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.

In order to explore the random nature of high-speed railway train operation safety indices, the pseudo-excitation method, extreme value theory, and non-stationary harmonic superposition theory are used in this paper to study the statistics of train operation safety indices. The pseudo-excitation load formulation for track irregularity is obtained by the pseudo-excitation method, and the resulting non-stationary random vibration problem is transformed into a deterministic time history problem. The pseudo-excitation method is used to establish the dynamic equations of motion, and the separation iteration method is used to solve the equations, so as to obtain the power spectral density of the wheel-rail interaction forces. The wheel-rail interaction forces are obtained by using a modulation function and the harmonic superposition method. By fitting an extreme value distribution, the maximum values of the train running safety indices are explored. The proposed numerical approach is validated experimentally using the data from a 24.6 m long simply supported concrete bridge by studying the extreme value distributions of driving safety indices. Additional numerical simulation are conducted for varying train speeds and bridge spans. The results show that the Gumbel distribution can fit the extreme value of driving safety parameters for different speeds and different bridge span lengths. It is observed that the higher the speed, the sharper the extreme value distribution of train running safety indices, and the larger the train running safety index values corresponding to 99.87% confidence level. The corresponding extreme values at the 99.87% confidence level are greater than the maximum value of each time-domain sample.

To better understand the randomness of the derailment factor of the high-speed trains, its probability distribution function (PDF) and extreme value PDF are studied using the identical probability distribution evolution method (IPDEM). The standard deviation and mean of wheel-rail contact forces are obtained by pseudo-excitation method (PEM) and time history simulations. Then, based on the mean and standard deviation, the PDF and extreme value PDF of the derailment factor are obtained by IPDEM. The PDF and extreme value PDF of the derailment factor are validated by Monte Carlo method (MCM) simulations. Using a 6-car Pioneer EMU train passing over a 3-span 32-m long simply supported bridge as a case study, the applicability of the PDF and extreme value PDF of the derailment factor under varying train speeds, different track irregularities and different bridge deflection/span ratios are discussed. Finally, experimental data obtained from a field test are also used to verify the accuracy of the theoretical approaches. The results show that the PDF and extreme value PDF of the derailment factor proposed in this paper can provide a foundation for statistical evaluations.

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