SOLUTIONS OF SOME ELASTIC SHALLOW SHELL PROBLEMS

Abstract

In this thesis the well known Marguerre's equations for thin shallow spherical elastic shells are taken as the basis for obtaining general expressions for stresses and displacements in polar coordinates involving Kelvin functions and elliptic coordinates involving Mathieu functions. It is shown how these can be used to obtain solutions under fairly general prescribed edge displacements or loadings of shells with circular boundary or boundary with elliptic base projection. In the case of a circular boundary the method is illustrated by solving two problems, one with specified edge deformations and the other with equal and opposite concentrated forces acting along a diameter of the circular base. The results in the former case are compared with results obtained under the same boundary conditions but certain simplifying assumptions discussed by Reissner. The comparison shows that the more accurate analysis attempted here is worthwhile. In the case of the shell with elliptic base projection, the method is illustrated by solving a problem with specified edge deformations as boundary conditions. The problems with specified edge deformations in both circular and elliptic cases are also solved by an entirely different method involving the numerical solution of finite difference equations derived from a modified form of Marguerre's equations. The results are sufficiently close to those by the earlier method to make either method reliably workable.