Library / Mechanical / MEC-36
Mechanical
Second moment of area, rectangle
Resistance of a rectangular section to bending about its centroid.
- Ref
- MEC-36
- Rev
- v1
- Field
- Mechanical
- Result
- I [mm⁴]
b is the width and h the depth of the rectangle, with h measured in the direction of bending. I is taken about the centroidal axis parallel to b. The depth is cubed, so orientation matters far more than width.
Resistance of a rectangular section to bending about its centroid. Depth counts as a cube, so a deeper section is much stiffer.
Calculator
- Section modulus I/(h/2)83,330 mm³
- I with width and depth swapped, bh³ → hb³/121,042,000 mm⁴
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.
- A 50 × 100 mm plank on edge has I = 4.17 × 10⁶ mm⁴, but laid flat it is only 1.04 × 10⁶ mm⁴, four times less stiff.
- For the same amount of material, making a section deeper is the cheapest way to resist bending.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mm⁴) | Local step | Change in result |
|---|---|---|---|---|---|---|
| h Depth | 1 to 1,000 mm | 4.167 | 4.167e9 | 4.167e9 | +10% | +33.1 % |
| b Width | 1 to 1,000 mm | 83,330 | 8.333e7 | 8.325e7 | +10% | +10 % |
Sweep of Depth
| Depth (mm) | Second moment of area (mm⁴) |
|---|---|
| 1 | 4.167 |
| 2.371 | 55.56 |
| 5.623 | 740.9 |
| 13.34 | 9,881 |
| 31.62 | 131,800 |
| 74.99 | 1,757,000 |
| 177.8 | 2.343e7 |
| 421.7 | 3.125e8 |
| 1,000 | 4.167e9 |
Sweep one variable
Variables and how each one contributes
Second moment of area
What the equation gives you. Everything else feeds it.
Width
The width of the rectangle, perpendicular to the direction of bending, in mm. Directly proportional: doubling the width doubles the stiffness.
Depth
The depth, in the direction of bending, in mm. Cubed: doubling the depth makes the section 8 times stiffer. A 10 % rise in h raises I by about 33 %.
Worked example with the values above
- Cube the depth: h³ = (100)³ = 1,000,000.
- Multiply by the width: b h³ = 50 × 1,000,000 = 5×107.
- I = 5×107 / 12 = 4,167,000 mm⁴.
Assumptions and limits
- A solid rectangular section, bending about the axis through its centroid.
- Dimensions are the actual sizes after finishing, not nominal sizes.
Common mistakes
- Swapping b and h, which gives the stiffness about the weak axis.
- Using nominal sizes (such as ‘2 × 4’) in place of the actual size.
Statistics and uncertainty
I ∝ b h³, so a 1 % error in depth gives about 3 % in I and in the deflection. Measuring timber or plate depth carefully matters far more than the width.
Where it is used
- Why a plank is stiffer on edge than flat.
- Finding I for the stress and deflection formulas of beams.
Maths you need
- Cubes: h³
- Area moments: how area is spread about an axis