Finite element steady state solution of a beam on a frictionally damped foundation under a moving load

The application of beams on foundation models subjected to moving loads in the engineering analysis of high-speed railway tracks requires a realistic characterization of the foundation behavior. In particular, the dissipative frictional character of the substructure of a railway track should be taken into account. In this paper, the steady state responses of a Euler-Bernoulli beam under a moving load on a foundation composed of a continuous distribution of linear elastic springs associated in parallel with a continuous distribution of Coulomb frictional dampers are computed. The steady state of the beam is governed by a partial differential inclusion that is semi-discretized in space, using a Discontinuous Least-Squares Finite Element Method (DLSFEM), as a system of ordinary differential inclusions, and integrated using a special implementation of the Non-Smooth Contact Dynamics method (NSCD) adapted to distributed Coulomb friction. The steady state solutions are then obtained for different values of the maximum force per unit length of the frictional dampers and for different values of the load velocity at both subcritical and supercritical regimes. It is found that the NSCD-DLSFEM produces consistent and accurate numerical outcomes in a wide range of the mechanical parameters that may be of interest in high-speed railway tracks engineering.