Elastic stability of columns with variable flexural rigidity under arbitrary axial load using the finite difference method

In this paper, the finite difference method (FDM) is applied to investigate the stability analysis and buckling load of columns with variable flexural rigidity, different boundary conditions and subjected to variable axial loads. Between various mathematical techniques adopted to solve the equilibrium equation, the finite difference method, especially in its explicit formulation, requires a minimum of computing stages. This numerical method is therefore one of the most suitable and fast approaches for engineering applications where the exact solution is very difficult to obtain. The main idea of this method is to replace all the derivatives presented in the governing equilibrium equation and boundary condition equations with the corresponding forward, central and backward second order finite difference expressions. The critical buckling loads are finally determined by solving the eigenvalue problem of the obtained algebraic system resulting from FDM expansions. In order to illustrate the correctness and performance of FDM, several numerical examples are presented. The results are compared with finite element results using Ansys software and other available numerical and analytical solutions. The competency and efficiency of the method is then declared.