Stability and free vibration analyses of non-prismatic columns using the combination of Power series expansions and Galerkin’s method

Abstract

As a first endeavor, a mixed power series expansions and Galerkin’s method in the context of linear buckling and free vibration analyses of non-uniform beams is presented. For this aim, the governing equilibrium and motion equations are first obtained from the stationary condition of the total potential energy. The power series approximation is then applied to solve the fourth order differential equilibrium equation, since in the presence of variable cross-section, geometrical properties are variable. For this purpose, displacement component and cross-section properties are expanded in terms of power series of a known degree. The critical buckling loads can be acquired by imposing the boundary conditions and solving the eigenvalue problem. Regarding aforementioned method, the expression of deflected shape of the buckled member is also identified. The buckling mode shapes of an elastic member are similar to the vibrational ones. Therefore, the obtained deformation shape of the considered non-prismatic columns under linear stability analysis can be used as vibrational shape of member. The natural frequencies of non-prismatic beams can be then estimated by adopting Galerkin’s method based on the energy principle. In order to illustrate the correctness and performance of the method, several numerical examples of non-uniform beams under different circumstances are presented. The results are compared with the finite element ones using Ansys software and other available numerical and analytical solutions in terms of stability and free vibration analyses. The proposed method can be applied for the buckling load and natural frequency computations of prismatic members as well as non-prismatic ones.