Effective Capacity of Diagonal Strut for Shear Strength of Reinforced Concrete Beams without Shear Reinforcement

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ACI Structural Journal/March-April 2012 139
Title no. 109-S13
ACI STRUCTURAL JOURNAL TECHNICAL PAPER
ACI Structural Journal, V. 109, No. 2, March-April 2012.
MS No. S-2009-211.R1 received December 26, 2010, and reviewed under Institute
publication policies. Copyright © 2012, American Concrete Institute. All rights
reserved, including the making of copies unless permission is obtained from the
copyright proprietors. Pertinent discussion including author’s closure, if any, will be
published in the January-February 2013 ACI Structural Journal if the discussion is
received by September 1, 2012.
Effective Capacity of Diagonal Strut for Shear Strength of
Reinforced Concrete Beams without Shear Reinforcement
by Sung-Gul Hong and Taehun Ha
The appropriate evaluation of the effective capacity of a concrete
strut is an important factor in the analysis and design of concrete
members using the strut-and-tie model. Current design codes for
the strut-and-tie model introduce this factor using the effective
compressive strength of concrete in a strut or nodal zone. This
study considers that the mechanism of diagonal cracking reduces
the width of a concrete strut and hence causes a reduction in the
capacity of the concrete strut. Based on this approach, models
for predicting the diagonal cracking strength and ultimate shear
strength of simply supported beams without shear reinforcement
are proposed, with the concrete strength, shear span-depth ratio
(a/h), and longitudinal reinforcement ratio as the primary parame-
ters. The predicted values are compared with proven test data from
various published experiments and codes of practice to show the
validity of the proposed models.
Keywords: bond strength; concrete strut; diagonal crack; effective strength;
reinforced concrete beam; shear strength; strut-and-tie model; strut width.
INTRODUCTION
The shear behavior of reinforced concrete (RC) members
has been a challenging issue over the past century. This area
is noted for its lack of consensus regarding an explanation
of the essence of member behavior failing in shear. Despite
the rigorous studies that have piled up over the past decades,
only a small number of researchers have adopted theoretical
methods in assessing the shear strength of RC members.
Attempts to solve the problems by means of purely empir-
ical approaches have been insufficient thus far, as the conse-
quences are usually restricted to the type of member used in
the particular test series from which the empirical solutions
are derived and calibrated.
According to many previous studies, the behavior of
RC members under applied shear force is the relative
combination of the arch action and the beam action of the
member.
1
For cracked members without shear reinforce-
ment, these two actions are made possible by transmitting
the shear with the following contributions: 1) an uncracked
region in the flexural compression zone; 2) aggregate inter-
lock between the cracked surface; and 3) the dowel action of
the longitudinal reinforcement. Important parameters influ-
encing the shear capacity include the strength of concrete,
the shear span-depth ratio (a/h) and support conditions, the
longitudinal reinforcement ratio, the axial force, the light-
weight aggregate concrete, the depth of the member or the
size effect, and the coarse aggregate size.
As many types of shear strength models exist as the
number of shear-transfer mechanisms for RC members
without shear reinforcement. Regarding empirical models,
Zsutty’s
2
regression equation provides a more comprehensive
understanding of the shear-related parameters compared to
ACI’s simple lower-bound equation. Most theoretical models
for design purposes can be classified into the following three
categories
3
: 1) mechanical or physical models for structural
behavior and failure; 2) fracture mechanics approaches; and
3) nonlinear finite element analysis.
The strut-and-tie model is a representative design instru-
ment based on the plasticity theory for RC members; hence,
it belongs to the first aforementioned category. Although
the history of the strut-and-tie model goes back to the late
19th century, when Ritter and Mörsch used truss analogy in
the analysis of RC structures,
3
it was finally included in the
Appendix of ACI 318-02
4
for the design of what is known
as the D-region. Although the application of the plasticity
theory to materials of limited ductility, such as concrete,
are limited in terms of plastic idealization, current strut-
and-tie models successfully predict the ultimate strength
of RC members, such as beams and corbels, by adopting
the concept of the effective strength of concrete for a strut
and a node in the strut-and-tie model. One of the general
concepts of the effective compressive strength of concrete
involves multiplying a certain reduction factor in the form
of a coefficient to the strength of concrete. This is usually
termed an “efficiency factor” or an “effectiveness factor” for
