What are compact, noncompact, and slender elements in steel design?

AISC 360-22 Table B4.1a classifies steel plate elements into three categories based on width-to-thickness ratio (b/t for flanges, h/tw for webs). Compact elements have b/t ≤ λp and can reach full plastic moment (Mp) with sufficient rotation capacity for moment redistribution — required for SMF seismic connections and plastic design. Noncompact elements have λp λr and buckle elastically before reaching yield — requiring the Q-factor reduction method for columns per AISC Section E7. For unstiffened elements (flanges, angles) at Fy=50 ksi: λp = 0.56√(E/Fy) = 13.49, λr = 1.03√(E/Fy) = 24.08. For stiffened elements (webs under uniform compression): λp = λr = 1.49√(E/Fy) = 35.88. Higher Fy results in more restrictive limits because the plate must be stockier to reach a higher yield stress without buckling. A section compact in A36 (Fy=36 ksi) may be noncompact or slender in A572 Gr 65 (Fy=65 ksi). Most standard W-shapes have compact flanges at 50 ksi but noncompact webs, particularly in lighter sections like W16x26 (h/tw=56.8).

How does the Q-factor method work for columns with slender elements?

The Q-factor method in AISC 360-22 Section E7 reduces column compressive strength when slender elements are present. Instead of using Fy directly, the column curve uses Q × Fy where Q = Qs × Qa. Qs is the reduction factor for slender unstiffened elements (flange outstands, angle legs): for b/t ≤ λp, Qs = 1.0; for λp λr, Qs = 0.69E/(Fy × (b/t)²). Qa is the reduction factor for slender stiffened elements (webs, cover plates): Qa = Aeff/Ag where Aeff is the effective area computed by summing effective widths of slender elements with the full area of non-slender elements. The effective width for a uniformly compressed stiffened element is be = 1.92t√(E/f) × [1 − 0.38/(b/t) × √(E/f)] ≤ b, where f is taken as Fcr (the column critical stress, computed iteratively since Qa depends on f and f depends on Qa). This iteration typically converges in 2-3 cycles. For example, a W14x column fabricated from thin plates with b/t=30 (slender for Fy=50 ksi) would have Qs = 0.69 × 29,000/(50 × 30²) = 0.444, reducing the effective yield strength to 22.2 ksi for buckling calculations.

How do I calculate the effective width of a slender stiffened element?

The effective width method (AISC E7.2) replaces the actual width b of a slender stiffened element with a reduced effective width be that accounts for post-buckling strength through the von Kármán effective width concept. For a uniformly compressed stiffened element: be = 1.92t√(E/f) × [1 − 0.38/(b/t) × √(E/f)] ≤ b. Here f is the compressive stress at which buckling is being checked — for columns, f = Fcr (iterative), and for beam flanges, f = Fy. Example: a 12 in. wide × 3/8 in. thick cover plate (b/t = 32) on a column, Fy = 50 ksi, E = 29,000 ksi, with f = 30 ksi (first iteration). Effective width: be = 1.92 × 0.375 × √(29,000/30) × [1 − 0.38/32 × √(29,000/30)] = 0.72 × 31.09 × [1 − 0.0119 × 31.09] = 22.38 × [1 − 0.370] = 14.10 in. But be ≤ b = 12 in., so be = 12 in. The element is fully effective at this stress level. If f were 50 ksi: be = 1.92 × 0.375 × √(29,000/50) × [1 − 0.38/32 × √(29,000/50)] = 0.72 × 24.08 × [1 − 0.0119 × 24.08] = 17.34 × [1 − 0.287] = 12.36 in. > 12 in., still fully effective. For a thinner plate (b/t=60): at f=50 ksi, be = 1.92t × 24.08 × [1 − 0.38/60 × 24.08] = 46.23t × 0.847 = 39.2t, so effective width is 65% of the gross width. The ineffective width at the center of the plate carries no stress, and the load is concentrated in two strips of width be/2 adjacent to the stiffened edges.

Which W-shapes have slender elements at Fy=50 ksi?

