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


Geometry
---------

Lengths are expressed in meters.


.. image:: images/Object_1.svg
    :width: 495
    :height: 374


.. _RefImage_Object_1.svg:

The cylindrical panel is embedded along its 4 sides and subjected to surface force.


* given in the global frame of reference for modeling A:

:math:`\mathrm{f}\mathrm{=}\mathrm{-}{f}_{z}{e}_{z};{f}_{z}>0`


* given in the local coordinate system for modeling B:

:math:`\mathrm{f}\mathrm{=}\mathrm{-}{f}_{z}n;{f}_{z}>0`

which leads to membrane compression of the panel, accompanied by localized flexions. In the case of B modeling, :math:`n` is normal to the updated geometry of the panel.

Because of the geometric and physical symmetry of the problem, only quarter :math:`{P}_{1}{P}_{2}{P}_{3}{P}_{4}` of the panel is modeled, taking into account symmetry conditions.


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

Elastic behavior:

:math:`E\mathrm{=}450000\mathit{MPa};\nu \mathrm{=}0.3`


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


.. csv-table::

    "Boundary conditions", ":math:`\mathrm{P2P3}`: :math:`\mathrm{DX}=0.` "," :math:`\mathrm{DY}=0.` "," "," :math:`\mathrm{DZ}=0.` "," :math:`\mathrm{DRX}=0` "," :math:`\mathrm{DRY}=0` "," :math:`\mathrm{DRZ}=0`"
    "", ":math:`\mathrm{P3P4}`: :math:`\mathrm{DX}=0.` "," :math:`\mathrm{DY}=0.` "," :math:`\mathrm{DZ}=0.` "," "," :math:`\mathrm{DRX}=0` "," :math:`\mathrm{DRY}=0` "," :math:`\mathrm{DRZ}=0`"
    "", "", "", "", "", "", ""
    "Symmetry:", ":math:`\mathrm{P1P2}`:", ":math:`\mathrm{DY}=0.` "," ", "", ":math:`\mathrm{DRX}=0.` ", "", ":math:`\mathrm{DRZ}=0.`"
    "", ":math:`\mathrm{P4P1}`: :math:`\mathrm{DX}=0.` ", "", "", "", "", ":math:`\mathrm{DRY}=0.` "," :math:`\mathrm{DRZ}=0.`"


We look for the successive states of equilibrium under the load consisting of the surface force given in the global coordinate system:

:math:`{f}_{z}(t)\mathrm{=}t`

:math:`t` being the nickname.

We are interested in the only vertical component of displacement in :math:`{P}_{1}`.