Semi-analytic solution in the time domain for non-uniform multi-span Bernoulli–Euler beams traversed by moving loads

This paper presents a semi-analytic solution of the moving load problem which is of great interest for the analysis of multi-span uniform and non-uniform beams subjected to moving forces, such as high-speed trains. The solution is based on the response of the structure to a unit load circulating at a constant speed of the train. Viscous modal damping is considered. Using Bernoulli–Euler beam elements with variable cross-sectional properties, the structure is discretized and the mode shapes are computed using standard procedure. The moving load is represented by a unitary Dirac Delta function, and the modal loads are obtained in terms of cubic Hermitian polynomials. This leads in a straightforward manner to the closed-form solution for the unit load in the time domain. The solution is expressed in terms of 10 coefficients per element and per mode, the values of which are independent of the speed of the moving load. Finally, the response to a series of loads is built simply by adding the contribution of each. The overall procedure is fast and accurate, depending only on the spatial discretization and the time step selected for evaluating the solution without the need of any integration step. Numerical tests have been included in order to show the efficiency of this technique.