Using time weighted residual method in forced dynamic vibration analysis of timoshenko beam under moving load

In this paper, formulation of a recently proposed time-weighted residual method has been developed for the vibration analysis of Timoshenko beams under moving loads. Employing pre-integration relations as well as equilibrium equations, is the main idea of this method. In the first step of the proposed method, the time interval is subdivided into a number of sub-intervals and then the acceleration field in each sub-interval is defined as the combination of an unknown function and an exponential series with constant coefficients. Finally, the solution of the problem is estimated by the time-weighted residual method along with exact satisfaction of the initial and the boundary conditions at the two ends of the beam. Storing the information of solution at each time step on the exponential coefficients, is the most important advantage of this method, so that the solution is progressed in time without the need to discretize beams and only by using an appropriate recursive relation to update the exponential coefficients. In order to investigate the accuracy and efficiency of the proposed method, the results of solving three sample problems of constant and accelerated moving load on the beams with different boundary conditions, are compared with the results of finite element method. This comparison illustrates the speed and accuracy of the proposed method in estimating the internal shear forces and bending moments rather than those obtained by the finite element method.