Compactness Ratio (λ, λp, λr) — Width-to-Thickness Limits in Steel Design
The compactness ratio (symbol λ, sometimes written as b/t or h/tw) is the width-to-thickness ratio of a plate element in a structural steel cross-section. It is the single most important local stability parameter in steel design — controlling whether a flange or web can reach its yield strength, its plastic moment capacity, or buckles elastically before either.
At its core, a thin plate element (high λ) buckles sooner than a thick one (low λ). The AISC 360 specification codifies this behavior via Table B4.1b, which defines two critical limits — λp (compact limit) and λr (non-compact limit) — for every common plate element configuration.
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The Three Limits: λ, λp, λr
| Symbol | Meaning | Purpose |
|---|---|---|
| λ (lambda) | Actual width-to-thickness ratio | Measured or calculated from section geometry |
| λp (lambda-p) | Compact limit | Maximum λ for the section to reach full plastic moment Mp |
| λr (lambda-r) | Non-compact limit | Minimum λ at which elastic local buckling governs |
Classification Decision Tree
Is λ ≤ λp for ALL elements?
YES → COMPACT section. Mn = Mp = Fy × Z.
NO → Is λ ≤ λr for ALL elements?
YES → NON-COMPACT section. Mp > Mn ≥ My = Fy × S.
NO → SLENDER section. Mn < My. Elastic local buckling governs.
A section is classified based on the worst (most slender) element. One slender flange makes the entire cross-section slender, even if the web is compact.
AISC 360-22 Table B4.1b — Key Limits
I-Shape Flange Limits
| Loading Condition | λ | λp | λr (Fy = 50 ksi) |
|---|---|---|---|
| Flexure | bf/(2tf) | 0.38√(E/Fy) = 9.15 | 1.0√(E/Fy) = 24.1 |
| Uniform compression | bf/(2tf) | 0.56√(E/Fy) = 13.5 | 1.0√(E/Fy) = 24.1 |
I-Shape Web Limits
| Loading Condition | λ | λp | λr (Fy = 50 ksi) |
|---|---|---|---|
| Flexure | h/tw | 3.76√(E/Fy) = 90.5 | 5.70√(E/Fy) = 137 |
| Uniform compression | h/tw | 1.49√(E/Fy) = 35.9 | 1.49√(E/Fy) = 35.9 |
Note: For uniform compression, λr equals λp — meaning webs under uniform compression are either compact (≤ 35.9) or slender (> 35.9) with no non-compact transition zone. This reflects the brittle nature of web local buckling under pure compression.
HSS (Hollow Structural Section) Limits
| Element | λp | λr (Fy = 50 ksi) |
|---|---|---|
| Rectangular HSS flange (flexure) | 1.12√(E/Fy) = 27.0 | 1.40√(E/Fy) = 33.7 |
| Rectangular HSS web (flexure) | 2.42√(E/Fy) = 58.3 | 5.70√(E/Fy) = 137 |
| Round HSS (flexure) | 0.07E/Fy = 40.6 | 0.31E/Fy = 180 |
Round HSS limits use a different form (D/t ratio, not √(E/Fy) proportionality) because cylindrical shell buckling mechanics differ from plate buckling.
Tee Stem Limits
| Element | λ | λp | λr (Fy = 50 ksi) |
|---|---|---|---|
| Stem in flexural compression | d/tw | 0.84√(E/Fy) = 20.2 | 1.03√(E/Fy) = 24.8 |
Tee sections are common in built-up girders and truss chords. The stem limits are tighter than web limits because the free edge provides less rotational restraint than a web connected to two flanges.
Root of the Limits: Plate Buckling Theory
The compactness limits derive from classical plate buckling theory. For a rectangular plate of width b and thickness t, simply supported on all four edges, under uniform compressive stress, the elastic buckling stress is:
Fcr = (k × π² × E) / (12 × (1 − ν²) × (b/t)²)
Where k is the buckling coefficient depending on:
- Edge support conditions (simply supported vs. fixed)
- Aspect ratio (length/width)
- Stress gradient (uniform compression vs. bending)
Setting Fcr = Fy and rearranging gives the form:
(b/t)limit = constant × √(E/Fy)
The multiplier (0.38, 1.0, 3.76, etc.) accounts for the particular k-factor and edge restraint of each element type. For example, a web in flexure has k ≈ 23.9 (stress gradient provides partial tension restraint), yielding λp = 3.76√(E/Fy) — much more generous than a web in uniform compression (k = 4.0, λp = 1.49√(E/Fy)).
Effect of Fy on Compactness
Increasing Fy reduces λp and λr because √(E/Fy) decreases:
| Fy (ksi) | √(E/Fy) | Flange λp | Flange λr | Web λp (flexure) |
|---|---|---|---|---|
| 36 | 28.4 | 10.8 | 28.4 | 107 |
| 50 | 24.1 | 9.15 | 24.1 | 90.5 |
| 65 | 21.1 | 8.03 | 21.1 | 79.4 |
| 70 | 20.3 | 7.73 | 20.3 | 76.3 |
Practical implication: A section that is compact at Fy = 50 ksi may become non-compact or slender if the same geometry is used with higher-strength steel (Fy = 65 or 70 ksi). Conversely, A36 steel (Fy = 36 ksi) permits thinner elements while still classifying as compact.
