Rapid seismic damage prediction of prestressed concrete bridge columns using validated machine learning models

Abstract

This study presents a machine learning–based framework for predicting seismic damage states in partially prestressed reinforced concrete bridge piers. A database of 2163 numerically simulated bridge columns was developed using Latin Hypercube Sampling, covering wide variations in geometry, material strengths, axial load ratios, and prestressing levels representing a diverse range of demand–capacity conditions. Ten supervised learning algorithms including support vector machines, neural networks, random forests, gradient boosting methods, and CatBoost were trained and evaluated through multiple statistical metrics. Model predictions were validated against experimental results for selected column specimens, demonstrating strong agreement between simulations and physical behavior. Among the evaluated models, the Deep Neural Network (DNN) exhibited the highest overall predictive accuracy, achieving Pearson correlation coefficients of 0.982, 0.998, 0.9996, 0.988, and 0.9989 for residual displacement, residual force, force at collapse drift, hysteretic energy, and maximum base-shear, respectively. The corresponding RMSE values were 0.0275, 0.0023, 0.0036, 0.0281, and 0.0293. Although, CatBoost and Artificial Neural Network showed slightly better performance than the DNN in predicting hysteretic energy, the DNN remained the most reliable model across the structural responses examined. Overall, it provides a robust and consistent tool for damage estimation and performance assessment of prestressed Reinforced Concrete bridge piers subjected to seismic loading.

Graphical abstract

Keywords

prestressed reinforced concrete columns / machine learning / seismic response / damage limit

1 Introduction

Bridges are recognized as one of the most important engineering structures, as they connect different parts of cities and countries. During earthquakes, bridges may experience large deformations and dissipate earthquake-induced energy at the expense of permanent structural damage, such as the formation of plastic hinges. Consequently, the damage resulting from the nonlinear behavior of bridge components and materials can compromise bridge functionality, imposing direct costs associated with repair and retrofitting, as well as indirect costs due to disruptions in transportation, particularly in the immediate aftermath of an earthquake, when emergency vehicles need to access the affected area [1,2]. Bridge piers are among the most critical components of a bridge’s structural system, as they largely govern the overall performance of the bridge under seismic loading. Therefore, accurately identifying and assessing their probable performance is a fundamental element of performance-based seismic design. In highly seismic-prone regions, bridge piers must be capable of sustaining inelastic deformations, exhibiting high ductility, and dissipating energy in order to prevent severe damage and structural collapse [3–5].

Many efforts have been conducted to improve seismic performance of bridge piers and reduce the vulnerability of the overall bridge system [6–8]. Among those, Self-centering post-tensioned (SCPT) piers have been introduced as an effective solution to enhance bridge performance by improving post-earthquake functionality. Their behavior is governed by a controlled rocking mechanism combined with the self-centering action provided by post-tensioned (PT) tendons and the weight of the superstructure. In SCPT piers, softening and nonlinear response originate primarily from the rocking joints at the pier base, representing geometric rather than material nonlinearity, which helps prevent structural damage and significantly reduces residual deformations. When supplemented with an energy mechanism, SCPT piers demonstrate adequate energy-dissipation capacity and favorable seismic performance. Nonetheless, the incorporation of energy dissipators can introduce a limited amount of residual deformation in these systems [9–11]. In self-centering systems with unbonded PT strands during crack propagation, the strands return to their original position while energy dissipation is sustained through bonded reinforcement [12–14]. Experimental works have confirmed the efficiency of unbonded PT strands in reducing residual displacement at the cost of ultimate strength [15]. Sakai et al. [16] conducted pseudodynamic and seismic experimental analyses on Unbonded Bar Reinforced Concrete (UBRC) specimens previously developed by Iemura et al. [17]. The results exhibited stable cyclic behavior and limited permanent deformation of UBRC specimens. In other studies, self-centering behavior was examined in precast segmental piers, demonstrating good drift capacity and repeatability in re-centering for various tendon configurations of multi-segment piers [18–20]. These findings were later extended by experimental and numerical research on residual drift and crack width by Sun et al. [21].

Despite notable advancements, physical testing cannot cover the full range of column geometries, material properties, corrosion levels, and loading conditions relevant to self-centering bridge piers. High-fidelity finite-element (FE) simulations are often utilized to explore this range of parametric space; however, they remain computationally expensive and frequently encounter convergence challenges when modeling complex PT–Reinforced Concrete (RC) interactions across large data sets. Although early studies applied FE analysis for seismic vulnerability assessment, the high computational cost limits their practicality for large-scale evaluations. Moreover, numerous investigations have relied on FE simulations to examine the seismic behavior of self-centering reinforced concrete bridge piers, yet these approaches still face the same limitations in efficiency and scalability [22].

To overcome these challenges, machine learning (ML) data-driven solutions have been created as high-performance replacements to FE analysis capable of rapid and precise prediction of structural response and capacity. Other areas of civil engineering have already taken advantage of the computational efficiencies provide by ML-based models that delete the need for traditionally intensive analytical procedures. ML models have been used to predict earthquake-related building shifts, damage classification measure and stiffness capacity [23–27]. Wakjira et al. [28] proposed a data-driven approach for predicting the load-bearing and flexural capacities of RC beams strengthened with fabric-reinforced cementitious matrix (FRCM) composites. Seven ML models, including kernel ridge regression, k-nearest neighbors (KNN), support vector regression, classification and regression trees (CARTs), random forest (RF), gradient-boosted trees, and extreme gradient boosting were evaluated to identify the most accurate predictive model for FRCM-strengthened beams. Moreover, Wakjira and Alam [29] developed a ML powered predictive model for the drift ratio limit states of ultra-high-performance concrete (UHPC) bridge piers through four damage states. They generated a comprehensive database of 10000 UHPC bridge columns, accounting for uncertainties across a wide range of design parameters, including column geometry, material properties, reinforcement ratio, and axial load ratio. The drift ratios corresponding to the onset of four distinct damage states were determined. The performance of the developed model was evaluated using multiple statistical measures, including the composite fitness score, coefficient of determination, root mean squared error (RMSE), normalized RMSE, and mean absolute error. The results demonstrated that the model accurately predicted drift ratio limit states for both the training data and previously unseen test data sets. Furthermore, Wakjira et al. [30] proposed a novel hybrid ML-based stress-strain constitutive model for UHPC confined with normal-strength steel (NSS) or high-strength steel (HSS) spirals. The proposed hybrid ML model showed high accuracy as well as high coefficients of determination (R²) in predicting stress-strain behavior of UHPC with NSS or HSS spirals. In another study, ML has been used to evaluate earthquake risks and post-event recovery potential in UHPC bridges, to predict hoop strain in fiber-reinforced polymer-wrapped concrete columns, and to forecast peak strength and deformation behavior in confined UHPC using synthetic data sets generated by conditional tabular GANs and optimized with intelligent search strategies [31–34].

