Fire Design Guidelines for Steel Beam-Columns: Part II, Beams

NIST GCR 11-952
Guidelines for Fire Design of Steel
Beam-Columns: Part II, Beams
NIST GCR 11-952
Guidelines for Fire Design of Steel
Beam-Columns: Part II, Beams
M.M.S. Dwaikat
V.K.R. Kodur
Department of Civil & Environmental Engineering
Michigan State University
East Lansing, MI 48824
Report No. ECC-RR-2011-1
Grant 60NANB7D6120
November 2011
U.S. Department of Commerce
John E. Bryson, Secretary
National Institute of Standards and Technology
Patrick D. Gallagher, Under Secretary of Commerce for Standards and Technology and Director
Notice
This report was prepared for the Engineering Laboratory of the
National Institute of Standards and Technology under Grant number
60NANB7D6120. The statement and conclusions contained in this report
are those of the authors and do not necessarily reflect the views of the
National Institute of Standards and Technology or the Engineering
Laboratory.
NIST REPORT
Guidelines for Fire Design of Steel Beam-Columns:
Part II: Beams
by
M.M.S DWAIKAT
V.K.R KODUR
Research Associate
Dept. of Civil & Env. Engg.
Michigan State University
East Lansing, MI 48824
Professor
Dept. of Civil & Env. Engg.
Michigan State University
East Lansing, MI 48824
Submitted to
National Institute of Standards & Technology
100 Bureau Drive
Gaithersburg, MD 20899
Report No. CEE-RR-2011-1
Project Title - Collaborative Research: Fire Engineering
Guidelines for the Design of Steel Beam-Columns
Department of Civil and Environmental Engineering
Michigan State University
East Lansing, Michigan
Acknowledgements
The authors of this report wish to acknowledge the following sources of financial support:

The National Science Foundation (NSF) through the grant No. CMMI 065292.

The National Institute of Standards and Technology (NIST) through the Fire Grant Program (No.
60NANB7D6120).

Additional office support and contract accounting at Michigan State University/College of
Engineering/ Department of Civil & Environmental Engineering.
The research carried out at Michigan State University and presented in this report was undertaken as a
part of a larger research project in collaboration with Princeton University on beam-columns. Princeton
University is separately developing a project report that specifically addresses fire resistance of
perimeter beam-columns.
Disclaimer
"Any opinions, findings, and conclusions expressed in this material are those of the authors and do not
necessarily represent the views and opinions of the National Science Foundation, National Institute of
Standards and Technology and Michigan State University."
EXECUTIVE SUMMARY
Steel structures when exposed to fire develop significant fire induced forces and
deformations, and these effects are to be properly accounted for in evaluating realistic response
of structural systems. The current design methodologies are based on prescriptive approach
which does not take into consideration the fire induced forces that develop in steel beamcolumns. To overcome these drawbacks, a performance based fire design guidelines for steel
beam-columns and restrained beams is developed as part of a collaborative research project
between Michigan State University (MSU) and Princeton University (PU). MSU’s research was
focused on beams that transform into beam-columns, while PU focus was on columns that
transform into beam-columns. This report presents guidelines for fire design of restrained
beams. A companion report entitled, Guidelines for Predicting the Capacity and Demand of
Steel Beam-Columns under Fire: Part 1, Perimeter Columns, is submitted by authors at
Princeton University (Drs. Garlcok M., Quiel S., Paya-Zaforteza I.).
The fire design guidelines for restrained beams are derived based on equilibrium and
compatibility principles and fire resistance is evaluated by applying deflection or strength
limiting criteria. The proposed approach takes into consideration critical factors
governing fire response, including fire scenario, end restraints, connection configuration
(location of axial restraint force), thermal gradient, load level, beam geometry, and failure
criterion. The validity of the proposed approach is established by comparing predictions
from the proposed approach with results obtained through validated finite element
analysis. The proposed approach are simple to use and provide efficient alternative to
complex numerical simulations for evaluating fire response of steel beam-columns and
restrained beams.
The guidelines presented in this document are based on the Ph.D. dissertation of M.M.S.
Dwaikat (2010). This document is a compilation of information from a number of
published journal and conference papers written by the authors which are cited
throughout this report.
i
Table of Contents
Guidelines for Fire Design of Steel Beam-Columns ....................................................................... i
Executive Summary ......................................................................................................................... i
1. Introduction ................................................................................................................................. 3
2. Response of Restrained Beams ................................................................................................... 5
3. Previous Studies .......................................................................................................................... 8
4. Experimental and Numerical Studies.......................................................................................... 9
5. Design Guidelines For steel Beam-Columns ............................................................................ 10
5.1 Method for Evaluating Steel Temperature .......................................................................... 11
5.1.1 Proposed methodology .................................................................................................. 11
5.1.2 Verification of the proposed temperature evaluation equations.................................... 12
5.2 Method for Evaluating Plastic P-M Diagrams .................................................................... 13
5.2.1 Proposed methodology .................................................................................................. 14
5.2.2 Verification of the proposed P-M evaluation approach ................................................ 16
5.3 Method for Evaluating Deflection of Restrained Steel Beams............................................ 17
5.3.1 Proposed methodology .................................................................................................. 17
5.3.2 Verification of the proposed deflection approach ......................................................... 20
6. Design Applicability ................................................................................................................. 21
7. Design Implications .................................................................................................................. 25
8. Summary and Conclusions ....................................................................................................... 25
9. References ................................................................................................................................. 25
2
1. INTRODUCTION
When a restrained steel beam or a restrained steel column are exposed to fire, the temperatures
attained in steel lead to both degradation in material properties and generation of additional
internal forces that do not exist at room temperature. These fire induced restraint forces (axial
force & bending moment), which can often reach 30% of the room-temperature capacity, transfer
the behavior of the beam or column to that of a beam-column (Kodur and Dwaikat 2009).
Current provisions in codes and standards do not take this transformation into consideration in
fire design of structures.
To overcome these drawbacks a collaborative research project between Michigan State
University (MSU) and Princeton University (PU) was undertaken to study the fire response of
steel beam-columns. This project was jointly funded by National Science Foundation and
National Institute of Standards and Technology. The main objective of this project is to advance
the state-of-the-art on the fire response of beam-columns and to develop simple design tools for
practicing engineers. To achieve this objective, the work was divided between the two groups
with Michigan State University focusing on beams that transform into beam-columns and
Princeton University focusing on columns that transform into beam-columns.
From a fire resistance perspective, generally two types of beam-columns can be identified. The
first type is a beam-column that is transferred from a restrained beam. The second type is a
beam-column that developed as a result of fire effects on a column. Though both can be treated
as beam-columns, there is a clear distinction between these members in terms of levels of “P”
and “M” and also failure criteria. For example, in case of a column transferring to beam-column,
generally, strength failure criteria may dominate the design, while for a beam transferring to
beam-column deflection failure criteria may dominate the design. In fire conditions, where
deflections can be significant, failure criteria based on limiting deflection (or rate of deflection)
become very important for limiting the spread of fire from one compartment to another, and to
maintain functional integrity of the structure.
The two cases of beam-columns are illustrated in Fig. 1, which shows a typical steel frame in a
building exposed to fire. In current fire design practice, this type of structure is modeled as a
simply supported beam (or column) (Fig. 1(b)). However, due to the restraint offered by the
adjacent members, the beams and columns are likely to behave as restrained members (Fig. 1(c,
d)). When the beam (or column) expands under increasing temperatures, the adjacent members
3
impose both axial and rotational restraints (Ka and Kr) on the beam (or column) that prevent it
from free expansion.
Ka
Kr
(a) Layout of typical Steel frame
w
w
Ka, Kr
L
P
Ka,
Kr
L
Simply supported beam
Perimeter column
Restrained beam
Δ
L
θ
θ
Δ
Δ
P
Deflected shapes
M = Kr× θ
P
P
M = wL2/8
M = wL2/8 + P×Δ
Bending moment and axial force
(b)Simply supported
(c) Restrained beam
P
(d)Restrained column-beam
Fig. 1: Development of fire induced restraint forces in restrained steel beams and columns.
This generates axial force (P) and bending moment (M) as seen in Fig. 1 (c, d). Also, with the
increase of temperature, steel loses its strength and stiffness properties. Therefore, both the
midspan deflection (Δ) and the end-rotation (θ) of the beam (or column) gradually increase with
the increase in steel temperature. Further, the fire induced axial force (P) causes additional
bending moment on the beam (or column) due to P-Δ effect, as seen in Fig. 1(c, d). The increase
4
in the bending moment (M) produces additional deflections and hence further deterioration in the
response of the beam (or column) under fire. Furthermore, the increasing temperature and stress
in steel accelerate the development of high temperature creep deformations leading to rapid
increase in deflection and ultimately failure of the member.
As part of the collaborative project, MSU is taking the lead with respect to restrained beams
transferring to beam-columns, while PU is taking the lead on columns transferring to beamcolumns. The common factor between the two types is the role of the interaction between P and
M on the fire response of restrained beams or columns. However, since both types differ in
failure mode and behavioral trends, each one of these requires separate and detailed analysis.
This report (Part II) deals with fire design guidelines for restrained steel beams. The design
guidelines relevant to columns are presented in a companion report (Part I) prepared by PU.
2. RESPONSE OF RESTRAINED BEAMS
The response of the beam-column primarily depends on the interaction between bending moment
(M) and axial force (P). The combination of P and M affects the load bearing capacity of beamcolumns which is generally expressed using the following P-M interaction relationship at
ambient conditions:

