Further considerations of stress concentration problems for twisted or sheared shallow spherical shells

Abstract

We extend earlier studies of closed-form and asymptotic solutions of the problems of transverse twisting and membrane shearing of shallow spherical shells with small circular tractionfree holes or rigid inserts. Our first extension concerns a modified form of the relevant exact solution of the shell equations, so as to take advantage of the static-geometric duality property of linear shell theory. As a consequence we observe that the results for two of the four previously considered problems can be seen as the static-geometric duals of the other two. The second extension concerns the formulation of a general boundary value problem, with the problems of the hole and the rigid insert as special cases. Furthermore, on the basis of this formulation we find, as a particularly simple boundary value problem of physical interest, the problem of a hole the edge of which is transversely fixed. The asymptotic properties of the twisting and shearing solutions for this case are shown to be intermediate, in a specific sense, to the previously established order of magnitude properties for the free hole or rigid insert cases.