Single Web Shear Element Model for Shear Strength of RC Beams with Stirrups
International Journal of Concrete Structures and Materials
Single Web Shear Element Model for Shear Strength of RC Beams with Stirrups
Esa Obande Jude
In this paper, an analytical model to rationally evaluate the shear strength of reinforced concrete beams with stirrups has been developed. In the developed model, a shear critical section has been idealized with a single web element for shear and the top and bottom chords for flexure, respectively. With the longitudinal strain at the mid-depth in the shear critical section, evaluated from the flexural analysis, the web element has been analyzed, based on the analysis procedure modified from the modified compression field theory finite element formulation. Through the comparison with the test results of 201 reinforced concrete beams with stirrups exhibiting shear failure before flexural yielding, it was investigated that the developed model well predicted the actual shear strength of reinforced concrete beams with stirrups. In addition, it was investigated that the developed model rationally considered the effect of main parameters such as concrete compressive strength, shear span-depth ratio, stirrup ratio, and member depth. Through simplification, the developed model can be useful to develop more rational shear design provisions for reinforced concrete members with stirrups.
shear; shear strength; stirrup; MCFT; diagonal crack angle
Although a number of researches have been conducted to
evaluate shear strength of reinforced concrete beams for the
last century, it is still hard to well predict actual shear
strength because shear behavior of reinforced concrete
beams is very complicating due to many parameters such as
concrete compressive strength, stirrup ratio, shear
span-todepth ratio, longitudinal reinforcement ratio, and so on
2004; Lee et al. 2010; Labib et al. 2013; Russo et al. 2013;
Mofidi and Chaallal 2014; Jeong and Kim 2014; Chiu et al.
2016; El-Sayed and Shuraim 2016)
. On evaluating shear
strength of reinforced concrete beams, even shear design
provisions around the world are much different through each
other, even from theoretical basis, specifically for reinforced
concrete beams with stirrups
(Eurocode 2 2004; ACI 318
2014; CSA A23.3 2014)
For reinforced concrete beams with stirrups and with not
small shear span-to-depth ratio, ACI 318-14 (2014) simply
evaluates shear strength as sum of concrete and stirrup
contributions (Vc and Vs, respectively), based on 45 truss model. In
this provision, Vc is affected neither by stirrup ratio nor by
deformation due to flexure since only force equilibrium is
considered. Eurocode 2 (2004) acknowledges only contribution
of stirrups on evaluating shear strength of reinforced concrete
beams with stirrups, based on variable angle truss model. In
Eurocode 2, only force equilibrium is considered with variable
diagonal angle of compression strut while deformation due to
flexure is not considered. CSA A23.3 (2014) is based on the
variable truss angle model as like Eurocode 2, but it considers
effect of deformation due to flexure on evaluating contribution
of concrete and stirrups for shear strength. CSA A23.3
considers that deformation of web due to flexure affects shear
strengths provided by concrete and stirrups since it was
developed through simplifying the Modified Compression Field
(Vecchio and Collins 1986)
, which can consider
compatibility and equilibrium together.
For reinforced concrete beams with small shear
span-todepth ratio, ACI 318-14 recommends to employ strut-and-tie
model. On the other hand, Eurocode 2 provides a simple
method; applied shear force within short shear span can be
artificially reduced so that more shear force can be resisted
by acknowledging effect of arch action.
As summarized in the above and Table 1, even theoretical
basis is quite different through the shear design provisions,
so predictions show relatively lots of scatter on shear
strength of reinforced concrete beams with stirrups.
Therefore, this paper focuses on development of an analytical
model which can rationally considers effect of arch action,
contribution of concrete, and diagonal angle of compression
strut in reinforced concrete web all together. In order to
simply evaluate effects of the main parameters, the analytical
model is developed to be much simpler than section analysis
with layers model
(Bentz 2001; Guner and Vecchio 2008)
2. Development of Single Web Shear
2.1 Idealization of a RC Beam for Modelling
To evaluate shear strength of a reinforced concrete beam
with stirrups subjected to flexural shear, the reinforced
concrete beam is idealized to have three layers; top
compression chord, single web shear element, and bottom
tension chord, as illustrated in Fig. 1.
