Exact Solution of the Multi-Cracked Euler-Bernoulli Column

The use of distributions (generalized functions) is a powerful tool to treat singularities in structural mechanics and, besides providing a mathematical modelling, their capability of leading to closed form exact solutions is shown in this paper. In particular, the problem of stability of the uniform Euler–Bernoulli column in presence of multiple concentrated cracks, subjected to an axial compression load, under general boundary conditions is tackled. Concentrated cracks are modelled by means of Dirac’s delta distributions. An integration procedure of the fourth order differential governing equation, which is not allowed by the classical distribution theory, is proposed. The exact buckling mode solution of the column, as functions of four integration constants, and the corresponding exact buckling load equation for any number, position and intensity of the cracks are presented. As an example a parametric study of the multi-cracked simply supported and clamped–clamped Euler–Bernoulli columns is presented.