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::
 :label: eq-1

    {P} _ {\ mathrm {CR}}} =\ frac {E} {12. (1- {\nu} ^ {2})}\ mathrm {.} {n} ^ {2}\ mathrm {.} {(\ frac {e} {R})}} ^ {3}

with :math:`n` the mode number (here :math:`n=\mathrm{2,4}\mathrm{,6}`)


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

The critical pressures (in :math:`\mathrm{Pa}`) are:


.. csv-table::

    "**Mode (n)**", "**Reference**"
    "2", "73260"
    "4", "293040"
    "6", "659340"



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.