Convergence of Galerkin truncation for dynamic response of finite beams on nonlinear foundations under a moving load

The present paper investigates the convergence of the Galerkin method for the dynamic response of an elastic beam resting on a nonlinear foundation with viscous damping subjected to a moving concentrated load. It also studies the effect of different boundary conditions and span length on the convergence and dynamic response. A train–track or vehicle–pavement system is modeled as a force moving along a finite length Euler–Bernoulli beam on a nonlinear foundation. Nonlinear foundation is assumed to be cubic. The Galerkin method is utilized in order to discretize the nonlinear partial differential governing equation of the forced vibration. The dynamic response of the beam is obtained via the fourth-order Runge–Kutta method. Three types of the conventional boundary conditions are investigated. The railway tracks on stiff soil foundation running the train and the asphalt pavement on soft soil foundation moving the vehicle are treated as examples. The dependence of the convergence of the Galerkin method on boundary conditions, span length and other system parameters are studied.

► Convergence of Galerkin method for beams on a nonlinear foundation is studied.
► Three types of conventional boundary conditions are compared.
► Effect of span length of beams on dynamic response is examined.
► Effects of boundary conditions and span length on the convergence are investigated.