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Civil
Hollow rectangular section second moment of area
Second moment of area about the horizontal centroidal axis of a rectangle of outer width B and depth H with a concentric rectangular void of width b and depth h.
- Ref
- CIV-08
- Rev
- v1
- Field
- Civil
- Result
- Ix [m⁴]
A hollow rectangle is the outer rectangle with an inner one cut out, so its second moment of area is the difference BH³/12 − bh³/12. Because depth is cubed, the material far from the axis does most of the work: a thin-walled box keeps most of the stiffness of the solid section with a fraction of the material. Use the true geometry for rounded corners.
Second moment of area about the horizontal centroidal axis of a rectangle of outer width B and depth H with a concentric rectangular void of width b and depth h. It is the outer rectangle’s inertia minus the void’s. Valid for sharp corners; use the true geometry for rounded corners or fillets.
Calculator
- Solid outer rectangle inertia, BH³/124.5×10−4 m⁴
- Void inertia, bh³/122.343×10−4 m⁴
- Elastic section modulus, I/(H/2)0.001438 m³
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.
- The hollow section keeps 47.92 % of the solid section’s inertia with only 30.67 % of its area.
- Its elastic section modulus is 0.001438 m³, so a bending moment of 100 kN·m would produce a peak stress of 69.56 MPa.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m⁴) | Local step | Change in result |
|---|---|---|---|---|---|---|
| H Outer depth | 0.27 to 3 m | 9.37e-5 | 0.4498 | 0.4497 | +10% | +69.1 % |
| B Outer width | 0.17 to 2 m | 1.482e-4 | 0.004266 | 0.004117 | +10% | +20.9 % |
| h Void depth | 0 to 0.29 m | 1.248e-4 | 4.5e-4 | 3.252e-4 | +10% | −36 % |
| b Void width | 0 to 0.19 m | 1.717e-4 | 4.5e-4 | 2.783e-4 | +10% | −10.9 % |
Sweep of Outer depth
| Outer depth (m) | Second moment of area (m⁴) |
|---|---|
| 0.27 | 9.37e-5 |
| 0.6113 | 0.003572 |
| 0.9525 | 0.01417 |
| 1.294 | 0.03586 |
| 1.635 | 0.07261 |
| 1.976 | 0.1284 |
| 2.317 | 0.2072 |
| 2.659 | 0.313 |
| 3 | 0.4498 |
Sweep one variable
Variables and how each one contributes
Second moment of area
What the equation gives you. Everything else feeds it.
Outer width
The outer width of the section, in m. Raises the inertia in direct proportion for a fixed depth. Raising it by 10 % (to 0.22) moves the result by +20.9 %, from 2.157×10−4 to 2.607×10−4 m⁴.
Outer depth
The outer depth, in m, measured perpendicular to the bending axis. Raises the inertia with the cube of the depth, so depth is the most effective dimension. Raising it by 10 % (to 0.33) moves the result by +69.1 %, from 2.157×10−4 to 3.646×10−4 m⁴.
Void width
The width of the void, in m. A wider void removes material and lowers the inertia. Raising it by 10 % (to 0.176) moves the result by −10.9 %, from 2.157×10−4 to 1.922×10−4 m⁴.
Void depth
The depth of the void, in m. A deeper void lowers the inertia strongly, with the cube of its depth. Raising it by 10 % (to 0.286) moves the result by −36 %, from 2.157×10−4 to 1.381×10−4 m⁴.
Worked example with the values above
- Outer rectangle: BH³/12 = 0.2 × 0.3³ / 12 = 4.5×10−4 m⁴.
- Void: bh³/12 = 0.16 × 0.26³ / 12 = 2.34347×10−4 m⁴.
- I = 4.5×10−4 − 2.34347×10−4 = 2.15653×10−4 m⁴.
Assumptions and limits
- Linear elastic material, plane sections staying plane, and perfect bond between components.
- Sharp-cornered ideal geometry with no fillets, holes or local buckling.
Common mistakes
- Mixing millimetres with metres, which changes I by a factor of 10¹² when it is raised to the fourth power.
- Using the inertia about the wrong axis (width and depth swapped).
- Forgetting that composite properties depend on the modulus of each part, not only its area.
Statistics and uncertainty
These are exact geometric results for the stated shapes, so the uncertainty is in the dimensions. Because the second moment of area scales with depth cubed, a 2 % error in depth gives about a 6 % error in stiffness. Rolled and fabricated sections vary by a few per cent from nominal, and composite sections carry extra uncertainty from the modulus ratio.
Where it is used
- Calculating stiffness, stresses and capacity of beams and built-up members.
- Spacing connectors in composite and built-up sections.
Maths you need
- Second moment of area: I = ∫y² dA about the centroidal axis
- Parallel axis theorem: I = I₀ + Ad² about a parallel axis
- Neutral axis: the line of zero bending stress in an elastic section