A displacement-based Timoshenko finite element of multi-cracked circular arch

This study presents a Displacement-Based (DB) finite element formulation of the multi-cracked planar circular arch within the Timoshenko theory. The cracks presence is simulated with the localised flexibility approach, considering the tangential, radial and rotational kinematic jumps at the damaged sections. The kinematic discontinuities are modelled with Dirac's deltas in the generalised governing equations defined over a unique domain for a generic number of damaged sections, which represents a significant computational advantage over classic approaches requiring enforcement of continuity conditions at the discontinuous sections. The governing equations of the statics of the multi-cracked Timoshenko circular arch are integrated, leading to the closed-form response as a function of six boundary conditions only. Then, the static discontinuous shape functions are inferred and exploited to obtain the stiffness and consistent mass matrices of the multi-cracked circular arch, considering just three degrees of freedom for each end of the member irrespective of the amount of cracks along the axis of the element. The obtained circular multi-cracked element, which is completely locking-free, is implemented in a finite element environment that encompasses an analogous finite element introduced for cracked straight members, thus allowing the study of general planar damaged frames both in the static and dynamic contexts.