Exact shell solutions for conical springs

In this article, the equations of equilibrium of conical disk springs of thin and moderate thickness are obtained through the variational principles for thin-walled and thick-walled conical shells. The closed form analytical solutions based on the common deformation hypotheses for the equations of thin- and thick-walled truncated conical shells were achieved. The results of calculations of reaction forces, based on analytical formulae, were compared with the results of finite element analysis, demonstrating the good accuracy of the derived formulae. The theory is extended to incorporate the anisotropy of the material. The problem for optimal anisotropy is solved. The minimal stiffness of the spring is achieved, if the upmost modulus of the orthotropic material is in the meridional direction. Analogously, the highest stiffness is attained, if the maximal elastic modulus circumferentially oriented. Engineering applications of the current theory potentially include Bellville springs and slotted disk springs with moderate flatness for automotive and industrial applications.