Shear Behaviour of RC Beams without Stirrups Made of Normal Strength
and High Strength Concretes
by
M. Hamrat, B. Boulekbache, M. Chemrouk and S. Amziane
Reprinted from
Advances in Structural Engineering
Volume 13 No. 1 2010
MULTI-SCIENCE PUBLISHING CO. LTD.
5 Wates Way, Brentwood, Essex CM15 9TB, United Kingdom
Shear Behaviour of RC Beams without Stirrups Made
of Normal Strength and High Strength Concretes
M. Hamrat1,*, B. Boulekbache1, M. Chemrouk2 and S. Amziane3
1University
of Science and Technology Houari Boumediene, Algeria/LIMATB, France
of Science and Technology Houari Boumediene, Algeria
3University Blaise Pascal, France
2University
(Received: 1 December 2008; Received revised form: 25 March 2009; Accepted: 14 April 2009)
Abstract: The paper describes a study in which sixteen reinforced concrete beams
without transverse stirrups were tested to failure to investigate the influences of the
shear-span/depth ratio (a/d), the longitudinal steel ratio ( ρ) and the compressive
strength of concrete ( f c′ ) on the global behaviour in shear for beams made of normal
strength concrete (NSC) as well as high strength concrete (HSC). Crack development
and propagation were studied through continuous monitoring of the shear cracking
using digital video recording. The test results show that the shear capacity depends
more on the shear-span/depth ratio (a/d) and the longitudinal steel ratio ( ρ) and
relatively less so on the compressive strength ( f c′ ) in the case of HSC.
Based on this study and on data collected from the literature, a proposition for
computing the shear stength of concrete beams without stirrups is made; it covers
compressive strengths in the range of 40 MPa to 104 MPa. Using the proposed shear
formulae, a comparison is made between the different modelling approaches used in
the major design codes concerning the contribution of concrete to the shear resistance
of beams. In the light of the test results obtained in the present investigation, the
applicability of these code approaches to high strength concrete is examined.
Key words: high strength concrete, shear-span, longitudinal steel ratio, compressive strength, tensile strength, shear
strength, arch action, beam action.
1. INTRODUCTION
The shear behaviour of reinforced concrete elements,
particularly those made of high strength concrete (HSC),
is still a subject of interest in research despite the
numerous experimental and theoretical work papers
published on normal strength concrete (NSC). According
to this literature, the compressive strength of concrete
( f c′ ), the shear-span/depth ratio (a/d) and the longitudinal
steel ratio (ρ) are thought to be the most important
parameters affecting the shear resistance of normal
strength concrete. Little is known, however, about the
influence of these parameters for high strength concrete.
High strength concrete has a number of advantages;
its use in the construction industry is ever increasing in
many regions of the world. This is due to such improved
physico-mechanical properties as compressive strength,
stiffness and long term durability and also to the
economic gains which can be achieved with reductions
in geometrical sections and gain in the architectural
space to be exploited. Hence, technical, economic as
well as aesthetic criteria point towards the choice of
HSC instead of NSC. In effect, with the advances in
materials technology, particularly the development of
superplaticisers and colour admixtures, concrete
*Corresponding author. Email address: [email protected]; Fax: 0021327625285; Tel: 00213772868195.
Advances in Structural Engineering Vol. 13 No. 1 2010
29
Shear Behaviour of RC Beams without Stirrups Made of Normal Strength and High Strength Concretes
a
Vex
C (Concrete
compression)
Vc
d
Va
T(Steel tension)
Vd
V = Vc + Va + Vd
Figure 2. Internal forces in a beam without stirrups
Figure 1. The cracking crossing the aggregates in the case of HSC
construction aesthetics are continuously improving
through the use of high strength concrete.
On the other hand, HSC has proved to be a brittle
material. Indeed, during the tests of this study, it was
observed that cracking was sudden, transversing
completely aggregate particles, as in Figure 1 above,
producing relatively smooth fracture planes. The cracks
do not go around the aggregate particles as is usually
the case with NSC. Smooth fracture planes reduce the
concrete shear strength by reducing the contribution
of the aggregate interlock between the fracture
surfaces (Carrasquillo et al. 1981; Martinez et al. 1982;
Elzanaty et al. 1986; Collins and Kuchma 1999).
The rupture mechanism of a beam element without
stirrups can be reasonably thought of as generating three
internal contributing forces to shear resistance (Figure 2).
These consist of the contribution of concrete in the
compression zone (Vc), the shear contribution due to
aggregate interlock (Va) and the shear contribution due to
the dowel action of the longitudinal reinforcement (Vd).
As a result, the total shear resistance can be written as:
V = Vc + Vd + Va
(1)
With an excessive opening of a diagonal crack, the
components Va and Vd become ineffective. Consequently,
the component Vc carries all the acting shear. This leads
to the collapse of the beam, initiated by crushing of
concrete in compression (Taylor 1974).
