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


Geometry
---------

We consider a hollow pipe with a cross-section varying homothetically, with an external radius varying from :math:`10\mathit{mm}` to :math:`5\mathit{mm}`, with a thickness varying from :math:`5\mathit{mm}` to :math:`2.5\mathit{mm}` and with a length :math:`2m`.


.. image:: images/100002010000020200000072517CB7DC8A612B8C.png
    :width: 5.3535in
    :height: 1.1866in


.. _RefImage_100002010000020200000072517CB7DC8A612B8C.png:


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

The material is isotropic elastic whose properties are:


* :math:`E=210\mathit{GPa}`


* :math:`\nu =0.3`


* :math:`\mathrm{{\rm P}}=7800\mathit{kg}/{m}^{3}`


* :math:`\alpha =1.2E-5{K}^{-1}`

Ramberg Osgood's law used for post-treatments is written as:

:math:`{ϵ}_{p}=K{(\frac{\sigma }{E})}^{1/n}`

With:


* :math:`K=0.01`


* :math:`n=2`


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

A set of elastic calculations are performed independently on the beam-like structure.


.. image:: images/100002010000019C000000785C63C96ABAD54C29.png
    :width: 4.2909in
    :height: 1.2492in


.. _RefImage_100002010000019C000000785C63C96ABAD54C29.png:


Net weight
~~~~~~~~~~~~

A :math:`g=9.81m/{s}^{2}` gravity field is imposed in the :math:`-Z` direction. Point :math:`A` is embedded.


Imposed displacement
~~~~~~~~~~~~~~~~~~~

Point :math:`A` is embedded. The structure is torsified at point :math:`B` via an imposed angle :math:`\theta =0.02\mathit{rad}` along the axis :math:`X`.


Thermal expansion
~~~~~~~~~~~~~~~~~~~~~~

Point :math:`A` is embedded. Point :math:`B` is stuck along the :math:`X` axis. A variation in the temperature field :math:`\mathrm{\Delta }T=100°K` generates compression in the beam.


Pressure imposed
~~~~~~~~~~~~~~~~~

The pressure inside the pipe :math:`P=1\mathit{MPa}` is taken into account.


Spectral seismic
~~~~~~~~~~~~~~~~~~

Point :math:`A` is embedded. Point :math:`B` is stuck along the :math:`Z` axis. Spectral acceleration is imposed along the 3 axes :math:`X`, :math:`Y` and :math:`Z` with the respective ratios :math:`1.`, :math:`1.` and :math:`0.5`:


.. image:: images/1000020100000280000001E0D438E8C97734A253.png
    :width: 4.0134in
    :height: 3.0102in


.. _RefImage_1000020100000280000001E0D438E8C97734A253.png:

The first 4 dynamic natural modes and the static correction are taken into account via pseudo-modes along the 3 axes. Reduced amortization is set to :math:`2\text{\%}`.