Closed-form solutions for vibrations analysis of cracked Timoshenko beams on elastic medium: An analytically approach

Abstract

In this paper a novel analytically method for analysing the dynamic behaviour of Timoshenko beam model, resting on Winkler-type elastic soil, under simply-supported boundary condition and with a crack is proposed. The beam is also supposed to be constrained at the ends by elastically flexible springs, with transverse stiffness and rotational stiffness. Applying the Timoshenko beam theory and employing the auxiliary functions, the equation of motion is derived. The natural frequencies are obtained by applying the Euler–Bernoulli method and are derived by the corresponding auxiliary functions of the governing equation of the Euler–Bernoulli beam in free vibration. For different values of soil parameter, taking into account the effects of rotational and shear deformation and considering the presence of crack, typical results are presented, in order to demonstrate the efficiency of the proposed approach. Finally the obtained results are compared with some results available in the literature. It is shown that very good results are obtained. This approach is very effective for the study of vibration problems of Timoshenko beams. The novelty of the proposed approach is that although the auxiliary functions, used to find the solution to the dynamic problem of a Timoshenko beam, are different for the two theories applied, in both cases, the dynamic problem is traced to the study of an Euler - Bernoulli beam subjected to an axial load.