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


Geometry
---------



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+ .. image:: images/1000000000000523000002386D0FC4CCF9683083.png                                                                                             + .. image:: images/1000000000000182000002368E0686A7F2FBD1E2.png                                                                                             +
|    :width: 3.9882in                                                                                                                                        |    :width: 1.2319in                                                                                                                                        |
+    :height: 1.6693in                                                                                                                                       +    :height: 1.7799in                                                                                                                                       +
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+ Figure 1.1-1: Problem geometry                                                                                                                             + Figure 1.1-2: Problem geometry                                                                                                                             +
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Height: :math:`h=0.2\text{m}`

Width: :math:`b=0.1\text{m}`

Length: :math:`L=2\text{m}`

Section: :math:`A=b\times h=0.02\text{m}`

Inertia: :math:`I=\frac{b\times {h}^{3}}{12}=1.66\times {10}^{-5}\text{m}`

Section reduction coefficient :math:`k\text{'}=\frac{5}{6}`


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


.. csv-table::

    "Young's module", ":math:`E=2.1\times {10}^{11}\mathit{Pa}`"
    "Poisson's Ratio", ":math:`\nu =0.3`"
    "Density", ":math:`\rho =7800.0{\mathit{kg.m}}^{-3}`"
    "Sliding module", ":math:`G=\frac{E}{2(1+\nu )}=8.076\times {10}^{10}\text{Pa}`"



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

Only the flexure in the :math:`\mathit{XY}` plane and the extension along the :math:`X` axis are allowed. Since the model is solid, the boundary conditions differ somewhat from those that would be imposed on a beam model.

Imposed displacement:

.. csv-table::

    "In :math:`X=0`, :math:`Y=h/2` "," :math:`\mathit{DX}=0`, :math:`\mathit{DY}=0`"
    "In :math:`X=L`, :math:`Y=h/2` "," :math:`\mathit{DY}=0`"
    "In :math:`Z=b/2` "," :math:`\mathit{DZ}=0`"


To the previous conditions, we add the flatness constraint of the sections in :math:`X=0` and :math:`X=L`. This constraint can be expressed as follows. Let's designate :math:`{x}^{T}=(X,Y,Z)` the coordinate vector and :math:`{u}^{T}=(\mathit{DX},\mathit{DY},\mathit{DZ})` the displacement vector; the position of a point is identified by the vector :math:`x{\text{'}}^{T}={x}^{T}+{u}^{T}=(X\text{'},Y\text{'},Z\text{'})`. Let :math:`A`, :math:`B`, and :math:`C` be three non-aligned points in the section. Any point :math:`P` is subject to the condition:

:math:`∣\begin{array}{cccc}X{\text{'}}_{P}& Y{\text{'}}_{P}& Z{\text{'}}_{P}& 1\\ X{\text{'}}_{A}& Y{\text{'}}_{A}& Z{\text{'}}_{A}& 1\\ X{\text{'}}_{B}& Y{\text{'}}_{B}& Z{\text{'}}_{B}& 1\\ X{\text{'}}_{C}& Y{\text{'}}_{C}& Z{\text{'}}_{C}& 1\end{array}∣=0`