Two-Way Shear Resistance of FRP Reinforced-Concrete Slabs: Data and a Comparative Study

This study aims to investigate the two-way shear strength of concrete slabs with FRP reinforcements. Twenty-one strength models were briefly outlined and compared. In addition, information on a total of 248 concrete slabs with FRP reinforcements were collected from 50 different research studies. Moreover, behavior trends and correlations between their strength and various parameters were identified and discussed. Strength models were compared to each other with respect to the experimentally measured strength, which were conducted by comparing overall performance versus selected basic variables. Areas of future research were identified. Concluding remarks were outlined and discussed, which could help further the development of future design codes. The ACI is the least consistent model because it does not include the effects of size, dowel action, and depth-to-control perimeter ratio. While the EE-b is the most consistent model with respect to the size effect, concrete compressive strength, depth to control perimeter ratio, and the shear span-to-depth ratio. This is because of it using experimentally observed behavior as well as being based on mechanical bases.


Introduction
In 2021, victims of the collapse of a condominium building [1] that is shown in Figure 1a totaled 98 people. In addition, a parking garage collapsed suddenly on a playground in Spain [2], as shown in Figure 1b. Moreover, most of the two-way shear designs of reinforced concrete (RC) slabs are empirical or semi-empirical. Thus, extensive research efforts are direct towards understanding the two-way shear types. However, the mechanism of the two-way shear of the slabs is complicated; thus, it is still open for investigation [3][4][5][6]. The two-way shear resistance of concrete slabs that are without shear reinforcements is composed of several resistance mechanisms, as follows: (1) flexure reinforcements resist shear through using dowel shear; (2) aggregates resist shear across the sides of the diagonal concrete crack through using aggregate interlock and friction; (3) uncracked concrete resists shear through using direct shear [7][8][9].
To avoid corrosion problems, replacing the conventional reinforcement with fiberreinforced polymers (FRP) reinforcements in concrete slabs is a common solution [10]. In addition, FRP reinforcements are magnetic neutral and have a high strength-to-weight ratio. Thus, they are the best choice to use in buildings that are subjected to severe environmental conditions including, and not limited to, wet-dry cycles, de-icing salts, and freeze-thaw cycles. Many researchers have investigated the behavior of new and existing beams and slabs that are reinforced with FRP bars or fabrics under one-way and two-way shear as well as torsion, mostly through experimental investigations [11][12][13][14][15]. Many research studies have tackled the two-way shear of concrete FRP reinforcements, while very few mechanical models were developed for this case [8]. The FRP's failure is brittle; thus, before failure, the FRP-reinforced concrete cracks are wider when compared to those in conventional RC [16][17][18]. Wider cracks significantly affect the various mechanisms of the two-way shear strength. Figure 1. Collapsed of (a) condominium building [1] and (b) a parking garage on a playground [2].
The traditional two-way shear design equations for RC slabs are based on theories that were developed in the early 1960s. These models were based on studies of that period's tested specimens; however, over the last few decades, much more testing was conducted which show several drawbacks of these methods including, and not limited to, size effect and those models being severely unconservative in many situations. Hence, there is a room for improvements to the two-way shear design models, which could help design code developments [19,20].
This study aims to assess the available methods for study of the two-way shear strength of FRP-reinforced concrete slabs. A state-of-the-art review of design codes, guides, and models for the study of the two-way shear strength of FRP-reinforced concrete slabs was outlined. An extensive review of the experimental testing of the FRP-reinforced concrete slabs that were tested under two-way loads was compiled. The studies that used to extract the models and their experimental testing were collected through various engineering search engines including, and not limited to: Google Scholar, Science Direct, MDPI Hub, and Engineering Village. The strengths calculated using all models were compared with those that were measured by testing. Concluding remarks were outlined and discussed.

Research Significance
Many researchers have proposed design models that address the two-way shear strength of RC slabs with FRP. Although safety is the main goal for the design purposes, evaluating these design models is a necessity. The accuracy of these models was assessed based on data from a limited experimental database. Thus, this study provides the community with an extensive collection of models and tested specimens as well as a comparison between the accuracy of each of these models. These results can help to improve the code developments.

