# Shear Strength Prediction Model of FRP Bar-Reinforced Concrete Beams without Stirrups

Authors: Danying Gao; Changhui Zhang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

The shear strength prediction model for fiber-reinforced polymer (FRP) bar-reinforced concrete beams without stirrups in ACI440.1R-2015 does not consider the “size effect” and the effect of shear span-to-depth ratio and predicts the zero-shear strength for concrete members without longitudinal reinforcement. A modified shear strength prediction model for FRP bar-reinforced concrete beams without stirrups was presented in this paper. The proposed model takes into account the effect of concrete strength, size of the beam, shear span-to-depth ratio, reinforcement ratio, and modulus of elasticity of the longitudinal reinforcement and the “size effect.” The superiority of the proposed model has been evaluated by comparing the calculated shear strength of FRP bar-reinforced concrete beams without stirrups by the proposed model with the experimental results and calculated values by the models in design codes, respectively. It confirmed that the shear strength of FRP bar-reinforced concrete beams without stirrups by the proposed model was in better agreement with the experimental results.


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## Body

## 1. Introduction

Fiber-reinforced polymer (FRP) bars have gained the acceptance as an alternative to conventional steel bars for concrete structures due to their corrosion resistance, high strength-to-weight ratio, and magnetic neutrality. It has been recognized that the flexural capacity of FRP bar-reinforced concrete beams can be predicted by the traditional assumptions used in steel bar-reinforced concrete beams. However, while FRP bars have two drawbacks including brittle failure and low modulus of elasticity compared with the steel bars, the shear behavior (including shear strength, deformation, and crack width) of concrete beams reinforced with the FRP bar is different from those reinforced with similar amount of steel reinforcement [1–4]. For example, the stiffness [3] and dowel action [4] of FRP bar-reinforced beams are smaller compared to the concrete beams with steel bars, subsequently resulting in the larger deformation and lower shear strength. Therefore, the existing shear strength prediction models for steel bar-reinforced beams cannot be directly applied to FRP bar-reinforced beams.The shear strength prediction model of FRP bar-reinforced concrete beams in ACI440.1R-2015 [5] was based on the research by Tureyen and Frosch [6]. The ACI440.1R-2015 model was a function of the width (b) and effective depth (d) of the beam, concrete compressive strength (fc′), reinforcement ratio (ρf), and modulus of elasticity (Ef) of FRP bars, and the shear span-to-depth ratio (a/d) was not included [7, 8]. Several investigations confirmed that shear strength Vc of FRP bar-reinforced concrete beams decreases as the a/d increases [7, 9–13]. Some investigators [7] revealed that Vc decreases almost linearly with a/d2/3, while the others [10] reported that it decreases linearly with a/d1/3. In addition, the ACI440.1R-2015 model did not take into account the effect of concrete in the tensile zone of the beam on the shear strength for FRP bar-reinforced concrete beams without longitudinal reinforcement and predicted the zero-shear strength for concrete beams without longitudinal reinforcement [8]. Thirdly, the ACI440.1R-15 model did not take into account the “size effect” [14–17] which explains the phenomenon that the normalized shear strength of FRP bar-reinforced concrete beams decreases with the increasing of beam depth.It has been recognized thatfc′, b, d, a/d, ρf, and Ef are the important parameters affecting the shear strength of FRP bar-reinforced concrete beams without stirrups. An equation that cannot predict the effects of known parameters would lack generality, and its applicability to general design situations would be uncertain [18]. Thus, there is a need to develop a modified shear strength prediction model for properly reflecting the effects of important parameters, which are known to affect the shear strength of FRP bar-reinforced beams without stirrups.In this paper, a database of published test results on shear strength of 369 beams reinforced with FRP bars without stirrups was compiled. Considering the effects ofa/d on the shear strength of FRP bar-reinforced concrete beams without stirrups and the “size effect” and the contribution of the concrete to the shear strength for concrete beams without longitudinal reinforcement, a more accurate and rational-modified prediction model was proposed based on the ACI440.1R-2015 model. The efficiencies of the proposed model and ACI440.1R-2015, CAN/CSA-S806-2012 [19], JSCE-1997 [20], AASHTO LRFD-2017 [21], and CNR-DT203-2006 [22] models were evaluated by comparing the calculated values with the experimental values in the database and the test data which are not contained in the database.


## 2. Experimental Database

To present a new shear strength prediction model of FRP bar-reinforced concrete beams without stirrups, a relatively large database including 369 beams reinforced with FRP bars without stirrups from 42 different investigations was established. The criteria for collecting the data of beam specimens were as follows: (1) rectangular cross sections; (2) simply supported; (3) tested under one- or two-point loading; (4) statically loaded; and (5) failed in shear. This database included various parameters which are known to affect the shear strength of FRP bar-reinforced concrete beams without stirrups, such asa/d, fc′, b, d, ρf, and Ef. The variation range of each parameter and the corresponding shear strength of the beam used in this study are given in Table 1, which is collected from the original source.Table 1
Specimen details.
Investigator
No. of specimens
a
/
d
f
c
′ (MPa)
b (mm)
d (mm)
E
f (GPa)
ρ
f (%)
V
c (kN)
Ashour and Kara [15, 23]
18
2.5∼5.9
23∼50.2
150∼200
163∼371
32∼142
0.12∼1.39
9∼36.1
Gross [4, 24, 25]
42
4.1∼6.5
36.3∼81.4
65∼279
141∼225
41∼139
0.33∼2.56
8.8∼51
El-Sayedand El-Salakawy [26–28]
18
3.1∼6.5
40∼63
250∼1000
155∼326
39∼135
0.39∼2.63
60∼190
Razaqpur et al. [10]
7
1.8∼4.2
40.5∼49
200
225
145
0.25∼0.88
36.1∼96.2
Benmokrane [29]
12
2.8∼3.7
34.1∼43.2
130∼160
310∼346
42∼120
0.72∼1.54
42.7∼63.7
Tureyen and Frosch [30]
6
3.4
39.7∼42.6
457
360
38∼47
0.96∼1.92
94.7∼177
Nanni [31]
3
2.6∼2.7
24.1
178
279∼287
40
0.77∼2.3
36.1∼53.4
Deitz and Harik [32]
5
4.5∼5.8
27∼30.8
305
158
40
0.73
26.8∼29.2
Mizukawa [33]
1
2.7
34.7
200
260
130
1.3
62.2
Duranovic [34]
3
3.7
32.9∼38.1
150
210
45∼130
1.31∼1.36
26.2∼62.2
Swamy [35]
2
3.2∼4.1
38∼39
154∼305
192∼222
34∼42
0.36∼1.55
19.5∼26.7
Suzuki [36]
3
3
34.3
150
250
105
1.51∼3.02
40.5∼46
Alam and Hussein [7, 17]
37
1.5∼3.5
34.5∼88.3
250∼300
291∼744
47∼144
0.18∼1.47
43.7∼155.8
Nakamura and Higai [37]
2
3
22.7∼27.8
300
150
29
1.3∼1.8
33∼36
Bentz et al. [14]
6
3.3∼4.1
35∼46
450
188∼937
37
0.51∼2.54
54.5∼232
Baz̆ant and Yu [38]
1
3.1
40
450
970
40
0.46
136
Wakui and Tottori [39]
4
3.2
44.6∼46.9
200
325
58∼192
0.7∼0.9
87∼118
Nagasaka and Fukuyama [40]
2
3.1
22.9∼34.1
250
265
56
1.9
83∼113
Issa et al. [41]
6
5.7∼7
35.9
300
165∼170
48∼53
0.8∼4.12
29.3∼51.5
Tomlinson and Fam [42]
3
4.1∼4.5
56.5∼60
150
245∼270
70
0.39∼0.85
20.9∼29.2
Abed et al. [3]
9
1∼1.5
43∼65
200
230∼330
51
0.92∼1.84
116.6∼373.9
Guadagnini et al. [43]
3
1.1∼3.3
42.8∼47.7
150
223
45
1.28
27.2∼81
Kim and Jang[44]
40
1.5∼4.5
30∼40.3
150∼200
214∼216
40∼148
0.33∼0.79
16.6∼85.1
Olivito and Zuccarello [45]
20
5.6
20.4∼27.2
150
180
115
0.87∼1.45
16.6∼29.9
Matta et al. [46]
12
3.1
29.5∼59.7
114∼457
146∼883
41∼49
0.12∼0.28
17.9∼220.7
Thomas and Ramadass [9]
8
0.5∼1.8
40.6∼65.3
100∼170
270∼416
40
1.16∼1.75
30∼300
Wegian and Abdalla [47]
6
6.5∼9.5
32.5
1000
105∼155
42∼147
0.23∼0.96
23.5∼127
Andermatt and Lubell [11]
12
1.1∼2.1
39.9∼68.5
300∼310
257∼891
38∼43
1.47∼2.13
96∼1134.5
Huaxin and Genjin [13]
13
1.1∼2
34.9∼54.6
200
260∼360
52∼210
0.76∼1.16
83.8∼309.7
Xiaoliang and Wenjun [48]
3
2.5∼2.7
47.3∼50.4
300
300∼315
45
2.14∼5.62
85.3∼122.7
El Refai and Abed [8]
8
2.5∼3.3
49
152
195∼215
50
0.31∼1.53
16.9∼31.6
Chang and Seo [49]
14
5.8∼8
30
1200
130∼182
44∼50
0.24∼1.22
26.3∼159
Abdul-Salam et al. [50]
16
5.7∼6.3
41.3∼86.2
1000
134∼150
41∼148
0.51∼3.78
94∼213
Farghaly and Benmokrane [51]
4
1.1
38.7∼49.3
300
1088∼1111
48∼144
0.26∼1.24
595.5∼953
Ali et al. [12]
12
2.3∼3
13∼33.5
130
196∼200
52
0.3∼0.91
12.7∼39.4
Omeman et al. [1]
8
1.4∼2.3
34.7∼63.1
150
387∼662
134
1.13∼2.26
45.8∼234.1
Total
369
0.5∼9.5
13∼88.3
65∼1200
105∼1111
29∼210
0.12∼5.62
8.8∼1134.5
Note: if the concrete cylinder compressive strengthfc′ and modulus of elasticity of concrete Ec were not provided by the investigator while the concrete cube compressive strength fcu was only measured, it is assumed that fc′=0.8fcu and Ec=4733fc′. If the unit of data was BS provided by the investigator, it is converted by the following factors: 1 ksi = 6.895 MPa, 1 in = 25.4 mm, and 1 kip = 4.448 KN.


