Data-Driven Interpretable Machine Learning Prediction Method for the Bond Strength of Near-Surface-Mounted FRP-Concrete

Abstract

The Near-Surface-Mounted (NSM) technique for Fiber-Reinforced Polymer (FRP) strengthening is widely applied in the seismic retrofitting of concrete structures. The key aspect of the NSM technique lies in the adhesive performance between the FRP, adhesive layer, and concrete. In order to accurately predict the bond strength of embedded reinforced NSM FRP–concrete, this study constructs the relationship between the influencing factors of bonding performance and bond strength based on four machine learning (ML) algorithms: Decision Tree (DT), Support Vector Machine (SVM), Random Forest (RF), and eXtreme Gradient Boosting (XGB). A unified and interpretable prediction method for FRP–concrete interface bond strength based on SHAP values and ML algorithms is proposed. The results indicate that the ML models exhibit good predictive performance, with the R² of the test set ranging from 0.8190 to 0.9621, showing higher accuracy than empirical calculation formulas. Among them, the RF algorithm demonstrates the highest overall accuracy and optimal performance. Additionally, the SHAP (Shapley additional explanations) method quantitatively confirms that the width of the FRP strip has the most significant impact on bond strength. The newly developed hybrid ML model has the potential to become a new choice for accurately assessing the bond strength of NSM FRP strengthening technology.

1. Introduction

The corrosion of steel reinforcement and deterioration of concrete have always been issues of concern in bridge engineering and concrete reinforcement projects [1,2,3,4]. In recent years, Fiber-Reinforced Polymer (FRP) has become a powerful choice for concrete repair due to its high strength, excellent corrosion resistance, and other advantages [5,6,7,8]. In particular, the NSM FRP reinforcement technology for concrete structures is more effective in utilizing the high-strength properties of FRP compared to traditional external FRP application techniques, and demonstrates excellent durability and fire resistance [9]. NSM reinforcement involves embedding FRP strips into grooves on the concrete surface and achieving the synergistic load-bearing behavior of FRP and concrete through resin bonding [9,10]. The results indicate that FRP enhances the flexural, shear, axial, and seismic performance of concrete [10,11]. However, practical engineering applications have shown that FRP-strengthened concrete can experience interface debonding (delamination failure), leading to structural failure under low loads [12]. The occurrence and timing of debonding failure largely depend on the interfacial bond performance between FRP and concrete [13,14]. Therefore, it is necessary to conduct in-depth research on the bond behavior of FRP and concrete to ensure the durability and reliability of the strengthening effect.

In recent years, numerous scholars have studied the interfacial bond behavior between FRP and concrete based on lap shear tests. These studies have covered the influence of factors such as concrete compressive strength, bond length, FRP strip width, FRP strip thickness, FRP strip elastic modulus, groove depth, groove width, groove spacing, and member section width on the interfacial bond performance between FRP and concrete [15,16,17,18,19,20,21,22,23,24,25,26,27,28]. Based on these studies, many scholars have proposed empirical and semi-empirical models to evaluate the interfacial bond performance between FRP and concrete, incorporating relevant theoretical research [29,30,31,32,33,34,35,36]. Empirical models are primarily based on experimental phenomena and data, incorporating bond degradation mechanisms, and constructing empirical models of bond strength through fitting experimental data. Semi-empirical models are mainly based on stress analysis or fracture mechanics to construct models, with model coefficients calibrated using experimental data. Despite the widespread application of the aforementioned bond strength models in the interfacial bond performance between FRP and concrete, they still have limitations: (i) Experimental induction methods focus on the analysis of specific experimental phenomena, lacking generalizability of the theory. (ii) The prediction of bond strength is a high-dimensional problem involving multiple variables, and neither experimental nor theoretical methods can fully consider the multivariate correlations among factors.

Currently, the use of artificial intelligence-based ML techniques to address complex engineering problems has gradually attracted widespread attention across various fields, particularly within civil engineering [37,38,39,40,41,42,43]. Su et al. [44] used three machine learning methods—MLR, SVM, and ANN—to predict the bond performance between FRP and concrete. The SVM-ML method showed the best performance. Jahangir et al. [45] developed an artificial neural network (ANN) model and a combination model of an ANN with the artificial bee colony optimization algorithm (ABC-ANN) for predicting the bond strength between FRP and concrete based on experimental results. The models offered short prediction times and acceptable errors. Vu et al. [46] developed a machine learning model for the ultimate shear capacity of FRP-reinforced slabs using least squares support vector machines (LS-SVM). The prediction results were excellent and superior to existing methods. Mahjoubi et al. [47] integrated data-driven methods with domain knowledge to develop an intelligent prediction model for interface bond performance, which outperformed other models. Haddad et al. [48] collected experimental research data on the bond strength between FRP and concrete from the existing literature and developed a prediction model based on an ANN. The prediction results significantly outperformed empirical formulas. Nasrollahzadeh et al. [49] collected experimental research data on the bond strength between NSM FRP and concrete from the existing literature and developed Mamdani and Takagi–Sugeno models using fuzzy logic methods for training and testing. The model’s prediction accuracy is superior to that of published models. Golafshani et al. [50] employed an artificial neural network and genetic programming to predict the bond strength between FRP and concrete. The model’s prediction performance significantly surpasses that of multiple linear regression and existing standard formulas. Yasmin et al. [51] collected 770 sets of experimental data and established and tested a Gene Expression Programming (GEP) model. The results show that the GEP model aligns well with existing mathematical models and experimental data.

