Seismic Deformation Capacity Prediction of Steel-Reinforced Concrete (SRC) Columns Based on Test Database and Machine Learning

Abstract

Seismic resilience assessment of high-rise buildings heavily relies on the deformation limits and fragility data of structural components, yet such data is still lacking for steel-reinforced concrete (SRC) columns, which are widely used in high-rise structures. To address this gap, this study establishes a test database consisting of 312 SRC column specimens, including 17 input parameters and three key experimental results, i.e., failure mode, yielding drift ratio θ_y, and ultimate drift ratio θ_u. Two machine learning (ML) frameworks are proposed and four ML models are trained and compared. It is found that the two-stage framework incorporating a failure mode classification shows only a slight improvement in the model performance. Thus, an end-to-end framework is recommended due to its simplicity and avoidance of error propagation, and RF and XGBoost models are adopted and tuned for θ_y and θ_u prediction for their optimal performance. Model interpretation is carried out using permutation importance (PI) and SHAP analyses to verify consistency with domain knowledge, with the key influencing factors identified as longitudinal reinforcement strength (f_yl) and axial load ratio (n_t) for deformation capacity, and shear-span ratio (λ) for failure mode classification. The performance of ML models is compared with conventional data-fitting approaches, and it is proven that ML models outperform conventional formulas, with the R² for predicting θ_y and θ_u improved by 26.5% and 32.9%, the RMSE reduced by 30.0% and 30.4%, and the MAPE reduced by 18.5% and 48.4%, respectively, thus providing a powerful data-driven tool for the seismic resilience assessment of SRC columns and expanding the fragility data of composite components.

1. Introduction

Seismic design is one of the most important aspects in the design of high-rise buildings. Various theories and technologies have been put forward to improve the seismic performance of high-rise buildings, e.g., conceptual design and performance-based design method [1], the application of energy dissipation devices [2], steel–concrete composite structures [3], etc. At the same time, the assessment of the seismic performance of the design outcome is of equal importance, as it serves as a criterion to judge whether the design target can be achieved under specific seismic scenarios.

The next-generation seismic performance assessment method was published by the US Federal Emergency Management and Agency (FEMA) as FEMA P-58 [4] in 2012. This methodology incorporates the probabilistic approach to characterize the performance at the component level, considering both structural and non-structural components, and the seismic performance of the whole structure is quantified with economic and social losses such as repair cost, repair time, injuries and casualties through the loss functions of component damage. Following the FEMA P-58 methodology, several assessment standards were developed, including the REDi Rating System by ARUP [5], the USRC Rating System by US Resiliency Council [6], and the Chinese Standard for seismic resilience assessment of buildings (GB/T 38591-2020) [7]. These standards enable detailed analyses of the consequences of earthquakes, and provide useful tools for risk management in designing new buildings or renovating existing buildings [8].

It is noted that in the current seismic resilience assessment systems, the determination of damage states of structural components and the corresponding losses primarily relies on their deformation under seismic actions. Consequently, establishing the deformation limits and developing the fragility curves for components under different damage states are essential for this procedure [9]. In FEMA P-58 and GB/T 38591-2020, the values for reinforced concrete components and steel components have been provided, based on a collection of experimental results. A methodology for developing such parameters using a test database is presented in FEMA P-58.

The methods for predicting deformation capacity of reinforced concrete (RC) members date back to the 1950s, known as the plastic hinge analogy [10], which is still used in the New Zealand Engineering Assessment Guidelines [11]. Then, regression-based empirical formulas were proposed by Pujol et al. [12], Haselton [13], etc. These mechanics-based or empirical formulas generally exhibit significant scatter when predicting the drift ratio [14]. Moreover, for steel–concrete composite columns (with steel section encased in concrete) widely used in high-rise structures, such data is still lacking.

Earlier research has employed a parametric regression approach to study this problem, based on either a large-scale parametric finite element analysis or a collection of test databases. For example, Hu [15] established a database of 6379 concrete-filled steel plate composite shear walls by finite element analysis and derived simplified formulas for calculating the ultimate curvature with data regression; Cui [16] established a database of 103 RC beams, 469 RC columns and 236 RC shear walls, and developed failure mode classification and deformation capacity regression formulas based on data-fitting. Fu [17] investigated the deformation capacity of SRC members by establishing a database including 246 SRC columns. However, due to the complex and high-dimensional coupling effects of the structural parameters, these semi-empirical models often lead to a compromise between physical simplicity and predictive accuracy.

Machine learning (ML) algorithms are capable at multi-dimensional nonlinear classification and fitting problems, providing a powerful tool for analysis of the highly complex and stochastic experimental results in civil engineering. In recent years, ML methods have been extensively investigated for multiple purposes, including failure mode classification [18,19,20,21], strength prediction [22,23,24], deformation capacity prediction [25,26], backbone curve prediction [27,28], analysis of full loading process [29,30,31], etc. It has been proven that ML models not only offer higher predictive accuracy, but also provide deep insights into the underlying failure mechanisms, effectively bridging the gap between data-driven black-box models and structural mechanics.

To facilitate the seismic resilience assessment of steel–concrete composite structures, this paper investigates the failure mode classification and lateral deformation capacity prediction of the most widely used type of composite component, i.e., steel-reinforced concrete columns. In this study, a comprehensive experimental database of SRC columns is established, covering a wide range of design parameters such as axial load ratio, encased steel ratio, and confinement configurations. Two ML frameworks and four mainstream ML models are developed to handle the complex nonlinear mappings between structural features and deformation at critical characteristic points. Statistical analysis, feature selection and hyperparameter optimization are conducted to ensure the predictive accuracy and robustness of models. Furthermore, the SHapley Additive exPlanations (SHAP) and feature importance analysis are employed to investigate the governing factors across different loading stages. This research offers a rich dataset of experimental results concerning the deformation capacity of SRC columns and a robust and reliable tool for the seismic resilience evaluation of high-rise composite structures.

