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Section knowledge · I/H shapes

I/H shapes: understanding and calculating section properties

Everything about I-shapes and H-shapes (wide-flange shapes): how to calculate A, I, Z, J and Cw and what they are used for in design, shown with a typical shape series of your region. Every value is calculated live and compared with the catalog value.

  • Example series: W (United States)
  • AISC Manual (ASTM A6)
All series of I/H shapes (page in German)

The I/H shapes

The I/H profile form covers all doubly symmetric hot-rolled shapes made of two parallel flanges and a web. In the United States they are W-shapes (wide-flange shapes). In Europe, narrow sections such as IPE are much deeper than they are wide, while HEA, HEB and HEM are about as wide as they are deep; the UK uses UB and UC. The formulas and terms apply to all of them.

In a narrow, deep shape almost all of the material sits in the flanges, as far from the strong axis as possible. That makes it the most economical choice for bending about the strong axis. About the weak axis it is flexible, and as an open thin-walled section it has very little torsional stiffness, so long beams without lateral bracing often fail by lateral-torsional buckling.

Wide shapes (column shapes such as W14 or HEB) have a much larger weak-axis moment of inertia and radius of gyration. They are the typical column sections and are less prone to lateral-torsional buckling, but they weigh more for the same depth.

Shapes with sloped flanges (S-shapes, IPN) and built-up welded sections are not covered here: their fillets and flanges need other formulas.

Typical uses

  • Floor beams, girders, purlins and girts (narrow, deep shapes)
  • Columns and frame posts (wide column shapes)
  • Beams braced laterally by a deck or bracing

Less suited for

  • Members under significant torsion
  • Long beams without lateral bracing (especially narrow shapes)
  • Compression members that can buckle about both axes (narrow shapes)

Which section for which load?

LoadNarrow I (IPE · UB · W)Wide H (HEB · UC · W14)Round HSSRectangular HSSReason
Bending about the strong axis●●●●●●●●●●●●material far from the strong axis
Bending about the weak axis●●●●●●●●●●●●narrow flanges give little stiffness
Compression with buckling●●●●●●●●●●●●the smaller radius of gyration governs; it is small for narrow shapes
Torsion●●●●●●●●●●●●closed sections have a many times larger J
Lateral-torsional buckling (long unbraced beams)●●●●●●●●●●●●depends on Iy
Simple connections●●●●●●●●●●●●open shapes are easy to bolt

Rules of thumb (●●● = very well suited). They do not replace a design check.

The key section properties

Formula and catalog value for the selected shape. Click a property for its meaning, formula, design use and typical mistake.

A

Cross-sectional area

Formula, W 12x26: 7.641 in²Catalog: 7.650 in²Deviation: −0.1 %

What does it describe?

The area of the section. It governs weight, axial strength and the axial stiffness EA.

Calculation

A = 2·bf·tf + (d − 2tf)·tw + (4 − π)·r²

Share of the root fillets: 0.077 in² (1.0 % of the value).

What is it used for? (AISC 360)

  • Tensile yielding in the gross section Pn = Fy·Ag (AISC 360)
  • Self-weight w = A·ρ (steel ≈ 490 lb/ft³)
  • Axial elongation ΔL = P·L/(E·A)

Typical mistake

Leaving out the root fillets costs about 1 to 5 % of the area of a rolled shape.

Av

Shear area for the shear force Vz

Formula, W 12x26: 3.024 in²Catalog: not in the catalogDeviation: –

What does it describe?

The part of the area that carries the shear force in the direction of the web: essentially the web with its root fillets and the adjoining part of the flanges. The flange tips carry hardly any shear.

Calculation

Av = A − 2·bf·tf + (tw + 2r)·tf (Eurocode; AISC: Aw = d·tw)

Share of the root fillets: 0.305 in² (10.1 % of the value).

Catalog value A_v,z of the producers; the British and US catalogs do not list it.

What is it used for? (AISC 360)

  • AISC 360, Chapter G uses the web area Aw = d·tw for rolled I-shapes, not this value
  • Vn = 0.6·Fy·Aw·Cv1; for most rolled W-shapes Cv1 = 1.0 and φv = 1.0

Typical mistake

Simply taking h·tw. For rolled sections the Eurocode shear area is clearly larger because the fillets and part of the flanges contribute.

Ix

Moment of inertia about the strong axis

Formula, W 12x26: 203.5 in⁴Catalog: 204.0 in⁴Deviation: −0.2 %

What does it describe?

Measure of the bending stiffness about the strong axis. Every bit of area counts with the square of its distance from the axis, which is why I- and H-shapes put their material in the flanges.

Calculation

Ix = [bf·d³ − (bf − tw)·(d − 2tf)³] / 12 + 4·(Ir + Ar·yr²)

Share of the root fillets: 2.469 in⁴ (1.2 % of the value).

