Proposal and evaluation of new models for predicting the FRP contribution to shear strength in reinforced concrete beams using gene expression programming

Abstract

Fiber-reinforced polymers (FRP) have been widely used in shear strengthening applications of reinforced concrete (RC) beams. The accurate prediction of the FRP contribution to the shear strength of beams is essential for reliable design. Gene expression programming (GEP) has been widely utilized because it reliably expresses complex relationships between experimental variables. In this study, three new GEP models are proposed for three different strengthening configurations of FRP such as fully-wrapping, U wrapping, and side-bonding to predict the FRP contribution to shear strength. These models are developed using the most comprehensive database containing a total of 811 strengthened beams (350 fully-wrapped, 328 U-wrapped, and 133 side-bonded. Many variables have been considered in the proposed GEP models, including those that have been experimentally effective but are often neglected in existing literature equations, such as the shear span-to-effective depth ratio $(a/d)$ and the stirrup ratio (${\rho }_{w}$). Additionally, the reliability of existing equations in the literature and the proposed GEP models for predicting the FRP contribution to shear strength was statistically evaluated. As a result of this evaluation, the proposed GEP models for each strengthening configuration of FRP yielded the most accurate statistical results, with the lowest coefficient of variation (COV), and the highest coefficient of correlation (R).

1 Introduction

The strengthening of reinforced concrete (RC) structures to achieve the required performance is a widely preferred method. Recently, the strengthening method using fiber-reinforced polymer (FRP) has become popular in both academic studies and projects. FRP, an alternative composite material, is favored for strengthening structural elements such as columns, beams, and slabs due to its superior properties, including corrosion resistance, resistance to various chemicals, high strength-to-weight ratio, ease of application, and high tensile modulus. RC beams with insufficient shear strength could be strengthened using FRP by mainly four different strengthening configurations including fully-wrapped, U-wrapped with anchoraged, U-wrapped, and side-bonded.

Many previous experimental [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137], analytical [3, 36, 82, 119, 123, 131, 138, 139], and statistical [140,141,142,143,144] studies were conducted on FRP-strengthened beams in shear. Li and Leung [6, 91] and Sengun and Arslan [35, 36] elaborately investigated the effect of $a/d$ on the RC beams strengthened with various FRP configurations. Accordingly, it was stated that $a/d$ affects the FRP contribution to shear strength by altering the angle of the shear crack and the load-carrying mechanism of beam. Therefore, $a/d$ should be taken into account in the analytical equations. Bousselham and Chaallal [145] stated that the increase in shear capacity of the beams was limited by the increase in FRP reinforcement ratio (${\rho }_{f}$) in slender beams. However, no significant increase in shear capacity was observed in deep beams. The size effect on the shear strength of beams strengthened with FRP was experimentally examined by Godat et al. [4] and Benzeguir et al. [146]. Godat et al. [4] concluded that the effect of FRP decreased as the effective depth of beam ($d$) and the beam width (${b}_{w}$) increased. Benzeguir et al. [146] demonstrated that the size effect was more prominent in the FRP-strengthened beams without stirrups. The shear strip width-to-spacing ratio (${w}_{f}/{s}_{f})$ was one of the important variables effective on the behavior of the FRP-strengthened beams studied by various researchers [1, 2, 35, 36, 82]. Akkaya et al. [1, 2], Sengun and Arslan [35, 36], and Mofidi and Chaallal [82] pointed out that the FRP contribution to the shear strength of the beam increased as the spacing between the FRP strips decreased. Mofidi and Chaallal [82] emphasized that variable $({w}_{f}/{s}_{f})$ was not considered in the proposed equations. However, they suggested that it should be included. Additionally, several experimental studies indicated [16, 35,36,37, 121, 145] that the increase in the stirrups ratio (${\rho }_{w}$) was inversely proportional to the increase in the FRP contribution to shear strength. It was stated by various researchers [74, 119, 147,148,149,150,151,152] that the contribution of FRP to the shear strength depended on $({\rho }_{f})$, FRP thickness (${t}_{f}$) and the concrete compressive strength (${f}_{c}$).

