How Column Buckling Works
Push down on a long thin ruler. It does not crush. It bows sideways and suddenly snaps out of alignment. That is buckling. A long slender part in compression fails at a much lower load than its raw material strength suggests, because any tiny off-center push makes it bend, and bending makes the pushing force worse.
Euler vs Johnson
For long slender columns, use the Euler formula: Pcr = π² E I ÷ (KL)². It predicts the load at which the column snaps sideways. For short stocky columns, Euler overpredicts because the material yields first. The Johnson formula covers this short range. The cross-over is the critical slenderness Cc.
End Conditions Matter a Lot
The K factor in the formula depends on how the ends are held. Pinned-pinned is K=1.0. Fixed-fixed is K=0.5, which means 4 times more buckling strength. Fixed-free (cantilever) is K=2.0, which is only 1/4 the strength of a pinned column. Always check your assumptions about end support.
Pro tip: Real end conditions are rarely perfectly fixed or perfectly pinned. For bolted flanges treat as pinned. For welded gussets or clamped bases treat as fixed. When in doubt, use the more conservative pinned assumption.
Cross-Section Choice
Radius of gyration r is the key property. It is sqrt(I/A), a measure of how far the material sits from the centerline. Hollow tubes have the highest r per pound, so they resist buckling best. That is why bicycle frames and aerospace struts are tubes. Solid rectangles buckle about their weak axis, which can be 10 times weaker than their strong axis.
Safety Factors
Never design to the exact Euler or Johnson load. Real columns have imperfections: slight curvature, eccentric loading, residual stresses from welding. Apply a safety factor of at least 2.5 to 3 for machine parts and 3 to 5 for structural columns. Building codes like AISC prescribe specific factors for different load types.