Numerical Investigation of the Shear Strength of RC Deep

Beams Using the Microplane Model

Kamaran S. Ismail 1; Maurizio Guadagnini 2; and Kypros Pilakoutas 3
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Abstract: Although much work has been done on the shear behavior of RC elements, current design provisions are still based on empirical
data and their predictions, especially for deep beams, are not always reliable and can lead to unconservative results. This paper presents an
extensive numerical investigation on the role of key parameters on the shear performance of RC deep beams using the microplane M4 material
model. The model is validated against experimental results of 20 RC deep beams. A parametric study is then carried out to investigate the
effect of shear span to depth ratio and concrete compressive strength for RC deep beams with and without shear reinforcement. Although a
single strut mechanism is generally mobilized in deep beams, the presence of shear reinforcement can enable a more uniform distribution of
shear stresses within the shear span and enhance the effectiveness of concrete cracked in tension. The study confirms that both shear span to
depth ratio and concrete strength are the key parameters that affect the shear capacity of RC deep beams and should be taken into account in
code equations. DOI: 10.1061/(ASCE)ST.1943-541X.0001552. © 2016 American Society of Civil Engineers.
Author keywords: Deep beams; Finite-element analysis; Microplane model M4; Effectiveness factor of concrete strut; Code comparison;
Concrete and masonry structures.

Introduction this effect. EC2 recommends the use of an effectiveness factor that
can only account for the effect of concrete compressive strength,
Deep beams are structural members characterized by shear span while ACI 318-14 only accounts for the effect of shear reinforce-
to depth ratio smaller than 2 [Eurocode 2 (EC2) (British Standards ment. Although the STM concept as a lower-bound plasticity
Institution 2004), ACI 318-14 (2014)], and their analysis and approach should yield conservative results (Kuchma et al. 2008;
design cannot be carried out according to conventional bending Arabzadeh et al. 2009; Sagaseta and Vollum 2010; Tuchscherer
theory (Wight and MacGregor 2009; Mohammad et al. 2011). et al. 2014), there is evidence that its current implementation in
Generally, such members appear as transfer girders in tall building, the codes can lead to unsafe predictions (Brown and Bayrak
bridges, and offshore structures. Shear action is critical in these 2008a; Collins et al. 2008; Sagaseta and Vollum 2010). Therefore,
members and, if underestimated, could lead to catastrophic failure the code provisions need to be reassessed and improved to more
without warning. Thus, a reliable method to determine their struc- accurately account for the effect of all relevant parameters.
tural performance is needed. Experiments can provide key information on the behavior of RC
Codes of practice, such as EC2 and ACI 318-14, provide design deep beams, but such tests can be expensive, time-consuming, and
procedures, such as the strut-and-tie model (STM), to predict the sometimes impractical due to the limiting capabilities of structural
ultimate capacity of RC deep beams. The performance of STM laboratories, especially when dealing with large elements. Finite-
relies on (1) selecting appropriate strut-and-tie layout and sizes element analysis can provide a valid alternative to laboratory test-
of each element, and (2) estimating the maximum allowable stress ing, but its accuracy depends on the accuracy of the constitutive
in each element. The strength of the inclined strut is lower than the models implemented. Generally, the available material models
uniaxial concrete compressive strength due to the existence of can be classified into two categories: (1) macroscopic models, ac-
lateral tensile strain. Therefore, a reduction factor, known as the cording to which the material behavior is simplified from a com-
effectiveness factor, is used in the design process to account for plex microstructural stress transfer mechanism to a relationship
between average stress and strain at the continuum level; and
1 (2) microscopic models, which describe the material behavior as
Ph.D. Student, Dept. of Civil and Structural Engineering, Univ. of
a stress-strain relationship at the micro level. Although the latter
Sheffield, Sir Frederick Mappin Building, Mappin St., Sheffield S1 3JD,
U.K.; Lecturer, Dept. of Civil Engineering, Salahaddin Univ.—Erbil, is considered to be more accurate and can capture the microscopic
Kirkuk Rd., Erbil, Iraq (corresponding author). E-mail: ksismail1@ material behavior, such as cohesion, aggregate interlock, and
sheffield.ac.uk; ksi312ismail@gmail.com friction (Ožbolt et al. 2001), this approach has two main draw-
2 backs: (1) cracks are forced to follow a predefined path along
Senior Lecturer, Dept. of Civil and Structural Engineering, Univ. of
Sheffield, Sir Frederick Mappin Building, Mappin St., Sheffield S1 3JD, element edges, and (2) it needs remeshing throughout the solution
U.K. process due to change in nodal connectivity because of the crack
3
Professor of Construction Innovation and Director of the Centre for development. From a practical point of view, the implementation of
Cement and Concrete, Dept. of Civil and Structural Engineering, Univ. microscopic models is computationally extremely expensive;
of Sheffield, Sir Frederick Mappin Building, Mappin St., Sheffield S1
hence, macroscopic models are more widely used.
3JD, U.K.
Note. This manuscript was submitted on October 2, 2015; approved on
In numerical analysis, modeling of concrete and other quasi-
February 25, 2016; published online on April 29, 2016. Discussion period brittle materials has always been a challenging issue because of
open until September 29, 2016; separate discussions must be submitted for the complexity of their behavior, and different approaches have
individual papers. This paper is part of the Journal of Structural Engineer- been proposed based on the plasticity theory, the plastic-fracturing
ing, © ASCE, ISSN 0733-9445. theory, continuum damage mechanics, or their combinations

© ASCE 04016077-1 J. Struct. Eng.

