Shear Strength Response of HSRC deep beams without Stirrups
Part II Beam characteristics
Ahmed I. Ramadan*; Aly G. Abd-Elshafy; Mahmoud H. Ahmed and Atif M. Abdel-Hafez
Civil Engineering Department, Faculty of Engineering, Assiut University, Assiut, Egypt
*Corresponding email: a.i.h.r1978@gmail.com
Abstract: ACI code specifies the shear strength of deep beams based on the strength at the
first diagonal crack of NSC beams without the consideration of beam size effect. Therefore, it
is necessary to evaluate if the ACI design equation for deep beams is applicable to high
strength reinforced concrete (HSRC) deep beams with main reinforcement ratio less and more
than 1% with considering size effect or not. This paper is considered a supplement to the
companion paper (Part I Comparison of Design Equations). Eighteen simple span HSRC deep
beams without stirrups were tested to examine various parameters on the shear capacity;
f’cu=50 MPa, three values of main reinforcement, (ρs%), (0.73%,1.21% &1.83%) and four
values of shear span to overall depth ratio, (a/h), ( 0.84,1.3,1.7&2.3) were selected to mainly
study the characteristics of deep beams. The increase in overall depth (h) under the same
a/h=1.3&1.7, led to more brittle failure with wide diagonal cracks and high energy release
rate related to size effects. HSRC deep beams exhibited more remarkable size effects with
regard to brittle behavior. It was also shown that the ACI code gives similar safety factors on
the shear strength at the first diagonal crack of HSRC deep beams, but do not specify a high
enough safety factor on their ultimate strength due to the size effects.
Keywords: deep beams, high strength concrete, first diagonal crack, ultimate strength, Beam
characteristics.
1. INTRODUCTION
Deep beams are structural elements loaded as beams but having small shear span to effective depth
ratios. These have useful applications in many structures, such as tall buildings, foundations,
offshore structures, and several others. Many formulas in concrete design codes can of course be
developed purely empirically because it is possible to obtain sufficient test data for the entire range
of practical interest, and to sample that range statistically uniformly, without alignment; an example
is the ratio of tensile and compressive strengths, or the effect of reinforcement ratio in various code
specifications. Unfortunately, the size effect is not a problem of that kind. Thus, it is necessary to
investigate the shear behavior of deep beams according to various variables including beam size.
Based on the strength at the first diagonal cracks of NSC normal beams without the consideration of
size effects, the ACI code [1,2] specifies the shear strength of beams. The design code has been
developed from experimental results obtained from concrete beams without shear reinforcements,
and with concrete strength less than 40 MPa, an average overall depth of 340 mm, and an average
reinforcement ratio of 2%. Thus, it is necessary to investigate the validity of the ACI code when
applied to concrete deep beam in other ranges.
A number of papers reporting on the shear strength of normal beams have concluded that the ACI
code tends to overestimate the shear strength as concrete strength and overall depth increase.
Collins and Kuchma [3] showed experimentally that the ACI code overestimated the shear capacity
of reinforced concrete beams having large sections and small reinforcement ratios. Smith and
Vantsiotis [4] compared between test data obtained from an experimental investigation of 52 simply
supported deep beams and mathematical model. They reported that considerable increase in loadcarrying capacity occurs with increasing concrete strength and decreasing shear span to depth ratio.
The shear strength of deep beams is closely related to the beam size. There have been many studies
to investigate the size effects of normal beams with test variables like shear span to depth ratio (a/h)
and maximum diameter of aggregates by Kani [5], Taylor [6], Walraven and Lehwalter [7].
However, these experiments were confined to concrete strengths less than 40 MPa and the size
effects according to overall depth under the same shear span/depth ratio were investigated. Taylor
[6] reported that size effects diminish if the maximum diameter of aggregates increases in
proportion to the increase in overall section depth. Tan and Lu [8] made a study of size effects of
deep beams having a/h=1.0, concrete strength of 40 MPa, and reinforcement ratio of 2.6 %, and
evaluated and compared the shear behavior of deep beams as the overall depth increases. They
reported that the ACI code predictions are generally conservative for all sizes of beams even though
the conservatism decreases with increasing h and a/h, the CIRIA [9] predictions are unsafe for large
beams with h exceeding 1000 mm, and the Canadian CSA code [10] predictions provide uniform
safety margin for the beam specimens.
The experimental variables to investigate size effects on the shear behavior of deep beams have
been limited to aggregates less than 10 mm and concrete strength of 40 MPa. Thus, it is difficult to
apply the experimental results to members having HSC greater than 40 MPa and other aggregate
sizes. The shear behavior of HSC deep beams with reinforcement ratios less and more than 1%
should be evaluated according to the change of overall depth and the validity of the ACI code
should be examined.
The objective of this paper is to evaluate the shear strength as an index to predict the load carrying
capacity and behavior of 18 HSRC deep beams without web reinforcement and to investigate the
validity of the ACI code involving beam size effects. Test variables include compressive concrete
strength f’cu=50 MPa, tensile reinforcement ratio (0.73%, 1.21% &1.83%), the shear span to
effective depth ratio-(a/d)-(1, 1.5 &2), shear span to overall depth ratio-(a/h)-(0.84, 1.26, 1.3, 1.7 &
1.75), and overall depth (h=400 & 700 mm).
2. EXPRIMENTAL PROGRAM
2-1 Test Specimens
This paper consist of eighteen HSRC deep beams, two groups; nine deep beams each, without
stirrups, summarized in Table 1 and dimension details shown in Fig. 1, was tested. The anchorage
of the longitudinal bars was maintained according to ACI (318-11)-Section (12.5). The width of the
bearing plate at the two loading points, two support points and development length of main steel
bars were calculated, Fig. 2. Only the two support points were reinforced by three web
reinforcements to avoid bearing failure. The clear thickness of cover concrete was selected as 40
mm to prevent a splitting failure of concrete. Strain gages were used to measure the strain of the
longitudinal bars in the specimens. All the beams were tested to failure under two point symmetric
loading. The distance between the two loading points for the specimens (ah) were varied according
to a/h. However, the distance between the two loading points for the specimens of h = 600 mm was
chosen according to two a/h ratios of 0.5 and 1.0. Based on the study by Tan et al. [11] that (L/h)
does not affect the ultimate shear strength of deep beams, this study did not consider the effects of
(L/h).
Table 1. Specimen Details
a
ar
a/h a/d
S mm
mm
mm s
1224 1.7
2 326 200
1224 1.7
2 326 200
1224 1.7
2 326 200
918
1.3 1.5 332 800
918
1.3 1.5 332 800
918
1.3 1.5 332 800
1600 2.3
1 326 1000
1600 2.3
1 326 1000
1600 2.3
1 326 1000
10
11
12
13
14
15
16
17
18
h
mm
B700-2-50-r1 700
B700-2-50-r2 700
B700-2-50-r3 700
B700-1.5-50-r1 700
B700-1.5-50-r2 700
B700-1.5-50-r3 700
B700-1-50-r1 700
B700-1-50-r2 700
B700-1-50-r3 700
d
mm
660
660
660
660
660
660
660
660
660
28
29
30
31
32
33
34
35
36
B400-2-50-r1
B400-2-50-r2
B400-2-50-r3
B400-1.5-50-r1
B400-1.5-50-r2
B400-1.5-50-r3
B400-1-50-r1
B400-1-50-r2
B400-1-50-r3
360 670
1.7
2 330
360 670
1.7
2 330
360 670
1.7
2 330
360 502.5 1.3 1.5 348
360 502.5 1.3 1.5 348
360 502.5 1.3 1.5 348
360 335 0.84 1 365
360 335 0.84 1 365
360 335 0.84 1 365
Beam Name
400
400
400
400
400
400
400
400
400
P/2
800
800
800
1000
1000
1000
1000
1000
1000
Leff
ρs (%) Pu(test) KN
mm
2948 0.73
562.0
2948 1.21
997.0
2948 1.83
939.8
2936 0.73
1007.0
2936 1.21
1399.9
2936 1.83
1288.3
2948 0.73
1443.5
2948 1.21
1949.0
2948 1.83
2492.4
2340
2340
2340
2305
2305
2305
2270
2270
2270
0.73
1.21
1.83
0.73
1.21
1.83
0.73
1.21
1.83
P/2
CS
400&700mm
No.
Ss
Strain Gauge
Varies
LVDT
ar
a
2 LVDT
ah
LVDT
a
Fig. 1 Details of Specimen
ar
416.2
451.3
744.5
560.8
886.0
1075.4
862.7
1176.1
1530.4
Fig. 2 Development length calculations for Beam 10 & 28
2-2 Materials
The beams are constructed using HSC (50 MPa) provided by a local ready-mix supplier. The
strength of the concrete ranged from 48 MPa to 52 MPa with an average value of 50 MPa at the age
of 28 days. Four diameters of high strength deformed bars 10, 12, 14, 18, and 20 mm and of 765,
650, 670, 670, and 670 MPa proof strengths respectively were used for longitudinal reinforcement,
Fig. 3.
2-3 Test Procedure
Each specimen was tested at 28 days as a simply supported beam under four point bending, Fig. 4.
Two point loads were applied at constant rate by hydraulic jacks in a load frame. To monitor the
behavior of the tested beams, the crack widths, strains in the reinforcement and at the concrete
surface, and displacements are measured using different instruments. Instrumentation of the beams
includes electrical resistance strain gauges for strain measurements and two of linear variable
displacement transducers (LVDTs) for deflection and crack width measurements at middle of span
and one at each of middle of shear span.
Fig. 3 Stress-Strain of used steel bars
Fig. 4 Test Setup of test Specimens
3. RESULTS AND DISCUSSION
3-1 Results
All specimens exhibited similar overall behavior, which was characterized by a nearly bilinear
response. The measured load, deflection, crack development and failure of each of the eighteen
tested specimens were recorded. The tests results for the experimental program are summarized in
the companion paper.
3-2 Mode of Failures
Initial flexural cracks occurred at the mid-span of the specimen, (10–35%) of maximum load, and
then diagonal cracks were propagated, in range 25-85% of maximum load, to the mid-height of
beam rapidly. The specimen showed brittle failure mechanism with the appearance of the diagonal
crack carrying capacity by strut region. Regardless of the tensile reinforcement ratio, the crack
development and failure modes influence the shear span to overall depth ratio (a/h) and overall
depth of specimens (h). As in the companion paper, crack patterns of specimens with (a/h =
1.26&1.3) exhibited the combined failure modes of diagonal shear cracks and shear compression.
It is logically that the rigidity of tested beams decreased once the diagonal cracks started appear and
remarkably decrease with the increase in (a/h), . This can be explained as with increasing overall
depth, the beams behave more brittle and more energy release at the diagonal crack. Fig. 5
represents the strain values of the longitudinal bars along hall span for some tested beams, almost
symmetric with increasing strains towards mid-span of the beam. It can be shown that the
longitudinal bars did not reach the yield strain value, according to steel tension test, (0.46%) at the
maximum load. It was observed a sudden reduction in strain values of main reinforcement between
the inner edge of the support plate and the back side of the plate. Before failure occurs, strains in
excess of yield were recorded near the face of the bearing plate of support.
a) Beam 10
b) Beam 11
a) Beam 18
b) Beam 33
a) Beam 35
b) Beam 36
Fig. 5 Strain distribution of lower layer of reinforcement steel
Japanese electrical strain gages were used to measure concrete strains in the mid-span region and at
the middle of inclination line between load plate and bearing support plate (εST). Fig.5 (a-f)
illustrate the progression of surface concrete strains (εc), at failure in the flexural compression zone
ranged between 0.001 and 0.0035, and at the strut region, which simulates the main load-carrying
strut assumed in the STM design, with increasing the applied load for two locations.
