Buckling of non-uniform columns with an arbitrary number of cracks

The governing differential equation for buckling of a non-uniform column is expressed in the form of bending moment. The special solutions for 14 types of non-uniform columns are derived from the governing equation. Using a rotational spring and a hinge connection to describe the local flexibility induced by a crack in the column yields the eigenvalue equation for buckling. The main advantage of the proposed method is that the eigenvalue equation for buckling of a non-uniform column with an arbitrary number of cracks, any kind of two end supports, and an arbitrary number of rotational and translational spring supports at intermediate points can be conveniently determined from a second-order determinant based on the fundamental solutions and recurrence formula developed in this paper. As a consequence, the decrease in the determinant order as compared with previously developed procedures simplifies the analysis. A numerical example is given to illustrate the application of the proposed method and to study the effect of cracks on the critical buckling force of a non-uniform column.