Explainable machine learning and application-oriented tool for predicting effective hoop strain of fiber-reinforced polymer-confined concrete

Abstract

The reliable prediction of hoop strain of fiber-reinforced polymer (FRP)-confined concrete is crucial for assessing confinement efficiency and ensuring structural integrity. Existing empirical models often fall short as a result of idealized assumptions and limited generalizability across diverse materials and geometries. This study presents a novel, data-driven machine learning (ML) approach to estimate the effective hoop strain of FRP-confined circular concrete columns. A refined database comprising 309 experimental specimens, including Carbon, glass, and aramid FRPs, was used. Eight ML algorithms, encompassing both single (K-Nearest Neighbors, Kernel Ridge Regression, Support Vector Regression, Decision Tree) and ensemble (AdaBoost, Gradient Boosting Machine, Extreme Gradient Boosting, Random Forest) models, were trained and optimized using Optuna with 10-fold cross-validation. The top-performing models have coefficient of determination of greater than 95% as well as low residual variance and error on the full data set. Accordingly, SHapley Additive exPlanations were incorporated for global and local interpretability of the model predictions. The best-performing model was deployed in a user-friendly graphical interface, aiding an accurate and interpretable tool for practitioners. The proposed framework significantly outperforms conventional empirical models, offering a scalable solution for assessing hoop strain of FRP-confined concrete.

Graphical abstract

Keywords

column confinement / FRP confinement / hoop strain / ML / predictive modeling

1 Introduction

The usage of fiber-reinforced polymer (FRP) composites for confining concrete structure has revolutionized structural strengthening and seismic retrofitting. FRP-confined concrete columns display substantial improvements in strength, ductility, and energy dissipation capacity, making FRP confinement particularly effective in improving the performance of deficient or deteriorated structural members [1–9]. An essential mechanism behind these improvements is the hoop strain developed in the FRP wrap due to the lateral expansion of concrete. The effective hoop strain is an important parameter in evaluating the degree of confinement and the effectiveness of the FRP system [10]. Unlike conventional reinforcement, FRP materials such as carbon (CFRP), glass (GFRP), and aramid (AFRP) possess lightweight, corrosion-resistant, and high-strength alternatives with tailorable properties. These merits have placed FRP as a preferred material in both retrofit and new construction. The confinement procedure works by restricting the lateral dilation of concrete, thereby introducing tri-axial compressive stress states in the core that enhance the axial performance of the structural systems [8,11–13]. The hoop strain developed in the FRP wrap plays a crucial role in establishing the confinement pressure and in turn the ultimate strength and strain capacity of FRP-confined concrete. Hence, accurately calculating the hoop strain is vital to both design safety and material efficiency.

Over the past few decades, numerous analytical and semi-empirical models have been proposed to calculate the hoop strain in FRP-confined concrete. These models are normally grounded in mechanics-based relations, which combine force equilibrium, compatibility conditions, and idealized material behavior. Early works, such as Mander et al. [14], laid the basis for understanding confined concrete behavior that was later adapted to FRP materials. Later studies such as [10,15–19] introduced the concept of an efficiency factor, connecting the effective hoop strain to the ultimate FRP strain. Several models have since evolved to calculate this reduction factor based on parameters such as FRP stiffness and concrete strength [20–22]. Such models often treat fiber type explicitly, and allocate separate reduction factors or empirical coefficients for CFRP, GFRP, and AFRP. Meanwhile, from a purely mechanistic viewpoint, the fiber type alone should not govern confinement efficacy—rather, the mechanical interaction among FRP stiffness, thickness, rupture strain, and bond should dictates the performance. Despite the theoretical value, most of the existing analytical models have been calibrated on limited data sets, often derived from idealized or small-scale experiments. Hence, these models are constrained by simplifying assumptions such as perfect bonding, uniform confinement, and axisymmetric condition. Furthermore, many models fail to consider interdependencies between parameters or capture the nonlinear and coupled behavior observed in real-world specimens. Thereby, these models may suffer from limited generalizability, especially when applied to new materials, hybrid composites, or non-standard geometries.

In light of these limitations, machine learning (ML) has emerged as a powerful option for modeling complex engineering phenomena. ML models excel at revealing nonlinear, multivariate relationships from large data sets without needing rigid assumptions [23–34]. In the context of FRP-confined concrete, ML has been effectively applied to predict properties such as peak stress, ultimate strain, and ductility ratio. However, limited work has explicitly addressed the prediction of hoop strain, despite its central role in confinement mechanics and design of FRP-confined concrete. Accurately modeling of hoop strain through ML would enable designers to move beyond the use of conservative efficiency factors and instead use data-driven assessments tailored to specific materials, geometries, and loading scenarios. Such models can integrate multiple influencing factors simultaneously, such as diameter, height, concrete strength, FRP modulus, thickness, rupture strain, fiber type, and number of layers, capturing interactions that are infeasible to address analytically. Besides, ML models can be trained and validated on comprehensive experimental databases assembled from various sources, increasing robustness and generalization ability of ML-based models. When optimized using methods such as cross-validation (CV) and hyperparameter tuning, these models can outperform conventional techniques in accuracy. On the other hand, model explanation techniques like SHapley Additive exPlanations (SHAP), provide interpretability and enable researchers and practitioners to understand which predictors most strongly influence hoop strain.

So, this study aims to bridge the gap between empirical, mechanistic, and ML-based approaches by developing an optimized ML model trained on a curated experimental database of 309 FRP-confined circular columns, covering CFRP, GFRP, and AFRP specimens. Eight ML algorithms, including both ensemble methods and non-ensemble learners, are tuned and evaluated, with the best-performing model deployed into a user-friendly prediction interface. This framework not only improves the reliability of the predictions of hoop strain but also advances the integration of explainable ML into structural design practice.

