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Section properties

Pick a shape or a catalog section and get area, centroid, moments of inertia, S and Z moduli, radii of gyration, torsional constant and weight per foot, with a drawing of the section and its axes. For built-up sections, paste the rectangles from Excel.

Pre-sizing and estimating tool: results do not replace design, calculations or reports signed by a licensed professional, nor verification against the applicable codes and standards.

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Section

Catalog shapes fill in nominal dimensions (AISC Manual, EN 10365, EN 10056-1). Editing any dimension makes the section custom.

Rectangles of the built-up section

One row per rectangle: width b, height h and the x, y position of its lower-left corner, in the page units (in or mm). Do not overlap rectangles: shared area would be counted twice.

See what was read
—Moment of inertia Ix (horizontal axis)
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Section drawing

To scale. Dashed lines: x and y axes through the centroid (CG). Dash-dot: principal axes u and v, when inclined.

Section modulus versus weight, compared with catalog shapes

Each dot is a catalog shape computed by the same method. Higher for the same weight means more efficient in bending.

Tap the chart to see the values.

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Take this page’s results to another calculator with the fields already filled in.

Tension and compression stress with this area →Paint needed for this shape →Weight and cost of these bars →

The report gathers this page's inputs, results, equations, assumptions and references, with fields for project, engineer and revision. In the print dialog, choose Save as PDF.

Method, assumptions and references

Equations used

A = Σ Ai; x̄ = Σ Ai xi ÷ A; ȳ = Σ Ai yi ÷ A (holes as negative area)
Ix = Σ (Ix,i + Ai di²) (parallel axis); rectangle: b h³ ÷ 12; circle: π D⁴ ÷ 64; Ixy = Σ (Ixy,i + Ai dxi dyi)
S = I ÷ c_max; r = √(I ÷ A); Z = Σ |y − y_p| dA, with y_p on the equal-area line
Principal axes: I1,2 = (Ix + Iy) ÷ 2 ± √[((Ix − Iy) ÷ 2)² + Ixy²]; θ = ½ atan[−2 Ixy ÷ (Ix − Iy)]
Torsion: open sections J ≈ Σ b t³ ÷ 3; I-shapes with fillets, El Darwish and Johnston formula; closed tubes J = 4 Am² t ÷ p (Bredt); circle π D⁴ ÷ 32

Assumptions and limits

  • Constant-thickness flanges. Fillets are modeled as spandrels; angle toe radius is r ÷ 2 (EN 10056-1). Rectangular tubes with sharp corners (HSS corner radii reduce properties by a few percent).
  • The plastic modulus is integrated over 3,000 strips (error below 0.1%). For angles, use the principal-axis properties.
  • Weight per foot with steel at 490 lb/ft³. Values are for the gross section: no bolt-hole deductions and no local slenderness checks.

Standards and technical documents

  • Young, W. C.; Budynas, R. G.: Roark’s Formulas for Stress and Strain, 8th ed., McGraw-Hill
  • AISC Steel Construction Manual, 16th ed., Part 1: Dimensions and Properties
  • AISC Design Guide 9: Torsional Analysis of Structural Steel Members
  • EN 10365:2017: Hot rolled steel channels, I and H sections — Dimensions and masses
  • EN 10056-1:2017: Structural steel equal and unequal leg angles — Dimensions
  • Gere, J. M.; Goodno, B. J.: Mechanics of Materials, Cengage

U.S. regulations

  • ANSI/AISC 360-22: Specification for Structural Steel Buildings
  • ASTM A6/A6M: General Requirements for Rolled Structural Steel Bars, Plates, Shapes, and Sheet Piling
  • OSHA 29 CFR 1926 Subpart R: Steel Erection

Frequently asked questions

What is the moment of inertia of a W8×31?

About 110 in⁴ about the strong axis, with S = 27.5 in³ and Z = 30.4 in³. About the weak axis, 37.1 in⁴. It weighs 31 lb/ft.

What is the difference between S and Z?

S is the elastic modulus: the extreme fiber yields at M = S × Fy. Z is the plastic modulus: the whole section yields at M = Z × Fy. For a rectangle Z = 1.5 S; for a W shape about 1.1 S.

Why are angle axes inclined?

An angle has no symmetry about x and y. For an L2×2×1/4, Ix = 0.35 in⁴, but about the minor principal axis it is about 0.14 in⁴, and that is the buckling axis.

How do I calculate a built-up girder?

Choose Built-up section and paste one rectangle per row with width, height and lower-left corner. A girder with 12 × 1/2 in and 8 × 1/2 in flanges and a 3/8 × 19 in web has Ix of about 1,140 in⁴.