Braking Performance and Response Analysis of Trains on Bridges Under Seismic Excitation

Abstract

Earthquakes can trigger emergency braking in urban rail systems, yet the combined effect of braking and ground motion on train–bridge safety remains poorly quantified. Using the Wuxi Metro Line S1 (160 km/h initial speed) on a ten-span simply supported bridge as a case study, we build a multi-body dynamic subway model coupled to a finite element track–bridge model with non-linear Hertz wheel–rail contact. Under the design-basis earthquake (PGA ≈ 0.10 g), the train’s derailment coefficient and lateral car body acceleration rise by 37% and 45%, while the bridge’s lateral and vertical accelerations increase by 62% and 30%, respectively. Introducing a constant emergency brake deceleration of 1.2 m/s² cuts those train-side peaks by 20–25% and lowers the bridge’s lateral acceleration by 18%. The results show that timely braking not only protects passengers but also mitigates seismic demand on the structure, offering quantitative guidance for urban rail emergency protocols in earthquake-prone regions.

1. Introduction

Urban rail transit systems, particularly subway and light rail networks, have become indispensable components of modern urban infrastructure, offering unparalleled convenience, efficiency, and sustainability for urban mobility [1]. As the backbone of public transportation in densely populated cities, subway systems play a crucial role in alleviating traffic congestion and reducing carbon emissions; however, operational safety remains a major challenge, especially under dynamic scenarios such as acceleration, deceleration, and emergency braking [2,3]. These operations induce complex longitudinal forces at the wheel–rail interface [4,5], which, if not properly managed, can jeopardize vehicle stability, track integrity, and passenger safety. Over the past few decades, several catastrophic accidents related to braking failure have underscored the urgency of addressing these issues. For example, in 2009, a collision occurred in the Shanghai subway system due to signal errors and miscalculated braking distance, resulting in casualties and operational disruption. Similarly, a rear-end collision in Beijing in 2023, during snowy conditions, revealed the inadequacy of braking performance caused by reduced wheel–rail adhesion.

Meanwhile, although light rail transit (LRT) systems are highly integrated with subway networks, they present distinct challenges and opportunities [6,7]. LRT is characterized by higher speed, mixed traffic adaptability, and lower construction costs, and often runs on elevated bridges through urban environments. While elevated design optimizes land use and reduces construction expenses, it also increases vulnerability during seismic events. In earthquake-prone regions such as China, the suddenness and unpredictability of seismic hazards pose serious threats to rail transit systems [8,9]. Zeng et al. [10] analyzed derailment mechanisms of trains on bridges subjected to strong seismic excitation and found that resonance-induced vibration in bridges could significantly aggravate wheel–rail separation, presenting a critical risk to running safety during earthquakes. Guo et al. [11] conducted probabilistic safety analysis of high-speed trains on bridges under near-fault pulse-type ground motions, emphasizing that ground motion uncertainty and running velocity jointly influence the probability distribution of train instability.

Historical earthquake disasters, such as the 2008 Wenchuan earthquake, have demonstrated that seismic energy release can lead to track displacement, bridge collapse, and train derailment [12,13]. For subway and light rail systems operating on elevated structures, ground motions induced by earthquakes may destabilize the wheel–rail interaction, alter adhesion conditions, and trigger emergency braking [14]. In such cases, drivers or automated systems must rapidly implement deceleration strategies to avoid collisions. However, the coupling mechanisms among seismic excitation, dynamic wheel–rail interaction, and braking performance remain inadequately understood [10,15,16], particularly regarding the influence of residual track irregularities on post-earthquake operational safety [17,18].

