W-Beam Section Properties and Moment of Inertia

W-beam section properties are the numbers that describe how a wide-flange cross-section behaves: its area, its moment of inertia (Ix and Iy), its section moduli (Sx, Zx), its radius of gyration and its torsion constant. The one people search for most is the moment of inertia about the strong axis, Ix, because it controls how much a beam deflects. A W250x33 (W10x22) has an Ix of 48.9 x 10⁶ mm⁴; turned on its side, the same beam has an Iy of only 4.73 x 10⁶ mm⁴, about a tenth as much.

The chart below lists properties for common Canadian W-shapes, using values from the Canadian Institute of Steel Construction (CISC) section tables, cross-checked against the metric columns of the AISC Shapes Database. The rest of this guide explains what each column means and how designers use it.

What each section property means

A W-shape's properties depend only on its geometry. They don't change with the steel grade, which affects strength but not stiffness. Here's what you'll see in a section table:

Two things trip people up. First, the units: a metric table usually prints Ix as, say, "48.9" with a column header of 10⁶ mm⁴, so the real value is 48,900,000 mm⁴. Second, imperial tables give Ix in in⁴. To convert, 1 in⁴ is about 0.416 x 10⁶ mm⁴, so the W250x33's 48.9 x 10⁶ mm⁴ is about 117 in⁴, matching the US W10x22.

If you're not sure how the size name maps to the beam, our guide to reading steel beam sizes like W200x27 and W8x18 explains the naming.

Strong axis properties: W-beam moment of inertia chart

These are the values you'll use for a beam standing upright, loaded through its web. Mass and dimensions are rounded the way the CISC tables print them.

For the full list of sizes with flange and web thicknesses, see our Canadian W-shape size chart.

Weak axis and torsion properties

The weak axis (y-y) runs vertically through the web. These values matter for columns, for beams that aren't braced sideways, and for anything loaded horizontally.

The last column shows why orientation matters. Narrow, deep sections like the W360x45 are around 15 times stiffer standing up than lying flat. Square-ish sections like the W200x46 (a "column" shape, as wide as it is deep) are only about 3 times stiffer, which is exactly why they're used as columns: they buckle less readily about the weak axis.

How moment of inertia controls deflection

For a simply supported beam with a uniform load, mid-span deflection is 5wL⁴ / (384 E I), where E for steel is 200,000 MPa. Deflection is inversely proportional to I, so doubling Ix halves the sag.

Here's a beam spanning 5.0 m and carrying a specified live load of 7.6 kN/m (one floor of a house with 4 m joists on each side):

Two lessons:

• Depth beats weight. The W250x33 is about 29% lighter than the W200x46 yet slightly stiffer, because its material sits farther from the neutral axis. When headroom allows, a deeper, lighter beam is usually the efficient choice.
• Never install a beam on its side. The same W250x33 laid flat would sag about ten times as much and fail any reasonable limit.

Deflection limits and floor bounce are covered in more depth in steel beam deflection limits and floor bounce.

Section modulus and bending strength

Stiffness (Ix) and strength (Sx, Zx) are related but not the same.

• Sx = Ix / (d/2). Multiply by the yield strength and you get the moment at which the outer edge of the flange first yields.
• Zx assumes the whole section has yielded, half in tension and half in compression. For W-shapes, Zx is typically about 10% to 15% larger than Sx; for the W250x33 the ratio is 424 / 379, about 1.12.

Under CSA S16, a laterally supported beam with a Class 1 or 2 section (most W-shapes in 350W steel) has a factored moment resistance of Mr = 0.90 x Zx x Fy. For a W250x33 in 350W, that's 0.90 x 424 x 10³ mm³ x 350 MPa, about 134 kN·m. If the compression flange isn't braced, the resistance drops, and that calculation uses Iy, J and Cw as well. Our post on how much weight a steel beam can hold works a full example.

Grade changes Fy and therefore Mr. It doesn't change Ix, so a higher-strength beam is not a stiffer beam.

Checking the numbers by hand

You can approximate Ix for any I-shape by treating it as a solid rectangle minus the two empty rectangles beside the web: I = [bf x d³ − (bf − tw) x h³] / 12, where h is the clear depth between flanges.

For the W250x33 (d = 258 mm, bf = 146 mm, tf = 9.1 mm, tw = 6.1 mm, h = 239.8 mm), that gives about 48.2 x 10⁶ mm⁴, against the handbook's 48.9 x 10⁶ mm⁴. The small difference is the rolled fillets where web meets flange, which the simple formula ignores. Use the handbook value for design; the hand method is a sanity check, and it's how built-up sections such as plate girders are calculated.

For channels, angles and HSS, the properties work the same way but the axes differ (angles, for instance, have inclined principal axes). See section properties of HSS, channel and angle.

Getting the right section on your job

Section properties tell a designer which beam works; they don't replace the design. A real beam also has to be checked for lateral stability, shear, bearing, connections and the posts and footings below it, and in Ontario that work needs a qualified designer or engineer and usually a building permit.

Once the size is on the drawings, we supply and fabricate W-beams cut to length with the holes, plates and connections they need, and engineering services are available for projects that require them. Send your drawings or beam schedule for a quote, or browse our steel products.