Efficient Dynamic Performance Prediction of Railway Bridges Situated on Small-Radius Reverse Curves

Bridges situated on small-radius reverse curves play a pivotal role within some railway networks, exerting infuence over project-wide design progress. Typically, assessing the safety of bridge design parameters necessitates laborious vehicle-bridge dynamic coupling vibration numerical analysis or model experiments. To streamline the design process and enhance efciency during the preliminary design phase, we propose an efcient method to assess the dynamic performance of bridges on small-radius reverse curves. Tis approach enables direct prediction of bridge dynamic performance based on design parameters, eliminating the need for numerical simulations and model experiments. We frst develop a vehicle-bridge coupling vibration program grounded in train-curve bridge coupling vibration theory, validated using on-site measured data. Subsequently, through numerical simulation experiments, we evaluate 80 simply supported beam bridges on small-radius reverse curves under various operating conditions, generating ample dynamic response data for bridge pier tops and girders. Tese data are then compared with regulatory thresholds to assign dynamic performance labels. After identifying essential design parameters as data features using Fisher scores, we proceed to input these features into a support vector machine (SVM). Trough supervised training with dynamic performance labels, this process empowers the SVM model to predict the dynamic performance of the bridge. Our results demonstrate that this method circumvents the need for detailed vehicle-bridge interaction analysis, yielding an impressive 86.9% accuracy in predicting dynamic performance and signifcantly boosting computational efciency. Besides, the top fve design parameters that significantly infuence the prediction of bridge dynamic performance are obtained. Tis novel approach has the potential to expedite design assessments and enhance safety in railway bridge construction.


Introduction
Bridges situated on small-radius reverse curves hold a pivotal role in railway networks, shaping project design progress.Traditionally, ensuring the safety of these bridge design parameters involves time-consuming vehicle-bridge dynamic coupling vibration numerical analysis and model experiments.
Over the past decades, the dynamic interaction between moving vehicles and bridges has been extensively studied.Researchers have employed various numerical models and analysis methods to address vehicle-bridge interaction.Tese studies contribute to structural dynamics and encompass theoretical, numerical simulation, and real-world experimental aspects [1][2][3][4][5].Tese investigations often utilize diferent vehicle models, such as moving force models, moving mass models, quarter-vehicle models, and fullvehicle models [6][7][8][9][10][11][12][13][14][15].Tese models describe vehicle motion and response through theoretical deduction or fnite element modeling.Simultaneously, numerical simulation of bridge structures usually employs fnite element modeling to account for their intricate structural attributes.Te vehiclebridge interaction (VBI) system, a core concept in this feld, describes the dynamic interaction between vehicles and bridges [16,17].Tis system models vehicle and bridge subsystems while considering wheel-rail contact interactions.Te VBI system models, through theoretical analysis or numerical simulation, have demonstrated accurate simulations of vehicle and bridge dynamic responses.However, in some cases, considering the fullvehicle model and spatial dynamic interaction can entail high computational costs, particularly when extensive simulations or real-time online evaluations are required [2,18,19].
Recently, machine learning methods have emerged as promising computational algorithms in civil structural dynamics.Machine learning models demonstrate the ability to tackle complex structural dynamic challenges by leveraging vast structural data and vibrational information, providing efcient and accurate predictions.Notably, literature [20,21] employed fully connected neural networks for dynamic analysis of buildings under seismic excitations, reducing computational workload.Another study [22] predicted wind pressure time histories for tall buildings using machine learning, addressing the dynamic efects of wind loads.In addition, references [23,24] estimated aerodynamic forces and responses for long-span bridges, exploring deep learning algorithms for dynamic response prediction of wind-sensitive structures.In the realm of predicting vehiclebridge interaction, reference [25] developed an artifcial neural network as a proxy model for VBI systems in railway transportation, aiming to predict the time history of bridge responses caused by vehicles.Furthermore, reference [26] employed a nonlinear autoregressive exogenous input neural network model to predict bridge accelerations subjected to quarter-vehicle model loads in the vertical plane.Beyond time history estimation, machine learning methods have been employed to predict statistical variables.Reference [27] achieved maximum response prediction of building structures under artifcial seismic efects using structural parameters and multilayer perceptrons.Reference [28] predicted the average strain response of bridges by selecting total vehicle weight, truck length, vehicle speed, and truck axle spacing as neural network inputs.However, to the author's knowledge, no literature has been found that utilizes a machine learning approach to directly predict the dynamic performance of bridges on small-radius curves subjected to vehicle-induced vibrations and assess the importance of selected design parameters for the dynamic performance prediction.
Te objective of this paper is to propose an efcient method for predicting the dynamic performance of bridges on small-radius reverse curves, bypassing the cumbersome steps of numerical simulation or model experiments, and enhancing bridge design efciency.Te main process of the methodology proposed in this article is described in Figure 1.First, we establish a vehicle-bridge coupling vibration program based on train-curve bridge coupling vibration theory and validate it using on-site measured data.Subsequently, we conduct numerical simulation experiments on 80 simply supported beam bridges located on small-radius reverse curves under various operating conditions, generating ample dynamic response data for bridge pier tops and girders.Tis data is then compared with regulatory thresholds to assign dynamic performance labels.By selecting critical bridge design parameters as data features, we input them into a support vector machine (SVM) and train the model under the supervision of dynamic performance labels, empowering the SVM model with the ability to predict the dynamic performance of bridge piers and girders.In conclusion, our proposed approach ofers a novel and efcient method to predict the dynamic performance of bridges on small-radius reverse curves, alleviating the need for exhaustive numerical simulations or model experiments.Tis methodology holds the potential to accelerate design assessments and enhance safety in railway bridge construction.

