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

Compression elements in axial compression

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

1. 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.
2. 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.
3. 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%.
4. 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.
5. 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.

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Types of buckling in steel

Steel members can buckle in several distinct modes depending on their cross-section geometry, loading direction, boundary conditions, and slenderness. Understanding which mode governs is essential because each has different equations, limits, and design strategies. The table below summarizes the primary buckling types addressed by AISC 360-22.

In practice, the governing buckling mode is the one that produces the lowest capacity. For a typical W-shape column, flexural buckling about the weak axis governs. For a beam with a long unbraced compression flange, LTB governs. For a thin-walled HSS, local plate buckling may govern. Always check all applicable modes.

Euler buckling deep dive

Critical load

The fundamental Euler buckling equation gives the elastic critical load for a pin-ended column:

P_cr = pi^2 * E * I / (K * L)^2

Where E is the modulus of elasticity (29,000 ksi for steel), I is the moment of inertia about the buckling axis, K is the effective length factor (depends on end conditions), and L is the actual member length. The effective length factor K accounts for end restraint:

Critical stress

Expressed as stress rather than force, the Euler critical stress is:

F_cr = pi^2 * E / (K * L / r)^2

Here KL/r is the slenderness ratio, and r = sqrt(I/A) is the radius of gyration. This form makes it clear that buckling capacity depends on geometry (slenderness) rather than material strength. Doubling Fy does not increase the Euler buckling load if KL/r remains the same.

The column curve

Plotting F_cr versus KL/r on a log-log scale produces a hyperbolic curve that drops steeply at low slenderness and flattens asymptotically toward zero at high slenderness. At KL/r = 0, the critical stress equals the yield stress (squash load). At KL/r = 50, F_cr is still near Fy. At KL/r = 100, F_cr has dropped to roughly one-third of Fy. At KL/r = 200, F_cr is under 10% of Fy. This curve illustrates why stocky columns (KL/r < 50) are governed by yielding, while slender columns (KL/r > 100) are governed by stability.

Inelastic buckling — why real columns deviate from Euler

The Euler equation assumes perfect elastic behavior, but real steel columns deviate from the ideal Euler curve for three principal reasons:

1. Residual stresses. Rolling and welding introduce self-equilibrating residual stresses that can reach 10–20 ksi in compression at flange tips. These premature compressive stresses cause partial yielding at loads below Fy, effectively reducing the tangent modulus and lowering the buckling strength. Hot-rolled W-shapes typically have residual stresses of about 0.3·Fy at flange tips.

2. Initial crookedness (out-of-straightness). No column is perfectly straight. AISC's tolerance is L/1000, but even small initial bowing amplifies under compression via the P-delta effect, reducing the effective capacity. This geometric imperfection creates second-order moments that the Euler equation does not capture.

3. End restraint imperfections. Real connections are neither perfectly pinned nor perfectly fixed. Partial fixity, connection flexibility, and rotational stiffness all deviate from the idealized K factors, introducing uncertainty in the effective length.

AISC E3 tangent modulus approach

AISC 360-22 Section E3 uses a two-branch column curve that accounts for these effects. Rather than the pure Euler hyperbola, the code provides:

Inelastic range (when KL/r ≤ 4.71·sqrt(E/Fy), or equivalently when Fe ≥ 0.44·Fy):

F_cr = 0.658^(Fy/Fe) * Fy

This is a power-law transition between Fy (at KL/r = 0) and the Euler curve. At the transition point (Fe = 0.44·Fy), the inelastic equation gives F_cr = 0.877·Fe, matching the elastic branch.

Elastic range (when KL/r > 4.71·sqrt(E/Fy)):

F_cr = 0.877 * Fe

The 0.877 factor is a knockdown that accounts for initial crookedness. It reduces the theoretical Euler stress by approximately 12%, which aligns the code curve with experimental data from the Structural Stability Research Council (SSRC) Column Curve 2.

The transition slenderness 4.71·sqrt(E/Fy) equals approximately 113.4 for Fy = 50 ksi and 100.3 for Fy = 65 ksi. Columns with KL/r below this threshold are in the inelastic range where residual stresses and partial yielding reduce capacity. Columns above this threshold follow elastic Euler behavior with the 0.877 reduction.

Lateral-torsional buckling (LTB) deep dive

Lateral-torsional buckling is the critical limit state for beams with long unbraced compression flanges. AISC Chapter F organizes the LTB check into three zones based on the unbraced length Lb relative to two limiting lengths: Lp (plastic limit) and Lr (inelastic limit).

