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


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

The material properties and the characteristics specific to the RCC -M calculation are as follows:


1. Young's modulus: :math:`E\mathrm{=}2.E+05\mathit{MPa}`;


2. material constants for the calculation of :math:`\mathrm{Ke}`: :math:`n=0.2`, :math:`m=\mathrm{2 }`;


3. Young's modulus of reference: :math:`{E}_{\mathit{REFE}}\mathrm{=}2.E+05\mathit{MPa}`;


4. allowable stress: :math:`\mathit{Sm}=2000\mathit{MPa}`.

The Wöhler curve is defined analytically: :math:`{N}_{\mathrm{adm}}=\frac{{5.10}^{5}}{{S}_{\mathrm{alt}}}`


Evolution of constraints
-------------------------

The constraints on the analysis segment are not calculated but read directly from a table. The only non-zero component of the stress tensor is :math:`{\sigma }_{\mathrm{yy}}`. Two situations are considered. These situations do not aim to represent a specific real transient, but to cover all possible constraints (constant, linear or non-linear evolution of the stress in thickness).


+-------+-----------------------+---+----+--------------------------+---+----+---------------------------+----+----+
|Instant|**Thermal constraints**         |**Stress** due to pressure         |**Thermal stresses+pressure          |
+       +-----------------------+---+----+--------------------------+---+----+---------------------------+----+----+
|       |*Abscissor*                     |*Abscissor*                        |*Abscissor*                          |
+       +-----------------------+---+----+--------------------------+---+----+---------------------------+----+----+
|       |0                      |1  |2   |0                         |1  |2   |0                          |1   |2   |
+-------+-----------------------+---+----+--------------------------+---+----+---------------------------+----+----+
|1, 5   |90                     |100|110 |90                        |100|110 |180                        |200 |220 |
+-------+-----------------------+---+----+--------------------------+---+----+---------------------------+----+----+
|2, 5   |0                      |100|0   |0                         |100|0   |0                          |200 |0   |
+-------+-----------------------+---+----+--------------------------+---+----+---------------------------+----+----+
|3, 5   |100                    |-50|-100|100                       |-50|-100|200                        |-100|-200|
+-------+-----------------------+---+----+--------------------------+---+----+---------------------------+----+----+
|4, 5   |0                      |0  |0   |0                         |0  |0   |0                          |0   |0   |
+-------+-----------------------+---+----+--------------------------+---+----+---------------------------+----+----+

**Table** 1.2-1 **:** Definition of constraints :math:`{\mathrm{\sigma }}_{\mathit{yy}}` (in :math:`\mathit{MPa}`) for the moments of situation 1 as a function of the curvilinear abscissa


+-------+-----------------------+---+----+--------------------------+-+-+---------------------------+---+----+
|Instant|**Thermal constraints**         |**Stress** due to pressure    |**Thermal stresses+pressure         |
+       +-----------------------+---+----+--------------------------+-+-+---------------------------+---+----+
|       |*Abscissor*                     |*Abscissor*                   |*Abscissor*                         |
+       +-----------------------+---+----+--------------------------+-+-+---------------------------+---+----+
|       |0                      |1  |2   |0                         |1|2|0                          |1  |2   |
+-------+-----------------------+---+----+--------------------------+-+-+---------------------------+---+----+
|1      |90                     |100|90  |0                         |0|0|90                         |100|90  |
+-------+-----------------------+---+----+--------------------------+-+-+---------------------------+---+----+
|2      |0                      |100|0   |0                         |0|0|0                          |100|0   |
+-------+-----------------------+---+----+--------------------------+-+-+---------------------------+---+----+
|3      |100                    |-50|-100|0                         |0|0|100                        |-50|-100|
+-------+-----------------------+---+----+--------------------------+-+-+---------------------------+---+----+
|4      |0                      |0  |0   |0                         |0|0|0                          |0  |0   |
+-------+-----------------------+---+----+--------------------------+-+-+---------------------------+---+----+

Table 1.2-2: Definition of the constraints :math:`{\mathrm{\sigma }}_{\mathit{yy}}` (in :math:`\mathit{MPa}`) for the moments of situation 2 as a function of the curvilinear abscissa

In ZE200, the moments are defined according to two twists (in ze200a, the pressure also)


.. csv-table::

    "", ":math:`{P}_{A}` "," :math:`{P}_{B}` "," :math:`{M}_{\mathit{xA}}` "," "," :math:`{M}_{\mathit{yA}}` "," "," :math:`{M}_{\mathit{zA}}` "," :math:`{M}_{\mathit{xB}}` "," :math:`{M}_{\mathit{yB}}` "," :math:`{M}_{\mathit{zB}}`"
    "Situation 1", "201", "1", "21", "21", "0", "0", "1", "0", "0"
    "Situation 2", "0", "0", "0", "1", "0", "0", "61", "0", "0"


Table 1.2-3: Definition of torsors on moments (in N.mm) and pressure (in MPa) for situations 1 and 2

 In ZE200, the characteristics of the pipe (thickness, radius, moment of inertia) are necessary to calculate the quantities, as are the stress indices. In this example, we select*arbitrarily select*

:math:`e=1\mathit{mm}`

:math:`R=\mathrm{0,5}\mathit{mm}`

:math:`I=1{m}^{4}`

:math:`{K}_{1}=1` and :math:`{C}_{1}=1`

:math:`{K}_{2}=1` and :math:`{C}_{2}=2`

:math:`{K}_{3}=1` and :math:`{C}_{3}=1`


.. csv-table::

    ":math:`{M}_{\mathit{xS}}` "," :math:`{M}_{\mathit{yS}}` "," :math:`{M}_{\mathit{zS}}`"
    "21", "0", "0"


Table 1.2-4: Definition of torsors on moments (in N.mm) for the earthquake