In the idealized reinforced concrete beam under the
concentrated load (P) in Fig. 1, the followings are
fundamentally assumed; (1) the top compression and bottom tension
chords fully resist the flexural moment caused by the
concentrated load; (2) the web shear element along with the top
compression chord supports shear stress developed in the
RC beam. Based on the assumption that the top compression
chord is not cracked while the web shear element is cracked,
shear stress exhibits parabolic distribution through the
noncracked region based on
Gere and Goodno (2013)
exhibits a uniform distribution through the web shear
element, as presented in
Vecchio and Collins (1988)
shear stress distribution was analyzed for RC beams with
rectangular section; and (3) critical section is located at the
middle of the main diagonal crack based on
Collins et al.
. Figure 2 illustrates the idealized shear stress
distribution along depth.
2.2 Flexural Behavior with Top and Bottom
Adopting Bernoulli’s hypothesis, linear longitudinal strain
distribution along the cross section of the RC beam can be
considered. To satisfy force equilibrium to the longitudinal
Vs ¼ Avfvyd=s
Vs ¼ Avfvyz=s
Vs ¼ Avfvyz cot h=s
Main diagonal angle (h)
direction, compression at the top of the section is assumed to
be fully resisted by the concrete section while the tension at
the bottom of the section is resisted by the longitudinal
tensile reinforcement. Thus, the following equation can be
ec Ec Ac;top ¼ es Es As
where ec and es are the longitudinal strains at the top and
bottom chords due to the bending moment. In satisfying the
above equation, it should be noted that the concrete and
main longitudinal tensile reinforcement still exhibit a linear–
elastic behavior as most reinforced concrete beams in
literature exhibit shear failure before yielding. If nonlinearity is
exhibited due to significant flexural bending moment,
Eq. (1) should be modified accordingly to consider the
stress–strain response of the concrete and reinforcements.
After diagonal cracking, the web shear element is
subjected to additional longitudinal compressive force. The
longitudinal compressive force on the web shear element due
to shear force is defined as follows:
Nx ¼ rx bw z
where Nx is longitudinal compressive force on the web shear
element due to shear force.
The longitudinal compressive force on the web shear
element due to shear force is assumed to be equally shared
by the top and bottom chords in order to satisfy force
equilibrium through the section. Taking the flexural moment
applied on the critical section, M, and considering the shear
force contribution, the subsequent equations for strains at the
top and bottom chords are derived considering the effect of
the longitudinal compressive force on the web shear
Top compression chord web
C 0.5V cot
where Dec and Des are the additional longitudinal strains at
the top and bottom chords due to the shear force, and z is the
internal lever arm length which can be chosen to 0.9 times
the effective depth of the section. From Eqs. (1)–(4), the
strains at the top and bottom chords can be calculated for a
given flexural moment at the section.
2.3 Longitudinal Strain and Shear Stress at a Shear Critical Section
Using Eqs. (1)–(4), the longitudinal strain at the
middepth ex, can be calculated as following:
This calculation is done by employing numerical analysis
until the force equilibrium to the longitudinal direction is
satisfied. The calculated longitudinal strain serves as
representative longitudinal strain in the web shear element.
Therefore, it is taken into an account for shear analysis of the
web shear element.
By considering the shear stress distribution as described in
Fig. 2, the contribution of the top compression chord on the
shear force is calculated to 0.07 V by considering
geometrical shape of shear stress distribution. Therefore, 0.93 V is
resisted by the web shear element. Since the shear stress
distribution is constant through the web shear element, the
shear stress acting on the web shear element can be
evaluated as following;
sxy ¼ 0:93 b z
where V is the shear force in the critical section.
