A Thick Hollow Sphere Compressed by Equal and Opposite Concentrated Axial Loads: An Asymptotic Solution

We consider an elastic hollow sphere with midsurface radius R and thickness 2h which is subjected to two equal and opposite concentrated loads acting at the ends of a diameter. The three-dimensional linear elasticity solution to this problem consists of (i) a narrow Saint Venant component extending a distance of order O(h) from each load point, (ii) a wider “edge bending” component extending a distance of order O( √ Rh) from each load point, and (iii) a “membrane” component which permeates the whole sphere. Because of the stress singularities at the load points, the Saint Venant component is extremely complex and difficult to calculate. We determine the other two (outer) components of the solution without any explicit knowledge of the Saint Venant (or inner) component. This is achieved by a rigorously valid method which does not depend on any physical heuristic, such as Saint Venant’s principle. Our method depends on the fact, established here for the first time, that the eigenfunctions for the spherical shell satisfy bi-orthogonality relations. We use these bi-orthogonality relations to find a two-term asymptotic approximation to the outer solution of the problem; in principle, further terms could be obtained. The leading term of this approximation corresponds to the classical thin shell solution to within O(h/R). Our solution is an outer solution in the sense that it is valid outside an O(h) neighborhood of the load points. We make numerical comparisons between the predictions of our two-term refined theory, those of thin shell theory, and those of the Reissner–Wan theory (which generalizes thin shell theory by allowing for transverse shear deformation). All three solutions agree closely when h/R ≤ 1/20. When h/R = 1/10, however, differences become pronounced. Since our refined theory is still applicable to such moderately thick shells, the results we obtain enable us to analyze the limitations of the thin shell theory. Furthermore, we find that the predictions of the Reissner–Wan theory are considerably closer to the true values than those of classical thin shell theory.