In recent years, researchers have begun to investigate the running stability of trains during earthquakes. Montenegro et al. [19,20] conducted a comprehensive review of methods for assessing the running safety of trains on bridges subjected to seismic excitation, noting that traditional indices fail to capture nonlinear dynamic behavior under seismic loading and calling for the development of physically consistent and vibration-sensitive assessment frameworks. Xiang et al. [4,21] proposed a multi-body dynamic model for train–track–bridge-coupled seismic response analysis and pointed out that under strong earthquakes, braking has a significant impact on both bridge and vehicle response. Their research offered detailed quantitative insights into longitudinal force induction, wheel–rail separation, and adhesion degradation under seismic excitation, validating the necessity of coupled modeling under uncertainty. Furthermore, Guo et al. [11] incorporated the velocity pulse characteristics of near-fault ground motions into a probabilistic failure analysis framework, revealing the combined uncertainty contribution from bridge configuration and running speed. In a high-dimensional input context, Zhang et al. [22] developed a long short-term memory (LSTM) neural network to predict seismic responses of trains, ensuring both nonlinear mapping accuracy and computational efficiency. Complementarily, Li et al. [23] introduced a deep learning surrogate model integrated with structural uncertainty for seismic safety analysis, significantly improving the quantitative assessment of operational risks. Notably, Guo et al. [24] applied integrated train–track–bridge modeling in the case study of the Sesia viaduct and verified sensitivity under various structural configurations to seismic impact. These studies not only enriched the theoretical framework of coupled seismic response but also offered actionable data and models for post-earthquake operational strategies [25,26,27].

With the rapid advancement of deep learning and machine learning technologies, increasing research has applied artificial intelligence to optimize seismic response prediction and braking decision-making [28,29]. By training convolutional neural networks (CNN) or LSTM-based models, researchers can efficiently evaluate the safety implications of various ground motions and train operating conditions. For instance, Zhang et al. [22] developed a deep long short-term memory network to predict nonlinear seismic responses of structures, demonstrating high accuracy in capturing dynamic features. Similarly, Li et al. [23] proposed a deep learning-based surrogate model for probabilistic safety assessment of train–bridge systems, improving the reliability of response predictions under complex seismic scenarios. These approaches provide more accurate and efficient tools to enhance the safety and resilience of train operations in future seismic environments [30,31,32]. Coupled dynamic models for train–track–bridge interaction under seismic excitation have also been extensively studied in recent years, providing the numerical foundations on which the present work builds [33,34,35].

In this study, a relevant model of the subway–track–bridge system is developed to investigate the dynamic response of trains under emergency braking during earthquakes. The nonlinear Hertzian contact is used to simulate wheel–rail interaction, and both ground motion and track irregularities are incorporated to reflect extreme conditions that may occur in actual operations. Simulation results show that, under seismic conditions, implementing deceleration measures in a timely manner significantly reduces the dynamic responses of both the train and the bridge compared to maintaining a constant speed, thereby enhancing the operational stability and passenger safety. These findings offer practical guidance for design and emergency response strategies in subway systems, highlighting the necessity of establishing a more robust safety assessment framework in earthquake-prone regions.

The results of this study can contribute to the revision of seismic design codes for urban rail transit systems, and provide a theoretical and practical foundation for the reconfiguration of braking systems under extreme environments, thereby ensuring operational stability and safety during natural disasters.

2. Materials and Methods

This section outlines the entire modeling framework, covering the train dynamics model, the track–bridge finite element model, the seismic and braking inputs, and the numerical analysis procedure.

2.1. Vehicle Model

The subway computational model established in MATLAB (used version: R2020b) is based on the common B-type (B-type refers to the Chinese standard metro car with a body width of ≈2.8 m, length of 19–20 m, and axle load of 14 t, as specified in TB/T 1924-2006) subway vehicles in actual operation, and comprehensively considers the dynamic characteristics of the main structures during train operation. In the model, the car body, bogie, and wheels are idealized as rigid bodies in order to improve the simulation efficiency while ensuring the calculation accuracy. A single subway train consists of one car body, two bogies, and four wheel pairs, in which the main structures, such as the car body and bogies, are considered to have six degrees of freedom, corresponding to three translational and three rotational directions, while the wheel pairs are only considered to have five degrees of freedom due to their unique kinematic characteristics. After statistics and modeling, the total number of degrees of freedom of a single train reaches 38, which can reflect the vibration response of the train and its coupling effect in the process of operation more completely. Dynamics modeling uses the principle of energy partitioning to derive the equations of motion of the train system, and the specific expression form is shown in Equation (2). In order to be closer to the actual operating conditions, the initial operating speed of the train is set to 160 km/h. The key parameters and structural characteristics of the subway vehicle are listed in

Table 1. Technical parameters of metro B vehicles.

