Analytical solutions for the vibration response of thin-walled beams under bidirectional moving random loads

In this study, analytical solutions are presented for the flexural-torsional coupled vibration response analyses of thin-walled beams under bidirectional moving random loads. Based on classical Euler-Bernoulli and Vlasov beam theories, the governing dynamic equations considering the influence of additional torque have been established. The modal superposition method, the Laplace transform, and the Duhamel's integral technique have been employed to obtain the average value and standard deviation of beam displacements in vertical, lateral, and torsional directions. For the validation of the proposed formulations, the results obtained in this paper are compared with the results acquired by the Newmark-beta method and the Monte Carlo method. Comparisons of the results prove the accuracy of the suggested formulations. Through the parametric analysis, it is confirmed that the position where the average value reaches its maximum is related to the load velocity. But the maximum standard deviation always occurs at the end of the beam, which decreases with the growth of velocity and the drop in span. When the velocity does not exceed 30 m/s, the displacement response is mainly controlled by low-order modes.