Vehicle-Curve Bridge Coupling Vibration Model
2.1.Model Construction.Te interaction system between the vehicle and the bridge is comprised of two subsystems: the vehicle subsystem and the bridge subsystem.Tese two subsystems are connected through wheel-rail interactions, with irregularities in the given track serving as an additional excitation for the system.Te two subsystems utilize a common coordinate system where the x-axis aligns with the direction of train travel, the y-axis points upward, and the z-axis is defned according to the right-hand rule.Tis study employs the modal superposition method [29] to solve the coupled vehicle-bridge vibration equation.Te implementation encompasses both program development and the incorporation of modal analysis results obtained from the commercial software ANSYS.Subsequent sections provide an introduction to the spatial relationships of the vehicle and curved bridge in modal coordinates, as well as an explanation of the coupled vibration equations.

Train-Curved Bridge Spatial Relationship Processing.
Utilizing the method of moving coordinates and employing coordinate transformations, the precise position of a train as it traverses a curved bridge at a specifc moment can be ascertained.Te mobile coordinate system is tailored for each wheelset and moves uniformly along the centerline of the track.Te origin of this coordinate system is situated at the intersection point of the wheel axle and the track centerline.According to the right-hand rule, the positive direction of the x-axis aligns with the tangent to the curve, the positive direction of the y-axis points vertically upwards from the track plane, and the positive direction of the z-axis corresponds to the curve's normal direction.Tis is depicted in the coordinate systems denoted as o i x i y i z i and o j x j y j z j , as illustrated in Figure 2. Assuming the train is moving at a constant velocity, denoted as v, and at time t, the train has covered a distance of S i along the track.By considering the characteristics of the track curvature, the pertinent parameters at point S i (representing the train's position at time t) can be determined: (1) Te overall coordinates of the curve represented as (X Oi , Y Oi , Z Oi ); (2) Parameters at point S i including the angle α i between the curve's tangent and the OX axis, curvature ρ i , and cant angle c i ; 2 Shock and Vibration (3) Te employed moving coordinate systems, o i x i y i z i and o j x j y j z j .
Assuming the wheelset's center plane projection at time t coincides with the centroid line of the bridge at node B i , the transformation expression of the kth mode shape vector φ from the global coordinate system OXYZ to the moving coordinate system o i x i y i z i can be formulated as follows: In the provided equation, φ i,B denotes the displacement vector of the bridge's centroid mode shape in the moving coordinate system.Te matrix λ represents the coordinate transformation matrix, attainable through a sequence of two coordinate rotations.
Te formulation for Λ is given by After the coordinate transformation, the mode shape vector φ i,B centered around the bridge's centroid requires additional contemplation of the rigid arm constraint equations.When transitioning to the point along the track with a distance S i where the train is situated, the vector φ i undergoes a transformation characterized by the following relationship:

Wheel-j
Wheel-i

Center line of track
Center line of bridge Shock and Vibration 3 ( In the equation, x i,B , y i,B , and z i,B denote the positional coordinates of the bridge centroid node B i in the moving coordinate system.
If at time t the plane projection of the wheelset center does not coincide with the bridge centroid node B i , a solution can be achieved by utilizing a third-order cubic spline interpolation between the overlapping points.Tis approach allows for the determination of the bridge's mode shape displacement at any given point along the track's centerline.

Train-Curved Bridge Coupling Vibration Equation.
Using the modal superposition technique [30], the dynamic diferential equations of the coupled vehicle-bridge vibration system are established.Tis methodology involves segregating the vibration system into distinct vehicle and bridge subsystems.Te resultant set of diferential equations governing the overall system's motion is articulated as follows: where subscripts v and b, respectively, pertain to the vehicle and the bridge.Te dynamic interaction between the train and the bridge is realized by means of the coupling forces exerted at the contact interface between the two subsystems.Tese coupling forces serve as the mechanism for coupling and simultaneously act as internal excitations within the vibration system.Conversely, generalized forces represent external excitations applied to the vibration system.
(1) Solving Wheel-Rail Contact Force.Te contact forces at the wheel-rail interface for both the vehicle and bridge subsystems represent critical boundary conditions.Under spatial vibration conditions, these contact forces consist of the vertical force F yj and the lateral force F zj acting at the wheel-rail interface.Exploiting the assumptions of tight wheel-rail contact and neglecting yaw angles (assuming that the wheelset is always in radial position), the association between each contact force and the respective normal force N j and tangential force T zj in the coordinate system established at the contact point is formulated as follows: where θ ws represents the roll angle of wheelset.
(2) Calculation of Wheel-Rail Creep Force.Based on Kalker's theory of creep forces, the relationship between creep force, creep coefcient, and creepage rate is defned as follows [11]: where T xj , T zj , and M j represent the longitudinal creep force, lateral creep force, and rotational creep torque at the wheel-rail interface, respectively.f 11j , f 22j , f 23j , f 33j are the creep coefcients.ξ 1j , ξ 2j , and ξ 3j denote the longitudinal creepage rate, lateral creepage rate, and spin creepage rate of the wheel-rail contact, respectively, when the train is in straight-line motion.
Te wheel-rail creepage rate for wheelsets traversing curved tracks can be revised as follows: where ξ 1j ′ , ξ 2j ′ , and ξ 3j ′ denote the longitudinal, lateral, and spin creepage rates of the wheel-rail contact when the wheelset traverses a curve with curvature ρ i .r j stands for the actual rolling radius of the wheelset.δ j represents the wheelset contact angle.ψ ws represents the wheelset yaw angle.
Te infuence of curvature on the creep coefcient is minimal, and for the sake of simplicity, it is often treated as if in a straight-line scenario.Tis treatment only considers the efect of the lateral ofset of the wheelset from the track centerline when the wheelset traverses a curve.