2. Database of Seismic Tests on SRC Columns

2.1. Description of Test Database

The results of seismic tests on SRC columns are extracted from existing literature to establish the test database. The database includes data from 312 test specimens derived from 38 published articles in the literature [32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69], spanning the research period from 1994 to 2023. All the SRC columns were subjected to cyclic lateral loading under constant axial force, and the lateral force versus lateral displacement hysteretic curves were recorded.

The data for each specimen includes 17 design parameters (categorized into geometric dimensions, material properties, reinforcement/steel ratios, loading conditions, some composite parameters, detailing, etc.) and three experimental result parameters (including failure modes and deformation values at yielding and failure point on the skeleton curve obtained from tests, etc.). The collected parameters for each specimen are listed in

Table 1. Parameters collected for each specimen in the database.

Notation	Explanation	Category
b	Width of the section (perpendicular to the lateral loading direction)	Geometric dimensions
h	Depth of the section (parallel to the lateral loading direction)
d	Length of the shear span
fck	Measured compressive strength of concrete	Material properties
fyv	Measured yielding strength of transverse reinforcement
fyl	Measured yielding strength of longitudinal reinforcement
fak	Measured yielding strength of encased steel (if the strengths of different plates differ, the average value of the steel section is taken)
ρl	Ratio of longitudinal reinforcement	Reinforcement or steel ratio
ρsv	Area ratio of transverse reinforcement
ρv	Volumetric ratio of transverse reinforcement
ρa	Area ratio of the encased steel section
nt	Test axial load ratio, calculated as N/(fckAc + fakAa), where N is the applied axial force, Ac and Aa are areas of the concrete section and encased steel section	Loading conditions
λ	Shear-span-to-depth ratio, calculated as d/h	Composite parameters
αl	Characteristic value of longitudinal reinforcement, calculated as ρlfyl/fck
λv	Characteristic value of transverse reinforcement, calculated as ρvfyv/fck
αs	Characteristic value of encased steel, calculated as ρafya/fck
Shear connector	Whether shear connector exists between steel section and concrete	Detailing
mode	Failure mode, including shear-bond, shear-compression, flexural-shear failure and flexural failure, noted as SB, SC, FS and F, respectively	Test results
θy	Drift ratio at yielding point
θu	Drift ratio at failure point

. The distribution of the parameters of the collected test specimens is illustrated in Figure 1. As observed, the parameters of the collected specimens essentially cover the ranges commonly used in engineering practice, and the specimens’ failure modes comprise the four typical patterns (detailed classification criterion is provided in Section 2.2.1): shear-bond failure, shear-compression failure, flexural-shear failure, and flexural failure.

2.2. Definition of Failure Modes and Damage States of SRC Columns

2.2.1. Failure Modes

The failure modes reflect the damage mechanisms of the SRC column specimens and can potentially aid in predicting their deformation capacity. Therefore, the test database established in this study also includes the failure modes of the test specimens. This section briefly describes the classification principles for failure modes of SRC column specimens.

Based on experimental observations reported in the literature, the failure modes of SRC columns under axial compression and lateral cyclic loading are mainly classified into three types: shear failure, shear-flexural failure, and flexural failure. Among these, shear failure can be further categorized into two types: shear-compression (diagonal compression) and shear-bond failure. These failure modes can be distinguished by the crack patterns and strain development in both reinforcing bars and steel sections, as described in

Table 2. Key characteristics of the failure modes of SRC columns.

Failure Mode	Crack Pattern	Yielding States	Other Failure Phenomenon
Shear-compression failure	Wide diagonal cracks	Transverse rebar/steel web yield before longitudinal rebar/steel flanges.
Shear-bond failure	Vertical splitting cracks	No typical yielding characteristics.	Mainly occurred in specimens without shear connectors between steel section and concrete.
Flexural-shear failure	Horizontal and diagonal cracks	Both longitudinal rebar/steel flanges and transverse rebar/steel web can reach yielding stress.	Intermediate mode between flexural and shear-compression failure, exhibiting characteristics of both failure modes.
Flexural failure	Horizontal cracks	Longitudinal rebar/steel flanges yield at either tension or compression or both sides, and transverse rebar/steel web generally cannot reach yielding stress.	Plastic hinge forms at near the root of the column, accompanied by buckling of longitudinal rebars and/or steel flanges.

. And the typical characteristics of these failure modes are illustrated in Figure 2.

To ensure the accuracy of failure mode classification, this study carefully re-evaluated the failure modes of the specimens from source documents using the above-mentioned criterion, incorporating the descriptions of crack, strain development during testing, and phenomena at specimen failure. Meanwhile, the failure mode classifications judged by the original researchers are also considered as a reference.

2.2.2. Damage States and the Corresponding Deformation Limits

With the development of seismic performance assessment methods in recent decades, various codes have defined several limit states for structures under seismic action, serving as the objectives of seismic design. In seismic performance assessment methodologies, these damage states of structural components correspond to relatively distinct damage phenomena and are used to map structural response to seismic-induced economic losses and repair costs.

Some widely recognized damage states include Immediate Occupancy (IO), Life Safety (LS), and Collapse Prevention (CP) as proposed in ASCE/SEI 41-17 [70], and Damage Limitation (DL), Significant Damage (SD), and Near Collapse (NC) defined in Eurocode 8-3 [71], etc. Both codes adopt deformation or force as damage assessment indicators for each damage state of reinforced concrete members, depending on whether the failure mode is flexure-dominated (ductile failure) or shear-dominated (brittle failure). However, SRC columns exhibit considerable ductility even under shear failure, due to the encased steel section. Moreover, in the Chinese Standard for seismic resilience assessment of buildings [7], for members failing in shear, deformation can also be adopted as an assessment indicator for damage states. Therefore, this study adopts deformation limits as the criterion for determining damage states and performs machine-learning regression analysis on the collected experimental data.

