# Development of Shear Capacity Prediction Model for FRP-RC Beam without Web Reinforcement

Authors: Md. Arman Chowdhury; Zubayer Ibna Zahid; Md. Mashfiqul Islam
Journal: Advances in Materials Science and Engineering
 (2016)

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## Abstract

Available codes and models generally use partially modified shear design equation, developed earlier for steel reinforced concrete, for predicting the shear capacity of FRP-RC members. Consequently, calculated shear capacity shows under- or overestimation. Furthermore, in most models some affecting parameters of shear strength are overlooked. In this study, a new and simplified shear capacity prediction model is proposed considering all the parameters. A large database containing 157 experimental results of FRP-RC beams without shear reinforcement is assembled from the published literature. A parametric study is then performed to verify the accuracy of the proposed model. Again, a comprehensive review of 9 codes and 12 available models is done, published back from 1997 to date for comparison with the proposed model. Hence, it is observed that the proposed equation shows overall optimized performance compared to all the codes and models within the range of used experimental dataset.


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## Body

## 1. Introduction

Corrosion of steel reinforcement is one of the major issues for the deterioration of RC structures. Thus, it leads to the rehabilitation process of existing structures which is very costly and reduces the service life. To resolve the problem, fiber reinforced polymer (FRP) bar has been brought in as internal reinforcement in RC structures replacing conventional steel. FRP bar provides a proper combination of physical and chemical properties such as high resistance to corrosion, high strength to weight ratio, and being magnetically inert. Despite the advantages, FRP imparts lower modulus of elasticity and linear elastic-brittle stress-strain relationship up to rupture with no yield point.Previously proposed models using steel RC beams where the plastic behavior helps combining stirrup and concrete in the resistance of shear (independently) cannot directly be applicable to FRP-RC beams. FRP-RC beams become weaker in shear than steel reinforced beams despite having the same value of reinforcement ratio, which is opposite to the scenario where flexure is considered [1]. Unique material characteristics of FRP could visibly change the contribution of each resistance mechanism contributing to the total shear resistance [2, 3], even though the very basic mechanisms are similar to those of regular steel reinforced concrete members. The Joint ASCE-ACI Committee 445 has assessed that the value of concrete shear strength (Vc) can be read as a combination of five separate mechanisms that are activated consequentially to the formation of diagonal cracks (Figure 1). This has been summed up as follows: (i) the shear resisted by the compressed concrete chord; (ii) the residual tensile strength existing between inclined cracks, which works as a tie of the truss mechanism jointly with the compression chord, the tensile reinforcement, and the concrete compression struts; (iii) the shear strength provided by the longitudinal reinforcement (dowel action); (iv) the friction forces developed on the gap length, which are adverse to the relative displacement of both crack faces (aggregate interlock); (v) the shear strength provided by the transverse reinforcement (this is out of scope of this study).Figure 1
Shear mechanism of FRP reinforced concrete beams without stirrups [5].Previous studies show that current shear design equations are very conservative in calculating the shear capacity of FRP-RC beams [4]. Their accuracy seems limited as most of these equations are developed using predefined forms where some influential parameters are overlooked. Consequently, the excessive amount of FRP needed to resist shear could be costly and may create reinforcement congestion inside concrete. Accordingly, the purpose of this paper is to develop a simple but accurate model for initially predicting the shear strength of FRP-RC beams (a/d > 2.5) without stirrups where all the shear affecting parameters are considered.


## 2. Research Significance

Due to complex shear transfer mechanisms in reinforced concrete beams and various influencing parameters, our understanding of shear is still relatively limited. Since many of the shear design codes are principally empirically derived, the design and behavior of structural concrete to shear are an important and ongoing area of research in structural concrete [6]. Thus, the case is also true for FRP-RC beams. This paper represents a prediction model to calculate the shear strength of FRP-RC beams. The proposed model is developed using a large database containing 157 experimental results where all controlling parameters for shear strength measurement are also examined with respect to the model and experimental database. The information gathered in this research and the proposed model could be very helpful in predicting shear strength as a primary tool and an experimental database for future study.


## 3. Experimental Database

In order to study the behavior of FRP-RC beams and check the performance of the proposed model, a large database of 157 beams that failed in shear was compiled in Appendix. Among them, 116 data items are used for developing the proposed multilinear regression model and another 41 data items are used for testing the performance of the model. The shear design parameters used in this study are concrete compressive strength (fc′), beam width (bw), effective depth (d), beam shear span (a), shear span to depth ratio (a/d), reinforcement ratio of longitudinal FRP bars (ρf), and modulus of elasticity of the reinforcing bar (Ef). The shear span to depth ratio (a/d) ranged from 1.1 to 6.5. The beam compressive strength is varied from 24.1 MPa to 88.3 MPa. Beam effective depth, d, varied from 141 mm to 889 mm. Longitudinal reinforcement ratio (ρf) used in this study varies between 0.2% and 2.6%. In this study, modulus of elasticity of FRP bar (Ef) used is between 4.1 GPa and 145 GPa. The material and geometrical properties of the 116 members in the refined database, as well as their original sources, are given in Appendix. All specimens in the database are simply supported, are tested in either three- or four-point loading arrangement, had no transverse reinforcement, and failed in shear. The distribution of geometrical and mechanical properties of the 116 test specimens is given in Figure 2.Figure 2
Data distribution of experimental parameters.