At Fy=50 ksi, virtually all standard rolled W-shapes have compact flanges (bf/2tf < 13.49) because the hot-rolling process naturally produces stocky flanges relative to width. The common exception is W-shapes with very wide thin flanges like W14x22 (bf/2tf = 10.6, still compact) and W21x44 (bf/2tf = 8.65, compact). The limiting element for most W-shapes is the web: sections like W16x26 (h/tw = 56.8), W18x35 (52.7), W21x44 (53.6), W24x55 (56.0), W30x90 (50.1), W36x135 (52.5), and W44x230 (46.0) all have noncompact webs (h/tw > 35.88 at Fy=50 ksi). Noncompact webs reduce the flexural capacity for beam applications — the nominal moment is interpolated between Mp and My using the web slenderness parameter. For column applications where the web is in uniform compression, noncompact webs do not reduce capacity (the web is a stiffened element and λp = λr for uniform compression, so there is no noncompact range — it's either compact or slender). True slender element W-shapes are rare in standard rolled sections: built-up plate girders with thin webs (h/tw > 137 for Fy=50) and cold-formed steel shapes are the primary slender-element applications. The AISC Seismic Provisions (AISC 341) require compact sections for SMF beams and columns, further restricting the available shapes for seismic applications.

What is the difference between local buckling and global buckling?

Local buckling is the buckling of an individual plate element (flange or web) without overall member buckling, governed by the plate width-to-thickness ratio b/t. The buckled shape has wavelengths on the order of the plate width and occurs when the compressive stress reaches the plate critical buckling stress Fcr_plate = k × π²E/(12(1−ν²)(b/t)²), where k is the plate buckling coefficient (k=0.425 for a flange with one free edge, k=4.0 for a web with simply supported edges). Global buckling (flexural, torsional, or lateral-torsional) involves the entire member cross-section translating or rotating, governed by the member slenderness KL/r. The key distinction: local buckling is a cross-section failure mode that limits the stress the section can develop; global buckling is a member failure mode that limits the load the member can carry. The AISC column curve implicitly handles the interaction — slender-element columns use QFy as the effective yield stress in the column curve, reducing the plateau while maintaining the same Euler buckling curve at high KL/r. Members with slender elements can be designed safely but have lower capacity than compact sections of the same nominal area. For seismic applications (AISC 341), local buckling is prohibited in yielding regions because the sudden loss of compression flange capacity during cyclic loading would degrade the hysteresis loop — hence the compactness requirements for SMF, SCBF, and EBF members.

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)

Lambda values by steel grade

Fy (ksi)	E (ksi)	Unstiffened lambda_p	Unstiffened lambda_r	Stiffened lambda_p	Stiffened lambda_r
36	29,000	13.49	24.08	35.88	35.88
50	29,000	13.49	24.08	29.30	35.88
65	29,000	10.03	21.15	22.55	31.52

Higher Fy = more restrictive limits. A section that is compact in A36 may be noncompact or slender in A572 Gr 65.

Common W-shape slenderness classification (Fy = 50 ksi)

Section	bf/(2tf)	Status	h/tw	Status	Governs?
W8x31	9.19	Compact	28.0	Compact	No
W12x65	9.90	Compact	24.9	Compact	No
W14x82	5.92	Compact	21.7	Compact	No
W16x26	10.44	Compact	56.8	Noncompact	Web
W18x35	8.10	Compact	52.7	Noncompact	Web
W21x44	8.65	Compact	53.6	Noncompact	Web
W24x55	9.38	Compact	56.0	Noncompact	Web
W30x90	9.80	Compact	50.1	Noncompact	Web
W36x135	9.54	Compact	52.5	Noncompact	Web
W44x230	9.28	Compact	46.0	Noncompact	Web

Most standard W-shapes have compact flanges at Fy = 50 ksi. Noncompact and slender webs are more common in lighter sections.

The Q-factor method (AISC Section E7)

For columns with slender elements, the nominal compressive strength uses Q*Fy instead of Fy:

Fcr_modified based on Q*Fy
Q = Qs * Qa

Qs = reduction factor for slender unstiffened elements (flanges, angles)
Qa = reduction factor for slender stiffened elements (webs, cover plates)

Qs values for unstiffened elements (AISC Table E7-1)

b/t	Qs Formula
b/t <= 0.56*sqrt(E/Fy)	1.0
0.56sqrt(E/Fy) < b/t <= 1.03sqrt(E/Fy)	1.415 - 0.74sqrt(Fy/E)(b/t)
b/t > 1.03*sqrt(E/Fy)	0.69E/(Fy(b/t)^2)

Qa values for stiffened elements (AISC Eq. E7-16 to E7-18)

For uniformly compressed stiffened elements, Qa = Aeff / Ag where:

be = 1.92*t*sqrt(E/f)*[1 - (0.34/(b/t))*sqrt(E/f)]
Qa = (Ag - (b - be)*t) / Ag

Worked example — Q-factor for built-up column

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.

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):

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] be = 13.81 * [1 - 0.2038] = 13.81 * 0.796 = 11.0 in.

Step 3 — Qa factor:

Qa = A_eff / A_g = [A_g - (b - b_e) * t] / A_g

If A_g = 25.0 in^2: Qa = [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 (Qs = 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.

Worked example — W16x26 as compression member (Fy = 50 ksi)

Given: W16x26 column, KL = 14 ft. A = 7.68 in^2, bf = 5.50 in., tf = 0.345 in., h = 14.8 in., tw = 0.250 in.

Step 1 — Flange slenderness: bf/(2tf) = 5.50/(2*0.345) = 7.97 < 13.5 (lambda_p) => Compact. Qs = 1.0

Step 2 — Web slenderness: h/tw = 14.8/0.250 = 59.2 > 35.9 (lambda_r) => Slender

Step 3 — Effective web width: Assume Fcr = 40 ksi (first iteration). be = 1.92*0.250*sqrt(29000/40)*[1-(0.34/59.2)*sqrt(29000/40)] be = 0.48*26.92*[1-0.00574*26.92] be = 12.92*[1-0.1546] = 12.92*0.845 = 10.92 in.

Web area lost = (14.8 - 10.92)*0.250 = 0.97 in^2 Aeff = 7.68 - 0.97 = 6.71 in^2 Qa = 6.71/7.68 = 0.874

Step 4 — Modified capacity: Q*Fy = 0.874*50 = 43.7 ksi. Use this in the AISC column curve.

Effective width by h/tw ratio (Fy = 50 ksi, f = Fcr)

h/tw	f (ksi)	be/h	Qa (web only)	Capacity Reduction
36	50	1.000	1.000	0%
40	48	0.98	0.995	<1%
50	45	0.89	0.955	5%
60	40	0.74	0.874	13%
80	33	0.52	0.708	29%
100	27	0.38	0.556	44%
120	22	0.29	0.435	57%

The effective width drops sharply above h/tw = 60. For columns with h/tw > 80, consider a thicker web or different cross-section.

EN 1993 classification comparison

Class	EN 1993 Definition	AISC Equivalent	Rotation Capacity
Class 1	Plastic hinge formation	Compact (plastic)	>= 3
Class 2	Plastic moment capacity	Compact (compact)	>= 0
Class 3	Elastic capacity	Noncompact	0
Class 4	Effective width only	Slender	N/A

EN 1993 limits (epsilon = sqrt(235/fy))

Element	Class 1	Class 2	Class 3
Flange (rolled I)	9 epsilon	10 epsilon	14 epsilon
Web (compression)	33 epsilon	38 epsilon	42 epsilon
Flange (welded I)	9 epsilon	10 epsilon	14 epsilon
Web (bending)	72 epsilon	83 epsilon	124 epsilon

For fy = 355 MPa: epsilon = 0.814. Flange Class 3 limit = 14 x 0.814 = 11.4.

Code comparison for slender elements

Aspect	AISC 360-22	AS 4100:2020	EN 1993-1-1	CSA S16-19
Classification method	Q-factor (Qs, Qa)	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

AISI S100 Effective Width Method for Slender Walls

For cold-formed steel (CFS) stud walls with slender elements, AISI S100 provides the effective width method, which is analogous to the AISC Q-factor approach but uses the direct Winter equation for each plate element independently.

B2 Procedure: Effective Width of Stiffened Elements

AISI S100 Section B2.1 defines the effective width be of uniformly compressed stiffened elements (webs of C-studs, flanges of track sections):

For lambda <= 0.673:  be = b       (fully effective)
For lambda > 0.673:   be = rho * b  (partially effective)

Where:
  rho = (1 - 0.22/lambda) / lambda    (Winter reduction factor)
  lambda = 1.052 * (b/t) * sqrt(f/E)  (slenderness ratio)
  b = flat width of the element
  t = base metal thickness
  f = stress in the element
  E = modulus of elasticity (29,500 ksi for steel)

B3 Procedure: Effective Width of Unstiffened Elements

For unstiffened elements (flange tips of C-studs, lips of track sections):

For lambda <= 0.673:  be = b       (fully effective)
For lambda > 0.673:   be = rho * b  (partially effective)

Same Winter formula applies, but lambda uses the unstiffened element width.

Effective Width Calculation Example

Problem: A 6 in C-stud has a web flat width b = 5.25 in and thickness t = 0.033 in (33 mil, 20 ga). The web is subjected to uniform compression f = 33 ksi. Determine the effective width.

lambda = 1.052 * (5.25/0.033) * sqrt(33/29500)
       = 1.052 * 159.1 * 0.03344
       = 1.052 * 5.319
       = 5.596

Since lambda = 5.596 > 0.673:
  rho = (1 - 0.22/5.596) / 5.596
      = (1 - 0.0393) / 5.596
      = 0.961 / 5.596
      = 0.1717

  be = rho * b = 0.1717 * 5.25 = 0.901 in

Effective ratio: be/b = 0.901/5.25 = 17.2%

Only 17% of the web flat width is effective. This dramatic reduction is characteristic of very slender CFS elements and demonstrates why effective width calculations are essential for cold-formed steel design. A thicker stud (54 mil, t = 0.054 in) would give lambda = 3.42, rho = 0.262, be = 1.38 in -- still only 26% effective, but substantially better.

Common C-Stud Section Properties Table

Section	Depth (in)	Thickness (mil)	Flange (in)	Lip (in)	Gross Area (in2)	Ix (in4)	Sx (in3)
350S162-33	3.5	33	1.625	0.36	0.167	0.344	0.197
350S162-43	3.5	43	1.625	0.36	0.217	0.437	0.250
350S162-54	3.5	54	1.625	0.36	0.271	0.537	0.307
350S162-68	3.5	68	1.625	0.36	0.339	0.662	0.379
550S162-33	5.5	33	1.625	0.36	0.233	1.014	0.369
550S162-43	5.5	43	1.625	0.36	0.303	1.302	0.473
550S162-54	5.5	54	1.625	0.36	0.379	1.601	0.582
550S162-68	5.5	68	1.625	0.36	0.475	1.974	0.718
600S162-33	6.0	33	1.625	0.36	0.249	1.274	0.425
600S162-43	6.0	43	1.625	0.36	0.324	1.638	0.546
600S162-54	6.0	1.625	0.36	0.405	2.016	0.672
600S162-68	6.0	68	1.625	0.36	0.508	2.487	0.829
800S162-43	8.0	43	1.625	0.36	0.410	3.749	0.937
800S162-54	8.0	54	1.625	0.36	0.513	4.639	1.160
800S162-68	8.0	68	1.625	0.36	0.643	5.736	1.434
1000S162-43	10.0	43	1.625	0.36	0.496	7.106	1.421
1000S162-54	10.0	54	1.625	0.36	0.621	8.831	1.766
1000S162-68	10.0	68	1.625	0.36	0.778	10.959	2.192

Note: Properties shown are gross section values. Effective section properties depend on the applied stress level and must be calculated per AISI S100 Section B2 for each loading condition. The effective Sx and Ix are significantly lower than the gross values for thinner sections.

