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}=200\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:`{\mathrm{\sigma }}_{\mathit{yy}}`. Several 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** of situations 1 and 2         |Instant|**Thermal constraints** of situation 3         |
+       +---------------------------------------------+---+----+       +--------------------------------------+---+----+
|       |*Abscissor*                                           |       |*Abscissor*                                    |
+       +---------------------------------------------+---+----+       +--------------------------------------+---+----+
|       |0                                            |1  |2   |       |0                                     |1  |2   |
+-------+---------------------------------------------+---+----+-------+--------------------------------------+---+----+
|1      |50                                           |100|150 |1, 5   |50                                    |100|150 |
+-------+---------------------------------------------+---+----+-------+--------------------------------------+---+----+
|2      |0                                            |50 |-100|2, 5   |0                                     |50 |-100|
+-------+---------------------------------------------+---+----+-------+--------------------------------------+---+----+
|3      |0                                            |0  |50  |3, 5   |0                                     |0  |50  |
+-------+---------------------------------------------+---+----+-------+--------------------------------------+---+----+
|4      |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 and situation 2 as a function of the curvilinear abscissa

In this example, moments and pressure are defined by two torsors and four unit tensors. Again, only the :math:`{\mathrm{\sigma }}_{\mathit{yy}}` constraint is non-zero in these tensors.


+--------------------------+----------------------------------------+--+---+
|Unit tensor               |:math:`{\mathrm{\sigma }}_{\mathit{yy}}`       |
+                          +----------------------------------------+--+---+
|                          |*Abscissor*                                    |
+                          +----------------------------------------+--+---+
|                          |0                                       |1 |2  |
+--------------------------+----------------------------------------+--+---+
|:math:`{\mathit{Mom}}_{X}`|50                                      |0 |0  |
+--------------------------+----------------------------------------+--+---+
|:math:`{\mathit{Mom}}_{Y}`|0                                       |50|0  |
+--------------------------+----------------------------------------+--+---+
|:math:`{\mathit{Mom}}_{Z}`|0                                       |0 |100|
+--------------------------+----------------------------------------+--+---+
|:math:`\mathit{Pres}`     |50                                      |0 |0  |
+--------------------------+----------------------------------------+--+---+


.. 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", "1", "10", "10", "1", "1", "-1", "-1.5", "-10", "1", "0.1"
    "Situation 2", "1", "10", "10", "1", "1", "-1", "-1.5", "-10", "1", "0.1"
    "Situation 3", "0.4", "1", "0.4", "0.4", "0", "-0.6", "1", "-1", "-1.5"


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