The exact explicit dynamic stiffness matrix of multi-cracked Euler–Bernoulli beam and applications to damaged frame structures

In this paper the closed form expression of the exact dynamic stiffness matrix of an Euler-Bernoulli beam in the presence of an arbitrary number of concentrated cracks is derived. The procedure adopted for the evaluation of the dynamic stiffness matrix is based on the availability of the exact closed form solution of the vibration modes of the multi–cracked beam, derived by the same authors in a previous paper. The knowledge of the exact explicit dynamic stiffness matrix of the multi–cracked beam makes the direct evaluation of the exact global dynamic stiffness matrix of damaged frame structures possible. Furthermore, it allows the exact evaluation of the frequencies and the corresponding vibration modes, consistent with the distributed parameter model, through the application of the well–known Wittrick–Williams algorithm. Some numerical applications, relative to the evaluation of frequencies and the corresponding mode shapes of multi-cracked framed structure, are reported. Furthermore, the closed–form solution has been validated by comparing with some exact results available in the literature, for a simple single cracked frame. Finally, further new results for a multi–cracked frame have been compared with those obtained by a finite element simulation.