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


Geometry
---------


.. image:: images/1000020000000355000000A93F4BAE99F84FCD49.png
    :width: 5.861in
    :height: 1.389in


.. _RefImage_1000020000000355000000A93F4BAE99F84FCD49.png:

Beam length: :math:`L=1m`


.. csv-table::

    "Section:", "Inner radius :math:`=0.09m`"
    "", "Outer radius :math:`=0.10m`"



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


.. csv-table::

    ":math:`E={10}^{10}\mathrm{Pa}` "," :math:`\nu =0.3` "," :math:`\rho ={1.10}^{4}\mathrm{kg}/{m}^{3}`"


Modeling A: no amortization

Modeling B: Proportional Damping (RAYLEIGH):

:math:`C=\alpha K+\beta M` with :math:`\alpha =6.5{10}^{-6}s` and :math:`\beta =16.0{s}^{-1}`.

These values correspond to a depreciation reduced by 1% in the first mode of the structure.


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

On the whole structure we impose :math:`\mathrm{DY}=\mathrm{DZ}=\mathrm{DRX}=\mathrm{DRY}=\mathrm{DRZ}=0`.

In point :math:`A` we impose the embedding condition :math:`\mathrm{DX}=0`.

In :math:`\mathrm{N10}` we apply a constant force starting from the instant :math:`t=0`: :math:`{F}_{x}=–100N`.


.. image:: images/100002000000032E0000014844B5093A8C46BF22.png
    :width: 4.4043in
    :height: 1.7634in


.. _RefImage_100002000000032E0000014844B5093A8C46BF22.png:


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

The structure is initially at rest.