Applied Mathematics, 2013, 4, 1568-1582
Published Online November 2013 (http://www.scirp.org/journal/am)
http://dx.doi.org/10.4236/am.2013.411212
Open Access AM
Fracture Response of Reinforced Concrete Deep Beams
Finite Element Investigation of Strength and Beam Size
Guillermo A. Riveros1, Vellore Gopalaratnam2
1Information Technology Laboratory, US Army Engineer Research and Development Center, Vicksburg, MS, USA
2Department of Civil and Environmental Engineering, University of Missouri-Columbia, Columbia, MO, USA
Email: Guillermo.A.Riveros@us.army.mil
Received January 8, 2013; revised February 8, 2013; accepted February 15, 2013
Copyright © 2013 Guillermo A. Riveros, Vellore Gopalaratnam. This is an open access article distributed under the Creative Com-
mons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work
is properly cited.
ABSTRACT
This article presents a finite element analysis of reinforced concrete deep beams using nonlinear fracture mechanics.
The article describes the development of a numerical model that includes several nonlinear processes such as compres-
sion and tension softening of concrete, bond slip between concrete and reinforcement, and the yielding of the longitu-
dinal steel reinforcement. The development also incorporates the Delaunay refinement algorithm to create a triangular
topology that is then transformed into a quadrilateral mesh by the quad-morphing algorithm. These two techniques al-
low automatic remeshing using the discrete crack approach. Nonlinear fracture mechanics is incorporated using the fic-
titious crack model and the principal tensile strength for crack initiation and propagation. The model has been success-
ful in reproducing the load deflections, cracking patterns and size effects observed in experiments of normal and
high-strength concrete deep beams with and without stirrup reinforcement.
Keywords: Automatic Remeshing; Bond Slip; Concrete; Discrete Crack; Finite Element; Fracture Mechanics; Size
Effects; Tensile Softening
1. Introduction
Reinforced concrete (RC) deep beams have useful appli-
cations in tall buildings, offshore structures, foundations,
and military structures. A significant number of failures
in RC structures initiate in tension regions caused by
areas of high-stress concentrations or preexisting cracks.
Stable growth of these tensile cracks, until peak loads, is
associated with the development of large zones of frac-
ture (fracture process zone (FPZ)). The growth of the
FPZ, until peak load is reached, introduces the effect of
structural size on the failure loads. Hence, if one was to
design structures based on equations that were developed
based on strength analysis, as in current American Con-
crete Institute (ACI) code [1], the margin of safety pro-
vided would depend upon the size of the structure. The
margin of safety will be higher for smaller structures than
for larger ones. It is also conceivable that this approach
would lead to unconservative designs for some very large
structures, e.g., deep slabs for underground storage tanks.
Early attempts [2] to analyze failure in concrete struc-
tures caused by crack growth were not successful, even
though it was obvious that a fracture mechanics approach
would be realistic to model brittle crack propagation type
failures. The lack of success in the early attempts to ana-
lyze crack propagation failures was due to the use of lin-
ear elastic fracture mechanics (LEFM). LEFM assumes
that the fracture process is small and can be replaced, and
that the rest of the member volume remains elastic;
however, research in the last four decades has resulted in
modifications to LEFM to account for the distributed
nature of pre-peak micro-cracking and the presence of a
large FPZ in concrete [3-6]. These modifications have
produced better results in the application of fracture me-
chanics concepts to brittle failure in reinforced concrete.
Theories that allow tensile softening and FPZ of rela-
tively large sizes are classified as nonlinear fracture me-
chanics models.
A considerable effort has been committed to develop
numerical models to simulate the fracture behavior of
materials exhibiting tensile softening and FPZ, such as
mortar, concrete, rock, or bricks used in civil engineering
structures [4,7]. Two numerical methods to simulate frac-
ture are available; the smeared crack approach and dis-
G. A. RIVEROS, V. GOPALARATNAM 1569
crete crack approach. In the smeared crack approach, in-
troduced by [8], the crack is replaced by a continuous
medium with altered mechanical properties. Because the
crack is established through stress computations at inte-
gration points, a significant number of cracks with small
openings are imagined to be continually distributed over
the finite element. The constitutive laws, defined by stress-
strain relations, are nonlinear and may exhibit strain sof-
tening. Strain localization instabilities and spurious mesh
sensitivity of finite element calculations are likely, when
strain softening is modeled numerically. These difficul-
ties can be overcome by adopting appropriate mathe-
matical techniques [6].
In the discrete crack approach, the crack is formulated
as a geometrical change that requires remeshing each
time a crack is initiated or propagated. The computa-
tional demand, as a result, has been one of the biggest
drawbacks of the method; however, this article shows the
development of multicrack initiation and propagation pro-
cedures that enhance the method and make it less cum-
bersome. The cohesive crack model, developed by [2]
and discussed in [9], has been shown to be effective for
modeling the nonlinear fracture behavior of RC (fracture
process of quasi-brittle materials).
Several numerical models have been developed to study
the behavior of brittle failure (shear) of reinforced con-
crete beams [10-13]. Because these models differ in ma-
terial models, element formulations, and solution proce-
dures, a specific approach will be more suited for spe-
cific structures and/or loading situations and less suited
to others [13]; however, nonlinear fracture mechanics
models are capable of analyzing the complete behavior
of reinforced concrete beams of any size and loading
geometry.
This article presents a nonlinear fracture mechanics
finite element code that incorporates nonlinear fracture
mechanics analysis on reinforced concrete beams. The
system uses the discrete crack approach with the ficti-
tious crack model (FCM) [2,9,14,15] to represent the
tensile softening of concrete; the Shah-Fafitis-Arnold model
[16] to characterize compression softening; a nonlinear
bond-slip constitutive model to account for bond-slip
degradation observed when cracks cross the tensile rein-
forcement [12,17] and an elastic, perfectly plastic con-
stitutive model to represent the yielding of the tensile
reinforcement.