On the other hand, only a small number of studies have used ML-based predictive models as useful engineering tools through graphical user interfaces (GUI) or web-based platforms, despite the models’ explosive growth in structural and earthquake engineering [35–37]. By allowing engineers to make quick predictions without the need for sophisticated programming knowledge, a number of recent studies have shown that integrating trained ML models into approachable computational tools can greatly increase their accessibility and practical impact. However, a research gap exists in the use of ML regression methods for predicting the limit states of self-centering bridge columns. Considering the variability in design of self-centering bridge pier, including material and geometric properties as well as performance requirements, there is a clear need for predictive tools capable of providing reliable estimates of performance-based damage limit states of these components. The objective of this study is to implement different ML techniques to properly capture the different seismic response of partly prestressed reinforced concrete (PRC) bridge columns. For this purpose, several classes of ML techniques such as KNN, Catboost (CB), decision tree (DT), Artificial Neural Network (ANN), support vector machine (SVM), RF, Deep Neural Network (DNN), Multi-Layer Perceptron (MLP), Feedforward Neural Network (FNN) and Deep and Cross Network (DCN) were employed to properly capture the different seismic response of PRCs. First, finite element models of seven experimental specimens of RC and PRC columns were developed and validated against experimental results. Then, data set of 2163 RC and PRC columns were developed (after excluding extreme or unrealistic cases) using finite element simulation and the residual displacement, residual force, the force corresponding to collapse drift, energy dissipation capacity, and maximum base shear of the columns were predicted through ML models. To further evaluate the predictive capability of the ML models, an extended external validation was conducted using a supplementary data set consisting of 36771 data points derived from residual drift and residual force measurements across drift levels ranging from −4% to +4% for all RC and PRC specimens and compared to the experimental results. Figure 1 shows a schematic diagram of the overall research process.

2 Database development of prestressed column seismic response

2.1 Finite element modeling of prestressed reinforced concrete columns

A two-dimensional nonlinear FE model representing the PRC column was developed using OpenSees platform [38]. Displacement-based nonlinear Beam−Column elements with fiber-discretized sections were used for modeling the RC column. The column was divided into two elements; the bottom element utilized three integration points and whereas the upper element employed five integration points [39,40]. The column was assumed to be fixed at the end in all degrees of freedoms. It was assumed that the column exhibits the idealized cantilever behavior and the plastic hinges were formed at the end of the column, near the point of fixity (Fig. 2(a)). The length of the plastic hinge in the columns was estimated according to the equation recommended in Caltrans Seismic Design Criteria [41] which considers the effects of strain localization and softening:

(1)L_{\mathrm{P}}=0.08L+0.022f_{\text{ye}}{d}_{\text{bl}}\ge {0.044f}_{\text{ye}}{d}_{\text{bl}},

where L is the column length, f_{\text{ye}} is the expected yield strength of longitudinal reinforcement, d_{\text{bl}} is nominal bar diameter of longitudinal reinforcement (Fig. 2(b)).

The Concrete07 constitutive material model that considers tensile strength with slow post-peak softening was used for modeling concrete. The compressive strength and strain for the confined and unconfined concrete were defined according to the equation developed by Cheng and Mander [42]. As observed in Fig. 2(b), in the Concrete07 material model, the nonlinear behavior of concrete is characterized using the compressive strength, f_{\mathrm{c}}, the corresponding strain, {\epsilon }_{\mathrm{c}}, and the initial elastic modulus, E_{\mathrm{c}}. The crushing strain, {\epsilon }_{\text{cr}} defines the termination of the descending branch in compression, while the tensile strength, f_{\mathrm{t}}, and its corresponding strain {\epsilon }_{\mathrm{t}} represent the tensile response. In addition, the tension-softening parameter X_{\mathrm{n}} governs the post-peak tensile behavior, and the curvature parameter, r_{\mathrm{k}} controls the shape of the descending branch of the stress–strain curve. The steel reinforcements were modeled using Steel02 material based on the Giuffre-Menegotto-Pinto constitutive model, which provides a smooth transformation between the elastic phase and the after-yield [43] (Fig. 2(c)). For steel reinforcement, the Steel02 material model was defined by the yield stress, f_{\mathrm{y}}, the elastic modulus E, and the post-yield hardening modulus E_{\mathrm{p}}, allowing for an accurate representation of the elastoplastic behavior with strain hardening. Furthermore, the steel material constants shown in Fig. 2(d) include the yield stress, f_{\mathrm{y}}, ultimate stress, f_{\mathrm{u}}, yield strain {\epsilon }_{\mathrm{y}}, strain at the onset of strain hardening, {\epsilon }_{\text{sh}}, and the ultimate strain, {\epsilon }_{\mathrm{u}}. The elastic modulus in the initial linear range, E_{\mathrm{s}} and the strain-hardening modulus, E_{\text{sh}} together describe the full stress–strain response of the steel from the elastic region through yielding and hardening up to the ultimate stage. Corotational Truss element was used for modeling the prestressed strands, this element takes into account geometric nonlinearities. PT strands are tensile members so the constitutive material model, which is assigned should not include compression. An initial strain was incorporated into the material to account for the initial prestressing.

2.2 Validation of finite element modeling

For validation of finite element modeling, the experimental study conducted by Sun et al. [21] was utilized. Seven 1:4-scale RC bridge pier specimens, configured as cantilevers, were designed and tested to experimentally examine the seismic behavior of piers equipped with vertical unbonded strands under cyclic loading (see Fig. 3). The specimens had 300 mm-diameter circular sections and were anchored in heavy RC footings. As shown in Fig. 3(a), their height to the lateral loading point was 1100 mm, corresponding to an aspect ratio of 3.67, which generally results in flexural failure. As shown in Fig. 3(a) Specimen RC-1 represented a conventional RC bridge pier without prestressing strands and served as the reference specimen for comparison with the prestressed specimens. It was reinforced with eight 12-mm-diameter longitudinal mild steel bars evenly distributed around the perimeter, while 8-mm-diameter mild bars spaced at 75 mm were used as transverse reinforcement, corresponding to a transverse reinforcement ratio of 1%. Specimen PRC-1 was the standard prestressed specimen and was identical to RC-1 except for the inclusion of prestressing strands. Four 12.7-mm-diameter unbonded strands (each with a nominal area of 98.7 mm²) were used, providing a total prestressing force ratio of 0.1. The strands were arranged in a square configuration with a side length of 135 mm. The reinforcement layout is shown in Fig. 3(d). Specimen PRC-2 was identical to PRC-1 except for the longitudinal mild reinforcement. As shown in Fig. 3(e), eight 8-mm-diameter longitudinal mild bars were used to examine the influence of mild reinforcement on the seismic performance of prestressed piers. Specimen PRC-3 (Fig. 3(b)) was designed to assess the effect of the amount of prestressing. Only two strands were used, while all other parameters remained the same as PRC-1. Specimens PRC-4 and PRC-5 were intended to evaluate the influence of initial prestressing level. PRC-4 had a reduced prestressing force ratio of 0.05, whereas PRC-5 had an increased ratio of 0.15. Their reinforcement details were otherwise identical to PRC-1 (Fig. 3(d)). Specimen PRC-6 incorporated four prestressing strands concentrated at the center of the section to investigate the effect of strand arrangement, as shown in Fig. 3(c). In total, three different prestressing strand layouts were considered and are summarized in Fig. 3. It should be noted that the prestressing strands were placed inside the longitudinal mild bars, providing a substantially larger concrete cover compared with the mild reinforcement. Consequently, the strands were expected to be better protected against corrosion.

Table 1 summarized the design details of the tested specimens. As observed, the compressive strength of concrete material was 55.9 MPa and the yielding strength of longitudinal and transverse steel reinforcements were 517 and 453 MPa, respectively. The ultimate strength of the prestressed strand was 1939 MPa.

The column specimens were subjected to lateral cyclic loading while maintaining a constant axial load. Cyclic lateral displacements were applied corresponding to drift levels of 0.32%, 0.5%, 1%, 1.5%, 2%, 2.5%, 3%, and 3.5%. In the numerical models, the columns were analyzed under quasi-static, displacement-controlled loading following the same cyclic protocol as the experiments. The lateral load and residual drift results from the finite element models and experiments for RC and PRC specimens are compared in Figs. 4 and 5, respectively. The agreement between the results demonstrates that the numerical modeling within the finite element framework was effectively capable of simulating the nonlinear behavior of both the conventional RC column and the column reinforced with prestressed strands.

