2.1. Shear wall experimental database

The present study focuses exclusively on RC shear walls with rectangular and I-shaped cross-sections. Studies were performed using data derived from the ACI 445B Shear Wall Database (Pujol et al., Reference Pujol, Song, Wang, Puranam and Usta2019).

From this database, a processed subset of 231 RC shear wall specimens containing the parameters required for the present analysis was considered, including walls with rectangular, I-shaped, T-shaped, and general cross-sectional configurations. Only walls with rectangular and I-shaped sections were retained. Walls with T-shaped or general cross-sections were excluded due to their limited representation in the available experimental data, which could introduce bias and increase uncertainty in the training of machine learning models. This filtering resulted in a final database of 208 RC shear wall specimens, consisting of 106 rectangular walls and 102 I-shaped walls. The complete processed database, including the identifiers linking each specimen to the corresponding records in the ACI 445B Shear Wall Database and the assignment of each specimen to the development and hold-out test subsets, is provided as Supplementary Material and is also openly available on Zenodo (Kadhim and Craifaleanu, Reference Kadhim and Craifaleanu2026).

All selected specimens were subjected to quasi-static cyclic lateral loading, typically applied at the top of the wall while maintaining a constant axial load. Boundary conditions were generally idealized as fixed at the base, consistent with standard experimental configurations for cantilever-type shear wall testing. Only specimens with complete geometric, mechanical, and reinforcement data were included in the final database to ensure uniformity during model training and evaluation.

To provide a more direct physical characterization of the final dataset, Table 1 summarizes the ranges of the main geometric dimensions and reinforcement ratios separately for rectangular and I-shaped walls. For I-shaped specimens, flange dimensions and their proportions relative to the total wall length are also reported.

Table 1.

Geometric and reinforcement characteristics of the final RC shear wall database

Table 1. Long description

The table lists 12 properties with minimum-maximum ranges for two categories: Rectangular walls and I-shaped walls.

* Wall height, H (m): Rectangular 0.860 to 6.096; I-shaped 0.215 to 4.572.

* Wall length, L sub w (m): Rectangular 0.548 to 3.048; I-shaped 0.254 to 3.960.

* Web or wall thickness, t sub w (mm): Rectangular 60 to 300; I-shaped 16 to 160.

* Aspect ratio, H over L sub w: Rectangular 0.50 to 3.10; I-shaped 0.40 to 4.20.

* Gross section area, A sub g (m super 2): Rectangular 0.036 to 0.840; I-shaped 0.0132 to 0.648.

* Flange width, b sub f (mm): Rectangular is not applicable; I-shaped 60 to 914.

* Flange thickness, t sub f (mm): Rectangular is not applicable; I-shaped 30 to 360.

* Flange width ratio, b sub f over L sub w: Rectangular is not applicable; I-shaped 0.091 to 0.749.

* Flange thickness ratio, t sub f over L sub w: Rectangular is not applicable; I-shaped 0.054 to 0.250.

* Boundary longitudinal reinforcement ratio, rho sub l,b (percent): Rectangular 0.65 to 9.91; I-shaped 0.68 to 6.48.

* Web vertical reinforcement ratio, rho sub v,w (percent): Rectangular 0.17 to 1.76; I-shaped 0 to 1.47.

* Web horizontal reinforcement ratio, rho sub h,w (percent): Rectangular 0.23 to 1.76; I-shaped 0 to 2.45.

Note. Values are reported as minimum-maximum ranges. b_f and t_f denote flange width and flange thickness, respectively.

2.2. Input parameters and output variable

The input parameters used in this study were selected based on their documented influence on the seismic response and deformation capacity of RC shear walls, as observed in prior experimental and analytical research. The following variables were adopted as predictors:

• • ratio between wall length and thickness (L_w/t_w);

• • normalized shear coefficient (V/A_gf_c);

• • axial load ratio (P/A_gf_c);

• • maximum ratio of transverse reinforcement spacing to vertical bar diameter (s/d_b);

• • ratio of ultimate to yield strength of boundary longitudinal reinforcement (f_u/f_y);

• • longitudinal reinforcement ratio in boundary regions and web (ρ _l);

• • transverse reinforcement ratio (ρ_w);

• • cross-sectional shape (rectangular or I-shaped).