2 Existing empirical models

The effective hoop strain of FRP-confined concrete ({\epsilon }_{\mathrm{h},\mathrm{r}}) is usually lower than the ultimate tensile strain obtained from uniaxial coupon tests or manufacturer provided ultimate tensile strain ({\epsilon }_{\mathrm{f}}). This reduction is ascribed to various practical factors like premature rupture, stress concentrations, imperfect FRP application, as well as non-uniform strain distribution around the specimen perimeter [10]. To account for this differences, numerous models, empirical and semi-empirical, have been proposed to relate {\epsilon }_{\mathrm{h},\mathrm{r}} to {\epsilon }_{\mathrm{f}} using either reduction factors or strain efficiency coefficients, as summarized in Table 1. While some models appear to offer separate recommendations for different FRP types (e.g., CFRP, GFRP, AFRP), this categorization naturally stems from empirical calibration rather than fundamental differences in confinement mechanics. Theoretically, the confinement effect is governed by parameters such as FRP stiffness, thickness, and configuration, rather than FRP type alone. Thus, care must be taken to interpret the FRP “type-based” models as proxies for material properties rather than a suggestion that type alone dictates performance. Remarkably, most existing models calculate {\epsilon }_{\mathrm{h},\mathrm{r}} as a fixed fraction of {\epsilon }_{\mathrm{f}}, typically using the manufacturer-reported ultimate tensile strain [10,15–19]. Only a few studies [20–22] have proposed more adaptable approaches that estimate {\epsilon }_{\mathrm{h},\mathrm{r}} based on mechanical parameters, e.g., compressive strength (f_{\mathrm{c}\mathrm{o}}) and elastic modulus (E_{\mathrm{f}}), rather than depend solely on {\epsilon }_{\mathrm{f}}. Therefore, to ensure robustness and generalizability of the existing models, this study applies the empirical models listed in Table 1 to the assembled experimental database and subsequently assesses their predictive performance across a diverse set of samples.

3 Data aggregation

The initial stage of this study involved the gathering and fine-tuning of an experimental database focused on {\epsilon }_{\mathrm{h},\mathrm{r}}. The preliminary data set consist of 573 experimental specimens, 448 CFRP, 94 GFRP, and 31 AFRP samples, initially assembled by Lobo et al. [22] from various published sources [17,19,35–72]. However, many of the sourced studies included incomplete or partially reported experimental data, requiring rigorous data cleaning and validation procedures. After removing incomplete entries, a finalized data set of 309 specimens—228 CFRP, 59 GFRP, and 22 AFRP samples with complete hoop strain data was established for model training and validation. For each specimen, relevant geometric and mechanical parameters were extracted from the literature, including concrete properties, diameter (D), height (H), compressive strength (f_{\mathrm{c}\mathrm{o}}), and corresponding strain ({\epsilon }_{\mathrm{c}\mathrm{o}}), as well as FRP properties, elastic modulus (E_{\mathrm{f}}), ultimate tensile strain ({\epsilon }_{\mathrm{f}}), and thickness (t_{\mathrm{f}}). The selected input parameters are grounded in the mechanics of FRP-confinement behavior in concrete. Typically, {\epsilon }_{\mathrm{h},\mathrm{r}} is a direct result of the lateral expansion of concrete under axial load restrained by the externally bonded FRP jacket. The geometric parameters influence the lateral confinement efficiency and stress distribution within the concrete core, f_{\mathrm{c}\mathrm{o}} and {\epsilon }_{\mathrm{c}\mathrm{o}} characterize the stress–strain response of the unconfined concrete, which in turn governs the commencement and magnitude of radial expansion [73]. Meanwhile, the mechanical properties of FRP confinement govern the stiffness, deformability, and confining pressure exerted by the FRP wrap. Thus, these input variables collectively govern the interaction between the expanding concrete and the resisting FRP, ultimately affecting {\epsilon }_{\mathrm{h},\mathrm{r}}.

Figure 1 shows the numerical distribution of {\epsilon }_{\mathrm{h},\mathrm{r}} and the predictors while Fig. 2 presents a heatmap of descriptive statistics for the variables, including the mean, standard deviation, 25th percentile, median, and 75th percentile values. The heatmap elucidates a broad range of hoop strains, reflecting varied confinement conditions. The wide variability in geometric properties reflect a spectrum from small- to large-scale columns with varying concrete strengths. Meanwhile the spread in FRP parameters indicates a range of confinement capacities and strengthening levels. This diversity ensures that the data set comprehensively represents the influence of both concrete and FRP properties on hoop strain behavior.

Besides, Fig. 3 shows a pair plot matrix indicating the distribution and interrelationships among the variables. The 9 × 9 correlation matrix, filtered by fiber type, further confirms the heterogeneity of the data set. Collectively, the breadth and balance of the parameters provide a strong foundation for developing a robust and generalizable ML-based predictive model for {\epsilon }_{\mathrm{h},\mathrm{r}}.

4 Research methodology

The overall research methodology is outlined in the flowchart in Fig. 4, with detailed steps discussed in the following subsections.

4.1 Framework of machine learning model

The proposed ML framework aims to deliver precise, robust, and generalizable predictions of {\epsilon }_{\mathrm{h},\mathrm{r}} by leveraging a different single and ensemble learners. The model training procedure initializes with the setup of eight distinct algorithms: four single learners [K-Nearest Neighbors (KNN), Kernel Ridge Regression (KRR), Support Vector Regression (SVR), and Decision Tree (DT)] as well as four ensemble techniques [AdaBoost, Gradient Boosting Machine (GBM), Extreme Gradient Boosting (XGBoost), and Random Forest (RF)]. To ensure a comprehensive evaluation of regression performance, these ML algorithms were selected based on their diverse learning paradigms and established effectiveness in modeling nonlinear relationships in engineering data sets. KNN is a non-parametric, instance-based method that forecasts target values based on the average of k nearest samples in the feature space. The algorithm is effective for capturing localized patterns without needing a predefined functional form [74]. KRR integrates the regularization properties of ridge regression with the nonlinear mapping abilities of kernel functions, providing good generalization, especially in high-dimensional or collinear data scenarios [75]. Meanwhile, SVR, grounded in the theory of support vector machines, uses kernel transformations and an ε-insensitive loss function, making it appropriate for high-dimensional, nonlinear problems with strong resistance to overfitting [76]. DT is a tree-based learner that partitions the input space using hierarchical rules, offering high interpretability and the capability to model both linear and nonlinear interactions. However, the learner tendency to overfit motivates its use as a base learner in ensemble methods. AdaBoost enhances the predictive accuracy by adaptively reweighting misclassified instances across sequential weak learners, typically shallow DT [77]. GBM extends this concept by using gradient descent to reduce a loss function, allowing greater flexibility in controlling the bias-variance trade-off. XGBoost further optimizes GBM by incorporating regularization, parallel computation, and advanced tree pruning strategies, leading to enhanced computational efficiency and predictive performance [78]. RF, another ensemble of decorrelated DT built via bootstrap aggregation, reduces model variance and improves robustness against noise and overfitting [79]. The inclusion of both single and ensemble models allows for a balanced comparison across instance-based, kernel-based, tree-based, and ensemble-based approaches, allowing the identification of the most suitable predictive strategy for the targeted output. The configurations of the ML framework (Optuna-tuned learner selection with SHAP-based interpretability) are summarized below.