3 Database development

To develop a comprehensive database of PRC bridge columns representing typical single circular RC columns in highway bridges, ten variables were considered due to their significant influence on the structural and seismic behavior of RC columns. Initially, a larger set of simulations was generated to cover the full spectrum of geometric, material, axial load, and prestressing parameters. From this initial set, simulations with extreme or unrealistic combinations of parameters that could lead to non-physical behavior or numerical instability were excluded. This filtering resulted in a final data set of 2163 simulations, which ensures sufficient variability for reliable ML training while maintaining practical computational cost and data quality. A PRC column, which combines conventional reinforcement steel with prestressed strands, integrates the ductility of reinforced concrete with the stiffness and crack-control benefits of prestressing. Under cyclic loading, each variable affects residual displacement, residual force, force at collapse drift, energy dissipation capacity, and maximum base shear. Increasing concrete compressive strength delays crushing and enhances energy dissipation by improving compression capacity, stiffness retention, and confinement behavior. The column diameter and height influence its stiffness, stability, and curvature distribution: a larger diameter increases hysteretic energy and collapse resistance, while a taller column increases residual deformations and flexibility. The axial load governs the compression-flexure interaction, with excessive axial loads reducing ductility. The initial prestressing stress and strands area determine the total prestressing force, contributing to pre-compression, delayed cracking, and increased stiffness. Similarly, higher prestressing ratios reduce residual displacements and enhance collapse strength, but may limit hysteretic energy dissipation [15,16,21,44–46]. The ranges and probability distributions of these variables are summarized in Table 2. It should be noted that the ranges selected for the variables in Table 2 were chosen to ensure the development of a comprehensive data set representing the full spectrum of RC and prestressed concrete bridge columns commonly used in practice. The minimum and maximum values for geometric (column diameter, height, rebar diameter), material (concrete strength, prestressing stress and area), and loading parameters (axial force and axial force ratio) were defined to cover all plausible configurations, from small- to large-scale members. This wide range captures both typical and extreme design cases, improving the robustness and generalizability of the ML models for seismic performance assessment. Therefore, the ranges of variables listed in Table 2 were selected based on standard design practices for prestressed RC bridge piers and experimental data reported in Refs. [16,21,47].

A comprehensive data set comprising 2163 PRC bridge columns was generated using Latin Hypercube Sampling to ensure a representative and stratified distribution across the input space. Each column in the data set is defined by ten independent variables, sampled from assumed uniform distributions. Figure 6 presents histograms illustrating the statistical distribution of input parameters.

The displacement-controlled cyclic analyses were conducted up to a maximum drift ratio of ±5%. This ratio was chosen to represent large inelastic deformation demands in PRC columns with self-centering behavior. PRC columns are usually designed to handle significant drift by allowing controlled gap opening and tendon elongation while keeping residual deformation to a minimum when unloaded. Previous experimental and numerical studies on PRC and self-centering column systems have indicated that drift ratios of about 5% are needed to fully activate gap opening, develop prestressing force, and trigger energy dissipation mechanisms [15,16,21,47]. Therefore, the chosen target drift supports a thorough assessment of strength loss, energy dissipation, and re-centering ability during severe seismic loading. Figure 7 illustrates the displacement-controlled cyclic loading protocol applied at the top of the PRC column in terms of increasing drift ratios. For PRC columns, residual drift mainly shows the balance between prestressing restoring forces and inelastic deformation in the energy-dissipating parts. The residual drift noted in this study is not linked to one maximum drift level. Instead, it builds up over the entire cyclic loading history, with the final residual drift measured after completing the ±5% drift cycle.

Figure 8 presents a schematic force–displacement response for one loading cycle highlighting response parameters, including residual drift, residual force, energy dissipation per cycle, force at collapse, and maximum base shear. It should be noted that in a single loading cycle, the force at collapse is equal to the maximum base shear (as shown in Fig. 8), whereas under multiple loading cycles, these forces are not necessarily identical. The output responses within the developed database are displayed in Fig. 9.

4 Machine learning–based seismic response identification of PRC columns

The seismic response prediction in PRC columns is a challenging process, especially if prestressing methods are utilized. To address this issue, in this paper, 10 supervised ML classification algorithms such as SVM, ANN, DNN, KNN, FNN, RF, CB, DT, DCN and MLP were used to classify the seismic behavior of RC columns in terms of their material and geometric parameters obtained from experimental tests were preprocessed and normalized to input features for the proposed models. These algorithms were intentionally chosen to represent a broad spectrum of learning paradigms, including linear, nonlinear, tree-based, ensemble, and deep-learning approaches, allowing for a comprehensive and unbiased comparison of predictive performance and robustness across fundamentally different model families. For evaluating model performance on novel data, the data set was split into training and test subsets in an 80:20 ratio. Additionally, input features were scaled using a Min-MaxScaler for better performance of some models such as ANN and K-NN. The training set that consisted of the majority of data was used to train the regression models, while the test set was reserved for evaluating their predictive precision on novel data, providing an estimate of each model’s generalization capability. While hyperparameter tuning, 5-fold cross-validation was employed to provide stable model evaluation and prevent overfitting. Grid Search was used in combination with 5-fold cross-validation to systematically identify the optimal hyperparameters for each model, ensuring robust performance and generalization. During this process, the training data set was divided into five equal portions. Each model configuration was trained five times, with each iteration using four portions for training and the fifth portion as a validation set. The performance metric from all 5-folds was averaged with equal weighting to provide a representative evaluation score. This cross-validation method eliminates potential bias that could arise from tuning models against the ultimate test set directly, enhancing the generalizability of the selected hyperparameters. More importantly, this method is balanced between computational efficiency and performance.

4.1 K-nearest neighbors

The KNN is an algorithm for both classification and regression that calculates the distance between the nearest neighbors. K-nearest rule is a concept originally proposed in the context of pattern recognition and later formalized, demonstrating that KNN can asymptotically achieve the performance of a Bayesian classifier when provided with a sufficiently large number of training samples [48,49]. By exploiting the information of the K nearest neighbors of the query point, KNN regression approximates the underlying relationship between the input features and the target variable. To generate a prediction for a given data point x_i, the algorithm first calculates the distance between the query point and all training samples, and then ranks these distances. The K closest samples constitute the neighborhood N_0.

A weighted prediction for regression can then be formulated using the Euclidean distance as:

(2)D=\sum _{i=1}^k{(x_i-y_i)}^2.

Predicted values are calculated based on the (potentially weighted) average of the target responses within N_0. The choice of K plays a critical role in the performance of the model: small values of K tend to produce low bias but high variance, making the model sensitive to noise and prone to overfitting, whereas large values of K increase bias and reduce variance, potentially leading to underfitting. As a result, determining an optimal value for K is essential for achieving reliable predictive performance.

4.2 CatBoost

ML algorithm CB was developed by Yandex to handle categorical and heterogeneous data. Through ordered boosting and Ordered Target Statistics, which calculate target means sequentially for categorical features, the algorithm reduces data and label leakage. With CB, categorical features can be processed without manual transformations, such as One-Hot Encoding, and large data sets can be accelerated by graphics processing unit. Using symmetric trees improves performance and prevents overfitting; key parameters include tree number (iterations), learning rate, tree depth, and loss function. As well as incorporating greedy feature combinations to address gradient boosting issues like overfitting and label leakage, CB can also be tuned with grid or random search, depending on the data set. Its efficiency, accuracy, and ease of handling categorical data make it suitable for many real-world applications.

