Applicability Limit of Euler's Buckling Formula

Euler's critical stress can be written in terms of the slenderness ratio:

$$\sigma_{cr,E}=\frac{N_{cr}}{A}=\frac{\pi^2E}{\lambda^2}.$$

Euler's formula is used in the elastic range when the critical stress does not exceed the proportional limit $\sigma_p$. The limiting slenderness follows from $\sigma_{cr,E}=\sigma_p$:

$$\lambda_{lim}=\pi\sqrt{\frac{E}{\sigma_p}}.$$

Thus, for $\lambda\ge\lambda_{lim}$, the Euler model is consistent with elastic buckling. For lower slenderness, another relation or a code-based buckling curve is required.