Buckling of elastically restrained non-uniform columns

An exact solution approach is presented to study the buckling of a non-uniform column with spring supports under combined concentrated and distributed loads. In this paper, the expression for describing the distribution of flexural stiffness of a non-uniform column is arbitrary, and the distribution of axial forces of the column is expressed as a functional relation with the distribution of flexural stiffness and vice versa. The governing equation for buckling of a one-step non-uniform column is reduced to Bessel equations and other solvable equations for 13 cases, several of which are important in engineering practices, by means of the functional transformation. The exact buckling solutions of one-step non-uniform columns are thus found. Then the obtained exact solutions are used to derive the eigenvalues equation for buckling of a multi-step non-uniform column with spring supports by using the transfer matrix method. A numerical example shows that the critical buckling force of a non-uniform structure calculated by the proposed method is almost the same as that determined by FEM, but the present method takes much less computational time than the FEM, illustrating the proposed procedure is an exact and efficient method.