along top chord
Shear stress distribu on
Taking the shear span in the RC beam subjected to a
concentrated load, the flexural moment is observed to vary
while the shear force remains constant. Therefore, location
of shear critical section should be reasonably chosen for the
analysis. When the principal tensile direction in the web
shear element is h, it can be considered that the critical
section has a distance of 0.5d coth from the concentrated
(Collins et al. 1996)
, as illustrated in Fig. 2. In addition,
when the shear span-to-depth ratio is small, the shear force
acting on the web shear element can be significantly reduced
by arch action
(Park and Paulay 1975)
. Consequently, by
adopting the coefficient to consider the arch action
introduced in Eurocode 2 (2004), the actual shear force at the
critical section can be calculated as follows:
V ¼ a
0:5d cot h
where ba=d ¼ a=2d, not less than 0.25 and not larger than
When the shear span-to-depth ratio is too small, the
principal tensile direction in the web shear element is limited
to d cot h a. In this case, the shear critical section should
be chosen at the middle in the shear span, so the actual shear
force for a given flexural moment can be calculated as
V ¼ 0:5a
cot h [ a=d
2.4 Shear Analysis for the Web Shear Element
From the flexural analysis with the top and bottom chords,
two parameters are essential to the shear analysis for the web
shear element at the critical section; the shear stress acting
on the web shear element and the longitudinal strain at the
mid-depth. These are calculated for a given flexural moment.
Then, the subsequent shear analysis of the web shear
element can be done by employing the MCFT
, which adequately predicts structural behavior
of a reinforced concrete element subjected to bi-axial stress
while considering compatibility, equilibrium, and
constitutive relations of the materials together.
Taking the finite element implementation procedure with
the MCFT derived by
, which employs a
secant material stiffness formulation, the relationship
between the stresses and strains in the web shear element can
be expressed as:
frg ¼ ½D feg
where frg ¼ frx; ry; rxygT denotes the longitudinal,
transverse, and shear stresses, feg ¼ fex; ey; cxygT is longitudinal,
transverse, and shear strains, and [D] is a stiffness matrix of
the web shear element.
Since the web shear element can be considered as a plane
stress element, [D] can be expressed as a 3 9 3 matrix
which can be evaluated for given strains feg through
considering constitutive relations of the materials, concrete and
steel reinforcements. Therefore, Eq. (8) becomes
8 9 2 38 9
< rx = D11 D12 D13 5< ex =
: ry ; ¼ 4 D21 D22 D23 : ey ; ð9Þ
sxy D31 D23 D33 cxy
In the above equation, known variables are the longitudinal
strain at the mid-depth ex and the shear stress acting on the
web shear element sxy, which are calculated from the flexural
analysis. In addition, when the shear span-to-depth ratio is
not so small, the transverse stress of the web shear element
ry can be assumed to 0. Therefore, Eq. (8) has three known
variables ex; ry; cxy and three unknown variables
rx; ey; cxy , so it can be decomposed to two parts, then
transformed as follows:
rx ¼ D11ex þ D12ey þ D13cxy
solved for the unknown variables rx,ey, and cxy.
Consequently, stresses and strains in the web shear element at a
critical section can be calculated for a given flexural
2.5 Stiffness Matrix [D] for the Web Shear
As the web shear element is composed of both concrete
and reinforcement, the stiffness matrix [D] is evaluated
through the superposition of a stiffness matrix for concrete
[Dc] and a stiffness matrix for steel reinforcements [Ds]
described as follows:
½D ¼ ½Dc þ ½Ds
2.5.1 Development of the Stiffness Matrix
for the Steel Reinforcements, [Ds]
Since most reinforced concrete beams with stirrups have
longitudinal and transverse reinforcements without any
inclined steel reinforcements, the stiffness matrix for steel
reinforcements can be simply evaluated from the following
where qsx is an equivalent longitudinal reinforcement ratio in
the web shear element, qsy is a stirrup ratio, Esx and Esy are
secant stiffness for the longitudinal reinforcements and
stirrups. Regarding qsx, several researches assumed that the
main tensile longitudinal reinforcements excluding the area
required to resist the bending moment can be acknowledged
to have contribution on the longitudinal behavior of the web
(Paul et al. 1988; Lee and Kim 2004)
However, this assumption undesirably affects the
longitudinal stiffness in the web shear element, specifically
resulted by large bending moments. In this paper, therefore,
equivalent yield strength of the longitudinal reinforcement in
the web shear element is employed by excluding the tensile
stress of the main longitudinal reinforcement due to the
bending moment, instead of reducing the equivalent
Fig. 3 Principal compressive stress–strain response of
(Vecchio and Collins 1993)
Assume strain at the top, εc
Calculate strain at the bo om chord
Calculate force at the bo om chord
Calculate force at the top chord
Calculate strain at the top chord
Assume εy and γxy
εy== εx , γxy= εx
Calculate ε1, ε2, θ
Calculate rebar stresses
Calculate concrete stresses
Set-up the s ffness matrices
Calculate εy and γxy
Check convergence on
εy and γxy
longitudinal reinforcement ratio. Consequently, as the steel
reinforcements in the web shear element undergo an
elastoplastic behavior, the secant stiffness of the reinforcement in
the web shear element can be evaluated in both directions
with the followings:
where fsx and fsy are stresses of longitudinal and transverse
reinforcements in the web shear element, respectively. Esx
and Esy are the elastic moduli of the longitudinal and
Esx ¼ ex
Esy ¼ ey
transverse reinforcements, respectively, and fsxy and fsyy are
the yield strengths of the steel reinforcements.