This study focuses on the prediction of the deformation limits for yielding and ultimate states, which are determined based on the skeleton curves of the test specimens. The yield point refers to the equivalent yield point of the SRC column determined by Park’s method [72], where visible cracking has already appeared on the concrete surface, and the reinforcement or steel section begins to yield. This condition approximately aligns with the IO level defined in ASCE 41. The ultimate point corresponds to the state when the member’s load-bearing capacity drops to 85% of its peak strength, which approximately aligns with the CP level in ASCE 41. Other damage states can be considered as intermediate states between yielding and ultimate points, and their deformation limits can be obtained through linear interpolation [73]. The determination of the two limit states is illustrated in Figure 3.

3. Preliminary Analysis of Inter-Variable Correlation

3.1. Pearson’s Correlation Analysis

To preliminarily assess the interrelationships among various parameters in the database and the influence of input parameters on output parameters, the Pearson correlation coefficients between all parameter pairs were calculated with Equation (1):

$${\rho }_{X,Y}={\displaystyle \frac{\mathrm{cov}\left(X,Y\right)}{{\sigma }_X{\sigma }_Y}}={\displaystyle \frac{E\left[\left(X-{\mu }_X\right)\left(Y-{\mu }_Y\right)\right]}{{\sigma }_X{\sigma }_Y}}$$

(1)

where X and Y are the two parameters under consideration, cov(X,Y) denotes their covariance, σ_X and σ_Y are the standard deviations of X and Y, respectively, μ_X and μ_Y are the means of X and Y, and E represents the expectation operator. The Pearson coefficients are presented in the heatmap in Figure 4 to highlight significant parameter interactions (only variable pairs with absolute Pearson coefficients greater than 0.1 are displayed). A larger absolute value indicates a stronger linear correlation between two parameters, where positive and negative values represent positive and negative linear relationships respectively.

From the Pearson correlation coefficients, both the deformation limits for both yielding and ultimate points (θ_y and θ_u) increase with the strength of encased steel and longitudinal reinforcement (f_ak and f_yl), while decreasing with higher concrete strength and axial compression ratios (f_ck and n_t); moreover, enlarging cross-sectional dimensions (b and h) and increasing stirrup confinement (λ_v) are beneficial for improving the ultimate deformation capacity of SRC columns. These observations align with widely accepted domain knowledge, indicating that the experimental data collected from different literature sources can reflect a consistent influence pattern of input parameters on deformation capacity.

On the other hand, it is observed that the composite parameters formed by combining multiple input parameters often exhibit high correlation with their constituent parameters, and both composite and individual parameters may significantly influence output parameters. For example, the Pearson coefficients between α_l and its constituent parameters (ρ_l, f_yk and f_ck) are notably high, as α_l is mathematically derived from these three parameters. Meanwhile, similar to its constituent parameters, α_l also exhibits high correlation with deformation limits of both yielding and ultimate point. This phenomenon is known as multicollinearity in ML, where highly correlated features introduce redundancy in the training process of models, often adversely affecting model performance. To mitigate this issue, feature selection techniques must be employed for dimensionality reduction, as detailed in Section 4.2.

3.2. Statistical Analysis of the Influence of Failure Modes on Deformation Limits: Pearson’s Correlation Analysis

Since the failure mode is a categorical parameter rather than a numerical parameter, it cannot be directly used to calculate Pearson correlation coefficients. Therefore, this section separately analyzes the influence of failure modes on deformation limits. The boxplots of deformation limits of the specimens at yielding and ultimate points, categorized by failure modes, are plotted in Figure 5. It is observed that the deformation limits of shear-bond failure, shear-compression failure, and flexural-shear failure show minor differences, while the deformation limits of flexural failure exhibit significantly greater variation compared to the other three failure modes. Notably, the deformation limits of flexural-shear failure at yielding and ultimate points do not fall between those of shear failure and flexural failure, but instead are close to the values of shear failure; this is primarily because the encased steel section effectively enhances the shear capacity and energy dissipation capability of SRC columns, mitigating the brittle failure characteristics typically associated with shear-dominated failure modes.

Furthermore, quantitative analysis of the differences in deformation limits among various failure modes is conducted using analysis of variance (ANOVA) followed by Tukey’s honestly significant difference (HSD) post hoc test.

ANOVA is a parametric statistical procedure to compare means across three or more independent groups. By partitioning total variance into between-group and within-group components, the test statistic F is calculated as the ratio of between-group mean squares (MSB) to within-group mean squares (MSW), i.e.,

$$MSB={\displaystyle \frac{SSB}{d{f}_{between}}}, \mathrm{where} SSB={\displaystyle \sum _{i=1}^k{n}_i{\left(\overline{X_i}-\overline{X}\right)}^2}$$

(2)

$$MSW={\displaystyle \frac{SSW}{d{f}_{within}}}, \mathrm{where} SSW={\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^{n_i}{\left(X_{ij}-\overline{X_i}\right)}^2}}$$

(3)

$$F={\displaystyle \frac{MSB}{MSW}}$$

(4)

in which SSB is the sum of squares between groups, df_between is the degree of freedom between groups (df_between = k − 1, where k is the number of groups), n_i is the sample size of group i, $\overline{X_i}$ is the mean of group i and $\overline{X}$ is the mean of all data; SSW is the sum of squares within group, df_within is the degree of freedom within group (df_within = N − k, where N is the total sample size), and X_ij is the j-th sample in group i. A significantly large F-value (typically for p < 0.05) indicates rejection of the null hypothesis (H₀: μ₁ = μ₂ = … = μₖ) in favor of the alternative (H₁: at least one mean differs).

