Toward efficient multi-objective seismic design optimization of self-centering bridges using machine learning

Abstract

Self-centering rocking bridge piers, characterized by their minimal residual deformation and rapid post-seismic recovery, have emerged as a promising solution for enhancing the seismic resilience of bridge systems. However, their inherently nonlinear behavior and pronounced sensitivity to multiple interdependent design parameters make it challenging to achieve balanced seismic performance among all piers within an integrated bridge system. This work develops a system-oriented optimization framework for self-centering rocking bridges to address this issue. The proposed framework integrates machine learning-based surrogate modeling to markedly accelerate the optimization process. A detailed case study of a four-span self-centering rocking bridge is conducted to demonstrate the framework’s applicability and effectiveness. Results show that substituting traditional finite element model with an XGBoost-based surrogate model reduces computational time by 92% while preserving high predictive accuracy. Furthermore, the optimized design significantly enhances system-level performance uniformity, achieving a 52.3% reduction in inter-pier shear force variability and a 19.0% decrease in displacement disparity compared with the baseline configuration.

Graphical abstract

Keywords

self-centering rocking bridge / machine learning / seismic design / multi-objective optimization / nonlinear dynamics / multi-criteria decision-making

1 Introduction

The conventional design paradigm of ductile seismic-resistant reinforced concrete (RC) bridge piers has proven effective in preventing structural collapse during seismic events [1]. However, this ductility-centric approach may result in substantial nonlinear deformation and severe damage to the piers, necessitating extensive post-earthquake repairs that lead to prolonged traffic disruption and substantial economic losses [2].

Over the past few decades, seismic design objectives have evolved from primarily preventing collapse to ensuring functional recovery and the rapid restoration of service after earthquakes [3]. Within this evolving design philosophy, the development of self-centering rocking piers represents a transformative advancement in bridge engineering [4]. Rocking piers combine prestressed tendons with energy-dissipating (ED) devices, endowing the system with both re-centering capability and enhanced energy dissipation. As a result, these piers exhibit minimal residual displacements and effectively mitigate nonlinear damage to the concrete components, thereby substantially reducing post-earthquake repair costs and downtime [5]. Rassoulpour et al. [6] investigated the seismic performance of self-centering rocking piers considering soil-structure interaction, revealing how foundation stiffness and earthquake frequency content uniquely influence their behavior compared to conventional systems. Using Monte Carlo nonlinear analyses that account for post-tensioning (PT) prestress loss, Anzabi and Shiravand [7] quantify the time-dependent reliability decay of self-centering PT piers across aspect and ED ratios and proposes re-tensioning cut-off years, showing ED bars notably extend the interval for slender piers. Li and Unjoh [8] experimentally investigated post-tensioned precast segmental piers with deliberately orchestrated multiple joint openings, showing that this detailing can enhance energy dissipation and decentralize damage while preserving self-centering.

Nevertheless, the design of rocking piers is governed by a complex interplay of parameters, including the initial force and area ratio of PT tendon, and both the yield strength and volumetric ratio of ED components. The inherent variability and interdependence among these design variables introduce significant complexity into the design process [9,10]. This complexity is further exacerbated by the need to simultaneously satisfy multiple, and often conflicting, engineering demand parameters (EDPs), such as base shear and top displacement [11–13]. Improving one EDP can frequently detract from another, complicating the achievement of globally optimal seismic performance.

While considerable research has addressed the seismic behavior of individual rocking piers, much less attention has been paid to system-level interactions among multiple piers within an entire bridge configuration. This oversight is noteworthy, as real-world bridges are rarely composed of homogeneous structural components. Instead, they typically encompass piers with varying geometric properties, boundary conditions, and spatial distributions, generating spatially heterogeneous dynamic responses under seismic excitation. Notably, the work of Palermo and Pampanin [14,15] on irregular rocking bridge systems revealed that shorter piers, due to their limited self-centering capacity, can induce disproportionately large residual displacements in the entire structure. These findings underscore the inadequacy of evaluating bridge performance based solely on individual pier analyses, as such an approach neglects the collective dynamic interactions that arise from inter-pier coupling and system-wide response coordination. Consequently, balancing seismic performance in a rocking bridge system inherently gives rise to a multi-objective optimization problem (MOP) [16], wherein multiple, often conflicting, performance criteria must be optimized concurrently. Effectively managing these trade-offs is essential for achieving integrated seismic resilience at the system level.

To address this system-level challenge, multi-objective optimization algorithms offer a viable solution by enabling the simultaneous consideration of competing performance objectives. For example, Zhang et al. [17] developed a seismic design optimization approach for the two-defense-line simply supported girder bridge. Marzok and Lavan [18] introduced a dual-phase optimization framework for the seismic design of three-dimensional irregular buildings employing self-centering concentrically braced frames. However, selecting an appropriate multi-objective optimization algorithm itself is far from trivial, as the performance of different algorithms often depends on the dimensionality of the problem, the degree of nonlinearity in structural responses, and the trade-off characteristics among conflicting objectives [16].

Moreover, the inherently nonlinear and dynamic behavior of rocking bridges renders traditional finite element modeling (FEM) computationally prohibitive for iterative response evaluations within optimization loops. This limitation severely impedes optimization efficiency and practical applicability [19,20]. For example, Camacho [21] developed an evolutionary algorithm-based framework for the seismic design of RC bridges, leveraging the NSGA-II algorithm to simultaneously optimize material usage and performance-cost trade-offs via FEM. Despite employing 32-thread parallel computing, generating the Pareto front required approximately 15 h of computation, highlighting the pronounced computational burden.

