2. Benchmark solution#
2.1. Calculation method#
The reference result was obtained analytically with the following hypotheses:
The plate is assumed to be of infinite dimension,
Muskheliskvili and Kolosov method in polar coordinates.
\({\mathrm{\sigma }}_{\mathit{rr}}=\frac{P}{2}[(1-\frac{{a}^{2}}{{r}^{2}})-(1-\frac{4{a}^{2}}{{r}^{2}}+\frac{3{a}^{4}}{{r}^{4}})\mathrm{cos}2\mathrm{\theta }]\)
\({\mathrm{\sigma }}_{\mathrm{\theta }\mathrm{\theta }}=\frac{P}{2}[(1+\frac{{a}^{2}}{{r}^{2}})+(1+\frac{3{a}^{4}}{{r}^{4}})\mathrm{cos}2\mathrm{\theta }]\)
\({\mathrm{\sigma }}_{r\mathrm{\theta }}=\frac{P}{2}(1+\frac{2{a}^{2}}{{r}^{2}}-\frac{3{a}^{4}}{{r}^{4}})\mathrm{sin}2\mathrm{\theta }\)
2.2. Reference quantities and results#
The selected reference results relate to circumferential stress \({\mathrm{\sigma }}_{\mathrm{\theta }\mathrm{\theta }}\).
\({\mathrm{\sigma }}_{\mathrm{\theta }\mathrm{\theta }}(a,\mathrm{\theta })=P(1+2\mathrm{cos}2\mathrm{\theta })\)
Point |
Size |
Value (N/mm²) |
\(\text{A}(a\mathrm{,0})\) |
|
|
\(\text{F}(a,\frac{\mathrm{\pi }}{4})\) |
|
|
\(\text{E}(a,\frac{\mathrm{\pi }}{2})\) |
|
|
2.3. Uncertainties about the solution#
Semi-analytical solution
2.4. Bibliographical references#
Guide VPCS - 1990 edition.