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


Geometry
---------

The study concerns a pipe comprising two straight pipes and an elbow [:ref:`Figure 1.1-a <Figure 1.1-a>`].

The geometric data for the problem is as follows:


* the length :math:`{L}_{G}` of the two straight pipes is :math:`3m`,


* the initial radius :math:`\mathrm{Rc}` of the elbow is :math:`0.3m`,


* Elbow angle :math:`\theta` is 90 degrees,


* the thickness of the straight pipes and the elbow is :math:`0.02m`,


* and the outside radius :math:`\text{Re}` of the straight pipes and the elbow is :math:`0.2m`.


.. image:: images/100013EA0000288400001F36789D77AD3C40BA9C.svg
    :width: 522
    :height: 402


.. _RefImage_100013EA0000288400001F36789D77AD3C40BA9C.svg:

**Figure 1.1-a**

**Note:**

*The geometry of the problem is symmetric with respect to the plane* :math:`(A,X,Y)` *.*


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

Isotropic linear elastic material. The properties of the material are those of :math:`\mathrm{A42}` steel:


.. csv-table::

    "* Young's modulus:", ":math:`E=1.8{10}^{11}\mathrm{Pa}`"
    "* Poisson's ratio:", ":math:`\nu =0.3`"



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


* Boundary conditions: embedding at the level of section :math:`A`,


* Loading: constant force :math:`\mathrm{FY}` directed along the :math:`Y` axis and applied to section :math:`B`.

The value of :math:`\mathrm{FY}` is calculated from:


* of the average radius: :math:`\mathrm{RMOY}=0.19`,


* of the total force applied: :math:`\mathrm{FTOT}=500000N/{m}^{2}`,

His expression is as follows:

:math:`\mathrm{FY}=\mathrm{FTOT}/(2\pi \mathrm{RMOY})(\simeq 418828.8)`


* The Von Mises limit constraint is :math:`2.0E+09N/{m}^{2}`