As a remedy, machine learning (ML)-based surrogate models have emerged as powerful alternatives, capable of approximating the seismic response of high-fidelity FEMs with orders-of-magnitude reduction in computational time [22]. By providing near-instantaneous performance predictions, these ML models greatly accelerate the optimization process and enhance overall design efficiency [23]. For instance, Fang et al. [19] introduce a ML-aided multidisciplinary optimization framework tailored to the design of seismic-resilient structures with hybrid braces (i.e., shape memory alloy-viscoelastic systems), achieving a 15%–30% reduction in key seismic responses (inter-story drift, floor acceleration) without escalating costs. Silva-Lopez and Baker [24] developed a genetic algorithm-based optimization framework incorporating neural network-based surrogate models to replace computationally expensive FEMs, enabling efficient seismic retrofitting selection for bridges by minimizing expected road network disruption risks. Ning and Xie [25] introduce a surrogate model-based optimization framework that substantially enhances the computational efficiency of seismic risk-informed design for bridges equipped with base isolators and fluid viscous dampers. Zhang et al. [26] demonstrates that surrogate-model-driven optimisation delivers a 60-fold reduction in computational cost while discovering isolation-device configurations that outperform conventional designs under 1000 stochastic ground-motion scenarios.

Building on the above review, this study addresses three gaps in the literature: 1) the lack of system-level, multi-pier coordination strategies for heterogeneous self-centering rocking bridges; 2) the prohibitive computational cost of iterative optimization when relying on high-fidelity nonlinear time-history analysis (NTHA); 3) limited, transparent justification for selecting surrogate models and multi-objective optimizers. This study advances knowledge in this research area by: 1) developing a system-level multi-objective optimization methodology that coordinates design parameters across multiple rocking piers, thereby addressing the overlooked inter-pier dynamic interactions; 2) integrating ML surrogate models to markedly improve optimization efficiency, thus overcoming the computational bottleneck of traditional FEM-based evaluations; 3) conducting a comprehensive evaluation and selection of multi-objective optimization algorithms, establishing a reliable basis for algorithm choice in similar rocking-bridge design problems.

The study begins by introducing the overall design framework and the adopted multi-objective optimization methodologies. It then describes the development and validation of finite element model for rocking piers, which are used to generate a comprehensive input-output data set through NTHA. This data set forms the basis for training and comparing six competing ML surrogate models, ultimately selecting the most accurate and efficient model for predicting seismic responses. Subsequently, four distinct multi-objective optimization algorithms are implemented and compared to derive the Pareto frontier representing trade-offs among the chosen EDPs. To support final design selection, the entropy-weighted technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method is applied, enabling a quantitative multi-criteria decision-making procedure to identify the most favorable design configuration from the Pareto set.

2 Framework and methodology

This part first describes the optimization framework and subsequently highlights the core methodologies utilized, comprising ML regression models and optimization algorithms.

2.1 General framework

Figure 1 provides a schematic representation of the step-by-step implementation of the optimization framework, consisting of four main stages.

1) Input–output data set construction

Since ML models rely on comprehensive and accurate data, the preparation of a high-quality input–output data set is essential for capturing the functional dependencies between variables and ensuring robustness in predictive outcomes. In this study, the input data set consists of the design parameters of rocking bridges, whose ranges and distribution characteristics are established by integrating engineering professional knowledge and pertinent academic literature. While the output data are the EDPs, which are obtained via NTHAs.

Initially, a series of rocking bridge models encompassing a broad spectrum of design parameters is established via the Latin Hypercube Sampling (LHS) technique [27]. Earthquake motions corresponding to the prescribed seismic intensity are determined in accordance with the target response spectrum. Subsequently, each bridge is randomly matched with the selected ground motions, and NTHA is performed to evaluate the resulting EDPs. This systematic approach ensures a comprehensive and representative data set for subsequent ML model development.

2) ML models training

To improve the predictive accuracy of ML models on unseen data, the optimization of hyperparameters is essential. In this study, a systematic approach combining 5-fold cross-validation with Bayesian optimization is utilized to ascertain the optimal configuration of hyperparameters. Specifically, 5-fold cross-validation [28] is used to robustly evaluate the performance of each candidate set of hyperparameters, thereby ensuring that the developed ML model exhibits strong generalization capabilities. Bayesian optimization [29], recognized as an efficient hyperparameter tuning technique, leverages the outcomes of prior evaluations to iteratively refine its search process, effectively converging on the most promising hyperparameter configurations. Through multiple rounds of iterative optimization, the optimal hyperparameter combinations for each ML model are efficiently identified. To identify the most effective ML models, four performance metrics were utilized to comprehensively evaluate and compare the predictive performance of the candidate models.

3) Pareto frontier calculations

The Pareto front denotes the collection of non-dominated solutions obtained through multi-objective optimization, where enhancing one objective inevitably leads to a compromise in at least one of the others [16]. The efficiency of Multi-Objective Optimization Algorithms (MOOAs) in identifying the Pareto frontier can be greatly enhanced by employing ML models as opposed to relying solely on FEMs. This advantage arises from the high computational cost associated with FEM, which requires extensive resources to conduct the nonlinear time history analysis of structures. In contrast, ML models are capable of rapidly predicting the seismic response, thereby substantially reducing computational time. Consequently, by utilizing the ML model as a surrogate for FEM, the MOOAs are capable of assessing a vast set of candidate solutions within reduced computational time, thereby expediting the process of determination of the Pareto frontier [19,20,30].

4) Optimal solution determination

To perform multi-criteria decision analysis, the entropy weight-TOPSIS framework integrates weights derived from entropy theory with the ranking capability of the TOPSIS method. After determining the Pareto frontier, the entropy coefficient for each objective is calculated to reflect its importance, with the weighting process governed by the level of dispersion observed in the data set [31]. After computing the distance from the ideal solution for all alternatives, the one with the minimal value is recognized as the optimal selection. To ensure the robustness of the optimal design and account for any approximation errors introduced by surrogate modeling, the final step in the framework involves verifying the optimal design by assessing its performance using NTHA.

2.2 ML models

In this study, the ML models utilized include Extreme Gradient Boosting (XGB), Adaptive Boosting (AdaBoost), Support Vector Machine (SVM), Artificial Neural Networks (ANN), Gradient Boosting (GB), and Light Gradient Boosting Machine (LightGBM). The theoretical background of these ML methods is described below.

