Subido por Juan Sotomayor


MS No. S-2012-404.R1
Experimental Evaluation of Strut-and-Tie Model of
Indeterminate Deep Beam
by David B. Garber, José M. Gallardo, Guillermo D. Huaco, Vasileios A. Samaras, and
John E. Breen
An experimental study was performed to determine the accuracy of
designing a statically indeterminate deep beam with three openings
using the strut-and-tie modeling (STM) method in accordance with
Appendix A of ACI 318-08. In the present study, four STM models
were independently developed to closely match the flow of forces
according to a finite element analysis. Four specimens were fabricated based on the associated STM, and confining reinforcement
was provided at each load and support point. In all four specimens,
the failure load exceeded the factored nominal design strength,
demonstrating the conservatism of the STM method. The mode of
failure in each specimen was dependent on the stress concentrations revealed in the elastic analysis and the STM chosen for the
Keywords: indeterminate system; node confinement; strut-and-tie
A strut-and-tie model (STM) is a lower-bound plasticity
solution that idealizes a complex flow of stresses in reinforced concrete members as a collection of compression
elements (struts), tension elements (ties), and the intersection of such elements (nodes). While the use of STM is
applicable to all design scenarios for any type of reinforced
concrete member, this method is more widely used in deep
beams, complex structural members, and disturbed regions
where the Bernoulli hypothesis that plane sections remain
plane does not apply (Schlaich et al. 1987; Bergmeister et
al. 1993).
The test specimens examined in this study incorporated a
combination of point loads and large openings—requiring
the use of STM or nonlinear finite element analysis. The
pattern, location, and size of the openings and load point
locations were chosen to investigate the conservativeness of
STM when applied to structures with extreme discontinuities.
The support conditions applied were chosen to investigate
the conservativeness of STM when applied to indeterminate structures. Schlaich et al. (1987) suggests that plastic
methods of analysis, such as STM, are suitable for determining a realistic ultimate load capacity. The researchers
suggested that a safe solution for the ultimate load can also
be obtained by a linear or nonlinear analysis. In this investigation, a two-dimensional finite element model (FEM) was
used as guidance for the STM used for design and detailing
of the structure.
The specimen size was approximately 1:10.5 scale. A
similar example was given by Reineck (2002). While size
does have an effect on the shear behavior of unreinforced
ACI Structural Journal
specimens, it has been suggested that in reinforced concrete
specimens, “a more gentle size effect may be expected”
(Shah et al. 1995). With respect to STM, Muttoni et al.
(1997) observed that “a load system based on a statically
admissible stress field which nowhere violates the yield
criterion is a lower-bound to the collapse load.” This should
hold true regardless of the size of the specimen. Ley et al.
(2007) conducted testing to investigate the size effect on reinforced concrete structures (similar in size and type to those
investigated in this study) detailed using STM. Scaled specimens were designed using STM, detailed, and constructed
at 1:10.5 scale and 1:6 scale. Both specimens with reinforcement and without reinforcement were constructed and tested
at both scales. The researchers found that the scale had no
effect on the performance of the reinforced specimens, but
had a significant impact on the unreinforced specimens. For
these reasons, the findings from this small-scale investigation are valid to full-scale applications.
The objective of this project was to design and verify, by
laboratory testing, four different indeterminate small-scale
concrete members using the STM provisions of ACI 318-08
(ACI Committee 318 2008). An FEM was constructed to
examine the flow of forces through the structure. Based on
the elastic stresses, different STM models were developed
to determine the required reinforcement for each specimen.
In the recent past, a significant amount of experimental
research has been conducted, examining the applicability
and conservatism of STM. Despite this recent research, there
has been little experimental validation of STM for statically
indeterminate structures. The research conducted in this
study investigates the applicability of the provisions of ACI
318-08 for indeterminate structures.
Material properties
For each of the specimens, four 80 lb (36.3 kg) sacks of
commercially available, pre-mixed concrete (specified 4 ksi
[27.6 MPa]) were mixed with 8 fl oz (227 mL) of high-range
ACI Structural Journal, V. 111, No. 1-6, January-December 2014.
MS No. S-2012-404.R1 received June 10, 2013, and reviewed under Institute
publication policies. Copyright © 2014, American Concrete Institute. All rights
reserved, including the making of copies unless permission is obtained from the
copyright proprietors. Pertinent discussion including author’s closure, if any, will be
published ten months from this journal’s date if the discussion is received within four
months of the paper’s print publication.
