Hybrid support vector regression approaches for modeling punching shear strength of reinforced concrete flat plates

Abstract

Predicting the punching shear strength (PSS) of flat slabs is crucial for ensuring the safety and efficiency of reinforced concrete structures. This study presents novel hybrid approaches combining support vector regression (SVR) with advanced optimization algorithms to enhance the accuracy of PSS predictions. Four optimization algorithms, krill herd algorithm, biogeography-based optimization, equilibrium optimizer, and genetic algorithm (GA), were employed to optimize SVR parameters for improved PSS estimation. A data set of 264 samples with seven design parameters was used as input to model PSS. Sensitivity analysis and comparison to standard equations were conducted to evaluate the significance of input variables and the reliability of proposed models in predicting PSS. The results demonstrated that integrating optimization algorithms significantly improved the predictive performance of SVR models. Among the proposed approaches, the SVR-GA model achieved the highest accuracy, with a correlation coefficient of 0.95 and a mean absolute error of 132.28 kN in the testing phase. Sensitivity analysis revealed that slab thickness and depth, followed by concrete strength, were the most influential parameters for predicting PSS. The proposed SVR-GA model was found more accurate than American, European, and Canadian concrete code standards in modeling PSS. These findings underscore the effectiveness of hybrid SVR models in accurately modeling PSS and highlight the importance of optimizing input features to ensure robust predictions.

Graphical abstract

Keywords

flat slab / PSS / SVR / hybrid / GA

1 Introduction

Flat slabs are extensively utilized in comparison to alternative slab systems. They provide a multitude of advantages over conventional reinforced concrete (RC) options owing to their architectural adaptability, diminished construction duration, and ability to support heavier loads. Nevertheless, their structural integrity may be jeopardized by punching shear failure, a brittle failure mechanism that takes place at the slab–column connections under severe loading scenarios. Such failures often lead to catastrophic structural collapses, emphasizing the need for accurate punching shear strength (PSS) estimation in the design process [1]. Modeling the PSS of flat slabs is essential for understanding the behavior of RC structures under various loading conditions and ensuring that safety margins are adequately maintained [1,2]. Traditional approaches to PSS prediction rely on code-based empirical formulas or simplified analytical methods, which may lack precision due to the complex interactions among multiple design variables, such as structure geometry and materials characteristics [2]. Nowadays, advanced computational techniques, such as soft computing techniques, can be used for addressing these limitations.

1.1 Background of punching shear strength estimation

In RC flat plates, several features should be considered in estimating the PSS, such as material properties, geometrical factors, and loading conditions [2–5]. Figure 1 shows the main factors that can be considered in modeling the PSS. Meanwhile, it should be noted that standard codes, such as ACI 318-19 [6], BS 8110-97 [7], CEB-FIP 90 [8], and CSA-A23.3-04 [9] have different design equations to determine the PSS of flat slabs [2]. Table 1 proposes the equations for estimating the PSS of flat slabs based on previous studies and standard codes.

From Fig. 1 and Table 1, it can be seen that the most influential variables are the thickness of the slab, column diameter/width, depth of slab, main reinforcement ratio of the slab, and CS of concrete. In this study, besides these parameters, the column shape and yield strength (f_y) of steel are considered. In addition, previous studies found that depth of slab, main reinforcement ratio, column shape ratio, and concrete strength influenced the accuracy of PSS estimation [13–15].

1.2 Literature review on soft computing for punching shear strength estimation

Soft computing techniques have several applications in modeling concrete characteristics. For instance, support vector regression (SVR) was employed to extract the PSS of flat slabs reinforced by fiber-reinforced polymer (FRP) bars and compared to artificial neural network (ANN), random forest (RF), and gradient boosting regression tree (GBRT) models [13]. The accuracy of SVR, ANN, RF, and GBRT was 0.89, 90, 90, and 0.93, respectively, in terms of coefficient of determination (R²). SVR was also used to estimate the PSS of FRP concrete slab–column connections and compared to a developed deep neural network technique; the results showed the outperformance of SVR in modeling the PSS with R² = 0.96 [14]. gaussian process regression (GPR) and SVR were compared in estimating the PSS of flat slabs [16]. GPR outperformed SVR and both models were found better than EN in modeling PSS. PSS of the fiber-reinforced concrete flat slabs was modeled using the extreme gradient boosting (XGB) algorithm and compared to standard codes, such as ACI, BS, and EN [15]. XGB outperformed SVR, RF, ANN, and standard codes in modeling PSS. From these studies it can be shown the ability of SVR in modeling PSS due to its capability to handle nonlinear relationships and provide robust predictions with limited data sets. For further details, the following studies employed SVR in modeling the shear strength of concrete and the performance of structure members [17–21]. Although the performance of the SVR model is shown high in the estimation of concrete characteristics, its accuracy is highly dependent on the selection of hyperparameters [22]; thus, hybrid models are commonly used to address this limitation [22,23].

