Benchmark solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------


.. csv-table::

    ":math:`{u}_{r}(r,z,\theta )\mathrm{=}u(r,z)\mathrm{cos}\theta` ", "with :math:`u(r,z)\mathrm{=}\frac{M}{\mathrm{2EI}}{z}^{2}+\frac{v\stackrel{ˉ}{p}}{\mathrm{2ER}}{r}^{2}`"
    ":math:`{u}_{z}(r,z,\theta )\mathrm{=}v(r,z)\mathrm{cos}\theta` ", "with :math:`v(r,z)\mathrm{=}\mathrm{-}\frac{\stackrel{ˉ}{p}}{\mathrm{2EI}}rz`"
    ":math:`{u}_{\theta }(r,z,\theta )\mathrm{=}w(r,z)(\mathrm{-}\mathrm{sin}\theta )` ", "with :math:`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 :math:`{\sigma }_{\mathit{zz}}(r,z)\mathrm{=}\mathrm{-}\frac{\stackrel{ˉ}{p}}{R}r`.

The data was chosen in such a way as :math:`u(x)\mathrm{=}u(\mathrm{0,}l)\mathrm{=}1`.

The trips are therefore written here:

:math:`u(r,z)\mathrm{=}\frac{{z}^{2}}{144}+\frac{{r}^{2}}{480}`; :math:`v(r,z)\mathrm{=}\mathrm{-}\frac{rz}{72}`; :math:`w(r,z)\mathrm{=}\frac{{z}^{2}}{144}\mathrm{-}\frac{{r}^{2}}{480}`

and:

:math:`{\sigma }_{\mathit{zz}}(r,z)\mathrm{=}\mathrm{-}r`


Benchmark results
----------------------


.. csv-table::

    ":math:`u,v,w,{\sigma }_{\mathit{zz}}` ", "in :math:`\begin{array}{c}r\mathrm{=}0.,0.5,1.\\ z\mathrm{=}0.,6.,12.\end{array}`"
    ":math:`{u}_{r},{u}_{z},{u}_{\theta }` ", "in :math:`\begin{array}{c}r\mathrm{=}0.\\ z\mathrm{=}6.\\ \theta \mathrm{=}45°\end{array}`"



Uncertainty about the solution
---------------------------

Analytical solution.


Bibliographical references
---------------------------


1. PERMAS -HS. Axisymmetric Continua with arbitrary loads. Stuttgart 1985. INTES publication no. 224 pp 42 - 49.