1) Input predictors X and target variable y. Apply log-transform of y_{\mathrm{l}\mathrm{o}\mathrm{g}}=\mathrm{l}\mathrm{o}\mathrm{g}\left(1+y\right).

2) X_{\mathrm{c}\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{n}},y_{\mathrm{c}\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{n}}\leftarrow non-missing rows only.

3) X_{\mathrm{f}\mathrm{i}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{d}},y_{\mathrm{f}\mathrm{i}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{d}}\leftarrow retain only rows with .

4) X_{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}}\leftarrow \mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{n}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{F}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{s}\left(\mathrm{d}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}=2\right).\mathrm{f}\mathrm{i}\mathrm{t}\mathrm{\_}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\left(X_{\mathrm{f}\mathrm{i}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{d}}\right).

5) X_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}},X_{\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}},y_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}},y_{\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}},\leftarrow train_test_split.

6) X_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{\_}\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{d}},X_{\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{\_}\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{d}}\leftarrow MinMaxScaler().fit_transform.

7) For each models, M_k\in KNN, Kernel ridge, SVR, DT, Adaboost, GBM, XGBoost, RF. Define objective function for Optuna using 10-fold CV. Optimize hyperparameters using 50 trials to minimize mean MAE. Identify and record the best hyperparameters {\theta }_k^{\ast } and lowest overall MAE.

8) For each tuned model M_k\left({\theta }_k^{\ast }\right), train on full training set. Predict on both training and tests sets using \mathrm{e}\mathrm{x}\mathrm{p}\left(y_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}-1\right). Calculate MAE, RMSE, MAPE as well as R² for training, testing, and all data.

9) Identify model M_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}} with lowest overall MAE on combined data.

10) Return the trained model M_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}, optimal parameters {\theta }_k^{\ast }, and performance metrics.

11) Reconstruct predictor names from polynomial transformation: predictor names \leftarrow \mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}.\mathrm{g}\mathrm{e}\mathrm{t}\mathrm{\_}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{\_}\mathrm{n}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{\_}\mathrm{o}\mathrm{u}\mathrm{t}(X_{\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}. columns).

12) If M_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}} is tree-based (e.g., RF, XGBoost), initialize \mathrm{e}\mathrm{x}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{r}\leftarrow \mathrm{s}\mathrm{h}\mathrm{a}\mathrm{p}.\mathrm{T}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{E}\mathrm{x}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{r}\left(M_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}\right); else, use \mathrm{e}\mathrm{x}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{r}\leftarrow \mathrm{s}\mathrm{h}\mathrm{a}\mathrm{p}.\mathrm{K}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{e}\mathrm{l}\mathrm{E}\mathrm{x}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{r}\left(M_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}.\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t},\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{p}.\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\left(X_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{\_}\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{d}},100\right)\right).

13) Compute \mathrm{s}\mathrm{h}\mathrm{a}\mathrm{p}\mathrm{\_}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}\mathrm{s}\leftarrow \mathrm{e}\mathrm{x}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{r}.\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{p}\mathrm{\_}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}\mathrm{s}\left(X_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{\_}\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{d}}\right).

14) Generate global explanation (SHAP summary bar plot) and local explanation (force plot).

15) Output: Best-performing model, optimal parameters, and SHAP explanations.

The parameters of each model are optimized using a comprehensive 10-fold CV approach in combination with Optuna-based Bayesian optimization, allowing systematic exploration of hyperparameter spaces. This optimization process simultaneously determines the optimal algorithm type, model configuration, as well as the hyperparameters that minimize prediction errors. In this study, 10-fold CV was applied using random shuffling without stratification. The architecture of the models adopts an adaptive and iterative mechanism, enabling dynamic selection and fine-tuning of model components. This methodology effectively reduces overfitting and captures complex nonlinear interactions between predictors, permitting strong generalization across diverse regression tasks. The model includes interpretability via both global and local SHAP analysis, providing insight into feature contributions and instance-specific predictions. Meanwhile, performance metrics, the coefficient of determination (R²), mean absolute error (MAE), root mean squared error (RMSE), and mean absolute percentage error (MAPE), were calculated for each model. The final model was selected based on the criterion of lowest MAE on the combined data set.

4.2 Deployment of machine learning-based predictive model

To ensure practical user-friendliness and encourage the adoption of the proposed ML model for engineering applications, a graphical user interface (GUI) was developed using PySimpleGUI in Python named Hoop Strain of FRP-confined concrete Predictor. The GUI serves as a lightweight desktop tool allowing users to input design parameters and receive instant predictions of {\epsilon }_{\mathrm{h},\mathrm{r}} in FRP-confined circular concrete columns. The GUI integrates the complete trained ML model including, M_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}, \mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{n}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{F}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{s}\left(\mathrm{d}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}=2\right), MinMaxScaler, and \mathrm{e}\mathrm{x}\mathrm{p}\left(y_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}-1\right). The input interface is categorized into two main sections—concrete Geometric and parameter and FRP parameters. The categorical predictors (e.g., fiber type) are internally mapped to numerical codes before model inference For instance, fiber types are mapped as: CFRP = 1, GFRP = 2, AFRP = 3. Besides, the GUI presents value ranges obtained from the training data to guide users in order to prevent out-of-bound inputs. Accordingly, the trained M_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}} predicts the {\epsilon }_{\mathrm{h},\mathrm{r}} using \mathrm{e}\mathrm{x}\mathrm{p}\left(y_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}-1\right). The final predicted value is displayed on the GUI in percentage format. Also, built-in error handling is embedded to flag missing or invalid entries, and a “Clear” button resets all input fields. A “Cancel” button exits the program.

