional dimensions.
primary bond—A covalent, ionic, or metallic bond between two atoms.
primary response—A response in the same plane as or a parallel plane to
that of the appliance activation.
principal fibers—The shorter, periodontal-ligament fibers generally directed in a radial manner with regard to the long axis of the tooth
and embedded in the cementum and the alveolar bone.
process—The change of state experienced by a thermodynamic system
as energy is transferred to and/or from it.
projected root area—The plane area enclosed in the outline of a tooth
root as viewed from a specific direction.
projection (of a vector)—The magnitude of the vector component associated with a specified line or reference axis; the product of the vector
Glossaiy of Terms
magnitude and the cosine of the angle between the vector and the
specified line or axis.
property—A characteristic of a material or body that partially describes
proportional limit—The upper bound of the linear portion of a load-deformation or stress-strain plot for a Hookean material.
proton—A basic, positively charged particle within the nucleus of an
pure bending—Flexural action produced by couple loading such that no
shear stresses exist on right cross-sections of the beam or wire.
pure rotation—A displacement of a body, produced by couple action,
characterized by the center of rotation coinciding with the center of
pure shear—The existence of shear stresses in the absence of normal
stresses associated with a specified area within the activated body.
quasi-static—Descriptive of the configuration of the appliance and the
dentition in an instantaneous or short-term analysis between appointments: velocities, accelerations, and inertias are sufficiently
small to be negligible such that the principles of statics (mechanical
equilibrium) may be used in the analysis.
rear resorption—The decomposition of bone within the alveolus, beginning away from the bone-ligament interface where the compressive
stress (pressure) induced by the activated appliance is reduced
(through distribution) from the more substantial intensities delivered by the tooth root.
reference frame—A set of two or three intersecting axes, ordinarily mu-
tually perpendicular, used as a basis for locations of points and
directions of lines, graphically expressing relationships among variables, or undertaking vector analyses, in a plane or in space.
relaxation—The decrease over time of force internal to a body maintained under conditions of constant strain.
resilience—The amount of energy transferred by mechanical work to
take a body from an initial, passive state to its elastic limit; the area
under the load-deformation plot up to the elastic limit.
resultant—A force system—in its simplest form, a single concentrated
force, a couple, or a force and a couple—that is mechanically equivalent to the given, more complex system of forces.
Bioengineering Analysis of Orthodontic Mechanics
reverse-pull—Descriptive of an extraoral appliance that, when activated,
exerts anteriorly directed force on the dentition.
right cross-section—A cross-section of a beam, shaft, or wire that is per-
pendicular (at right angles) to the longitudinal reference axis (and
the external surface[s]) of the member at the location of the crosssection.
rigid body—A model of analytical mechanics that ignores potential deformations resulting from the application of force systems, focusing
attention on possible or actual whole-body movement.
root movement—The transverse tooth movement characterized by little
displacement of the crown and the center of rotation located on the
long axis and at or near bracket level.
rotation—Any orientation-changing, whole-body displacement; a displacement characterized by the change in angulation of some line(s)
in the body with respect to a specified reference frame.
scalar—A mathematical quantity possessing only magnitude.
scale—The means of conversion between the magnitude of a specific
variable and its graphical representation as a coordinate or length.
second moments of area—See moment of inertia and polar moment of
secondary bonds—Forces that hold molecules together in a solid, and
also known as van der Waals bonds; these bonds are weaker than
interatomic (primary) bonds.
secondary response—A response in a plane angulated to that of the
appliance activation, occurring in part because of the curvature of
the dental arch.
second-order clearance—The angle through which an engaged arch
wire may be tipped within the bracket slot, relative to the "slot-parallel" configuration, before making contact with the occlusogingival
slot surfaces.
section modulus—The ratio of a second moment of area to the depth of a
beam or wire; often symbolized by "Z" and the geometric parameter in the flexural-strength equation.
shaft—A long, slender, machine member, ordinarily straight and of constant cross-section (most often circular), that carries torque.
