Wheel Unloading Rate Research Articles

Dynamic Analysis of Rockfall Impact on Bridges: Implications for Train Safety

The threat of rockfall impacting bridges in mountainous areas poses a great risk to the safety of passing trains. This study delves into the dynamics of rockfall impact and its implications on the interaction between train vehicles and bridges. Leveraging LS-DYNA, this study first captured the force–time history of rockfall impact on bridge structures. Subsequently, there was a comparison with the impact forces generated at various speeds with those predicted by established formulas, validating the accuracy of simulations. Employing BANSYS software, the dynamic responses of both bridge structures and the vehicle–bridge coupling system to falling rocks were analyzed. The investigation encompassed parameters such as impact speed, position, and train location. The findings reveal that escalating impact speeds correlate with increased average and maximum impact forces from falling rocks. Notably, the average impact force does not linearly correspond with rock speed and often exceeds values calculated by conventional formulas. Impact position minimally affects maximum impact force, yet alterations in position prolong impact duration, consequently reducing average impact force. Rockfall-induced impacts precipitate notable spikes in train lateral acceleration, lateral wheelset force, wheel unloading rate, and derailment coefficient, albeit with a comparatively lesser impact on vertical acceleration. Increasing impact speed and altering position intensifies the vehicle’s response, particularly when the train is in close proximity to the impact site.

Safety of the express freight train running over a long-span bridge

PurposeExpress freight transportation is in rapid development currently. Owing to the higher speed of express freight train, the deformation of the bridge deck worsens the railway line condition under the action of wind and train moving load when the train runs over a long-span bridge. Besides, the blunt car body of vehicle has poor aerodynamic characteristics, bringing a greater challenge on the running stability in the crosswind.Design/methodology/approachIn this study, the aerodynamic force coefficients of express freight vehicles on the bridge are measured by scale model wind tunnel test. The dynamic model of the train-long-span steel truss bridge coupling system is established, and the dynamic response as well as the running safety of vehicle are evaluated.FindingsThe results show that wind speed has a significant influence on running safety, which is mainly reflected in the over-limitation of wheel unloading rate. The wind speed limit decreases with train speed, and it reduces to 18.83 m/s when the train speed is 160 km/h.Originality/valueThis study deepens the theoretical understanding of the interaction between vehicles and bridges and proposes new methods for analyzing similar engineering problems. It also provides a new theoretical basis for the safety assessment of express freight trains.

Running safety assessment method of trains under seismic conditions based on the derailment risk domain

The accurate assessment of running safety during earthquakes is of significant importance for ensuring the safety of railway lines. Currently, assessment methods based on a single index suffer from issues such as misjudgment of operational safety and difficulty in evaluating operational margin, making them unsuitable for assessing train safety during earthquakes. Therefore, in order to propose an effective evaluation method for the running safety of trains during earthquakes, this study employs three indexes, namely lateral displacement of the wheel–rail contact point, wheel unloading rate, and wheel lift, to describe the lateral and vertical contact states between the wheel and rail. The corresponding evolution characteristics of the wheel–rail contact states are determined, and the derailment forms under different frequency components of seismic motion are identified through dynamic numerical simulations of the train–track coupled system under sine excitation. The variations in the wheel–rail contact states during the transition from a safe state to the critical state of derailment are analyzed, thereby constructing the evolutionary path of train derailment and seismic derailment risk domain. Lastly, the wheel–rail contact and derailment states under seismic conditions are analyzed, thus verifying the effectiveness of the evaluation method for assessing running safety under earthquakes proposed in this study. The results indicate that the assessment method based on the derailment risk domain accurately and comprehensively reflects the wheel–rail contact states under seismic conditions. It successfully determines the forms of train derailment, the risk levels of derailment, and the evolutionary paths of derailment risk.