ML models have demonstrated excellent performance in regression tasks, particularly when dealing with correlated data. Considering the presence of multiple variables, it is crucial to investigate the feasibility of applying ML methods to predict the interfacial bond strength between NSM FRP and concrete. However, the decision-making process of ML models lacks transparency, and most ML methods suffer from the “black box” phenomenon, limiting their applicability in engineering practice and raising concerns about the reliability of the output results. Therefore, there is an urgent need to develop an interpretable ML method to address the gap in the practicality and interpretability of models for predicting the interfacial bond performance between NSM FRP and concrete.

This paper proposes an intelligent prediction method for the interfacial bond performance between NSM FRP and concrete based on ML techniques. The model is built upon ML algorithms, namely DT, SVM, RF, and XGB. Utilizing a dataset comprising 167 sets of experimental research data on the interfacial bond performance between NSM FRP and concrete, the model establishes the relationship between the influencing factors of bond performance and bond strength. The predictive performance of DT, SVM, RF, and XGB algorithms for bond strength is investigated. Furthermore, a comparison is made between the ML model and empirical models to validate the effectiveness and generality of the ML model in predicting the interfacial bond performance between NSM FRP and concrete. Additionally, to analyze the impact of input features on the final results, the SHAP method is employed in this model to assess the relationship between input features and output results. This analysis helps to evaluate the importance of input parameters and overcome the “black box” problem associated with ML methods.

2. Construction of Bond Strength Database

2.1. Parameter Selection and Data Analysis

In this study, a new experimental database for ML models was established by compiling the experimental data [18,19,20,21,22,23,24,25,26,27,28] on the interfacial bond performance between FRP and concrete. The database consists of 167 experimental data samples.

Table 1. Information on the calculated characteristic parameters.

Classification	Variable	ID	Unit	Statistics
				Min	Max	Mean	SD
Input	Lb	X1	mm	30.00	450.00	255.97	72.90
Input	tf	X2	mm	1.20	20.60	3.39	2.04
Input	bf	X3	mm	10.00	40.00	15.70	3.43
Input	Ef	X4	GPa	129.84	173.00	137.82	12.70
Input	fc	X5	MPa	11.85	64.80	33.76	8.79
Input	bc	X6	mm	150.00	220.00	180.65	29.81
Input	wg	X7	mm	3.00	35.00	11.17	5.93
Input	hg	X8	mm	11.00	51.00	26.60	6.96
Input	Ed	X9	mm	20.00	226.00	91.10	41.40
Output	Pu	Y	kN	13.00	205.10	65.53	27.73

presents the maximum, minimum, mean, and standard deviation of the recorded input and output feature values in the database.

The database records the main factors influencing the bond strength. Figure 1 and Table 1 display the distribution of input and output parameters. Nine effective features are considered as input variables, including more specifically bond length (L_b), width of the FRP strip (t_f), thickness of the FRP strip (b_f), elastic modulus of the FRP strip (E_f), compressive strength of the concrete (f_c), section width of the member (b_c), width of grooves (w_g), depth of grooves (h_g), and distance of the groove edges (E_d), which are in the ranges of 30.00 to 450.0 mm, 1.20 to 20.60 mm, 10.00 to 40.00 mm, 129.84 to 173.00 GPa, 11.85 to 64.80 MPa, 150.00 to 220.00 mm, 3.00 to 35.00 mm, 11.00 to 51.00 mm, and 20.00 to 226.00 mm, respectively. The output variable is bond strength (P_u) with a range of 13.00–205.10 kN. This ensures the reliability and representativeness of the database, facilitating the predictability assessment of bond strength by the ML models.

2.2. Sensitivity Analysis of Database Feature Parameters

Figure 2 displays the Pearson correlation coefficient matrix among the variables, with correlation coefficient values ranging from −1 to 1. From the graph, it can be observed that the correlation coefficient range between input and output variables is [−0.015:0.58], and the correlation coefficient range between input variables is [−0.62:0.69]. This indicates that there is no significant correlation between the database parameters, which can provide a reliable database for the construction of the ML model of interfacial bond strength between FRP and concrete.