Hybrid models have been successfully used in improving the performance of conventional machine learning techniques. A combination of SVR and grid search optimization algorithm was developed to model the CS of concrete, and the integration improved the accuracy of the SVR model from 0.85 to 0.93, in terms of R² [24]. Integration of SVR and firefly algorithm (FFA), gray wolf optimization (GWO), and particle swarm optimization (PSO) was evaluated in modeling the CS of concrete [25]. The results found that SVR-FFA outperformed conventional algorithms. SVR-genetic algorithm (SVR-GA) was examined to estimate the shear strength of RC deep beams and compared to classical SVR, ANN, and ensemble models; the SVR-GA achieved a high accuracy and improved the performance of SVR [26]. The equilibrium optimizer (EO) algorithm was integrated with ANN and least-square SVR (LSVR) models to enhance the modeling of the bond capacity of the FRP-to-concrete interface, and it improved the performance of proposed models and enhance the prediction results [27]. EO was also integrated with LSVR, ANN, multivariate adaptive regression splines, logit boosting, and XGB technique to model the concrete strength [28]. EO-LSVR outperformed other proposed models, in addition, EO improved the accuracy of the conventional techniques. Furthermore, RF, Adaptive Boosting, and gradient boosting (GB) were integrated with PSO to estimate the CS of concrete; GB-PSO successfully achieved a high accuracy in the modeling CS, R² = 0.98 [29]. GWO and whale optimization algorithm were combined with XGB for modeling the PSS, and GWO-XGB attained the best accuracy [30]. Integration of adaptive neuro-fuzzy inference systems (ANFIS) and GA achieved a correlation coefficient (R) of 0.91 in modeling the shear strength of RC beams [31]. Hybrid krill herd algorithm-ANN (ANN-KHA) was developed to estimate the strength and stiffness of bolt connection, and its accuracy was found better than ANN-GA [32]. Biogeography-based optimization (BBO) was integrated with ANN to estimate the CS of concrete and its achieved accuracy of 0.99 in terms of R² [33]. ANFIS-BBO achieved R² = 0.97 in modeling the heating load of buildings [34]. Based on our knowledge, EO, GA, BBO, and KHA optimization techniques are still limited in use to enhance the SVR for modeling the PSS of flat slabs. Therefore, it is imperative to examine these techniques for automatic tuning of SVR model hyperparameters in this study to estimate an accurate PSS of flat slabs.

1.3 Study objectives

Therefore, this investigation enriches the existing body of knowledge by presenting various novel viewpoints on predicting the PSS of flat slabs and examining the impact of input parameters. The primary objective is to establish dependable and accurate PSS prediction models using hybrid SVR techniques. To achieve this, optimization algorithms, including BBO, KHA, EO, and GA, were employed, as their high performance has been demonstrated efficacy in previous investigations. A thorough comparative analysis was performed to evaluate the accuracy of the proposed models against established formulas and the findings of earlier studies. Furthermore, the sensitivity of input variables was meticulously analyzed, with outcomes juxtaposed with prior research and normative equations, thereby yielding enhanced insights and validation. By integrating advanced computational techniques like SVR and hybrid optimization methods into the PSS prediction framework, this study presents notable progress for the structural engineering discipline. The developed hybrid SVR model effectively mitigate the constraints of conventional approaches, thereby guaranteeing accurate and reliable evaluations for the design of RC flat slabs.

2 Material and methods

2.1 Used data sets

Descriptive details of the employed data set are presented in Table 2. Notably, 281 data sets of PSS slabs were acquired from previous studies, see supplementary materials, and after successful data mining using boxplot, a total of 264 samples were extracted and utilized in this study after removing the outliers. As presented in Table 2, two-column shapes (S), circular and square, were considered. Additionally, the slab thickness (H), column diameter/width (C), slab depth (D), main reinforcement ratio of the slab (ρ), CS of concrete (f_c), and f_y were used as input parameters to model the PSS of flat slabs (V_p). Although these parameters incorporate more variables than those presented in Table 1, additional parameters could be considered in future studies to further enhance the model’s accuracy and comprehensiveness. The statistical evaluation of the variables in Table 2 highlights significant variation within the data set. This variation indicates the complexity of data sets and the nonlinear correlation between inputs and output variables. Figure 2 illustrates the linear correlation between the input parameters and V_p. From the figure, it is evident that slab depth and thickness exhibit the strongest correlation with PSS, followed by the CS of concrete, column dimensions, reinforcement ratio, column shape, and steel strength. The analysis suggests that slab depth or thickness alone can serve as a simple regression model for PSS, achieving an R² value of up to 0.80. However, relying solely on slab thickness is insufficient for the safe design of structures, as all parameters must be considered during the design stage. The relationship between input parameters and PSS, as well as the standard equations for PSS prediction, reveals a complex interplay of factors.

Furthermore, the distribution of input and output variables is non-normal, underscoring the complexity of the correlations. Given this complexity, machine learning approaches offer a promising solution to enhance the accuracy of PSS predictions and ensure reliable estimations within the range of the available data.

2.2 Machine learning techniques

2.2.1 Support vector regression

SVR is a type of machine learning algorithm based on support vector machines [35]. It is used for regression tasks, predicting continuous outcomes while maintaining the principles of maximizing the margin and minimizing overfitting. SVR is a versatile regression technique, especially useful for complex data sets, combining flexibility with strong theoretical foundations. The details of SVR can be found in Ref. [35]. Here, the summary of SVR can be proposed as follows [35–37]:

The regression of PSS can be expressed as follows:

(1)f\left(x\right)=w^{\mathrm{T}}x+b,

where x, w, and b are the input variables, weight vector, and bias, respectively. In this study, the nonlinear case is needed to overcome the complexity of used data sets. Thus, Eq. (1) can be reformed as follows:

(2)f\left(x\right)=y=w^{\mathrm{T}}\varnothing \left(x\right)+b,

where \varnothing \left(x\right) denotes nonlinear kernel. In this study, radial basis function (RBF) kernel (K\left(x_i,x\right)) is employed, which can be expressed as follows:

(3)K\left(x_i,x\right)=\mathrm{e}\mathrm{x}\mathrm{p}(-\gamma {\Vert x_i-x\Vert }^2),

where \gamma is the kernel width. To estimate the parameters in Eq. (2), the following equation can be used to minimize the model errors.

(4)\{\begin{array}{l}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}0.5w^{\mathrm{T}}+c\sum _{i=1}^n({\zeta }_i+{\zeta }_i^{\ast }),\\ y_i-w^{\mathrm{T}}\varnothing \left(x\right)-b\le \epsilon +{\zeta }_i,\\ w^{\mathrm{T}}\varnothing \left(x\right)+b-y_i\le \epsilon +{\zeta }_i^{\ast },\\ {\zeta }_i,{\zeta }_i^{\ast }\ge 0,\end{array}

where n and c are the number of samples and regularization parameter, respectively. {\zeta }_i and {\zeta }_i^{\ast } are slack variables.

Equation (4) represents the core optimization formulation of the SVR model using the ε-insensitive loss function. In this context, the SVR algorithm aims to identify a regression function that not only fits the training data with minimal error but also maintains a balance between prediction accuracy and model complexity. The objective function includes two main components: the term 0.5w^T which minimizes the magnitude of the weight vector and thereby enforces a flatter regression function, and the penalty term which starts with the second term which controls the total error exceeding the predefined ε-insensitive margin. Here, c is the regularization parameter that governs the trade-off between minimizing training error and ensuring model generalization. The constraints ensure that for each training sample, the predicted output either lies within an acceptable ε-deviation from the actual value or is penalized by the slack variables which represent the amount by which predictions fall outside the ε-tube on either side of the regression function. The function \varnothing(x) denotes a nonlinear transformation that maps the input features into a higher-dimensional space, enabling the SVR to capture nonlinear relationships. The bias term b adjusts the output of the regression function. Collectively, this formulation allows SVR to perform robust regression by minimizing structural complexity while tolerating small deviations and penalizing significant errors, making it particularly well-suited for modeling complex relationships such as the PSS of flat slabs. For modeling the PSS with RBF, the parameters of c and γ are the main factors affecting the performance of SVR. In this study, BBO, KHA, EO, and GA are employed to estimate SVR parameters.

2.2.2 Optimization algorithms

The KHA is a nature-inspired optimization algorithm developed by imitating the herding behavior of krill in oceans. It was introduced by Gandomi and Alavi [38] as a metaheuristic for solving optimization problems. The algorithm is particularly effective for complex, multidimensional, and nonlinear problems. Here, among marine species, Antarctic krill are particularly well-studied due to their ability to form dense populations. Mathematical models developed for krill behavior highlight key factors influencing their group formation, including food availability, population density, and random movements. To simulate this natural behavior for optimization problems, the KHA was proposed. It mimics three primary actions of krill. 1) Increasing group density (attraction to other krill) (Ni). 2) Moving toward food sources (foraging behavior) (Fi). 3) Random walking (exploration of the environment) (Di). In KHA, each krill represents a solution to the optimization problem. Their movements reflect the search for a globally optimal solution. Figure 3(a) presents the KHA flowchart, and details of it can be found in Refs. [32,37,39].

The BBO is an evolutionary algorithm inspired by biogeography, the study of species distribution over geographical areas [40]. It models how species migrate between habitats, survive, or go extinct, and applies these principles to solve optimization problems. In BBO, each habitat represents a potential solution and is evaluated based on its habitat suitability index (HSI), which reflects the quality of the solution [40]. Habitats with higher HSI values represent better solutions. The algorithm simulates the migration of features (known as suitability index variables from habitats with high HSI to those with low HSI. This exchange allows less-suitable habitats to improve by acquiring beneficial properties from more suitable ones. It was developed by Simon, the BBO algorithm is designed to solve optimization problems by iteratively improving solutions to maximize HSI. The workflow for implementing a genetic optimization code using BBO is depicted in Fig. 3(b). more details of BBO theory and applications can be in Refs. [34,41–43].

The EO was introduced by Faramarzi et al. [44]; it is an innovative physics-based metaheuristic algorithm, specifically designed to address complex engineering problems effectively. Similar to other metaheuristic algorithms, the EO operates with a population of particles, where each particle’s position (or concentration) corresponds to a potential solution to the optimization problem. The algorithm maintains an equilibrium pool comprising the four best solutions discovered so far, along with their average. During the optimization process, particles adjust their positions or concentrations based on equilibrium candidates selected randomly from this pool. This iterative adjustment guides the particles toward the equilibrium state, which represents the optimal solution to the problem [44]. The details of the EO algorithm can be found in Ref. [44]. Figure 3(c) proposes the workflow of the EO algorithm. \lambda, r, F, and G are random vector [0 1], random vector [0 1], exponential term, and generation rate, respectively. Generation-rate control parameter vector is the generation control parameter that determines the likelihood of including a term’s contribution in the concentration update process. EO previous applications can be found in Refs. [45–47].

The GA is an optimization technique inspired by the principles of natural evolution, including selection, crossover, and mutation [48]. It represents candidate solutions as chromosome-like data structures, which are evaluated based on a fitness function to determine their quality. The algorithm starts with a randomly generated population of chromosomes, and through iterative processes of reproduction, fitter solutions are combined and modified to explore the solution space effectively. Key genetic operators, such as crossover (combining solutions) and mutation (introducing diversity), ensure the population evolves toward an optimal or near-optimal solution. GA is widely used in various fields for solving complex, nonlinear, and multidimensional problems, especially regression and resource allocation tasks. The workflow for implementing GA is depicted in Fig. 3(d). GA applications and theory can be found in Refs. [41,49–51].

2.3 Accuracy assessment

In this study, different statistical indices were utilized to evaluate the proposed models. The correlation between the measured and predicted values was analyzed, along with error indices for the models. The R and Nash-Sutcliffe efficiency (NSE) were used to quantify the correlation, while root-mean-square error (RMSE), mean absolute error (MAE), and RMSE to observation’s standard deviation ratio (RSR) were employed to measure the models’ errors. These indices can be calculated as follows:

(5)R=\frac{{\displaystyle \sum _{i=1}^n(C_i-\overline{C})({Cp}_i-\overline{C}p)}}{\sqrt{{\displaystyle \sum _{i=1}^n{(C_i-\overline{C})}^2}}\sqrt{{\displaystyle \sum _{i=1}^n{({Cp}_i-\overline{C}p)}^2}}},

(6)NSE=1-\frac{{\displaystyle \sum _{i=1}^n{(C_i-{Cp}_i)}^2}}{{{\displaystyle \sum }}_{i=1}^n{(C_i-\overline{C})}^2},

(7)RMSE=\sqrt{\frac{{\displaystyle \sum _{i=1}^n{(C_i-{Cp}_i)}^2}}{n}},

(8)MAE=\frac{{\displaystyle \sum _{i=1}^n\left|\right(C_i-{Cp}_i\left)\right|}}{n},

(9)RSR=\frac{RMSE}{\sqrt{{\displaystyle \frac{1}{n}}\left({\displaystyle \sum _{i=1}^n{\left(C_i-\overline{C}\right)}^2}\right)}},

where C and C_p represent the measured and predicted PSS values, respectively; \overline{C} and {\overline{C}}_p are the average of measured and predicted PSS values, and n is the number of observed data sets.