When ANOVA indicates significant differences among the means of different failure modes, Tukey’s HSD test is employed to conduct pairwise comparisons of group means. The critical HSD value is calculated by

$$HSD=q_{\alpha ,k,N-k}\sqrt{{\displaystyle \frac{MSW}{n}}}$$

(5)

where q_α,k,N−k is the critical value from the studentized range distribution (depends on the number of group k, degrees of freedom df = N − k and significance level α; n is the sample size per group (harmonic mean is used if the sizes of the groups under comparison are unequal). If the absolute difference between the means of two failure modes exceeds the critical HSD value ($\left|\overline{X_i}-\overline{X_j}\right|>HSD$), the difference is statistically significant (p < α).

The results of the above analysis are presented in

Table 3. Results of ANOVA and Tukey’s HSD test.

Test Step	Failure Modes Compared	θy	θu
ANOVA	/	p= 0.000004 < 0.05	p= 0.000000 < 0.05
Tukey’s HSD	SB vs. SC	p = 1 > 0.05	p = 1 > 0.05
	SB vs. FS	p = 1 > 0.05	p = 0.77 > 0.05
	SB vs. F	p= 0.043 < 0.05	p = 0.084 > 0.05
	SC vs. FS	p = 1 > 0.05	p = 1 > 0.05
	SC vs. F	p= 0.000081 < 0.05	p= 0.000004 < 0.05
	FS vs. F	p= 0.0029 < 0.05	p= 0.000008 < 0.05

Note: bold signifies for p values inferior than 0.05 (which indicate significant difference between the two groups compared).

. The results of ANOVA indicate that at least one failure mode exhibits significantly different deformation limits for both yielding and ultimate points, compared to other failure modes. Furthermore, Tukey’s HSD test reveals that the deformation limits of flexural failure show statistically significant differences from those of shear-compression failure and flexural-shear failure, for both yielding and ultimate points. This result suggests that distinguishing failure modes may be beneficial for predicting deformation limits, especially for specimens exhibiting a flexural failure mode. For the other failure modes, although their deformation limits are relatively close, the fact that different failure modes reflect distinct failure mechanisms leads to the necessity to predict each failure mode for the classification model.

3.3. Integrated Analysis on the Influence of Input Parameters and Failure Modes on Deformation Capacity and Implications on the Prediction Procedure

To visualize the effects of input parameters and failure modes on the deformation capacity, the scatter plots of θ_y and θ_u with some of their highly correlated parameters are shown in Figure 6, in which the data points of different failure modes are distinguished by different markers. The Pearson correlation coefficient r between each input parameter and the target parameter is also displayed above each figure.

From Figure 6, it can be observed that, without distinguishing failure modes, although the data exhibit a certain degree of dispersion, the general variation trends of θ_y and θ_u with respect to the input parameters are relatively clear, reflecting the influence mechanisms of these variables on θ_y and θ_u.

On the other hand, however, it can be seen from each figure that distinctions of data points from different failure modes are comparatively minor, indicating that the influence of a single input parameter on deformation capacity seems weakly dependent on the failure mode, which is inconsistent with the conclusions drawn in the preceding section. Therefore, it is necessary to further discuss the importance of distinguishing failure modes when predicting deformation capacity.

Summarizing the results of preliminary statistical analysis in this section, two ML framework will be discussed in this study: the first is a two-stage framework, which predicts the failure mode of SRC columns using classification algorithms and then predicts deformation limits using regression algorithms, taking the predicted failure mode as an additional input; the second framework is an end-to-end framework which predicts the deformation limits directly from input parameters, skipping the failure mode classification step.

4. Machine Learning Framework

The flowchart of the ML framework implemented in this study is shown in Figure 7. First, feature selection is executed to eliminate redundant variables and prevent overfitting induced by multicollinearity. Then, different predicting procedures and ML models are evaluated to determine the best-performing configuration. Afterwards, the hyperparameters of the chosen model are optimized using grid search with five-fold cross-validation to improve its accuracy and stability. Finally, model interpretation is performed to verify the consistency of the data-driven predictions with domain knowledge.

4.1. Introduction to the Machine Learning Models and Performance Metrics

Considering the relatively small data volume (around 300 SRC column test specimens), numerous input parameters (17 input parameters, including both individual and composite ones) and ambiguous nonlinear relationships among variables (test results shows great dispersion due to coupled parameter interactions), four ML models, i.e., Support Vector Machine (SVM), Artificial Neural Network (ANN), eXtreme Gradient Boosting (XGBoost) and Random Forest (RF), are implemented in this study. The technical backgrounds of the models used in this study can be found extensively in ML-related research works and thus are not detailed here; only the main reasons for choosing the four typical models are listed:

• SVM [74] is a supervised learning algorithm for classification and regression, which tries to find an optimal hyperplane to maximize the margin between classes. It is efficient in dealing with high-dimensional small-scale datasets and can deal with nonlinear problems via kernel tricks.
• ANN [75] is included due to its capacity for modeling complex nonlinear relationships through layered architectures, which is suitable for automatically capturing the nonlinear relationships among various input parameters of SRC columns to predict the output variables.
• XGBoost [76] and RF [77] represent two main branches of ensemble learning, which seeks to improve the accuracy and generalization capability of models by combining multiple simple base learners. XGBoost is an optimized gradient boosting algorithm that sequentially trains decision trees to correct errors from previous models, thus improving the precision of prediction, while RF trains parallel decision trees independently via bootstrap sampling (random subsets of data with replacement), aggregating their predictions through voting or averaging to enhance robustness and reduce overfitting.

