The dynamic stiffness matrix (DSM) of axially loaded multi-cracked frames

The presence of axial load, due to mechanical loading or to temperature effects, represents a strong peculiarity of frame applications both in direct and inverse problems involving the presence of cracks. The lack of explicit formulations of the Dynamic Stiffness Matrix (DSM) for cracked beam elements accounting for the influence of axial loads inspired and motivated the present work and the calculations herein reported. In this paper, the transcendental, frequency dependent, exact explicit expression of the DSM of an Euler-Bernoulli beam-column in presence of an arbitrary number of open cracks is presented. The advantage of the derived expression consists in a fourth order DSM whose dimension stays consistent independently as the number of cracks along the beam increases. Assembling DSMs of single beam-column elements, according to any required frame layout, leads to a number of degrees of freedom of the overall damaged frame structure exactly the same as those of the equivalent undamaged structure. Among other recent achievements on the vibration of damaged frames, the presented formulation provides a further original contribution, in the context of the dynamic stiffness method, which, in conjunction with the Wittrick & Williams algorithm, allows the exact evaluation of the frequencies and vibration modes of damaged frame structures in presence of axially loaded elements.