Reference solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------

Finite element calculation with Samcef Version 7.0 (Mecanl).


Reference solution
---------------------

**Variation of tensions on the skin along the meridian section for internal pressure** :math:`{p}_{0}` [:ref:`bib2 <bib2>`]


.. image:: images/10047EE2000069D500003B88B51E55E8B4B9D0FF.svg
    :width: 595
    :height: 368


.. _RefImage_10047EE2000069D500003B88B51E55E8B4B9D0FF.svg:

**Variation of tensions on the skin along the meridian section in the vicinity of the ruin for internal pressure** :math:`{p}_{\mathrm{max}}` [:ref:`bib2 <bib2>`]


.. image:: images/1000000000000828000003C73BCD8A0883A5CCF3.png
    :width: 415
    :height: 320


.. _RefImage_1000000000000828000003C73BCD8A0883A5CCF3.png:

**Variation in azimuthal stress** :math:`{\sigma }_{\theta \theta }` **on the upper skin along the meridian section in the vicinity of the ruin for internal pressure** :math:`{p}_{\mathrm{max}}` **. Geometric linear calculation.**


.. image:: images/100034DA000069BB0000518CAD1F98B2B150FB9B.svg
    :width: 595
    :height: 368


.. _RefImage_100034DA000069BB0000518CAD1F98B2B150FB9B.svg:

**Variation in azimuthal stress** :math:`{\sigma }_{\theta \theta }` **on the lower skin along the meridian section in the vicinity of the ruin for internal pressure** :math:`{p}_{\mathrm{max}}` **. Geometric linear calculation.**


.. image:: images/100031AC000069BB000050692FF31C6E08C7BB3A.svg
    :width: 595
    :height: 368


.. _RefImage_100031AC000069BB000050692FF31C6E08C7BB3A.svg:

**Variation in azimuthal stress** :math:`{\sigma }_{\theta \theta }` **on the upper skin along the meridian section in the vicinity of the ruin for internal pressure** :math:`{p}_{\mathrm{max}}` **. Geometric nonlinear calculation.**


.. image:: images/100034B4000069D500004BA8CE7AD9CEC388FA2A.svg
    :width: 595
    :height: 368


.. _RefImage_100034B4000069D500004BA8CE7AD9CEC388FA2A.svg:

**Variation in azimuthal stress** :math:`{\sigma }_{\theta \theta }` **on the lower skin along the meridian section in the vicinity of the ruin for internal pressure** :math:`{p}_{\mathrm{max}}` **. Geometric nonlinear calculation.**


.. image:: images/10003442000069D500004E3D3F0B6F011C9F77F7.svg
    :width: 595
    :height: 368


.. _RefImage_10003442000069D500004E3D3F0B6F011C9F77F7.svg:


.. image:: images/10076266000069D500008539CD3B4C1C8558DB38.svg
    :width: 595
    :height: 368


.. _RefImage_10076266000069D500008539CD3B4C1C8558DB38.svg:

**Discretization used for the reference solution. Definition of axes**

**Geometric nonlinear calculation:** :math:`\mathrm{pression}={p}_{\mathrm{max}}`


+--------+------------------------------------------------------+-------------+-----------------------------------------------------------------------------------+
|Skin    |Meridian length                                       |Knot position|                                                                                   |
+        +                                                      +             + .. image:: images/Object_1.svg                                                    +
|        | :math:`(\mathrm{inch})`                              |             |    :width: 595                                                                    |
+        +                                                      +             +    :height: 368                                                                   +
|        |                                                      |             |                                                                                   |
+        +                                                      +             +                                                                                   +
|        |                                                      |             |                                                                                   |
+--------+------------------------------------------------------+-------------+-----------------------------------------------------------------------------------+
|        |0.0                                                   |1            |26.223                                                                             |
+--------+------------------------------------------------------+-------------+-----------------------------------------------------------------------------------+
|Superior|46.298                                                |73           |—26.688                                                                            |
+--------+------------------------------------------------------+-------------+-----------------------------------------------------------------------------------+
|        |137.9                                                 |137          |24.103                                                                             |
+--------+------------------------------------------------------+-------------+-----------------------------------------------------------------------------------+
|        |0.0                                                   |1            |25.231                                                                             |
+--------+------------------------------------------------------+-------------+-----------------------------------------------------------------------------------+
|        |16.491                                                |27           |27.909                                                                             |
+--------+------------------------------------------------------+-------------+-----------------------------------------------------------------------------------+
|Lower   |35.518                                                |57           |—12.711                                                                            |
+--------+------------------------------------------------------+-------------+-----------------------------------------------------------------------------------+
|        |52.707                                                |87           |—8.1652                                                                            |
+--------+------------------------------------------------------+-------------+-----------------------------------------------------------------------------------+
|        |134.76                                                |137          |25.103                                                                             |
+--------+------------------------------------------------------+-------------+-----------------------------------------------------------------------------------+


**Geometric linear calculation:** :math:`\mathrm{pression}={p}_{\mathrm{max}}`


+--------+------------------------------------------------------+-------------+-----------------------------------------------------------------------------------+
|Skin    |Meridian length                                       |Knot position|                                                                                   |
+        +                                                      +             + .. image:: images/Object_2.svg                                                    +
|        | :math:`(\mathrm{inch})`                              |             |    :width: 595                                                                    |
+        +                                                      +             +    :height: 368                                                                   +
|        |                                                      |             |                                                                                   |
+        +                                                      +             +                                                                                   +
|        |                                                      |             |                                                                                   |
+--------+------------------------------------------------------+-------------+-----------------------------------------------------------------------------------+
|        |0.0                                                   |1            |26.2                                                                               |
+--------+------------------------------------------------------+-------------+-----------------------------------------------------------------------------------+
|Superior|45.1                                                  |71           |—30.7                                                                              |
+--------+------------------------------------------------------+-------------+-----------------------------------------------------------------------------------+
|        |138                                                   |137          |24.0                                                                               |
+--------+------------------------------------------------------+-------------+-----------------------------------------------------------------------------------+
|        |0.0                                                   |1            |25.231                                                                             |
+--------+------------------------------------------------------+-------------+-----------------------------------------------------------------------------------+
|        |22.833                                                |37           |29.899                                                                             |
+--------+------------------------------------------------------+-------------+-----------------------------------------------------------------------------------+
|Inferior|32.981                                                |53           |—16.127                                                                            |
+--------+------------------------------------------------------+-------------+-----------------------------------------------------------------------------------+
|        |52.707                                                |87           |—13,756                                                                            |
+--------+------------------------------------------------------+-------------+-----------------------------------------------------------------------------------+
|        |134.76                                                |137          |25.008                                                                             |
+--------+------------------------------------------------------+-------------+-----------------------------------------------------------------------------------+


**Movement field (point** :math:`A` **, lower skin)**


.. csv-table::

    "Calculus... geometric", "Meridian length
    :math:`(\mathrm{inch})` ", "Node position", "Pressure", "Displacement :math:`{U}^{Z}` :math:`(\mathrm{inch})`"
    "Linear", "0.0", "1"," :math:`0.5\mathrm{pmax}` ", "0.100945"
    "", "0.0", "1"," :math:`\mathrm{pmax}` ", "0.370468"
    "Nonlinear", "0.0", "1", "1"," :math:`0.5\mathrm{pmax}` ", "0.0990524"
    "", "0.0", "1"," :math:`\mathrm{pmax}` ", "0.244347"



Uncertainty about the solution
---------------------------

Uncertainty less than :math:`\text{2 \%}` (linear regime), less than :math:`\text{5 \%}` (elasto-plastic regime).


Bibliographical references
---------------------------


* LARSEN, P.K., POPOV, E.P., E.P., Elastic-plastic analysis of thick-walled pressure vessels with sharp discontinuities, Trans. ASME Applied Mechanics, pp 1016-1019, 1971


* NYSSEN, C., Finite element modeling of the nonlinear behavior of aerospace structures, doctoral thesis, University of Liège, 1979