Shock and Vibration
As a result, when a vehicle undergoes uniform curved motion, the creep force at the wheel-rail interface can be expressed as follows: (3) Generalized Forces.In theory, when a train is in curved motion, the primary suspension force and secondary suspension force of the train body should be transformed into the moving coordinate system through coordinate transformations.For this study, assuming a minimum curve radius of 400 m, it can be inferred that the deviation angle between the train body coordinate system and the wheelset coordinate system is extremely small.Terefore, the calculation of primary and secondary suspension forces can be directly performed as if the train is moving in a straight line.
Centrifugal force and canting force are generalized forces that must be considered during curved train motion.Assuming the wheelset i is at a curvature of ρ i and a cant angle of c i , the centrifugal force and canting force experienced by the wheelset in the moving coordinate system are as follows: where C wi represents the centrifugal force along the o iz direction.H wi signifes the force along the o iz direction induced by canting.V wi denotes the force along the o iy direction induced by canting.m wi refers to the mass of the wheelset i.
Approximating by considering the average curvature and cant angle for each of the four wheelsets in a train section, the centrifugal force and canting force experienced by each train section are calculated as follows: where subscripts f and c, respectively, indicate the bogie and carbody.

Model Verifcation
2.2.1.Field Experiment.Due to geographical constraints, a cluster of railway bridges is situated along an "S"-shaped curve with relatively small radii.Figure 3  We selected several representative bridges on the railway line and installed velocity and displacement sensors at the mid-span and the top of the bridge piers.Te sensors were set to a sampling frequency of 50 Hz.Te layout of the sensors is illustrated in Figures 5 and 6.Te predominant train types on this route are C62 freight trains [31].Te dynamic responses of the bridges as the trains passed the sensor locations were recorded.
After completing the testing phase, we performed postprocessing and concise analysis of the test signals, primarily focused on determining the resonant frequencies of the main beam and bridge piers.Figure 7 displays the displacement curve of a specifc pier within 10 seconds after a train passes, along with its frequency spectrum representation.Te displacement curve is measured by the displacement gauge at the top of the pier.Te frequency spectrum is obtained through Fourier transform.Te spectrum analysis reveals a frst transverse vibration frequency of 3.6 Hz for the pier.Following this method, the resonant frequencies of both main beams and bridge piers for all tested bridges were determined.

Numerical Simulation.
In general, employing solid elements for bridge modeling provides a fner analysis of vehicle-bridge coupling vibrations.However, given the extensive number of bridges in this case study and the focus on dynamic responses at bridge midspans and pier tops, a beam element approach was adopted for bridge modeling.Te use of beam elements ofers a substantial reduction in computational costs while still meeting research requirements.Figure 8 depicts the beam element fnite element models for all bridges located on the S-shaped curve, along with the frst-order vertical bending mode and frst-order transverse bending mode for the 32 m bridge.With the bridge models in place, we proceeded to conduct vehicle-bridge coupling vibration analysis, following the principles outlined in Section 2.1.Given the limited data on track irregularities, the Shock and Vibration US six-level spectrum from the Federal Railroad Administration is utilized to simulate irregularities in the lateral, rotational, and vertical directions.We chose the level 5 irregularity as the track irregularity used in simulation by survey and trail.Tis enabled us to obtain dynamic response results for the bridge fnite element models corresponding to the feld test measurement points.

Results
Comparison.We utilized displacement and velocity signals obtained from feld measurements to calculate the natural frequencies of the main beam and the modal displacement ratios between the pier top and bottom.Tese results were then compared with the outcomes of a numerical simulation using a fnite element model of the bridge.Table 1    Additionally, a comparative analysis was conducted between the measured and simulated displacement timehistory curves of the identical bridge during the train's passage.However, achieving complete consistency between the vehicle parameters of the simulated train and those of the actual operational train poses signifcant challenges.Tis is due to the operational freight trains, whose load weights are typically unknown and whose speeds may not be constant.Moreover, real measured data often contains noise interference, making it difcult to perfectly align the measured and simulated results.

presents a comparison between the numerical
Taking into consideration that freight trains often operate empty, a simulation model of the C62 freight train (comprising a DF4 locomotive and 20 empty C62 cars) is established.After investigating the approximate speed of the freight train passing through this curved bridge (at 35 km/h), the lateral displacement response at the midspan of a 24 m simply supported beam bridge is computed.Tis simulated data is then compared with the displacement response obtained from the same bridge point during the actual passage of the train, as shown in Figures 9 and 10.It is apparent that the two responses in the time and frequency domains are largely consistent, afrming the efectiveness of the numerical simulation model.