Simplified Strength Models
For the study of the two-way shear strength of FRP-RC slabs, several simplified strength models have been proposed, either by modifications for conventional concrete slabs or empirically based on limited test data. The two-way shear design provisions of the North American design codes have neglected the effect of flexure reinforcement on strength. They focused on the direct shear resistance of the compression zone. This could be reasonable for conventional steel reinforcements with relatively high stiffness when compared to the FRP ones. Thus, the direct shear component governs the two-way shear strength. However, due to the relatively lower stiffness of the FRP when compared to the steel one, a dowel action could be a more significant contributor to its strength. Details and the background of various models are outlined in this section. V is the two-way shear failure load. E is the Young's modulus of the FRP. d is the effective depth. f c is the concrete compressive strength. ρ is the flexure reinforcement ratio. b and c are the column dimensions. A and B are the slab dimensions. E s is the Young's modulus of the steel. b 0.5d is the control perimeter at 0.5d, which is taken as 2(b + c + 2d). b 1.5d is the control perimeter at 1.5d, which is taken as 2(b + c + 6d). b 2.0d is the control perimeter at 2.0 d, which is taken as 2(b + c + 8d).

Gardner (1990)
In 1990, Gardner developed a strength model, which will be referred to herein as G [21]. It is based on an experimental database for two-way shear, and the existing design codes were assessed. Gardner concluded that considering the size effect and the flexure reinforcement ratio provides a more reliable design model; thus, when fitting it to the experimental database of that time, the two-way shear is calculated such that:

Japanese Approaches (1997), JSCE
In 1997, the JSCE [22] used a similar approach to the conventional North American design codes, and they implemented the assumption that the strength was proportional to the square root of the concrete compressive strength. Thus, they implemented the strain approach on the British Standard of that time and proposed that the two-way shear was calculated such that: where β d = 1000 , and e x and e y are the loading eccentricity in the x and y direction of the slabs, respectively.

Elghandour (2000), EG [23]
In 2000, Elghandour developed a strength model, which will be referred to as EG [23]. The model was developed using the strain and stress approaches to determine the steel area equivalent to the FRP area, and it can be used in the conventional two-way shear models. Thus, they implemented the strain approach, with a limit of the value of 0.0045 for the strain and the British Standard of the time, and proposed that the two-way shear was calculated such that:

Mattys and Taerwe (2000), MT
In 2000, Mattys and Taerwe, developed a strength model, which will be referred to herein as MT [24]. It was developed based on the observed behavior of the experimental testing of FRP-reinforced concrete slabs; their stiffness is less than that of conventional reinforced slabs. In addition, the depth and axial stiffness of FRP reinforcements have a significant effect on their strength; thus, they modified the British design code to be as follows: 3.5. Ospina (2003), O In 2003, Ospina [25] developed a model (O), which is based on their experimental observations, and it was found that the strength is affected by the axial stiffness of the FRP reinforcements and the bond they have with the concrete. Thus, when it is modified, the MT model is expressed as follows: 3.6. Zaghloul (2003), Z Zaghloul [26] has adapted the one-way shear design of the FRP-reinforced concrete of the Canadian design codes and multiplied it by a factor of two and introduced a perimeter size effect factor, such that: α s = 4, 3, and 2 for an inner connection, an edge connection, and a corner connection, respectively.

Jacbson (2005), Jb [27]
This is an empirical model which was developed through experimental testing.
3.8. ACI (2005), ACI [28] This ACI model accounts for the effect of the direct shear of the compression zone of the concrete, where the ACI equation for the conventionally reinforced concrete slabs (0.3 f c db 0.5d ) is multiplied by the factor (2.5 k). Thus, the shear strength is calculated such that: where k = 2ρn + (ρn) 2 − ρn, modular ratio (n) = E E c , concrete young's modulus, and E c = 4750 f c .

