What is the maximum torque TE that can be applied to a solid steel cylindrical shaft 8 cm in diameter, if the shaft is to remain elastic ? (Take the elastic limit in shear and the shear modulus τ₀ = 145 MPa and G = 76 GPa, respectively)
- A14,580 N-mCorrect
- B7,290 N-m
- C3,645 N-m
- D29,160 N-m
Solution
Concept
Torsion formula
Formula
T = τ · π d³ / 16
Step-by-step solution
1. J/r = π d³ / 16 = π × 80³ / 16 = 100 531 mm³. 2. T = 145 × 100 531 = 14.58 × 10⁶ N·mm = 14 580 N·m.
Explanation
The shaft remains elastic until the maximum shear stress at the surface reaches the elastic limit τ₀. Substituting in τ = T r / J gives T = τ₀ π d³/16. G is not needed.
Why the other options are wrong
7 290, 3 645 and 29 160 N·m are ½, ¼ and 2 times the correct value of 14 580 N·m.
Shortcut
T = τ π d³ / 16 (solid shaft).
Common mistake
Using π d³/32 (the bending section modulus).
Correct answer: A) 14,580 N-m