Prediction about Effect of Span-to-Depth Ratio on Shear Capacity for FRP Bar Reinforced Concrete Beams without Web Reinforcement

Most of the prediction models for the shear capacity of FRP bar reinforced concrete beams without web reinforcement in current codes are reported to be conservative and do not consider the effect of span-to-depth ratio. Grey relational analysis (GRA)method was used in this paper to investigate the relevance of span-to-depth ratio with the shear capacity, and the results shown that the span-to-depth ratio greatly affects the shear capacity of FRP bar reinforced concrete beams without web reinforcement. A prediction model which considers the effect of span-to-depth ratio was proposed for the shear capacity of FRP bar reinforced concrete beams without web reinforcement by regression analysis, which is more accurate and reasonable in comparison with the current models.


Introduction
For the concrete structures exposed to severe environments, the corrosion of steel bars has always been one of the main reasons for structural damage, and it will bring relatively expensive repair and reinforcement costs. To solve this problem, several different types of steel reinforcements have been used, such as stainless, epoxy coated, and galvanized steel, but none of them can fundamentally solve the problem. At the same time, comparing with traditional steel reinforcement, fiber reinforced polymer (FRP) bars have better specific strength, corrosion resistance, and weak electromagnetic properties [1,2]. Due to the aforementioned advantages, the FRP bars are considered to be an effective substitute for traditional reinforcement, and the application of FRP bars in reinforced concrete beams has been rapidly developed over the last few decades.
As the FRP bars have several differences in the mechanical properties from steel bars, such as lower elastic modulus, brittleness and lower transverse shear strength, a large number of experiments on the shear behavior of FRP bar reinforced concrete beams and one-way slabs have been carried out. In general, the reinforced concrete (RC) beams without web reinforcement resist the shear stresses by means of five possible mechanisms, including shear resistance of the uncracked concrete compression zone, aggregate interlock, residual tensile stresses across cracks, dowel action of the longitudinal reinforcement and arch action [3]. e previous studies showed that the FRP bar reinforced concrete beam typically has a smaller neutral axis depth in comparison with the steel bar reinforced concrete beam with equal areas of longitudinal reinforcement [4,5]. As a result, the shear capacity generated by FRP bar reinforced concrete beams through the shear resistance of the uncracked concrete compression zone is weaker than that generated by traditional steel bar reinforced concrete beams. Moreover, because of the lower elastic modulus of FRP bars, the axial stiffness of FRP bar reinforced concrete beams is relatively low, which makes the width and depth of the diagonal cracks in the beams increase accordingly, thereby leading to the lower shear capacity generated by the aggregate interlock and residual tensile stresses across cracks [6]. Additionally, owing to the lower transverse shear strength of FRP bars, the dowel action of the longitudinal reinforcement contribution to shear resistance of FRP bar reinforced concrete beams is smaller than that of steel bar reinforced concrete beams [7]. In addition, due to the differences in mechanical properties between FRP bars and steel bars, the failure mode of FRP bar reinforced concrete beams is different from that of traditional steel bar reinforced concrete beams. erefore, the prediction models of the shear capacity of steel bar reinforced concrete beams are not suitable for FRP bar reinforced concrete beams. e current researches [8,9] have confirmed that the shear capacity of FRP bar reinforced concrete beams without web reinforcement, as shown in Figure 1, is influenced by several factors, such as concrete compressive strength (f c ′ ), shear span-to-depth ratio (a/d), elastic modulus of FRP bars (E f ), reinforcement ratio of longitudinal FRP bars (ρ f ), width of the beam (b) and effective depth of the beam (d). However, the influence of span-to-depth ratio (L/d) on the shear capacity of FRP bar reinforced concrete beams without web reinforcement is rarely involved, which represents the ratio of two sizes of the specimen, including effective span (L) and effective depth of the beam (d), respectively. Existing experimental studies showed that the ultimate shear capacity of steel bar reinforced concrete beams without web reinforcement under uniformly distributed loads will gradually decrease as the L/d increases [10,11]. Because the uniform load can be equivalent to two concentrated loads, and shear span is L/4, where L is the effective span of the beam [12,13], the L/d affects the shear capacity of the steel bar reinforced concrete beams. Furthermore, the effect of L/d on the shear capacity of the reinforced concrete beams is also reflected in the calculation models in codes. e GB50010-2010 in China stipulates that the beams with L/d of less than 5 are collectively called the deep flexural member [14]. At the same time, GB50010-2010 provides a different calculation model for the calculation of the shear capacity of deep flexural members in comparison with the beams with L/d of more than 5, which considers the effect of L/d. e ACI318R-19 stipulates that the beams that satisfy the following two criteria are called deep beams: the effective span does not exceed four times the overall member depth (h) and the concentrated loads exist within a distance 2h from the face of the support [15]. e Eurocode 2 stipulates that the span of a beam is not less than 3 times the overall section depth, otherwise it should be considered as a deep beam [16]. Both ACI318R-19 and Eurocode 2 deem that the deep beams are designed with strut and tie models, which is different from the design model for slender beams. In recent years, Uday Naik has proved that the change of L/d has an effect on the shear capacity of steel bar reinforced concrete beams by the artificial neural network technology [17], which has been widely used in civil engineering because of its excellent performance in the development of accurate and reliable prediction models for the shear capacity of reinforced concrete beams without web reinforcement [18][19][20][21][22][23]. e previous experimental study indicated that although FRP bars and steel bars have the great differences in elastic modulus and some mechanical properties, FRP bar reinforced concrete beams and steel bar reinforced concrete beams are still similar in shear-carrying mechanism [1,9,24].
us, the L/d also plays a certain role in the structural analysis of shear capacity of FRP bar reinforced concrete beams. e calculation models of the shear capacity of FRP bar reinforced concrete beams without web reinforcement in codes, such as CSA/CAN-S806-12, ACI440.1R-15, CNR-DT203-2006, GB50608-2020, and a modified model by Gao and Zhang are summarized in Table 1 [9,[25][26][27][28]. e factors that have been considered are listed in Table 2. Obviously, some of the factors which affect the shear capacity, such as a/d and L/d, have not been considered by the codes listed. Previous research results have confirmed that the current shear capacity calculation models of FRP bar reinforced concrete beams without web reinforcement in codes are conservative [9,29].
is may be due to the lack of the proper consideration of a/d and L/d. e CSA/CAN-S806-12 model and the Gao and Zhang model consider the effect of a/d on shear capacity and are proven to be more accurate in comparing with other models [29]. In spite of this, the effect of L/d on the shear capacity is not fully embodied. e models that are unable to accurately predict the shear capacity will lead to excessive shear design, which results in high costs. erefore, establishing a more comprehensive and accurate prediction model, which can correctly reflect the relevance of L/d and other parameters on the shear capacity of FRP bar reinforced concrete beams without web reinforcement, is essential at present.
In the present investigation, a database which contains the experimental results of 314 FRP bar reinforced concrete beams without web reinforcement was compiled to propose a more accurate calculation model for the shear capacity of FRP bar reinforced concrete beams. e method of grey relational analysis was used to analyze the relevance of L/d and other parameters with the shear capacity of FRP bar reinforced concrete beams without web reinforcement. On this basis, a new model was proposed based on the CAN/ CSA-S806-12 model, which considered the influence of L/d.

