Which one of the following is the equation for extension of whole length of a uniformly tapering rectangular bars where b₁ and b₂ are the limits of widths (b₂ > b₁), length L, thickness t and the bar is subjected to an axial force P and elastic module E ?
- AΔ = PL / ((b₂ − b₁) t E) · logₑ(b₂/b₁)Correct
- BΔ = PL / ((b₂ − b₁) t E) · log₁₀(b₁/b₂)
- CΔ = PL / ((b₁ − b₂) t E) · log₁₀(b₂/b₁)
- DΔ = PLt / ((b₂ − b₁) E) · logₑ(b₂/b₁)
Solution
Concept
Elongation of a tapering rectangular bar
Formula
Δ = P L ln(b₂/b₁) / [(b₂ − b₁) t E]
Step-by-step solution
1. Width at x: b = b₁ + (b₂ − b₁) x / L. 2. Δ = ∫ P dx / (E t b) = P L / [E t (b₂ − b₁)] · ln(b₂/b₁).
Explanation
Integration gives a natural logarithm. The result is positive with b₂ > b₁ and ln(b₂/b₁).
Why the other options are wrong
Base-10 logarithms and inverted ratios are incorrect, and PLt in the numerator has the wrong dimensions.
Shortcut
Tapering rectangle: natural log of width ratio.
Common mistake
Using log₁₀.
Correct answer: A) Δ = PL / ((b₂ − b₁) t E) · logₑ(b₂/b₁)