4.3 Decision Tree

The DT is a nonlinear, rule-based algorithm that is available for both classification and regression tasks. It is commonly called CARTs. Based on a series of splitting rules arranged in a tree-like style, the CART framework divides the feature space into multiple smaller, non-overlapping regions with similar response values. Nodes define tests applied to data attributes, and branches correspond to the results of those tests. The root node identifies the most important feature. It is particularly useful when there is a nonlinear relationship between predictors and responses.

Given a set of training samples \{(x_i,y_i){\}}_{i=1}^n, where x_i\in X and y_i\in Y, DT regression recursively partitions the input feature space into disjoint regions R_{\mathrm{k}}, each associated with a leaf node of the tree, and estimates the response value within each region. Despite this, DTs often exhibit high variance and overfitting, making them inefficient. This limitation can be addressed by using various ensemble-based methods.

4.4 Artificial neural networks

The ANN is a computational model derived from biological neural networks that captures the complex interactions between inputs and outputs to support classification and regression tasks [50]. By adjusting the weights of connections between neurons, an ANN can detect hidden patterns in the data by learning by adjusting the input layer, hidden layer, and output layer.

Each neuron computes a weighted sum of its inputs plus a bias term, which is then passed through an activation function \phi:

(3)y_k=\phi \left[\sum _{j=1}^m{w}_{kj}{\mathit{x}}_j+b_k\right],

where w_{kj} are the connection weights, {\mathit{x}}_j are the input signals, b_k are the bias, and m is the number of inputs. Activation functions, such as ReLU, introduce nonlinearity, allowing the network to model complex relationships and avoid issues like vanishing gradients that occur with functions like sigmoid or tanh. Optimization algorithms, such as Adam, iteratively update the weights to minimize a loss function, improving convergence and learning efficiency. By stacking multiple hidden layers, ANN can extract hierarchical features and represent highly nonlinear mappings between input and output, making them powerful tools for predictive modeling

4.5 Support vector machines

ML algorithms Support Vector Machines (SVM) are used for classification and regression [51]. SVM uses the closest data points to construct hyperplanes that maximize the margin between classes. Data that is linearly inseparable can be mapped into higher-dimensional spaces by kernel functions like polynomials, Gaussian (RBF), and sigmoid.

In regression (SVR), the RBF kernel is commonly used for modeling nonlinear relationships. Key hyperparameters include C (model complexity), ε (insensitive loss margin), and γ (kernel influence). Lower C or higher ε simplifies the model, while appropriate γ controls the effect of individual points. The StandardScaler standardizes features and the GridSearchCV identifies the optimal hyperparameter combinations. SVR with RBF kernels can capture complex relationships with high accuracy and manageability.

SVR employs a mapping process to transform the input data into a high-dimensional feature space. Given \mathrm{n}number of training examples \{({\mathit{x}}_i,y_i){\}}_{i=1}^n\in {\mathbb{R}}^n\times \mathbb{R}, SVR estimates the regression function f(x) in Eq. (4) by minimizing the regularized risk function in Eq. (5) subject to Eqs. (6) and (7) [52]:

(4)f\left(x\right)=\mathit{w}\cdot \mathit{\varphi }\left(x\right)+b,

(5)\tau (\mathit{w},\xi ,{\xi }^{\ast })=\frac{1}{2}{\Vert \mathit{w}\Vert }^2+C\frac{1}{n}\sum _{i=1}^n\left(\xi +{\xi }_i^{\ast }\right),i=1,2,\dots ,n,

(6)\left(\mathit{w}\cdot \mathit{\varphi }\left(x\right)+b\right)-y_i\le \epsilon +{\xi }_i,i=1,2,\dots ,n,

(7)y_i\left(\mathit{w}\cdot \mathit{\varphi }\left(x\right)+b\right)\le \epsilon +{\xi }_i^{\ast },i=1,2,\dots ,n,

(8){\mathit{x}}_i\in X\subseteq R^n,y_i\in Y\subseteq R,

(9){\xi }_i,{\xi }_i^{\ast }\ge 0,

where {\xi }_i and {\xi }_i^{\ast } are slack variables, \mathit{w} and b are weight vector and bias, \epsilon is Vapnik’s insensitive loss that ignores errors smaller than \epsilon, and c is the regularization parameter. The prediction in SVR is given by [53,54]:

(10)\begin{array}{rl}f\left(x\right)=& \sum _{i\in SV}\left({\alpha }_i-{\alpha }_i^{\ast }\right)K\left({\mathit{x}}_i,x\right)\\ & +b\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}0\le {\alpha }_i\le C,0\le {\alpha }_i^{\ast }\le C,\end{array}

where K({\mathit{x}}_i,x) is the kernel function, {\alpha }_i and {\alpha }_i^{\ast } are the Lagrange multipliers, and SV denotes support vectors, which are subsets of training data.

4.6 Random Forest

For classification and regression tasks, RF combines multiple DTs to improve prediction accuracy. For classification and regression, trees are trained on randomly selected subsets of data and features, and predictions are aggregated based on a majority vote or an average. Each tree is trained on a bootstrap sample of the original data set, which is a random sample drawn with replacement. Through the introduction of diversity among trees, overfitting is reduced, generalization error is lower, and model robustness is increased. It is possible to split trees based on criteria like information gain (classification) or variance reduction (regression). At each node, a random subset of input features is selected from the bootstrap sample, and the best feature from this subset is used to perform the split. This two-step procedure introduces further randomness and diversity into the forest. Number of estimators, maximum depth, and minimum samples per node are key hyperparameters. This algorithm uses grid search to select optimal values, balancing complexity with generalization. RF uses bagging and feature randomness to achieve a powerful, flexible prediction. The final prediction of the RF is obtained by averaging the predictions of all trees (for regression) or by majority voting (for classification).

4.7 Deep Neural Network

The DNN captures highly nonlinear relationships in data and complex patterns by combining multiple hidden layers between the input and output layers. With forward propagation, each neuron in a layer communicates with neurons in the layer above and below, whose weights are initialized randomly. By comparing predicted outputs with actual values, the model adjusts weights to minimize errors during training. Unlike traditional neural networks, DNN are feedforward networks that can learn from mistakes. This makes them useful for the processing of unstructured data such as images, audio, and text [55]. For images, convolutional layers are used, Long Short-Term Memory for sequential data, attention is used in order to focus on important features, and normalization is used to stabilize training. Sequential architectures allow layers to be stacked, and optimizers like Adam improve convergence and accuracy for regression or classification tasks.

4.8 Multilayer Perceptron

MLP are neural networks containing multiple layers of neurons that combine to solve problems where there is no linear separation of data. They consist of several input layers, several hidden layers, and an output layer. The input layer feeds input features, the hidden layer processes them through weighted connections and biases, and finally the output layer transforms them into predictions. By using algorithms such as Gradient Descent, outputs are compared to actual values, and errors propagated backward are corrected. With activation functions like ReLU, the network can capture complex relationships. With optimizers like Adam, weights are adjusted efficiently, improving convergence. Pattern recognition, classification, and regression tasks can be performed with MLP, which can be configured according to the complexity of the problem by configuring the number of layers and neurons.

4.9 Feed forward Neural Network

In FNN, information flows unidirectionally from one layer to another, with an input layer, a hidden layer, and an output layer. Neurons in a layer are connected to neurons in adjacent layers, and their weights and biases can be trained. Nonlinear activation functions such as ReLU, Sigmoid, and Tanh are used to model complex relationships. In forward propagation, input data are passed through the network in order to produce predictions. Losses are calculated using loss functions, such as Mean Square Error and Cross-Entropy. Gradient descent techniques such as BatchNormalization, Dropout, and learning rate adjustment improve training stability, prevent overfitting, and accelerate training during backpropagation. Using FNN as a regression or classification algorithm, the output layer produces final predictions.

