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

Geometry
---------


.. image:: images/100000000000013E00000125CA8863A32DA1AE6D.png
    :width: 2.2972in
    :height: 2.2925in


.. _RefImage_100000000000013E00000125CA8863A32DA1AE6D.png:


.. image:: images/10000000000000A900000088AF956D0A6115A417.png
    :width: 1.6228in
    :height: 1.3008in


.. _RefImage_10000000000000A900000088AF956D0A6115A417.png:


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

The properties of the material from which the pipe is made are:

.. csv-table::

    ":math:`E\mathrm{=}193.{10}^{9}\mathit{Pa}` ", "Young's module"
    ":math:`\nu \mathrm{=}0.2642` ", "Poisson's ratio"



+-----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------+-------------------------------------------------------+
|                                                                                                                             |Constraint :math:`\mathrm{Pa}`                       |Plastic Deformation :math:`{\varepsilon }_{p}`         |
+ .. image:: images/10000000000001A7000000F34F313BB45E859DF7.png                                                              +-----------------------------------------------------+-------------------------------------------------------+
|    :width: 3.5138in                                                                                                         |2.72E+08 3.46E+08 3.79E+08 4.04E+08 4.24E+08 5.28E+08|0.00000 0.00473 0.01264 0.01264 0.02836 0.04910 0.10500|
+    :height: 2.0193in                                                                                                        +-----------------------------------------------------+-------------------------------------------------------+
|                                                                                                                             |                                                     |                                                       |
+                                                                                                                             +                                                     +                                                       +
|                                                                                                                             |                                                     |                                                       |
+-----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------+-------------------------------------------------------+


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

• Boundary conditions:

◦ Recessed section :math:`A`

◦ Rigid section :math:`D` (no deformation of the section)

• Loading: we are looking for the successive states of equilibrium under the following loads:

◦ Step A: :math:`0\mathrm{=}t<{t}_{1}`

▪ the pressure varies from :math:`0` to :math:`3.45{10}^{6}\mathrm{Pa}`

▪ The strength (shape effect) at point :math:`D` varies from :math:`0` to :math:`4.0414{10}^{5}N`

▪ The moment sucks

◦ Step B: :math:`{t}_{1}\mathrm{=}t<{t}_{2}`

▪ the pressure is constant and is :math:`3.45{10}^{6}\mathrm{Pa}`

▪ the force (shape effect) at point :math:`D` is constant and is equal to :math:`4.0414{10}^{5}N`

▪ the timing varies from :math:`0` to :math:`2.534{10}^{5}\mathit{N.m}`