Multi-cracked Euler–Bernoulli beams: Mathematical modeling and exact solutions

Localized flexibility models of cracks enable one for simple and effective representation of the behavior of damaged beams and frames. Important applications, such as the determination of closed-form solutions and the development of diagnostic methods of analysis have attracted the interest of many researchers in recent years. Nevertheless, certain fundamental questions have not been completely clarified yet. One of these issues concerns with the mechanical justification of the macroscopic model of rotational elastic spring commonly used to describe the presence of an open crack in a beam under bending deformation. Two main analytical formulations have been recently proposed to take into account the singularity generated by the crack. The crack is represented by suitable Dirac’s delta functions either in the beam’s flexural rigidity or in the beam’s flexural flexibility. Both approaches require some caution due to mathematical subtleties of the analysis. Motivated by these considerations, in this paper we propose a justification of the rotational elastic spring model of an open crack in a beam in bending deformation. We show that this localized flexibility model of a crack is the variational limit of a family of one-dimensional beams when the flexural stiffness of these beams tends to zero in an interval centered at the cracked cross-section and, simultaneously, the length of the interval vanishes in a suitable way. We also show that the static and dynamic problem for the flexibility model of cracked beam can be formulated within the classical context of the theory of distributions, avoiding the hindrances encountered in previous approaches to the problem. In addition, the proposed treatment leads to a simple and efficient determination of exact closed form solutions of both static and dynamic problems for beams with multiple cracks.