Table 1—Reinforcement properties summary
Reinforcement type
Bar area, in.2
Yield strength, ksi
1/8 in.-f Swedish bar
0.0143 (9.23)
70 (483)
5/32 in.- f Swedish bar
0.0190 (12.3)
82 (565)
7/32 in.- f Swedish bar
0.0450 (29.0)
75 (517)
14-gauge welded wire mesh
0.0050 (3.23)
46 (317)
Fig. 1—Test specimen.
water-reducing admixture and approximately 25 lb (11.3
kg) of water. The concrete was mixed in a 1 yd3 (0.76 m3)
electric mixing drum for the prescribed 10-minute length of
time or until a consistent mixture was achieved. Three 4 x
8 in. (101.6 x 203.2 mm) cylinders were cast with each of
the specimens to know the concrete strength at the time of
Three different types of steel reinforcing bars and one
type of welded wire mesh were used to construct the specimens. The areas and yield strengths of the reinforcement are
summarized in Table 1.
Specimen geometry
The geometry of the specimens is shown in Fig. 1. As
stated previously, the geometry of the specimens was chosen
to investigate the conservatism of STM when applied to:
1) structures with extreme discontinuities; and 2) indeterminate structures. Each of the specimens was 26 in. (660.4
mm) tall, 48 in. (1219.2 mm) wide, and 2.75 in. (69.9 mm)
thick. Moreover, each specimen had three 6 x 6 in. (152.4
x 152.4 mm) square openings. As shown in Fig. 1, two of
the square openings were located on the lower portion of
the specimen, whereas the third opening was located on the
top left corner of the specimens. The specimens were seated
on three supports and loaded with three concentrated point
loads on the top.
Design process
For the purpose of this study, each of the specimens was
detailed such that the controlling element in each model
would hold a factored design load of 48.0 kip (214 kN) (16.0
kip [71.2 kN] applied through each of the three hydraulic
rams). There were two aspects of the design that proved to be
challenging. The first challenge was the static indeterminacy
of the system. In statically indeterminate structures, when
failure occurs in one location, forces and moments can be
redistributed to other parts of the structure. With the knowledge that the forces and moments will be redistributed, the
designer has an additional level of freedom when choosing
the STM and reinforcement layout. The second challenge
of the prescribed structure was the location of the openings, specifically the central opening. This central opening
is located directly over the most highly loaded support,
creating a large stress concentration, which can be seen in
Fig. 2.
With the purpose of defining an appropriate design for the
concrete members, four STMs (A, B, C, and D) were developed by four independent groups. The first step in the development of the STM was to base the geometry of the model
on the elastic stress fields indicated by an FEM analysis, as
shown in Fig. 2. A two-dimensional planar FEM was developed using quadratic shell elements. This simple elastic
analysis gave a general idea of the stress flows through the
structure. From these results, a large concentration of both
tensile and compressive stresses was found in a relatively
small area between the center opening and the bearing pad.
It was assumed that the center support, located under the
middle opening, would be the most highly stressed region,
and would therefore govern design.
Fig. 2—Finite element model showing: (a) high tensile stresses; and (b) high compressive stresses in specimen.
ACI Structural Journal
Fig. 3—Strut-and-tie model for Specimens: (a) A; (b) B; (c) C; and (d) D.
Although the stress field from the elastic FEM does not
represent the state of stress in the specimen at the time
of failure, it does give the designer a sense of the state of
stress in the uncracked specimen. With this knowledge, the
basic goal of STM is to place ties in the location of high
tensile stresses (Fig. 2(a)) and struts in the location of high
compressive stresses (Fig. 2(b)). The structure will be more
efficient and perform better under service loads the more
closely the ties coincide with high tensile regions and struts
with high compression regions. This is a result of the structure carrying the force as it would naturally like (Schlaich et
al. 1987; Bergmeister et al. 1993).
The STMs developed for all four specimens are shown
in Fig. 3. All of the models were constructed based on the
elastic flow of stresses. The elements were laid out to ensure
that all struts and ties converged into nodes in a way that
equilibrium could be obtained and to ensure that the overall
idealized truss configuration was stable. An additional
consideration was the angle between struts and ties, which is
required by ACI 318-08 to be larger than 25 degrees.
