Shim analysis for spherical elastomeric bearings

This paper describes development of equations for linear elastic stress analysis of concentric shims (reinforcements) in spherical multilayer bearings under general three-dimensional loading and uniform temperature change. The theory is based on a generalization of thin shell theory and the Complementary Strain Energy Principle. Some examples are given to illustrate the shim stresses due to direct loading from the rubber pads and their agreement with finite element predictions. This loading is found using the linear elastic pad and bearing analysis in (Schapery, 2018). Although the shims are assumed perfectly rigid in the pad and shim analyses, the total strain energy in the shims can be found using their Young's modulus and Poisson's ratio, enabling estimation of the effect of shim deformation on bearing stiffness. These analytical models for pads and shims are intended to assist in rapid preliminary design of elastomeric bearings when utilized, for example, in a Mathcad computer program embedded in Excel that the author used. Final designs can be done with finite elements to account for rubber nonlinearity and shim deformation.