Punching shear Prediction of Flat slabs Reinforced with FRP Bars

1. Introduction

Flat slabs are widely used in constructions owing to their architectural and economical advantages. They provide clear space and demand less formwork as they directly supported on columns. This would increase the number of stories for a given height of a building, ease the construction process, and reduce the time and construction cost. However, due to their configuration, the bending and shear stresses are concentrated at the column-slab connections leading to punching shear (PS) failure. The latter is a sudden local failure characterized by the formation of diagonal cracks forming a truncated cone around the column. Similar to steel reinforced flat slabs, those reinforced with fiber reinforced polymer (FRP) bars are also experience the PS failure. Various experimental studies were conducted to examine the key parameters influencing the PS strength of FRP flat slabs. These studies suggested that the concrete strength $\left({\text{f}}_{\text{c}}^{\prime }\right)$ , the FRP reinforcement ratio (ρf), the elastic modulus of FRP bars (Ef), the column size (C1, C2), and the effective depth of a slab (d) have pronounced influence of the PS strength of FRP reinforced concrete (RC) flat slabs. Various predictive models have been proposed to estimate the punching shear capacity of FRP-reinforced slabs. Among these, (Matthys and Taerwe, 2000) presented a PS design expression that allowed for the reduced stiffness of FRP rebars compared to steel ones. (El-Ghandour et al., 2003) presented a design expression for GRFRP RC flat slabs. Their expression modified that of the British Standard BS8810 for RC steel slabs to account for the low stiffness and reduced bond characteristics of FRP reinforcement. (Ospina et al., 2003) suggested another PS expression that considered the axial stiffness and bond properties of FRP reinforcement. (Hemzah et al., 2019) proposed an empirical model that considered the effects of reinforcement characteristics, concrete compressive strength, and column rectangularity. (Ju et al., 2021) presented a model based on the probabilistic–deterministic framework that considered the reduced stiffness of FRP bars and the ratio of the effective depth to the critical section. Recently, (Dinh et al., 2025) carried out theoretical and experimental investigations and found that the axial stiffness of FRP reinforcement affects the PS strength of FRP RC flat slabs. From this, they introduced an empirical model that agreed well with the test results.

The current design codes, such as (Institute, 2022), (ACI 440.11-22), (CAN/CSA S806-12), and (JSCE-97), provide empirical equations for designing FRP slabs against PS failure. Whereas the Eurocode (EN 1992-1-1 (2023)) adopts the crack sliding theory (CST) for the PS design of steel-reinforced flat slabs, allowing for the lower modulus of elasticity of FRP reinforcement. All these codes allow for a lower modulus of elasticity of FRP reinforcement than that of steel.

Therefore, the codes’ PS equations provide different strength predictions and widely scattered. On the basis of the comprehensive literature review presented herein, it may be concluded that the punching shear capacity of interior flat slabs reinforced with FRP bars is governed by a number of critical parameters, namely the axial stiffness of the FRP reinforcement, slab thickness, concrete compressive strength, column dimensions, and column aspect ratio.

The development in computer programs, especially in the field of machine learning has provided accurate techniques to tackle various engineering problems such as symbolic regression (SR), artificial neural networks (ANN), extreme gradient Boosting (XGBoost), and random forest (RF). Unlike SR, the ANN, XGBoost, and RF do not require a predefined mathematical relationship between the input and output. However, these models generally operate as black-box models, lacking explicit mathematical formulations of the learned relationships. These methods are successfully implemented to tackle various structural concrete issues (Al-Bayati et al., 2025); (Taki & Al-Bayati, 2024); (Freiesleben et al., 2022); (Al-Bayati & Taki, 2024).

This paper presents three models for the calculations of the PS of FRP RC flat slabs. The proposed models are developed using SR, XGBoost, and RF techniques, and they are trained and tested using the test results of 204 FRP RC flat slabs from the literature. The strength predictions of the proposed models are compared with those of the existing ones for validation.

2. Existing Models

This section summarizes the PS expressions of the selected existing models along with those of the current American (ACI440.11-22), Japanese (JSCE-97), Canadian (CAN/CSA S806-12) and European (EN 1992-1-1 (2023)) design codes.