Compact limit — Lp

Lp = 1.76 * ry * sqrt(E / Fy)

When the unbraced length Lb is at or below Lp, the beam can develop its full plastic moment Mp = Fy · Zx. No LTB check is required. For a W16x36 (ry = 1.29 in, Fy = 50 ksi): Lp = 1.76 · 1.29 · sqrt(29000/50) = 54.7 in = 4.56 ft. This is a short distance — most practical beams require bracing within this interval to achieve full plastic capacity.

Inelastic limit — Lr

Lr = 1.95 * rts * (E / (0.7*Fy)) * sqrt(1 + sqrt(1 + 6.86*(Sx*ho/(J*ry^2)) * (0.7*Fy/E)))

Where rts^2 = sqrt(Iy · Cw) / Sx. The full AISC expression for Lr is:

Lr = pi * ry * sqrt(E / (0.7*Fy)) * sqrt(1 + sqrt(1 + 6.86 * (Sx*h / (J*ry^2)) * (0.7*Fy/E)))

When Lb falls between Lp and Lr, the beam is in the inelastic LTB zone. The nominal moment capacity interpolates linearly between Mp (at Lb = Lp) and 0.7·Fy·Sx (at Lb = Lr), scaled by the Cb factor:

Mn = Cb * [Mp - (Mp - 0.7*Fy*Sx) * ((Lb - Lp) / (Lr - Lp))]  ≤  Mp

Three zones summary

Cb factor

The moment gradient modifier Cb accounts for the beneficial effect of non-uniform moment distributions. For a beam with uniform moment (double curvature with equal end moments), Cb = 1.0 — this is the worst case. For other distributions:

Cb = 12.5 * Mmax / (2.5*Mmax + 3*MA + 4*MB + 3*MC)

Where MA, MB, and MC are moments at the quarter-point, midpoint, and three-quarter point of the unbraced segment. Typical values:

Using Cb > 1.0 increases the inelastic LTB capacity and extends the effective Lp distance. However, Cb is capped at: Mn ≤ Mp. Even with a high Cb, the capacity cannot exceed the plastic moment. The Cb factor does not apply in the plastic zone (Zone 1).

Local buckling limits — AISC Table B4.1b

Local buckling of individual plate elements (flanges, webs) is controlled through width-to-thickness ratio limits. AISC Table B4.1b defines three classification tiers for elements in flexural compression:

Flange slenderness for W-shapes in flexure

Web slenderness for W-shapes in flexure

Slenderness limits for axial compression

For uniform compression (columns), only two tiers exist: nonslender and slender. Elements exceeding the nonslender limit require effective area reductions per AISC E7.

What happens when limits are exceeded

When an element exceeds the compact limit but remains below the noncompact limit (for flexure) or nonslender limit (for compression), the capacity is reduced but a simplified reduction applies. When elements are slender:

• Qs reduction (flanges): For slender flanges in compression, AISC E7 applies a reduction factor Qs to the flange area. Qs depends on the actual slenderness ratio relative to λr and is typically 0.7–0.95 for marginally slender elements.

• Qa reduction (webs): For slender webs, AISC E7 computes an effective web area based on an effective width be = (actual width) · Qa. The reduced effective area Ag_eff = Ag - (actual web area - effective web area).

• Combined Q = Qs · Qa: Both reductions multiply together. The column capacity becomes phiPn = 0.90 · Fcr · (Q · Ag), where Fcr is computed using the reduced slenderness if applicable.

For flexure, slender flanges are addressed in AISC F3 and slender webs in AISC F4/F5, using effective section moduli and reduced moment capacities.

Worked example — Column buckling check

Given: W10x45, ASTM A992 (Fy = 50 ksi), pinned-pinned (K = 1.0), L = 16 ft, Pu = 200 kips.

Section properties (W10x45):

Step 1 — Local buckling check (nonslender):

Flange: bf/2tf = 6.23 < 13.5 (nonslender limit for Fy = 50 ksi). OK. Web: h/tw = 21.8 < 35.9 (nonslender limit for Fy = 50 ksi). OK.

All elements are nonslender. No Q reduction required. Q = 1.0.

Step 2 — Determine governing axis:

Weak axis (y-axis): KL/ry = 1.0 · (16 · 12) / 1.79 = 192 / 1.79 = 107.3 Strong axis (x-axis): KL/rx = 1.0 · (16 · 12) / 4.32 = 192 / 4.32 = 44.4

Weak axis governs with KL/r = 107.3.

