Torsion – comparative strength of solid vs. hollow circular shafts Let a solid circular shaft of diameter D be compared with a hollow circular shaft of external diameter D and internal diameter d. What is the ratio of their torsional moments of resistance (solid : hollow)?
-
AD^4 : (D^4 − d^4)
-
BD^3 : (D^3 − d^3)
-
CD^2 : (D^2 − d^2)
-
DD : (D − d)
-
Enone of these
Answer
Correct Answer: D^4 : (D^4 − d^4)
Explanation
Introduction / Context:Designers often select hollow shafts for weight savings while maintaining torsional capacity. Comparing the torsional “moment of resistance” (polar section modulus) clarifies why hollow shafts can be efficient.
Given Data / Assumptions:
- Solid shaft diameter = D.
- Hollow shaft external diameter = D, internal diameter = d.
- Elastic torsion (Saint-Venant) with circular sections.
Concept / Approach:The torsional moment T that produces a given extreme shear stress tau_max is:For solid: T = (pi / 16) * tau_max * D^3For hollow: T = (pi / 16) * tau_max * (D^4 − d^4) / DThe torsional “moment of resistance” Z_p is T / tau_max, so:Z_p(solid) = (pi / 16) * D^3Z_p(hollow) = (pi / 16) * (D^4 − d^4) / DTherefore the ratio (solid : hollow) is:[(pi/16) D^3] : [(pi/16) (D^4 − d^4) / D] = D^4 : (D^4 − d^4)
Step-by-Step Solution:Write Z_p(solid) and Z_p(hollow) as above.Cancel common factor (pi/16).Multiply both sides by D to clear denominator → ratio = D^4 : (D^4 − d^4).
Verification / Alternative check:Limiting cases: if d → 0, hollow → solid and ratio → D^4 : D^4 = 1, consistent. If d → D, hollow capacity → 0, ratio → infinite, also consistent.
Why Other Options Are Wrong:Options with powers 1–3 confuse torsion with bending section modulus patterns; torsion depends on polar properties leading to fourth-power terms for circular shafts.
Common Pitfalls:Using J (polar moment of inertia) directly without converting to Z_p = J / R; neglecting the outer radius R = D/2 in the hollow expression.
Final Answer:D^4 : (D^4 − d^4)