Deflection Criteria for Slender Wall Studs

Wall studs supporting cladding must meet serviceability deflection limits. For curtain wall studs with brittle finishes (brick veneer, stucco), the deflection limit is L/360. For flexible finishes (gypsum board), L/180 or L/240 is typical. For exterior insulation finish systems (EIFS), L/240 is common.

Finish Type	Deflection Limit	Typical Application
Stone veneer	L/600	Heavy cladding, crack-sensitive
Brick veneer	L/360	Masonry veneer walls
Stucco / plaster	L/360	Rigid cementitious finishes
EIFS	L/240	Exterior insulation finish systems
Metal panels	L/180	Flexible metal cladding
Gypsum board (interior)	L/240	Standard interior partitions
Gypsum board (premium)	L/360	High-end finish, visible walls

For a 10 ft (120 in) stud with gypsum board: allowable deflection = 120/240 = 0.50 in. For a 600S162-54 stud spanning 10 ft with a wind pressure of 20 psf at 16 in stud spacing: w = 20 * 16/12 = 26.7 plf = 0.0267 klf. Using effective Ix = 1.5 in4 (estimated for partial effectiveness): delta = 5wL^4/(384EI) = 50.0267120^4/(384*29500*1.5) = 0.579 in > 0.50 in. The stud is slightly over the limit -- either increase thickness to 68 mil or reduce stud spacing to 12 in.

Bracing Requirements for Slender Wall Studs

CFS wall studs require lateral bracing to prevent lateral-torsional buckling and to stabilize the slender compression flange. AISI S100 Section C2.2 addresses bracing design.

Bracing Type	Maximum Spacing	Force Requirement	Application
Straps (flat strap)	4 ft to 8 ft (depends on stud capacity)	1-2% of stud compression force	Most common for curtain walls
Channel bridging	4 ft to 8 ft	Similar to straps	Heavier walls, axial loads
Gypsum board attachment	Continuous (at each stud)	Per AISI B1.1	Interior walls only
Steel studs with glued sheathing	Per AISI S100	Composite action consideration	Structural walls with sheathing

For curtain wall applications, 1-1/2 in x 0.033 in (33 mil) strap braces at 4 ft on center is the most common specification. The strap must be capable of resisting 1% of the design compression force in the stud at each bracing point, transferred through the bridging channel or direct attachment.

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

Common mistakes

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.
Not checking both axes for beam-columns — a section with a compact flange but noncompact web must be checked for local buckling under combined axial + bending. The Q-factor applies to the axial component only.

Frequently asked questions

What is the difference between compact, noncompact, and slender? Compact sections can develop their full plastic moment. Noncompact sections can reach yield but not full plastic moment. Slender sections buckle locally before reaching yield, requiring a reduced effective area.

Do I need to check slenderness for beams? Yes, but differently than columns. For beams, check Table B4.1b (flexure). Flange slenderness affects LTB and moment capacity. Web slenderness affects shear capacity and bend-buckling.

What is the Winter formula? The effective width formula be = 1.92*t*sqrt(E/f)*[1 - C/(b/t)*sqrt(E/f)]. It was developed by George Winter in the 1940s for cold-formed steel and is the foundation of all modern plate buckling reduction methods.

Does the Q-factor reduce my column capacity much? For most standard W-shapes at Fy = 50 ksi, Q = 1.0 (no reduction). For built-up sections with slender webs or thin cover plates, Q can reduce capacity by 10-30%. For cold-formed sections, Q can be as low as 0.4-0.6.

What is the difference between Qs and Qa? Qs applies to unstiffened elements (flanges, outstanding legs). Qa applies to stiffened elements (webs, cover plates supported on both edges). The total Q = Qs * Qa.

When do I use Table B4.1a vs B4.1b? Table B4.1a is for members subject to axial compression (columns). Table B4.1b is for members subject to flexure (beams). For beam-columns, check both.

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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 the applicable standard and project specification before use. The site operator disclaims liability for any loss arising from the use of this information.