# A Model for Shear Strength of FRP Bar Reinforced Concrete Beams without Stirrups

Authors: Danying Gao; Changhui Zhang
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

The shear failure of a reinforced concrete beam generally occurs when the principal tensile stress near the neutral axis is equal to or greater than the tension strength of concrete. In order to set up a model for shear strength for FRP bar reinforced concrete beams without stirrups by the mechanical method, this paper equivalently transformed the FRP bar reinforced concrete rectangular beam with cracks as one composed of ideal elastic material to facilitate the analysis and proposed a new and more reasonable model of shear strength for FRP bar reinforced concrete beams without stirrups. Then, an experimental database including 235 FRP bar reinforced beams without stirrups was compiled to verify the validity of the proposed model. It was found that the values from the proposed model are in better agreement with the experimental results of shear strength of FRP bar reinforced concrete beams without stirrups in comparison with the models in codes.


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## Body

## 1. Introduction

Fiber-reinforced polymer (FRP) bars have been considered as an advantageous alternative to replace steel bars for reinforced concrete structures due to the high-tension strength, durability, and good fatigue properties [1]. Since the modulus of elasticity, surface characteristics, tensile strength of FRP bars, and the bonding properties between FRP bars and concrete are different from those of steel bars, the use of FRP bars as reinforcement may cause different properties of FRP bar reinforced concrete members compared with the steel bar reinforced concrete members. It has been proved by several experimental investigations [2–4] that the shear capacity of FRP bar reinforced concrete beams without stirrups is different from that of steel bar reinforced members. Hence, the shear strength prediction models for steel bar reinforced concrete beams without stirrups cannot be directly applied to those for FRP bar reinforced members.The shear design equations for shear strength of FRP bar reinforced concrete beams without stirrups were provided by many design codes or guidelines, including ACI 440.1R-15 [5], JSCE-97 [6], CAN/CSA S806-12 [7], GB50608 [8], and BISE-99 [9], as summarized in Table 1. It can be easily found that the calculation models are mainly based upon the empirical or semiempirical method or obtained from the modification of the existing design equations for concrete members reinforced with steel bars. Some of the models are expressed as a function of the square roots of concrete cylinder compression strength (fc′) [5, 8], whilst the others are functions of the cubic roots of concrete cylinder compression strength (fc′3) [6, 7, 9], but none of them is a function of the cubic roots of the square of concrete cylinder compression strength (fc′23). The shear strength calculation models provided by ACI 440.1R-15, JSCE-97, GB50608, and BISE-99 were reported to be conservative for the FRP bar reinforced concrete slender beams without stirrups, whilst the CAN/CSA S806-12 model did not well consider the effects of ρf nor Ef [10], and thus led to excessive reinforcement and also increasing the overall cost of construction [11].Table 1
Shear strength calculating models in codes.
CodeCalculating modelACI440.1R-15Vcaci=0.4fc′bkd,k=2ρfnf+ρfnf2−ρfnf,nf=Ef/EcJSCE-97Vcjsce=βdβρfvcdbd,βd=1000/d1/4≤1.5βρ=100ρfEf/Es1/3≤1.5,fvcd=0.2fc′3≤0.72MPaCAN/CSA S806-120.11fc′bd≤Vcs806=0.05kmkrkaksfc′3bd≤0.22fc′bdkm=d/a≤1.0,kr=1+Efρf3,1.0≤ka=2.5d/a≤2.5,ks=750/450+d≤1.0GB50608-10VcGB=0.86ftbkd, k=2ρfnf+ρfnf2−ρfnf, nf=Ef/EcBISE-99Vcbise=0.79100ρfEf/Es1/3400/d1/4fcu/251/3bdFor FRP bar reinforced concrete beams without stirrups and shear span-to-depth ratio of more than 2.7, a number of studies have shown that several parameters have the impacts on the shear strength (Vc) [12–15], such as concrete cylinder compression strength (fc′), beam width (b), effective depth (d), reinforcement ratio (ρf), and modulus of elasticity (Ef) of the longitudinal FRP bar. The effect of the concrete compression strength on the shear strength of FRP bar reinforced concrete beams has been studied by many researchers. Experimental results [2, 16] indicated that the shear strength of FRP bar reinforced concrete beams increases with the increase of the concrete compression strength. It has been known that, in a beam without the external axial forces, the principal tensile stress arises from the interaction of normal and shear stresses. When the principal tensile stress from this interaction exceeds the tensile strength of concrete ft, the crack occurs and the beam is finally damaged [17]. Thus, the shear strength Vc of FRP bar reinforced concrete beams is a function of ft. The concrete tension strength can be evaluated through the fc′23 in Eurocode 2 [18]; therefore, Vc can be expressed as a function of fc′23 to show the significant effect of fc′ on Vc.In this paper, the primary objective is to propose a new and more accurate shear strength model expressed as a function offc′23 for FRP bar reinforced concrete beams without stirrups by the theoretical method. A database of 235 FRP bar reinforced concrete beams without stirrups was collected to verify the validity of the proposed model. The efficiency of the proposed model and the ACI 440.1R-15, CAN/CSA S806-12, GB50608-2010, JSCE-97, and BISE-99 models were evaluated by comparing the predictions with experimental results in the database.