Additionally, violin plot and regression error characteristic (REC) curves for the proposed models are used to assess the accuracy of models in the training and testing stages. The violin plot combines the benefits of a boxplot and a density plot. It displays the distribution of predicted errors or model outputs, providing insights into variability, central tendency, and skewness. The REC curve, representing the cumulative distribution function of errors, along with the area over the curve (AOC), is employed to evaluate the accuracy of the proposed models. Models with smaller AOC values demonstrate higher accuracy [52]. Furthermore, the a^{20} index is examined to evaluate the reliability of the models in modeling PSS. This index is particularly valuable because it provides a clear physical and engineering interpretation, reflecting the proportion of samples with predictions deviating by no more than 20% from actual values [53].

2.4 Developed hybrid models for punching shear strength estimation

The framework for modeling PSS is outlined in Fig. 4. The data collected from literature was collected to consider the effect of two different kinds of column shapes. The data was collected to cover different ranges of input variables as presented in Table 2. Then, the data was normalized into 0 to 1. The data was divided into 70% and 30% for tuning and validating the proposed models, respectively. The BBO, KHA, EO, and GA algorithms was integrated the SVR to enhance the accuracy of the proposed models of PSS through optimize best values of SVR parameters, c, ϵ, and \mathcal{G}. The best integrated model is used to assess the impact of input variables in modeling PSS of flat slabs. In addition, the best model was compared to standard equations and previous studies.

3 Results and discussion

3.1 Design models

To optimize the performance of the SVR model for predicting the PSS of flat slabs, four advanced metaheuristic algorithms, KHA, BBO, EO, and GA were employed. These algorithms were selected due to their complementary search strategies, proven effectiveness in high-dimensional optimization problems, and limited prior application to PSS estimation, offering a novel perspective to this domain. The KHA, inspired by the herding behavior of krill swarms, offers a balanced exploration-exploitation mechanism that enables fast convergence in the early stages of optimization. BBO, based on the migration and mutation behavior of species in different habitats, has demonstrated robust global search capabilities and solution diversity through its unique use of habitat suitability indices. EO is a physics-inspired optimizer that updates candidate solutions using equilibrium concepts, offering adaptive behavior and competitive performance in engineering tasks. GA, a well-established evolutionary technique, mimics the process of natural selection and has consistently yielded reliable results in regression and structural parameter tuning. The parameter optimization process for SVR was implemented uniformly across all hybrid models to ensure fairness and reproducibility. The SVR hyperparameters tuned by each optimizer included the regularization constant (c), epsilon-insensitive loss function (ε), and the RBF kernel parameter (γ), with respective search ranges of 0.1–100, 0.001–1, and 0.0001–1, respectively. Each optimization algorithm was configured with a fixed number of iterations and population sizes, while their specific algorithmic parameters were adapted from existing literature. For instance, the KHA used a population of 20, 50 iterations, and additional coefficients including an inertia weight of 1, a diffusion coefficient of 0.005, and a foraging factor of 0.02. BBO operated with 50 individuals over 50 iterations using habitat-based migration and mutation. EO was configured with 20 particles over 100 iterations, utilizing an equilibrium pool for solution updates. GA employed a population of 20 over 10 generations, with 10 parents, a mutation rate of 0.1, and 10 offspring per generation.

The optimal SVR parameters derived from each algorithm were: for SVR-KHA, c = 88.75, ε = 0.0081, and γ = 0.0037; for SVR-BBO, c = 74.32, ε = 0.0059, and γ = 0.0029; for SVR-EO, c = 66.40, ε = 0.0073, and γ = 0.0055; and for SVR-GA, c = 92.10, ε = 0.0064, and γ = 0.0041. These optimized parameter sets were selected based on their performance in minimizing the mean squared error (MSE) during the training phase. All optimization procedures were performed with fixed random seeds to ensure reproducibility. Table 3 demonstrates the parameters of the proposed models. As presented in the Electronic Supplementary Material, the first 70% of data was used to estimate the proposed models. Figure 5 illustrates the convergence curves of the proposed comparative models, evaluated in terms of MSE. The figure demonstrates that the convergence of the SVR-GA model is superior to that of the other methods across all iterations. This enhanced convergence toward the optimal solution can be attributed to the synergistic combination of the GA and the local search strategy, which significantly enhances the performance of the SVR-GA algorithm. Additionally, a comparison of the performance of the KHA and EO optimization methods in estimating SVR parameters further emphasizes the superiority of KHA, particularly in the early iterations. This indicates that KHA provides a more efficient approach to parameter estimation than EO, especially in the initial stages of convergence.

3.2 Evaluation models

Table 4 and Fig. 6 present the results of the proposed models in estimating PSS of flat slabs. The results of training (Tr) and testing (Ts) stages are presented in Table 4 and Fig. 6. From Table 4, it can be seen that the performance of proposed models is superior to model the PSS. The accuracy of proposed models is greater than 0.97 in terms of R. In addition, the high MAE is 73.63 kN for SVR-BBO model. The comparison between the proposed models in the training stage shows the optimal model is SVR-GA to model the PSS. The SVR-KHA, -BBO, -EO, and -GA can obtained PSS with model error up to 1.78, 3.89, 2.54, and 1.46, respectively, in the training stage. Moreover, the scatter plot of the proposed models in the training stage shows a high correlation between observed and predicted PSS values. Most of estimated PSS is in between +20% interval confidence. This indicates the reliability of the proposed models is high into estimating the PSS of flat slabs.

In the testing stage, the outliers can be observed. However, the R between observed and measured PSS is seen high, higher than 92%. The accuracy of SVR-GA is seen the best in terms of correlation statistical indices, R and NSE are 95% and 90%, respectively. The SVR-BBO model results are shown the best after SVR-GA with R and NSE equal 94% and 87%. In the model error indices, the SVR-GA model is shown to be the best with estimating PSS with error reach to 7.60; while the model error of SVR-KHA, -BBO, and -EO is 9.07, 9.30, and 9.20, respectively. These results indicates that the PSS of flat slabs can be accurately estimated by SVR-GA model.

To assess the accuracy of the proposed models in correlation and model error indices. Different statistical indices are used to evaluate the ranking accuracy of the proposed models. Figure 7 proposes the accuracy ranking of the proposed models. From the figure, it can be observed that the accuracy of SVR-GA ranked a high accuracy in modeling PSS with high correlation indices and low model errors. SVR-BBO model can be ranked the second to estimate the PSS of the flat slabs.

For further analysis, the violin plot and REC curves are presented in Fig. 8 for both the training and testing stages. In the training stage, the violin plot of the proposed models reveals a strong correlation between the mean and median for the SVR-EO model, although it shows a high number of outliers. Similarly, the SVR-BBO model also displays a significant number of outliers. The best models for estimating PSS are SVR-KHA and SVR-GA, with the SVR-GA model exhibiting the lowest error range. The violin plot indicates that the SVR-GA model has a tight distribution with its median close to zero, suggesting it consistently predicts values close to the actual values with low variance. Furthermore, the SVR-GA model shows lower error, and a smaller, less skewed distribution compared to SVR-KHA, with fewer outliers, making SVR-GA a reliable model for estimating PSS. In the testing stage, the SVR-GA model demonstrates a more symmetric distribution than the other proposed models. The correlation between the mean and median is high and those are close to zero, further confirming its reliability. Overall, the SVR-GA model outperforms the other proposed models in terms of both accuracy and consistency, with its predictions showing greater reliability compared to the others.

AOC of REC curves are 42.37, 72.59, 38.99, 30.98 for the SVR-KHA, -BBO, -EO, and -GA, respectively, in the training stage. In the testing stage, the AOC are 1.47 × 10², 1.64 × 10², 1.45 × 10², 1.27 × 10² for the SVR-KHA, -BBO, -EO, and -GA, respectively. These results indicate the accuracy of SVR-GA is shown to be high in the training and testing stages. The accuracy of SVR-EO is become the second after SVR-GA. Thus, the SVR-GA model can be considered the best proposed model for estimating the PSS of flat slabs in this study.

To evaluate the reliability of the proposed models in predicting PSS, an uncertainty analysis was conducted on the testing data set. Initially, the absolute error between the observed and predicted PSS values was computed. Subsequently, several performance and uncertainty metrics were calculated, including the standard error (SE), MAE, STDev of the prediction error, and the lower and upper bounds (LB and UB) of the prediction interval. Based on these bounds, the width of the confidence bound (WCB) was determined at a 90% confidence level. Table 5 ranks the models according to their WCB values. The details of these uncertainty metrics can be found in Gholami et al. [54]. A model is considered more reliable when there is minimal variation between the LB and UB, indicating a narrower WCB. Thus, a smaller WCB value reflects higher model reliability and consistency in prediction performance.