P
M

 1.0
Pn
M n
[1]
where α and β are interaction coefficients, and ΦMn and ΦPn are the moment and axial
capacities, respectively.
The typical response of a restrained steel beam exposed to fire is illustrated in Fig. 2. The beam
is restrained with rotational and axial restraints of stiffnesses Kr and Ka, respectively. Under fire
exposure, the restrained beam produces varying response under three distinct stages. In Stage I,
elastic response dominates the behavior wherein the beam expands as a result of heating, and
compressive force (P) and bending moment (M) develop in the beam due to the prevention of
expansion by the end-restraints. Fire-induced internal forces and deflection of the beam continue
to increase until either yielding or buckling occurs in the beam. If buckling does not occur then
elasto-plastic response dominates the behavior in Stage II, as shown in Fig. 2. As steel
temperature continues to increase, softening of steel causes faster deflection rates. The beam
enters catenary phase in Stage III when the fire-induced compressive axial force vanishes as
5
shown in Fig. 2. In the catenary phase, tensile forces develop in the beam and the load bearing
mechanism gradually changes form flexural to cable (tensile) until failure occurs by rupture of
the beam (or in the connections).
w
Ka, Kr
w
Ka, Kr
L
L
Restrained
Simply supported
a) Steel beam
Axial force
Stage I
Stage II
Stage III
Yielding
Elastic
Elastoplastic
Catenary action
Steel temperature
b) Fire-induced axial force
Ty
Midspan deflection
Δe
Δy
Catenary
Δc action (P = 0)
Tc
Tf
Rupture of
steel
Restrained beam
Simply supported beam
c) Midspan deflection
Fig. 2: Typical fire response of simply supported and restrained beams that fail by yielding
The fire induced internal forces (axial force P and bending moment M) that develop in a
restrained beam depend mainly on the characteristics of the beam, fire scenario, and restraint
conditions. Rotational restraint is a function of many factors including the continuity of the beam
(to adjacent spans), rotational stiffness offered by the supporting columns, rotational stiffness of
the beam-to-column connection, and the extent of composite action that develops between
concrete slab and steel beam. The axial restraint is also a function of a number of factors
including the continuity of the beam, lateral stiffness of the supporting columns, and the
6
composite interaction between concrete slab and steel beam. If the influence of composite action
between concrete slab and the steel beam is neglected, then the computation of these restraint
stiffnesses (axial and rotational) at room temperature can be carried out using conventional
methods of analysis for indeterminate structures that can be found in structural analysis
textbooks (Ghali et al. 2003). In this study, we assume that the axial and rotational restraints on
the ends of the beam are elastic, constant, and symmetric as shown in the illustration in Fig. 2.
It is worth mentioning that the response described above is influenced, at each stage, by the
interaction between fire induced forces (P and M) in the restrained beam. This is in contrast to
the response of a simply supported beam where the response is influenced by the applied
bending moment (M) on the beam and failure occurs by developing a single plastic hinge under
fire. This is shown in Fig. 2 (c), where a run away failure occurs in the simply supported beam
upon developing a plastic hinge at the midspan of the beam (where moment in maximum).
However, the failure of restrained beam requires the development of a plastic mechanism and the
formation of three distinct plastic hinges (two at supports and one at midspan). Further, the
formation of these three plastic hinges is governed by the interaction between P and M.
The nonlinear behavior of the restrained beam is complex and requires an iterative procedure due
to the interaction between fire-induced restraint force, bending moment and deflection of the
restrained beam. In order to avoid these complexities and to provide a practical simplified tool
for analysis and design of restrained steel beams under fire, the nonlinear behavior of the beam
has been incorporated in an approximate approach. This will be shown in the following sections
wherein fire response of a restrained beam will be traced for a general case. Once the response is
obtained, either deflection or strength criteria can then be applied to evaluate the fire resistance.
Equilibrium and compatibility principles are applied to develop a simple methodology for
estimating fire-induced forces and midspan deflection. The proposed approach is valid for
standard and design fire scenarios and can be used for either restrained or simply supported
beams.
7
3. PREVIOUS STUDIES
Note: The basis for this chapter is based on the following paper:
 Kodur V.K.R, Dwaikat M.M.S., "Response of Steel Beam-Columns Exposed to Fire", Engineering
Structures, Vol. 31, Issue 2, Pages 369-379, Feb. 2009.
Restrained steel beams when exposed to fire develop significant internal forces (axial force and
bending moment) as illustrated above. These fire-induced restraint forces influence the behavior
of the beam and can alter its failure pattern. Laboratory tests and actual fire incidents have shown
that restrained steel beams develop significant tensile catenary forces that enhance the fire
resistance. If the beam-to-column connections can sustain the catenary forces induced in the
restrained beam, large deflections can be achieved before reaching ultimate load-bearing
capacity of the beam under fire (Skowronski 1988 and 1990).
Current fire design provisions in codes and standards assess failure of steel beams by applying
critical temperature or strength failure criteria (AISC 2005, EC3 2005). However, often in
restrained beams, strength failure is reached after achieving large deflections. The fire-induced
large deflections in beams can cause loss of functionality and integrity of the steel framed
structure. Such large deflections in beams might lead to the loss of compartmentation, and thus
facilitate the spread of fire between compartments either horizontally or vertically. Therefore,
using a strength limit state to assess the failure of restrained steel beams may not reflect realistic
conditions in fire scenarios.
Some standards limit the magnitude of beam deflection attained during fire tests in order to
ensure the safety of test furnaces (BS 1987). However, deflection limit states do not exist for the
fire design of actual framed structures. This is in contrary to ambient design where deflection is
one of the limit states to be considered in design process.
There have been limited studies on the behavior of restrained beams under fire conditions. Li et
al. (2000) Liu et al. (2002), and Li and Guo (2008) conducted fire tests on axially restrained steel