To compare the performance of the aforementioned models, the evaluation metrics shown in

Table 4. Evaluation metrics of classification and regression tasks.

Task	Evaluation Metrics	Symbols	Formula
Classification	Accuracy	Acc	$Acc={\displaystyle \frac{{\displaystyle \sum TP}}{n}}$
	Precision	P	$P={\displaystyle \frac{TP}{TP+FP}}$
	Recall	R	$R={\displaystyle \frac{TP}{TP+FN}}$
	Macro F1-Score	Macro-F1	$\mathit{macro-}F1={\displaystyle \frac{1}{N_{\mathrm{class}}}}{\displaystyle \sum _{i=1}^{N_{\mathrm{class}}}\left(2\times \frac{P_i\times R_i}{P_i+R_i}\right)}$
Regression	Coefficient of determination	R2	$R^2=1-{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{\left(y_i-\widehat{y_i}\right)}^2}}{{\displaystyle \sum _{i=1}^n{\left(y_i-\overline{y}\right)}^2}}}$
	Root mean square error	RMSE	$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(y_i-\widehat{y_i}\right)}^2}}$
	Mean absolute percentage error	MAPE	$MAPE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|\frac{y_i-\widehat{y_i}}{\widehat{y_i}}\right|}$

Note: for multi-class classification, TP denotes the number of correct classifications, FP and FN denotes the number of false positive (predicting the specimen failed in other modes as the mode under consideration) and false negative (predicting the specimen failed in the mode under consideration as other modes); the precision and recall are calculated for each class, and Macro-F1 score is the harmonic mean of F1 scores of each class; N_class denotes the total number of classes. For the regression task, $y_i$, $\widehat{y_i}$ and $\overline{y}$ denote the true value, the predicted value of the i-th sample, and the average of all true values. n denotes the number of samples.

are employed.

4.2. Feature Selection

As mentioned in Section 3.1, since the original input parameter set of the collected experimental data includes both single features and composite features, in order to avoid a reduction in model accuracy caused by multicollinearity, feature selection combining Pearson correlation coefficients and variance inflation factor (VIF) is conducted.

Based on the Pearson correlation coefficients shown in Figure 4, pairwise selection is conducted. For any pair of features exhibiting correlation coefficients exceeding a predefined threshold (0.85 is used in this study), only the feature with the higher correlation coefficient with the target parameter is retained. This step aimed to identify information redundancy pairwise, where two variables provide nearly identical information. Then, VIF is calculated for the remaining features to detect complex redundancy. The VIF is calculated with the following steps: for the considered input feature X_i, supposing it can be fitted by a linear combination of the rest of the input features, the coefficient of determination of this linear combination $R_i^2$ is calculated, and then VIF is obtained by

$$VIF={\displaystyle \frac{1}{1-R_i^2}}$$

(6)

Features exhibiting a high VIF can be considered to be largely explainable by linear combinations of other features, thereby constituting redundant information. Removing such features is therefore a reasonable strategy. This strategy can be applied iteratively to obtain a more concise feature set. Figure 8a shows the VIF of all input features before selection.

Combining the information from Figure 4 and Figure 8a, a systematic feature selection is conducted. Although d, h and b provide complete geometric information, d and b are excluded due to their excessive VIF (>10), which indicates redundant geometric information. Similarly, ρ_sv and α_l are excluded, as their physical roles were effectively represented by ρ_v and the combination of ρ_l and f_yl, respectively. The final feature set prioritizes dimensionless mechanical indices (e.g., n_t and λ), ensuring the model is both statistically robust and physically consistent. After feature selection, 10 input features are left to train the ML classification and regression model. These features are listed in

Table 5. Input feature set after selection.

Category	Features
Geometric dimension	h
Material properties	fck, fyl, fak
Reinforcement or steel ratio	ρl
Loading conditions	nt
Composite parameters	λ, αs, λv
Detailing	Shear connector

. The VIF of the remaining features is shown in Figure 8b. It is observed that the VIF values for all the parameters have been reduced to below 5.0, indicating negligible multicollinearity.

4.3. Training and Results of Machine Learning Models

4.3.1. Failure Mode Classification

In this study, the ML models are implemented with Scikit-learn (sklearn) [78], a robust and comprehensive open-source framework for predictive data analysis. For the classification models, the hyperparameters listed in

Table 6. Hyperparameters of models for classification.

Model	Hyperparameter
SVM	kernel = ‘rbf’, probability = True
ANN	hidden_layer_sizes = (100, 50), max_iter = 1000
XGBoost	n_estimators = 200, learning_rate = 0.05, max_depth = 6
RF	n_estimators = 200

are initially used, and the rest are kept at the default value of the program.

It is noted that the distribution of failure modes in the test database is highly imbalanced (Figure 1), with nearly half of the specimens exhibiting flexural failure, while the number of specimens exhibiting shear-bond failure and flexure-shear failure is relatively small. This is consistent with the actual proportion in engineering practice, where engineers prefer flexural failure with good ductility. To address this problem, the Synthetic Minority Over-sampling Technique (SMOTE) is employed on the training set. This technique generates new instances of minority classes (e.g., SB and FS) between a real minority sample and one of its nearest neighbors by linear interpolation, thereby filling the feature space of the minority class, so that the classifier can learn a more robust decision boundary.

For the SVM and ANN algorithms, distance calculations or gradient-based optimization is conducted in model training. Therefore, these models are dependent on the geometry of the feature space, and standardization on the input and output features is necessary to prevent the features with larger numerical scales from dominating the loss function or causing gradient saturation. Z-score standardization is conducted with the StandardScaler function provided in the sklearn package to transform the data of each parameter to have a zero mean and unit variance, i.e., $Z=\left(x-\mu \right)/\sigma$ where Z and x are the data after and before standardization, μ and σ are the mean and variance of the data before standardization.