Performance Evaluation Metric.
In order to assess the dynamic performance of bridges on small-radius reverse curves, a performance evaluation criterion must be devised.Critical indicators afecting bridge safety encompass the lateral displacement, lateral acceleration, and vertical acceleration at the midspan of the main beam and the top of bridge piers.Evaluation of whether these parameters meet safety requirements involves adherence to the railway bridge inspection specifcation of China [32], with specifc threshold values or recommendations as follows: (1) For situations where the train speed is less than 80 km/h, the recommended maximum lateral displacement at the midspan of a concrete bridge is defned by where L represents the calculated span of the bridge, and B 2 denotes either the transverse center-to-center distance of multi-girder systems or the spacing between supports.
For bridge piers with pile foundations, the suggested maximum lateral displacement can be expressed as follows: where the corrected height ∆h accounts for soil characteristics.(2) Te lateral acceleration of bridges is limited to 1.4 m/s 2 .(3) Te vertical acceleration of bridges is restricted to 3.5 m/s 2 .
Tese standards, derived from the railway bridge inspection specifcation, guide the assessment of lateral displacement, lateral acceleration, and vertical acceleration in order to ensure the safety of bridges situated on small-radius reverse curves.Compliance with the specifcation is pivotal in enhancing the dynamic performance and safety of such structures.Shock and Vibration

Shock and Vibration
Tis paper introduces a comprehensive method for evaluating bridge dynamic performance based on the aforementioned threshold values.Tis approach integrates with labels used in classifcation tasks in machine learning, facilitating the direct prediction of the dynamic performance level of bridges through design parameters.Figure 11 illustrates the process of grading bridge dynamic performance using this comprehensive evaluation method.Initially, displacement and acceleration metrics for evaluating bridge dynamic performance are computed based on numerical simulation models, totaling four distinct metrics.Te simulated values for these four metrics are calculated in relation to the limits or recommended values specifed in regulations, measuring the surplus capacity of bridge dynamic performance.A higher proportion signifes a smaller surplus capacity in the bridge's dynamic performance.If the proportion exceeds 100%, it indicates a need for enhancement in the bridge's dynamic performance.Subsequently, the maximum value among these four proportions is selected, compared against the ratio intervals corresponding to the dynamic performance levels proposed in this study.Based on this comparison, the bridge is assigned its respective dynamic performance level.It is worth noting that these dynamic performance levels correspond to the labels used in the subsequent classifcation tasks.