The aggregate interlocking mechanism is a
significant contributor to the shear capacity of beams
and hence has a predominant influence on the ultimate
load carrying capacity of beams failing in shear
(Sarkar et al. 1999). The following quantification for
the different contributions to shear capacity at ultimate
load has been suggested to be (Taylor 1970):
For beams having strengths in the range of 26 to 49
Va = 33 − 50%

MPa: Vc = 20 − 40%
V = 15 − 25%
 d
30
(2)
However, at higher concrete strengths, the aggregate
interlocking does not seem to contribute greatly towards
shear; this is supported by the smooth fracture planes
and the straight cracks which do not go around the
aggregate particles as stated above. Indeed, according to
Mphonde (1988), the following contributions are made
in the case of HSC:
 Va = 0 %
→ fc' > 62 MPa

'
Vc = 16 − 26% → 21 < fc < 90 MPa
V = 31 − 74% → 21 < f ' < 90 MPa
c
 d
(3)
The aggregate contribution to the shear capacity
seems to be insignificant at higher concrete strengths.
Indeed, no aggregate interlocking was observed for the
high strength concrete beams of the tests in this study
and the shear capacity was not significantly increased
even when compressive strengths had almost doubled,
from 44.2 MPa to 85.5 MPa.
The state-of-the-art concerning shear in reinforced
concrete is that there is no unifying rational theory
explaining the interactions of the factors which contribute
to the shear resistance of this composite material.
Current Codes such as the ACI-318 (2002), the BS8110 (1997), the Eurocode 2 (2002) and the BAEL (1999)
contain various design equations for the shear strength of
concrete beams as functions of the longitudinal steel
ratio, the shear-span/depth ratio, the concrete compressive
strength and the sizes of the beam specimens (Table 1).
Although most of these four major universal codes take
into account the three parameters considered for
investigation in the authors’ work, the change from
normal strength concrete to high strength concrete may
induce changes in the way these parameters contribute
to the shear capacity; the influence of these parameters
may be different in the two types of concrete. Thus, the
application of these equations to RC beams with higher
compressive strengths ( f c′ > 40 MPa) needs to be
carefully examined.
Advances in Structural Engineering Vol. 13 No. 1 2010
M. Hamrat, B.Boulekbache, M.Chemrouk and S.Amziane
Table 1. Empirical equations and code provisions for shear strength of concrete beams
Codes
Equation
V=
ACI-318 (American)
1
d
fc′ + 120 ρ  ⋅ bd
7 
a
1
BS-8110 (British)
0.79  100 As  3  400 
V=


γ m  bv d   d 
1
1
4
 fcu 
 25 
3
 d 0.79  100 As   400 
V =  2 


 a  γ m  bv d   d 
Eurocode-2
(European)
Comments
1
4
1
3
⋅ bd
 fcu 
 
25
f c′ cylinder compressive strength of
concrete,
d effective depth of the section
b width of the section
a /d ≥ 2
1
3
⋅ bd
a /d < 2
V=
0, 0525
( fc′)2 / 3 (2, 5d /a) ⋅ (1, 6 − d ) ⋅ (1.2 + 40 ρ ) ⋅ bd
γc
a /d < 2.5
V=
0, 0525
( fc′) 2 / 3 ⋅ (1, 6 − d ) ⋅ (1.2 + 40 ρ ) ⋅ bd
γc
a /d ≥ 2.5
V = 0.3 ft K ·b·d
BAEL (French)
fcu cube compressive strength
of concrete,
As section of the longitudinal
steel
γm safety factor for material
f ′c = 0,8 fcu
ρ longitudinal steel ratio
γc safety factor for concrete
material
a/d shear span /depth ratio
K = 1 (for shear through simple
flexural loading)
ft tensile strength of concrete
strengths (Table 3), producing two groups of eight
concrete beams C40 and C80. For each group, two
longitudinal steel ratios were used ( ρl = 1.16 and 2.31%)
and four different shear-span/depth ratios were tested
(a/d = 1, 1.5, 2 and 3).
The notation of the specimens (C40-1-1 or C80-3-2)
is such that C40 and C80 designate normal strength
concrete (NSC) and high strength concrete (HSC),
2. TEST PROGRAM
2.1. Test Specimens
Table 2 and Figure 3 show the details of the beam
specimens which had a constant width of 100 mm and a
constant depth of 160 mm. The beams were tested under
different shear-spans. A total of sixteen beams were
constructed and loaded to failure under two-point loads.
