The Dynamic Analysis of Beams under Distributed Loads Using Laplace-Based Spectral Element Method

The finite element method (FEM) is certainly one of the most popular methods used for structural analysis. However, it is well known that a sufficiently large number of finite elements are necessary in order to obtain reliable structural dynamic responses owing to their high flexibility and large size, especially at high frequency. Since the shape functions of the spectral element method (SEM) are exact solutions of the governing differential equations of structures, the number of elements is significantly decreased. Unfortunately, conventional SEM can only be applied to structures subjected to concentrated loads. This paper presents the dynamic analysis of beams under distributed loads using Laplace-based SEM. Distributed loads are equivalent to concentrated nodal forces based on the principle of linear superposition. A continuous Bernoulli-Euler beam subjected to uniform vertical dynamic distributed load is analyzed here to show the effect of SEM. Both internal viscoelastic damping and external viscous damping are considered in this paper. The numerical results obtained from SEM are compared with those from FEM. It has been found that the SEM provides good dynamic results under distributed loads. The SEM has been proved to be an efficient method to analyze the dynamic responses of structures while the number of elements can greatly decrease.