The effect of FRP type on the shear behavior of beam was studied by Cao et al. [8] and Baggio et al. [57]. Chen and Teng [138, 139] expressed that there were mainly two different failure modes named as FRP debonding and rupture depending on the FRP strengthening configuration and proposed equations to calculate the FRP shear contribution for each of these failure modes considering effective FRP strain. The effects of different variables on the prediction accuracies of the equations were statistically analyzed by Kar and Biswal [140]; Kotynia et al. [141]; Lima and Barros [142]; Oller et al. [143]; Pellegrino and Vasic [144]. It was stated by Oller et al. [143] that the accuracies of the predicted equations were lower in the beams with stirrups. Kar and Biswal [140] concluded that (1) the experimental FRP shear contribution increased for the specimens without stirrups by the increase in beam depth; (2) the strengthening efficiency of the external strengthening system increased with an increase in a/d ratio, whereas the prediction accuracies of different design guidelines for the prediction of FRP shear contribution become lower with an increase in $a/d$ ratio.

Researchers widely employ various computational methods [153, 154] and intelligent design techniques, such as utilizing origami structures [155,156,157], for solving engineering problems. In addition, the use of artificial intelligence for solving engineering problems is increasing day by day. With the rapid development of technology, it has become preferable to utilize artificial intelligence in the construction industry due to its advantage of easily learning complex datasets and making predictions with high accuracy. Machine learning is a subset of artificial intelligence that can be used in many different engineering fields. Machine learning has become a main topic in recent publications [158,159,160]. Various machine learning approaches such as gene expression programming, random forest, support vector regression, multiple expression programming, Gaussian process regression, artificial neural networks, and adaptive neuro-fuzzy interface system have been frequently applied in modeling complex problems in structural engineering [161,162,163,164].

Gene expression programming (GEP) is one of the widely used machine learning programs. GEP is a genetic algorithm using populations to derive analytical expressions, taking into account the complex relationships between the relevant variables. Model outputs are easier to interpret and analyze in GEP method compared to other machine learning and nonlinear regression methods due to the tractability and adaptability. In addition, unlike other machine learning methods, the GEP method derives an equation by considering the complex relationships among different variables in the dataset. There are studies providing different GEP models for the calculation of shear strength, deflection, and strains of different structural components/materials using the GEP method and examining the validity of these equations. Alshboul et al. [165], Gandomi et al. [166], and Kara [167] first determined the relevant input variables considering the previous experimental investigations conducted by various researchers on beams without stirrups and then gave an equation for the shear strength of beams without stirrups using the GEP method. Al-Ghrery et al. [168] dealt with the concrete cover separation on RC beams strengthened with FRP in flexure and proposed a model by using a total of 127 beams. Ism and Rabie [169] performed an experimental study to examine the flexural behavior of CFRP-strengthened beams and proposed an equation with the GEP method to calculate the rupture strain of FRP. Anvari et al. [170] gave a GEP model to calculate the total shear strength in RC beams strengthened with externally bonded FRP in shear and verified the validity of the model. The statistical results of the equations predicting the FRP contribution to shear strength carried out in the literature were discussed.

When previous experimental studies are evaluated, it could be expressed that the behavior of FRP-strengthened specimens varies significantly depending on many variables such as stirrups ratio (${\rho }_{w}$), $a/d$, and the FRP reinforcement ratio (${\rho }_{f}$). Furthermore, as the limited number of the statistical studies conducted using the limited number of the equations proposed to calculate the FRP contribution is analyzed, the predictions of the equations are not in good agreement with the experimental results. This is due to the fact that many important variables such as stirrups ratio (${\rho }_{w}$) and $a/d$ are not effectively taken into account in the proposed equations. Besides, the number of studies examining the FRP contribution using the extensive database by the GEP method is quite limited. The aim of this study is to propose equations that provide better predictions than those obtained using existing equations in the literature. Therefore, three different GEP models were proposed in this study using an extensive dataset containing a total of 811 specimens to determine the FRP contribution for each strengthening configuration. In the proposed models, a total of 12 variables whose effects have been shown experimentally have been tried to be effectively considered. The prediction accuracies of the different equations proposed in the literature to calculate the contribution of FRP to the shear strength were statistically evaluated and compared with the prediction results of the proposed GEP models.

2 Description of experimental database

Most of the experimental studies performed to date were attempted to be considered in collecting dataset. However, data quality is crucial for data mining projects. Some data may contain missing or incorrect values, which can negatively affect the accuracy and reliability of the model. Therefore, such corrupt or misleading data were removed from the datasets to increase the reliability of the models, their generalization ability, and the accuracies of the results. A database of 811 FRP reinforced beams with or without stirrups was considered in this study.

