In this paper, the dynamic response of a simply supported travelling beam subjected to a transverse load is investigated in the super-critical speed range. The well-known axially moving beam theory is considered and a simple viscous damping mechanism has been introduced. The displacement field is expanded in a series of the buckling modes, a sine series, and different techniques have been used in analyzing the response of the dynamical system. Periodic oscillations are studied by means of continuation techniques, while non-stationary dynamics are investigated through direct simulations. A comparison with the literature and a convergence test on the series expansion are performed. A sample case of a physical beam is developed and numerical results are presented concerning bifurcation analysis and stability, and direct simulations of global postcritical dynamics. A complex scenario of alternate regular and chaotic motions is found in a large range of the main parameters.