Stability and Free Vibration Analyses of Non-prismatic Columns using the Combination of Power Series Expansions and Galerkin’s Method

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. Regarding aforementioned method, the expression of deflected shape of the buckled member is identified. Afterwards, the critical buckling loads can be acquired by imposing the boundary conditions and solving the eigenvalue problem. Note that 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 beams with varying cross-section can be estimated by adopting Galerkin’s method based on the energy principle. In order to illustrate the correctness and performance of this method, one comprehensive example of non-uniform beams with various end conditions is presented.