Large Deformations of Elastic Conical Shells

Axisymmetric conical shells under axial forces are investigated. A geometrically nonlinear approach leads to the REISSNER-MEISSNER equations, which allow the calculation of large deformations. These two nonlinear second order equations have been integrated by a matrix method, suggested by E.L. AXELRAD. Another effective solution, suitable for a small computer, uses the RUNGE-KUTTA-integration combined with an iteration program for the unknown boundary values. Useful results for the practical design of conical springs have been published, which give slight corrections to the famous ALMEN-LASZLO-paper of 1936. While conical springs are rather flat and thick, the presented theory can be used for steep and thin shells. Then the “spring-characteristics”, the force-deflection curves, become very complicated and their stability must be discussed. A simple criterion of stability is derived here out of the DIRICHLET definition. Presented as a problem of catastrophe theory, interesting curves in the parameter plane are obtained.