Classic Plate Equation: Introduction
Classical Plate Equation 
The small transverse (outofplane) displacement w of a thin plate is governed by the Classical Plate Equation,
where p is the distributed load (force per unit area) acting in the same direction as z (and w), and D is the bending/flexual rigidity of the plate defined as follows,
in which E is the Young's modulus, is the Poisson's ratio of the plate material, and t is the thickness of the plate. Furthermore, the differential operator is called the Laplacian differential operator ,
If the bending rigidity D is constant throughout the plate, the plate equation can be simplified to,
where is called the bihamonic differential operator.
A plate is called thin when its thickness t is at least one order of magnitude smaller than the span or diameter of the plate.

Origin of the Plate Equation 
The classical plate equation arises from a combination of four distinct subsets of plate theory: the kinematic, constitutive, force resultant, and equilibrium equations. The outcome of each of these segments is summarized here: 
Kinematics:  where w_{0} is the displacement of the middle plane in z direction.  
Constitutive:  
Resultants:  
Equilibrium:  
To relate the plate's outofplane displacement w_{0} to its pressure loading p_{z}, we combine the results of the four plate subcategories in this order: 
Kinematics  >  Constitutive  >  Resultants  >  Equilibrium  =  Plate Equation 
The plate is assumed to be constructed by isotropic material and subjected to transverse loading. Also, the Cartesian coordinate system is used. We'll demonstrate this hierarchy by working backwards. We first combine the 3 equilibrium equations to eliminate Q_{xz} and Q_{yz},
Next, replace the moment resultants with its definition in terms of the direct stress,
Note that uniform thickness is assumed. Use the constitutive relation to eliminate stress in favor of the strain,
and then use kinematics to replace strain in favor of the normal displacement w_{0},
The equation of equilibrium can then be expressed in terms of the normal displacement w_{0}
which yields
Note that homogeneous material across the plate (x and y directions) is assumed. As a final step, assuming homogeneous material along the thickness of the plate, the bending stiffness of the plate can be written as
We then arrive at the Classical Plate equation,
or a slimmer form
where w_{0} is replaced by w and p_{z} replaced by p to be consistent with the notations in most published literatures. 
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