2.1. Data construction from experimental database

In this study, a database containing 238 experimental data points was established to investigate the bond behavior between steel reinforcement and geopolymer concrete (Fig 1). The collected data were used to develop and evaluate predictive models of bond strength. The database was assembled from 23 peer-reviewed studies reporting laboratory tests on the bond performance of steel bars embedded in geopolymer concrete [2,10,11,15,17,18,26–41] and [42], as summarized in Table 2. Most of the specimens were tested using standard pull-out tests. A smaller number of studies adopted beam-end or lap-splice test configurations (Table 1).

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Table 1. Statistical information for parameters in the databases.

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Table 2. Optimal Hyperparameters of Six Machine Learning Models.

https://doi.org/10.1371/journal.pone.0352645.t002

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Fig 1. Schematic illustration of the bond between steel reinforcement and geopolymer concrete in the pull-out test.

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The data collection process included numerical values reported in experimental tables, parameter descriptions provided in the text, and values digitized from published figures when tabulated data were not directly available. After data collection, the compiled database was organized into input and output variables for subsequent modeling. Because the database was compiled from multiple independent literature sources, a data consistency-control procedure was conducted before model development. All extracted data were first checked for consistency in variable definitions, units, and admissible value ranges. Reported values were converted into common units when necessary. Derived geometric variables were then recalculated when sufficient original information was available. These variables included the concrete cover-to-bar diameter ratio (c/d_b) and the embedment length-to-bar diameter ratio (l_b/d_b). Duplicated records were removed from the database. Incomplete records were also excluded when essential variables were missing. Major sources of experimental variability were retained as input variables in the machine learning models. These sources included mixture composition, curing temperature, curing duration, testing age, bar diameter, concrete cover, embedment length, and bar type. This procedure improved data consistency. It also allowed the models to account for inter-study differences in materials, curing conditions, and geometric conditions. The final dataset contains 19 numerical input variables describing material properties, geopolymer mixture composition, specimen geometry, and reinforcement characteristics. One additional categorical input variable was used to identify the bar type, namely ribbed or plain.

The selected input variables were chosen to represent the main physical, chemical, and geometric factors governing the bond behavior between steel reinforcement and geopolymer concrete. These variables can be grouped into four categories: binder and mixture composition, activator characteristics, curing conditions, and reinforcement/specimen geometry. Binder-related variables, such as fly ash content, slag content, CaO content, and the SiO₂/Al₂O₃ ratio, were included because they influence geopolymerization, reaction products, matrix density, and mechanical strength development. Activator-related variables, including NaOH concentration, Na₂SiO₃/NaOH ratio, alkaline solution-to-binder ratio, and water-to-binder ratio, were considered because they affect dissolution of aluminosilicate precursors, reaction kinetics, workability, porosity, and microstructural formation. Curing-related variables were included because temperature, duration, and age control the extent of reaction and the degree of strength gain. Mechanical and geometric variables, such as compressive strength, bar diameter, reinforcement yield strength, concrete cover-to-bar diameter ratio, and embedment length-to-bar diameter ratio, were included because they directly govern adhesion, friction, mechanical interlock, confinement, and stress transfer along the steel–concrete interface.

To incorporate this categorical variable into the machine learning models, a one-hot encoding technique was employed, whereby the bar type was transformed into binary indicator variables representing the two categories. This encoding approach enables the learning algorithms to properly interpret categorical information without introducing artificial ordinal relationships between categories. The output variable corresponds to the bond strength between steel reinforcement and geopolymer concrete. To characterize the dataset, descriptive statistics were computed for all input and output variables. As presented in Table 1 and Fig 2, the statistical summary includes the minimum, maximum, mean, median, standard deviation, skewness, and kurtosis of each variable. These indicators were used to examine the distribution, spread, and heterogeneity of the collected data before model development. The statistical results show that the database covers a broad range of geopolymer concrete mixtures and test conditions, which is important for developing predictive models that are adequately representative.

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Fig 2. Distribution of 19 numerical input and 01 numerical output features.

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The descriptive statistics also indicate substantial variability in several key parameters. The fly ash content (FA) varies from 80 to 525.30 kg/m³, with an average value of 400.94 kg/m³. This indicates that most mixtures are predominantly fly ash–based. In contrast, the ground granulated blast furnace slag (Slag) content ranges from 0 to 240 kg/m³ with a highly skewed distribution (skewness = 2.98). The slag was incorporated only in a limited portion of the mixtures.

The chemical composition of the binder shows substantial variation across the collected studies. The SiO₂/Al₂O₃ ratio ranges from 1.20 to 5.08, with a mean value of 2.26. The CaO content ranges from 0.40% to 33.81%. These results reflect the diversity of precursor materials used in geopolymer concrete. Binder chemistry plays a key role in the geopolymerization process. It also affects the resulting microstructure and bond behavior of the material.