J. Struct. Eng., 2016, 142(10): 04016077

 Table 1. Summary of Beams Used for Model Validation
fc Failure
Researcher Identifier L (mm) h (mm) b (mm) a (mm) (MPa) As ðmm2 Þ ρv (%) ρh (%) load (kN) V exp =V FEM Δexp =ΔFEM
Current work E1 1,400 400 100 425 58 1,206 0 0 415 1.01 1.26
G1 1,400 400 100 550 31 1,206 0.56 0.22 292 0.90 1.02
G2 1,400 400 100 425 33 1,206 0.59 0.22 372 1.00 1.00
G3 1,400 400 100 300 31 1,206 0.67 0.22 489 1.06 1.07
Foster and Gilbert (1998) B2.0-1 1,900 700 125 825 83 1,880 0.6 0.33 1,590 1.05 0.80
B2.0-2 1,900 700 125 825 120 1,880 0.6 0.33 1,650 0.95 0.85
B2.0-3 1,900 700 125 825 78 1,880 0.6 0.33 1,400 1.06 0.95
B2.0A-4 1,900 700 125 675 86 1,880 0.6 0.33 1,900 1.11 0.82
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B2.0B-5 1,900 700 125 825 89 1,880 0 0 1,170 1.06 1.01

B2.0C-6 1,900 700 125 825 93 1,880 0.9 0 1,460 0.99 0.83
B2.0D-7 1,900 700 125 825 104 1,880 0.6 0 1,440 0.95 1.02
B3.0-1 2,600 700 125 1,175 80 1,880 0.6 0.33 1,020 0.92 1.07
B3.0-2 2,600 700 125 1,175 120 1,880 0.6 0.33 1,050 0.92 1.06
B3.0-3 2,600 700 125 1,175 77 1,880 0.6 0.33 1,050 0.89 0.97
B3.0A-4 2,600 700 125 925 88 1,880 0.6 0.33 1,550 1.04 1.08
B3.0B-5 2,600 700 125 1,175 89 1,880 0 0 870 0.92 1.02
Aguilar et al. (2002) ACI-I 4,470 915 305 915 33 2,940 0.31 0.46 2,713 1.07 1.56
STM-I 4,470 915 305 915 33 2,940 0.31 0.15 2,268 1.05 1.04
STM-H 4,470 915 305 915 28 2,940 0.31 0.15 2,571 1.02 1.66
STM-M 4,470 915 305 915 28 2,940 0.15 0 2,553 0.97 1.46
Average — — — — — — — — — — 1.00 1.08
Coefficient of variation — — — — — — — — — — 0.064 0.214

(Bazant et al. 2000). In these models, the constitutive relationships Numerical Analysis
are written in terms of tensors on one or two loading surfaces; how-
ever, in reality many simultaneous loading surfaces intersect at Model Description
every point (Bazant et al. 2000). Hence, these models have limited
success in predicting realistically the behavior of concrete (Ožbolt A total of 20 RC deep beams [four specimens tested as part of this
et al. 2001). The more sophisticated microplane model (Bazant et al. research program and 16 from literature (Foster and Gilbert 1998;
2000; Caner and Bazant 2000) has been shown to capture the mi- Aguilar et al. 2002)] are analyzed. The beam details and their fail-
croscopic behavior of concrete in a more reliable manner and it has ure load are summarized in Table 1 and the geometry of the beams
been successfully implemented in finite-element analysis to simu- is shown in Fig. 1. Because failure is expected to develop in the
late the nonlinear behavior of concrete and capture shear behavior shear span, taking advantage of symmetry in geometry and loading
conditions, only half-specimen are modeled (Fig. 1) to reduce overall
of RC elements (Caner and Bazant 2000; Ožbolt and Li 2001;
computational time. The loading is applied through prescribed
Červenka et al. 2005; Di Luzio 2007). The main difference between
displacement at the loading points to capture the failure load and post-
microplane and other material models is that the constitutive law is
peak response. The concrete is modeled using 4-noded plane stress
written in terms of vectors on microplanes rather than tensors at the
elements (CPS4R). These elements, though not aligned with the
macro level (Bazant et al. 2000). Therefore, inelastic physical phe-
expected shear failure, were chosen over triangular elements which
nomena such as slip and friction can be characterized directly in
have been reported to be overstiff in Abaqus. The thickness of the
terms of stress and strain on the microplanes. The model utilizes
plane stress elements is taken as the width of the tested specimens.
a frictional yield surface that can account for the effect of shear The steel reinforcement is modeled using 2-noded linear two-
cracking (Bazant et al. 2000). The microplane model M4 (Bazant dimensional (2D) truss elements (T2D2). The reinforcement is
et al. 2000; Caner and Bazant 2000) is adopted in this paper embedded in the concrete and perfect bond between concrete
and implemented in a commercially available finite-element pack- and reinforcement is assumed. In the present study, the elastic-per-
age, Abaqus 6.9-2, to simulate and analyze the behavior of RC fect plastic stress-strain relationship is used to simulate the behavior
deep beams. The concrete material models of Abaqus, including of reinforcing steel in both tension and compression. For the
the smeared crack model (SCM) and the damaged plasticity model simulation of the studied beams, the yield strength of 365, 448,
(DPM), are also examined and their performance is assessed and 577 MPa for main longitudinal reinforcement, vertical stirrups,
against experimental results. and horizontal shear reinforcement, respectively, are used for
On the basis of a series of finite-element analyses and a larger beams E1, G1, G2, and G3 as determined from tensile tests.
numerical parametric study, the effect of shear span to depth ratio, Because explicit analysis is more robust, Abaqus/Explicit is
concrete compressive strength, and shear reinforcement on the adopted; however, Abaqus/Standard is used when comparisons
shear strength of RC deep beams is investigated. Particular atten- are made with the smeared crack model because this cannot be used
tion is paid to the development and distribution of principal stresses with explicit procedure. Although Abaqus/Explicit is a true dy-
within the shear span of the modeled beams to gain deeper insight namic platform, it can also be used for quasi-static analysis; how-
into the development and capacity of shear carrying mechanism. ever, special consideration is required to change the procedure from
The results of this study are compared to the provisions of EC2 dynamic to quasi-static. The quasi-static analysis is achieved in
and ACI 318-14 and recommendations are given to improve Abaqus by using mass scaling or by changing the loading rate
existing design models. of Abaqus or a combination thereof. The simulation is carried

© ASCE 04016077-2 J. Struct. Eng.

J. Struct. Eng., 2016, 142(10): 04016077

 125

300
6
Steel plate
100x30 mm

8 6

h = 700
h = 400
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As a 200

L/2=700 4@75mm

300
As
(a)

L/2 = a 125

(b)
Steel plate
Steel plate
305x30 mm
100x30 mm
6
10
h = 700

h = 915
300

As 4@75mm

As a = 915 202,5

a 125 L/2 = 2235

L/2

(c) (d)

Fig. 1. Detail of the analyzed beams: (a) current research; (b) beams tested by Foster and Gilbert (1998) (except for B2.0 A-4 and B3.0 A-4 beams);
(c) B2.0 A-4 and B3.0 A-4 beams; (d) beams tested by Aguilar et al. (2002)

out in displacement control with the smooth amplitude option. The smeared cracking approach first developed by Rashid (1968).
kinetic and internal energy are monitored to ensure the kinetic energy Cracks in concrete can be detected at any location when the
does not exceed 5–10% of its internal energy throughout most of the concrete stresses reach one of the failure surfaces (crack detection
analysis process in Abaqus. When this is not achieved through the surfaces) in the biaxial tension region or combined tension-
implementation of the loading rate, mass scaling is also used along compression region. The smeared crack model does not track each
with loading rate to keep the kinetic energy within the required limits. macrocrack, but it performs independent constitutive calculations at
The element size is chosen on the basis of a systematic mesh sen- each integration point of the finite-element model by using de-
sitivity analysis, as explained subsequently and to maintain the balance graded stiffness. This model accounts for the effect of shear through
between kinetic energy and internal energy. The element size is four a shear retention factor, which can specify the amount of shear
times the maximum aggregate size for beams with overall depth less stresses that can be transferred after cracking. The DPM is a con-
than 500 mm, while an approximate global element size of 100 mm tinuum damage plasticity-based model proposed by Lubliner et al.
is used for beams with overall depth greater than 500 mm. These (1989). This model represents the inelastic behavior of concrete by
element sizes are 10–15% of the total height of the beams. The relevant using the concepts of isotropic damaged elasticity in combination
element sizes were used to calibrate the performance of the microplane with isotropic tensile and compressive plasticity. The model de-
model before conducting any subsequent analysis. scribes the irreversible damage in concrete due to the fracturing
process by a combination of nonassociated multihardening plastic-
ity and scalar damaged elasticity. Tensile cracking and compressive
Evaluation of Concrete Material Models Available in
crushing are assumed to be the two main failure mechanisms of
Abaqus
concrete. The evolution of the failure surface is controlled by
The concrete constitutive models implemented in Abaqus include two hardening variables describing the failure mechanisms under
the SCM and the DPM. The smeared crack model is based on the tension and compression.

© ASCE 04016077-3 J. Struct. Eng.

J. Struct. Eng., 2016, 142(10): 04016077

 600
G1 more realistic way can compensate for this assumption to a reason-
500 able degree, as will be discussed subsequently.
The inclusion of a shear retention factor in the smeared crack
Applied Load (kN) 400 model of Abaqus can theoretically account for the effect of shear
300
cracking. However, previous studies (Guadagnini 2002) have
shown that even the use of a wide range of shear retention values
200 Exp. could not affect significantly the global behavior and the calculated
ABAQUS-SCM
shear strength did not vary significantly for the analyzed beams.
100
Hence, more accurate constitutive models are required to simulate
ABAQUS-DPM
0 the behavior of RC deep beams, such as the microplane model.
0 2 4 6 8
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Deflection (mm)
Microplane Material Model M4 for Concrete
Fig. 2. Experimental and predicted load-deflection curves (Specimen
G1) The microplane material model is a macroscopic material model
that defines the relation between the stress and strain vectors on
planes of various orientations (microplanes). These microplanes
can be assumed as cracked planes or weak planes, such as the con-
These two models are examined first to assess their accuracy tact faces between aggregate particles in concrete. The basic idea of
in capturing the shear behavior of RC deep beams. The models the microplane model can be traced back to the pioneering idea of
had been earlier optimized using mesh sensitivity and the chosen Taylor (1938), which was later developed by Batdorf and Budian-
mesh size that is four times the maximum aggregate size was found sky (1949) for polycrystalline metals and became known as the slip
to best approximate overall structural response of the examined theory of plasticity. This model was later extended by Bazant
beams. Figs. 2 and 3 show the load-deflection response and strain and Gambarova (1984), Bazant and Prat (1988), Bazant and Ožbolt
in both horizontal and vertical shear reinforcement for one of the (1990), Carol et al. (1992), Bazant et al. (1996a, b, 2000), Caner
beams that were tested. The numerical responses are much stiffer and Bazant (2000), and Di Luzio (2007), who added extra features
than the experimental responses after cracking. This can be attrib- to better represent the behavior of quasi-brittle materials, including
uted mainly to the inability of the implemented concrete models to concrete. These features can be briefly summarized as follows:
realistically predict the behavior of the concrete subjected to shear • The static micro-macro constraint should be replaced by kine-
stress and lateral tensile stresses. In reality, high shear stresses and matic micro-macro constraint to stabilize the postpeak strain
lateral tensile stresses develop in the shear span of RC deep beams softening; that is, the strain vectors on microplanes are the
and this makes the concrete softer than under uniaxial conditions. projection of the strain tensors.
This softening reduces the overall stiffness of the member and • Elastic strain is included at the microplane level instead of
eventually leads to shear failure in the shear span. However, this adding it at the macro level due to the replacement of the static
softening behavior accompanied by increasing shear deformation micro-macro constraint with the kinematic.
cannot be realistically estimated by either approach. The analysis • The principle of virtual work is used instead of simple super-
of the experimental results shown in Fig. 3 shows that shear position of the microplane stresses to relate the stresses on the
cracking occurred at earlier loading stages compared with the microplane, which can have any possible orientation to the
numerical predictions and strain in shear reinforcement is vastly stress at macro level.
underestimated after cracking at similar load levels. This means Because the kinematic constraint is used, the microplane strain
that for the same applied load, concrete is more damaged in vector εNi is determined as the projection of the strain tensor εij.
the experiments and this is the main reason the numerical re- The normal strain εN and both shear strains εM and εL on the micro-
plane can then be found according to the following equations:
sponse is stiffer than the experimental and the failure load is
overestimated. εN ¼ N ij εij ; εM ¼ Mij εij ; εL ¼ Lij εij ð1Þ
Another reason for this discrepancy between experimental and
numerical results is the adoption of average stress and strain. How- where N ij ¼ ni nj ; M ij ¼ ðmi nj þ mj ni Þ=2; and Lij ¼
ever, this is not necessarily a major issue because an accurate con- ðli nj þ lj ni Þ=2; and n, m, and l are direction cosines, the values
stitutive law that can capture the behavior of concrete material in a of which can be found elsewhere (Bažant and Oh 1986).

600 600
G1 G1
500 500
Applied Load (kN)

Applied Load (kN)

400 400

300 300

200 200
Exp. Exp.
100 ABAQUS-SCM 100 ABAQUS-SCM
ABAQUS-DPM ABAQUS-DPM
0 0
0 1000 2000 3000 4000 5000 0 500 1000 1500 2000
Stirrup strain, Microstrain Horizonatal shear reinforcement strain, Microstrain

Fig. 3. Experimental and predicted shear reinforcement strain (Specimen G1)

© ASCE 04016077-4 J. Struct. Eng.

J. Struct. Eng., 2016, 142(10): 04016077

 500 1,800
450 G1 B2-0-1
1,600
400 1,400

Applied Load (kN)

Applied load (kN)

350 1,200
300
1,000
250
800
200 Exp.
600 Exp.
150 50x50mm 50x50mm
100 75x75mm 400 75x75mm
100x100mm 200 100x100mm
50
125x125mm 125x125mm
0 0
0 1 2 3 4 5 6 7 0 2 4 6 8
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Deflection (mm) Deflection (mm)

Fig. 4. Effect of element size on the predicted load-deflection response

Bazant et al. (2000) calculated the stress at the continuum level strain softening, finite-element modeling results are sensitive to
from the microplane stresses applying the principle of virtual work, mesh size. When the model mesh is refined, the fracture energy
which can be approximated by optimal Gaussian integration for a dissipated during brittle failure in the critical regions of strain-
spherical surface. softening damage can decrease considerably, thus affecting overall
The constitutive law of microplane M4 is characterized by an failure load. Mesh sensitivity techniques such as crack band
elastic stress-strain relationship up to a defined set of limits, which (Červenka et al. 2005; Slobbe et al. 2013) can be implemented
is called stress-strain boundaries, followed by a strain-softening in a material model to control crack propagation and compressive
behavior. The stresses are never allowed to exceed the boundaries, post peak behavior. However, these are not easy to apply to com-
however traveling along the boundaries is allowed if the strain plex material model such as the microplane model M4 in its current
increment and the total stress have the same sign, otherwise, un- form because it is not easy to identify which material parameters
loading occurs. need to be adjusted according to the element size to ensure correct
Bazant et al. (2000) and Caner and Bazant (2000) split the nor- energy dissipation in the softening regime. Therefore, in the current
mal stress and strain components into volumetric and deviatoric study, mesh sensitivity analysis is performed and the best mesh
parts (σN ¼ σV þ σD and εN ¼ εV þ εD ) to realistically model sizes are chosen on the basis of this analysis.
the compression failure and to control the value of Poisson’s ratio. To examine the effect of element size on global behavior, four
The microplane elastic moduli can be used in the case of loading element sizes were considered. Fig. 4 shows the effect of element
as well as reloading. Additionally, they can be used for unloading if size on the load-deflection response and failure load for Beams G1
the sign of σΔε is positive, otherwise stiffness degradation occurs and and B2-0-1. It can be seen that for beam G1, with an overall depth
the value of the tangential stiffness modulus is used for unloading. of 40 0 mm, the 50 × 50 mm element size, which is equal to four
The microplane model M4 was implemented in general finite- times the maximum aggregate size, is in better agreement with the
element package Abaqus using a VUMAT subroutine. This allowed experimental results. However, for Beam B2-0-1, with an overall
the development of a more robust numerical platform that could depth of 700 mm, the use of larger elements yields slightly better
be used to obtain an invaluable insight on the behavior of RC el- correlation with the experimental results. The result from all 20 RC
ements. This also allowed a more systematic and reliable analysis deep beams showed that for beams with overall depth less than
of the effect of different parameters on the structural behavior of the 500 mm, using an element size equivalent to four times the maxi-
examined specimens. The number of microplanes adopted in each mum aggregate size is in good agreement with the experimental
integration points is 21, which is the minimum number of micro- results. The use of elements with a size of 100 × 100 mm seem
planes needed to yield acceptable results (Bažant and Oh 1986). to better approximate experimental results of deep beams with
With the exception of the adjustable material parameter (k1 ), which an overall depth greater than 500 mm regardless of their maximum
controls the concrete uniaxial tensile and compressive peak aggregate size. These mesh sizes are within 10–15% of the height
strength, the value of the other adjustable material parameters of the specimens.
k2 , k3 , and k4 were optimized to best represent the biaxial compres-
sive and tensile stress-strain relationships and had values of 200,
Load-Deflection Response
15, and 100, respectively. A value of 0.0003 was used for the
adjustable material parameter k1 for beams with shear span to depth The load-deflection curves obtained from the numerical analyses of
ratio greater than 1.0, while a value of 0.0004 was used for beams eight of the examined beams are presented in Fig. 5 along with
with shear span to depth ratio less than 1.0. All other fixed param- the experimentally measured load-deflection responses. As can
eters were used as defined by Caner and Bazant (2000). In all cases be seen, the results show an overall good agreement with the ex-
for which the experimental modulus of elasticity was not available perimental data, although in some beams the finite-element results
the modulus of elasticity was determined according to EC2. still exhibit a slightly stiffer response, as also seen from the strain
results. Table 1 shows the ratio of experimental to predicted failure
load and deflection at failure, respectively, for all 20 analyzed
Model Validation beams. Fig. 6(a) shows the ratio of experimental to finite-element
failure load as a function of shear span to depth ratio. From the
analysis of the load-deflection response of beams G1, G2, and
Effect of Element Size
G3 with shear span to depth ratio of 1.67, 1.29, and 0.91, respec-
Concrete and other quasi-brittle materials exhibit strain softening tively (Fig. 5), it can be seen that the numerical prediction for
in the postpeak response in tension and compression. Due to this beams with smaller shear span to depth ratio is in better agreement

© ASCE 04016077-5 J. Struct. Eng.

J. Struct. Eng., 2016, 142(10): 04016077

 450 350
E1 G1
400 300
350

Applied Load (kN)

Applied Load (kN)

250
300
250 200
200 150
150
100
100 Exp. Exp.
50 50
FEM FEM
0 0
0 1 2 3 4 5 0 1 2 3 4 5 6
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Deflection (mm) Deflection (mm)

400 600
G2 G3
350
500
300
Applied Load (kN)

Applied Load (kN)

400
250
200 300
150
200
100
Exp. 100 Exp.
50
FEM FEM
0 0
0 1 2 3 4 5 0 1 2 3 4 5
Deflection (mm) Deflection (mm)

1800 2000
B2.0-1 1800 B2.0-2
1600
1400 1600
Applied Load (kN)

Applied Load (kN)

1400
1200
1200
1000
1000
800
800
600
600
400 Exp. 400 Exp.
200 FEM 200 FEM
0 0
0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
Deflection (mm) Deflection (mm)

3000 2500
ACI-I STM-I
2500
2000
Applied Load (kN)

Applied Load (kN)

2000
1500
1500
1000
1000

500 Exp.
500 Exp.
FEM FEM
0 0
0 5 10 15 20 25 30 35 0 5 10 15 20 25 30
Deflection (mm) Deflection (mm)

Fig. 5. Experimental and predicted load-deflection curves

with the experimental results. This can again be attributed to the with shear reinforcement and without shear reinforcement. Fig. 7
inability of a smeared crack approach to capture realistically the shows the comparison between the experimental crack pattern at
behavior of the member in terms of cracking and local strain dis- failure and predicted failure by finite-element analysis for two
tribution. For beams with shear span to depth ratio less than 1.0, the of the analyzed beams.
applied load is directly transferred to the support through a strut and
the shear span has less discrete cracks. This can be better repre-
sented by the smeared crack approach than for shear span to depth Strain in Longitudinal and Shear Reinforcement
ratios greater than 1, which are typically characterized by multiple The load-strain plots for the main flexural, vertical, and horizontal
discrete cracks. shear reinforcement at specified locations are shown in Figs. 8–10,
Fig. 6(b) shows the ratio of experimental and predicted failure respectively. For the elastic stage, up to the formation of cracking,
load as a function of the shear reinforcement ratio. The results show the results from the finite-element analysis show a good agreement
that the model can provide reasonable predictions for both beams with the experimental data. After the formation of flexural cracks,

© ASCE 04016077-6 J. Struct. Eng.

J. Struct. Eng., 2016, 142(10): 04016077

 1.4 1.4
1.3 1.3
1.2 1.2
1.1 1.1

VExp./VFEM
VExp./VFEM
1 1
0.9 0.9
0.8 0.8
0.7 0.7
0.6 0.6
0.6 0.9 1.2 1.5 1.8 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
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(a) a/d (b) v (%)

Fig. 6. Effect of (a) shear span to depth ratio; (b) shear reinforcement ratio on the predicted capacity of the analyzed beams

Fig. 7. Experimental and numerical failure: (a) Beam G1; (b) Beam B2.0-2 (data from Foster and Gilbert 1998)

the experimental response in terms of strain in the main flexural • To prevent local concrete crushing at the location of application
reinforcement is generally softer than the finite-element results; of point loads and supports, steel spreader plates are used in both
however, the trends are generally comparable. This discrepancy experimental and numerical analysis. In numerical analysis the
in the load-strain response after crack formation can be attributed spreader plates are rigidly connected to the concrete to prevent
to the following: them from moving. This means that at the location of the tie,
• Concrete tensile strains are highly localized at crack locations concrete and the steel plates share the same nodes; as a result,
and the intact area of concrete between cracks can still contri- the stiffness of the element in the surrounding area is increased
bute to the load-resisting mechanism by means of tension stif- by the high stiffness of the steel plates.
fening; however, in numerical analysis, due to the use of average Despite some inherent modeling deficiencies, overall the imple-
stress and strain, such tension stiffening and strain localization mentation of the microplane model gives very good predictions of
cannot be properly modeled. This can result in the inaccurate the behavior of RC deep beams and can be a useful tool in under-
prediction of the stiffness of cracked concrete and strain distri- standing the accuracy of design equations used in codes of practice.
bution within the member, which in turn directly affects overall
stiffness.
• Perfect bond between concrete and reinforcing bars is used in Parametric Study
the simulations and thus the numerical model is unable to cap-
ture the stiffness degradation due to debonding and local slip of On the basis of the numerical model described previously, a para-
the reinforcement at the location of the cracks. metric study was carried out to investigate the effect of different

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 450 350
E1 G1
400 300
350

Applied Load (kN)

Applied Load (kN)
250
300
250 200
200 150
150
100
100 Exp. Exp.
50 50
FEM FEM
0 0
0 500 1000 1500 2000 0 500 1000 1500 2000
Microstrain Microstrain
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400 500
G2 G3
350 450
400
300

Applied Load (kN)

Applied Load (kN)

350
250 300
200 250
150 200
150
100
Exp. 100 Exp.
50 50
FEM FEM
0 0
0 500 1000 1500 2000 0 500 1000 1500 2000
Microstrain Microstrain

Fig. 8. Experimental and predicted main flexural reinforcement strain

350 400
G1 G2
300 350
300
Applied Load (kN)

Applied Load (kN)

250
250
200
200
150
150
100
100
Exp. Exp.
50 50
FEM FEM
0 0
0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 5000
Stirrup strain, Microstrain Stirrup strain, Microstrain

500
G3
450
400
Applied Load (kN)

350
300
250
200
150
100 Exp.
50 FEM
0
0 500 1000 1500 2000 2500 3000
Stirrup strain, Microstrain

Fig. 9. Experimental and predicted vertical shear reinforcement strain for Beams G1, G2, and G3

design parameters on the shear capacity of RC deep beams. The minimum amount of shear reinforcement (according to ACI 318-14
details of the beams used in this parametric study are given in and EC2) were considered in this study. The width of support and
Table 2. The design parameters are shear span to depth ratio of loading plates are 150 mm.
0.75, 1.3, and 2.0 and concrete compressive strength of 30, 55,
and 80 MPa. As shear in RC deep beams is primarily resisted
Results and Discussion
through the development of a single strut, the overall behavior
is not greatly affected by an increasing amount of shear reinforce- Fig. 11 shows the effect of shear span to depth ratio and concrete
ment. Thus, only beams without shear reinforcement and with the compressive strength on the capacity of RC deep beams. The

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 350 400
G1 G2
300 350
300

Applied Load (kN)

Applied Load (kN)

250
250
200
200
150
150
100
100
Exp. Exp.
50 50
FEM FEM
0 0
0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 3000
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Horizonatal shear reinforcement strain, Microstrain Horizonatal shear reinforcement strain, Microstrain

