Moment of Inertia of an I-Beam
Iₓ, Iₖ, Section Modulus & Radius of Gyration • Wide-Flange Sections
Σ Live equations — formulas + values for the current shape
💡 What-if coach — design insights from current values
Moment of Inertia of an I-Beam — Formula and Worked Example
An I-beam concentrates material in the flanges, far from the neutral axis, which is exactly where the parallel-axis term Ad² does the most work. That is why it is the default shape for bending members.
Formulas
| Quantity | Expression |
|---|---|
| Area | A = BH − (B − t₷)(H − 2tₜ) |
| Second moment about x | Iₓ = [BH³ − (B − t₷)(H − 2tₜ)³] / 12 |
| Second moment about y | Iₖ = [2tₜB³ + (H − 2tₜ)t₷³] / 12 |
| Section modulus | Sₓ = 2Iₓ / H |
| Radius of gyration | rₓ = √(Iₓ/A) |
Because the section is doubly symmetric, the centroid sits at mid-height and mid-width, and x and y are the principal axes — no rotation is needed.
Worked Example — A 100 × 200 mm I-section with an 8 mm web and 12 mm flanges
| Quantity | Value |
|---|---|
| B (flange width) | 100 mm |
| H (overall depth) | 200 mm |
| t₷ (web) | 8 mm |
| tₜ (flange) | 12 mm |
| A | 3,808 mm² |
| Iₓ | 24.87 × 10⁶ mm⁴ |
| Iₖ | 2.01 × 10⁶ mm⁴ |
| Sₓ | 248.7 × 10³ mm³ |
| rₓ | 80.8 mm |
Values computed by the calculator above. Change any dimension and every property updates live.
What is the moment of inertia of an I-beam?
For a doubly symmetric I-section it is Ix = [BH³ − (B − tw)(H − 2tf)³]/12, where B is the flange width, H the overall depth, tw the web thickness and tf the flange thickness. The expression is simply the full bounding rectangle minus the two voids either side of the web.
Why is an I-beam stronger than a rectangle of the same area?
Bending stiffness depends on how far material sits from the neutral axis, and the transfer term in the parallel-axis theorem grows with the square of that distance. Moving area out into the flanges therefore buys far more Ix than the same area spread uniformly, which is why the web can be thin.
Which axis should I use for a beam?
Use Ix (bending about the strong, horizontal axis) when the beam is loaded in its usual upright orientation. Iy is much smaller and governs weak-axis bending and lateral-torsional buckling, so it still matters for stability checks.
Other Cross-Sections
The same calculator handles all eight standard sections:
- I-Beam (this page)
- Channel
- Angle
- T-Section
- Rectangle
- Circle
- Hollow Circle
- Hollow Rect
For the full section-property reference and the parallel-axis theorem, see the Moment of Inertia Calculator hub.