Distributional Green’s functions for the vibrations of multi-cracked Timoshenko beams

In this study the forced vibrations of cracked beams in presence of damping are analysed. The adopted beam model is in accordance with the Timoshenko beam model and the presence of multiple bending and shear concentrated flexibilities, commonly used to model cracks, is accounted for. The strong discontinuities derived by the localised flexibilities are dealt with by means of a distributional approach avoiding the need of enforcing continuity conditions at the discontinuous sections. First, the exact Green’s functions, that is the steady-state response in the case of concentrated harmonic loads, are obtained via the presented distributional approach. The presented exact solutions are a computationally advantageous evaluation of the steady state response alternative to the direct time integration, as well as to a beam span sub-division. In addition, the presented distributional Green’s functions are employed to evaluate the response of multi-cracked beams subjected to arbitrary loading conditions (i.e. generic spatial distribution and time dependency), via convolution integral equation combined with an appropriate frequency domain analysis.