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


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

The reference solution is a digital solution [:ref:`1 <1>`].

If :math:`a\mathrm{-}b\mathrm{\le }r<a+b`


* Displacement: :math:`{\delta }_{r}\mathrm{=}\frac{\mathit{pb}}{\mathrm{2Eh}}(r\mathrm{-}\nu (r+a))`


* Constraints: :math:`{\sigma }_{11}\mathrm{=}\frac{\mathit{pb}}{\mathrm{2h}}\mathrm{\times }\frac{r+a}{r}` :math:`{\sigma }_{22}\mathrm{=}\frac{\mathit{pb}}{\mathrm{2h}}`

These formulas are only applicable to thin tori, such as :math:`\frac{b}{h}>10` and with a radius of curvature such as :math:`r\mathrm{\times }\pi <100\sqrt{\frac{{I}_{x}}{A}}` with :math:`{I}_{x}` =moment of inertia and :math:`A` = area of the geometric cross section of the torus


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


* Travel


.. csv-table::

    "Dot", ":math:`\mathit{DX}(m)`"
    ":math:`r\mathrm{=}a\mathrm{-}b` "," :math:`{\delta }_{r}\mathrm{=}1.19\mathrm{\times }{10}^{\mathrm{-}7}`"
    ":math:`r\mathrm{=}a+b` "," :math:`{\delta }_{r}\mathrm{=}1.79\mathrm{\times }{10}^{\mathrm{-}6}`"



* Constraints


.. csv-table::

    "Point", "Stresses (Pa)"
    ":math:`\mathrm{\forall }r` "," :math:`{\sigma }_{22}\mathrm{=}\mathrm{2,5}\mathrm{\times }{10}^{5}`"
    ":math:`r\mathrm{=}a\mathrm{-}b` "," :math:`{\sigma }_{11}\mathrm{=}7.5\mathrm{\times }{10}^{5}`"
    ":math:`r\mathrm{=}a+b` "," :math:`{\sigma }_{11}\mathrm{=}4.17\mathrm{\times }{10}^{5}`"



Uncertainties about the solution
----------------------------

Analytical solution


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


1. Guide VPCS - 1990 edition.