**Finite Element Analysis
of a Square Plate**

__Exact
results of a square plate__

__Generation
of the structure model__

__Input
of the boundary conditions__

__Exact
results of a square plate__

__1.) Simply supported square plate__

The following results should be calculated for a simply supported plate at the boundaries:

the maximum deflection in Z-direction, the maximum moments Mxx
and Myy, the polar moment Mxy all located

in the plate center and the sum of the reaction forces.

We consider the cases:

- poisson ratio = 0
- steel concrete with a poisson ratio of 0.15
- steel with a poisson ratio of 0.3

The following results come from the tables (1), which are not exact, but are good approximations:

__2.) Exact Results of a square plate
clamped on 3 edges and simply
supported on one edge__

All quantities are the same as in the previous case, but the
plate is clamped on 3 sides, the 4^{th} side has rotation
dof about the x- and y-axes. Instead of the moment Mxx the
reaction moments Mxs and Mys at the fixed edges are requested.

__The following tables have been used to get the approximate
results:__

(1)Berechnungstafeln für Platten und Wandscheiben, Richard Bares, Bauverlag GmbH, Wiesbaden 1969

__Generation
of the structure model__

- In the main icon bar choose

to call the CAD icon bar and to create a line model

- In the icon bar with the icon enter "1" for the element group

- Select to enter the data of rectangle via keyboard:

1. end point: X=5, Y=0, Z=0

2. end point: X=5, Y=6, Z=0

3. end point: X=0, Y=6, Z=0

4. end point: X=0, Y=0, Z=0

- Save the line model under
**Square_Plate_Line Model.LIN**in the directory \MEANS\STRUKTUR

- Using the icon two meshes should be generatet with the
element PLA8S and two mesh densities

of 5 and 15 which result in two models with 25 and 224 PLA8S elements resp.

Complete the dialog box with the menu **Editor** and **Material
Datas** in NETGEN as shown below:

The material data are:

Thickness H1, H2, H3, H4 = 0.1 m

Young's modulus = 20000000 kN/m^{2}

Poisson ratio = 0.3/0.15/0.0

__Input
of the boundary conditions__

__1.) Simply supported square plate__

The square plate is simply supported, i.e. the degrees of freedom at the plated edges are suppressed in the Z-direction.

- With the icon call the icon bar for the boundary conditions

- In the icon bar for the boundary conditions select for boundary conditions in the Z-direction.

__Marking with a polygon__

The outer boundary is marked by a polygon consisting of 10 points as shown below:

__2.) Clamped square plate with one
edge simply supported__

The square plate is clamped on 3 edges and simply supported on
one edge. This means that the translation degree

of freedom in z-direction is suppressed at all edges and the
rotation degrees of freedom about the x- and y-axes are

suppressed on 3 edges.

Use the x icons
and the y icon
and the buttons **Create** and **Marking with a polygon**.

The distributed load is generated as follows:

- With call the icon bar for the loads

- In the menu "2D distributed loads" a dialog box opens, where the magnitude for the distributed load "-6" and the load case number "1" must be entered.

- The distribution region of the load can be outlined:

- Input via keyboard
- Dragging with curser a rectangle
- Surface Model

Both structure files are now complete and are saved as **Square_Plate_PLA8S_simply_supported.fem
**and

**Square_Plate_PLA8S_clamped.fem**

As a prerequisite of the analysis select the result quantities
in the menu **FEM-Analysis** and and **Statics**. Otherwise

only the default quantities are calculated, the displacements and
averaged nodal stresses.

The analysis is started from the menu **FEM-Analysis** and **FEM-Analysis**.

The analysis can be performed in 2 steps.

The displacement in Z-direction and the distribution of the
components of the moment Mxx and Myy

result with the icon .

Comparison of the results with different element types and mesh densities:

__1.) Simply supported square plate__

**2.) **__Clamped square
plate with one edge simply supported__