## 4. Parameters Affecting Shear Strength

Due to the complex behavior of FRP-RC beam in terms of shear strength, it is still a challenge to solve. Substantial research effort is given to understand and predict the shear strength more accurately [7]. Based on the analysis on different influencing parameters and critical review of the available database, some parameters are found to be liable for the overall contribution in shear strength. These influencing parameters are (i) shear span to depth ratio (a/d), (ii) effective depth of beam (d), (iii) reinforcement ratio (ρf), (iv) modulus of elasticity of FRP bar (Ef), (v) concrete tensile strength in terms of compressive strength (fc′), and (vi) width of beam (bw). Different studies considered different parameters and all of them are summarized in Table 1. These parameters and their influence are thoroughly investigated and appended in Table 1.Table 1
Parameter considered in available models and codes.
Equations or models Year Design parameters f c ′ b w d ρ f a / d E f Proposed model 2015 ● ● ● ● ● ● S. Lee and C. Lee 2014 ● ● ● ● ● ● Marí et al. 2014 ● ● ● ● ● ● Nasrollahzadeh and Basiri 2013 ● ● ● ● ● ● Kim et al. 2013 ● ● ● ● ● ● Alam and Hussein 2012 ● ● ● ● ● ● Kara 2011 ● ● ● ● ● ● CSA S6-09 2009 ● ● ● ● × ● Hoult et al. 2008 ● ● ● × × × Nehdi et al. 2007 ● ● ● × × × Nehdi et al. 2006 ● ● ● ● ● ● Razaqpur and Isgor 2006 ● ● ● × × ● CSA S6-06 2006 ● ● ● ● × ● CNR DT-203 2006 ● ● ● ● × ● ACI 440.1R-06 2006 ● ● ● × × × El-Sayed et al. 2005 ● ● ● ● × ● Wegian and Abdalla 2005 ● ● ● ● ● ● Tureyen and Frosch 2003 ● ● ● ● × ● CSA-S806-02 2002 ● ● ● × ● × ISIS-M03 2001 ● ● ● × × ● Deitz et al. 1999 ● ● ● × × × JSCE 1997 ● ● ● × × ×


### 4.1. Shear Span to Depth Ratio (a/d)

The shear strength of reinforced concrete beams is influenced significantly by the shear span to effective depth ratio,a/d [8]. Based on this ratio, the reinforced concrete beams are divided into two categories with different shear behavior and strength [9, 10]. The first category is “slender beams” having a shear span to depth ratio, a/d, greater than or equal to 2.5, while the second category is “short or deep beams” having a shear span to depth ratio, a/d, less than 2.5. In slender concrete beams, the behavior of the beams is governed by the beam action, and the contribution of the arch action to the strength and behavior of such beams is insignificant [11, 12]. Different researchers considered different coefficient for shear span to depth ratio (a/d). Evaluation of the appropriate coefficient may lead to accurate prediction. Shear strength decreases as the shear span to depth increases, but the trend of decrease is not clear. It may be linear or nonlinear [13]. Nehdi et al. [14] and CSA S806-02 [15] suggest that (d/a) to the power of (1/3) is proportional to the concrete shear strength. Kara [16] showed good result with a power of (1/9) for d/a ratio. Razaqpur et al. [17] and Alam and Hussein [18] suggest that (d/a) to the power of 0.3 is proportional to the concrete shear strength. Wegian and Abdalla [19] used (d/a) to the power of (1/3) for their study. El Chabib et al. [20] used (d/a) to the power of 0.23. Kim and Jang [21] proposed a linear relation between concrete shear strength and (a/d) ratio. Effect of shear span to depth ratio is not considered in the study of El-Sayed et al. [11, 12], JSCE [22], BISE design guideline [23], ISIS-M03-01 [24], Deitz et al. [25], CNR DT-203/206 [26], and Hoult et al. [27]. In this study, a coefficient of (a/d) to the power of (2/3) is found to be the most appropriate coefficient to predict the shear strength accurately. To see the trend line predicted by different researchers, five proposed models are plotted in Figure 3(a). As all models are not good enough to predict shear strength accurately, best five models are selected based on their statistical parameters and plotted with proposed model to see the trend.Figure 3
(a) Shear strength of FRP-RC beam with varying shear span to depth ratio (a/d). (b) Shear strength of FRP-RC beam with varying depth of beam (d). (c) Shear strength of FRP-RC beam with varying longitudinal reinforcement ratio (ρf). (d) Shear strength of FRP-RC beam with varying modulus of elasticity of FRP bar (Ef). (e) Shear strength of FRP-RC beam with varying concrete compressive strength (fc′). (f) Shear strength of FRP-RC beam with varying beam width (bw).
(a) (b) (c) (d) (e) (f)


### 4.2. Effective Depth of Beam (d)

Zhang et al. [28] found that the angle of the critical diagonal crack β CDC is strongly correlated with a/d ratio, which is equivalent to the ratio of M/Vd. By keeping all other parameters constant, if we increase the depth of the beam, shear span to depth ratio will be decreased. Increased depth will also increase the cracking angle with the horizontal plane, which can be calculated from (1) given by Zhang et al. [28].(1)βCDC=-15.0MaVad+89.7;MaVad≤3.14βCDC=42.6;MaVad>3.14,where the unit for βCDC is in degree.This phenomenon is also evident from the work of Alam [29] where the beam with the same materials and varying depth shows increasing cracking angle with increasing depth. This is due to the principle tensile stress that is effective at the shear critical zone, and it is maximum when depth is minimum. According to Nilson et al. [30], this combined effect of shear stress and bending stress at any section can be calculated by using (2)t=f2+f24+v2,where f=My/I and v=VQ/Ib.This is a clear indication that cracking angle increases with the increasing depth and thereby decreases the principal tensile stress at the shear critical zone of the beam. So, with increasing depth concrete shear strength will also increase, which is well supported by our proposed equation and also by another study [31]. Some code and model assumed a linear relation between concrete shear strength (Vc) and effective depth of the beam (d) [2, 11, 12, 16, 21, 22, 24, 27, 32–34] whereas according to CSA S806-02 [15], concrete shear strength is a function of effective depth (d) where the degree of depth is 1/3. Nehdi et al. [14, 20] used two different coefficients as 0.23 and 0.3. Wegian and Abdalla [19] used 0.3 as Nehdi et al. [14] used. So, there is a direct relation between effective depth and concrete shear strength. In this study, 4 is used to the power of d and a constant of 1.63×10-10 is used which normalizes the value and predicts the effect of depth accurately as shown in Figure 3(b).


### 4.3. Reinforcement Ratio (ρf)

The shear strength of beams without web reinforcement is a function of the longitudinal reinforcement ratio (ρf). Figure 3(c) indicates that the longitudinal reinforcement ratio is also an influencing parameter for shear strength. A lower ratio of longitudinal reinforcement leads to the formation of wider and deeper cracks compared to beams with high reinforcement ratio. Interface shear is reduced due to wider crack by reducing aggregate interlock in the cracked section. Deeper crack also decreases the depth of the uncracked concrete section which eventually decreases the concrete contribution to the shear strength. Dowel action contribution depends largely on the reinforcement ratio, as the ratio decreases the dowel action contribution decreases due to the formation of wider crack. According to ACI-ASCE 445 [35], flexural failure may govern rather than shear failure if the beam is slender and the longitudinal reinforcement ratio is very low.


### 4.4. Modulus of Elasticity of FRP Bar (Ef)

Modulus of elasticity of FRP bar is an influencing parameter for predicting the concrete shear strength. The influence of FRP elastic modulusEf is presented in Figure 3(d). The figure shows that concrete shear strength (Vc) increases with increasing the modulus of elasticity of FRP bar (Ef). The difference between the modulus of elasticity of steel and FRP bar has become the point of interest for the researchers. ACI 440.1R-03 guidelines assumed a linear relationship between concrete shear strength and the modulus of elasticity of FRP bar [32], but most of the research shows a cubic root relationship between them [11, 12, 15–19, 22, 26] and some others assumed a square root relationship (CSA S6-06 [36], CNR DT 203 [26], and ISIS-M03-01 [24] design manual).


### 4.5. Concrete Tensile Strength in terms of Compressive Strength (fc′)

The shear failure of the beam without web reinforcement occurs when the inclined crack forms or shortly after the formation of inclined cracks. The effect of concrete compressive strength on shear strength is shown in Figure3(e). These cracks occur when principle tensile strength exceeds its capacity. The tensile strength of concrete is considered as a function of the compressive strength of concrete (fc′). Concrete shear strength is usually taken to be proportional to fc′ to the power of 0.5 [21, 24, 27, 32, 37] and this study also found fc′ to the power of 0.5 as appropriate to predicting good result. Nehdi et al. [14, 20] used fc′ to the power of 0.23 and 0.3 in the consecutive study. CSA S806-02 [15], Kara [16], and JSCE [22] assume fc′ to the power of 1/3 as concrete tensile strength. In ACI 440.1R-06 [33], concrete compressive strength is used as a function of concrete shear strength, where concrete shear strength is proportional to fc′ to the power of 1/2; Tureyen and Frosch [34] assume a linear relation between concrete compressive strength and shear strength, where the power of concrete compressive strength is 0.5. CNR DT-203 [26] and El-Sayed et al. [11, 12] considered fc′ to the power of (1/2) and (1/6), respectively.


