**1. Introduction**

Reinforced concrete slabs are one of the common horizontal load-carrying members in civil engineering, and widely applied in bridges, ports and hydro-structures. Since there is no beam in the flat slab under longitudinal load, the punching failure of reinforced concrete slab occurred easily. Many researchers have carried out numerous experiments on the punching shear resistance of reinforced concrete slabs, and obtained successful results [1–3]. However, steel bars are prone to corrosion, which will result in the shortening of the actual service life [4]. In recent years, with the development of technology, the durability of structure has attracted people's attention. For coastal areas and the areas of using chlorides such as deicing salt, the actual service life of structures is often much lower than their design service life, resulting in massive losses [5].

Fiber reinforced polymer (FRP) is a material which has many advantages such as light, high strength and corrosion resistance. In a corrosion environment, to solve the problem of short actual service life of structure, FRP bar can be applied as an alternative to steel bars in concrete structures. This is because FRP bar can be appropriate for service

**Citation:** Shen, Y.; Sun, J.; Liang, S. Interpretable Machine Learning Models for Punching Shear Strength Estimation of FRP Reinforced Concrete Slabs. *Crystals* **2022**, *12*, 259. https://doi.org/10.3390/ cryst12020259

Academic Editors: Yifeng Ling, Chuanqing Fu, Peng Zhang and Peter Taylor

Received: 26 January 2022 Accepted: 13 February 2022 Published: 14 February 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/).

demands of concrete structure in severe environments which contain plenty of chloride ion and sulfate ion. However, FRP bar has a low elastic modulus, which will result in FRP reinforced concrete slabs being more prone to punching failure than reinforced slabs (Figure 1). Many experimental studies show that [6–11] under the same conditions, the punching shear strength and stiffness of the cracked FRP reinforced concrete slabs decrease faster than reinforced concrete slabs. Matthys and Taerwe [6] concluded that the crack development of slabs and brittleness of punching failure were significantly affected by the bond performance of FRP grid reinforcement through experimental results. The experimental results of Ospina et al. [7] indicated that the bond behavior between FRP bar and concrete have a significant impact on punching shear capacity of FRP reinforced concrete slabs. It is shown that the slab thickness and the concrete compressive strength were of considerable influence on the punching shear capacities of FRP reinforced concrete slabs by Bouguerra et al. [9]. Ramzy et al. [12] reported that the punching shear behavior of flat slab was related to the size effect of construction and the Young's modulus of FRP reinforcement.

**Figure 1.** Punching shear failure mode of FRP reinforced concrete slabs. (**a**) Stereogram; (**b**) Profile.

In terms of theoretical models, most of the punching shear strength computational formulas of FRP reinforced concrete slabs were derived from traditional reinforced concrete flat with modifications to account for FRP [13]. Some current design specifications such as the GB 50010-2010 [14] and the ACI 318-14 [15] regard the eccentric shear stress model as theoretical basis. Based on the ACI 318-11, El-Ghandour et al. [16] suggested that one considers the impact of elastic modulus of FRP bars, and come up with an improved equation for punching shear strength. After that, El-Ghandour et al. [17] used the same method to modify the design formula of the BS 8110-97. Matthys and Taerwe [6] considered the effect of the equivalent reinforcement ratio, and proposed a modification of the BS 8110-97. On this basis, Ospina et al. [7] proposed an empirical model for computing the punching shear strength of FRP slabs by modifying the relation between the punching shear capacity and the equivalent reinforcement ratio. On the basis of the probability of exceedance, Ju et al. [18] proposed a new approach for analyzing the punching shear strength of FRP reinforced two-way concrete slabs by using Monte Carlo simulations. However, the aforementioned empirical models adopted some simplifications during theoretical derivations, thus the empirical models were unable to consider all of the influential factors. What's more, the parameters in the aforementioned empirical models were determined by the traditional regression analyses from experimental results. Therefore, the accuracy of the models is highly dependent on the choices of theoretical models and quality of the databases.

In recent years, with the development of artificial intelligence, some algorithms with data at the core have emerged [19]. Among these algorithms, machine learning has received remarkable attention of researchers, and there have been many successful examples [20–24]. In structure engineering, Hoang et al. [25] constructed machine learning based alternatives for estimating the punching shear capacity of steel fiber reinforced concrete (SFRC) flat slabs. Hoang et al. [26] presented the development of an ensemble machine learning model to predict the punching shear resistance of R/C interior slabs. Mangalathu et al. [27] build an explainable machine learning model to predict the punching shear strength of flat slabs without transverse reinforcement. In addition, some researchers [28,29] even use the atomistic simulations as the input parameters of machine learning to predict the performance of materials and structures, which has also seen success. To the best of the authors' knowledge, however, no study examined the interpretable machine learning models in predicting the punching shear strength of FRP reinforced concrete slabs. Therefore, this study intends to fill this research gap. The deep relationship between material properties and punching shear strength will be found if the machine learning research on FRP reinforced concrete slabs is carried out. It is helpful to refine the traditional empirical models and design codes and make them more accurate than before. In this paper, an experimental database for the punching shear strength of FRP reinforced concrete slabs is first compiled, and used for training, validating, and testing machine learning models.

Four machine learning algorithms, namely artificial neural network (ANN), support vector machine (SVM), decision tree (DT) and adaptive boosting (AdaBoost), are employed for punching shear strength prediction. Then, by comparing the performance of machine learning models, their efficiency and accuracy can be determined. These comparisons are valuable for identifying the efficiency and prediction ability of the machine learning models. However, there is problems which need to be solved; for example, the values of hyper-parameters in machine learning are often difficult to determine. Therefore, to assist the improvement of the predicted performance of machine learning models, optimization methods such as particle swarm optimization (PSO) and empirical method will be used. In addition, since previous models based on machine learning found it difficult to explain the predicted mechanism, the predicted results of models were somehow unconvincing [27]. This paper uses SHapley Additive exPlanation (SHAP) [30] to explain the predicted result of model. Distinct from other machine learning papers explained by SHAP, the kernel explainer will be used in this paper, which is appropriate for all the machine learning models. Not only can SHAP carry out analysis of feature importance for input factors, it can also determine whether the impact of input features on predictions is positive or negative. In addition, SHAP can make researchers realize the interrelation between input features, and how each input feature will influence the final predicted value for a single sample. Consequently, the emergence of SHAP renders the predicted results of machine learning more convincing than before.

### **2. Experimental Database of FRP Reinforced Concrete Slab**

To ensure the accuracy of predicted results, a database with adequate samples is required for model training. In this paper, 121 groups of experimental results [6–10,12,31–49] of FRP reinforced concrete slabs under punching shear tests were collected, and made it into a database. In the data set, 80% of the data was used as a training set (10% of this was used as validation set), and remaining 20% were used as test set. In this study, 6 critical parameters are selected to characterize the punching shear strength of FRP slabs such as the types of column section, cross-section area of column (*A*/cm<sup>2</sup> ), slab's effective depth (*d*/mm), compressive strength of concrete (*f'*c/Mpa), Young's modulus of FRP reinforcement (*E*f/Gpa), and reinforcement ratio (*ρ*f/%). According to the findings of Ju et al. [18], these 6 parameters have a great influence on punching shear strength of FRP slabs. Therefore, these 6 parameters are considered as input parameters and the punching shear strength of FRP slabs is considered as an output parameter of machine learning algorithms. To simplify the names of the 6 input parameters and the 1 output parameter, we used *x*<sup>1</sup> to *x*<sup>6</sup> and *y* to represent them, respectively. The column section has 3 types: square (*x*<sup>1</sup> = 1), circle (*x*<sup>1</sup> = 2), rectangle (*x*<sup>1</sup> = 3). The distribution of the dataset is shown as Table 1 and Figure 2, and the detailed information is shown as Appendix A.


**Table 1.** Parameters in the punching shear strength samples.

**Figure 2.** Histograms of input and output variables. (**a**) *x*<sup>1</sup> ; (**b**) *x*<sup>2</sup> ; (**c**) *x*<sup>3</sup> ; (**d**) *x*<sup>4</sup> ; (**e**) *x*<sup>5</sup> ; (**f**) *x*<sup>6</sup> ; (**g**) *y*.

Before starting the machine learning procedure, the data of input variables should be preprocessed. In this paper, we used deviation standardization as the preprocessing method; this can be written as:

$$\mathbf{x}\_{i}^{\*} = \frac{\mathbf{x}\_{i} - \min\_{1 \le j \le m} \{ \mathbf{x}\_{j} \}}{\max\_{1 \le j \le m} \{ \mathbf{x}\_{j} \} - \min\_{1 \le j \le m} \{ \mathbf{x}\_{j} \}} \tag{1}$$

where *x* is the sample before feature scaling; *x* \* is the sample after feature scaling; *m* is the total number of samples. Since the sample *x* contains features of several input parameters, it can also be named the feature vector.

### **3. Machine Learning Algorithms**

Generally, the implementation of machine learning has four stages: (a) divide the database into training set and test set; (b) apply the training set to model training; (c) check whether the accuracy requirements are met; d) output the predicted model for test or adjust the values of hyper-parameters. The flowchart for this procedure is shown as Figure 3. A total of 4 machine learning models, namely, ANN, SVM, DT and Adaboost, are selected

in this study. As for ANN, an input layer, hidden layers and an output layer are formed for the predicted results, where several neurons are applied. As for SVM, a prediction equation is formed based on the predicted output in which the error should be smaller than a fixed value. As for DT, the whole database is separated into tree-like decisions which are based on one or several input characteristics, and the output is the tree which falls into the data. As for the AdaBoost, a group of weak learners are collected to form a strong learner which achieves better predicted results. Notably, the predicted performance of model has relations with the value of hyper-parameters. Therefore, in this paper, some methods will be selected to better determine the value of hyper-parameters.

**Figure 3.** Flowchart for implementation of machine learning algorithm.