3. Experimental database

A total of 204 test results of FRP RC flat slabs subject to concentric PS are collected from the literature. The data base consists of (11, 12, 5, 3, 5, 6, 3, 4, 5, 5, 7, 6, 5, 2, 6, 5, 7, 5, 3, 8, 4, 7, 6, 4, 3, 6, 4, 3, 1, 3, 5, 3, 5, 1, 10, 1, 5, 4, 4, 3, and 9) reported by, (Louka, 1999), (Matthys & Taerwe, 2000), (Rahman et al., 2000), (Hassan et al., 2000), (El-Ghandour et al., 2003), (Zaghloul et al., 2003), (Ospina et al., 2003), (Hussein et al., 2004), (Jacobson et al., 2005), (El-Gamal et al., 2005a), (El-Gamal et al., 2005b), (Zhang et al., 2005), (El-Tom, 2007), (Zaghloul, 2007), (El-Gamal et al., 2007), (Lee et al., 2009), (Zhu et al., 2011), (Bouguerra et al., 2011), (Zhu et al., 2012), (Nguyen-Minh & Rovňák, 2013), (Dulude et al., 2013), (Hassan et al., 2013a), (Hassan et al., 2013b), (El-Gendy & El-Salakawy, 2015), (Tharmarajah et al., 2015), (Mostafa, 2016), (Elgabbas et al., 2016), (Gouda & El-Salakawy, 2016b), (Gouda & El-Salakawy, 2016a), (Oskouei et al., 2017), (Ju et al., 2018), (Al Ajami, 2018), (Hussein & El-Salakawy, 2018), (Hemzah et al., 2019), (Huang et al., 2020), (Gu, 2020), (Eladawy et al., 2020), (Salihi & Hamad, 2023), (Alkhattabi et al., 2024), (Elnady et al., 2024), (Khdir & Yaseen, 2024), and (Dinh et al., 2025) respectively. All these slabs were constructed without shear reinforcement and failed by the PS. It is randomly divided into two sets for training and testing. The training set includes 80% (163) slabs, and the remaining 20% (41) are for testing. The former set is used to form a relationship between the input parameters and the output to develop the models, and the latter is for unbiased evaluation of the developed models. The statistical descriptions of each parameter are summarized as illustrated in Table 1.

The frequency histograms and normal distribution curves are presented in Figure 1. The investigated variables are moderately asymmetrically distributed, but they are not symmetrically distributed in terms of normal distribution. The majority of these parameters follow a bell-shaped trend but deviate from normality at varying levels of positive skewness. The majority are distributed around the middle ranges, and only a few approaches the extreme values. It should be noted that the histogram class intervals shown in Figure 1 were determined using the Freedman–Diaconis rule to obtain statistically optimal bin widths (Freedman & Diaconis, 1981). The calculated bin widths were subsequently rounded to practical engineering intervals to improve the readability of the graphical presentation without affecting the underlying statistical representation. Figure 2 presents the violin plots of the considered parameters. These plots show distribution of the data considering the region of concentration, dispersion, variability, skewness and potential outliers.

Figure 1:

Figure 2:

4. Methodology

Three machine learning algorithms, namely eXtreme Gradient Boosting (XGBoost), Random Forest (RF), and Symbolic Regression (SR), were employed to develop the predictive models. These algorithms were selected due to their accuracy and consistency. The SR is regarded as a conventional machine learning approach, whereas XGBoost and R.F belong to the category of tree-based learning models. Among them, XGBoost has gained considerable attention because of its superior predictive performance and robustness across diverse datasets.

The ML models were developed using Python and the database is divided into training and testing sets. The performance of the proposed models is measured using the coefficient of determination (R²), coefficient of variance (CV), mean absolute error (MAE), and root mean square error (RMSE). Figure 3 shows the general workflow for the proposed models and the evaluation of their performance

Figure 3:

Where:

• R²− Coefficient of determination [-],

• V_test − Experimental punching shear strength [kN],

• V_pred. − Predicted punching shear strength [kN],

• N − No. of test results [-],

• C. O. V − Coefficient of Variation [-],

• MAE − Mean absolute error [kN],

• RMSE − Root mean square error [kN].

4.1. eXtreme Gradient Boosting (XGBoost)

Boosting (XGBoost) is an advanced ensemble predictive learning algorithm based on the Gradient Boosting framework. XGBoost supports parallel processing, enabling efficient model training on large datasets. The algorithm iteratively constructs a sequence of decision trees, with each subsequent tree developed to minimize the prediction errors of the preceding ones. The outputs of all individual trees are then assembled to produce the final prediction. Figure 4 shows a typical flowchart of the XGBoost algorithm technique. In this figure, the XGBoost algorithm utilizes an iterative procedure. Accordingly, the parameters in the subsequent step are updated using residual obtained from the previous step in order to optimize the objective function. Here xi, ɸ and Fk shown in Figure 4, represent the i-th input variable, the trained parameters, and the prediction function of an individual tree, respectively (Nguyen et al., 2021).