3. ML-Based Predictive Models

This study aims to establish a predictive model for the bond strength between FRP reinforcement and concrete using ML methods. Due to the complex bonding mechanism and numerous factors influencing the bond, we selected four representative ML algorithms for bond strength prediction: DT, SVM, RF, and XGB algorithms. All of these ML principles and techniques are well known, and detailed explanations can be found in the references [37,38,42,52,53,54]. Therefore, in this study, we will provide a brief introduction to these ML techniques.

3.1. DT

The DT model is illustrated in Figure 3. DT is a tree structure model, usually consisting of tree nodes and directed edges. There are generally three types of nodes in a DT, namely the root, child, and leaf nodes. The decision-making step of the model begins at the root node, where a characteristic is tested, and based on the result, it is split into sub-nodes. This recursive process of testing and splitting the samples continues until reaching a leaf node, which represents the final decision outcome. The path to each leaf node from the root node corresponds to a decision path. The key factors of the model include the max depth of the tree, the max node number of leaves, and so on. The prediction principle of the DT method can be represented by Equation (1) [37,52].

$$h(x)={{\displaystyle {\sum }_{j=1}^J{b}_{j}I}}_{\{x\in Rj\}}$$

(1)

where R_j represents the j-th disjoint region assigned to a leaf of the tree, b_j is the predicted value for region R_j, and I_(x∈Rj) is the indicator function equal to the unit when x ∈ R_j.

3.2. SVM

The SVM model is illustrated in Figure 4. SVM is an intelligent learning algorithm based on the principle of minimizing structural risk, which has incomparable advantages compared to the empirical risk minimization principle of traditional ML algorithms. Moreover, SVM is a convex quadratic optimization problem, ensuring that the obtained extreme value solution is the global optimal solution. Figure 4 illustrates the principle of SVM. SVM maps low-dimensional data nonlinearly to a high-dimensional feature space through a kernel function and then seeks the regression function within this high-dimensional feature space. The principles of the SVM algorithm can be represented by Equations (2)–(6) [38,53].

The expression of the hyperplane $f(x)$ can be expressed as Equation (2).

$$f(x)={\omega }^T\phi (x_i)+b$$

(2)

where $\omega$ is the weight; $\phi (x_i)$ is the mapping function; and $b$ is the bias term.

The formulation of the optimal regression hyperplane into a quadratic programming problem can be expressed as Equation (3).

$$\min {\displaystyle \frac{1}{2}}{\Vert \omega \Vert }^2+c{\displaystyle \sum _{i=1}^l({\xi }_i+{\xi }_i^{*})}$$

(3)

In the equation, $c$ represents the penalty factor; ${\xi }_i$ and ${\xi }_i^{*}$ represent the slack variable.

By utilizing optimization methods, it can be transformed into its dual form, and ultimately, the Lagrange regression function can be represented by Equation (4).

$$f\left(x,a\right)={\displaystyle \sum _{i=1}^n({\alpha }_i^{*}-{\alpha }_i)K(x_i,x_j)+b}$$

(4)

In the equation, ${\alpha }_i$ and ${\alpha }_i^{*}$ represent the Lagrange multiplier vector corresponding to each sample, while $K(x_i,x_j)$ represents the kernel function used to replace inner product operations in a high-dimensional space.

$$\{\begin{array}{c}{\displaystyle \sum _{i=1}^n({\alpha }_i-{\alpha }_i^{*})=0}\\ 0\le {\alpha }_i\le c\\ 0\le {\alpha }_i^{*}\le c\end{array}$$

(5)

In this study, we selected the RBF (Radial Basis Function) kernel function, which has good generalization ability. The specific expression can be found in Equation (6).

$$K(x_i,x_j)={\mathrm{e}}^{-{\Vert x-x_i\Vert }^2/(2{\sigma }^2)}$$

(6)

3.3. RF

RF is a bagging ensemble learning model, which is built upon DT by introducing randomization in the feature selection process during training. The schematic diagram of the RF model is shown in Figure 5. When constructing trees in RF, it randomly selects M samples as subsets from the entire database using bootstrap, and randomly selects B features as splitting nodes to build the DT model using these subsets. The above process is repeated B times to generate B decision trees corresponding to the training sample subsets. For the output results of the B decision trees, the values obtained from all the leaf nodes in the regression decision trees are averaged, and this is output as the regression prediction result Y, as shown in Formula (7) [38,42].

$$\mathrm{Y}={\displaystyle \frac{1}{B}}{\displaystyle {\sum }_{j=1}^b{Y}_b}\left(X^{’}\right)$$

(7)

3.4. XGB

XGB is an advanced algorithm based on the Boosting technique for iterative decision trees. The schematic diagram of the XGB model is shown in Figure 6. XGB originated from the Gradient Boosting Decision Tree (GBDT) algorithm. In comparison, XGB applies pre-pruning to each individual tree during the training process, which avoids algorithm redundancy. Moreover, XGB optimizes the handling of the loss function using second-order Taylor expansion, incorporating first- and second-order derivative information, while GBDT only utilizes the first-order derivative in its loss function expansion. Additionally, XGB introduces a regularization term in the objective function to consider model complexity, thereby preventing overfitting and enhancing the algorithm’s generalization ability and predictive performance. The objective function of XGB is shown in Formula (8) [38,54].

$$Obj={\displaystyle \sum _{\mathrm{i}=1}^{n}l(y_i,{\widehat{y}}_i)}+{\displaystyle \sum _{j=1}^t\Omega (f_j)}$$

(8)

In the formula, Obj represents the objective function; n is the total number of samples; $l(y_i,{\widehat{y}}_i)$ is the loss function, mainly used to calculate the difference between the true value $y_i$ and the predicted value ${\widehat{y}}_i$ for the i-th sample; and ${\displaystyle \sum _{j=1}^t\Omega (f_j)}$ represents the regularization term that indicates the model complexity.

The training objective is to minimize the objective function by expanding the loss function to the second order using Taylor series. By setting the first derivative of the objective function obtained during the t-th round of iteration to 0, the minimum value of the objective function is achieved, leading to the optimal algorithm.

4. Model Operation and Analysis

4.1. Data Normalization

To avoid the significant impact of certain features on the model training, data normalization is performed to scale all feature values within a similar numerical range. The following formula is used to scale the data to [0,1].

$$Y={\displaystyle \frac{T-T_{\min }}{T_{\max }-T_{\min }}}$$

(9)

In the equation, T_max and T_min represent the maximum and minimum value in the dataset, respectively.

4.2. Evaluation Indicators

ML model prediction accuracy is comprehensively compared using five evaluation indicators to assess the performance of each model. These indicators include R², RMSE, MAE, MAPE, and the a20-index for quantitative measurement. The expressions are shown in

Table 2. Statistical indicators.

Parameters	Formula	Ideal Value
Coefficient of determination	${\mathrm{R}}^2={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n{\left(x_i-p_i\right)}^2-{\displaystyle {\sum }_{i=1}^n{\left(x_i-y_i\right)}^2}}}{{\displaystyle {\sum }_{i=1}^n{\left(x_i-{\mu }_i\right)}^2}}}$	1
Root mean square error	$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle {\sum }_{i=1}^n{\left(x_i-y_i\right)}^2}}$	0
Mean absolute error	$\mathrm{MAE}={\displaystyle \frac{1}{n}}{\displaystyle {\sum }_{i=1}^n\left|x_i-y_i\right|}$	0
Mean absolute percentage error	$\mathrm{MAPE}={\displaystyle \frac{1}{n}}{\displaystyle {\sum }_{i=1}^n\left|\frac{x_i-y_i}{x_i}\right|}\times 100\%$	0
a20-index	$\mathrm{a}20-\mathrm{index}={\displaystyle \frac{m20}{n}}$	1

Note: X_i represents the experimental result, Y_i represents forecasted result, P_i is the average experimental value, ${\mu }_i$ is the average forecasted value, n represents the number of calculated samples, and m20 represents the number of samples where the bond strength predicted values fall within ±20% error range.

.

4.3. Model Building Process

This section explains the construction method of ML models for predicting the bond strength between NSM, FRP reinforcement, and concrete. Figure 7 illustrates the overall architecture of the ML model. The method framework consists of four stages: data collection, model training, model validation, and model interpretation.

Step 1—Data Collection: The data from the existing literature on the experimental research of bond performance between NSM FRP reinforcement and concrete are collected to train and test the ML models. Additionally, preprocessing is performed on the input and output sample data, including data normalization and partitioning. The data are normalized to the range of [0,1]. Then, the 167 data samples are randomly divided into two datasets. Among them, 133 data samples (80%) are selected as the training set, and 34 data samples (20%) are selected as the testing set.

Step 2—Model Training: The DT, SVM, RF, and XGB models are trained using the training dataset. Evaluation metrics such as R2, RMSE, MAPE, MAE, and the a20-index are used to assess the prediction stability of the ML models. After multiple programming tests, the optimal hyperparameters for each model’s main parameters are obtained. The specific values of the main parameters are shown in

Table 3. Parameters of machine learning models.

Model	Hyperparameters
	Title	Value
DT	random_state	65
	max_depth	43
SVM	Kernel functions	RBF
	Regularization parameter	1000
	Gamma	0.00009
RF	N estimators	15
	Max depth	20
XGB	N estimators	75
	Learning rate	0.3

.

Step 3—Model Validation: The performance of the models is evaluated using the testing dataset. Based on the obtained values of R², RMSE, MAPE, MAE, and the a20-index, further analysis and discussion of the models are conducted.

Step 4—Model Interpretation: In order to better understand the relationship between bond strength and input variables, a method combining SHAP value analysis is used to provide the importance ranking of features and their specific impact patterns from the perspective of the entire dataset. Additionally, the influence and polarity of features are provided from the perspective of individual samples.

5. Model Training and Discussion of Results

5.1. Model Training

As described in Section 4.2, five statistical indicators are used to evaluate the predictive performance of the established models.