4.10 Deep and Cross Network

The DCN combines two main components: the Cross Network and the Deep Network, which capture explicit and implicit interactions in high-dimensional structured data. Through inner product operations, Cross Network explicitly models interaction between previous layers and the original input, allowing higher-order interactions. Similarly to a MLP, Deep Networks capture complex, implicit patterns using fully connected layers and nonlinear activation functions. DCN architectures can incorporate Attention Mechanisms to highlight important features, improving performance when the data are noisy or multi-dimensional. By combining the outputs of Cross and Deep networks either in parallel or stacked, the model can simultaneously learn low-level and high-level interactions. In DCN, the final output is generated by a dense layer, typically for regression tasks. Its design makes it suitable for recommender systems, user behavior analysis, and targeted applications based on complex relationships.

5 Grid search and cross-validation for hyperparameter tuning

Hyperparameters are essential in ML because they dictate a model’s structure and learning behavior. To create a model that is reliable and able to produce accurate predictions, these hyperparameters must be precisely adjusted. In this study, hyperparameter optimization was conducted using the Grid Search method, which owing to its systematic and exhaustive nature is regarded as one of the most reliable and commonly used approaches for fine-tuning ML models. To enhance the stability of the results and prevent overfitting, 5-fold cross-validation was employed alongside Grid Search. This process divides the training data into five roughly equal subsets for each combination of hyperparameters. The model is trained on four subsets and validated on the fifth in each iteration. This process is repeated for all five subsets, and the final performance of each hyperparameter configuration is computed as the average of the five validation results. To guarantee that the final assessment of the model is carried out on unseen data, the data set was also divided into training and testing sets using an 80/20 ratio. This framework was used to determine the hyperparameters listed in Tables 3–7.

Each hyperparameter examined in this study significantly influences model complexity and generalization capability. For instance, the number of neighbors K in KNN governs the balance between bias and variance; the depth of the tree in DT and CB directly affects the degree of overfitting or underfitting; and the number of layers and neurons in neural networks defines the learning capacity and the model’s ability to capture complex patterns. By choosing the best hyperparameter settings, overfitting and underfitting are simultaneously reduced and predictive accuracy is increased. Although default, recommended, or even arbitrary hyperparameter values can be applied, systematic tuning through Grid Search combined with cross-validation generally leads to improved model performance and greater generalization to unseen data.

6 Performance metric

To evaluate the effectiveness of the optimized ML model in predicting residual displacement, residual force, force at collapse drift, hysteretic energy, and maximum base shear, four commonly used performance metrics Pearson’s correlation coefficient (r), coefficient of determination (R²), mean absolute percentage error (MAPE), and root mean squared error (RMSE) were employed. The metrics are defined as bellows:

(11)\mathrm{P}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{n}\text{'}\mathrm{s}r=\frac{{\displaystyle \sum _{i=1}^n\left(Y_i^{\prime }-\overline{Y^{\prime }}\right)\left(Y_i-\overline{Y}\right)}}{\sqrt{{\displaystyle \sum _{i=1}^n\left(Y_i^{\prime }-\overline{Y^{\prime }}\right)}}\sqrt{{\displaystyle \sum _{i=1}^n\left(Y_i-\overline{Y}\right)}}},

(12)R^2=1-\frac{{\displaystyle \sum _i{\left(Y_i-Y^{\prime }\right)}^2}}{{\displaystyle \sum _i{\left(Y_i-\overline{Y}\right)}^2}},

(13)MAPE=\frac{100\mathrm{\%}}{n}\sum _{i=1}^n\frac{\left|Y_i-Y_i^{\prime }\right|}{\left|Y_i\right|},

(14)RMSE=\sqrt{\frac{1}{n}\sum _{i=1}^n{\left(Y_i^{\prime }-Y_i\right)}^2},

where Y_i^{\prime } and Y_i are the actual and predicted values from the regression model, respectively, \overline{Y^{\prime }} and \overline{Y} are their corresponding mean values, and n is the total number of observations.

Every one of the aforementioned metrics offers a different perspective on how well the model performs. Pearson’s r measures the strength and direction of the linear relationship between observed and predicted values. A value of r near 1 or −1 denotes a strong positive or negative linear correlation, while a value near 0 denotes little to no linear relationship. This metric is particularly useful when assessing how well the model captures the overall trend and variability of the data.

R² measures the proportion of variance in the response variable that can be predicted from the input features, serving as an indicator of the model’s explanatory power. A higher R² value, approaching 1.0, indicates a strong correlation between observed and predicted values, suggesting that the model captures a significant portion of the variance in the data. It is crucial to remember that although R² and Pearson’s r are both helpful measures of model fit and correlation [56].

RMSE measures the average magnitude of error by calculating the square root of the mean of the squared differences between the expected and actual response values. Furthermore, MAPE gives the assessment an additional viewpoint. Unlike RMSE, MAPE measures the absolute percentage error between predicted and actual values, providing an easy-to-understand measure of prediction accuracy. Due to its relative error expression, MAPE is particularly useful since it is less dependent on data scale.

7 Results of hyperparameter optimization

According to the Tables 3–7, optimal hyperparameter configurations and accuracy values for each algorithm can be found. Predictive models were chosen based on these configurations to minimize overfitting and underfitting, thereby enhancing their reliability and generalizability.

As shown in Table 3, the optimized parameter settings for each model differ significantly in terms of model complexity. Despite the DNN deep architecture (512, 256, 128, 64, 32), which makes it possible to extract high-level nonlinear features, it has the best performance (92.43%). Strong learning capabilities are also shown by the ANN with three hidden layers (256, 128, 64). With its configuration of depth 6, learning rate 0.05, and 200 estimators, CB achieves high accuracy 87.26% among tree-based models, while RF achieves comparable performance 88.34% with 100 trees and a small feature subset (m = 5), avoiding overfitting. Simpler models like SVM (RBF kernel, C = 100, ε = 0.01) and KNN (k = 7) perform noticeably worse, suggesting that they are not flexible enough to capture post-elastic displacement behavior. Residual displacement prediction is more accurate with deeper and hierarchical neural representations (MLP, FNN, and DCN) than shallower neural structures (MLP, FNN, and DCN).

According to Table 4, boosting and ensemble methods are superior for optimizing residual force. It achieves the highest accuracy of 93.45% with deeper trees (depth 8), an aggressive learning rate of 0.1, and 250 estimators enabling effective gradient-based refinement. Similarly, RF performs well with 150 trees and m = 6, reducing variance to 92.16%. The DNN and ANN architectures exhibit stable convergence with consistent layer structures, while deep networks exhibit strong performance but are marginally less powerful. On the other hand, SVM produces a poor accuracy of 56% even after tuning C = 150 and ε = 0.001, indicating the model’s sensitivity to noise and residual force nonlinearities. Further, FNN and DCN models confirm the necessity for greater depth and estimator counts due to their lower accuracies. Optimized residual-force predictions benefit from deeper trees and larger ensemble sizes.

According to Table 5, predicting the force at collapse drift involves higher nonlinearity, reflected in the hyperparameter trends. DNNs maintain the highest accuracy, 89.77%, owing to their multiple layers, which effectively model collapse-level nonlinearities. A calibration of CB with depth 5 and learning rate 0.1, combined with 180 estimators, demonstrates the appropriateness of moderate tree depth for this application. An 80-tree RF converges to an accuracy of 80.99%, highlighting the need for a reduced number of trees. ANN and MLP also benefit from their consistent 256–128–64 architectures. Because collapse drift involves discontinuous behavior that the hyperparameters are unable to adequately capture, models like KNN (k = 3) and SVM (C = 100, ε = 0.1) perform poorly. Limited depth limits the model’s capacity to generalize close to the collapse threshold, as further confirmed by shallower networks like FNN and DCN.