The tensile stresses along the bottom of the specimens and
above the lower two openings were handled with similarly
located ties in all of the models. There are also two highly
loaded struts in all of the models located in the regions of
high compression stress, which deliver the load into the
central support.
Besides these similarities, the models are fairly unique
from one another. Ties were primarily kept orthogonal in
three of the models (A, B, and D) to make construction
easier. In the other model (C), struts and ties were laid out to
closely follow the stresses observed in the FEM. It has been
shown in previous research (Maxwell and Breen 2000) that
the highest efficiency ratio (or load capacity-to-weight of
steel ratio) was obtained by nonorthogonal models; this led
to the selection of the truss proposed in Model C. The other
ACI Structural Journal
models were chosen to closer represent field specimens,
which are almost exclusively constructed with orthogonal
Reinforcement layout
The nominal strength of all the struts and nodes in the
models were checked, and ties were designed according
to the provisions of ACI 318-08, Appendix A. The design
forces were obtained from an elastic truss analysis of the
STMs. All struts in the models were designed as bottleshaped struts, with the exception of those clearly bound by
the edge of the structure. These bottle-shaped struts were
provided with adequate reinforcement to account for the
tension that resulted from the spreading of the load.
Most of the struts, in all of the models, satisfied the nominal
strength requirements without any compression reinforcement or confining steel. The two diagonal struts funneling
the load to the center support had limited available widths
due to the center opening and due to their proximity to each
other. This limited width required the use of confining reinforcement spirals in two of the specimens (C and D), and
compression steel in the other two specimens (A and B).
The addition of the compression steel and confining reinforcement allowed the compressive struts to behave similarly to reinforced columns in compression. Due to the large
compressive stresses present at the supports, confinement
was provided at all of the supports in all of the specimens.
The layout of the confining reinforcement and compression
steel is shown in the reinforcement layouts in Fig. 4. Spirals
of 2-1/4 in. (57.2 mm) diameter and 5/8 in. (15.9 mm) pitch
were constructed out of 70 ksi (482.6 MPa) deformed wire,
1/8 in. (3.4 mm) in diameter, to provide confinement at all
supports and in the cages using confinement reinforcement
at the two struts above the central support. The compression
Fig. 4—Reinforcement layout for Specimens: (a) A; (b) B; (c) C; and (d) D.
steel in Cages A and B was 82 ksi (565.4 MPa) deformed
wire, 5/32 in. (4 mm) in diameter.
When laying out reinforcement for an STM, detailing is
of utmost importance. Reinforcement is placed in the location of the tension ties in the model so that the centroid of
the reinforcement coincides with the centroid of the tensile
tie (Schlaich et al. 1987). This reinforcement should be able
to resist the tensile force in the ties and should have sufficient development lengths. The details of the reinforcement
provided to resist the forces in the tension ties are shown in
Fig. 4. Both 82 ksi (565.4 MPa) deformed wire, 5/32 in. (4
mm) in diameter, and 75 ksi (517.1 MPa) deformed wire,
7/32 in. (6 mm) in diameter, were used in construction of
the cages.
Three possible configurations were considered for
the development of the reinforcement: straight, bent (90
degrees), and hooked (180 degrees) bars. Because of the
limited size of the specimens, it was impossible to achieve
the proper development length for many of the bars without
90-degree bends or hooks. To avoid detailing issues, most of
the bars in the designs were made continuous with splices
located in noncritical regions. Laying out the reinforcement
in this fashion increased the weight of the steel, but ensured
that anchorage would not govern the capacity of the test
Two layers of 14-gauge welded wire mesh were provided
in the specimens in accordance with ACI 318-08, Appendix
A.3.3.1, to allow for the stress spreading of bottle-shaped
struts. Additionally, this mesh prevented the widening of any
shrinkage cracking that may have occurred in the specimens.
Fig. 5—Test setup and location of instrumentation.
Testing apparatus
The test setup, shown in Fig. 5, was comprised of three
loading rams and three supports. All bearing areas consisted
of steel plates (3 x 3 x 1 in. thick [76.2 x 76.2 x 25.4 mm
thick]) and neoprene bearing pads (3 x 3 x 1 in. thick [76.2 x
76.2 x 24.5 mm thick]). The load points were located along
the top, spaced at 10.5 in. (266.7 mm) apart and centered on
the specimen. Similarly, the supports were centered under
the specimen and spaced at 21 in. (533.4 mm) on center. A
strap was secured around the top of the specimen to help
with stability during testing.
Three linear potentiometers were used to measure deflection at the bottom face of the specimen. Two were placed at
the midpoints of each clear-span, while the third was located
directly above the center support to measure the support
settlement of the bearing pad. The measurement of support
settlement allowed for the isolation of the behavior of the
ACI Structural Journal
Table 2—Summary of test results
Reinforcing bar cage
weight, lb (kg)
Concrete strength,
ksi (MPa)
Factored design
load, kip (kN)
Cracking load,
kip (kN)
Failure load,
kip (kN)
Efficiency ratio,
15.5 (7.03)
4.0 (27.6)
48.0 (214)
30.0 (133)
68.6 (305)
11.6 (5.26)
4.0 (27.6)
48.0 (214)
20.0 (89.0)
84.4 (375)
9.50 (4.31)
4.0 (27.6)
48.0 (214)
20.0 (89.0)
77.8 (346)
13.2 (5.99)
4.0 (27.6)
48.0 (214)
20.0 (89.0)
Fig. 6—Cracking pattern for Specimens: (a) A; (b) B; (c) C; and (d) D.
member; this was done by subtracting the support settlement from the midpoint deflection measurements. The load
was applied evenly by the three hydraulic rams, and was
measured by a pressure transducer. Load was applied in
5 kip (22.2 kN) intervals until the maximum load. At the end
of each load step, cracks were traced and measured, and a
photograph was taken.
Performance of specimens
A summary of the test results is presented in Table 2.
Each of the specimens tested was detailed such that the
controlling element in each model would hold a load of 48.0
kip (214 kN) (16.0 kip [71.2 kN] applied through each of
the three hydraulic rams), which was the factored design
load chosen for this study. All of the measured failure loads
were significantly higher than this factored design load,
reflecting the conservative, lower-bound nature of STM. It
should also be noted that all the specimens exhibited satisfactory serviceability performance, as none showed signs of
cracking at loads smaller than 20.0 kip (89.0 kN).
The cracking pattern at the time of failure and the failure
crack are shown in Fig. 6. In each of the specimens, the
ACI Structural Journal
load should be transferred from each of the loading points
to each of the supports via some load path; that is, a combination of struts, ties, and nodes. In a statically determinate
system, global failure of the specimen will occur with a local
failure in one of the load paths. In a statically indeterminate
system, such as the one investigated in this study, a local
failure in one load path will cause load redistribution but not
a global failure; two local failures are required for a global
failure mechanism to form. In all of the specimens, one of
these local failures was located immediately over the center
support. This failure location was expected due to the large
concentration of stresses in this location. Steps (the use of
confining and compressive steel) were attempted to delay
this failure, but the stress concentration in this central region
was extremely high, and failure could not be prevented.
In three of the four specimens (Specimens A, B, and C),
the second local failure location was on the region that transferred loads (mainly the left load) into the left support of the
structure, which involved the area around the uppermost of
the openings. Specimen D was the only specimen where the
second failure location was on the right side of the structure.
From the reinforcement layout, Specimen D was the only
layout with continuous reinforcement in the upper half of the
right span of the structure. Specimen D is the only specimen
Fig. 7—Load-deflection plots for all specimens.
with a significant amount of reinforcement crossing the location of the local failure crack on the left side of the structure
in the other three specimens. The potential failure crack may
have been prevented by the use of this additional reinforcement. A combination of these two features resulted in failure
in the right span rather than the left span.
Three of the four specimens had similar ultimate capacities (Specimens B, C, and D) of approximately 80.0 kips
(355 kN), while Specimen A had an ultimate capacity of
only 68.6 kips (305 kN). The main behavioral difference
between the specimens is the order in the occurrence of the
local failures, which led to global failure of the specimens.
In Specimens B, C, and D, the two local failures required to
form a global failure mechanism occurred nearly simultaneously; this would suggest that the two failure load paths
were similarly loaded compared with their corresponding
capacities; that is, the specimen did not have one significantly under-designed section. In these specimens, the left
side (or right side for Specimen D) of the beam held together
longer than the center support region. This is due to these
specimens having a larger amount of steel crossing the
local failure crack and the larger tensile stress in the central
region. When failure of the center support occurred, the
specimens could no longer hold load, and the second local
failure occurred immediately. In Specimen A, the first local
failure occurred over the uppermost opening followed by the
region over the center support, and then the remainder of the
left side failed. The failure in Specimen A suggests that the
uppermost opening was under-reinforced compared with the
rest of the specimen.
The total load versus the average of the two midspan
deflections is shown in Fig. 7. It can be seen that three specimens (Specimens B, C, and D) had similar ultimate loads
as well as deflections at ultimate load. Specimen A had a
slightly larger observed ductility, which can be attributed
to the slightly different failure mechanism, previously
Confinement of Nodes (Specimen D)
Previous research has shown that confinement can be used
in nodal regions to allow for the safe development of high
compressive stresses (Bergmeister et al. 1993). Confinement
of the node is typically achieved through the use of steel reinforcement. In structures where the bearing pad area is smaller
than the concrete specimen being loaded, the surrounding
concrete can act as confinement to the node (Tuchscherer
et al. 2010). In the specimens examined in this study, the
bearing width (3 in. [76.2 mm]) was slightly greater than
specimen width (2.75 in. [69.9 mm]). For this support configuration, there is no surrounding concrete to help confine the
support node; this means that the only confinement of the
node comes from any provided reinforcement.
To investigate the effect of confinement reinforcement on
the node capacity, two tests were conducted on specimens
with the reinforcement layout of Specimen D, with one specimen having confining reinforcement at all the nodes (Fig.
8(c)), and one specimen only have confining reinforcement
over the bottom-center and bottom-left supports (Fig. 8(a)).
Similar testing was conducted on the other three specimens,
and produced similar results to those presented for Specimen D. The failure cracks and spalling locations are shown
alongside the reinforcement layouts for these two specimens
in Fig. 8.
The main difference between specimens is the difference
in ultimate capacities: 44.0 kips (196 kN) for the specimen
without confining reinforcement, and 80.4 kips (358 kN) in
the specimen with fully confined load and support points. By
comparing the FEM in Fig. 2(b) to the location of spalling
in Fig. 8(b), it can be seen that spalling occurred at the
unconfined regions of high compressive stress concentrations. When these regions were fully confined, the failure
mechanism was changed, and the ultimate capacity greatly
increased. The importance of proper detailing and confinement of compressive struts in regions of high stress concentrations is emphasized by this comparison.
Optimization versus constructibility
Previous research has suggested that closely following the
elastic stress flow will help minimize the total amount of
reinforcement required and lead to a more efficient design
(Schlaich et al. 1987). Closely following the stress field can
lead to a significant amount of nonorthogonal reinforcement,
which may make a specimen difficult to fabricate (Bergmeister et al. 1993).
As suggested by Schlaich et al. (1987), the specimen
with the highest efficiency ratio (ultimate load-to-specimen
weight ratio) was designed with tension ties in high tensile
stress regions, shown in Fig. 9(a), and compressive struts
located in high compressive regions, shown in Fig. 9(b). To
complete the model, the struts and ties obtained from the
FEM were modified to intersect at nodes in a way that equilibrium could be obtained. This process resulted with an
STM that closely represented the natural stress flow through
the structure.
The reinforcement layout necessary for this model required
the use of nonorthogonal tie reinforcement. Nonorthogonal
reinforcement is difficult to construct due to complicated
bend locations and bend angles. In practice, STMs are
developed to keep all reinforcement orthogonal to ease the
construction and inspection processes.
ACI Structural Journal
Fig. 8—Reinforcement configuration and cracking pattern at failure for Specimen D: (a) and (b) with confining reinforcement
at two supports; and (c) and (d) with confining reinforcement at all supports and loading locations.
Fig. 9—Optimized layout of: (a) ties; and (b) struts based on finite element modeling.
The purpose of this paper was to: 1) investigate the conservatism of STM when applied to a structure with extreme
discontinuities; 2) investigate the conservatism of STM
applied to statically indeterminate structures; and 3) demonstrate the applicability of STM with widely different reinforcement details.
