Moment of inertia of an I-beam
The closed-form Ix formula for symmetric I-sections, plus a calculator that handles any custom geometry or standard W / IPE / HEA / HEB section. Returns Ix, Iy, polar J, section moduli S and Z, radii of gyration, and area. Reviewed by a licensed PE.
Use the I-beam moment of inertia calculator
The calculator opens with the I-beam form selected. Either type your custom dimensions (h, b, tw, tf) or pick a named standard section like W14x22 or IPE200 from the "Standard section" dropdown — dimensions auto-fill.
Pick an AISC W / IPE / HEA / HEB / HSS / channel / angle to auto-fill dimensions, or leave on "Manual entry" for a custom section.
Rectangle: simplest case. Ix = b·h³/12 about centroidal axis parallel to base.
Hollow rectangle (centred). Ix = (b·h³ − bi·hi³)/12. Ensure inner dims < outer.
Solid circle. Ix = Iy = π·D⁴/64. Polar J = π·D⁴/32.
Hollow circle / pipe. Ix = π(D⁴ − d⁴)/64. d = inside diameter.
Half-disc with flat edge at bottom. Centroid sits 4r/(3π) above the flat edge.
I-beam (W / IPE / UB). Symmetric flanges and centred web.
T-beam: top flange + vertical stem. Centroid is below the flange — origin is bottom of stem.
C-channel: web on left, flanges open to the right. Asymmetric — centroid offset from web.
L-angle: legs of width leg1 (horizontal) and leg2 (vertical), uniform thickness t.
Right triangle: right-angle at the origin, base b along x, height h along y. Centroid at (b/3, h/3).
Isosceles trapezoid: a = top, b = bottom, h = height. ȳ = h(b + 2a)/(3(a+b)) from bottom.
- Polar J
- — mm⁴
- Sx (elastic)
- — mm³
- Sy (elastic)
- — mm³
- Zx (plastic)
- — mm³
- rx
- — mm
- ry
- — mm
- Area
- — mm²
- Centroid (x̄, ȳ)
- —
- I about offset
- —
The I-beam moment of inertia formula
The closed form treats the I-section as the bounding rectangle minus two web-side rectangular notches. Both flanges and the web are accounted for in one expression — no need to split into pieces if the section is symmetric.
- I_x
- second moment of area about the strong axis, mm⁴ / in⁴
- h
- overall depth (top of top flange to bottom of bottom flange), mm / in
- b
- flange width, mm / in
- t_w
- web thickness, mm / in
- t_f
- flange thickness, mm / in
For the weak axis, the formula reduces to two flange contributions plus the very thin web:
- I_y
- second moment of area about the weak axis, mm⁴ / in⁴
- b, h, t_w, t_f
- as above, mm / in
Iy is always smaller than Ix for a typical I-beam, often by a factor of 5–20 — that asymmetry is why orientation matters: lay the beam with the web vertical (loads in the strong direction) or it will be much more flexible than designed.
Worked example, W14x22
The AISC W14x22 is one of the most common floor and roof beams in commercial construction. Compute its strong-axis Ix step by step, with units kept in millimetres throughout. Dimensions from the AISC Manual, converted from inches.
| Step | Substitution | Result |
|---|---|---|
| Given dimensions | h = 348, b = 127, tw = 5.84, tf = 8.51 mm | — |
| Outer rectangle bh³ | 127 × 348³ | 5.36 × 10⁹ mm⁴· |
| Inner depth (h − 2tf) | 348 − 2 × 8.51 | 331 mm |
| Inner width (b − tw) | 127 − 5.84 | 121 mm |
| Subtracted notches | 121 × 331³ | 4.39 × 10⁹ mm⁴· |