Predicting the strength of fiber reinforced polymer materials externally bonded to masonry using artificial intelligent techniques

Abstract

Fiber reinforced polymer (FRP) retrofits are widely used to strengthen structures due to their advantages such as high strength-to-weight ratio and durability. However, the bond strength between FRP and masonry is crucial for the success of these retrofits. Limited data exists on the shear bond between FRP composites and masonry substrates, necessitating the development of accurate prediction models. This study aimed to create machine learning models based on 1583 tests from 56 different experiments on FRP-masonry bond strength. The researchers identified key factors influencing failure load and developed machine learning models using three algorithms. The proposed models outperformed an existing model with up to 97% accuracy in predicting shear bond strength. These findings have significant implications for designing safer and more effective FRP retrofits in masonry structures. The study also used Sobol sensitivity analysis and SHapley Additive exPlanations (SHAP) analysis to understand the machine learning models, identifying key input features and their importance in driving predictions. This enhances model transparency and reliability for practical use.

Graphical abstract

Keywords

fiber reinforced polymer retrofits / bond strength / masonry substrate / shear pull out tests / machine learning model

1 Introduction

Unreinforced masonry structures are highly weak to seismic forces, often resulting in significant damage or complete collapse during earthquakes [1]. As a result, there is a growing demand for retrofit solutions that can enhance the structural capacity of existing masonry buildings. One such solution gaining popularity is the use of fiber reinforced polymer (FRP) composites. These composites offer favorable properties such as low weight, high strength-to-weight ratio, and ease of installation, making them an attractive choice for strengthening masonry structures [2]. The most commonly used FRPs for retrofitting structures include carbon fiber reinforced polymer (CFRP), glass fiber reinforced polymer (GFRP), basalt fiber reinforced polymer (BFRP), and aramid fiber reinforced polymer (AFRP) [2].

Evaluating the performance of FRP in structural engineering is crucial, and understanding the bond mechanism between FRP and masonry is a key aspect of assessing the effectiveness of FRP strengthening. The bond interface between FRP and masonry units is often a weak point in tensile strengthened masonry structures, and debonding at this interface is a critical failure mode for externally bonded reinforcement (EBR) systems using fibers [3,4]. The bond mechanism’s complexity stems from the mechanical characteristics of masonry blocks, mortar joints, adhesive materials, and FRP reinforcement.

The use of FRP composites for retrofitting masonry structures has gained significant popularity in recent years, particularly in Italy, where there is a considerable amount of masonry buildings and a rich heritage of monuments [5]. The application of FRP laminates as external bonded materials for retrofitting masonry buildings has been extensively studied due to the favorable properties of this technology, such as durability, reversibility, and minimal mass and stiffness increment [6].

Research has shown that FRP plating can boost a wall’s strength against in plane shear and against out of plane flexure [7]. The ability to transfer shear load effectively across the bond between FRP and masonry is crucial for the successful use of FRP strengthening. Pull out tests have been widely employed to investigate bond behavior and provide insights into the shear deformation.

While the literature includes numerous experimental and theoretical studies analyzing the bond between individual units and various types of FRP materials, there are relatively fewer studies focusing on the bond between masonry prisms and FRP composite materials. Various studies have explored the use of FRP for seismic retrofitting of masonry structures [8–15]. Others have concentrated on strengthening masonry arches and columns using FRP composites [16–28]. Additionally, specific attention has been given to investigating different aspects of bond behavior between FRP composites and masonry substrates, with a particular emphasis on the influence of mortar joints [29–35].

Latest research has confirmed a clear correlation between the microstructural and chemical−physical characteristics of masonry units and mortar, and the bonding performance of FRP composite materials. For instance, a recent study [36] examined the capability of a polymeric matrix to penetrate the substrate, which is evidently affected by the masonry surface properties. Furthermore, the surface roughness of the unit can be improved through sandblasting pressure treatment, leading to enhanced resin penetration and approximately a 13% increase in debonding load compared to untreated units.

The utilization of FRP strengthening techniques in masonry structures has gained significant recognition due to its effectiveness in enhancing structural performance. However, the development of reliable design formulas that accurately predict the bond strength between FRP and masonry is essential to ensure the long-term durability and safety of these reinforced structures. Conventional design approaches often rely on simplified empirical formulas that may overlook the intricate interactions between FRP and masonry materials. This limitation can result in inaccurate predictions and potentially compromise the safety of the designs. To tackle this challenge, this study investigates the potential of utilizing machine learning (ML) methods for predicting the bond strength of FRP in masonry structures. ML algorithms have excelled in diverse engineering domains, supported by compelling research [37–39], offering several advantages over traditional data-fitting methods. These advantages include the ability of ML models to continuously learn and improve their predictive accuracy as they are exposed to more data, their robustness against noisy or incomplete data, making them suitable for real-world applications where data quality may not be perfect, and their ability to capture complex, nonlinear relationships between input variables and the target variable. By leveraging an extensive experimental database comprising shear pull-out tests on FRP materials bonded to masonry, this study aims to develop and evaluate ML models for predicting the bond capacity. The study employs three different algorithms, namely the CatBoost Regressor, Extra Trees Regressor, and XGBRegressor, to explore their effectiveness in this specific context.