Elgamal (2005), E [29]
Elgamal developed a model, which was based on the experimentally observed fact that the two-way shear is affected by the FRP axial stiffness, and the slab end conditions are in terms of their continuity or them having an edge beam. Thus, the strength was proposed, such that: where n = 0, 1, and 2 represents a simple slab, a continuous, one-sided slab, and a continuous, two-sided slab, respectively.
3.10. Zhang (2006), Zg [30,31] Zhang developed a design model which included the following assumptions: (1) that two-way shear failure occurs after the critical diagonal shear crack passes through the compression zone; (2) that failure is related to concrete tensile strength; (3) that dowel action contributes to the strength. In addition, the model was calibrated using the experimental database that was available at that time, such that: 3.11. Theodorkopoulos and Swamy (2008), TS [32] The proposed model was based on moment-shear interaction, which determined the compression zone depth using the tensile elastic stiffness of the FRP reinforcements and the bond between FRP reinforcements and the concrete.
where λ f = 0.55 6 −1 + 3.12. CSA-S806-12 (2012), CSA [33] The design model was developed, based on the conventional concrete design code, however, it was modified for FRP reinforcements instead of steel ones. (12) β c = ratio between the long and short side of the loading area; α s = 4, 3, and 2 for an inner connection, an edge connection, and a corner connection, respectively. b 0.5d = 4(c + d).
where a is the slab's shear span.

Hassan, et al. (2017), H [35]
The model is a modification of the CSA which combines the three equations into a single formula. Then, it used a multi-linear regression technique to fit the 69 specimens in the experimental database using a power equation, such that: 3.15. Kara and Sinani (2017), KS [36] The KS model is a modification of the MT model that replaces the coefficient with 0.46 instead of 1.36 and removes the d parameter, such that: 3. 16. Oller, et al. (2018), CCCM [37] The CCCM model was developed, based on the model by Mari and co-workers [38], and it is a unified model for two-way shear; thus, it applies the following assumptions: (1) the strains are higher due to the lower modulus of elasticity of the FRP bars; (2) the cracks are wider; (3) the basic perimeter at the point of failure is lower in an FRP-reinforced concrete (RC) slab than it is in a conventional RC slab. Thus, the shear capacity is calculated such that: where Using numerical modeling and an experimental database, a two-way shear formula which considers the flexure reinforcement ratio and type, the compressive strength of concrete, and the shape of the column was developed, such that: k = 0.77 and 0.55 for circular and rectangular columns, respectively.

Elgendy and Elsalakawy (2020), EE [40]
Considering the eccentricity of the slab-column joint, the H model and the EG model were modified, such that: α s = 4, 3, and 2 for an inner connection, an edge connection, and a corner connection, respectively. N = 1, 2, and 3 for a simple slab, a continuous, one-sided slab, and a continuous, two-sided slab, respectively.

Ju, et al. (2021), Ju [41]
To guarantee the lowest probability of failure, the design strength was calculated with the probabilistic method with 95% confidence; thus, the Monte Carlo Simulation (MCS) was used to develop the probability distribution with key uncertain factors, such that:

Alrudaini (2022), A [42]
A rational model is developed, which considers the following: (1) concrete compressive strength, elastic properties of reinforcement, reinforcement ratio, and slab depth to the effective perimeter. Each parameter was fitted to the measured strength, such that: Table 1 shows a comparison between the various design models, where it is clear that there is a lack of agreement among researchers regarding the considered parameters and methodology used to account for it. All design methods included the effect of concrete compressive strength in terms of f c 1 /3 or f c 1 /2 . Most of the methods included the dowel action in terms of the flexure reinforcement, which was taken as (ρ) 1 /3 or (ρ) 1 /2 . More than half of the methods included the FRP type in terms of Young's modulus, which was taken as (E) 1 /3 ., or (E) 1 /2 . About half of the methods included the size effect in terms of and included the ratio between the critical perimeter and the depth in terms of (0.44 On the other hand, very few models included the compression zone and the shear span-to-depth ratio.