These parameters collectively describe the geometric slenderness, loading conditions, reinforcement detailing, and material strength characteristics governing the nonlinear response of RC shear walls under cyclic loading.

Failure mode was not included as an input variable in the present model. Although the deformation capacity of RC shear walls is strongly influenced by whether the response is predominantly flexural, shear-controlled, or governed by a mixed mechanism, a consistent three-class failure-mode classification was not available for the complete dataset. The source database provides a binary “Shear Damage” field indicating whether the original researcher attributed the cause of failure to shear, but this information is unavailable for a substantial portion of the retained specimens and does not distinguish consistently between flexural and mixed flexure-shear mechanisms.

The output variable considered in this study is the drift capacity, defined as the lateral displacement at the intersection of the horizontal line corresponding to 80% of the maximum lateral strength, 0.8Vmax, with the descending branch of the load–displacement envelope, as defined in the ACI 445B Shear Wall Database (Pujol et al., Reference Pujol, Song, Wang, Puranam and Usta2019). The displacement capacity reported in the source database was normalized by the corresponding lateral loading height and expressed as a percentage.

2.3. Statistical ranges and exploratory data analysis

The statistical characteristics and variability of the input parameters were investigated using histograms and scatter plots illustrating their relationship with the drift capacity, as summarized in Figures 1 and 2.

Figure 1.

Statistical distributions of the input parameters for RC shear walls, shown as histograms.

Figure 1. Long description

The figure consists of eight panels labeled a through h. Each panel plots Frequency on the y-axis against a specific structural parameter on the x-axis.

* Panel a: L w forward slash t w. A bimodal distribution with peaks near 12 and 22, ranging from 5 to 45.

* Panel b: Shear coefficient. A right-skewed distribution peaking between 0.05 and 0.10, extending to 0.25.

* Panel c: Axial ratio. A highly skewed distribution with a dominant peak near 0.00, tapering off significantly by 0.15.

* Panel d: Maximum s forward slash d b. An irregular distribution with peaks at 0.0, 5.0, 8.0, and a final spike at 20.0.

* Panel e: f u forward slash f y vertical reinforcement. A distribution with a primary peak at 1.4 and smaller clusters between 1.1 and 1.7.

* Panel f: Boundary vertical reinforcement ratio. A right-skewed distribution with a major peak at 0.015 and a secondary peak at 0.025, trailing off toward 0.10.

* Panel g: Web vertical reinforcement ratio. A right-skewed distribution peaking at 0.0025, with most data points below 0.0100.

* Panel h: Web horizontal reinforcement ratio. A right-skewed distribution peaking between 0.002 and 0.006, with a long tail extending to 0.025.

Figure 2.

Relationships between the input parameters and drift capacity for RC shear walls, shown as scatter plots.

Figure 2. Long description

The figure consists of eight panels arranged in a two-column by four-row grid. All panels share a common x-axis labeled Drift capacity percentage, ranging from 0 to 9.

* Panel a: Y-axis is L w forward slash t w, ranging from 5 to 45. Data points are concentrated between 0 and 4 percent drift, showing a slight downward trend as drift increases.

* Panel b: Y-axis is Shear coefficient, ranging from 0.00 to 0.25. Data is densest at low drift values with a wide vertical spread.

* Panel c: Y-axis is Axial ratio, ranging from 0.00 to 0.25. Many points are clustered at the 0.00 baseline.

* Panel d: Y-axis is Maximum s forward slash d b, ranging from 0 to 20. Data points are distributed in horizontal bands corresponding to integer values.

* Panel e: Y-axis is f u forward slash f y vertical reinforcement, ranging from 1.1 to 1.7. The distribution is relatively uniform across the drift range.

* Panel f: Y-axis is Boundary vertical reinforcement ratio, ranging from 0.00 to 0.10. Most points are below 0.06.

