Static analysis of a thick ring on a unilateral elastic foundation

• A very common structure: Ring on Elastic Foundation (REF) is extended to nonlinear case with unilateral (collapsible or tensionless) elastic foundation.
• An efficient iterative compensation method to analyze this unilateral foundation problem is presented which is based on the analytical solutions of linear foundation case and avoids solving complex and coupled nonlinear differential equations. Detailed derivations are outlined for all aspects of the problem and example results are given.
• The ring is modeled as extensible and orthotropic using Timoshenko beam theory; both linear and unilateral foundation problems are solved.
• The solutions for this general mechanics model are applicable for many practical structures such as automotive tires, bearings and gears.

A thick ring on a unilateral elastic foundation can be used to model important applications such as non-pneumatic tires or bushing bearings. This paper presents a reduced-order compensation scheme for computing the static deformation response of a thick ring supported by a unilateral elastic foundation to an arbitrary applied force. The ring considered is an orthotropic and extensible ring that can be treated as a Timoshenko beam. The elastic foundation is a two-parameter foundation with a linear torsional stiffness but a unilateral radial stiffness whose value vanishes when compressed or tensioned. The paper first derives the deformation response for the linear foundation case for which Fourier expansion techniques can be applied to obtain an analytical solution. Then, the nonlinear unilateral foundation problem is solved via an iterative compensation scheme that identifies regions with vanishing radial stiffness and applies a compensation force to the linear foundation model to counteract the excessive foundation forces that would not be there with the unilateral foundation. This scheme avoids the need for solving the complex set of nonlinear differential equations and gives a computationally efficient tool for rapidly analyzing and designing such systems. Representative results are compared with Finite Element Analysis (FEA) results to illustrate the validity of the proposed approach.