Elastic instability of non-prismatic Timoshenko beams by the power series method

Abstract

This paper presents a numerical approach based on the power series method for linear stability analysis of non-prismatic Timoshenko beams subjected to a constant axial load tangential to the beam axis. The governing system of equilibrium equations are derived from principle of stationary total potential energy. For this purpose, the total potential energy is derived from the elastic strain energy and the potential energy due to effects of the initial stresses resultants. Then the equilibrium equations lead to a unique homogeneous second-order differential equation in term of bending rotation, since in the presence of flexural and shear rigidities of cross-section of the considered Timoshenko beam, the obtained system of stability equations are coupled and simultaneous. In the case of non-uniform members, all stiffness coefficients are variable along the beam’s length. The power series approximation is then adopted to ease the solution of the differential equation with variable coefficient. Finally, the critical buckling loads are determined by solving the eigenvalue problem of the obtained algebraic system. In order to illustrate the correctness and performance of proposed numerical method, one comprehensive example of Timoshenko beam with non-uniform section is presented. The obtained results are compared with available numerical or analytical solutions. The accuracy of the method is then remarked.