Experimental Database Profile
Over the last 30 years, a significant number of experimentally tested specimens failed due to the effect of two-way shear. The most comprehensive experimental database, when compared to those of previous studies [2,39,41,43], was produced with a total of 248 slabs with FRP reinforcements that were collected from 50 different research studies. All the gathered slabs were subjected to two-way shear loading and failed, suddenly, under the application of two-way shear, as shown in Figure 2. Table 2 shows a detailed description of the experimental database, where E is the Young's modulus of FRP, d is the effective depth, f c is the concrete compressive strength, ρ is the flexure reinforcement ratio, b and c are the column dimensions, A and B are the slab dimensions, and FRP type including carbon FRP (CFRP), glass FRP (GFRP), and Basalt FRP (BFRP) are listed. Although FRP reinforcements could have several shapes and configurations, these variations were considered in terms of ρ and E. Figure 3 shows the frequency and the range of each variable. All variables cover a wide range of values, while also being normally distributed.

Behavior Patterns
Based on the existing models and previous studies of slabs, the relationship between the shear stress (V/b o d) and the effective parameters including d, E, ρ, fc', d/b o , and a/d is a power relationship. Thus, Figure 4 shows the scatter plots for the pattern of the log of the shear stress versus the log of the effective parameters. The scatter plots do not allow a straightforward interpretation of the data because of the significant dispersion and poor distribution of the test parameters; thus, the best regression fit line and the Pearson correlation coefficients (r) are shown in Figure 4. The inclination of the best fit lines between the stress and basic variables d, E, ρ, fc', d/b o , and a/d were the values of −0.19, 0.19, 0.05, 0.34, 0.33, and 0.41, respectively. Comparing these values to those that were used in selected models, as shown in Table 1, it can be shown that variables ρ and fc' have quite similar power coefficients, while E, d/b o , and a/d are significantly different, and d is only like that of one selected model.

Pearson Correlation
The correlation coefficients between the shear stress and the basic variables d, E, ρ, fc', d/b o , and a/d were calculated, as shown in Figure 4, where their values are −0.19, 0.21, 0.07, 0.23, −0.34, and −0.43, respectively. Therefore, the evidence is sufficient to say that the shear strength is correlated to the basic variables in a reasonable manner, except for the flexure reinforcement ratio. This could be because the effect of the flexure reinforcement varies significantly based on its value [83]. Since the experimental database covered a wide range of flexure reinforcement ratio values, it provided a misleading value for the correlation coefficient.