5 Results and discussion

5.1 Predictive performance of existing empirical models

The scatter plots showing the predicted {\epsilon }_{\mathrm{h},\mathrm{r}} values in FRP-confined concrete using widely adopted empirical models (summarized in Table 1) are shown in Fig. 5. A close analysis of the plots indicates a significant disparity between the predicted and experimentally observed values. Across all models, substantial deviations from the ideal 45° line are evident, signifying poor predictive precision and inconsistency. Even those models explicitly developed to address a broad spectrum of fiber types, such as those by Refs. [18,20], exhibiting a tendency to either overestimate or underestimate the {\epsilon }_{\mathrm{h},\mathrm{r}}, underscoring the limited ability to generalize the models across different confinement circumstances and material properties. Therefore, the inability of the empirical models to capture the complex nonlinear relationship inherent in {\epsilon }_{\mathrm{h},\mathrm{r}} in FRP-confined concrete undermines the reliability of the models in both design and assessment contexts.

In addition to the visual analysis, the performance of the models was quantitatively assessed using multiple error metrics. As shown in the accompanying heatmap (Fig. 5), the empirical models consistently produced high error values. The MAE ranged from 0.35% to 0.52%, while RMSE spanned 0.48% to 0.95%, demonstrating the presence of considerable residual variance in the predictions. Most strikingly, the MAPE values were exceedingly high, ranging from 40% to 73% that further reflects the inconsistency and lack of robustness of the empirical formulations when applied to diverse data sets. These findings highlight a critical limitation of conventional empirical equations—the reliance on simplified assumptions and linear regressions limits the ability of the models to capture the multi-dimensional and nonlinear behavior of {\epsilon }_{\mathrm{h},\mathrm{r}}. The formulation of these empirical models often depends on small, fiber-specific experimental data sets and is highly sensitive to changes in material parameters, such as E_{\mathrm{f}}, t_{\mathrm{f}}, and f_{\mathrm{c}\mathrm{o}}. Given these limitations, there is a compelling necessity to explore alternative modeling strategies capable of capturing the complicated behavior of FRP-confined concrete systems. In this context, ML offers a promising data-driven approach. Unlike empirical models, ML algorithms can learn complex relationships from large and diverse data sets without presuming a predefined functional form, making the ML-based model better suited for generalizing across a wide range of material properties and confinement configurations. Consequently, the next phase of this study investigates the development and validation of robust ML-based models to overcome the shortcomings of conventional empirical approaches.

5.2 Hyperparameters optimization for the models

To ensure optimal predictive performance, the ML models were fine-tuned using Optuna with 10-fold CV. The best hyperparameters obtained for the models are summarized in Table 2.

These hyperparameters were selected to minimize validation error while enhancing the generalization capability of each model. For nonlinear models such as SVR and KRR, the chosen settings effectively capture intricate patterns in the data while ensuring robustness through appropriate regularization. In ensemble-based methods like AdaBoost and XGBoost, the configured parameters facilitate the iterative correction of residual errors by weak learners, thereby improving accuracy without overfitting. Overall, the selected configurations achieve a balanced trade-off between model complexity, training efficiency, and predictive performance.

5.3 Predictive performance of machine learning-based models

Figure 6 presents scatter plots comparing the observed and predicted {\epsilon }_{\mathrm{h},\mathrm{r}}values obtained from the eight ML models. Each plot includes a 45° dashed reference line, signifying the ideal agreement between predicted and observed values. A close alignment of data points along this line represents strong model performance. Also, R² is reported for the training, testing, and entire data sets, presenting a quantitative assessment of each model’s predictive precision and generalization capacity. Among the non-ensemble models, KNN and the DT models demonstrated the most favorable results. Explicitly, KNN achieved R² values of 0.981 and 0.880 on the training and testing data sets, respectively, while the DT model achieved R² values of 0.922 and 0.862. The predicted values for these models were compactly clustered around the 45° fit line, reflecting high predictive accuracy and minimal overfitting. Overall, the R² values for all single models on the testing data set ranged from 0.862 to 0.891—which, while slightly lower than the training performance, still indicated strong generalization to unseen data. In contrast, the ensemble models, such as GBM, XGBoost, and RF, exhibited superior performance, with R² values consistently above 90% on both training and testing data sets. The exception was AdaBoost that slightly underperformed relative to other ensemble methods. These findings show the strength of ensemble learning in capturing the nonlinear and interactive effects among the predictors influencing {\epsilon }_{\mathrm{h},\mathrm{r}}.

A more comprehensive evaluation of model performance was performed using error metrics, as visualized in the heatmap in Fig. 7. For the training data set, the majority of models achieved MAE values below 0.08%, with the exception of KRR, SVR, DT, and AdaBoost that recorded slightly higher MAEs above 0.10%. Similarly, RMSE and MAPE values were below 0.14% and 12.32%, respectively, for the top-performing ML models. For the testing data set, error values were slightly elevated, with MAE ranging from 0.13% to 0.18%, RMSE between 0.18% and 0.24%, and MAPE spanning 15.38% to 21.35%. Across the entire data set, the best models maintained consistently low error metrics, confirming the robustness and generalizability of the models.

Collectively, the results confirm that both single and ensemble ML models substantially outperform the conventional empirical models (Fig. 5), mostly in the ability of the models to capture complex, nonlinear relationships inherent in {\epsilon }_{\mathrm{h},\mathrm{r}} prediction. The superior performance of ML models shows the value as reliable, data-driven tools in structural modeling. Consequently, the model yielding the lowest MAE across the full data set was selected for further interpretation and deployment, comprising feature importance and explainability analysis.

On the other hand, a comparison between observed and model-predicted {\epsilon }_{\mathrm{h},\mathrm{r}} values for data set used for testing and training the XGBoost model is presented in Fig. 8. The close alignment between the plots indicates a strong predictive capability of a typical developed ML-based model. The residuals of the observed and model-predicted {\epsilon }_{\mathrm{h},\mathrm{r}} values are color-coded to distinguish under-predictions from over-predictions. Generally, the residuals are concerted moderately around zero with no significant clustering, indicating that the ML-based model does not show systematic bias across the prediction range. Thus, this further validates the efficiency of ML-based models in apprehending the nonlinear relationships between the {\epsilon }_{\mathrm{h},\mathrm{r}} and its underlying predictors.

5.4 Model interpretation

To enhance the interpretability of the developed ML-based model, SHAP were adopted to analyze predictor contributions at both the global and local levels. SHAP, rooted in cooperative game theory, assigns an importance value, known as a SHAP value, to each predictor, indicative of the contribution of such predictor to a given prediction [80]. This framework allows for a robust, model-agnostic interpretation and transparency of complex ML algorithms by fairly allocating the prediction difference (between the baseline and actual prediction) among the input predictors. In this study, SHAP values were computed for all samples in the data set, offering both an aggregate understanding and individualized insight into the decision-making process of model. The global interpretability was examined using a SHAP bar chart, as illustrated in Fig. 9. The bar chart ranks predictors based on the mean absolute SHAP value, thus quantifying the overall influence of the predictors on model predictions across the data set. The SHAP summary plot offers a more nuanced view by integrating a color gradient that represents the original predictor value ranging from low (dark blue) to high (bright yellow). The results of the SHAP analysis divulged that FRP-related parameters dominate the prediction of {\epsilon }_{\mathrm{h},\mathrm{r}}. Specially, E_{\mathrm{f}}, fiber type, and {\epsilon }_{\mathrm{f}}appeared as the top three contributors, showing the central role of the confining material in governing strain behavior. Subsequently, critical concrete properties, including {\epsilon }_{\mathrm{c}\mathrm{o}}, D, and f_{\mathrm{c}\mathrm{o}}, were identified as the next most influential predictors. These findings align with existing structural mechanics theory, which proposes that both confinement material properties and the mechanical response of the core concrete substantially influence confinement effectiveness [6,14]. Conversely, predictors such as the number of FRP layers, t_{\mathrm{f}}, and H were found to have fairly minor impacts on the model’s predictions, suggesting limited sensitivity to these predictors within the data set.

The local interpretability was further examined using the information presented in Table 3 and the result is illustrated in Fig. 10. Scenarios 1 and 2 correspond to GFRP confinement systems with one and two layers, respectively, and vary in E_{\mathrm{f}}. The local SHAP waterfall plots offer a transparent breakdown of the model’s reasoning, clearly differentiating the positive and negative contributions of each predictors to the predicted {\epsilon }_{\mathrm{h},\mathrm{r}}. In Scenario 1, all predictors, except f_{\mathrm{c}\mathrm{o}}, {\epsilon }_{\mathrm{c}\mathrm{o}}, and H, contribute positively to increasing the predicted {\epsilon }_{\mathrm{h},\mathrm{r}}. In contrast, in Scenario 2, all predictors, except for the number of layers and t_{\mathrm{f}}, positively affect the predicted {\epsilon }_{\mathrm{h},\mathrm{r}}. The absolute residual errors are 0.042 and 0.108 for Scenarios 1 and Scenario 2, respectively. This level of insight is mainly valuable for structural engineers seeking to understand not only what the model predicts but also why the model predicts a certain value for a given case. Generally, the SHAP analysis confirms that the developed ML-based predictive model not only achieves high precision but also maintains interpretability, thus making it a practical tool for both research and design applications involving FRP-confined concrete systems.

5.5 Deployment of machine learning-based predictive model

To support practical implementation and promote the use of ML in structural engineering applications, a user-accessible software tool, named Hoop Strain of FRP-Confined Concrete Predictor, was established. This GUI-based tool is powered by the developed and optimized best-performing ML model identified through broad model evaluation and error analysis. The tool enables users to efficiently predict the {\epsilon }_{\mathrm{h},\mathrm{r}} in FRP-confined concrete systems by simply entering relevant concrete geometric and parameters as well as FRP parameters, without the necessity for coding expertise or access to the original data set or model code. A visual representation of the tool interface is presented in Fig. 11. Once the parameters are inserted, the trained ML model embedded within the tool instantly returns the predicted {\epsilon }_{\mathrm{h},\mathrm{r}}. This streamlined functionality bridges the gap between complex ML methods and the real-world application of models in structural design, particularly in the field of FRP confinement and retrofit systems.

To verify the accuracy of the prediction tool, a test case was performed using experimental data reported by Lobo et al. [22]. The model predicted {\epsilon }_{\mathrm{h},\mathrm{r}} of 0.91%, which closely aligned with the experimentally recorded value of 0.96%, indicating the practical reliability of the prediction tool. This case study serves to validate the predictive abilities of the prediction tool and reinforces its potential as a reliable aid for structural engineers. Typically, the prediction tool is intended to complement, not replace, conventional design checks and expert evaluation. Overall, the inclusion of this GUI-based tool enhances the transparency, usability, and applicability of the developed ML model. The tool empowers practitioners to harness the predictive power of advanced data-driven techniques without requiring in-depth programming knowledge, thereby supporting more informed decision-making in the design and assessment of FRP-strengthened concrete structures.

6 Limitations and future work

The predictive performance of data-driven models is fundamentally constrained by the quality, comprehensiveness, and variety of the training data set. Although the final data set used in this study included 309 experimentally validated specimens across CFRP, GFRP, and AFRP types, and certain parameters, such as failure modes, environmental exposure effects, etc., were unavailable or sparsely reported in the literature. Thus, the exclusion of such parameters may limit the ML-based model’s generalizability in real-world applications. On the other hand, the developed model assumes uniform circular confinement and ignores effects related to load eccentricity, FRP wrapping procedure (e.g., continuous vs. discontinuous wrapping), and non-circular geometries. These features may influence {\epsilon }_{\mathrm{h},\mathrm{r}} development and failure mechanisms in approaches not apprehended by the current input predictors. While the current study focuses on point predictions, future work should integrate uncertainty quantification (e.g., prediction intervals, conformal prediction) to further improve the model’s utility in risk-sensitive applications. Besides, the GUI tool was established as a static input form, suitable for single-scenario assessments. Meanwhile, batch prediction, model updating and/or retraining, and graphical visualization of confinement effects (e.g., interaction diagrams) are not supported in the current version of the tool.

Thus, future studies should aim to integrate additional experimental data, specifically for underrepresented FRP types and non-circular sections. Synthetic data generation methods such as CTGAN or physics-informed generative models could also be adopted to augment the data set while maintaining physical consistency of {\epsilon }_{\mathrm{h},\mathrm{r}} in FRP-confined concrete. The ML-based model could be extended to predict {\epsilon }_{\mathrm{h},\mathrm{r}} in rectangular or eccentrically loaded columns, where stress distributions are non-uniform. Besides, integrating Bayesian ML or quantile regression may permit for confidence interval predictions, providing users with uncertainty bounds around {\epsilon }_{\mathrm{h},\mathrm{r}} estimates, critical for safety-driven design decisions. A future version of the GUI could be deployed as a web-based application with real-time input validation, batch-processing abilities, and interactive visualization features (e.g., SHAP plots). This would improve user-friendliness for engineers and researchers worldwide.