2.5.2 Stiffness Matrix for Concrete, [Dc]
To develop the stiffness matrix for concrete [Dc], the
principal strains and its angles in the web shear element should be
calculated, subsequently the principal stresses associated with
these strains can then be evaluated using the constitutive
relations for concrete. For the strains ex, ey, and cxy in the web
shear element, the principal strains e1 and e2, and principal
tensile direction h can be calculated as followings:
h ¼ 21 cos 1 ey
where R is a radius of strain Mohr’s circle, which can be
compressive behavior of concrete is softened by increasing
lateral tensile strain in a cracked reinforced concrete as
presented in Fig. 3
(Vecchio and Collins 1993)
. Based on
, the stress–strain
response of concrete along the principal compressive
direction can be calculated as following:
The principal compressive stress of concrete fc2 can be
evaluated from the principal compressive strain e2, with the
consideration of the compression softening effect that the
where fp ¼ bpfc0 for the peak compressive stress of concrete,
ep ¼ bpec for the strain corresponding to fp, and bp is a factor
to consider the compression softening effect, which depends
on the principal compressive and tensile strains together
presented in the following equation
(Vecchio and Collins
Considering the principal tensile direction, the tensile
response of the concrete behaves differently before and after
cracking. The stress–strain relationship follows a linear–
elastic behavior before cracking, while tensile stress decays
slowly as the tensile strain increases after cracking due to the
bond interaction between the concrete and the
reinforcement; this is known as the tension stiffening effect. By
adopting the tension stiffening model proposed by
and Collins (1982)
in Fig. 4, the principal tensile stress of
concrete in the web shear element can be calculated as
where fcr is the cracking strength of concrete, and ecr is the
cracking strain calculated from ecr ¼ fcr=Ec where Ec is the
elastic modulus of concrete.
In Eq. (22b), fc1;max is the maximum limit of the tension
stiffening effect, which considers local yielding of the steel
reinforcements along crack. For the web shear element
containing longitudinal and transverse steel reinforcements,
fc1;max is calculated as following:
Upon deriving the principal stresses in the concrete, the
stiffness matrix for the concrete in the web shear element can
then be developed using the secant stiffness formulation
where EC1 ¼ fc1=e1, EC2 ¼ fc2=e2, and Gc ¼
EC1EC2= EC2 þ EC2 as illustrated in Figs. 3 and 4.
