Stability of non-uniform columns under the combined action of concentrated follower forces and variably distributed loads

An exact approach for stability analysis of a non-uniform column subjected to concentrated tangential follower (non-conservative) forces and variably distributed (conservative) loads along the column is proposed in this paper. The governing differential equation for such a stability problem is established first. Then, the closed-form solutions are derived for three important cases. In order to simplify the analysis for the title problem, the fundamental solutions and recurrence formulas developed in this paper are adopted to establish the eigenvalue equation for the non-conservative stability problem. With this proposed procedure, the eigenvalue equation for stability of a multi-step non-uniform column with any kind of two end supports including the case of two spring supports at the end of the column and any number of concentrated masses can be conveniently determined from a second order determinant. As a consequence, the decrease in the determinant order as compared with previously developed procedures leads to significant savings in the computational effort. Numerical examples show that the critical buckling forces of non-uniform columns calculated by the proposed method are in good agreement with those determined by the Finite Element Method (FEM), but the present method takes less computational time than FEM, illustrating the proposed procedure is an exact and efficient method.