beam and found noticeable axial forces building up in the beam with fire exposure time. Usmani
et al. (2001) presented a general discussion of the basic principles that govern fire response of
8
restrained steel beams. In this study, the deflected shape was approximated as a sine wave of
length L(1+εT), where L is the span length of the beam and (εT) is the free thermal expansion of
the beam. This assumption greatly overestimates the deflection because in restrained beams,
unlike in simply supported beams, the thermal expansion is not free (due to presence of axial
restraint) and can be significantly less than (εT). Also in this study, the authors did not consider
the influence of gravity loads on the beam.
Tan and Huang (2005) investigated the development of fire-induced restraint forces in steel
beams using one dimensional single span beams. The effects of slenderness ratio, load utilization
factor, thermal gradient across the steel section, and axial and rotational restraints on the fire
response of steel beams were evaluated. Yin and Wang (2003) and Wang and Yin (2006)
applied finite element method to study the behavior of restrained steel beams under fire
conditions and then develop a method for predicting the fire behavior of restrained steel beams.
In this method, the user assumes an initial deflected profile for the beam. Equilibrium principles,
coupled with moment-axial force interaction formulas, are then used to iterate for the internal
forces and midspan deflection of the beam. This method is laborious and requires lengthy
calculations and numerical integration.
4. EXPERIMENTAL AND NUMERICAL STUDIES
Note: The basis for this chapter is based on the following paper:
 Dwaikat, M.M.S., Kodur, V.K.R., Quiel, S.E., Garlock, M.E.M., (2011) “Experimental Behaviour of
Steel Beam-Columns Subjected to Fire-Induced Thermal Gradients”, Journal of Constructional
Steel Research, 67(1), pp. 30-38.
 Dwaikat M.M.S., Kodur V.K.R, "Effect of Location of Restraint on Fire Response of Steel Beams,",
Journal of Fire Technology, 46(1), pp. 109-128, 2010.
 Kodur V.K.R, Dwaikat M.M.S., "Effect of High Temperature Creep on Fire Response of Restrained
Steel Beams", J. of Materials and Structures (RILEM), V. 43, No. 10, pp. 1327-1341, 2010
 V.K., Garlock M.E., Dwaikat M.S., Selamet S., and Quiel S., "Collaborative Research: Fire
Engineering Guidelines for the Design of Steel Beam-Columns", Proceedings of 2009 NSF-CMMI
Engineering Research and Innovation Conference, Honolulu, Hawaii, 2009.
To overcome some of the current drawbacks, both experimental and numerical studies were
carried out on beam-columns with the aim of developing rational fire design guidelines (Dwaikat
2010, Kodur et al. 2009). The experimental studies comprised of fire resistance tests on four
9
beam-columns. The critical parameters that were varied in these tests included load level, fire
scenario, insulation thickness, thermal gradients, and axis orientation of the beam-columns. The
results of the tests indicated that axis orientation, fire induced thermal gradients, and load ratio
have major effect on the capacity of beam-columns (Dwaikat 2010, Kodur et al. 2009).
The test results were used to validate numerical models which in turn were used to conduct a
series of parametric studies. The numerical analysis carried out using ANSYS program (ANSYS
2007) included two steps, namely thermal and structural analyses. In the thermal analysis, solid
plane elements were used to discretize the cross section and the boundary conditions simulated
both radiation and convection that would result in a fire exposure. In the structural analysis, 8noded shell elements were used and the boundary conditions were simulated based on the test
setup (Dwaikat 2010, Kodur et al. 2009). Steel temperatures recorded in the fire tests were used
for validating thermal analysis results. Load-displacement and time-moment histories obtained
from the fire tests through LVDT’s and high quality strain gauges were used for validating
structural analysis results. Once validated, the finite element models were used to conduct a
series of parametric studies to further identify the critical factors governing fire response. Results
from parametric studies indicated that load level, end restrained, orientation and magnitude of
thermal gradients, and fire scenarios, have significant influence on fire response of steel beamcolumns (Dwaikat 2010, Dwaikat and Kodur 2009). The results from the parametric studies were
used to develop rational design methodologies and are presented below.
5. DESIGN GUIDELINES FOR STEEL BEAM-COLUMNS
Based on the numerical and experimental studies undertaken as part of this research, a set of
guidelines have been developed for fire design of restrained steel beams and beam-columns.
First, a simple method for computing steel temperature was developed, and that method is used
to evaluate steel temperature and the fire induced thermal gradient in the section. Once the steel
temperature and thermal gradient are evaluated, the plastic P-M diagram for the steel beamcolumns is computed based on a simplified approach that takes into account the effect of thermal
gradient on the plastic P-M diagrams. Also, a simplified approach was developed to compute the
fire induced deflection in restrained steel beams. Both approaches serve as guidelines for
evaluating fire resistance of restrained beams based on strength or deflection limit states.
10
5.1 Method for Evaluating Steel Temperature
Note: The basis for this chapter is based on the following paper:
 Dwaikat, M.M.S and Kodur, V.K.R. (2010) “A Simplified Approach for Predicting
Temperatures in Fire Exposed Steel Members”, Submitted, Fire Safety Journal.
 Dwaikat, M.M.S. and Kodur, V.K.R., "A Simplified Approach for Predicting Steel
Temperatures under Design Fires” Application of Structural Fire Engineering, Prague, Czech
Republic, 2011.
5.1.1 Proposed methodology
For accurate evaluation of fire resistance of steel structural members, temperatures in the cross
section of the member are required. The current simplified methods for temperature evaluation in
steel members have number of drawbacks and they do not account for a number of factors. As
part of this research, a simplified approach for evaluating cross sectional temperature in steel
members exposed to fire was developed (Dwaikat and Kodur 2010a). The approach is derived
utilizing simplifying assumptions to the general heat transfer equation for standard fire and is
then extended for application under design fire scenarios. The proposed approach is applicable
for both protected and unprotected steel sections.
For a section exposed to fire with growth phase (e.g. a standard fire), steel temperature can be
obtained using the following equation:
[1]
T (t )  T 1  e  st 
s
f

