2. Benchmark solution#
2.1. Calculation method used for the reference solution#
\({u}_{r}(r,z,\theta )\mathrm{=}u(r,z)\mathrm{cos}\theta\) |
with \(u(r,z)\mathrm{=}\frac{M}{\mathrm{2EI}}{z}^{2}+\frac{v\stackrel{ˉ}{p}}{\mathrm{2ER}}{r}^{2}\) |
\({u}_{z}(r,z,\theta )\mathrm{=}v(r,z)\mathrm{cos}\theta\) |
with \(v(r,z)\mathrm{=}\mathrm{-}\frac{\stackrel{ˉ}{p}}{\mathrm{2EI}}rz\) |
\({u}_{\theta }(r,z,\theta )\mathrm{=}w(r,z)(\mathrm{-}\mathrm{sin}\theta )\) |
with \(w(r,z)\mathrm{=}\frac{M}{\mathrm{2EI}}{z}^{2}\mathrm{-}\frac{v\stackrel{ˉ}{p}}{\mathrm{2ER}}{r}^{2}\) |
All constraints are zero except \({\sigma }_{\mathit{zz}}(r,z)\mathrm{=}\mathrm{-}\frac{\stackrel{ˉ}{p}}{R}r\).
The data was chosen in such a way as \(u(x)\mathrm{=}u(\mathrm{0,}l)\mathrm{=}1\).
The trips are therefore written here:
\(u(r,z)\mathrm{=}\frac{{z}^{2}}{144}+\frac{{r}^{2}}{480}\); \(v(r,z)\mathrm{=}\mathrm{-}\frac{rz}{72}\); \(w(r,z)\mathrm{=}\frac{{z}^{2}}{144}\mathrm{-}\frac{{r}^{2}}{480}\)
and:
\({\sigma }_{\mathit{zz}}(r,z)\mathrm{=}\mathrm{-}r\)
2.2. Benchmark results#
\(u,v,w,{\sigma }_{\mathit{zz}}\) |
in \(\begin{array}{c}r\mathrm{=}0.,0.5,1.\\ z\mathrm{=}0.,6.,12.\end{array}\) |
\({u}_{r},{u}_{z},{u}_{\theta }\) |
in \(\begin{array}{c}r\mathrm{=}0.\\ z\mathrm{=}6.\\ \theta \mathrm{=}45°\end{array}\) |
2.3. Uncertainty about the solution#
Analytical solution.
2.4. Bibliographical references#
PERMAS -HS. Axisymmetric Continua with arbitrary loads. Stuttgart 1985. INTES publication no. 224 pp 42 - 49.