Linear and nonlinear analysis of shells of revolution with asymmetrical stiffness properties

Abstract

A method is described for the linear and nonlinear elastic analysis of shells of revolution, under arbitrary loading, with variable thickness properties in the circumferential direction as well as in the meridional direction. The method of approach is the matrix displacement method of structural analysis. The shell of revolution is idealized as an assemblage of curved elements connected together at their nodal circles or nodes. The variable thickness in the circumferential direction of the shell element is expanded in a Fourier series. This asymmetrical thickness variation results in the coupling of all Fourier harmonics. The element stiffness matrix is obtained by applying Castigliano's theorem and observing the coupling of the harmonics. To account for the geometric nonlinearities the total internal energy is separated into a sum of two parts corresponding to the internal energy based on linear theory and to the internal energy based on nonlinear theory. The nonlinear terms are regarded as additional generalized forces and are obtained by applying Castigliano's theorem resulting in the equations of equilibrium. To solve the equations of equilibrium, three methods are used: a) incremental approach; b) iterational approach; and c) a combination of the two first methods. The primary applications of this method are to the linear, nonlinear and stability analyses of the Apollo scalloped after heat shield under asymmetric loading due to water impact as well as to the linear analysis of the Apollo subsystems test bed. In the analysis of the Apollo subsystems test bed the proposed numerical method is extended, and a combination of the matrix displacement and force methods is used. Both meridional and circumferential rib stiffeners are considered. The method is further illustrated by application to a number of problems including hemispheres, shallow caps and circular plates. A comparison of the results with known exact solutions and with the results obtained by other methods shows excellent agreement and demonstrates that the proposed method yields realistic answers for shells of revolution of complex construction.