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


Geometry
---------

We study the buckling of a beam of length :math:`L\mathrm{=}1000\mathit{mm}` and height :math:`h\mathrm{=}100\mathit{mm}`.


.. image:: images/Shape1.gif


.. _RefSchema_Shape1.gif:


.. image:: images/100011CA000069D500004697973F4A11015B160D.svg
    :width: 288
    :height: 192


.. _RefImage_100011CA000069D500004697973F4A11015B160D.svg:


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

The material is assumed to be elastic. The material characteristics are as follows:

Young's module :math:`E\mathrm{=}20000\mathit{MPa}`

Poisson's ratio :math:`\nu \mathrm{=}0.3`

Density :math:`\rho \mathrm{=}{10}^{\mathrm{-}6}{\mathit{kg.mm}}^{\mathrm{-}3}`

Large rotations are taken into account (DEFORMATION =' GREEN ').


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

The left side (:math:`\mathit{OC}`) is recessed (:math:`\mathit{DX}\mathrm{=}\mathit{DY}\mathrm{=}0`).

Moreover, gravity applies (:math:`{g}_{y}\mathrm{=}9810{\mathit{mm.s}}^{\mathrm{-}2}`) and a compression force is imposed in the direction :math:`\mathrm{-}x` on the right side :math:`\mathit{AB}`. The control is applied by controlling the movement of node :math:`A`:


* a maximum displacement of :math:`–1\mathit{mm}` according to :math:`y` between each increment when applying gravity


* a move of :math:`50\mathit{mm}` following :math:`x` and :math:`y` between each increment when compression is imposed.