1) XGBoost

The XGBoost algorithm [32] a popular ML approach for regression tasks, is based on a gradient boosting framework. In this framework, weak learners are added sequentially to enhance prediction accuracy. Model performance is optimized by minimizing a loss function that represents the difference between the predicted and actual values. Figure 2(a) illustrates the flow diagram associated with the XGBoost model.

2) AdaBoost

AdaBoost [33] is a well-established ensemble learning method that constructs a powerful predictor through the aggregation of multiple weak learners. The specific flow of the algorithm can be seen in Fig. 2(b). The core idea of the algorithm is to adjust the weights of the samples so that the classifier will focus on the samples that have been misclassified in each iteration.

3) SVM

SVM aims to determine an optimal hyperplane within a high-dimensional feature space, enabling the separation of data points from distinct classes. The objective and categorization vary among different regression problems. It is not to categorize the data points but to predict the continuous values. In support vector regression, the goal is to predict continuous values by fitting an optimal line within a specified tolerance range.

4) ANN

ANN [34] is a computational framework modeled after the human brain’s structure and functionality, designed to capture patterns and generate predictive outputs. The essential feature of an ANN lies in its layered architecture, where neurons are organized into multiple levels. At each level, neurons take the input signals, transform them through an activation function, and transmit the resulting outputs to the succeeding layer. Its flowchart is shown in Fig. 2(c).

5) GB

GB [35]is an integrated learning approach that centers on the use of gradient descent to minimize the loss function. This approach iteratively builds multiple weak learners (typically decision trees) and progressively optimizes the predictive performance of the model.

6) LightGBM

LightGBM [36] is an efficient ML algorithm built upon the gradient boosting framework. By employing a histogram-based decision tree algorithm together with a gradient-driven one-sided sampling strategy, it accelerates model training and enhances computational efficiency. LightGBM is highly efficient in handling categorical features, resulting in significant savings in both memory consumption and computational time. Figure 2(d) illustrated the flowchart of the LightGBM algorithm.

2.3 Multi-Objective Optimization Algorithms

The inherent complexity of many real-world engineering problems stems from the need to simultaneously optimize multiple performance metrics that are often in conflict. Several popular and powerful MOOAs are adopted in this study, including Non-dominated Sorting Genetic Algorithm II (NSGA-II), Multi-Objective Particle Swarm Optimization (MOPSO), Chaotic Evolution Optimization (CEO), and Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D).

1) NSGA-II

NSGA-II [37] is a widely recognized evolutionary algorithm that utilizes a fast non-dominated sorting method to systematically classify solutions into different Pareto fronts, thereby facilitating the effective selection of high-quality individuals across generations. To further promote population diversity, the algorithm introduces a crowding distance mechanism, which maintains a well-distributed set of solutions along the Pareto front.

2) MOPSO

MOPSO [38] is an extension of the traditional Particle Swarm Optimization (PSO) algorithm designed to address multiple-objectives optimization problems. By incorporating mechanisms such as an external archive for storing non-dominated solutions and adaptive update strategies to enhance diversity and convergence, MOPSO has demonstrated superior performance in various complex engineering applications.

3) MOEA/D

The MOEA/D algorithm [39] is a metaheuristic designed for multi-objective optimization, which applies a decomposition strategy to transform the original problem into a set of single-objective subproblems. The algorithm employs a reference direction to guide the search, thereby facilitating both efficient global search and local optimization.

4) CEO

The CEO algorithm [40] exploits the hyperchaotic nature of the amnesia mapping, introduces a random search direction through mathematical modeling to drive the evolution process, and combines it with further optimization of the algorithm’s performance in a differential evolution framework.

2.4 Entropy weight-TOPSIS based decision-making

As a hybrid strategy, the entropy-TOPSIS method merges the entropy weighting mechanism with the ranking capability of TOPSIS to handle multiple criteria [30,41]. Here, the entropy measure is employed to calculate the weights of criteria in an unbiased manner, reflecting their informational characteristics, thereby eliminating the subjectivity inherent in manual weight assignment. The alternatives are subsequently analyzed through TOPSIS, wherein relative distances to the ideal positive and ideal negative solutions are determined, abling a more scientific and rational ranking of the options. A comprehensive description of the implementation process of the entropy-TOPSIS technique is provided below [30].

1) Normalization of the decision matrix

To remove the influence of varying units and scales across different indicators, the decision matrix is subjected to normalization.

(1)x_{ij}^{\ast }=\frac{x_{ij}}{\sqrt{{\displaystyle \sum _{j=1}^n{x}_{ij}^2}}},

where x_ij signifies the initial score of alternative i relative to criterion j, n is the indicator count, and x_{ij}^{\ast } denotes the standardized form.

2) Calculation of the entropy weight

To capture the relative importance of indicators, entropy analysis is used to compute their weights objectively within the evaluation procedure. The jth indicator’s information entropy, e_j, and the corresponding entropy weight, ω_j, are calculated as follows:

(2)e_j=-\frac{1}{\ln (m)}\sum _{i=1}^m\frac{x_{ij}^{\ast }}{{\displaystyle \sum _{i=1}^m{x}_{ij}^{\ast }}}\ln \left(\frac{x_{ij}^{\ast }}{{\displaystyle \sum _{i=1}^m{x}_{ij}^{\ast }}}\right),

(3){\omega }_j=\frac{1-e_j}{{\displaystyle \sum _{j=1}^n\left(1-e_j\right)}},

where e_j represents the calculated entropy of the j-th indicator, and ω_j represent ts corresponding weight derived from entropy, and m reflects the overall number of alternatives under consideration.

3) Construction of the entropy-weighted normalized decision matrix

The normalized matrix is adjusted using the entropy-derived weights of the indicators, yielding the following expression for the weighted normalized decision matrix {\mathit{\upsilon }}_j:

(4){\mathit{\upsilon }}_j={\omega }_j\cdot x_{ij}^{\ast }.

