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The response of the train–bridge system has an obvious random behavior. A high traffic density and a long maintenance period of a track will result in a substantial increase in the number of trains running on a bridge, and there is small likelihood that the maximum responses of the train and bridge happen in the total maintenance period of the track. Firstly, the coupling model of train–bridge systems is reviewed. Then, an ensemble method is presented, which can estimate the small probabilities of a dynamic system with stochastic excitations. The main idea of the ensemble method is to use the NARX (nonlinear autoregressive with exogenous input) model to replace the physical model and apply subset simulation with splitting to obtain the extreme distribution. Finally, the efficiency of the suggested method is compared with the direct Monte Carlo simulation method, and the probability exceedance of train responses under the vertical track irregularity is discussed. The results show that when the small probability of train responses under vertical track irregularity is estimated, the ensemble method can reduce both the calculation time of a single sample and the required number of samples.

Random dynamic analysis of vertical train–bridge systems under small probability by surrogate model and subset simulation with splitting