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


Geometry
---------

Beam in three-point bending, defined by:


.. image:: images/Object_1.svg
    :width: 222
    :height: 113


.. _RefImage_Object_1.svg:

With a duplicate section :math:`T`:


.. image:: images/Object_2.svg
    :width: 222
    :height: 113


.. _RefImage_Object_2.svg:

In this diagram, :math:`O` is located halfway up the section.

The total cross section of the upper steels is :math:`{3.10}^{\mathrm{-}4}{m}^{2}` and that of the lower steels is :math:`{4.10}^{\mathrm{-}4}{m}^{2}`.


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


* concrete: :math:`E\mathrm{=}2.{10}^{10}\mathit{Pa}`; :math:`\nu \mathrm{=}0.2`; :math:`\rho \mathrm{=}2400{\mathit{kg.m}}^{\mathrm{-}3}`; :math:`\alpha \mathrm{=}10\mathrm{-}5{K}^{\mathrm{-}1}`


* steel: :math:`E\mathrm{=}\mathrm{2,1}\mathrm{.}{10}^{11}\mathit{Pa}`; :math:`\nu \mathrm{=}0.33`; :math:`\rho \mathrm{=}7800{\mathit{kg.m}}^{\mathrm{-}3}`; :math:`\alpha \mathrm{=}10\mathrm{-}5{K}^{\mathrm{-}1}`


Boundary conditions
----------------------

Simple press in :math:`B`: :math:`\mathit{dy}\mathrm{=}0`

Press "double" in :math:`A`: :math:`\mathit{dx}\mathrm{=}\mathit{dy}\mathrm{=}\mathit{dz}\mathrm{=}0` as well as :math:`\mathit{rx}\mathrm{=}\mathit{ry}\mathrm{=}0`.


Loads
-----------

Three load cases are tested in succession:


* Load 1: concentrated effort in the middle of the beam, :math:`F\mathrm{=}10000N`


* Loading 2: self-weight of the beam, :math:`g\mathrm{=}\mathrm{9,8}{\mathit{m.s}}^{\mathrm{-}2}`


* Load 3: homogeneous heating of the :math:`\Delta T\mathrm{=}100K` beam