The Statics of the Multi-cracked Bickford–Reddy Beam

The presence of concentrated damage, abrupt changes in the geometry, or coupling different materials, or any kind of discontinuities in beams implies the presence of singularities that produce peaks or jumps in the kinematic or force entities. According to classic theories, when a three-dimensional modelling is to be avoided to reduce the required computational burden, uniaxial models are usually employed and continuity conditions at the sections with singularities are enforced. However, the presence of singularities can be effectively modelled by means of a distributional approach, and the governing equations can be formulated over a unique domain such that the equations can be integrated despite the presence of generalised functions. In this study, the distributional approach is applied to the statics of beams in presence of singularities, representative of multiple cracks, within the Higher order Shear Deformation Theory (HSDT) by Bickford and Reddy. The governing equations are formulated and an effective integration strategy able to infer the relevant closed-form solution is presented. Classic boundary conditions only are enforced, any continuity condition at the discontinuous sections is avoided. Several comparisons are presented with regard to the static response of beams with singularities against results obtained with the First-order Shear Deformation Theory (FSDT).