2. Benchmark solution#

2.1. Calculation method used for the reference solution#

Two reference solutions can be used to calculate the deformation, depending on the plate theory used:

  • the Love-Kirchhoff theory, commonly used for so-called « thin » plates, which will be used for models \(A\), \(B\) and \(E\),

  • Reissner’s theory, including the effects of shear for so-called « thick » plates, which will be used for models \(F\), \(G\) and \(H\).

At any point distant from \(r\) from the center of the plate \(r\le R\), we have for the calculation of the arrow:

\(w(r)\mathrm{=}\mathrm{-}P\frac{{R}^{4}}{64D}(1\mathrm{-}\frac{{r}^{2}}{{R}^{2}})(1+\frac{{r}^{2}}{{R}^{2}}\mathrm{-}\frac{2(3+\nu )}{1+\nu }\mathrm{-}\varphi )\) with \(D\mathrm{=}\frac{E{t}^{3}}{12(1\mathrm{-}{\nu }^{2})}\)

and \(\varphi \mathrm{=}0(\text{Love-Kirchhoff})\) or \(\varphi \mathrm{=}\frac{16}{5}{(\frac{t}{R})}^{2}\frac{1}{1\mathrm{-}\nu }(\text{Reissner})\)

For the calculation of moments, the two theories lead to the same expressions:

\({M}_{\mathit{rr}}(r)\mathrm{=}\frac{{\mathit{PR}}^{2}}{16}(3+\nu )\left[{(\frac{r}{R})}^{2}\mathrm{-}1\right]{M}_{\theta \theta }(r)\mathrm{=}\frac{{\mathit{PR}}^{2}}{16}(3+\nu )\left[1\mathrm{-}\frac{1+3\nu }{3+\nu }{(\frac{r}{R})}^{2}\right]\)

In the center of the plate:

\(\begin{array}{c}w(0)\mathrm{=}\mathrm{-}\frac{{\mathit{PR}}^{4}}{64D}(\frac{5+\nu }{1+\nu })(\text{Love}\mathrm{-}\text{Kirchhoff})\text{ou}w(0)\mathrm{=}\mathrm{-}\frac{{\mathit{PR}}^{4}}{64D}(\frac{5+\nu }{1+\nu }+\varphi )(\text{Reissner})\\ {M}_{\mathit{rr}}(0)\mathrm{=}{M}_{\theta \theta }(0)\mathrm{=}\mathrm{-}\frac{{\mathit{PR}}^{2}}{16}(3+\nu )\end{array}\)

Note:

Code_Aster calculates the moments at the nodes of each finite element in the reference frame defined by the outer normal and the reference axes defined on the shell (see AFFE_CARA_ELEM in the user documentation) .

The current value \({M}_{\mathrm{xx}}\) (or \({M}_{\mathrm{yy}}\) ), taken from the field “ EFGE_ELNO “, in a node belonging to several finite elements can be considered to be the average of the values calculated on the elements that have this node in common. This average can be obtained by the procedure POST_RELEVE [U4.74.03].

For each node, we have: \(({M}_{\text{rr}}+{M}_{\theta \theta })\mathrm{=}({M}_{\text{xx}}+{M}_{\text{yy}})\mathrm{=}\text{Sm}\)

For the point \(O\) \({M}_{\text{xx}}\mathrm{=}{M}_{\text{yy}}\mathrm{=}{M}_{\text{rr}}\mathrm{=}{M}_{\theta \theta }\)

For points \(A\mathit{et}D\) \({M}_{\text{xx}}\mathrm{=}{M}_{\text{rr}}\mathit{et}{M}_{\text{yy}}\mathrm{=}{M}_{\theta \theta }\)

For points \(C\mathit{et}E\) \({M}_{\text{xx}}\mathrm{=}{M}_{\theta \theta }\mathit{et}{M}_{\text{yy}}\mathrm{=}{M}_{\mathit{rr}}\)

For points \(B\mathit{et}F\) \({M}_{\text{xx}}\mathrm{=}{M}_{\text{yy}}\mathrm{=}({M}_{\text{rr}}+{M}_{\theta \theta })\mathrm{/}2\)

2.2. Benchmark results#

Arrow and moments in points: \(O,A,BC,DE,F\).

2.3. Uncertainty about the solution#

Analytical solution

2.4. Bibliographical references#

  1. TIMOSHENKO and WOINOWSKY - KRIEGER, Plates and shells, Béranger Edition - (1961).