### 4.6. Width of Beam (bw)

All available models including the proposed model considered a linear relationship between concrete compressive strength and beam width and it is shown in Figure3(f) [2, 4, 11, 12, 14–16, 18–22, 24–27, 33, 34, 36, 38–40].


## 4.1. Shear Span to Depth Ratio (a/d)

The shear strength of reinforced concrete beams is influenced significantly by the shear span to effective depth ratio,a/d [8]. Based on this ratio, the reinforced concrete beams are divided into two categories with different shear behavior and strength [9, 10]. The first category is “slender beams” having a shear span to depth ratio, a/d, greater than or equal to 2.5, while the second category is “short or deep beams” having a shear span to depth ratio, a/d, less than 2.5. In slender concrete beams, the behavior of the beams is governed by the beam action, and the contribution of the arch action to the strength and behavior of such beams is insignificant [11, 12]. Different researchers considered different coefficient for shear span to depth ratio (a/d). Evaluation of the appropriate coefficient may lead to accurate prediction. Shear strength decreases as the shear span to depth increases, but the trend of decrease is not clear. It may be linear or nonlinear [13]. Nehdi et al. [14] and CSA S806-02 [15] suggest that (d/a) to the power of (1/3) is proportional to the concrete shear strength. Kara [16] showed good result with a power of (1/9) for d/a ratio. Razaqpur et al. [17] and Alam and Hussein [18] suggest that (d/a) to the power of 0.3 is proportional to the concrete shear strength. Wegian and Abdalla [19] used (d/a) to the power of (1/3) for their study. El Chabib et al. [20] used (d/a) to the power of 0.23. Kim and Jang [21] proposed a linear relation between concrete shear strength and (a/d) ratio. Effect of shear span to depth ratio is not considered in the study of El-Sayed et al. [11, 12], JSCE [22], BISE design guideline [23], ISIS-M03-01 [24], Deitz et al. [25], CNR DT-203/206 [26], and Hoult et al. [27]. In this study, a coefficient of (a/d) to the power of (2/3) is found to be the most appropriate coefficient to predict the shear strength accurately. To see the trend line predicted by different researchers, five proposed models are plotted in Figure 3(a). As all models are not good enough to predict shear strength accurately, best five models are selected based on their statistical parameters and plotted with proposed model to see the trend.Figure 3
(a) Shear strength of FRP-RC beam with varying shear span to depth ratio (a/d). (b) Shear strength of FRP-RC beam with varying depth of beam (d). (c) Shear strength of FRP-RC beam with varying longitudinal reinforcement ratio (ρf). (d) Shear strength of FRP-RC beam with varying modulus of elasticity of FRP bar (Ef). (e) Shear strength of FRP-RC beam with varying concrete compressive strength (fc′). (f) Shear strength of FRP-RC beam with varying beam width (bw).
(a) (b) (c) (d) (e) (f)


## 4.2. Effective Depth of Beam (d)

Zhang et al. [28] found that the angle of the critical diagonal crack β CDC is strongly correlated with a/d ratio, which is equivalent to the ratio of M/Vd. By keeping all other parameters constant, if we increase the depth of the beam, shear span to depth ratio will be decreased. Increased depth will also increase the cracking angle with the horizontal plane, which can be calculated from (1) given by Zhang et al. [28].(1)βCDC=-15.0MaVad+89.7;MaVad≤3.14βCDC=42.6;MaVad>3.14,where the unit for βCDC is in degree.This phenomenon is also evident from the work of Alam [29] where the beam with the same materials and varying depth shows increasing cracking angle with increasing depth. This is due to the principle tensile stress that is effective at the shear critical zone, and it is maximum when depth is minimum. According to Nilson et al. [30], this combined effect of shear stress and bending stress at any section can be calculated by using (2)t=f2+f24+v2,where f=My/I and v=VQ/Ib.This is a clear indication that cracking angle increases with the increasing depth and thereby decreases the principal tensile stress at the shear critical zone of the beam. So, with increasing depth concrete shear strength will also increase, which is well supported by our proposed equation and also by another study [31]. Some code and model assumed a linear relation between concrete shear strength (Vc) and effective depth of the beam (d) [2, 11, 12, 16, 21, 22, 24, 27, 32–34] whereas according to CSA S806-02 [15], concrete shear strength is a function of effective depth (d) where the degree of depth is 1/3. Nehdi et al. [14, 20] used two different coefficients as 0.23 and 0.3. Wegian and Abdalla [19] used 0.3 as Nehdi et al. [14] used. So, there is a direct relation between effective depth and concrete shear strength. In this study, 4 is used to the power of d and a constant of 1.63×10-10 is used which normalizes the value and predicts the effect of depth accurately as shown in Figure 3(b).


