Moment of Inertia of an Angle Section
Iₓ, Iₖ, Product of Inertia & Principal Axes • L-Sections
Display Controls
Σ Live equations — formulas + values for the current shape
💡 What-if coach — design insights from current values
Moment of Inertia of an Angle Section — Formula and Worked Example
An angle has no axis of symmetry parallel to its legs, and that changes the problem. Iₓ and Iₖ alone are not enough — the section also has a non-zero product of inertia Iₓₖ, so the geometric axes are not the principal axes and an angle loaded vertically will deflect sideways as well.
Formulas
| Quantity | Expression |
|---|---|
| Centroid | xᶜ = Σ(Aᵢxᵢ)/ΣAᵢ , yᶜ = Σ(Aᵢyᵢ)/ΣAᵢ |
| Product of inertia | Iₓₖ = ΣAᵢ(xᵢ − xᶜ)(yᵢ − yᶜ) |
| Principal moments | I₁,₂ = (Iₓ+Iₖ)/2 ± √[((Iₓ−Iₖ)/2)² + Iₓₖ²] |
| Principal axis rotation | tan 2θ = −2Iₓₖ / (Iₓ − Iₖ) |
For an equal-leg angle Iₓ = Iₖ, which makes the rotation exactly 45°. That is the classic trap: the two equal values look reassuring, but the true minimum stiffness I₂ is far lower than either of them, and it is I₂ that governs buckling.
Worked Example — A 100 × 100 × 10 mm equal-leg angle
| Quantity | Value |
|---|---|
| Legs | 100 × 100 mm |
| t (thickness) | 10 mm |
| A | 1,900 mm² |
| Centroid xᶜ = yᶜ | 28.7 mm from the heel |
| Iₓ = Iₖ | 1.800 × 10⁶ mm⁴ |
| Iₓₖ | -1.066 × 10⁶ mm⁴ |
| I₁ (max) | 2.866 × 10⁶ mm⁴ |
| I₂ (min) | 0.734 × 10⁶ mm⁴ |
| Principal rotation θ | 45° |
Values computed by the calculator above, which shows Ixy, Iu, Iv, θp and the minimum radius of gyration rv for any angle and draws the u and v axes on the section. Sharp corners are assumed: rolled angles have a root fillet, so steel tables list slightly larger values.
Why do angle sections need principal axes?
Because an angle has no axis of symmetry parallel to its legs, its product of inertia Ixy about those axes is not zero (an equal-leg angle is symmetric only about its 45° diagonal, which is why that diagonal is a principal axis). Ix and Iy are then just the values about two arbitrary axes, not the maximum and minimum. The true extremes are the principal moments I1 and I2, found by rotating the axes until Ixy vanishes.
What is the principal axis angle of an equal-leg angle?
Exactly 45 degrees. Since Ix = Iy for an equal-leg angle, tan 2θ = −2Ixy/(Ix − Iy) has a zero denominator and the rotation resolves to 45°. For a 100 × 100 × 10 angle this gives I1 ≈ 2.87 × 10⁶ mm⁴ and I2 ≈ 0.73 × 10⁶ mm⁴.
Which value should I use for buckling?
The smaller principal moment I2, because a column buckles about its weakest axis. Using Ix or Iy for an equal-leg angle would overestimate the capacity substantially — here I2 is only about 40% of Ix.
Other Cross-Sections
The same calculator handles all eight standard sections:
- I-Beam
- Channel
- Angle (this page)
- T-Section
- Rectangle
- Circle
- Hollow Circle
- Hollow Rect
For the full section-property reference and the parallel-axis theorem, use the moment of inertia calculator for any cross-section.
Updated