Feature Selection.
In order to directly predict the dynamic performance of bridges on small-radius reverse curves, it is crucial to select the most relevant and signifcant parameters from the multitude of variables.Feature selection serves to reduce computational costs, mitigate overftting risks, and enhance the interpretability of computational outcomes.Tis study opts for design parameters closely linked to or directly correlated with dynamic responses as features.Te selected parameters encompass (1) train speed, (2) bridge span, (3) pier height, (4) radius of the curve where the main beam is located, 10 Shock and Vibration ( 5)-( 7) frst to third-order vertical natural frequency of the main beam, ( 8)-( 10) First to third-order lateral natural frequency of the main beam, ( 11)-( 13) frst to third-order torsional natural frequency of the main beam, ( 14)-( 16) frst to third-order longitudinal natural frequency of the pier, ( 17)-( 19) frst to third-order lateral natural frequency of the pier, and ( 20)-( 22) frst to third-order torsional natural frequency of the pier.
To eliminate redundant features and undertake dimensionality reduction, principal component analysis (PCA) [33] is oftentimes employed.PCA is a classic statistical method that transforms correlated variables into a set of unrelated composite variables, or principal components.Tese new variables, obtained as linear combinations of the original variables, efectively reduce data dimensions.While PCA is capable of projecting high-dimensional data into a lower-dimensional space, preserving as much original data information as possible while reducing redundancy, it falls short in discerning the importance of original features (i.e., their contribution to machine learning model performance) due to the nature of feature mapping.To address this limitation, we turn to the Fisher Score method [34,35] to quantify the distinctiveness between features, thereby further exploring which design parameters within the original set hold greater relevance for predicting bridge dynamic performance.
Te Fisher score is a powerful technique used for reducing the dimensions of features in a dataset.Its primary objective is to identify a subset of features that, within a data space formed by these selected features, maximizes the distances between data points belonging to diferent classes while minimizing the distances between data points within the same class.Specifcally, given a training dataset X ∈ R m×n associated with c distinct classes, the Fisher score of the ith feature is calculated using the following equations [34,35]: where S b (f i ) represents the between-class scatter of the ith feature, n k stands for the number of samples in the kth class, μ (k)   i denotes the mean of the ith feature within the kth class, μ i signifes the mean of the ith feature in the dataset X, S (k)  t (f i ) represents the within-class scatter matrix of the ith feature related to the kth class, and x (k)  ij denotes the value of the ith feature for the jth sample in the kth class.
Higher Fisher scores indicate larger feature disparities between the classes, thus highlighting signifcant contributions for the dynamic performance prediction tasks in this study.In addition, reducing feature dimensions by selecting a small number of features associated with high Fisher Scores can enhance the training efciency of classifcation models and mitigate the risk of overftting.
3.3.Support Vector Machine.Support vector machine (SVM) [36] stands as a classical classifcation algorithm with the core objective of identifying the optimal classifcation Shock and Vibration hyperplane when the sample data is linearly separable, as illustrated in Figure 12.Te samples in proximity to the optimal hyperplane are termed support vectors, while the dashed lines parallel to the hyperplane and passing through these sample points represent the margin boundaries.SVM centers its focus on these support vector points, seeking to maximize the margin between the boundary lines through the identifcation of an optimal hyperplane.Let x ∈ R n represent the feature vector data, and y ∈ −1, 1 { } denote the class labels.Te discriminant function (hyperplane) can be expressed as f(x) � ωx + b, where ω represents the weight vector normal to the hyperplane, and b is the bias term.At f(x) � 0, the sample x lies on the classifcation hyperplane; when f(x) > 0, x is classifed under y � 1; and when f(x) < 0, x falls within y � −1.Terefore, if y(ωx + b) > 0 holds for all samples x, it implies correct classifcation for all samples.
Generally, the two margin boundaries on which the support vectors reside are defned as ωx + b � ± 1. Consequently, the distance from a support vector to the classifcation hyperplane becomes 1/‖ω‖, while the classifcation margin equates to 2/‖ω‖.To the classifcation margin of 2/‖ω‖, and to facilitate optimization, this is often reformulated to minimize (1/2)‖ω‖ 2 .Tus, the SVM algorithm's objective function for discovering the optimal hyperplane is obtained as follows: Te standard approach for solving the objective function employs the Lagrange dual method.Initially, the objective function is transformed into Lagrange form: where α i ≥ 0 represents Lagrange multipliers.Taking derivatives of function L with respect to ω and b, and setting them to zero, yields the following equation: Tus, function L can be deduced as follows: Ultimately, the optimization problem is formulated as follows: Trough this, the algorithm can compute α that maximizes the objective function, thereby obtaining corresponding ω and b values, establishing the optimal classifcation hyperplane.
Te aforementioned outlines the SVM algorithm's implementation for linearly separable datasets.However, in practical applications, many problems are not linearly separable.For such nonlinearly separable scenarios, a nonlinear mapping function is often employed to map the dataset into a higher-dimensional space, rendering it linearly separable in that space.Tis approach facilitates the construction of an SVM classifer to seek the optimal hyperplane."It is noteworthy that this study utilizes the radial basis function kernel function in SVM to address nonlinearly separable scenarios efectively." Te original SVM method, initially designed for binary classifcation, has been adapted in this study using the one-vs-rest strategy for multiclass classifcation.Tis approach involves training individual binary classifers for each class, designating the specifc class as the positive class, and grouping the remaining classes as the negative class.During the prediction phase, these classifers assign decision scores, and the class associated with the classifer yielding the highest score is selected as the ultimate predicted class.Tis strategic adoption efciently transforms the multiclass problem into a sequence of binary decisions.

