Dynamic response of damaged beams with uncertain crack depth

In many engineering applications detection of cracks and identification of their size for structures having beam form is of crucial importance. Dynamic response of damaged beams changes due to the change of its mechanical characteristics. In particular the stress field is influenced only in the region adjacent to the crack and such perturbation determines a local reduction in the flexural rigidity. In this study the analysis is performed adopting a finite element model for the damaged beam extended to the case where multiple non-propagating cracks are presented. Undamaged elements of the beam are modelled by Euler-type finite elements. Moreover, in the deterministic fracture mechanics analysis, the crack characteristics (crack size, crack position, applied stress etc.) are assumed to be known. However they possess considerable scatter or uncertainty. The aim of this paper is to evaluate the time varying upper and lower bounds of the response of damaged beams with multiple cracks subjected to deterministic and stochastic excitations with uncertain cracks depth assumed as interval variables. In order to evaluate the bounds of the response the authors propose an approach developed in the time domain starting from the derivation of the bounds of the interval eigenvalues evaluated as solution of two appropriate deterministic eigenvalue problems. Results provided by the analysis of a damaged cantilever aluminum beam for uncertain-butbounded crack depth model in terms of bounds of the vertical displacement of the free end of the beam compared with the exact solution via vertex method asses the efficiency and the accuracy of the presented procedure.