A multicrack initiation and propagation routine incor-
porating the Delaunay refinement algorithm [18] to cre-
ate a triangular topology necessary to obtain a high-
quality mesh when multiple cracks are generated in RC
beams that are transformed into a quadrilateral mesh by
the quad-morphing algorithm [19] has also been devel-
oped.
The primary motivation for this investigation is to
study brittle shear failures in reinforced concrete beams
from a nonlinear fracture mechanics and finite element
point of view and to study the implications for shear de-
sign practices. A secondary motivation is to study brit-
tleness in a more general context that includes structural
(size and loading geometry) and material contributions to
brittleness. The model has been successful in reproduc-
ing the load deflections, cracking patterns and size ef-
fects observed in experiments of normal and high-strength
concrete deep beams with and without stirrup reinforce-
ment [20] with shear-span-to-depth ratios a/d of 1.5 and
2.5.
2. Experimental Evaluation
[21] conducted systematic experiments to characterize
the structural and material response of over 150 rein-
forced concrete deep beams with and without shear rein-
forcement. Four beam sizes and three different concrete
mixes were used. The effective depths d used were 50,
100, 200, and 800 mm (2, 4, 8, and 32 in.). Shear-span-
to-depth ratios a/d of 1.5 and 2.5 were also used to char-
acterize different failure modes. Results from the two
larger beams with and without shear reinforcement were
used in this numerical study for comparisons (Figure 1
and Tables 1 and 2).
Compressive response of concrete was obtained using
specimens cored (75-mm (3 in.) diameter, 150-mm (6 in.)
length) and tested under specimen displacement controlled
conditions to obtain the complete (including post-peak
softening) stress strain response [22]. Mode I fracture
parameters were obtained testing notched beams on a
(a)
(b)
Figure 1. Details of beam geometry and loading configura-
tion for beam (a) without stirrup reinforcement and (b)
with stirrup reinforcement.
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G. A. RIVEROS, V. GOPALARATNAM
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Table 1. Dimensional details of the reinforced concrete
beams (dimensions are in mm (inches)).
Size A Size B
L 5486.4
(216.0)
1422.4
(56.0)
S 4876.8
(192.0)
1219.2
(48.0)
H 914.4
(36.0)
241.3
(9.5)
b 152.4
(6.0)
152.4
(6.0)
d 812.8
(32.0)
203.2
(8.0)
1219.2 [1.5]
(48.0)
304.8 [1.5]
(12.0)
a [a/d]
2032.0 [2.5]
(80.0)
508.0 [2.5]
(20.0)
Table 2. Material properties from tests.
Mix
NSC HSC
Property
28 Days Test 28 Days Test
c
f
, MPa
(psi)
32.2
(4668)
43.0
(6238) - 72.9
(10,570)
E, MPa
(psi)
19,289
(2,797,650)
29,320
(4,252,520) - 31,354
(4,547,560)
ft, MPa
(psi)
4.3
(618)
4.6
(664)
Gf, N/mm
(lb/in.)
0.10028
(0.57267)
0.09100
(0.51967)
three-point loading configuration. Material properties and
fracture energy for normal-strength concrete (NSC) and
high-strength concrete (HSC) are presented in Table 2.
For beam sizes A and B, [20] used two ram displace-
ments controlled by a dual ramp command function. For
size A beams he used an initial load ratio of 25 mm (1
in.)/hour for the first half hour and a ratio of 76 mm (3.0
in.)/hour thereafter. For size B beams he used an initial
load ratio of 7.1 mm (0.28 in.)/hour and a ratio of 25.0
mm (1.0 in.)/hour until failure. Load displacement be-
haviors obtained in the experiments were then used in the
numerical model to determine the force boundary condi-
tions needed to predict a similar response.
3. Fracture Mechanics Analysis of
Reinforced Concrete Beams (FMARCB)
A finite element system, Fracture Mechanics Analysis of
Reinforced Concrete Beams (FMARCB) [23], has been
developed to perform nonlinear fracture mechanics analy-
sis on reinforced concrete beams. The system consists of
a graphic input interface, analysis routines using finite
element techniques, and graphic output interface. FMA-
RCB is a two-dimensional finite element program with
triangular (3 and 6 nodes), isoparametric (4 and 8 nodes),
bar (truss), and interface elements (bond-link). The sys-
tem uses the discrete crack approach with the FCM
[2,9,14,15] to represent the tensile concrete softening; the
Shah-Fafitis-Arnold model [16] to characterize the com-
pression softening; a nonlinear bond-slip constitutive
model for the bond-slip phenomenon, which is degraded
when cracks cross the tensile reinforcement [12,17] and
an elastic perfectly plastic constitutive model to represent
the yielding of the tensile reinforcement.
The analysis begins with the definition of the finite
element model of the continuum in the elastic state. Once
the elastic analysis of the system is completed for the
first load step and the principal stresses are extrapolated
at the nodes, cracking criteria based on the principal
tensile stresses are verified. If the principal tensile stress
exceeds the tensile strength, a fictitious crack is incorpo-
rated at the location and automatic remeshing is under-
taken. Once the system has cracked, the nonlinear solver
is activated. If new cracks and extensions are required
after the nonlinear problem satisfies equilibrium for an
unbalanced tolerance, the system is remeshed with new
cracks and the existing crack extensions. It is then cali-
brated again for the same load step until no new cracks or
extensions are required. This iterative process is repeated
for each load step.
FMARCB incorporates the Delaunay refinement algo-
rithm [8] to create a triangular topology that then is
transformed into a quadrilateral mesh by the quad-morph-
ing algorithm [19]. The Delaunay refinement mesh gen-
eration algorithm constructs meshes of triangular ele-
ments. The algorithm operates by imposing a Delaunay
or constrained Delaunay triangulation that is refined by
inserting additional vertices until the mesh meets con-
straints on element quality and size. These algorithms
simultaneously offer theoretical bounds on element qual-
ity, edge lengths, and spatial grading of element sizes.