7 Conclusions

This study proposed an innovative, interpretable ML framework consisting of both ensemble and non-ensemble models for the precise prediction of effective hoop strain in FRP-confined concrete. Using a curated database of experimental test results, eight ML algorithms were developed and optimized using the Optuna hyperparameter tuning framework. The performance of the models were thoroughly assessed using standard regression metrics on both training and testing data sets, and further interpreted using SHAP to enable transparency and explainability. Generally, improved predictions allow more accurate estimation of confinement efficacy that in turn supports optimized FRP layout and material usage. Thus reducing the risk of under- or over-design, improves structural reliability, and contributes to more cost-efficient strengthening strategies in practice. The key findings and contributions of the study are summarized as follows.

1) The conventional empirical models demonstrated poor predictive performance for hoop strain in FRP-confined concrete, even those explicitly established for a wide range of fiber types. These models often either overestimate or underestimate the hoop strain, reflecting limited generalizability and accuracy.

2) The top-performing ML models achieved outstanding results across both training and testing data sets. Specifically, the MAE and RMSE remained below 0.14%, and the MAPE was consistently under 15%, while R² values exceeded 0.90, demonstrating strong predictive accuracy and good generalization on unseen data.

3) Unlike conservative black-box ML methods, the developed ML framework maintains high interpretability. Accordingly, FRP-related parameters, such as elastic modulus, fiber type, and tensile strain, were identified as the most influential predictors governing hoop strain in FRP-confined concrete, thereby aligning with fundamental structural mechanics principles.

4) To promote practical adoption, the best-performing ML model was incorporated into a standalone, user-friendly GUI tagged Hoop Strain of FRP-Confined Concrete Predictor. This tool allows engineers to enter relevant design parameters and instantly obtain hoop strain predictions, providing valuable support for FRP design, assessment, and quality control tasks.

5) Despite the promising capabilities of the ML framework, the study acknowledges the following limitations: the current observed experimental data set may not completely encompass rare or unconventional design scenarios, possibly constraining the model’s extrapolative ability; the GUI currently operates on a static model, without integrating real-time retraining or updates based on new data inputs.

6) Future enhancements should target: expanding the training data set using advanced synthetic data generation methods such as CTGAN; including uncertainty quantification to provide prediction confidence intervals; upgrading the GUI to support dynamic model updates, real-time analytics, and interactive visualizations for deeper user engagement.

In conclusion, this study presents a robust, explainable, and practical ML-based prediction framework, realized through the Hoop Strain in FRP-Confined Concrete Predictor, that enables accurate assessment of hoop strain in FRP-confined concrete. This tool offers substantial value for both researchers and practicing structural engineers, nurturing more informed and data-driven decision-making in FRP retrofit and confinement design.

References

[1] Wang Y F , Wu H L . Size effect of concrete short columns confined with aramid FRP jackets. Journal of Composites for Construction, 2011, 15(4): 535–544

[2] Wu Y F , Zhou Y W . Unified strength model based on hoek-brown failure criterion for circular and square concrete columns confined by FRP. Journal of Composites for Construction, 2010, 14(2): 175–184

[3] Wei Y Y , Wu Y F . Unified stress-strain model of concrete for FRP-confined columns. Construction and Building Materials, 2012, 26(1): 381–392

[4] Ilki A , Peker O , Karamuk E , Demir C , Kumbasar N . FRP retrofit of low and medium strength circular and rectangular reinforced concrete columns. Journal of Materials in Civil Engineering, 2008, 20(2): 169–188

[5] Wu Y F , Jiang C . Effect of load eccentricity on the stress-strain relationship of FRP-confined concrete columns. Composite Structures, 2013, 98: 228–241

[6] Lam L , Teng J G . Design-oriented stress-strain model for FRP-confined concrete in rectangular columns. Journal of Reinforced Plastics and Composites, 2003, 22(13): 1149–1186

[7] Wu Y F , Jiang C . Quantification of bond-slip relationship for externally bonded FRP-to-concrete joints. Journal of Composites for Construction, 2013, 17(5): 673–686

[8] Tijani I A , Jiang C , Lim C W , Wu Y F . Eccentrically loaded concrete under nonuniform passive confinement. Journal of Structural Engineering, 2022, 148(1): 04021247

[9] Tijani I A , Jiang C , Lim C W , Wu Y F . Effect of load eccentricity on the mechanical response of FRP-confined predamaged concrete under compression. Journal of Composites for Construction, 2020, 24(5): 04020057

[10] Lam L , Teng J G . Design-oriented stress–strain model for FRP-confined concrete. Construction and Building Materials, 2003, 17(6–7): 471–489

[11] Mohammadi M , Wu Y F . Triaxial test for concrete under non-uniform passive confinement. Construction and Building Materials, 2017, 138: 455–468

[12] Tijani I A , Wu Y F , Lim C W . Aggregate size effects and general static loading response on mechanical behavior of passively confined concrete. Construction and Building Materials, 2019, 205: 61–72

[13] Chen Y , Li P , Zhang S . Experimental investigation on triaxial mechanical properties of coral coarse aggregate-sea sand seawater concrete. Construction and Building Materials, 2023, 409: 134213

[14] Mander J B , Priestley M J N , Park R . Theoretical stress‐strain model for confined concrete. Journal of Structural Engineering, 1988, 114(8): 1804–1826

[15] Benzaid R , Mesbah H . FRP-confined concrete cylinders: Axial compression experiments and strength model. Journal of Reinforced Plastics and Composites, 2010, 29(16): 2469–2488

[16] Rousakis T C , Karabinis A I , Kiousis P D . FRP-confined concrete members: Axial compression experiments and plasticity modelling. Engineering Structures, 2007, 29(7): 1343–1353

[17] Silva Lobo P , Faustino P , Jesus M , Marreiros R . Design model of concrete for circular columns confined with AFRP. Composite Structures, 2018, 200: 69–78

[18] Toutanji H A , Han M , Gilbert J , Matthys S . Behavior of large-scale rectangular columns confined with FRP composites. Journal of Composites for Construction, 2010, 14(1): 62–71

[19] Vincent T , Ozbakkaloglu T . Influence of fiber orientation and specimen end condition on axial compressive behavior of FRP-confined concrete. Construction and Building Materials, 2013, 47: 814–826

[20] Lim J C , Ozbakkaloglu T . Design model for FRP-confined normal- and high-strength concrete square and rectangular columns. Magazine of Concrete Research, 2014, 66(20): 1020–1035

[21] Arabshahi A , Gharaei-Moghaddam N , Tavakkolizadeh M . Development of applicable design models for concrete columns confined with aramid fiber reinforced polymer using Multi-Expression Programming. Structures, 2020, 23: 225–244

[22] Silva Lobo P , Jesus M , Freitas P S A . Prediction of the failure strain of FRP sheets on confined concrete columns with circular cross-section. International Journal of Structural Integrity, 2023, 14(1): 19–43

[23] Eshaghi M S , Anitescu C , Thombre M , Wang Y , Zhuang X , Rabczuk T . Variational Physics-informed Neural Operator (VINO) for solving partial differential equations. Computer Methods in Applied Mechanics and Engineering, 2025, 437: 117785

[24] Es-haghi M S , Anitescu C , Rabczuk T . Methods for enabling real-time analysis in digital twins: A literature review. Computers and Structures, 2024, 297: 107342

[25] Liu B , Liu P , Wang Y , Li Z , Lv H , Lu W , Olofsson T , Rabczuk T . Explainable machine learning for multiscale thermal conductivity modeling in polymer nanocomposites with uncertainty quantification. Composite Structures, 2025, 370: 119292

[26] Liu B , Lu W , Olofsson T , Zhuang X , Rabczuk T . Stochastic interpretable machine learning based multiscale modeling in thermal conductivity of Polymeric graphene-enhanced composites. Composite Structures, 2024, 327: 117601

[27] Liu B , Vu-Bac N , Zhuang X , Fu X , Rabczuk T . Stochastic integrated machine learning based multiscale approach for the prediction of the thermal conductivity in carbon nanotube reinforced polymeric composites. Composites Science and Technology, 2022, 224: 109425

[28] Liu B , Vu-Bac N , Zhuang X , Fu X , Rabczuk T . Stochastic full-range multiscale modeling of thermal conductivity of Polymeric carbon nanotubes composites: A machine learning approach. Composite Structures, 2022, 289: 115393

[29] Liu B , Vu-Bac N , Rabczuk T . A stochastic multiscale method for the prediction of the thermal conductivity of Polymer nanocomposites through hybrid machine learning algorithms. Composite Structures, 2021, 273: 114269

[30] Liu B , Vu-Bac N , Zhuang X , Lu W , Fu X , Rabczuk T . Al-DeMat: A web-based expert system platform for computationally expensive models in materials design. Advances in Engineering Software, 2023, 176: 103398

[31] Zhou C , Wang W , Zheng Y . Data-driven shear capacity analysis of headed stud in steel-UHPC composite structures. Engineering Structures, 2024, 321: 118946

[32] Tang Y , Zhou C , Wang W , Zhang W , Cheng Y . Machine learning driven bond performance prediction between FRP bars and coral aggregate concrete. Construction and Building Materials, 2024, 442: 137684

[33] Zhou C , Xie Y , Wang W , Zheng Y . Machine learning driven post-impact damage state prediction for performance-based crashworthiness design of bridge piers. Engineering Structures, 2023, 292: 116539

[34] TijaniI ADaudaJ AKareemM ALawalA I. Genetic programming-based model for estimating maximum pull load of fiber-reinforced polymer-to-concrete bond interfaces with graphical user interface implementation. Structural Concrete, 2025

[35] Lam L , Teng J G . Ultimate condition of fiber reinforced polymer-confined concrete. Journal of Composites for Construction, 2004, 8(6): 539–548

[36] Lam L , Teng J G , Cheung C H , Xiao Y . FRP-confined concrete under axial cyclic compression. Cement and Concrete Composites, 2006, 28(10): 949–958

[37] Lee J Y , Yi C K , Jeong H S , Kim S W , Kim J K . Compressive response of concrete confined with steel spirals and FRP composites. Journal of Composite Materials, 2010, 44(4): 481–504

[38] Liang M , Wu Z M , Ueda T , Zheng J J , Akogbe R . Experiment and modeling on axial behavior of carbon fiber reinforced polymer confined concrete cylinders with different sizes. Journal of Reinforced Plastics and Composites, 2012, 31(6): 389–403

[39] KonoSInazumiMKakuT. Evaluation of confining effects of CFRP sheets on reinforced concrete members. In: Second International Conference on Composites in Infrastructure. Washington, D.C.: Transportation Research Board, 1998, 343–355

[40] Abbasnia R , Ziaadiny H . Experimental investigation and strength modeling of CFRP-confined concrete rectangular prisms under axial monotonic compression. Materials and Structures, 2015, 48(1–2): 485–500

[41] Al-Salloum Y A . Influence of edge sharpness on the strength of square concrete columns confined with FRP composite laminates. Composites Part B: Engineering, 2007, 38(5–6): 640–650

[42] BulloS. Experimental study of the effects of the ultimate strain of fiber reinforced plastic jackets on the behavior of confined concrete. In: International Conference on Composite Construction. Cosenza, Italy: University of Calabria, 2003, 465–470

[43] Cui C , Sheikh S A . Experimental study of normal- and high-strength concrete confined with fiber-reinforced polymers. Journal of Composites for Construction, 2010, 14(5): 553–561

[44] Eid R , Roy N , Paultre P . Normal- and high-strength concrete circular elements wrapped with FRP composites. Journal of Composites for Construction, 2009, 13(2): 113–124

[45] Micelli F , Modarelli R . Experimental and analytical study on properties affecting the behaviour of FRP-confined concrete. Composites Part B: Engineering, 2013, 45(1): 1420–1431

[46] Elsanadedy H M , Al-Salloum Y A , Alsayed S H , Iqbal R A . Experimental and numerical investigation of size effects in FRP-wrapped concrete columns. Construction and Building Materials, 2012, 29: 56–72

[47] Harries K A , Carey S A . Shape and “gap” effects on the behavior of variably confined concrete. Cement and Concrete Research, 2003, 33(6): 881–890

[48] Saenz N , Pantelides C P . Short and medium term durability evaluation of FRP-confined circular concrete. Journal of Composites for Construction, 2006, 10(3): 244–253

