# Classical Plate Equation Notes

# 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.