Plates - Mechanics of Structures: Variational and Computational Methods - page 755

13

Plates

Plates are flat structural elements with thicknesses much smaller than the other dimensions.
Familiar examples of plates include flat roofs, doors, table tops, and manhole covers.
In analyzing a plate (Fig. 13.1), it is common to consider the plate to be divided into
equal halves by a plane, the midplane or middle surface , parallel to the flat faces. The plate
thickness t is measured normal to the middle surface. The fundamental equations of plate
theory relate the displacements and forces of the middle surface to the applied loading.
Plate equations will appear similar to beam equations, since beam theory can be regarded
as a special case of plate theory.
Plates can be classified as being thin or thick . Most of the material presented here applies
to thin plates. Roughly, for a plate to be considered as being thin, the ratio of the thickness
to the shortest span length should be less than about 1/10. The plates treated here are made
of materials that are homogeneous and isotropic. If material properties are the same at all
locations, the material is said to be homogeneous , while material properties identical in all
directions as viewed from a particular point are isotropic .

13.1 In-Plane Deformation (Stretching)

Although the primary subject of this chapter will be the bending of plates, we choose to
begin this chapter with a summary of the equations developed in Chapters 1 and 2 for the
in-plane deformation of a flat element (plate). This is the two-dimensional equivalent of
the extension of a bar, a one-dimensional problem. If the applied loads are tensile, the plate
is sometimes referred to as a membrane.

13.1.1 Cartesian Coordinate System

Kinematical Relationships
The two displacements for a thin element lying in the xy plane

u y ] T

u = [ u x (13.1a)

and three strains

xy ] T

= γ

[ (13.1b)

x y

757