They also possess the ability to triangulate general
straight-line domains.
Quad-morphing [19] is a technique used for generating
quadrilaterals from an existing triangle mesh. Beginning
with an initial triangulation, triangles are systematically
transformed and combined. Quad-morphing can be cate-
gorized as an unstructured, indirect method that utilizes
an advancing front algorithm to form an all-quad mesh.
As an indirect method it is able to take advantage of local
topology information from the initial triangulation. Unlike
other indirect methods it is able to generate boundary-
Open Access AM
G. A. RIVEROS, V. GOPALARATNAM 1571
sensitive rows of elements, with few irregular nodes.
4. Nonlinear Fracture Mechanics Using the
Fictitious Crack Model
The FCM assumes that there is an inelastic zone (FPZ)
ahead of the crack tip [2,14]. Along the FPZ, the stress
carrying capacity decreases as a function of the crack
opening displacement (COD). The FPZ is characterized
by a normal stress versus COD response (Figure 2),
which is considered a material property. This model
adopts the tensile strength criterion for crack initiation
and subsequent growth. It also assumes that a stress-free
crack occurs when the COD is larger than the critical
COD wc.
5. Material Properties Characterization
Tension softening curve. FMARCB has the capability
to use either a linear or bilinear softening curve (Figure
2). The fictitious crack model is incorporated into the
finite element analysis by employing interface elements.
For a linear softening curve, the critical COD value wc is
2
f
c
t
G
w
f
(1)
where, Gf is the fracture energy, ft is the tensile strength,
and wc is the COD, when the tensile capacity is reduced
to zero.
Figure 2 also shows the bilinear softening curve pro-
posed by [24], where wc is
3.6
f
c
t
G
w
f
(2)
and w1 is
10.8
f
t
G
w
f
(3)
where w1 is the COD at the kink of the bilinear curve, wc
is the COD when the tensile carrying capacity is com-
pletely lost, and the stress at the kink is 1/3 ft. In the
FCM the interface element is a nonlinear function of the
(a) (b)
Figure 2. FMARCB tension softening models: (a) Linear; (b)
Nonlinear.
COD as shown in Figure 2. When interface elements are
used to model the FPZ, care must be taken to avoid di-
vergent numerical behavior. Figure 2 shows that when
the COD is small, the stiffness of the interface element is
large, which requires a small load step for convergence
[25]. A finite initial stiffness has to be used as shown in
Figure 2. An initial stiffness corresponding to wc/20 to
wc/30 has been used by [25] with success.
Compression softening. The compression softening
model used in this work is the one proposed by [16]. The
model describes well, the stress-strain relation for con-
fined and unconfined concrete. The ascending part of the
model is described by
0
0
11
A
ff




(4)
and the descending part as

1.15
0
expff k


0
(5)
where f is the stress corresponding to the predefined
strain ε, and the peak stress f0 and peak strain ε0 for later-
ally confined concrete are defined as
0
2100 KP
0
.15 a1
c
c
ff f
f

 


r
(6a)

0
3048
p
si1.15
c
c
ff f
f

 


r
(6b)
8
00.0291.60.00195 KPa491 r
c
c
f
Ef f

(7a)

7
00.021960.00195 ps.0 i27 r
c
c
f
Ef f
 (7b)
where c
f
is the compressive strength for unconfined
concrete and fr is the confinement pressure.
The confinement pressure fr is then defined as
2vy
r
c
f
f
d
(8)
where Av is the area of the lateral reinforcement, fy is the
yield strength of the stirrups, s is the spacing between
stirrups, and dc is the diameter of the concrete core.
Parameters A and k are constants that were statistically
evaluated from experimental data of unconfined and con-
fined concrete subjected to monotonically increasing load-
ing [16] and are defined as
0
0
c
E
Af
(9)
 
0.025 exp0.00145KPa
cr
kf f
 (10a)
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G. A. RIVEROS, V. GOPALARATNAM
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
0.17 exp0.01psi
cr
kf f

(10b)
where Ec is the secant modulus of elasticity.
Bond-slip curve. The bond between concrete and re-
inforcement is one of the most important factors influ-
encing the capacity of a reinforced concrete beam. Bond
is the load-carrying mechanism between concrete and
reinforcement in the longitudinal direction of the rein-
forcing bar. In regions of high stress at the contact inter-
face, the bond stresses are related to relative displace-
ments, usually called bond-slip, which are caused by
different average strains in the concrete and reinforce-
ment [26].
The bond stress-slip relationship depends on a consid-
erable number of influencing factors including bar rough-
ness (relative rib area), concrete strength, position and
orientation of the bar during casting, state of stress,
boundary conditions, and concrete cover [12].
Figure 3 shows the bond-slip curve used in FMARCB;
here the ascending part of the curve refers to the stage in
which the ribs penetrate into the mortar matrix, charac-
terized by local crushing and microcracking. The de-
scending part that starts at the maximum bonding strength
τmax of the curve refers to the reduction of bond resistance
from the occurrence of splitting cracks, transverse to the
bars. The horizontal part characterizes a residual bond
capacity τmin, which can be attributed to frictional slip
based load transfer.
The following considerations apply to the generation
of bond stresses. Reinforcement and concrete have the
same strain (εs = εc) in those areas of the structure under
compression and in the uncracked parts of the structure
under tension. Bond stresses are generated between the
concrete and the reinforcing steel by the relative dis-
placement ss = us uc where us is the displacement of the
steel and uc is the concrete displacement. The magnitude
of these bond stresses depends predominantly on the steel
stresses, the slip s, the concrete compressive strength c
f
,
and the position of the reinforcement during placing (top
cast or bottom cast). Tension stiffening, a term needed to
describe the contribution of the concrete between cracks
Figure 3. Bond-slip model.
to the stiffness of the cracked concrete beam, is also ef-
fective as a result of the interface bond between steel and
concrete.
