Section Classification Framework — AISC 360-22 Section B4
AISC 360 classifies cross-section elements (flanges and webs) into three categories based on the width-to-thickness ratio lambda relative to the limiting values lambda_p and lambda_r:
Compact (lambda <= lambda_p): The section can reach the full plastic moment Mp and sustain inelastic rotation beyond Mp. The plastic bending distribution coefficient applies for flexure, and the full plastic stress distribution applies for compression. All standard A992 W-shapes are compact for flexure.
Non-compact (lambda_p < lambda <= lambda_r): The section can reach the yield moment My at the extreme fibre but local buckling of one or more elements occurs before full plastification is achieved. The flexural strength is interpolated linearly between Mp and My using AISC 360-22 Equation F3-1 (flange) or F4-9 (web).
Slender (lambda > lambda_r): Elastic local buckling of the slender element occurs before first yield. The section capacity is governed by an effective width approach per Section E7 for compression members, and Appendix 1 for slender-element flexural members.
Critical principle: The element with the highest lambda/lambda_limit ratio governs the overall section classification. A section with a compact flange and slender web is slender, not compact.
W-Shape Flange Limits — Table B4.1b Cases 1 and 2
For I-shaped rolled sections in flexure (flanges under uniform compression), the flange slenderness parameter is:
lambda_f = b_f / (2 t_f), where b_f is the total flange width and t_f is the flange thickness.
Limits for A992 (Fy = 50 ksi):
| Condition | lambda_p | lambda_r | Formula (Table B4.1b) |
|---|---|---|---|
| Flexure | 9.15 | 24.1 | lambda_p = 0.38 sqrt(E/Fy), lambda_r = 1.0 sqrt(E/Fy) |
| Axial | 13.5 | 24.1 | lambda_p = 0.56 sqrt(E/Fy) per Case 2 (uniform compression) |
For flexure (Case 1): lambda_p = 0.38 x sqrt(29,000/50) = 0.38 x 24.08 = 9.15, lambda_r = 1.0 x 24.08 = 24.1. For axial compression (Case 2): lambda_p = 0.56 x sqrt(29,000/50) = 0.56 x 24.08 = 13.5.
Standard W-Shape Flange Compactness (A992, Fy = 50 ksi):
| Shape | b_f (in.) | t_f (in.) | b_f/2t_f | lambda_p | Status |
|---|---|---|---|---|---|
| W12x22 | 4.030 | 0.425 | 4.74 | 9.15 | Compact |
| W16x31 | 5.525 | 0.440 | 6.28 | 9.15 | Compact |
| W21x50 | 6.530 | 0.535 | 6.10 | 9.15 | Compact |
| W24x76 | 8.990 | 0.680 | 6.61 | 9.15 | Compact |
| W27x94 | 9.990 | 0.745 | 6.71 | 9.15 | Compact |
| W30x108 | 10.475 | 0.760 | 6.89 | 9.15 | Compact |
| W33x130 | 11.510 | 0.855 | 6.73 | 9.15 | Compact |
| W36x160 | 12.005 | 1.020 | 5.88 | 9.15 | Compact |
All standard A992 W-shapes have compact flanges. Non-compact W-shape flanges would require b_f/2t_f > 9.15, which occurs only in very shallow but wide flange sections that are generally not economically produced. Welded plate girders with thin wide flanges are the primary case where flange slenderness limits govern.
W-Shape Web Limits — Table B4.1b Cases 9 and 10
For webs in flexural compression, the slenderness parameter is:
lambda_w = h / t_w, where h is the clear distance between flanges less the fillet radius (for rolled shapes, approximately h = d - 2 k_des, where k_des is the design fillet distance tabulated in the AISC Manual).
Web Limits for A992 (Fy = 50 ksi):
| Condition | lambda_p | lambda_r | Formula (Table B4.1b) |
|---|---|---|---|
| Flexure | 90.6 | 137.3 | lambda_p = 3.76 sqrt(E/Fy), lambda_r = 5.70 sqrt(E/Fy) |
For E = 29,000 ksi: lambda_p = 3.76 x 24.08 = 90.6. lambda_r = 5.70 x 24.08 = 137.3.
Web Compactness for Selected W-Shapes:
| Shape | h/t_w | lambda_p | Status |
|---|---|---|---|
| W12x22 | 41.2 | 90.6 | Compact |
| W16x31 | 51.6 | 90.6 | Compact |
| W18x35 | 53.3 | 90.6 | Compact |
| W21x50 | 49.5 | 90.6 | Compact |
| W24x62 | 50.1 | 90.6 | Compact |
| W24x55 | 54.6 | 90.6 | Compact |
| W30x90 | 51.7 | 90.6 | Compact |
Standard W-shapes in A992 are compact in flexure. Web non-compact sections occur primarily in deep slender plate girders where the web depth is large relative to the web thickness (h/t_w > 90.6).