the strength of concrete. It is denoted as n in the literature.
Among the possible physical reasons for adopting a reduc-
tion factor, it is assumed in this study that the capacity of a
concrete strut is reduced due to the development of diag-
onal cracks in the shear span of RC members. The reduced
capacity of the concrete strut is first derived based on the
mechanics and the geometry of diagonal cracks. It is then
used to predict the shear strength of RC beams without shear
reinforcement, which is verified by comparing the predicted
strength with proven experimental results from the literature.
RESEARCH SIGNIFICANCE
This study regards the physical phenomenon of diag-
onal cracking as a main feature of the shear failure of RC
members without shear reinforcement and derives a model
for the effective capacity of a concrete strut for a reliable
application of strut-and-tie models. The proposed model is
suitable for estimating the shear strength of RC beams with
an intermediate a/h, as their governing failure modes are
determined by a critical diagonal crack.
www.modiriat-sakht.blogfa.com
140
ACI Structural Journal/March-April 2012
ACI member
Sung-Gul Hong
is a Professor in the Department of Architecture at
Seoul National University, Seoul, South Korea. He received his BS and MS from
Seoul National University and his PhD from Lehigh University, Bethlehem, PA. His
research interests include strut-and-tie models for bond transfer, shear strength of
reinforced concrete members, shear friction with creep, anchorage of multiple bars,
and deformation of reinforced concrete columns in shear.
Taehun Ha
is a Principal Researcher in the Architectural Engineering Research
Team at the Daewoo Institute of Construction Technology, Suwon, South Korea. He
received his BS, MS, and PhD from Seoul National University in 1997, 1999, and
2004, respectively. His research interests include shear strength of reinforced concrete
members, strut-and-tie models for shear-critical members, shear friction, openings in
flat-plate slabs, and construction stage analysis of high-rise buildings.
INFLUENCE OF DIAGONAL CRACKING ON
CONCRETE STRUT
Principal compressive stress trajectories drawn from
nonlinear finite element models for both slender and deep
beams are shown in Fig. 1.
5
A downward concentrated load
on a slender beam is mainly resisted by compressive flexural
stress at the upper half of the beam (beam action), whereas in
the case of a deep beam, the load is directly transferred to the
support by diagonal compression (arch action). These two
distinctive mechanisms of force transfer coexist in most RC
structural members, and the governing mechanism is usually
determined by the geometric characteristics of the members,
the reinforcement details, and the patterns of applied loads.
For RC beams without web reinforcement, diagonal cracks
that develop in the shear span deteriorate the capacity of the
direct load transfer and consequently lead to unexpected
brittle shear failure of the beams before the horizontal
reinforcement has reached its yield strength.
6
The behavior of concrete members in compression is
generally considered to be described adequately on the
basis of experimental information obtained from tests on
concrete specimens such as cubes, prisms, or cylinders.
There are fundamental differences, however, between the
strength of concrete derived from a material-level test and
the strength of concrete used in a structural member-level
test. The principal parameters that are supposed to have an
influence on the strength of concrete in compression include
the direction of the applied stresses, the boundary condi
-
tions surrounding the concrete strut, and the existence of
transverse stresses or strains.
The ideal strut-and-tie model shown in Fig. 2(a) consists of
an inclined concrete strut and a horizontal steel reinforcing
bar with tension
T
. When the reinforcing bar is buried in
the concrete, as in Fig. 2(b), the tension in the reinforcing
bar is transferred to the surrounding concrete with possible
crack development; hence, the concrete strut is affected by
this surrounding stress condition. Neither the exact distribu
-
tion of the tension in the steel and that of the bond stress
along the reinforcing bar between cracks nor the shape of the
concrete strut is known in general. The compressive force
flow in this stress condition can be modeled by a curved arch
or by a series of deviated concrete struts.
6,7
The single diagonal crack in Fig. 2(c) develops due to
the diagonal tension in the concrete. This is induced by the
flexural stress and/or bond stress between the concrete and
the reinforcing bars. The crack penetrates into an adjacent
concrete strut, which consequently weakens the transfer
capability of diagonal compression to the support. A
concrete strut with a reduced width in the middle, as shown
in Fig. 2(c), is assumed in this study to consider the effect
of a diagonal crack without sacrificing much of the original
simplicity of the strut-and-tie model. The minimum reduced
width of the concrete strut is denoted as
w
s

in Fig. 3 as
compared with
w
s
, which is the width of an undamaged
concrete strut.
The reduction in the width (area) of the compression load
path may explain why the shear strength of RC members
is usually less compared to the theoretical values. If the
degree of reduction in the strut width is properly evalu
-
Fig. 1—Compression stress trajectories of RC beams without
shear reinforcement. (Note: 1 MPa = 145 psi.)
Fig. 2—Inclined compressive strut with reduced strut width.
Fig. 3—Original strut width versus effective strut width.
ACI Structural Journal/March-April 2012 141
ated, this concept can be used to predict the shear strength
of RC members based on the original stress-strain curve of
the concrete specimens, as shown in the stress distribution
in Fig. 3. The effect of using this premise may be identical
to using the reduced compressive strength as the effective
strength; however, it is more theoretically convincing, as it is
based on the physical background.
Three instances of strut-and-tie models for D-regions are
shown in Fig. 4 with traces of (diagonal) tension cracks for
each member. The cracks start at the extreme tension fiber
of the maximum flexural stress and gradually propagate into
concrete struts in all cases. Deep beams are typical exam
-
ples in which the reduced width of a concrete strut can be
applied, as shown in Fig. 4(a). Corbels and dapped ends
are also considered to have reduced-width struts that are
affected by flexural cracks, as shown in Fig. 4(b) and (c).
Any other strut-and-tie model can be simulated using a strut
with a reduced width if penetration of the strut by flexural
cracks cannot be prevented.
EFFECTIVE STRENGTH OF CONCRETE STRUT
Diagonal crack development
A series of straight crack segments is shown overlapped on
a simple strut-and-tie model in Fig. 5. The crack segments
represent the trajectory of a critical diagonal crack, where
the term “critical” implies that it leads to the failure of
the member among the possible crack developments. It is
divided into three segments according to the process of crack
development. The first segment appears at Point
A
when
the extreme tension fiber equals the tensile strength of the
concrete and progresses vertically until it reaches the loca
-
tion of the horizontal reinforcing steel. Only flexural tension
is exerted on the concrete during this stage.
Once the reinforcing steel is engaged in the flexural
resistance, the state of stress in the concrete becomes more
complicated. The shear stress from the stress gradient of
reinforcing bars develops bond stress at the interface of the
concrete and reinforcing bars. Under the combination of the
flexural tension and bond stress between the concrete and the
reinforcing steel, successive progress of the crack segment
develops in a somewhat deviated direction, as has been shown
in many tests on RC beams without web reinforcement. The
term “diagonal” refers to the deviated direction of the flexural
shear crack; therefore, a diagonal crack in this stage refers to
a flexural shear crack. The second crack segment then meets
the extension line of the neutral axis at Point
B
, above which
the flexural compression exists in the area of the concrete
strut. The remaining segment of the diagonal crack follows
the original yield line of the concrete strut.
Whereas the first and second segments of the crack repre
-
sent the diagonal cracking process, the third segment of the
crack develops due to the direct load-resisting mechanism
of the concrete strut. Therefore, the effective capacity of
a concrete strut can be explained in terms of the reduced
Fig. 4—Strut-and-tie models with their compressive strut being disturbed by
cracks from outside of strut.
Fig. 5—Development of critical diagonal crack and reduced
width of compressive strut.
142
ACI Structural Journal/March-April 2012
width of the concrete strut at the stage of the third segment
of crack development.
Evaluation of neutral axis
Given that the second segment of a diagonal crack is
limited by the extension line of the neutral axis, the loca
-
tion of the neutral axis must be examined. This can be deter
-
mined by assuming any reasonable stress block in the flex
-
ural compression zone above the neutral axis, such as a rect
-
angular or triangular stress block. A parabolic distribution
of concrete stress along the compression depth is adopted in
this study to match the parabolic stress distribution across
the width of the concrete strut. For the rectangular section
of RC members shown in Fig. 6(a), a horizontal equilibrium
condition for the stress distribution in Fig. 6(d) yields
s s s n
A E bcε = σ
(1)
where
A
s
is the total area of longitudinal tension steel;
E
s

is the modulus of elasticity of steel;
e
s
is the steel strain;
b
is the width of the member;
c
n
is the flexural compression
depth; and
s
is the average concrete stress on concrete. For a
parabolic stress-strain relationship with strain in the extreme
compression fiber set to
e
cu
= 0.003 and the maximum stress

f
c

for
e
c
0
= 0.002,
s
is calculated as dependent on the value
of the concrete strain
e
c
as follows
0
0
1 1
3
6
c
c
c c c c
c c
d E
ε
 
ε
σ = σ ε = ε −

 
ε ε 
(2)
Inserting
s
in Eq. (2);
r
=
A
s
/
bd
, the longitudinal
reinforcement ratio; and
n
=
E
s
/
E
c
, the ratio of the modulus
of elasticity for the reinforcing steel and concrete, into
Eq. (1) results in the following equation for
c
n
/
d
as a func
-
tion of
n
,
r
, and
e
c
( )
2
0
0
1
2 1
3
1
1
3
c
c
n
c
c
n n n
c
d
 
ε
ρ + ρ − − ρ
 
ε
 
=
ε

ε
(3)
where
d
is the effective depth. Finally, inserting
e
c
=
e
cu
=
1.5
e
c
0
yields
( )
2
2
n
c
n n n
d
 
= ρ + ρ − ρ
 
 
(4)
For a practical range of the strength of concrete, the value
of
n
varies between 5 and 10 and the corresponding increase
in the compression depth is approximately 30%. When a
lower bound of
n
= 5 is used in Eq. (4), the compression
depth is dependent only on the magnitude of the longitu
-
dinal reinforcement ratio. This expression can be further
simplified by mathematical inference and graphical review,
as follows
8
3
1.7
n
c
d
= ρ
(5)
It should be noted that the compression depth in Eq. (5) is
expressed as normalized to the effective depth. The vertical
dimension of the node in the strut-and-tie model shown in
Fig. 5 is set as equal to the compression depth, as the para
-
bolic distribution of the compressive stress across the strut
width is assumed, as shown in Fig. 3.
Orientation of flexural shear cracks and diagonal
cracking strength
The stress field in the concrete below the neutral axis neigh
-
boring a vertical flexural crack is represented as a combined
tension and shear stress, as shown in Fig. 7. Flexural tensile
stress not only comes from uncracked concrete but is also
transmitted from the stress gradient of the reinforcing steel
bars—the latter being the primary source. In contrast, the
shear stress is developed from bond stress at the interface of
the concrete and reinforcing bars.
The bond stress distribution is usually determined from
the variation in the strain or stress of embedded steel.
According to
fib
Bulletin 51,
9
the distribution of bond stress
over transfer length is nonlinear and can be computed using
the differential equation of the bond and postulated bond-
slip relationship. For practical applications, however, it is
Fig. 6—Strain distribution and stress block in rectangular concrete member.
Fig. 7—Combined tension and shear stress field neighboring
flexural crack.
ACI Structural Journal/March-April 2012 143
suitable to use a constant mean bond stress value, which is
conservatively equal to the tensile strength of concrete for
normal-sized ribbed bars and for cases other than good bond
condition. Using the traditional mechanics dealing with
bond, the inclined stresses due to bond on the ribs of the
reinforcing bars develop radial and circumferential tensile
stresses on concrete. The splitting bond failure occurs when
the circumferential tensile stress reaches the tensile strength
of concrete. If the inclined angle of the stress on the bar is
assumed to be 45 degrees, the longitudinal bond stress (shear
stress) equals the tensile strength of concrete.
1
As a lower
bound for the bond stress,
t
in Fig. 7(b) is set equal to the
flexural tensile stress
s
. The magnitude and orientation of
the principal stresses according to the varying magnitude of
the flexural tensile stress and shear stress can then be found
from the Mohr’s circle and the separation failure condition
of the modified Coulomb criteria,
8
as shown in Fig. 8(b).
This leads to the relationship between
s
,
t
, and the tensile
strength of concrete
f
t

, derived as follows
( )
2
2
2
2
t
f
σ
+ σ + τ = ′
(6)
Inserting
s
=
t
in Eq. (6) and organizing it in terms of
s

or
t
yields
2
0.365
1 2 5
t t
f f
τ = =
′ ′
+
(7)
From the free-body diagram of Fig. 7(b), the equilibrium
of the entire system of an infinitesimal element of concrete
and steel bars in the horizontal direction requires
b dx b dx dTτ⋅ ⋅ + σ⋅ ⋅ =
(8)
where
b
is the width of the element;
dx
is the infinitesimal
length of the element; and
dT
is the differential tension force
in the reinforcing steel bar. Setting
s
equal to
t
and using the
relationships
dT
=
dM
cr
/
jd
and
dM
cr
/
dx
=
V
cr
sequentially lead
to the following expression of the diagonal cracking strength.
( )
0.2 in MPa
cr c c
V f bd f= ′ ′
(9)
When
f
c

is in psi, the coefficient 0.2 should be changed
to 2.4.
Here,
V
cr
and
M
cr
are the diagonal cracking strength and
corresponding moment, respectively. In deriving Eq. (9), the
lever arm of coupled horizontal forces
jd
is set equal to 0.9
d
,
whereas the tensile strength of concrete
= 0.1 ( in MPa) (when is in psi, = 3.8 ),
t c c c t c
f f f f f f′ ′ ′ ′ ′ ′
as proposed by Nielsen.
10
It should be noted that although
the diagonal cracking strength obtained from Eq. (9) is theo
-
retically derived, it has a mathematical form that is identical
to the form of ACI 318-02,
4
Eq. (11-3), for the shear strength
provided by concrete
4
; the difference in this value is within
20%. Therefore, Eq. (9) is automatically verified by the
experimental data used to formulate the ACI equation and,
at the same time, indicates that the shear strength recom
-
mended by the ACI code is conservative.
The angle between the flexural shear crack and hori
-
zontal axis, designated as
b
in Fig. 5, is also calculated
from the Mohr’s circle as 58.3 degrees. The actual crack
patterns observed in various test results confirm this condi
-
tion.
11
It follows that several flexural shear cracks at an
angle of approximately 60 degrees develop in the flexural
tension zone until the final crack is determined by the inter
-
section of the flexural shear crack and the yield line of the
concrete strut.
Reduced width of concrete strut
After flexural shear cracks have fully developed in the
flexural tension zone, an additional load can be resisted by
the compression in the concrete strut and the tension in the
horizontal steel if the capacity is greater than the diagonal
cracking strength calculated by Eq. (9). Given that the
vertical dimension of the node in the strut-and-tie model
is set equal to the flexural compression depth
c
n
, the intact
width of the concrete strut
w
s
, which is not disturbed by the
diagonal crack, is easily obtained by the principle of similar
triangles of Fig. 5 as follows
2
1
n
s n
h c
w c
a

 
= +
 
 
(10)
where
h
is the overall depth of the member. Due to the tensile
stress along the surface of the diagonal crack, the intact
width given by Eq. (10) is reduced to
w
s

at the tip of the
diagonal crack, as shown in Fig. 5. To find the relationship of
w
s
and
w
s

, the location of the intersection point of the critical
diagonal crack and the yield line of the concrete strut must
be identified. The location can be determined from the inter
-
action between the flexural resistance mechanism (beam
action) and the direct load-transfer mechanism (arch action).
Three straight lines (
k
=
k
1
x
,
k
=
k
2
x
, and
k
= 0) origi
-
nating from the neutral axis at the midspan in the diagrams
of Fig. 9(a) are assumed to represent the possible upper
boundaries of the flexural tension zone along the member
Fig. 8—Stress condition in flexural tension zone.
144
ACI Structural Journal/March-April 2012
2
2 2
n
n n
ac
k
a c c h
=
− +
(12)
As the diagonal crack and strut yield line are assumed to
represent the actual propagation of the crack, crack patterns
from test observations are used to determine the slope of the
flexural boundary line. The horizontal crack projections of
the diagonal crack, denoted as
x
in Fig. 5 and Fig. 9(a), were
measured by Peng
12
from a total of 105 test specimens of RC
beams without shear reinforcement. From the notations in
Fig. 5,
x
is calculated for a general
k
as follows
( )
cot
B B n
x x y d c= + + − β
(13)
where
b
= 58.3 degrees; and
x
B
and
y
B
are also given from the
geometries of Fig. 5 and Fig. 9(a) by
( )
( )
( )
2 2
2 2
n n n
n
B
n n
a k h c a c c h
c
x
a
ah k a c c h
 
− − + −
 
=
+ + −
(14)
( )
n n
B B
c h c
y k x
a
 

= +
 
 
(15)
The theoretical values of horizontal projection calculated
from Eq. (13), (14), and (15) in terms of the normalized
projection to the overall member depth
x
/
h
are compared
with those of experimental observations with varying
a
/
h

in Fig. 9(b). As clearly shown in the graph, the projection
lengths predicted by
k
=
k
1
and
k
=
k
2
constitute the upper
and lower bounds of the experimental observation, respec
-
tively. The test observations generally follow the tendency
of the calculated value of
x
based on
k
=
k
1
for deep beams
(
a
/
h
< 2.0), gradually moving to
k
=
k
2
for slender beams
(4.0 <
a
/
h
). The assumption of
k
= 0 is generally in good
agreement with intermediate beams (2.0 <
a
/
h
< 4.0);
however, it was not significantly different from
k
=
k
2
. There
-
fore, the reduced width of a concrete strut is evaluated for
members with 2.0 <
a
/
h
on the assumption that diagonal
cracks rise to the level of the horizontal neutral axis (
k
= 0).
From the principle of similar triangles for the concrete strut
in Fig. 5, the following equation holds
0
s n B
w c x
w h a x

= =

(16)
The reduced width of a concrete strut is then obtained by
inserting Eq. (10) into the first equation of Eq. (16).
2
2
1
n n
s
c h c
w
h a

 
= +′
 
 
(17)
This expression is understood more easily if it is presented
as a normalized form as follows
2
2
1
1
s
w
h
′ −ζ
 
= ζ +
 
 
α
(18)
axis, above which flexural compression exists. The left ends
of the straight lines near the support region range vertically
from the level of the horizontal reinforcement to the extreme
compression fiber, with the action of either governing the
load transfer—depending on the case—the beam action
for slender beams or the arch action for deep beams. The
adoption of varying boundary lines can be justified by the
different compression stress trajectories for slender and deep
beams, as shown in Fig. 1. In contrast to the curved stress
trajectories shown in Fig. 1, the boundary lines in Fig. 9(a)
are assumed to be straight so that the calculations are simple
and efficient. According to the coordinate system adopted in
Fig. 9(a), the boundary line is defined by
y
=
kx
, where
k
1



k



k
2
. For
k
=
k
1
, the direct load-transfer mechanism governs,
and most of the original width of the concrete strut is used as
the effective width. The width of the concrete strut decreases
as
k
becomes
k
2
due to the increasing action of the flexural
resistance. When
k
= 0, the horizontal boundary line repre
-
sents the conventional neutral axis of the flexural stresses.
Slopes
k
1
and
k
2
can be evaluated from the geometries of
Fig. 9(a) and the notations of Fig. 5.
( )
( )
( )
1
2 2
n
n n n n
ah d c
k
c h h c d a c c h
− −
=
− + + −
(11)
Fig. 9—Determination of location of critical diagonal crack.
ACI Structural Journal/March-April 2012 145
a
represents the mechanism of a direct load transfer via a
concrete strut, whereas the term
z
represents the influence
of the flexural behavior of the beams as a function of the
longitudinal reinforcement ratio.
COMPARISON OF PREDICTIONS AND
EXPERIMENTAL RESULTS
The test results of Kani et al.’s
11
50 beams without web
reinforcement are compared with the theoretical predic
-
tions from Eq. (21) with a varying
a
/
h
. This comparison
is shown in Fig. 11. Also shown is the diagonal cracking
strength calculated by Eq. (9). Both the tested and predicted
values show a tendency in which the strength decreases as
the
a
/
h
increases. The predicted shear strength given by the
reduced capacity of the concrete strut shows a reasonable
lower bound of the test results for beams when 2.0


a


4.0
(intermediate beams) but largely underestimates the shear
strength of beams when
a


2.0 (deep beams) or 4.0


a

(slender beams).
For deep beams with
a


2.0, the effect of diagonal cracks
and, hence, the reduction in the width of the concrete strut
is not as significant as it is for more slender beams (refer to
Fig. 9(a)) that the effectiveness of the strut width needs to
be increased. This can be done by interpolation between the
values of the original strut width and the reduced strut width
where
a
=
a
/
h
; and
z
=
c
n
/
h
=
3
1.53 ρ
. These represent the
ratio of the shear span and flexural compression depth to the
overall depth of the member, respectively. If the reduced
width of the concrete strut is used as a measure for the intact
-
ness of the strut under loading, it can be said from Eq. (18)
that the compressive strut exerts more of its capacity with an
increasing amount of longitudinal reinforcement but gradu
-
ally weakens as the
a
/
h
increases due to the penetration of
the diagonal crack into the strut.
SHEAR STRENGTH OF RC BEAM WITHOUT
SHEAR REINFORCEMENT
An example application of the reduced capacity of a
concrete strut is presented for the shear strength of RC
beams without shear reinforcement. It is assumed that the
shear failure of the beam is induced by the crushing of the
concrete strut after a critical diagonal tension crack has fully
developed. Furthermore, the failure might well precede the
yielding of the horizontal reinforcement due to the reduced
capacity of the concrete strut.
The force
P
in Fig. 10 can be derived as the vertical
component of a strut force
sin
c s
P f w b
= θ

(19)
where
q
is the inclined angle of the concrete strut; and
f
c

is the average concrete stress over the reduced width of the
concrete strut. For the parabolic stress-strain relationship
and when
e
cu
= 0.003 for the assumed ultimate strain,
f
c
=
0.75
f
c

for the original full-width strut. The variable
f
c
for the
reduced-width strut involves some factor for the degree of
reduction of the width. From the stress distribution in Fig. 3,
the average concrete stress over the reduced strut width can
be calculated by averaging the integrated stress for
w
s

only.
Using the first equation in Eq. (16), the following integration
holds true for calculating
f
c
( )
( )
( )
2
0
2
0
2
1
2
1
3
1
4
cu
cu
cu
cu
c c c
cu cu
c
c c
c cu
c
f d
f
d
f
ε
ζε
ε
ζε
= σ ε

ε −ζε

= ε ε − ε ε

ε ε −ζ
= +ζ −ζ′
(20)
Dividing Eq. (19) by the area of the beam section and
substituting Eq. (18) for
w
s

/
h
and Eq. (20) for
f
c
yield the
shear strength
v
=
P
/
bh
, as in the following equation
( )
( )
2 2
1 1
3
4
c
v f
ζ −ζ +ζ −ζ
= ′
α
(21)
The derived equation for the shear strength is expressed
in terms of the strength of concrete
f
c

; the ratio of the flex
-
ural compression depth to the overall depth of the member
z
, which in turn represents the longitudinal reinforcement
ratio; and the ratio of shear span to the overall depth of the
member
a
. The shear strength increases as the value of the
strength of the concrete and the longitudinal reinforcement
ratio increase but decreases when the
a
/
h
increases. The term
Fig. 10—Strut-and-tie model for simply supported RC beam
with critical diagonal crack.
Fig. 11—Comparison of predicted shear strength of RC beams
without shear reinforcement and test results by Kani et al.
11
146
ACI Structural Journal/March-April 2012
in the range of 0


a


2.0. By the linear interpolation of the
strut width given by Eq. (10) and (18), the following expres
-
sion is obtained
( )
2
1
1
1 1 for 2.0
2
s
w
h
 α −ζ
′ −ζ
 
= − ζ + α ≤
   
 
α
 
(22)
Equation (21) for the shear strength of reinforced beams is
also modified as follows
( )
( )
( )
2
1 1
1
3
1
4 2
for 2.0
c
v f
ζ −ζ +ζ −ζ
 α −ζ
= −′
 
α
 
α ≤
(23)
For slender beams when 4.0


a
, the shear strength
predicted by Eq. (21) is smaller than the diagonal cracking
strength given by Eq. (9), as shown in Fig. 11, and the
test results are more accurately predicted by the diagonal
cracking strength than by the strut capacity. This is true
because the diagonal crack has nearly penetrated into the full
width of the concrete strut for these beams and the remaining
capacity of the concrete strut cannot resist any additional
load applied after the diagonal crack has completely devel
-
oped. Therefore, the diagonal cracking strength determines
the ultimate shear strength of slender beams with an
a
/
h

that exceeds 4.0. In Fig. 12(a), the predictions of the shear
strength are compared with the test results of 374 RC beams
without shear reinforcement that have failed by shear.
13
The
data bank of these test results is referred to as the evalu
-
ation shear data bank (ESDB), which was agreed upon
by Joint ACI-ASCE Committee 445 to satisfy the criteria
to be used for assessing the capability of design rules or
expressions derived from models for structural behavior. It
should be noted that the test results from members with a
T-shaped section were not included in the comparison. In
Fig. 12(b), (c), and (d), the equations from ACI 318-02
4
and
Eurocode 2
14
and the predictions by the modified compres
-
sion field theory (MCFT),
15
respectively, are also compared
with the ESDB to exhibit the difference of the comparison
results. As in Reference 13, the strength notations in the
graphs in Fig. 12 are expressed in dimension-free shear force
to exclude the effect of the member size and the strength of
the concrete. For the test results, the ultimate strength values
were divided by
bdf
1
c
, where
f
1
c
is the uniaxial compressive
strength of the concrete and can be calculated from
f
c

by
f
1
c

= 0.95(
f
c

+ 2.4 MPa) (
f
1
c
= 0.95(
f
c

+ 348 psi)).
13
The calcu
-
lated values are given by Eq. (23), (21), and (9) for members
with
a


2.0, 2.0


a


4.0, and 4.0


a
, respectively. They
are divided only by
f
c

because the calculated strengths are
already in the unit of stress. For the ACI 318, Eurocode 2,
and MCFT values, the original equation values were divided
Fig. 12—Test values versus calculated values of dimension-free shear strength of RC
beams without shear reinforcement: (a) suggested equations in this paper; (b) Eq. (11-3) of
ACI 318; (c) Eq. (6.2.a) of Eurocode 2; and (d) MCFT.
ACI Structural Journal/March-April 2012 147
by
bdf
c

,
bdf
ck
, and
bdf
c

, respectively, where
f
ck
is the charac
-
teristic compressive cylinder strength of concrete at 28 days
and is related with
f
1
c
by
f
1
c
= 0.95(
f
ck
+ 4 MPa) (
f
1
c
= 0.95(
f
ck
+ 580 psi)).
13
The statistical results for the test/calculated strength are
shown in Table 1, which implies a good correlation between
the predicted and experimental values as compared with
the cases of the code equations and MCFT. Although both
Eurocode 2 and MCFT result in a slightly greater mean value
as compared with ACI 318, the standard deviation and coef
-
ficient of variation are smaller, which leads to consistently
conservative calculation results, as shown in Fig. 12(c) and
(d). The proposed shear strength is not as conservative as
Eurocode 2 or MCFT because the equations are not derived
as a lower-bound solution. By introducing a certain value of
the safety factor, however—say, 0.8—the conservatism can
be secured while maintaining a better co-relationship with
the test results.
The influences of three parameters in the proposed
equations—the
a
/
h
, longitudinal reinforcement ratio, and
compressive strength of concrete—are investigated in the
three graphs of Fig. 13, respectively. The y-values in each
graph are the ratios of the test/calculated values, defined
as the model safety factor. The ACI 318, Eurocode 2, and
MCFT values are also shown simultaneously. It is easily
shown in the three graphs that the values of the safety factor
of the proposed model is constantly maintained, regard
-
less of the varying parameters, whereas those of the code
equations or MCFT show unsafe or wide distributions for a
certain range of parameters. For members with a small
a
/
h

(2 <
a
< 2.5), the ACI 318, Eurocode 2, and MCFT predicted
values may be much lower compared with the actual capacity
of the member. This is because these methods were devel
-
oped under the assumption that the plane section remains
plane, which is true for slender beams but is generally
conservative for deep beams. As already indicated by other
researchers,
13
ACI 318, Eq. (11-3), is unsafe for members
with low reinforcement ratios, as shown in Fig. 13(b). It
should be noted that Eq. (23) is not used in calculating the
shear strength of ESDB because the data bank does not
currently have the test results for members with
a
< 2.0.
CONCLUSIONS
Based on the developed model for the influence of a diag
-
onal crack on the effective capacity of a concrete strut, the
following conclusions are drawn:
1. Diagonal cracking is initiated by a combination of flex
-
ural tension and bond stress between the concrete and the
longitudinal reinforcing steel. The penetration of a diagonal
crack into the concrete strut impairs the capacity of the
concrete strut, which can be expressed by the reduced width
of the strut.
2. The reduced width of the concrete strut is evaluated
based on the undamaged width of the strut, considering the
trajectory of the diagonal crack that has developed into the
Table
1—Statistical results for test/calculated strength for different approaches
Quantity
Proposed
ACI 318, Eq. (11-3)
Eurocode 2, Eq. (6.2.a)
MCFT
Average value
1.052
1.417
1.588
1.462
Standard deviation
0.25
0.562
0.435
0.4
Coefficient of variation
0.238
0.351
0.274
0.274
Fig. 13—Influence of important parameters on model safety
factor: (a)
a
/
h
; (b) longitudinal reinforcement ratio; and
(c) compressive strength of concrete. (Note: 1 MPa = 145 psi.)
148
ACI Structural Journal/March-April 2012
strut. The degree of penetration of the diagonal crack into the
concrete strut is determined by the intersection point of the
diagonal crack and the yield line of the strut, each of which
represents the beam action and the arch action of RC beams,
respectively. Diagonal cracks are assumed to develop only to
the level of the neutral axis for beams with an
a
/
h
of 2.0


a

from comparisons with observations of the trajectories of
actual diagonal cracks from the existing test results. The
derived expression for the reduced width of a concrete strut is
a function of the
a
/
h
and the longitudinal reinforcement ratio.
3. The proposed model for the reduced width of a
concrete strut is used to predict the shear strength of
reinforced beams without shear reinforcement. As a result
of comparisons with the test results of 374 beams from the
ESDB, the predicted shear strength based on the reduced
strut model combined with the diagonal cracking strength
is in good agreement with the test results for beams with
an
a
/
h
of 2.0


a
. It shows a better co-relationship than the
ACI 318 and Eurocode 2 equations and keeps a constant
value of the model safety factor for a wide range of key
parameters, such as the
a
/
h
, the longitudinal reinforcement
ratio, and the compressive strength of concrete.
NOTATION
a
= shear span (horizontal distance between applied force and reaction)
b
= width of member
c
n
= flexural compression depth
d
= effective depth of member
dT
= differential tension force in reinforcing steel
dx
= infinitesimal length of member
E
c
= modulus of elasticity of concrete
E
s
= modulus of elasticity of steel
f
c
= average compressive stress of concrete
f
c

= compressive strength of concrete
f
ck
= characteristic compressive cylinder strength of concrete at 28 days
f
t

= tensile strength of concrete
f
1
c
= uniaxial compressive strength of concrete
h
= overall height of member
jd
= lever arm of horizontal coupled force
k
= slope of boundary line for flexural tension
k
1
= lower bound for
k
k
2
= upper bound for
k
M
cr
= diagonal cracking moment
n
= ratio of modulus of elasticity of steel to concrete (=
E
s
/
E
c
)
P
= applied vertical force
T
= flexural tension
V
cr
= diagonal cracking load
v
= predicted shear strength of RC beams without shear reinforcement
v
t
= measured shear strength of RC beams without shear reinforcement
w
s
= width of concrete strut
w
s

= reduced width of concrete strut
x
= horizontal projection of diagonal crack and strut yield line
x
B
= x-coordinate of intersection point of diagonal crack and strut yield line
x
0
= horizontal width of nodal zone
y
B
= y-coordinate of intersection point of diagonal crack and strut yield line
a
= ratio of shear span to overall depth of member (=
a
/
h
)
b
= inclined angle of diagonal crack with respect to horizontal axis
e
cu
= ultimate concrete strain at failure (=0.003)
e
c
0
= concrete strain at maximum stress (=0.002)
n
= effectiveness factor for compressive strength of concrete
q
= inclined angle of concrete strut with respect to horizontal axis
r
= longitudinal reinforcement ratio
s
= normal stress
s

= average compressive stress on concrete
s
1
= first principal stress
t
= shear stress
z
= ratio of flexural compression depth to overall depth of member
(=
c
n
/
h
)
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