Beams on elastic foundation: A variable reduction approach for nonlinear contact problems

Beams on elastic foundations are applied to a vast number of engineering problems. Several elastic foundation models are available, from the simplest Winkler element with one parameter to complex models with more parameters and nonlinear characteristics. Analytical and numerical approaches have been developed in the literature for the solution of this problem, often specialized for a particular application. In this paper, a novel numerical approach that can be applied to any combination of beam and foundation models is presented. The method is based on independent meshes for the beam and for the foundation. The independent discretization of the foundation opens the possibility to model any kind of foundation behaviour, including nonlinearities, discontinuities and space-dependent properties. The two meshes are then connected by a variable reduction approach, formulated by standard finite element procedures. Such an approach allows to refine the discretization of the foundation without affecting the dimension of the solving system, i.e. with a limited effect on the computational time. Additionally, a relevant advantage of the presented method is that, contrary to most approaches described in the literature, gaps between the beam and the foundation can be straightforwardly included by an energy-based formulation. Examples of applications to linear, nonlinear, and foundation with gaps are reported in the paper. This innovative approach not only simplifies the modelling process but also offers significant computational advantages, making it a versatile and efficient tool for a wide range of engineering applications involving beam–foundation interactions.