On the response bounds of damaged Euler–Bernoulli beams with switching cracks under moving masses

Abstract

The response of damaged Euler–Bernoulli beams with any number of unilateral cracks and subjected to one or more moving masses which remain always in contact with the beam are investigated in this paper. A flexibility switching crack model is employed here to represent a crack as either open or closed at a particular instant depending on the sign of the axial strain at the crack centre and taking into account the location of the crack either at the top or bottom of the beam. This model leads to closed-form solutions of the damaged beam mode shapes as a function of four integration constants, which can be computed by enforcing the boundary conditions, and a Boolean switching crack array which identifies the open cracks. Based on this switching crack model, an efficient computational procedure is proposed to tackle the problems of multiple masses moving on a multi-cracked beam with generic boundary conditions. The proposed procedure is applied to three beams with switching cracks under moving masses, and the results obtained are compared to the widely adopted always open cracks model, showing significant differences. In particular, it is shown that: (i) Flexible boundary conditions, multiple switching cracks and multiple moving masses can be easily accounted for with the proposed approach; (ii) the two limiting conditions of always open cracks and always closed cracks (undamaged) models do not always provide bounds on the dynamic response of a beam with switching cracks subject to moving masses; (iii) the side where each crack is located can largely affect the response.