The beams were cast in two batches of two concrete
Table 2. Specification of test specimens and material properties
Longitudinal tensile steel
Beams
C40-1-1
C40-1-2
C40-1.5-1
C40-1.5-2
C40-2-1
C40-2-2
C40-3-1
C40-3-2
C80-1-1
C80-1-2
C80-1.5-1
C80-1.5-2
C80-2-1
C80-2-2
C80-3-1
C80-3-2
f c′ (MPa)
f t (MPa)
44.2
3.37
85.5
4.50
Advances in Structural Engineering Vol. 13 No. 1 2010
d (mm)
a/d
As (mm2)
ρ (%)
135
133
135
133
135
133
135
133
135
133
135
133
135
133
135
133
1
1
1.5
1.5
2
2
3
3
1
1
1.5
1.5
2
2
3
3
157
308
157
308
157
308
157
308
157
308
157
308
157
308
157
308
1.16
2.31
1.16
2.31
1.16
2.31
1.16
2.31
1.16
2.31
1.16
2.31
1.16
2.31
1.16
2.31
31
Shear Behaviour of RC Beams without Stirrups Made of Normal Strength and High Strength Concretes
P/2
P/2
1
Area investigated using
Gom-Aramis system
10
d 16
a
13.5
16
2∅14
10
1
a
30
2∅10
10
ρ = 1.16 %
13.3
10
1-1
ρ = 2.31 %
L = 77 (a/d = 1), L = 90 (a/d = 1.5), L = 104 (a/d = 2), L = 131(a/d = 3)
(all dimensions are in cm)
Figure 3. Dimensions and reinforcement of the test beams (C40 and C80)
Table 3. Proportions of the NSC and HSC concrete
mixtures
Material
Units
NSC
HSC
Cement type CEMI-52.5 N-CPJ
Silica fume
Aggregate (3–10 mm)
Sand (0–4 mm)
Water
Filler (limestone)
Superplasticiser (Chrysofluid
Optima 206)
W/C
kg/m3
−
kg/m3
kg/m3
L
kg/m3
275
−
990
740
170
44
425
42.5
980
700
144
−
L
−
−
0.62
6.4
0.34
respectively. The first number after the first hyphen
gives the a/d ratio and the number after the second
hyphen is the longitudinal steel percentage (ρ = As /bd).
2.2. Instrumentation
The load was applied using a 250 kN servo-controlled
hydraulic jack. One LVDT was attached to the bottom
surface at midspan of the test specimen to measure the
midspan displacement of the beam. Electrical strain
gauges were attached to the surface of the longitudinal
steel to record the bar strains. Strains gages were covered
with silicone gel to prevent damage during and after
casting. The applied load, the corresponding displacement
and the strains were recorded automatically through a
data logger.
A Video Gom-Aramis system was used to measure
crack widths and to monitor the development of the
diagonal cracking. The system follows the movements
of some reference points indicated beforehand in each
beam (Figure 4) as the load increases. The analysis of
the numerical image is carried out with the help of GomAramis software which enables the strain fields to be
calculated over the whole zone studied. Moreover, this
enables the detection of the initial crack opening with
great precision.
32
Mouchtis
Video camera
(Gom-Aramis system)
Figure 4. Experimental set-up and video camera (Gom-Aramis)
3. TEST RESULTS AND DISCUSSION
3.1. Global Response
The load-deflection behaviour consists of two stages
(see Figure 5):
(1) The first stage corresponds to the elastic
behaviour where no cracking exists in the tension
zone of the concrete beam;
(2) The second stage starts with the appearance of
flexural cracking in the central section of the beam
which reduces in stiffness due to the cracking.
At further increases of load, the existing cracks
develop in length and new cracks would appear
within the shear spans. One of the flexural cracks
in the shear spans close to the supports would
eventually depart from the vertical and bend
diagonally towards the loading point.
Alternatively, depending on the length of the
shear span, a diagonal crack might develop
independently within the shear span and extend
towards the loading and support points. On
further increase in load, concrete would crush in
the compression zone near the loading point.
Advances in Structural Engineering Vol. 13 No. 1 2010
M. Hamrat, B.Boulekbache, M.Chemrouk and S.Amziane
70
Applied load (kN)
60
50
40
C40-3-2 (ρ = 2.31%)
f ′c = 44.2 MPa
a/d = 3.0
60
50
40
30
30
20
20
10
C40-3-1 ( ρ = 1.16%)
C40-3-2 (steel)
C80-3-2 ( ρ = 2.31%)
C40-3-1 (steel)
C80-3-1 ( ρ = 1.16%)
−6
−4
−2
Mid-span deflection (mm)
C80-3-2 (steel)
10
C80-3-1 (steel)
0
0
−8
f ′c = 85.5 MPa
a/d = 3.0
Applied load (kN)
70
0
2
4
Steel strain (%)
6
8
−8
−6
−4
−2
Mid-span deflection (mm)
0
2
4
Steel strain (%)
6
8
Figure 5. Mid-span deflections and steel strains against applied loads for beams without stirrups (C40 and C80)
Table 4. Test results of reinforced concrete beams
Beams
Pfl*
(kN)
Pcr
(kN)
Pu
(kN)
Pu/Pcr
δu**
(mm)
W***
(mm)
Number and type of cracks
(along the beam)
C40-1-1
C40-1-2
C40-1.5-1
C40-1.5-2
C40-2-1
25
30
22
26
17
133
140
96
112
67
187
195
129.4
143.6
85,1
1.45
1.39
1.35
1.28
1.27
2.84
1.80
3.04
2.12
3.77
0.46
0.39
0.53
0.27
0,81
C40-2-2
C40-3-1
C40-3-2
C80-1-1
C80-1-2
C80-1.5-1
C80-1.5-2
C80-2-1
C80-2-2
23
11
14
31
55
27
34
20
27
85
44
55
132
152
95
109
64
83
100.5
47.3
55
210.9
224
144.3
158.3
95.5
113.6
1.18
1.07
1.00
1.60
1.47
1.52
1.45
1.49
1.37
2.65
4.70
3.29
3.04
1.65
3.1
1.97
3.82
2.71
0.63
1.10
0.37
0.51
0.22
0.45
0.39
1.53
0.74
C80-3-1
C80-3-2
15
18
40
48.5
50.7
59
1.27
1.22
5.72
3.19
1.20
0.84
1 Diagonal shear + 4 flexural
2 Diagonal shear + 3 flexural
2 Diagonal shear + 5 flexural
2 Diagonal shear + 3 flexural
1 Diagonal shear+1 flexural
shear+ 5 flexural
2 Diagonal shear + 4 flexural
4 Flexural shear + 6 flexural
3 Flexural shear + 5 flexural
2 Diagonal shear + 6 flexural
1 Diagonal shear + 5 flexural
2 Diagonal shear + 7 flexural
1 Diagonal shear + 5 flexural
1 Diagonal shear + 6 flexural
2 Diagonal shear + 2 flexural
shear + 4 flexural
7 Flexural shear + 9 flexural
5 Flexural shear + 7 flexural
*Load
at first flexural crack, **Mid-span deflection corresponding to the ultimate load, ***Diagonal crack width.
Failure has always occurred during this second
load-deflection behaviour stage since no
confining lateral reinforcement was provided as
clearly shown by the load-deflection curves.
In these tests, the load versus deflection response for
RC beams seemed to be less dependent on the concrete
compressive strength. It was, however, more dependent
on the longitudinal steel ratio and the shear-span/depth
ratio as clearly seen in Table 4.
3.1.1. Effect of compressive strength of concrete
Figure 5 shows the applied load versus defection
response for the beams tested. When both a/d and ρ
were constant, the shear capacity varied from 187 kN to
211 kN as the compression strength varied from 44.2 MPa
to 85.5 MPa as in beams C40-1-1 and C80-1-1
Advances in Structural Engineering Vol. 13 No. 1 2010
respectively; an increase of 13%. This could indeed
be considered as only a small increase in relation to the
almost doubling of the compressive strength of the
concrete.
Shear strength increases, however, relates more to
tensile strength increases as clearly expressed in some
codes such as the BAEL. Previous work has shown that
tensile strength does not increase in proportion to
compressive strength (Chemrouk and Hamrat 2002). The
test results show that while the compressive strength
almost doubled, the tensile strength increased by an
average of only 30%. The second reason for the small
increase in shear capacity is related to the aggregate
interlock mechanism which is thought to be absent in
HSC since the crack goes smoothly right through the
aggregates particles and hence shear resistance
33
Shear Behaviour of RC Beams without Stirrups Made of Normal Strength and High Strength Concretes
3.1.2. Effect of longitudinal steel ratio
An increase in the longitudinal steel ratio increases the
ultimate shear capacity and reduces the deflection at
mid-span as illustrated in Figure 5; an increase of 19%
was recorded between beam C80-2-1 (Pu = 95.5 kN) and
beam C80-2-2 (Pu = 113.6 kN) as the steel percentage
was increased from 1.16 to 2.31%. The increase is
mainly due to the dowel action which improves with the
amount of longitudinal steel crossing the cracks. The
dowel contribution to the shear capacity is higher for
high strength concrete beams since the bond between
the dowel reinforcement and concrete is stronger
(Abdelmadjid and Michel 1995; Michel 1996).
Moreover, the increase in the amount of longitudinal
steel leads to an increase in the effective depth of the
ε (mm/mm)
0,009 0,008 0,007 0,006 0,005 0,004 0,003 0,002 0,001
0 −0,001 −0,002 −0,003
0
f ′c = 44.2 MPa
compression zone as shown in Figure 6 below and this
also improves the shear capacity, even though, not in
proportion with the increase in concrete strength.
The tests have also demonstrated that the beam
specimens reinforced with ρ = 2.31% exhibited less
strains in the longitudinal steel than those reinforced
with ρ = 1.16% (Figure 5). This is probably due to the
fact that beams containing 1.16% steel are more underreinforced and hence exhibit more deformation than
those with a 2.31% steel. The balanced steel ratio for the
present beams is ρb = 2.7%.
3.1.3. Effect of shear span /depth ratio (a/d)
Figure 7 shows the mid-span deflections against the
applied loads for beams having a constant steel ratio of
ρ = 1.16% and varying a/d ratios.
The load-deflection curves for beams with a/d = 1 are
steeper than those with a/d of 1.5 and 2. The ultimate
deflections of beams with a/d of 1.5 and 2 are greater
than those when a/d = 1. Thus stiffness, as represented
by the load deflection curves, reduces as a/d increases.
