THE AXISYMMETRICAL DEFORMATION OF A THIN, OR MODERATELY THICK, ELASTIC SPHERICAL CAP

A refined shell theory is developed for the elastostatics of a moderat ely thick spherical cap in axisymmetric deformation. This is a two-ter m asymptotic theory, valid as the dimensionless shell thickness tends to zero. The theory is more accurate than ''thin shell'' theory, but i s still much more tractable than the full three-dimensional theory. A fundamental difficulty encountered in the formulation of shell (and pl ate) theories is the determination of correct two-dimensional boundary conditions, applicable to the shell solution, from edge data prescrib ed for the three-dimensional problem. A major contribution of this art icle is the derivation of such boundary conditions for our refined the ory of the spherical cap. These conditions are more difficult to obtai n than those already known for the semi-infinite cylindrical shell, si nce they depend on the cap angle as well as the dimensionless thicknes s. For the stress boundary value problem, We find that a Saint-Venant- type principle does not apply in the refined theory, although it does hold in thin shell theory. We also obtain correct boundary conditions for pure displacement and mixed boundary data. In these cases, convent ional formulations do not generally provide even the first approximati on solution correctly. As an illustration of the refined theory, we ob tain two-term asymptotic solutions to two problems, (i) a complete sph erical shell subjected to a normally directed equatorial line loading and (ii) an unloaded spherical cap rotating about its axis of symmetry .