* Panel g: Y-axis is Web vertical reinforcement ratio, ranging from 0.0000 to 0.0175. Data is concentrated below 0.0075.

* Panel h: Y-axis is Web horizontal reinforcement ratio, ranging from 0.000 to 0.025. Data shows a slight positive correlation with drift capacity.

The wall length-to-thickness ratio (L_w/t_w) ranges approximately from 5 to 45, with most of the specimens exhibiting values between 10 and 25 (Figure 1a). The corresponding scatter plot indicates a highly dispersed relationship with drift capacity, which ranges from below 1% to values exceeding 8–9% across nearly the entire L_w/t_w interval (Figure 2a). No clear monotonic trend is observed, indicating that wall slenderness alone does not control deformation capacity.

The normalized shear coefficient (V/A_gf_c) varies roughly between 0.02 and 0.26, with most specimens clustered below 0.10 (Figure 1b). Higher shear coefficients are generally associated with lower drift capacities, typically below 3%, while low shear demand corresponds to a wide range of drift values, from <1% up to ~9%, possibly indicating significant interaction effects with other parameters (Figure 2b).

The axial load ratio (P/A_gf_c) is predominantly concentrated at very low values, with a large number of specimens tested at axial ratio ≈ 0, and fewer walls subjected to axial load ratios up to ~0.28 (Figure 1c). The scatter plot (Figure 2c) suggests a weak inverse relationship with drift capacity, as higher axial load ratios are mostly associated with drift values below 4%, while low axial load cases exhibit a broad dispersion.

The maximum transverse reinforcement spacing-to-bar diameter ratio (s/d_b) ranges from ~0 to 21, with several clusters between 2 and 8 (Figure 1d). Larger drift capacities, including values exceeding 5%, are predominantly observed for spacing ratios below 8, whereas higher spacing ratios are mainly associated with reduced drift capacity, although substantial scatter remains (Figure 2d).

The ultimate-to-yield strength ratio of vertical reinforcement (f_u/f_y) spans from ~1.07 to 1.72, with most values concentrated between 1.3 and 1.5 (Figure 1e). The corresponding scatter plot (Figure 2e) shows no clear linear relationship with drift capacity, as similar drift levels are observed across the full range of strength ratios.

The boundary region vertical reinforcement ratio varies approximately between 0.007 and 0.10, with most specimens concentrated below 0.05 (Figure 1f). High drift values are observed at both low and moderate reinforcement ratios, indicating that increased boundary reinforcement does not directly translate into increased drift capacity (Figure 2f). Similarly, the web vertical reinforcement ratio (Figure 1g) ranges from near 0 up to ~0.018, while the web horizontal reinforcement ratio (Figure 1h) spans roughly from 0 to 0.025. For both parameters, the scatter plots (Figure 2g and Figure 2h) reveal significant dispersion and no evident monotonic relationship with drift capacity.

To further quantify the statistical relationships between the input parameters and the drift capacity, a Pearson correlation matrix was computed and is presented in Figure 3. The results confirm that the linear correlations between individual predictors and drift capacity are generally weak to moderate. Among the geometric and loading-related parameters, the wall length-to-thickness ratio (L_w/t_w) shows a moderate negative correlation with drift capacity (r = −0.29), while the normalized shear coefficient exhibits a slightly stronger negative correlation (r = −0.34). The axial load ratio presents an almost negligible linear relationship with drift capacity (r = 0.02), indicating that its influence cannot be captured through linear dependence alone.

Figure 3.

Pearson correlation matrix illustrating the linear relationships between input parameters and drift capacity for RC shear walls.

Figure 3. Long description

A square correlation matrix with 11 variables on both the x and y axes. A vertical color scale on the right ranges from negative 1.00 in dark blue to 1.00 in dark red, with 0.00 in light gray.