Degradation of bond-slip caused by cracking. Bond
behavior has the same important influence on the re-
sponse to applied loads of reinforced concrete beams as
the properties of reinforcement and concrete. Bond stiff-
ness and maximum bond stresses deteriorate near the
cracks in proportion to the distance to the crack and the
bar diameter [17]. [12,17] have reported that bond deg-
radation occurs in the vicinity of flexure cracks. To ac-
count for this degradation of bonding, they recommend
the calculation of the reduction factor α, which is then
applied to the bond stresses of the original bond-slip
function. The reduction factor proposed by [12] is deter-
mined as follows:
0.20 1
s
x
ad
(11)
where x is the distance from the crack-rebar intersection
center line to the desirable location, and ds is the bar di-
ameter.
6. Interface Element
To model tensile softening in concrete and bond-slip for
the steel-concrete interaction, the bond-link element [27]
was implemented in FMARCB. These elements can cal-
culate the stresses generated between any two surfaces
(steel and concrete (bond-slip) or concrete to concrete
(softening)) as a function of the relative displacements
between the surfaces. This type of element relies on
normal and shear stiffness to simulate the strength be-
tween the two surfaces. The constitutive models used for
the concrete tensile softening and the bond-slip are shown
in Figures 2 and 3. As seen in the constitutive models,
the bond-link element requires a nonlinear solver in con-
trast to the linear behavior first proposed by [27].
7. Model Validation
Numerical analyses were conducted on two sizes of
geometrically proportionate reinforced concrete beams
[20] with normal and high compressive strengths with
and without shear reinforcement. The beams were ana-
lyzed with a/d ratios of 2.5 and 1.5. Figure 1 and Table
1 show the beam size and loading configurations, while
Table 3 lists the parameters used in the numerical com-
putations for beams with and without shear reinforce-
ment. Results from numerical models were then com-
pared to experimental results [20]. Further analyses were
also conducted on the larger size beams with shear rein-
forcement. Results of load displacement, cracking pat-
terns, size effects, and concrete strength are discussed in
the following section.
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G. A. RIVEROS, V. GOPALARATNAM
Open Access AM
1573
Table 3. Numerical model parameters for beams without shear reinforcement.
Beam
Parameter ANW21
and
ANW11
AHW21
and
AHW11
BNW21
and
BNW11
BHW21
and
BHW11
ANS22
and
ANS11
AHS22
and
AHS11
E, GPa
(psi)
29
(4.25E6)
31
(4.50E6)
29
(4.25E6)
31
(4.50E6)
29
(4.25E6)
31
(4.50E6)
c
f
, MPa
(psi)
44.8
(6500)
68.9
(10,000)
44.8
(6500)
68.9
(10,000)
44.8
(6500)
68.9
(10,000)
ν 0.18 0.18 0.18 0.18 0.18 0.18
ft, MPa
(psi)
4.1
(600)
4.3
(625)
4.1
(600)
4.3
(625)
4.1
(600)
4.3
(625)
As 2#8 4#8 2#4 4#4 2#8 4#8
As 2#4 2#4
Es, GPa
(psi)
209
(30E6)
209
(30E6)
209
(30E6)
209
(30E6)
209
(30E6)
209
(30E6)
fy, MPa
(psi)
462
(67,000)
462
(67,000)
441
(64,000)
441
(64,000)
462
(67,000)
462
(67,000)
wc, mm
(in.)
0.0484
(0.0019)
0.0422
(0.0016) 0.0484
(0.0019) 0.0422
(0.0016) 0.0484
(0.0019)
0.0422
(0.0016)
τmax, MPa
(psi)
5.5
(800)
5.5
(800)
5.5
(800)
5.5
(800)
5.5
(800)
5.5
(800)
u1, mm
(in.)
0.0127
(0.0005)
0.0127
(0.0005)
0.0127
(0.0005)
0.0127
(0.0005)
0.0127
(0.0005)
0.0127
(0.0005)
u2, mm
(in.)
1.02
(0.04)
1.02
(0.04)
1.02
(0.04)
1.02
(0.04)
1.02
(0.04)
1.02
(0.04)
8. Numerical Solution without Shear
Reinforcement curve toward midspan at beam midheight and continue to
grow. This is shown as Point 3 in Figure 5. Longitudinal
steel yielding initiates at Point 4 in Figure 5. Ultimate
failure occurs after reinforcement reaches failure.
Typical load deflection response. Load deformation
responses are discussed for sizes A and B. The overall
load deformation behavior, observed in size A beams, is
representative of the other sizes. Aspects of the response,
unique to size B members, are also discussed. In the
analysis presented herein, the results shown included the
members’ self-weight.
Figure 6 presents results from the test of an HSC
beam without lateral reinforcement (Beam AHW22). A
shear-span-to-depth ratio of 2.5 was used in the numeri-
cal model. The beam failed from diagonal tension failure.
Once again, the failure was driven by the unstable crack
growth of a flexure shear crack combined with debond-
ing of the longitudinal reinforcement; however, no yield-
ing of the longitudinal reinforcement was observed prior
to failure.
Results from the numerical analysis of an NSC beam
without lateral reinforcement (Beam ANW21) and a shear-
span-to-depth ratio of 2.5, indicated a diagonal tension
failure after yielding of the longitudinal steel reinforce-
ment. This type of failure was driven by an unstable
growth of a flexure shear crack (Figure 4).
