Slender Element Design — Plate Buckling and Effective Width
When a plate element of a steel section (flange or web) has a width-to-thickness ratio exceeding the slenderness limit, the plate may buckle locally before the section reaches full yield. AISC 360-22 Chapter B classifies elements as compact, noncompact, or slender, and Chapter E7 provides the Q-factor method for reducing column capacity when slender elements are present.
Slenderness classification per AISC 360-22 Table B4.1a
For unstiffened elements (flanges of W-shapes, legs of angles):
| Class | Limit (uniform compression) | Formula |
|---|---|---|
| Compact | b/t <= lambda_p | 0.56 * sqrt(E / Fy) |
| Noncompact | lambda_p < b/t <= lambda_r | 1.03 * sqrt(E / Fy) |
| Slender | b/t > lambda_r | Reduction required |
For stiffened elements (webs of W-shapes under uniform compression):
| Class | Limit (uniform compression) | Formula |
|---|---|---|
| Compact | h/t_w <= lambda_p | 1.49 * sqrt(E / Fy) |
| Slender | h/t_w > lambda_r | 1.49 * sqrt(E / Fy) |
With E = 29,000 ksi and Fy = 50 ksi:
- Unstiffened lambda*r = 1.03 * sqrt(29000/50) = 1.03 _ 24.08 = 24.8
- Stiffened lambda*r = 1.49 * sqrt(29000/50) = 1.49 _ 24.08 = 35.9
The Q-factor method — worked example
Given: A built-up column made from two C15x33.9 channels with a 3/8 in. cover plate on each flange, A572 Gr. 50 steel (Fy = 50 ksi). The cover plate is 12 in. wide. Effective column length KL = 20 ft.
Step 1 — Check cover plate slenderness:
The cover plate is welded along both edges (stiffened element under uniform compression).
b/t = 12 / 0.375 = 32.0
lambda*r = 1.49 * sqrt(E / Fy) = 1.49 _ sqrt(29000 / 50) = 35.9
Since 32.0 < 35.9, the cover plate is not slender. Check the channel flanges:
b_f / t_f = 3.40 / 0.650 = 5.23 (well below lambda_r = 24.8, not slender).
Now consider a thinner cover plate scenario (1/4 in.):
b/t = 12 / 0.25 = 48.0 > 35.9 => Slender
Step 2 — Effective width (AISC 360 Eq. E7-18):
b*e = 1.92 * t _ sqrt(E / f) _ [1 - (0.34 / (b/t)) _ sqrt(E / f)]
Using f = Fcr (assume initial Fcr = 35 ksi for iteration):
be = 1.92 * 0.25 _ sqrt(29000 / 35) _ [1 - (0.34 / 48.0) _ sqrt(29000 / 35)] be = 0.48 * 28.78 _ [1 - 0.00708 * 28.78] b_e = 13.81 _ [1 - 0.2038] = 13.81 _ 0.796 = 11.0 in.
Step 3 — Q_a factor:
Q_a = A_eff / A_g = [A_g - (b - b_e) * t] / A_g
If A_g = 25.0 in.^2: Q_a = [25.0 - (12.0 - 11.0) * 0.25] / 25.0 = 24.75 / 25.0 = 0.990
Step 4 — Modified column capacity:
Q = Qs * Qa = 1.0 * 0.990 = 0.990 (Q_s = 1.0 because flanges are not slender)
The column nominal strength is then calculated using Q * Fy in the AISC column curve equations (Chapter E), which slightly reduces the elastic-inelastic transition.
Code comparison for slender elements
| Aspect | AISC 360-22 | AS 4100:2020 | EN 1993-1-1 | CSA S16-19 |
|---|---|---|---|---|
| Classification method | Q-factor (Q_s, Q_a) | Effective section (Clause 6.2) | Effective width (EN 1993-1-5) | Q-factor (same as AISC) |
| Unstiffened limit | 0.56 sqrt(E/Fy) to 1.03 sqrt(E/Fy) | b/t limits in Table 6.2.4 | c/t <= 14 epsilon (Class 3) | Same as AISC |
| Stiffened limit | 1.49 sqrt(E/Fy) | b/t limits in Table 6.2.4 | c/t <= 42 epsilon (Class 3) | Same as AISC |
| Plate buckling standard | Winter formula embedded | AS/NZS 4600 for CFS | EN 1993-1-5 (detailed) | CSA S136 for CFS |
| epsilon factor | Not used; Fy explicit | Not used; fy explicit | epsilon = sqrt(235/fy) | Not used |
EN 1993 classifies sections as Class 1 through Class 4 rather than compact/noncompact/slender. A Class 4 section requires effective width calculation per EN 1993-1-5, which uses the same Winter formula foundation as AISC but with different notation (rho reduction factor, k_sigma buckling coefficient).
Key clause references
- AISC 360-22 Table B4.1a / B4.1b — Width-to-thickness limits for compression and flexure
- AISC 360-22 Section E7 — Q-factor method for slender-element columns
- AISC 360-22 Eq. E7-18 — Effective width formula (Winter equation)
- EN 1993-1-5 Section 4.4 — Effective width of plate elements
- AS 4100 Clause 6.2.4 — Element slenderness limits
- CSA S16-19 Clause 13.3.5 — Slender cross-section compression members
Topic-specific pitfalls
- Using the wrong boundary condition for lambda_r — an outstanding flange tip is an unstiffened element (one free edge), while a web plate between flanges is stiffened (two supported edges). Applying stiffened limits to an unstiffened element unconservatively overestimates capacity.
- Ignoring the iteration in the effective width calculation — the effective width depends on Fcr, which in turn depends on the effective area. An iterative approach (or conservative first-pass using f = Fy) is required for accuracy.
- Mixing slenderness limits from different load cases — AISC Table B4.1a (members subject to compression) and Table B4.1b (members subject to flexure) have different lambda_p and lambda_r values. A beam-column must be checked against both sets of limits for the applicable flange and web.
- Forgetting that slender elements affect connection capacity too — a slender web in the connection region reduces block shear and bolt bearing capacities because the effective material is less than the gross area.
Run this calculation
Related references
- How to Verify Calculations
- compact section limits table
- plate girder slender web design
- LTB with noncompact sections
- steel beam capacity calculator
- structural engineering unit converter
- Cold Formed Reference
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