HSS Limits — Table B4.1b Cases 12 and 15
For rectangular HSS (A500 Gr. C, Fy = 50 ksi), the limiting ratios differ from W-shapes:
| Element | lambda_p | lambda_r | Parameter |
|---|---|---|---|
| HSS flange (b/t) | 28.1 | 39.9 | lambda_p = 1.12 sqrt(E/Fy), lambda_r = 1.40 sqrt(E/Fy) |
| HSS web (h/t) | 90.6 | 137.3 | Same as W-shape web in flexure |
For round HSS (A500 Gr. C):
| Element | lambda_p | lambda_r | Formula |
|---|---|---|---|
| D/t | 43.3 | 180.0 | lambda_p = 0.07 E/Fy, lambda_r = 0.31 E/Fy |
Effect of Yield Strength on Lambda Limits
All lambda limits in Table B4.1b depend on E/Fy (or sqrt(E/Fy)). Higher yield strength results in more restrictive (lower) lambda limits, meaning a section that is compact at 50 ksi may be non-compact at 65 ksi. This is because higher-strength steel reaches its yield strain without a corresponding increase in elastic modulus, so local buckling initiates at a lower slenderness.
For A913 Grade 65 (Fy = 65 ksi):
lambda_p_flange = 0.38 x sqrt(29,000/65) = 0.38 x 21.12 = 8.03 (vs 9.15 for 50 ksi) lambda_p_web = 3.76 x sqrt(29,000/65) = 3.76 x 21.12 = 79.4 (vs 90.6 for 50 ksi)
The reduction in lambda_p is approximately 12.3%. A section with b_f/2t_f = 8.5 is compact in A992 (50 ksi) but non-compact in A913 Grade 65.
Stiffened vs Unstiffened Elements
The classification rules distinguish between stiffened elements (supported along two edges parallel to the compression, e.g., webs) and unstiffened elements (supported along one edge only, e.g., flange outstands). The lambda parameter definition depends on this distinction.
For unstiffened elements (flanges), the parameter is based on the outstand (b/2 for I-shapes, b for angles), representing the free-edge-to-supported-edge distance that controls local buckling. For stiffened elements (webs, HSS walls), the parameter uses the full clear depth or width between supports.
Frequently Asked Questions
What does lambda_p and lambda_r mean in AISC 360 Table B4.1b?
Lambda_p (compact limit) defines the maximum width-to-thickness ratio for which the element can reach full plastification without local buckling. Lambda_r (non-compact/slender limit) defines the maximum width-to-thickness ratio for which the element can reach yield stress at the extreme fibre without elastic local buckling. Elements with lambda <= lambda_p are compact. Elements with lambda_p < lambda <= lambda_r are non-compact. Elements with lambda > lambda_r are slender.
Are all standard W-shapes compact per AISC 360?
For flexure in A992 (Fy = 50 ksi), yes — all standard rolled W-shapes in the AISC Manual are compact. The hot-rolling process naturally produces relatively stocky flanges and webs. Non-compact or slender W-shapes for flexure are not produced in the standard range. However, for axial compression, some lightly loaded W-shapes (especially W4, W5, and W6 sections used as secondary members) may have slender webs or flanges depending on the specific profile.
How does section classification affect flexural capacity per AISC 360?
- Compact sections (Chapter F2): Mn = Mp = Fy x Zx (plastic moment). The full plastic section modulus is achieved.
- Non-compact flange sections (Chapter F3): Mn is interpolated between Mp and 0.7 Fy Sx (yield moment at the flange tips) using the flange slenderness ratio.
- Slender web sections (Chapter F4/F5): The web bend-buckling coefficient Rpg reduces the nominal moment capacity below Mp, captured via a hybrid factor or a reduced effective section modulus. The design is governed by lateral-torsional buckling or flange local buckling whichever is lower.
What is the difference between flange local buckling (FLB) and web local buckling (WLB)?
FLB occurs when the compression flange buckles locally about its own axis (out-of-plane wave in the flange). WLB occurs when the web in the compression zone buckles in its plane (lateral waving between flanges). Both are prevented when the element slenderness is below lambda_p. When lambda_p < lambda <= lambda_r, the section can reach My but not Mp. Beyond lambda_r, elastic local buckling governs and the capacity falls below My.
Educational reference only. All design values are per AISC 360-22 and the AISC Steel Construction Manual 16th Edition. Verify section property limits against the actual mill certificates for the steel supplied to the project. Designs must be independently verified by a licensed Professional Engineer (PE) or Structural Engineer (SE). Results are PRELIMINARY — NOT FOR CONSTRUCTION without independent professional verification.