Cb Factor — Moment Gradient Modifier for Lateral-Torsional Buckling

The Cb factor (moment gradient modifier) accounts for the beneficial effect of non-uniform moment distribution on lateral-torsional buckling resistance. A beam with uniform moment (Cb = 1.0) is the most susceptible to LTB. Any non-uniform moment distribution increases the effective LTB resistance, captured by Cb > 1.0.

AISC 360-22 Equation F1-1

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

All moments are absolute values within the unbraced segment: Mmax = maximum moment, MA = moment at 1/4 point, MB = moment at midpoint, MC = moment at 3/4 point. This equation was developed by Kirby and Nethercot and adopted by AISC as a simpler alternative to the older double-end-moment formula.

Cb values for standard loading cases

Simply supported beams

Loading Moment Shape Cb
Uniform distributed load Parabolic 1.14
Concentrated load at midspan Triangular 1.32
Two equal loads at third-points Trapezoidal 1.01
Equal end moments, single curvature Uniform 1.00
Equal end moments, reverse curvature Linear crossing zero 2.27
One end moment only (M to 0) Linear 1.75

Cantilever beams

Use Cb = 1.0 for all cantilever cases. The standard Cb equation does not properly model cantilever LTB because the compression flange varies, load height matters, and the free-end boundary condition is fundamentally different.

Continuous beams (negative moment region)

For unbraced segments near interior supports, Cb depends on the specific moment shape. Gravity loads typically give Cb = 1.0-1.3 for the bottom-flange-compression region. Always compute from the actual moment diagram.

Worked example — braced beam with uniform load

Given: W18x50, span = 30 ft, uniform load, brace points at third-points (Lb = 10 ft each segment).

Middle segment: Moment is nearly uniform in the middle third. Cb approximately 1.01.

End segment: Moment rises from 0 at support to maximum at third-point. MA = 0.44Mmax, MB = 0.75Mmax, MC = 0.94*Mmax. Cb = 12.5/(2.5 + 1.32 + 3.0 + 2.82) = 12.5/9.64 = 1.30. The end segment has significantly higher Cb because the moment varies from zero to max, making LTB less critical.

Effect on beam capacity

The LTB capacity is multiplied by Cb in the inelastic and elastic zones, but Mn cannot exceed Mp. For a beam near the LTB transition (Lb close to Lp or Lr), even Cb = 1.14 can make the difference between a compact-zone and inelastic-zone design, potentially saving one beam size.

Multi-code equivalents

AS 4100-2020 (alpha_m): alpha_m = 1.7*Mmax / sqrt(MA^2 + MB^2 + MC^2). Slightly different formula, similar results.

EN 1993-1-1 (C1): Tabulated in National Annexes. Common values: 1.00 (uniform), 1.13 (UDL), 1.35 (point load), 1.75 (linear M-to-0).

CSA S16-19 (omega_2): omega_2 = 4Mmax / sqrt(Mmax^2 + 4MA^2 + 7MB^2 + 4MC^2). Generally close to AISC and Eurocode values.

When Cb does not apply

Use Cb = 1.0 for: cantilever beams, beams loaded at the bottom flange (destabilizing loads), beams with significant axial compression (use Chapter H interaction instead), and uniform moment (Cb = 1.0 exactly).

Common mistakes

  1. Using Cb > 1.0 for cantilevers. The equation was derived for segments between brace points, not free-ended cantilevers.

  2. Applying Cb to the wrong segment. Each unbraced segment has its own Cb based on its own moment diagram.

  3. Letting Cb amplify Mn above Mp. Mn is always capped at Mp regardless of Cb.

  4. Forgetting absolute values. All moment values in the Cb equation must be absolute values, not signed.

  5. Assuming Cb = 1.0 for all cases. While conservative, this can lead to beams one or two sizes heavier than necessary.

Frequently asked questions

What is Cb in steel design? Cb is the lateral-torsional buckling modification factor per AISC 360. It accounts for the beneficial effect of non-uniform moment distribution. Cb = 1.0 for uniform moment (worst case), Cb > 1.0 for non-uniform distributions.

When does Cb matter most? When Lb is in the inelastic LTB zone (Lp < Lb <= Lr). In this zone, Cb = 1.14 can increase capacity by up to 14%. When Lb < Lp, Cb is irrelevant (Mn = Mp regardless).

Can Cb be less than 1.0? No. The minimum Cb from the AISC equation is 1.0 (uniform moment). Any variation produces Cb > 1.0.

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Related references

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 Section F1 and the governing project specification. The site operator disclaims liability for any loss arising from the use of this information.