2 Shear bond strength of fiber reinforced polymer externally bonded reinforcement over a brittle support

The Italian Guideline [40] propose the subsequent expressions for estimating the design load value of debonding in masonry members that are externally reinforced with FRP composites:

(1)F_{\mathrm{d}\mathrm{e}\mathrm{b},\mathrm{t}\mathrm{h}}=P_{\mathrm{m}\mathrm{a}\mathrm{x}}=\frac{b_{\mathrm{p}}}{{\gamma }_{\mathrm{f},\mathrm{d}}}\sqrt{2E_{\mathrm{p}}{t}_{\mathrm{p}}{\mathrm{\Gamma }}_{\mathrm{F}\mathrm{d}}},

(2){\mathrm{\Gamma }}_{\mathrm{F}\mathrm{d}}=\frac{k_{\mathrm{G}\mathrm{k}}{k}_{\mathrm{b}}}{FC}\sqrt{f_{\mathrm{u}\mathrm{c}}\cdot f_{\mathrm{t}\mathrm{m}}}.

The calculation involves determining the mean value of the tensile strength for each individual masonry block as f_{\mathrm{t}\mathrm{m}} = 0.1f_{\mathrm{u}\mathrm{c}}. The value of k_{\mathrm{b}} is obtained from Eq. (3). FC accounts for material knowledge confidence. k_{\mathrm{G}\mathrm{k}} is the 0.05 percentile corrective factor from a statistical regression of bond test data. Specifically, k_{\mathrm{G}\mathrm{k}} = 0.031 mm for clay brick, 0.048 mm for tuff, and 0.012 mm for limestone masonry (including calcarenite and Lecce stone). {\gamma }_{\mathrm{f},\mathrm{d}} = 1.2 accounts for model uncertainty. Average k_{\mathrm{G}\mathrm{k}} values are 0.093 mm for clay brick, 0.157 mm for tuff, and 0.022 mm for limestone masonry. These k_{\mathrm{G}\mathrm{k}} values came from FRP-masonry bond tests on single bricks or natural stones without plaster. The Young’s modulus, and thickness of the FRP reinforcement is represented by E_{\mathrm{p}}, and t_{\mathrm{p}}, respectively. The width of the reinforcement is b_{\mathrm{p}}. The width factor k_{\mathrm{b}} is calculated according to Eq. (3) as:

(3)k_{\mathrm{b}}=\sqrt{\frac{3-{\displaystyle \frac{b_{\mathrm{p}}}{b_{\mathrm{s}}}}}{1+{\displaystyle \frac{b_{\mathrm{p}}}{b_{\mathrm{s}}}}}},

where b_{\mathrm{s}} is the width of the substrate. Evaluating k_{\mathrm{b}} is important for reliably accounting for the non-uniform distribution of bond shear stresses across the element width in design considerations. These effects depend not only on the width ratio b_{\mathrm{p}}/b_{\mathrm{s}}, but also the absolute b_{\mathrm{p}} value, as stated for FRP-masonry joints by Ref. [41]. Drawing upon existing literature on FRP-concrete joints [42,43], debonding strength increases with b_{\mathrm{p}} when b_{\mathrm{p}}/b_{\mathrm{s}} is sufficiently low under 0.4. It is anticipated that there will be an enhancement in debonding strength as there is an increase in b_{\mathrm{p}}.

3 Experimental database

To ensure that the best possible performance of the ML models was achieved, the study employed an extensive database, which is thoroughly described in this section. The database’s intricacies and features are extensively discussed, along with an in-depth analysis of the input parameters utilized in the study.

3.1 Collected database from literature

A scientific database consisting of 1583 shear pull out tests on masonry samples was collected by Vaculik et al. [44] by conducting a thorough examination of the current literature on 56 published studies [45–98]. The collected data was meticulously analyzed and organized in a structured format for ease of use and accessibility.

The database contains an assortment of substrate materials that have undergone testing, including clay brick, limestone, tuff, concrete block, calcium silicate brick, sandstone, and mortar specimens. These materials have been evaluated for their mechanical properties, specifically tensile and compressive strength. Tensile tests are further classified into direct, flexural, splitting, or unspecified tensile tests. Additionally, the database encompasses retrofits using externally bonded (EB) and near-surface-mounted (NSM) techniques. Various reinforcement shapes have been utilized in these retrofits, such as EB sheets, pre-formed rectangular strips, and round bars. Composite materials, including CFRP, GFRP, BFRP, SRP, AFRP, and natural flax FRP, have been employed in conjunction with epoxy, polyurethane, or cementitious adhesive. The tests conducted for this database include both single-lap and double-lap arrangements. The double-lap arrangement has been further subdivided into single block and double block variants. While the majority of tests utilize monotonic loading, some also employ cyclic loading.

The database comprises tests performed under standard conditions in which the FRP plate is directly bonded to the masonry substrate and subsequently subjected to quasi-static loading. Vaculik et al. [44] excluded from the database tests performed under situations that deviate from the standard, such as tests with additional anchorage, different bonding, non-quasi-static loading conditions, or different plate materials. However, control specimens from these studies are still included in the database. Tests that do not report the failure load for interfacial debonding are also excluded. Duplicate tests are excluded from the database. However, Vaculik et al. [44] included tests conducted under special conditions, such as confining pre-compression, curved specimens, and repaired specimens that were re-tested. For a more detailed and comprehensive analysis of the experimental database, it is recommended to consult the database [44].