Variables include I shape, Rectangular, L w over t w, Shear coefficient, Axial ratio, Maximum s over d b, f u over f y Vertical Reinforcement, Boundary Region Vertical Reinforcement Ratio, Web Vertical Reinforcement Ratio, Web Horizontal Reinforcement Ratio, and Drift percentage.

Key data points:

* The diagonal from top-left to bottom-right shows a perfect positive correlation of 1.00 for each variable with itself.

* I shape and Rectangular have a perfect negative correlation of negative 1.00.

* Web Vertical Reinforcement Ratio and Web Horizontal Reinforcement Ratio show a strong positive correlation of 0.81.

* Drift percentage shows its strongest positive correlation with Web Horizontal Reinforcement Ratio at 0.36 and its strongest negative correlation with Maximum s over d b at negative 0.35.

* Most other intersections show weak correlations between negative 0.30 and 0.30, indicated by lighter shades of blue, red, and gray.

Regarding reinforcement-related parameters, the maximum transverse reinforcement spacing-to-bar diameter ratio (s/d_b) shows a moderate negative correlation with drift capacity (r = −0.35), consistent with reduced deformation capacity for less effective confinement. In contrast, weak positive correlations are observed for the boundary region vertical reinforcement ratio (r = 0.18), web vertical reinforcement ratio (r = 0.19), and web horizontal reinforcement ratio (r = 0.36), although none of these parameters exhibits a dominant linear influence. The ultimate-to-yield strength ratio of vertical reinforcement shows an almost negligible correlation with drift capacity (r = 0.05).

Strong correlations are observed between some input parameters themselves, most notably between the web vertical and web horizontal reinforcement ratios (r = 0.81), indicating multicollinearity within the reinforcement detailing variables. However, these strong inter-parameter correlations do not translate into similarly strong relationships with the drift capacity.

Overall, the exploratory distributions and scatter plots presented in Figure 1 and 2, together with the Pearson correlation analysis in Figure 3, confirm that no single parameter governs the drift capacity of RC shear walls through a simple linear relationship. The generally low correlation coefficients, combined with strong interdependencies among predictors, highlight the inherently nonlinear and multivariate nature of drift capacity. These findings further justify the adoption of advanced nonlinear machine learning models capable of capturing complex interaction effects and higher-order dependencies that cannot be adequately represented through linear statistical analysis.

3.1. Tree-based regressors

Tree-based methods are particularly suited for structural engineering applications due to their ability to capture nonlinear relationships, interaction effects between variables, and nonuniform data distributions without requiring explicit functional assumptions. In the authors’ previous work on RC columns (Kadhim et al., Reference Kadhim, Craifaleanu and Lozincă2025), Random Forest (RF) and Extreme Gradient Boosting (XGBoost) showed superior performance compared to single learners, especially when dealing with heterogeneous experimental data and multicollinearity among predictors.

Random Forest is a bagging-based ensemble algorithm that aggregates the predictions of multiple decision trees trained on bootstrap samples of the dataset. This approach reduces variance and improves model stability, making RF particularly robust for experimental databases characterized by large scatter and measurement uncertainty (Verikas et al., Reference Verikas, Gelzinis and Bacauskiene2011).

XGBoost is a gradient boosting algorithm that constructs trees sequentially, with each tree correcting the residuals of the previous ensemble. Through regularization mechanisms such as shrinkage and column subsampling, XGBoost effectively controls overfitting while maintaining high predictive accuracy (Chen and Guestrin, Reference Chen and Guestrin2016).

In addition to RF and XGBoost, the Extremely Randomized Trees (Extra Trees) regressor was included in this study. Extra Trees (ET) is a tree-based ensemble method that shares conceptual similarities with Random Forest (RF) but introduces a higher degree of randomness during model construction. As discussed by Yahyapour et al. (Reference Yahyapour, Azhari and Hamidia2026), while RF relies on bootstrap sampling and locally optimized split selection, ET typically uses the entire training dataset for each tree and enforces randomness by selecting split thresholds at random rather than through optimization. For each candidate feature, multiple random split points are generated, and the best among these randomly proposed splits is selected.