Comparison between Selected Models
All collected models were used to calculate the strengths of the slab column connections that were in the experimental database. Three categories of comparison were defined: graphical, central tendency, and statistical goodness-of-fit. The ratio between the measured and calculated strength was taken as the safety ratio (SR). An SR value that is close to unity means that the prediction is accurate. An SR value that is more than the unity indicates that the prediction is conservative. An SR value that is less than the unity mean that the shear strength was overestimated and so, the prediction is conservative. Statistical measures in terms of the coefficient of determination (R 2 ), the root mean square error (RMSE), the mean average error (MAE), the mean, the coefficient of variation (C.O.V.), the lower value with a 95% confidence level (Lower 95%), the maximum value, and minimum value were applied on the SR for each selected model, as shown in Table 3 and Figure 5. Table 3 shows central tendency and statistical goodness-of-fit for all the selected models, which is helpful for the future development of the design models. The JSCE, the ACI, and the H models are over-conservative, with an average value of 2.71, 2.18, and 2.16, respectively. The Zg, the EE-b, the Ju, and the A models are more consistent with respect to other models, where the coefficient of variation values of these were 35%, 35%, 36%, and 36%, respectively.  Figure 5 shows a Box plot for all the selected models. A large dispersion and extreme values are observed in the ACI model. Also, severely un-conservative predictions resulted from the application of the Gd and NR models. The recent models (i.e., Ju, A, Hz, and EE-a) provide accurate predictions for the strength (when the mean is close to the unity), as shown in Figure 6. However, the consistency in these models is still lacking (i.e., C.O.V. is higher than 35%), as shown in Table 3. Models considering basic variables in a power form equation seem to be the most accurate and consistent when they are compared with the mechanically based model (CCCM) or the fracture-based model (NR). In addition, from Figure 5, it is clear that each method was developed or calibrated for a nonsystematic margin of safety which was defined by the judgment and experience of each developer(s). This should be managed by a reliability assessment that includes the resistance and load uncertainty. Although it is an interesting topic, it is not in the scope of this study, but it can be the subject of further studies. Moreover, there is a need for further improved mechanically based models that make physical sense, while being simple in their design. Figures 5-11 shows the SR value, which was calculated using different models, versus the value of the selected effective variable. Although the SR value is affected by the values of the various variables and not only the specific variable in the figure, it is assumed that the presence of the noise, because of the other variables, is insignificant with respect to the specific variable that is in the figure. It is worth noting that this approach was implemented in several pioneering studies as a base for international design codes [11,19,28,33,37,38,42]. In addition, some models do not include that specific variable, however, the experimentally measured strength includes the effect of that variable. Thus, the model's ability to represent the true value of the strength can be evaluated properly with respect to the effect of that specific variable.        Figure 6 shows the SR value that was calculated using the ACI model; the JSCE model; the CSA model; the CCCM model; the Ju model; the EE-b model versus the effective depth. In addition, the best fit line was plotted, whose slope was 0.0011, −0.003, −0.0016, −0.0019, −0.0003, and −0.0025 for the JSCE model, the ACI model, the CSA model, the CCCM model, the EE-b model, and the Ju model, respectively. The safety of the selected models decreases with the increase in the depth, except for the JSCE. The best fit line for the SR value that was calculated using the EE-b model is the lowest, thus, it is the most consistent with respect to the depth. However, using the ACI model resulted in the highest SR value, thus, it is the least consistent one. This could be due to the ACI model not having a size effect factor. Figure 7 shows the SR value that was calculated using the ACI model, the JSCE model, the CSA model, the CCCM model, the Ju model, and the EE-b model versus the concrete compressive strength. In addition, the best fit line was plotted, whose slope was 0.0019, 0.0046, 0.0174, −0.0031, −0.0017, and −0.0027 for the JSCE model, the ACI model, the CSA model, the CCCM model, the EE-b model, and the Ju model, respectively. The safety of the CCCM model, the EE-b model, and the Ju model decreases with the increase in the concrete compressive strength. On the other hand, the safety of the JSCE model, the ACI model, and the CSA model increases with the increase in the concrete compressive strength. The best fit line for the SR value that was calculated using the EE-b model is the lowest; thus, it is the most consistent with respect to the concrete compressive strength. However, using the CSA model resulted in the highest SR value; thus, it is the least consistent one. Figure 8 shows the SR value that was calculated using the ACI model, the JSCE model, the CSA model, the CCCM model, the Ju model, and the EE-b model versus the flexure reinforcement ratio. In addition, the best fit line was plotted, whose slope was 0.2529, −0.6244, −0.2229, −0.0328, −0.1817, and −0.1844 for the JSCE model, the ACI model, the CSA model, the CCCM model, the EE-b model, and the Ju model, respectively. The safety of the selected models decreases with the increase in flexure reinforcement ratio, except for the JSCE model. The best fit line for the SR value that was calculated using the CCCM model is the lowest; thus, it is the most consistent with respect to the flexure reinforcement ratio. However, using the ACI model resulted in the highest SR value; thus, it is the least consistent one. This could be due to the ACI model not including the flexure reinforcement ratio in the model. Figure 9 shows the SR value that was calculated using the ACI model, the JSCE model, the CSA model, the CCCM model, the Ju model, and the EE-b model versus the Young's modulus. In addition, the best fit line was plotted, whose slope was 0.0075, 0.0012, 0.0013, −0.002, −0.0011, and 0.0002 for the JSCE model, the ACI model, the CSA model, the CCCM model, the EE-b model, and the Ju model, respectively. The safety of selected models increases with the increase in Young's modulus, except for the CCCM model and the EE-b model. The best fit line for the SR value that was calculated using the Ju model is the lowest; thus, it is the most consistent with respect to the Young's modulus. However, using the JSCE model resulted in the highest SR value; thus, it is the least consistent one. Figure 10 shows the SR value that was calculated using the ACI model, the JSCE model, the CSA model, the CCCM model, the Ju model, and the EE-b model versus the depth-to-control perimeter ratio. In addition, the best fit line was plotted; whose slope was 8.5935, 13.86, 6.8699, −2.4433, −0.8117, and −1.6327 for the JSCE model, the ACI model, the CSA model, the CCCM model, the EE-b model, and the Ju model, respectively. The safety of the CCCM model, the EE-b model, and the Ju model decreases with the increase in the depth-to-control perimeter ratio. On the other hand, the safety of the JSCE model, the ACI model, and the CSA model increases with the increase in the depth-to-control perimeter ratio. The best fit line for the SR value that was calculated using the EE-b model is the lowest; thus, it is the most consistent with respect to the depth-to-control perimeter ratio. However, using the ACI model resulted in the highest SR value; thus, it is the least consistent one. This could be because the ACI model does not include the effect of this parameter. 7.6. Shear Span-to-Depth Ratio Figure 11 shows the SR value that was calculated using the ACI model, the JSCE model, the CSA model, the CCCM model, the Ju model, and the EE-b model versus the shear span-to-effective depth ratio. In addition, the best fit line was plotted, whose slope was −0.0507, −0.0475, −0.0224, 0.1353, 0.023, and 0.0213 for the JSCE model, the ACI model, the CSA model, the CCCM model, the EE-b model, and the Ju model, respectively. The safety of the JSCE model, the ACI model, the CSA model decreases with the increase in the shear span-to-effective depth ratio. On the other hand, the safety of the CCCM model, the EE-b model, and the Ju model increases with the increase in the shear span-to-effective depth ratio. The best fit line for the SR value that was calculated using the CSA model, the Ju model, and the EE-b model is the lowest; thus, they are the most consistent with respect to the shear span-to-effective depth ratio. However, using the CCCM model resulted in the highest; thus, it is the least consistent one.

