Benchmark solution
=====================


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

The constituent material verifies the vonMises criterion, with the threshold being

.. image:: images/Object_4.svg
    :width: 22
    :height: 24


.. _RefImage_Object_4.svg:

. The structure is subjected to pressure at the horizontal edges

.. image:: images/Object_5.svg
    :width: 22
    :height: 24


.. _RefImage_Object_5.svg:

and vertical

.. image:: images/Object_6.svg
    :width: 22
    :height: 24


.. _RefImage_Object_6.svg:

with

.. image:: images/Object_7.svg
    :width: 22
    :height: 24


.. _RefImage_Object_7.svg:

(

.. image:: images/Object_8.svg
    :width: 22
    :height: 24


.. _RefImage_Object_8.svg:

in 2D,

.. image:: images/Object_9.svg
    :width: 22
    :height: 24


.. _RefImage_Object_9.svg:

in 3D). In 2D plane, we consider two ways of doing this: on the one hand by amplifying the two pressures together, on the other hand by amplifying only the horizontal pressure, and by leaving the vertical pressure constant. In axisymmetry, the solid is subject to internal pressure alone

.. image:: images/Object_10.svg
    :width: 22
    :height: 24


.. _RefImage_Object_10.svg:

. We obtain the exact limit load and that by the regularization method [:ref:`R7.07.01 <R7.07.01>`] in this loading direction, for the vonMises criterion, with the threshold

.. image:: images/Object_11.svg
    :width: 22
    :height: 24


.. _RefImage_Object_11.svg:

.


Case plan
--------

The structure is subjected to pressures on the horizontal edges:

.. image:: images/Object_12.svg
    :width: 22
    :height: 24


.. _RefImage_Object_12.svg:

and vertical:

.. image:: images/Object_13.svg
    :width: 22
    :height: 24


.. _RefImage_Object_13.svg:

, with:

.. image:: images/Object_14.svg
    :width: 22
    :height: 24


.. _RefImage_Object_14.svg:

, and a block is exerted by

.. image:: images/Object_15.svg
    :width: 22
    :height: 24


.. _RefImage_Object_15.svg:

. Two ways of controlling the load are considered:


* cas1: both horizontal and vertical pressures are set by

    .. image:: images/Object_16.svg
        :width: 22
        :height: 24


    .. _RefImage_Object_16.svg:

,


* cas2: the horizontal pressure is set by

    .. image:: images/Object_17.svg
        :width: 22
        :height: 24


    .. _RefImage_Object_17.svg:

, while the vertical pressure is constant

    .. image:: images/Object_18.svg
        :width: 22
        :height: 24


    .. _RefImage_Object_18.svg:

, with

    .. image:: images/Object_19.svg
        :width: 22
        :height: 24


    .. _RefImage_Object_19.svg:

.


Boundary analysis solution
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The solution is homogeneous (biaxial stresses)

.. image:: images/Object_20.svg
    :width: 22
    :height: 24


.. _RefImage_Object_20.svg:

:

.. image:: images/Object_21.svg
    :width: 22
    :height: 24


.. _RefImage_Object_21.svg:

,

.. image:: images/Object_22.svg
    :width: 22
    :height: 24


.. _RefImage_Object_22.svg:

, plane deformations

.. image:: images/Object_23.svg
    :width: 22
    :height: 24


.. _RefImage_Object_23.svg:

). We obtain [:ref:`bib2 <bib2>`] the limit load in these loading directions, for the vonMises criterion, in plane deformations, with the threshold

.. image:: images/Object_24.svg
    :width: 22
    :height: 24


.. _RefImage_Object_24.svg:

:

case 1:

.. image:: images/Object_25.svg
    :width: 22
    :height: 24


.. _RefImage_Object_25.svg:


.. _RefEquation 2.2.1-1:

                              eq 2.2.1-1

case 2:

.. image:: images/Object_26.svg
    :width: 22
    :height: 24


.. _RefImage_Object_26.svg:


.. _RefEquation 2.2.1-2:

                      Eq 2.2.1-2

We check that if we take

.. image:: images/Object_27.svg
    :width: 22
    :height: 24


.. _RefImage_Object_27.svg:

in case 2, we then find case 1.


