Euler's Formula for Columns

For columns under load, what does Euler's formula determine?

Hint
$$P_{cr}=\frac{\pi ^2EI}{(Kl)}^2$$$where $$P_{cr}$$ is the critical axial load, $$l$$ is the unbraced column length, $$E$$ is the material's elastic modulus, $$I$$ is the moment of inertia, and $$K$$ is the effective length factor to adjust for end supports. Hint 2 Critical stress: $$\sigma_{cr}=\frac{P_{cr}}{A}=\frac{\pi^2E}{(Kl/r)^2}$$$
where $$r=\sqrt{I/A}$$ which is the radius of gyration, $$A$$ is the area, and $$Kl/r$$ is the column's effective slenderness ratio.
The Euler formula is used for relatively long columns subjected to concentric axial load compressions to determine elastic stability. It is used to determine elastic buckling:
$$P_{cr}=\frac{\pi ^2EI}{(Kl)}^2$$$where $$P_{cr}$$ is the critical axial load, $$l$$ is the unbraced column length, $$E$$ is the material's elastic modulus, $$I$$ is the moment of inertia, and $$K$$ is the effective length factor to adjust for end supports. Theoretical effective length factors for columns include: Pinned-pinned - $$K=1.0$$ Fixed-fixed - $$K=0.5$$ Fixed-pinned - $$K=0.7$$ Fixed-free - $$K=2.0$$ To find the critical buckling stress for long columns: $$\sigma_{cr}=\frac{P_{cr}}{A}=\frac{\pi^2E}{(Kl/r)^2}$$$
where $$r=\sqrt{I/A}$$ which is the radius of gyration, $$A$$ is the area, and $$Kl/r$$ is the column's effective slenderness ratio.
Elastic Buckling