What is it used for? (AISC 360)

  • Deflection, e.g. simple span Δ = 5wL⁴/(384·E·Ix)
  • Elastic buckling load Pe = π²·E·I/Lc²
  • Internal forces of statically indeterminate systems

Typical mistake

Forgetting the parallel-axis term A·d² of the flanges. It is almost the whole value.

Iy

Moment of inertia about the weak axis

Formula, W 12x26: 17.33 in⁴Catalog: 17.30 in⁴Deviation: +0.2 %

What does it describe?

Bending stiffness about the weak axis. Almost only the flanges contribute, so the value is much smaller than about the strong axis for narrow shapes and considerably larger for wide column shapes.

Calculation

Iy = [2·tf·bf³ + (d − 2tf)·tw³] / 12 + 4·(Ir + Ar·xr²)

Share of the root fillets: 0.003 in⁴ (0.0 % of the value).

What is it used for? (AISC 360)

  • Buckling about the weak axis
  • Lateral-torsional buckling through rts² = √(Iy·Cw)/Sx (AISC 360, Chapter F2)
  • Warping constant Cw = Iy·ho²/4

Typical mistake

Mixing up the axis names. The weak axis is z in Eurocode notation and y in AISC notation.

rx

Radius of gyration about the strong axis

Formula, W 12x26: 5.161 inCatalog: 5.170 inDeviation: −0.2 %

What does it describe?

The distance at which the whole area would have to be concentrated to give the same moment of inertia: r = √(I/A). It turns stiffness and area into a length and governs the slenderness.

Calculation

rx = √(Ix / A)

Share of the root fillets: 0.005 in (0.1 % of the value).

What is it used for? (AISC 360)

  • Slenderness Lc/rx for buckling about the strong axis (AISC 360, Chapter E3)
  • Recommended limit Lc/r ≤ 200 for compression members

Typical mistake

Combining the buckling length with the radius of gyration of the wrong axis. Each axis has its own buckling length and its own radius.

ry

Radius of gyration about the weak axis

Formula, W 12x26: 1.506 inCatalog: 1.510 inDeviation: −0.3 %

What does it describe?

The distance at which the whole area would have to be concentrated to give the same moment of inertia: r = √(I/A). It turns stiffness and area into a length and governs the slenderness.

Calculation

ry = √(Iy / A)

Share of the root fillets: -0.008 in (-0.5 % of the value).

What is it used for? (AISC 360)

  • Slenderness Lc/ry, usually governing for column buckling (AISC 360, Chapter E3)
  • Limiting length Lp = 1.76·ry·√(E/Fy) for lateral-torsional buckling (Chapter F2)

Typical mistake

Combining the buckling length with the radius of gyration of the wrong axis. Each axis has its own buckling length and its own radius.

Sx

Elastic section modulus about the strong axis

Formula, W 12x26: 33.36 in³Catalog: 33.40 in³Deviation: −0.1 %

What does it describe?

Moment of inertia divided by the distance to the extreme fiber: it links bending moment and extreme-fiber stress, σ = M/S. The linear stress distribution holds as long as the material stays elastic.

Calculation

Sx = Ix / (d/2)

Share of the root fillets: 0.405 in³ (1.2 % of the value).

What is it used for? (AISC 360)

  • Yield moment My = Fy·Sx; design of noncompact sections per AISC 360, Chapter F
  • Elastic lateral-torsional buckling, Mn = Fcr·Sx
  • Shape factor Zx/Sx

Typical mistake

Using a single value for sections that are not doubly symmetric. Here both extreme fibers are equally far away; otherwise there are two different section moduli.

Zx

Plastic section modulus

Formula, W 12x26: 37.11 in³Catalog: 37.20 in³Deviation: −0.2 %

What does it describe?

Twice the first moment of half the area about the neutral axis: the bending resistance when the whole section has yielded. The elastic modulus Sx = Ix/(d/2) applies until the extreme fiber reaches Fy.

Calculation

Zx = bf·tf·(d − tf) + tw·(d − 2tf)²/4 + 4·Ar·yr

Share of the root fillets: 0.437 in³ (1.2 % of the value).

What is it used for? (AISC 360)

  • Plastic moment Mp = Fy·Zx, design strength φb·Mn with φb = 0.90 (LRFD)
  • Applies to compact sections; otherwise reduced per AISC 360, Chapter F
  • Shape factor Zx/Sx

Typical mistake

Using the plastic value for noncompact or slender sections (Eurocode classes 3 and 4).

Zy

Plastic section modulus about the weak axis

Formula, W 12x26: 8.168 in³Catalog: 8.170 in³Deviation: −0.0 %

What does it describe?

Bending resistance about the weak axis when the whole section has yielded. Almost only the rectangular flanges contribute, so the plastic reserve is large: Zy is about 1.5 times Sy.

Calculation

Zy = 2·tf·bf²/4 + (d − 2tf)·tw²/4 + 4·Ar·xr

Share of the root fillets: 0.014 in³ (0.2 % of the value).

