Non-stationary random vibration analysis for train–bridge systems subjected to horizontal earthquakes

This paper investigates the use of PEM (the pseudo-excitation method) and PIM (the precise integration method), to compute the non-stationary random responses of three-dimensional train–bridge systems subjected to lateral horizontal earthquakes. A method for estimating their maxima is also proposed and evaluated. Each vehicle of the train is described by 27 degrees of freedom and the bridge deck is modeled by three-dimensional Euler beam elements. It is assumed that the lateral horizontal earthquake is a uniformly modulated non-stationary random process, while the excitations due to track irregularity are multi-phase stationary random ones. PEM is first proved to be still valid for such time-dependent systems, and then applied to transform the random excitations of the lateral horizontal earthquakes and track irregularities into a series of deterministic pseudo-excitations, respectively. Furthermore, PIM is extended to solve the corresponding pseudo-responses efficiently and precisely. Hence the time-dependent power spectral density functions and standard deviations of the system responses can be obtained conveniently. Finally, a new formula for estimating the maxima of such non-stationary random responses is suggested, based on the first-passage failure criterion. As a case study, the China-Star high-speed train running on a five-span bridge subjected to lateral horizontal earthquakes is investigated. The results show the effectiveness and accuracy of the proposed method and the reliability of the maximum estimation formula, by comparison with the time-history method. The influences of site soils and train speeds on system random responses are discussed.