with the coefficient s is computed as follows:
s
Fp / As 
 c p  p Fp t p 
n  1
cs s 1/ hcon  t p / k p 1 

c  A m

s s
s
[2]

where hcon is the convective heat transfer coefficient, csρs and cpρp are the heat capacity of steel
and insulation, respectively, kp is conductivity of the insulation material, As is the cross sectional
area of steel section, tp is insulation thickness, Fp is the heated perimeter of the section, m is a
constant used for averaging temperature in insulation, and is generally taken as 2.
In deriving Eq. [1], the growth phase of the fire was assumed to follow an power function, i.e.;
Tf = atn , Tf is the temperature (in ˚C) of the growth phase of the fire as a function of time t (in
minutes), a and n are a regression coefficients: For ISO 834 standard fire growth a = 469.9 and n
11
= 0.1677 (R2 = 0.995), and for ASTM E119 standard fire growth a = 496.5 and n = 0.1478 (R2 =
0.989).
In case of a design fire with maximum fire temperature Tf,max occurring at t = t1 and followed by
a decay phase with cooling rate equal to r, the steel temperature in growth phase can be
computed using Eq. [1]. In the cooling phase of the design fire, the maximum temperature
attained in steel and its corresponding time can be computed using the following equations:
T
 2t r
f , max
1
T

