2. Reference solution

2.1. 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 [bib1].

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

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

The plus sign corresponds to positive moments as shown in the figure of [§1.1].

2.2. Benchmark results

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

Fashion	Critical Moment ( \(\mathit{Nm}\) )
1	2.86074
2	8.63207
3	—8.78382
4	14.4147
5	—14.5551

2.3. Uncertainty about the solution

Analytical solution

2.4. Bibliographical references

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