As shown in Table 5, while the SVR-BBO model reported a higher MAE compared to SVR-EO and SVR-KHA, it achieved the second-lowest WCB. This indicates that although the point predictions are less accurate, the prediction intervals are more consistent, justifying its second-place rank based on uncertainty evaluation. The SVR-EO and SVR-KHA models showed similar uncertainty profiles, with WCB values of 75.91 and 75.98, respectively. However, the slightly better MAE of SVR-EO gives it a slight advantage over SVR-KHA. The overall reliability statistical evaluation demonstrates that the SVR-GA model outperforms the other proposed hybrid SVR models in modeling PSS. It achieved the lowest MAE of 132.28, indicating superior prediction accuracy, and also exhibited the lowest STDev and SE, reflecting high stability and precision. Most notably, SVR-GA recorded the narrowest WCB (WCB = 62.71) at a 90% confidence level, confirming its high reliability and consistency in the prediction of PSS.

3.3 Sensitivity analysis

To assess the impact of input variables, such as column shape (S), slab thickness (H), column diameter/width (C), slab depth (D), main reinforcement ratio of the slab (ρ), CS of concrete (f_c), and yield strength of steel (f_y) in modeling PSS, the sensitivity analysis (SA), the commonly known Cosine Amplitude method was implemented [55]. The strength between inputs and V_p is calculated and presented in Fig. 9(a) in the training and testing stages. Here, the SVR-GA model is used to assess the impact of input variables on modeling the PSS. From the figure, it can be seen that the slab depth and thickness are the most influential variables in modeling PSS, and this is consistent with standard equations, Table 1, and previous studies. The column dimensions and CS of concrete are shown influenced followed the slab characteristics. The column shape is insignificant in this study, this could be due to the numerical assumption of this variable. Nowadays, SHapley Additive exPlanations (SHAP) has been used to assess the feature importance. Figure 9(b) presents the global SHAP values of input variables on modeling the PSS using the SVR-GA model. The global SHAP values illustrate how each feature contributes, either positively or negatively, to the prediction of PSS. In this figure, each point represents a sample, with red points indicating high eigenvalues and blue points indicating low eigenvalues. For instance, considering the slab depth feature, high eigenvalues (red) positively impact the model's output is shown lower compared to low eigenvalues (blue) which contribute negatively to modeling PSS. Lower eigenvalues (blue) correspond to higher SHAP values, suggesting that column and slab geometries positively influence PSS. For the main reinforcement ratio of the slab, high eigenvalues (green) positively affect the model output, while low eigenvalues (blue) have a negative impact. In this case, higher eigenvalues (red) result in greater SHAP values, highlighting the positive effect of slab thickness and effective depth followed by the CS of concrete on PSS.

3.4 Comparison with standard equation and previous studies

Figure 10 illustrates a comprehensive comparison between the proposed model and existing approaches, including previously published studies and standard equations (refer to Table 1 for the details of these equations). The statistical indices employed in this evaluation include maximum (Max), minimum (Min), mean, STDev, and coefficient of variance (COV). These metrics were computed for the ratio of measured to predicted values (V_m/V_p), where a distribution of the ratio close to 1.0 signifies optimal model performance. The proposed SVR-GA model demonstrated superior performance in comparison to several standard codes, such as ACI, EN, and CSA, particularly in terms of its maximum and mean values. Notably, the mean value of the SVR-GA predictions was closer to the ideal compared to the ACI, EN, CEB, and Rizk et al. equations. Among the standard equations, the BS approach showed the best overall agreement with the experimental data. The variability of the prediction accuracy around the optimal value (V_m/V_p = 1.0) was analyzed using STDev and COV. The SVR-GA model exhibited lower variability than the ACI, EN, and CSA equations, indicating greater reliability and robustness. However, certain formulas, including CEB, the work by Elshafey et al. [11], Rizk et al. [12] and BS [7], were observed to be highly effective in modeling the PSS of flat slabs, particularly for cases that align closely with their assumptions and parameter ranges. These findings emphasize the potential of the SVR-GA model as a reliable tool for estimating PSS in flat slabs, particularly when additional variables, such as column geometry and the yield strength of reinforcement steel, are incorporated. The adaptability and accuracy of the SVR-GA approach make it a promising alternative for scenarios where traditional standard equations exhibit limitations.

