A circular log of timber has diameter d. The ratio of breadth b to depth h (b : h) of the rectangular beam that can be cut from a circular log for strongest section in bending is
- A1 : 1
- B1 : √2Correct
- C1 : √3
- D1 : 2
Solution
Concept
Maximum section modulus from a circular log
Formula
Z = b h² / 6, b² + h² = d²
Step-by-step solution
1. Z = b h²/6 with h² = d² − b², so Z = b (d² − b²)/6. 2. dZ/db = 0 ⇒ d² − 3b² = 0 ⇒ b = d/√3. 3. h² = d² − d²/3 = 2d²/3, so h = d√(2/3). 4. b : h = 1/√3 : √2/√3 = 1 : √2.
Explanation
The strongest beam has maximum section modulus. Maximising b(d² − b²) gives b = d/√3 and h = √2 b. (The stiffest section would need b = d/2.)
Why the other options are wrong
A square (1:1) is not the strongest. 1:√3 and 1:2 are not the result of the maximisation.
Shortcut
Strongest: b : h = 1 : √2. Stiffest: b : h = 1 : √3.
Common mistake
Mixing up the strongest and stiffest sections.
Correct answer: B) 1 : √2