Test Data.
is study collected 314 the experimental data of FRP bar reinforced concrete beams without web reinforcement failed in shear from the literature. e effects of a/d, E f , ρ f , f c ′ , b, d and L/d on the shear capacity of FRP bar reinforced concrete beams are considered. In addition, all Advances in Civil Engineering test specimens in the database are simply supported and subjected to concentrated loads, and the shape of the cross section is rectangular. e details of the database and the minimum, maximum, average and range of each parameter are shown in Table 3.

Data Processing Principles.
If the cubic compression strength of concrete (f cu ) corresponding to the beams is only provided in the original paper, it can be converted to the cylinder compression strength of concrete (f c ′ ) according to the following formula [30].
If the elastic modulus (E c ) and tensile strength (f t ) of concrete corresponding to the beams are not provided in the original paper, they can be obtained according to the following formula [15]. (2)

Grey Relational Analysis (GRA) Method.
Grey relational analysis (GRA) method [57,58], as one of multi-factor statistical analysis methods, is mainly used to investigate the relevance between the reference sequence and comparison sequences by calculating the grey relational grade. e purpose of GRA method is to determine the main factors that affect the target value. e advantage of GRA method is that it does not require high sample size, and the workload is relatively small. e grey correlation degree can be calculated as following.

Models
Design parameters Step 1. Determining the reference sequence and the comparison sequences. e data sequence that reflects the characteristics of the system behavior is called reference sequence and expressed as follows.
e data sequence composed of the factors affecting system behavior is called comparison sequence and expressed as follows.
Step 2. Making the reference sequence and comparison sequences being dimensionless as follows.
Step 3. Calculating the grey correlation coefficient ξ i (k) of the reference sequence and the comparison sequences as follows.
where ρ is the grey resolution coefficient, and generally ρ � 0.5.
Step 4. Calculating the grey relational grade c i as follows.
Step 5. Evaluating the relevance as follows. e grey relational grade is a manifestation of the degree of the relevance between the reference sequence and comparison sequences. Basically, the greater the grey relational grade, the higher the degree of the influence factors affecting the characteristics of the system behavior. It can be considered that the characteristics of the system behavior is greatly influenced by the influence factor while c ≥ 0.8 or c � 0.5 ∼ 0.8, not influenced by the influence factor while c < 0.5 [59]. In this paper, to further illustrate the function of the factors, especially the L/d on the shear capacity of FRP bar reinforced concrete beams without web reinforcement, the GRA method will be used.