## 2. Proposed Model

According to the mechanics theory, the shear failure of a reinforced concrete beam generally occurs when the principal tensile stress near the neutral axis is equal to or greater than the tension strength of concrete. For FRP bar reinforced concrete beams without stirrups, the shear strength in shear bending section of the beam is mainly provided by the uncracked concrete and the aggregate interlock of the cracked concrete. Based on the reasoning similar to the original model by Tureyen and Frosch [17], an actual FRP bar reinforced concrete rectangular beam with cracks shown in Figure 1(a) can be equivalently transmitted to that shown in Figure 1(b), which is composed of an ideal elastic material to facilitate the analysis and modeling of the shear strength. In the ideal elastic material beam, the elastic modulus and tension strength of the ideal elastic material are equal to Ec and ft of concrete. The effect of the longitudinal FRP bars and the aggregate interlock of the cracked concrete on the shear strength can be considered by the tension zone of the equivalent rectangular beam with the ideal elastic material, and the depth of concrete tension zone is equal to that of concrete compression zone c, as shown in Figure 2.Figure 1
Transition of the FRP bar reinforced concrete beam to the ideal elastic material beam. (a) FRP reinforced beam. (b) Equal ideal elastic material beam.
(a)(b)Figure 2
Stress distribution in the equivalent rectangular beam composed with ideal elastic material.The depth of the compression zone of ideal elastic material beam can be calculated, as follows, according to ACI 440.1R-15 [5]:(1)c=kd,where k=2ρfnf+ρfnf2−ρfnf, nf=Ef/Ec.For a rectangular section beam, as shown in Figure2, the maximum shear stress occurs at the neutral axis and can be calculated by the material mechanics as follows:(2)τmax=3V2b2c.The normal stress at the neutral axis is(3)σ=0.The principal stress shown in Figure3 can be determined by the normal stress and shear stress as follows:(4)σ1=σ2+σ22+τ2.Figure 3
Principal stress.Substituting the values ofτ and σ in equations (2)–(3) into equation (4), the principal tensile stress σ1 at the neutral axis can be rewritten as follows:(5)σ1=3V2b2c.It is assumed that the shear failure occurs when the principal tensile stressσ1 is equal to or greater than the tension strength of concrete:(6)σ1≤ft.From equations (5) and (6), the shear strength of the FRP bar reinforced concrete beam without stirrups can be predicted as follows:(7)Vc=43ftbc.The concrete tension strength can be evaluated by Eurocode 2 [18] as follows: (8)ft=0.3fc′23.Then, the equation for predicting the shear strength of the FRP bar reinforced concrete beam without stirrups can be changed as follows:(9)VcProp=0.4fc′23bc.


