On the Determination of Stresses and Displacements for Unsymmetrical Deformation of Shallow Spherical Shells

摘要

The report is concerned with problems of bending of thin, elastic, shallow spherical shells of uniform thickness. Explicit expressions are derived, in terms of stress function and axial displacement component, for radial and circumferential displacement components. It is shown that certain uni-valued portions of the solution for stress function and axial displacement components give rise to multi-valued expressions for radial and circumferential displacement components. A new type of solution of the differential equations is derived which is multi-valued insofar as the stress function is concerned but given uni-valued expressions for all quantities which should be uni-valued. It is shown that this type of solution is needed for the analysis of problems for which a resultant side force is acting along the edge of a symmetrical circular boundary of the shell. The nature and form of this side-force solution differs in important respects from the corresponding known solution for the case of a flat plate. No direct transition is possible from the shell solution to the corresponding flat-plate solution. As a specific application, a shell is considered which is supported at its outer edge and which has a small symmetrical rigid insert.