Reference solution
=====================


Calculation method
-----------------

We use the theory of LOVE - KIRCHHOFF to calculate the analytical solution of this problem.

The arrow in the axisymmetric coordinate system is written:

:math:`{u}_{x}=\{\begin{array}{cc}\frac{P}{8{\mathrm{\alpha }}^{4}D}(2-{\mathrm{e}}^{\mathrm{\alpha }y}\mathrm{cos}(\mathrm{\alpha }y))& \forall y\le 0\\ \frac{P}{8{\mathrm{\alpha }}^{4}D}{\mathrm{e}}^{-\mathrm{\alpha }y}\mathrm{cos}(\mathrm{\alpha }y)& \forall y\ge 0\end{array}`

with :math:`D=\frac{E{t}^{3}}{12(1-{\mathrm{\nu }}^{2})}` and :math:`4{\mathrm{\alpha }}^{4}=\frac{\mathit{Et}}{{\mathit{DR}}^{2}}`.

The rotation is written as:

:math:`{\mathrm{\beta }}_{s}=\{\begin{array}{cc}\frac{P}{8{\mathrm{\alpha }}^{3}D}{\mathrm{e}}^{\mathrm{\alpha }y}(\mathrm{cos}(\mathrm{\alpha }y)-\mathrm{sin}(\mathrm{\alpha }y))& \forall y\le 0\\ \frac{P}{8{\mathrm{\alpha }}^{3}D}{\mathrm{e}}^{-\mathrm{\alpha }y}(\mathrm{cos}(\mathrm{\alpha }y)+\mathrm{sin}(\mathrm{\alpha }y))& \forall y\ge 0\end{array}`

The widespread efforts are:

:math:`{N}_{\mathrm{\theta }\mathrm{\theta }}=\frac{\mathit{Et}}{R}{u}_{x}\left(y\right)`,

:math: `{M} _ {\ mathit {ss}} = {\ mathit {ss}} = {\ mathit {ss}}} = {\ mathit {ss}} =\ frac {p} {4 {\ mathrm {\ alpha}} {4 {\ mathrm {\ alpha}}} ^ {\ alpha}} ^ {-|y|} {4 {\ mathrm {\ alpha}}} ^ {-|y|} {4 {\ mathrm {\ alpha}}} ^ {-|y|} {4 {\ mathrm {\ alpha}}} ^ {-|y|} {4 {\ mathrm {\ alpha}}} ^ {-|y|}\ mathrm {sin}}} (\ mathrm {\ alpha} y)`

The three-dimensional constraints are:

:math:`{\mathrm{\sigma }}_{\mathrm{\theta }\mathrm{\theta }}=\frac{{N}_{\mathrm{\theta }\mathrm{\theta }}}{t}+12\frac{{M}_{\mathrm{\theta }\mathrm{\theta }}(x-R)}{{t}^{3}}`,

:math:`{\mathrm{\sigma }}_{\mathit{ss}}=12\frac{{M}_{\mathit{ss}}(x-R)}{{t}^{3}}`,

either:

:math:`{\mathrm{\sigma }}_{\mathrm{\theta }\mathrm{\theta }}(y,x)=\{\begin{array}{cc}\frac{\mathit{pR}}{t}\left(1-\frac{{\mathrm{e}}^{\mathrm{\alpha }y}}{2}\left(\mathrm{cos}(\mathrm{\alpha }y)+2\mathrm{\nu }\frac{R-x}{t}\sqrt{\frac{3}{1-{\nu }^{2}}}\mathrm{sin}(\mathrm{\alpha }y)\right)\right)& \forall y\le 0\\ \frac{\mathit{pR}}{t}\cdot \frac{{\mathrm{e}}^{-\mathrm{\alpha }y}}{2}\left(\mathrm{cos}(\mathrm{\alpha }y)-2\mathrm{\nu }\frac{R-x}{t}\sqrt{\frac{3}{1-{\nu }^{2}}}\mathrm{sin}(\mathrm{\alpha }y)\right)& \forall y\ge 0\end{array}`,

and:

:math:`{\mathrm{\sigma }}_{\mathit{ss}}(y,x)=\{\begin{array}{cc}\frac{\mathit{pR}}{t}\cdot \frac{x-R}{t}\sqrt{\frac{3}{1-{\mathrm{\nu }}^{2}}}{\mathrm{e}}^{\mathrm{\alpha }y}\mathrm{sin}(\mathrm{\alpha }y)& \forall y\le 0\\ \frac{\mathit{pR}}{t}\cdot \frac{x-R}{t}\sqrt{\frac{3}{1-{\mathrm{\nu }}^{2}}}{\mathrm{e}}^{-\mathrm{\alpha }y}\mathrm{sin}(\mathrm{\alpha }y)& \forall y\ge 0\end{array}`.


Reference quantities and results
-----------------------------------

The following values are tested:


* the arrow :math:`\mathit{DX}` at points :math:`A`, :math:`B` and :math:`C`,


* rotation :math:`\mathit{DRZ}`, at points :math:`A` and :math:`B`,


* the generalized effort :math:`\mathit{NYY}` at points :math:`B` and :math:`{B}_{1}`,


* the generalized moment :math:`\mathit{MXX}` at point :math:`{B}_{1}`


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


1. S. ANDRIEUX - F. VOLDOIRE: Shell models. Linear static applications. Summer School CEA - EDF - INRIA of Numerical Analysis 1992.


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

     There is no step