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.