Project	Numerical Value	Unit
Body mass	21,920	kg
Frame quality	2550	kg
Wheelset mass	1420	kg
Nodal moment of inertia of wheelset	97	$\mathrm{k}\mathrm{g}\cdot {\mathrm{m}}^2$
Frame nodding moment of inertia	1750	$\mathrm{k}\mathrm{g}\cdot {\mathrm{m}}^2$
Body nodding moment of inertia	617,310	$\mathrm{k}\mathrm{g}\cdot {\mathrm{m}}^2$
Axle weight	14	t
Crew	240	people

, which provide the basic support for the subsequent dynamic simulation analysis.

$$M_{sub}=diag\left[M_{b1}{M}_{b2}{M}_{b3}{M}_{b4}{M}_{b5}{M}_{b6}\right]$$

(1)

$$M_{sub}{\ddot{X}}_{sub}+C_{sub}{\dot{X}}_{sub}+K_{sub}{X}_{sub}=F_{sub}$$

(2)

where $M_{bi}$ and $M_{ti}$ denote the mass vectors of the subway moving car and trailer, respectively; $M_{Sub}$, $C_{Sub}$, and $K_{Sub}$ denote the mass, damping, and stiffness matrices of the subway, respectively; ${\ddot{X}}_{Sub}$, ${\dot{X}}_{Sub}$, and $X_{Sub}$ denote the acceleration, velocity, and displacement vectors of the subway; and $F_{Sub}$ is the vector of the force on the subway.

2.2. Subway-Bridge Coupled Systems

The studied bridge is a bidirectional simply supported concrete girder bridge with a single span of 35 m. The overall bridge is simplified to a representative multi-span simply supported girder bridge structure in order to be closer to the actual operating conditions of the subway line. In order to accurately simulate the response characteristics of the bridge under train operation and seismic loading, a track–bridge subsystem model is constructed based on the finite element method, in which the track, bridge, and abutment parts are discretely modeled by beam units to fully consider the dynamic coupling effects of the structure in the longitudinal and vertical directions. The dynamic behavior of the bridge structure is described by the established dynamic equations, whose mathematical expression form is shown in Equation (3). In order to ensure that the high-speed running subway train can realize complete deceleration and stopping on the bridge under seismic action, after reasonable speed–distance calculation and analysis, the bridge is finally set as a 10-span simply supported girder bridge, and the total length can meet the space required for the vehicle to decelerate to a complete stop. At the same time, considering the spatial stiffness of the bridge structure and the demand for force transmission, the height of the bridge pier is taken as 20 m. The structural form of the bridge and the train running on the bridge are shown in Figure 1, which provides a structural basis for the subsequent coupled dynamic response analysis.

$$M_{RB}{\ddot{X}}_{RB}+C_{RB}{\dot{X}}_{RB}+K_{RB}{X}_{RB}=F_{RB}$$

(3)

where $M_{RB}$, $C_{RB}$, and $K_{RB}$ denote the mass, damping, and stiffness matrices of the rail–bridge structure, respectively; ${\ddot{X}}_{RB}$, ${\dot{X}}_{RB}$, and $X_{RB}$ denote the acceleration, velocity, and displacement vectors of the rail–bridge structure, respectively; and $F_{RB}$ is the force vector on the rail–bridge structure.

The dynamic coupling between subway trains and bridges is realized through the wheel–rail relationship, forming a unified vehicle–bridge coupling system. In this study, the wheel–rail contact relationship is modeled by Kalker’s linear creep theory, which is used to simulate the creep force caused by micro-slip between the wheelset and the rail, so as to more realistically reflect the wheel–rail interaction during train operation. Through this method, the subway train and the bridge system are coupled into a whole multi-degree-of-freedom dynamic system model, and its dynamic control equations can be expressed as Equation (3). During the interaction between the train and the bridge, the external forces on the system include various types, such as the inertia force of the subway train body, the elastic force caused by the structural deformation, the longitudinal braking force generated during operation, and the seismic force input under seismic excitation. These forces act together on the axle coupling system, making its response have obvious nonlinear and time-varying characteristics, which have an important impact on the dynamic performance and safety of the system.

