For the strongest rectangular beam that can be cut from a circular log, the ratio of width to depth is:
Try it first: pick an option, then read the explanation.
Asked in SSC JE 04/03/2017 (Evening)
Detailed Explanation
"Strongest" in bending means the largest section modulus. The rectangle's corners must touch the log, so width b, depth h and log diameter d are linked by b² + h² = d².
\[ Z = \frac{bh^2}{6} = \frac{b(d^2-b^2)}{6} \]
\[ \frac{dZ}{db} = \frac{d^2 - 3b^2}{6} = 0 \Rightarrow b = \frac{d}{\sqrt3},\quad h = \sqrt{\frac{2}{3}}\,d \Rightarrow \frac{b}{h} = \frac{1}{\sqrt2} = 0.707 \]
Why “0.707” Is Correct
The strongest beam has b/h = 0.707 (depth = √2 × width). Note the difference: the stiffest beam (maximum I) has b/h = 1/√3 = 0.577.
Why the Other Options Are Wrong
- A. 0.303: 0.303 is not an optimum of Z or I.
- B. 0.404: 0.404 is not an optimum of Z or I.
- C. 0.505: 0.505 is not an optimum either; the stiffest beam gives 0.577.
Key Point
Strongest beam from a log: b/h = 1/√2 = 0.707; stiffest: b/h = 1/√3 = 0.577.
Key Concepts for Students
- Strength ∝ Z; stiffness ∝ I.
- b = d/√3, h = d√(2/3) for the strongest beam.
- Square cut from a log: b/h = 1, which is weaker than the optimum.
Practice more: this is question 29 in Bending and Shear Stress MCQs – Part 2. Try the whole set and check every answer instantly.