2. Benchmark solution#
2.1. Calculation method used for the reference solution#
The analytical reference solution is based on the Love-Kirchhoff theory, commonly used for so-called « thin » plates [bia1].
Taking into account the problem and at any point on the plate, we have for the calculation of the arrow:
\(w\mathrm{=}\frac{{f}_{0}{a}^{4}}{4{\pi }^{4}D}\mathrm{sin}\pi \frac{x}{a}\mathrm{sin}\pi \frac{y}{a}\)
with:
\(D\mathrm{=}\frac{E{h}^{3}}{12(1\mathrm{-}{\nu }^{2})}\), \({f}_{0}\mathrm{=}1\), \(a\mathrm{=}1\), and \(\nu \mathrm{=}0.25\)
For the calculation of moments, the theory leads to the following expressions:
\(\begin{array}{ccc}{M}_{\mathit{xx}}& \mathrm{=}& \alpha (1+\nu )\mathrm{sin}\pi \frac{x}{a}\mathrm{sin}\pi \frac{y}{a}\\ {M}_{\mathit{yy}}& \mathrm{=}& {M}_{\mathit{xx}}\\ {M}_{\mathit{xy}}& \mathrm{=}& \mathrm{-}\alpha (1\mathrm{-}\nu )\mathrm{cos}\pi \frac{x}{a}\mathrm{cos}\pi \frac{y}{a}\end{array}\)
For the calculation of curvatures, the theory leads to the following expressions:
\({\kappa }_{\mathit{xx}}\mathrm{=}\mathrm{-}\frac{{f}_{0}{a}^{2}}{4{\pi }^{2}D}\mathrm{sin}(\pi \frac{x}{a})\mathrm{sin}(\pi \frac{y}{a})\)
\({\kappa }_{\mathit{yy}}\mathrm{=}\mathrm{-}\frac{{f}_{0}{a}^{2}}{4{\pi }^{2}D}\mathrm{sin}(\pi \frac{x}{a})\mathrm{sin}(\pi \frac{y}{a})\)
\({\kappa }_{\mathit{xy}}\mathrm{=}\frac{{f}_{0}{a}^{2}}{4{\pi }^{2}D}\mathrm{cos}(\pi \frac{x}{a})\mathrm{cos}(\pi \frac{y}{a})\)
with \(\alpha \mathrm{=}\frac{{f}_{0}{a}^{2}}{4{\pi }^{2}}\)
For shear forces, we get:
\(\begin{array}{c}{T}_{x}\mathrm{=}\frac{{f}_{0}a}{2\pi }\mathrm{cos}\pi \frac{x}{a}\mathrm{sin}\pi \frac{y}{a}\\ {T}_{y}\mathrm{=}\frac{{f}_{0}a}{2\pi }\mathrm{sin}\pi \frac{x}{a}\mathrm{cos}\pi \frac{y}{a}\end{array}\)
For a homogeneous plate, the plane stresses are given by:
\((\begin{array}{c}{\sigma }_{\mathit{xx}}\\ {\sigma }_{\mathit{yy}}\\ {\sigma }_{\mathit{xy}}\end{array})\mathrm{=}z\mathrm{[}A\mathrm{]}(\begin{array}{c}{M}_{\mathit{xx}}\\ {M}_{\mathit{yy}}\\ {M}_{\mathit{xy}}\end{array})\)
with \(\mathrm{[}A\mathrm{]}\mathrm{=}\frac{12}{{h}^{3}}\mathrm{[}I\mathrm{]}\) and \(z\) the position in the thickness of the plate
and the transverse shear stresses by:
\((\begin{array}{c}{\sigma }_{x}\\ {\sigma }_{y}\end{array})\mathrm{=}\mathrm{[}{D}_{1}(z)\mathrm{]}(\begin{array}{c}{T}_{x}\\ {T}_{y}\end{array})\),
with \(\mathrm{[}{D}_{1}(z)\mathrm{]}\mathrm{=}\frac{6}{{h}^{3}}({(\frac{h}{2})}^{2}\mathrm{-}{z}^{2})\)
For the flexural deformation energy, we obtain:
\({E}_{\mathit{flexion}}\mathrm{=}\frac{1}{2}{\mathrm{\int }}_{S}\mathrm{[}({M}_{\mathit{xx}}\mathrm{.}{\kappa }_{\mathit{xx}}+{M}_{\mathit{yy}}\mathrm{.}{\kappa }_{\mathit{yy}}+{M}_{\mathit{xy}}\mathrm{.}{\kappa }_{\mathit{xy}})\mathrm{]}\mathit{dS}\)
2.2. Benchmark results#
For each of the models, we calculate:
in the center of the plate, the displacement,
in the center of the plate and in the middle of the \(\mathit{AB}\) side, the constraints \({\sigma }_{\mathit{xx}}\), \({\sigma }_{\mathit{yy}}\),
,
,
on the plans:
lower, middle and upper part of the plate in the case of monolayer,