The test database is randomly partitioned into a training set and a testing set with a ratio of 8:2. The ML models are trained using the training set (249 samples), while the remaining 20% of the data (63 samples), previously unseen by the models, is utilized to independently validate the models’ performance. The confusion matrices of four models on the testing set are shown in Figure 9, with the accuracy and macro-F1 score marked on top of each matrix.

From Figure 9, it is observed that all four models can reach a relatively high global accuracy and micro-F1 score, indicating that ML models can effectively distinguish the failure modes of SRC columns with the selected input features. Notably, the XGBoost model demonstrated the best predictive capability, achieving the highest classification accuracy of 0.968 and a macro-F1 score of 0.957. Specifically, the ensemble models (XGBoost and RF) exhibited exceptional precision in identifying ductile failure (F), with a 100% accuracy. For the brittle failure modes (SB and SC), the XGBoost model outperformed others, accurately capturing the minority class features facilitated by the SMOTE algorithm. The trained XGBoost model is therefore used in the two-stage framework, whose prediction result will be used as an additional input feature in the regression models of the second step.

4.3.2. Deformation Limit Prediction

For the regression models, the hyperparameters listed in

Table 7. Hyperparameters of models for regression.

Model	Hyperparameter
SVM	kernel = ‘rbf’, C = 10, gamma = ‘scale’, epsilon = 0.01
ANN	hidden_layer_sizes = (100, 50), max_iter = 200
XGBoost	n_estimators = 200, learning_rate = 0.05, max_depth = 6
RF	n_estimators = 200

are used. The standardization process for both input and output features, and the random partition ratio of training and testing sets, are identical to those of the classification step.

Figure 10 and

Table 8. Evaluation metrics of the regression models in two-stage framework (testing set).

Model	θy			θu
	R2	RMSE	MAPE	R2	RMSE	MAPE
SVM	0.837	0.00176	14.83%	0.845	0.00655	17.93%
ANN	0.839	0.00175	17.07%	0.837	0.00672	18.64%
XGBoost	0.869	0.00158	15.85%	0.893	0.00544	16.05%
RF	0.882	0.00150	15.85%	0.861	0.00620	18.42%

show the regression results of θ_y and θ_u generated with the four models in the two-stage framework. The red dashed line in the figure represents where the predicted values are equal to the experimental values. Overall, the proposed two-stage framework exhibits high predictive performance, with the majority of data points located closely around the equality line (y = x). For θ_y, all four models achieve R² values on testing set ranging from 0.837 to 0.882, and RF model outperforms others with the highest R² of 0.882 and lowest RMSE of 0.00150; for θ_u, the models also show high predictive capacity, with R² values on testing set exceeding 0.837 across all the models, and XGBoost model can reach R² of 0.893 and MAPE of 16.40%, providing a highly reliable tool for ultimate deformation capacity analysis.

The robustness of the two optimal models is further evaluated by analyzing the R² degradation between the training and testing sets. For θ_y, the RF model exhibited an R² degradation of 0.085, and for θ_u, the XGBoost model exhibited an R² degradation of only 0.101; both show satisfactory robustness. Together with the high testing performance (R² near 0.9), it is confirmed that the model has effectively captured the mechanical laws governing the yielding and ultimate limit of SRC columns rather than overfitting to specific training samples.

4.3.3. End-to-End Framework

This study also proposes an end-to-end framework to directly predict the deformation capacity of SRC columns from input parameters, and the results are shown in Figure 11 and

Table 9. Evaluation metrics of the regression models in end-to-end framework (testing set).

Model	θy			θu
	R2	RMSE	MAPE	R2	RMSE	MAPE
SVM	0.843	0.00173	16.44%	0.843	0.00659	18.32%
ANN	0.823	0.00183	19.01%	0.863	0.00616	16.90%
XGBoost	0.867	0.00159	15.80%	0.895	0.00539	15.86%
RF	0.879	0.00152	16.01%	0.862	0.00617	18.28%

. Again, the models showed high predictive accuracy and satisfactory robustness. Similar to the case of the two-stage framework, the RF and XGBoost models are the optimal models for θ_y and θ_u with R² values on the testing set of 0.879 and 0.895, respectively.

4.4. Comparison of the Two Frameworks

The performance of optimal models in two-stage and end-to-end frameworks is compared in

Table 10. Comparison of optimal model performance between two frameworks.

Model	θy Prediction with RF				θu Prediction with XGBoost
	R2	RMSE	MAPE	R2 Degradation	R2	RMSE	MAPE	R2 Degradation
Two-stage	0.882	0.00150	15.85%	0.085	0.893	0.00544	16.05%	0.101
End-to-end	0.879	0.00152	16.01%	0.087	0.895	0.00539	15.86%	0.099
Difference	−0.34%	1.33%	1.01%	2.35%	0.22%	−0.92%	−1.18%	−1.98%

. In contrast to the existing literature where a two-stage framework often yields significant accuracy improvement, it is observed that only a slight improvement occurs in the two-stage framework in this study. This can be explained by the fact that the existence of encased steel sections largely enhances the deformation capacity under shear failure, thereby blurring the boundary between brittle and ductile failure modes. On the other hand, an end-to-end framework avoids the error propagation from the classification to the regression stage, and is also capable of implicitly capturing the underlying mechanism of deformation, thus giving highly accurate and robust predictive results. Therefore, the end-to-end framework developed in this study is recommended for the prediction of the deformation capacity of SRC columns, and is discussed in the following part.

