A Decoupled Method for Efficient Probabilistic Train–Bridge Interaction Analysis Based on a Multi-Variate Stochastic Wheel–Rail Force Model

Based on a decoupling assumption made for approximating the wheelset responses, this paper presents a physically meaningful wheel–rail force model along with an efficient decoupled method for random vibration analysis of the train–bridge interaction problems. Specifically, the wheel–rail forces are treated as a multi-variate stationary random process statistically characterized by its spectral density matrix, obtained through stochastic analysis of the train equation incorporating only the track irregularity while excluding the train gravity. Next, the obtained spectral density matrix is decomposed through the spectral representation technique, yielding a discrete formulation of the wheel–rail forces with respect to a pair of orthogonal stochastic vectors (OSVs). Then, the Duhamel integral is employed to derive explicit relations between the bridge response and the OSVs, with the associated coefficient matrices efficiently determined using a recursive scheme derived based on the Newmark integration algorithm. The resulting response-OSVs expression allows for statistical calculation of the bridge responses in a decoupled manner, avoiding iterative computations or repeated system matrix updates typically required in coupled stochastic analyses of train–bridge systems. Lastly, the proposed method is numerically demonstrated through stochastic analysis of a three-span continuous bridge subjected to train loads, with comparisons made to the Monte Carlo simulation, pseudo-excitation method, and a fully coupled approach. The numerical results suggest that, compared with traditional coupled approach, the proposed decoupled strategy significantly improves the computational efficiency, particularly for cases involving a large number of vehicles.