The proportions of the mixture also vary widely. The coarse aggregate content ranges from 912.77 to 1411 kg/m³. The fine aggregate content ranges from 471 to 893 kg/m³. The activator-related parameters also show significant variability. The sodium hydroxide concentration ranges from 8 to 20 M. The sodium silicate-to-sodium hydroxide ratio ranges from 0 to 7.94 and exhibits strong right skewness (skewness = 2.73). The alkaline solution-to-binder ratio ranges from 0.265 to 0.75. The water-to-binder ratio ranges from 0.067 to 0.30. These parameters strongly influence the reaction kinetics of the geopolymer system. They also affect the resulting mechanical properties of geopolymer concrete.

The curing conditions included in the database are also diverse. The curing temperature (T^o) ranges from 0 to 90°C with a mean of 40.17°C, while the curing duration varies substantially from 0 to 168 hours, showing extremely high skewness (skewness = 4.75), indicating that most specimens were cured for relatively short periods with a few cases of prolonged curing. The testing age ranges from 1 to 180 days, reflecting both early-age and long-term bond behavior.

Regarding mechanical and geometric parameters influencing bond performance, the compressive strength of geopolymer concrete (fc’) ranges from 16 to 73.41 MPa with a mean value of 37.27 MPa. The bar diameter (d_b) varies from 8 to 25 mm, while the yield strength of reinforcement (f_y) ranges between 305.09 and 649.88 MPa. Additionally, the concrete cover-to-bar diameter ratio (c/d_b) spans from 1.04 to 7.83, and the embedment length-to-bar diameter ratio (l_b/d_b) varies widely from 2.0 to 30.13, with strong positive skewness (3.85). These geometric parameters are well known to significantly affect the bond stress transfer mechanisms between steel reinforcement and surrounding concrete.

The target variable, namely the ultimate bond-related load capacity (τ_u), ranges from 2.49 to 32.0 MPa with an average value of 16.61 MPa and moderate variability (standard deviation = 7.59 MPa). The relatively wide distribution of this parameter reflects the significant influence of mixture composition, curing conditions, and reinforcement detailing on the bond performance of geopolymer concrete.

The statistical characteristics reveal substantial variability and non-normal distributions across several key parameters. Such complexity highlights the limitations of conventional empirical formulations and motivates the application of advanced machine learning techniques capable of capturing nonlinear relationships and complex interactions among material composition, curing conditions, and reinforcement parameters governing the bond behavior between steel reinforcement and geopolymer concrete.

Fig 3 presents the correlation matrix of the numerical input variables included in the database. A preliminary overview of the linear relationships among the parameters under consideration is provided. The observed correlation patterns indicate that the input variables exhibit different levels of association rather than a uniform trend. This result suggests that bond behavior is governed by multiple interacting factors, including material composition, curing conditions, and specimen geometry. The use of machine learning approaches can capture the complex relationships among the variables more effectively.

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Fig 3. Correlation matrix for the numerical input parameters of the dataset.

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2.2. Machine learning models

This section introduces the machine learning models employed in the present study. These models include Support Vector Regression (SVR), Random Forest (RF), Extremely Randomized Trees (ETR), Gradient Boosting Machine (GBM), XGBoost, and CatBoost.

These algorithms represent both ensemble tree-based learning approaches and a kernel-based regression method. Their predictive performances were systematically evaluated and compared to identify the most reliable model for estimating the bond strength of steel reinforcement embedded in geopolymer concrete.

All machine learning models were implemented as supervised regression models, with the ultimate bond stress (τ_u) used as the output variable. The numerical input variables were used after data preprocessing and consistency checking. The categorical variable representing bar type was converted into binary variables using one-hot encoding. For the SVR model, the input variables were standardized because kernel-based methods are sensitive to feature scale. In contrast, the tree-based ensemble models, including RF, ETR, GBM, XGBoost, and CatBoost, were trained using their original numerical scales because these algorithms are relatively insensitive to monotonic feature scaling. The SVR model was implemented using a kernel-based regression formulation. The RF and ETR models were implemented as bagging-based tree ensembles, where the final prediction was obtained by averaging the outputs of multiple decision trees. The GBM, XGBoost, and CatBoost models were implemented as boosting-based ensembles, where decision trees were sequentially constructed to improve prediction accuracy. XGBoost further included regularization, shrinkage, and subsampling mechanisms, while CatBoost used ordered boosting and symmetric trees to improve generalization and reduce overfitting. Each model was trained on the training subset and optimized using Bayesian optimization combined with 5-fold cross-validation. After identifying the optimal hyperparameters, the final model was retrained using the full training set and evaluated on the hold-out testing set. This implementation strategy ensured a consistent comparison among all machine learning models.

2.3. Hyperparameter tuning using Bayesian optimization

Hyperparameter optimization was performed to improve the predictive performance of the developed machine learning models. In this study, Bayesian optimization was adopted to identify the optimal hyperparameter combination for each model. Compared with conventional search methods Bayesian optimization provides a more efficient strategy for exploring the hyperparameter space. It does so by building a probabilistic surrogate model of the objective function and sequentially selecting promising candidate solutions [24].

The collected dataset was first divided into two subsets. The training set comprised 80% of the total samples and was used for model development and hyperparameter optimization. The remaining 20% of the samples were reserved as an independent test set for the final performance evaluation. This data partitioning strategy was adopted to ensure that the generalization capability of the optimized models could be assessed on unseen data.

Within the training set, Bayesian optimization was coupled with a 5-fold cross-validation procedure. In each optimization iteration, the training data were randomly divided into five subsets of approximately equal size. Four subsets were used for model training, and the remaining subset was used for validation. This process was repeated five times so that each subset served once as the validation set. The average cross-validation performance across the five folds was used as the objective value to guide the optimization process [50].

For each machine learning algorithm, a predefined search space was specified for the corresponding hyperparameters. Bayesian optimization then iteratively evaluated candidate hyperparameter combinations and updated the surrogate model according to the observed cross-validation performance. An acquisition function was used to balance the exploration of unexplored regions and the exploitation of promising regions in the search space. Through this iterative procedure, the hyperparameter combination yielding the best average cross-validation performance was selected for each model. After optimization, the selected hyperparameters were used to retrain each model on the full training set. The optimized models were then evaluated on the independent test set. This final step was performed to examine the predictive accuracy and generalization ability of the developed models under unseen conditions.

2.4. Model evaluation metrics

The predictive performance of the ML models was assessed using four statistical indicators. The coefficient of determination (R²), root mean square error (RMSE), mean absolute percentage error (MAPE), and mean absolute error (MAE) were used. These metrics are commonly used in regression-based structural engineering studies. A higher R² value reflects stronger predictive capability. By contrast, lower values of RMSE, MAPE, and MAE indicate smaller prediction errors and better model accuracy. The mathematical formulations of these performance metrics are presented as follows:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

These evaluation metrics were used to assess the predictive performance of the developed models from different perspectives. Their combined use provides a more comprehensive evaluation of prediction accuracy, error magnitude, and model robustness. This approach also supports the identification of the most reliable model for bond strength prediction.

2.5. Shapley additive explanation (SHAP)

Machine learning models can achieve high predictive accuracy. However, their internal prediction process is often difficult to interpret. This limitation reduces model transparency and makes it difficult to understand how individual input variables influence the final output. In structural engineering applications, such interpretability is important because reliable prediction should be accompanied by a clear explanation of the governing factors. For this reason, Shapley Additive Explanations (SHAP) was employed in this study to interpret the developed machine learning model [25].

The SHAP value assigned to a variable represents its contribution to the prediction. This value is calculated by considering the marginal contribution of the variable across different combinations of input features. SHAP provides a consistent and quantitative measure of feature importance. It also allows both the magnitude and direction of each variable’s effect on the prediction to be examined. SHAP analysis was applied to the best-performing machine learning model to investigate the influence of input variables on the predicted bond strength between steel reinforcement and geopolymer concrete. Both global and local interpretability analyses were conducted. Global interpretation was performed using SHAP summary plots to identify the most influential variables affecting bond strength prediction. Local interpretation was further carried out to explain individual predictions by quantifying the contribution of each input parameter.

Using SHAP provides a clearer understanding of the relationships the model learns. It also improves the transparency and credibility of the prediction framework. The developed model can be interpreted not only in terms of predictive accuracy but also in terms of the relative importance and directional influence of the governing variables. This is useful for identifying the key factors that control the bond behavior between steel reinforcement and geopolymer concrete.

3.1. Model training and test procedure

Fig 4 presents the workflow of the proposed machine learning framework for predicting the bond strength between steel reinforcement and geopolymer concrete. The workflow consists of database preparation, data preprocessing, model development, hyperparameter optimization, performance evaluation, and model interpretation. After preprocessing, the dataset is split into training and test sets at an 80/20 ratio. Six machine learning algorithms, including SVR, RF, ETR, GBM, XGBoost, and CatBoost, are implemented for model development. Bayesian optimization combined with cross-validation is then used to determine the optimal hyperparameters for each model. The predictive performance of the optimized models is assessed using statistical evaluation metrics. The best-performing model is finally interpreted using SHAP to quantify the influence of the input variables on bond strength prediction.

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Fig 4. Model training and test procedure.

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The computational framework was implemented in Python using established open-source libraries. The SVR, RF, ETR, and GBM models were implemented using Scikit-learn. The XGBoost and CatBoost models were implemented using the XGBoost and CatBoost Python libraries, respectively. SHAP-based model interpretation was performed using the SHAP library. Bayesian optimization was conducted using a Python-based optimization package. These libraries provided the core algorithmic implementations of the machine learning models. The original code developed by the authors was used to construct the modeling pipeline, including data preprocessing, one-hot encoding of categorical variables, train–test splitting, cross-validation, Bayesian optimization setup, model evaluation, residual analysis, repeated train–test validation, confidence interval estimation, empirical-model comparison, and visualization of results.

To further improve the statistical robustness of the model evaluation, a repeated train–test splitting procedure was additionally conducted. Instead of relying on a single 80/20 data partition, the dataset was repeatedly divided into training and testing subsets using different random seeds while maintaining the same training-to-testing ratio. For each repetition, the models were retrained and evaluated using the same optimized hyperparameter settings. The mean value and 95% confidence interval of each performance metric were then calculated to quantify the variability in model performance caused by data partitioning. This repeated evaluation strategy provides a more reliable assessment of model stability and reduces the risk that the reported performance depends on a particular random split. The 95% confidence interval was computed as:

(8)

where is the mean value of the evaluation metric over repeated runs, s is the standard deviation, and n is the number of repetitions and is the critical value of Student’s t-distribution.

3.2. Evaluation of model

The optimal hyperparameters obtained for each model are summarized in Table 2. These configurations correspond to the best results achieved during the Bayesian optimization process with 5 cross-validation. This optimization procedure enables efficient exploration of the hyperparameter space and improves model generalization. The selected hyperparameters also reflect the specific learning characteristics of each algorithm. These results highlight the importance of proper hyperparameter optimization for improving bond strength prediction.

Following the optimization stage, the predictive performance of all models is evaluated using several statistical indicators. The comparative results are presented in Table 3. A comprehensive assessment of the performance of RF, ERT, GBM, XGBoost and CatBoost were provide. The combined presentation of the optimized hyperparameters in Table 2 and the corresponding performance metrics in Table 3 enhances transparency and supports a fair comparison of the developed models in terms of accuracy, stability, and generalization.

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Table 3. Average Predictive Performance of Models.

https://doi.org/10.1371/journal.pone.0352645.t003

To evaluate the model performance on the training dataset using the K-fold technique, the data were partitioned into five subsets (5-K Fold). In each iteration, four folds were used for training while the remaining fold served as the validation set. The averaged results and the variability across the 5-K Fold are illustrated in the Fig 5.

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Fig 5. Relationship between actual and predicted values using the GBM and CatBoost models.

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After completing the model evaluation on the training portion of the dataset (constituting 80% of the total samples) and determining the optimal hyperparameters through the combined use of Bayesian optimization and 5-fold cross-validation, each optimized model was further subjected to validation on the independent test set comprising the remaining 20% of the data. This additional evaluation step was undertaken to examine the generalization capability of the models beyond the cross-validation framework and to ensure that the selected hyperparameters did not lead to overfitting on the training folds. The resulting performance metrics, including accuracy and error-based indicators, are systematically compiled in Table 3.

All models in this study showed strong fitting performance on the training dataset. The corresponding R² values ranged from 0.968 to 0.986. These values indicate that the selected input variables effectively capture the relationship between the governing parameters and bond strength. CatBoost outperformed the other models in terms of training performance with an R² of 0.986 and the lowest errors (MAE = 0.684, RMSE = 0.897, and MAPE = 5.084%). XGBoost and GBM are following, with training R² values of 0.981 and 0.980, respectively. The remaining models, including SVR, RF, and ETR, also showed satisfactory fitting capability, with R² values above 0.96.

The predictive performance was further examined using the independent testing dataset. The accuracy decreased slightly on unseen data. However, the overall results remained strong for all models. CatBoost again provided the best performance on the test set, with the highest R² of 0.950 and the lowest prediction errors (MAE = 1.173, RMSE = 1.669, and MAPE = 11.608%). GBM ranked second with a testing R² value of 0.946. ETR and XGBoost followed with R² values of 0.940 and 0.938, respectively. RF and SVR showed slightly lower test accuracies, with R² values of 0.928 and 0.923.

The ensemble boosting models, especially CatBoost and GBM, provided the most reliable predictive performance for bond strength prediction. These models achieved high accuracy on both the training and testing datasets. The results suggest that they offer good internal generalization capability within the range of the compiled experimental database. Among them, the CatBoost model consistently achieved the best performance across both training and testing datasets. Therefore, the CatBoost model was selected for the subsequent interpretability analysis. To quantitatively summarize the variability of model performance under repeated random data partitions, the mean values and 95% confidence intervals of the evaluation metrics are presented in Table 4.

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Table 4. Repeated train–test validation results with 95% confidence intervals for the optimized machine learning models.

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As shown in Table 4, the repeated train–test validation provides a quantitative assessment of the sensitivity of model performance to random data partitioning. The CatBoost model achieved the best overall performance across the repeated evaluations, with high R2 and low error values. The relatively small variations in the evaluation metrics indicate that the predictive performance of the optimized CatBoost model remained stable under different training and testing subsets.

The repeated train–test evaluation further confirmed the stability of the proposed machine learning framework. Across repeated data partitions, the CatBoost model consistently achieved the best overall performance, with only small variations in R², MAE, MAPE, and RMSE. These results indicate that the predictive accuracy of the proposed model is not substantially affected by the random selection of training and testing samples. Therefore, the reported performance of the BO-CatBoost model can be considered statistically stable for internal validation within the range of the compiled experimental database.

The relatively small dataset size compared with the number of input variables may increase the risk of overfitting, especially for complex ensemble models. Therefore, several measures were adopted to improve model generalization. First, hyperparameter optimization was conducted using Bayesian optimization combined with 5-fold cross-validation on the training set. Second, the optimized models were evaluated using a hold-out testing set that was not involved in model training or hyperparameter tuning. Third, the training and testing performances were compared to identify possible overfitting. The relatively small performance gap between the training and testing results of the CatBoost model indicates that severe overfitting was not observed. Moreover, the repeated train–test evaluation further confirmed that the predictive performance remained stable across different random data partitions.

