---
title: "Torsional rigidity; hollow shaft vs solid shaft strength"
subject: "Strength of Materials"
chapter: 6
key: "semester_3/strength_of_materials:torsional-rigidity-hollow-vs-solid-300rpm"
source: https://www.bctengineer.com/q/semester_3/strength_of_materials/torsional-rigidity-hollow-vs-solid-300rpm
site: BCT Engineering
---

# Torsional rigidity; hollow shaft vs solid shaft strength

Strength of Materials · Chapter 6: Torsion

- Asked in: 2080 Baisakh, 8 marks
- Video: [Example: Hollow vs Solid Torsional  Shaft](https://www.youtube.com/watch?v=rjGtp4FROSQ) — Calvin Rans
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/strength_of_materials/torsional-rigidity-hollow-vs-solid-300rpm
- Whole chapter: https://www.bctengineer.com/md/semester_3/strength_of_materials/chapter-6

## Question

Define torsional rigidity. A hollow steel shaft of 10 cm outer diameter and 7 cm internal diameter is rotating with a speed of 300 rpm. If the permissible shearing stress for the material is $80\ \text{MN/m}^2$, determine the torque and compare the strength of this hollow shaft with that of solid shaft of same material, weight and length.

## Short answer

Torsional rigidity is the torque for unit twist per unit length, equal to GJ. The hollow shaft torque is 11.94 kN·m. Compared to a solid shaft of equal weight, it is 2.09 times stronger.

## Answer

**Torsional rigidity** is the torque required to produce a twist of one radian per unit length of the shaft; it equals the product $GJ$ of the modulus of rigidity and the polar moment of inertia (from $\dfrac{T}{\varphi} = \dfrac{GJ}{L}$). Unit: N·mm² (or N·m²).

#### Torque of the hollow shaft
**Data:** $D = 100$ mm, $d = 70$ mm, $\tau = 80\ \text{MN/m}^2 = 80\ \text{N/mm}^2$, $N = 300$ rpm (the speed is not needed for the torque).

$$
J = \frac{\pi}{32}(100^4 - 70^4) = 7.4603\times10^{6}\ \text{mm}^4
$$

$$
T_h = \frac{\tau J}{R} = \frac{80\times7.4603\times10^{6}}{50} = 1.1936\times10^{7}\ \text{N·mm} = 11.936\ \text{kN·m}
$$

(Power at 300 rpm would be $P = \frac{2\pi N T}{60} = 375.0$ kW.)

#### Comparison with a solid shaft of the same weight
Same material, weight and length means equal area:

$$
\frac{\pi}{4}D_s^2 = \frac{\pi}{4}(100^2 - 70^2) \Rightarrow D_s = \sqrt{5100} = 71.41\ \text{mm}
$$

$$
T_s = \frac{\pi}{16}\tau D_s^3 = \frac{\pi}{16}(80)(71.41)^3 = 5.7210\times10^{6}\ \text{N·mm} = 5.721\ \text{kN·m}
$$

$$
\frac{T_h}{T_s} = \frac{11.936}{5.721} = 2.086
$$

**Answer:** torque of the hollow shaft $= 11.94$ kN·m; it is **2.09 times as strong** as the solid shaft of equal weight (solid shaft torque $= 5.72$ kN·m).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is torsional rigidity?
   A. EI
   B. GJ, torque for one radian twist per unit length
   C. Torque per unit area
   D. tau/R
   Answer: B. From T/phi = GJ/L. (Key point: Definition)

2. Is the 300 rpm needed for the torque?
   A. Yes, T = 2 pi N
   B. Yes, for J
   C. No, torque depends on tau and section only; speed gives power
   D. Yes, for tau
   Answer: C. T = tau J/R. (Key point: Speed)

3. What equal-weight condition finds the solid shaft?
   A. Equal diameters 100 mm
   B. Equal J
   C. D_s = 30 mm
   D. Equal areas: D_s^2 = 100^2 - 70^2, D_s = 71.41 mm
   Answer: D. Same material and length means same area. (Key point: Equal weight)

4. What is the strength ratio?
   A. Hollow is 2.09 times as strong
   B. Equal
   C. 0.48 times
   D. 5 times
   Answer: A. 11.936/5.721 = 2.086. (Key point: Comparison)

## Related in Strength of Materials

- [Hollow shaft 10/7 cm: power transmitted](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Ahollow-shaft-power-300rpm-13-mean) — Strength of Materials, chapter 6, asked 2×
- [Solid shaft 300 kW; hollow shaft saving](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Asolid-shaft-300kw-percent-saving) — Strength of Materials, chapter 6
- [Torsion vs bending; hollow shaft replaced by solid](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Ahollow-to-solid-shaft-replacement) — Strength of Materials, chapter 6
- [Solid shaft 300 kW, 120 rpm with twist limit](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Asolid-shaft-300kw-120rpm-twist-limit) — Strength of Materials, chapter 6
- [Hollow shaft 1670 kW at 4500 rpm: size](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Ahollow-shaft-1670kw-4500rpm) — Strength of Materials, chapter 6
- [Hollow shaft 30 kW at 240 rpm: diameters](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Ahollow-shaft-30kw-240rpm) — Strength of Materials, chapter 6

## Related in other subjects

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- [Hollow shaft against solid shaft](https://www.bctengineer.com/question.md?key=semester_6%2Fmachine_design_i%3Ahollow-versus-solid-shaft) — Machine Design I, chapter 5
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