Variational Considerations for Elastic Beams and Shells

A general variational theorem for stresses and displacements in small deflection shell theory, including the effect of transverse shear deformation, is presented. The theorem is used, in conjunction with the Lagrange multiplier method, to establish appropriately contracted systems of stress boundary conditions for shells rigid in regard to (1) transverse shear and (2) transverse shear and midsurface strain. An analogous variational theorem for stresses and displacements of space curved beams is also presented. Appropriate formulations of shell theory and of beam theory are combined to formulate boundary conditions for shells with edge loads applied through the intermediary of reinforcing edge beams. It is then found that contracted systems of boundary conditions for edge -reinforced shells rigid in regard to (1) transverse shear and (2) transverse shear and midsurface strain are an immediate consequence of introducing rigidity conditions for the edge beams, of the same nature as the rigidity conditions for the shell, into the original uncontracted system of boundary conditions. These contracted systems of boundary conditions for the edge-reinforced shell contain, as limiting case and in a simpler form than through use of Lagrange multipliers, the corresponding stress boundary conditions for the shell without edge reinforcement. The significance of the boundary condition contraction here obtained is that for many problems of shell theory it makes possible the explicit determination of “interior” membrane and bending states, without consideration of the more complete problems that also include “edge zone” bending states.