G-03 Steel Column Design

Column capacity depends on slenderness — the ratio of effective length to radius of gyration (KL/r). Short columns yield at Fy. Long columns buckle elastically at the Euler load. The AISC column curve blends these two regimes into one smooth equation that gives the critical stress Fcr at any slenderness.

key concepts

overviewSlenderness, buckling, and the AISC column curve

A column's axial capacity depends on its slenderness ratio KL/r. Short columns (KL/r < 4.71√(E/Fy)) yield; long columns buckle elastically per Euler (Fe = π²E/(KL/r)²). AISC E3 blends these into a single curve: for low slenderness Fcr = [0.658^Fy/Fe]·Fy; for high slenderness Fcr = 0.877·Fe. Design strength: φ_cPn = φ_c·Fcr·Ag where φ_c = 0.9.

why columns are different from beamsStability vs. strength — buckling dominates

Beams fail by bending — the material yields. Columns fail by buckling — a sudden sideways deflection that can happen well below the yield stress. The shorter and stockier a column, the more it behaves like a beam (yields). The taller and thinner, the more buckling dominates. Column design is fundamentally a stability problem, not just a strength problem.

the slenderness ratioK, L, and r combined into one governing number

The slenderness ratio KL/r captures this in one number. K is the effective length factor (depends on end conditions — 1.0 for pinned-pinned, 0.65 for fixed-fixed in theory). L is the unbraced length. r is the radius of gyration (√(I/A)) — a measure of how spread out the cross-section is. Higher KL/r = more slender = lower capacity. AISC limits KL/r to 200 for compression members.

the two-branch curveInelastic and elastic buckling equations from AISC E3

AISC E3 uses two equations that blend together into a single curve. For stocky columns (KL/r below ~133 for 50 ksi steel): Fcr = (0.658^Fy/Fe) × Fy — an inelastic curve where residual stresses and imperfections reduce capacity below Fy. For slender columns (KL/r above ~133): Fcr = 0.877 × Fe where Fe = π²E/(KL/r)² — elastic Euler buckling with a 12.3% reduction for imperfections.

reading the column curveFinding your column's operating point on the curve

The interactive tool plots this curve. The x-axis is slenderness (KL/r), the y-axis is critical stress (Fcr). Your column's operating point sits somewhere on this curve. Left side = short column, high capacity. Right side = tall column, capacity drops fast. The design strength is φ_cPn = 0.90 × Fcr × Ag. If the demand Pu exceeds φ_cPn, the column fails.

Test Your Understanding

1. Using a W8x28 (Ag=8.25 in², r=1.87") with Fy=50 ksi, K=1.0, and L=12 ft, the tool shows φ_cPn ≈ 241 kips. If you change K to 2.0, what happens?

φ_cPn drops to roughly 79 kips — the column shifts from inelastic to elastic buckling φ_cPn stays about the same because Ag hasn't changed φ_cPn increases because K amplifies the capacity

2. In the equation Fe = π²E/(KL/r)², what does Fe represent?

The factored axial demand on the column The elastic (Euler) buckling stress — the stress at which a perfect column would buckle The yield strength of the steel

3. Unlike beams, columns can fail well below the yield stress. What makes column design fundamentally different?

Columns use a higher resistance factor than beams Columns always fail by shear, not bending Column design is a stability problem — buckling can occur before yielding

short columns crush. tall columns buckle.

pick a W-shape, set the length, and watch the column curve respond.

G-03 Steel Column Design