It should be noted that the hold-out testing set used in this study represents internal validation because it was randomly selected from the compiled database. Therefore, it should not be interpreted as a fully external validation dataset. Nevertheless, the combined use of 5-fold cross-validation, independent testing, and repeated train–test evaluation provides a more reliable assessment of model stability than a single data split. The generalization capability reported in this study should therefore be understood within the range of material, curing, and geometric conditions represented in the compiled experimental database.

Although the optimized CatBoost model achieved a high testing R², this result should be interpreted with caution. The compiled database includes samples extracted from multiple literature sources, and some records may share similar mixture designs, testing protocols, or experimental conditions. Therefore, data dependency or hidden correlations among samples may partly influence the observed predictive performance. The reported accuracy should be regarded as internal predictive performance within the range of the compiled database rather than definitive evidence of external generalizability. Further validation using independent external datasets, study-wise validation, or newly generated experimental results is required to confirm the broader predictive capability of the proposed model.

3.3. ML models prediction performance of best machine learning model

As illustrated in Fig 5, the horizontal axis represents the experimentally measured bond strength (τ_u), while the vertical axis denotes the predicted values obtained from two best model: GBM and CatBoost model for both the training and testing datasets. The red dashed line corresponds to the ideal prediction line (y = x), while the green and blue dashed lines indicate the ± 10% and ±20% error bands, respectively.

Fig 5 compares the predicted and experimental bond-strength values from the GBM and CatBoost models. In the training dataset, the predicted values from both models are closely distributed around the ideal line. CatBoost shows slightly better performance than GBM, as reflected by its higher R² value. A similar trend is observed in the testing dataset.

The GBM model achieved a coefficient of determination of R² = 0.9801, while the CatBoost model exhibited slightly higher accuracy with R² = 0.9861. In both cases, the majority of the training data points are concentrated within the ± 10% error band, demonstrating the strong capability of these ensemble learning models to capture the nonlinear relationships between the governing parameters and the bond strength response. A similarly strong predictive capability is observed for the testing dataset. The GBM model achieved R² = 0.9458, whereas the CatBoost model provided a slightly improved performance with R² = 0.9503, confirming its better generalization capability when applied to unseen data. Most of the testing data points are distributed within the ± 20% error bounds, with a considerable number still falling inside the ± 10% band, indicating that the predicted bond strengths remain in close agreement with the experimental values across a wide range of bond strength levels. Although the scatter widens slightly for unseen data, most points remain within ±20% error bounds, and many still fall within ±10% error bounds. These results confirm that both models provide reliable predictions, while CatBoost demonstrates slightly better accuracy and generalization performance.

Fig 6 presents the residual distributions of the GBM and CatBoost models. For the CatBoost model, the residuals in the training dataset exhibit an approximately normal distribution centered very close to zero, with a mean error of μ = 0.0047 and a standard deviation of σ = 0.8966. The model predictions are essentially unbiased and that the majority of errors remain relatively small, confirming the strong fitting capability of the model for the training data. For the test dataset, the residual distribution also follows an approximately Gaussian pattern, with a mean of μ = −0.0890 and a standard deviation of σ = 1.6671. The slightly negative mean value indicates a slight tendency for the model to underestimate bond strength. However, the magnitude of this deviation remains small and does not indicate any systematic bias in predictions. The larger standard deviation compared with the training dataset reflects the expected increase in prediction uncertainty when the model is applied to previously unseen data. Across the entire dataset, the residuals are symmetrically distributed around zero, with μ = −0.0142 and σ = 1.0971, confirming the overall stability and robustness of the CatBoost model. Most residuals are concentrated within a relatively narrow range around zero, and no pronounced skewness or extreme outliers are observed. The results obtained from the predicted-versus-experimental plots and the residual analysis consistently demonstrate the strong predictive capability of the developed models. Both GBM and CatBoost show close agreement between predicted and observed bond strengths in the training and test datasets, indicating that the main patterns in the data were effectively captured. CatBoost exhibits slightly better performance in terms of prediction accuracy, error dispersion, and generalization to unseen data.

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Fig 6. Distribution of prediction residuals for the GBM and CatBoost models in the training, testing, and entire datasets.

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To further quantify the uncertainty associated with individual predictions, an approximate residual-based prediction interval was estimated for the optimized CatBoost model. The residual distribution of the testing dataset was used because it reflects the prediction error on unseen samples within the compiled database. Assuming that the testing residuals are approximately distributed around zero, the 95% prediction interval can be expressed as:

(9)

where is the predicted bond strength and is the standard deviation of the testing residuals. For the CatBoost model, the testing residual standard deviation was =1.6671. Therefore, the approximate 95% prediction interval was:

(10)

in the same unit as the target variable. This interval provides an approximate uncertainty range for individual predictions. It should be noted that this interval was derived from the residual distribution of the internal testing dataset. Therefore, it should be interpreted as an internal uncertainty estimate rather than an externally validated prediction interval.