Compared to RF, this additional randomization reduces variance more aggressively and decreases the correlation among individual trees, which can improve generalization performance in problems governed by strong nonlinearities and interaction effects. Furthermore, because ET avoids both bootstrap resampling and computationally intensive split optimization, it is generally faster to train than RF, especially for medium to large-sized datasets.

3.2. Stacking ensemble framework

While individual ensemble regressors can achieve strong predictive performance, combining multiple models through stacking has been shown to further improve accuracy and robustness. Stacking ensemble learning integrates the predictions of several base learners by using them as inputs to a higher-level meta-model. This strategy allows the ensemble to exploit the complementary strengths of different algorithms, particularly when individual models capture different aspects of the underlying nonlinear behavior.

Following the general framework adopted in the authors’ previous study on RC columns (Kadhim et al., Reference Kadhim, Craifaleanu and Lozincă2025), RF, XGBoost, and Extra Trees were employed as base learners, while a Ridge regression model was selected as the meta-learner. Ridge regression was chosen due to its L2 regularization capability, which mitigates multicollinearity among the predictions of base learners and ensures stable coefficient estimation (Scikit-learn Developers, 2025a). This property is particularly important in stacking frameworks relying on tree-based models, whose outputs are often highly correlated.

4.1. Data preparation and preprocessing

All analyses were conducted using the reinforced concrete (RC) shear wall database described in Section 2, with drift capacity adopted as the prediction target. The input feature set consists primarily of continuous variables describing geometry, loading, and reinforcement detailing, and includes a categorical variable describing the cross-sectional shape (rectangular or I-shaped). In the database, the cross-sectional shape was numerically encoded and treated as a binary input feature, and the same representation was preserved throughout model training and evaluation to ensure consistent processing across all regressors.

Given that all the selected base regressors are tree-based algorithms (Extra Trees, Random Forest, and XGBoost), no scaling or normalization was applied to the predictors. Tree-based models do not require standardized inputs and are robust to different variable magnitudes. Consequently, all numerical predictors were passed directly to the models without transformation.

Prior to model development, the dataset was split once into a development subset comprising 80% of the specimens and an independent hold-out test subset comprising the remaining 20%. To ensure that low, moderate, and high drift-capacity values were adequately represented in both subsets, the train-test split was stratified using five quantile-based groups of the target variable. These groups were used only for data partitioning and were not included as predictors. A fixed random state of 42 was used to ensure reproducibility, resulting in 166 specimens in the development subset and 42 specimens in the independent test subset. The test subset was kept completely separate from all subsequent hyperparameter tuning and model-selection procedures and was used only for the final evaluation of the trained models.

4.2. Nested cross-validation framework

A nested cross-validation (CV) strategy was adopted to reduce bias in model selection and to obtain reliable generalization estimates under limited experimental data. Model development was performed exclusively within the 166-specimen development subset, using a two-level CV framework:

• • Outer loop (fivefold KFold): used to estimate the expected generalization performance during model selection

• • Inner loop (fivefold KFold): used for hyperparameter tuning via exhaustive grid search.

Both CV loops were implemented with shuffled KFold partitions (Scikit-learn Developers, 2025b). The inner CV loop optimized hyperparameters by maximizing the coefficient of determination R ². For each regressor, the outer loop produced five validation predictions and corresponding scores, and the mean outer-loop R ² value was recorded as the nested CV performance indicator.

After nested evaluation, the final hyperparameter configuration for each model was selected using grid search fitted on the full development subset (with the same inner fivefold CV), and the resulting optimized model was then evaluated once on the independent test set. The test subset was not involved in either hyperparameter optimization or model selection. This separation between model development and final testing is consistent with recent machine-learning applications in structural engineering in which hyperparameter tuning is conducted within the training data and performance is subsequently assessed on a separate test set (Calò et al., Reference Calò, Nettis, Di Mucci, Nettis, Ruggieri, Uva and Dall’Asta2026).