The ultimate load decreased as the a/d increased. This is
due to the strut and tie action (tied-arch action) effect
which becomes greater as the a/d gets smaller. The
ultimate loads for beams C80-1-1, C80-1.5-1, C80-2-1,
and C80-3-1 were 210.9 kN, 144.3 kN, 95.5 kN, and
ε (mm/mm)
0,009 0,008 0,007 0,006 0,005 0,004 0,003 0,002 0,001
0 −0,001 −0,002 −0,003
0
f ′c = 85.5 MPa
2
2
4
6
8
10
ρρ = 1.16%
12
ρρ = 2.31%
C40-1-1
C40-1-2
C40-2-1
C40-2-2
14
Overall height (cm)
4
6
8
10
ρ = 1.16%
12
ρ = 2.31%
Overall height (cm)
contributed by aggregate interlocking is close to nil. The
shear contribution in HSC is due mainly to the dowel
effect and, at a lesser extent, to the compression zone,
which is not so deep in the case of higher compressive
strengths. Indeed, the vertical flexural cracks were
longer in beams made of HSC than in the case of NSC.
This explains the relatively low increase of shear
capacity in high strength concrete even though the
compressive strength of the material is increased
considerably.
14
16
16
C40-3-1
C40-3-2
C80-1-1
C80-1-2
C80-2-1
C80-2-2
C80-3-1
C80-3-2
Figure 6. Evolution of the depth of the compression zone with the longitudinal steel ratio
270
270
C40-1-1
240
C40-1.5-1
150
120
90
150
120
90
60
30
30
0.0
0.5
1.0
1.5
2.0
2.5
Mid-span deflection (mm)
3.0
3.5
4.0
C80-2-1
180
60
0
f ′c = 85.5 MPa
C80-1.5-1
210
C40-2-1
180
Applied load (kN)
Applied load (kN)
210
C80-1-1
240
f ′c = 44.2 MPa
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Mid-span deflection (mm)
Figure 7. Mid-span deflection against applied load for beams without stirrups (C40 and C80)
34
Advances in Structural Engineering Vol. 13 No. 1 2010
M. Hamrat, B.Boulekbache, M.Chemrouk and S.Amziane
50.72 kN respectively. Similarly, for beams made of
NSC and having the same longitudinal steel ratios, C401-1, C40-1.5-1, C40-2-1, and C40-3-1, the ultimate
loads were 187 kN 129.38 kN, 85.1 kN and 47.25 kN
respectively. The results for both types of concrete clearly
reveal the decrease in the ultimate load as a/d increases.
Moreover, the mid-span deflection at ultimate load
increased as the value of a/d increased, revealing the
above pronounced flexural behaviour with greater a/d.
For the smaller values of a/d, shear dominated the
beam’s behaviour and, thus, the deflection is smaller.
Indeed, it is clearly known from the literature (Park and
Paulay 1975; Chung and Ahmad 1994) that for beams
with a/d ≥ 2.5, flexure dominates the loading behaviour,
resulting in a significant increase in deflection compared
with beams having smaller values of a/d.
It is to be noted that the maximum deflection for
beams made of HSC (group C80) is 20% higher than for
the corresponding beams made with NSC (group C40).
This is thought to be mainly due to the better quality of
the bonding between concrete and the reinforcing steel
in the case of HSC, enabling a better transmission of
internal stresses and strains from concrete to the
reinforcing steel.
The tests also demonstrated that the ratio Pu /Pcr
(ultimate load/load causing diagonal cracking)
decreased as a/d increased. This ratio, which represents
the reserve of strength beyond diagonal cracking, was
also seen to increase as the compressive strength
increased. For a/d = 1, 1.5 and 2, the beams possessed a
moderate strength reserve after diagonal cracking. For
a/d = 3, however, the ultimate load was sensibly equal
to the diagonal cracking load, with no strength reserve
for beams made of NSC (C40), whereas for HSC (C80),
the ultimate load was 22 to 27% higher than the
diagonal cracking load (see Table 4).
3.2. Crack Development and Modes of Failure
3.2.1. Crack development
Typical crack propagation for the beams tested under
the various shear span/depth ratios (a/d) is shown in
Figure 8. Due to zooming restrictions of the recording
equipment, the shear-span was monitored only at one
end of beam and the crack pattern assumed to be
symmetrical for both shear-spans.
The flexural cracks were the first to develop at the
central part of the beams. They widened on subsequent
loading but were never harmful enough to precipitate
failure and tended to close up with the development of
inclined and diagonal cracking in the shear span zone.
Their formation, however, did reduce the stiffness of the
beams. Such a reduction is clearly seen in the flattening
of the load-deflection curves of the test specimens.
Advances in Structural Engineering Vol. 13 No. 1 2010
For specimens made of NSC, the first crack occurred
vertically at a load level of 13% - 25% of ultimate. A
similar crack was observed at a load level of 15% - 30%
of ultimate for the beams made of HSC. In effect, HSC
has a higher modulus of rupture than NSC which results
in the delayed flexural cracking observed in these tests.
However, the beams made of HSC showed more flexural
cracking than the beams made of NSC (see Table 4).
This might be explained by the better quality of the
bond between concrete and steel in the case of HSC,
resulting in the steel reinforcement restraining the crack
opening better.
With further increase in load, new flexural cracks
formed in the shear spans curving toward the loading
area. The development of diagonal shear cracking in the
beams, indicating inclined strut action, depended
strongly on the a/d ratio:
(1) For small values of a/d (a/d ≤ 2), the cracks
develop diagonally within the shear span
extending from the loading point to the support,
clearly indicating an inclined strut;
(2) For larger values of a/d, the diagonal cracking,
which is shorter, is rather the development of a
flexural crack that bends towards the loading
point (see Figure 8).
