Finite Element Method
2DPlate is calculated using Finite Element Method with the basic presumptions as following:
Element 
Background 
Convergence of solution 
Convex plain 4nodal thin plate/shell element used for static and free vibrations calculation.

Convex plain modified element designed by Fraeijs de Veubeke which correspond theory Kirchhoff (thin plates) is used. Element represents five nodal displacements: X’, Y’, Z’, X’r, Y’r in the local element space and, after global transformation, six nodal displacements: X, Y, Z, Xr, Yr, Zr in the global coordinate space.
Element stiffness matrix is build separately for X’Y’ (Strain – Stress) and Z’X’rY’r (Bending). Integration in the element – numerical.
Calculation of results as σx, σy, σxy (StressStrain), Mx, My, Mxy, Vx, Vy (Bending) is represented in the center of gravity of an element. 
u  displacements; σ  stresses/forces; u  square root from square of the integral of u in domain of an element;
Means, by refine of meshing net for two times possible error for displacements is decreasing four times, error for stresses (forces)  two times. 
Theories of plates
Depending on the thicknesstolength ratio several theories of plates have been developed:

Moderately thick 
Thin 
Very thin 
t/l_{x} , t/l_{y} 
1/5 to 1/10 
1/5 to 1/50 
< 1/50 
Description 
With transverse shear deformation 
Without transverse shear deformation, mostly used for practical applications 
Geometrically nonlinear, with membrane deformation 
Theory 
Reissner, Mindlin 
Kirchhoff 
von Karman 
Related beam theory 
Timoshenko 
Euler, Bernoulli 
Theory of second order 
Most of the practical applications deal with thin plates. Within the valid range of linear behavior a pure bending theory will be good enough and shear deformation can be neglected (Kirchhoff theory).
Assumptions of the Kirchhoff plate theory:
 geometrically linear (small strains, small deflections)
 linear material (linear elastic (Hooke), in the most simple case homogeneous and isotrop)
 thin plate
Bernoulli a+b: Kirchhoff theory
Bernoulli a: ReissnerMindlin theory