Dynamic Stability of Planar Frames Supported by Elastic Foundation

Abstract

An exact analytical solution for a vibrating beam-column element on an elastic Winkler foundation is derived. The solution covers all cases comprised of constant compressive and tensile axial force with restrictions of ks -mω1 > 0 and ks -ma>2 < 0. Closed form solutions of dynamic shape functions are explicitly derived for each case and they are used to obtain frequency-dependent dynamic stiffness terms. Governing dynamic equilibrium equations are not only enforced at element ends, but also at any point along the element. To this end, derived stiffness terms are exact and they include distributed mass effects and geometric nonlinear effects such as axial-bending coupling. For this reason, the proposed solution eliminates the need of further element discretization to obtain more accurate results. In absence of elastic foundation (i.e., ks → 0), exact dynamic stiffness terms for beam-columns are also derived and presented in this study. Derived stiffness terms are implemented in a software program and several examples are provided to demonstrate the potential of the present study.