All structures are three-dimensional, and the exact analysis of stresses in them presents formidable difficulties. However, such precision is seldom needed, nor indeed justified, for the magnitude and distribution of the applied loading and the strength and stiffness of the structural material are not known accurately. For this reason it is adequate to analyse certain structures as if they are one- or two-dimensional. Thus the engineer's theory of beams is one-dimensional: the distribution of direct and shearing stresses across any section is assumed to depend only on the moment and shear at that section. By the same token, a plate, which is characterized by the fact that its thickness is small compared with its other linear dimensions, may be analysed in a two-dimensional manner. The simplest and most widely used plate theory is the classical small-deflexion theory which we will now consider.

The classical small-deflexion theory of plates, developed by Lagrange (1811), is based on the following assumptions:

1. (i) points which lie on a normal to the mid-plane of the undeflected plate lie on a normal to the mid-plane of the deflected plate;

2. (ii) the stresses normal to the mid-plane of the plate, arising from the applied loading, are negligible in comparison with the stresses in the plane of the plate;

3. (iii) the slope of the deflected plate in any direction is small so that its square may be neglected in comparison with unity;

4. […]

The exact large-deflexion analysis of plates generally presents considerable difficulties, but there are three classes of plate problems for which simplified theories are available for describing their behaviour under relatively high loading. These ‘asymptotic’ theories are membrane theory, tension field theory (sometimes called wrinkled membrane theory) and inextensional theory. All are described below. For a plate of perfectly elastic material, the error involved in using these theories tends to zero as the loading is increased or as the thickness is reduced. In any practical material, however, there is a limit to the elastic strain that may be developed, and this in turn limits the range of validity of these asymptotic theories to plates which are very thin. For steel and aluminium alloys, a typical limit to the elastic strain is 0.004, and this restricts the range of validity of the asymptotic theories as follows. For membrane theory and tension field theory the thickness must be less than about 0.001 of a typical planar dimension, while for inextensional theory the thickness must be less than about 0.01 of a typical planar dimension.

Membrane theory (considered by Föppl 1907)

When a thin plate is continuously supported along the boundaries in such a manner that restraint is afforded against movement in the plane of the plate, the load tends to be resisted to an increasing extent by middle-surface forces.

In the first edition of this book, I attempted to present a concise and unified introduction to elastic plate theory. Wherever possible, the approach was to give a clear physical picture of plate behaviour. The presentation was thus geared more towards engineers than towards mathematicians, particularly to structural engineers in aeronautical, civil and mechanical engineering and to structural research workers. These comments apply equally to this second edition. The main difference here is that I have included thermal stress effects, the behaviour of multi-layered composite plates and much additional material on plates in the largedeflexion régime. The objective throughout is to derive ‘continuum’ or analytical solutions rather than solutions based on numerical techniques such as finite elements which give little direct information on the significance of the structural design parameters; indeed, such solutions can become simply number-crunching exercises that mask the true physical behaviour.