ABSTRACT
The objective of this chapter is to develop the governing equations of the classical theory of plates. A plate is a structural element with planform dimensions that are large compared to its thickness and is subjected to loads that cause bending deformation in addition to stretching. In most cases, the thickness is no greater than one-tenth of the smallest in-plane dimension. Because of the smallness of thickness dimension, it is often not necessary to model them using 3D elasticity equations. Simple 2D plate theories can be developed to study the deformation and stresses in plate structures. Plate theories are developed using the semi-inverse method wherein an educated
guess is made as to the form of the displacement field or stress field, leaving enough freedom in the assumed field to satisfy the equations of elasticity. In the case of beams, plates, and shells, the displacement field is in terms of unknown functions ϕji of the surface coordinates (x, y) and time t:
ui(x, y, z, t) = NX j=0
(z)jϕji (x, y, t) (3.1.1)
and equations governing ϕji are determined such that the principle of virtual displacements is satisfied. The specific form of Eq. (3.1.1) depends on the kinematics of deformation to be included. For example, the Euler-Bernoulli beam theory is based on the hypothesis that the beam deforms such that a straight line normal to the beam axis before bending remains (1) straight, (2) inextensible, and (3) normal to the beam axis after deformation. This three-part hypothesis provides a clue to the form of the displacement field of the Euler-Bernoulli beam theory
u1(x, y, z, t) = u0(x, t)− z ∂w0 ∂x
, u2 = 0, u3(x, y, z, t) = w0(x, t)
where u0 and w0 are functions to be determined such that the equations of elasticity or its equivalent, namely, the principle of virtual displacements is satisfied, as shown in Examples 2.1.1 and 2.2.1. The extension of the Euler-Bernoulli beam theory to plates is known as the
Kirchhoff plate theory or classical plate theory (CPT). The present chapter is dedicated to the development of the classical plate theory.
