On a special class of boundary-value problems in the theory of shells and plates

This paper is concerned with problems in shell theory in which displacements are specified on a major surface of a three-dimensional shell-like body. It is unclear whether the theory of shells is in fact applicable to such problems; in virtually any treatment of shells, the equations of motion are derived by assuming that the tractions on both major surfaces are known. These known tractions manifest themselves as body forces in the resulting two-dimensional set of equations. We demonstrate in this paper that provided certain modifications are made, the latter set can indeed be used to solve problems in which displacements are specified on a major surface. These modifications are essential; they in fact reduce the number of differential equations and change the nature of the boundary conditions. We outline a clear procedure to treat such problems. The procedure is general in that it is valid for finite deformations of shells made of any material. We illustrate the efficacy of the final set of equations by presenting some examples from the linear theory of elastic plates.