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

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

The reference solution is obtained analytically for an Euler-Bernoulli beam. Theoretical aspects are developed in the reference [:ref:`bib1 <bib1>`].

Using the notations in paragraph [:ref:`§1 <§1>`], the critical values are given by the expression:

:math:`{M}_{\mathrm{CR}}=-\frac{{\mathrm{EI}}_{x}+\mathrm{GJ}}{\mathrm{2R}}\pm \sqrt{{(\frac{{\mathrm{EI}}_{x}-\mathrm{GJ}}{\mathrm{2R}})}^{2}+{\mathrm{4n}}^{2}\frac{{\mathrm{EI}}_{x}\mathrm{GJ}}{{R}^{2}}}n=\mathrm{1,2}\mathrm{,3},\mathrm{...}`

The plus sign corresponds to positive moments as shown in the figure of [:ref:`§1.1 <§1.1>`].


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

The first 5 critical loads are ranked in order of increasing modulus.

.. csv-table::

    "**Fashion**", "**Critical Moment (** :math:`\mathit{Nm}` **)**"
    "1", "2.86074"
    "2", "8.63207"
    "3", "—8.78382"
    "4", "14.4147"
    "5", "—14.5551"



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

Analytical solution


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

[:ref:`1 <1>`] TIMOSHENKO Stephen P., GERE James M., Theory of Elastic Stability, McGraw-Hill, International Edition, 1963, pp. 313-318.