What is it used for? (AISC 360)

  • Mp = Fy·Zy ≤ 1.6·Fy·Sy for weak-axis bending (AISC 360, Chapter F6)
  • Biaxial bending per Chapter H

Typical mistake

Using the large weak-axis plastic reserve unchecked. The flange slenderness (section class) decides whether it is available.

J

Torsional constant (St. Venant)

Formula, W 12x26: 0.300 in⁴Catalog: 0.300 in⁴Deviation: −0.0 %

What does it describe?

Resistance to twisting without warping. For open thin-walled parts it is approximately Σ b·t³/3. Thickness enters to the third power, which is why the value is small.

Calculation

J ≈ [2·bf·tf³ + (d − 2tf)·tw³] / 3 + 2·α·D⁴ − 0.420·tf⁴

Share of the root fillets: 0.016 in⁴ (5.4 % of the value).

Junction term for the web-flange junction, usually attributed to El Darwish and Johnston; source not yet checked. It matches the catalog values of EN, BS and AISC shapes.

What is it used for? (AISC 360)

  • St. Venant torsion T = G·J·θ′
  • Lateral-torsional buckling, limiting length Lr and Mn (AISC 360, Chapter F2)
  • Torsional parameter a = √(E·Cw/(G·J)) (AISC Design Guide 9)

Typical mistake

The simple formula Σ b·t³/3 clearly underestimates rolled shapes (by 16 to 28 % for IPE), because the root fillets at the web-flange junction add a lot of torsional stiffness.

Cw

Warping constant

Formula, W 12x26: 605.2 in⁶Catalog: 607.0 in⁶Deviation: −0.3 %

What does it describe?

Resistance to warping, the out-of-plane displacement of the section under torsion. In an I-shape it arises because the flanges bend laterally against each other.

Calculation

Cw = Iy · (d − tf)² / 4

Share of the root fillets: 0.098 in⁶ (0.0 % of the value).

What is it used for? (AISC 360)

  • Lateral-torsional buckling through rts (AISC 360, Chapter F2)
  • Warping torsion and warping normal stresses (AISC Design Guide 9)
  • Torsional parameter a = √(E·Cw/(G·J))

Typical mistake

Overlooking the unit (in⁶ or cm⁶). Converting cm⁶ to mm⁶ is a factor of 10⁶.

Colors of the deviation from the catalog: up to 1 % up to 5 % above 5 %

Section class: may the section be designed plastically?

Thin, wide parts buckle locally before the section becomes fully plastic. The slenderness of flange and web decides whether Z (compact), only S (noncompact) or a reduced section (slender) applies; Eurocode 3 calls these classes 1 and 2, 3 and 4.

  • Flexure (strong axis)compact
  • Axial compressionslender
ElementLoadb/t or h/twλp / λrResult
Flange (unstiffened)Flexure (strong axis)8.59.2 / 24.1compact
Web (stiffened)Flexure (strong axis)47.190.6 / 137.3compact
Flange (unstiffened)Axial compression8.513.5nonslender
Web (stiffened)Axial compression47.135.9slender

√(E/Fy) = 24.08 with E = 29,000 ksi, Fy = 50 ksi

AISC 360-16, Table B4.1a (compression) and B4.1b (flexure), with h = d − 2tf − 2r for rolled shapes.

Animation: from the elastic to the plastic moment

Until the extreme fibers reach the yield stress Fy, stresses grow linearly with the distance from the neutral axis: My = S·Fy. Then the section yields from the outside in until it is fully plastic: Mp = Z·Fy. The ratio Z/S is the plastic reserve that compact sections may use.

−fy+fyCompressionTensionW 12x26: orange = yieldedStress σ
M / Mel1.000
Shape factor Z/S1.113

Idealized elastic-perfectly plastic, without residual stresses and without local buckling.

Formula check against the catalog

Every formula on this page is evaluated for all 289 shapes of the W series, in the Dlubal library before every release and here once more, live.

Section propertyLargest deviationat shapeup to 1 %up to 5 %above 5 %
A Cross-sectional area+0.7 %W 18x19228900
Av Shear area for the shear force Vznot in the catalog
Ix Moment of inertia about the strong axis+1.0 %W 18x13028900
Iy Moment of inertia about the weak axis+1.4 %W 36x135266230
rx Radius of gyration about the strong axis+0.6 %W 24x19228900
ry Radius of gyration about the weak axis−0.7 %W 30x13228900
Sx Elastic section modulus about the strong axis+0.8 %W 44x26228900
Zx Plastic section modulus−1.0 %W 14x6828810
Zy Plastic section modulus about the weak axis+1.1 %W 33x15228630
J Torsional constant (St. Venant)+1.1 %W 40x33128630
Cw Warping constant+2.1 %W 16x89253360

Deviation formula ↔ catalog in %. Catalog values are rounded; deviations up to about 0.5 % are rounding.

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