Introduction

The main task of this chapter is to investigate some purely geometrical aspects of the distortion of curved surfaces. In general, if a given surface is distorted from its original configuration, every point on the surface will under-go a displacement; and at every point the surface will experience strain (‘stretching’) and change of curvature (‘bending’). Clearly the components of strain and change of curvature are, in general, functions not only of the components of displacement but also of the geometry of the surface in its original configuration.

In the present chapter (and indeed throughout the book) we shall consider only the limited class of distortions in which displacements, strains and changes of curvature are regarded as small, just as they are in the classical theory of simpler structures. In consequence of this simplification, the functional relationships between strain, change of curvature, and displacement will be relatively simple, and indeed linear.

In chapter 3 we have already established ad hoc expressions for change of curvature and hoop strain in terms of radial displacement for symmetrical deformations of a cylindrical shell surface. Our present task includes not only the investigation of more general types of distortion of cylindrical shells but also the consideration of distortion of other kinds of surface.

The arrangement of the chapter is as follows. First we investigate some aspects of the distortion of initially plane and cylindrical surfaces: these have the advantage that they may be described in terms of simple Cartesian coordinates.