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

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

Taking warpage into account, the calculations made by V. De Ville De Goyet [:ref:`bib1 <bib1>`] give:

either:

:math:`{I}_{y}={\int }_{A}{z}^{2}\mathrm{dA}` :math:`{I}_{y}={\int }_{A}{y}^{2}\mathrm{dA}` :math:`{I}_{{\mathrm{yr}}^{2}}={\int }_{A}y({y}^{2}+{z}^{2})\mathrm{dA}` :math:`{I}_{{\mathrm{yr}}^{2}}={\int }_{A}z({y}^{2}+{z}^{2})\mathrm{dA}`

:math:`{P}_{\mathrm{cry}}=\frac{{\pi }^{2}E{I}_{z}}{{L}^{2}}` :math:`{P}_{\mathrm{crz}}=\frac{{\pi }^{2}E{I}_{y}}{{L}^{2}}` :math:`{P}_{\mathrm{crx}}=(\frac{GJ+{\pi }^{2}E{I}_{\omega }}{{L}^{2}}){\mathrm{Ar}}_{a}`

:math:`{\mathrm{Ar}}_{c}=\frac{({I}_{y}+{I}_{z})}{A}+{y}_{c}^{2}+{z}_{c}^{2}+{y}_{c}(\frac{{I}_{\mathrm{yrz}}}{\mathrm{Iz}}-2{y}_{c})+{z}_{c}(\frac{{I}_{{\mathrm{zr}}^{2}}}{\mathrm{Iz}}-2{z}_{c})`

:math:`{\mathrm{Ar}}_{a}=\frac{({I}_{y}+{I}_{z})}{A}+{y}_{c}^{2}+{z}_{c}^{2}+{y}_{a}(\frac{{I}_{\mathrm{yrz}}}{\mathrm{Iz}}-2{y}_{c})+{z}_{a}(\frac{{I}_{{\mathrm{zr}}^{2}}}{\mathrm{Iz}}-2{z}_{c})`

with:

:math:`({y}_{a},{z}_{a})`: coordinates of the point of application of the effort

:math:`({y}_{c},{z}_{c})`: coordinates of the center of torsion

Cases 1, 2, 3:

We get 3 critical loads by solving the 3rd degree equation in :math:`P`:

:math:`{\mathrm{Ar}}_{a}({P}_{\mathrm{cry}}-P)({P}_{\mathrm{crz}}-P)({P}_{\mathrm{crx}}-P)-{P}^{2}({P}_{\mathrm{crz}}-P){({z}_{c}-{z}_{a})}^{2}-{P}^{2}({P}_{\mathrm{cry}}-P){({y}_{c}-{y}_{a})}^{2}=0`

Case 4:

The critical moment :math:`\mathrm{Mcr}` (around the :math:`y` axis) is equal to:

:math:`\mathrm{Mcr}=\pm {((GJ+\frac{{\pi }^{2}E{I}_{\omega }}{{L}^{2}}){P}_{\mathrm{cry}})}^{1/2}`

Overlooking warping: the analytical reference solution is given in [:ref:`bib2 <bib2>`] [:ref:`bib3 <bib3>`].


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

Critical load values corresponding to the first buckling modes for the various load cases.


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

Analytical solution. The reference values are obtained using :math:`\mathit{NAG}` (routine :math:`\mathit{C0SAGF}`, :math:`\mathit{EPS}\mathrm{=}{10}^{\mathrm{-}8}`).


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


1. V. DE VILLE DE GOYET "Nonlinear static analysis by the finite element method of spatial structures formed by beams with a non-symmetric cross section" - Doctoral thesis University of Liège, MSM, academic year (1988-1989).


2. P. PENSERINI "Elastic instability of open thin profile beams: theoretical and numerical aspects" Note EDF/DER/HM77 /112.


3. J. CERISIER "Propagation of two test cases for modeling the calculation of elastic buckling beams in *Code_Aster*" HM77 /184