## 3. Experimental Verification and Comparison

To demonstrate the validity of the proposed model, a database of 235 FRP bar reinforced concrete beams without stirrups was collected based on the following criteria: the specimens had rectangular cross-sections, simply supported, tested under one- or two-point loading, statically loaded, failed in shear, and the shear span-to-depth ratioa/d of specimens was greater than 2.7.The parameters in this study included the concrete cylinder compression strength (fc′), beam width (b), effective depth (d), reinforcement ratio (ρf), and modulus of elasticity (Ef) of the longitudinal FRP bars. The changing ranges of each parameter and the corresponding shear strengths of beams are given in Table 2, which are identical to the original data collected from the literatures.Table 2
Beam specimen details and shear strengths.
InvestigatorsQuantity of specimensa/dfc′ (MPa)b (mm)d (mm)Ef (GPa)ρf (%)Vc (kN)Ashour et al. [19, 20]142.7∼5.923∼50.2150∼200163∼37132∼1420.12∼1.399∼36.1Yost et al. [4, 21, 22]424.1∼6.536.3∼81.465∼279141∼22541∼1390.33∼2.568.8∼51El-Sayed et al. [16, 23, 24]183.1∼6.540∼63250∼1000155∼32639∼1350.39∼2.6360∼190Razaqpur et al. [25]23.6∼4.240.52002251450.538.5∼49.7Tariq and Newhook [26]122.8∼3.734.1∼43.2130∼160310∼34642∼1200.72∼1.5442.7∼63.7Tureyen and Frosch [27]63.439.7∼42.645736038∼470.96∼1.9294.7∼177Deitz et al. [28]54.5∼5.827∼30.8305158400.7326.8∼29.2Duranovic [29]33.732.9∼38.115021045∼1301.31∼1.3626.2∼62.2Swamy [30]23.2∼4.138∼39154∼305192∼22234∼420.36∼1.5519.5∼26.7Suzuki et al. [31]3334.31502501051.51∼3.0240.5∼46Alam and Hussein [32]23.534.5∼39.8250305∼31047∼1440.42∼0.8643.7∼58.9Nakamura and Higai [33]2322.7∼27.8300150291.3∼1.833∼36Bentz et al. [34]63.3∼4.135∼46450188∼937370.51∼2.5454.5∼232Baz̆ant and Yu [35]13.140450970400.46136Wakui and Tottori [36]43.244.6∼46.920032558∼1920.7∼0.987∼118Nagasaka et al. [37]23.122.9∼34.1250265561.983∼113Issa et al. [38]65.7∼735.9300165∼17048∼530.8∼4.1229.3∼51.5Tomlinson and Fam [39]34.1∼4.556.5∼60150245∼270700.39∼0.8520.9∼29.2Guadagnini et al. [40]13.342.8150223451.2827.2Kim and Jang [41]223.1∼4.530∼40.3150∼200214∼21640∼1480.33∼0.7916.6∼28.9Olivito and Zuccarello [42]205.620.4∼27.21501801150.87∼1.4516.6∼29.9Matta et al. [43]123.129.5∼59.7114∼457146∼88341∼490.12∼0.2817.9∼220.7Wegian and Abdalla [44]66.5∼9.532.51000105∼15542∼1470.23∼0.9623.5∼127El Refai and Abed [45]53.349152195∼215500.31∼1.5316.9∼29.9Chang and Seo [46]145.8∼8301200130∼18244∼500.24∼1.2226.3∼159Abdul-Salam et al. [47]165.7∼6.341.3∼86.21000134∼15041∼1480.51∼3.7894∼213Ali et al. [48]6313∼33.5130200520.3∼0.9112.7∼23.6Total2352.7∼9.513∼86.265∼1200105∼97029∼1920.12∼4.128.8∼232Note 1: if the concrete cylinder compression strength fc′ and modulus of elasticity of concrete Ec are not provided by the investigator while the concrete cube compression strength fcu is only measured, it is assumed that fc′=0.8fcu and Ec=4733fc′. Note 2: if the data are provided for BS by the investigator, they are converted as follows: 1 ksi = 6.895 MPa, 1 in = 25.4 mm, and 1 kip = 4.448 kN.


