Library / Mechanical / MEC-22
Mechanical
Second moment of a built-up section
Combined second moment of area of two parts about the centroid of the whole.
- Ref
- MEC-22
- Rev
- v1
- Field
- Mechanical
- Result
- I [10⁶ mm⁴]
Use parallel centroidal axes for the components, with di the offset from each component’s centroid to the composite centroid ȳ. Subtract void contributions consistently. Area inertia has units m⁴ (here 10⁶ mm⁴). The defaults are a T section: a 100×20 mm flange on a 20×100 mm web.
To find the stiffness of a section built from parts, find its centre, then add each part’s own inertia plus a term for how far its centre is from the whole’s centre.
Calculator
- Centroid ȳ80 mm
- Total area4 10³ mm²
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Material far from the centroid contributes most, which is why I-beams put the metal in the flanges.
- Beam stress σ = My/I falls in proportion as I rises.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (10⁶ mm⁴) | Local step | Change in result |
|---|---|---|---|---|---|---|
| y₁ Centroid of part 1 | 20 to 300 mm | 1.737 | 64.23 | 62.5 | +10% | +27 % |
| Ic1 Own inertia of part 1 | 0.001 to 50 10⁶ mm⁴ | 5.268 | 55.27 | 50 | +10% | +0.125 % |
| Ic2 Own inertia of part 2 | 0.001 to 50 10⁶ mm⁴ | 3.668 | 53.67 | 50 | +10% | +3.13 % |
| y₂ Centroid of part 2 | 5 to 150 mm | 1.735 | 12.76 | 11.02 | +10% | −10.8 % |
| A₁ Area of part 1 | 0.1 to 50 10³ mm² | 2.076 | 8.656 | 6.58 | +10% | +3.21 % |
| A₂ Area of part 2 | 0.1 to 50 10³ mm² | 2.076 | 8.656 | 6.58 | +10% | +3.21 % |
Sweep of Centroid of part 1
| Centroid of part 1 (mm) | Second moment of area (10⁶ mm⁴) |
|---|---|
| 20 | 2.633 |
| 55 | 1.758 |
| 90 | 3.333 |
| 125 | 7.358 |
| 160 | 13.83 |
| 195 | 22.76 |
| 230 | 34.13 |
| 265 | 47.96 |
| 300 | 64.23 |
Sweep one variable
Variables and how each one contributes
Second moment of area
What the equation gives you. Everything else feeds it.
Area of part 1
Area of the first part. Increases I through its offset.
Centroid of part 1
Height of its centroid above a datum. Moves the part away from the others.
Own inertia of part 1
Second moment of area of part 1 about its own centroid. Adds directly.
Area of part 2
Area of the second part. Increases I through its offset.
Centroid of part 2
Height of its centroid above the same datum. Moves the part away from the others.
Own inertia of part 2
Second moment of area of part 2 about its own centroid. Adds directly.
Worked example with the values above
- Centroid: ȳ = (2×110 + 2×50) / 4 = 80 mm.
- Offsets: d₁ = 30 mm, d₂ = −30 mm.
- I = 0.06667 + 1.8 + 1.667 + 1.8 = 5.333 × 10⁶ mm⁴.
Assumptions and limits
- Components share one parallel set of axes.
- Rigid connection between parts.
Common mistakes
- Forgetting the A d² terms.
- Using different datums for the two parts.
Statistics and uncertainty
Dimensions are squared or cubed in I, so a 1 mm error on a 100 mm depth can change I by 3%.
Where it is used
- Beam and girder section properties.
- Stiffener and frame design.
Maths you need
- Centroid: weighted mean of positions
- Second moment of area: I = ∫y² dA