On Parametric Response Characteristics of Beams With Multiple Transverse Cracks

Abstract

This study deals with the parametric instability of a beam with multiple cracks. The variation of buckling load and dynamic stability regions, with respect to relative crack depth and position of cracks, are analysed using FEM. The loading on the beam is considered to be axial with a simple harmonic fluctuation with respect to time. The equation of motion represents a system of second order differential equations with periodic coefficients of the Mathieu-Hill type. The development of the regions of instability arises from Floquet’s theory, and the periodic solution is obtained by Bolotin’s approach using the finite element method. The stiffness matrix of the cracked beam element is obtained from the flexibility matrix of the intact beam, and the additional flexibility matrix due to the crack. The frequencies of vibration and buckling loads of the cracked cantilever beams reduce with the increase in crack depth and number of cracks. The onset of instability occurs earlier with the introduction of more cracks. The instability region for the crack location nearer to the fixed end occurs at a lower excitation frequency of the cracked beam. The vibration and instability results can be used as a technique for structural health monitoring or testing of structural integrity, performance, and safety.