The database includes three different strengthening configurations such as fully-wrapped (350 beams), U-wrapped (328 beams), and side-bonded (133 beams). Since it is difficult to strengthen T-section RC beams with FRP due to the RC slabs, U-wrapped strengthening configuration with a proper anchoring system can be used in the strengthening applications. It is indicated in ACI 440.2R [171] that anchorage systems used in U-wrapped and side-bonded beams can result in higher FRP strains compared to U-wrapped and side-bonded configurations without anchorage. According to ACI 440.2R [171], the effective strain limits to account for the FRP contribution to shear strength in U-wrapped beams with anchored systems can be taken as the same as for fully-wrapped beams. In addition, fib TG 9.3 [172] has similar provisions regarding the calculation of the contribution of FRP to the shear strength for both fully-wrapped and anchored U-wrapped. Therefore, RC beams strengthened with anchored U-wrapped were considered as fully-wrapped beams to determine the contribution of FRP to the shear strength of the beam in this study. The number of beams in terms of FRP type in each strengthening configuration is given in Fig. 1. Test specimens where FRP strips were applied perpendicular to the beam axis were considered. Furthermore, 20 equations for fully-wrapped beams, 13 equations for U-wrapped beams, and 10 equations for side-bonded beams provided by various codes and researchers were statistically compared with the GEP equations.

Fig. 1

The distribution of beams by FRP type

The variables considered as input variables in GEP models are as follows: effective depth of beam ($d$), the effective depth of FRP (${d}_{{\text{fe}}}$), the shear span-to-effective depth ratio $(a/d)$, the beam width (${b}_{w}$), the FRP width-to-spacing ratio ${(w}_{f}/{s}_{f})$, the FRP thickness $\left({t}_{f}\right),$ the FRP reinforcement ratio (${\rho }_{f}$), the ultimate strain (${\varepsilon }_{{\text{fu}}}$), the modulus of elasticity (${E}_{f}$), the concrete compressive strength (${f}_{c}$), the stirrups ratio (${\rho }_{w}$), and the yield strength of stirrups (${f}_{{\text{yw}}}$).

Within the scope of this study, the maximum and minimum values of each variable for references in the databases gathered to derive GEP models of three different strengthening configurations are given in Table A1-3. The histograms for all input variables were separately given for each strengthening configuration in Figs. 2, 3 and 4.

Fig. 2

Histograms of the input variables for fully-wrapped beams

Fig. 3

Histograms of the input variables for U-wrapped beams

Fig. 4

Histograms of the input variables for the side-bonded beams

The rose diagrams of the input variables consisting of degrees of a circle to display the frequency of each class are presented in Fig. 5 for each strengthening configuration. Each spoke has a proportional length to indicate its quantity. θ, $R$ bar, and $v$ represent the mean direction, mean resultant length, and circular standard deviation in the rose diagrams. If the R bar is close to one, it indicates a high concentration. v also refers to the circular analog of the linear standard deviation. The summary of the statistical results of the rose diagrams is shown in Table 1 for each strengthening configuration separately.

Fig. 5

Rose diagrams of the input variables

Table 1 Summary statistics of the rose diagrams

Additionally, the Pearson correlation coefficients matrices were given as heat map in Fig. 6 to measure the linear correlation between the two variables for each strengthening configuration. The correlation coefficient could be different values between -1 and 1 and indicate the relationship between two variables in terms of force and direction. Positive values of correlation coefficients mean indicate a positive correlation between variables, and negative values indicate a negative correlation. In addition, the correlation coefficients close to zero demonstrate the poor relationship between the variables.

Fig. 6

Heat maps of the Pearson correlation matrix for each strengthening configuration

3 Derivation of equations with GEP

Gene expression programming (GEP) invented by Ferreira [173] is one of the machine learning tools frequently used to derive related prediction models for various engineering problems in recent years. The most important feature of gene expression programming is the generation of chromosomes representing any parse tree [173]. The GEP model consists of fixed length genes and chromosomes forming an expression tree (ET). The expression tree (ET) representation of a typical chromosome expressing the mathematical expression $\sqrt[3]{\left(a*b\right)/(c+d)}$ is presented in Fig. 7. ET consists of nonlinear entities of different sizes and shapes. A new language was created to read and express the information encoded in chromosomes. This reading of the expression tree is called the Karva language and is written in the format Q/* + abcd, where “Q” represents the cube root function of Fig. 7.