## 4.3. Reinforcement Ratio (ρf)

The shear strength of beams without web reinforcement is a function of the longitudinal reinforcement ratio (ρf). Figure 3(c) indicates that the longitudinal reinforcement ratio is also an influencing parameter for shear strength. A lower ratio of longitudinal reinforcement leads to the formation of wider and deeper cracks compared to beams with high reinforcement ratio. Interface shear is reduced due to wider crack by reducing aggregate interlock in the cracked section. Deeper crack also decreases the depth of the uncracked concrete section which eventually decreases the concrete contribution to the shear strength. Dowel action contribution depends largely on the reinforcement ratio, as the ratio decreases the dowel action contribution decreases due to the formation of wider crack. According to ACI-ASCE 445 [35], flexural failure may govern rather than shear failure if the beam is slender and the longitudinal reinforcement ratio is very low.


## 4.4. Modulus of Elasticity of FRP Bar (Ef)

Modulus of elasticity of FRP bar is an influencing parameter for predicting the concrete shear strength. The influence of FRP elastic modulusEf is presented in Figure 3(d). The figure shows that concrete shear strength (Vc) increases with increasing the modulus of elasticity of FRP bar (Ef). The difference between the modulus of elasticity of steel and FRP bar has become the point of interest for the researchers. ACI 440.1R-03 guidelines assumed a linear relationship between concrete shear strength and the modulus of elasticity of FRP bar [32], but most of the research shows a cubic root relationship between them [11, 12, 15–19, 22, 26] and some others assumed a square root relationship (CSA S6-06 [36], CNR DT 203 [26], and ISIS-M03-01 [24] design manual).


## 4.5. Concrete Tensile Strength in terms of Compressive Strength (fc′)

The shear failure of the beam without web reinforcement occurs when the inclined crack forms or shortly after the formation of inclined cracks. The effect of concrete compressive strength on shear strength is shown in Figure3(e). These cracks occur when principle tensile strength exceeds its capacity. The tensile strength of concrete is considered as a function of the compressive strength of concrete (fc′). Concrete shear strength is usually taken to be proportional to fc′ to the power of 0.5 [21, 24, 27, 32, 37] and this study also found fc′ to the power of 0.5 as appropriate to predicting good result. Nehdi et al. [14, 20] used fc′ to the power of 0.23 and 0.3 in the consecutive study. CSA S806-02 [15], Kara [16], and JSCE [22] assume fc′ to the power of 1/3 as concrete tensile strength. In ACI 440.1R-06 [33], concrete compressive strength is used as a function of concrete shear strength, where concrete shear strength is proportional to fc′ to the power of 1/2; Tureyen and Frosch [34] assume a linear relation between concrete compressive strength and shear strength, where the power of concrete compressive strength is 0.5. CNR DT-203 [26] and El-Sayed et al. [11, 12] considered fc′ to the power of (1/2) and (1/6), respectively.


## 4.6. Width of Beam (bw)

All available models including the proposed model considered a linear relationship between concrete compressive strength and beam width and it is shown in Figure3(f) [2, 4, 11, 12, 14–16, 18–22, 24–27, 33, 34, 36, 38–40].