Glossary of Terms
shear deformation—A change in shape as a result of loading (activation);
often characteristically, a rectangular element assumes the form of a
shear strain—A unit shear deformation; the change in angle (in radians)
between two intersecting, passively perpendicular lines.
shear stress—The intensity (force per unit area) of the tangential component of internal force.
simple tipping—The tooth movement produced by a labiolingually directed, concentrated crown force.
sliding jig—An auxiliary guided by an arch wire that transfers a mesiodistal component of force, delivered by a J-hook, directly to a posterior site unreachable by the J-hook itself.
spring-back—-The recovery exhibited by a beam, shaft, wire, or wire
loop upon its unloading (deactivation) from a state at or beyond its
elastic limit.
stabilizing wire—A stiff, rectangular arch wire that "fills" the bracket slot
and is used ordinarily to distribute a responsive force system
throughout a dental segment.
state—Defined by the collective values of the properties (characteristics)
of a thermodynamic system.
statics—That portion of mechanics (of physics) that concerns force systems exerted on bodies in mechanical-equilibrium states.
stiffness—The ratio of change in load to accompanying change in deformation of a member fabricated of a Hookean material and activated
within its elastic limit; the slope of the load-versus-deformation plot
beneath the elastic limit.
stop—An auxiliary affixed to the arch wire that prevents relative mesiodistal movement upon contact with a bracket; also a means of transferring mesiodistal force between wire and bracket.
straight-pull—Descriptive of a type of extraoral appliance having the activating forces parallel to the occiusal plane.
strain—Unit deformation.
strain energy—Energy of a system characterized by constraints maintaining the system in a deformed configuration compared to a reference (passive) state.
Bioengineering Analysis of Orthodontic Mechanics
hardening—A process in which a body is subjected to inelastic
(permanent) deformation resulting in the material incurring me-
chanical as well as geometric property changes (e.g., an increase in
surface hardness).
strand—A longitudinal element of a multistrand wire.
stress—The intensity of internal force; internal force per unit of associated area.
stress-relief—A heat-treatment process intended to reduce the magnitudes of residual ("locked in") stresses induced by inelastic deformation.
structural analysis—An engineering procedure in which a structure is
examined analytically, graphically, or through modeling to determine how large a load it will carry without failure or if it will carry
the known, desired load without failing.
structure—An assembly of members designed to withstand a specific
loading and to transfer that loading into a foundation (e.g., a frame-
work, bridge, or building).
surroundings—The collection of bodies or objects that might interact,
through energy transfer, with a defined thermodynamic system.
tensile deformation—Deformation directly associated with a pulling (extensional) force.
tensile strain—Tensile deformation per unit length at an internal location
of the loaded (activated) body.
tensile stress—Intensity of internal force (force per unit area) pulling
perpendicular to an
within the loaded (activated) body.
thermal energy—Energy possessed by a body because its temperature is
higher than that of its surroundings.
thermodynamic system—The body or bodies, separated from interacting
surroundings by a defined boundary, that experience changes in
energy levels during a process.
thermodynamics—That portion of physics that concerns interactions
between physical systems and their surroundings, specifically processes involving energy exchanges through heat and/or mechanical
third-order clearance—The angle through which an engaged rectangular
arch wire may be rotated about its longitudinal axis, relative to the
Glossary of Terms
"aligned-surfaces" configuration, before the edges of the wire make
diagonal contacts with the occlusogingival slot surfaces.
third-order couple—A couple, located in a mesiodistal plane, transmitted
by a rectangular arch wire or torquing spur to an orthodontic
threshold force—The minimum magnitude of force needed to produce a
desired orthodontic displacement.
torque—An internal force system, carried longitudinally through a shaft
or wire, and its resultant at any location is a couple in the plane of
the right cross-section.
torquing auxiliary—An element of an orthodontic appliance, separate
from the arch wire, used to transmit third-order action to the dentihon.
torquing displacement—The rotational movement of a tooth resulting
from the existence of an active second- and/or third-order couple in
the crown force system.
torquing spur—A loop formed in round wire, either integral with or
affixed to the arch wire, that when activated imparts third-order
action to the tooth crown.
torsional couple—An active or responsive external couple exerted on a
shaft or wire and directly associated with the turning and/or twisting of the member about its longitudinal axis.
toughness—The maximum amount of energy, referenced to a passive
state, transferred to a body in the form of mechanical work, that the
body can absorb prior to structural failure (ordinarily by fracture or
rupture); the total area under the load-deformation plot.
traction band—A rectangular band of woven elastic materials, activated
by longitudinal stretching, and an element of some extraoral appliances.
trailing root surface—That portion of the root surface subjected to perio-
dontal-ligament tension in a transverse tooth movement.
translation—Any whole-body movement in which no line of the body
changes orientation (angulation) with respect to a specified, stationary, reference frame.
transmissibility—The principle that permits the moving of a concentrated force to any convenient location along its line of action without changing the mechanical effect of the force.
