Closed-form solution for the mode superposition analysis of the vibration in multi-span beam bridges caused by concentrated moving loads

In this paper, a closed-form solution for evaluating the dynamical behavior of a general multi-span Bernoulli–Euler beam is derived. The natural frequencies of vibration and corresponding mode shapes are obtained by applying the boundary conditions to the characteristic function of a beam. A Laplace transformation is applied to the governing differential equation which is then solved for each normal mode in the frequency domain. The main contribution of this paper is to provide a closed-form solution for the vibration of continuous stepped beams under constant moving loads. Several numerical examples are included.