Thin Plate Theory

Abstract

Introduction A thin plate is a structural element, just as a long beam is a structural element. The thin plate also is characterized by its special geometry. While a long beam has two dimensions very much shorter than the length dimension, a thin plate, as is pictured in Fig. 1.11, has one dimension that is very much less than the other two dimensions. The thin plate's least dimension is, of course, called the plate thickness. The thickness is usually a constant in most vehicular structures, but the possibility of a gradually changing thickness can be incorporated into thin plate bending theory (Ref. [16]). The difference between a thin plate and a membrane is that a thin plate can resist both stretching and bending, while a membrane can only resist lateral or inplane loads by stretching. There are two goals for this chapter. The first and lesser goal is to develop the equations of classical thin plate bending theory. More than an introductory discussion of the extensive body of classical thin plate bending solutions is outside the purposes of this textbook. The second and more important goal is to develop the equations necessary to describe a simple plate bending finite element comparable to the previously developed beam bending finite element. The greater importance assigned to the second, rather limited, goal not only suggests the present relative importance of the two topics but also suggests that certain simplifications like constant plate thickness are appropriate for the thin plate theory to be presented here.