The input variables used in the XGBoost model and the subsequent global feature contribution (SHAP) analysis were selected based on their physical significance and their established influence on the punching shear capacity of FRP-reinforced slab-column connections, as reported in previous experimental studies and design

Figure 4:

4.2. Random forest (RF)

RF is an ensemble learning algorithm that consists of a large number of decision trees to increase prediction accuracy and robustness. The RF algorithm is viewed as a variation of the bootstrap aggregation (bagging) method created to minimize variance and overfitting of decision trees by stabilizing them, improving their accuracy, or forcing extreme values. Each decision tree is built independently on a bootstrap subsample of the training dataset. The process employs a random subset of features at each split rather than considering all input variables. This increases diversity within the models and reduces correlation among individual trees. Every tree is constructed by taking a random subset of features and then just using a classical split-selection procedure. Then, the RF utilizes the predictions of many independent trees to improve performance and reduce overfitting (Mangalathu & Jeon, 2019). Figure 5 shows a typical flowchart of the RF algorithm technique.

Figure 5:

4.4. SHAP Analysis

To examine the contribution of the most effective parameter to the PS strength to adopt in the SR model. This method examines all possible combinations of inputs to find the average effect of each one, and the more crucial input parameter achieves the higher SHAP value. Figure 6 plots the input parameters on the vertical axis and ranks them according to their relative importance. The SHAP values are plotted along the horizontal axis and show how much each variable contributes to the model prediction. Each dot represents a database instance, and the strength of each parameter is indicated by a color scale ranging from blue to red. The red-colored points represent higher parameter values, whereas the blue-colored points correspond to lower values. It should be noted that the parameters with positive SHAP values increase the predicted PS strength, and those with negative SHAP values reduce the predicted response. Furthermore, the broad horizontal spread of the SHAP values for the dominant variables indicates their substantial sensitivity and strong influence on the model prediction. The SHAP analysis indicates that the combined parameters ${\left({\rho }_f{f}_c^{\prime }\beta \right)}^{1/4}$ have the strongest influence on the PS strength. Therefore, it is adopted in the SR model, and the PS strength is calculated by multiplying the combined parameter by the shear area of the critical section (bo.d).

Figure 6:

Based on the SHAP analysis, the combined parameters ${\left({\rho }_f{f}_c^{\prime }\beta \right)}^{1/4}$ are found to have the strongest influence of the PS of FRP RC flat slabs. Hence, it is adopted in the proposed SR model, which takes the following form:

(18)

$$V_c=0.3134{\left(E_f\right)}^{\frac{1}{3}}{\left({\rho }_f{f}_c^{\prime }{\beta }_c\right)}^{\frac{1}{4}}{b}_{0\left(1.5d\right).}d$$

Where:

• β_c − Ratio of long side to short side of column’s cross section ${\beta }_c=\frac{max\left(c_1,c_2\right)}{min\left(c_1,c_2\right)}\left.\text{-}\right]$ .

All other parameters of this expression are explained earlier.

The above expression utilizes the influence of the mechanical properties of concrete and FRP reinforcement in the PS calculation. It combines the effects of reinforcement ratio, concrete compressive strength, column aspect ratio, and the effective shear-resisting area. It is hardly surprising since most existing models adopted the same parameters.

5. Statistical and Comparative Results

Figure 7 illustrates the Heatmap visualization comparing the PS predictions of the proposed and existing models. Among the empirical formulations, the equations proposed by (Hemzah et al., 2019), (Ospina et al., 2003), (El-Ghandour et al., 2003), and (El-Gamal et al., 2005) demonstrate the most satisfactory predictive performance among the conventional equations proposed by other researchers, exhibiting prediction ratios closest to unity with relatively lower variability. The equations proposed by (Matthys & Taerwe, 2000), (Jacobson et al., 2005), and (Ju et al., 2021), together with the provisions of (JSCE-97), (CSA S806-12) and EN 1992-1-1 (2023)) design codes, exhibit modest accuracy and widely scatter. In contrast, the provisions of ACI 440.11-22 design code show the least satisfactory performance, characterized by highly conservative predictions and significant variability relative to the experimental database. This behavior may be attributed to the inability of the existing formulations to adequately represent the interaction among concrete properties, reinforcement characteristics, slab geometry, and stiffness-related mechanisms governing the punching shear strength capacity.

