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


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

The material properties are as follows:


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


2. Poisson's ratio: :math:`\nu =0.3`;


3. Thermal expansion coefficient: :math:`\alpha =1.E–05\mathrm{m.}°{C}^{-1}`.

The characteristics specific to the calculation RCC -M are:


1. material constants for calculating :math:`\mathrm{Ke}`: :math:`n=0.2`, :math:`m=\mathrm{2 }`;


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


3. allowable stress: :math:`\mathrm{Sm}=200\mathrm{MPa}`.

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

**Note:**


.. csv-table::

    "*To validate the consideration of the elastoplastic concentration factor* :math:`\mathrm{Ke}` *, some calculations are carried out with a lower or higher allowable stress:* :math:`\mathit{Sm}=50\mathit{MPa}` (C modeling) *and* :math:`\mathit{Sm}=2000\mathit{MPa}` (D and E modeling) *.*"



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 transients are considered:


+-------+-----------------------+---+----+--------------------------+---+----+-------------------------------------------+---+---+----------------------------+---+----+
|Instant|**Thermal constraints**         |**Stress** due to pressure         |**Mechanical stress** (moments and efforts)        |**Constraints** total **es**         |
+       +-----------------------+---+----+--------------------------+---+----+-------------------------------------------+---+---+----------------------------+---+----+
|       |*Abscissor*                     |*Abscissor*                        |*Abscissor*                                        |*Abscissor*                          |
+       +-----------------------+---+----+--------------------------+---+----+-------------------------------------------+---+---+----------------------------+---+----+
|       |0                      |1  |2   |0                         |1  |2   |0                                          |1  |2  |0                           |1  |2   |
+-------+-----------------------+---+----+--------------------------+---+----+-------------------------------------------+---+---+----------------------------+---+----+
|1      |50                     |100|150 |40                        |0  |40  |0                                          |0  |-80|90                          |100|110 |
+-------+-----------------------+---+----+--------------------------+---+----+-------------------------------------------+---+---+----------------------------+---+----+
|2      |0                      |50 |-100|0                         |50 |0   |0                                          |0  |10 |0                           |100|-90 |
+-------+-----------------------+---+----+--------------------------+---+----+-------------------------------------------+---+---+----------------------------+---+----+
|3      |0                      |0  |50  |10                        |-10|-200|90                                         |-40|50 |100                         |-50|-100|
+-------+-----------------------+---+----+--------------------------+---+----+-------------------------------------------+---+---+----------------------------+---+----+
|4      |0                      |0  |0   |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           |**Mechanical stress** (moments and efforts)        |**Constraints** total **es**           |
+       +-----------------------+---+----+--------------------------+---+------+-------------------------------------------+---+---+----------------------------+---+------+
|       |*Abscissor*                     |*Abscissor*                          |*Abscissor*                                        |*Abscissor*                            |
+       +-----------------------+---+----+--------------------------+---+------+-------------------------------------------+---+---+----------------------------+---+------+
|       |0                      |1  |2   |0                         |1  |2     |0                                          |1  |2  |0                           |1  |2     |
+-------+-----------------------+---+----+--------------------------+---+------+-------------------------------------------+---+---+----------------------------+---+------+
|1, 5   |50                     |100|150 |40                        |0  |2**0**|0                                          |0  |-80|90                          |100|9**0**|
+-------+-----------------------+---+----+--------------------------+---+------+-------------------------------------------+---+---+----------------------------+---+------+
|2, 5   |0                      |50 |-100|0                         |50 |0     |0                                          |0  |10 |0                           |100|-90   |
+-------+-----------------------+---+----+--------------------------+---+------+-------------------------------------------+---+---+----------------------------+---+------+
|3, 5   |0                      |0  |50  |10                        |-10|-200  |90                                         |-40|50 |100                         |-50|-100  |
+-------+-----------------------+---+----+--------------------------+---+------+-------------------------------------------+---+---+----------------------------+---+------+

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

These transients 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).

Two other situations are taken into account in this test case. They are provided in the form of two torsors and a unit stress tensor.

Situation 3 is composed of a state A that corresponds to instant 3 of situation 1 and of a state B that corresponds to instant 2 of situation 1. It also consists of a thermal transient which is that of situation 1.

Situation 4 is composed of a state A that corresponds to the instant 1.5 of the situation 2 and of a state B that corresponds to the instant 3.5 of the situation 2. It also consists of a thermal transient which is that of situation 2.


+-----+--------------------------+---+----+-------------------------------------------+---+--+---------------------------------------------+---+----+
|State|**Stress** due to pressure         |**Mechanical stress** (moments and efforts)       |**Restraints** total mechanical constraints**         |
+     +--------------------------+---+----+-------------------------------------------+---+--+---------------------------------------------+---+----+
|     |*Abscissor*                        |*Abscissor*                                       |*Abscissor*                                           |
+     +--------------------------+---+----+-------------------------------------------+---+--+---------------------------------------------+---+----+
|     |0                         |1  |2   |0                                          |1  |2 |0                                            |1  |2   |
+-----+--------------------------+---+----+-------------------------------------------+---+--+---------------------------------------------+---+----+
|A    |10                        |-10|-200|90                                         |-40|50|100                                          |-50|-100|
+-----+--------------------------+---+----+-------------------------------------------+---+--+---------------------------------------------+---+----+
|B    |0                         |50 |0   |0                                          |0  |10|0                                            |100|-90 |
+-----+--------------------------+---+----+-------------------------------------------+---+--+---------------------------------------------+---+----+

Table 1.2-3: Definition of the constraints :math:`{\mathrm{\sigma }}_{\mathit{yy}}` (in :math:`\mathit{MPa}`) of situation 3 as a function of the curvilinear abscissa


+-----+--------------------------+---+------+-------------------------------------------+---+---+---------------------------------------------+---+------+
|State|**Stress** due to pressure           |**Mechanical stress** (moments and efforts)        |**Restraints** total mechanical constraints**           |
+     +--------------------------+---+------+-------------------------------------------+---+---+---------------------------------------------+---+------+
|     |*Abscissor*                          |*Abscissor*                                        |*Abscissor*                                             |
+     +--------------------------+---+------+-------------------------------------------+---+---+---------------------------------------------+---+------+
|     |0                         |1  |2     |0                                          |1  |2  |0                                            |1  |2     |
+-----+--------------------------+---+------+-------------------------------------------+---+---+---------------------------------------------+---+------+
|A    |40                        |0  |2**0**|0                                          |0  |-80|90                                           |100|9**0**|
+-----+--------------------------+---+------+-------------------------------------------+---+---+---------------------------------------------+---+------+
|B    |10                        |-10|-200  |90                                         |-40|50 |100                                          |-50|-100  |
+-----+--------------------------+---+------+-------------------------------------------+---+---+---------------------------------------------+---+------+

**Table** 1.2-4: Definition of constraints :math:`{\sigma }_{\mathit{yy}}` (in :math:`\mathit{MPa}`) of situation 4 as a function of the curvilinear abscissa