2. Benchmark solution#
2.1. Calculation method used for the reference solution#
The elastic relationships, connecting membrane forces \(N\) and bending forces \(M\) to membrane strains \(\varepsilon\) and curvatures \(\kappa\) and taking into account two symmetric grids, are written:
\(N\mathrm{=}(\frac{{E}_{b}h}{1\mathrm{-}{\nu }_{b}^{2}}\left[\begin{array}{ccc}1& {\nu }_{b}& 0\\ {\nu }_{b}& 1& 0\\ 0& 0& \frac{1\mathrm{-}{\nu }_{b}}{2}\end{array}\right]+2{E}_{a}\left[\begin{array}{ccc}{a}_{x}& 0& 0\\ 0& {a}_{y}& 0\\ 0& 0& 0\end{array}\right])\epsilon\)
\(M\mathrm{=}(\frac{{E}_{b}{h}^{3}}{12(1\mathrm{-}{\nu }_{b}^{2})}\left[\begin{array}{ccc}1& {\nu }_{b}& 0\\ {\nu }_{b}& 1& 0\\ 0& 0& \frac{1\mathrm{-}{\nu }_{b}}{2}\end{array}\right]+2{E}_{a}{e}_{s}^{2}\left[\begin{array}{ccc}{a}_{x}& 0& 0\\ 0& {a}_{y}& 0\\ 0& 0& 0\end{array}\right])\kappa\)
As it is a slab configuration, the concrete is assigned the usual Poisson’s ratio \({\nu }_{b}=\mathrm{0,22}\). The slab is simply supported on the four edges:
|
The equivalent bending stiffness is:
|
Size |
Expression |
Center arrow under pressure [2] |
\(w\mathrm{=}\frac{\mathrm{0,0464}{\mathit{pl}}^{4}}{12(1\mathrm{-}{\nu }_{\mathit{éq}}^{2}){D}_{\mathit{éq}}}\) |
Center curve [2] |
\({\kappa }_{\mathit{xx}}\mathrm{=}{\kappa }_{\mathit{yy}}\mathrm{=}\frac{\mathrm{0,04784}{\mathit{pl}}^{2}}{(1+{\nu }_{\mathit{éq}}){D}_{\mathit{éq}}}\) |
Global moment in the center [2] |
\({M}_{\mathrm{xx}}={M}_{\mathrm{yy}}=\mathrm{0,04784}{\mathrm{pl}}^{2}\) |
2.2. Benchmark results#
For A and B models in which we validate the GLRC_DAMA law with the DKTG elements:
Center arrow under pressure: \(w=\mathrm{6,926}{.10}^{-5}\text{m}\)
Curvature at the center: \({\kappa }_{\mathit{xx}}\mathrm{=}{\kappa }_{\mathit{yy}}\mathrm{=}\mathrm{2,193}{.10}^{\mathrm{-}4}{\text{m}}^{\mathrm{-}1}\)
Global moment at the center: \({M}_{\mathit{xx}}\mathrm{=}{M}_{\mathit{yy}}\mathrm{=}1550\text{Nm/ml}\)
For C and D models in which we validate the ELAS law with the DKTG elements:
Center arrow under pressure: \(w=\mathrm{7,895}{.10}^{-5}\text{m}\)
Curvature at the center: \({\kappa }_{\mathit{xx}}={\kappa }_{\mathit{yy}}=\mathrm{2,351}{.10}^{-4}{\text{m}}^{-1}\)
Global moment at the center: \({M}_{\mathit{xx}}\mathrm{=}{M}_{\mathit{yy}}\mathrm{=}1550\text{Nm/ml}\)
2.3. Uncertainty about the solution#
Analytical solution.
2.4. Bibliographical references#
[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.
[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