Reference problem
=====================


Geometry
---------

Cylindrical tank of medium radius :math:`R=5.7m`, :math:`e=0.04m` thick and :math:`L=16m` high,, and in height, simply supported at its base (rotation is free), and subjected to internal hydrostatic pressure.


.. image:: images/Shape1.gif


.. _RefSchema_Shape1.gif:


.. csv-table::

    "Average radius:", ":math:`R=5.7m`"
    "Thickness:", ":math:`e=0.04m`"
    "Height:", ":math:`L=16m`"



Material properties
-----------------------

The properties of the materials constituting the plate are:


+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------+
|*Material 1*: isotropic elastic:                                                                                                                                                                          |Young's modulus                                                                                                                                                                                                                                                                                                                                                                        |:math:`E=2.1{10}^{11}\mathrm{Pa}`|
+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------+
|                                                                                                                                                                                                          |                                                                                                                                                                                                                                                                                                                                                                                       |                                 |
+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------+
|                                                                                                                                                                                                          |Poisson's ratio                                                                                                                                                                                                                                                                                                                                                                        |:math:`\nu =0.3`                 |
+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------+
|                                                                                                                                                                                                          |                                                                                                                                                                                                                                                                                                                                                                                       |                                 |
+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------+
|*Material 2*: orthotropic elastic: In order to get rid of the dependence of the notations on the orthotropy coordinate system, the material characteristics are given in the cylindrical coordinate system|:math:`\begin{array}{c}{E}_{r}\mathrm{=}2.1E11\mathit{Pa}\\ {E}_{\theta }\mathrm{=}2.1E11\mathit{Pa}\\ {E}_{z}\mathrm{=}4E11\mathit{Pa}\\ {\nu }_{r\theta }\mathrm{=}0.075\\ {\nu }_{rz}\mathrm{=}0.075\\ {\nu }_{\theta z}\mathrm{=}0.075\\ {G}_{r\theta }\mathrm{=}0.35E10\mathit{Pa}\\ {G}_{rz}\mathrm{=}0.45E10\mathit{Pa}\\ {G}_{\theta z}\mathrm{=}0.45E10\mathit{Pa}\end{array}`                                  |
+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------+


Boundary conditions and loads
-------------------------------------

Base :math:`z=0` simply pressed

Internal pressure varying linearly as follows :math:`z`: :math:`p(z)=\mathrm{P0.}(L-z)/L`

with :math:`\mathrm{P0}=15000\mathrm{Pa}`.


Initial conditions
--------------------

Not applicable.