## 3. Review of Shear Strength Prediction Models in Design Codes

The shear strength prediction models of FRP bar-reinforced concrete beams without stirrups were highly valued by many countries, such as America, Japan, Canada, and Italy. The “size effect” and effects ofa/d, ρf, and Ef on the shear strength of FRP bar-reinforced concrete beams without stirrups were mainly considered in the shear strength prediction models of different design codes.


### 3.1. ACI440.1R-2015 Guidelines [5]

The shear strength calculating model of FRP bar-reinforced concrete beams without stirrups recommended by ACI Committee 440 is as follows:(1)Vcaci=25fc′bkd,where k=2ρfnf+ρfnf2−ρfnf, nf=Ef/Ec.It can be clearly seen that equation (1) of the ACI440.1R-2015 model predicts zero-shear strength for concrete beams without longitudinal reinforcement and does not consider the effect of a/d on the shear strength of FRP bar-reinforced concrete beams without stirrups and the “size effect.”


### 3.2. JSCE-1997 Design Recommendations [20]

The shear strength calculating model of FRP bar-reinforced concrete beams without stirrups recommended by Japan Society of Civil Engineering (JSCE-1997) is as follows:(2)Vcjsce=βdβρβnfvcdbd,where βd=1000/d1/4≤1.5, βρ=100ρEf/Es1/3≤1.5, and fvcd=0.2fc′3≤0.72MPaIt is the same as equation (1) of the ACI440.1R-15 model that equation (2) of the JSCE-1997 model predicts zero-shear strength for concrete beams without longitudinal reinforcement and does not consider the effect of a/d on the shear strength of FRP bar-reinforced concrete beams without stirrups although it takes into account the “size effect” through βd.


### 3.3. CAN/CSA-S806-2012 Design Provisions [19]

The shear strength calculating model of FRP bar-reinforced concrete beams without stirrups recommended by the Technical Committee on Design and Construction of Building Structures with Fibre-Reinforced Polymers of Canadian Standards Association (CAN/CSA-S806) is as follows:(3)Vcs806=0.05kmkrkaksfc′3bd,(4)0.11fc′bd≤Vcs806≤0.22fc′bd,where km=d/a≤1.0, kr=1+Efρf3, 1.0≤ka=2.5d/a≤2.5, and ks=750/450+d≤1.0.Equation (3) of the CAN/CSA-S806-2012 model considers the effects of nearly all parameters on the shear strength of FRP bar-reinforced concrete beams without stirrups.


### 3.4. AASHTO LRFD-2017 Design Guide Specifications [21]

The shear strength calculating model of FRP bar-reinforced concrete beams without stirrups recommended by the American Association of State Highway and Transportation Officials is as follows:(5)Vclrfd=0.0676fc′+4.6ρfdabd≤0.126fc′bd.Equation (5) of the AASHTO LRFD-2017 model considers the effects of a/d and ρf on the shear strength of FRP bar-reinforced concrete beams without stirrups while the effect of Ef and the “size effect” are not included.


### 3.5. CNR-DT203-2006 [22]

The shear strength of FRP bar-reinforced concrete beams without stirrups recommended by Advisory Committee Technical Recommendations Construction of Italian National Research Council is as follows:(6)Vccnr=1.3Ef/Es1/2τrkd1.2+40ρfbd,where 1.3Ef/Es1/2≤1, τr=0.25ft, kd=1.6−d/1000≥1, and ρf≤0.02.Equation (6) of the CNR-DT203-2006 model considers the “size effect” through kd and does not consider the effect of a/d on the shear strength of FRP bar-reinforced concrete beams without stirrups.


## 3.1. ACI440.1R-2015 Guidelines [5]

The shear strength calculating model of FRP bar-reinforced concrete beams without stirrups recommended by ACI Committee 440 is as follows:(1)Vcaci=25fc′bkd,where k=2ρfnf+ρfnf2−ρfnf, nf=Ef/Ec.It can be clearly seen that equation (1) of the ACI440.1R-2015 model predicts zero-shear strength for concrete beams without longitudinal reinforcement and does not consider the effect of a/d on the shear strength of FRP bar-reinforced concrete beams without stirrups and the “size effect.”


## 3.2. JSCE-1997 Design Recommendations [20]

The shear strength calculating model of FRP bar-reinforced concrete beams without stirrups recommended by Japan Society of Civil Engineering (JSCE-1997) is as follows:(2)Vcjsce=βdβρβnfvcdbd,where βd=1000/d1/4≤1.5, βρ=100ρEf/Es1/3≤1.5, and fvcd=0.2fc′3≤0.72MPaIt is the same as equation (1) of the ACI440.1R-15 model that equation (2) of the JSCE-1997 model predicts zero-shear strength for concrete beams without longitudinal reinforcement and does not consider the effect of a/d on the shear strength of FRP bar-reinforced concrete beams without stirrups although it takes into account the “size effect” through βd.


## 3.3. CAN/CSA-S806-2012 Design Provisions [19]

The shear strength calculating model of FRP bar-reinforced concrete beams without stirrups recommended by the Technical Committee on Design and Construction of Building Structures with Fibre-Reinforced Polymers of Canadian Standards Association (CAN/CSA-S806) is as follows:(3)Vcs806=0.05kmkrkaksfc′3bd,(4)0.11fc′bd≤Vcs806≤0.22fc′bd,where km=d/a≤1.0, kr=1+Efρf3, 1.0≤ka=2.5d/a≤2.5, and ks=750/450+d≤1.0.Equation (3) of the CAN/CSA-S806-2012 model considers the effects of nearly all parameters on the shear strength of FRP bar-reinforced concrete beams without stirrups.


## 3.4. AASHTO LRFD-2017 Design Guide Specifications [21]

The shear strength calculating model of FRP bar-reinforced concrete beams without stirrups recommended by the American Association of State Highway and Transportation Officials is as follows:(5)Vclrfd=0.0676fc′+4.6ρfdabd≤0.126fc′bd.Equation (5) of the AASHTO LRFD-2017 model considers the effects of a/d and ρf on the shear strength of FRP bar-reinforced concrete beams without stirrups while the effect of Ef and the “size effect” are not included.