Results
Tis study utilizes a C62 freight train comprising 21 cars, inclusive of a DF4 locomotive, 10 loaded C62 cars, and 10 empty C62 cars.Te speed conditions vary from 40 km/h to 80 km/h, increasing in increments of 10 km/h.Te dynamic responses of the train passing through 80 simply supported beam bridges on small-radius curves are computed.Tus, the resulting dataset comprises 400 data samples.For each sample, the lateral displacement, lateral acceleration, and vertical acceleration at the midspan of the girder and the lateral displacement at the top of the pier are compared with the corresponding code limits or recommended values.Te performance evaluation values, representing the proportions of simulation values to standard values, as discussed in Section 3.1, are derived.Figure 13 displays the maximum evaluation values for all bridges under the fve train speed conditions.
Observing Figure 13 reveals that, with the exception of certain bridges under the 70 km/h and 80 km/h train speeds, where the evaluation values surpass 100%, all other bridges meet the specifed limit requirements.Considering that the actual operating speeds of freight trains on this curve line typically do not exceed 60 km/h, the dynamic performance of all bridges meets the regulatory requirements.Further analysis of the dynamic performance computation results indicates that the majority of bridges exhibiting high evaluation values do so because the lateral displacement of the pier top increases with higher train speeds, often exceeding the recommended values stipulated in design codes.As a result, when designing curved bridges, enhancing the lateral and torsional stifness of piers becomes crucial.
Te 400 data samples are labeled from 0 to 10 according to the outcomes depicted in Figure 13 and in accordance with the dynamic performance levels proposed in Section 3.1.However, in this study, only 5 labels correspond to the data samples.Te distribution of these labels is depicted in Figure 14.Tis fgure highlights that label 8 constitutes the majority among all the labels, while label 6 holds the smallest proportion.
In this study, the predictive outcomes of bridge dynamic performance are evaluated across six distinct tasks based on diferent speed divisions, as outlined in Table 2.With the exception of the fnal task, which encompasses all speed conditions in both training and test sets with independent data, tasks 1-5 involve distinct speed conditions.Tasks 1-5 specifcally evaluate the model's efcacy in handling diverse and unfamiliar vehicle speeds.Evidently, the frst fve tasks establish the sample ratio of 4 : 1 for the training and test sets based on the division of speed conditions.Tis same sample allocation ratio is maintained in the fnal task, where the training and test sets follow a 4 : 1 ratio, randomly selected from the entire dataset.Tese tasks primarily assess the predictive models' ability to generalize bridge dynamic performance from known to unknown train speeds, as well as the model's performance when trained and applied using data from identical train speeds.During the training phase, a 5-fold cross-validation technique is utilized to determine the optimal hyperparameters for the support vector machine (SVM).In the application phase, the SVM with optimal hyperparameters is employed on the test data to predict dynamic performance outcomes.
Figure 15 showcases the importance of various design parameters obtained through Fisher Score computation across the six tasks.Te feature numbers correspond to the design parameter numbers in Section 3.2, ensuring consistency.Te signifcance of diferent features remain consistent across various tasks, as illustrated by the average values depicted in Figure 15 across the six tasks.Notably, the top fve contributing features to the predictive model's performance are features numbered 1, 17, 18, 8, and 4. Tey correspond to Train speed, First-order lateral natural frequency of the pier, Second-order lateral natural frequency of the pier, First-order lateral natural frequency of the main beam, and Radius of the curve where the main beam is located, respectively.Tis fnding emphasizes the paramount importance of aligning these fve crucial design parameters during the bridge design phase.In other words, if the predicted outcomes for bridge dynamic performance do not meet the regulatory requirements, adjusting these fve design parameters should be considered the primary course of action.
Figure 16 illustrates the results of dynamic performance prediction utilizing features sorted by the Fisher Score.Te fgure clearly indicates that as the count of features increases, the predictive accuracy for all six tasks steadily enhances.Upon reaching eight features, the predictive accuracy has essentially reached the level of accuracy achieved using all features.Tis validates the efectiveness of the Fisher score in   Te comparative results of these six tasks underscore the importance of comprehensively incorporating all operational conditions within the training set.Furthermore, the 86.9% predictive accuracy achieved in Task 6 highlights the efectiveness of the proposed method and its signifcant potential for application in the design of bridges on smallradius curves.Tis method involves training a bridge dynamic performance prediction model based on established numerical simulation models.It directly provides predicted values for the dynamic performance level of similar bridges, facilitating the optimization of design parameters in the initial design phase.In addition, it allows for a rough but rapid safety assessment of already constructed bridges, saving signifcant costs and time required to establish vehicle-bridge coupled vibration analysis models.

