Finite element procedures for nonlinear structures in moving coordinates. Part II: Infinite beam under moving harmonic loads

This paper presents a numerical approach to the stationary solution of infinite Euler–Bernoulli beams posed on Winkler foundations under moving harmonic loads. The procedure proposed in Part 1 [Nguyen V-H, Duhamel D. Finite element procedures for nonlinear structures in moving coordinates. Part I: infinite bar under moving axial loads. Comput Struct 2006;84(21):1368–80], which has been applied to consider the longitudinal vibration of rods under constant amplitude moving loads in moving coordinates, is enhanced herein for the case of moving loads with time-dependent amplitudes. Firstly, the separation of variables is used to distinguish the convection component from the amplitude component of the displacement function. Then, the stationary condition is applied to the convection component to obtain a dynamic formulation in the moving coordinates. Numerical examples are computed with a linear structure to validate the proposed method. Finally, nonlinear elastic foundation problems are presented.