Benchmark 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])\epsilon`

: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}_{s}^{2}\left[\begin{array}{ccc}{a}_{x}& 0& 0\\ 0& {a}_{y}& 0\\ 0& 0& 0\end{array}\right])\kappa`

As it is a slab configuration, the concrete is assigned the usual Poisson's ratio :math:`{\nu }_{b}=\mathrm{0,22}`. The slab is simply supported on the four edges:


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|                                                                                                                   |The equivalent bending stiffness is:                                                                                                                                                                                                                                                                                                                                                                         |
+ .. image:: images/100013D6000069BB00005245D61C86F33C5274F5.svg                                                    +                                                                                                                                                                                                                                                                                                                                                                                                             +
|    :width: 218                                                                                                    | :math:`{D}_{\mathit{éq}}\mathrm{=}\frac{{E}_{b}{h}^{3}}{12(1\mathrm{-}{\nu }_{b}^{2})}+2{E}_{a}{e}_{s}^{2}{a}_{x}`, or here: :math:`{D}_{\mathrm{éq}}=\mathrm{5,8786}\text{MNm}`; In addition: :math:`{\nu }_{\mathrm{éq}}=\frac{{\nu }_{b}{E}_{b}{h}^{3}}{12(1-{\nu }_{b}^{2}){D}_{\mathrm{éq}}}`, that is: :math:`{\nu }_{\mathrm{éq}}=\mathrm{0,2022}`                                                   |
+    :height: 169                                                                                                   +                                                                                                                                                                                                                                                                                                                                                                                                             +
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.. csv-table::

    "Size", "Expression"
    "Center arrow under pressure [:ref:`2 <2>`]", ":math:`w\mathrm{=}\frac{\mathrm{0,0464}{\mathit{pl}}^{4}}{12(1\mathrm{-}{\nu }_{\mathit{éq}}^{2}){D}_{\mathit{éq}}}`"
    "Center curve [:ref:`2 <2>`]", ":math:`{\kappa }_{\mathit{xx}}\mathrm{=}{\kappa }_{\mathit{yy}}\mathrm{=}\frac{\mathrm{0,04784}{\mathit{pl}}^{2}}{(1+{\nu }_{\mathit{éq}}){D}_{\mathit{éq}}}`"
    "Global moment in the center [:ref:`2 <2>`]", ":math:`{M}_{\mathrm{xx}}={M}_{\mathrm{yy}}=\mathrm{0,04784}{\mathrm{pl}}^{2}`"



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

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


* Center arrow under pressure: :math:`w=\mathrm{6,926}{.10}^{-5}\text{m}`


* Curvature at the center: :math:`{\kappa }_{\mathit{xx}}\mathrm{=}{\kappa }_{\mathit{yy}}\mathrm{=}\mathrm{2,193}{.10}^{\mathrm{-}4}{\text{m}}^{\mathrm{-}1}`


* Global moment at the center: :math:`{M}_{\mathit{xx}}\mathrm{=}{M}_{\mathit{yy}}\mathrm{=}1550\text{Nm/ml}`

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


* Center arrow under pressure: :math:`w=\mathrm{7,895}{.10}^{-5}\text{m}`


* Curvature at the center: :math:`{\kappa }_{\mathit{xx}}={\kappa }_{\mathit{yy}}=\mathrm{2,351}{.10}^{-4}{\text{m}}^{-1}`


* Global moment at the center: :math:`{M}_{\mathit{xx}}\mathrm{=}{M}_{\mathit{yy}}\mathrm{=}1550\text{Nm/ml}`


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

Analytical solution.


Bibliographical references
---------------------------

[: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.

[:ref:`2 <2>`] J.Dulac, "Elasto-plastic dynamic behavior of reinforced concrete slabs. Essays CEMETE — December 1979 — Slabs 8 to 12": Note EDF: ESE /GC/82/13/A