Conclusion
Tis paper proposes an efcient method to predict the dynamic performance of bridges on small-radius reverse curves.By establishing a vehicle-bridge coupling vibration program based on train-curve bridge coupling vibration theory and conducting numerical simulation experiments, we generate ample a dynamic response data of bridges and label them with dynamic performance level.Te bridge design parameters related to these dynamic response data are fed into a SVM to train a model that can directly predict the bridge's dynamic performance.
Te outcomes reveal the highest predictive accuracy for bridge dynamic performance to be 86.9%,demonstrating the efectiveness of our proposed methodology and its potential utility in the design of bridges on small-radius curves.Te relationship between feature number and Fisher score is also explored.Notably, the top fve design parameters that signifcantly afect bridge dynamic performance prediction are train speed, frst-order lateral natural frequency of the pier, second-order lateral natural frequency of the pier, frst-order lateral natural frequency of the main beam, and radius of the curve where the main beam is located.Tese fndings underscore the critical role of harmonizing these parameters during the bridge design process.When predicted bridge dynamic performance falls short of safety standards, prioritizing adjustments to parameters directly related to these fve parameters is recommended.
Te results from our study underscore the potential of our approach for facilitating the design and evaluation of bridges, particularly those located on small-radius curves.As the railway industry continues to evolve, our research contributes valuable insights into the fast optimization of bridge designs, ultimately ensuring safer and more efcient railway systems.Shock and Vibration

Figure 1 :
Figure 1: Main process of the methodology.
illustrates the plan layout of the railway route situated in East China, depicting curve radii of 400 m, 500 m, and 500 m.Transition curves of lengths 50 m, 40 m, and 60 m are respectively used between three sections of circular arc curves.Te visual representation of the railway line and bridges can be found in Figure 4. Te bridge types encompass: (1) 80 prestressed concrete simply supported beam bridges: Spans include 16 m, 20 m, 24 m, and 32 m.(2) Reinforced concrete continuous rigid frame bridges: Span confgurations of 18.3 + 24.0 + 18.3 m and 17.3 + 30 + 30 + 19.3 m.Tis study exclusively focuses on the investigation of the 80 simply supported beam bridges.

Figure 3 :
Figure 3: Plan layout of the railway line.

Figure 7 :Figure 8 :
Figure 7: Residual vibration curve in time and frequency domain.(a) Time domain.(b) Frequency domain.

Figure 8 :
Figure 8: Bridge fnite element models and typical displacement modes.(a) Finite element models of all bridges.(b) First-order vertical bending mode.(c) First-order lateral bending mode.

Figure 11 :
Figure 11: Determination of bridge dynamic performance level.

Figure 14 :
Figure 14: Distribution of data sample labels.

Figure 16 :
Figure 16: Predictive accuracy of bridge dynamic performance for diferent tasks.

Table 1 :
Comparison of modal results between numerical simulation and feld measurements.