4.3. Hyperparameter tuning of individual regressors

The three above-mentioned base regressors were considered: Extra Trees (ET), Random Forest (RF), and Extreme Gradient Boosting (XGBoost). Each model was embedded in a scikit-learn Pipeline consisting of the preprocessing transformer followed by the corresponding regressor, ensuring that the same feature selection and data flow were preserved during cross-validation and grid search.

Hyperparameters were optimized using GridSearchCV (Scikit-learn Developers, 2025c) with fivefold inner CV, using R ² (the coefficient of determination) as the scoring function. The tuned hyperparameters were selected to control both model complexity and generalization, as follows.

• • Extra Trees (ET): number of trees, maximum depth, minimum samples required to split, and number of features considered at each split.

• • Random Forest (RF): the same hyperparameter set shown above was tuned, enabling a consistent comparison with ET under similar model capacity controls.

• • XGBoost: number of boosting rounds, maximum depth, learning rate, subsampling ratio, and column subsampling ratio.

The final optimal hyperparameter configuration for each regressor was determined using the complete development subset and was stored and reported for transparency, together with the nested CV and independent test-set performance metrics.

4.4. Stacking ensemble implementation

To leverage complementary predictive strengths across tree-based learners, a stacking ensemble was constructed using the optimized ET, RF, and XGBoost models as base estimators. The stacking model was implemented through StackingRegressor (Scikit-learn Developers, 2025d), where the base learners generate first-level predictions that serve as inputs to a second-level meta-model.

A Ridge regression meta-learner (RidgeCV) (Scikit-learn Developers, 2025a) was selected to combine base model predictions. Ridge regression provides L2 regularization, which is beneficial because base learner outputs are typically correlated; the regularization stabilizes coefficient estimation and improves generalization.

The stacking model was trained exclusively using the development subset and was subsequently evaluated on the same independent 42-specimen hold-out test subset used for the individual regressors, enabling a direct comparison under identical testing conditions. The hold-out test subset was not used during the construction or training of the stacking ensemble.

4.5. Performance metrics and visual assessment

Model performance was assessed on the independent test set using three complementary metrics: the coefficient of determination, R ², the mean absolute error, MAE, and the root mean squared error, RMSE. The R ² coefficient was defined as:

(1) $$ {R}^2=1-\frac{{\sum \limits}_{i=1}^n{\left(\;{y}_i-{\hat{y}}_i\right)}^2}{{\sum \limits}_{i=1}^n{\left(\;{y}_i-\overline{y}\right)}^2} $$

where $ {y}_i $ is the true value; $ {\hat{y}}_i $ is the predicted value; $ \overline{y} $ is the mean of the observed values.

MAE and RMSE were computed as:

(2) $$ MAE=\frac{1}{n}\;{\sum \limits}_{i=1}^n\left|{y}_i-{\hat{y}}_i\right| $$

(3) $$ RMSE=\sqrt{\frac{1}{n}\;{\sum \limits}_{i=1}^n{\left({y}_i-{\hat{y}}_i\right)}^2} $$

For each regression model, R ², MAE, and RMSE were computed on the same independent 42-specimen hold-out test subset, using the implementations available in scikit-learn (Scikit-learn Developers, 2025e, 2025f, 2025g). The reported test metrics therefore correspond to a single evaluation on data that remained unseen throughout model development, hyperparameter optimization, and model selection.

To further assess whether model performance remained consistent across the range of observed drift capacities, prediction errors on the independent test subset were also evaluated within four drift-capacity intervals: ≤1%, 1–2%, 2–3%, and >3%. For each interval, the number of specimens, MAE, RMSE, and mean prediction bias were determined. In addition to the numerical evaluation, model behavior was examined visually using observed-versus-predicted scatter plots.

Using the fixed independent test subset, the observed-versus-predicted responses of Extra Trees, Random Forest, XGBoost, and the stacking ensemble were combined into a single 2 × 2 composite panel, enabling a direct visual comparison between individual regressors and the ensemble model. A complementary graphical assessment of prediction errors across the defined drift-capacity intervals is presented in the Results section to examine possible systematic variations in model accuracy over the response range.