Rectangular Area Moment of Inertia
Second moment of area for a rectangular cross-section.
Formula
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Example
Depth matters far more than width
How it works
This formula gives the bending resistance of a solid rectangular section about its neutral axis, and it shows how area is distributed relative to that axis. The h^3 term appears because material placed farther from the neutral axis contributes disproportionately more to resisting bending, so increasing height is far more effective than increasing width. It comes from integrating y^2 over the rectangle’s area, which is the definition of second moment of area.
When to use it
Use this when analyzing beams, joists, brackets, or any rectangular cross-section under bending, especially in civil and structural design. It is also useful when comparing how changing a section’s dimensions affects stiffness and bending stress.
Worked example, step by step
Find the second moment of area for a rectangular timber beam with width b = 80 mm and height h = 240 mm. I = (b h^3) / 12 I = (80 × 240^3) / 12 240^3 = 13,824,000 I = (80 × 13,824,000) / 12 I = 1,105,920,000 / 12 I = 92,160,000 mm^4 Final answer: I = 9.216 × 10^7 mm^4
Common mistakes to avoid
- Using the wrong dimension as h and forgetting that the vertical depth is cubed
- Mixing up units and leaving dimensions in mixed mm and m without converting consistently
- Confusing second moment of area with mass moment of inertia, which is a different quantity
Did You Know?
This is why beams are oriented tall, not wide.
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