Classical Plate Equation Notes
The small transverse (out-of-plane) 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 bi-harmonic differential operator.
This small deflection theory assumes that w is small in comparison to the thickness of the plate t, and the strains and the mid-plane slopes are much smaller than 1.
- 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.