## 3.5. CNR-DT203-2006 [22]

The shear strength of FRP bar-reinforced concrete beams without stirrups recommended by Advisory Committee Technical Recommendations Construction of Italian National Research Council is as follows:(6)Vccnr=1.3Ef/Es1/2τrkd1.2+40ρfbd,where 1.3Ef/Es1/2≤1, τr=0.25ft, kd=1.6−d/1000≥1, and ρf≤0.02.Equation (6) of the CNR-DT203-2006 model considers the “size effect” through kd and does not consider the effect of a/d on the shear strength of FRP bar-reinforced concrete beams without stirrups.


## 4. Proposed Shear Strength Prediction Model

According to the experimental database mentioned above, the normalized shear strengthVcexp/fc′bd was plotted against ρfnf (nf=Ef/Ec) as shown in Figure 1(a). It can be seen that the relationship between the normalized shear strength Vcexp/fc′bd and ρfnf fits into the trendline in Figure 1(a) except some plots. To express this trendline with a reasonable equation, the normalized shear strength Vcexp/fc′bd was also plotted against k (k=2ρfnf+ρfnf2−ρfnf) as shown in Figure 1(b), where k can be defined as the relative depth of the compressive zone because the depth of compressive zone c can be calculated by multiplying d by k (c=kd). From the regression of the results of FRP bar-reinforced concrete beams without stirrups, there is a linear equation between Vcexp/fc′bd and k, which can be written as(7)Vcexpfc′bd=0.4k+0.1.Figure 1
Effect ofρfnf and k on Vcexp/fc′bd.
(a)
(b)Equation (7) can take into account the effects of ρf and Ef on the shear strength for FRP bar-reinforced concrete beams without stirrups and the contribution of concrete on shear strength for FRP bar-reinforced concrete beams without longitudinal reinforcement.


### 4.1. Effect of Shear Span-to-Depth Ratio (a/d) on the Shear Strength

In order to eliminate the influence ofρf and Ef on shear strength of FRP bar-reinforced concrete beams without stirrups, the normalized shear strength was taken as Vcexp/0.4fc′bk+0.1d on the basis of equation (7), which was plotted against a/d according to the experimental database mentioned above, as shown in Figure 2. Evidently, Vcexp/0.4fc′bk+0.1d decreases as a/d increases, and there is a trendline between Vcexp/0.4fc′bk+0.1d with a/d. The relationships between Vcexp/0.4fc′bk+0.1d and a/d can be obtained by the regressions based on the following three criteria:(1)
The regressions can be described in four forms depending upon the value ofa/d or determined by the types of failure referred to those for steel bar-reinforced beams [52].Diagonal compression failure (when0≤a/d<1):(8)Vcexp0.4fc′bk+0.1d=f1a/d.Shear compression failure (when1≤a/d<3):(9)Vcexp0.4fc′bk+0.1d=f2ad.Diagonal tension failure (when3≤a/d<6):(10)Vcexp0.4fc′bk+0.1d=f3ad.Bending failure (whena/d≥6):(11)Vcexp0.4fc′bk+0.1d=f4ad.(2)
The regressions remain coordinated on boundary conditions(12)f11=f21,f23=f33,f36=f46.(3)
According to Reinforced Concrete Design to Eurocode 2 [53], when ρf=0 and a/d=0, the shear strength can be calculated by the following equation:(13)Vcρf=0,a/d=0=τbd.Figure 2
Effect ofa/d on Vcexp/0.4fc′bk+0.1d.The shear strength of concreteτ can be expressed by the tension strength of concrete ft as follows [52]:(14)τ=αft,where α is a constant, generally, α=1.5−2.5. Conservatively, here, α=1.6.The concrete tension strengthft can be evaluated with the equation in ACI318-14 [54] as follows:(15)ft=58fc′.Substituting the values offt in equation (15) into equation (14), the equation for shear strength of concrete can be rewritten as(16)τ=fc′.Substituting the values ofτ in equation (16) into equation (13), when ρf=0 and a/d=0, the equation for the shear strength of the FRP bar-reinforced concrete beams without stirrups can be rewritten as(17)Vcρf=0,a/d=0=fc′bd.Based on the regressions of the database aforementioned, as shown in Figure2, the functions between Vcexp/0.4fc′bk+0.1d and a/d can be obtained as follows:(18)f2ad=2.5a/d−0.5,f3ad=1,f4ad=6a/d.As there are only two specimens in the database whose shear span-to-depth ratio is below 1.0, no more data can be employed for regression; therefore, the function off1a/d could be set up according to the criterions (2) and (3) as follows:(19)f1ad=1−a/d/1+a/d0.4k+0.1+5ad.Then, the equation for shear strength of FRP bar-reinforced concrete beams without stirrups, which considers the effect of shear span-to-depth ratio (a/d), can be modified as follows:(20)Vc=kfadfc′bd,where(21)kfad=1−a/d1+a/d+2k+0.1ad,whilead≤1,k+0.1a/d−0.5,1<ad≤3,0.4k+0.1,while3<ad≤6,2.4k+0.1a/d,whilea/d>6.