### 3.1. Effect of Longitudinal FRP Bars

Based on the collected experimental data of 235 beams reinforced with FRP bar without stirrups in Table1, the relation of the normalized shear strength Vcexp/fc′23bd with ρfnfnf=Ef/Ec is shown in Figure 4(a). It obviously indicates that Vcexp/fc′23bd increases as the ρfnf increases like the trendline, which illustrates that the shear strength of FRP bar reinforced concrete beams without stirrups is highly dependent on the reinforcement ratio (ρf) and modulus of elasticity (Ef) of the longitudinal FRP bars.Figure 4
Effect of longitudinal reinforcement onVcexp/fc′23bd.
(a)(b)The relation between normalized shear strengthVcexp/fc′23bd and k is also shown in Figure 4(b). It can be clearly seen that Vcexp/fc′23bd increases as k increases. Through the regression of the experimental data, the relation between normalized shear strength Vcexp/fc′23bd and k was obtained as follows:(10)Vcexpfc′23bd=0.4k.From the comparison of equation (10) with equation (9), it confirms that the effect of longitudinal FRP bars on the shear strength of FRP bar reinforced concrete beams without stirrups can be involved in equation (9) through the depth of the compression zone of the beam.


### 3.2. Effect of Concrete Compression Strength

The relation between normalized shear strengthVcexp/0.4bkd and fc′ is shown in Figure 5. It can be observed that the normalized shear strength Vcexp/0.4bkd increases as the concrete compression strength increases, and the trendline between Vcexp/0.4bkd and fc′23 is in better agreement with the experimental data than that between Vcexp/0.4bkd and fc′ or fc′3. It proves that it is suitable by using fc′23 in equation (9) to express the effect of fc′ on the shear strength of FRP bar reinforced concrete beams without stirrups.Figure 5
Effect of concrete compression strength onVcexp/0.4bkd.


### 3.3. Comparison and Verification

To verify the efficiency that equation (9) captures the effects of the concrete compression strength and the longitudinal FRP bars on shear strength of the FRP bar reinforced concrete beams without stirrups, the values Vcexp/Vcpred, which are the ratios of the experimental shear strength to prediction values according to the proposed model, and models recommended by ACI440.1R-15, JSCE-97, CAN/CSA S806-12, GB50608-2010, and BISE-99 are shown in Figures 6 and 7 with fc′ and ρfEf/Ec, respectively. It is apparent that the values of Vcexp/VcProp, Vcexp/VcACI, Vcexp/VcJSCE, Vcexp/VcS806, Vcexp/VcGB, and Vcexp/VcBISE scatter in a range of 0.46–3.11, 0.82–5.54, 0.44–3.41, 0.24–3.13, 0.81–5.48, and 0.74–7.53, respectively, as the changing of fc′ and ρfEf/Ec. It can be seen that the most values of Vcexp/VcProp are scattered around the line of Vcexp/Vcpred=1. Additionally, there is a declining trend of Vcexp/VcBISE with the increasing of fc′ and an increasing trend of Vcexp/VcS806 with the increasing of ρfnf, which illustrates that the BISE-99 model does not well capture the influence of concrete compression strength, and the CAN/CSA S806-12 model does not well capture the effect of FRP bars.Figure 6
Comparison ofVcexp/Vcpred with the change of fc′.Figure 7
Comparison ofVcexp/Vcpred with the change of ρfEf/Ec.Figure8 presents the correlations among the experimental shear strength Vcexp of all 235 specimens and VcACI, VcJSCE, VcS806, VcGB, VcBISE, and VcProppredicted by ACI 440.1R-15, JSCE-97, CAN/CSA S806-12, GB50608-2010, and BISE-99 models and the proposed model. A line with tolerance of 0% has been represented in the graph, which indicates that the exact prediction Vcexp/Vcpred=1 of the shear strength. It can be seen that the ACI 440.1R-15, JSCE-97, CAN/CSA S806-12, GB50608-2010, and BISE-99 models provide the conservative predictions for the shear strength of the most FRP bar reinforced concrete beams without stirrups compared with the experimental results. Moreover, the larger the shear strength of beam, the greater the conservative degree of predictions. The ACI 440.1R-15, GB50608-2010, and BISE-99 models give the most conservative predictions. The values calculated by the proposed model in this study are more consistent with the experimental results.Figure 8
Comparison ofVcpred with Vcexp.To further demonstrate the efficiency of the proposed model and compare it with the ACI 440.1R-15, CAN/CSA S806-12, GB50608-2010, JSCE-97, and BISE-99 models for shear strength prediction of FRP bar reinforced concrete beams without stirrups, the mean, standard deviation (SD), and coefficient of variation (COV) ofVcexp/Vcpred for all of the beams in the database are shown in Table 3. The mean values of Vcexp/VcProp are closer to 1 than that of Vcexp/VcACI, Vcexp/VcJSCE, Vcexp/VcS806, Vcexp/VcGB, and Vcexp/VcBISE, while the SD values of Vcexp/VcProp are smaller than that of Vcexp/VcACI, Vcexp/VcJSCE, Vcexp/VcS806, Vcexp/VcGB, and Vcexp/VcBISE. It confirms that the proposed model provides more accurate predictions for the shear strength of the FRP bar reinforced concrete beams than ACI 440.1R-15, CSA S806-12, GB50608-2010, JSCE-97, or BISE-99 models.Table 3
Mean, SD, and COV for ratio ofVcexp/Vcpred.
ChecksVcexp/VcACIVcexp/VcJSCEVcexp/VcS806Vcexp/VcBISEVcexp/VcGBVcexp/VcPropMean1.961.361.232.281.921.07SD0.650.430.470.960.640.37COV0.330.310.380.420.330.34


