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

Geometry
---------

Square tile, :math:`l\mathrm{=}1.8m` long, :math:`h=0.12m` thick, with simple support on all four edges. The flexure reinforcement is parallel to the edges; it is identical on each of both faces and in each of the two directions (:math:`\mathrm{dx}`, :math:`\mathrm{dy}` being the spacings of the bars in the directions :math:`x` and :math:`y`). The coating of the longitudinal bars closest to the faces is :math:`22\mathrm{mm}`. The coating of the irons in relation to the lateral edges of the :math:`2\mathrm{cm}` slab is neglected. The table below summarizes the reinforcement data. The geometric percentage of steel :math:`\mu` is given for one face in one direction.


.. csv-table::

    "Reinforcement diameter", "Spacing", "Steel section/concrete section", "grid distance/mean slab area"
    ":math:`\Phi =\mathrm{0,01}\text{m}` "," :math:`\mathrm{dx}=\mathrm{dy}=\mathrm{0,1}\text{m}` "," :math:`\mu =\mathrm{0,65}\text{}` "," :math:`{e}_{s}=\pm \mathrm{0,038}\text{m}`"



.. image:: images/Object_4.svg
    :width: 423
    :height: 211


.. _RefImage_Object_4.svg:

We denote :math:`{a}_{x}=\frac{{A}_{x}}{{d}_{x}}` and :math:`{a}_{y}=\frac{{A}_{y}}{{d}_{y}}` the reinforcement rates (here: :math:`{a}_{x}={a}_{y}=\mathrm{7,854}{.10}^{-4}\text{m}`), with :math:`{A}_{x}` (:math:`{A}_{y}`) being the area of the section of an iron bar in the :math:`x(y)` direction; :math:`{e}_{s}` is the distance of the sheets from the mean surface.


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

The mechanical properties of steels are as follows:


.. csv-table::

    "Young's Modulus :math:`{E}_{a}` ", "Poisson's Ratio", "Elastic limit at 0.2% :math:`{\sigma }_{y}` ", "Fracture limit :math:`{\sigma }_{r}` ", "Work hardening slope", "Elongation at break"
    ":math:`210000\text{MPa}` ", "0.3"," :math:`500\text{MPa}` "," "," :math:`570\text{MPa}` "," :math:`473\text{MPa}` ", "15%"


Those of concrete are as follows:


.. csv-table::

    "Young's Module :math:`{E}_{b}` ", "Poisson's Ratio", "Compressive Strength :math:`{\sigma }_{c}` ", "Tensile Strength :math:`{\sigma }_{t}`"
    ":math:`35700\text{MPa}` ", "0.22"," :math:`\mathrm{52,5}\text{MPa}` "," :math:`\mathrm{4,4}\text{MPa}`"



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


* The boundary conditions boil down to simple supports: blocked vertical movement and free rotations on the four edges of the slab.


* Uniform pressure :math:`p\mathrm{=}\mathrm{0,01}\mathit{MPa}`


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

Not applicable.