The slopes of the diagonal cracks were considerably
different for the four series of beams. For beams with
a/d = 1, the diagonal crack was often unique and
inclined at more than 45° to the longitudinal axis of the
beam (Figure 8), whereas for a/d = 3, the diagonal crack
was more horizontal (about 38°).
The same crack pattern was observed for both
types of concrete as in Figure 8. A great difference,
however, is noticed in the trajectories of the cracks
which are straighter in beams made of HSC material.
The cracks go right througth the aggregate particles
and do not deviate around them, a cracking trajectory
which is completely different from that observed in
NSC beams.
The diagonal crack width varied from 0.27 mm to
1.10 mm for beams made of ordinary concrete. Those in
beams made of HSC were wider, ranging from 0.22 mm
to 1.5 mm. This is probably due to the greater brittleness
of the material, liberating more energy when cracking.
Such a crack opening, cutting through an aggregate
particle, is shown in Figure 1.
3.2.2. Modes of failure
All beams failed in shear. Beams with shorter shear
spans (a/d ≤ 2) failed in shear-compression as in
Figure 8. In this type of failure, the diagonal crack,
which appears independently within the shear-span and
not as a result of prior flexural cracks, extends towards
35
Shear Behaviour of RC Beams without Stirrups Made of Normal Strength and High Strength Concretes
C40-1-1
C40-2-1
C40-3-1
45°
40°
Deviation
cracking
trajectory
C80-1-1
Support
Support
C40
C80-3-1
C80-1.5-1
45°
38°
Support
Support
Linear
cracking
trajectory
Support
C80
Figure 8. Crack development in beams without stirrups
(Photos obtained by digitizing video GOM ARAMIS system)
the compression zone at the support or to the loading
point resulting in crushing of the concrete at those
locations. The crushing of concrete in the compression
zone at the loading point occurs often in an explosive
36
manner in HSC as a result of a sudden release of
accumulated energy.
On the other hand, for beams having an a/d ratio equal
to 3, the inclined crack resulting in failure develops from
Advances in Structural Engineering Vol. 13 No. 1 2010
M. Hamrat, B.Boulekbache, M.Chemrouk and S.Amziane
a flexural one within the shear-span and extends towards
the support point (see Figure 8). At subsequent loading,
the diagonal crack widens to split the beam into two
parts. This type of failure is referred to as diagonal
tension. At the lower end of the diagonal crack, a
horizontal crack may develop along the longitudinal
reinforcement, destroying the bond between concrete
and the reinforcing steel. This failure mode is less
explosive. Beams with a/d ≥ 3 display more cracks
during loading. These cracks release the stored enegy
gradually so that, at the ultimate state, the final amount
of energy liberated is not as great as cases with a/d ≤ 2,
and is therefore a less explosive failure (Bazant and
Kazemi 1991).
4. PREDICTION OF THE ULTIMATE SHEAR
STRENGTH OF RC BEAMS
Many equations have already been proposed to estimate
the concrete contribution to the shear resistance, Vc, which
is considered as independent of shear reinforcement.
Two Formulae similar to that proposed by Zsutty
(1968), but applicable to high strength reinforced
concrete beams, have been developed during this study
using multiple regression analysis. The proposed
expressions are based on a total of 247 test results, some
tested by the authors (Hamrat et al. 2009) and the rest
extracted from the literature such as those of Mphonde
and Frantz (1984), Ahmad and Lue (1987), Mphonde
(1988), Xie et al. (1994), Kim and Park (1994), Shin et al.
(1999), Ozcebe et al. (1999), Pendyala and Mendis
(2000), Khaldoun and Khaled (2004), Cladera and Mari
(2005), Shah and Ahmad (2007), and finally Lee and
Kim (2008). The test beams considered have various
concrete strengths, steel ratios, shear-span/depth ratios
(a/d) and geometrical sizes (Table 5 ).
The proposed expressions, below, are based on a
regression analysis and take into consideration the main
parameters that are found to influence the shear resistance
of concrete as discussed above, namely the concrete
compressive strength, the main longitudinal steel ratio
and the shear-span/depth ratio. Since the behaviour of
the concrete beams in shear was found to be influenced
by a “strut and tie” action for the shorter shear-spans,
the proposed formulae take this into consideration by
Table 5. Dimensions and properties of test specimens and measured shear stress
Authors
Mphonde
(1984)
Ahmad
(1987)
Mphonde
(1988)
Xie
(1994)
Kim
(1994)
Shin
(1999)
Ozcebe
(1999)
Pendyala
(2000)
Khaldoun
(2004)
Cladera
(2005)
Shah
(2007)
Lee
(2008)
Authors
Beam data
Total
ν
*
test
)
*
νtest (ν test
=
V
bd
Number
of beams
f c′
MPa
b
mm
d
mm
h
mm
a/d
ρ
%
12
42 to 94
152
298
337
1.5 to 3.6
3.36
1.12 to 6.85
49
66 to 73
127
184 to 213
254
1 to 4
1.8 to 6.6
0.49 to 9.32
10
40 to 84
152
298
337
3.6
3.36
1.68 to 2.95
14
40 to 104
127
198 to 216
254
1 to 4
2.07 to 4.54
1.34 to 8.8
14
54
170
255 to 272
300
1.5 to 6
1.01 to 4.68
1.22 to 4.69
30
52 and 73
125
215
250
1.5 to 2.5
3.77
1.95 to 5.29
12
58 to 82
150
310
360
3 and 5
2.6 to 4.4
1.54 to 1.83
16
55 to 91
80
140
160
2 and 5
2
1.42 to 1.99
07
61 to 66
200
300 to 330
370
2.7 and 3
2.2 and 4
1.9 to 2.8
17
50 to 87
200
351 to 359
400
3 to 3.1
2.24 to 2.99
1.2 to 1.95
20
56.5
230
245 to 253
300
3 to 6
1 to 2
0.58 to 1.86
12
40.8
350
385 to 410
450
2 to 4
1.79 to 4.76
0.78 to 1.67
34
44.2 to 85.5
100
127 to 135
160
1 to 3
1.2 and 2.3
1.76 to 8.36
247
40 to 104
80 to 350
127 to 410
160 to 450
1 to 6
1 to 6.6
0.49 to 9.32
MPa
: Shear stress at diagonal cracking for beams having stirrups.
Advances in Structural Engineering Vol. 13 No. 1 2010
37
Shear Behaviour of RC Beams without Stirrups Made of Normal Strength and High Strength Concretes
10
40 MPa ≤ f ′c ≤ 104 Mpa
9
νtest
νpred
8
νtest, νpred (MPa)
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
a/d
Figure 9. Comparison between predicted and measured shear stresses
providing two distinct expressions depending on the
shear-span/depth ratio. These expressions are valid for
reinforced concrete beams having strengths in the range
40 ≤ f′c ≤ 104 MPa. The correlation coefficients are 0.74
and 0.86 for Eqns 4 and 5 respectively.
d

= 3.88  fc′⋅ ρ ⋅ 

a
0.47
d

Beams with a/d ≥ 2.5: ν prop = 2.40  fc′⋅ ρ ⋅ 

a
0.41
Beams with a/d < 2.5: ν prop
(4)
(5)
Figure 9 presents a comparison between the predicted
shear stresses using the proposed formulae and the
measured ones. A good correlation exists between the
experimental values and the predicted values over a
80
f ′c = 44.2 MPa
a/d = 1.5
5. COMPARISON OF THE MEASURED SHEAR
CAPACITY WITH THOSE PREDICTED
Figure 10 shows the predicted shear capacity for beams
without stirrups according to most of the main models
in the literature such as ACI-318, BS-8110, BAEL,
Eurocode 2 and those predicted by the equation proposed
by the authors. For comparison purposes, the measured
shear capacities for the tested beams of this study are
also shown. For ease of comparison, all the safety factors
have been taken as equal to 1.0.
80
ρ = 1.16%
70
a/d = 2
60
a/d = 3
a/d = 1.5
a/d = 2
a/d = 3
Prop. Eq
Test res
Euroc- 2
BAEL 99
BS-8110
ACI -318
Test res
Prop. Eq
Euroc- 2
BAEL 99
BS-8110
ACI -318
Test res
Prop. Eq
Prop. Eq
Test res
Euroc- 2
BS-8110
ACI -318
BAEL 99
Test res
Prop. Eq
Euroc- 2
BS-8110
ACI -318
BAEL 99
Test res
0
Prop. Eq
10
0
Euroc- 2
20
10
BS-8110
20
ACI -318
30
Euroc- 2
40
30
BS-8110
40
BAEL 99
V(kN)
50
BAEL 99
V (kN)
50
ρ = 1.16%
f′c = 85.5 MPa
70
ACI -318
60
wide range of shear-span/depth ratios. The correlation is
even better for a/d values higher than 1.5, where beam
action dominates the shear behaviour. For shorter shear
span where a/d < 1.5, the strut and tie action dominates
the loading behaviour and the proposed expression
needs more refinement.
Figure 10. Shear capacity: comparison of test results with the various predictions
38
Advances in Structural Engineering Vol. 13 No. 1 2010
M. Hamrat, B.Boulekbache, M.Chemrouk and S.Amziane
It can be seen from Figure 10 that for a/d = 1.5, great
differences exist between the different predictions and
the experimental values. The values predicted by ACI318, BAEL and, to a lesser extent BS-8110, are
excessively conservative. Those predicted by Eurocode 2
and by the formulae proposed by authors yield
relatively better results though they are still relatively
conservative. The difference between the experimental
shear strengths and the predicted values reveals that
most of the theoretical models, particularly those of
ACI-318 and BAEL, do not take into account in a
rational manner the increase in shear capacity of beams
having shorter shear spans. Even the best predictions do
not adequately reflect the strut and tie action behaviour
that is exhibited by beams with shorter shear-spans, a
mechanism similar to that applying to deep beams. For
NSC and HSC, both Eurocode 2 and the authors’
proposed formulae seem to give better predictions,
though the Eurocode 2 may slightly overestimate the
shear capacity of HSC beams having higher shear-spans
of a/d = 3 as illustrated in Figure 10. In contrast, both
the ACI-318 code and the BAEL code are excessively
conservative. It is to be noted however, that Figure 10
shows clearly that the four design models for shear
capacity prediction are more conservative for NSC than
they are for HSC, particularly in the case of Eurocode 2.
