The theory of elastic plates and shells deals with a class of problems in Solid Mechanics characterized by the restriction that the structures concerned are thin, signifying that, in the typical geometric representation, the particular diameter measuring the thickness is substantially smaller than the other length scales of the configuration. There is a direct correlation between this common mensural feature and certain mechanical properties shared by such structures, namely an enhanced flexibility in the thickness direction and the occurrence of boundary layer phenomena reflecting the fact that the influence of some of the conditions applied to the edge surface is confined to a relatively narrow neighborhood of their area of application.
In order to treat the problem of the straight elastic beam within the framework of a two-dimensional formulation, we consider the boundary value problem for the plane figure defined by the side-view projection of the three-dimensional body. Corresponding to the midplane of the beam the figure has an axis of symmetry, called the centerline or axis, which immediately defines a pair of reference directions in the plane; namely, the axial and transverse directions respectively parallel and normal to the centerline.
In its general form, a plate is a three-dimensional right cylindrical body having a plane of symmetry called the midplane. The bounding surface, in three distinct parts, consists of a cylindrical component known as the edge-surface, normal to the midplane, together with a pair of faces which, symmetrically placed with respect to the midplane, close the ends of the cylinder. We shall refer to the intersection of the edge-surface with the midplane as the edge-curve: for a finite plate this consists of one or more simple closed curves. The distance between the faces, taken along the normal to the midplane at a given point, measures the thickness of the plate at that point.
The configuration of a plate, namely, a three-dimensional figure cut from a right cylinder by two mutually reflecting surfaces symmetrically placed with respect to a plane normal to the generators of the cylinder, is generalized in the shell by replacing the plane of symmetry by an arbitrary base surface, which, by analogy, is termed the midsurface. For a figure defined on the base surface by one or more simple closed curves, which we collectively call the edge-curve, the ruled surface, generated by the normals to the midsurface along the edge-curve, defines a region of space. Introducing the two faces, namely two surfaces mutually reflecting with respect to the midsurface, so that, on every normal to the latter, the intercept between the faces is bisected by the midsurface, then the portion of the ruled surface lying between the faces will be referred to as the edge-surface. The figure enclosed by the edge-surface and the two faces is a shell and for any point on the midsurface the normal intercept between the faces measures the shell-thickness at that point.