In Table 6, the optimized hyperparameters for hysteretic energy demonstrate the strong performance of CB, which achieves 92.57% by using deeper trees (depth = 7), a moderate learning rate (0.05), and a large ensemble of 300 estimators. RF, using 120 trees with m = 5, also performs well 91.53%. While DNN performs marginally worse, ANN achieves 89.12%, indicating that extreme depth is not required for this target. KNN (k = 10) shows reasonable accuracy, reflecting smoother response surfaces for energy-based prediction. On the other hand, lower accuracy is produced by MLP, FNN, and DCN, indicating that simpler architectures are unable to adequately capture cyclic hysteretic behavior.

As reported in Table 7, maximum base shear exhibits uniformly high predictability across models. Using 120 trees and m = 5, the RF model achieves its best accuracy at 97.43%. This result stems from an optimal balance between depth and feature sampling. ANN and DNN follow closely with accuracies of 97.26% and 97.34%, respectively, largely due to the stability provided by their multi-layer architectures. CB configured with depth = 6, η = 0.05, and 400 estimators achieves 96.30%, showing the value of large boosting ensembles. Classical models such as KNN (k = 5), SVM (C = 120, ε = 0.01), and DT (depth = 10) also perform strongly but remain below ensemble and deep-learning methods. FNN and DCN deliver competitive performance, albeit slightly lower. Trends in hyperparameters indicate that maximum base shear is primarily influenced by features effectively captured by deeper ensembles and multi-layer neural networks.

8 Performance evaluation of machine learning algorithms

To comprehensively assess the performance of the tuning configurations, four metrics were used including Pearson’s r, R², MAPE, and RMSE. These metrics were computed on both training and test data sets to monitor for potential overfitting or underfitting as presented in Figs. 10 and 11.

A 3D bar chart was employed to compare the predictive accuracy and reliability of various ML models in estimating structural response parameters of PRC columns under cyclic loading (Fig. 10). As seen, the DT shows the highest accuracy in predicting residual force and hysteretic energy, with R² = 0.996, MAPE = 0.066 and R² = 0.999, MAPE = 0.074, respectively. As a result of this superior performance, the DT model clearly captured nonlinear cyclic behavior, such as energy dissipation, stiffness degradation, and strength degradation. For residual displacement and force at collapse, the KNN model showed the weakest performance, R² = 0.636 and 0.810, it is highly sensitive to data noise and difficult to handle multi-dimensional nonlinear interactions among mechanical parameters, such as concrete compressive strength and prestressing level. Despite these limitations, KNN achieved relatively acceptable results in hysteretic energy, R² = 0.9623, MAPE = 0.2381, RMSE = 0.02158 and residual force (R² = 0.9473, MAPE = 0.440), indicating a limited ability to approximate the energy absorption and post-yield response of the column. For hysteretic energy, the SVM model showed moderate accuracy, with R² = 0.9012 and MAPE = 0.3881, and high accuracy for force at collapse drift (R² = 0.9930). Its residual displacement performance (R² = 0.7045, MAPE = 0.2609) and residual force performance (R² = 0.9735, MAPE = 0.440) were less consistent, due to its reliance on kernel function type and hyperparameter calibration, and its limited ability to capture the transition between flexural and shear-dominated behavior, the model has limitations. The DNN model provided the most accurate prediction of force at collapse drift (R² = 0.999, MAPE = 0.102), it demonstrated its ability to learn and represent complex relationships among axial stiffness, prestressing stress, and reinforcement yielding. Similarly, the ANN achieved high accuracy in predicting residual displacement (R² = 0.9394, MAPE = 0.093) and residual force (R² = 0.9934, MAPE = 0.1159), effectively capturing nonlinear cyclic behavior, pattern recognition, and the stiffness degradation trends typical of PRC columns under seismic-type loading. According to these results, model selection should take into consideration the target seismic response parameter. ANN showed strong ability to detect cyclic behavior, while DT and SVM need to be tuned carefully to represent post-elastic structural response optimally. In terms of maximum base shear, the RF, ANN, and CB models exhibited the highest predictive accuracy, exhibiting R² above 0.995 and MAPE below 0, while the KNN model had the lowest performance, showing R² of 0.924 and MAPE of 0.144. Similar to the top models, the DT model performed well, but was not as accurate as the top models. These models are superior because they capture complex nonlinear interactions among key structural parameters, such as concrete compressive strength, column geometry, rebar diameter, axial load, and prestressing level. As a result of these parameters, cyclic loading of PRC columns determines their peak lateral resistance, stiffness degradation, and post-yield behavior. KNN is less accurate due to its sensitivity to data noise, and its inability to model multidimensional nonlinear effects, which are critical for accurately predicting maximum base shear.

The comparative analysis of ML models is represented in Fig. 11 through two complementary visualization methods. The diagrams at the right side illustrate the error distribution for MAPE, Pearson’s r, R², and RMSE, revealing their central tendency, variability, and range for five prediction cases, whereas at the left side illustrates corresponding metric values for each model, facilitating direct comparison of each model’s prediction performance. Collectively, these plots provide a detailed assessment of the accuracy and error characteristics for the ML models. The analysis of the statistical indices in Fig. 11 reveals a direct relationship between the ML model type and its ability to reproduce the nonlinear seismic behavior of RC columns. MAPE results indicate that DNN and RF models provide the most robust and accurate estimation of the structural parameters, such as residual displacement, residual forces, force at collapse drift, hysteretic energy, and maximum base shear. According to these results, the models can simulate realistic nonlinear responses to seismic loading as well as post-elastic cyclic behavior. CB and DCN models also show strong performance in capturing stiffness degradation, strength deterioration, and cyclic force–deformation trends, reflecting the effectiveness of their adaptive learning structures in representing complex structural behavior. SVM, on the other hand, showed the highest mean errors and error dispersion, suggesting limitations in generalizing nonlinear responses. The KNN approach exhibits reduced accuracy near collapse conditions due to its sensitivity to noise and geometric variability. The performance of these methods was moderate, capturing general cyclic trends appropriately, but with reduced precision when considering nonlinear phenomena. From Pearson’s r perspective, deep neural architectures (DNN, MLP, FNN, DCN) achieved the strongest correlations between experimental and predicted responses, highlighting their ability to capture the interdependence of key structural parameters such as axial force, column stiffness, hysteretic energy, and maximum base shear. Flexural and shear responses, as well as energy dissipation, were effectively captured by tree-based models (RF, CB, and DT). In contrast, KNN and SVM performed less well in reproducing combined flexural-shear behavior under seismic loading due to their simpler structures. As measured by R², the DNN, RF, CB, and DCN models performed well, confirming their high reliability and stability. FNN, MLP, ANN, and DT showed acceptable accuracy but were less capable of reproducing post-elastic nonlinear responses. SVM performed inconsistently due to sensitivity to outliers and edge cases. DCN and DNN excelled in learning complex nonlinear dependencies, while RF and CB models achieved the lowest RMSE, indicating stable and reliable computational performance, delivering precise predictions of PRC column hysteretic behavior, energy dissipation, maximum base shear, and collapse mechanisms.