This latter design code should be considered with care,
therefore, when used with high strength concrete,
considering the catastrophic nature of shear failures in
the case of HSC and the brittlness of this material.
For beams having a/d of 2 and more, the predictions
improve for all the models, particularly so for HSC
beams. For a/d = 2, the Eurocode 2 procedure together
with the proposed formulae correlate better with the
experimental results for beams made of HSC than is the
case with the ACI-318, BAEL and BS-8110 formulae.
In general, the prediction of shear strength values is
better in comparison to the predictions when a/d is
smaller than 1.5 for the five models. This is probably
due to the influence of the strut action behaviour which
decreases in effect as a/d increases leading to more
pronounced beam-action.
When a/d is equal to 3, the predictions of the five
models compare well with the measured ones, revealing
that for beam-action behaviour, the four codes and the
proposed formula are satisfactory for both NSC and for
HSC. It is to be noted that the French code BAEL is the
only one that does not explicitly take into account either
the main longitudinal steel ratio or the a/d ratio. This
explains why it is so conservative.
To sum up, it can be argued that the four shear design
code provisions discussed in this paper seem to lead to
safe design against shear forces for both NSC and for
Advances in Structural Engineering Vol. 13 No. 1 2010
HSC. The formulae developed by the authors in this
study lead to results which are as good as the best
predictions of those four codes. For beams with shorter
shear spans, however, the behaviour of the beam
approaches that of a deep beam with more strength
reserve. This strength reserve is related to the strut and
tie action behaviour following the development of the
diagonal crack. Such behaviour is clearly revealed by
the crack patterns defining inclined compression struts
within the shear spans and a tension tie region at the
bottom where the cracks are uniformly distributed. In
practical cases, this mechanism is present when the load
is applied close to the support resulting in shorter shearspans. None of the models being used take this
mechanism into account. The formulae proposed by
authors also needs some improvement to better model
the behaviour of beams with shorter shear spans.
6. CONCLUSIONS
In this study sixteen RC beams were tested to evaluate
the contributions of a/d, ρ and f′c on the global
behaviour in shear for beams made of NSC and for
beams made of HSC. Based on the experimental results
obtained, the following conclusions are drawn:
(1) Beams without stirrups, particularly those made
of HSC, exhibit a brittle behaviour;
(2) An increase of 13% in strength capacity was
recorded as the compressive strength of the
concrete increased from 44.2 MPa to 85.5 MPa.
This is a small benefit given that the concrete
strength had almost doubled;
(3) An increase in longitudinal steel ratio increases
the ultimate shear capacity and reduces the
deflection at mid-span; an increase of 19% was
recorded between beam C80-2-1 and beam
C80-2-2 where the steel percentage increased
from 1.16 to 2.31%;
(4) Ultimate load decreases as a/d increases. In the
same manner, mid-span deflections at ultimate
load increase as the values of a/d increase,
revealing the tendency for a pronounced flexural
behaviour that is more associated with a beamaction as a/d increases. The tests also
demonstrated that the ratio Pu /Pcr decreased as
a/d increased. For a/d = 3, the ultimate load was
effectively equal to the load at which diagonal
cracking occured, without any strength reserve
for beams made of NSC (C40). For those made
of HSC (C80), the ultimate load was 22 to 27%
greater than the diagonal cracking load;
(5) The formulae proposed by authors give better
predictions than do most of the models used in
the existing codes. The formulae yield similar
39
Shear Behaviour of RC Beams without Stirrups Made of Normal Strength and High Strength Concretes
predictions as Eurocode 2 estimates, considered
to be the most rational document in use today,
even though, this design document might
overestimate the shear capacity of HSC beams at
the higher a/d ratios. The proposed formulae are
safe for the design of both NSC beams and HSC
beams provided the relevant safety factors are
used. The formulae take into consideration the
main parameters which most influence the shear
capacity of reinforced concrete for both normal
strength and high strength concretes;
(6) The shear capacity of HSC does not seem to
increase at the same rate as the compressive
strength of the material. The four major code
provisions for shear in NSC are safe for use also
in the case of HSC with the exception that
Eurocode 2 should be used with care as, in the
case of HSC, it might have a tight safety margin
against brittle shear failures;
(7) The different models considered in this study do
not accurately reflect the increase in shear
capacity of beams with shorter shear spans
(a/d = 1.5). For these types of beams, most of
the design models are excessively conservative,
and the code predictions only seem to be more
accurate as a/d increases beyond a value of 2.0.
It seems that the guideline models for shear
design given in the four codes considered above
are better suited to NSC and HSC beams which
exhibit beam action behaviour.
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41