The Influence of CRTS II Slab Ballastless Track Upper Arch Deformation on the Wheel Jumping Law of High-Speed Vehicle

To study the impact of upwarp deformation in the ballastless track on the jumping behavior of the high-speed vehicle, utilizing UM and ANSYS joint simulation, a vertical vibration model of high-speed vehicle on CRTS II slab ballastless track was developed based on the non-Hertz wheel–rail contact model of virtual penetration theory. By using the single-wave cosine curve simulating the characteristics of upwarp deformation in the track slab, we calculated the whole process of wheel jumping. This allowed us to analyze how the amplitude and wavelength of the track slab upward deformation influence the vibration response of the vehicle–track system. Our findings indicate that when a wheel passes through the arch section of the track slab, the entire wheel jumping process consists of distinct stages: “wheel–rail bonding, wheel–rail separation, wheel–rail impact (one or more times), and wheel–rail bonding.” As the amplitude of upwarp deformation increases and the wavelength decreases, significant changes occur in several parameters, including the vertical force between the wheel and rail, wheel unloading rate, wheel jump height, frequency, duration, and vertical displacement of the rail. Additionally, when the wavelength is between 2 and 6[Formula: see text]m and the amplitude is 8[Formula: see text]mm, the vertical force between the wheel and rail becomes zero, the wheel load reduction rate is one, and the wheel jumps. When the wavelength is less than 3[Formula: see text]m, the wheel jump height exceeds the flange height, increasing the risk of derailment. Meanwhile, during the first wheel–rail impact, the wheel–rail vertical force and the rail vertical displacement reach their maximum, potentially impacting rail service performance negatively. Finally, compared to the amplitude of the track slab camber deformation, its wavelength has a greater impact on the entire process of wheel jumping. It is recommended that attention be paid to the change in the wavelength of the track slab camber during the maintenance and repair of the ballastless track.

A hybrid reliability assessment method for slab track with changing structural parameters

Changes in the dynamic properties of high-speed rail (HSR) slab track structures can have a great impact on train safety and ride quality. However, the dynamic response of wheel–rail systems, which is related to operational safety, has rarely been considered in existing service reliability assessments of track structures. To solve this problem, a hybrid method was developed, with the serviceability limit state (SLS) first defined with respect to the derailment coefficient (DC) and wheel unloading rate (WUR). In calculating the reliability index, the response surface method and the first-order reliability method were used to solve the implicit expression of wheel–rail force in the SLS equation. To reduce the computation cost, a surrogate model expressing the non-linear mapping of the wheel–rail interaction is proposed. The computation efficiency and accuracy of the hybrid method were compared with Monte Carlo simulation and a back-propagation neural network (BPNN). The computation time of the hybrid method was only 1/8.4 of the BPNN method, while the accuracy of the reliability index was 98% for the DC and 97% for the WUR. The hybrid method was used to assess the reliability of a typical slab track structure under changing stiffness and damping coefficients of the fasteners, cement asphalt mortar and foundation. The stiffness and damping of the fasteners had the most impact on wheel–rail dynamics and track reliability. This research provides new insights into reliability assessments for HSR slab track with respect to train operational safety.

Safety of an express freight train running over a bridge in crosswind

Owing to the blunt car body and high design speed of express freight trains, the running safety is significantly affected by crosswinds when the train runs over a bridge; moreover, severe accidents, such as derailment or overturning may occur. Therefore, adequate measures should be adopted to avoid the occurrence of such accidents. In this study, a finite element model of the train, ballastless track, and parameterized 5-span simply supported girder bridge along with modal superposition method are adopted to establish the train-track-bridge coupling dynamic system. Numerical simulation of stochastic wind speed, wind tunnel tests, and computational fluid dynamics (CFD) simulation of vehicle-bridge scale model were conducted to calculate the wind load acting on the train and bridge. The dynamic responses of the bridge and train were evaluated to present the diagrams of wind and train speed limits for safe operation. The results indicated that the running safety of the vehicle declined with the wind and train speeds, and the wheel unloading rate was the most sensitive to wind speed. For safe operation, the train speed must be restricted if the mean wind speed is greater than 22.7 m/s, and an extreme scenario of derailment may occur for a mean wind speed of 34.0 m/s. In addition, the wind speed limit should be appropriately adjusted and improved according to the pier height of the bridge and terrain category. This research provides a theoretical basis and data support for the assessment of safety of express freight trains in actual operation.

