Dynamics of multiple cracked prismatic beams with uncertain-but-bounded depths under deterministic and stochastic loads

Dynamic response of cracked beams changes with respect to undamaged ones due to the change in mechanical properties. Moreover, usually, the crack characteristics (crack size, crack position, applied stress etc.) are assumed to be known. However, in practice they possess considerable scatter of 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 or stochastic excitations assuming uncertain crack depths as interval variables. 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. In order to provide 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. To assess the efficiency and the accuracy of the presented method a damaged cantilever prismatic aluminum beam with multiple uncertain-but-bounded cracks is examined. The bounds of the free-end displacement and variance are calculated and compared with the exact solution via combinatorial procedure. The proposed results show a high level of accuracy for both increasing amplitude value of interval deviation and increasing number of uncertain parameters