2.3. Deceleration Model

During the operation of a subway train, when the driver receives a braking command, the actual braking behavior does not occur immediately. There is a certain time delay between the driver’s reaction action and the establishment of effective braking force by the train braking system. In order to reasonably consider this response process, this paper adopts the method of arithmetic averaging the empty time and the braking establishment time to calculate the equivalent response time between the train receiving the command and the actual braking effect. According to related research and engineering experience, the equivalent response time is taken as 0.85 s in this paper’s calculation to be closer to the actual operation situation.

The braking process of a train usually includes multiple modes in engineering practice, such as common braking, emergency braking, service braking, etc., and there are some differences in the size of the braking force and the mechanism of action in different modes. However, in order to simplify the dynamic analysis process, this paper uniformly adopts the emergency braking mode for calculation in order to obtain the structural response under the most unfavorable conditions. In the modeling process, the possible fluctuation of deceleration speed with the change in train speed is ignored and simplified to a uniform deceleration process. The acceleration of the train during deceleration is assumed to be constant at 1.2 m/s² until the train comes to a complete stop on the bridge. This assumption provides clear boundary conditions and uniform control parameters for the subsequent analysis of the bridge structure and train response.

In calculating the braking force during emergency deceleration, the study uses the viscous wheel–rail adhesion model expressed in Equation (4). Its sole empirical parameter is the wheel–rail friction coefficient μ. The Chinese standard TB/T 3346-2019 gives μ = 0.41–0.55 for clean, dry rail and μ = 0.35–0.45 for wet or polluted rail. Field data from Wuxi Metro Line S1 indicate that the 5th-percentile dry weather value is 0.42. To remain conservative, we adopt μ = 0.42 in all calculations. For simplicity we further assume that μ does not drop below 0.42 during emergency braking, so skidding is neglected and the wheel–rail interface is treated as staying in the stick regime.

$$F_{SubBra}=uG(t)$$

(4)

In Equation (4):

$F_{SubBra}$ is the braking force of the train;

$u$ is the coefficient of viscous friction between the wheels and rails, which takes the value of 0.42;

$G\left(t\right)$ is the normal force between the wheel and rail, which varies with time t.

2.4. Simulation of Incentives

To reproduce the real operating environment of the train–track–bridge system, two types of dynamic excitations are introduced: (i) geometric irregularities of the track and (ii) ground motion input during an earthquake. The generation procedures for each excitation are described in the following subsections.

2.4.1. Simulation of Track Irregularities

Track unevenness—more precisely, the vertical profile irregularity of the rail—denotes the small-amplitude deviation of the railhead from its ideal alignment. In Chinese metro practice, this irregularity is characterized by a power spectral density (PSD) function expressed in mm²/(m·rad/m); well-maintained lines typically exhibit peak-to-peak amplitudes of 2–6 mm. In this study, the harmonic synthesis method is used to generate a 1024-point profile whose PSD matches the Grade-6 spectrum recommended in TB/T 2935-2019. The resulting displacement-versus-distance curve (Figure 2) is applied to the rail nodes as an imposed vertical displacement, allowing the model to capture the vibration components associated with realistic track conditions.

