A slender column loaded in compression can fail long before its material yields: at a critical load it bows sideways and loses capacity abruptly. Euler’s formula gives that elastic critical load as Pcr = π²EI/(KL)², where K is the effective length factor set by the end conditions — 1.0 for pinned-pinned, 0.5 for fixed-fixed, 0.7 for fixed-pinned, and 2.0 for the fixed-free cantilever, which buckles as if it were twice its actual length. The I in the formula is the least second moment of area, because an unbraced column always buckles about its weakest axis.
Whether buckling or plain yielding governs comes down to slenderness, λ = KL/r, with r = √(I/A) the radius of gyration. The critical stress π²E/λ² falls off with the square of slenderness, so long thin members buckle elastically while stocky ones reach yield first — the Euler expression is only valid while its predicted stress stays below the yield or proportional limit. A worked case: a pinned 6 m steel column with I = 2.0 × 10⁷ mm⁴ and A = 6000 mm² has λ = 103.9 and a critical stress of 182.8 MPa, well under its 355 MPa yield, so elastic buckling governs.
The theoretical Pcr is an upper bound on real behaviour. Actual columns carry imperfections — initial crookedness, residual stresses, load eccentricity — so design codes replace the raw Euler value with buckling curves and resistance factors that reduce the usable capacity below the theoretical one.