3000
2500
Load (kN)
2000
Relationship between applied load &
strains of concrete & SG of strut for B16
& 17 & 18
1500
1000
500
CS1 B16
CS2 B16
0
Strain ε-750
(μm)
-2250-2000-1750-1500-1250-1000
-500 -250
0
250
a) Beams 10, 11 & 12
Load (kN)
1100
Relationship between applied load &
1000
strains of concrete & SG of strut for
900
800
B10 & 11 & 12
700
600
500
400
300
200
100
CS1 B10
CS2 B10
0
(μm)
-2900
-2650
-2400
-2150
-1900
-Strain
1650
-1400
-ε1150
-900-650-400-150 100
b) Beams 13, 14& 15
Load (kN)
1000
c) Beams 16,
Relationship between applied load &
strains of concrete & SG of strut for B28
& 29 & 30
CS1 B28
CS2 B28
0
ε (μm)
-2000 -1750 -1500 -1250Strain
-1000 -750
-500 -250
0
250
17& 18
1500
1250
Load (kN)
1000
Relationship between applied load &
strains of concrete & SG of strut for B13
& 14 & 15
750
500
250
CS1 B13
CS2 B13
0
Strain
(μm) -250
-1750 -1500 -1250 -1000
-750 ε-500
0
250
d) Beams 28, 29& 30
Load (kN)
2000
Relationship between applied load &
strains of concrete & SG of strut for B31
& 32 & 33
CS1 B31
CS2 B31
0
ε (μm)
-2000 -1750 -1500 -1250Strain
-1000 -750
-500 -250
0
250
e) Beams 31, 32& 33
Load (kN)
2000
Relationship between applied load &
strains of concrete & SG of strut for B34
& 35 & 36
CS1 B34
CS2 B34
0
Strain
ε (μm)-500 -250 0 250
-2500-2250-2000-1750-1500
-1250-1000-750
f) Beams 34, 35& 36
Fig. 6 (a-f) Relationship of (Papp.- εc & εST)
3-2 Vision on Diagonal crack and ultimate strength Stresses
After the first diagonal crack, the tested beams exhibited much load-resisting capacity by
compressive strut action. The difference between the ultimate strength (Vu) and the diagonal crack
load (Vcr) is defined as reserve strength which was analyzed according to the experimental variables.
This study considered Vu/Vcr as a way to measure the reserve strength. At Table 3, the values of
Vu/Vcr are within the range of 2.2–4.7 at the specimens of a/h = 1.3 and 1.6–5.7 for a/h = 1.7. It can
be understood that the increase in overall depth of beam section led to the decrease in load-carrying
capacity after the diagonal crack, [12].
Based on (a/h), Vu and Vcr were investigated according to tensile steel ratio (ρs%); where r1,r2 and r3
indicate to 0.78%, 1.21% and 1.83% respectively, and is illustrated in Fig. 7,8,9&10. With, at
constant f’c =50MPa for all tested beams, increasing (ρs%), 0.78%, 1.21% & 1.83%, for same (a/h)
and the overall depth increases from 400 to 700 mm, ѵcr which is the shear stress at the first
diagonal crack increase about 40,52&60% respectively at a/h=1.3 and 40,70&90% respectively at
a/h=1.7. While on the contrary, with increasing (ρs%), 0.78%, 1.21% & 1.83%, for same (a/h) and
the overall depth increases from 400 to 700 mm, the ultimate shear stress ѵu reduced about 5,16&53%
respectively at a/h=1.3 and 35,40&45% respectively at a/h=1.7. The increase in overall depth led to
the reduction of the resisting capacity. The reduction of the resisting capacity with increasing
overall depth is caused by the decreased aggregate interlocking action due to large crack widths and
high energy release rate due to a number of cracks at the concrete strut.
The specimens of 700 mm overall depth showed several flexural cracks but did not show much
diagonal cracks. In the specimens of 400 mm overall depth, flexural cracks gradually extended to
compression region of beam section, and diagonal cracks were considerably developed at the
concrete strut. It can be understood that deep beams after the first diagonal crack resist the load by
compressive strut action and the crack width gradually increases with higher energy release rate due
to the reduction of reserve strength.
r1,h400
r2,h400
r3,h400
1.00
r1,h700
0.00
r2,h700
0.00 0.50 1.00 1.50 2.00 2.50
Fig. 7 (ѵcr - a/h)
a
10.00
Fig. 8
ѵu a/h
5.00
MPa
MPa
ѵcr
2.00
ѵcr a/h
ѵu
3.00
r1,h400
r2,h400
r3,h400
r1,h700
0.00
r2,h700
0.00 0.50 1.00 1.50 2.00 2.50
a
(ѵu - a/h)
r2,ѵcr
a
r2,ѵu
r3,ѵcr
2.00
6.00
ѵcr , ѵu - a/h ,
h=700 mm
r1,ѵcr
r1,ѵu
4.00
MPa
r1,ѵu
8.00
ѵcr, ѵu
r1,ѵcr
MPa
ѵcr, ѵu
10.00
ѵcr , ѵu - a/h ,
8.00
h=400 mm
6.00
4.00
2.00
0.00
0.00
1.00
r2,ѵcr
0.00
r3,ѵcr
0.0 0.5 1.0 1.5 2.0 2.5
2.00
r2,ѵu
a
Fig. 9 (ѵcr ,ѵu - a/h) for h=400 mm
(ѵcr ,ѵu - a/h) for h=700 mm
Fig. 10
It can be observed that the increase in (ρs%) cause good development in ultimate strength of deep
beams. Comparing results of a/h=0.84,1.3,1.7& 2.3, it can be observed that the specimens of
a/h=0.84&1.3, which show high energy release rate, are less sensitive than that of a/h=1.7&2.3.