Practical Example: W10x12 Compactness Check
Given W10x12, A992 (Fy = 50 ksi):
- bf = 4.00 in, tf = 0.210 in → λ_flange = 4.00/(2×0.210) = 9.52
- d = 9.87 in, tw = 0.190 in → h/tw ≈ (9.87−2×0.210)/0.190 = 49.7
Flange check: λ = 9.52 > λp = 9.15 → non-compact flange Web check: λ = 49.7 < λp = 90.5 → compact web
Worst case governs → W10x12 is NON-COMPACT (due to flange). It cannot reach Mp.
Contrast with W10x26
- bf = 5.77 in, tf = 0.440 in → λ = 6.56 < 9.15 → compact flange
- h/tw ≈ 42.1 < 90.5 → compact web
→ W10x26 is fully COMPACT and can reach Mp = Fy × Zx.
Compactness and Flexural Strength: How λ Affects Mn
The slenderness classification directly determines the nominal flexural strength formula from AISC 360 Chapter F:
Compact Section (λ ≤ λp for all elements)
Mn = Mp = Fy × Zx
Applicable when Lb ≤ Lp (no LTB). Full plastic hinge rotation capacity available.
Non-Compact Flange (λp < λ ≤ λr)
Mn = Mp − (Mp − 0.7FySx) × (λ − λpf)/(λrf − λpf)
Linear interpolation between Mp at λp and 0.7FySx at λr.
Slender Flange (λ > λr)
Mn = 0.9 × E × kc × Sx / λ²
Elastic local buckling governs. kc accounts for web-flange interaction:
kc = 4 / √(h/tw) (0.35 ≤ kc ≤ 0.76)
Non-Compact Web
Similar interpolation per AISC 360 Section F4, accounting for web local buckling interaction with flange local buckling and LTB in members with non-compact or slender webs.
AS 4100 (Australian) Plate Slenderness
AS 4100 uses the plate element slenderness λe instead of width-to-thickness ratio:
| Element | λe = (b/t)√(fy/250) | λey (yield limit) | λep (plastic limit) |
|---|---|---|---|
| Flat flange (Grade 300) | b/t × √(300/250) | 14 | 9 |
| Flat flange (Grade 350) | b/t × √(350/250) | 15 | 10 |
| Web (bending) | d1/tw × √(fy/250) | 82 | 45 |
The inclusion of √(fy/250) normalizes the slenderness to a reference yield strength, so a single set of λey and λep values works across all steel grades.
EN 1993-1-1 (European) — c/t Limits
EN 1993 uses the c/t ratio (c = flat width between fillets) and a material factor ε = √(235/fy):
| Class | I-shape flange (c/tf) | I-shape web (c/tw, bending) |
|---|---|---|
| Class 1 (plastic) | ≤ 9ε | ≤ 72ε |
| Class 2 (compact) | ≤ 10ε | ≤ 83ε |
| Class 3 (semi-compact) | ≤ 14ε | ≤ 124ε |
For S355 (fy = 355 MPa), ε = √(235/355) = 0.814. Therefore Class 1 flange limit = 9×0.814 = 7.32, web limit = 72×0.814 = 58.6.
Frequently Asked Questions
Why are compactness limits different for flanges in flexure vs. uniform compression?
A flange in flexure has a linear strain gradient (maximum compression at the tip, zero at the neutral axis when the neutral axis is in the web), which stiffens the plate against buckling. A flange in uniform compression experiences the same stress across its entire width and buckles at a lower stress. Hence λp = 0.38√(E/Fy) for flexure but 0.56√(E/Fy) for uniform compression — the flexure limit is 47% more generous.
Can a section be compact in flexure but slender in compression?
Yes. Consider a section used as a beam-column: the flange may be compact for flexure (λ ≤ 9.15 at Fy = 50 ksi) but the web could be slender under the combined axial compression (λ = h/tw > 35.9). Each element must be checked against the λp and λr applicable to its stress state in that particular loading condition.
How does compactness interact with lateral-torsional buckling?
Compactness and LTB are independent checks. A section can be compact (no local buckling, can reach Mp) but still undergo lateral-torsional buckling if the unbraced length Lb exceeds Lp. In that case, the flexural strength is limited by LTB, not local buckling. Conversely, a non-compact section cannot reach Mp regardless of how well-braced it is.
Related Terms and Pages
- Compact Section — Lambda Limits & AISC Table B4.1
- Lateral Torsional Buckling — LTB Explained
- Plastic Modulus — Definition & Formula
- Elastic Section Modulus — Definition & Formula
- Yield Strength (Fy) — Definition & Values
- Modulus of Elasticity (E) — Definition & Values
- Beam Capacity Calculator — Free Online Tool
- Column Buckling Equations — Reference Guide
Educational reference only. Width-to-thickness ratios and compactness classification must be verified per the governing design code (AISC 360 Table B4.1b, AS 4100 Table 5.2, EN 1993-1-1 Table 5.2) by a licensed Professional Engineer for all construction applications.
Disclaimer: This content is for educational purposes only. Results must be verified by a licensed professional engineer. Steel Calculator provides preliminary design tools — NOT a substitute for professional engineering judgment.