E modeling
==============


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

The load is of the tensile — pure compression type.


.. image:: images/1000000000000320000001E18AA1E5624797F771.jpg
    :width: 3.3354in
    :height: 1.9992in


.. _RefImage_1000000000000320000001E18AA1E5624797F771.jpg:

**Figure** 7.1-a **: mesh and boundary conditions.**

Modeling: DKTG

Boundary conditions:


* Embedding in :math:`{A}_{1}`;


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


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

where :math:`{U}_{0}=1.0\times {10}^{-3}m` and :math:`f(t)` represent the amplitude of cyclic loading as a function of the (pseudo-time) parameter :math:`t`. To properly verify the model, we consider a loading function as follows:


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


.. _RefImage_10000000000001F8000001201AA96929C8D1ABF2.png:

**Figure** 7.1-b **: loading function: traction, then compression**

*Note: the extreme deformation is:* :math:`1.0\times {10}^{-3}` *, or about a third of the deformation during the transition to plasticity of steels.* *No* *integration time:* :math:`0.025` *.*


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

Number of knots: 9.

Number of stitches: 8 TRIA3; 8 SEG2.


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

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


.. csv-table::

    "**Identification**", "**Reference type**", "**Reference value**", "**Tolerance**"
    "**TRACTION -** **ELASTIQUE** :math:`t=\mathrm{0,25}` ", "", "", ""
    "Relative difference :math:`\mathit{FX}` "," NON_REGRESSION ", "-", "1 10-6"
    "*Relative difference* :math:`\mathrm{DY}` "," NON_REGRESSION "," ", "-", "1 10-6"
    "**TRACTION -** **ENDOMMAGEMENT** :math:`t=\mathrm{1,0}` ", "", "", ""
    "*Relative difference* :math:`\mathrm{FX}` "," NON_REGRESSION "," ", "-", "1 10-6"
    "**TRACTION -** **DECHARGEMENT** :math:`t=\mathrm{1,5}` ", "", "", ""
    "*Relative difference* :math:`\mathrm{FX}` "," NON_REGRESSION "," ", "-", "1 10-6"
    "**COMPRESSION - CHARGEMENT** :math:`t=\mathrm{2,5}` ", "", "", ""
    "*Relative difference* :math:`\mathrm{FX}` "," NON_REGRESSION "," ", "-", "1 10-6"
    "**COMPRESSION -** **DECHARGEMENT** :math:`t=\mathrm{3,5}` ", "", "", ""
    "*Relative difference* :math:`\mathrm{FX}` "," NON_REGRESSION "," ", "-", "1 10-6"
    "", "", "", ""


**Comparative effort graphs** :math:`{N}_{\mathrm{xx}}` **— displacement** :math:`\mathrm{DX}` ****in****traction —****compression for load**:math:`f`**: **:**


.. image:: images/10000201000002DC00000236E2455D1AFB076A0F.png
    :width: 4.6457in
    :height: 3.7283in


.. _RefImage_10000201000002DC00000236E2455D1AFB076A0F.png:

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


.. image:: images/10000201000002E3000002355DFBBB8EF5C9DA8E.png
    :width: 4.6457in
    :height: 3.311in


.. _RefImage_10000201000002E3000002355DFBBB8EF5C9DA8E.png:


.. _RefNumPara__30473097:


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 traction for laws BETON_REGLE_PRet ENDO_ISOT_BETON under load: the differences appear for a significant compression due to the parabolic response of law BETON_REGLE_PR. In traction, after the concrete ruins, we find the stiffness corresponding to the resumption of forces by the steel.

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

The deformation due to the Poisson effect is not taken into account by law BETON_REGLE_PR. For law ENDO_ISOT_BETON, DY displacements are not zero before the appearance of concrete ruins around 0.1 s.

Since the convergence of the model with law BETON_REGLE_PRest is difficult around time 0.6s (stiffness recovery) with the matrix TANGENTE, the matrix ELASTIQUE is then used.




.. _refnumpara__12085_1477809071: