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


Geometry
---------

The geometry used in this test case is a reinforced concrete plate with a thickness of :math:`e=0.1\text{m}` and a length of :math:`l=1\text{m}`.


.. image:: images/Object_4.svg
    :width: 338
    :height: 207


.. _RefImage_Object_4.svg:

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

The characteristics of the steel sheets of the reinforced concrete plate are:


* Upper sheet: section per linear meter :math:`=0.05{m}^{\mathrm{²}}/\mathrm{ml}`; eccentricity with respect to the middle sheet: :math:`+0.03m`,


* Lower sheet: section per linear meter :math:`=0.05{m}^{\mathrm{²}}/\mathrm{ml}`; eccentricity compared to the middle sheet: :math:`–0.03m`.


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

The material characteristics for multi-layer concrete modeling with steel reinforcements (DKT and GRILLE_EXCENTRE) are summarized in the following table.


.. csv-table::

    "**Modeling**", "**Young Module** :math:`N/{m}^{\mathrm{²}}` ", "**Fish Coefficient**", "**Density** :math:`\mathrm{kg}/{m}^{3}`"
    "Concrete (plate DKT)", "1. 1010", "0.0", "2500"
    "Steel (GRILLE_EXCENTRE)", "1. 1011", "0.0", "7800"



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

On the :math:`A` (:math:`\mathrm{B0X}`) side of the plate are embedded the movements :math:`{u}_{x}={u}_{y}={u}_{z}=0`, as well as the rotations :math:`{\theta }_{x}={\theta }_{y}={\theta }_{z}=0`. During the modal calculation, displacement :math:`{u}_{y}=0` is blocked everywhere on the plate. A linear force is applied to the :math:`\mathrm{B1X}` side (side opposite :math:`\mathrm{B0X}`) in the direction :math:`(0.0\mathrm{,0}.0\mathrm{,1}\mathrm{,0})` and depends on the value :math:`{F}_{0}={10}^{6}N`.

In the case of dynamic calculation, a sinusoidal linear force is applied on the :math:`\mathrm{B1X}` side in the :math:`(0.0\mathrm{,0}.0\mathrm{,1}.0)` direction. The sine wave frequency is :math:`20\mathrm{Hz}`. The duration of the request is :math:`\mathrm{0,1}s`.


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

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