Table 4. Statistical evaluation of the multiple models developed.

Data Type	Model	R2	RMSE	MAE	MAPE	a20-Index
Training data	DT	0.9435	6.8209	3.3922	0.0548	0.9474
	SVM	0.9265	7.7989	4.0032	0.0611	0.9097
	RF	0.9695	4.9608	3.1583	0.0519	0.9774
	XGB	0.9507	6.3696	4.0759	0.0777	0.9624
Testing data	DT	0.9043	7.0523	5.5013	0.0958	0.9117
	SVM	0.8190	8.7680	6.1346	0.1013	0.8823
	RF	0.9621	4.4779	3.6252	0.0732	0.9705
	XGB	0.9157	6.4673	5.0721	0.0935	0.9411

presents the comparative results of four models (DT, SVM, RF, and XGB) on R², RMSE, MAE, MAPE, and the a20-index. From the table, it can be further observed that the training set R² based on the RF and XGB algorithms reaches 0.9774 and 0.9705, significantly higher than the DT and SVM algorithms with 0.9435 and 0.9265, an improvement of more than 2.86%. The training set a20-index based on the RF and XGB algorithms reaches 0.9774 and 0.9624, significantly higher than the DT and SVM algorithms with 0.8474 and 0.9097, an improvement of more than 5.79%. This indicates that the RF and XGB models have better fitting to the samples. Additionally, the training set RMSE reaches 3.1583 and 4.0759, and the MAE reaches 0.0519 and 0.0777 for the RF and XGB algorithms respectively. It can be seen that the predicted values of RF and XGB have relatively small differences compared to the actual values, and the prediction accuracy is better than for DT and SVM. For the testing dataset, the R2 values of the DT, SVM, RF, and XGB models are 0.9043, 0.8190, 0.9621, and 0.9157, respectively, and the a20-index values are 0.9117, 0.8823, 0.9705, and 0.9411, respectively. The results further confirm that among the four models (DT, SVM, RF, and XGB), the RF and XGB models have better predictive capabilities.

Figure 8 is a scatter plot showing the regression prediction results of the four ML models, illustrating the correlation analysis between the predicted bond strength values of the ML models on the training and testing datasets and the actual bond strength values obtained from experiments. As shown in the figure, it can be intuitively observed that for both the training and testing datasets, the bond strength values predicted by the RF and XGB models are closer to the experimental results. Specifically, compared to DT and SVM, the bond strength predictions based on RF and XGB are more concentrated within the ±20% error limits, and the proportion within the ±20% range is noticeably higher. This indicates that the FRP–concrete interfacial bond strength prediction model based on the RF and XGB algorithms has a better predictive performance for the samples. Furthermore, by comparing the predictive performance of the RF and XGB models based on Table 4 and Figure 8, it can be concluded that the RF model has relatively better predictive capabilities among all the models.

Figure 9 shows the error results between the predicted values and experimental values of the FRP–concrete interfacial bond strength prediction models under the DT, SVM, RF, and XGB ML algorithms. It can be visually observed from the figure that for the training dataset, the majority of the predicted values by the models fall within the ±20% error limit. Most of the 133 predicted results are close to the experimental values, with only 5.26%, 9.03%, 2.26%, and 3.76% of the data ratios greater than 1.2 or less than 0.8 for the DT, SVM, RF, and XGB models, respectively. Therefore, this emphasizes that the number of samples with such errors can be neglected, and the prediction results are accurate and reliable. Overall, all four ML algorithms show good bond strength prediction performance, with the RF and XGB models exhibiting better prediction accuracy. For the testing dataset, 91.17%, 88.23%, 97.05%, and 94.11% of the 34 predicted results by the DT, SVM, RF, and XGB models, respectively, are close to the experimental values. It can be seen that the RF and XGB models also demonstrate better prediction accuracy on the testing dataset. This further confirms the effectiveness of the RF and XGB models in predicting the FRP–concrete interfacial bond strength. Additionally, the experimental results indicate that the prediction accuracy during the training phase is higher than the testing phase, which is beneficial for avoiding overfitting.

Figure 10 displays the comparison results between the predicted values and experimental values of the FRP–concrete interfacial bond strength prediction models under the DT, SVM, RF, and XGB ML algorithms. It can be visually observed from the figure that the predicted values of the models align well with the true values. In both the training and testing datasets, the SVM model performs the worst, consistent with the measurements in Table 4. From a statistical perspective, the closer the R² value is to 1, the lower the corresponding error measurement, indicating better model performance. From an engineering perspective, the closer the a20-index is to 1, the stronger the predictive ability of the model. By comparing Table 3 and Figure 10, it can be observed that the ensemble models RF and XGB outperform the individual models DT and SVM in terms of prediction. This suggests that the prediction of bond strength is a complex problem influenced by multiple parameters. Particularly, the RF and XGB models achieve relatively better results in both the training and testing datasets.