Figure 7:

Furthermore, the proposed machine-learning-based models, particularly XGBoost and RF, significantly outperform the existing models by achieving substantially higher prediction accuracy, lower variability, and stronger agreement with the experimental results. Similarly, the proposed SR model demonstrates highly competitive predictive capability compared with the existing empirical equations, providing improved agreement with the experimental database while maintaining a simplified mathematical formulation. The enhanced predictive performance of the proposed XGBoost and Random Forest (RF) models, together with the improved performance of the Symbolic Regression (SR) model, is attributed not merely to their data-driven nature, but rather to their ability to capture the interaction-controlled behavior governing punching shear resistance. Unlike conventional empirical equations and design-code provisions, which assume fixed functional relationships among the governing parameters, the proposed models identify the evolving interactions between concrete strength, reinforcement characteristics, slab geometry, stiffness, and critical shear perimeter directly from the experimental data. The SHAP analysis further confirms that punching shear resistance is governed primarily by coupled interaction terms rather than isolated variables. Consequently, the proposed models provide a more realistic representation of the underlying failure mechanism, resulting in substantially improved predictive accuracy and reduced scatter.

The performance evaluation metrics of the three proposed prediction models for both the training and testing stages are summarized in Table 2. Overall, all models demonstrate satisfactory predictive capability in estimating the punching shear strength of FRP-reinforced slabs. However, clear differences in prediction accuracy, stability, and generalization capability can be observed among the investigated approaches.

Figures 8 and 9 illustrate Box and violin plot comparisons of the punching shear strength ratio for existing design equations and ML models. The proposed ML models (XGBoost and RF) showed higher prediction accuracy than that of the SR model. That is, the coefficient of determination for the overall dataset obtained using XGBoost, RF, and SR are 0.966, 0.949, and 0.806, respectively. In general, a noticeable increase in data scatter is observed at higher load levels, as illustrated in Figure 10, with the dispersion starting to increase at approximately (150–200)kN and becoming significantly more pronounced beyond (250–300)kN. This behavior can be attributed to several interacting parameters. At elevated loads, the structural response transitions from a relatively stable, quasi-linear behavior to a highly nonlinear response characterized by crack localization, concrete softening, and potential shifts in failure mode (from flexural to punching shear or combined mechanisms) (Muttoni & Simões, 2023); (El-Gamal et al., 2005a). In addition, the bond degradation between concrete and reinforcement, particularly in the case of FRP bars, may reduce shear transfer efficiency. The system becomes increasingly sensitive to minor variations in material and geometric properties, amplifying prediction errors (Ospina et al., 2003). These factors, combined with the inherent limitations of empirical equations and possible experimental uncertainties, result in significantly higher dispersion at higher load levels.

Figure 8:

Figure 9:

Figure 10:

6. Conclusions

This paper has presented three empirical models developed based on the XGBoost, RF, and SR techniques to predict the PS of FRP RC slabs. A comprehensive database of slab columns with a wide range of test parameters was used to train and test the proposed models. The following can be reported:

1. The proposed ML models (XGBoost and RF) showed stronger agreement with the test results than the SR model. The coefficients of determination for the overall dataset obtained using XGBoost, RF, and SR are 0.966, 0.949, and 0.806. This higher predictive accuracy of XGBoost and Random Forest suggests that the PS strength of FRP-reinforced slabs is governed by multiple interacting mechanical mechanisms rather than a single global mathematical relationship, enabling ensemble-learning models to capture the underlying structural behavior more effectively than Symbolic Regression.

2. The proposed models were validated by comparing their punching shear predictions with the existing model (Hemzah et al., 2019), (Ospina et al., 2003), (El-Ghandour et al., 2003), and (El-Gamal et al., 2005). The comparisons revealed that the PS calculations of the proposed model overperformed those provided by the existing ones.

3. The SHAP analysis revealed that the combined parameters ${\left({\rho }_f{f}_c^{\prime }\beta \right)}^{1/4}$ and the effective shear-resisting area (b0.d) as the dominant factors governing the punching shear capacity of FRP-reinforced slabs.

4. The provisions of ACI 440.11-22 show the poorest predictive performance, characterized by highly conservative predictions and considerably greater variability than the experimental database and the other international design codes, including the JSCE-97, CAN/CSA S806-12, and EN 1992-1-1 (2023).

5. Overall, most punching shear prediction equations proposed in the literature exhibited limited predictive accuracy, often yielding conservative estimates of experimental punching shear capacity, particularly at higher load levels and for large-scale flat slab specimens.