s, max
1  2t r / T
s,1
1
t
 T
 T 

f , max
s,1
 t 1 

s, max 1
1 T  rt 

s
,
1
1
2


[3]
[4]
where Ts,1 is the steel temperature computed using Eq. [1] at the time of maximum fire
temperature (t1).
5.1.2 Verification of the proposed temperature evaluation equations
The validity of the approach is established by comparing predicted temperatures against data
from fire tests and against results from finite element analysis generated via ANSYS. Fig. 3
shows a sample of comparison for protected (tp ≠ 0) and unprotected (tp = 0) steel section. In
addition, predictions from the proposed method are also compared with the temperatures
predicted by “best-fit” method. Fig. 4 shows that maximum steel temperature and the
corresponding time as predicted by the proposed approach and by finite element analysis for
different steel sections exposed to different design fire scenarios. The comparisons shown in Fig.
3 and 4 indicate that the proposed equations are capable of predicting the temperature in steel
section exposed to different fire scenarios. The simplicity of the proposed method makes it
attractive for use in design situations. Further details on the proposed approach can be found in
Dwaikat and Kodur (2010a)
12
1200
1000
800
600
ISO 834 fire curve
Finite element method
Proposed method
Best-fit method
400
200
0
0
50
100 150 200
Time, min
250
Temperature, ˚C
Temperature, ˚C
1000
800
600
ISO 834 fire curve
Finite elment method
Proposed method
Best-fit method
400
200
0
0
300
25
50
75 100
Time, min
125
150
900
250
+10% margin
t s ,m a x (m in .) F ro m F .E .A .
T s ,m a x ( o C ) F ro m fin ite e le m e n t a n a ly s is
(a) Protected steel section
(b) unprotected steel section
Fig. 3: Steel temperature obtained by the proposed approach to those obtained using different methods
200
700
150
500
300
300
o
500
700
50
0
0
900
50
t
Ts,max ( C) From proposed approach
(a) Maximum steel temperature
- 10% margin
100
-10% margin
100
100
+10% margin
s,max
100
150
200
(min.) From proposed approach
250
(b) Time for reaching maximum steel temperature
Fig. 4: Comparing predictions from proposed equations and from finite element analysis for Ts,max and ts,max
5.2 Method for Evaluating Plastic P-M Diagrams
Note: The basis for this chapter is based on the following papers:
 Dwaikat, M.M.S and Kodur, V.K.R. (2010) “A Simplified Approach for Evaluating
Plastic Axial and Moment Capacity Curves for Beam-Columns with Non-uniform
Thermal Gradients”, Engineering Structures, 32(5), pp.1423-1436
 Dwaikat, M.M.S and Kodur, V.K.R. (2011) “Strength Design Criteria for Steel Beam-Columns
with Fire Induced Thermal Gradients”, In Press, Engineering Journal, AISC.
 Dwaikat, M.M.S. and Kodur, V.K.R., (2010) " Performance-based Methodology for Tracing the
Response of Restrained Steel Beams Exposed to Fire”, COST Action C26 International
Conference: "Urban Habitat Constructions under Catastrophic Events", pp. 1-12, Naples, Italy.
13
5.2.1 Proposed methodology
The underlying mechanics of the distortion of P-M diagrams that is induced by thermal gradients
was first studied by Garlock and Quiel (2007; 2008) as part of a collaboration research project
with MSU. These studies showed that a thermal gradient in a steel section causes the center of
stiffness (CS) of the cross section to migrate towards the cooler (stiffer) regions and away from
the heated (softer) regions, as shown in Fig. 5. This migration of the center of stiffness generates
an eccentricity (e) between the geometric center (CG) and the center of stiffness of the cross
section. As a result of this eccentricity, bending moment is generated since the axial force will
now act eccentrically with respect to the new center of stiffness of the section. This generated
bending moment may counteract the bending moment that results due to thermal bowing.
Therefore, this migration of center of stiffness causes a distortion in the plastic P-M interactive
diagram (Dwaikat and Kodur 2009; Garlock and Quiel 2007 and 2008).
Thermal
gradient
M
P / Pu
P
Eccentricity (e)
e × P u = M TG
Hotter side
Cooler side
ΔT
A'
M
ΔT
C'
B
M TG
C
P
A
M / Mu
B'
Center of
stiffness (CS)
Geometric
center (GC)
a) Development of thermal gradient
T = T ave
b) Effect of thermal gradient on P-M diagrams
Fig. 5: Characterizing plastic P-M diagram for a WF section with thermal gradient in the strong direction
Based on these studies, Garlock and Quiel proposed a numerical procedure to compute the
resulting distorted P-M diagrams for an I-shaped cross section subjected to any thermal gradient
of any shape (Garlock and Quiel 2008). The proposed method requires intensive use of
numerical programs, such as MATLAB or Spreadsheets. For example, lengthy algorithms are
required to compute the lumped temperature in each steel plate of the section to numerically
14
integrate the temperature-dependent ultimate stresses along the depth of the cross section. This
makes the method laborious and not straightforward. Further research by Dwaikat and Kodur
(2010b and 2011a) on the influence of fire induced thermal gradient on P-M diagrams has led to
modifications to the current interaction P-M equations in codes and standards.
The basic features of the distorted plastic P-M diagram for a WF section, with thermal gradient
in the strong direction, is compared to the case of a uniform temperature in Fig. 5. The figure
shows that the value of moment capacity under peak axial capacity (point A in Fig. 5) moves
back and forth (to point A’ in Fig. 5) depending on the eccentricity (e) between center of
stiffness “CS” and center of geometry “GC” that is caused by the thermal gradient in a WF
section. The magnitude of the shift (MTG) in the P-M capacity envelope (Fig. 5) is assumed to be
numerically equal to the ultimate axial capacity (Pu,Tave) of the section multiplied by the
eccentricity (e) between the center of geometry (YGC) and of the center of stiffness (YCS) of the
section as shown in Fig. 2 (Dwaikat and Kodur 2010b). The ultimate capacity is computed based
on the average temperature of the section, i.e:
MTG = e × Pu,Tave = e × ky(Ts,Ave)×Fu×As
[5]
where the eccentricity (e) between YGC and YCS can be calculated as follows:
e
d  2BF t F k E (Ts,CF )  t w dkE (Ts, Ave ) 
1

2 
k E (Ts, Ave )(2BF t F  t w d )

[6]
In Eq. [6] d, BF, , tF, and tw are beam depth, flange width, thickness of flange and web of the
cross section, respectively. kE and ky are the reduction factors for elastic modulus and yield
strength at steel temperature T.
With the calculation of MTG using Eq. [5], an adjustment for the P-M interaction curves can now
be proposed. The adjustment of the P-M interaction curves is based on using the average
temperature of steel section with a shift MTG that occurs as a result of thermal gradient in the
section. The adjustment of P-M diagram is aimed at preserving the room-temperature shape of
the P-M diagram and only introducing the shift MTG to account for the thermal gradient effect.
The adjusted equations of the plastic P-M diagrams for wide flange section with linearized
thermal gradient in the strong direction can be written as (Dwaikat and Kodur 2010b):
M  M TG

M u Ts, ave
P
  Pu Ts, ave   1.0
[7a]
15
and
M  M TG

M u Ts, ave


P
 1.0
Pu Ts, ave

[7b]

