**1. Introduction**

Reinforced concrete (RC) slab-column structures comprised of slabs and columns are susceptible to punching shear, because the beams are not arranged for the considerations of structural layout under slabs [1]. Under excessive punching shear loads, the interior slab-column joint is usually destroyed first, the rest of the joints are destroyed in succession, and the progressive collapse of overall structure takes place [2]. Accidents (Figure 1), such as the collapse of a 16-storey apartment building [3] in Boston, US and Skyline Plaza [4] in Virginia, US, have caused severe damage, which arouse the researchers' attention regarding the reliability analysis of RC slab-column joints.

**Citation:** Shen, L.; Shen, Y.; Liang, S. Reliability Analysis of RC Slab-Column Joints under Punching Shear Load Using a Machine Learning-Based Surrogate Model. *Buildings* **2022**, *12*, 1750. https:// doi.org/10.3390/buildings12101750

Academic Editors: Ming-Hung Hsu, Zheng-Yun Zhuang and Ying-Wu Yang

Received: 12 September 2022 Accepted: 18 October 2022 Published: 20 October 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

**Figure 1.** Collapse of slab-column structures: (**a**) a 16-storey apartment building, Boston [3]; (**b**) Skyline Plaza, Virginia [4].

To assess the performance of slab-column structures, especially the slab-column joints, a number of experimental studies have been conducted on the punching shear resistance. With the experimental results, some empirical models [5–15] have been proposed based on a variety of mechanical theories. Kinnunen and Nylander [5] analyzed the experimental data of circle slab-circle columns, and created the sector model. Based on this, Broms [6,7] proposed a modified model considering the impact of size effect, which obtained the solution of the ultimate angle of the slabs. Tian et al. [8] proposed a prediction model considering the impact of reinforcement strength (*ρfy*). According to the eccentric shear stress model proposed by Stasio et al. [9], an improved model with stronger applicability was proposed by Moe [10], which became the theory basis of both GB 50010-2010 [11] and ACI 318-19 [12]. After analyzing the critical cracks of slab-column joints and considering the impact of aggregate size, the critical shear crack theory (CSCT) was proposed by Muttoni [13]. Based on the modified compression field theory (MCFT), Wu et al. [14] developed a prediction model; its prediction performance was validated by many experimental data. According to the regression analysis of the experimental data, a prediction model was proposed by Chetchotisak et al. [15].

However, the aforementioned mechanical or empirical models possess the problem of prediction precision [16,17]. As a typical data-driven model with advantages such as superior prediction performance and high computational efficiency, machine learning (ML) is applied to many engineering fields successfully [18–25]. In the resistance prediction of slab-column joints, Nguyen et al. [16] established a prediction model using extreme gradient boosting (XGBoost), the performance of which was validated by empirical models and other two ML models. Mangalathu et al. [17] also constructed XGBoost models, and used SHapley Additive exPlanation (SHAP) to illustrate the prediction process of XGBoost. Shen et al. [23] established an ML model to predict the punching shear resistance of fiberreinforced polymer (FRP) -reinforced concrete slabs, the performance of which was better than that of the compared empirical models. Truong et al. [24] studied the punching shear strength of FRP-RC slab column connections with the assistance of ML models.

The objective of reliability analysis is to evaluate the safety of structures by considering how their performances are affected by the uncertainties, which are introduced by random material properties or stochastic loads [26]. There are two types of methods for reliability analysis, namely the gradient-based method and the simulation-based method [27]. The first method contains the first-order reliability method (FORM), and the second-order reliability method (SORM) aims to find the most likely failure point through the limit state function estimation. Such a method has a high computational efficiency, but it introduces approximations that are sometimes unacceptable from a precision point of view [28]. As the main simulation-based method, the Monte Carlo sampling method is conventional, clear, and easy to use, but such a method requires numerous samples [29,30]. Nassim et al. [31] studied the reliability of two cases by using the response surface method (RSM) as well as Monte Carlo simulation (MCS). Olmati et al. [32] proposed a simplified analysis framework and used MCS to analyze the reliability of an office building. Chetchotisak et al. [15]

studied the structural reliability within two kinds of concrete (normal-strength concrete and high-strength concrete) by using MCS. Ricker et al. [33] utilized three reliability analysis techniques, such as the mean-value first-order second moment method (MVFOSM), the first-order second moment method (FOSM), and MCS, to assess the safety levels of the punching shear resistance of flat slabs without shear reinforcement. However, the relatively low prediction accuracy of the aforementioned mechanical or empirical models led to unsatisfying results of the reliability analysis. To obtain more accurate reliability analysis results, the finite element method (FEM) is popularly applied as the surrogate model of structural response under stochastic material properties or loading conditions [34]. The complexity and nonlinearity existing in structures, as well as the randomness produced by influential factors of a structure itself, prove that FEM becomes a fine choice. However, the mechanical property-based analysis restricts the computational efficiency of FEM, which is inapplicable to practical projects [35]. Furthermore, as the most commonly used parallel analysis method in a stochastic context, MCS has a problem of inadequate computational efficiency, because the number of samples needed for analysis is considerably large [36]. The ML model is a prospective solution for the contradiction between computational efficiency and accuracy, and has been applied in the reliability analyses of RC structures in the latest studies [37].

To the best knowledge of the authors, there is no available example combining reliability analysis of RC slab-column joints and ML; thus, this paper establishes an ML-MCS model for reliability analysis to meet the requirements of practical projects. The candidate ML models selected in this paper are artificial neural network (ANN), decision tree (DT), random forest (RF), and XGBoost. The final prediction model is screened from these four ML models, and the performance comparison between them is implemented through three performance measures: root mean squared error (RMSE), mean absolute error (MAE), and coefficient of determination (R2). To display the advantages of the ML models, two design provisions (GB 50010-2010 [11] and ACI 318-19 [12]), as well as three prediction models proposed by Tian et al. [8], Wu et al. [14], and Chetchotisak et al. [15], are used for prediction performance comparison with ML models. Furthermore, SHAP is introduced for model explanation and analysis of influential factors; the prediction process can be visualized to facilitate the understanding [22]. Based on the established ML model, a slab-column structure in an actual engineering application is used for reliability analysis through MCS. Moreover, the safety assessment of the structure is discussed through sensitivity analysis.