lower, middle and upper of each slice in the case of multi-layer (5 layers),
at the center, at the corners and in the middle of the sides \(\mathit{AB}\) and \(\mathit{AD}\), the membrane forces
,
,
, the shear forces \({T}_{x}\), \({T}_{y}\) and the moments \({M}_{\mathit{xx}}\), \({M}_{\mathit{yy}}\), and \({M}_{\mathit{xy}}\),
at the center (point O), at point A, the membrane deformations \({e}_{\mathit{xx}}\), \({e}_{\mathit{yy}}\), \({e}_{\mathit{xy}}\) and the curvatures \({\kappa }_{\mathit{xx}}\), \({\kappa }_{\mathit{yy}}\), \({\kappa }_{\mathit{xy}}\),
For the V modeling, the deformation energy \(\mathit{TOTALE}\), \(\mathit{MEMBRANE}\) and \(\mathit{FLEXION}\) is calculated at the center (point O).
Expressing these quantities at points \(O,A,B,C,D\) gives:
\(w\) |
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\(O\) |
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0 |
0 |
0 |
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\(A\) |
0 |
0 |
0 |
|
\(\mathrm{-}\alpha (1\mathrm{-}\nu )\) |
0 |
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0 |
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||
\(B\) |
0 |
0 |
0 |
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\(\alpha (1\mathrm{-}\nu )\) |
0 |
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\(\mathit{B1}\) |
0 |
0 |
0 |
0 |
0 |
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\(C\) |
0 |
0 |
0 |
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\(\mathrm{-}\alpha (1\mathrm{-}\nu )\) |
0 |
|||||
\(D\) |
0 |
0 |
0 |
|
\(\alpha (1\mathrm{-}\nu )\) |
0 |
Digital application:
\(\frac{3(1\mathrm{-}{\nu }^{2})}{{\pi }^{4}E{h}^{3}}\mathrm{=}1.154923\)
\(\alpha (1+\nu )\mathrm{=}0.0316629\)
\(\alpha (1\mathrm{-}\nu )\mathrm{=}0.0189972\)
\(1\mathrm{/}2\pi \mathrm{=}0.159155\)
The distribution of plane and shear stresses at points \(O\) and \(\mathit{B1}\) inside the plate is as follows:
\(O\) |
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\(h\mathrm{/}2\) |
18.9972 |
18.9972 |
18.9972 |
0 |
0 |
|
\(\mathrm{3h}\mathrm{/}10\) |
11.3983 |
11.3983 |
11.3983 |
0 |
0 |
|
\(h\mathrm{/}10\) |
3.7994 |
3.7994 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
\(\mathrm{-}h\mathrm{/}10\) |
—3.7994 |
—3.7994 |
0 |
0 |
0 |
0 |
\(\mathrm{-}\mathrm{3h}\mathrm{/}10\) |
—11.3983 |
—11.3983 |
0 |
0 |
0 |
0 |
\(\mathrm{-}h\mathrm{/}2\) |
—18.9972 |
—18.9972 |
0 |
0 |
0 |
0 |
\(\mathit{B1}\) |
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|
|
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|
\(h\mathrm{/}2\) |
0 |
0 |
0 |
0 |
0 |
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\(\mathrm{3h}\mathrm{/}10\) |
0 |
0 |
0 |
0 |
1.5278 |
|
\(h\mathrm{/}10\) |
0 |
0 |
0 |
0 |
2.3777 |
|
0 |
0 |
0 |
0 |
0 |
2.3873 |
|
\(\mathrm{-}h\mathrm{/}10\) |
0 |
0 |
0 |
0 |
2.3777 |
|
\(\mathrm{-}\mathrm{3h}\mathrm{/}10\) |
0 |
0 |
0 |
0 |
1.5278 |
|
\(\mathrm{-}h\mathrm{/}2\) |
0 |
0 |
0 |
0 |
0 |
2.3. Uncertainty about the solution#
Analytical solution.
2.4. Bibliographical references#
BATOZ and DHATT. Modeling of structures by finite elements. Beams and Plates. Volume 2 HERMES, 1990.