Initial stiffness differences result from the higher
modulus for the HSC matrix and the larger steel content
used in the HSC beam. The increased load and deflection
capacity between diagonal cracking and ultimate capac-
ity depends upon the geometry and material characteris-
tics. For the beam geometry and material properties used
in this investigation, ultimate capacity in all the modes of
failure and for all beam sizes was distinct from diagonal
cracking.
Typically, the load deflection response is linear until
the first flexural crack appears in the tension face (Point
1 in Figure 5). Flexural cracks in the inner span of the
beam grow in number and size with continued loading.
Further loading produces diagonal cracks at the mid-
height of the beam. This stage in the load deflection re-
sponse is denoted as Point 2 in Figure 5. At this load
level, debonding of the steel begins; with additional load,
the bond capacity deteriorates, reflecting added nonlinear
behavior that causes deflections to increase more rapidly.
Also, some flexural cracks that develop in the shear span
In the case of NSC and HSC beams without stirrup re-
inforcement analyzed at an a/d ratio of 1.5 (Beams
ANW11 and AHW11 in Table 3 ), multiple diagonal ten-
sion cracks in each shear span (as shown in Figure 7, to
be discussed later) were observed at incipient failure. A
G. A. RIVEROS, V. GOPALARATNAM
1574
7153.00 7348.00
(a)
(b)
1668.00 1668.00
(c)
(d)
Figure 4. Final cracking pattern: (a) ANW21 numerical model; (b) ANW21 experiment; (c) BNW21 numerical model; (d)
BNW21 experiment.
0
5
10
15
20
25
30
35
40
45
0.0 0.2 0.4 0.6 0.8
CENTER DEFLECTION (IN.)
LOAD (KIPS)
0
50
100
150
200
0510 15 20
P
LEFT
CENTER DEFLECTION (MM)
LOAD (KN)
Experimental
FMARCB
12
3
4
a
0
5
10
15
20
25
30
35
40
45
0.00.20.40.60.8
CENTER DEFLECTION (IN.)
LOAD (KIPS)
0
50
100
150
200
05101520
P
RIGHT
CENTER DEFLECTION (MM)
LOAD (KN)
Experimental
FMARCB
1
2
3
4
b
Figure 5. Left and right load displacement responses, respectively, for ANW21.
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G. A. RIVEROS, V. GOPALARATNAM 1575
0
10
20
30
40
50
60
70
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
CENTER DEFLECTION (IN.)
LOAD (KIPS )
0
50
100
150
200
250
300
0510 15 20
P
LEFT
CENTER DEFLE CTION (MM)
LOAD (KN)
Experimental
FMARCB
a
0
10
20
30
40
50
60
70
0.0 0.1 0.2 0.30.4 0.5 0.6 0.7 0.8
CENTER DISPLACEMENT (IN.)
LOAD (KIPS)
0
50
100
150
200
250
300
0510 15 20
P
RIGHT
CENTER DEFLECTION (MM)
LOAD (KN)
Experimental
FMARCB
b
Figure 6. Left and right load displacement responses, respectively, for AHW22.
18787.00 23000.00
(a)
(b)
6840.00 6840.00
(c)
(d)
Figure 7. Final cracking pattern: (a) AHW11 numerical model; (b) AHW11 experiment; (c) BHW11 numerical model; (d)
BHW11 experiment.
Open Access AM
G. A. RIVEROS, V. GOPALARATNAM
Open Access AM
1576
combination of ultimate diagonal tension failure and
shear compression failures resulted from the catastrophic
growth of these diagonal cracks. Shear compression fail-
ure occurred when diagonal cracks penetrated the com-
pression region and compressive strength was reached
(Figure 7). Reinforcement yielding began prior to the
ultimate failure (Figures 8 and 9, Point 1).
The load displacement responses for Size B are shown
in Figures 10 and 11. For these beams, analyzed with an
a/d ratio of 1.5 and 2.5, a diagonal compression and a
diagonal shear failure similar to the one discussed for the
Size A beam were observed. However, a fewer number
of cracks were observed prior to failure.
General observations on the crack patterns. Fig-
ures 4 and 7 include numerical and experimental crack-
ing patterns of the two different beam sizes without stir-
rup reinforcement for NSC beams with an a/d ratio of 2.5
and HSC beams with an a/d ratio of 1.5. NSC beams
with an a/d ratio of 1.5 and HSC beams with an a/d ratio
of 2.5 were observed to have a somewhat similar crack-
ing pattern in contrast to the beams with similar a/d ra-
tios.
0
20
40
60
80
100
0.0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8
CENTER DEFLECTION (IN.)
LOAD (KIPS)
0
100
200
300
400
0510 15 20
P
LEFT
CENTER DEFLECTION (MM)
LOAD (KN)
Experimental
FMARCB
a
1
0
20
40
60
80
100
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
CENTER DEFLECTION (IN.)
LOAD (KIPS)
0
100
200
300
400
0510 15 20
P
RIGHT
CENTER DFLECTION (MM)
LOAD (KN)
Experimental
FMARCB
b
1
Figure 8. Left and right load displacement responses, respectively, for ANW11.
0
20
40
60
80
100
120
140
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
CENTER DEFLECTION (IN.)
LOAD (KIPS)
0
100
200
300
400
500
0510 15 20
P
LEFT
CENTER DEFLECTION (MM)
LOAD (KN)
Experimental
FMARCB
a
1
0
20
40
60
80
100
120
140
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
CENTER DEFLECTION (IN.)
LOAD (KIPS)
0
100
200
300
400
500
0510 15 20
P
RIGHT
CENTER DEFLECTION (MM)
LOAD (KN)
Experimental
FMARCB
b
1
Figure 9. Left and right load displacement responses, respectively, for AHW11.
0.0
2.0
4.0
6.0
8.0
10.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
CENTER DEFLECTION (IN.)