3.2 Processed database

The success of ML models heavily depends on the quality of data used for training. Therefore, having a well-structured database is crucial for the model to accurately predict outcomes. In the case of predicting the strength of FRP materials EB to masonry using artificial intelligent techniques, experimental pull out tests have been extensively collected from the literature and were used to establish input parameters [44]. Successfully trained ML models can predict the failure load that an FRP material EB to masonry can withstand (P_{\mathrm{m}\mathrm{a}\mathrm{x}}) with minimum errors.

The original collected database for the FRP pull out tests included two types of retrofit configurations: NSM and EB. However, NSM was removed from the database as it was out of scope for this paper. The researchers removed any experimental tests with missing data. The database was cleaned by removing specimens tested under cyclic loading. The remaining data set includes a limited number of tests with FRP laminates passing through perpendicular joints, specimens subjected to precompression, and specimens that were tested and repaired. Most of the adhesive used in these tests was epoxy, with a few tests using polyurethane. The original collected database had many parameters, which made it challenging to determine which inputs were essential for accurate predictions of P_{\mathrm{m}\mathrm{a}\mathrm{x}}. To address this, researchers performed trial and error tests with engineering judgment, and the database could be reduced to only seven essential input parameters. The filtered database led to a table with eight columns for the seven essential parameters and one outcome, the failure load that an FRP material EB to masonry can withstand. The table included 1099 rows for pulling out tests.

The seven essential input parameters selected for the database were: compressive strength of substrate unit, f_{\mathrm{u}\mathrm{c}} (in MPa), width of prism, b_{\mathrm{s}} (in mm), FRP ultimate tensile stress, f_{\mathrm{p}\mathrm{u}} (in MPa), FRP modulus of elasticity, E_{\mathrm{p}} (in MPa), width of FRP plate, b_{\mathrm{p}} (in mm), thickness of FRP plate, t_{\mathrm{p}} (in mm), and plate bonded length along FRP axis, L_{\mathrm{b}} (in mm). Most of these parameters are related to the mechanical properties of the FRP plates.

This restructuring of the pull out tests for FRP materials EB to masonry database highlights the importance of selecting the right parameters for training ML models. By doing so, the models can accurately predict the failure load that an FRP material EB to masonry can withstand, which is crucial for the design of safe and effective structures. Tab.1 provides an overview of the range of variations for the key parameters within the reorganized database.

Tab.1 shows a wide range of values observed for each parameter. The compressive strength of the substrate unit, represented by f_{\mathrm{u}\mathrm{c}}, ranged from a minimum of 2.10 MPa to a maximum of 86.5 MPa with a mean value of 20.17 MPa. The substrate width, b_{\mathrm{s}}, ranged from a minimum of 51 mm to a maximum of 910 mm with a mean value of 145.18 mm. The ultimate tensile strength of the FRP material, represented by f_{\mathrm{p}\mathrm{u}}, ranged from a minimum of 542 MPa to a maximum of 4840 MPa with a mean value of 2390.63 MPa. The elastic modulus of the FRP material, represented by E_{\mathrm{p}}, ranged from a minimum of 45000 MPa to a maximum of 330000 MPa with a mean value of 157968.88 MPa. The bonded length of the plate along the FRP axis, represented by L_{\mathrm{p}}, ranged from a minimum of 10 mm to a maximum of 740 mm with a mean value of 169.49 mm. Finally, the maximum load values for the FRP materials, represented by P_{\mathrm{m}\mathrm{a}\mathrm{x}}, ranged from a minimum of 0.75 kN to a maximum of 79 kN with a mean value of 8.13 kN.

The percentiles of the parameters revealed that there was a significant variation in values, with wide ranges observed for each parameter. For example, the 25th percentile of f_{\mathrm{u}\mathrm{c}} was 17.4 MPa, while the 75th percentile was 20.2 MPa, indicating a considerable spread of compressive strength values across the experiments. Similarly, the 25th percentile of P_{\mathrm{m}\mathrm{a}\mathrm{x}} was 4.52 kN, while the 75th percentile was 9.07 kN, indicating a broad range of maximum load values for the FRP materials. The majority of the experiments also used a substrate width (b_{\mathrm{s}}) of 120 mm and a bonded length of plate (L_{\mathrm{p}}) along the FRP axis of 143–200 mm, as indicated by the percentiles. Overall, the results highlight the need for careful consideration of the range of values observed for each parameter in FRP experiments to ensure accurate and reliable outcomes.

The distribution of the input parameters selected for this study has been analyzed. The variability of both the input and targeted outputs of the filtered data set has been evaluated through histogram plots. Additionally, the random distribution and variance of the following values have been included: compressive strength of the substrate unit, width of the prism, modulus of elasticity and ultimate tensile stress of the FRP, width of the FRP plate, thickness of the FRP plate, bonded length of the plate along the FRP axis, and the output of maximum force in Fig.1.

To develop an accurate predictive model for pull out tests, it is crucial to analyze the correlations between input and output features in the data set. One commonly used method for calculating the correlation coefficient between features is Pearson’s method, which provides a measurable relationship between various characteristics. In statistical analysis, the coefficient is employed to statistically quantify the linear relationship between multiple variables. This coefficient is bounded within the range of −1 to 1, where a value of 0 represents the absence of correlation between the two specific characteristics being examined. Pearson’s method can calculate the correlation coefficient using the following formula:

(4)\begin{array}{rl}& \mathrm{P}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{n}\text{’}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{t}\\ =& \frac{{\sigma }_{xy}}{{\sigma }_x\times {\sigma }_x}=\frac{{\sum }_i^n(x_i-\overline{x})\times (y_i-\overline{y})}{\sqrt{{\sum }_i^n{\left(x_i-\overline{x}\right)}^2}\times \sqrt{{\sum }_i^n{\left(y_i-\overline{y}\right)}^2}}.\end{array}

In the given equation, y_i denotes the experimental values, \widehat{y_{\mathrm{i}}} represents the regression values, and \overline{y} signifies the average of the simulation values. Here, x and y are two distinct features, while the overhead bar and subscript i indicate the average value and the ith observation, respectively.