3-3 Comparison between ACI 318-11 code and experimental shear strength results
According to ACI code specification, the shear strength is the sum of the shear strength of concrete
contribution and those of the stirrups. The ultimate shear strength of deep beams without stirrups is
expressed by multiplying the diagonal crack strength of concrete beams (Vcr)test by the reserve
strength factor (Vu)test /(Vcr)test.
According to the specification of ACI code, there is a restriction on material properties in shear
design equation, as concrete strength < 40 MPa, the overall depth in range of (300–500) mm, and
2.2% for the main reinforcement ratio. Therefore, it is necessary to investigate the validity of the
ACI code on HSRC deep beams without stirrups with considering the size effect and the reserve
strength. Table 2 compares experimental results and analytical results calculated by ACI eqs. (1),(2)
for shear design code of deep beams:
(Vcr)ACI = [0.156 √f’c +17.96ρ (Vu d/Mu ] bwd
(1)
(Vc)ACI = [3.5-2.5 Mu/Vu d ] [ 0.156 √f’c +17.96ρ (Vu d/Mu) bwd
(2)
;where; [3.5-2.5 Mu/Vu d ]≤2.5 ,
(Vc)ACI ≤ 0.5 √f’c bwd
(Vcr)ACI denotes the shear strength at the first diagonal crack, and (Vc)ACI the shear strength
contributed by concrete.
a/h
Pcr
(KN)
(Vcr)test
(KN)
(Vu)test
(KN)
(Vu)R
(KN)
1.7
1.7
1.7
1.3
1.3
1.3
2.3
2.3
2.3
1.7
1.7
1.7
1.3
1.3
1.3
0.84
0.84
0.84
122.5
125
130
137.5
175
245
200
200
260
34.63
37.17
37.84
35.76
38.81
35.91
31.82
34.49
33.07
175
200
240
175
265
300
300
325
450
50
65
65
60
75
110
120
115
150
281.0
498.5
469.9
503.5
699.9
644.1
721.8
974.5
1246.2
208.1
225.7
372.3
280.4
443.0
537.7
431.4
588.0
765.2
106.0
298.5
229.9
328.5
434.9
344.1
421.8
649.5
796.2
158.1
160.7
307.3
220.4
368.0
427.7
311.4
473.0
615.2
ACI (318-11) code Values
(Vu)tes
t
/(Vcr)te
st
1.6
2.5
2.0
2.9
2.6
2.2
2.4
3.0
2.8
4.2
3.5
5.7
4.7
5.9
4.9
3.6
5.1
5.1
ValueTest
/ValueACI
ѵu/√f'c
(Vcr)ACI
(KN)
(Vu)ACI
(KN)
(Vu)test
/(Vu)ACI
10
11
12
13
14
15
16
17
18
28
29
30
31
32
33
34
35
36
Experimental Values
(Vcr)test
/(Vcr)ACI
Beam No.
Table 2. Test results and ACI values of Vcr ,Vu & ѵu/√f'c
0.2
0.4
0.4
0.4
0.6
0.6
0.6
0.8
1.1
0.3
0.2
0.6
0.4
0.7
0.8
0.9
1.2
1.2
193.7
201.3
211.3
197.6
207.8
221.0
190.9
196.8
204.4
105.6
109.8
115.2
107.7
113.3
120.5
112.0
120.3
131.1
197.7
205.0
214.4
201.4
211.1
223.6
195.1
200.7
207.9
107.8
111.8
116.9
109.8
115.1
121.9
113.8
121.7
131.9
0.9
1.0
1.1
0.9
1.3
1.4
1.6
1.7
2.2
0.5
0.6
0.6
0.6
0.7
0.9
1.1
1.0
1.1
1.4
2.4
2.2
2.5
3.3
2.9
3.7
4.9
6.0
1.9
2.0
3.2
2.6
3.8
4.4
3.8
4.8
5.8
3-4 Shear strength at the first diagonal crack & Ultimate shear strength
Fig. 11 illustrates the relation of (Vcr)test / (Vcr)ACI and a/h according to ρs% & h. It can be shown,
the experimental result is (1.3-1.4) & (1.6-2.2) times as large as the ACI code in case of h=400 mm
and a/h=1.3& 2.3 respectively. However, the safety factor decreases with the increase in the depth,
the specimens of overall depth 700 mm exhibit an inadequate safety factor. It can be concluded that
the calculation of predicted load at the diagonal crack according to ACI 318-11 for HSRC tested
deep beams, 700 mm, should include the size effects.
