For a given stress, the ratio of the moment of resistance of a square beam placed with two sides horizontal to that with a diagonal horizontal is:
Try it first: pick an option, then read the explanation.
Asked in SSC JE 2012
Detailed Explanation
For a square the moment of inertia is the same about every centroidal axis (b⁴/12). Only the distance to the extreme fibre changes when it is turned.
\[ Z_1 = \frac{b^4/12}{b/2} = \frac{b^3}{6},\qquad Z_2 = \frac{b^4/12}{b/\sqrt2} = \frac{b^3}{6\sqrt2} \]
\[ \frac{M_1}{M_2} = \frac{Z_1}{Z_2} = \sqrt2 \]
Why “√2” Is Correct
With sides horizontal the beam resists √2 ≈ 1.414 times the moment it resists with a diagonal horizontal, because the corner fibres in the diamond position are farther away and are highly stressed.
Why the Other Options Are Wrong
- A. 1/2: 1/2 is not the ratio of the two Z values.
- B. 1: I is the same, but Z is not, because y_max changes.
- C. 1/√2: 1/√2 is the inverse — diagonal-horizontal to sides-horizontal.
Key Point
Square: Z (sides horizontal) / Z (diagonal horizontal) = √2.
Key Concepts for Students
- Cutting off the sharp corners of the diamond can actually increase its Z slightly.
- I of a square is the same about any centroidal axis.
- The diamond's max shear stress is 9/8 τavg, away from the N.A.
Practice more: this is question 48 in Bending and Shear Stress MCQs – Part 4. Try the whole set and check every answer instantly.