Closed form solutions of a multi-cracked circular arch under static loads

Generalised functions have been widely adopted in structural mechanics to treat singularities of beam-like structures. However due to the curved geometry, that couples axial and transversal displacements, their use has never been explored for curved beams. In this paper the capability of distributions of leading to closed form exact solutions for multi-cracked circular arch is shown. The exact closed-form solution of a circular Euler arch in presence of any number of discontinuities due to concentrated damage and subjected to an arbitrary distribution of static loads is obtained. Damage, under the form of concentrated cracks, has been modeled through the widely adopted and validated equivalent elastic hinge concept and has been introduced in the governing differential equations by making use of Dirac's delta functions. The resulting nontrivial generalised six order differential equations have been derived and solved in closed form. Independently of the number of along arch concentrated cracks, the solution is expressed as a function of six integration constants only in which the damage positions and intensities are given data appearing explicitly in the solution expression. This latter aspect constitutes a fundamental aid towards the resolution of the static damage inverse identification problem. The results have been validated through some comparisons with finite element numerical simulations: examples referred to multi-cracked Euler arches with different boundary conditions, damage and load scenarios are presented.