Hi Lisa_247
From an applied theory standpoint this is actually pretty well defined. We should be cognizant that most engineering methods are semi-empirical and are intended to save us from the labor associated with solving equations of the theory of elasticity directly. Unlike the theory of elasticity, where equilibrium is imposed on an infinitesimal element's stresses and body forces, engineering plate & membrane theory imposes equilibrium on the stresses and applied forces.
In engineering theory of thin plates and membranes, stresses in the thickness (transverse) direction are generally ignored. The equilibrium equations for the stress resultants and applied forces are for 2 dimensions only. That is, thin plate equations are set up under the assumption of plane stress. Any transverse shearing deformations are neglected.
You can think if this the same way you might think of Euler-Bernoulli beam theory. This is an engineering method which assumed the neutral plane of the beam remains plane, and that sections normal to this plane remain normal during deflection. Obviously transverse shear Tau = VQ/Ib can be predicted in this method, but shear deformations of the cross section are neglected.
Situations where you might need to apply Timoshenko beam theory are: deep beams, beams of non-uniform cross section or built-up beams, etc.
This is analogous to plates in that there are also situations where a plate's geometry will inherently necessitate a treatment as a "thick" plate. Thin plate theory makes similar assumptions (normals to the surface remain normal during deflection).
Like Timoshenko theory, there is a more detailed method for plates that accounts for these issues and can be applied to thick plates. This is called Mindlin-Reissner plate theory. The plate equivalent of Euler-Bernoulli is called Kirchoff-Love theory
There are a few ways in which you could determine when treatment as a "thick" plate is appropriate.
For example, if you are using an approximate method for thin plates such as Rayleigh-Ritz and you are approximating loads as concentrated forces, the "real" dimensions of the area on which the force is applied should be less than the thickness of the plate. Or, maybe if you are predicting large deformations using thin plate theory.
Another good litmus test would be to check the plane strain transition thickness for the material in question. If the plate is thick enough to act in plane strain, you should probably use a thick plate method. I have seen a couple articles which generally state Kirchoff-Love is best for plates whose thickness < about 10% of the average side length. This would generally be more restrictive than simply checking the plane strain transition thickness.
So there is no hard and fast rule of thumb, but the basis of the answer is whether or not it is acceptable to ignore shear deformations through the thickness of the plate.
A caveat: the above is specific to bending and extension of plates. "Thick" vs "thin" with reference to instability would be different
Keep em' Flying
//Fight Corrosion!
