Torsion comparison: What is the ratio of torsional strength (torque capacity) of a hollow circular shaft to that of a solid shaft when both have the same outside diameter D, and the hollow shaft has inside diameter D/2? Assume the same material and allowable shear stress.

Introduction / Context:
Designers often compare the torsional strength of solid and hollow shafts to save material while maintaining capacity. Torsional strength (allowable torque) is proportional to the polar section modulus Zp = J / r_max for a given allowable shear stress.

Given Data / Assumptions:

• Outside diameter of both shafts: D.
• Inside diameter of hollow shaft: D/2.
• Same material and allowable shear stress.
• Torque capacity T_allow ∝ Zp.

Concept / Approach:
For a solid circular shaft: Zp_solid = π D^3 / 16. For a hollow circular shaft: Zp_hollow = [π (D^4 − d^4)] / (16 D), where d is the inside diameter. The ratio hollow-to-solid equals (1 − (d/D)^4). Substituting d/D = 1/2 gives 1 − (1/2)^4 = 1 − 1/16 = 15/16.

Step-by-Step Solution:
1) Solid: Zp_solid = π D^3 / 16.2) Hollow: Zp_hollow = π (D^4 − d^4) / (16 D) = (π D^3 / 16) * [1 − (d/D)^4].3) Ratio (hollow/solid) = Zp_hollow / Zp_solid = 1 − (d/D)^4.4) With d/D = 1/2: ratio = 1 − (1/2)^4 = 15/16.

Verification / Alternative check:
Numerically, 15/16 = 0.9375. Thus, a hollow shaft with inside diameter D/2 retains about 93.75 percent of the torsional capacity of a solid shaft with the same outside diameter, showing efficient material use.

Why Other Options Are Wrong:
Fractions like 1/4, 1/2, 1/16, or 3/8 severely underestimate capacity and do not follow from the Zp relationship.

Common Pitfalls:
Confusing the asked ratio direction (solid-to-hollow vs hollow-to-solid); using polar moment J instead of Zp without dividing by r_max; or substituting radius ratio instead of diameter ratio.

Final Answer:
15/16