## 3.1. Effect of Longitudinal FRP Bars

Based on the collected experimental data of 235 beams reinforced with FRP bar without stirrups in Table1, the relation of the normalized shear strength Vcexp/fc′23bd with ρfnfnf=Ef/Ec is shown in Figure 4(a). It obviously indicates that Vcexp/fc′23bd increases as the ρfnf increases like the trendline, which illustrates that the shear strength of FRP bar reinforced concrete beams without stirrups is highly dependent on the reinforcement ratio (ρf) and modulus of elasticity (Ef) of the longitudinal FRP bars.Figure 4
Effect of longitudinal reinforcement onVcexp/fc′23bd.
(a)(b)The relation between normalized shear strengthVcexp/fc′23bd and k is also shown in Figure 4(b). It can be clearly seen that Vcexp/fc′23bd increases as k increases. Through the regression of the experimental data, the relation between normalized shear strength Vcexp/fc′23bd and k was obtained as follows:(10)Vcexpfc′23bd=0.4k.From the comparison of equation (10) with equation (9), it confirms that the effect of longitudinal FRP bars on the shear strength of FRP bar reinforced concrete beams without stirrups can be involved in equation (9) through the depth of the compression zone of the beam.


## 3.2. Effect of Concrete Compression Strength

The relation between normalized shear strengthVcexp/0.4bkd and fc′ is shown in Figure 5. It can be observed that the normalized shear strength Vcexp/0.4bkd increases as the concrete compression strength increases, and the trendline between Vcexp/0.4bkd and fc′23 is in better agreement with the experimental data than that between Vcexp/0.4bkd and fc′ or fc′3. It proves that it is suitable by using fc′23 in equation (9) to express the effect of fc′ on the shear strength of FRP bar reinforced concrete beams without stirrups.Figure 5
Effect of concrete compression strength onVcexp/0.4bkd.