The heatmap is a useful tool for visualizing the relationships between input and output features, as shown in Fig.2. which can provide valuable insights for developing an accurate predictive model for pull out tests.

The analysis of the correlation matrix can reveal strong correlations between certain features. For example, there is a strong correlation (0.78) between the bonded length of the plate along the FRP axis (L_{\mathrm{p}}) and the failure load that an FRP material EB to masonry can withstand (P_{\mathrm{m}\mathrm{a}\mathrm{x}}), compared to 0.76 for substrate width (b_{\mathrm{s}}). Additionally, the width of the FRP plate (b_{\mathrm{p}}) showed a strong correlation with 0.71. This finding is not surprising since a larger contact area between the masonry unit and FRP material will lead to more shear bond.

Scatter matrix plots serve as a valuable tool for visualizing data and identifying linear correlations among multiple variables. These plots offer insights into the data and aid in the selection of appropriate ML models. By visually analyzing the correlations between variables, it is possible to identify any similarities in their relationships with the data. Fig.3 presents a scatter matrix plot of the filtered data set.

The scatter plot indicates a moderate positive correlation between the compressive strength of the substrate and the failure load. Furthermore, the width of the substrate parameter displays a positive correlation with the failure load. Specifically, as the width of the substrate increases, there is a corresponding increase in the failure load. On the other hand, the FRP tensile strength and modulus do not exhibit a clear positive correlation with the failure load, considering the scattered nature of the data. However, it is noteworthy that the FRP width and bonded length demonstrate a more distinct positive relationship with the failure load. This implies that wider and longer FRP plates have the ability to distribute stresses more effectively, resulting in higher debonding loads. In contrast, the FRP thickness does not show a clear correlation with the maximum load. Overall, the observed parameters demonstrate a nonlinear correlation, suggesting that traditional analytical approaches may have limitations in fully capturing their intricate relationships. However, this presents a promising opportunity for ML algorithms, which can effectively analyze the interplay between the various parameters and uncover hidden patterns or subtle dependencies that might not clearly apparent.

4 Artificial intelligent techniques

ML systems are designed to learn from observations by using data to make decisions based on patterns they find. These systems are specifically programmed to improve independently. The most critical step in an ML algorithm is training, during which the model makes predictions and identifies patterns in prepared data. By learning from data, the model can accomplish the task and improve over time. To conduct this study, training data was randomly selected for 80% of data set, while testing data was randomly selected for the remaining 20%. In this section, a concise introduction is provided to the principle behind the three ML algorithms employed in this study. The algorithms, which consist of the CatBoost regressor, Extra Tree Regression (ETR), and XGBRegressor, were implemented using the Python programming language. To accomplish hyperparameter tuning for the ML models, the grid search method was utilized. The process entailed generating a comprehensive list of values for each hyperparameter and subsequently assessing the model’s performance for every possible combination of these values.

4.1 CatBoost regressor

CatBoost is a recently developed gradient boosting technology [99] that has proven to be a powerful ML technique. Its effectiveness has led to its application in a variety of fields, including driving style recognition [100], and diabetes prediction [101]. As well, CatBoost is gradually being utilized in the field of finance for fraud detection [102,103].

CatBoost is a gradient boosting algorithm specifically designed for handling categorical features in an unbiased manner. It offers various solutions for dealing with categorical features, optimizing and applying them during the tree splitting process rather than the pre-processing stage. When confronted with a limited number of classes, a classifier in CatBoost utilizes a technique called “one hot encoding” to convert categorical features into numeric ones. This transformation is based on the frequency of occurrences for each category. In the case of composite features, the classes are replaced with the average target value. To prevent overfitting, the classifier calculates the average risk for each sample by considering the target values.

CatBoost is a highly efficient algorithm used for training boosting models based on random forests (RFs). It incorporates a distinctive training mechanism known as Minimal Variance Sampling , which is a variant of the sampling technique used for regularization, but with weights assigned to the samples. CatBoost encompasses various parameters required for constructing each decision tree, configuring the RF model, and specifying the hyperparameters essential for constructing the boosting method.

In the training process, CatBoost optimizes the hyper-parameters for the boosting model. Once the model is trained, it undergoes validation, and the parameters used during training are saved. The saved parameters are significant in determining the threshold parameters for constructing the RF model. Additionally, the hyperparameters of the CatBoost algorithm are stored and continuously optimized as additional data points become accessible.

4.2 Extra trees regressor

Geurts et al. [104] introduced the ETR method. The RF model was expanded with the introduction of ETR, an ensemble ML technique. Using the traditional top-down approach, ETR creates unpruned regression or decision trees. In contrast to other models, the RF model employs bootstrapping and bagging techniques to perform regression. In this approach, each decision tree is created using a sample from the training data set selected randomly during the bootstrapping phase. Subsequently, the bagging step is utilized to divide the nodes within the decision tree. The RF model consists of multiple decision trees, with the predicted outcome derived from the tree of results. Additionally, a uniform independent distribution vector is given prior to extending the tree.

The RF model, as described by Breiman [105], consists of a series of decision trees. In this model, the predicting tree represents the tree of results, and the predicting vector is the uniform independent distribution vector assigned prior to expanding the tree. The construction of a forest involves combining and averaging all the trees. The ETR approach utilizes cutting points to divide nodes effectively. It also aims to minimize bias by training the trees on the entire learning sample. The ETR split process is governed by two parameters: the number of randomly sampled features in each node, and by the minimum number of features required to separate each node. These parameters determine the strength of attribute selection and the level of average output noise. Adjusting the parameters appropriately improves the model’s precision and reduces the risk of overfitting.

4.3 XGBRegressor

XGBoost, also known as Extreme Gradient Boosting, is a ML model based on boosting trees. It was initially proposed by Geurts et al. [104] and has since undergone continuous optimization and improvement by various researchers. The conventional Boosting Tree model relies solely on first-order derivative information, which poses challenges for distributed training due to its use of residuals from previous trees. In contrast, XGBoost utilizes a second-order Taylor expansion on the loss function, allowing it to leverage the CPU’s multithreading capabilities for parallel computing. Additionally, XGBoost employs diverse techniques to mitigate overfitting, making it a more precise alternative to simplistic statistical approaches like using mean, median, or mode to handle missing values.

XGBRegressor is a popular ML algorithm used for regression tasks, and it is an implementation of the gradient boosting decision tree algorithm, similar to LightGBM and CatBoost regressor. By using an ensemble of decision trees, where each tree learns from the errors of the previous tree in the sequence, XGBRegressor fits a regression model to the data and adds additional models to correct the errors of the previous models. This approach has several advantages over other gradient boosting algorithms, including scalability, the ability to handle large data sets with high dimensionality, a regularization term that helps prevent overfitting and improves generalization performance, and parallel processing, making it faster than other gradient boosting algorithms. XGBRegressor also offers several hyperparameters that can be tuned to optimize model performance, such as the learning rate, number of trees, and maximum depth of each tree. Overall, XGBRegressor is a powerful ML algorithm for regression tasks that offers great performance and scalability, with the added benefit of being able to optimize hyperparameters for improved accuracy.

5 Results and discussion

In the preceding section, the utilization of ML methods was discussed for the determination of the maximum load capacity (P_{\mathrm{m}\mathrm{a}\mathrm{x}}) of FRP material EB to masonry. The author employed ML models to convert the material and geometric properties of the FRP-masonry composite into seven input parameters, as elaborated in Subsection ‎3.2. The ML codes employed for the models discussed in Section ‎4 were developed using a Python package that is freely available and open source. To establish the predictive model, 80% of the compiled data was allocated as training data, while the remaining 20% served as test data for assessing the model’s performance. The partitioning of the data set into training and test sets was conducted randomly, ensuring that the model’s performance on the test set reflected its ability to generalize to unseen data. To summarize, the ML models were developed following the methods outlined in Section ‎4, based on 80% of the compiled data. Evaluating accuracy is crucial for assessing the performance of ML models during training.

During model optimization, the coefficient of determination (R^2) was employed, which is calculated using the formula:

(5)R^2=1-\frac{{\sum }_{i=1}^n{\left(y_i-\widehat{y_i}\right)}^2}{{\sum }_{i=1}^n{\left(y_i-\overline{y}\right)}^2}.

The coefficient of determination, ranging from 0 to 1, provides a measure of how well the regression model fits the simulation data. An R^2 value of 0 indicates a poor fit, suggesting that the regression model does not accurately represent the replication data. Conversely, an R^2 value of 1 signifies a perfect fit, indicating that the regression model precisely captures the simulation data.

To evaluate the predictive performance of a ML model, accuracy assessment is necessary. Various metrics are usually employed to estimate the accuracy of regression models, offering valuable insights into the ML model’s ability to predict output values. By comparing these metrics across different models, it becomes possible to identify the most accurate model. The metrics commonly used for evaluating the performance of regression models include Mean Squared Error (MSE), Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and R^2.

MAE is calculated by taking the average of the absolute difference between the predicted and actual values over the data set, as shown in Eq. (6). MSE is calculated by squaring the average difference between the predicted and actual values over the data set, as shown in Eq. (7). RMSE is obtained by taking the square root of MSE, as shown in Eq. (8).

(6)MAE=\frac{1}{n}\times \sum _{i=1}^n\left|\left(y_i-\widehat{y_{\mathrm{i}}}\right)\right|,

(7)MSE=\frac{1}{n}\times \sum _{i=1}^n{\left(y_i-\widehat{y_{\mathrm{i}}}\right)}^2,

(8)RMSE=\frac{1}{n}\times \sqrt{\sum _{i=1}^n{\left(y_i-\widehat{y_{\mathrm{i}}}\right)}^2},

where y is predicted value of y and \hat{y} is mean value of y. Tab.2 provides a comprehensive analysis of three different regression models: CatBoost regressor, Extra Trees Regressor, and XGBRegressor. The table presents statistical indicators such as R², MAE, MSE, and RMSE, which are used to evaluate the models’ performance on both training and test data sets. These indicators allow us to assess the models’ predictive capabilities and compare their effectiveness.

The CatBoost Regressor exhibits a remarkable performance on the training data, achieving an R² value of 0.966, demonstrating that approximately 96.6% of the variance in the target variable can be explained by this model. Additionally, it demonstrates a low MAE of 0.939, implying that the average difference between the predicted and actual values is relatively small. MSE and RMSE values for the training data are 2.188 and 1.479, respectively, further indicating the model’s ability to capture the data patterns accurately. On the test data, the CatBoost Regressor still performs well, although there is a slight decrease in the R² value to 0.898. The MAE and RMSE values also remain relatively low at 1.073 and 1.934, respectively. However, the MSE value increases to 3.74, indicating a slightly higher average squared difference between the predicted and actual values compared to the training data.

The Extra Trees Regressor demonstrates a comparable performance to the CatBoost Regressor. It achieves a high R² value of 0.968 on the training data. Moreover, it exhibits a lower MAE value of 0.848 compared to the CatBoost Regressor, suggesting a slightly better accuracy in predicting the target variable. The MSE and RMSE values for the training data are 2.098 and 1.449, respectively, further emphasizing the model’s effectiveness. However, on the test data, the Extra Trees Regressor shows a slight decrease in performance compared to the CatBoost Regressor. It achieves an R² value of 0.873. The MAE and RMSE values increase to 1.149 and 2.159, respectively. Notably, the MSE value significantly rises to 4.662, indicating a higher average squared difference between the predicted and actual values compared to both the CatBoost Regressor and the XGBRegressor.

The XGBRegressor also demonstrates strong predictive capabilities, similar to the CatBoost and Extra Trees models. With an R² value of 0.966 on the training data, it explains approximately 96.6% of the variance, matching the performance of the CatBoost Regressor. The model achieves a low MAE value of 0.932, suggesting accurate predictions. The MSE and RMSE values for the training data are 2.187 and 1.479, respectively, which are comparable to the other models. On the test data, the XGBRegressor performs admirably, with an R² value of 0.903, explaining approximately 90.3% of the variance. The MAE and RMSE values remain low at 1.076 and 1.89, respectively. However, the MSE value decreases to 3.571, indicating a slightly lower average squared difference compared to the CatBoost Regressor but higher than the Extra Trees Regressor.

Fig.4 illustrates the coefficient of determination for all models, highlighting the disparity between the predicted outcomes of the ML models and the test data.

Fig.5 exhibits MAE, MSE, and RMSE values for all the models, illustrating the contrasting results between the predicted outcomes of the ML regression model and the actual observed data.

The figures provided, namely Fig.6–Fig.8, showcase the comparisons between the predicted values generated by each model and the corresponding experimental data.

In conclusion, all three regression models, namely CatBoost Regressor, Extra Trees Regressor, and XGBRegressor, demonstrate strong predictive accuracy on both the training and test data. The CatBoost Regressor and XGBRegressor exhibit similar performance, while the Extra Trees Regressor slightly lags behind in terms of predictive accuracy.

6 Comparison between the proposed artificial intelligent techniques and existing formulas

The Italian Guideline (CNR-DT 200 R1) [40] provides equations for estimating the design debonding load of masonry elements externally reinforced with FRP materials. The determination of average tensile strength for individual blocks is given by f_{\mathrm{t}\mathrm{m}} = 0.1f_{\mathrm{u}\mathrm{c}}. The value of k_{\mathrm{b}} is obtained from Eq. (3), where the material knowledge confidence factor (FC) is taken as 1 for this section. The Italian Guideline proposes two approaches to determine the k_{\mathrm{G}\mathrm{k}} parameter: 1) as the 0.05% corrective factor from statistical regression of bond test data, and 2) as average k_{\mathrm{G}\mathrm{k}} values. Tab.3 compares these k_{\mathrm{G}\mathrm{k}} values recommended by CNR-DT 200 R1 for different FRP-reinforced masonry materials.

The experimental database collected from literature includes clay brick, limestone, tuff, concrete block, calcium silicate brick, sandstone, and mortar specimens. However, some data tests were removed from the compilation due to a lack of recommended values for k_{\mathrm{G}\mathrm{k}} in the Italian code. Specifically, a total of 45 tests were removed, comprising of 20 calcium silicate brick tests, 20 concrete block tests, 3 mortar specimen tests, and 2 sandstone tests.

The precision of the strength equations described in Section 2 has been evaluated and visualized in Fig.9 for both the 0.05% and average values according to CNR-DT 200 R1 for the entire database after removing the 45 tests mentioned above. Additionally, the ability of CNR-DT 200 R1 is examined in Fig.10 by splitting the database based on substrate materials such as clay brick, tuff, and limestone specimens. Statistical values summarizing the results are reported in Tab.4 for the 0.05% values and Tab.5 for the average values as per CNR-DT 200 R1.

Based on the statistical metrics provided in Tab.4 and Tab.5, the predictive performance of the ML models can be further compared to the CNR-DT 200 R1 design equation across different substrate materials.

For the 0.05% predictions (Tab.4), all ML algorithms achieved superior R^2 values ranging from 0.87 to 0.90 compared to the CNR-DT 200 R1 (0.36–0.74) depending on the substrate. They also exhibited substantially lower MAE, MSE and RMSE, indicating reductions in prediction error relative to CNR-DT 200 R1.

Analysis of average value predictions (Tab.5) tells a similar story, with ML algorithms maintaining R^2 > 0.87 versus 0.62 for CNR-DT 200 R1 across all data. Reductions in MAE, MSE and RMSE were slightly more modest, though still exceptionally improved over the design equation. Most notably, while MAE ranges for CNR-DT 200 R1 from 2.9 to 3.4, all ML models achieved an MAE of around 1.1. MSE and RMSE errors were also at least half of those from CNR-DT.

Breaking performance down by individual substrates again reinforces the dramatic predictive enhancements provided by the ML techniques. Therefore, given their consistent, quantitative statistical superiority over CNR-DT 200 R1, these algorithms demonstrated clear potential for advancing FRP retrofit analysis through sophisticated, data-driven modeling.

The predictions generated by the CNR-DT 200 R1 equation for the filtered database, with the exclusion of 45 tests, are depicted in Fig.9. The equation utilizes both the 0.05 percentile and the average k_{\mathrm{G}\mathrm{k}} values specified in the CNR-DT 200 R1 guidelines to estimate the bond strength for the entire data set. The results indicate that employing the 0.05 percentile predictions leads to more cautious designs compared to using the average k_{\mathrm{G}\mathrm{k}} values for all types of substrate materials. Additionally, Fig.10 illustrates the predictions for various substrate materials, such as clay brick, limestone, and tuff specimens. It is noteworthy that the 0.05 percentile predictions exhibit a conservative trend.

7 Models’ explanation

The ability to comprehend and trust the decision-making processes of intricate ML models is contingent upon a thorough understanding of the underlying model. Although these models frequently demonstrate exceptional performance, the absence of transparency can foster uncertainty and impede their widespread adoption. Consequently, model explanation assumes paramount importance, particularly in the case of “black box” models, as it not only enables the refinement of their decision-making processes, but also enhances their dependability, resilience, and user acceptance.

This section explores the concept of model explanation, which is categorized into two primary approaches: global analysis and local analysis. The global analysis will be accomplished through the application of sensitivity analysis and SHAP analysis to examine the behavior of three distinct ML models: the CatBoost regressor, Extra Trees Regressor, and XGBRegressor. In contrast, local explanatory analysis by SHAP will be employed to provide an individualized understanding of the predictions generated by each of these models, thereby enabling a more nuanced comprehension of their decision-making processes.

Global explanatory analysis seeks to amalgamate and scrutinize extensive data sets in order to reveal the connections between individual attributes and the overall model output. By incorporating multiple approaches, a more thorough and precise comprehension of the model can be achieved, thereby facilitating its effective deployment and utilization.

Sensitivity analysis investigates how the model’s output responds to changes in its input. This method, employed by Refs. [106,107], provides insight into the relative importance of different input parameters. It assesses the stability of model outputs in the face of inaccurate or fluctuating input data and facilitates feature selection by identifying and excluding features with minimal impact on the output.

Among the various sensitivity analysis techniques employed in engineering, Sobol’s method is particularly prevalent. This approach, grounded in total variance decomposition, dissects the variance of the model outputs into contributions from individual input variables and their interdependencies. By calculating the variance of main and interaction effects, Sobol sensitivity analysis evaluates the importance of input variables, utilizing a range of sensitivity indices, including first-order, second-order, and total-order indices, to facilitate a comprehensive analysis. First-order sensitivity (S{1}_i) would reflect the direct effect of changes in a single feature on the model output. Second-order sensitivity (S{2}_{i,\mathrm{t}}) would reflect the effect of interactions between two features on the model output. Where reflects of the overall effect of a single feature reflects on the model output, both directly and indirectly through interactions with other features are total-order sensitivity (S{T}_i). The formulas for calculating these indices are provided by Saltelli et al. in Ref. [108], where \mathrm{V}\mathrm{a}\mathrm{r}\left(Y\right) represents the variance of all numbers in Y.

(9)S{1}_i=\frac{\mathrm{V}\mathrm{a}{{\mathrm{r}}_x}_i\left(E_{X_{\sim i}}\left(Y|X_i\right)\right)}{\mathrm{V}\mathrm{a}\mathrm{r}\left(Y\right)},

(10)S{2}_{i,\mathrm{t}}=\frac{{{\mathrm{V}\mathrm{a}\mathrm{r}}_x}_i\left(E_{X_{\sim i,t}}\left(Y|X_i,X_t\right)\right)}{\mathrm{V}\mathrm{a}\mathrm{r}\left(Y\right)},

(11)S{T}_i=\frac{E\left(\mathrm{V}\mathrm{a}{{\mathrm{r}}_x}_i\left(E_{X_{\sim i}}\left(Y|X_i\right)\right)\right)}{\mathrm{V}\mathrm{a}\mathrm{r}\left(Y\right)}.

In this section, we will utilize S{1}_i and S{T}_i to explain the three models. These indices will be calculated using the Salib library [109], a Python library specifically designed for sensitivity analysis.

The sensitivity analysis of three ML models, CatBoost, Extra Trees Regressor, and XGBRegressor, was performed to identify the most influential features in predicting the shear bond strength between FRP materials and masonry substrates. The results, presented in Figures Fig.11–Fig.13, respectively, show that the feature b_{\mathrm{p}} consistently exhibits the highest first-order and total-order sensitivity across CatBoost regressor and XGBRegressor models. This indicates that the width of FRP reinforcement has a significant direct and indirect impact on the model’s prediction of shear bond strength.

However, a closer examination of the sensitivity indices reveals some interesting differences between the models. While all models agree on the importance of b_{\mathrm{p}}, the relative importance of other features varies.

