Analytical solutions for buckling of multi-step non-uniform columns with arbitrary distribution of flexural stiffness or axial distributed loading

In this paper, the function for describing the distribution of flexural stiffness K(x) of a non-uniform column is arbitrary, and the distribution of axial distributed loading N(x) acting on the column is expressed as a function of K(x) and vice versa. The governing equation for buckling of a one-step non-uniform column is reduced to a differential equation of the second-order without the first-order derivative by means of variable transformation. Then, this kind of differential equation is reduced to Bessel equations and other solvable equations for 14 cases. The analytical buckling solutions of one-step non-uniform columns are thus found. Then the obtained analytical solutions are used to derive the eigenvalue equation for buckling of a multi-step non-uniform column for several boundary supports by using the transfer matrix method. A numerical example shows that the proposed procedure is an efficient method for buckling analysis of multi-step non-uniform columns.