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


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

The critical pressure is given in [:ref:`bib1 <bib1>`] or [:ref:`bib2 <bib2>`] by the following expression:

:math:`{P}_{\mathit{cr}}\mathrm{=}\frac{\mathit{Eh}}{R}\frac{1}{({n}^{2}+\frac{{b}^{2}}{2})}\left[\frac{1}{{(\frac{{n}^{2}}{{b}^{2}}+1)}^{2}}+\frac{{h}^{2}}{12{R}^{2}(1\mathrm{-}{\nu }^{2})}{({n}^{2}+{b}^{2})}^{2}\right]`

.. csv-table::

    "with", ":math:`b\mathrm{=}\frac{\pi R}{L}`"
    "", ":math:`n` represents the number of circumferential modes"


This formula is valid in the case where :math:`N\mathrm{=}0.5R{p}_{\mathit{cr}}`.


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


.. image:: images/Object_6.svg
    :width: 445
    :height: 230


.. _RefImage_Object_6.svg:


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

Analytical solution


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


1. S.P. TIMOSHENKO, J.M. GERE: Elastic Stability Theory, page 500, second edition, DUNOD 1966.


2. BO O. ALMROTH, D.O. BRUSH: Buckling of bars, plates and shells, page 173, Mc Graw-Hill, New York, 1975.