An analytical dynamic model for single-cracked beams including bending, axial stiffness, rotational inertia, shear deformation and coupling effects

Abstract

This paper develops an analytical dynamic model for cracked beams including bending, axial stiffness, rotational inertia, shear deformation and the coupling of the last two effects. The damage is modelled using a rotational spring that simulates the crack based on fracture mechanics theory. The developed model is used to predict variations on natural frequencies for several crack sites and damage magnitude along the beam. The importance of this work lies in the development of an analytical model that has no approximation due to discretization of the displacement field. This initial theoretical approach describes the expected behaviour for changes in the natural frequencies for simply-supported and clamped-free beams with the precision that only analytical methods allow. The results provide a useful benchmark to compare with approximate numerical methods that can be used to model and analyse the problem. The model showed similar results for long span beams, but the inclusion of rotational inertia and shear deformation effects rendered improvements in the dynamic behaviour mainly in the case of slender and short span beams when compared with the simplified Euler–Bernoulli model.