To express ½Dc 0 along the longitudinal direction, the
principal tensile stress angle, h, is used to develop a
transformation matrix given as:
2 cos h sin h
2 cos h sin h
cos h sin h
cos h sin h
cos2 h sin2 h
This is then combined with the local stiffness matrix for
concrete ½Dc 0 to give the global stiffness matrix for concrete
[Dc] using the equation:
2.6 Analysis Algorithm
In Fig. 5, the analysis algorithm describing the overall
process of the proposed model to evaluate shear strength of
RC beams is presented. The algorithm can be divided into
two parts; the flexural analysis part and the shear analysis
In the analysis algorithm, the longitudinal strain ex at
middepth in a shear critical section is initially given. With the
given ex, the compressive strain at the top chord ec is
assumed, then forces at the top and bottom chords can be
calculated. Through iterative procedure, strain at the top
chord satisfying force equilibrium along the critical section
can be found. Then, shear stress along the web shear element
sxy can be calculated, which will be used as input for the
shear analysis part. For the shear analysis part, the transverse
strain ey and shear strain cxy are initially assumed. Through
the shear analysis procedure for the web shear element, ey
and cxy can be checked for convergence. If convergence is
attained, it is indicated that the reinforced concrete beam can
resist the applied load. To find the ultimate shear capacity,
the longitudinal strain at the mid-depth ex is gradually
increased, and the whole procedure is repeated again until ey
and cxy diverge. When divergence is occurred, the shear
force at the last analysis step just before the divergence can
be defined as ultimate shear capacity of the reinforced
concrete beam. The analysis was conducted with Matlab
3. Verification of the Proposed Model
3.1 Database for the Verification
Verification of the proposed model was conducted with
201 test results of RC beams with a rectangular cross section
and stirrups exhibiting shear failure before yielding of the
main longitudinal reinforcement
(Clark 1951; Bresler and
Scordelis 1963; Delbaiky and Elniema 1982; Smith and
Vantsiotis 1982; Mphonde and Franz 1984; Hsiung and
Frantz 1985; Elzanaty et al. 1986; Narayanan and Darwish
1987; Johnson and Ramirez 1989; Mau and Hsu 1989;
Roller and Russell 1990; Saram and Al-Musawi 1992; Xie
et al. 1994; Kriski 1996; Yoon et al. 1996; Shin et al. 1996;
McGormley et al. 1996; Tan et al. 1997; Kong and Rangan
1998; Collins and Kuchma 1999; Peng 1999; Oh and Shin
2001; Angelakos et al. 2001; Tompos and Frosch 2002; Cho
. The shear strength based on Eurocode 2 (2004), ACI
318-14 (2014) and CSA A23.3 (2014) are also evaluated on
these specimens, and compared against the results from the
Parameters known to affect the shear strength of RC
beams are also investigated to check whether the proposed
model adequately accounts for their contribution in
calculating the shear strength of the RC beams. These parameters
are shown in the Table 2. Noted that the database in the
table consists of 129 beams with a=d [ 2:0 and 72 beams
with a=d 2:0.
3.2 Comparison Results
The results with the proposed model for 201 RC beams
showed a mean value of 1.16 and a coefficient of variance
(CoV) of 0.22 on the comparison between the experimental
results and predicted values. A graphical representation of
this result is presented in Fig. 6. To investigate the adequacy
of the proposed model on prediction of shear strength for RC
beams, comparison results with the predictions by the ACI
318-14 (2014), Eurocode 2 (2004) and CSA A23.3 (2014)
were also presented in the figure. ACI 318-14 gave a mean
value of 1.37 and a CoV of 0.31 while those of the Eurocode
2 gave a mean value of 1.39 and a CoV of 0.36, and those of
CSA A23.3 was 1.39 and 0.32, respectively; the three shear
design provisions showed larger values for the mean as well
as CoV than those of the proposed model as presented in
Significant difference among the proposed model and the
code provisions can also be found through Figs. 7, 8, 9, and
10 where effects of main parameters on shear strength were
investigated. As can be seen in Fig. 7 and 8, no clear
tendency was investigated with the effect of concrete
v fvy (MPa)
rv fvy (MPa)
rv fvy (MPa)
compressive strength and member depth on the predictions
for shear strength. On the other hand, as compared in Fig. 9,
the effect of shear span-to-depth ratio was quite clear; ACI
318-14 and CSA A23.3 significantly underestimated shear
strength of reinforced concrete beams with shear
span-todepth ratio less than 2. Unlike these two design provisions,
predictions by Eurocode 2 were not significantly affected
although it showed lots of scatter from the test results. In the
case of the proposed model, predictions showed good
agreement with the test results regardless of shear
span-todepth ratio as it considered the effect of arch action with
In addition, clear tendency was investigated with the effect
of shear reinforcement ratio on the predictions for shear
strength. As can be seen in Fig. 10, Eurocode 2 significantly
underestimated shear strength of reinforced concrete beams
with small shear reinforcement ratio because concrete
contribution was not acknowledged. On the other hand, the
proposed model still showed good agreement with the test
results since concrete contribution was rationally
acknowledged through the analysis of web shear element by
employing the MCFT. It is noted that scatters with ACI
318-14 and CSA A23.3 were mainly due to the effect of
shear span-to-depth ratio as investigated in Fig. 9, not due to
the effect of shear reinforcement ratio.