3.5 Implementation and limitation

From the proposed results, while this study highlights improvements in prediction accuracy through hybrid SVR models, their practical significance becomes more evident when implemented through a graphical user interface (GUI). Enhancements in statistical indicators such as an increase in the R from 0.92 to 0.95 and NSE from 0.85 to 0.90 reflect a more reliable alignment between predicted and experimental PSS values. Simultaneously, reductions in RMSE and MAE (e.g., a drop in MAE from 154.5 to 132.3 kN) mean that design predictions are not only statistically improved but also more structurally meaningful. In real-world terms, these reductions help engineers avoid overly conservative designs, reduce material costs, and enhance safety margins by providing more precise estimations of critical load capacities. The incorporation of these high-performing models into a GUI framework magnifies their real-world applicability. A GUI offers an accessible platform where users particularly practicing engineers can input basic design parameters (e.g., slab depth, concrete strength, reinforcement ratio) and instantly obtain accurate PSS predictions without needing to understand the underlying machine learning algorithms. It transforms a complex model into a functional engineering tool suitable for preliminary design checks, sensitivity assessments, or educational demonstrations. However, some implementation constraints should be acknowledged. These include the need for consistent input data formatting, computational load for model inference in large data sets, and limitations when extrapolating beyond the trained data range. Nonetheless, the GUI-based deployment of these hybrid models addresses a critical gap between research and practice, enabling rapid and informed decision-making while maintaining model transparency and reliability.

The models that are both proposed and existing, as examined in this investigation, are predominantly conservative in their predictions regarding the PSS of flat slabs; however, they demonstrate significant variability, indicating inherent limitations in accurately capturing the variability. While design standards furnish simplified methodologies for estimating PSS, their dependability is hampered by a constrained set of parameters. To augment accuracy and consistency, it is advisable to integrate supplementary factors such as shear area, load capacity, and span length into forthcoming formulations. The proposed models, which employ a wider array of input variables and are partially informed by experimental data, present enhanced predictive accuracy with a concomitant reduction in variability. Nevertheless, for effective practical design applications, a broader spectrum of input variables and safety factors necessitates incorporation into the hybrid SVR models, which will be the subject of further investigation in subsequent research.

4 Conclusions

This study examines novel hybrid techniques in modeling the PSS of RC flat slabs. SVR was integrated with KHA, BBO, EO, and GA to model the PSS. The column shapes, circular and square, slab thickness, column diameter/width, slab depth, main reinforcement ratio of the slab, CS of concrete, and yield strength of steel were used as input variables to model the PSS of flat slabs. ACI 318-1, BS 8110-97, CEB-FIP 90, CSA-A23.3-04, and EN 1992-1-1 formulas were compared to the proposed models.

The comparison of proposed models showed the R of the proposed models is higher than 92%. The accuracy of SVR-GA is seen the best in terms of correlation statistical indices, R and NSE are 95% and 90%, respectively. The SVR-BBO model results are shown the best after SVR-GA with R and NSE equal 94% and 87%, respectively. In the model error indices, the SVR-GA model is shown to be the best with estimating PSS with an error reach to 7.60; while the model errors of SVR-KHA, -BBO, and -EO are 9.07, 9.30, and 9.20, respectively. This reveals the PSS of flat slabs can be accurately estimated by the SVR-GA model.

The comparison of the SVR-GA model with standard equations showed SVR-GA model has a superior performance in comparison to ACI, EN, and CSA, particularly in terms of its maximum and mean values. The variability of the prediction accuracy around the optimal value showed the SVR-GA model exhibited lower variability than the ACI, EN, and CSA equations, indicating greater reliability and robustness. However, certain formulas, including CEB, the work by Elshafey et al. [11], Rizk et al. [12], and BS [7], were observed to be highly effective in modeling the PSS of flat slabs. This indicates the potential of using the SVR-GA model as a reliable tool for estimating PSS in flat slabs, particularly when additional variables, such as column geometry and the yield strength of reinforcement steel, are incorporated.

The SA and SHAP evaluation showed that slab depth and thickness are the most influential variables in modeling PSS, and this is consistent with standard equations. The column shape, reinforcement ratio of the slab, and yield strength of steel positively affect the model output and can be considered in modeling the PSS of flat slabs. However, for practical design applications, more input variables and safety factors must be integrated into the hybrid SVR models, which will be explored in future work.

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