### 4.2. “Size Effect” on the Shear Strength

According to the experimental database mentioned above and equation (21), the normalized shear strength Vcexp/kfadfc′bd was plotted against a/d, fc′, b, d, ρf, and Ef as shown in Figure 3, respectively. It can be seen that Vcexp/kfadfc′bd does not have the obvious variation with the increase of b, d, fc′, ρf, Ef, and a/d, respectively, but there is a relation between Vcexp/kfadfc′bd and d. It indicates that the influence of b, d, fc′, ρf, Ef, and a/d upon the shear strength of FRP bar-reinforced concrete beams without stirrups has been embodied reasonably well by equation (20) except the influence of d. This phenomenon demonstrates the “size effect” exists on the shear strength of FRP bar-reinforced concrete beams without stirrups mentioned in the literature review.Figure 3
Effect of parameters onVcexp/kfadfc′bd.
(a)
(b)
(c)
(d)
(e)
(f)Then, referring to Eurocode 2 for the shear capacity of concrete reinforced with the steel bar, the equation for shear strength of FRP bar-reinforced concrete beams without stirrups, which has taken into account the “size effect” by a linear reduction form [15], can be established as follows:(22)Vc=kfadkdfc′bd,where(23)kd=1.35−d1000≥0.75.


## 4.1. Effect of Shear Span-to-Depth Ratio (a/d) on the Shear Strength

In order to eliminate the influence ofρf and Ef on shear strength of FRP bar-reinforced concrete beams without stirrups, the normalized shear strength was taken as Vcexp/0.4fc′bk+0.1d on the basis of equation (7), which was plotted against a/d according to the experimental database mentioned above, as shown in Figure 2. Evidently, Vcexp/0.4fc′bk+0.1d decreases as a/d increases, and there is a trendline between Vcexp/0.4fc′bk+0.1d with a/d. The relationships between Vcexp/0.4fc′bk+0.1d and a/d can be obtained by the regressions based on the following three criteria:(1)
The regressions can be described in four forms depending upon the value ofa/d or determined by the types of failure referred to those for steel bar-reinforced beams [52].Diagonal compression failure (when0≤a/d<1):(8)Vcexp0.4fc′bk+0.1d=f1a/d.Shear compression failure (when1≤a/d<3):(9)Vcexp0.4fc′bk+0.1d=f2ad.Diagonal tension failure (when3≤a/d<6):(10)Vcexp0.4fc′bk+0.1d=f3ad.Bending failure (whena/d≥6):(11)Vcexp0.4fc′bk+0.1d=f4ad.(2)
The regressions remain coordinated on boundary conditions(12)f11=f21,f23=f33,f36=f46.(3)
According to Reinforced Concrete Design to Eurocode 2 [53], when ρf=0 and a/d=0, the shear strength can be calculated by the following equation:(13)Vcρf=0,a/d=0=τbd.Figure 2
Effect ofa/d on Vcexp/0.4fc′bk+0.1d.The shear strength of concreteτ can be expressed by the tension strength of concrete ft as follows [52]:(14)τ=αft,where α is a constant, generally, α=1.5−2.5. Conservatively, here, α=1.6.The concrete tension strengthft can be evaluated with the equation in ACI318-14 [54] as follows:(15)ft=58fc′.Substituting the values offt in equation (15) into equation (14), the equation for shear strength of concrete can be rewritten as(16)τ=fc′.Substituting the values ofτ in equation (16) into equation (13), when ρf=0 and a/d=0, the equation for the shear strength of the FRP bar-reinforced concrete beams without stirrups can be rewritten as(17)Vcρf=0,a/d=0=fc′bd.Based on the regressions of the database aforementioned, as shown in Figure2, the functions between Vcexp/0.4fc′bk+0.1d and a/d can be obtained as follows:(18)f2ad=2.5a/d−0.5,f3ad=1,f4ad=6a/d.As there are only two specimens in the database whose shear span-to-depth ratio is below 1.0, no more data can be employed for regression; therefore, the function off1a/d could be set up according to the criterions (2) and (3) as follows:(19)f1ad=1−a/d/1+a/d0.4k+0.1+5ad.Then, the equation for shear strength of FRP bar-reinforced concrete beams without stirrups, which considers the effect of shear span-to-depth ratio (a/d), can be modified as follows:(20)Vc=kfadfc′bd,where(21)kfad=1−a/d1+a/d+2k+0.1ad,whilead≤1,k+0.1a/d−0.5,1<ad≤3,0.4k+0.1,while3<ad≤6,2.4k+0.1a/d,whilea/d>6.