Regularized limit analysis solution
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The solution is homogeneous. The plane deformations are necessarily of the form:


.. image:: images/Object_28.svg
    :width: 22
    :height: 24


.. _RefImage_Object_28.svg:


.. _RefEquation 2.2.2-1:

                     eq 2.2.2-1

By Norton-Hoff's law, the coefficient

.. image:: images/Object_29.svg
    :width: 22
    :height: 24


.. _RefImage_Object_29.svg:

given, we obtain the deviatory constraints:


.. image:: images/Object_30.svg
    :width: 22
    :height: 24


.. _RefImage_Object_30.svg:


.. _RefEquation 2.2.2-2:

     eq 2.2.2-2

The normalization (unit power) of the load for which the limit load parameter is sought, cf. [:ref:`R7.07.01 <R7.07.01>`] [:ref:`§1.2 <§1.2>`], leads to:

case 1:

.. image:: images/Object_31.svg
    :width: 22
    :height: 24


.. _RefImage_Object_31.svg:


.. _RefEquation 2.2.2-3:

                         Eq 2.2.2-3

case 2:

.. image:: images/Object_32.svg
    :width: 22
    :height: 24


.. _RefImage_Object_32.svg:


.. _RefEquation 2.2.2-4:

                                 Eq 2.2.2-4

The terms of the suite

.. image:: images/Object_33.svg
    :width: 22
    :height: 24


.. _RefImage_Object_33.svg:

Approximations by excess of the limit load in these two load settings are then:

case 1:

.. image:: images/Object_34.svg
    :width: 22
    :height: 24


.. _RefImage_Object_34.svg:


.. _RefEquation 2.2.2-5:

                     eq 2.2.2-5

case 2:

.. image:: images/Object_35.svg
    :width: 22
    :height: 24


.. _RefImage_Object_35.svg:


.. _RefEquation 2.2.2-6:

              eq 2.2.2-6

The invariance as a function of

.. image:: images/Object_36.svg
    :width: 22
    :height: 24


.. _RefImage_Object_36.svg:

observed here (which is a particular case) results from the fact that we are in an isostatic situation. In case 1, we can also use the series of approximations by default to the limit load

.. image:: images/Object_37.svg
    :width: 22
    :height: 24


.. _RefImage_Object_37.svg:

:

case 1:

.. image:: images/Object_38.svg
    :width: 22
    :height: 24


.. _RefImage_Object_38.svg:


.. _RefEquation 2.2.2-7:

                              eq 2.2.2-7

So we get the limit load

.. image:: images/Object_39.svg
    :width: 22
    :height: 24


.. _RefImage_Object_39.svg:

Exact when

.. image:: images/Object_40.svg
    :width: 22
    :height: 24


.. _RefImage_Object_40.svg:

.

In case2, where the vertical pressure is constant, the power of this "permanent" load in the displacement solution is:

case 2:

.. image:: images/Object_41.svg
    :width: 22
    :height: 24


.. _RefImage_Object_41.svg:


.. _RefEquation 2.2.2-8:

                              eq 2.2.2-8


Axisymmetric case
-----------------

In axisymmetric 2D, the same geometry is considered, but the solid, on which a complete axial locking is imposed, is only subjected to pressure on the internal wall:

.. image:: images/Object_42.svg
    :width: 22
    :height: 24


.. _RefImage_Object_42.svg:

set by

.. image:: images/Object_43.svg
    :width: 22
    :height: 24


.. _RefImage_Object_43.svg:

.


Boundary analysis solution
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

We obtain [:ref:`bib2 <bib2>`] the limit load in this loading direction, for the vonMises criterion, in axisymmetry and zero axial deformations, with the threshold

.. image:: images/Object_44.svg
    :width: 22
    :height: 24


.. _RefImage_Object_44.svg:

:


.. image:: images/Object_45.svg
    :width: 22
    :height: 24


.. _RefImage_Object_45.svg:


.. _RefEquation 2.3.1-1:

                                 eq 2.3.1-1


Regularized limit analysis solution
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The solution is homogeneous. Since the displacement is only radial, the isochoric deformations are necessarily of the form:


.. image:: images/Object_46.svg
    :width: 22
    :height: 24


.. _RefImage_Object_46.svg:


.. _RefEquation 2.3.2-1:

         eq 2.3.2-1

By Norton-Hoff's law, the coefficient

.. image:: images/Object_47.svg
    :width: 22
    :height: 24


.. _RefImage_Object_47.svg:

given, we obtain the deviatory constraints:


.. image:: images/Object_48.svg
    :width: 22
    :height: 24


.. _RefImage_Object_48.svg:


.. _RefEquation 2.3.2-2:

 eq 2.3.2-2

The axial and radial equilibrium equations lead to the determination of the mean stress:


.. image:: images/Object_49.svg
    :width: 22
    :height: 24


.. _RefImage_Object_49.svg:


.. _RefEquation 2.3.2-3:

             Eq 2.3.2-3

where

.. image:: images/Object_50.svg
    :width: 22
    :height: 24


.. _RefImage_Object_50.svg:

is a constant, which is calculated from the condition at the zero pressure limits on the outer wall. The components of the constraints are then obtained:


.. image:: images/Object_51.svg
    :width: 22
    :height: 24


.. _RefImage_Object_51.svg:


.. _RefEquation 2.3.2-4:

         eq 2.3.2-4

The normalization (unit power) of the load for which the limit load parameter is sought, cf. [:ref:`R7.07.01 <R7.07.01>`] [:ref:`§1.2 <§1.2>`], leads to:

.. image:: images/Object_52.svg
    :width: 22
    :height: 24


.. _RefImage_Object_52.svg:

.

The terms of the suite

.. image:: images/Object_53.svg
    :width: 22
    :height: 24


.. _RefImage_Object_53.svg:

Approximations by excess of the limit load for this load are then:


.. image:: images/Object_54.svg
    :width: 22
    :height: 24


.. _RefImage_Object_54.svg:


.. _RefEquation 2.3.2-5:

             eq 2.3.2-5

The terms of the suite

.. image:: images/Object_55.svg
    :width: 22
    :height: 24


.. _RefImage_Object_55.svg:

Some approximations by default to the load limit for this load are:


.. image:: images/Object_56.svg
    :width: 22
    :height: 24


.. _RefImage_Object_56.svg:


.. _RefEquation 2.3.2-6:

     eq 2.3.2-6

En

.. image:: images/Object_57.svg
    :width: 22
    :height: 24


.. _RefImage_Object_57.svg:

, we find:

.. image:: images/Object_58.svg
    :width: 22
    :height: 24


.. _RefImage_Object_58.svg:

, that is, the same value as

.. image:: images/Object_59.svg
    :width: 22
    :height: 24


.. _RefImage_Object_59.svg:

and

.. image:: images/Object_60.svg
    :width: 22
    :height: 24


.. _RefImage_Object_60.svg:

.


Three-dimensional case
-------------------

In 3D, we consider the same geometry, but the solid, of unit thickness, is free in the antiplane direction.

.. image:: images/Object_61.svg
    :width: 22
    :height: 24


.. _RefImage_Object_61.svg:

. The solid is subjected to pressures on the horizontal walls:

.. image:: images/Object_62.svg
    :width: 22
    :height: 24


.. _RefImage_Object_62.svg:

and vertical:

.. image:: images/Object_63.svg
    :width: 22
    :height: 24


.. _RefImage_Object_63.svg:

, with:

.. image:: images/Object_64.svg
    :width: 22
    :height: 24


.. _RefImage_Object_64.svg:

. Both horizontal and vertical pressures are set by

.. image:: images/Object_65.svg
    :width: 22
    :height: 24


.. _RefImage_Object_65.svg:

.


Boundary analysis solution
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The solution is homogeneous (biaxial stresses)

.. image:: images/Object_66.svg
    :width: 22
    :height: 24


.. _RefImage_Object_66.svg:

:

.. image:: images/Object_67.svg
    :width: 22
    :height: 24


.. _RefImage_Object_67.svg:

,

.. image:: images/Object_68.svg
    :width: 22
    :height: 24


.. _RefImage_Object_68.svg:

,

.. image:: images/Object_69.svg
    :width: 22
    :height: 24


.. _RefImage_Object_69.svg:

, deformations

.. image:: images/Object_70.svg
    :width: 22
    :height: 24


.. _RefImage_Object_70.svg:

). We obtain the limit load in this loading direction [:ref:`bib2 <bib2>`], for the vonMises criterion, with the threshold

.. image:: images/Object_71.svg
    :width: 22
    :height: 24


.. _RefImage_Object_71.svg:

:


.. image:: images/Object_72.svg
    :width: 22
    :height: 24


.. _RefImage_Object_72.svg:


.. _RefEquation 2.4.1-1:

                             eq 2.4.1-1


Regularized limit analysis solution
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The solution is homogeneous. Isochoric deformations are necessarily of the form:


.. image:: images/Object_73.svg
    :width: 22
    :height: 24


.. _RefImage_Object_73.svg:


.. _RefEquation 2.4.2-1:

                 eq 2.4.2-1

By Norton-Hoff's law, the coefficient

.. image:: images/Object_74.svg
    :width: 22
    :height: 24


.. _RefImage_Object_74.svg:

given, we obtain the deviatory constraints:


.. image:: images/Object_75.svg
    :width: 22
    :height: 24


.. _RefImage_Object_75.svg:

;

.. image:: images/Object_76.svg
    :width: 22
    :height: 24


.. _RefImage_Object_76.svg:

eq 2.4.2-2

We deduce from

.. image:: images/Object_77.svg
    :width: 22
    :height: 24


.. _RefImage_Object_77.svg:

:

.. image:: images/Object_78.svg
    :width: 22
    :height: 24


.. _RefImage_Object_78.svg:

. Hence the constraints:

.. image:: images/Object_79.svg
    :width: 22
    :height: 24


.. _RefImage_Object_79.svg:

.

The balance of the solid requires that

.. image:: images/Object_80.svg
    :width: 22
    :height: 24


.. _RefImage_Object_80.svg:

. The parameter is deduced

.. image:: images/Object_81.svg
    :width: 22
    :height: 24


.. _RefImage_Object_81.svg:

.

The normalization (unit power) of the load for which the limit load parameter is sought, cf. [:ref:`R7.07.01 <R7.07.01>`] [:ref:`§1.2 <§1.2>`], leads to:


.. image:: images/Object_82.svg
    :width: 22
    :height: 24


.. _RefImage_Object_82.svg:


.. _RefEquation 2.4.2-3:

                             Eq 2.4.2-3

The terms of the suite

.. image:: images/Object_83.svg
    :width: 22
    :height: 24


.. _RefImage_Object_83.svg:

upper limit load limits in this charging case are thus identical to:


.. image:: images/Object_84.svg
    :width: 22
    :height: 24


.. _RefImage_Object_84.svg:


.. _RefEquation 2.4.2-4:

                     Eq 2.4.2-4


Benchmark results
----------------------


.. _OLE_LINK1:


+---------------------------------------------------------------------------------------------------------+---------+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------+
|Modeling                                                                                                 |sightings|                                                                                   |                                                                                   |                                                                                   |Power                                                                                                       |
+                                                                                                         +         + .. image:: images/Object_85.svg                                                   + .. image:: images/Object_86.svg                                                   +                                                                                   +                                                                                                            +
|                                                                                                         |         |    :width: 22                                                                     |    :width: 41                                                                     |                                                                                   |                                                                                                            |
+                                                                                                         +         +    :height: 24                                                                    +    :height: 21                                                                    +                                                                                   + .. image:: images/Object_88.svg                                                                            +
|                                                                                                         |         |                                                                                   |                                                                                   |                                                                                   |    :width: 48                                                                                              |
+                                                                                                         +         +                                                                                   +                                                                                   +                                                                                   +    :height: 17                                                                                             +
|                                                                                                         |         |                                                                                   |                                                                                   |                                                                                   |                                                                                                            |
+                                                                                                         +         +                                                                                   +                                                                                   +                                                                                   +                                                                                                            +
|                                                                                                         |         |                                                                                   |                                                                                   |                                                                                   |                                                                                                            |
+                                                                                                         +         +                                                                                   +                                                                                   +                                                                                   +                                                                                                            +
|                                                                                                         |         |                                                                                   |                                                                                   |                                                                                   |                                                                                                            |
+---------------------------------------------------------------------------------------------------------+---------+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------+
|A (case 1)                                                                                               |2D map   |                                                                                   |                                                                                   |                                                                                   |                                                                                                            |
+                                                                                                         +         + .. image:: images/Object_89.svg                                                   + .. image:: images/Object_90.svg                                                   + .. image:: images/Object_91.svg                                                   + .. image:: images/Object_92.svg                                                                            +
|                                                                                                         |         |    :width: 22                                                                     |    :width: 41                                                                     |    :width: 74                                                                     |    :width: 48                                                                                              |
+                                                                                                         +         +    :height: 24                                                                    +    :height: 21                                                                    +    :height: 22                                                                    +    :height: 17                                                                                             +
|                                                                                                         |         |                                                                                   |                                                                                   |                                                                                   |                                                                                                            |
+                                                                                                         +         +                                                                                   +                                                                                   +                                                                                   +                                                                                                            +
|                                                                                                         |         |                                                                                   |                                                                                   |                                                                                   |                                                                                                            |
+---------------------------------------------------------------------------------------------------------+---------+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------+
|Avis (cas2)                                                                                              |2D map   |                                                                                   |                                                                                   |nought                                                                             |                                                                                                            |
+                                                                                                         +         + .. image:: images/Object_93.svg                                                   + .. image:: images/Object_94.svg                                                   +                                                                                   + .. image:: images/Object_95.svg                                                                            +
|                                                                                                         |         |    :width: 22                                                                     |    :width: 41                                                                     |                                                                                   |    :width: 74                                                                                              |
+                                                                                                         +         +    :height: 24                                                                    +    :height: 21                                                                    +                                                                                   +    :height: 22                                                                                             +
|                                                                                                         |         |                                                                                   |                                                                                   |                                                                                   |                                                                                                            |
+                                                                                                         +         +                                                                                   +                                                                                   +                                                                                   +                                                                                                            +
|                                                                                                         |         |                                                                                   |                                                                                   |                                                                                   |                                                                                                            |
+---------------------------------------------------------------------------------------------------------+---------+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------+
|B                                                                                                        |3D       |                                                                                   |                                                                                   |                                                                                   |                                                                                                            |
+                                                                                                         +         + .. image:: images/Object_96.svg                                                   + .. image:: images/Object_97.svg                                                   + .. image:: images/Object_98.svg                                                   + .. image:: images/Object_99.svg                                                                            +
|                                                                                                         |         |    :width: 22                                                                     |    :width: 41                                                                     |    :width: 74                                                                     |    :width: 48                                                                                              |
+                                                                                                         +         +    :height: 24                                                                    +    :height: 21                                                                    +    :height: 22                                                                    +    :height: 17                                                                                             +
|                                                                                                         |         |                                                                                   |                                                                                   |                                                                                   |                                                                                                            |
+                                                                                                         +         +                                                                                   +                                                                                   +                                                                                   +                                                                                                            +
|                                                                                                         |         |                                                                                   |                                                                                   |                                                                                   |                                                                                                            |
+---------------------------------------------------------------------------------------------------------+---------+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------+
|C                                                                                                        |2D AXIS  |                                                                                   |                                                                                   |                                                                                   |                                                                                                            |
+                                                                                                         +         + .. image:: images/Object_101.svg                                                  + .. image:: images/Object_102.svg                                                  + .. image:: images/Object_103.svg                                                  + .. image:: images/Object_104.svg                                                                           +
|                                                                                                         |         |    :width: 22                                                                     |    :width: 41                                                                     |    :width: 74                                                                     |    :width: 48                                                                                              |
+ .. image:: images/Object_100.svg                                                                        +         +    :height: 24                                                                    +    :height: 21                                                                    +    :height: 22                                                                    +    :height: 17                                                                                             +
|    :width: 39                                                                                           |         |                                                                                   |                                                                                   |                                                                                   |                                                                                                            |
+    :height: 15                                                                                          +         +                                                                                   +                                                                                   +                                                                                   +                                                                                                            +
|                                                                                                         |         |                                                                                   |                                                                                   |                                                                                   |                                                                                                            |
+                                                                                                         +         +                                                                                   +                                                                                   +                                                                                   +                                                                                                            +
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+                                                                                                         +         +                                                                                   +                                                                                   +                                                                                   +                                                                                                            +
|                                                                                                         |         |                                                                                   |                                                                                   |                                                                                   |                                                                                                            |
+---------------------------------------------------------------------------------------------------------+---------+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------+



