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

Geometry
---------


.. image:: images/10000000000001D000000178AF9AEF317C64DD08.png
    :width: 2.9681in
    :height: 2.3701in


.. _RefImage_10000000000001D000000178AF9AEF317C64DD08.png:


.. csv-table::

    "Radius of curvature", ":math:`R\mathrm{=}0.3m`"
    "Profile height", ":math:`h\mathrm{=}0.015m`"
    "Profile width", ":math:`b\mathrm{=}0.002m`"
    "Section", ":math:`S\mathrm{=}\mathit{bh}`"
    "1st flexural inertia", ":math:`{I}_{X}\mathrm{=}{\mathit{bh}}^{3}\mathrm{/}12`"
    "2nd flexural inertia", ":math:`{I}_{Y}\mathrm{=}{\mathit{hb}}^{3}\mathrm{/}12`"
    "Torsional inertia", ":math:`J\mathrm{=}{\mathit{hb}}^{3}\mathrm{/}3`"



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

.. csv-table::

    "Young's module", ":math:`E\mathrm{=}7.E10N\mathrm{/}{m}^{2}`"
    "Poisson's Ratio", ":math:`\nu \mathrm{=}0.3`"
    "Sliding module", ":math:`G\mathrm{=}E\mathrm{/}2(1+\nu )`"



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

The beam is double-supported. The section is prevented from twisting at the :math:`A` and :math:`B` ends. To respect the hypotheses of the theoretical model taken as a reference, it is important that the moment is constant and that the normal force is zero along the beam. This is why we leave the :math:`u` movement free according to :math:`X` at point :math:`B`. The boundary conditions are:

• At point :math:`A`: :math:`u\mathrm{=}v\mathrm{=}w\mathrm{=}0`; :math:`{\Phi }_{Y}\mathrm{=}0`

• At point :math:`B`: :math:`v\mathrm{=}w\mathrm{=}0`; :math:`{\Phi }_{X}\mathrm{=}0`

The initial state of stress that allows the stability analysis to be carried out is obtained by imposing a moment of bending around the :math:`Z` axis, at points :math:`A` and :math:`B`: :math:`M\mathrm{=}1\mathit{Nm}`


Initial conditions
--------------------

Not applicable in static stability analysis.