Buckling analysis of multi-step non-uniform columns

An analytical procedure for the elastic buckling of a multi-step non-uniform column subjected to concentrated axial compressive forces is presented in this paper. The governing differential equation for the buckling problem is established. Special solutions of the governing equation for five different types of non-uniform columns are given. The procedure for determining the fundamental solutions that satisfy the normalization conditions is proposed. Using the fundamental solutions and a recurrence formula developed in this paper, the bending moment function that includes only one unknown parameter for each step column can be easily determined. Thus, it is convenient to establish the eigenvalue equation for buckling of a multi-step non-uniform column in terms of the normalized fundamental solutions of the governing equation. The eigenvalue equations for three general boundary conditions are given. A numerical example shows that the proposed procedure for determining the critical buckling force of a multi-step non-uniform column with general boundary conditions is a simple, efficient and exact method.