Moment of Inertia — Why a 2×6 on Edge is 13× Stiffer
If stress tells you whether something will break, moment of inertia tells you whether it will bend. It’s the single most important concept in structural design, and it explains why beams are shaped the way they are.
The Formula That Matters
For a rectangle (which covers most lumber and bar stock):
I = bh³ / 12
Where b = width (horizontal) and h = height (vertical, the direction of loading).
Notice: height is cubed. That’s the magic. Doubling the height increases stiffness 8×. Tripling it: 27×.
The 2×6 Demo
A nominal 2×6 is actually 1.5″ × 5.5″.
| Orientation | b × h | I = bh³/12 | Relative Stiffness |
|---|---|---|---|
| ▯ On edge (tall) | 1.5 × 5.5 | 20.80 in⁴ | 13.4× |
| = Flat (wide) | 5.5 × 1.5 | 1.55 in⁴ | 1.0× |
Same board. Same wood. Same weight. 13.4× stiffer on edge. This is why floor joists, rafters, and headers are always oriented tall. The h³ term dominates everything.
Common Cross-Section Formulas
| Shape | Ix (about horizontal centroidal axis) |
|---|---|
| Rectangle | bh³/12 |
| Circle | πd⁴/64 |
| Hollow Circle (tube) | π(d₀⁴ – dᵢ⁴)/64 |
| I-Beam (approx) | bH³/12 – (b-tw)(H-2tf)³/12 |
| Triangle | bh³/36 |
The Parallel Axis Theorem — Building Complex Shapes
What if your cross-section isn’t a simple shape? Break it into simple parts, calculate each I about its own centroid, then shift to the overall centroid:
Itotal = Σ(Ilocal + A × d²)
Where d is the distance from each piece’s centroid to the overall centroid. The Ad² term is often larger than Ilocal — which is exactly why I-beams work. The flanges have small I on their own, but they’re far from the centroid, so Ad² is huge.
Example: Built-Up Box Beam
Two 2×6 boards with ½” plywood top and bottom, 5.5″ apart (like a box beam):
- Plywood caps: Ilocal ≈ 0 (thin), but A × d² = (5.5 × 0.5) × 3.0² = 24.75 in⁴ each
- Side boards: Ilocal = 20.80 in⁴ each
- Total: 2(24.75) + 2(20.80) = 91.1 in⁴
More than 4× stiffer than a single 2×6, using cheap plywood to space the material apart.
Key Takeaways
- Stiffness depends on h³ — height matters much more than width
- This is why lumber goes on edge, why I-beams exist, and why tubes are more efficient than solid bars
- Parallel axis theorem lets you analyze any complex shape by breaking it into rectangles
- Deflection = load × length³ / (E × I) — so I directly controls how much a beam bends