F modeling
==============


Characteristics of modeling
-----------------------------------

The loading is of the pure alternating bending type.


.. image:: images/10000000000001A9000001269D6B8497D226D3CE.png
    :width: 3.7043in
    :height: 2.2953in


.. _RefImage_10000000000001A9000001269D6B8497D226D3CE.png:

**Figure** 8.1-a **: mesh and boundary conditions**

Modeling: DKTG

Boundary conditions:


* :math:`\mathrm{DRY}=0.0` on the :math:`{A}_{1}-{A}_{3}` ridge


* :math:`\mathrm{DRY}={R}_{0}\times f(t)` on the :math:`{A}_{2}-{A}_{4}` edge,

where :math:`{R}_{0}=3.0\times {10}^{-2}` and :math:`f(t)` is the magnitude of the cyclic loading as a function of the (pseudo-time) parameter :math:`t`.

To verify the model, consider the following loading function:


.. image:: images/10000000000001F8000001201AA96929C8D1ABF2.png
    :width: 3.5398in
    :height: 2.022in


.. _RefImage_10000000000001F8000001201AA96929C8D1ABF2.png:

**Figure** 8.1-b **: loading function**


Characteristics of the mesh
----------------------------

Number of knots: 9.

Number of stitches: 8 TRIA3; 8 SEG2.


Tested sizes and results
------------------------------

The reaction moments along the :math:`\mathit{Oy}` axis in :math:`\mathit{A1}\mathrm{-}\mathit{A3}` and the rotations along the :math:`\mathit{Ox}` *en* :math:`\mathit{A4}` axis obtained by multilayer modeling with the ENDO_ISOT_BETON law and by the one based on the BETON_REGLE_PR law are compared, in terms of relative differences; the tolerances are taken in absolute value:


.. csv-table::

    "**Identification**", "**Reference type**", "**Reference value**", "**Tolerance**"
    "**FLEXION POSITIVE -** **ELASTIQUE** :math:`t=\mathrm{0,25}` ", "", "", ""
    "Relative difference :math:`\mathit{MY}` "," NON_REGRESSION ", "-", "1 10-6"
    "*Relative difference* :math:`\mathit{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6"
    "**FLEXION POSITIVE -** **ENDOMMAGEMENT** :math:`t=\mathrm{1,0}` ", "", "", ""
    "Relative difference :math:`\mathit{MY}` "," NON_REGRESSION ", "-", "1 10-6"
    "**FLEXION POSITIVE -** **DECHARGEMENT** :math:`t=\mathrm{1,5}` ", "", "", ""
    "Relative difference :math:`\mathit{MY}` "," NON_REGRESSION ", "-", "1 10-6"
    "**FLEXION NEGATIVE — ELASTIQUE** :math:`t=\mathrm{2,25}` ", "", "", ""
    "Relative difference :math:`\mathit{MY}` "," NON_REGRESSION ", "-", "1 10-6"
    "FLEXION NEGATIVE - ENDOMMAGEMENT :math:`t=\mathrm{3,0}` ", "", "", ""
    "Relative difference :math:`\mathit{MY}` "," NON_REGRESSION ", "-", "1 10-6"
    "**FLEXION NEGATIVE** **-** **** DECHARGEMENT ** :math:`t=\mathrm{3,5}` ", "", "", ""
    "Relative difference :math:`\mathit{MY}` "," NON_REGRESSION ", "-", "1 10-6"


**Comparative moment graphs** :math:`\mathrm{MY}` **— rotation** :math:`\mathrm{DRY}` **in alternating flexure for load** :math:`f` **:**: **


.. image:: images/10000201000002CD00000235931637CEAC41C4CE.png
    :width: 4.6457in
    :height: 3.5783in


.. _RefImage_10000201000002CD00000235931637CEAC41C4CE.png:

**Comparative rotation graphs** :math:`\mathit{DRX}` **(due to the Poisson effect) as a function of time:**


.. image:: images/10000201000002ED00000235542842B37B17D9A9.png
    :width: 4.6457in
    :height: 3.1299in


.. _RefImage_10000201000002ED00000235542842B37B17D9A9.png:


notes
---------

The test case carried out here aims to test model BETON_REGLE_PRsous stresses that are significant enough for steels to effectively recover their stiffness.

The behavior is similar in bending for the laws BETON_REGLE_PRet ENDO_ISOT_BETON under load: the differences appear for large loads due to the difference in behavior under compression.

The landfill response is not taken into account by law BETON_REGLE_PR (elastic response).

We observe a symmetry of the response for law BETON_REGLE_PR.

The :math:`\mathrm{DRX}` rotation is zero with the BETON_REGLE_PR law because the Poisson effect is not taken into account.


.. _refnumpara__6727_1772789992: