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


Geometry
---------


.. image:: images/100004240000181500000C81B3C9BDB981A2B23F.svg
    :width: 310
    :height: 161


.. _RefImage_100004240000181500000C81B3C9BDB981A2B23F.svg:

Or a plate whose characteristics are as follows:


.. csv-table::

    "length: :math:`a=1.5m` ", "width: :math:`b=1m` ", "thickness: :math:`t=0.01m`"


The characteristic points of the plate have the following coordinates:


.. csv-table::

    "", ":math:`A` "," :math:`B` "," :math:`C` "," :math:`D`"
    ":math:`x` ", "0. ", "0. ", "1. ", "1."
    ":math:`y` ", "0. ", "1.5", "1.5", "1.5", "0."
    ":math:`z` ", "0. ", "0. ", "0. ", "0."



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

The parameters characterizing the properties of the material are:


.. csv-table::

    ":math:`E=2.1{10}^{\mathrm{11 }}\mathrm{Pa}` "," :math:`\nu =0.3` "," :math:`\rho =7800\mathrm{kg}/{m}^{3}`"



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


Flexion problem
~~~~~~~~~~~~~~~~~~~~~

The plate is easily supported on all sides: for any point :math:`P` on the edge we have: :math:`w=0`.


Membrane problem
~~~~~~~~~~~~~~~~~~~~~~

For all points on the plate, we block the movement in :math:`z` and the three degrees of rotation, that is to say:

:math:`w=0.` :math:`{\theta }_{x}={\theta }_{y}={\theta }_{z}=0.`

On the sides :math:`\mathrm{AD}` and :math:`\mathrm{BC}` we block the movement in :math:`y`: for :math:`y=0.` or :math:`y=a` we have :math:`v=0`.

At points :math:`A,B,C,D`, springs with a stiffness of :math:`k` are attached. The axis of these springs is the :math:`x` direction.


.. image:: images/10000712000018CE00000B0F086A97A8D3B0E151.svg
    :width: 310
    :height: 161


.. _RefImage_10000712000018CE00000B0F086A97A8D3B0E151.svg:

The numeric value for :math:`k` is :math:`k=25N/m`.