Extended modal superposition for forced damped vibrations of multi-cracked shear deformable beams

Abstract Shear deformable beams modelled in accordance with the first-order Timoshenko theory in presence of multiple cracks are the object of this study. Contrarily to the Euler–Bernoulli theory, forced vibrations of the latter multi-cracked beam model have not been satisfactorily studied in the literature. The contribution to the above issue provided by this work consists, first, in the derivation of the “extended” mode shapes explicitly formulated, by means of a distributional approach, to include the effect of multiple cracks on bending and shear stiffness in presence of external proportional damping. Then, it is proved that the orthogonality property of the proposed mode shapes embedding the effect of multiple cracks is preserved and the relevant conditions are explicitly formulated. On this ground, forced vibrations of the multi-cracked beam are analysed by making use of modal superposition exploiting explicit expressions of the Impulse Response Function.