The objective of this research is to comprehensively investigate the vibration response of multi-span beams with unequal spans under moving loads and propose an efficient approach for obtaining the maximum acceleration with fewer calculations. To decrease the volume of computations, apart from employing analytical solution of the bridge motion equation, an efficient approach for determining the crucial trains is adopted. To do so, the governing equation is obtained and solved analytically using Duhamel integral method. To examine the solution, the results are compared with those of the Newmark Beta method and finite element method. Then, the vertical accelerations of the beam caused by equidistant loads are calculated, and their maximum values are determined. The results show that for finding the maximum acceleration, it is enough to calculate the responses for the first two modes at resonant speeds. After that, the beam responses under ten high-speed trains are computed, and the challenging conditions are determined. Finally, the accelerations due to trains and equidistant loads are compared, and a good correlation is observed between the results of two loadings. Therefore, it is possible to distinguish the challenging trains and their critical conditions using the former results and without calculating the vibrations caused by trains. This method enables the designers to concentrate on the most critical loading cases and can be applied in designing or upgrading railway bridges.

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