Buckling of an elastically restrained multi-step non-uniform beam with multiple cracks

Abstract

A model of a massless rotational spring is adopted to describe the local flexibility induced by cracks in a beam. The governing differential equation for buckling of a multi-step non-uniform beam with spring supports, each step of which has an arbitrary number of cracks, is expressed in the terms of the bending moment. Linearly independent solutions of the governing equation are derived for five different types of non-uniform beams. The main advantage of the proposed method is that the eigenvalue equation of a multi-step non-uniform beam with any kind of two-end supports, any finite number of cracks and spring supports at intermediate points can be conveniently determined from a second-order determinant based on the fundamental solutions developed in this paper. The decrease in the determinant's order, as compared with previously developed procedures, leads to significant savings in the computational effort. Two numerical examples are given to illustrate the application of the proposed method and to study the effect of cracks on the critical buckling force. The accuracy of the proposed method is verified through numerical examples.