[49] Shahawy M , Mirmiran A , Beitelman T . Tests and modeling of carbon-wrapped concrete columns. Composites Part B: Engineering, 2000, 31(6–7): 471–480

[50] Tamuzs V , Valdmanis V , Tepfers R , Gylltoft K . Stability analysis of CFRP-wrapped concrete columns strengthened with external longitudinal CFRP sheets. Mechanics of Composite Materials, 2008, 44(3): 199–208

[51] Toutanji H A . Stress-strain characteristics of concrete columns externally confined with advanced fiber composite sheets. ACI Materials Journal, 1999, 96(3): 397–404

[52] Wu G , Wu Z S , Lu Z T , Ando Y B . Structural Performance of Concrete Confined with Hybrid FRP Composites. Journal of Reinforced Plastics and Composites, 2008, 27(12): 1323–1348

[53] Valdmanis V , De Lorenzis L , Rousakis T , Tepfers R . Behaviour and capacity of CFRP-confined concrete cylinders subjected to monotonic and cyclic axial compressive load. Structural Concrete, 2007, 8(4): 187–200

[54] Wang L M , Wu Y F . Effect of corner radius on the performance of CFRP-confined square concrete columns. Engineering Structures, 2008, 30(2): 493–505

[55] Watanabe K , Nakamura H , Honda Y , Toyoshima M , Iso M , Fujimaki T , Kaneto M , Shirai N . Confinement effect of FRP sheet on strength and ductility of concrete cylinders under uniaxial compression. In: Third International Symposium on Non-Metallic (FRP) Reinforcement for Concrete Structures. Sapporo, Japan: Japan Society of Civil Engineers, 1997, 1: 233–240

[56] RousakisTTepfersR. Experimental investigation of concrete cylinders confined by carbon FRP sheets, under monotonic and cyclic axial compressive load. KTH Technical Report KTH-R-01-44. 2001

[57] Shehata I A E M , Carneiro L A V , Shehata L C D . Strength of short concrete columns confined with CFRP sheets. Materials and Structures, 2002, 35(1): 50–58

[58] Song X , Gu X , Li Y , Chen T , Zhang W . Mechanical behavior of FRP-strengthened concrete columns subjected to concentric and eccentric compression loading. Journal of Composites for Construction, 2013, 17(3): 336–346

[59] Xiao Y , Wu H . Compressive behavior of concrete confined by carbon fiber composite jackets. Journal of Materials in Civil Engineering, 2000, 12(2): 139–146

[60] Aire C , Gettu R , Casas J R . Study of the compressive behavior of concrete confined by fiber reinforced composites. Journal of Composites for Construction, 2001, 5(4): 239–243

[61] Harries K A , Kharel G . Behavior and modeling of concrete subject to variable confining pressure. ACI Materials Journal, 2002, 99: 180–189

[62] Silva M A G , Rodrigues C C . Size and relative stiffness effects on compressive failure of concrete columns wrapped with glass FRP. Journal of Materials in Civil Engineering, 2006, 18(3): 334–342

[63] Dai J G , Bai Y L , Teng J G . Behavior and modeling of concrete confined with FRP composites of large deformability. Journal of Composites for Construction, 2011, 15(6): 963–973

[64] Rochette P , Labossière P . Axial testing of rectangular column models confined with composites. Journal of Composites for Construction, 2000, 4(3): 129–136

[65] Toutanji H , Deng Y . Strength and durability performance of concrete axially loaded members confined with AFRP composite sheets. Composites Part B: Engineering, 2002, 33(4): 255–261

[66] Vincent T , Ozbakkaloglu T . Influence of concrete strength and confinement method on axial compressive behavior of FRP confined high- and ultra high-strength concrete. Composites Part B: Engineering, 2013, 50: 413–428

[67] PicherFRochettePLabossiereP. Confinement of concrete cylinders with CFRP. In: 1st International Conference on Composites in Infrastructures, ICCI’96. Tucson, Arizona: American Concrete Institute, 1996, 829–841

[68] ParrettiRNanniA. Axial testing of concrete columns confined with carbon FRP: Effect of fiber orientation. In: 2nd International Conference on Composites in Infrastructures, ICCI. Tucson, Arizona: American Concrete Institute, 2002: 1–10

[69] Ozbakkaloglu T , Akin E . Behavior of FRP-confined normal- and high-strength concrete under cyclic axial compression. Journal of Composites for Construction, 2012, 16(4): 451–463

[70] Micelli F , Myers J J . Durability of FRP-confined concrete. Proceedings of the Institution of Civil Engineers—Construction Materials, 2008, 161(4): 173–185

[71] Berthet J F , Ferrier E , Hamelin P . Compressive behavior of concrete externally confined by composite jackets: Part B: Modeling. Construction and Building Materials, 2006, 20(5): 338–347

[72] Matthys S , Taerwe L , Audenaert K . Tests on axially loaded concrete columns confined by fiber reinforced polymer sheet wrapping. Special Publication, 1999, 188: 217–228

[73] Wu Y F , Jiang J F . Effective strain of FRP for confined circular concrete columns. Composite Structures, 2013, 95: 479–491

[74] HastieTTibshiraniRFriedmanJ. The Elements of Statistical Learning. New York: Springer, 2009

[75] Evgeniou T , Pontil M , Poggio T . Regularization Networks and Support Vector Machines. Advances in Computational Mathematics, 2000, 13(1): 1–50

[76] Cortes C , Vapnik V . Support-vector networks. Machine Learning, 1995, 20(3): 273–297

[77] Freund Y , Schapire R E . A decision-theoretic generalization of on-line learning and an application to boosting. Journal of Computer and System Sciences, 1997, 55(1): 119–139

[78] ChenTGuestrinC. XGBoost: A scalable tree boosting system. In: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, California: Association for Computing Machinery, 2016, 13–17

[79] KeGMengQFinleyTWangTChenWMaWYeQLiuT. LightGBM: A highly efficient gradient boosting decision tree. In: Advances in Neural Information Processing Systems. Long Beach, California: Neural Information Processing Systems Foundation, 2017

[80] Lundberg S M , Lee S I . A unified approach to interpreting model predictions. Advances in Neural Information Processing Systems, 2017, 30: 4766–4775

RIGHTS & PERMISSIONS

Higher Education Press