## 5. Review of Current Shear Design Provisions for FRP-RC Beams

Most of the shear design provisions incorporated in these codes and models along with the shear capacity of FRP-RC beams have focused on modifying existing shear design equations for steel reinforced concrete beams to account for the significant differences between FRP and steel reinforcement [16]. These provisions are mostly grounded in the parallel truss model with 45° constant inclination diagonal shear cracks. In this paper, total 9 codes and 12 models suggested by different researchers are reviewed thoroughly. ACI shear design guideline published in 2003 is modified and again published in 2006 as ACI 440.1R-06 [33] that is adopted from the design method proposed by Tureyen and Frosch [34]. Modified equation considered the effect of concrete compressive strength, longitudinal reinforcement ratio and modulus of elasticity of FRP bar, beam width, and effective depth but they did not include the parameter shear span to depth ratio. This model cannot predict the change in shear strength in case of deep or very slender beams. So, they underestimate the strength for deep beams and, on the other hand, overestimate it in the case of long and slender beams. This lacking is due to a simple modification in ACI-318 shear equation by simply studying the difference in axial stiffness of FRP reinforcement with steel. ISIS-M03-01 also considered the change in modulus of elasticity between FRP bar and steel without considering all the influencing parameter such as shear span to depth ratio and the longitudinal reinforcement ratio which deviates its result from actual data thereby increasing the conservation highly [24]. Although JSCE-1997 gives a proper balance in predicting the shear force [22], its accuracy is undermined due to the absence of shear span to depth ratio. Alam [29] considered all the influencing parameters yet unable to predict shear strength accurately due to the inappropriate coefficient for each parameter which is placed through our work. From the study of El-Sayed et al. [11, 12], it is again evident that coefficient selection besides the parameter selection is also significant as each influencing parameter behaving in different ways and can lead to an over conservative result if not selected accurately. CNR DT-203 [26], Deitz et al. [25], CSA S6-09 [39], and Hoult et al. [27] shear capacity approach did not consider the shear span to depth ratio and gives unrealistic results in the case of deep or very slender beams. S. Lee and C. Lee [2] proposed an equation which has a large scatter with the complex equation for the calculation of shear strength. Equation provided by the study of Nasrollahzadeh and Basiri [4] showed good scatter but it is impractical to use such lengthy formula for handy usage. On the other hand proposed model gives reasonable value without sacrificing the intensity and likewise easy to get an initial idea about the shear strength of beams without web reinforcement. All examples and codes up to this time are conceived of this study and exhibited in Table 2 in a systematic manner to present an overview of overall advancement in this specific part of the research. The model proposed by this study is simple yet effective to predict the shear strength of FRP-RC beams and clearly adopted by the influencing parameters. It is to be noted that, in these design guidelines and equations, all safety elements were ignored, that is, assigned to 1.0. In reality, safety factor would be employed to make shear capacity predictions more conservative and acceptable for design use [8, 57, 58].Table 2
Analysis of statistical parameter of all available models.
Model Year V e x p/Vpred Mean Max Min SD COV (%) AAE (%) R 2 Proposed model 2015 1.03 1.83 0.70 0.22 20.9 15.8 0.89 S. Lee and C. Lee 2014 1.37 6.11 0.30 0.92 67.6 45.7 0.38 Marí et al. 2014 1.04 2.92 0.63 0.30 29.0 16.3 0.88 Nasrollahzadeh and Basiri 2013 1.13 3.67 0.43 0.48 43.0 28.0 0.66 Kim et al. 2013 0.91 1.64 0.51 0.23 23.4 25.3 0.80 Alam and Hussein 2012 1.71 3.74 0.92 0.40 23.5 38.8 0.90 Kara 2011 1.05 2.58 0.61 0.26 25.1 16.6 0.87 CSA S6-09 2009 1.64 4.46 0.76 0.53 32.3 35.1 0.78 Hoult et al. 2008 0.92 2.37 0.47 0.28 30.3 27.1 0.85 Nehdi et al. 2007 1.14 2.03 0.64 0.24 20.8 17.4 0.83 Nehdi et al. 2006 1.33 4.78 0.07 1.08 31.1 149.9 0.82 Razaqpur and Isgor 2006 2.23 5.75 1.17 0.57 25.7 52.6 0.89 CSA S6-06 2006 3.06 9.30 1.11 1.18 38.5 61.9 0.67 CNR DT-203 2006 0.97 4.09 0.32 0.43 44.1 31.1 0.71 ACI 440.1R-06 2006 1.90 5.23 1.09 0.59 30.9 43.7 0.87 El-Sayed et al. 2005 1.35 3.76 0.81 0.36 27.0 24.2 0.87 Wegian and Abdalla 2005 1.30 2.49 0.71 0.29 22.7 22.7 0.83 Tureyen and Frosch 2003 1.29 3.50 0.70 0.37 28.56 22.62 0.84 CSA-S806-02 2002 1.38 2.50 0.79 0.34 25.0 25.4 0.87 ISIS-M03 2001 1.33 3.86 0.46 0.51 38.8 33.5 0.65 Deitz et al. 1999 1.02 6.88 0.22 0.74 72.7 73.1 0.35 JSCE 1997 1.26 3.46 0.55 0.41 32.3 23.6 0.74


## 6. Model Development

Influencing parameters are identified and measured against a large database of 157 experimental results. There are mainly linear and nonlinear coefficients in the regression which is shown below in(3)Y=β0+β1X1α1+β2X2α2+β3X3α3+⋯.All probable combinations of nonlinear coefficient of influencing parameters with different coefficient using coefficient selection network are selected which is shown in Figure 4(b). A total of 72 combinations are found from the network. Linear regression is done afterward by considering the nonlinear effect of each influencing parameter. The algorithm that is shown in Figure 4(a) explains the steps adopted during the establishment of the proposed model. After regression analysis, all constants are determined, whether each parameter’s sign is coherent and meaningful. If all sign is rational and acceptable, the reasonable magnitude of constants and their statistical significance are then evaluated. Ultimately, the best example is chosen based on adjusted R-square from the 72 models that is found from regression.Figure 4
(a) Algorithm to develop proposed model and (b) network used for coefficient selection (NI: not included).
(a) (b)


