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


Geometry
---------

The configuration studied consists of two coaxial cylindrical shells :math:`4m` in height.


.. image:: images/10001BB4000069D500003B6EB99021CCE3B62D62.svg
    :width: 411
    :height: 231


.. _RefImage_10001BB4000069D500003B6EB99021CCE3B62D62.svg:

The inner shell has an average radius :math:`{R}_{1}` of :math:`\mathrm{0,98}m` and a thickness :math:`{e}_{1}` of :math:`4\mathrm{cm}`.

The outer shell has an average radius :math:`{R}_{2}` of :math:`\mathrm{1,105}m` and a thickness :math:`{e}_{2}` of :math:`1\mathrm{cm}`.

**Note:**


+--------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------+
|*The thickness and the average radius of the annular space between the two shells are given*                                                                                                                                                                            |
+--------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------+
|*by*                                                                                        |:math:`H\mathrm{=}{R}_{\text{ext}}\mathrm{-}{R}_{\text{int}}\mathrm{=}\mathrm{0,1}m`|:math:`R\mathrm{=}\frac{{R}_{\text{int}}+{R}_{\text{ext}}}{2}\mathrm{=}\mathrm{1,05}m`|
+--------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------+
|*with*                                                                                      |:math:`{R}_{\text{int}}\mathrm{=}R1+\frac{{e}_{1}}{2}\mathrm{=}1m`                  |:math:`{R}_{\text{ext}}\mathrm{=}{R}_{2}–\frac{{e}_{2}}{2}\mathrm{=}\mathrm{1,1}m`    |
+--------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------+


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

The material that makes up the two shells is steel. Its physical characteristics are


.. csv-table::

    ":math:`\rho \mathrm{=}7800\mathit{kg}\mathrm{/}{m}^{3}` "," :math:`E\mathrm{=}2.{10}^{11}\mathit{Pa}` "," :math:`\nu \mathrm{=}\mathrm{0,3}`"



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

The outer shell is supposed to be rigid: all the knots are embedded.

Regarding the internal shell, the support conditions are as follows: embedded end in the lower part (:math:`z\mathrm{=}0`) and free in the upper part (:math:`z\mathrm{=}\mathrm{4m}`).