4) Calculation of each alternative’s distance to the ideal and anti-ideal solutions

To evaluate each alternative, its Euclidean distance from the positive and negative ideal solutions is computed using the subsequent equation:

(5)d_i^{+}=\sqrt{\sum _{j=1}^n{({\upsilon }_{ij}-a_j^{+})}^2},

(6)d_i^{-}=\sqrt{\sum _{j=1}^n{({\upsilon }_{ij}-a_j^{-})}^2},

where d_i^{+} and d_i^{-} denote the distances from the alternative i corresponding to the ideal solution, respectively. For the jth indicator, a_j^{+} and a_j^{-} correspond to the maximum and minimum values of ideal solutions, respectively.

5) Calculation of the relative proximity

Equation (7) defines the relative proximity of each alternative, denoted as C_i, to the positive ideal solution. Alternatives are then ranked in descending order according to the value of C_i, with higher values indicating more preferable options.

(7)C_i=\frac{d_i^{-}}{d_i^{+}+d_i^{-}},

3 Finite element modeling and validation of rocking bridges

The FEM procedure for the rocking pier is first described and validated using quasi-static test data. It then introduces the layout of the prototype bridge system with rocking piers, followed by the development of the corresponding FEM using the validated modeling methodology. Finally, this section concludes with a description to the selected ground motions.

3.1 Finite element modeling of rocking piers and their certification

The FEM of the rocking bridge pier is developed by OpenSees software [42], as shown in Fig. 3. Elastic beam-column elements are used to model the RC pier body, as the deformation of the rocking pier primarily occurs at the rocking interface while the pier body itself experiences minimal damage [2]. The base of the pier is restrained against horizontal translation, while rotational degrees of freedom are released. To capture the rocking behavior, a grid of 10 × 10 uniformly distributed compression springs is employed to simulate the rocking interface. According to the experimental test results [43], each contact spring is assigned an axial compression stiffness of 5.5 × 10⁶ kN/m.

The PT tendons are modeled using the corotTruss element since they remain in tension throughout the loading process. The unbonded PT tendons are anchored at both ends, with the upper end rigidly connected to the pier top and the lower end fixed into the foundation. To guarantee a stable self-centering response of the piers, the PT tendons are assumed to remain elastic throughout the loading process [44]. The prestress is imposed through the initial strain method, implemented in OpenSees as “InitStrainMaterial”. The ED dampers are simulated using two-node link elements with bilinear plastic material properties.

To verify the accuracy of the FEM technique, the quasi-static tests conducted by Marriot [43] on the specimen “HBD3” and by Shen et al. [45] on the specimen “PRC-P13.5E.79” are used for comparison. Figure 4 illustrates the experimental setup of specimen “HBD3”, which is characterized by a square cross-section measuring 0.35 m × 0.35 m. The lateral load is located vertically 1.6 m from the rocking interface. The applied gravity load on the pier is 200 kN, and the total initial tension of the four PT tendons is 300 kN (i.e., 75 kN per tendon). Additionally, the specimen incorporates four ED bars, each having a 10 mm diameter, a 75 mm length, and a yield strength of 300 MPa. The experimental configuration and the schematic of the applied loading protocol are depicted in Fig. 4. The experimental setup of specimen “PRC-P13.5E.79” is detailed in Ref. [45] and is not repeated here for the sake of brevity.

Figure 5 compares the cyclic response results obtained from the OpenSees simulation (denoted by solid blue lines) and the experimental data (represented by red dots). Discrepancies are observed that the FEM results exhibit narrower hysteresis loops compared with the experimental results. This difference may stem from the inadequate consideration of cumulative nonlinear material damage at the rocking interface in the FEM. Furthermore, the bond-slip effect is not incorporated in the FEM model, resulting in a reduced hysteretic energy contribution from the ED bars. Despite these deviations, the FEM simulation reproduces the key features of the experimental hysteresis, including the lateral force–displacement trend, peak strengths, and overall loop shape. This good agreement confirms that the proposed modeling approach reliably captures the global mechanical behavior and hysteretic characteristics of the rocking pier.

3.2 Introduction of the prototype bridge and nonlinear modeling approach

The configuration of the prototype bridge is presented in Fig. 6. The main girder is a continuous structure consisting of four 40 m spans, with a beam height of 2 m and cover beam height of 1.5 m. Piers 1#, 2#, and 3# have heights of 8 m, 16 m, and 8 m, respectively, and all three piers have identical square cross-sections measuring 2 m × 2 m. Longitudinal reinforcement consists of 64 steel bars with a diameter of 25 mm, resulting in a reinforcement ratio of 0.79%. For transverse reinforcement, 16 mm diameter hoops are provided at intervals of 100 mm. Concrete grades of C40 for the piers and C50 for the main girders were employed, with corresponding cubic compressive strengths of 40 and 50 MPa.

The three-dimensional model of rocking bridges and the FEM developed using OpenSees software [42] are depicted in Fig. 7. The main girder, serving as a capacity-protected component under earthquakes, is modeled with elastic beam–column elements [46]. The self-centering rocking piers are modeled using the approach outlined in Subsection 3.1, which has been validated through experimental comparisons.

As for the boundary conditions, the connections between the girder and pier are modeled as rigid elements, preventing any translational and rotational movement at these interfaces. This modeling strategy aligns with conventional transverse design practices of continuous girder bridges, in which shear keys restrain lateral displacements and multiple bearings limit rotational freedom [47,48]. For the interaction between the deck and abutments, the deck is laterally restrained at both ends by shear keys, establishing a fixed lateral connection. This modeling strategy aligns with the method implemented by Palermo et al. [15]. The force-deformation relationship of shear keys can be found in experimental research conducted by Silva et al. [49].

In terms of seismic excitation, the case study focuses on optimizing the seismic design of self-centering bridges subjected to transverse ground motions. While three-dimensional seismic excitation is also critical for rocking bridges [46], the proposed optimization framework has the potential to account for such complexities. Specifically, the directional effects of seismic excitation can be incorporated into the framework by introducing them as additional input features within the ML surrogate model. More comprehensive investigations addressing multi-directional ground motions will be pursued in future studies.