As shown in Table 5, the testing dataset produced a wider prediction interval than the training dataset, which reflects the expected increase in uncertainty when the model is applied to unseen data. Nevertheless, the mean testing residual remained close to zero, indicating that the optimized CatBoost model did not show a strong systematic bias. The residual-based prediction interval provides a practical estimate of the reliability range of individual predictions within the compiled database.

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Table 5. Residual-based prediction uncertainty of the optimized CatBoost model.

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3.4. Comparison with existing empirical models

The empirical equations considered in this comparison were selected based on their direct applicability to the available experimental database. These models use variables that were consistently reported or could be derived from the collected studies, such as concrete compressive strength, bar diameter, concrete cover, and embedment length. International design standards, such as Eurocode, Model Code, Japanese, Chinese, and ACI provisions, were not included in the quantitative comparison because many of these provisions are primarily intended for development length or anchorage design rather than direct prediction of ultimate bond stress from pull-out tests. In addition, some code-based equations require design-specific parameters, including bar position, bond condition, confinement condition, transverse reinforcement, casting condition, and safety or modification factors. These parameters were not consistently available in the compiled database. Therefore, including such standards would require additional assumptions, which could undermine the fairness and consistency of the comparison. A broader evaluation involving international design provisions should be considered in future studies when more complete experimental information becomes available.

Table 6 summarizes several empirical models proposed in previous studies for estimating the bond strength between steel reinforcement and concrete. These analytical expressions typically relate bond strength to a limited number of parameters, most commonly the compressive strength of concrete (fc′), the concrete cover-to-bar diameter ratio (c/d_b), and the embedment length-to-bar diameter ratio (lb/db). For instance, the model proposed by Orangun et al. [8] incorporates the effects of compressive strength, concrete cover, and embedment length, reflecting the influence of both material properties and geometric parameters on bond performance. Similarly, the design provisions in AS 3600 [51] estimate bond strength as a function of compressive strength and the concrete cover-to-bar diameter ratio.

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Table 6. Empirical models proposed by different research.

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Other simplified models, such as those proposed by Topark-Ngram [11] et al. and Dahou [18], assume that bond strength is primarily governed by the compressive strength of concrete, resulting in equations that depend only on fc′. While these simplified relationships are convenient for engineering calculations, they may not fully capture the complex interactions among multiple parameters affecting the bond behavior. In contrast, the model proposed by Kim and Park [19] incorporates additional geometric variables, providing a more comprehensive representation of the bond mechanism.

Despite their simplicity and practical applicability, these empirical formulations are generally derived from limited experimental datasets and rely on predefined functional relationships between variables. As a result, their predictive capability may be restricted when applied to datasets with broader ranges of material compositions and geometric configurations, such as those encountered in geopolymer concrete systems. In the present study, the predictive performance of these traditional analytical equations is compared with that of an optimized CatBoost model, which is capable of capturing complex nonlinear relationships between input parameters and bond strength without requiring predefined mathematical forms (Fig 7).

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Fig 7. Comparison of the predictive performance of the proposed ML model and five empirical equations for bond strength.

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The bond strength between steel reinforcement and geopolymer concrete predicted by the proposed machine learning (ML) model and five existing empirical equations is compared in Fig 8 and Table 7. The performance of each model is evaluated using statistical indicators, namely the mean ratio (μ), standard deviation (σ), and coefficient of variation (COV). In addition, a graphical comparison between predicted and experimentally measured bond strengths is presented to provide a comprehensive assessment of the predictive capability of each model.

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Table 7. Statistical comparison of the predictive accuracy of the CatBoost model and existing empirical equations for estimating the bond strength between steel reinforcement and geopolymer concrete.

https://doi.org/10.1371/journal.pone.0352645.t007

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Fig 8. Box plot comparison of the proposed ML model and five empirical equations for bond strength.

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The statistical comparison of the predicted-to-experimental bond strength ratios for different models is summarized in Table 7. The proposed BO-CatBoost model demonstrates the best overall performance, with a mean ratio (µ) of 1.046, which is very close to the ideal value of 1.0. This indicates that the machine learning model provides an almost unbiased prediction of bond strength. In addition, the standard deviation (σ = 0.314) and coefficient of variation (COV = 0.300) are significantly lower than those of the empirical models. The proposed model produces more stable and consistent predictions.

The empirical models generally underestimate the bond strength. For instance, the Orangun model yields a mean ratio of 0.842, indicating a systematic underestimation of approximately 16%. The variability of this model is relatively high (σ = 0.501 and COV = 0.595), demonstrating a wide dispersion between predicted and experimental values. Similar trends are observed for the Dahou and Topark models, which produce mean ratios of 0.511 and 0.418, respectively.

The Kim & Park and AS 3600 models show even larger deviations from the experimental values, with mean ratios of 0.216 and 0.240, respectively. These very low mean values indicate substantial underestimation of bond strength. Additionally, their high coefficients of variation (COV = 0.698 and 0.730) reveal considerable prediction uncertainty. This discrepancy may be attributed to the fact that these empirical formulations were originally developed based on specific experimental conditions and material compositions that differ from those in the present database. The statistical indicators clearly demonstrate the superiority of the BO-CatBoost model over the conventional empirical equations. The machine learning approach not only achieves predictions closer to the experimental results but also significantly reduces prediction variability. Therefore, the proposed model provides a more reliable and robust tool for estimating the bond strength of geopolymer concrete reinforced with steel bars.

4.1. SHAP-based analysis

Based on the comparative performance assessment, the CatBoost model was selected for SHAP-based interpretability analysis. This choice was primarily motivated by its consistently high predictive accuracy on the independent test dataset, accompanied by relatively low error metrics, indicating strong generalization capability. In addition, CatBoost demonstrated stable behavior across different data partitions, minimizing the risk of overfitting observed in several alternative models.

The relative importance of the input variables for predicting the bond strength between steel reinforcement and geopolymer concrete (T_u) was analyzed using SHAP global importance, as illustrated in Fig 9. The results reveal that both the mechanical properties of geopolymer concrete and the geometric characteristics of reinforcement play dominant roles in governing the bond behavior.

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Fig 9. Relative importance of input features for bond between steel and geopolymer concrete.

https://doi.org/10.1371/journal.pone.0352645.g009

Among all variables, the compressive strength of geopolymer concrete (f′c) is identified as the most influential parameter, contributing 16.7% to the prediction of bond strength. This result is consistent with the fundamental mechanism of bond development, where higher concrete strength enhances adhesion, friction, and mechanical interlock between the steel surface and the surrounding matrix. Consequently, stronger geopolymer matrices provide greater resistance against bond slip and failure.

The embedment length-to-bar diameter ratio (l_b/d_b) represents the second most influential factor, with a relative importance of 12.3%. This parameter directly controls the available anchorage length and the distribution of bond stresses along the steel–concrete interface. A larger embedment length generally improves the efficiency of stress transfer and increases the ultimate bond capacity.

Another important geometric parameter is the cover-to-bar diameter ratio (c/d_b), which contributes 9.2% to the prediction. Adequate concrete cover enhances confinement around the reinforcing bar, delaying splitting cracks and improving the bond resistance. The alkaline solution-to-binder ratio also exhibits significant influence (8.5%), highlighting the importance of mixture design in determining the microstructural characteristics of geopolymer concrete.

Parameters related to the geopolymer chemistry further contribute to bond behavior. In particular, curing temperature (6.6%), CaO content in the binder (5.9%), and the SiO₂/Al₂O₃ ratio (5.9%) play important roles in controlling the geopolymerization process and the formation of reaction products. These chemical factors affect the development of gel phases such as N–A–S–H and C–(A)–S–H, which determine the density and mechanical strength of the geopolymer matrix. A denser microstructure improves the quality of the interfacial transition zone (ITZ) surrounding the reinforcing bar, thereby enhancing adhesion and bond performance.

Moderate contributions are also observed for parameters associated with reinforcement characteristics and mixture composition. The type of reinforcement bar, including plain bars (4.3%) and ribbed bars (3.2%), influences the mechanical interlocking mechanism at the steel–concrete interface. Deformed (ribbed) bars generally provide stronger interlock compared with plain bars, which improves bond resistance. Other variables such as superplasticizer dosage (3.0%), bar diameter (3.0%), fine aggregate content (2.9%), and steel yield strength (2.9%) show moderate but noticeable effects on bond behavior.

Finally, several mixture parameters exhibit relatively smaller contributions, including fly ash content (2.6%), NaOH concentration (2.5%), coarse aggregate content (2.5%), water-to-binder ratio (2.4%), Na₂SiO₃/NaOH ratio (1.9%), curing age (1.7%), curing duration (1.3%), and slag content (0.9%). Although their individual impacts are limited, these variables collectively influence the geopolymer matrix formation and may indirectly affect bond performance.

The SHAP analysis demonstrates that the bond strength of steel reinforcement in geopolymer concrete is governed by a complex interaction between concrete mechanical properties, reinforcement geometry, and geopolymer mixture chemistry. While compressive strength and anchorage geometry dominate the macroscopic bond mechanism, the chemical composition of the geopolymer binder plays a crucial role in controlling the microstructure and the quality of the steel–concrete interface. These findings provide valuable insights for optimizing both mixture design and reinforcement detailing to enhance the bond performance of geopolymer concrete structures.

4.2. Feature dependency analysis

The SHAP beeswarm plot shown in Fig 10 was used to further examine the influence of individual variables on the predicted bond strength. Each point represents an individual observation, while the color gradient indicates the feature value ranging from low (blue) to high (red). Positive SHAP values indicate that the variable increases the predicted bond strength, whereas negative values suggest a reduction in the predicted output.

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Fig 10. SHAP Beeswarm Plot.

https://doi.org/10.1371/journal.pone.0352645.g010

Consistent with the global SHAP importance results, the compressive strength of geopolymer concrete (fc’) exhibits the strongest influence on the predicted bond strength. In the beeswarm distribution, higher values of fc’ (red points) are predominantly located on the positive SHAP side, indicating that increasing compressive strength generally leads to higher predicted bond capacity. This behavior is consistent with fundamental bond mechanics, where a stronger concrete matrix enhances adhesion, friction, and mechanical interlocking at the steel–concrete interface.

