For shafts designed to transmit the same torque, which statement is generally true when comparing a solid shaft to a hollow shaft of the same material and outer diameter?
The hollow shaft has a lower torsional stiffness but lower weight.
This is a comparison of a solid shaft and a hollow shaft made of the same material and having the same outer diameter D. The two properties in question are torsional stiffness and weight, and both are governed by how much material lies far from the axis.
Torsional stiffness depends on the polar moment of inertia J through k = GJ/L (G = modulus of rigidity, L = length). For the two sections:
- Solid: Jsolid = πD⁴/32
- Hollow: Jhollow = π(D⁴ − d⁴)/32, where d is the inner (bore) diameter.
Since removing the central core subtracts the positive term πd⁴/32, we have Jhollow < Jsolid. With the same G and L, this means the hollow shaft has lower torsional stiffness. Note, however, that the reduction is modest — the removed core is close to the axis where it contributes little torsional resistance — so the stiffness penalty is small.
Weight is proportional to the cross-sectional area of material. The hollow shaft has a bored-out centre, so it contains less material and is therefore lighter. Because most of the material that carries torsion sits near the outer surface anyway, the hollow shaft keeps most of its strength/stiffness while shedding significant weight — this is exactly why hollow shafts are preferred where weight (or a high strength-to-weight ratio) matters.
Putting the two results together, the correct statement is that the hollow shaft has a lower torsional stiffness but lower weight. The other options contradict the mathematics: it is impossible for the hollow shaft to have higher stiffness and higher weight when material has been removed; the solid shaft cannot have higher stiffness and lower weight because more material makes it heavier, not lighter; and the two clearly cannot be equal in both stiffness and weight when their cross-sections differ.
Similar Questions
A hollow shaft is designed to transmit a torque of 40000 N-m. The polar moment of inertia (J) is 0.004 m⁴. The ratio of inside diameter to outside diameter of the hollow shaft (Dᵢ/Dₒ) is 0.8. The inside diameter is 400 mm. What is the maximum induced shear stress (τₘₐˣ) at the outer fiber of the shaft?
Important Questions from Equation of Torsion
What is the maximum torque transmitted by a hollow shaft of external radius ‘R’, internal radius ‘r’ and maximum allowable shear stress τ?
The maximum torque that can be safely applied to a shaft of 100 mm diameter if the permissible angle of twist is 1 degree in a length of 3 m and the permissible shear stress is 30 N/mm2. Take G = 0.8 × 105 N/mm2.
Which of the following assumptions are True for torsion theory for axisymmetric sections?
The magnitude of shear stress induced in a shaft due to applied torque varies from:
A circular shaft is subjected to a torque of 50 kN-m. If the permissible shear stress is 40 MPa, then the maximum permissible diameter of the shaft is ______.
