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

Geometry
---------

Straight beam with variable circular cross section
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


.. image:: images/1000000000000165000000485356E243AEEE10BE.png
    :width: 3.7189in
    :height: 0.75in


.. _RefImage_1000000000000165000000485356E243AEEE10BE.png:

**Figure** 1.1.1-a **: Beam with variable circular cross section.**


.. csv-table::

    "Length", ": :math:`1m`"
    "Built-in radius", ": :math:`\mathrm{0,1}m`"
    "Free End Radius", ": :math:`\mathrm{0,05}m`"



Straight beam with variable rectangular cross section
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


.. image:: images/10000000000001F30000008EC41478BE35E47508.png
    :width: 5.1846in
    :height: 1.4638in


.. _RefImage_10000000000001F30000008EC41478BE35E47508.png:

**Figure** 1.1.2-a **: Beam with variable rectangular section**


.. csv-table::

    "Length", ": :math:`1m` ", ""
    "embedded", ": :math:`\mathit{Hy}\mathrm{=}\mathrm{0,05}m` "," :math:`\mathit{Hz}\mathrm{=}\mathrm{0,10}m`"
    "at the free end", ": :math:`\mathit{Hy}\mathrm{=}\mathrm{0,05}m` "," :math:`\mathit{Hz}\mathrm{=}\mathrm{0,05}m`"



Straight beam with variable general cross section
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


.. image:: images/10000000000001E70000006AA8DFB717622898B0.png
    :width: 5.0665in
    :height: 1.0953in


.. _RefImage_10000000000001E70000006AA8DFB717622898B0.png:

**Figure** 1.1.3-a **: Beam with variable general section**


.. csv-table::

    "Length", ": :math:`1m` ", ""
    "embedded", ": :math:`A\mathrm{=}{10}^{\mathrm{-}2}{m}^{2}` "," :math:`\mathit{Iy}\mathrm{=}\mathrm{8,3333}{10}^{\mathrm{-}6}{m}^{4}`"
    "at the free end", ": :math:`A\mathrm{=}{\mathrm{2,510}}^{\mathrm{-}3}{m}^{2}` "," :math:`\mathit{Iy}\mathrm{=}\mathrm{5,20833}{10}^{\mathrm{-}7}{m}^{4}`"



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

.. csv-table::

    "Young's module:", ":math:`E\mathrm{=}2.{10}^{11}\mathit{Pa}`"
    "Poisson's ratio:", ":math:`\nu \mathrm{=}\mathrm{0,3}`"
    "Density:", ":math:`\rho \mathrm{=}7800{\mathit{kg.m}}^{\mathrm{-}3}`"



Boundary conditions and loading
------------------------------------

**Boundary condition:**

Recessed end: :math:`\mathit{DX}\mathrm{=}\mathit{DY}\mathrm{=}\mathit{DZ}\mathrm{=}\mathit{DRX}\mathrm{=}\mathit{DRY}\mathrm{=}\mathit{DRZ}\mathrm{=}0`

**Charging:**

On the right beam of variable circular cross section and on the right beam of variable rectangular cross section, the following are applied successively:

.. csv-table::

    "**Load case**", "**Nature**"
    "1", "a one-time effort following :math:`X` at the free end, :math:`\mathit{Fx}\mathrm{=}100N`"
    "2", "a one-time effort following :math:`Y` at the free end, :math:`\mathit{Fy}\mathrm{=}100N`"
    "3", "a point moment around the :math:`X` axis at the free end, :math:`\mathit{Mx}\mathrm{=}100\mathit{m.N}`"
    "4", "a point moment around the :math:`Z` axis at the free end, :math:`\mathit{Mz}\mathrm{=}100\mathit{m.N}`"
    "5", "a load distributed over the entire beam, :math:`\mathit{fx}\mathrm{=}100{\mathit{N.m}}^{\mathrm{-}1}`"
    "6", "a load distributed over the entire beam, :math:`\mathrm{fy}=100{\mathrm{N.m}}^{-1}`"


On the right beam of variable general cross section, the following are applied:

.. csv-table::

    "**Load case**", "**Nature**"
    "7", "a heavy-handed effort following :math:`z` with :math:`g=\mathrm{9,81}{\mathrm{m.s}}^{-2}`"