Calculation method

Finite Element Method

2D-Plate is calculated using Finite Element Method with the basic presumptions as following:



Convergence of solution

Convex plain 4-nodal thin plate/shell element used for static and free vibrations calculation.


Convex plain 4-nodal thin plate/shell element


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’r-Y’r (Bending). Integration in the element – numerical.


Calculation of results as σx, σy, σxy (Stress-Strain), Mx, My, Mxy, Vx, Vy (Bending) is represented in the center of gravity of an element.  

Convergence of solution


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 thickness-to-length ratio several theories of plates have been developed:


Moderately thick


Very thin

t/lx , t/ly

1/5  to  1/10

1/5  to  1/50

< 1/50


With transverse shear deformation

Without transverse shear deformation, mostly used for practical applications

Geometrically non-linear, with membrane deformation


Reissner, Mindlin


von Karman

Related beam theory


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:

  1. geometrically linear (small strains, small deflections)
  2. linear material (linear elastic (Hooke), in the most simple case homogeneous and isotrop)
  3. thin plate


Bernoulli a+b: Kirchhoff theory  Bernoulli a+b: Kirchhoff theory

Bernoulli a+b: Kirchhoff theory


Bernoulli a: Reissner-Mindlin theory  Bernoulli a: Reissner-Mindlin theory

Bernoulli a: Reissner-Mindlin theory