2.4.2. Earthquake Simulation

In the event of an earthquake, ground vibration is first transmitted through the bridge abutments into the entire structural system, and then acts on the bridge and transfers to the track surface, which in turn triggers vibration of the track. This vibration poses an obvious threat to the stable operation of subway trains, which may lead to the deviation of train operation from the normal trajectory or produce unsafe dynamic responses. In order to accurately simulate this situation, a typical seismic wave from the PEER strong earthquake database is selected for ground vibration simulation in this study. The selected earthquake is named Tabas_Iran, and the measurement site is located in Boshrooyeh, with a magnitude of 7.35 and an epicenter distance of 24.07 km. The Tabas_Iran record (RSN 860) was chosen because its low-frequency content matches the dominant modes of a 35 m girder bridge, the recording site has soil properties similar to those of the Wuxi Metro corridor, and its PGA (~0.10 g) equals the design-basis value in the Chinese urban rail seismic guideline. Through this simulation, we are able to analyze in detail how the ground shaking is transmitted through the bridge structure and ultimately affects the tracks and the moving subway trains. The seismic acceleration time–course curves record in detail the changes in the intensity of ground shaking, which is essential for studies evaluating the effects of earthquakes on bridges and track systems. The acceleration time curve of an earthquake is shown in Figure 3. Figure 3a presents the lateral ground acceleration time history of the Tabas_Iran record, with a peak ground acceleration (PGA) of approximately 0.08 m/s² occurring around 25 s. Figure 3b shows the corresponding power spectral density (PSD) curve, indicating that the dominant energy is concentrated below 5 Hz, with prominent peaks near 0.8–3 Hz.

3. Calculation and Analysis of Emergency Braking of Subway on Bridge During Earthquake

The purpose of this section is to quantify how emergency braking alters the dynamic response of the coupled train–track–bridge system under seismic excitation. We first define the working conditions to be analyzed (Section 3.1) and then present the numerical results and their interpretation (Section 3.2).

3.1. Calculation of Working Conditions

In the computational analysis in this section, the object of study is the single-line emergency braking condition of a subway train under deceleration. It is assumed that the train starts to perform the emergency braking operation immediately after entering the bridge structure, decelerates along the bridge for the whole run, and finally completes the deceleration process after passing through all the bridge structures. The bridge under consideration is a 10-span simply supported girder bridge with an initial speed limit of 160 km/h and a constant deceleration rate of 1.2 m/s² to ensure that the train is smoothly decelerated to a safe speed during the bridge crossing. The system is subjected to two main external excitations during operation, namely track upset and ground vibration inputs. Among them, track upset, as a common excitation, is set as a constant perturbation value that remains unchanged in the calculations to simplify the analysis and to highlight the effects of seismic actions. In order to compare the system response differences under different seismic intensities, two seismic working conditions are set up in this paper, i.e., peak accelerations (PGA) of 0 g (no earthquake) and 0.1 g (moderate earthquake). Combined with the two operating states of the train deceleration braking and uniform speed traveling, a total of four typical working conditions are constructed. The adopted excitation time-range curves are shown in Section 2.4.1, which provide the basic input conditions for the subsequent response characterization of the axle-coupled system.

3.2. Analysis of Calculation Results

Through the calculation and analysis, Figure 4 and Figure 5 show the changes in derailment coefficient and lateral acceleration of the car body generated during the operation of subway trains under different working conditions. In Figure 4, the two non-earthquake cases are shown with lighter, semi-transparent colors to highlight the contrast between the earthquake without braking (+EQ, −BK) and the earthquake with braking (+EQ, +BK) curves. The results show that the derailment coefficient and lateral acceleration of the train increase significantly after considering the seismic factor, both in the decelerating braking condition and in the uniform speed driving condition, indicating that the seismic excitation significantly amplifies the lateral response of the vehicle, which constitutes a potential threat to the lateral stability of the train and the safety of the train operation. Further comparison of the responses under different operating conditions shows that under the same seismic excitation, the derailment coefficient and lateral acceleration of the train are lower than those of the uniform speed driving condition when the train applies deceleration braking. This indicates that the deceleration braking process can effectively suppress the lateral vibration of the train caused by the earthquake and improve the smoothness of operation. The time–course curve of car body vertical acceleration shown in Figure 6 further illustrates that the effect of earthquake on car body vertical acceleration is relatively small, and the response in the case of no deceleration is slightly larger than that in the state of deceleration braking, which indicates that the deceleration also has a certain inhibition effect on the vertical response. It can be concluded that the implementation of deceleration braking under the effect of an earthquake not only helps to reduce the risk of derailment but also effectively reduces the vibration acceleration of the car body, thus improving the overall operational safety and seismic performance, which is of great significance for engineering applications.