Future Research
Several areas of potential for future research studies were identified as follows: • Experimental testing of high strength slabs with a compressive strength of more than 45 MPa; • Experimental testing of ultra-high-performance concrete slabs with a compressive strength of more than 100 MPa; • Experimental testing of non-slender concrete slabs with a shear span-to-depth ratio of less than 2.5; • Reliability-based analysis for the safety of the design which includes the variability in the loads, the geometry, the material, and the construction; • A more reliable and consistent mechanically based model that makes physical sense, while being simple in its design.

Conclusions
The accuracy of twenty-one selected methods to predict the two-way shear strength of the concrete slabs was assessed. Each method's ability to predict the two-way strength of concrete slabs without shear reinforcement was studied by comparing predictions against their measured strength from an extensive experimental database comprising a total of 248 slabs from over 50 research studies. Several statistical measures were applied, and the effect of the various basic variables was discussed. The following conclusions were reached:

•
The JSCE, the ACI, the H models are over-conservative, with an average value of 2.71, 2.18, and 2.16, respectively. The Zg, the EE-b, the Ju, the A models are more consistent with respect to other models, where the coefficient of variation value was 35%, 35%, 36%, and 36%, respectively. • The ACI model is the least consistent with respect to the size effect, the dowel action, and the depth-to-control perimeter ratio. This could be due to the fact that the ACI model does not consider these factors in the model.

•
The EE-b model is the most consistent with respect to size effect, concrete compressive strength, depth-to-control perimeter ratio, and the shear span-to-depth ratio. This is because of it using experimentally observed behavior as well as it being based on mechanical bases.