+-------------------------------------------------------------------------------------------------------------------------------+---------+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------+
|Modeling                                                                                                                       |sightings|                                                                                   |                                                                                   |                                                                                                                                                                         |(                                                                                    |Power                                                                                                        |
+                                                                                                                               +         + .. image:: images/Object_105.svg                                                  + .. image:: images/Object_106.svg                                                  + .. image:: images/Object_107.svg                                                                                                                                        + .. image:: images/Object_110.svg                                                    +                                                                                                             +
|                                                                                                                               |         |    :width: 22                                                                     |    :width: 41                                                                     |    :width: 74                                                                                                                                                           |    :width: 41                                                                       |                                                                                                             |
+                                                                                                                               +         +    :height: 24                                                                    +    :height: 21                                                                    +    :height: 22                                                                                                                                                          +    :height: 21                                                                      + .. image:: images/Object_111.svg                                                                            +
|                                                                                                                               |         |                                                                                   |                                                                                   |                                                                                                                                                                         |                                                                                     |    :width: 39                                                                                               |
+                                                                                                                               +         +                                                                                   +                                                                                   + (                                                                                                                                                                       +)                                                                                    +    :height: 15                                                                                              +
|                                                                                                                               |         |                                                                                   |                                                                                   | .. image:: images/Object_108.svg                                                                                                                                        |                                                                                     |                                                                                                             |
+                                                                                                                               +         +                                                                                   +                                                                                   +    :width: 22                                                                                                                                                           +                                                                                     +                                                                                                             +
|                                                                                                                               |         |                                                                                   |                                                                                   |    :height: 24                                                                                                                                                          |                                                                                     |                                                                                                             |
+                                                                                                                               +         +                                                                                   +                                                                                   +                                                                                                                                                                         +                                                                                     +                                                                                                             +
|                                                                                                                               |         |                                                                                   |                                                                                   |)                                                                                                                                                                        |                                                                                     |                                                                                                             |
+-------------------------------------------------------------------------------------------------------------------------------+---------+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------+
|A                                                                                                                              |2D map   |11.547                                                                             |11.547                                                                             |9.6225                                                                                                                                                                   |11.5458                                                                              |0                                                                                                            |
+-------------------------------------------------------------------------------------------------------------------------------+---------+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------+
|Abis                                                                                                                           |2D map   |14.6837                                                                            |14.6837                                                                            |nought                                                                                                                                                                   |nought                                                                               |0.25                                                                                                         |
+-------------------------------------------------------------------------------------------------------------------------------+---------+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------+
|B (                                                                                                                            |3D       |13.8675                                                                            |13.8675                                                                            |11.5562                                                                                                                                                                  |13.8661                                                                              |0                                                                                                            |
+                                                                                                                               +         +                                                                                   +                                                                                   +                                                                                                                                                                         +                                                                                     +                                                                                                             +
|                                                                                                                               |         |                                                                                   |                                                                                   |                                                                                                                                                                         |                                                                                     |                                                                                                             |
+ .. image:: images/Object_112.svg                                                                                              +         +                                                                                   +                                                                                   +                                                                                                                                                                         +                                                                                     +                                                                                                             +
|    :width: 22                                                                                                                 |         |                                                                                   |                                                                                   |                                                                                                                                                                         |                                                                                     |                                                                                                             |
+    :height: 24                                                                                                                +         +                                                                                   +                                                                                   +                                                                                                                                                                         +                                                                                     +                                                                                                             +
|                                                                                                                               |         |                                                                                   |                                                                                   |                                                                                                                                                                         |                                                                                     |                                                                                                             |
+                                                                                                                               +         +                                                                                   +                                                                                   +                                                                                                                                                                         +                                                                                     +                                                                                                             +
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+                                                                                                                               +         +                                                                                   +                                                                                   +                                                                                                                                                                         +                                                                                     +                                                                                                             +
|                                                                                                                               |         |                                                                                   |                                                                                   |                                                                                                                                                                         |                                                                                     |                                                                                                             |
+                                                                                                                               +         +                                                                                   +                                                                                   +                                                                                                                                                                         +                                                                                     +                                                                                                             +
|                                                                                                                               |         |                                                                                   |                                                                                   |                                                                                                                                                                         |                                                                                     |                                                                                                             |
+)                                                                                                                              +         +                                                                                   +                                                                                   +                                                                                                                                                                         +                                                                                     +                                                                                                             +
|                                                                                                                               |         |                                                                                   |                                                                                   |                                                                                                                                                                         |                                                                                     |                                                                                                             |
+-------------------------------------------------------------------------------------------------------------------------------+---------+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------+
|C                                                                                                                              |2D AXIS  |12.6857                                                                            |12.6857                                                                            |8.5545                                                                                                                                                                   |12.6830                                                                              |0                                                                                                            |
+-------------------------------------------------------------------------------------------------------------------------------+---------+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------+


**Note:**

*Code_Aster calculates the opposite value of the power of the "permanent" load in the displacement solution*

.. image:: images/Object_113.svg
    :width: 22
    :height: 24


.. _RefImage_Object_113.svg:

*.*


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


1. F. VOLDOIRE, E. LORENTZ, J.M. PROIX, E. VISSE: Limit load calculation by the Norton-Hoff-Friaâ method. [:ref:`R7.07.01 <R7.07.01>`].


2. F. VOLDOIRE, Fracture calculation and structural limit analysis, note EDF HI-74/93/082.