Lateral-Torsional Buckling (LTB) — AISC 360 Chapter F Reference
Lateral-torsional buckling is the governing failure mode for most unbraced steel beams. When a beam is loaded in bending, the compression flange acts like a column and can buckle laterally while the beam simultaneously twists. The unbraced length Lb -- the distance between points of lateral support for the compression flange -- determines whether the beam reaches its full plastic moment or is limited by LTB.
Three zones of flexural behavior
AISC 360-22 Chapter F divides flexural capacity into three zones based on Lb relative to Lp and Lr:
Zone 1 — Plastic (Lb <= Lp): Full plastic moment available. Mn = Mp = FyZx. Lp = 1.76rysqrt(E/Fy). For Fy = 50 ksi: Lp = 42.4ry (inches).
Zone 2 — Inelastic LTB (Lp < Lb <= Lr): Capacity decreases linearly from Mp to 0.7FySx. Mn = Cb*[Mp - (Mp - 0.7FySx)*(Lb - Lp)/(Lr - Lp)], capped at Mp.
Zone 3 — Elastic LTB (Lb > Lr): Capacity governed by elastic critical moment. Mn = FcrSx, capped at Mp. Fcr = (Cbpi^2E/(Lb/rts)^2)sqrt(1 + 0.078(Jc/(Sxho))(Lb/rts)^2).
The Cb factor — moment gradient modifier
Cb accounts for non-uniform moment distribution. Uniform moment (Cb = 1.0) is the worst case.
Cb = 12.5*Mmax / (2.5*Mmax + 3*MA + 4*MB + 3*MC)
All values are absolute moments at quarter, mid, and three-quarter points of the unbraced segment.
| Moment Diagram | Cb |
|---|---|
| Uniform moment | 1.00 |
| Simple beam, uniform load | 1.14 |
| Simple beam, midpoint load | 1.32 |
| Linear (M to 0) | 1.75 |
| Reverse curvature, equal end moments | 2.27 |
| Cantilever (any loading) | 1.00 (conservative) |
Lp and Lr values for common W-shapes (Fy = 50 ksi)
| Section | ry (in) | Lp (ft) | Lr (ft) | Mp (kip-ft) |
|---|---|---|---|---|
| W10x12 | 0.785 | 2.77 | 7.3 | 55.0 |
| W14x22 | 1.04 | 3.67 | 10.3 | 131 |
| W16x40 | 1.57 | 5.55 | 16.0 | 295 |
| W18x50 | 1.65 | 5.83 | 16.4 | 400 |
| W21x62 | 1.77 | 6.25 | 17.1 | 540 |
| W24x76 | 1.92 | 6.78 | 18.8 | 716 |
| W12x26 | 1.51 | 5.33 | 13.4 | 164 |
Lp is typically 4-7 feet for common sizes. Metal deck at 2-3 ft spacing prevents LTB entirely.
Worked example — W18x50, Lb = 12 ft
Given: W18x50, A992, uniform load, Lb = 12 ft. Zx = 101 in^3, Sx = 88.9 in^3, ry = 1.65 in.
Step 1: Lp = 5.83 ft, Lr = 16.4 ft. Since 5.83 < 12 < 16.4, inelastic LTB (Zone 2).
Step 2: Cb = 1.14 (simply supported, uniform load).
Step 3: Mp = 50101/12 = 420.8 kip-ft. 0.7FySx = 0.750*88.9/12 = 259.3 kip-ft.
Mn = 1.14*[420.8 - (420.8-259.3)(12-5.83)/(16.4-5.83)] = 1.14[420.8-94.2] = 1.14*326.6 = 372.3 kip-ft. Check: 372.3 < 420.8 OK.
phiMn = 0.90*372.3 = 335 kip-ft (vs full phiMp = 379 kip-ft, a 12% reduction from the 12-ft unbraced length).
Multi-code comparison
AS 4100-2020: Uses phiMb = phialpha_malpha_sZe*fy, where alpha_m = moment modification factor (like Cb) and alpha_s = slenderness reduction factor based on reference buckling moment Mo.
EN 1993-1-1: Uses chi_LT reduction factor: Mb,Rd = chi_LTWyfy/gamma_M1, determined from non-dimensional slenderness and buckling curves (a-d).
CSA S16-19: Uses Mr = phi*Mu with inelastic transition similar to AISC.
Common mistakes
Confusing bracing point with support point. A brace point is any location where the compression flange is restrained laterally, including intermediate braces, not just supports.
Ignoring Cb when it helps. Using Cb = 1.0 for all cases can be 14-30% overly conservative for typical loading.
Using Cb > 1.0 for cantilevers. The AISC Cb equation does not capture cantilever LTB behavior. Use Cb = 1.0.
Bracing the tension flange instead of the compression flange. LTB involves the compression flange. Bracing the tension flange does not prevent LTB.
Not checking negative moment regions. Bottom flange is in compression at supports of continuous beams. Ensure bottom flange is braced in these regions.
Frequently asked questions
What is lateral-torsional buckling? LTB is a failure mode where a beam's compression flange displaces laterally and the cross-section twists simultaneously. It is analogous to column buckling but involves both lateral translation and rotation.
How do I prevent LTB? Provide lateral bracing to the compression flange at intervals no greater than Lp. Metal deck or concrete slab with shear studs provides continuous bracing to the top flange.
What is the unbraced length? Lb is the distance between points where the compression flange is restrained against lateral displacement. A 30-ft beam with bracing at third points has Lb = 10 ft.
Run this calculation
Related references
- Cb Factor
- Compact Section Limits
- Beam Sizes
- Beam Formulas
- Deflection 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 Chapter F and the governing project specification. The site operator disclaims liability for any loss arising from the use of this information.