The exact distributional model for free vibrations of shear-bending multi-cracked Timoshenko beams

The influence of shear deformation in the dynamics of beam-like problems is usually studied by means of a First order Shear Deformation Theory (FSDT) due to the original Timoshenko formulation. In comparison to higher order theories, the FSDT is able to embed additional concentrated flexibilities due to the presence of cracks by making use of the principle of the fracture mechanics. Precisely, flexural and shear concentrated flexibilities due to the first and second mode of the crack growth, respectively, can be included in the latter model. Despite various procedures suggested in the literature, the formulation of closed form solutions of free vibrations of multi-cracked Timoshenko beams is available for the case of concentrated flexural flexibilities only. In this work an original distributional model, to account for both flexural and shear concentrated flexibilities due to multiple cracks, is presented. New governing equations of the Timoshenko beam, enriched by suitable distributional terms, over a single integration domain are formulated. An integration procedure of the governing equations by making use of the generalised function theory is presented and an original closed form solution of vibration modes and frequency equation is obtained. The closed form solution allows a parametric study to discuss the influence of the concentrated shear and flexural stiffness decay due to the presence of multiple cracks.