Train/track coupled dynamics model of long heavy haul train based on substructure and parallel computing

To consider the coupled effect on the running safety between elastic track and longitudinal impulse of Long Heavy Haul Train(LHHT), a train/track coupled dynamics model is established by using connection substructure theory. The ballasted track is divided into several segments called sub-tracks: a sub-track includes rail, sleepers and ballast. In the sub-track model, the sleepers and ballast are modelled as lumped mass. The rail is divided into the contact and connection rail. The contact rail is modelled as an Euler beam to reflect the wheel/rail interaction and the flexible vibration of the rail. The connection rail is modelled as a super element to reflect the interaction between adjacent contact rail. To increase the simulation speed, a new parallel computing method is proposed: a train/track coupled dynamics model is divided into different submodule, a submodule includes a sub-track and a vehicle on the sub-track. A submodule is calculated by a single computer core. The submodule is connected by connection rail, couplers and ballast. The advantage of this parallel method is that the load of each computer core is almost uniform. The simulation speed depends on the number of parallel computing cores instead of one core with a particularly large load. Finally, taking the 10,000-ton train as an example, the distribution of coupler force, the derailment coefficient and wheel unloading rate are given during the train braking on a curve, which shows the application and necessity of the train/track coupled dynamics model based on substructure and parallel computing.

Effect of liquid sloshing in lateral and longitudinal directions on railway tank cars based on an improved equivalent model

Additional slosh forces and moments are generated owing to the movement of the liquid inside a partly filled tank car. The loads may have a great influence on the structure reliability and dynamic performance of the vehicle. An improved equivalent model of liquid sloshing is put forward to obtain the forces and moments of sloshing liquid acting on the tank during the lateral and longitudinal sloshing processes. An improved method is proposed to obtain the equivalent masses and stiffness for the improved equivalent model. The equivalent model is integrated into the multi-body dynamic model of a railway tank car with 68 degrees of freedom. Furthermore, the equivalent liquid-vehicle model is used to investigate the effects of fluid sloshing on dynamic characteristics. The influence of the filling ratio on the dynamic response of the tank car is discussed based on the forces and the corresponding moments between the tank car body and the liquid in different layers. The railway tank car is obviously influenced by the liquid sloshing during passing the curved and sloped tracks. The influence of liquid sloshing on the pitching motion of the tank is heavy if the tank is about half filled. The derailment coefficient and the wheel unloading rate are slightly influenced by the filling ratio of tank cars on curved tracks while the indices of the tank cars on sloped tracks are the greatest if the filling ratio is 40%.

Service reliability assessment of ballastless track in high speed railway via improved response surface method

The spectral feature of the track irregularity has a great impact on the running stability of high-speed trains. This study assesses the service reliability of the high-speed rail (HSR) ballastless track structure considering the effect of wavelength distribution characteristics of the track irregularities. The serviceability limit state (SLS) function of the ballastless track is established with respect to the derailment coefficient and wheel-unloading rate. To overcome the problem of high computational cost in the iteration of reliability assessment for the vehicle-track system, this study proposes an improved response surface method (RSM) and the explicit solution of the SLS function of the ballastless track can be realized. Compared with the traditional RSM, the improved RSM can adaptively adjust the variable interpolation coefficient to increase the iterative convergence speed. To obtain the vertical and lateral wheel-rail forces, a 3-D vehicle-track coupling model is established based on multibody dynamics and the finite element method (FEM). To analyze the wavelength effect of the track irregularities, a binary wavelet-based inversion algorithm is proposed to generate time series of both the vertical and alignment irregularities from the track irregularity spectrum (TIS). By comparing it with the MCS method, it is found that by using the improved RSM method, the convergence condition of the reliability index can be satisfied only by tens of iterations and the total computation time is 1.92 × 104 shorter than using MCS. Finally, the effect of different wavelengths of the track irregularity on the service reliability of track structure is discussed. The results show that the wavelength of 32 to 64 m is the main unfavorable wavelength range affecting the track service reliability under the normal operation condition of HSR.

Speed Limit of Linear Induction Motor Subway Trains Running through 65 m Radius Curves on Yard Line

Linear induction motor (LIM) subway yard line often have very-small-radius curves and small turnouts with “S”-shaped curves in the middle. The lateral stability of the train is low, and wheel-climb (slipway) derailments often occur at a speed of 10 km/h. To study the derailment safety and speed limit of a linear motor subway train running through yard line, a coupling dynamic model of the train-track system is established, fully considering the driving characteristics of the linear motor, characteristics of the linear motor train, and track structure. The safety of train derailment is judged by the over-limit time of the derailment coefficient. Under the premise of a safe operation, the variations in the derailment coefficient, out-of-limit time, wheel unloading rate, and wheel-axle lateral force with the train running speed are studied. When the train runs at speeds below 25 km/h, for both new and worn rails, the derailment coefficient and lateral force of the wheel axle at each speed exceed the safety limits, and are less affected by changes in the speed. The duration of the derailment coefficient is within the safety limit because the speed of crossing the curve is low, and thus the train will not overturn and derail. Rail wear reduces the train derailment safety through the yard line, while the wheel wear improves the train derailment safety through the yard line. Considering the actual state of the yard line (wheel-rail wear, irregularity, etc.), the speed of the linear motor subway train running through the yard line should be limited to below 15 km/h.

Effects of different wheel wear positions on dynamic performance of metro vehicles

PurposeMetro wheels running on different lines can undergo wear at different positions. This paper aims to investigate the effects of wheel wear at two typical positions, i.e. wheel flange and tread, on the dynamic performance of metro vehicles and analyzes the differences, with an aim of providing theoretical support on wheel reprofiling for different metro lines.Design/methodology/approachWheel profile data were measured on two actual metro lines, denoted A and B. It was observed that wheel wear on Lines A and B was concentrated on flanges and treads, respectively. A metro vehicle dynamics model was built using multibody dynamics software SIMPACK. Then it was applied to analyze the differences in effects of wheel wear at different positions on vehicle dynamic performance (VDP) for various speeds (50, 60 and 70 km/h) and line conditions (straight line, R1000m, R600m and R300m curves). Critical speed and vibration acceleration were used as indicators of VDP during linear motion (on straight track), while VDP during curvilinear motion (on curved track) was evaluated in terms of wheel/rail lateral force, wheel/rail vertical force, derailment coefficient and wheel unloading rate.FindingsFirst, compared to wheel profile with tread wear, wheel profile with flange wear showed better performance during linear motion. When the distance traveled reached 8 × 104 and 14 × 104 km, the vehicle’s critical speed was 12.2 and 21.6% higher, respectively. The corresponding vertical and lateral vibration accelerations were 59.7 and 74.8% lower. Second, compared to wheel profile with flange wear, that with tread wear showed better performance during curvilinear motion, with smaller wheel/rail lateral force, derailment coefficient and wheel unloading rate. When the vehicle speed was 50, 60 and 70 km/h, the maximum difference in the three indicators between the two wheel profiles was 40.2, 44.7 and 23.1%, respectively. For R1000m, R600m and R300m curves, the corresponding maximum difference was 45.7, 69.0 and 44.4%, respectively.Practical implicationsThe results of the study can provide a guidance and theoretical support on wheel reprofiling for different metro lines. On lines with large proportions of curved sections, metro vehicles are more prone to wheel flange wear and have poorer dynamic performance during curvilinear motion. Therefore, more attention should be paid to flange lubrication and maintenance for such lines. On lines with higher proportions of straight sections, metro vehicles are more prone to tread wear and have poorer performance on straight sections. So, tread maintenance and service requires more attention for such lines.Originality/valueExisting research has focused primarily on the effects of wheel wear on VDP, but fails to consider the differences in the effects of wheel wear at different positions on VDP. In actual metro operation, the position of wheel wear can vary significantly between lines. Based on measured positions of wheel wear, this paper examines the differences in the effects of wheel wear at two typical positions, i.e. tread and flange, on VDP in detail.

Estimation of Vehicle Dynamic Response from Track Irregularity Using Deep Learning Techniques

To improve the quality of track maintenance work, it is a desire to estimate vehicle dynamic behavior from track geometry irregularities. This paper proposes a deep learning model to predict vehicle responses (e.g., vertical wheel-rail forces, wheel unloading rate, and car body vertical acceleration) using deep learning techniques. In the proposed CA-CNN-MUSE model, convolutional neural networks (CNNs) are used to learn features of track irregularities, and multiscale self-attention mechanisms (MUSE) are employed to capture the long-term and short-term trends of sequences. Coordinate attention (CA) is introduced into CNN to focus on important interchannel relationships and important spatial mileage points. The experiments were performed on a multibody simulation model of the vehicle system and the measured data of the actual high-speed line. The results show that the CA-CNN-MUSE has high prediction accuracy for vertical vehicle responses and fast computation speed. The predicted time-domain waveforms and power spectral densities (PSDs) agree well with the actual vehicle responses. The main features of the lateral vehicle responses can also be captured by the proposed method, yet the results are not as good as the vertical ones.