Bioengineering Analysis of Orthodontic Mechanics
transseptal fibers—Periodontal-ligament fibers, located occlusogingi-
vally near the cementoenamel junction, that circumscribe roots and
extend mesiodistally, interconnecting adjacent teeth.
transverse displacement—An orthodontic tooth movement characterized by displacements of points of the tooth at right angles (perpendicular) to the long-axis orientation.
transverse rotation—A nontranslational tooth movement characterized
by the axis of rotation positioned perpendicular to the long axis of
the tooth.
triangle law—A corollary to the parallelogram law; the two concurrent
point forces and their resultant coincide with the three sides of a
true strain—Strain computed with reference to the deformed state of the
true stress—Internal force per unit of associated deformed area within a
twist—See angle of twist.
twisting deformation—The rotation about the longitudinal axis of the
member of one cross-section of a shaft or wire with respect to another cross-section.
ultimate strength—The maximum load imparted to a structural or machine member (or maximum stress induced in a material) preceding
failure by fracture or rupture.
undermining resorption—See rear resorption.
unit bending stiffness—The product of the modulus of elasticity and the
second moment of the right cross-sectional area with respect to the
neutral axis for a beam or wire.
units—The completion of a dimensional measurement or property value
(e.g., 4 oz, 7 mm, 15 lb per square in.).
utility arch—An arch-wire appliance ordinarily engaging the four or six
anterior teeth and the terminal molars, skipping the premolars (and
often the canines) to create flexible, bilateral levers between anterior
and posterior teeth.
variable—A mathematical quantity capable of assuming more than one
value in a problem or during a discussion.
Glossary of Terms
vector—A mathematical quantity exhibiting magnitude, direction, and
work—See mechanical work.
working wire—An orthodontic arch wire, often rectangular, used to exert
crown force systems capable of controlling or imparting root torque
and the associated displacements in all three planes of space.
yield point—The intersection of the stress-strain (or load-deformation)
curve and a reference line with slope equal to that of the initial part
of the cited curve and meeting the strain (deformation) axis at a
specified offset location.
yield strength—The value of the load (or stress) at the yield point.
yielding—A substantial increase in strain with little or no increase
in stress, occurring just above the elastic limit in some metallic
List of Symbols
anterior; apical; distance from the center of resistance to
crown center; mesiodistal loop deformation
internal stressed area
dimension of a rectangular beam, shaft, or wire cross-section; moment arm; a transverse inner-bow dimension
dimension of a rectangular shaft or a wire cross-section;
one-half of the depth of a beam or wire; a transverse
face-bow dimension
carbon; couple
center of a tooth crown
center of resistance of a dental arch
center of resistance of a tooth
center of rotation
deformation; diameter; dimension from the center of re-
sistance to crown center; displacement; moment arm
lever arm
distance; eccentricity; elongation; extensional deformation;
moment arm; strain
modulus of elasticity (Young's modulus)
elastic limit
force exerted by an elastic (appliance element)
total energy
facial; frictional component of force; lever force
a concentrated, contact force
gingival; grams
center of gravity; modulus of rigidity
List of Symbols
dimension of a rectangular beam or a wire cross-section;
intensity of a distributed force; a vertical (height) dimension
heat; horizontal component of force; hydrogen; a vertical
(height) dimension
headgear force
heat treatment
inferior; inner-bow dimension
unit vectors associated with the x, y, z reference frame
second moment of area (moment of "inertia")
polar second moment of area (polar moment of "inertia")
lingual; length
lateral force; length; losses (of energy)
passive length
passive length
i, j, k
moment of a force or force system
external first- or second-order couple exerted on a beam or
normal component of force
occiusal; outer-bow dimension
origin of a reference frame; oxygen
intensity of a distributed force; posterior; pressure
a concentrated force
proportional limit
radius; radius of curvature
position vector
radius of a circular shaft or wire
