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


Geometry
---------

Lee's gantry is a 2-armed gantry, articulated at its ends:


.. image:: images/10000000000002000000011EE65F9E57AFD56532.png
    :width: 3.9689in
    :height: 2.252in


.. _RefImage_10000000000002000000011EE65F9E57AFD56532.png:

Coordinates of the points (in :math:`\mathrm{cm}`):


.. csv-table::

    "", ":math:`A` "," :math:`B` "," :math:`C` "," :math:`X`"
    ":math:`x` ", "0", "0", "0", "120", "24"
    ":math:`y` ", "0", "120", "120", "120"



Characteristics of the section
------------------------------

The arms have a rectangular cross section of :math:`3\mathrm{cm}` by :math:`2\mathrm{cm}`. The section is oriented in such a way that the weakest inertia axis is perpendicular to the plane of the gantry.


.. csv-table::

    ":math:`A=6{\mathrm{cm}}^{2}`"
    ":math:`{I}_{y}=2.0{\mathrm{cm}}^{4}`"
    ":math:`{I}_{z}=4.5{\mathrm{cm}}^{4}`"
    ":math:`{A}_{y}=1.2`"
    ":math:`{A}_{z}=1.2`"



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


.. csv-table::

    ":math:`E=720N/{\mathrm{cm}}^{2}`"
    ":math:`\nu =0.3`"
    ":math:`{E}_{t}=72N/{\mathrm{cm}}^{2}`"
    ":math:`{\sigma }_{e}=10.44N/{\mathrm{cm}}^{2}`"



Boundary conditions and loading
------------------------------------


.. csv-table::

    "1)", "Boundary conditions are imposed at points :math:`A` and :math:`C` (articulated around :math:`\mathrm{Oz}`):
    :math:`\mathrm{DX}=\mathrm{DY}=\mathrm{DZ}=\mathrm{DRX}=\mathrm{DRY}=0`"
    "2)", "We impose a vertical loading :math:`F` on point :math:`X`:
    :math:`\mathrm{FY}=C\times t`
    The constant :math:`C` corresponds to the steering coefficient determined for example by an arc-length method. Indeed, this structure has a complex response under non-monotonic loading."