The forced vibrations of an elastic beam resting on a non-linear tensionless Winkler foundation subjected to a concentrated dynamic load at its centre are described in this paper. The problem is a non-linear one because of the tensionless character of the foundation and the non-linear term in the foundation model. A non-linear governing differential equation for the forced vibrations of the beam is derived in matrix form by employing the Galerkin method. The free vibration mode functions of the completely free beam are adopted as the co-ordinate functions of the displacement function of the beam. Firstly, the static solution is obtained and the contact length is determined. This is then used as an initial configuration of the forced vibrations. The results which represent the static and dynamic responses of the beam for linear and non-linear cases are presented in the Figures. (C) 2000 Academic Press.