The embedment length-to-bar diameter ratio (l_b/d_b) also shows a strong positive influence on the model output. Higher values of l_b/d_b are mainly associated with positive SHAP values, confirming that a larger embedment length improves anchorage capacity and facilitates more efficient stress transfer along the steel–concrete interface. Similarly, the cover-to-bar diameter ratio (c/d_b) demonstrates a noticeable positive contribution, indicating that greater concrete cover enhances confinement around the reinforcement and delays splitting failure, thereby improving bond resistance.

For mixture-related parameters, the alkaline solution-to-binder ratio and curing temperature exhibit clear positive trends in the SHAP distribution. Higher values of these parameters generally correspond to positive SHAP values. This result indicates that enhanced geopolymerization conditions and improved reaction kinetics favor the development of a denser matrix and a stronger interfacial transition zone (ITZ). The CaO content in the binder and the SiO₂/Al₂O₃ ratio also exert moderate influences on the predicted bond strength. Their effects reflect the important role of binder chemistry in governing the chemical composition and microstructural evolution of the geopolymer matrix.

In contrast, some variables such as reinforcement type and aggregate contents exhibit more scattered SHAP distributions around zero. This dispersion indicates that their effects on bond strength are relatively moderate and may interact with other parameters in a nonlinear manner. The broader spread of SHAP values observed for several variables highlights the capability of the CatBoost model to capture complex interactions among mixture composition, reinforcement geometry, and curing conditions when predicting the bond strength between steel reinforcement and geopolymer concrete.

4.3. ICE and PDP

ICE–PDP (Individual Conditional Expectation and Partial Dependence) plots were further employed to examine the effects of individual variables on the predicted bond strength (Fig 11). The thin blue lines show the prediction trends for individual samples. The red dashed line indicates the average partial dependence. It represents the overall trend of each variable with respect to the predicted bond strength.

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Fig 11. ICE (Individual Conditional Expectation) and PDP – (Partial Dependence Plot) analyses.

https://doi.org/10.1371/journal.pone.0352645.g011

The results clearly indicate that the compressive strength of geopolymer concrete (fc’) exerts a strong positive influence on the predicted bond strength. As the compressive strength increases, the average predicted bond strength (τ_u) increases noticeably. This behavior is consistent with classical bond mechanics, where a stronger concrete matrix enhances adhesion, friction, and mechanical interlocking between the steel reinforcement and the surrounding concrete. The geometric parameter representing confinement conditions, namely the cover-to-bar diameter ratio (c/d_b), also exhibits a positive relationship with the predicted bond strength. As c/d_b increases, the PDP curve shows a gradual increase followed by a slight stabilization at higher values, indicating that increased concrete cover improves confinement and delays splitting cracks around the reinforcing bar. Conversely, the embedment length-to-bar diameter ratio (l_b/d_b) demonstrates a decreasing trend in the partial dependence curve. The predicted bond strength decreases sharply at lower values of l_b/d_b and then gradually stabilizes, suggesting that once sufficient embedment length is provided, further increases contribute less effectively to improving the bond capacity.

Several parameters related to geopolymer mixture composition also exhibit noticeable effects. The alkaline solution-to-binder ratio shows a moderate increase, indicating that a higher alkaline activator content enhances geopolymerization and improves matrix densification. Similarly, curing temperature positively influences the predicted bond strength, reflecting its role in accelerating reaction kinetics and promoting the formation of geopolymer gels. The CaO content in the binder also demonstrates a mild positive trend before stabilizing, suggesting that moderate calcium content contributes to the formation of additional reaction products that improve matrix strength and the quality of the interfacial transition zone (ITZ).

Other mixture parameters such as fly ash content, slag content, NaOH molarity, Na₂SiO₃/NaOH ratio, water-to-binder ratio, and aggregate contents exhibit relatively mild variations in the PDP curves. These variables generally produce small fluctuations in the predicted bond strength, suggesting that their effects are secondary to the dominant mechanical and geometric parameters. Slight nonlinear responses can still be observed for certain variables, particularly the SiO₂/Al₂O₃ ratio and aggregate content, suggesting that complex interactions may exist within the geopolymer mixture.

The ICE curves further reveal variability among individual samples, suggesting that the influence of each variable may differ across specific combinations of input parameters. This spread of individual trajectories highlights the nonlinear interactions captured by the machine learning model. Nevertheless, the overall trends represented by the PDP curves remain physically meaningful and consistent with established bond behavior mechanisms.

The ICE–PDP analysis enhances the interpretability of the machine learning model by illustrating how variations in individual variables influence the predicted bond strength. The results confirm that the bond behavior of steel reinforcement in geopolymer concrete is primarily governed by concrete compressive strength, reinforcement geometry, and confinement conditions, while geopolymer mixture chemistry and curing parameters contribute secondary but still meaningful effects on the overall bond performance.