2. Benchmark solution#
2.1. Calculation method used for the reference solution#
Taking warpage into account, the calculations made by V. De Ville De Goyet [bib1] give:
either:
\({I}_{y}={\int }_{A}{z}^{2}\mathrm{dA}\) \({I}_{y}={\int }_{A}{y}^{2}\mathrm{dA}\) \({I}_{{\mathrm{yr}}^{2}}={\int }_{A}y({y}^{2}+{z}^{2})\mathrm{dA}\) \({I}_{{\mathrm{yr}}^{2}}={\int }_{A}z({y}^{2}+{z}^{2})\mathrm{dA}\)
\({P}_{\mathrm{cry}}=\frac{{\pi }^{2}E{I}_{z}}{{L}^{2}}\) \({P}_{\mathrm{crz}}=\frac{{\pi }^{2}E{I}_{y}}{{L}^{2}}\) \({P}_{\mathrm{crx}}=(\frac{GJ+{\pi }^{2}E{I}_{\omega }}{{L}^{2}}){\mathrm{Ar}}_{a}\)
\({\mathrm{Ar}}_{c}=\frac{({I}_{y}+{I}_{z})}{A}+{y}_{c}^{2}+{z}_{c}^{2}+{y}_{c}(\frac{{I}_{\mathrm{yrz}}}{\mathrm{Iz}}-2{y}_{c})+{z}_{c}(\frac{{I}_{{\mathrm{zr}}^{2}}}{\mathrm{Iz}}-2{z}_{c})\)
\({\mathrm{Ar}}_{a}=\frac{({I}_{y}+{I}_{z})}{A}+{y}_{c}^{2}+{z}_{c}^{2}+{y}_{a}(\frac{{I}_{\mathrm{yrz}}}{\mathrm{Iz}}-2{y}_{c})+{z}_{a}(\frac{{I}_{{\mathrm{zr}}^{2}}}{\mathrm{Iz}}-2{z}_{c})\)
with:
\(({y}_{a},{z}_{a})\): coordinates of the point of application of the effort
\(({y}_{c},{z}_{c})\): coordinates of the center of torsion
Cases 1, 2, 3:
We get 3 critical loads by solving the 3rd degree equation in \(P\):
\({\mathrm{Ar}}_{a}({P}_{\mathrm{cry}}-P)({P}_{\mathrm{crz}}-P)({P}_{\mathrm{crx}}-P)-{P}^{2}({P}_{\mathrm{crz}}-P){({z}_{c}-{z}_{a})}^{2}-{P}^{2}({P}_{\mathrm{cry}}-P){({y}_{c}-{y}_{a})}^{2}=0\)
Case 4:
The critical moment \(\mathrm{Mcr}\) (around the \(y\) axis) is equal to:
\(\mathrm{Mcr}=\pm {((GJ+\frac{{\pi }^{2}E{I}_{\omega }}{{L}^{2}}){P}_{\mathrm{cry}})}^{1/2}\)
Overlooking warping: the analytical reference solution is given in [bib2] [bib3].
2.2. Benchmark results#
Critical load values corresponding to the first buckling modes for the various load cases.
2.3. Uncertainty about the solution#
Analytical solution. The reference values are obtained using \(\mathit{NAG}\) (routine \(\mathit{C0SAGF}\), \(\mathit{EPS}\mathrm{=}{10}^{\mathrm{-}8}\)).
2.4. Bibliographical references#
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).
PENSERINI « Elastic instability of open thin profile beams: theoretical and numerical aspects » Note EDF/DER/HM77 /112.
CERISIER « Propagation of two test cases for modeling the calculation of elastic buckling beams in Code_Aster » HM77 /184