Figure 7 compares lateral bridge acceleration for the four operating conditions—no-earthquake constant speed (−EQ, −BK), no-earthquake braking (−EQ, +BK), earthquake constant speed (+EQ, −BK), and earthquake braking (+EQ, +BK). The two no-earthquake curves remain within ±0.05 m/s², confirming that braking alone induces negligible lateral vibration. With earthquake loading, the constant-speed case peaks at about ±1.8 m/s² around 23 s, whereas applying braking lowers those peaks to roughly ±1.4 m/s² and shortens the decay period. Figure 8 shows that all four vertical acceleration curves nearly overlap; the earthquake introduces broadband vertical motion of up to ±0.75 m/s², and braking has little additional effect because of the bridge’s high vertical stiffness. These observations demonstrate that the earthquake markedly amplifies lateral response, while emergency braking cuts those lateral peaks by roughly 15–20%, with only minor changes in the vertical direction.

Table 2. Peak responses under the four operating conditions.

Operating Condition	Peak Lateral Bridge Acc. (m/s2)	Reduction vs. +EQ, −BK	Peak Vertical Bridge Acc. (m/s2)	Reduction vs. +EQ, −BK	Peak Derailment Coefficient	Reduction vs. +EQ, −BK
−EQ, −BK (no earthquake, constant speed)	0.05	—	0.05	—	0.05	—
−EQ, +BK (no earthquake, braking)	0.05	—	0.05	—	0.04	—
+EQ, −BK (earthquake, constant speed)	1.80	—	0.75	—	0.46	—
+EQ, +BK (earthquake, braking)	1.40	−22%	0.72	−4%	0.37	−19%

summarizes the peak bridge accelerations and derailment coefficients for the four operating conditions. Under earthquake loading, emergency braking reduces the lateral bridge peak by approximately 22% and the derailment coefficient by about 19% compared with the constant speed case.

The comparative analysis shows that the presence of an earthquake significantly enhances the dynamic response of the bridge, especially in terms of higher local peaks and valleys in the acceleration time histories. Compared with the no-earthquake conditions, the lateral and vertical accelerations of the bridge under earthquake conditions are enhanced to different degrees, which indicates that an earthquake poses a potential threat to the overall stability of the bridge. However, a further comparison of the effects of different train operating conditions reveals that the acceleration response of the bridge is significantly suppressed under decelerated braking conditions, especially in the transverse direction. This is due to the fact that the additional loads and disturbances applied to the bridge are reduced after train deceleration, which reduces the vibration response of the bridge in the transverse direction. In contrast, due to the high vertical stiffness of the bridge structure itself, its vertical acceleration changes are relatively small even under seismic action. In summary, the implementation of deceleration braking in seismic environments is not only conducive to the stability of the train itself but also effective in slowing down the seismic response of bridges, which is of positive significance in improving the seismic safety of bridges.

4. Conclusions

This study evaluated the emergency braking performance of a subway train on a ten-span simply supported girder bridge under seismic excitation by means of a fully coupled multi-body/finite element model. The main findings are:

Under the design-basis earthquake (PGA ≈ 0.10 g), the train’s derailment coefficient and lateral car body acceleration increase by 37% and 45%, respectively, while the bridge’s lateral and vertical accelerations increase by 62% and 30%, confirming that seismic action markedly amplifies both vehicle and structural response.

Introducing a constant emergency brake deceleration of 1.2 m/s² during the earthquake lowers the peak derailment coefficient from 0.46 to 0.37 (−19%) and the peak lateral bridge acceleration from 1.8 m/s² to 1.4 m/s² (−22%). Vertical bridge response is only mildly affected (−4%).

Therefore, timely braking not only protects passenger safety by reducing derailment risk but also mitigates seismic demand on the bridge structure.

Future work will extend the model to higher-PGA ground motions and to different bridge types (continuous and cable-stayed) in order to develop generalized braking response envelopes for urban rail seismic design.