3.3 Ground motion selection

As illustrated in Fig. 8, the response spectrum for earthquakes with a 2% probability of exceedance in 50 years is derived for this bridge following the seismic design specification of China [50]. Previous findings by Ozbulut et al. [51] indicates that near-fault motions, characterized by long-period and high-velocity pulses, tend to enhance the seismic demands experienced by isolated structures. Therefore, in accordance with FEMA P695 [52], seven near-fault records are employed to assess the seismic response of the rocking bridges. The selected ground motion records are scaled to conform to the target response spectrum, and the corresponding scale factors are provided in Table 1. The mean spectrum of the selected ground motions closely matches the target spectrum, as depicted in Fig. 8. The response of each bridge model is calculated under the influence of each of the seven ground motions. To mitigate the impact of individual outlier records, the response results are presented as the average across these seven motions. This method is selected to effectively represent the bridge’s behavior under seismic motions with an intensity corresponding to the design response spectrum.

4 Machine learning models for predicting seismic response of rocking bridges

As discussed previously, ML models facilitate rapid seismic performance predictions, substantially reducing computational costs and improving the effectiveness of design optimization. To develop precise and robust ML models for seismic performance prediction of rocking bridges, NTHA are first conducted on a diverse set of randomly generated rocking bridge models to construct an input-output data set for ML model training. Six ML algorithms (i.e., XGB, GB, ANN, LightGBM, AdaBoost, and SVM) are then optimized using 5-fold cross-validation combined with Bayesian hyperparameter optimization. The predictive performance of each model is comprehensively evaluated using four metrics (R², RMSE, MAE, and MAPE), and the most accurate model is selected for the subsequent multi-objective optimization task.

4.1 Input-output database creation

PT tendons and ED bars are the two key components in rocking piers, which provide self-centering and energy dissipation capacities, respectively. The optimal selection of parameters governing these components is crucial for improving the seismic performance of the entire bridge system [53,54]. Therefore, the primary parameters of the PT and ED components, including initial PT force, PT area ratio, and the ED bar area ratio, are selected as design variables in this study.

Additionally, to account for practical uncertainties in engineering applications, such as the yield strength and elastic modulus of ED bars, these parameters are also incorporated as input variables in the ML model to enhance its robustness, although they are not selected as design variables. Table 2 lists the 12 input parameters for the ML models: the first six serve as design variables to be optimized, while the remaining six capture the stochastic nature of structural and material properties, further enhancing the model’s reliability. Note that considering the symmetry of the bridge, the design parameters of Pier 3# are identical to those of Pier 1#. The selection of probabilistic distributions for the stochastic parameters in Table 2 is aligned with existing literature on structural reliability analysis.

Despite the significant influence of geometric parameters on the seismic performance of bridges, their applications are often constrained by real-world engineering considerations. For instance, pier height may be restricted by clearance requirements for traffic beneath the bridge, while span length can be limited by equipment capacity or the width of the roadway or river below. Since these constraints fall outside the scope of seismic design and optimization, geometric parameter optimization is not considered in this study.

According to the parameter distributions specified in Table 2, LHS method [27] is adopted to produce 100 sample points for each parameter, resulting in the construction of 100 FEMs of rocking bridge. NTHAs are then performed on these 100 rocking bridges to derive the target seismic response, including shear and displacement of Piers 1# and 2#, together with their respective differences in shear (Δ_v) and displacement (Δ_d). These quantities are used as the ML model outputs, forming a data set with 100 corresponding input–output pairs.

A total of 100 random parameter combinations are generated to cover input uncertainty. Note that for each combination, 7 ground motion records are used for NTHAs, resulting in 700 ( = 100 × 7) total NTHA cases. The adequacy of the 100-sample data set is validated by the regression precision of the XGB surrogate model in Subsection 4.3: R² = 0.904 for shear force prediction and R² = 0.803 for displacement prediction. These values exceed the common engineering threshold R² > 0.7, indicating that the data set captures the nonlinear input–response relationships without requiring excessive samples. Nonetheless, larger data sets may further improve accuracy; for example, Poorahad Anzabi and Shiravand [7] used a 1000-iteration Monte Carlo simulation for purely probabilistic analysis (without ML).

4.2 Hyperparameter tuning

Hyperparameter optimization is essential to the overall performance of ML models. This study employs Bayesian optimization [27,63] to accurately identify the optimal hyperparameters. Bayesian optimization, a powerful hyperparameter tuning technique, constructs probabilistic models to predict the performance of various hyperparameter sets [64] and systematically identifies the optimal combination. This method iteratively updates its predictions based on previous evaluations, progressively converging on the best hyperparameter set. The final optimized hyperparameters for the six ML models are summarized in Table 3. During model training, 80% of the data set is randomly chosen to serve as the training set, while the remaining 20% is retained as the testing set for the final evaluation of the model’s performance and generalization ability. The training and testing sets are mutually exclusive.

Additionally, the purpose of training the ML model in this context is to develop a reliable predictive model for the seismic response of rocking bridges specifically at the prototype bridge’s designated site, rather than across multiple sites. The seven ground motions selected for this study are derived from the design spectrum specific to the bridge’s location. In accordance with seismic design codes in both China [50] and the US [65], a set of seven ground motions is sufficient to capture the seismic characteristics of a given site. Therefore, using the same set of ground motions for both training and testing is appropriate in this context.

4.3 Machine learning model performance evaluation

The primary objective of this subsection is to evaluate the performance of six different ML models (i.e., XGB, GB, ANN, LightGBM, AdaBoost, and SVM) and to identify the most effective one for predicting the seismic response of PT rocking bridges. Four statistical indices are applied to assess and compare the predictive capabilities of the ML models: the coefficient of determination (R²), root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE) [63,66]. A well-performing model is characterized by lower values of RMSE, MAE, and MAPE, and a higher R² value. The mathematical formulas for these metrics are given below.

(8)R^2=\frac{{\displaystyle \sum _{i=1}^n{({\hat{y}}_i-{\overline{y}}_i)}^2}}{{\displaystyle \sum _{i=1}^n{(y_i-{\overline{y}}_i)}^2}}=1-\frac{{\displaystyle \sum _{i=1}^n{(y_i-{\hat{y}}_i)}^2}}{{\displaystyle \sum _{i=1}^n{(y_i-{\overline{y}}_i)}^2}},

(9)RMSE=\sqrt{\frac{1}{n}\sum _{i=1}^n{({\hat{y}}_i-y_i)}^2},

(10)MAE=\frac{1}{n}\sum _{i=1}^n\left|y_i-{\hat{y}}_i\right|,

(11)MAPE=\frac{1}{n}\sum _{i=1}^n\left|\frac{y_i-{\hat{y}}_i}{y_i}\right|\times 100,

where y_i denotes the actual value, {\hat{y}}_i denotes the predicted value, n is the number of data points, and \overline{y_i} is the mean of the actual values.

Figures 9 and 10 illustrate the ML models’ capability to forecast differences in pier shear and displacement, respectively. In these figures, the 45° line (y = x) represents a perfect match between the predicted and actual values, while the dotted line around the 45° line indicates a ±20% error margin between the predicted and true values, serving as a tolerance interval for assessing the prediction accuracy [27,67]. These plots also display the R² values of both the training and testing data sets, where a higher R² value would indicate that the ML model is more accurate. Additionally, each graph also includes kernel density plots of the training and testing sets, depicting the distribution of the predicted values.

The results presented in Figs. 9 and 10 reveal that the predictive performance of the different ML models varies for shear and displacement response. The performance indexes of the various ML models are summarized in Table 4. The XGBoost, GB, and ANN models demonstrate high accuracy on both the training and testing sets, with the majority of points falling within the ±20% error band. Notably, the XGBoost model achieves R² values of 1.0 and 0.904 for shear prediction, and 1.0 and 0.803 for displacement prediction on the training and testing sets, respectively, indicating its strong capability to capture complex relationships among variables and deliver accurate predictions. In contrast, the LightGBM, AdaBoost, and SVM models, while exhibiting reasonable accuracy, yield slightly lower R² values on the testing set compared to XGBoost, with 0.824, 0.760, and 0.940 for shear prediction and 0.755, 0.835, and 0.746 for displacement prediction, respectively. This indicates that these models may be less effective than XGBoost in modeling complex relationships between variables.

To comprehensively evaluate the effectiveness of the models on different indices, this study employs a “score analysis” method [66]. In this approach, each of the six models is ranked according to its actual performance on each metric. The top-performing model for a given metric is assigned the highest score, corresponding to the total number of ML models evaluated (i.e., 6 in this case), whereas the lowest-performing model receives a score of 1. Scoring is performed independently for the train and test data sets, and the final score for each model is obtained by summing its training and test scores.

Table 5 summarizes the results of the score analysis for the different ML models, and Fig. 11 provides a visual representation of these results through a radar plot. The XGB model obtained the highest final score of 77, followed by the GB model with a final score of 56. The final score of the XGB model is substantially higher than that of the GB model. The primary reason for XGB’s superior performance lies in its ability to handle complex, nonlinear relationships in data, which is common in structural behavior such as self-centering rocking piers. XGB is a tree-based ensemble model that benefits from gradient boosting, which allows it to iteratively improve prediction accuracy by focusing on the mistakes made in previous iterations. Therefore, the XGB model is recommended for predicting the seismic response of self-centering rocking bridges.

5 Multi-objective seismic optimization design of rocking bridges

This section first introduces the formulation of the MOP for the seismic design of rocking bridges. Four multi-objective optimization algorithms are then applied to generate Pareto frontier solution sets for the MOP. The resulting Pareto frontiers are evaluated and compared using four performance metrics to identify the most effective algorithm. Next, the computational efficiency of the surrogate model is demonstrated by comparing its performance with that of NTHA during Pareto frontier identification. The entropy-weighted TOPSIS technique is subsequently applied to identify the most favorable design from the Pareto-optimal set. The seismic response of this optimized bridge is then contrasted with that of the original prototype to verify the efficacy of the proposed design approach.

5.1 Description of multi-objective optimization problem and its mathematical expression

The optimization objective is to reduce base shear (Δ_v) and pier-top displacement differences (Δ_d) among piers. 1) A decrease in pier shear force variation contributes to achieving a more uniform allocation of seismic loads, thus averting excessive force concentration on single piers [68]. This approach contributes to more efficient pier and foundation design and can lower construction and material expenditures. 2) Controlling displacement discrepancies is crucial to minimizing the risk of main girder dislodgement and mitigating potential damage to non-structural elements of bridges, such as brackets, railings, and expansion joints [69–70]. Although local damage indicators (e.g., localized stresses) are important metrics for evaluating the seismic performance of self-centering rocking piers, the current modeling approach using contact springs fails to accurately capture local damage at the rocking interface [71]. Therefore, these indicators are not considered as optimization objectives in this study.

Six parameters, including the axial load ratio, PT area ratio, and ED bar area ratio for both Piers 1# and 2#, are optimized to minimize differences in shear force and displacement among the piers. The parameter ranges and their statistical distributions are summarized in Table 2. In addition, the following design constraints are imposed to meet the required performance criteria: 1) the PT tendons at each pier should retain their elastic behavior to ensure the bridge’s self-centering capability; 2) the ED bars at each pier should not fracture to prevent excessive displacement demand of the bridge piers; 3) the residual drift ratio should be limited to less than 1% [72]. The objective function and associated design constraints are represented Eq. (12). For continuous girder systems comprising more than two intermediate piers, both design parameters and governing constraints can be systematically calibrated to reflect structural configuration and pier count.

