A DQEM for transverse vibration analysis of multiple cracked non-uniform Timoshenko beams with general boundary conditions

In this paper, a differential quadrature element method (DQEM) for free transverse vibration analysis of multiple cracked non-uniform Timoshenko beams with general boundary conditions is proposed. Governing equations, the compatibility conditions at the damaged cross-sections and implementation of the external boundary conditions are derived and formulated by the differential quadrature analogue. The accuracy, convergence, and versatility of the proposed method are confirmed by the exact solution of the uniform beam which has been presented by other authors, and 2D finite element method (FEM) numerical results for non-uniform beam. After the validation of the presented method, the effect of quantity, depth and location of the cracks on the frequency values of vibrations are investigated. The achieved results show that the existence of the crack leads to a decrease in the frequencies of the vibrations through decrease in the stiffness of the beam. Meanwhile, the compatibility conditions at the damaged section is considered as a discontinuity in slope and vertical displacement where the effect of the discontinuity in the slope is more considerable as many authors have neglected the discontinuity in the vertical displacement. As it will be shown, consideration of the discontinuity in the vertical displacement causes more decrease in the frequencies.