References
Amount

Grey Relational Grade for Each Experimental Parameter.
As a systematic analysis method, GRA can overcome the shortcomings of conventional analysis methods in analyzing the correlation relationship of factor sequences. In this paper, the GRA method is used to analyze the sensitivity factors of shear capacity. Based on the database newly collected in Table 3, the V exp c is taken as the reference sequence (x 0 ); the experimental parameters including a/d( 6 ) and L/d(x 7 ) are taken as the comparison sequences, respectively. en, the new dimensionless sequences were obtained according to equation (5), and the grey relational coefficient and grey relational grade were calculated by equations (6) and (7), respectively. e calculated values of the grey relational grade are shown in Table 4.
From the data in Table 4, the grey relational grade values of seven experimental parameters are all above 0.8, which means that the seven experimental parameters are well correlated with shear capacity of FRP bar reinforced concrete beams without web reinforcement. Among them, the grey relational grade values of

Prediction Model.
Considering the effect of L/d, a calculation model for the shear capacity of FRP bar reinforced concrete beams without web reinforcement can be proposed by modifying the model in CAN/CSA-S806-12. Based on the regression analysis for the database newly collected, the model for the shear capacity of FRP bar reinforced concrete beams without web reinforcement has been obtained as follows.
V c � 0.07k l k m k r k a k s f c ′ 1/3 bd, (8) where the coefficients of k l , k m , k r , k a and k s are given as follows, of which k l represents the effect of L/d on the shear capacity.
Considering the shear capacity of plain concrete members or members with extremely low reinforcement ratios, equation (10) needs to be used as the lower limit of the shear capacity for the proposed model as follows [5].

Evaluation of the Models.
To evaluate whether the models in Table 1 and equation (8)  which represents that the calculated value is equal to the experimental value is also plotted. When the data is located above the line, the experimental value is greater than the calculated value; when the data is located below the line, it means that the experimental value is less than the calculated value.
As shown in Figure 2, it is clear to find out that there is a declining trend for V values of the proposed model is smaller than that of other models. It is apparent that the results of the proposed model for the shear capacity is more consistent with the experimental value compared with the four calculation models in codes and the Gao and Zhang model.

Performance Checking and Sensitive Analysis.
To check the performance of the proposed model, the calculation values by the models are compared with the experimental values. As shown in Figure 3, the proposed model in this study has a better prediction function on the shear capacity of FRP bar reinforced concrete beams without web reinforcement than the CAN/CSA-S806-12, ACI440.1R-15, CNR-DT203-2006, GB50608-2020 and Gao and Zhang models.  To further investigate the superiority of the proposed model, the mean (MEAN), standard deviation (SD), and coefficient of variation (COV) were employed as the evaluation indicators, which can be calculated by the following formula.   Table 5.
It is obvious that the proposed model of equation (8) can well predict the shear capacity with MEAN, SD and COV of 1.000, 0.273 and 27.3%. Meanwhile, the MEAN of the proposed model is closer to 1, and the SD and COV of the proposed model are smaller than CAN/CSA-S806-12, ACI440.1R-15, CNR-DT203-2006, GB50608-2020 and Gao and Zhang models, respectively. us, the proposed model yields more scientific and accurate results for predicting the shear capacity in comparison with the CAN/CSA-S806-12, ACI440.1R-15, CNR-DT203-2006, GB50608-2020 and Gao and Zhang models, which did not consider the effect of L/d. is may be due to the effect of L/d is considered properly by the proposed model.
To analyze the sensitivity of every parameter for the proposed model, a set of data corresponding to the averages in Table 3 is selected as the starting data. When each parameter increases by 20%, the change rate of the calculated shear capacity is considered as the sensitivity index. e calculating results of sensitivity index are shown in Figure 5. It can be seen from the figure that the most important parameter impacting the shear capacity is d. e L/d has less effect on the shear capacity than the other six parameters. Under the fixed values of other parameter, the value of L/d varied from the minimum to maximum, the sensitivity index is 12.08%, which indicates the L/d cannot be ignored for   predicting the shear capacity for FRP bar reinforced concrete beams without web reinforcement.

Conclusions
Based on the collected experimental data of 314 FRP bar reinforced concrete beams without web reinforcement, the GRA method was used to analyze the relevance of experimental parameters with the shear capacity, and a new shear capacity prediction model was proposed for FRP bar reinforced concrete beams without web reinforcement. e main conclusions can be drawn as follows.
(1) e GRA method can be used to analyze the relevance of experimental parameters on the shear capacity of FRP bar reinforced concrete beams without web reinforcement. e span-to-depth ratio (L/d) affects the shear capacity as the same as the shear span-to-depth ratio

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.