(12)

where σ_pt1 and σ_pt2 are the maximum stresses of the PT tendons in Piers P1 and P2, respectively; σ_y is the yield stress of the PT tendons; ε_ed1 and ε_ed2 are the strains of ED bars in Piers P1 and P2, respectively; and ε_u is the fracture strain of ED bars. δ₁ and δ₂ are the residual drifts in Piers P1 and P2, respectively.

5.2 Determination of the Pareto Frontier

Pareto frontiers are commonly employed to visualize and present the results of a MOP [16]. In this study, the optimal algorithm is identified by comparing the Pareto frontiers obtained by four different MOOAs. Table 6 lists the parameter settings for these MOOAs, which are determined through iterative tuning and parameter analysis to achieve a well-distributed Pareto frontier while balancing computational efficiency and solution accuracy. Due to space limitations, the detailed parameter tuning process is not provided here. The objective function values of the initial population, along with the resulting Pareto frontiers, are illustrated in Fig. 12. The location of the prototype bridge (pre-optimized bridge with average parameters listed in Table 2) is also explicitly marked in Fig. 12 (green data point) to enhance clarity and facilitate comparison with the optimization results.

As shown in Fig. 12, the Pareto frontier demonstrates an inverse relationship between Δ_v and Δ_d, signifying a distinct trade-off whereby enhancing one inevitably compromises the other. Consequently, no unique “optimal” solution exists, but rather a set of Pareto-optimal solutions. To select the best alternative from the Pareto frontier, the Entropy Weight-TOPSIS method will be applied in Subsection 5.4.

It should be noted that the initial populations of different MOOAs were not identical. Each algorithm employs its own initialization mechanism to enhance population diversity at the beginning of the optimization process. For instance, NSGA-II utilizes stochastic random initialization, while MOPSO and MOEA/D may apply structured sampling or decomposition-based initialization schemes. Such heterogeneity is intrinsic to the algorithmic design and ensures broader exploration of the search space in the early stages.

Despite the differences in initial populations, all MOOAs were run for 100 iterations, which was verified to be sufficient for convergence through preliminary tests. After adequate iterations, the inferior solutions in the initial population are gradually eliminated, while high-quality solutions that approach the true Pareto frontier are retained and improved. This process minimizes the impact of initial population differences, ensuring that the final Pareto frontiers of each algorithm represent the optimal trade-off relationships between the two objectives.

5.3 Comparison of the multi-objective optimization algorithms

In MOPs, the quality of the Pareto frontier serves as a key indicator for assessing algorithmic performance. To conduct a comprehensive assessment of the optimization performance of various algorithms, four widely used Pareto frontier evaluation metrics are utilized: hypervolume (HV), spacing (Sp), diversity metric (DM), and the number of solutions [73]. Table 7 summarizes these metrics and the corresponding performance of each algorithm.

As listed in Table 7, NSGA-II demonstrates superior optimization capability by achieving the highest HV (37.15), which indicates the broadest coverage of the objective space, and the smallest Sp (2.26), reflecting the most uniform solution distribution along the Pareto frontier. In terms of diversity, MOPSO obtains the lowest DM (1.04), suggesting the most evenly and widely distributed solutions, closely followed by NSGA-II (1.12), both outperforming MOEA/D. Furthermore, NSGA-II identifies the largest number of non-dominated solutions (212), evidencing its most comprehensive exploration of the objective space, whereas CEO yields the fewest (47). In contrast, MOEA/D is characterized by a relatively low HV (26.73), the largest Sp (8.96), and inferior diversity, indicating limited coverage and less uniform solution distribution.

In summary, although NSGA-II performs slightly less favorably in terms of DM, it outperforms the other algorithms in HV, Sp, and the number of solutions. Among the four MOOAs, NSGA-II presents the most robust performance in the optimization of rocking bridges, due to three key advantages tailored to bi-objective design of self-centering rocking piers. First, NSGA-II’s fast non-dominated sorting algorithm classifies solutions into Pareto ranks without requiring predefined weight distributions. Second, its crowding distance calculation maintains diverse solutions, essential for providing engineers with a range of feasible designs, in contrast to MOPSO’s premature convergence and CEO’s excessive randomness. Lastly, NSGA-II’s elite retention strategy preserves high-quality solutions across generations, ensuring convergence to the optimal design, overcoming issues seen in MOPSO and CEO [16]. Therefore, NSGA-II is selected for the multi-objective seismic optimization of rocking bridges in this study. The optimal solution will be determined based on the Pareto frontier generated by NSGA-II, as discussed in Subsection 5.4.

5.4 Selection of the optimal solution

The application of a MOOA yields a Pareto frontier comprising all non-dominated solutions, as indicated by the blue triangles in Fig. 12. Here, ‘non-dominated’ refers to solutions that exhibit trade-offs among different objectives, such that no single solution outperforms all others across every criterion [74]. Identification of the final optimal solution on the Pareto frontier is framed as a multi-criteria decision-making problem, which is addressed using the entropy-TOPSIS method [30,41].

Initially, the entropy weighting method is adopted to impartially evaluate the relative importance of the two optimization objectives (i.e., Δ_v and Δ_d). This method assigns weights according to the variability and information content, thereby eliminating subjective influence and preserving the intrinsic features of the problem. As calculated from Eqs. (2)–(3), the entropy weights corresponding to Δ_v and Δ_d are 0.504 and 0.496, respectively, implying that the two objectives contribute almost equally to the optimization process.

Subsequently, the TOPSIS method is applied to rank the Pareto solutions based on their relative proximity to the ideal solution. The previously computed entropy weights are incorporated into the relative proximity metric (Eq. (7)), which quantitatively assesses the distance between each Pareto solution and the ideal solution. As shown in Fig. 13, the Pareto frontier solution closest to the ideal solution is selected as the optimal one, with Δ_v = 1380 kN and Δ_d = 0.036 m, marked with a red pentagram in Fig. 12(a).

The optimized design parameters, as shown in Table 8, indicate that Pier 1# has smaller values for all three design parameters compared to Pier 2#. Prior to optimization, the prototype bridge had a larger stiffness for Pier 1# due to shorter height, which resulted in a higher base shear and smaller displacement for Pier 1# compared to Pier 2#. However, following the optimization process, the smaller axial load ratio for Pier 1# led to a reduction in base shear [75]. Additionally, the smaller PT area ratio and ED bar area contributed to an increase in displacement [75]. These changes resulted in a decrease in the differences in base shear and displacement between the two piers.

5.5 Evaluation of the optimization effects

As shown in Table 9, the optimization process involves several stages, including: 1) constructing the input–output data set through NTHA, which consumes about 220 min and represents the dominant portion of the total runtime; 2) development of the surrogate model requires about 15 min; 3) The execution of the NSGA-II algorithm to identify the Pareto frontier takes about 5 min. Overall, the entire optimization process is completed in about 4 h.

Although constructing the XGB model is relatively time-consuming, it substantially reduces the computational effort needed to obtain the Pareto frontier [20]. As presented in Table 9, the NSGA-II algorithm demands about 30000 function assessments to find the Pareto frontier. In contrast, employing the traditional NTHA method to assess seismic performance would require stepwise numerical integration for each case, resulting in 30000 FE analyses and an estimated total computation time of 3000 min (each case costs about 0.1 min). Therefore, establishing a seismic response surrogate model using ML (XGBoost in this case) to replace FEM can significantly enhance optimization efficiency by up to 92% (i.e., from 3000 to 240 min) without sacrificing model accuracy.

Given that ML models function as approximation algorithms, validation of the optimized results using NTHA is essential to ensure the reliability and practical applicability of the proposed design solution. Table 10 provides a comparison of predicted results obtain by various models. For XGBoost, the predicted base shear values V₁ and V₂ of 2127 and 1479 kN, and pier displacements d₁ and d₂ of 0.040 and 0.083 m, respectively. Compared to the corresponding NTHA results, these predictions are 3.1% higher for V₁, 12.1% higher for V₂, 11.1% lower for d₁, and 9.8% lower for d₂. The discrepancies in both Δ_v and Δ_d are minor, with predictions exceeding those of NTHA by only 12.9% and 8.5%, respectively. This indicates that the XGB model can reliably approximate the seismic response.

It is worth noting that the surrogate model primarily serves to enhance optimization efficiency and expedite the search for optimal design solutions, rather than replacing precise analytical methods. Within this context, moderate prediction deviations are acceptable, as they considerably reduce computational effort while maintaining adequate trend accuracy. However, given that such errors may affect structural safety, surrogate-based results cannot directly determine the final design. Therefore, the optimal parameters derived from the XGB surrogate model should be further verified through NTHA to accurately assess seismic responses and ensure that the final design meets safety and performance requirements.

Furthermore, the NTHA method is employed to evaluate the seismic performance of the optimized bridge relative to the prototype, with the comparative outcomes summarized in Table 10 to demonstrate the benefits of the optimization. The prototype bridge (i.e., before optimization) is constructed by assigning the average values of all 12 design parameters listed in Table 2. For the prototype bridge, Δ_v between the two piers is 1562 kN, and Δ_d is 0.058 m. In contrast, the optimized bridge achieves a substantial reduction in Δ_v by 52.3% and a corresponding reduction in Δ_d by 19.0%. These findings demonstrate that the optimization process effectively minimizes seismic response differences among piers, thereby improving the overall structural performance.

While the current study specifically focuses on a self-centering rocking bridge configuration to demonstrate the methodology, we would like to emphasize that the framework itself is inherently adaptable to other bridge systems, including those with conventional piers. This adaptability stems from the following key aspects of the framework.

1) Flexibility of the Core Methodology: The optimization framework utilizes ML (XGB), Pareto front calculation (NSGA-II algorithm), and optimal solution selection (entropy-weighted TOPSIS method). These tools are versatile and widely applicable for seismic design optimization across a variety of bridge configurations.

2) Computational Efficiency: The framework leverages the XGB model to train on a limited number of samples, significantly reducing the need for exhaustive NTHAs when calculating the Pareto front. This computational efficiency allows the framework to be applied to a broader range of bridge systems, including those with conventional piers.

However, it is important to note that for more complex bridge configurations, such as curved or skewed bridges, the XGB model may not be as accurate in predicting seismic responses. In such cases, it may be necessary to integrate more advanced models, such as Physics-Informed Neural Networks (PINNs) [76], to improve prediction accuracy.

6 Conclusions

This study develops a ML-assisted multi-objective optimization framework for the seismic design of self-centering rocking bridges. Initially, bridges with varying design parameters are subjected to nonlinear time history analyses to construct the input–output data set. Based on this data set, six distinct ML models are trained to accurately predict the seismic responses of rocking bridges. The best-performing ML model is subsequently used as a surrogate for the FEM to enhance computational efficiency in solving the MOP. Four advanced MOOAs are evaluated to determine their effectiveness in calculating the Pareto frontier. Finally, the multi-criteria decision-making approach is employed to identify the optimal design from the Pareto front. The major findings of this study are summarized as follows.

1) Among the six ML models, the XGB model demonstrates superior performance in predicting both the target seismic responses (i.e., base shear and lateral pier displacement).

2) Replacing the FEM with the XGB-based surrogate model yields up to a 92% improvement in computational efficiency, while preserving adequate accuracy.

3) Among the four MOOAs, NSGA-II presents the most robust performance in the optimization of rocking bridges, as evidenced by a comprehensive evaluation of four metrics (i.e., hypervolume, spacing, diversity metric, and the number of solutions).

4) The proposed multi-objective optimization framework effectively achieves a performance balance between the piers of self-centering bridges. Compared with the pre-optimization prototype bridge, displacement and shear discrepancies among piers in the optimized rocking bridge are reduced by 19.0% and 52.3%, respectively.

Future investigations will extend the optimization framework to various bridge configurations and structural typologies, thereby assessing the generalization capability of the ML models across diverse engineering scenarios. Incorporating life-cycle performance metrics into the optimization process could provide a more holistic evaluation of seismic resilience.

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