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

Geometry
---------

A system of 3 bars in :math:`U`, drawn here in a local :math:`(x,y)` coordinate system:


.. image:: images/10000000000001D30000011DAED2F2AF82E5748E.png
    :width: 3.8807in
    :height: 2.3673in


.. _RefImage_10000000000001D30000011DAED2F2AF82E5748E.png:

The area of the cross sections is :math:`A\mathrm{=}{\mathrm{1m}}^{2}`. The length of each of the 3 bars is :math:`L\mathrm{=}\mathrm{10m}`.

The coordinate system in which is drawn here is rotated by :math:`60°` in relation to the laboratory coordinate system :math:`(X,Y)`:


.. image:: images/1000020000000164000000D7903DA24EA29E41FA.png
    :width: 3.7134in
    :height: 2.2366in


.. _RefImage_1000020000000164000000D7903DA24EA29E41FA.png:


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

:math:`E\mathrm{=}2.{10}^{11}\mathit{Pa}` for the 3 bars.

:math:`\rho \mathrm{=}8000\mathit{kg}\mathrm{/}{m}^{3}` only for the :math:`\mathit{CD}` bar. For the other 2 bars, :math:`\rho \mathrm{=}0`.


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

Embedding in :math:`A` and :math:`B`.

To avoid rigid body movements, :math:`\mathit{DZ}\mathrm{=}0` for all knots, and :math:`\mathit{DX}\mathrm{=}0` in :math:`C` and :math:`D`.

A single load is applied: gravity. Gravity is obviously linked to the laboratory's coordinate system, so in the :math:`\mathrm{-}Y` direction; we take a virtual acceleration of :math:`g\mathrm{=}\mathrm{20m}\mathrm{/}{s}^{2}`. In the frame of reference of the structure, gravity is therefore expressed :math:`(\mathrm{sin}(60)g,\mathrm{-}\mathrm{cos}(60)g)\mathrm{=}(0.866\mathrm{\times }g,\mathrm{-}0.5\mathrm{\times }g\mathrm{,0})`, which is equivalent to :math:`g\mathrm{=}\mathrm{10m}\mathrm{/}{s}^{2}`, in the direction :math:`–y`.