responsive force; resultant force
radius of the elastic core of a circular shaft or wire
arc length; curvilinear displacement; stress; superior
concentrated force
Bioengineering Analysis of Orthodontic Mechanics
strain energy
temperature; torsional couple (torque)
thermal energy
occiusogingival loop dimension; speed
internal shear force resultant; vertical component of force
x, y; x, y, z
mutually perpendicular reference axes; rectangular coordinates
transverse deflection of a beam or wire
empirically determined coefficient of torsion theory for a
rectangular shaft
direction angles
a, /3, y
coordinate on a beam or wire cross-section, locating
points perpendicular to the neutral axis
empirically determined coefficient of torsion theory for a
rectangular shaft
transverse deflection of a beam or wire; small amount of;
small change in
change in
normal stress
shear stress
Ox, Oy,
angle between two lines; rotational deformation of a beam
or wire cross-section
direction angles
angle between two lines; angle of twist of a shaft or wire in
normal strain
List of Symbols
shear strain
anterior; apical
anterior left
anterior right
canine; center of resistance; compressive; crown
chin cap
elastic; elastic limit
facial; final
hinge; horizontal
hg, HG
incisal; initial; intrusive
incisal left
incisal right
lateral; left; lever; ligature; lingual
middle segment (between anterior and posterior segments)
initial; occlusal
associated with a force P; periodontal; posterior
Bioengineering Analysis of Orthodontic Mechanics
posterior left
posterior right
associated with a force Q
resultant; right
sagittal; shear; spring; strap; stress
tensile; terminal molar
ligature tie
temporomandibular joint
initial; passive
0, 1
initial, final
1, 2
1, 2, 3
1, 2
initial, final; left, right
first-, second-, third-order
8 central incisor, lateral incisor,
left central incisor
right central incisor
left lateral incisor
right lateral incisor
premolar segment
Class IT elastic
Class III elastic
third molar
forces or force components; comparable
lengths; elements of a force system; responsive couple; responsive force; successive force values; successive positions of a point or a particle
numbers in italics indicate illustrations; page numbers followed by t indicate tables.
Abscissa, 4
Acceleration, 27
Action, 26
ADA Specification No. 32, 99—100, 100,
101, 258
Aging of elastomers, 109
Allotropism, crystalline, 76
Alveolar bone, anatomy of, 170
in shearing deformation, 79
of triangle, 10—11
of twist, 81, 82, 277, 283, 284—286
Annealing heat treatment, 139—140
Anode, 107
Apical fibers, 173, 174
Appearance of materials, 111
Appliances, extraoral, 322—371
apposition of, 175—176
canine retraction, with headgear,
deformation of, with tooth movement,
344—348, 346, 347, 348
cervical-pull, face-bow, 325—344
buccal view analysis of, 335—343,
336, 337, 338, 339, 340, 341, 342
coronal plane view of, 343—344,
density of, 184
orthodontic vs orthopedic forces on,
periodontal ligament and, 169, 175
remodeling of, 175—176
with continuous orthodontic force,
179, 180
with intermittent orthodontic force,
resorption of, 175—176
stress distribution in, 170—171, 171
Alveolar process, 169—170
Alveolus, 169
American Iron and Steel Institute, 2
American Society for Testing Materials,
Amorphous matter, 77
Anchorage, 187—188
Anchorage preparation, 412—416
Angle, conversions between unit
systems, 22t
cosine of, 11
direction, 14—15
measurement of, 10, 10
occlusal plane analysis of, 326—335,
327, 328, 329, 330, 332, 333, 334
reactive force system in, 326, 326
chin-cap assembly, 362—365, 363, 364
dassification of, 322—323
with delivery of force to anterior
segment or entire arch, 348—357
in mandible, 356—357, 357
in maxilla, 348—356, 349, 350, 352,
353, 354, 355
with delivery of force to mandibular
arch, 358—361, 359, 360, 361
reverse-pull, 365—369, 365, 366, 367,
interconnections and contact force
systems, 64t
intraoral, activation of, bending in,
control of, 229
deformation with, 79
Appliances (Continued)
elastics in, 215—216, 215, 216, 217—
219, 218
elements of, 211—223
energy in, 125—137
force exerted in, 71
minimum energy configuration
with, 212
sites of, 211
soft tissue deformations with,
spacers in, 212—213, 213
springs in, 213—215, 214
strain energy in, 211
with tip-back bends, 224—228, 225,
226, 227
wire in, arch, 219—223, 220, 221,
wire in, ligature, 216—217, 217
anchorage location for, 187
attachment of, to dentition, 197—205