where Mu(Ts,ave) and Pu(Ts,ave) are the plastic bending and axial capacities computed assuming
uniform average steel temperature.
5.2.2 Verification of the proposed P-M evaluation approach
A comparison between the proposed equations, finite element analysis, and test results is shown
in Fig. 6, which plots the development of the fire-induced P and M in a beam-column that was
tested at MSU. The P-M envelope as predicted using finite element analysis and simplified
approach (Eq. [7]) at two fire exposure times is also plotted in Fig. 6. It is seen in the figure that
at different time steps, the beam-column experiences different thermal gradients which cause
different shifts in the P-M diagrams as seen in Fig. 6(a and b).
Based on the results in Fig. 6(a), the failure time of the beam-column is conservatively predicted
by the detailed finite element analysis at t = 216 min. which matches well with predicted failure
envelope using Eq. [7]. The actual failure of the beam-column was observed in the fire test at t =
220 min., as shown in Fig. 6(b). The actual failure in test is said to occur at the time after which
the beam-column is no longer capable of carrying the applied axial load and thus starts to deflect
at an accelerated rate. While, the predicted failure is assumed to be the time at which the fire
induced P and M exceed the capacity envelope given in Eq. [7]. The measured failure point is
inside the capacity envelope mainly due to the experimental parameters. For instances the fact
that the temperature is not exactly uniform along the length of the beam-column can result in
shifting the failure point towards the inside (more conservative) of the capacity envelope. The
comparison presented in Fig. 6 shows that the proposed equations are capable of predicting the
P-M envelop of beam-columns subjected to thermal gradients.
1
Pre dicte d failure
(t = 216 min.)
Measured failure
(t = 220 min.)
1.2
1
Finite e le me nt
e nve lope
0.8
Equation
Proposed 26
Eq.
0.2
0.4
0.2
0
0
-0.75 -0.5
-0.25
Finite element
envelope
0.6
0.4
-1
t = 220 min.
Tave = 670°C (1238°F)
ΔT = 200°C (360°F)
0.8
0.6
Equation
26
Proposed Eq.
1.4
P/P u,Tave
P/Pu,Tave
t = 216 min.
1.4
Tave = 655°C (1211°F)
1.2
ΔT = 150°C (270°F)
0
0.25
M/M u,Tave
0.5
0.75
1
-1
-0.75 -0.5 -0.25
0
0.25
M/M u,Tave
0.5
0.75
1
16
(a) At t = 216 min. of fire exposure
(b) At t = 220 min. of fire exposure
Fig. 6: Measured and predicted P-M response for the tested beam-column
5.3 Method for Evaluating Deflection of Restrained Steel Beams
Note: The basis for this chapter is based on the following papers:
 Dwaikat, M.M.S and Kodur, V.K.R. (2011) “An Engineering Approach for Predicting Fire
Response of Restrained Steel Beams.” ASCE Journal of Engineering Mechanics, 37(7), pp.
447-461, 2011.
 Dwaikat, M.M.S and Kodur, V.K.R. (2011) “A performance-based methodology for fire
design of restrained steel beams”, Journal of Constructional Steel Research, 67(3), 510-524,
2011.
 Kodur V.K., Dwaikat M.S. (2011) "An approach for evaluating fire resistance of restrained
steel beams based on strength and deflection limit states", Proceedings of 2011 NSF
Engineering Research and Innovation Conference, Atlanta, Georgia.
 Dwaikat, M.M.S. and Kodur, V.K.R. (2010), "Lateral-torsional buckling of steel beam-columns under
fire exposure", SSRC (AISC) Annual Stability Conference, Orlando, FL.
5.3.1 Proposed methodology
The influence of end restraints on the fire response of a steel beam can be accounted for through
a performance based design approach. The detailed derivation of the necessary equations has
been presented by the authors elsewhere (Dwaikat and Kodur 2011b,c). In this approach, the fire
response of the restrained beam is traced and the failure of the beam is assessed based on
performance criteria. By referring to Fig. 1, the transition in the fire response of a restrained
beam between elastic and elastoplastic stages is marked by the occurrence of yielding (if the
beam section is designed to be a compact section and local buckling effect is neglected). Since
the restrained beam will experience fire induced axial force (P(T)) and bending moment due to
restraint of non uniform thermal expansion (MG(T)), the temperature (Ty) at which yielding
occurs can be computed using the following yield P-M interaction equation.
P(T )
M (T )  M G (T )

 1.0
k y (T ) Py
k y (T ) M y
[8]
where Py and My are the room-temperature yield axial and moment capacity of the steel section,
respectively. In the elastic regime, where the deflection is small, it can be shown (Dwaikat and
Kodur 2011b,c) that the axial force P can be written as FA× Py and MG as FR× 0.5ΔT×My. If
ky(T), the temperature-dependent reduction factor for yield strength, is approximated as a linear
17
function (i.e. ky(T) =1 – a2×T), then by substitution into Eq. [8] the critical value of Ty can be
obtained as (Dwaikat and Kodur 2011b,c):
Ty 
1  M o / M y  0.5FR T
FA  a 2
,
Ty < 600°C
[9]
where Mo is the maximum bending moment in the beam due to gravity load. ΔT is the thermal
gradient between the top and bottom flanges of the section. FR and FA are the rotational and axial
restraint factors, respectively, and are defined as:
FR 
E s  a1 K r L /( E s I ) 


F y  2a1  K r L /( E s I ) 
and
FA 
E s  a1 K a L /( E s As )



F y  2a1  K a L /( E s As ) 
(10a and 10b)
Ka and Kr in Eq. [10] are the axial and rotational restraint stiffnesses, respectively. L, I, As are the
total length, second moment of area and cross sectional area of the steel beam, respectively. Es,
Fy and α are the room-temperature values of elastic modulus, yield strength and coefficient of
thermal expansion of steel, respectively. The coefficients a1 and a2 result from the linearization
of temperature-dependent reduction factors for yield strength (ky(T)) and elastic modulus (kE(T)),
and thus, a1 and a2 are dependent on the material model of steel which vary depending on the
codes (Dwaikat and Kodur 2011c):
- For steel properties as specified in Eurocode 3 (EC3 2005): a1 = 0.6 and a2 = 0.0013
- For steel properties as specified in ASCE manual (ASCE 1992): a1 = 0.6829 and a2 = 0.0008
As plasticity spreads throughout the section, the axial compressive force reduces until it reaches
zero at temperature Tc as shown in Fig 2(b). Using the condition P = 0 at M = ky(T)×Mu, at the
instant of catenary action (T = Tc), and assuming ky(T) =a3 – a4×T, the temperature Tc can be
computed as (Dwaikat and Kodur 2011c):
. Tc  1 1  M   M y FR T 


a2 
Mu
Mu
2
[11]

where Mu = FuZx and My = FySx.
18
The plastic deflection of the restrained beam at the instant of catenary action (P = 0 at T = Tc)
can be approximated as:
c 
L
2 (Tc  20)
2
[12]
Using linear interpolation, as shown in Fig 1, the elasto-plastic deflection between Ty and Tc can
be obtained as:
  y 
c   y
Tc  Ty
T Ty ,
Ty < T < Tc
[13]
Using Eq. [13], a limiting temperature based on a deflection limiting criterion Δ = LF can be
obtained as:
TDLS  T y 
LF   y Tc  T y 
 c   y 
[14]
where TDLS is steel temperature at deflection limit state, LF is the deflection limit state and is
usually taken to be in the range between L/30 and L/20.
Also, strength failure is assumed to occur at temperature (Tf) where the catenary tensile force in
the beam is maximal and is equal to the tensile capacity of the beam cross section. Based on this
approach, the temperature at which strength failure is eminent can be computed as (Dwaikat and
Kodur 2011c)
Tf 

