2. Benchmark solution#
2.1. Calculation method#
The position of the nodes, integration points and integration sub-points is calculated from their coordinates in the local axes in the final position of the plate, and from the transition matrices between the local axes and the global axes
The rotation around the \(Z\) axis is based on the following matrix:
\(\mathrm{Tz}(\alpha )=\left[\begin{array}{}\mathrm{cos}(\alpha )\\ \mathrm{sin}(\alpha )\\ 0\end{array}\begin{array}{}-\mathrm{sin}(\alpha )\\ \mathrm{cos}(\alpha )\\ 0\end{array}\begin{array}{}0\\ 0\\ 1\end{array}\right]\)
The rotation around the \(X\text{'}\) axis is based on the following matrix:
\(\mathrm{Tx}\text{'}(\beta )=\left[\begin{array}{}1\\ 0\\ 0\end{array}\begin{array}{}0\\ \mathrm{cos}(\beta )\\ \mathrm{sin}(\beta )\end{array}\begin{array}{}0\\ -\mathrm{sin}(\beta )\\ \mathrm{cos}(\beta )\end{array}\right]\)
For any point with initial coordinates \((X,Y,Z)\) we can calculate its coordinates expressed in the global coordinate system \((X\text{'},Y\text{'},Z\text{'})\) after rotations with the following transformation:
\(\left[\begin{array}{}X\text{'}\\ Y\text{'}\\ Z\text{'}\end{array}\right]=\left[\mathrm{Tz}(\alpha )\right]\left[\mathrm{Tx}\text{'}(\beta )\right]\left[\begin{array}{}X\\ Y\\ Z\end{array}\right]\)
2.2. Reference quantities and results#
The position of the integration sub-points in the global coordinate system is calculated knowing its position expressed in the local axes.
Here we have: \(\mathrm{Tx}\text{'}(\beta )=\left[\begin{array}{}1\\ 0\\ 0\end{array}\begin{array}{}0\\ 0.5\\ 0.866\end{array}\begin{array}{}0\\ -0.866\\ 0.5\end{array}\right]\) and \(\mathrm{Tz}(\alpha )=\left[\begin{array}{}0.866\\ 0.5\\ 0\end{array}\begin{array}{}-0.5\\ 0.866\\ 0\end{array}\begin{array}{}0\\ 0\\ 1\end{array}\right]\)
For a plate element QUA4 of length \({L}_{X}=2.0m\) and width \({L}_{Y}=1.0m\), the positions in the plane of the integration points are, for models A and B (DKT and DST) which have 4 integration points (see R3.01.01):
Dot |
\(x\) |
|
1 |
0.42264973081037416 |
0.21132486540518708 |
2 |
1.5773502691896257 |
0.21132486540518708 |
3 |
1.5773502691896257 |
0.78867513459481287 |
4 |
0.42264973081037416 |
0.78867513459481287 |
And for the C modeling (COQUE_3D) which has 9 integration points (see R3.01.01):
Dot |
\(x\) |
|
1 |
0.22540333075851704 |
0.11270166537925852 |
2 |
1.774596669241483 |
0.11270166537925852 |
3 |
1.774596669241483 |
0.88729833462074148 |
4 |
0.22540333075851704 |
0.88729833462074148 |
5 |
1 |
0.11270166537925852 |
6 |
1.774596669241483 |
0.5 |
7 |
1 |
0.88729833462 074148 |
8 |
0.22540333075851704 |
0.5 |
9 |
1 |
0.5 |
Thickness \(\mathit{EP}\mathrm{=}0.5m\) is discretized into 4 layers, which makes 12 sub-points whose heights with respect to the mean plane (except in the case of modeling D GRILLE_EXCENTREE) are:
Sub-point |
\(z\) |
Sub-point |
\(z\) |
1 |
-0.250 |
7 |
0.000 |
2 |
-0.1875 |
8 |
0.0625 |
3 |
-0.125 |
9 |
0.125 |
4 |
-0.125 |
10 |
0.125 |
5 |
-0.0625 |
11 |
0.1875 |
6 |
0.000 |
12 |
0.250 |
For the case of modeling D (GRILLE_EXCENTRE), with an eccentricity of 0.05, the position of the points and the sub-point in the plane of the integration points is:
Point |
Sub-Point |
\(x\) |
|
|
1 |
1 |
0.42264973081037416 |
0.21132486540518708 |
0.05 |
2 |
1 |
1.5773502691896257 |
0.21132486540518708 |
0.05 |
3 |
1 |
1.5773502691896257 |
0.78867513459481287 |
0.05 |
4 |
1 |
0.42264973081037416 |
0.78867513459481287 |
0.05 |
2.3. Uncertainties about the solution#
None, exact solution.