# Calculation method

#### Finite Element Method

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

 Element Background Convergence of solution Convex plain 4-nodal 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’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. 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 Thin Very thin t/lx , t/ly 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 non-linear, 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:

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: Reissner-Mindlin theory