Figure 11 presents a comparison of evaluation parameters such as R², RMSE, MAE, MAPE, and the a20-index for each model for the training and testing datasets, using a bar chart format. It can be visually observed from the figure that the performance metrics of the ensemble models (RF and XGB) surpass the other individual models (DT and SVM). Among them, the RF model exhibits the best overall prediction performance, followed closely by the XGB model in terms of prediction accuracy, while the SVM model ranks the lowest in terms of prediction accuracy among the proposed models. For the training dataset, the RF model has R², RMSE, MAE, MAPE, and a20-index values of 0.9695, 4.9608, 3.1583, 0.0519, and 0.9774, respectively. For the testing dataset, the RF model has R2, RMSE, MAE, MAPE, and a20-index values of 0.9621, 4.4779, 3.6252, 0.0732, and 0.9705 respectively.

5.2. Comparison Analysis with Existing Empirical Models

To determine the effectiveness of the ML algorithms studied, the results obtained from the RF and XGB models are compared with the calculations from eight empirical formulas proposed in references [29,30,31,32,33,34,35,36]. Specific information on the computational formulas is presented in

Table 5. Existing prediction models.

Researchers	Note	Formula	Equation
Reference [29]	M1	$P_{\mathrm{u}}=0.64{\beta }_l{k}_w{b}_f\sqrt{f_{ct}{E}_f{t}_f}$	(10)
Reference [30]	M2	$P_{\mathrm{u}}={\tau }_{\max }{b}_f{L}_b$	(11)
Reference [31]	M3	$P_{\mathrm{u}}=0.427b_f{L}_e{\beta }_l{k}_w\sqrt{f_c^{\prime }}$	(12)
Reference [32]	M4	$P_{\mathrm{u}}=b_f{\beta }_l\sqrt{{\displaystyle \frac{{\tau }_{\max }{E}_f{t}_f}{3}}}$	(13)
Reference [33]	M5	$P_{\mathrm{u}}=b_f{\beta }_L\sqrt{2E_f{t}_f{\Gamma }_{fd}}$	(14)
Reference [34]	M6	$P_{\mathrm{u}}=k_m{k}_w{\beta }_L{b}_f\sqrt{2E_f{t}_f{f}_{cm}^{2/3}}$	(15)
Reference [35]	M7	$P_{\mathrm{u}}={\tau }_a{b}_f{L}_e$	(16)
Reference [36]	M8	$P_{\mathrm{u}}=1.1f_c^{\prime 0.2}{b}_f{L}_e$	(17)

Note: P_u—bond strength; L_e—effective bond length; b_f—width of FRP strip; t_f—thickness of FRP strip; E_f—elastic modules of FRP strip; L_b—bond length; b_c—width of concrete specimen; $f_{\mathrm{c}}^{\prime }$—elastic modules of FRP strip; f_cm—mean concrete compressive strength; f_ct—splitting tensile strength; f_ctm—mean concrete tensile strength; G_f—fracture energy; τ_max—maximum shear stress; ${\delta }_f$—slip at τ_max; ${\delta }_1$—ultimate slip; ${\beta }_l$—reduction factor of bond length; ${\kappa }_w$—geometrical factor related to width of bonded strip; ${\Gamma }_{fd}=k_w{k}_G\sqrt{f_{cm}{f}_{ctm}}$.

.

The comparison results between the empirical models and ML models are shown in Figure 12 and

Table 6. Evaluation of different models for p_e/p_u.

Model	Max	Min	Mean	SD	C.V (%)
RF	1.1709	0.6535	0.9895	0.0762	7.71
M1	3.4081	0.4145	1.7508	0.4751	27.14
M2	35.2400	4.6300	10.1311	3.4268	33.82
M3	15.1620	2.6178	6.7444	2.3414	34.72
M4	14.9916	2.7017	7.0281	2.3348	33.22
M5	2.8159	0.3425	1.4466	0.3925	27.14
M6	14.6710	1.3167	6.4836	1.8455	28.46
M7	6.1448	0.7474	3.1567	0.8566	27.14
M8	11.3512	1.2672	1.2672	1.4835	26.22

. Table 6 presents the comparison results of the predicted bond strength to actual bond strength ratio for the eight empirical models and the RF model, including the mean, standard deviation, and coefficient of variation. From Figure 12 and Table 6, it can be observed that although the mean of the predicted-to-experimental value ratio for most empirical models is far away from 1 and empirical models exhibit significant variability in terms of the coefficient of variation. In contrast, the mean value of the predicted/tested value ratios of the RF model is close to 1 and the RF model has a smaller coefficients of variation, which is 7.71%, far less than the minimum of the coefficients of variation of the empirical models, which is 27.14%. This indicates that the proposed models in this study have high computational accuracy and low variability.

Table 7. Performance evaluation metrics of different models.

