Free vibration analysis of a cracked shear deformable beam on a two-parameter elastic foundation using a lattice spring model

Abstract

Abstract The free vibration of a shear deformable beam with multiple open edge cracks is studied using a lattice spring model (LSM). The beam is supported by a so-called two-parameter elastic foundation, where normal and shear foundation stiffnesses are considered. Through application of Timoshenko beam theory, the effects of transverse shear deformation and rotary inertia are taken into account. In the LSM, the beam is discretised into a one-dimensional assembly of segments interacting via rotational and shear springs. These springs represent the flexural and shear stiffnesses of the beam. The supporting action of the elastic foundation is described also by means of normal and shear springs acting on the centres of the segments. The relationship between stiffnesses of the springs and the elastic properties of the one-dimensional structure are identified by comparing the homogenised equations of motion of the discrete system and Timoshenko beam theory. The effects of the transverse open cracks are modelled by increasing the flexibility of the rotational springs in the discrete model at crack locations. In this manner, the cracked section is modelled by a massless rotational spring combined in series with the rotational spring that represents the flexural stiffness at that point. Numerical examples are provided to show the versatility and convergence of the present method, and to investigate the effects of geometrical and physical parameters on free vibration of a cracked beam. An analytical approach is also developed based on the transfer matrix method (TMM) to examine and validate the results obtained by the LSM against the corresponding analytical solution.