LOAD (KIPS)
0
10
20
30
40
00.511.522.53
BNW21
CENTER DEFLECTION (MM)
LOAD (KN)
Experimental
FMARCB
0.0
2.0
4.0
6.0
8.0
10.0
12.0
0.00 0.02 0.04 0.06 0.08 0.10
CENTER DEFLECTION (IN.)
LOAD (KIPS)
0
10
20
30
40
50
00.511.52
BHW21
CENTER DEFLECTION (MM)
LOAD (KN)
Experimental
FMARCB
Figure 10. Load displacement responses for BNW21 and BHW21.
G. A. RIVEROS, V. GOPALARATNAM 1577
0.0
5.0
10.0
15.0
20.0
25.0
30.0
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
CENTER DEFLECTION (IN.)
LOAD (KIPS)
0
25
50
75
100
125
012345678
BNW11
CENTER DEFLECTION (MM)
LOAD (KN)
Experimental
FMARCB
0.0
10.0
20.0
30.0
40.0
50.0
0.00 0.05 0.10 0.15 0.20 0.25 0.30
CENTER DEFLECTION (IN.)
LOAD (KIPS)
0
50
100
150
200
01234567
BHW11
CENTER DEFLECTION (MM)
LOAD (KN)
Experimental
FMARCB
Figure 11. Load displacement responses for BNW11 and BHW11.
Figure 7 shows a unique type of failure; shear com-
pression after yielding of longitudinal steel (AHW11).
For the a/d ratio used (1.5) and the material parameters
chosen for HSC (Table 2), the shear capacity and flex-
ural capacity are nearly comparable. The diagonal crack
penetrated the compression region, and compressive
strength was reached. Failure in NSC beams without lat-
eral reinforcement was observed to be often accompanied
by debonding of the longitudinal reinforcement (Figure
4). For both the NSC and HSC beams, general inclination
of diagonal cracks was influenced by a/d ratios used. For
an a/d ratio of 1.5, main diagonal cracks appeared to
span from the support to the load point in each shear span
(Figure 7). For an a/d ratio of 2.5, the diagonal cracks
were generally z-shaped, often connected with debonding
of the longitudinal reinforcement (Figure 4). Debonding
started when the first flexural crack crossed the longitu-
dinal reinforcement and ended at catastrophic diagonal
tension shear failure. The general crack and failure pat-
terns obtained from the numerical analysis in each case
correlated well with those obtained from the experi-
ments.
Crack patterns and failure modes in the smaller size
(Size B) of NSC and HSC beams without stirrup rein-
forcement for a/d ratios of 1.5 and 2.5 are similar to
those for Size A beams analyzed at the same a/d ratios.
Generally, the numerical model predicted fewer cracks,
which is similar to experimental observations for smaller
beams (Size B) [20].
Influence of specimen size. Brittle fractures [3,6,28]
are responsible for size effects observed in concrete struc-
tures. Shear failures in reinforced concrete beams with-
out shear reinforcement have been observed to be more
sensitive to beam size. Since many factors such as mate-
rial property, reinforcement content, and loading geome-
try affect brittleness of shear failure, it is expected that
these parameters will likely influence the size effect as
well. While the understanding of size effect in the failure
of plain concrete is good, only limited conclusive data
are available on how reinforcement affects size effect. It
is generally believed that if the reinforcement remains
elastic and bonded to concrete, size effect similar to that
observed for plain concrete will also be observed for re-
inforced concrete; however, if reinforcement yields or
slips, the size effect is expected to become milder or
stronger, respectively. Also, the presence of lateral rein-
forcement is expected to make size effect insensitive.
Even though voluminous data on shear failure of rein-
forced concrete beams are available in the literature, only
a limited number of these investigations provide all the
information needed for systematic fracture analysis. It is
hoped that the data obtained from the numerical model-
ing in this investigation would be a modest beginning in
providing additional answers to questions on size effect
in failure of reinforced concrete.
Diagonal crack initiation has been reported to be less
size dependent than ultimate failure in shear failure [29].
This observation is also valid based on the analysis com-
pleted for this investigation; however, the differences are
less significant in magnitude. A closer examination of the
numerical models and the experimental results showed
that size effect at diagonal crack initiation was only mar-
ginally less size dependent than that at ultimate failure
for very deep beams without stirrup reinforcement. A 42
percent reduction in strength at diagonal crack initiation
for an increase in effective depth from 0.2 to 0.8 m (8 to
32 in.) compared to an approximately 47 percent drop in
the ultimate capacity for a corresponding increase in
specimen depth (strut and tie action in the post-diagonal
cracking regime reported for these specimens). It should
be noted that conclusions on the extent of size effect at
ultimate capacity are strongly dependent on the failure
mode.
Size effect is milder for an a/d ratio of 1.5 than for one
of 2.5 for both the NSC and HSC beams. Strength reduc-
tion caused by shear failure as a function of a/d ratios can
be treated as a geometry or structural-configuration-re-
lated brittleness. If brittleness and size effect are implic-
itly related as implied in fracture mechanics analysis, it is
not surprising that an a/d ratio of 2.5 would exhibit a
Open Access AM
G. A. RIVEROS, V. GOPALARATNAM
1578
stronger size effect. Ultimate shear strength presents a
marginally milder size effect than that observed for di-
agonal cracking. It should be pointed out that these ob-
servations are from failures where yielding of longitudi-
nal steel preceded ultimate failure in shear, implying that
shear capacity may be comparable to the flexural capac-
ity. Ultimate shear strength for HSC also exhibits only a
mild size effect in spite of the fact that most failures are
shear failures that occur prior to yielding of the longitu-
dinal steel. Size effect is not significantly different from
that observed at diagonal crack initiation. This is some-
what similar to the deep beam test reported by [29]. Cau-
tion should be exercised in making generalizations re-
garding the influence of size effect on the ultimate ca-
pacity of reinforced concrete beams, particularly when
comparing failure types that are not exactly identical.
Influence of concrete strength. The stress at flexural
cracking, diagonal crack initiation, and ultimate capacity
are all larger for the HSC beams than for NSC beams. It
was expected that the size effect for HSC would be
stronger than for the NSC beams. No conclusive obser-
vations could be made concerning size effect either in
diagonal crack initiation or at ultimate capacity. The size
effect with regard to diagonal crack initiation was ob-
served to be comparable in the two concrete materials.
The size effect at ultimate capacity even with the slightly
different failure modes (for beams without stirrup rein-
forcement) was again comparable. One possible explana-
tion for the lack of distinct difference in size effect be-
tween the two concrete materials is that even though the
compressive strength ratio is 1.7, the tensile strength ra-
tio is approximately 1.3. Perhaps if the compressive
strengths differed by a greater amount, one could have
possibly seen stronger size effect for the HSC material.
9. Numerical Solution with Shear
Reinforcement
Numerical analyses of the Size A beams with shear rein-
forcement were conducted. The spacing and shear rein-
forcement content followed that was specified in [20].
Because of the confinement introduced by the shear re-
inforcement, a plane strain assumption was utilized in the
analysis. Beam geometry is shown in Figure 1 and Ta-
ble 1, and material properties for Size A beams are
shown in Tables 2 and 3.
Typical load deflection response. Results from the
numerical analysis of an NSC beam with lateral rein-
forcement (Beam ANS22) and a shear-span-to-depth ratio
of 2.5 indicated a flexural failure after yielding of the
longitudinal steel reinforcement. One of the factors of
this ductile type of failure is the confinement pressure
provided by the shear reinforcement, which reduces the
initiation and growth of tension shear cracks. Further-
more, the confinement pressure provides additional bond-
ing capacity, limiting the debonding of the tensile rein-
forcement. Load displacement curves for Size A beams
with shear reinforcement are shown in Figures 12-15.
The capacity of this beam increased by 10 percent com-
pared with that of the beam without shear reinforcement;
however, the main contribution was that the failure mode
changed from a brittle to a ductile failure.
In the case of NSC and HSC beams with stirrup rein-
forcement analyzed at an a/d ratio of 1.5, flexure failure
occurred after yielding of the tensile reinforcement prior
to crushing of the concrete. Once again, the confinement
pressure provided by shear reinforcement delayed the
initiation and catastrophic propagation of the diagonal
tension cracks in each shear span (Figures 14 and 15).
In general, numerical results show that the presence of
confinement pressure equivalent to the shear reinforce-
ment does not make a significant difference in the per-
formance of the Size A beams until the initiation of the
diagonal cracks. Furthermore, the amount of confinement
pressure equivalent to the stirrup spacing will greatly alter
the failure mode in reinforced concrete beams. The gen-
eral load deflection curves obtained from the numerical
analysis in each case are comparable with those obtained
from the experiments.
0
10
20
30
40
50
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
CENTER DEFLECTION (IN.)
LOAD (KIPS)
0
50
100
150
200
0510 15 20
P
LEFT
CENTER DEFLECTION (MM)
LOAD (KN)
Experime ntal
FMARCB
a
0
10
20
30
40
50
0.0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9
CENTER DEFLEC TI ON (I N.)
LOAD (KIPS)
0
50
100
150
200
0510 15 20
P
RIGHT
CENTER DEFLECTION (MM)
LOAD (KN)
Experimental
FMARCB
b
Figure 12. Left and right load displacement respons es, respectively, for ANS22.
Open Access AM
G. A. RIVEROS, V. GOPALARATNAM 1579
0
20
40
60
80
100
0.00 0.20 0.400.60 0.80 1.00
CENTER DEFLECTION (IN.)
LOAD (KIPS)
0
100
200
300
400
0510 15 20 25
P
LEFT
CENTER DEFLECTION (MM)
LOAD (KN)
Experimental
FMARCB
a
0
20
40
60
80
100
0.00 0.20 0.40 0.60 0.80 1.00
CENTER DEFLECTION (IN)
LOAD (KIPS)
0
100
200
300
400
0510 15 20 25
P
RIGHT
CENTER DEFLECTION (MM)
LOAD (KN)
Experimental
FMARCB
b
Figure 13. Left and right load displacement respons es, respectively, for AHS22.
0
20
40
60
80
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
CENTER DEFLECTION (I N.)
LOAD (KIPS)
0
100
200
300
400
0510 15 20
PRIGHT
CENTER DEFLECTION (MM)
LOAD (KN)
Experimental
FMARCB
b
0
20
40
60
80
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
CENTER DEFLECTION (IN.)
LOAD (KIPS)
0
100
200
300
400
051015 20
P
LEFT
CENTER DEFLECTION (MM)
LOAD (KN)
Experimental
FMARCB
a
Figure 14. Left and right load displacement respons es, respectively, for ANS11.
0
20
40
60
80
100
120
140
160
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
CENTER DEFLECTION (IN.)
LOAD (KIPS)
0
100
200
300
400
500
600
700
0510 15 20
P
LEFT
CENTER DEFLECTION (MM)
LOAD (KN)
Experimental
FMARCB
a
0
20
40
60
80
100
120
140
160
0.00 0.100.20 0.30 0.400.500.60 0.700.80 0.90
CENTER DEFLECTION (IN.)
LOAD (KIPS)
0
100
200
300
400
500
600
700
051015 20
P
RIGHT
CENTER DEFLECTION (MM)
LOAD (KN)
Experimental
FMARCB
b
Figure 15. Left and right load displacement respons es, respectively, for AHS11.
General observations on the crack patterns. Figure
16 includes numerical cracking patterns of Size A beams
with stirrup reinforcement (confined pressure) for NSC
and HSC beams with a/d ratios of 2.5 and 1.5, respec-
tively.