Fig. 7

The schematic representation of the expression tree of a chromosome [173]

In this study, a computer program called GeneXproTools 5.0 [174] was used to derive the models for the prediction of the FRP contribution to the shear strength in fully-wrapped, U-wrapped, and side-bonded beams. The input variables such as $d, {d}_{fe},{b}_{w}, {t}_{f},{E}_{f},$ ${f}_{c },{{f}_{{\text{yw}}},a/d,{w}_{f}{/s}_{f},{\rho }_{f}, \rho }_{w },$ and ${\varepsilon }_{fu}$ were used to obtain the output variable (${V}_{f})$. Root-mean-square error (RMSE) was taken as a fitness function in the derivation of GEP models. The GEP variable settings are summarized in Table 2 by considering the previous studies [175,176,177].

Table 2 GEP variable settings

The database was randomly divided into training (75%) and validation (25%) sets to overcome the problem of overfitting. For fully-wrapped, U-wrapped, and side-bonded beams, 262, 246, and 100 data points (75%) were used for the training sets, respectively, while the remaining 88, 82, and 33 data points (25%) were employed in the validation datasets. The proposed GEP model consisted of four different sub-expression trees linked by the linkage function for each strengthening configuration as shown in Figs. 8, 9 and 10. The GEP-based formulations for the contribution of FRP to the shear strength (${V}_{f}$) were given in Eqs. (1)–(3) for fully-wrapped, U-wrapped, and side-bonded beams, respectively. The proposed models are valid for three different FRP types such as CFRP, GFRP, and AFRP whose properties are between the limit values in the databases.

Fig. 8

Expression trees (ETs) of the GEP model for fully-wrapped beams

Fig. 9

Expression trees (ETs) of the GEP model for U-wrapped beams

Fig. 10

Expression trees (ETs) of the GEP model for side-bonded beams

For fully-wrapped beams:

$$V_{f} = \frac{{7.74\left[ {3.4 - \left( {a/d} \right)} \right]\left( {w_{f} /s_{f} } \right)}}{{\left( {a/d} \right)f_{c} \left( {\rho_{f} - \rho_{w} } \right)\rho_{f}^{1/2} }} + \left[ {b_{{\text{w}}} + \frac{{\varepsilon_{{{\text{fu}}}} E_{f} t_{f}^{1/3} }}{18.75}} \right]d_{{{\text{fe}}}} - \left[ {\frac{{0.12\left( {w_{f} /s_{f} } \right)b_{{\text{w}}} d_{{{\text{fe}}}} }}{{\varepsilon_{{{\text{fu}}}} \left( {a/d} \right)\left( {d_{{{\text{fe}}}} - f_{{{\text{yw}}}} } \right)}}} \right] - \frac{0.13}{{\rho_{f} }}\left[ {\frac{d}{{f_{c} t_{f} }} - \left( {\frac{d}{{t_{f} }}} \right)^{1/2} } \right]$$

(1)

For U-wrapped beams:

$$V_{f} = \frac{{b_{{\text{w}}} \left[ {d + 21.08\left( {a/d} \right)} \right]}}{{\left[ { - 0.93\left( {w_{f} /s_{f} } \right)f_{c} } \right]^{1/3} }} + \frac{{9.42d_{{{\text{fe}}}} E_{f}^{1/3} \left[ {\rho_{f} + (w_{f} /s_{f} )} \right]}}{{\left[ {0.87 + 3.31\left( {w_{f} /s_{f} } \right)} \right]}} + b_{{\text{w}}} \left[ {d_{{{\text{fe}}}} - \frac{{4.16\left( {\rho_{w} f_{{{\text{yw}}}} } \right)^{1/4} )}}{{\left( {t_{f} \varepsilon_{fu} } \right)^{1/2} }}} \right] + \left[ {\frac{{ - 60.39\varepsilon_{{{\text{fu}}}} t_{f} d}}{{f_{{{\text{yw}}}} - \left( {\frac{a/d}{{E_{f} }}} \right)^{1/3} }}} \right]$$

(2)

