A unified analysis of isotropic and composite Belleville springs

This article discusses a method developed for predicting the deformation behavior of a Belleville spring under axial loading by using the minimum potential energy principle. Strain energy and work done of the Belleville spring are formulated based on the classical thin shell theory in a conical coordinate system. The von Kármán and Reissner approximations to the nonlinear strain-displacement relations take the geometrical effects of the moderately and very large axial deflection into consideration, respectively. The Ritz method is used to solve for the deformation and force characteristics of isotropic springs and the solutions are compared with the seminal equation of Almen and Laszlo, experiments, and finite element analyses. The present energy model can capture the effect of a geometric parameter that has been missing from the formulation of Almen and Laszo. The comparison shows that the developed method gives very good agreement with the results from the testing and finite-element method, whereas in most cases, owing to their limited assumptions, Almen and Laszlo's equation overpredicts the applied load at a given deflection. This energy model is also applicable for composite Belleville springs with orthotropic material properties. A load factor and a orthotropic factor are defined to readily analyze load-deflection relationship of the composite springs based on the existing relationship of the isotropic elastic spring.