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 Qxz and Qyz,
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 w0,
The equation of equilibrium can then be expressed in terms of the normal displacement w0
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 w0 is replaced by w and pz replaced by p to be consistent with the notations in most published literatures.