As shown in Figs. 12 and 13, the circles and triangles represent the training and testing data, respectively, for the evaluation of the predictive quality of many different ML models. An ideal prediction line is a dashed black line that perfectly aligns predicted and observed values. Models that achieve high accuracy have points highly concentrated near this line, mostly within the ±30% error band. Models that achieve lower prediction reliability have wider dispersion, and some points extend well into or beyond the ±60% error band. Of the models tested, DNN and ANN achieve the best level of agreement with observed values, and most of their predictions fall well within the ±30% error band. This provides a strong indication of effective generalization behavior and a minimal chance of overfitting or noise propagation. RF model also provides steady and reliable performance, as there is a prominent accumulation of points around the ideal line. Although FNN and SVM models have relatively less dispersion in total, there are some of their predictions that extend well beyond the ±60% error band. FNN and SVM models actually perform relatively poorly in relation to FNN model, and this is because FNN model achieves greater flexibility in modeling complex nonlinear interrelations in data. The KNN model has the maximum drift from the ideal prediction line. This arises mainly from the sensitivity of KNN model to noise and because it is a victim of the curse of dimensionality that impede its prediction accuracy in higher dimensional spaces of features. In addition, the corresponding histograms at the top and the main-right margins of the chart indicate a prominent similarity in the distribution of observed and predicted values for models like DNN, ANN, RF, and CB. This provides more evidence for their superior generalization capacity for both the learning and test data sets.

9 Preprocessing and scaling of the reinforced concrete and prestressed reinforced concrete column database

To effectively train the ML algorithms adopted in this study, the primary database, comprising 2163 RC and PRC specimens, was normalized using the scale factors summarized in Table 8. This preprocessing step ensured that the feature and target variable ranges of the current data set were consistent with those of comparable column specimens [21]. The application of these scale factors not only enhanced the numerical stability of the learning process but also enabled a direct and meaningful comparison between the predictive outcomes of the present models and the benchmark experimental results available in the referenced study.

Table 9 summarizes the minimum, maximum, and mean values of the 2163 column specimens obtained after applying the scaling factors listed in Table 8. As a result, the normalized domain of the present data set is in good agreement with the experimental results of [21]. Based on the alignment, the scaled data captures the variability and behavioral characteristics of the reference study, making comparisons between the two databases simple and straightforward. Consequently, the trained ML models developed in this research can be reliably employed to predict and evaluate the residual responses of the columns [21], ensuring both statistical and physical consistency across data sets.

10 Comparison of the experimental results and machine learning predictions

A final comparison between the experimental data and the ML predictions generated by the best-tuned models was conducted. For each RC and PRC column of the primary database, which includes 2163 specimens, residual drift and residual force values were systematically recorded across lateral drift levels ranging from −4% to +4%. In this way, a new data set comprising 36771 data points was generated to evaluate the predictive capabilities of the top-performing algorithms identified in this study, RF, CB, and DNN in estimating residual drift at the specified drift levels for corresponding RC and PRC columns [21]. A dispersion analysis and corresponding visualizations of these measurements are presented in Fig. 14. As shown in Fig. 14(a), although lateral drift values partially return toward zero following unloading, significant residual forces persist in many samples. This observation is indicative of inelastic structural behavior and the retention of residual stresses. The persistence of internal forces, even at zero drift, suggests that the structure does not fully recover its original configuration, pointing to permanent deformation. Additionally, the scattered distribution of samples with both high residual drift and residual force reflects the presence of localized and severe damage in specific structural regions. Figure 14(b) illustrates the relationship between peak drift and residual force. A clear trend is observed: as the initial drift increases, particularly beyond ±4%, both the magnitude and dispersion of residual forces increase substantially. While the majority of observations are focused in lower drift ranges, in which residual forces are zero, localized plastic deformation leads to high residual responses at individual components. This implies that, at the system level, displacements are maintained as moderate while individual components experience large inelastic excursions. A plot for residual drift vs. peak lateral drift is illustrated in Fig. 14(c), and a practically linear relation is observed. The positive slope of the trend line of the data implies that residual drift tends to increase linearly proportional for peak drift demand. In lower drift levels, quasi-elastic, self-centering behavior dominates, and there exist high data concentrations about the origin. For increasing drift amplitudes, the data are increasingly scattered, and this is a hallmark of loss of capacity for self-centering and accumulation of irreversibility. High density clusters characterized by dark (purple) regions in the plots are indicative of behavioral consensus among corresponding drift intervals. They are a sign of the system exhibiting stable and repeatable inelastic response trends, and this is significant for structural reliability evaluation. Such results are very significant when formulated in a performance-based seismic evaluation framework because this sheds light into residual deformation behavior drivers controlling post-earthquake functionality and repair needs.

Residual drift is a significant parameter for bridge operational performance evaluation after an earthquake. This parameter has commonly been used for determining bridge functionality once seismic shakes have stopped. In the laboratory work, residual drift for each specimen was assessed at the end of the first cycle once the lateral load was brought to zero for a variety of displacement levels. A comparative plot between residual drift measured in the experiments and that predicted by all of the ML models is given by Fig. 15, while a broader comparison of the results from the best models (CB, RF, and DNN) for all Specimens and experimental data are given by Fig. 16.

It can be seen that specimen RC-1 exhibits the maximum residual drift among all other specimens at comparable levels of drift, highlighting the effectiveness of prestressed strands in minimizing residual drift for bridge piers. In addition, it can be observed that among the prestressed specimens, PRC-4 exhibits maximum residual drift, while PRC-2 exhibits minimum. Aside from PRC-4, specimen PRC-3 also exhibits much higher residual drift than other prestressed specimens for positive loading. In contrast, for negative loading conditions, the residual drift for PRC-3 is found higher than that for specimens PRC-2 and PRC-6. This pattern can be attributed to the observation that for specimen PRC-2, there was a maximum value for the mechanical prestressing ratio for all tested specimens, and it reduced from 0.76 to 0.70 after one of its prestressed strands failed. In contrast, specimen PRC-3 has the lowest level for the mechanical prestressing ratio, about 0.44, implying that higher prestressing ratio corresponds to less residual drift. The increase in residual drift for specimen PRC-4 can essentially be related to the relatively lower level for prestressing stress for its strands. Additionally, a comparison between experimental data and ML predictions exhibits a very strong correspondence, supporting that the ML-based models accurately retained the residual drift behavior for all the specimens. This similarity exhibits the reliability for the prediction method in reflecting experimental trends. Specimen PRC-2 was constructed with reduced longitudinal mild steel bars when compared for specimen PRC-1, along with its level for the mechanical prestressing ratio. Consequently, a remarkable level in residual drift ratio was observed, as illustrated from Fig. 16(b). Similarly, the ML-based predictions also exhibited very close correspondence for those experimental results, accurately reflecting the influence for a higher level of prestressing ratio for a reduced level for residual drift. In contrast, specimen PRC-3, that had fewer prestressing strands compared to PRC-1, exhibited a significantly reduced mechanical prestressing ratio (0.44). This result led both empirical estimates and ML predictions to indicate a rise in residual drift.

In specimens PRC-5 and PRC-4, the initial prestressing stresses were varied and studied for their influence on residual drift, as shown in Fig. 16(d). The results from both experimental observations and ML (ML)-based predictions always suggest that PRC-4 has a significantly higher residual drift ratio when compared with PRC-1, while PRC-5 has a slightly higher residual drift ratio when compared to PRC-1. This agreement between experimental results and ML predictions suggests that the influence of the initial prestressing stress on residual drift is relatively limited. Furthermore, in the case of PRC-4, the prestressing stress in strand PS1 reduced to 8 MPa at significant lateral displacements, essentially nullifying its contribution toward the restoring force and consequently increasing residual drift. From this combined analysis involving experimental data and ML-based assessments, it can thus be reasoned that the influence of the initial prestressing stress on residual drift is low under the presumption that the prestressing strands will function during earthquakes. As evidenced from Fig. 16(f), both experimental outcomes and ML-based predictions suggest that the residual drift for specimen PRC-6 is very close to that for PRC-1. This similarity mainly arises from their common mechanical prestressing ratio (0.61) and comparable total prestressing forces that remained consistent throughout the testing campaign. The close association between the experimental results and ML predictions further confirms that under these very specific conditions, the prestressing strand arrangement has a minimal influence toward the residual drift behavior of the piers.

