An efficient and accurate FEM based method is proposed for studying non-stationary random vibration of structures subjected to moving loads. The loads are assumed to be a stationary process with constant mean value. The non-stationary power spectral densities (PSD) and the time dependent standard deviations of dynamic response are derived by using the pseudo excitation method (PEM) to transform this random excitation problem into a deterministic harmonic excitation one. The precise integration method (PIM) is extended to solve the equation of motion of beams under moving harmonic loads by enhancing the very recent consistent decomposition procedure, in order to simulate the movement of the loads. Six numerical examples are given to show the very high efficiency and accuracy of the method and also to deduce some useful preliminary conclusions from investigation of the dynamic statistical characteristics of a simply supported beam and of a symmetrical three span beam with its centre span unequal to the outer ones.