Steel Buckling — Flexural, Local, Lateral-Torsional, and Torsional Modes
Buckling is the dominant failure mode for most steel compression and flexural members. Unlike yielding, which depends only on stress magnitude, buckling depends on geometry, boundary conditions, and loading pattern. A steel column or beam may buckle at stresses well below the yield stress if its slenderness is high enough. AISC 360-22 addresses four primary buckling modes: flexural (Chapter E), lateral-torsional (Chapter F), local plate buckling (Table B4.1), and torsional/flexural-torsional (Section E4).
Flexural buckling (column buckling)
The most common buckling mode for doubly symmetric compression members. The member deflects laterally about the weak axis without twisting. Governed by the Euler equation modified for inelastic behavior:
Fe = pi^2*E/(KL/r)^2 [elastic critical stress]
AISC uses a two-equation column curve (Section E3):
- Inelastic: Fcr = 0.658^(Fy/Fe) * Fy, when KL/r <= 4.71*sqrt(E/Fy)
- Elastic: Fcr = 0.877 * Fe, when KL/r > 4.71*sqrt(E/Fy)
The transition slenderness for Fy = 50 ksi is KL/r = 113.4. Design strength: phiPn = 0.90FcrAg.
For W-shapes, weak-axis flexural buckling almost always governs because ry < rx. Check both axes and use the lower capacity.
Local buckling (plate buckling)
Thin plate elements (flanges, webs) can buckle locally before the member reaches its full global buckling or yielding capacity. AISC classifies cross-section elements as compact, noncompact, or slender based on width-to-thickness ratios (Table B4.1):
Compression elements in flexure
| Element | lambda = b/t | lambda_p (compact) | lambda_r (noncompact) |
|---|---|---|---|
| Flange of W-shape | bf/(2*tf) | 0.38*sqrt(E/Fy) = 9.15 | 1.0*sqrt(E/Fy) = 24.1 |
| Web of W-shape | h/tw | 3.76*sqrt(E/Fy) = 90.6 | 5.70*sqrt(E/Fy) = 137.3 |
Compression elements in axial compression
| Element | lambda = b/t | lambda_r (nonslender) |
|---|---|---|
| Flange of W-shape | bf/(2*tf) | 0.56*sqrt(E/Fy) = 13.5 |
| Web of W-shape | h/tw | 1.49*sqrt(E/Fy) = 35.9 |
| Wall of rectangular HSS | b/t | 1.40*sqrt(E/Fy) = 33.7 |
| Wall of round HSS | D/t | 0.11*E/Fy = 63.8 |
All lambda values above are for Fy = 50 ksi, E = 29,000 ksi.
Impact on design:
- Compact sections can develop full plastic capacity (Mp for beams, full Fcr for columns)
- Noncompact sections have reduced flexural capacity between Mp and My
- Slender sections require effective width or effective area reductions per AISC Sections E7 and F7
Most standard W-shapes are compact for flexure. Some lighter sections (W14x22, W12x14) are noncompact. HSS sections can be slender at common wall thicknesses.
Lateral-torsional buckling (LTB)
Beams loaded in flexure can buckle laterally (compression flange displaces sideways) while simultaneously twisting. This is the primary buckling mode for unbraced beams. Governed by the unbraced length Lb relative to Lp and Lr:
- Lb <= Lp: Full plastic moment (Mn = Mp). Lp = 1.76rysqrt(E/Fy).
- Lp < Lb <= Lr: Inelastic LTB. Linear interpolation between Mp and 0.7FySx, multiplied by Cb.
- Lb > Lr: Elastic LTB. Mn = FcrSx, where Fcr depends on (Lb/rts)^2 and the J/(Sxho) ratio.
The Cb factor accounts for non-uniform moment (Cb = 1.0 for uniform moment, up to 2.27 for reverse curvature). See the Cb Factor reference for values and worked examples.
Prevention: Provide lateral bracing to the compression flange at intervals no greater than Lp. Metal deck with shear connectors provides continuous bracing to the top flange.
Torsional and flexural-torsional buckling
Pure torsional buckling (Section E4)
Doubly symmetric sections (W-shapes, HSS) can theoretically buckle in a pure torsional mode, but this is rarely critical because flexural buckling governs at a lower load. Exception: short, stocky cruciform or built-up sections with very low torsional stiffness.
