Objectives  In order to overcome the difficulties caused by boundary and elastic transverse supports in studying the vibrational characteristics of a continuous multi-span beam, this paper establishes an analytical model of the free vibration of a multi-span beam based on the Euler-Bernoulli beam theory.

  Methods  First, a new improved Fourier series form is constructed to represent the lateral displacement function of the multi-span beam over the entire segment. Second, the series expression of the displacement function is substituted into the Lagrangian function and combined with the Rayleigh-Ritz method, and the problem of free vibration is transformed into the form of eigenvalues of a standard matrix. The natural frequencies of an elastically supported multi-span beam can be solved.

  Results  In selected numerical examples, by changing the value of the translational elastic stiffness at the elastic support, the vibrational characteristics of the multi-span beam with any elastic support can be obtained. The feasibility and accuracy of the proposed method are fully verified by a comparison with existing results in the literature.

  Conclusions  Through the numerical simulation of the vibrational characteristics of a multi-span beam based on an Improved Fourier Series Method（IFSM）, this paper provides effective preliminary assessment for the dynamic performance of a multi-span beam.