Exact solution of the multi-cracked Euler–Bernoulli column

Abstract

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.

The non-linear dynamic response of the Euler–Bernoulli beam with an arbitrary number of switching cracks

“…In previous papers the authors [30,31] Appropriate initial conditions at the instant of transition between two successive linear phases have been considered and an energy control criteria has been enforced in order to establish the minimum number of modes that must be taken into account in order to obtain accurate results. Some numerical applications are presented with the aim to illustrate efficiency of the proposed approach for beams with multiple switching cracks under different boundary and loading conditions.…”

“…In equation (12) A is a constant which does not influence the damage modelling since the damage intensity can be correlated directly to the damage parameters i O as discussed in reference [30].…”

“…In order to perform the comparison, the values of damage intensity parameter O, adopted in this work, corresponding to the relative crack depths is defined as follows [30]:…”

The Hu–Washizu variational principle for the identification of imperfections in beams

“…Moreover, the presented procedure will be shown to be suitable for identification of concentrated damages that is a subject to which a great interest has been devoted in the specific literature [3,7,8,10,11,14,15,24,25]. Recently, a model of the Euler-Bernoulli beam with singularities able to model the presence of concentrated damages has been proposed [20,[29][30][31]. According to the latter approach, the concentrated damages are cracks treated as singularities modelled by means of Dirac's delta distributions (x − x 0i ) superimposed onto a uniform inertia moment I 0 of the undamaged cross section as follows:…”

“…The relationships between the classical models for concentrated damages and the model with Dirac's delta singularities have been reviewed and clarified in [31].…”