For side-bonded beams:

$$\begin{aligned} V_{f} = & - 30819.53 + f_{c} \left[ {9.9\left( {d - 9.9} \right) + \left( {0.88 - a/d} \right)\left( {f_{{{\text{yw}}}} + d_{{{\text{fe}}}} } \right)} \right] \\ & + \left[ {\left( {d_{{{\text{fe}}}} - \frac{{d_{{{\text{fe}}}} }}{{\varepsilon _{{{\text{fu}}}} }}} \right) + \left( {\frac{{2431.7t_{f} }}{{w_{f} /s_{f} }}} \right)} \right]\left( {\sqrt {\rho _{f} } - \varepsilon _{{{\text{fu}}}} b_{w} } \right) \\ & + \left[ {\left( {\frac{{f_{{{\text{yw}}}}^{3} \rho _{w} }}{{3.73}}} \right) - \left( {\frac{{(w_{f} /s_{f} )d_{{{\text{fe}}}}^{2} }}{{2.63}}} \right) + \frac{{\left( {d_{{{\text{fe}}}} - b_{{\text{w}}} } \right) + f_{c} }}{{\sqrt {\rho _{f} } }}} \right] \\ & - \,44.19\left( {\sqrt {E_{f} } - 9.98b_{{\text{w}}} } \right) \\ \end{aligned}$$

(3)

4 Results and discussion

The experimental FRP contribution (${V}_{f}$) to the shear strength was calculated by subtracting the shear strength of the reference beams from the shear strength of the strengthened beams. The predicted FRP contribution (${V}_{f}$) to shear strength was calculated by substituting the necessary variables (Tables A1-3) in both investigated equations and proposed GEP models. The accuracies of the considered equations were statistically interpreted by comparing the experimental (${V}_{f}$) and predicted FRP contribution (${V}_{f}$). Statistical variables such as the mean value (M), standard deviation (SD), mean absolute percentage error (MAPE), root-mean-square error (RMSE), coefficient of correlation (R), and coefficient of variation (COV) were considered in evaluation of the equations. The number of evaluated beams might differ due to the limitations of some equations to determine FRP contribution to shear strength as shown in Tables 3, 4 and 5. In order for the proposed models to be statistically in agreement with the experimental results, the MAPE, RMSE, and COV values should be low and the correlation coefficient (R) should be close to one value.

Table 3 Statistical values of each equation for the fully-wrapped beams

Table 4 Statistical values of each equation for the U-wrapped beams

Table 5 Statistical values of each equation for the side-bonded beams

The Pearson correlation matrix (Fig. 6) shows that ${d}_{{\text{fe}}}$ and ${b}_{{\text{w}}}$ were the most effective variables for the experimental FRP contribution (${V}_{f}$) of fully-wrapped beams, while ${f}_{c}$ and ${w}_{f}{/s}_{f}$ are less effective. In the proposed model for fully-wrapped beams, the same results related to the variables were obtained as shown in Fig. 11. For U-wrapped beams, $d, {d}_{fe},$ and ${b}_{{\text{w}}}$ were highly effective on the experimental performance in contrast to ${f}_{c}$, ${\rho }_{f}$ and ${\rho }_{w}$ as shown in Fig. 6. However, ${d}_{{\text{fe}}}$,${t}_{f}$, ${f}_{{\text{yw}}}$, and ${\varepsilon }_{{\text{fu}}}$ had a significant impact on the prediction accuracy of the proposed GEP model in U-wrapped beams (Fig. 11). In addition, for side-bonded beams, $d, {d}_{{\text{fe}}},$ and ${b}_{{\text{w}}}$ were highly effective on the experimental performance in opposition to ${\rho }_{f}$,${E}_{f}$,${\varepsilon }_{{\text{fu}}}$, and ${w}_{f}{/s}_{f}$. However, $d$ and ${b}_{{\text{w}}}$ had a strong effect on the prediction accuracy of the proposed GEP model in side-bonded beams (Fig. 11). To summarize, it was evaluated that the geometric dimensions such as $d, {d}_{{\text{fe}}},$ and ${b}_{{\text{w}}}$ are more effective on the FRP contribution to the shear strength and these variables were effectively considered in the proposed GEP models.

Fig. 11

The contribution of each input variable in GEP models for different strengthening configurations

The comparisons between FRP contribution to shear strength calculated by the proposed GEP models and experimental results were given for each FRP strengthening configuration in Figs. 12 and 13. As a result of this comparison, it was seen that the density of the distribution between the predicted and the experimental results of ${V}_{f}$ was in the prediction limit of the 5%. As shown in Table 3 for fully-wrapped beams, compared to the other equations, the GEP model was the most reliable with COV and R values of 0.492 and 0.880, respectively. It was seen that the equations given the statistically closest predictions to the GEP model in terms of COV and R values were Sengun and Arslan [36], Akkaya et al. [3], Khalifa et al. [178], Zhang and Hsu [123] and NCHRP Rep. No. 678 [183]. On the contrary, it was also seen that the COV and R values of the Chen and Teng [139], ACI 440.2R [171], DAfStf [182], CSA-S806-12 [184], CIDAR [185], EN 1998-3 [190], CECS 146 [187], and TEC-18 [188] equations were between 1.060–1.160 and 0.418–0.527, respectively. Thus, it was understood that the prediction accuracies of these equations were low. The predictions by NCHRP Project No. 678 and CNR-DT 200 R1 for fully-wrapped beams were not economical due to the lowest mean values.

