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

Geometry
---------

Straight beam of length :math:`L`, with direction vector :math:`(\mathrm{4,}\mathrm{3,}0)`.

+----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+
|                                                                                                                            |                                                                                                                             |
+ .. image:: images/10000000000001DE00000098DC1BD62BB1B66370.png                                                             + .. image:: images/100000000000009B0000006E5FDFFE913280E971.png                                                              +
|    :width: 3.978in                                                                                                         |    :width: 1.2807in                                                                                                         |
+    :height: 1.2602in                                                                                                       +    :height: 0.9075in                                                                                                        +
|                                                                                                                            |                                                                                                                             |
+                                                                                                                            +                                                                                                                             +
|                                                                                                                            |                                                                                                                             |
+----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+
|                                                                                                                            |**Pipe section**                                                                                                             |
+----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+


Tubular section with outer radius :math:`a\mathrm{=}\mathrm{0.04m}`, inner radius :math:`b\mathrm{=}\mathrm{0.032m}`, inner radius, and thickness :math:`e\mathrm{=}0.008m`.


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

The material used has elastic behavior. The material parameters take the following values:

• Young's module :math:`E=2.{10}^{11}\mathrm{Pa}`,

• Poisson's ratio :math:`\nu =0.3`,

• Density :math:`\rho =7800\mathrm{kg}/{m}^{3}`,

• Thermal expansion coefficient :math:`\alpha ={10}^{-5}`.


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

• Embedding in :math:`O`

• 6 basic loads at the :math:`B` end

• in coordinate :math:`(x,y,z)` linked to the beam:

.. csv-table::

    ":math:`{F}_{x}\mathrm{=}{5.10}^{2}N` "," :math:`{M}_{x}\mathrm{=}{5.10}^{2}\mathit{Nm}`"
    ":math:`{F}_{y}\mathrm{=}{5.10}^{2}N` "," :math:`{M}_{y}\mathrm{=}{5.10}^{2}\mathit{Nm}`"
    ":math:`{F}_{z}\mathrm{=}{5.10}^{2}N` "," :math:`{M}_{z}\mathrm{=}{5.10}^{2}\mathit{Nm}`"


• or, in the global frame of reference :math:`(X,Y,Z)`:

• 1 traction load: :math:`{F}_{X}={4.10}^{2}N` and :math:`{F}_{Y}={3.10}^{2}N`

• 2 sharp efforts: in the :math:`(\mathrm{oxy})` :math:`{F}_{X}=–3.102N` and :math:`{F}_{Y}={4.10}^{2}N` plan and in the :math:`(\mathrm{oyz})` :math:`{F}_{Z}={5.10}^{2}N` plan

• 1 torsional moment: :math:`{M}_{X}={4.10}^{2}\mathrm{Nm}` and :math:`{M}_{Y}={3.10}^{2}\mathrm{Nm}`

• 2 sharp efforts: in the :math:`(\mathrm{oxy})` :math:`{M}_{X}\mathrm{=}–{3.10}^{2}\mathit{Nm}` and :math:`{M}_{Y}\mathrm{=}{4.10}^{2}\mathit{Nm}` plan and in the :math:`(\mathrm{oyz})` :math:`{M}_{Z}\mathrm{=}{5.10}^{2}\mathit{Nm}` plan

• Internal pressure: :math:`P={10}^{7}\mathrm{Pa}`

• Gravity, with :math:`g=\mathrm{10m}/{s}^{2}`, in the :math:`-Z` direction

• Line load, :math:`\mathrm{Fz}=-141.146N/m` (which corresponds to the load due to gravity: :math:`\mathrm{Fz}=\mathrm{mg}`)

• Thermal expansion: :math:`\mathrm{Temp}=100°C`


Notation of cross-sectional characteristics
--------------------------------------------------------

The geometric characteristics of the cross sections are noted:

• :math:`S`: area of the section

• :math:`{I}_{y},{I}_{z}`: geometric moments of inertia in relation to the main axes of inertia of the section

• :math:`{J}_{x}`: torsional constant