Exact solutions for the statics of the multi-cracked circular arch

Cracked straight beams have been deeply investigated both with classic approaches, i.e. by enforcing continuity conditions at the cracked sections, and by means of the adoption of generalised functions which lead to closed-form solutions and treat the discontinuities in an effective way. The use of the distributional theory proved to be worth, since it opened a window onto more complicated problems (e.g. identification problems) that cannot be easily treated with classic approaches which require the enforcing of continuity conditions at the cracked sections. When it comes to curved cracked beams, the curved geometry couples the axial and transversal forces and in the literature only the classic approach which requires the enforcing of continuity conditions is adopted to account for the presence of cracks. In this paper the capability of the generalised functions to lead to closed-form exact solutions is extended to the case of multi-cracked Euler circular arches. Concentrated cracks, representative of a damage distribution, have been modeled through the equivalent elastic hinge concept and have been introduced in the governing differential equations by making use of Dirac's delta functions. As a consequence a generalized six order differential equations is inferred and solved in closed form. The solution is provided as a function of six integration constants only (depending on the boundary conditions), irrespectively of the number of along arch concentrated cracks. The proposed solution is preparatory for solving the static damage identification problem in arches. Validations of the proposed model have been performed by comparing the results with finite element numerical simulations, by considering different boundary conditions and load scenari.