How Columns Buckle: Length, End Conditions, and Euler

Push down on a short, stout post and it crushes. Push down on a long, skinny one and something else happens first: it bows sideways and gives way at a load far below what the material could carry. That sideways failure is buckling, and for slender columns it — not crushing — sets the limit.

End conditionsPinned – pinned (K=1)Fixed – pinned (K=0.7)Fixed – fixed (K=0.5)Fixed – free (cantilever) (K=2)

Unbraced length: 10 ft

At 10 ft with Pinned – pinned ends, the effective length is 10.0 ft (K = 1) — about 1× the buckling load of a pinned column the same height.

Buckling load vs. a pinned column the same length and section — equal to (1/K)².

• Pinned – pinned1×
• Fixed – pinned2.04×
• Fixed – fixed4×
• Fixed – free (cantilever)0.25×

Pcr = π²EI / (KL)². Capacity drops with the square of length — double the unbraced height and it falls to a quarter — and rises as you fix the ends (smaller K → shorter effective length KL). These are the theoretical K-factors; design codes (e.g. AISC) use higher recommended values. A teaching model — size real columns with an engineer.

Euler’s critical load

For an ideal slender column, the load at which it buckles is:

Pcr = π²·E·I / (K·L)²

Three levers control it:

• Stiffness (E·I) — a stiffer material (E) or a fatter section (I) raises the load. It’s the same moment of inertia from why joists stand on edge — a column buckles about its weakest axis, so the smaller I is what matters.
• Length (L) — it’s squared, so length punishes hard. Double the unbraced height and the buckling load drops to a quarter.
• End conditions (K) — how the ends are held changes the effective length.

Effective length and end conditions

A column doesn’t buckle over its full length if its ends are restrained — it buckles over its effective length, K·L. Fixing the ends forces the buckled shape into a shorter wave, so the column carries more:

• Pinned–pinned: K = 1.0 — the reference.
• Fixed–pinned: K = 0.7 — about 2× the pinned load.
• Fixed–fixed: K = 0.5 — about 4× the pinned load.
• Fixed–free (cantilever): K = 2.0 — only ¼ the pinned load.

These are the theoretical factors; design codes such as AISC use slightly higher recommended values (0.65, 0.80, 2.10) because real ends are never perfectly fixed.

Why bracing wins

Because capacity scales with 1/(K·L)², the cheapest way to strengthen a column is usually to shorten its unbraced length — add a brace at mid-height and each segment is half as long, so it buckles at roughly four times the load. It’s why wall studs (slender on their weak axis) are fine once sheathing braces them, and why long columns get intermediate bracing.

This is an idealized elastic model — it ignores initial crookedness, inelastic buckling, and material yield, which all lower real capacity. Size real columns with an engineer and the adopted code.