Through the experimental research conducted, several
observations and conclusions made based on the investigation of statically determinate, simple structures (Schlaich et
al. 1987; Bergmeister et al. 1993; Reineck 2002) can be reaffirmed for statically indeterminate structures with extreme
1. Strut-and-tie modeling is a conservative lower-bound
theory of plasticity. The specimens all held at least 1.43
times the factored design load. This conservatism shows that
strut-and-tie modeling still offers a conservative design solution in statically indeterminate structures;
2. Load path will vary based on provided reinforcement.
The local failure locations within the specimens depended
ACI Structural Journal
on the planned load path in the strut-and-tie model and the
subsequent reinforcement layout. The model in Specimen D
carried more load through the right portion of the specimen,
and experienced a local failure in this location;
3. Properly detailed and confined struts in high compressive stress regions help prevent local fragile failures. Locations of high compressive stress concentrations should be
properly confined to prevent strut crushing. The yielding of
tension ties is a more ductile failure mechanism; and
4. Models closely following elastic stress distribution
will offer more efficient designs. Specimen C, which was
constructed based on a model that most closely followed
the elastic stress distribution, offered the most efficient reinforcement layout.
ACI member David B. Garber is a PhD Candidate at The University of
Texas at Austin, Austin, TX. He received his BS from Johns Hopkins University, Baltimore, MD, and his MS from The University of Texas at Austin. His
research interests include plasticity in structural concrete and behavior of
prestressed concrete members.
ACI member José M. Gallardo is a PhD Candidate at The University of
Texas at Austin. He received his BS and MS from the Technological University of Panama. His research interests include time-dependent behavior of
concrete and monitoring of prestressed concrete members.
ACI member Guillermo D. Huaco is a PhD Candidate at The University of
Texas at Austin, where he completed his MS degree. He obtained his BS from
the National University of Engineering, Lima, Peru. He is a member of ACI
Committee 369, Seismic Repair and Rehabilitation.
Vasileios A. Samaras is a PhD Candidate at the University of Texas at
Austin. He received his BS from the University of Thessaly at Volos, Greece,
and his MS from The University of Texas at Austin. His research interests include structural health monitoring of structures, bridges, and steel
ACI Honorary Member John E. Breen holds the Nasser I. Al-Rashid Chair
Emeritus in Civil Engineering at The University of Texas at Austin.
The authors would like to thank B. Stasney, A. Valentine, D. Fillip, and
A. Avendaño for their assistance during the construction and testing of the
specimens. The authors would also like to thank the members of the design
teams for their contributions. Those members include: A. Abu Yousef, B.
Bowden, J. Clayshulte, A. Ghiami, M. Homer, J. Kim, K. Kreitman, N.
Larson, A. Moore, E. Nakamura, E. Reynolds, D. Santino, N. Satrom, A.
Wahr, and C. Williams.
ACI Committee 318, 2008, “Building Code Requirements for Structural
Concrete (ACI 318-08) and Commentary,” American Concrete Institute,
Farmington Hills, MI, 473 pp.
Bergmeister, K.; Breen, J. E.; Jirsa, J. O.; and Kreger, M. E., 1993,
“Detailing for Structural Concrete,” Research Report 1127-3F, Center for
Transportation Research, The University of Texas at Austin, Austin, TX,
300 pp.
Ley, M. T., and Riding, K. A., Widianto; Bae, S.; and Breen, J. E., 2007,
“Experimental Verification of Strut-and-Tie Design Method,” ACI Structural Journal, V. 104, No. 6, Nov.-Dec., pp. 749-755.
Maxwell, B. S., and Breen, J. E., 2000, “Experimental Evaluation of
Strut-and-Tie Model Applied to Deep Beam with Opening,” ACI Structural
Journal, V. 97, No. 1, Jan.-Feb., pp. 142-148.
Muttoni, A.; Schwartz, J.; and Thurlimann, B., 1997, Design of Concrete
Structures with Stress Fields, Birkhauser Verlag, Switzerland, 147 pp.
Reineck, K. H., ed., 2002, Examples for the Design of Structural
Concrete with Strut-and-Tie Models, SP-208, American Concrete Institute,
Farmington Hills, MI, 242 pp.
Schlaich, J.; Schafer, K.; and Jennewein, M., 1987, “Toward a Consistent Design of Structural Concrete,” PCI Journal, V. 32, No. 3, May-June,
pp. 74-150.
Shah, S. P.; Swartz, S. E.; and Ouyang, C., 1995, Fracture Mechanics of
Concrete: Applications of Fracture Mechanics to Concrete, Rock and Other
Quasi-Brittle Materials, John Wiley, New York, 591 pp.
Tuchscherer, R.; Birrcher, D.; Huizinga, M.; and Bayrak, O., 2010,
“Confinement of Deep Beam Nodal Regions,” ACI Structural Journal,
V. 107, No. 6, Nov.-Dec., pp. 709-717.
ACI Structural Journal