All Size A beams with shear reinforcement failed in
flexure. For the a/d ratio of 2.5 and the material parame-
ters chosen for NSC and HSC (Table 1), a reduction of
the amount of debonding was observed in addition to a
delay in the formation of the flexure shear cracks. For
both the NSC and HSC beams, general inclination of
diagonal cracks was influenced by a/d ratios used; how-
ever, diagonal cracks did not propagate in an unstable
manner, allowing the tensile reinforcement to yield prior
to the crushing of the concrete. For an a/d ratio of 1.5,
main diagonal cracks appeared to span from the support
to the load point in each shear span (Figure 16). For an
a/d ratio of 2.5, the diagonal cracks were generally z-
shaped, often connected with reduced amount of debond-
ing of the longitudinal reinforcement (Figure 16). Debond-
ing started when the first flexural crack crossed the lon-
gitudinal reinforcement; however, the bonding capacity
Open Access AM
G. A. RIVEROS, V. GOPALARATNAM
1580
7581.00 7422.00 ANS22
(a)
15271.00 13733.00 ANS22
(b)
12598.00 13416.00 ANS11
(c)
22864.00 23645.00 ANS11
(d)
Figure 16. Final nu merical cracking pattern for beams with shear rein forcement.
was larger because of the confinement pressure provided
by the shear reinforcement, which allows a ductile type
of failure.
10. Conclusions
Size effect in strength and deformation capacity.
Brittle shear failure in NSC and HSC beams without stir-
rup reinforcement, exhibited effects of size on ultimate
strength as well as corresponding deflections for effec-
tive beam depths of 0.2 and 0.8 m (8 and 32 in.). Stress at
diagonal crack initiation was observed to be less size
dependent. In this investigation, size effect at the ulti-
mate shear capacity was only marginally more size-de-
pendent than that observed for diagonal crack initiation.
Size effect on the deflection capacity at diagonal crack
initiation observed in the numerical analysis may be of
practical relevance in design. Although direct compari-
son of size effect in the deflection value at the ultimate
capacity was not made because of the differences in fail-
ure mechanisms, size effect similar to that at diagonal
crack initiation was observed at this loading.
Shear-span-to-depth ratio and geometry-related bri t-
tleness. For the two shear-span-to-depth ratios investi-
gated (a/d of 1.5 and 2.5), the failure in beams without
stirrup reinforcement was due predominantly to diagonal
tension and shear compression. Reduction in shear ca-
pacity compared to the flexural capacity in all cases in-
vestigated was more severe for the a/d ratio of 2.5. This
is in line with Kani’s [30] shear valley concept. Distinct
changes in crack patterns and resultant mode of failure
also accompanied changes in the a/d ratio. Size effect
was greater at an a/d ratio of 2.5. This observation can be
Open Access AM
G. A. RIVEROS, V. GOPALARATNAM 1581
treated as geometry-related brittleness in analytical mod-
els.
Concrete compressive strength. The shear strength
of HSC beams (compressive strength of 70 MPa (10,000
psi)) was markedly higher than that of NSC beams (com-
pressive strength 43 MPa (6250 psi)) at diagonal crack
initiation and at the ultimate capacity. Even while the
HSC was more brittle than the NSC, no noticeable dif-
ferences in the size effect on failure loads were observed.
Nonlinear fracture mechanics-based model. The
nonlinear fracture mechanics-based numerical model
described herein has unique features including automated
crack initiation and propagation, automated remeshing,
and solution of several nonlinear phenomena (concrete
softening in tension and compression, bond slip, and
yielding of reinforcement).
The numerical model developed to study the shear be-
havior of reinforced concrete deep beams has been vali-
dated with eight beams of two different sizes with dif-
ferent material properties and loading geometries. The
model successfully predicted the ultimate capacity of the
beams described herein. The model shows good correla-
tion between the predicted cracking pattern and the ex-
perimental cracking pattern. It also predicted the load
displacement response successfully. Bond-slip character-
istics exert significant influence on load deflection char-
acteristics of the reinforced concrete deep beams and
should be implicitly incorporated into any numerical
fracture model for the flexural behavior of reinforced
concrete beams. The model also shows no need to use the
shear capacity for the tension softening.
11. Acknowledgements
The development of the numerical model was funded by
the Computer-Aided Structural Engineering (CASE) Pro-
ject of the US Army Corps of Engineers and the Infor-
mation Technology Laboratory (ITL) of the US Army
Engineer Research and Development Center (ERDC). Mr.
Amos Chase, Scientific Applications International Cor-
poration, and Mr. Barry White, ITL, made significant
contributions to writing the computer code. Permission
was granted by the Chief of Engineers to publish this
information.
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Notation Gf = fracture energy
H = beam height
The following symbols are used in this paper: k = constant that was statistically evaluated from ex-
perimental data of unconfined and confined concrete
subjected to monotonically increasing loading
A = constant that was statistically evaluated from ex-
perimental data of unconfined and confined concrete
subjected to monotonically increasing loading L = beam length
As = tension steel reinforcement area S = distance between beam supports
s
A
= compression steel reinforcement area s = spacing between stirrups on beam geometry defini-
tion
Av = area of the lateral (shear) reinforcement
a/d = shear-span-to-depth ratio = slip on bond-slip curve definition
b = width of beam t = thickness
d = effective depth uc, us = displacement of concrete and steel, respec-
tively
dc = concrete core diameter
ds = bar diameter wc = critical crack opening displacement value
E = modulus of elasticity x = distance from the crack-rebar intersection center
line to the desirable location
Ec = secant modulus of elasticity
Es = module of elasticity of the reinforcement
= reduction factor
f = concrete stress at a predefined strain ε = predefined concrete strain
c
fr = confinement pressure
f = compressive strength ε0 = peak strain
ν = Poisson’s ratio
ft = tensile strength τmax, τmi n = maximum and minimum bonding strengths,
respectively
fy = yield strength of the reinforcement
f
0 = peak stress