For example, the CatBoost model shows a higher sensitivity to L_{\mathrm{p}} compared to the other two models. This suggests that CatBoost might be more sensitive to variations in plate bonded length along FRP axis, potentially due to its ability to capture complex interactions between features. This could be particularly relevant when dealing with data sets where FRP properties exhibit significant variability. On the other hand, the Extra Trees Regressor model exhibits a higher sensitivity to b_{\mathrm{s}} compared to the other models.

While other features, such as FRP tensile strength (f_{\mathrm{p}\mathrm{u}}), FRP modulus of elasticity (E_{\mathrm{p}}), and FRP thickness (t_{\mathrm{p}}), showed lower first-order and total-order sensitive indices values, suggest a very low indirect influence on the model’s predictions. This indicates that these features may not affect the model’s output.

These differences in feature sensitivity between the models highlight the importance of considering the specific characteristics and capabilities of each ML algorithm when analyzing the results of sensitivity analysis. While b_{\mathrm{p}} consistently emerges as the most influential feature, the relative importance of other features can vary depending on the chosen model. This underscores the need for a comprehensive understanding of the strengths and limitations of different ML algorithms for accurate interpretation of sensitivity analysis results.

Tree SHAP emerges as a valuable tool for interpretability analysis, providing insights into the contributions of individual features to model predictions. This method, rooted in the concept of SHAP values, quantifies the impact of each feature on the model’s output. Unlike traditional sensitivity analysis, Tree SHAP goes beyond mere feature importance, offering a nuanced understanding of feature-prediction relationships. Positive or negative SHAP values indicate the direction of correlation between a feature and the model’s output, revealing whether increasing the feature value leads to higher or lower predictions. The applicability of Tree SHAP extends to models built upon tree-based structures, making it particularly relevant for algorithms like CatBoost, Extra Trees Regressor, and XGBRegressor. This paper leverages Tree SHAP to elucidate the workings of these models, enhancing their transparency and reliability.

The feature importance and SHAP value distribution for three ML models were analyzed using the SHAP package in Python. The results, presented in Fig.14–Fig.16, reveal consistent patterns across all models. All three models consistently identify b_{\mathrm{p}} as the most influential feature. This aligns with the sensitivity analysis, where it exhibited a high total-order sensitivity, indicating its significant impact on model predictions.

The relative importance of other features varies slightly between models. For example, in Catboost, L_{\mathrm{p}} is the second most important feature, while in Extra Trees Regressor and XGBRegressor, b_{\mathrm{s}} takes the second spot. The results obtained are consistent with the Italian Guideline [40] recommendations for determining the design load value of debonding in masonry members reinforced with FRP composites. The guideline emphasizes the significance of accurately estimating k_{\mathrm{b}}, as it is crucial for accurately accounting for the non-uniform distribution of bond shear stresses across the element width in design calculations. The effects of k_{\mathrm{b}} are influenced not only by the width ratio b_{\mathrm{p}}/b_{\mathrm{s}}, but also by the absolute value of b_{\mathrm{p}}, as previously noted by Ref. [41] in the context of FRP-masonry joints.

8 Conclusions

A local SHAP value analysis was conducted for a single test case from the database, utilizing the CatBoost, Extra Trees Regressor, and XGBRegressor models. The results indicate a consistent pattern of importance across the models, with features b_{\mathrm{p}}, L_{\mathrm{p}}, b_{\mathrm{s}}, and t_{\mathrm{p}} emerging as key contributors. Notably, the feature consistently exhibits the highest SHAP value, as depicted in Fig.17. The XGBRegressor model uniquely highlights t_{\mathrm{p}} feature negatively influences the prediction, pushing it away from its current value. That is not evidently considered in the CatBoost and Extra Trees Regressor models. These insights into feature importance provide a deeper understanding of how each model utilizes different features to make predictions, thereby enhancing model interpretability and transparency.

This study developed ML models to predict the shear bond strength between FRP materials EB to masonry substrates. An extensive database of 1583 pull out tests were compiled from previous experimental work and analyzed to identify the key parameters influencing failure load. The database was then filtered to establish seven essential input parameters for the ML models. Three algorithms: CatBoost regressor, Extra Trees Regressor, and XGBRegressor were trained and validated on the experimental data.

The performance of the ML models was evaluated using statistical indicators such as R², MAE, MSE, and RMSE on both training and test data sets. All three models demonstrated high accuracy in predicting failure loads, with R² values exceeding 0.87 on the test set. The CatBoost and XGBRegressor models achieved the best results, explaining over 90% of the variance in failure loads and exhibiting low MAE and RMSE values close to 1 kN.

A key finding of this study was that the developed ML models significantly outperformed the existing CNR-DT 200 R1 model for estimating FRP-masonry bond strength. The AI-based predictions achieved less error compared to the CNR-DT 200 R1 model. This highlights the capability of ML to capture complex nonlinear relationships in large experimental databases. With further refinement, the developed models can be employed in design practice to reliably predict shear bond strengths. Overall, this research established a novel approach for predicting FRP-masonry bond strength using artificial intelligence techniques. By leveraging extensive experimental data, highly accurate ML models were developed that can aid safer and more effective design of FRP retrofitted masonry structures.

The study also provided a comprehensive explanation of the developed ML models through global sensitivity analysis and SHAP analysis. These techniques helped identify the key input features, such as the width of the FRP reinforcement, and their relative importance in driving the models’ predictions. The findings offer valuable insights into the inner workings of the models, enhancing their transparency and reliability for practical applications in the design of FRP-strengthened masonry structures.

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