Response of beams with crack of uncertain-but bounded depth subjected to deterministic or stochastic loads

Detection of cracks in structural components and identification of their size for structures having beam form is of crucial importance in many engineering applications. For damaged structures the dynamic response changes with respect to the undamaged ones due to the changes produced on their mechanical properties by the presence of the crack. In this paper the deterministic behavior of a beam with a transverse on edge non-propagating crack is first studied. Moreover the deterministic and stochastic setting pertaining the case in which the crack has an uncertain depth is investigated. Undamaged elements of the beam are modeled by Euler-Type finite elements. The uncertain crack depth is modeled as an interval variable and the cracked beam is subjected to both deterministic and zero-mean nonstationary Gaussian random excitations. In the latter case the equation governing the evolution of the main statistics of the response are derived by means of Kronecker algebra. Once the mathematical model of the beam is defined, the dynamic response is evaluated by applying a numerical procedure based on the philosophy the Improved Interval Analysis via Extra Unitary Interval. In particular the proposed procedure is based on the following main steps: i) to define a finite element model of the beam in which the model of fully open crack is used to represent the damaged element; ii) to model the crack depth as an interval variable; iii) to evaluate in time domain the response for deterministic and stochastic excitation, by adopting an unified approach.