Figure 11 shows more detailed investigations focused on
angle of concrete compressive strut or diagonal crack which
means how many stirrups have contribution on shear
strength. As can be seen in the figure, the diagonal crack
angle was constant to 45 according to ACI 318-14 while it
was evaluated to 22 in most cases according to Eurocode 2.
From these results, it can be inferred that shear strength
provided by stirrups is generally underestimated by ACI
318-14 while overestimated by Eurocode 2. On the other
hand, the proposed model and CSA A23.3 showed similar
evaluation results on the diagonal crack angle since they
were developed through simplification of the MCFT,
although more scatter was found with the proposed model.
In addition to the diagonal crack angle, contribution of
concrete on the shear strength was also investigated with a
coefficient b, which is used in CSA A23.3 to evaluate the
concrete contribution. The coefficient b can be evaluated
from the following equation;
Figure 12 shows b values evaluated from the proposed
model and the code provisions. As can be seen in the figure,
b was constant by ACI 318-14 and Eurocode 2; 0.18 and
0.00, respectively. CSA A23.3 evaluated some variation on
b mainly due to the flexural effect on the web. b was
predicted within some range regardless of shear span-to-depth
ratio. Since the effect of arch action was not considered,
Vpn ffiffiffi Vs
rv fvy (MPa)
rv fvy (MPa)
CSA A23.3 generally tended to underestimate shear strength
of reinforced concrete beams with short shear span-to-depth
ratio. On the other hand, the proposed model evaluated that
b increased as shear span-to-depth ratio decreased when the
shear span-to-depth ratio was smaller than 2, since it took the
effect of arch action into the account.
As investigated in Figs. 11 and 12, it can be concluded
that the proposed model well captured stirrup and concrete
contributions together through rational evaluation on
diagonal crack angle and b coefficient. Consequently, this was
resulted in comparison results that the proposed model
showed good agreement with the test results on shear
strength of reinforced concrete beams with stirrups.
rv fvy (MPa)
rv fvy (MPa)
In this paper, a rational analysis model to evaluate shear
strength of reinforced concrete beams with stirrups has been
developed. To develop the analysis model, a reinforced
concrete beam was idealized with top and bottom chords and
a single web shear element, which were designated to resist
flexural bending moment and shear force, respectively. With
consideration of the top and bottom chords, simple flexural
analysis was employed, which gave longitudinal strain at the
mid depth in a shear critical section. With the strain at the
mid-depth, shear analysis was conducted for the single web
shear element which was treated as a cracked orthotropic
reinforced concrete element. For the shear analysis, finite
element analysis procedure based on the MCFT was
modified accordingly to consider the strain at the mid-depth.
Thus, interaction between flexural and shear behaviors in a
reinforced concrete beam could be rationally considered.
For verification of the developed analysis model, 201
reinforced concrete beams with stirrups exhibiting shear
failure were analyzed. The comparison between the test
results and predictions showed that the proposed analysis
model well predicted the actual shear capacities of reinforced
concrete beams while the shear design provisions such as
ACI 318-14, Eurocode 2, and CSA A23.3 showed
significant scatter on the predictions. Furthermore, contributions of
stirrups and concrete, which were noted to h and b,
respectively, were evaluated with the proposed analysis
model, and compared with the shear design provisions. It
was investigated that the proposed analysis model well
captured effect of main parameters on h as like CSA A23.3,
and it well reflected concrete contribution on shear strength,
especially for beams with small shear span-to-depth ratio.
Although the developed analysis model is more
complicating than the shear design provisions, contributions of
concrete and stirrups at the ultimate can be rigorously
evaluated, so it might be useful to develop more rational
shear design provisions through simplification. In addition,
through simple modification, the proposed analysis model
would be useful to evaluate shear strength of reinforced
concrete beams with advanced materials such as steel fibers,
FRP sheets, and so on.
Since the developed analysis procedure considers
linearelastic behavior for the top and compression chords, the
developed analysis procedure should be adequately modified
to more precisely evaluate shear strength of reinforced
concrete beams exhibiting yielding of main longitudinal
rebars or crushing of top compression chord.
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