## 7. Proposed Model

A multilinear regression model based on an experimental database collected from literature is suggested and examined with different statistical parameters such asR2, t-test, skewness, and kurtosis for the statistical stability of the model. This model did well in all statistical analysis. In order to predict the shear strength of FRP-RC beams more accurately, the experimental shear strength of FRP-RC beams was analyzed with all the six parameters stated before. A general pattern of multilinear regression analysis, as shown in (4), was conducted in optimizing the regression parameters of β1, β2, β3, β4, β5, and β6. The interactions among the parameters had been neglected as suggested by other researchers [59]. The skewness and kurtosis are found to be 0.986 and 1.045, respectively.(4)Y=β0+β1X1α1+β2X2α2+β3X3α3+β4X4α4+β5X5α5+β6X6α6,where β0=-0.223, α1=1, α2=1/2, α3=4, α4=1/2, α5=2/3, α6=1, β1=0.19, β2=9.433,β3=1.63×10-10, β4=2.63, β5=-37.571, and β6=12.996.Finally, this model can be written as follows:(5)Vpred=-0.223+0.19×bw+9.433×fc′+1.63×10-10×d4+2.63×Ef-37.571×ad2/3+12.966×ρf,where bw is width of beam, fc′ is concrete tensile strength in terms of compressive strength, d is effective depth of beam, Ef is modulus of elasticity of FRP bar, a/d is shear span to depth ratio, and ρf is longitudinal reinforcement ratio.This model is valid only for slender beams where shear span to depth ratio (a/d) is greater than 2.5. For deep beams, this model may not show good consistency.


### 7.1. Training and Testing Data

Verification of the proposed model is done by using a large experimental database of 157 beams. Among them, 116 beams are used for developing the model and later this model is tested with 41 beams and slabs from the literature. This model showed a good prediction during the testing phase and it is shown in Figure5. This model can be used in 95% confidence level which is evident from Tables 3 and 4.Table 3
Statisticalt-test result.
Proposed model β 0 β 1 β 2 β 3 β 4 β 5 β 6 t-test −0.008 17.359 2.464 6.390 3.309 −7.294 4.00Table 4
Confidence interval based ont-test.
Confidence t-test value 90% 1.64 95% 1.96 99% 2.58 99.9% 3.29Figure 5
Training and testing of proposed model.


## 7.1. Training and Testing Data

Verification of the proposed model is done by using a large experimental database of 157 beams. Among them, 116 beams are used for developing the model and later this model is tested with 41 beams and slabs from the literature. This model showed a good prediction during the testing phase and it is shown in Figure5. This model can be used in 95% confidence level which is evident from Tables 3 and 4.Table 3
Statisticalt-test result.
Proposed model β 0 β 1 β 2 β 3 β 4 β 5 β 6 t-test −0.008 17.359 2.464 6.390 3.309 −7.294 4.00Table 4
Confidence interval based ont-test.
Confidence t-test value 90% 1.64 95% 1.96 99% 2.58 99.9% 3.29Figure 5
Training and testing of proposed model.


## 8. Result and Discussion

To compare the operation of several codes and models in predicting the shear strength of FRP-RC beams, a number of performance checks are used. Mean, standard deviation, Average Absolute Error (AAE), and Coefficient of Variation (COV) indicate the overall functioning of the design algorithm and are calculated by (6), (7), (8), and (9). The execution of the design equations in predicting the concrete contribution to shear strength is shown in Table 5 and Figures 6(a)–6(v) and 7(a)–7(v). The shear design equation proposed by this study had the most accurate prediction with an average of 1.03, standard deviation of 0.22, Coefficient of Variation (COV) of 20.9%, and Average Absolute Error (AAE) of 15.8% as indicated in Table 2.Table 5
Range of parameters and statistical comparison of experimental database.
Parameter Range Number of beams Mean SD COV (%) d (mm) <250 45 0.96 0.22 22.51 250–500 59 1.07 0.21 19.97 >500 12 1.10 0.17 15.51  f c ′ (MPa) <35 23 1.04 0.26 24.90 35–60 74 1.02 0.19 18.91 >60 19 1.04 0.25 24.26  ρ f (%) <0.75 39 1.01 0.24 23.92 0.75–1.5 44 1.01 0.16 15.85 >1.5 33 1.08 0.25 22.82  E f (GPa) <50 76 1.02 0.21 20.73 >100 40 1.04 0.23 21.68  a / d <3.5 66 1.03 0.20 19.71 3.5–5 34 1.05 0.24 23.25 >5 16 1.00 0.22 22.05  b w (mm) <250 49 1.05 0.24 22.82 250–500 59 1.03 0.21 20.04 >500 8 0.96 0.12 12.34Figure 6
V e x p / V p r e d versus number of beams.
(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l)  (m) (n) (o) (p) (q) (r) (s) (t) (u) (v)Figure 7
Ratio of predicted and experimental shear strength.
(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l)  (m) (n) (o) (p) (q) (r) (s) (t) (u) (v)Kara [16] and Marí et al. [38] have slightly more scattered results with an average of 1.05 and 1.04, respectively. In terms of COV, Kara [16] shows better results than Marí et al. [38] to 25.1% than 29%. Although Deitz et al. [25] show an average of 1.02, the results are much more scattered and more than 50% of total data overestimates shear strength. For the given database in Appendix, Tureyen and Frosch [34] show an average of 1.29 with an Average Absolute Error (AAE) of 22.62%. Nehdi et al. [14] optimized the previous model and its result also shows a balanced scatter with an average of 1.14 and for the previous study El Chabib et al. [20] it is 1.47. Standard deviation (SD), COV (%), and AAE (%) improved significantly by Nehdi et al. [14] and they are 0.24, 20.8%, and 17.4%, respectively. JSCE [22] guideline had the best balance of scattering out of all established design guidelines presented in this study. Razaqpur and Isgor [40] show scattered result with an average of 2.23 which is the second worst result from the last. This may have happened due to a small number of test results used to develop this model.Mean(6)μ=∑i=1nbVc,iexp/Vc,iprednb;standard  deviation(7)σ=∑i=1nbVc,iexp/Vc,ipred-μ2nb;Average  Absolute  Error(8)AAE=1nb∑Vcexp-VcpredVcexp×100;Coefficient of Variation(9)COV=∑i=1nbVc,iexp/Vc,ipred-μ2/nb∑i=1nbVc,iexp/Vc,ipred/nb.El-Sayed et al. [11, 12] show good accuracy in predicting the shear strength with a little conservatism and have also low COV and AAE as 27% and 24.2%, respectively. The CSA S6-06 equation was highly conservative with an average of 3.06 and standard deviation of 1.18 [36].The CSA S6-09 [39] addendum improved the performance of the CSA S6-06 [36] formulation by an average of 1.64 and it shows less scatter than before with COV and AAE of 32.3% and 35.1%, respectively. Wegian and Abdalla [19] and ISIS-M03-01 [24] show a similar average of 1.30 and 1.33 but ISIS-M03-01 is more conservative than Wegian and Abdalla with a COV of 38.8% and AAE of 33.5% which is much greater than 22.7% and 22.7%, respectively. Nasrollahzadeh and Basiri [4] used Fuzzy Inference System (FIS) to predict the shear strength of FRP-RC beams with and without shear reinforcement or stirrups. It exhibits a high COV of 43% in comparison to its average of 1.13; besides its proposed model is not much user-friendly when a quick approximation is required. Proposed model by guidelines and other researchers are compared in one platform and an overview is shown in Figures 8 and 9.Figure 8
Statistical parameter comparison of available models.Figure 9
Experimental to predicted shear strength comparison of models and codes.


## 9. Conclusion

This study is investigated using regression approach for predicting the shear strength of FRP-RC beams without stirrups and compared such predictions with those of the 21 available models and codes. The following conclusions can be drawn from this study:(1)
The result of this research suggests that regression approach can provide a more precise and reliable alternative method for the shear design of FRP concrete beams.(2)
By assessing the proposed regression model against a large database containing the test results of 157 FRP-RC beams that exhibited shear failure and were examined by several researchers, it was shown that shear strength results predicted using the proposed model corresponded well with the observational outcomes.(3)
The proposed model outperformed the shear design provisions considered in this work. The proposed model used in this study developed an average of theVexp/Vpred ratio equal to unity, hence leading to an economic use of FRP reinforcement.(4)
All existing shear provisions considered in this study provided unnecessarily conservative result except Kara [16] in estimating the shear strength of FRP-RC beams without shear reinforcement. Kara [16] showed an overall good balance In terms of scattering and standard deviation though there is the scope of improvement such as COV and AAE, which is satisfactorily reduced by our suggested model.(5)
This field could also suffice as a state-of-the-art review of all available models and can attend to render an overall performance idea, founded on different statistical parameters.The proposed model is also trained and tested to predict the shear strength within the range of input variables considered. Even so, they may not demonstrate accuracy when extrapolating beyond this scope.



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*Source: 4356967-2016-10-20.xml*