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Sheet S-122
PPDPDD CE

Structureconcept

Column buckling and effective length (K-factor)

One-line orientation

Slender columns fail by buckling (lateral instability) before they crush — and the effective-length factor K, set by the end conditions, is the single number the exam uses to ask “which column buckles first?”

Key points

  • Buckling = a compression member fails laterally due to instability rather than material failure. It governs for slender columns; stocky columns are limited by material strength.
  • Effective-length factor K converts actual length into the effective length that drives buckling. The more an end resists rotation/translation, the lower the K and the more buckling capacity.
  • Key K values (end conditions):
    • Fixed-fixed ≈ 0.5 — most stable.
    • Pinned-pinned = 1.0 — the standard/reference condition.
    • Fixed-free = 2.0 — least stable (a cantilevered / flagpole column).
  • Implication: higher K → greater effective length → greater buckling risk under the same load.
  • Buckling depends mainly on material strength and column slenderness, commonly written as KL/r. K is the end-condition factor, L is the unbraced length, and r reflects stiffness about the weaker axis.
  • Soft story — a story markedly weaker/softer than those above; a seismic-zone hazard. Avoid top-heavy buildings that drive demand into the weak story.

End fixity sets effective length — which decides who buckles first

PROPORTIONAL

Effective length KL = K × the real column height. A bigger K means a longer KL. Read each blue stem against the shared scale — the column with the tallest stem buckles first.

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Four column end conditions compared by effective length against a shared scale rail Four columns of identical real height stand on one baseline beside a shared vertical scale rail graduated in multiples of the real height: 0.5L, 0.7L, 1.0L, and 2.0L. Each column is drawn as its buckled shape with statics end supports — a hatched clamp for a fixed (built-in) end, a triangle for a pinned hinge, and a free unsupported top where the column simply terminates — with a blue effective-length bar rising to a cap at its true height on the shared scale. Fixed-fixed, clamped at both ends, bows tightly and its effective length reaches only 0.5L; K = 0.5, the most stable. Fixed-pinned, clamped base and pinned top, reaches 0.7L; K = 0.7, between the two. Pinned-pinned, hinged at both ends, bows as a single half-wave to the real-height datum, its effective-length bar set just left of the column axis and capped at 1.0L; K = 1.0, the standard reference. Fixed-free, clamped base and free top, sways like a flagpole and is the only column whose effective length rises above the real-height datum, to 2.0L; K = 2.0, the least stable, with an amber bracket marking its full effective length as twice its real height. REAL HEIGHTKL00.5L0.7L1.0L2.0LFixed–fixedK = 0.5most stableFixed–pinnedK = 0.7between the twoPinned–pinnedK = 1.0standard referenceFixed–freeK = 2.0least stable (flagpole)2× real heightbuckled shape

Higher K means MORE buckling risk, not more strength: the fixed-free flagpole's effective length is twice its real height, so it buckles first.

Confusions / comparison

End conditionK (effective-length factor)Stability
Fixed–fixed≈ 0.5Most stable (shortest effective length)
Fixed–pinned≈ 0.7Between the two
Pinned–pinned1.0Standard reference
Fixed–free (cantilever)2.0Least stable (longest effective length)

→ End connections (this topic): fixed vs pinned defines the K value · Structural system selection (this topic): steel HSS/W columns chosen to resist buckling · Beam basics (this topic): the bending counterpart to compression behavior.

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Related units

Beam basicsEnd connectionsSystem selection