TcTy FA  a1a3 Tc  Ty
Ty FA  a1a4 Tc  Ty



[15]
where a3 and a4 are regression coefficients that are dependent on steel properties:
- For steel properties as specified in Eurocode 3: a3 = 1.139; a4 = 0.0013
- For steel properties as specified in ASCE manual: a3 = 1.329; a4 = 0.0014
It will be shown through a design example that Eqs. [14] and [15] can effectively be used as a
performance-based guideline for fire design of restrained beam based on deflection criteria.
19
It can be shown through direct substitution that for the case of simply supported beams (Ka = Kr
= 0), and using elastic-perfectly plastic stress strain curves for steel (i.e. Fy = Fu), Eq. [14]
reduces to TDLS = Ty, which is the yielding temperature. This result conforms very well with the
current provisions of fire design of simply supported beams which assumes the failure to occurs
once the bending moment exceeds the plastic bending capacity of the beam under fire (Mo >
ky(T)×FyZx). Thus, Eq. [14] provides a general design methodology not only for beams under
axial and rotational restraint conditions, but also for simply supported steel beams.
Also, Eq. [14] can be used under design fire scenarios with cooling phase. Once the maximum
temperature attained in steel (Ts,max) due to exposure to design fire is evaluated, that temperature
(Ts,max) can then be compared to TDLS computed using Eq. [14]. If Ts,max > TDLS , the beam will
fail under that design fire by exceeding the deflection limit state, and thus the beam needs to be
redesigned. If Ts,max < TDLS , then no failure will occur and the beam will sustain the design fire
exposure.
Based on the discussion above, Eq. [14] accounts for various factors influencing response of a
restrained beam, including fire exposure (cooling phase) and provides an attractive alternative to
detailed nonlinear fire resistance analysis.
5.3.2 Verification of the proposed deflection approach
The proposed design equations have been verified by comparing its predictions against results
from rigorous finite element analysis carried out using ANSYS (ANSYS 2007). The details of
the finite element modeling and validation can be found elsewhere (Dwaikat and Kodur
2011b,c). The validation covered a wide range of beams with varying factors, such as end
restraint, connection configuration, load level, slenderness, and thermal gradient. The results
presented in Fig. 7 are generated for all the combinations of parameters according to Table 1.
Figure 7 compares the results from the proposed approach (Eq. [14]) to results from finite
element analysis on beams with different load, restraint and fire exposure conditions. Two
deflection criteria were used, LF = L/20 and LF = L/30. These deflection criteria are commonly
used in fire tests (BS 1987) and are chosen for comparison. As shown in Fig. 7, the approach
predicts the temperature at deflection limit state (TDLS) within 10% margin of error.
20
1000
1000
900
+10% margin
TDLS(°C) FromF.E.A
TDLS (°C) FromF.E.A
900
800
700
600
500
-10% margin
400
+10% margin
800
700
600
500
-10% margin
400
400
500
600
700
800
900
T DLS (°C) From simplified approach
1000
a) Deflection limit state (LF) = L/20
400
500
600
700
800
900
T DLS (°C) From simplified approach
1000
b) Deflection limit state (LF) = L/30
Fig. 7: Comparing temperatures at deflection limit state as predicted by finite element analysis and from Eq.
10 for LF = L/20 and L/30
Table 1: Characteristics of the beams used in parametric study of finite element analysis
Parameter
Values
Comments
Span length (L)
6m, 9m, 12m
Total length
Load ratio (LR)
30%, 50%,70%
LR  M max / F y Z x
Thermal gradient (ΔT)
0°C, 200°C
ΔT = TTF – TBF
Axial stiffness (Ka)
0, 0.1EsAs/L
Rotational stiffness (Kr)
0, EsI/L
Relative to roomtemperature beam
stiffnesses


6. DESIGN APPLICABILITY
The proposed simplified approach for predicting fire response of restrained steel beams can be
applied in fire design. An example is presented here to demonstrate the applicability and
rationality of the proposed approach. Step-by step design procedure in analyzing a typical
restrained beam (shown in Fig. 8(a)) under fire is presented below:
Problem:
-
Design the beam for 2 hours of fire exposure under ASTM E119 (2008) standard and
specified design fire. Use deflection limit state of LF = L/30.
Beam characteristics:
-
Beam length and section: 7000 mm, W24x76.
-
Loading: uniformly distributed dead and live service loads: wD = 35 kN/m, wL = 70
kN/m.
-
Axial restraint stiffness (Ka): 41.3 kN/mm (≈ 0.1EsAs/L).
-
Rotational restraint stiffness (Kr): 50 kN.m/milirad (≈ 2.0EsI/L )
-
Initial thermal gradient (ΔT) = 150°C.
21
-
Steel properties: Grade 50 steel; Fy = 355 MPa and Fu = 445 MPa.
-
High temperature properties: as per ASCE specified temperature-dependent reduction
factors (ASCE 1992).
Response parameters:
Load combination under fire (ASCE-07 2005)
-
wf = 1.0 wD + 0.5wL = 70 kN/m. (≈ 30% of ultimate load capacity at 20°C)
Axial and rotational restraint factors (as per Eq. [10])
-
E s  a1K a L /( E s As ) 


F 
A
F  2 a  K L /( E A ) 
y  1
a
s s 
14  10  6  2  10 5  0.6829  0.1 


  0.00037
355
 2  0.6829  0.1 
F
R

E s  a1K r L /( E s I ) 
F


 2a  K L /( E I ) 
r
s 
y  1
14  10 6  2  10 5  0.6829  2 

  0.0032
355
 2  0.6829  2 

-
Yield temperature (as per Eq. [9] and using Mo = max (Ms, Mm) = 285.8 kN.m)
Ty 
1  M o / M y  0.5 FR T
FA  a 2

1  285 .8 / 1023 .8  0.5  0.0032  150
 411 C
0.00037  0.0008
-
Catenary temperature (as per Eq.[11])
M F T 

M
1    y R


M u M u 2 

1 
285.8 1023.8 0.0032  150 




  878.8 C
1 
0.0008  1308.9 1308.9
2

Tc 
-
1
a2
Deflection at onset of yielding
 5wL4
L2 T  FR Fy 
1

y  

 384k E I
8 d  a1 Es 
E s



7 2 150   0.0032
355
5  70  7 4
  15 mm
 14  10 6 


 1 


6
5
 384  0.79  2  10 5  8.7  10 4

8
0
.
61
0
.
6829
14

10

2

10




-
Deflection at catenary point (as per Eq. [12])
c 
-
L
2
2   (Tc  20 ) 
7000
2
2  14  10  6 (878 .8  20 )  542 mm
Temperature at deflection limit state (as per Eq. [7.26])
22
TDLS  Ty 
-
LF   y Tc  Ty   411  7000 / 30  15878.8  411  605 C
 c   y 
542  15
Temperature at ultimate strength failure (tensile capacity) (as per Eq. [15])
Tf 


TcT y FA  a1a3 Tc  T y

T y FA  a1a4 Tc  T y


878.8  411 0.00037  0.6829  1.329  878.8  411
 931 C
411 0.00037  0.6829  0.0014  878.8  411
Design for standard fire exposure
After computing the limiting temperature (TDLS) based on deflection limit state, the fire
resistance duration can be computed Eq. [1]. The insulation material is assumed to have thermal
conductivity of 0.1 W/m.oC and heat capacity of 375 kJ/m3/°C. The insulation thickness is
increased and the average steel temperature is calculated using Eq. [1] at t = 120 min. of fire
exposure until the Ts ≈ TDLS. This occurred when an insulation thickness of 25 mm is used. For
the same insulation thickness, the limiting temperature based on strength failure (Tf) is reached at
218 min. of standard fire exposure
wL = 70 kN/m, wD = 35
W24 ×76
L=7m
i) Beam loading and
Ms = 143 kN.m
Mm = 285.8 kN.m
ii) Bending moment diagram under
(a) Properties of the continuous beam used in the design example
23
1200
Design fire
Temperature, °C
1000
Standard fire
800
600
T DLS = 605°C
400
200
Steel temperature
0
0
50
100
150
200
250
Time, min.
(b) Fire scenarios and steel temperature
0
Fire exposure time, min.
50
100
150
200
250
Midspan deflection, mm
0
-100
-200
-300
-400
-500
-600
Deflection limit
state "L F "
Strength limit
state
Standard fire (F.E.A.)
Standard fire (Proposed app.)
Design fire (F.E.A.)
Design fire (Proposed app.)
(c) Midspan deflection
Fig. 8: Steel temperature and midspan deflection as a function of fire exposure time for restrained beam
Design for design fire exposure
If the beam is exposed to a design fire scenario shown in Fig. 8(b), then the maximum
temperature attained in steel (Ts,max) should not exceed the limiting temperature based on
deflection limit state (TDLS = 605°C). Using Equations [3] and [4], the required insulation
thickness at which Ts,max ≈ TDLS is 15 mm, for which the maximum steel temperature under fire
is Ts,max = 597°C < TDLS = 605°C (at 90 min. of fire exposure). The average steel temperature as a
function of exposure time is plotted in Fig. 8(b) with the applied insulation thickness of 15 mm.
24
7. DESIGN IMPLICATIONS
The current approaches for evaluating fire resistance through standard fire tests on full-scale
steel members are expensive, time consuming and have a number of drawbacks. An alternative is
to use calculation methods for predicting fire resistance. However, such calculation methods are
not widely available at present. Further, the current fire resistance provisions in codes and
standards (ASCE 1992, EC3 2005) are prescriptive and do not account for realistic conditions
such as end restraint, loading and fire scenarios. Thus, the current design approaches may not be
fully applicable for undertaking performance-based design which provides rational and costeffective fire safety solutions.
The proposed design approach provides a convenient way of obtaining fire response and fire
resistance of restrained steel beams, and thus can be used for estimating fire resistance in lieu of
full-scale standard fire resistance tests. The proposed design approach can be applied to evaluate
fire resistance under realistic fire, loading and restraint scenarios. The proposed equations
express fire resistance in terms of structural parameters, and thus the approach is attractive for
incorporation in codes and standards. In summary, the use of the proposed approach will
facilitate a rational fire design under a performance-based code environment. Such a rational
design approach will contribute to reduced loss of life and property damage in fire incidents.
8. SUMMARY AND CONCLUSIONS
A set of simple, efficient, and sufficiently accurate guidelines have been developed for fire
design of steel beam-columns and restrained beams. The proposed guidelines provide attractive
alternative to complex finite element simulations. The approach is based on engineering
principles and utilizes engineering parameters that can be optimized for fire design under various
load, restraint and fire conditions. Also, the proposed approach accounts for realistic fire,
loading, restraint conditions and different failure limit states and thus can be applied for
undertaking fire design under performance based code environment.
9. REFERENCES

AISC. Steel Construction Manual. 13th Ed. (2005), American Institute of Steel Construction, Inc., Chicago, IL.

ANSYS. (2007). ANSYS Multiphysics, Version 11.0 SP1 ANSYS Inc., Canonsburg, PA. USA.
25

ASCE, (1992) “Structural Fire Protection”, manual No.78,, ASCE committee on fire protection, structural
division, American Society of Civil Engineers, New York.

ASCE-07 (2005), "Minimum Design Loads for Buildings and Other Structures", American Society of Civil
Engineers, Reston, VA.

ASTM E119a (2008), "Standard Methods of Fire Test of Building Construction and Materials", American
Society for Testing and Materials, West Conshohocken, PA.

BS 476-3:1987. Fire Tests on Building Materials and Structures – Part 20: Method for determination of the fire
resistance of elements of construction (General Principles). London, UK.

Dwaikat, M.M.S and Kodur, V.K.R. (2010a) “A Simplified Approach for Predicting Temperatures in Fire
Exposed Steel Members”, Submitted, Fire Safety Journal.

Dwaikat, M.M.S and Kodur, V.K.R. (2010b) “A Simplified Approach for Evaluating Plastic Axial and Moment
Capacity Curves for Beam-Columns with Non-uniform Thermal Gradients”, Engineering Structures, 32(5),
pp.1423-1436

Dwaikat, M.M.S., (2010c) Response of restrained steel beam subjected to fire induced thermal gradients, PhD
Thesis, Michigan State University, East Lansing, MI, US.

Dwaikat M.M.S., Kodur V.K.R, (2010d) "Effect of location of restraint on fire response of steel beams,",
Journal of Fire Technology, 46(1), pp. 109-128.

Dwaikat, M.M.S. and Kodur, V.K.R., (2010e) " Performance-based Methodology for Tracing the Response of
Restrained Steel Beams Exposed to Fire”, COST Action C26 International Conference: "Urban Habitat
Constructions under Catastrophic Events", pp. 1-12, Naples, Italy.
Dwaikat, M.M.S. and Kodur, V.K.R. (2010f), "Lateral-torsional buckling of steel beam-columns under fire
exposure", SSRC (AISC) Annual Stability Conference, Orlando, FL.
Dwaikat, M.M.S and Kodur, V.K.R. (2011a) “Strength Design Criteria for Steel Beam-Columns with Fire
Induced Thermal Gradients”, In Press, Engineering Journal, AISC.



Dwaikat, M.M.S and Kodur, V.K.R. (2011b) “An engineering approach for predicting fire response of
restrained steel beams.”, ASCE Journal of Engineering Mechanics, 37(7), pp. 447-461, 2011.

Dwaikat, M.M.S and Kodur, V.K.R. (2011c) “A performance-based methodology for fire design of restrained
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