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


Geometry
---------


.. image:: images/1000000000000622000002BC70A17BFD81DECE83.png
    :width: 4.7681in
    :height: 2.122in


.. _RefImage_1000000000000622000002BC70A17BFD81DECE83.png:

Figure 1.1-1: Problem geometry

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}`"
    "Shear modulus", ":math:`G=\frac{E}{2(1+\nu )}=8.076\times {10}^{10}\text{Pa}`"
    "Rayleigh damping coefficients", ":math:`\alpha =1.6\times {10}^{-5}\text{s}` and :math:`\beta =16{\text{s}}^{-1}`"



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{DX}=0`, :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`

Imposed force:

In :math:`Y=h`, for all :math:`X`, the load is defined by:

:math:`p(x,t)=p(x)\mathrm{sin}(\omega t)`

with :math:`p(x)={p}_{0}=5.0\times {10}^{4}{\text{N.m}}^{-1}` and :math:`\omega =2000\pi {\text{rd.s}}^{-1}`.