4.5. Hyperparameter Optimization

To ensure the predictive stability and to mitigate the risk of overfitting, the hyperparameter optimization process is conducted on the optimal machine learning models determined in the end-to-end framework, i.e., the RF model for θ_y and the XGBoost model for θ_u prediction. The tuning strategy employs a combination of five-fold cross-validation (CV) and grid search.

For each model, a multi-dimensional search space (grid) is defined, including key hyperparameters such as learning rates, tree depths, and regularization coefficients. During the five-fold CV process, the training dataset is partitioned into five subsets; for each combination of hyperparameter, four subsets are used for model fitting while the remaining subset serves as a validation set. This procedure is repeated five times for every parameter combination to obtain a stable average performance metric (R² is used in this study as the target of optimization). Finally, the hyperparameters leading to better performance are used to train a regression model using the whole training set. The search space and finally chosen hyperparameter for the two models are listed in

Table 11. Combination of hyperparameters for grid search.

Model	Hyperparameter	Values of Grid Search	Value of Optimal Model in 5-Fold CV	Value of Initial Model
RF for θy prediction	n_estimators	100, 200, 300	100	200
	max_depth	3, 5, 8, None	None	None
	min_samples_split	5, 10	5	2
	min_samples_leaf	1, 4, 6	1	1
	max_features	‘sqrt’, ‘log2’, ‘auto’	‘sqrt’	‘auto’
XGBoost for θu prediction	n_estimators	100, 200, 300	200	200
	max_depth	3, 6, 10	3	6
	learning_rate	0.03, 0.05, 0.1	0.05	0.05
	subsample	0.8, 1	0.8	1.0
	colsample_bytree	0.8, 1	1	1.0
	min_child_weight	1, 3, 5, None	1	None
	reg_lambda	1, 5, None	1	None

.

The performance of models with tuned hyperparameters is compared with that of initial hyperparameters (baseline model) and the main results are listed in

Table 12. Comparison of performance between baseline and tuned models.

RF Model for θy	Performance in CV	Performance on Testing Set
	Average R2	R2	RMSE	MAPE
Baseline	0.711	0.878	0.00152	16.04%
Tuned	0.731	0.856	0.00165	15.78%
Difference	2.81%	−2.51%	8.55%	−1.62%
XGBoost Model for θu	Performance in CV	Performance on Testing Set
	Average R2	R2	RMSE	MAPE
Baseline	0.775	0.895	0.00539	15.86%
Tuned	0.822	0.885	0.00564	16.67%
Difference	6.06%	−1.12%	4.64%	5.11%

. It is found that the tuned model yields a higher R² in CV, and a slight degradation in predictive accuracy on the testing set (e.g., decrease in R² and increase in RMSE). In fact, the reduction in the R² observed on the testing set (e.g., from 0.878 to 0.856 for θ_y) indicates a successful mitigation of initial overfitting. The narrowing gap between CV and testing performance suggests that the tuned models are more robust against the inherent stochasticity of the SRC experimental database. With a MAPE remaining consistently below 17%, the tuned models demonstrate a high level of predictive accuracy. Therefore, the tuned models are finally used to predict the seismic deformation capacity of SRC columns.

Moreover, from Table 12, it is also observed that both the RF and XGBoost models can achieve a higher R² on the independent testing set compared to the average R² obtained during the five-fold CV on the training set. This typically suggests that some of the models trained in a five-fold partition may have been adversely affected by experimental outliers, thereby the average CV score is penalized, i.e., the evaluation remains sensitive to the specific data split. To mitigate the bias introduced by a single K-fold partition and provide a thorough assessment of the model’s stability, ShuffleSplit CV is subsequently implemented. By executing 100 independent random shuffles and splits of testing sets, ShuffleSplit provides a sufficiently large sample size to observe the probability distribution of the model’s performance on diverse data combinations.

The results of ShuffleSplit are shown in Figure 12. The RF and XGBoost models show satisfactory predictive accuracy, with a mean R² of 0.777 and 0.855, respectively. Especially for the tuned XGBoost model for θ_u, while occasional unfavorable data partitions yield R² values as low as 0.65, the majority of the 100 iterations are concentrated within the 0.82 to 0.90 range, indicating a high accuracy and generalization capacity. For the RF model for θ_y, the wider distribution of R² indicates that the prediction of yielding drift is more sensitive to noise in data.

5. Discussion on the Machine Learning Results

5.1. Model Interpretation

5.1.1. Feature Importance

To quantitatively assess the contribution of each input parameter to the model’s predictive performance, the permutation importance (PI) method is employed. PI is a model-agnostic technique that evaluates feature importance by measuring the decrease in a model’s performance score when the values of a single feature are randomly shuffled. The results of the PI analysis are shown in Figure 13.

The comparison of two-stage and end-to-end frameworks is explained here with the feature importance. A significant divergence between the classification task and the regression tasks is observed: for the XGBoost model in classification, the shear span ratio (λ) dominates the contribution (0.593), indicating its role as the primary determinant of failure mode transitions (e.g., from shear to flexural failure); in contrast, the regression models, both within the two-stage and end-to-end frameworks, show highly consistent reliance on material strength (f_yl) and axial load ratio (n_t). This observation indicates that while λ governs the type of failure, the deformation capacity is predominantly controlled by material properties and loading conditions. Moreover, the high similarity between two-stage and end-to-end regression rankings and the low importance of predicted failure mode in the regression model of the two-stage framework further explain the negligible improvement of the two-stage approach: high-performance regressors like XGBoost inherently capture the categorical boundaries within their decision structures, rendering explicit classification guidance less important for accuracy.