## 4.2. “Size Effect” on the Shear Strength

According to the experimental database mentioned above and equation (21), the normalized shear strength Vcexp/kfadfc′bd was plotted against a/d, fc′, b, d, ρf, and Ef as shown in Figure 3, respectively. It can be seen that Vcexp/kfadfc′bd does not have the obvious variation with the increase of b, d, fc′, ρf, Ef, and a/d, respectively, but there is a relation between Vcexp/kfadfc′bd and d. It indicates that the influence of b, d, fc′, ρf, Ef, and a/d upon the shear strength of FRP bar-reinforced concrete beams without stirrups has been embodied reasonably well by equation (20) except the influence of d. This phenomenon demonstrates the “size effect” exists on the shear strength of FRP bar-reinforced concrete beams without stirrups mentioned in the literature review.Figure 3
Effect of parameters onVcexp/kfadfc′bd.
(a)
(b)
(c)
(d)
(e)
(f)Then, referring to Eurocode 2 for the shear capacity of concrete reinforced with the steel bar, the equation for shear strength of FRP bar-reinforced concrete beams without stirrups, which has taken into account the “size effect” by a linear reduction form [15], can be established as follows:(22)Vc=kfadkdfc′bd,where(23)kd=1.35−d1000≥0.75.


## 5. Comparison of Predicted Shear Strength and Experimental Results

Figure4 presents the correlations of the experimental shear strength Vcexp of all 369 specimens in the database mentioned above and another 18 test data of specimens from Jumaa and Yousif [55] and Ovitigala [56] with the calculated shear strengths Vcaci, Vcjsce, Vcs806, Vclrfd, Vccnr, and Vcprop, respectively. A line with tolerance of 0% has been represented in the graph, which indicates that the exact prediction Vcexp/Vccalc=1 of the shear strength. It can be seen that the ACI440.1R-2015, JSCE-1997, and AASHTO LRFD-2017 models provide conservative predictions of the shear strengths of the most specimens (Vcexp/Vccalc>1); the predictions of the CNR-DT203-2006 model are highly unconservative for many specimens; the CAN/CSA-S806-2012 model shows better agreement with the experiment results than other models in the design codes aforementioned; and the predictions of the proposed model by equation (22) fit better with the experiment results than all the models in the design codes aforementioned.Figure 4
Comparison ofVcexp and Vccalc.
(a)
(b)
(c)
(d)
(e)
(f)Figure5 illustrates the relationships among the shear strength ratio of the experimental shear strength to the calculated shear strength Vcexp/Vccalc and a/d, k, and d, respectively. As shown in Figure 5, the ratio of Vcexp/Vccalc decreases as a/d increases, while a/d≤2.5 for all the models in the design codes aforementioned; both ratios of Vcexp/Vcs806 and Vcexp/Vclrfd increase as k increases, for the CAN/CSA-S806-2012 model does not well consider the effects of ρf nor Ef, and the AASHTO LRFD-2017 model does not consider the effect of Ef; and the ratio of Vcexp/Vcprop does not have obvious variation with the increase of a/d, k, or d. Moreover, the proposed model has the least scatter range of Vcexp/Vcprop from 0.39 to 2.63, whilst the scatter range of Vcexp/Vccalcfor ACI440.1R-2015, JSCE-1997, CAN/CSA-S806-2012, AASHTO LRFD-2017, and CNR-DT203-2006 models is from 0.82 to 17.38, 0.44 to 12.88, 