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


Geometry
---------

The geometry used in this test case is a reinforced concrete plate with a thickness of :math:`e\mathrm{=}0.1\text{m}` and a trapezoidal shape.


.. image:: images/Object_4.png
    :width: 4.7925in
    :height: 2.7299in


.. _RefSchema_Object_4.png:

**Figure** 1.1-a **: Geometry studied**

The characteristics of the reinforced concrete section are:


* Upper sheet: section per linear meter following :math:`x` and :math:`y` :math:`=5.65{10}^{-4}{m}^{\mathrm{²}}/\mathrm{ml}`; eccentricity with respect to the following mean sheet :math:`x` and :math:`y`: :math:`+0.0475m` (i.e. 95% of the thickness),


* Lower sheet: section per linear meter following :math:`x` and :math:`y` :math:`=5.65{10}^{-4}{m}^{\mathrm{²}}/\mathrm{ml}`; eccentricity with respect to the following mean sheet :math:`x` and :math:`y`: :math:`-0.0475m` (i.e. 95% of the thickness),


* Prestress cables: section per linear meter following :math:`x=4.56{10}^{-3}{m}^{\mathrm{²}}/\mathrm{ml}` and :math:`y=1.32{10}^{-2}{m}^{\mathrm{²}}/\mathrm{ml}`; no eccentricity compared to the middle sheet; pretension following :math:`x` and :math:`y` :math:`=-3\mathrm{MN}`,


* Liner: the thickness of the liner is :math:`6\mathrm{mm}` and is positioned on the underside.


.. image:: images/10000000000004BC00000244BF10CC0BD09D7C5B.jpg
    :width: 4.9709in
    :height: 2.35in


.. _RefImage_10000000000004BC00000244BF10CC0BD09D7C5B.jpg:

**Figure** 1.1-b **: Reinforced concrete plate section**


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

The characteristics of the various materials for modeling GLRC_DAMAGE are summarized in the following table.


.. csv-table::

    "**Material**", "**Young's Module** :math:`\mathrm{MPa}` ", "**Fish Coefficient**", "**Density** :math:`\mathrm{kg}/{m}^{3}` ", "**Work hardening slope**", "**Elastic tensile limit** ", "**Elastic tensile limit** :math:`\mathrm{MPa}` ", "**Elastic compression limit** :math:`\mathrm{MPa}`"
    "Concrete", "30,000. ", "0.2", "2500", "0", "0", "5", "-35"
    "Reinforcement steel", "200000", "-", "-", "-", "0", "3000", "-3000", "-3000"
    "Liner steel and pretension cables", "200000", "200000", "-", "-", "0", "500", "-500"


To complete the law of behavior GLRC_DAMAGE, it is necessary to define the globalized parameters of homogenized law.


.. csv-table::

    "**Settings**", "**Values**"
    ":math:`\mathit{Gamma}` "," :math:`0`"
    ":math:`\mathit{QP1}` "," :math:`0.15`"
    ":math:`\mathit{QP2}` "," :math:`0.15`"
    ":math:`{C}_{N}` "," :math:`87.3\mathit{MPa}`"
    ":math:`{C}_{M}` "," :math:`14.8\mathit{MPa}`"



.. image:: images/Object_3.svg
    :width: 250
    :height: 137


.. _RefImage_Object_3.svg:

**Figure** 1.2-a **: Moment—curvature curve of the behavior of a reinforced concrete plate under bending.**

The material characteristics for modeling GLRC_DM are summarized in the following table.


.. csv-table::

    "**Settings**", "**Values**"
    ":math:`{E}_{\mathit{éq}}^{m}` "," :math:`30000\mathit{MPa}`"
    ":math:`{\nu }_{m}` "," :math:`0.22`"
    ":math:`{E}_{\mathit{éq}}^{f}` "," :math:`73000\mathit{MPa}`"
    ":math:`{\nu }_{f}` "," :math:`0.24`"
    ":math:`{\gamma }_{\mathit{mt}}` "," :math:`0.02`"
    ":math:`{\gamma }_{f}` "," :math:`0.05`"
    ":math:`{N}_{D}` "," :math:`470000N/m`"
    ":math:`{M}_{D}` "," :math:`16000N`"



Boundary conditions and loads
-------------------------------------

On corner :math:`\mathrm{A1}` of the plate, we embed the movements :math:`{u}_{x}={u}_{y}={u}_{z}=0`, as well as the rotations :math:`{\theta }_{x}\mathrm{=}{\theta }_{y}\mathrm{=}{\theta }_{z}\mathrm{=}0`. Travel is blocked following :math:`x` and :math:`z` on the sides :math:`\mathrm{A1A3}` *and* :math:`\mathrm{A2A4}`. Pressure is applied to the entire slab in the direction :math:`(0.0\mathrm{,0}.0\mathrm{,1}\mathrm{,0})` and is equal to :math:`{F}_{0}=20.{10}^{7}N` for modeling A. For models B and C, a nodal force is applied to the whole slab :math:`1500N`. This force is applied progressively by following the multiplier function shown in the following figure.


.. image:: images/10000000000001E3000001234BCE459B0C8DE952.png
    :width: 3.9366in
    :height: 2.3661in


.. _RefImage_10000000000001E3000001234BCE459B0C8DE952.png:

Figure 1.3-a: Load multiplier function for B and C models


Initial conditions
--------------------

In the initial state, displacements and speeds are zero everywhere.