500
G3
450
400
Applied Load (kN) 350
300
250
200
150
100 Exp.
50 FEM
0
0 1000 2000 3000 4000
Horizonatal shear reinforcement strain, Microstrain

Fig. 10. Experimental and predicted horizontal shear reinforcement strain for Beams G1, G2, and G3

Table 2. Details of the Beams Used in Parametric Study load is transferred directly by a single inclined strut, increasing con-
crete strength leads to an increase in resistance capacity of this strut
L h d b a fc ρv and as shown in Fig. 13(a). However, the compressive stress at failure in
Specimen (mm) (mm) (mm) (mm) (mm) a=d (MPa) ρ (%) ρh (%)
the strut of the beam with concrete strength of 80 MPa was only
BN-S-30 2,965 750 710 200 532.5 0.75 30 1.3 0 30% higher than that developed in the same beam with concrete
B-S-30 2,965 750 710 200 532.5 0.75 30 1.3 0.25 strength of 30 MPa. The maximum principal compressive stress
BN-S-55 2,965 750 710 200 532.5 0.75 55 1.3 0 in the beam with concrete strength of 30 MPa was approximately
B-S-55 2,965 750 710 200 532.5 0.75 55 1.3 0.25 75% of its concrete strength, while values of less than 40% were
BN-S-80 2,965 750 710 200 532.5 0.75 80 1.3 0
observed in the beams with concrete strength of 80 MPa. This is
B-S-80 2,965 750 710 200 532.5 0.75 80 1.3 0.25
BN-M-30 3,746 750 710 200 923 1.3 30 1.3 0 probably due to the fact that (1) the tensile strength of concrete
B-M-30 3,746 750 710 200 923 1.3 30 1.3 0.25 increases at a lower rate than its compressive strength, and (2) with
BN-M-55 3,746 750 710 200 923 1.3 55 1.3 0 increasing strength and load capacity, the lateral tensile stress in the
B-M-55 3,746 750 710 200 923 1.3 55 1.3 0.25 strut also increases, which reduces further the effective compressive
BN-M-80 3,746 750 710 200 923 1.3 80 1.3 0 strength.
B-M-80 3,746 750 710 200 923 1.3 80 1.3 0.25 Because strut and tie action is the primary mechanism of shear
BN-B-30 4,740 750 710 200 1,420 2 30 1.3 0 stress transfer in RC deep beams, shear reinforcement is not
B-B-30 4,740 750 710 200 1,420 2 30 1.3 0.25 expected to have a significant effect on the shear capacity. The pres-
BN-B-55 4,740 750 710 200 1,420 2 55 1.3 0 ence of shear reinforcement, however, is important because it
B-B-55 4,740 750 710 200 1,420 2 55 1.3 0.25
increases ductility and limits the propagation of inclined cracks,
BN-B-80 4,740 750 710 200 1,420 2 80 1.3 0
B-B-80 4,740 750 710 200 1,420 2 80 1.3 0.25 thus changing the shear stress distribution in the shear span as
shown in Fig. 14. A difference of approximately 15–20% between
the shear capacity of beams with and without shear reinforcement
was found in the parametric study. Fig. 13(b) shows the effect of
results of the analysis on beams without shear reinforcement (spec- shear reinforcement on the principal tensile strain in the shear span.
imens NSR) are shown along with those of their counterparts with For beams with shear reinforcement, the strain developed at a given
shear reinforcement (specimens SR). It can be seen that with in- applied load is lower due to the contribution of shear reinforcement
creasing shear span to depth ratio, the shear strength decreases. in resisting and controlling the development of cracks. Thus, by
This can be attributed to the fact that in beams with low shear span reducing the principal tensile strain, the presence of shear reinforce-
to depth ratios, the applied load is directly transferred through one ment can enable the development of higher concrete compressive
strut, which means that the concrete is more directly loaded in com- stresses and increase the effectiveness of the concrete strut.
pression, while with increasing shear span to depth ratio the angle
of the strut becomes shallower [Fig. 12(b)], which leads to larger
Effectiveness Factor
lateral tensile strains that weaken the compressive strut.
With increasing concrete compressive strength, as expected, the A key parameter in the strut-and-tie model as used in design is the
shear capacity of the RC deep beams increases. Because the applied definition of the effectiveness factor (v), which is used to calculate

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 1400 1400
NSR-fck=30MPa NSR-a/d=0.75
1200 NSR-fck=55MPa 1200 NSR-a/d=1.3
NSR-fck=80MPa NSR-a/d=2.0
1000 SR-fck=30MPa 1000
SR-a/d=0.75
SR-fck=55MPa SR-a/d=1.3
800 800

V (kN)

V (kN)
SR-fck=80MPa SR-a/d=2.0
600 600

400 400

200 200

0 0
0 0.5 1 1.5 2 2.5 0 15 30 45 60 75 90
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(a) a/d (b) fc (MPa)

Fig. 11. Effect of (a) shear span to depth ratio; (b) concrete compressive strength on shear strength of RC deep beams

Fig. 12. Shear stress transfer in beams with different shear span to depth ratio

1600 1400
1400 1200
1200
Applied Load (kN)

Applied Load (kN)

1000
1000
800
800
600
600
400
400
fc = 30MPa Shear Reinforcement
200 200
fc = 80MPa No Shear Reinforcement
0 0
0 10 20 30 40 0 2500 5000 7500 10000 12500 15000
(a) Principal Compressive Stress (MPa) (b) Principal Tensile Strain (Micro Strain)

Fig. 13. Effect of (a) concrete strength on principal compressive strength in the shear span; (b) shear reinforcement on principal tensile strain in
shear span

Fig. 14. Effect of shear reinforcement on shear stress distribution in shear span

the strut strength. This factor accounts for the reduced compressive and shear reinforcement ratio (Brown and Bayrak 2008b). The
strength of concrete when subjected to lateral tensile strains, as in effect of these variables are investigated numerically using the mi-
the web of deep beams, and can be expressed as a ratio of the maxi- croplane M4 model and are compared to the provisions of EC2 and
mum principal compressive stress to the uniaxial compressive ACI 318-14.
strength. The main variables that affect the effectiveness factor Fig. 15 shows that increasing either shear span to depth ratio or
are shear span to depth ratio, compressive strength of concrete, concrete compressive strength can reduce the effectiveness factor.

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 0.9 0.9
0.8 0.8
0.7 0.7
0.6 0.6
0.5 0.5 NSR-a/d=0.75

v'
v'
0.4 NSR-fck=30MPa 0.4 NSR-a/d=1.3
NSR-fck=55MPa NSR-a/d=2.0
0.3 0.3
NSR-fck=80MPa SR-a/d=0.75
0.2 SR-fck=30MPa 0.2
SR-fck=55MPa SR-a/d=1.3
0.1 0.1
SR-fck=80MPa SR-a/d=2.0
0 0
0 0.5 1 1.5 2 2.5 0 20 40 60 80 100
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(a) a/d (b) fc (MPa)

Fig. 15. Effect of (a) shear span to depth ratio; (b) concrete compressive strength on the effectiveness of concrete

1.8 1.8
EC2 EC2
1.6 1.6
1.4 1.4
v'FEM /v'ACI318-14

v'FEM /v'ACI318-14
1.2 1.2
1 1
0.8 NSR-a/d=0.75 0.8
NSR-fck=30MPa
0.6 NSR-a/d=1.3 0.6 NSR-fck=55MPa
NSR-a/d=2.0 NSR-fck=80MPa
0.4 SR-a/d=0.75 0.4 SR-fck=30MPa
0.2 SR-a/d=1.3 0.2 SR-fck=55MPa
SR-a/d=2.0 SR-fck=80MPa
0 0
0 20 40 60 80 100 0 0.5 1 1.5 2 2.5
(a) fc (MPa) (b) a/d

1.8 1.8
ACI 318-14 ACI 318-14
1.6 1.6
1.4 1.4
v'FEM /v'ACI318-14

v'FEM /v'ACI318-14

1.2 1.2
1 1
0.8 NSR-a/d=0.75 0.8
NSR-a/d=1.3 NSR-fck=30MPa
0.6 0.6 NSR-fck=55MPa
NSR-a/d=2.0
0.4 0.4 NSR-fck=80MPa
SR-a/d=0.75 SR-fck=30MPa
0.2 SR-a/d=1.3 0.2 SR-fck=55MPa
SR-a/d=2.0 SR-fck=80MPa
0 0
0 20 40 60 80 100 0 0.5 1 1.5 2 2.5
(c) fc (MPa) (d) a/d

Fig. 16. Comparison of code predictions with the numerically obtained effectiveness factors

The presence of a minimum amount of shear reinforcement (shown Fig. 17 shows the performance of the strut-and-tie model in pre-
with a dotted line in the figure) can lead to an increase in the ef- dicting the shear capacity of specimens used in the parametric study
fectiveness factor by approximately 15%. Nonetheless, the provi- by using the effectiveness factors from EC2 and ACI 318-14 along
sions of EC2 only account for the effect of concrete compressive with the effectiveness factor obtained from the numerical analysis.
strength, while ACI 318-14 only accounts for the effect of shear The calculated effectiveness factor leads to more accurate predic-
reinforcement. tions with lower standard deviations. The poorer performance of
Fig. 16 shows the ratio of the calculated effectiveness factor to EC2 and ACI 318-14 can be attributed to the fact that key param-
that obtained according to the provisions of EC2 and ACI 318-14 as eters in estimating the effectiveness factor are neglected.
a function of concrete compressive strength and shear span to depth
ratio. It can be seen that the EC2, even though it accounts for
concrete strength, still slightly overestimates the effectiveness fac- Conclusion
tor for beams with higher strength concrete. This is because on the
one hand it does not account accurately for strength, and on the Based on the results of the numerical investigation and parametric
other because it neglects the effect of shear span to depth ratio study discussed, the following conclusions can be drawn:
on the effectiveness factor. ACI 318-14 ignores both variables • Smeared crack and damaged plasticity models are unable to rea-
and yields conservative results only for concrete strengths of ap- listically simulate the shear behavior of RC deep beams. This is
proximately 30 MPa, becoming unsafe for higher concrete strength because these models cannot predict realistically shear deforma-
as well as higher shear span to depth ratios. tion in discontinuity regions.

© ASCE 04016077-11 J. Struct. Eng.

J. Struct. Eng., 2016, 142(10): 04016077

 3 3
EC2 Average = 1.42 ACI 318-14 Average = 1.18
2.5 STD = 0.41 2.5 STD = 0.41

2 2

VFEM /VSTM

VFEM /VSTM
1.5 1.5

1 1

0.5 0.5

0 0
0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5
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(a) a/d (b) a/d

3
Current research Average = 1.10
2.5 STD = 0.13

VFEM /VSTM 2

1.5

0.5

0
0 0.5 1 1.5 2 2.5
(c) a/d

Fig. 17. Effect of different effectiveness factor on the shear capacity prediction by strut-and-tie model

• The implementation of the microplane material model M4 in Batdorf, S. B., and Budiansky, B. (1949). “A mathematical theory of
Abaqus led to reasonably accurate predictions of the overall plasticity based on the concept of slip.” Tech. Note No. 1871, National
behavior of the RC deep beams analyzed; however, local strain Advisory Committee for Aeronautics, Washington, DC.
values were generally underestimated because a smeared ap- Bazant, Z. P., Caner, F. C., Carol, I., Adley, M. D., and Akers, S. A. (2000).
“Microplane model M4 for concrete—I: Formulation with work-
proach is unable to realistically simulate the tension stiffening
conjugate deviatoric stress.” J. Eng. Mech., 10.1061/(ASCE)0733-
of the concrete.
9399(2000)126:9(944), 944–953.
• Concrete compressive strength and shear span to depth ratio are Bazant, Z. P., and Gambarova, P. G. (1984). “Crack shear in concrete:
the key parameters affecting the concrete effectiveness factor in Crack band microplane model.” J. Struct. Eng., 10.1061/(ASCE)
the shear span of deep beams. 0733-9445(1984)110:9(2015), 2015–2035.
• Minimum shear reinforcement can enable a better distribution of Bazant, Z. P., and Ožbolt, J. (1990). “Nonlocal microplane model for frac-
stresses within the shear span, control tensile strain, and increase ture, damage, and size effect in structures.” J. Eng. Mech., 10.1061/
the effectiveness of the concrete by up to 15%. (ASCE)0733-9399(1990)116:11(2485), 2485–2505.
• An accurate model to estimate the effectiveness factor should Bazant, Z. P., and Prat, P. C. (1988). “Microplane model for brittle-plastic
include the effect of concrete strength, shear span to depth ratio, material—I: Theory.” J. Eng. Mech., 10.1061/(ASCE)0733-9399(1988)
and shear reinforcement. 114:10(1672), 1672–1688.
Bazant, Z. P., Xiang, Y., Adley, M. D., Prat, P. C., and Akers, S. A. (1996a).
“Microplane model for concrete—II: Data delocalization and verification.”
Acknowledgments J. Eng. Mech., 10.1061/(ASCE)0733-9399(1996)122:3(255), 255–262.
Bazant, Z. P., Xiang, Y., and Prat, P. C. (1996b). “Microplane model
The authors acknowledge the financial support of the Human for concrete—I: Stress-strain boundaries and finite strain.” J. Eng.
Capacity Development Program (HCDP) of the Ministry of Higher Mech., 10.1061/(ASCE)0733-9399(1996)122:3(245), 245–254.
Education and Scientific Research, Kurdistan Regional Govern- Bažant, P., and Oh, B. (1986). “Efficient numerical integration on the
ment for the Ph.D. studies of Kamaran S. Ismail. surface of a sphere.” J. Appl. Math. Mech., 66(1), 37–49.
British Standards Institution. (2004). “Design of concrete structures—Part
1-1: General rules and rules for buildings.” Eurocode 2, London.
References Brown, M. D., and Bayrak, O. (2008a). “Design of deep beams using
strut-and-tie models—Part I: Evaluating U.S. provisions.” ACI Struct.
Abaqus [Computer software]. Dassault Systèmes, Waltham, MA. J., 105(4), 395–404.
ACI (American Concrete Institute). (2014). “Building code requirements Brown, M. D., and Bayrak, O. (2008b). “Design of deep beams using
for structural concrete.” ACI 318M–14, Farmington Hills, MI. strut-and-tie models—Part II: Design recommendations.” ACI Struct.
Aguilar, G., Matamoros, A. B., Parra-Montesinos, G. J., Ramírez, J. A., and J., 105(4), 405–413.
Wight, J. K. (2002). “Experimental evaluation of design procedures for Caner, F. C., and Bazant, Z. P. (2000). “Microplane model M4 for concrete
shear strength of deep reinforced concrete beams.” ACI Struct. J., 99(4), —II: Algorithm and calibration.” J. Eng. Mech., 10.1061/(ASCE)0733-
539–548. 9399(2000)126:9(954), 954–961.
Arabzadeh, A., Rahaie, A. R., and Aghayari, R. (2009). “A simple strut- Carol, I., Prat, P. C., and Bažant, Z. P. (1992). “New explicit microplane
and-tie model for prediction of ultimate shear strength of RC deep model for concrete: Theoretical aspects and numerical implementation.”
beams.” Int. J. Civ. Eng., 7(3), 141–153. Int. J. Solids Struct., 29(9), 1173–1191.

© ASCE 04016077-12 J. Struct. Eng.

J. Struct. Eng., 2016, 142(10): 04016077

 Červenka, J., Bažant, Z. P., and Wierer, M. (2005). “Equivalent localization stress-strain distribution in high strength concrete deep beams.” Proc.
element for crack band approach to mesh-sensitivity in microplane Eng., 14(2011), 2141–2150.
model.” Int. J. Numer. Methods Eng., 62(5), 700–726. Ožbolt, J., Li, Y., and Kožar, I. (2001). “Microplane model for concrete
Collins, M. P., Bentz, E. C., and Sherwood, E. G. (2008). “Where is shear with relaxed kinematic constraint.” Int. J. Solids Struct., 38(16),
reinforcement required? Review of research results and design proce- 2683–2711.
dures.” ACI Struct. J., 105(5), 590–600. Ožbolt, J., and Li, Y.-J. (2001). “Three-dimensional cyclic analysis of com-
Di Luzio, G. (2007). “A symmetric over-nonlocal microplane model M4 for pressive diagonal shear failure.” ACI Spec. Publ., 205(4), 61–80.
fracture in concrete.” Int. J. Solids Struct., 44(13), 4418–4441. Rashid, Y. (1968). “Ultimate strength analysis of prestressed concrete
Foster, S. J., and Gilbert, R. I. (1998). “Experimental studies on high- pressure vessels.” Nucl. Eng. Des., 7(4), 334–344.
Sagaseta, J., and Vollum, R. (2010). “Shear design of short-span beams.”
strength concrete deep beams.” ACI Struct. J., 95(4), 382–390.
Mag. Concr. Res., 62(4), 267–282.
Guadagnini, M. (2002). “Shear behaviour and design of FRP RC beams.”
Slobbe, A. T., Hendriks, M. A. N., and Rots, J. G. (2013). “Systematic
Downloaded from ascelibrary.org by UEM - Universidade Estadual De Maringa on 09/20/20. Copyright ASCE. For personal use only; all rights reserved.

Ph.D. thesis, Univ. of Sheffield, Sheffield, U.K.

assessment of directional mesh bias with periodic boundary conditions:
Kuchma, D., Yindeesuk, S., Nagle, T., Hart, J., and Lee, H. H. (2008). Applied to the crack band model.” Eng. Fract. Mech., 109, 186–208.
“Experimental validation of strut-and-tie method for complex regions.” Taylor, G. I. (1938). “Plastic strain in metals.” J. Inst. Met., 62, 307–324.
ACI Struct. J., 105(5), 578–589. Tuchscherer, R. G., Birrcher, D. B., Williams, C. S., Deschenes, D. J., and
Lubliner, J., Oliver, J., Oller, S., and Onate, E. (1989). “A plastic-damage Bayrak, O. (2014). “Evaluation of existing strut-and-tie methods and
model for concrete.” Int. J. Solids Struct., 25(3), 299–326. recommended improvements.” ACI Struct. J., 111(6), 1451–1460.
Mohammad, M., Jumaat, M. Z. B., Chemrouk, M., Ghasemi, A., Hakim, Wight, J. K., and MacGregor, J. G. (2009). Reinforced concrete: Mechanics
S., and Najmeh, R. (2011). “An experimental investigation of the and design, Pearson Education, Upper Saddle River, NJ.

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