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?
A hollow shaft has an outer diameter twice its inner diameter of a solid shaft, both constructed from the same material, and having identical outer diameters. The ratio of their torsional strength is:
Important Questions from Equation of Torsion
- If two aluminum bar have a different length (L1 = 2L2) and diameter (d1 = 2d2) with an identical angle of a twist then, find torque value for bar 1, If bar 2 torque value is 50 N-m
Which of the following assumptions are True for torsion theory for axisymmetric sections?
A tubular shaft, having an inner diameter of 30 mm and an outer diameter of 40 mm, is to be used to transmit 80 kW of power. The speed of rotation of the shaft so that the shear stress will not exceed 50 MPa is
A circular solid shaft of span L = 5 m is fixed at one end and free at the other end. A torque T = 100 kN.m is applied at the free end. The shear modulus and polar moment of inertia of the section are denoted as G and J, respectively. The torsional rigidity GJ is 50,000 kN.m2 /rad. The following are reported for this shaft:
Statement i) The rotation at the free end is 0.01 rad
Statement ii) The torsional strain energy is 1.0 kN.m
With reference to the above statements, which of the following is true?
A solid circular shaft of diameter d and length L is fixed at one end and free at the other end. A torque T is applied at the free end. The shear modulus of the material is G. The angle of twist at three free ends is
