Reference solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------

The elastic relationships, connecting membrane forces :math:`N` and bending forces :math:`M` to membrane strains :math:`\varepsilon` and curvatures :math:`\kappa` and taking into account two symmetric grids, are written:

:math:`N\mathrm{=}(\frac{{E}_{b}h}{1\mathrm{-}{\nu }_{b}^{2}}\left[\begin{array}{ccc}1& {\nu }_{b}& 0\\ {\nu }_{b}& 1& 0\\ 0& 0& \frac{1\mathrm{-}{\nu }_{b}}{2}\end{array}\right]+2{E}_{a}\left[\begin{array}{ccc}{a}_{x}& 0& 0\\ 0& {a}_{y}& 0\\ 0& 0& 0\end{array}\right])\varepsilon`

:math:`M\mathrm{=}(\frac{{E}_{b}{h}^{3}}{12(1\mathrm{-}{\nu }_{b}^{2})}\left[\begin{array}{ccc}1& {\nu }_{b}& 0\\ {\nu }_{b}& 1& 0\\ 0& 0& \frac{1\mathrm{-}{\nu }_{b}}{2}\end{array}\right]+2{E}_{a}{e}_{z}^{2}\left[\begin{array}{ccc}{a}_{x}& 0& 0\\ 0& {a}_{y}& 0\\ 0& 0& 0\end{array}\right])\kappa`

In the case of a beam configuration, a Poisson's ratio equal to 0 is assigned to the concrete to cancel out any deflection in the perpendicular direction. Two opposite edges of the slab are simply pressed, with two others remaining free:


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|                                                                                                                             |The equivalent bending stiffness that takes steels into account is: :math:`{(\mathit{EI})}_{\mathit{éq}}\mathrm{=}\frac{{E}_{b}l{h}^{3}}{12}+2{E}_{a}{a}_{x}l{e}_{z}^{2}`, or here: :math:`{(\mathit{EI})}_{\mathit{éq}}\mathrm{=}10.111{\mathit{MNm}}^{2}`.|
+ .. image:: images/100000000000031C00000202AE88E7C796C3E04E.png                                                              +                                                                                                                                                                                                                                                            +
|    :width: 2.6591in                                                                                                         |                                                                                                                                                                                                                                                            |
+    :height: 1.6571in                                                                                                        +                                                                                                                                                                                                                                                            +
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The elastic solution is calculated in beam theory for an equivalent pressure value :math:`p\text{'}=\mathrm{pl}`. The values of the moments in the "plate" configuration are obtained by division by the width of the slab :math:`l`.


.. csv-table::

    "Size in the center", "Expression"
    "Center arrow under surface pressure", ":math:`w(l\mathrm{/}2)\mathrm{=}\frac{5p\text{'}{l}^{4}}{384{(\mathit{EI})}_{\mathit{éq}}}`"
    "Curvature", ":math:`{\kappa }_{\mathit{xx}}(l\mathrm{/}2)\mathrm{=}\frac{p\text{'}{l}^{2}}{8{(\mathit{EI})}_{\mathit{éq}}}`"
    "Deformation", ":math:`{\varepsilon }_{\mathit{xx}}\mathrm{=}{\kappa }_{\mathit{xx}}\frac{h}{2}`"
    "Global moment (in addition)", ":math:`M(l\mathrm{/}2)\mathrm{=}p\text{'}{l}^{2}\mathrm{/}8`"
    "Global moment (in plate)", ":math:`M(l\mathrm{/}2)\mathrm{=}p{l}^{2}\mathrm{/}8`"



Benchmark results
----------------------

For A and B models in which we validate the GLRC_DAMA law with the DKTG elements:


* Arrow in the center under surface pressure: :math:`w=\mathrm{2,433}\mathrm{.}{10}^{-4}m`


* Curvature: :math:`k=\mathrm{7,210}\mathrm{.}{10}^{-4}{m}^{-1}`


* Deformity: :math:`{\epsilon }_{\mathit{xx}}=-0.4326\mathrm{.}{10}^{-4}` on the lower skin


* Global moment (in addition): :math:`M=7290\mathit{Nm}`


* Global moment (in plate): :math:`M=4050\mathit{Nm}/\mathit{ml}`


*

For C and D models in which we validate the ELAS law with the Q4GG elements:


* Arrow in the center under surface pressure: :math:`w=\mathrm{2,658}\mathrm{.}{10}^{-4}m`


* Curvature: :math:`k=\mathrm{7,878}\mathrm{.}{10}^{-4}{m}^{-1}`


* Deformity: :math:`{\epsilon }_{\mathit{xx}}=-0.47269\mathrm{.}{10}^{-4}` on the lower skin


* Global moment (in addition): :math:`M=7290\mathrm{Nm}`


* Global moment (in plate): :math:`M=4050\mathrm{Nm}/\mathrm{ml}`


Uncertainty about the solution
---------------------------

Analytical solution.


Bibliographical reference
-------------------------

[:ref:`1 <1>`] KOECHLIN P., MOULIN S., "Global behavior model of reinforced concrete plates under dynamic flexural loading: Law GLRC ", Note EDF /R &D/ AMA HT-62/01/028A.