Finite element procedures for nonlinear structures in moving coordinates. Part 1: Infinite bar under moving axial loads

Abstract

This paper deals with analyzing nonlinear structures under high-speed moving loads by use of the finite element method. The stationary response of an infinite bar posed on a Winkler foundation under constant moving loads is investigated. Instead of the transient analysis, the stationary solution of this problem is obtained by solving a static system in a reference frame which moves with the load for reducing the computation cost. To overcome the difficulty due to numerical instabilities when considering very fast loads (supersonic loads), a new procedure to govern the finite element formulation in moving coordinates is proposed. Comparing numerical solutions with analytical ones shows that the proposed method is valid for all values of load speed. Last, an example of nonlinear elastic foundation is considered to outline the nonlinear effects.

Vibration of Timoshenko beam on hysteretically damped elastic foundation subjected to moving load

“…Numerical approaches, such as the moving element method (MEM), have been applied to deal with complex structures [1- 3,20]. As the standard finite element method used for wave propagation problems in unbounded media, MEM also poses difficulties in computational efficiency and in absorbing spurious reflections from the truncated boundary [11].…”

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

“…The advantage of this approach is that it requires smaller finite element meshes as the load is fixed in the moving coordinates. However, it has been noticed that when the speed of the load reaches the critical velocity, the stiffness matrix formulated in the moving coordinates becomes ill-conditioned because of an important convected term which occurs from the variable transformation (this convected term depends on the speed of the load and on the mass of the structure) [1,9]. In the previous paper [1], we have proposed a novel finite element formulation for the problem of an infinite bar under a constant moving load.…”

“…However, it has been noticed that when the speed of the load reaches the critical velocity, the stiffness matrix formulated in the moving coordinates becomes ill-conditioned because of an important convected term which occurs from the variable transformation (this convected term depends on the speed of the load and on the mass of the structure) [1,9]. In the previous paper [1], we have proposed a novel finite element formulation for the problem of an infinite bar under a constant moving load. The finite element formulation has been obtained through 2 steps: the dynamic equation is firstly discretized in time by using the generalized-a method and a variable transformation is then applied to this time-discretized equation for obtaining a static equation in the moving coordinates.…”

“…Based on the similar technique as presented in [1], we will propose an enhanced procedure to consider the moving harmonic load problem. Since the load amplitude is no longer constant, the equation established in the moving coordinates becomes a time-dependent equation.…”

Analytical Model of the Dynamics of Railway Sleeper

“…In addition, the sleeper dynamic response is important because it affects the stability of the railway track and the maximum speed of the train. Substantial researches using analytic methods for rail track have been carried out, for example : the effect of wheel-rail contact forces [17,18] or the vibration of the railway track (by considering the model of a ballasted track with discrete supports [8,9,13,14,1] or the model of the infinite beam placed on a continuous foundation [10,7,6,11]). The studies of each component of the rail track were performed on the rail [19,16,15], or on the ballast [20,22].…”