2. Benchmark solution#
2.1. Benchmark solution#
The reference is software SAMCEF. For information, in paragraph [§2.2], the theoretical results related to a thin-shell hypothesis are presented. Then, the results obtained with SAMCEF are presented depending on whether a thick shell hypothesis or a volume type hypothesis is chosen. It is the latter that is taken into consideration for the evaluation of code_aster.
2.2. Analytical solution and reference results#
The following formula gives arrow \({w}_{0}\) in the center of the plate:
\(\frac{{w}_{0}}{h}+A{(\frac{{w}_{0}}{h})}^{3}=\frac{\mathrm{Bp}}{E}{(\frac{r}{h})}^{4}\) with \(A=1.852\) and \(B=0.696\)
The half-thickness stresses are equal to:
\(\begin{array}{ccc}{\sigma }_{\mathrm{rr}}& =& {\alpha }_{r}E\frac{{w}_{0}}{{r}^{2}}\\ {\sigma }_{\theta \theta }& =& {\alpha }_{t}E\frac{{w}_{0}}{{r}^{2}}\end{array}\)
The lower skin constraints are worth
\(\begin{array}{ccc}{\sigma }_{\mathrm{rr}}\text{'}& =& {\beta }_{r}E\frac{{w}_{0}h}{{r}^{2}}\\ {\sigma }_{\theta \theta }\text{'}& =& {\beta }_{t}E\frac{{w}_{0}h}{{r}^{2}}\end{array}\)
The coefficients are equal to:
In the center of the plate:
\({\alpha }_{r}={\alpha }_{t}=0.905\)
\({\beta }_{r}={\beta }_{t}=1.778\)
At the edge of the plate:
\({\alpha }_{r}=0.610\)
\({\alpha }_{t}=0.183\)
\({\beta }_{r}=0\)
\({\beta }_{t}=0.755\)
For a pressure of \(222.72\mathrm{MPa}\), the arrow \({w}_{0}\) is equal to \(1.5\mathrm{mm}\) and the following constraints are obtained:
Position |
\({\sigma }_{\mathrm{rr}}\) \((\mathrm{MPa})\) |
|
|
|
Center |
4072.5 |
4072.5 |
5334.0 |
5334.0 |
These results correspond to a thin shell hypothesis.
The following table shows the results obtained by SAMCEF for thick shell and volume modeling.
Identification |
Thick shell |
Volume |
Arrow \(\mathrm{w0}\) \((\mathrm{mm})\) |
—1.43041 E—3 |
—1.441838 E—3 |
SIXX \((\mathrm{MPa})\) center, half—thickness |
3899.88 |
3850.88 |
SIYY \((\mathrm{MPa})\) center, half—thickness |
3899.53 |
3850.91 |
SIXX \((\mathrm{MPa})\) center, lower skin |
8085.81 |
8133.60 |
SIYY \((\mathrm{MPa})\) center, lower skin |
8083.32 |
8133.65 |
SIXX \((\mathrm{MPa})\) \(r=R/2\), half—thickness |
3596.91 |
3512.79 |
SIYY \((\mathrm{MPa})\) \(r=R/2\), half—thickness |
3056.37 |
2947.55 |
SIXX \((\mathrm{MPa})\) \(r=R/2\), lower skin |
7798.18 |
7815.73 |
SIYY \((\mathrm{MPa})\) \(r=R/2\), lower skin |
7264.69 |
7307.04 |
Constraint values are per-element values extrapolated to the nodes.
The results obtained with SAMCEF are similar to each other.
The differences in results between the thin shell theory and the finite element calculation using the thick shell hypothesis are significant.
We choose to take as a reference the volume calculation obtained by SAMCEF.
2.3. Bibliographical references#
Theory of Plates and Shells, Timoshenko S.P.,2nd edition, p 412