Discussion
In the limitation of Euler's formula, the critical stress σc which is defined as an average stress over the cross-section for a standard case is -
Answer: B
The critical stress is \(\sigma_{c} = \pi^{2} E / (l_{e}/K)^{2}\).
Euler's critical load is \(P = \pi^{2} EI / l_{e}^{2}\). Dividing by the area A and writing \(I = AK^{2}\), where K is the least radius of gyration:
\[ \begin{aligned} \sigma_{c} &= \frac{P}{A} = \frac{\pi^{2} E A K^{2}}{A\, l_{e}^{2}} \\ &= \frac{\pi^{2} E}{(l_{e}/K)^{2}} \end{aligned} \]
The ratio \(l_{e}/K\) is the slenderness ratio. The first option keeps I, which belongs in the load, not the stress. The options with \(K/l_{e}\) invert the slenderness ratio, which would make stress rise with length.
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