5.1.2. SHAP Analysis

To move beyond the “black-box” nature of ML models and ensure the physical consistency of the developed models, SHAP (SHapley Additive exPlanations) is utilized for post hoc interpretation. Grounded in cooperative game theory, SHAP provides a mathematically rigorous framework to attribute the contribution of each input feature to the final prediction. By calculating the SHAP value for each input feature, the method quantifies the change in the model output when a specific feature is observed versus when it is absent.

The SHAP plots for the two regressors are visualized in Figure 14. It is found that θ_y is predominantly governed by the strength of longitudinal reinforcement f_yl, which could be explained by the fact that the yielding of longitudinal reinforcement leads to the stiffness beginning of significant stiffness degradation; θ_u is primarily controlled by the axial compression ratio n_t, as the ultimate deformation capacity is mainly controlled by concrete crushing, which tends to happen at lower drift level for columns under higher axial compression; concrete compressive strength f_ck has a negative influence on both θ_y and θ_u, indicating that although columns with high-strength concrete (HSC) may behave higher capacity, their deformation capacity may be compromised because of the brittleness of HSC. These observations are generally consistent with engineering knowledge and prove that the ML models have effectively learned the underlying mechanisms of the deformation capacity of SRC columns.

5.1.3. Partial Dependence Plots

Partial dependence is also plotted in Figure 15 for θ_y and θ_u against three input parameters with the highest permutation importance. The same phenomenon as shown in SHAP analysis is observed: θ_y primarily increases with f_yl, which governs the onset of plasticity, and the ultimate drift θ_u decreases with the axial load ratio n_t, reflecting how high compression accelerates P-Δ effects and structural collapse. Additionally, both θ_y and θ_u show a threshold effect for section height h, where deformation capacity stabilizes once the member size is sufficient to mitigate localized instability.

5.2. Comparison with Conventional Approaches

The ML-based deformation capacity prediction models developed in this study are compared with the conventional data-fitting-based model results reported in the literature. The conventional models include both end-to-end models [73] and two-stage models [17], i.e., failure mode classification and deformation capacity regression, and detailed formulas can be found in the corresponding literature. Since both studies provided methods for calculating the drift ratio at 80% of the peak load in the post-peak stage, the ultimate deformation limit defined in this study is derived by linearly interpolating between the drift at the peak load and the drift at 80% of the peak load. Test results from [73] are used for the comparison.

Figure 16 and

Table 13. Statistics of performance of different models.

Model	θy			θu
	R2	RMSE	MAPE	R2	RMSE	MAPE
This study	0.834	0.0007	6.82%	0.824	0.0032	7.52%
[73]	0.659	0.0010	8.37%	0.620	0.0046	14.57%
[17]	−3.445	0.0036	26.87%	−1.005	0.0087	23.99%

show the results of the comparison. It is observed that the ML models proposed in this study achieve the best performance in both θ_y and θ_u prediction, followed by conventional end-to-end formulas, while the traditional two-stage model significantly underestimates the deformation capacity of SRC columns in shear failure modes, yielding the poorest fit.

5.3. Uncertainty Evaluation

When constructing fragility curves, in addition to accurately predicting the deformation capacity of components, it is also necessary to assess the uncertainty of the predictions. Therefore, the proposed framework should provide not only deterministic point predictions but also quantified uncertainty. This is achieved by the following: (1) for the RF model, the variance across individual decision trees is used; (2) for the XGBoost model, the variance of prediction is obtained through a bootstrapping approach with 100 iterations.

Figure 17 shows the comparison of the predicted values in the testing set, together with 95% prediction intervals (PIs), compared with the true value. It is found that the majority of true values fall in the 95% prediction interval, demonstrating the reliability of the model for stochastic resilience assessment.

Figure 18 displays the distribution of residuals. It is observed that most of the residuals are centered around zero, and their distribution is approximately normal, indicating reliable and stable prediction performance.

6. Conclusions

In this paper, an ML-based framework is proposed for the prediction of the deformation capacity of SRC columns, based on a comprehensive experimental database. The main conclusions are as follows:

• A test database of 312 SRC column specimens is constructed; the failure modes and range of parameters cover the four typical failure modes and the common range of engineering practice, providing a solid data foundation for ML modeling.
• Feature selection combining Pearson correlation analysis and VIF effectively eliminates redundant variables and mitigates multicollinearity among initial parameters. A final set of 10 core features is identified, ensuring physical interpretability and significantly enhancing models’ accuracy and generalization capability.
• Four ML models (SVM, ANN, XGBoost and RF) and two frameworks (two-stage and end-to-end) are compared. It is found that the performance improvement in the two-stage framework is insignificant; thus, the end-to-end framework is finally adopted to avoid classification error propagation. RF for θ_y and XGBoost for θ_u prediction demonstrated optimal overall performance, reaching an R² of 0.856 and 0.885 respectively. Hyperparameter optimization and ShuffleSplit cross-validation confirmed the model’s robustness and generalization ability, effectively avoiding overfitting.
• Model interpretation is conducted with the PI and SHAP analyses, revealing that the models’ decision logic aligns with engineering principles. Shear span ratio λ is the key factor determining failure modes, while longitudinal reinforcement strength f_yl and axial load ratio n_t dominate the deformation capacity. This confirms that the ML models are capable of capturing the underlying mechanical mechanisms.
• Compared with conventional data-fitting formulas (both end-to-end and two-stage) from the literature, the proposed ML models show significantly improved prediction accuracy. The R² values for predicting θ_y and θ_u are improved by 26.5% and 32.9%, the RMSE is reduced by 30.0% and 30.4%, and the MAPE is reduced by 18.5% and 48.4%, respectively, highlighting the power of data-driven approaches in handling complex nonlinear problems.
• The proposed model is applicable to practical engineering, as the parameters cover geometric and mechanical parameters commonly used in actual SRC columns. However, for parameters that are beyond the range of this database, the applicability of the trained model still needs further validation.