Model	R2	RMSE	MAE	MAPE
RF	0.9690	4.8664	3.2534	0.0563
M1	−0.2253	33.2769	28.2303	0.4108
M2	−0.7670	60.8606	55.3953	0.8315
M3	−0.7581	60.3494	55.0615	0.8331
M4	−0.7641	60.9050	55.5905	0.8413
M5	0.0522	26.5398	21.2689	0.3136
M6	−0.6049	48.0253	42.2556	0.6282
M7	−0.6070	48.1467	48.1467	0.6303
M8	−0.7438	59.0602	53.8324	0.8084

provides the comparative results of the eight empirical models and the RF model in terms of R², RMSE, MAE, and MAPE. From the table and figure, it can be further observed that the RF model achieves R² values of 0.9690, indicating a good fit with the samples. Among the eight empirical models, the M₅ performs the best. However, the R² value of this empirical model is only 0.0522, much lower than the RF model with 0.9690. The RMSE value of this empirical model is 26.5398, which is 5.45 times higher than the RF model (4.8664). Similarly, the MAE value of this empirical model is more than six times higher than the RF model (3.2534), with a value of 21.2689. The MAPE value of this empirical model is more than three times higher that of the RF model (0.0563), with a value of 0.3136. Therefore, it is evident that the ML models proposed in this chapter exhibit better predictive performance compared to the empirical models.

6. Interpretability Analysis Based on SHAP

6.1. Introduction to SHAP

Lundberg et al. [55] proposed the SHAP method as a feature attribution technique that combines traditional methods with game theory and local explanations, aiming to achieve consistency and local accuracy through expected representations. SHAP constructs an additive explanation model that considers all features as contributors and calculates their respective contribution values. The sum of all feature contribution values represents the ultimate forecast result of the model, and can be represented as Equation (18).

$$g(x^{\prime })={\phi }_0+{\displaystyle \sum _{i=1}^M{\phi }_i{x}_i^{\prime }}=f(x)$$

(18)

In the equation, g(x′) represents the explanation model, which is a function of $\phi$ and $x^{\prime }$. f(x) represents the model being explained, i.e., the ML model for bond strength prediction. ${\phi }_0$ represents the average predicted bond strength for all samples. $\phi$ can be either a positive or a negative value. $x^{\prime }$ takes a value of 0 or 1, where $x^{\prime }$ equals 1 when feature i is observed and 0 otherwise. If feature i is involved in the prediction process, then M denotes the number of characteristics. ${\phi }_i$ represents the SHAP value of the i-th feature, expressed as shown in Formula (19) [56]:

$${\phi }_{\mathrm{i}}={\displaystyle \sum _{S\subseteq N\backslash \mathrm{i}}\frac{\left|S\right|!(M-|S|-1)!}{M!}}[f(S\cup \{i\})-f(S)]$$

(19)

In the equation, N represents the set of all input variables; S represents the set containing the non-zero indices in $x^{\prime }$.

6.2. Feature Parameter Impact Analysis

ML algorithms are limited in their practical applications due to their lack of interpretability. Understanding how the model predicts the bond performance between FRP and concrete based on the feature values is crucial for assessing its predictive reliability. In this paper, the SHAP method is used to explain the model’s prediction results. Based on the established model, the prediction contribution analysis based on Shapley values can be divided into two levels. At the global level, the distribution of Shapley values can be used to explain the patterns and correlations between features. At the local level, it can provide quantitative contributions of each feature to the prediction of each sample. Figure 13 and Figure 14 show the global-level explanations of the RF model.

Figure 13 illustrates the feature importance of the nine input parameters. From the graph, it can be visually observed that, overall, width of the FRP strip (X₂) and thickness of the FRP strip (X₃) have the most significant impact on the interfacial bond strength between FRP and concrete, while the influence of section width of the member (X₆) and elastic modulus of the FRP strip (X₄) is relatively small. The interfacial bond strength between FRP and concrete (X₁) is the third most important parameter, with an average absolute SHAP value approximately 74% that of width of the FRP strip (X₂). Next are the depth of grooves (X₈), distance of the groove edges (X₉), compressive strength of concrete (X₅), and width of grooves (X₇), with their importance being approximately 66%, 47%, 34%, and 13% that of the key parameter, the width of the FRP strip (X₂), respectively. In comparison, the importance of the elastic modulus of the FRP strip (X₄) and section width of the member (X₆) is approximately 7% that of the width of the FRP strip (X₂).

Figure 14 displays the SHAP summary plot for the nine input parameters. The scatter points in the graph represent the distribution of SHAP values for individual features, and the color gradient from blue (low) to red (high) indicates the variation in feature values. The SHAP values indicate the positive or negative correlation between the feature variables and the output result. Therefore, Figure 14 not only showcases the importance of the input parameters but also reveals the impact patterns of each individual feature parameter on the bond strength. For example, width of the FRP strip (X₂) has the most significant influence on the bond strength, as the SHAP value increases with the increase in X₂, corresponding to an increase in bond strength. Similarly, for feature parameters such as bond length (X₁), depth of grooves (X₈), and compressive strength of the concrete (X₅), there is a tendency for the bond strength to increase with the increase in their values. As the feature parameter values increase, the SHAP values also increase, indicating a stronger positive effect.