Fig. 12 shows the variation of (Vu)test / (Vu)ACI to compare the ACI code 318-11 and the
experimental results with variation of (h). The safety factor represents a tendency to abruptly
decrease with the increase in the overall depth of beam. Based on this experimental study, it can be
observed that the ACI code prediction 318-11 overestimates the ultimate shear strength of HSRC
deep beams with overall depth 700 mm. Thus, it can be expected that the ACI code prediction
regarding the ultimate shear strength of deep beam is conservative for HSRC deep beams with
(Vcr)test
/(Vcr)ACI
2.00
(Vcr)test
/(Vcr)ACI a/h
r1,h400
r2,h400
r3,h400
r1,h700
1.00
(Vu)test
/(Vu)ACI - a/h
1.50
r1,h400
r2,h400
1.00
(Vu)test
/(Vu)ACI
3.00
r3,h400
r1,h700
0.50
r2,h700
0.00
0.00
1.00
a
2.00
r3,h700
3.00
r2,h700
0.00
0.00
1.00
a
2.00
r3,h700
3.00
overall depth 400 & 700 mm.
Fig. 11 ( ѵcr,test/ ѵcr,ACI - a/h)
Fig. 12 ( ѵu,test/ѵu,ACI - a/h)
7. CONCLUSIONS
This experimental study investigated the shear strength response of HSRC deep beams by studying
the beam characteristics as size effects and evaluated the validity of the ACI code 318-11. As a
conclusion of the experimental results, it was observed that:
1) The tested beams of a/h =0.84&1.3 show less sensitive size effects than the ones of a/h=1.7&2.3,
with more brittle failure at strut region and higher energy release rate as the results of size effects
with the decrease in a/h.
2) At constant f’c =50MPa, increasing (ρs%) 0.78%, 1.21% & 1.83%, for same (a/h) and (h)
increases from 400 to 700 mm, ѵcr increase about 40,52&60% respectively at a/h=1.3 and
40,70&90% respectively at a/h=1.7.
3) On contrary, at constant f’c =50MPa, increasing (ρs%) 0.78%, 1.21% & 1.83%, for same (a/h)
and (h) increases from 400 to 700 mm, ѵu reduced about 5,16&53% respectively at a/h=1.3 and
35,40&45% respectively at a/h=1.7.
4) a/h=0.84 show high energy release rate, it can be concluded that the specimens represent more
brittle failure of concrete strut and higher energy release rate as the results of size effects with the
decrease in a/h.
5) Depending on the results of size effect, the safety factor decreases with increasing in the overall
depth, especially 700 mm depth exhibit an inadequate safety factor.
6) The ACI code prediction overestimates the ultimate shear strength of deep beams with overall
depth 700 mm. Thus, it can be concluded that the calculation of predicted load at the diagonal crack
according to ACI 318-11 for HSRC tested deep beams, 700 mm, should include the size effects.
REFERENCES
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NOTATIONS
a = Shear span, distance between concentrated load and face of support, in mm
ah = distance between two concentrated loads , in mm
ar = distance between end of beam and face of support, in mm
b = Beam width, in mm
d = Effective beam depth, in mm
do = maximum aggregate size, in mm
a/h = shear span to overall depth ratio
f’c = Cylindrical compressive strength of concrete, in MPa
ρs = Ratio of Longitudinal reinforcement ratio; = As /bd
Ss = distance between two stirrups under concentrated loads, in mm
Vu = the ultimate strength load
Vcr = the diagonal crack load
(Vcr)ACI = the shear strength at the first diagonal crack according to ACI code
(Vc)ACI = the shear strength contributed by concrete according to ACI code
ѵcr = the shear stress at the first diagonal crack
ѵu = the ultimate shear stress