Simulation of dynamic performance and wheel-rail wear of railway wagon under deviation of cargo gravity center

A multibody dynamic model of railway wagon is established by Simpack software, and the derailment coefficient, wheel load reduction rate, impact angle and wear index of freight car are calculated under different gravity center offsets and different curve conditions. The simulation results show that the lateral and longitudinal offsets of cargo gravity center have a great influence on the derailment coefficient, wheel unloading rate, impact angle and wear index of the vehicle. The larger the offset of the cargo gravity center is, the worse the operation performance of the wagon is, the more serious the wheel-rail wear is, and the shorter the transport life of the wagon is. The smaller the radius of the curve is, the greater the impact is.

Determination of settlement limit values for a new concrete box subgrade with ballasted track

To obtain the settlement limits of ballasted track-concrete box subgrade (BTCBS), the influences of ground settlement on track deformation, force and vehicle dynamic responses are investigated in this paper. Firstly, the new type of BTCBS structure is introduced, and four settlement types are proposed according to the structural characteristics, including box faulting, trapezoid settlement, V-shape settlement and box rolling. Then, the influences of ground settlement on track deformation and fastener force are investigated based on the nonlinear finite element model of BTCBS. Furthermore, based on the train-ballasted track-concrete box subgrade (TBTCBS) dynamic model, the influences of settlement on train dynamic responses are studied by adopting a real-time co-simulation method which considers the nonlinearity of voided sleepers. On this basis, the settlement thresholds of BTCBS are determined according to China’s codes. Numerical results show that except that the settlement threshold with box faulting at the speed of 350 km/h is determined by wheel unloading rate, the settlement thresholds of all the other settlement conditions are controlled by 10 m chord versine of vertical irregularity. The settlement thresholds of BTCBS with box faulting, trapezoid settlement, V-shape settlement and box rolling are suggested as 4.02 mm, 17.63 mm, 9.17 mm and 5.53 mm, respectively.

Probability Distribution Functions of Maximum Wheel Unloading Rate Based on Extremal Index

To study the probabilistic distribution of maximum wheel unloading rate of high-speed trains and its temporal correlation when a train passes over a bridge, a method for the estimation of the extremal index is proposed. Using the time series threshold theory, the maximum value cumulative distribution function (CDF) when the wheel unloading rate is regarded as a time series is derived and validated. This approach can also address dependent series, which the traditional probability distribution function formulas could not. Then, the difference between treating the wheel unloading rate as a time series and independent series is investigated using Monte Carlo simulations. Finally, the influence of the number of calculation steps on the threshold is studied, and the differences between thresholds calculated by different extremal indices when considering the number of trains running during the service period of the bridge are explored. The maximum value CDFs of the wheel unloading rate for different track irregularities, bridge lengths, and vehicle speeds are investigated for a three-span simply-supported bridge. The results show that the differences in the maximum value probability density functions (PDFs) obtained by considering the wheel unloading rate as time series and independent random series cannot be ignored. However, when studying a high-confidence level problem, such as the threshold of the wheel unloading rate, the difference between the two approaches is small enough. As the number of calculation steps increases, the extremal index will gradually decrease. When considering a long-distance high-speed rail line, its shorter segment can be used to study the threshold of the wheel unloading rate.

Study on the Vertical Stiffness of Bridge Expansion Joints Based on the Vehicle-Track-Bridge Coupled Model

Bridge expansion joints (BEJs) are equipped at the girder end of long-span railway bridges to ensure the reliable transition of the track. BEJs should have suitable dynamic stiffness to ensure the running safety and stability of the vehicle. To explore the influence of dynamic vertical stiffness of BEJs on the dynamic response of vehicle, track, and bridge, a three-dimensional vehicle-track-bridge coupled model is established, and the influence of sleeper spacing and vehicle speed are taken into consideration. Taking a high-speed railway line in China as a case study, the dynamic response of the whole system is calculated. The results show that (1) the dynamic response of BEJs and the vehicle is affected by the vehicle speed and sleeper spacing. With the increase of vehicle speed and sleeper spacing, the dynamic response of the system increases; (2) the vertical stiffness of BEJs will affect the dynamic response of BEJs and the interaction between the wheel and rail: the vertical dynamic displacement of movable sleeper will exceed the limit if the vertical stiffness of pressure-bearing is insufficient, and the wheel unloading rate may exceed the limit if the vertical stiffness of the cushion plate is insufficient. Based on the results, the larger vertical stiffness of the cushion plate and pressure-bearing is recommended. These results have been applied to the design of BEJs on the studied railway line, which are in good service condition at present.