Flexural-torsional buckling (Section E4)
Singly symmetric sections (channels, tees, double angles) can buckle in a combined flexural-torsional mode where the member twists and translates simultaneously. The critical load is:
Fe_FT = [(Fex + Fez)/(2*H)] * [1 - sqrt(1 - 4*Fex*Fez*H/(Fex+Fez)^2)]
Where Fex = flexural buckling stress about the axis of symmetry, Fez = torsional buckling stress, H = 1 - (xo^2+yo^2)/ro^2. For channels and tees, flexural-torsional buckling can reduce capacity by 10-30% compared to flexural-only analysis.
Plate buckling (web shear buckling)
Thin webs in beams and plate girders can buckle in shear before reaching the shear yield strength. AISC Chapter G addresses this with the Cv shear coefficient:
- Stocky webs (h/tw <= 2.24*sqrt(E/Fy)): Shear yielding governs, Cv1 = 1.0, phi = 1.00
- Intermediate webs: Inelastic shear buckling, Cv1 < 1.0
- Slender webs: Elastic shear buckling, tension field action may be used (plate girders)
Most rolled W-shapes have stocky enough webs that shear buckling is not critical. Plate girders and deep built-up sections commonly require shear buckling checks.
Worked example -- buckling modes for W12x14
Given: W12x14, Fy = 50 ksi, Lb = 12 ft (unbraced), KL = 12 ft (compression).
Properties: A = 4.16 in^2, ry = 0.753 in, Zx = 17.4 in^3, Sx = 14.9 in^3, bf/(2tf) = 8.82, h/tw = 54.3.
Local buckling check: bf/(2tf) = 8.82 < 9.15 (compact for flexure) but close to the limit. h/tw = 54.3 -- compact. For compression: bf/(2tf) = 8.82 < 13.5 -- nonslender. The W12x14 is compact for flexure and nonslender for compression.
Flexural buckling: KL/ry = 144/0.753 = 191.2. Fe = pi^229000/191.2^2 = 7.83 ksi. Since 191.2 > 113.4, elastic buckling: Fcr = 0.8777.83 = 6.87 ksi. phiPn = 0.906.874.16 = 25.7 kips. This column retains only 14% of its squash load due to high slenderness.
LTB check: Lp = 1.760.753sqrt(29000/50) = 31.9 in = 2.66 ft. Since 12 ft >> 2.66 ft, this beam is well into the elastic LTB zone. Significant capacity reduction from LTB.
This example shows why the W12x14 is one of the lightest W-shapes that engineers should use cautiously -- its slenderness makes it vulnerable to multiple buckling modes.
Practical tip: avoiding buckling problems in design
The simplest way to prevent buckling issues is to: (1) brace compression flanges at close intervals (metal deck, girts, kickers), (2) use sections with low b/t ratios (compact sections), (3) avoid very slender columns (target KL/r < 100 for main columns, < 120 for bracing), and (4) check both axes for columns and verify the unbraced length assumption for beams.
Common mistakes
- Checking only flexural buckling for channels and tees. Singly symmetric sections require a flexural-torsional buckling check per AISC E4, which often gives a lower capacity.
- Assuming Lb = 0 when metal deck is present. Metal deck braces only the flange it is attached to (usually the top flange). For negative moment regions where the bottom flange is in compression, Lb is the distance between bottom-flange braces.
- Ignoring local buckling effects on column capacity. Slender elements (thin HSS walls, wide flanges) require effective area reductions per AISC E7 that can reduce column capacity by 10-20%.
- Not checking web shear buckling for deep beams. Deep W-shapes (W24+ with thin webs) and coped beams may have web shear buckling govern over shear yielding.
- Using Cb = 1.0 for all cases. While conservative, this can oversize beams by 1-2 sizes. Computing the actual Cb for the specific moment diagram is worth the effort.
Run this calculation
Related references
- Column Buckling Equations
- Column Curve
- Lateral-Torsional Buckling
- Cb Factor
- Compact Section Limits
- How to Verify Calculations
Disclaimer
This page is for educational and reference use only. It does not constitute professional engineering advice. All design values must be verified against AISC 360-22 Chapters B, E, F, and G and the governing project specification. The site operator disclaims liability for any loss arising from the use of this information.