2. Benchmark solution#
2.1. Calculation method#
The position of the nodes, integration points and integration sub-points is calculated from its coordinates in the local axes of the plate and the transition matrix between the local axes and the global axes.
\(T(\alpha )\mathrm{=}\left[\begin{array}{c}\mathrm{cos}(\alpha )\\ \mathrm{sin}(\alpha )\end{array}\begin{array}{c}\mathrm{-}\mathrm{sin}(\alpha )\\ \mathrm{cos}(\alpha )\end{array}\right]\)
For any point with initial coordinates \((X,Y)\) we can calculate its coordinates expressed in the global coordinate system \((X\text{'},Y\text{'})\) after rotation with the following transformation:
\(\left[\begin{array}{c}X\text{'}\\ Y\text{'}\end{array}\right]\mathrm{=}T(\alpha )\left[\begin{array}{c}X\\ Y\end{array}\right]\)
2.2. Reference quantities and results#
The positions of the integration sub-points in the global coordinate system are calculated knowing their positions expressed in the local axes.
Here we have: \(\mathrm{cos}(\alpha )=\frac{4}{5}\) and \(\mathrm{sin}(\alpha )=\frac{3}{5}\)
For a mesh SEG4 of pipe length \(L\mathrm{=}5m\), the distance of the integration points from the first node are (see R3.01.01):
Dot |
\(x\) (\(m\)) |
1 |
3.3499526089621403 |
2 |
1.6500473910378599 |
3 |
4.6528407789851318 |
4 |
0.34715922101486746 |
Thickness \(\mathit{EP}\mathrm{=}0.5m\) is discretized into 4 layers, which makes 12 sub-points whose heights with respect to the mean plane 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 |
2.3. Uncertainties about the solution#
None, exact solution.