On the dynamics of a beam partially supported by an elastic foundation: an exact solution-set

Free vibrations of straight beams which are partially supported by an elastic foundation are analyzed. For the sake of simplicity, only the Euler-Bernoulli beam model coupled with a Winkler-type elastic foundation is considered. This structural system can be used to study, in a rather accurate way, the dynamic response of partially embedded piles (like those used for telecommunications) when dealing with the problem of identifying their mechanical properties during operative conditions. The study makes clear that different kinds of vibration modes may occur in the part of the beam which is supported by the continuous elastic foundation: indeed apart from the classical modes, corresponding to the dynamics of a free beam, it is possible to have vibration modes which are similar to the static deflection of a beam on an elastic support or even corresponding to rigid-body modes. For the same beam it is shown that transition between these vibration modes can appear when switching from the fundamental natural frequency to subsequent ones. This effect is the focus of the presented numerical examples. In particular, the analytic expression of the transcendental functions governing the vibration modes, and of the coefficients of the eigenfunctions for all occurring cases, are given here — to the best of the author’s knowledge — for the first time. From the practical point of view, the reported results allow to define a suitable range of the elastic stiffness parameter such that the behavior of a partially supported beam can be conveniently approximated with that of a single-span beam, having one built-in end and the other free.