arch wires in, 201—205, 203, 204
brackets in, 197—198, 198, 199, 201,
buccal tubes in, 199
buttons, eyelets, and staples in,
199, 199
for distolingual rotation, 200, 200
for extrusion, 201, 201
force systems in, 200—202, 200
Begg torquing, 308, 308, 308t
continuous arch-wire model of, 205—
208, 206, 207
control of, 228—231
deactivation of elements of, 211—223
strain energy in, 212
force through connections between
parts of, 59
force through connections between
tooth and, 59
function of, 196
for mandibular growth control, 191
materials for. See Materials, appliance
in mechanical equilibrium, 57, 58
for palate splitting, 190
passive, 79
relaxation of, 212
structural analysis of, 194—232
as structure vs machine, 196—197
with tip-back bends, 224—228, 225,
226, 227
torque delivery by, 272—321
activating, isolation of, 288—289
in anterior segment, 299—310
with circular wire, 273—288
dentition response to, 289—294,
290, 291, 292, 293, 294
force systems and, 295—299, 298,
inelastic behavior in third-order
mechanics and, 310—314, 311,
with rectangular wire, 299—305,
300, 301t, 303, 304t
shaft theory and, 273—288
with spurs, 305—310, 306, 307, 308,
308t, 309
with wire ioops and springs, 314—
318, 314, 315, 317, 318
Arch(es), anterior segment of,
displacement of, 397
extraoral force delivery to, 348—357
retraction of, 409—412, 410, 411
torquing of, 272, 299—310
torsional activation of, 290, 290
bilateral action on, for tooth alignment,
extraoral force on entire, 348—357
opposing, in tooth malahgnment
correction, 388—391, 390, 391
posterior segment of, anchorage in,
412—416, 413, 414, 415
retraction mechanics within, 402—412,
404, 406, 407, 410, 411
utility, 392
mandibular, 392, 393, 394—397, 396,
maxillary, 394, 395
vertical positioning within, 392—402,
393, 395, 396, 398, 399, 400, 401
Atomic number, 73
Atomic weight, 73
Atoms, 73—74
bonding characteristics of, 74, 75
chemical composition of, 75
in crystalline structure, 76, 77
in density, 75
displacement of, 78
equilibrium position of, in solids, 78
forces between, 74, 75
in stress corrosion, 108
Available energy, 124—125
Availability of materials, 111
Axis, beam, 233, 235
long, in tooth displacement, 148, 149,
149, 150
moment, 38
reference, 3—8, 3
Basal bone, 169
Beam, arch-wire, activation of, 234, 235,
bending force systems on, 233
bending stress in, 243
cantilevered, 236, 237, 255—257, 255
characteristics of, 233
cross-sectional rotation of, 233
deformation of, 236—237, 237
elastic range and strength in analysis
of, 253
"fibers" of, 238—239, 238
force systems within, 241—248
framework for analysis of, 235—236
inelastic behavior of, in bending, 261—
264, 263
in-plane bending of, 236
longitudinal shortening of, 236, 237
neutral surface of, 239
section modulus for, 246
split-anchorage, 234—235, 234, 249
subject to in-plane couple loading,
253—255, 254
subject to transverse point load,
250, 251
stiffnesses of, 248—252, 249, 251
transverse deflection of, 233
Beam axis, 233, 235
Beam deformations, strains and stresses,
235—241, 237, 238, 240, 241
Beam stiffnesses, 248—256, 251
Begg therapy, 307—308
Bending, 80—81, 81
fatigue failure from, 103
load-deformation diagram for, 121
plane of, 236
of wire, activation process in, 130—135,
132, 133, 134
beam deformations, strains and
stresses within, 235—241, 237,
238, 240, 241
beam stiffness in, 248—256, 251
cantilevered beam with, 253, 255—
257, 255
couple loading of split-anchorage
beam with, 253—254, 254
deformation from, 82—85, 83
elastic beam theory application to,
257—261, 259t, 260t, 261t
elastic range with, 252—253
elastic strength in, 247
force systems within beam with, 241—
248, 242, 244, 245, 246, 247
force-deformation diagram for, 142—
143, 142, 143
inelastic behavior in, 261—264, 263
loops and, 264—269, 264, 265, 266,
268, 269t
material behavior with, 233—271
modulus of elasticity and yield
strength for, 99—100, 102t
springback for, 100, 102t
Bernoulli, Jean, 113
Bioengineering, orthodontic,
mathematical topics in, 1—23
constants, variables, and functions,
dimensions and units, 20—21, 22t
displacements of particles and solid
bodies, 8—10, 9
frames of reference, 3—8, 5, 6, 7
measurements, computations, and
numerical accuracy, 21—23
trigonometry, 10—16, 10, 11, 12t,
13, 14, 15
vector algebra, 16—20, 17, 18, 19
Bodily movement, tooth, 149, 152—155,
153, 154
Body forces, 25
Bonding in bracket-crown attachment,
Brackets, accessibility in placement of,
attachment of, to tooth, 197