Square Plate Subjected to Force at Center

Let us consider a rigidly supported plate subjected to a force applied at the center.
Only a quarter of the plate is analyzed due to symmetry conditions on the corresponding edges (restrained displacements in the direction of axis of the local coordinate system normal to the face plane).

The finite element model with applied loads and restraints

Analytical solution for the deflection at the point under the force is given by formula:
.
Let us use the following data: the length and width of the plate a = 500 mm, plate thickness is h = 3 mm, applied concentrated force is P = 5 kgf (or P = 49.03325 N ).
Elastic properties of the material are taken as: E = 2.1E+011 Pa, ν = 0.28.
Analytical solution can be expressed as: w = 5.3557E-004 m.
After carrying out calculations with the help of AutoFEM, the following results are obtained:

Table 1.Parameters of finite element mesh

Finite Element Type	Number of Nodes	Number of Finite Elements
quadratic tetrahedron	3698	10584

Table 2.Result "Displacement, magnitude"*

Numerical Solution Displacement w*, m	Analytical Solution Displacement w, m	Error δ = 100% * \|w* - w\| / \|w\|
5.3593E-004	5.3557E-004	0.07

Square Plate Subjected to Force at Center, Result "Displacement, magnitude" of finite element analysis

Conclusions:

The relative error of the numerical solution compared to the analytical solution is equal to 0.07% for quadratic finite elements.

*The results of numerical tests depend on the finite element mesh and may differ slightly from those given in the table.