The SHAP method can also quantify the impact of each individual input variable on the output result, as shown in Figure 15. The graph illustrates the change in bond strength with the variation in input parameters X₁ to X₉, represented by the corresponding SHAP values. For example, the SHAP values increase with the increase in feature values X₂, X₁, X₈, and X₅, indicating a positive correlation with the bond strength. The impact trends of these nine parameters on the bond strength are consistent with Figure 14. Specifically, as shown in Figure 15a, an increase in X₂ leads to an increase in bond strength, and the fitted curve approximates a linear function. When X₂ has relatively small values, the SHAP value is negative, indicating a negative impact on the bond strength. As X₂ increases beyond a certain value, the SHAP value becomes positive, indicating an increasingly positive influence on the bond strength. Similarly, Figure 15b–i display the changing trends of the Shapley values for the remaining feature parameters. These results suggest that, as SHAP provides detailed insights into the influence patterns of different features on the interfacial bond strength between FRP and concrete, it can be used to explore the design of new testing experiments to detect the interfacial bond behavior of FRP and concrete in the most optimal way.

The SHAP method provides both global and local explanations, allowing interpretation of the entire dataset as well as providing explanations for each individual sample. As shown in Figure 16, SHAP decomposes the predicted value into the sum of feature contributions. The baseline value is the average predicted value of the RF model for the database, which is 65.37 kN. Red indicates a positive impact of a feature on the model’s output, while blue indicates a negative impact. The length of the color bar represents the degree of positive or negative correlation. As shown in the visualization results of Figure 16, considering the baseline values and SHAP values for all feature parameters, the predicted result for sample A-1 is 56.43 kN, sample A-2 is 63.57 kN, and sample A-3 is 69.55 kN. Among the input variables, X₁, X₂, and X₈ have a favorable impact on the results, while X₃, X₅, X₆, X₇, and X₉ have an unfavorable impact on the bond strength in some samples. This aligns with the experimental observations and further demonstrates the reliability of ML methods in predicting the interfacial bond strength between FRP and concrete.

7. Conclusions

FRP strengthening of concrete structures is closely related to the mechanical performance and bond strength between FRP and concrete in service environments. Accurately predicting the bond strength at the interface between rusted FRP and concrete is essential in practical engineering environments. This paper presents a predictive model for the bond strength at the interface between rusted FRP and concrete using a hybrid ML algorithm based on SHAP values. The aim is to utilize ML algorithms (DT, SVM, RF, and XGB) to accurately predict the bond strength at the interface between FRP and concrete based on given input parameters. A bond strength database consisting of 167 experimental data points is collected from the literature to train and test the ML models, considering nine factors. Additionally, empirical or semi-empirical formulas for predicting the bond strength at the interface between FRP and concrete, proposed by previous researchers, are evaluated using the collected database. Furthermore, a comparison is made between the formula-based models and the ML models. The SHAP method, proposed in recent years, is utilized to reveal how specific variables influence the bond strength at the interface between FRP and concrete. Several important conclusions can be drawn:

(1)

By utilizing the database to train ML algorithms, an intelligent predictive model for the bond strength at the interface between FRP and concrete is established. The results demonstrate that the four ML models perform well, with the R2, RMSE, MAE, MAPE, and a20-index ranges for the training set being 0.9265–0.9695, 4.9608–7.79897, 3.1583–4.0759, 0.0519–0.0777, and 0.9097–0.9774, respectively. For the testing set, the ranges are 0.8190–0.9621, 4.4779–8.7680, 3.6252–6.1376, 1.0732–0.1013, and 0.8823–0.9705, respectively.

(2)

Among the selected four representative classical ML algorithms, namely DT, SVM, RF, and XGB, the performance of ensemble models (RF and XGB) is superior to that of individual models (DT and SVM), showing higher accuracy. Among all the models, the RF model performs the best.

(3)

Compared to the traditional empirical or semi-empirical models for predicting the bond strength at the interface between FRP and concrete proposed in the literature, the ML models consider the correlation between influencing parameters and can adapt to different experimental results, exhibiting higher accuracy and generality in bond strength prediction.

(4)

The SHAP method provides explanations for the prediction results from a global and local perspective. The results show that the width of the FRP bar (X₂) and the thickness of the FRP bar (X₃) have the most significant influence on the bond strength at the interface between FRP and concrete, while the width of the member section (X₆) and the elastic modulus of the FRP bar (X₄) have relatively smaller influences.

(5)

The SHAP method quantifies the influence of each input parameter on the prediction results. The SHAP feature dependence plots reveal how each individual feature affects the bond strength prediction. This provides useful ranges of values for individual features and their different combinations in influencing the bond strength, facilitating the development of improved bond strength models.