Complementary Energy Theorem for Symmetric Shells

A variational theorem is presented for the stresses and meridional slopes in axially symmetric thin shells with large strains, displacements, and slope changes, made of material having linear or nonlinear elasticity. This theorem represents a generalization of the stationary complementary energy theorem for infinitesimal deformations. The shell may have orthotropy consistent with axial symmetry and its elastic constants may vary meridionally. The following loadings are considered: On the shell middle surface, distributed radial and axial forces of prescribed magnitude per unit undeformed middle-surface area plus radial centrifugal forces; on the shell boundaries, prescribed radial force of displacement, prescribed axial force of displacement, and prescribed meridional bending moment or rotation. The theorem represents a particular application of a general stationary complementary energy theorem developed by the writer previously. In the present paper, the shell theorem is proved independently of the earlier paper by means of the calculus of variations. As a further check, the theorem is shown to reduce to the conventional form when the deformations are infinitesimal.