11 Conclusions

This study presented an integrated numerical, experimental, and machine-learning (FE–ML) framework for predicting the seismic behavior of PRC bridge columns. Five seismic response parameters including residual displacement, residual force, force at collapse drift, hysteretic energy, and maximum base shear were selected as primary prediction targets. Seven experimentally tested specimens (one RC and six PRC columns) were modeled and validated in OpenSees, providing a reliable foundation to generate a data set of 2163 samples for training, testing, and comparing ten supervised ML algorithms. Rigorous model validation, including systematic train-test partitioning, cross-validation, and external validation against experimental results confirmed that the trained ML models could generalize beyond the numerical data set and accurately capture seismic behavior. Extended validation using 36771 additional data points across drift levels from −4% to +4% further demonstrated strong agreement between ML predictions and experimental observations, effectively reproducing critical behavioral trends such as persistent inelastic residual forces, the near-linear relationship between residual and peak drift, and the influence of prestressing ratio and initial strand stress on post-earthquake deformation. The main results are summarized as follows.

1) Among all algorithms, deep learning models, particularly the DNN, demonstrated the highest predictive performance, whereas ensemble methods such as CB and RF effectively captured both force-related and energy-based responses. Overall, the findings indicate that ML-based prediction offers a reliable and computationally efficient approach for evaluating nonlinear seismic responses, with practical potential for performance-based seismic assessment.

2) The DNN and ANN models achieved the maximum accuracies, with DNN reaching 92.43% and ANN 90.7%. Moreover, most predictions lay within ±30% error of actuals, indicating strong generalization capability and resistance to overfitting. The RF (92.16%) and CB (93.45%) models also performed consistently, and there was a sharp clustering of predictions near the ideal prediction line.

3) The SVM (73.91%) and FNN (59.46%) exhibited moderate accuracy, and SVM performance was further affected by kernel sensitivity and convergence issues. The KNN model with 70.34% accuracy exhibited the weakest performance as it was highly affected by noise and high-dimensional feature space, and thus it generated very wide prediction errors.

4) The DNN model performed the best for residual displacement, with an accuracy of 92.43% and an RMSE of 0.0275. The ANN model came in second with an accuracy of 90.7% and an RMSE of 0.035. Both models were able to effectively capture the complex nonlinear relationships between column geometry and material properties, resulting in accurate displacement predictions. The superior performance of the DNN, in particular, suggests a strong capability to extract deeper hierarchical features, which is critical in modeling post-yield seismic behavior. These findings highlight the potential of neural network-based frameworks for accurately estimating residual demands, a crucial sign of structural damage and reparability.

5) For residual force, the best performance was achieved by CB, which attained an accuracy of 93.45% with an RMSE of 0.0025, followed by RF with 92.16% accuracy and a comparably low RMSE. These ensemble-based approaches effectively captured the complex interactions between forces and displacements, demonstrating strong capability in modeling nonlinear post‐earthquake response and minimizing prediction uncertainty. In contrast, the DCN (53.83% accuracy) and FNN (59.46% accuracy) models exhibited the weakest performance, generating unstable and highly variable predictions for residual column forces. This instability indicates limited learning of force-dependent features, particularly under high nonlinearity and data sparsity. Overall, the findings demonstrate how crucial reliable feature aggregation and regularization techniques are to precise residual force demand estimation.

6) For force at collapse drift, the best performance was achieved by the DNN model, which attained an accuracy of 89.77% with an RMSE of 0.0036, followed by the MLP model with 86.97% accuracy. Both models demonstrated a strong capability to predict force demands at the collapse stage of columns, where nonlinear behavior and strength degradation become dominant. Their performance suggests that neural network architectures are well-suited to capturing the complex interactions governing collapse mechanisms, particularly when conventional regression approaches fail to represent such highly nonlinear responses. These results indicate that data-driven deep learning methods can provide reliable estimates of critical force thresholds, supporting more accurate evaluation of structural capacity under extreme seismic demands.

7) Among the best algorithms for estimating hysteretic energy, CB and RF achieved high accuracies of 97.95% and 91.53%, respectively, with correspondingly low RMSE values. These ensemble-based methods effectively captured the nonlinear interactions between displacement and force, enabling accurate estimation of energy dissipation in columns. Their impressive performance demonstrates how well tree-based learners can simulate complicated hysteretic behaviors, especially under cyclic loading where the response is determined by stiffness and strength degradation. Overall, these results suggest that ensemble learning provides a robust and data-efficient framework for predicting hysteretic energy demands, supporting enhanced seismic performance assessment and design optimization.

8) For maximum base shear, the best predictive performance was achieved by the RF, ANN, and DNN models, each attaining accuracies exceeding 97% with very low RMSE values. These models provided highly precise estimates of base shear forces in columns, demonstrating strong capability in representing nonlinear strength characteristics under seismic loading. Their performance indicates that both ensemble learning and neural network architectures can effectively capture the combined influence of geometry, material properties, and deformation capacity on peak lateral resistance. Overall, the results underscore the suitability of advanced ML approaches for reliably predicting critical demand parameters, contributing to more accurate evaluation of structural capacity and seismic safety.

9) Overall, the ML findings as well as experimental results showed that the prestressing ratio had a big effect on the residual drift. High ratios (e.g., PRC-2) significantly diminished residual drifts, thereby improving post-seismic re-centering capacity, whereas lower ratios (e.g., PRC-3) resulted in increased residual drifts, underscoring the necessity for sufficient prestressing levels. In the early stages of prestressing, stress had little effect on residual drift if strands stayed locked under lateral loads (for example, PRC-4 vs. PRC-5). This shows that overall deformation is more important than initial prestress when it comes to re-centering. Strand placement had little effect when total prestressing force and mechanical ratio were constant (e.g., PRC-1 vs. PRC-6), showing that global prestressing demand governs post-yield behavior. These findings highlight the role of key prestressing parameters in minimizing residual deformations and demonstrate the potential of ML-based surrogate models as efficient tools for performance-based seismic design and post-earthquake decision support in bridges.

10) The strong positive correlation between ML predictions and observed data confirmed the ability of the proposed framework to accurately replicate real seismic behavior. This agreement was particularly evident for residual drift trends, where the models effectively captured both the magnitude and variability of post-yield deformations. The results demonstrate that ML-based approaches can reliably model complex nonlinear structural responses, providing a computationally efficient and scalable tool for performance-based seismic assessment. Such predictive capabilities are critical for informing design decisions, evaluating post-earthquake reparability, and supporting rapid decision-making in risk mitigation for infrastructure systems.

However, future work should focus on expanding the scope and applicability of the proposed FE, ML framework. In particular, increasing the size and diversity of experimental data sets would improve model generalizability and reduce bias from limited physical testing. Incorporating additional structural parameters, such as longitudinal and transverse reinforcement ratios, confinement characteristics, and loading protocols, can further improve predictive accuracy. Finally, developing a user-friendly GUI or web-based platform would support broader use of the framework by engineers and practitioners, enabling practical implementation in seismic risk assessment and performance-based bridge design.

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