Compact Section Limits — AISC 360 Table B4.1b Width-Thickness Ratios
Section compactness determines whether a steel member can develop its full plastic moment capacity (Mp) or is limited by local buckling. AISC 360-22 Table B4.1b classifies flexural members as compact, noncompact, or slender based on width-to-thickness ratios. This classification directly affects available flexural strength.
Why compactness matters
- Compact sections develop the full plastic moment Mp = Fy*Zx. Local buckling does not occur before the full cross-section yields. Most standard rolled W-shapes with Fy <= 50 ksi are compact.
- Noncompact sections develop partial yielding but local buckling limits capacity to between My (= Fy*Sx) and Mp. Strength varies linearly based on the width-thickness ratio.
- Slender sections buckle locally before yielding initiates. Capacity is significantly less than My. Uncommon in hot-rolled W-shapes but occurs in built-up sections and HSS.
AISC 360-22 Table B4.1b — flexural members
Flanges of rolled I-shapes and channels
Width-thickness ratio lambda = bf/(2*tf):
| Classification | Limit | Value (Fy=50 ksi) |
|---|---|---|
| Compact | lambda <= lambda_p | lambda_p = 0.38*sqrt(E/Fy) = 9.15 |
| Noncompact | lambda_p < lambda <= lambda_r | lambda_r = 1.0*sqrt(E/Fy) = 24.1 |
| Slender | lambda > lambda_r | Elastic local buckling governs |
Most standard W-shapes satisfy this (e.g., W16x40 has bf/(2tf) = 6.93). The lightest W-shapes approach the noncompact limit.
Webs of doubly symmetric I-shapes in flexure
Width-thickness ratio lambda = h/tw:
| Classification | Limit | Value (Fy=50 ksi) |
|---|---|---|
| Compact | lambda <= lambda_p | lambda_p = 3.76*sqrt(E/Fy) = 90.6 |
| Noncompact | lambda_p < lambda <= lambda_r | lambda_r = 5.70*sqrt(E/Fy) = 137.3 |
| Slender | lambda > lambda_r | Web local buckling governs |
All standard rolled W-shapes have compact webs for Fy <= 50 ksi. Web compactness only becomes an issue for built-up plate girders or high-strength steel.
Flanges of rectangular HSS
Width-thickness ratio lambda = b/t (flat width to wall thickness):
| Classification | Limit | Value (Fy=46 ksi, A500 Gr.C) |
|---|---|---|
| Compact | lambda <= lambda_p | lambda_p = 1.12*sqrt(E/Fy) = 28.1 |
| Noncompact | lambda_p < lambda <= lambda_r | lambda_r = 1.40*sqrt(E/Fy) = 35.2 |
| Slender | lambda > lambda_r | Elastic local buckling governs |
Round HSS (pipes)
Width-thickness ratio lambda = D/t:
| Classification | Limit | Value (Fy=42 ksi) |
|---|---|---|
| Compact | lambda <= lambda_p | lambda_p = 0.07*E/Fy = 48.3 |
| Noncompact | lambda_p < lambda <= lambda_r | lambda_r = 0.31*E/Fy = 214 |
Summary table — lambda_p and lambda_r
| Element | lambda | lambda_p (Fy=36) | lambda_p (Fy=50) | lambda_r (Fy=50) |
|---|---|---|---|---|
| I-shape flange | bf/(2tf) | 10.8 | 9.15 | 24.1 |
| I-shape web | h/tw | 107 | 90.6 | 137 |
| Rect. HSS wall | b/t | 31.8 | 28.1* | 35.2* |
| Round HSS | D/t | 56.4** | 48.3** | 214** |
*HSS Fy = 46 ksi standard. **Round HSS Fy = 42 ksi.
Checking compactness — step by step
Example: W21x44, Fy = 50 ksi. bf = 6.50 in, tf = 0.450 in, h/tw = 53.6.
Flange: lambda = 6.50/(20.450) = 7.22 < 9.15 -- COMPACT. Web: lambda = 53.6 < 90.6 -- COMPACT. W21x44 can develop full Mp = FyZx.
Example: HSS 12x6x1/4, Fy = 46 ksi. b/t = 22.8 (short wall), h/t = 46.5 (long wall). Short wall: 22.8 < 28.1 -- COMPACT. Long wall: 46.5 < 60.8 (web limit for rect. HSS) -- COMPACT.
High-strength steel considerations
For Fy >= 65 ksi, compact limits tighten significantly. I-shape flange lambda_p drops from 9.15 (Fy=50) to 8.03 (Fy=65). Several light W-shapes become noncompact at higher grades. Always verify compactness when using A913 Gr. 65 or A709 Gr. 70 steel.
Effect on flexural capacity
| Class | Mn (no LTB) | AISC Equation |
|---|---|---|
| Compact | Mn = Mp = Fy*Zx | F2-1 |
| Noncompact (flange) | Mn = Mp - (Mp - 0.7FySx)(lambda-lambda_p)/(lambda_r-lambda_p) | F3-1 |
| Slender (flange) | Mn = 0.9Ekc*Sx/lambda^2 | F3-2 |
Common mistakes
Assuming all rolled W-shapes are compact. Nearly all are for Fy = 50 ksi, but a few very light sections (W6x8.5, W8x10) are noncompact. Always verify.
Using wrong lambda formula for HSS. Rectangular HSS uses flat width b/t, not full outside dimension. Flat width = outside - 3*t.
Forgetting to check both flange and web. A section is classified by its most limiting element.
Applying compact equations to built-up sections. Plate girders with slender webs require AISC Chapter F4/F5.
Not adjusting for actual Fy. Higher actual Fy tightens compact limits. Use specified minimum Fy for design.
Frequently asked questions
What does compact mean for a steel section? A compact section has flanges and web stocky enough to develop Mp = Fy*Zx before local buckling occurs. All compression elements must have width-thickness ratios below limits in AISC 360 Table B4.1b.
Are all W-shapes compact? Nearly all for Fy = 50 ksi. The vast majority of standard rolled W-shapes in the AISC Manual are compact. No standard rolled W-shape has a slender web at Fy = 50 ksi.
What happens if my section is noncompact? Flexural capacity is reduced below Mp but remains above 0.7FySx. Use AISC Eq. F3-1 to interpolate.
Run this calculation
Related references
- Beam Sizes
- HSS Section Properties
- Steel Grades Reference
- Lateral-Torsional Buckling
- 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 Table B4.1b for the specific section and steel grade. The site operator disclaims liability for any loss arising from the use of this information.