Fig. 12

The comparison of experimental results versus GEP models

Fig. 13

The illustration of experimental and predicted values of FRP contribution to the shear strength

For U-wrapped beams shown in Table 4, the GEP model was the most reliable as fully-wrapped beams with COV and R values of 0.594 and 0.882, respectively, whereas the CAN/CSA [192] was the worst result with COV and R values of 2.124 and 0.115, respectively. In accordance with COV and R values, the equations given the closest predictions to the GEP model were Khalifa et al. [178], Khalifa and Nanni [74], Mofidi and Chaallal [82] and Sengun and Arslan [36]. However, it was also seen that the COV and R values of the ACI 440.2R [171], fib TG 9.3 [172], Bukhari et al. [32], CNR-DT 200 [191], and CNR-DT 200 R1 [180] equations were between 0.849–1.025 and 0.279–0.486, respectively. Thus, it was understood that the prediction accuracies of these equations were low.

As for side-bonded beams as shown in Table 5, the GEP model had the best prediction accuracy with COV and R values of 0.381 and 0.922, respectively. The equation by Sengun and Arslan [36], with a COV value of 0.468 and an R value of 0.667, had the closest predictions to the GEP model statistically. Contrary, the COV and R values of the CAN/CSA [192] equation were 1.047 and 0.315, respectively. Therefore, it was understood that the prediction accuracy of CAN/CSA [192] equation was incompatible for side-bonded beams. Considering the COV and R values among the three models derived by the GEP method, it was seen that the model with the highest predictive accuracy was for side-bonded beams. The predictions by Bukhari et al. [32], Chen and Teng [138, 139], and CNR-DT 200 R1/2013 for side-bonded beams were not economical due to the lowest mean values.

5 Conclusions

In this study, three different GEP models were proposed to assess the FRP contribution to the shear strength of the RC beams for each strengthening configuration. The prediction accuracies of the GEP models proposed and the existing equations commonly used in the literature were compared utilizing a comprehensive dataset with a total of 811 RC beams strengthened with FRP. Furthermore, the relative contribution of each input variable in GEP models proposed for different strengthening configurations was investigated. The main results obtained were as follows:

The effects of 12 experimental variables on the shear behavior of RC beams strengthened with FRP were considered to propose GEP models. Experimental variables with the most influence in the proposed GEP models were ${d}_{{\text{fe}}}$ and ${b}_{{\text{w}}}$ for fully-wrapped beams, ${d}_{{\text{fe}}}$ and ${f}_{{\text{yw}}}$ for U-wrapped beams, $d$ and ${b}_{{\text{w}}}$ for side-bonded beams.

The proposed GEP models outperformed equations commonly used in the literature, exhibiting higher R values and lower COV values. These models incorporated variables such as $a/d$ and ${\rho }_{w}$, which significantly influence shear behavior. Among the derived GEP models, it was observed that the model yielding the best statistical results was for side-bonded beams.

The GEP models proposed in this study are valid for beams where the shear failure occurs, effective depth of beam greater than 100 mm, and $a/d$ ratio greater than 0.71. Strengthened with carbon, glass and aramid strips was applied perpendicular to the beam axis. In future research, it is recommended to derive equations with higher accuracy by gathering a wider data set, including beams strengthened with different inclined type of FRP strips such as carbon, glass, aramid, and basalt.

Finally, this study proved that GEP models can be used effectively and reliably in various scientific fields, thanks to their robustness and ability to adapt to different conditions and problems. With these features, it can be said that GEP models have the potential to make significant contributions to research studies.

Source codes of the GEP

Appendix A

See Tables 6, 7 and 8.

Table 6 The minimum and maximum values of the variables for fully-wrapped references

Table 7 The minimum and maximum values of the variables for U-wrapped references

Table 8 The minimum and maximum values of the variables for side-bonded references