Benchmark solution
=====================

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.

:math:`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 :math:`(X,Y)` we can calculate its coordinates expressed in the global coordinate system :math:`(X\text{'},Y\text{'})` after rotation with the following transformation:

:math:`\left[\begin{array}{c}X\text{'}\\ Y\text{'}\end{array}\right]\mathrm{=}T(\alpha )\left[\begin{array}{c}X\\ Y\end{array}\right]`


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: :math:`\mathrm{cos}(\alpha )=\frac{4}{5}` and :math:`\mathrm{sin}(\alpha )=\frac{3}{5}`

For a mesh SEG4 of pipe length :math:`L\mathrm{=}5m`, the distance of the integration points from the first node are (see R3.01.01):

.. csv-table::

    "Dot", ":math:`x` (:math:`m`)"
    "1", "3.3499526089621403"
    "2", "1.6500473910378599"
    "3", "4.6528407789851318"
    "4", "0.34715922101486746"


Thickness :math:`\mathit{EP}\mathrm{=}0.5m` is discretized into 4 layers, which makes 12 sub-points whose heights with respect to the mean plane are:

.. csv-table::

    "Sub-point", ":math:`z` ", "Sub-point", ":math:`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"



Uncertainties about the solution
----------------------------

None, exact solution.