Continuous model for the transverse vibration of cracked Timoshenko beams

A continuous model for the transverse vibrations of cracked beams including the effect of shear deformation is derived. Partial differential equations of motion and associated boundary conditions are obtained via the Hu-Washizu-Barr variational principle, which allows simultaneous and independent assumptions on the displacement, stress and strain fields. The stress and strain concentration caused by the presence of a crack are represented by so-called crack disturbance functions, which modify the kinematic assumptions used in the variational procedure. For the shear stress/strain fields, a quadratic distribution over the beam depth is assumed, which is a refinement of the typical constant shear stress distribution implicit in the Timoshenko model for uncracked beams. The resulting equations of motion are solved by a Galerkin method using local B-splines as test functions. As a numerical verification, natural frequencies of the linear, open-crack model are computed and the results are compared to analytical results from similar models based on Euler-Bernoulli assumptions and experimental results found in the literature. For short beams, results from a 2-D finite element model are used to confirm the advantages of the proposed model when compared with previous formulations.