Global reliability analysis of running safety of a train traversing a bridge under crosswinds

The derailment of any wheel of a running train results in failure of the safe operation of the train. Therefore, the running risk of each wheel must be comprehensively considered when calculating the running safety and reliability of a train. In this study, a methodology for calculating the global reliability of the running safety of a train on a bridge under crosswinds is proposed using the probability density evolution method (PDEM)-based equivalent extreme value principle. First, a refined 3-D train-track-bridge interaction model is constructed to accurately obtain the dynamic response of a train on a bridge under crosswinds. Second, a dimension-reduced scheme is developed to simulate the representative samples of turbulent winds and track irregularities so that the stochastic processes can be expressed using only two random variables. Third, the PDEM-based equivalent extreme value principle is proposed to efficiently calculate the global reliability of the running safety of trains, where four running safety indexes, namely the derailment coefficient, wheel unloading rate, and wheel-rail vertical and lateral relative displacements, are used to evaluate the running safety of the train. Using numerical examples, the accuracy of the proposed methodology is verified through comparison with the field measurement data and Monte Carlo method, and the influence of the number of representative points on the accuracy is investigated. Furthermore, the differences between the global reliability and single-wheel reliability as well as those between the time-dependent and time-independent reliabilities of running safety are discussed in detail.

On the safety threshold of trains running on Sichuan-Tibet Railway bridges under the influence of floating ice

To investigate the running safety threshold of trains on the Sichuan-Tibet Railway bridges under the floating ice impact loads, a train-track-bridge dynamics model is established based on the theory of the train-track-bridge dynamic interaction and the design method of tuned mass damper (TMD), which uses the fluid-structure coupling method to calculate the floating ice impact loads. Taking a 5 × 32 m typical simply-supported beam bridge on the Sichuan-Tibet Railway as an example, the critical train speeds corresponding to floating ice impact loads of varying intensity are evaluated, and the effect of TMD on the running safety of the trains under extreme ice loads is investigated. The results show that the fluid effect has a significant influence on the impact loads of ice hitting the bridge pier, which cannot be ignored in the calculation. The ice impact intensities significantly reduce the train's safe speeds. Under the action of floating ice impact loads, the TMD attached to the bridge can effectively reduce the train's derailment factor and wheel unloading rate. The greater the TMD mass ratio, the more pronounced the vibration control effect.

In-situ testing and model updating of a long-span cable-stayed railway bridge with hybrid girders subjected to a running train

This study investigates the dynamic responses of a long-span cable-stayed railway bridge with steel–concrete hybrid girders in different train load scenarios. The investigated scenarios included driving and braking of running trains. The bridge was instrumented with different types of sensors to measure strains, displacements, and accelerations during train load testing. Based on the measurement results, this study evaluated the effects of train speed on the vertical and transverse accelerations, impact factor, derailment coefficient, and wheel unloading rate, which are critical parameters that affect safety and riding comfort. Assessment results revealed that the bridge had adequate dynamic behavior to ensure safe operation and riding comfort of the train. Based on test data, a finite element model was developed, and a finite element model updating framework was proposed to improve the prediction accuracy. This study provides real-life test data and an assessment tool for promoting design and evaluation of cable-stayed railway bridges.

Safety evaluation of a vehicle\u2013bridge interaction system using the pseudo-excitation method

A method for analysing the vehicle–bridge interaction system with enhanced objectivity is proposed in the paper, which considers the time-variant and random characteristics and allows finding the power spectral densities (PSDs) of the system responses directly from the PSD of track irregularity. The pseudo-excitation method is adopted in the proposed framework, where the vehicle is modelled as a rigid body and the bridge is modelled using the finite element method. The vertical and lateral wheel–rail pseudo-excitations are established assuming the wheel and rail have the same displacement and using the simplified Kalker creep theory, respectively. The power spectrum function of vehicle and bridge responses is calculated by history integral. Based on the dynamic responses from the deterministic and random analyses of the interaction system, and the probability density functions for three safety factors (derailment coefficient, wheel unloading rate, and lateral wheel axle force) are obtained, and the probabilities of the safety factors exceeding the given limits are calculated. The proposed method is validated by Monte Carlo simulations using a case study of a high-speed train running over a bridge with five simply supported spans and four piers.