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

+e 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. +en, 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.


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][3][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. e 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 ( �� f c ′ ) [5,8], whilst the others are functions of the cubic roots of concrete cylinder com-reinforcement and also increasing the overall cost of construction [11].
For 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 (V c ) [12][13][14][15], such as concrete cylinder compression strength (f c ′ ), beam width (b), effective depth (d), reinforcement ratio (ρ f ), and modulus of elasticity (E f ) of the longitudinal FRP bar. e 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 f t , the crack occurs and the beam is finally damaged [17]. us, the shear strength V c of FRP bar reinforced concrete beams is a function of f t . e concrete tension strength can be evaluated through the in Eurocode 2 [18]; therefore, V c can be expressed as a function of to show the significant effect of f c ′ on V c . In this paper, the primary objective is to propose a new and more accurate shear strength model expressed as a function of 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. e 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.

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 E c and f t of concrete. e 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. e depth of the compression zone of ideal elastic material beam can be calculated, as follows, according to ACI 440.1R-15 [5]: For a rectangular section beam, as shown in Figure 2, the maximum shear stress occurs at the neutral axis and can be calculated by the material mechanics as follows: . (2) e normal stress at the neutral axis is e principal stress shown in Figure 3 can be determined by the normal stress and shear stress as follows:  (0, -τ)

Advances in Civil Engineering
Substituting the values of τ and σ in equations (2)-(3) into equation (4), the principal tensile stress σ 1 at the neutral axis can be rewritten as follows: It is assumed that the shear failure occurs when the principal tensile stress σ 1 is equal to or greater than the tension strength of concrete: From equations (5) and (6), the shear strength of the FRP bar reinforced concrete beam without stirrups can be predicted as follows: e concrete tension strength can be evaluated by Eurocode 2 [18] as follows: en, the equation for predicting the shear strength of the FRP bar reinforced concrete beam without stirrups can be changed as follows:

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 ratio (a/d) of specimens was greater than 2.7. e parameters in this study included the concrete cylinder compression strength (f c ′ ), beam width (b), effective depth (d), reinforcement ratio (ρ f ), and modulus of elasticity (E f ) of the longitudinal FRP bars. e 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.

Effect of Longitudinal FRP Bars.
Based on the collected experimental data of 235 beams reinforced with FRP bar without stirrups in Table 1, the relation of the normalized Figure 4(a). It obviously indicates that increases as the ρ f n f 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 (ρ f ) and modulus of elasticity (E f ) of the longitudinal FRP bars.
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.

Effect of Concrete Compression Strength.
e relation between normalized shear strength (V exp c /0.4b(kd)) and f c ′ is shown in Figure 5. It can be observed that the normalized shear strength (V exp c /0.4b(kd)) increases as the concrete compression strength increases, and the trendline between

Investigators
Quantity of specimens (a/d)

Summary and Conclusions
is 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 of   .

Data Availability
All data included in this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest.