The moment of inertia of a rectangle of width d and depth b about its horizontal axis at mid-depth is:
Try it first: pick an option, then read the explanation.
Correct answer: A. db³/12
Asked in SSC JE 24/01/2018 (Morning), CWC & MES 2011, SSC JE 2007
Detailed Explanation
The letters are swapped on purpose. The rule is: width × depth³ / 12 about the centroidal axis parallel to the width.
\[ I_{xx} = \frac{(\text{width})(\text{depth})^3}{12} = \frac{d\,b^3}{12} \]
Why “db³/12” Is Correct
Here the width is d and the depth is b, so I = db³/12. The dimension perpendicular to the axis is cubed.
| Section | I about centroidal axis | y max | Section modulus Z |
|---|---|---|---|
| Rectangle b × d (axis ∥ b) | bd³/12 | d/2 | bd²/6 |
| Square side a (sides horizontal) | a⁴/12 | a/2 | a³/6 |
| Square side a (diagonal horizontal) | a⁴/12 | a/√2 | a³/(6√2) |
| Solid circle, diameter d | πd⁴/64 | d/2 | πd³/32 |
| Triangle b × h (axis ∥ base) | bh³/36 | 2h/3 | bh²/24 |
| Hollow circle D, d | π(D⁴ − d⁴)/64 | D/2 | π(D⁴ − d⁴)/(32D) |
Why the Other Options Are Wrong
- B. bd³/12: bd³/12 cubes the width — it is I about the vertical axis.
- C. bd³/3: bd³/3 is about an edge, with the letters also swapped.
- D. db³/3: db³/3 is about the bottom edge, not the mid-depth axis.
Key Point
I = (width)(depth)³/12; cube the dimension perpendicular to the axis.
Key Concepts for Students
- About the base: bd³/3.
- Depth matters far more than width for I.
- Always identify which side is perpendicular to the axis.
Practice more: this is question 42 in Bending and Shear Stress MCQs – Part 3. Try the whole set and check every answer instantly.