Large amplitude vibrations of slender, uniform beams on elastic foundation

The large amplitude vibrations of slender, uniform beams, with axially immovable ends, on an elastic foundation are described in this paper. The governing differential equation given by Woinowsky-Krieger is generalised to include the effect of the elastic foundation. The space variable in the differential equation is eliminated, by assuming suitable admissible functions in space to represent the simply supported or clamped boundary conditions moment at the ends, using the standard Galerkin method. The resulting temporal equation is solved by a numerical integration scheme to obtain the linear and nonlinear frequencies for a specific maximum amplitude. The results obtained, in terms of the ratio of the nonlinear frequency to the linear frequency, when the foundation stiffness is zero, agree well with those available in literature. It is observed from the numerical results that the effect of the elastic foundation reduces the nonlinearity for both the simply supported and the clamped beams.