Step 3 — Elastic critical stress:

Fe = pi^2 * E / (KL/r)^2 = pi^2 * 29000 / (107.3)^2 = 24.87 ksi

Step 4 — Transition check:

4.71 * sqrt(E/Fy) = 4.71 * sqrt(29000/50) = 113.4

Since KL/r = 107.3 ≤ 113.4, the column is in the inelastic range.

Step 5 — Critical stress (AISC E3-2):

F_cr = 0.658^(Fy/Fe) * Fy = 0.658^(50/24.87) * 50
     = 0.658^2.010 * 50
     = 0.4229 * 50
     = 21.14 ksi

Step 6 — Design capacity:

phiPn = 0.90 * F_cr * Ag = 0.90 * 21.14 * 13.3 = 253.1 kips

Step 7 — Demand-to-capacity ratio:

D/C = Pu / phiPn = 200 / 253.1 = 0.79

Result: D/C = 0.79 < 1.0. The W10x45 column at KL = 16 ft is adequate for Pu = 200 kips. The column operates at 79% of its design capacity. The critical stress of 21.14 ksi is only 42% of Fy, showing that buckling (not yielding) governs this design. If the unbraced length were reduced to 12 ft, KL/r would drop to 80.4, and the capacity would increase significantly to approximately 330 kips.

Buckling prevention strategies

Lateral bracing

The most effective way to prevent LTB is to brace the compression flange at regular intervals. Common bracing methods:

• Metal deck with shear connectors provides essentially continuous bracing to the top flange in composite construction. This is the standard solution for floor beams.
• Cross-beams or kickers brace the compression flange at discrete points. The brace must have adequate strength (2% of the compression flange force) and stiffness per AISC Appendix 6.
• Girts and purlins in metal building systems brace the inner flange of columns and rafters.
• Floor/roof diaphragms provide bracing through direct attachment.

The bracing interval should be at or below Lp to achieve full plastic moment capacity. For strength checks between brace points, use the actual Lb with Cb.

Torsional bracing

Unlike lateral bracing, which prevents lateral displacement of the compression flange, torsional bracing prevents twist of the cross-section. Examples include:

• Cross-frames and diaphragms in bridge girders
• Stiffener plates with through-bolts in built-up sections
• U-frame restraints in through-girder bridges

Torsional bracing is effective even when attached to the tension flange, making it useful for continuous beams where the compression flange is not accessible.

Stiffeners

• Transverse stiffeners (web): Closely spaced vertical plates on the web reduce the effective web panel aspect ratio, increasing shear buckling capacity. Required when h/tw exceeds 2.46·sqrt(E/Fy) and the full shear capacity is needed. Also used to develop tension field action in plate girders per AISC G3.

• Longitudinal stiffeners (web): Horizontal plates attached to the web at approximately 0.2·d from the compression flange. These divide the web into two sub-panels, each with a lower h/tw. Used in deep plate girders to control web bend-buckling.

• Bearing stiffeners: Required at concentrated loads and reactions to prevent web crippling and local web yielding per AISC Chapter J. These are distinct from stability stiffeners but serve a related role in preventing local buckling.

Section selection

Choosing the right cross-section is the first line of defense against buckling:

• Prefer compact sections. Standard W-shapes W12 and heavier are almost always compact for Fy ≤ 50 ksi. Light sections (W14x22, W12x14, W10x12) may be noncompact and should be checked.
• HSS sections. Square and rectangular HSS have high torsional stiffness and equal-axis properties, making them resistant to LTB and torsional buckling. However, thin-walled HSS can have slender elements — check b/t and D/t.
• Built-up sections. When using built-up I-shapes or plate girders, design the flanges and web to be compact or nonslender. If slender elements are unavoidable, apply the Q reductions in E7.
• Avoid extreme slenderness. For columns, target KL/r ≤ 100 for primary members and ≤ 120 for secondary members and bracing. AISC does not prohibit KL/r up to 200, but practical designs stay well below this.

Reducing unbraced length

When the unbraced length is too long for the available section, consider these strategies:

• Add intermediate braces. Even one mid-span brace cuts Lb in half, which can double the LTB capacity in the elastic range.
• Use continuous lateral support. Metal deck, slab on deck, or horizontal bracing can reduce Lb to effectively zero for the supported flange.
• Revise the structural layout. Shorter spans directly reduce KL/r and Lb.
• Increase section size. A heavier section with larger ry or rx reduces the slenderness ratio without changing the geometry.
• Use stronger steel. While Fy does not affect the Euler stress, a higher Fy increases the transition slenderness 4.71·sqrt(E/Fy), pushing the column deeper into the inelastic range where the 0.658^(Fy/Fe) curve gives higher F_cr.

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.

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