0.24 to 4.58, 0.39 to 13.91, and 0.18 to 4.87, respectively. Hence, the effect of a/d, ρf, Ef, and d on shear strength normalized by the proposed model of equation (22) is captured reasonably well. It should be mentioned that the predicted values by the proposed model for the two specimens with shear span-to-effective depth less than 1 are in good agreement with the experimental values (Vcexp/Vcprop=1) though the function of the proposed model (a/d≤1) was not obtained by the regression method.Figure 5
Influence ofa/d, k, and d on shear strength predictions.The ratios ofVcexp/Vcprop are presented graphically using a histogram as shown in Figure 6. The horizontal axis of the figure shows the ratio of Vcexp/Vcprop, and the vertical axis represents the frequency of the specimens for a certain Vcexp/Vcprop ratio. It can be seen that the values of Vcexp/Vcprop follow a normal distribution, and 60% of the values is in a narrow range 0.8 to 1.2.Figure 6
Histogram of the predictions of the proposed model.To further investigate the superiority of the proposed shear strength prediction model of FRP bar-reinforced concrete beams without stirrups, a total of three performance checks were adopted. The mean, standard deviation (SD), and coefficient of variation (COV) of the ratioVcexp/Vccalc are given in Table 2. It can be observed that the proposed model, as a whole, predicts the shear strength of FRP bar-reinforced concrete beams without stirrups with smaller SD and COV values than ACI440.1R-2015, JSCE-1997, AASHTO LRFD-2017, CNR-DT203-2006, and CAN/CSA-S806-2012 models.Table 2
Mean, SD, and COV of the ratioVcexp/Vccalc.
Performance checks
Regression database
Test data
Mean
SD
COV
Mean
SD
COV
V
c
exp
/
V
c
aci
2.61
1.84
0.70
1.95
0.55
0.28
V
c
exp
/
V
c
lrfd
2.55
1.73
0.68
2.09
0.64
0.30
V
c
exp
/
V
c
jsce
1.92
1.45
0.76
1.63
0.50
0.30
V
c
exp
/
V
c
s
806
1.15
0.52
0.45
1.12
0.19
0.17
V
c
exp
/
V
c
cnr
0.90
0.69
0.77
0.69
0.12
0.18
V
c
exp
/
V
c
prop
1.09
0.31
0.28
1.14
0.19
0.17


## 6. Conclusions

The impacts of shear span-to-depth ratio upon the shear strength of FRP bar-reinforced concrete beams without stirrups and the “size effect” were investigated by analyzing the collected experimental results. A new model for the shear strength prediction of FRP bar-reinforced concrete beams without stirrups was proposed by using the regression method based on the experimental database. The main conclusions of this research are summarized as follows:(1)
The proposed model considers the “size effect” and the effect of shear span-to-depth ratio on the shear strength for FRP bar-reinforced concrete beams without stirrups, which decrease as the shear span-to-depth ratio and effective depth increase, respectively.(2)
The values of experimental results for the shear strength of FRP bar-reinforced concrete beams without stirrups to calculated values by the proposed model follow a normal distribution, and 60% of the values distribute in a narrow range 0.8 to 1.2.(3)
The proposed model has more reasonable and reliable predictions for the shear strength of FRP bar-reinforced concrete beams without stirrups in comparison with the models in design codes mentioned.



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*Source: 7516502-2020-04-09.xml*