Dynamic analysis of railway bridges with random vertical rail irregularities

The dynamic response of bridges subjected to the passage of high-speed trains has long been a topic of great interest in the field of railway engineering. A comprehensive review of the history and literature on this subject can be found in Fryba (1996). The problem of dynamic interaction between the train and the railway bridge deserves special attention when the safety of the bridge structure and the riding comfort of rail cars are to be assessed. The studies concerning railway bridge dynamics pointed out that the dynamic response of the bridge and rail cars is influenced by several parameters, such as rail irregularities, vehicle speed, number of coaches, damping of vehicles and bridge, elastic properties of rails and bridge deck, etc. In particular, the track irregularities (elevation, alignment, superelevation, gauge) are considered as one of the most important vibration sources during the passage of trains over railway bridges. The irregularities are usually modeled either as known quantities, exploiting measured data (see e.g., Biondi et al. 2004, Xia and Zhang 2005), or as stationary and ergodic Gaussian random processes characterized by an assigned Power Spectral Density (PSD) function (see e.g. Lei and Noda 2002). The present paper focuses on the effects of vertical track irregularities on the vibrations of the coupled train-track-bridge system. Specifically, the dynamic response of a multi-span continuous bridge crossed by a running train is studied. The train-track-bridge system is modeled by applying a recently proposed substructure approach (Biondi et al. 2004, Biondi et al. 2005), which enables to handle simultaneously the dynamic responses of the vehicles, rails and bridge, taking into account the interaction among the three subsystems. The basic idea is to treat the running train, the rail and the bridge deck as three substructures. Vertical track irregularities are properly included in the model in order to investigate their influence on both bridge and vehicle vibrations. In particular, the rail irregularities are modeled as stationary and ergodic Gaussian random processes in space, characterized by an assigned PSD function. By properly coupling the discrete (train) and continuous (rail-deck system) substructures, a set of ordinary differential equations with time-dependent coefficients and stochastic excitation, governing the motion of the train-track-bridge system, is derived. In view of the stochastic variability of rail irregularities, the dynamic response of each substructure is described by a random process in time. In the present study, a second-order analysis of the response processes is carried out resorting to Monte Carlo simulation technique. As case study, an Italian five-span continuous bridge crossed by a train consisting of a locomotive and five identical carriages is considered (Fig. 1). The time-varying mean and variance functions of both bridge and vehicle response are evaluated for different classes of vertical rail irregularities. As expected, numerical results demonstrate that track irregularities have a strong influence on vehicle vibrations, while their effects on bridge response are practically negligible.