## 3.3. Comparison and Verification

To verify the efficiency that equation (9) captures the effects of the concrete compression strength and the longitudinal FRP bars on shear strength of the FRP bar reinforced concrete beams without stirrups, the values Vcexp/Vcpred, which are the ratios of the experimental shear strength to prediction values according to the proposed model, and models recommended by ACI440.1R-15, JSCE-97, CAN/CSA S806-12, GB50608-2010, and BISE-99 are shown in Figures 6 and 7 with fc′ and ρfEf/Ec, respectively. It is apparent that the values of Vcexp/VcProp, Vcexp/VcACI, Vcexp/VcJSCE, Vcexp/VcS806, Vcexp/VcGB, and Vcexp/VcBISE scatter in a range of 0.46–3.11, 0.82–5.54, 0.44–3.41, 0.24–3.13, 0.81–5.48, and 0.74–7.53, respectively, as the changing of fc′ and ρfEf/Ec. It can be seen that the most values of Vcexp/VcProp are scattered around the line of Vcexp/Vcpred=1. Additionally, there is a declining trend of Vcexp/VcBISE with the increasing of fc′ and an increasing trend of Vcexp/VcS806 with the increasing of ρfnf, which illustrates that the BISE-99 model does not well capture the influence of concrete compression strength, and the CAN/CSA S806-12 model does not well capture the effect of FRP bars.Figure 6
Comparison ofVcexp/Vcpred with the change of fc′.Figure 7
Comparison ofVcexp/Vcpred with the change of ρfEf/Ec.Figure8 presents the correlations among the experimental shear strength Vcexp of all 235 specimens and VcACI, VcJSCE, VcS806, VcGB, VcBISE, and VcProppredicted by ACI 440.1R-15, JSCE-97, CAN/CSA S806-12, GB50608-2010, and BISE-99 models and the proposed model. A line with tolerance of 0% has been represented in the graph, which indicates that the exact prediction Vcexp/Vcpred=1 of the shear strength. It can be seen that the ACI 440.1R-15, JSCE-97, CAN/CSA S806-12, GB50608-2010, and BISE-99 models provide the conservative predictions for the shear strength of the most FRP bar reinforced concrete beams without stirrups compared with the experimental results. Moreover, the larger the shear strength of beam, the greater the conservative degree of predictions. The ACI 440.1R-15, GB50608-2010, and BISE-99 models give the most conservative predictions. The values calculated by the proposed model in this study are more consistent with the experimental results.Figure 8
Comparison ofVcpred with Vcexp.To further demonstrate the efficiency of the proposed model and compare it with the ACI 440.1R-15, CAN/CSA S806-12, GB50608-2010, JSCE-97, and BISE-99 models for shear strength prediction of FRP bar reinforced concrete beams without stirrups, the mean, standard deviation (SD), and coefficient of variation (COV) ofVcexp/Vcpred for all of the beams in the database are shown in Table 3. The mean values of Vcexp/VcProp are closer to 1 than that of Vcexp/VcACI, Vcexp/VcJSCE, Vcexp/VcS806, Vcexp/VcGB, and Vcexp/VcBISE, while the SD values of Vcexp/VcProp are smaller than that of Vcexp/VcACI, Vcexp/VcJSCE, Vcexp/VcS806, Vcexp/VcGB, and Vcexp/VcBISE. It confirms that the proposed model provides more accurate predictions for the shear strength of the FRP bar reinforced concrete beams than ACI 440.1R-15, CSA S806-12, GB50608-2010, JSCE-97, or BISE-99 models.Table 3
Mean, SD, and COV for ratio ofVcexp/Vcpred.
ChecksVcexp/VcACIVcexp/VcJSCEVcexp/VcS806Vcexp/VcBISEVcexp/VcGBVcexp/VcPropMean1.961.361.232.281.921.07SD0.650.430.470.960.640.37COV0.330.310.380.420.330.34


## 4. Summary and Conclusions

This paper focused on the modeling of the shear strength prediction of FRP bar reinforced concrete beams without stirrups statically loaded. A simple and more accurate shear strength prediction model expressed as a function offc′23 was proposed by theoretical analysis. An experimental database of 235 FRP bar reinforced concrete beams without stirrups was established to evaluate the efficiency of the proposed model by comparing the calculated shear strengths with the experimental results and with that of ACI 440.1R-15, CAN/CSA S806-12, GB50608-2010, JSCE-97, and BISE-99 models. The main conclusions of this research are summarized as follows:(1)
An actual FRP bar reinforced concrete beam with cracks can be equivalently transmitted to a rectangular beam which composes of an ideal elastic material to propose the model of shear strength.(2)
The effect of longitudinal FRP bars on the shear strength of FRP bar reinforced concrete beams without stirrups can be involved in the new model through the depth of the concrete compression zone.(3)
It is more reasonable to be expressed as a function offc′23 for considering the effect of concrete strength on the shear strength of FRP bar reinforced concrete beams without stirrups than that of fc′ or fc′3.(4)
The calculating values by the proposed model are in better agreement with the experimental results of shear strength of FRP bar reinforced concrete beams without stirrups, comparing with the models in codes.



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*Source: 8881463-2020-10-29.xml*