G modeling
==============


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

Loading is a combination of traction — compression and flexure.


.. image:: images/1000020000000189000000EB5E0D602A193B4B87.png
    :width: 4.272in
    :height: 2.5339in


.. _RefImage_1000020000000189000000EB5E0D602A193B4B87.png:

**Figure** 9.1-a **: mesh and boundary conditions**

Modeling: DKTG

Boundary conditions: Traction — Compression and Flexion coupling:


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


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

where :math:`{U}_{0}\mathrm{=}1.\times {10}^{\mathrm{-}3}m`, :math:`{R}_{0}\mathrm{=}3.\times {10}^{\mathrm{-}2}\mathit{rad}`, and :math:`f(t)` is the magnitude of the cyclic loading as a function of the (pseudo-time) parameter :math:`t`.

The following load is considered:

The same :math:`f` loading function for membrane and flexure:


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


.. _RefImage_10000000000001F8000001201AA96929C8D1ABF2.png:

**Figure** 9.1-b **: loading function**


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

Number of knots: 9.

Number of stitches: 8 TRIA3; 8 SEG2.


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

We compare the forces along the axis :math:`\mathrm{Ox}` in :math:`\mathit{A1}\mathrm{-}\mathit{A3}`, the displacements along the axis :math:`\mathrm{Oy}` in :math:`\mathit{A4}`, the moments along the axis :math:`\mathrm{Oy}` in :math:`\mathit{A1}\mathrm{-}\mathit{A3}` and the rotations along the axis :math:`\mathrm{Ox}` *en* :math:`\mathit{A4}` obtained by the multilayer modeling (reference) and by those based on the model ENDO_ISOT_BETON, in in terms of relative differences; the tolerance is taken as an absolute value based on these relative differences:


.. 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"
    "Relative difference :math:`\mathit{FX}` "," NON_REGRESSION ", "-", "1 10-6"
    "*Relative difference* :math:`\mathit{DY}` "," NON_REGRESSION "," ", "-", "1 10-6"
    "**FLEXION POSITIVE -** **ENDOMMAGEMENT** :math:`t=\mathrm{1,0}` ", "", "", ""
    "Relative difference :math:`\mathit{MY}` "," NON_REGRESSION ", "-", "1 10-6"
    "Relative difference :math:`\mathit{FX}` "," NON_REGRESSION ", "-", "1 10-6"
    "**FLEXION POSITIVE -** **DECHARGEMENT** :math:`t=\mathrm{1,5}` ", "", "", ""
    "Relative difference :math:`\mathit{MY}` "," NON_REGRESSION ", "-", "1 10-6"
    "Relative difference :math:`\mathit{FX}` "," NON_REGRESSION ", "-", "1 10-6"
    "**FLEXION NEGATIVE — ELASTIQUE** :math:`t=\mathrm{2,25}` ", "", "", ""
    "Relative difference :math:`\mathit{MY}` "," NON_REGRESSION ", "-", "1 10-6"
    "Relative difference :math:`\mathit{FX}` "," NON_REGRESSION ", "-", "1 10-6"
    "FLEXION NEGATIVE - ENDOMMAGEMENT :math:`t=\mathrm{3,0}` ", "", ""
    "Relative difference :math:`\mathit{MY}` "," NON_REGRESSION ", "-", "1 10-6"
    "Relative difference :math:`\mathit{FX}` "," NON_REGRESSION ", "-", "1 10-6"
    "**FLEXION NEGATIVE** **-** **** DECHARGEMENT ** :math:`t=\mathrm{3,5}` ", "", "", ""
    "Relative difference :math:`\mathit{MY}` "," NON_REGRESSION ", "-", "1 10-6"
    "Relative difference :math:`\mathit{FX}` "," NON_REGRESSION ", "-", "1 10-6"



.. image:: images/10000201000002DC0000021F48A439D54A05F1F9.png
    :width: 4.6457in
    :height: 3.1299in


.. _RefImage_10000201000002DC0000021F48A439D54A05F1F9.png:

**Force comparative graphs** :math:`\mathrm{FX}` **— displacement** :math:`\mathrm{DX}` **for load** :math:`f` **:**

**Comparative moment graphs** :math:`\mathrm{MY}` ****— rotation**:math:`\mathrm{DRY}`**for loading**:math:`f`**: **


.. image:: images/10000201000002CD0000021F04D981AD7BB92418.png
    :width: 4.6457in
    :height: 3.4217in


.. _RefImage_10000201000002CD0000021F04D981AD7BB92418.png:

 **Comparative graphs** displacement :math:`\mathrm{DY}` (due to the Poisson effect) as a function of time:


.. image:: images/10000201000002E30000023567B941E3D007A585.png
    :width: 4.6457in
    :height: 3.311in


.. _RefImage_10000201000002E30000023567B941E3D007A585.png:

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


.. image:: images/10000201000002EC000002356409F4341D846AB7.png
    :width: 4.6457in
    :height: 3.2445in


.. _RefImage_10000201000002EC000002356409F4341D846AB7.png:


notes
---------

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

The behavior is similar in bending and traction 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).

Rotation :math:`\mathrm{DRX}` and displacement :math:`\mathit{DY}` are zero with law BETON_REGLE_PR because the Poisson effect is not taken into account.

.. _refnumpara__30394423: