---
title: "Stepped shaft: angle of twist at free end"
subject: "Strength of Materials"
chapter: 6
key: "semester_3/strength_of_materials:shaft-angle-of-twist-free-end"
source: https://www.bctengineer.com/q/semester_3/strength_of_materials/shaft-angle-of-twist-free-end
site: BCT Engineering
---

# Stepped shaft: angle of twist at free end

Strength of Materials · Chapter 6: Torsion

- Asked in: 2067 Asar (old course), 8 marks
- Video: [Angle of Twist of Shaft with Torsion](https://www.youtube.com/watch?v=ogaJRMjItjA) — The Ryder Project
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/strength_of_materials/shaft-angle-of-twist-free-end
- Whole chapter: https://www.bctengineer.com/md/semester_3/strength_of_materials/chapter-6

## Question

For the following shaft, obtain the angle of twist at free end. [Figure: shaft fixed at the left end; $G = 0.9\times10^5\ \text{N/mm}^2$; first segment of diameter 110 mm, length 3 m, with a 10 kNm torque applied at its right end; second segment of diameter 60 mm, length 2 m, with a 5 kNm torque at the free end.]

## Short answer

Internal torques: 15 kN·m (segment 1), 5 kN·m (segment 2). Using φ=TL/GJ, φ₁=0.03479 rad, φ₂=0.08733 rad. Total twist at free end = 0.1221 rad = 7.00°.

## Answer

**Reading of the figure:** fixed at the left; segment 1: $d = 110$ mm, $L = 3$ m, with a 10 kN·m torque at its right end; segment 2: $d = 60$ mm, $L = 2$ m, with a 5 kN·m torque at the free end, both in the same sense. $G = 0.9\times10^5\ \text{N/mm}^2$.

#### Internal torques
- Segment 2: $T_2 = 5$ kN·m
- Segment 1: $T_1 = 10 + 5 = 15$ kN·m

#### Polar moments
$$
J_1 = \frac{\pi}{32}(110)^4 = 1.4374\times10^{7}\ \text{mm}^4, \qquad J_2 = \frac{\pi}{32}(60)^4 = 1.2723\times10^{6}\ \text{mm}^4
$$

#### Angle of twist
$$
\varphi_1 = \frac{T_1L_1}{GJ_1} = \frac{(15\times10^6)(3000)}{(0.9\times10^5)(1.4374\times10^{7})} = 0.03479\ \text{rad}
$$

$$
\varphi_2 = \frac{T_2L_2}{GJ_2} = \frac{(5\times10^6)(2000)}{(0.9\times10^5)(1.2723\times10^{6})} = 0.08733\ \text{rad}
$$

$$
\varphi = \varphi_1 + \varphi_2 = 0.12211\ \text{rad} = 6.997^\circ
$$

**Answer: angle of twist at the free end $= 0.1221$ rad $= 7.00^\circ$.**

## Check yourself

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

1. What is the internal torque in the 110 mm segment next to the wall?
   A. 5 kN·m in the opposite sense
   B. 15 kN·m
   C. 10 kN·m
   D. 5 kN·m
   Answer: B. Both torques act in the same sense, so segment 1 carries 10 + 5 = 15 kN·m. (Key point: Torque in segment 1)

2. Which formula gives J for a solid circular segment?
   A. pi d^3/16
   B. pi d^2/4
   C. pi d^4/64
   D. pi d^4/32
   Answer: D. J = (pi/32)d^4 for a solid circular section. (Key point: Polar moment)

3. How is the twist at the free end obtained?
   A. Only phi2
   B. T total * L total/(G J of larger segment)
   C. phi1 * phi2
   D. phi1 + phi2, each from T L/(G J)
   Answer: D. Twists of the segments in series add: 0.0348 + 0.0873 rad. (Key point: Total twist method)

4. Which segment contributes more twist and what is the total?
   A. Segment 1; total 0.0348 rad
   B. Segment 2 (thin one); total about 0.122 rad = 7.0°
   C. Both equal; total 0.07 rad
   D. Segment 1; total 0.122 rad
   Answer: B. The thin 60 mm segment twists 0.0873 rad vs 0.0348 rad, total 0.1221 rad = 7.0°. (Key point: Result)

## Related in Strength of Materials

- [Stepped shaft, three torques: twist at free end](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Astepped-shaft-three-torques-twist-at) — Strength of Materials, chapter 6
- [Fixed shaft: end couples, diameter, section of no twist](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Ashaft-fixed-ends-couples-no-twist) — Strength of Materials, chapter 6
- [Brass-steel stepped shaft: max torque, rotation](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Astepped-shaft-brass-steel-max-torque) — Strength of Materials, chapter 6
- [Stepped shaft: maximum torque at C](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Astepped-shaft-fixed-max-torque-at) — Strength of Materials, chapter 6
- [Steel shaft 200 hp: torque, shear, twist](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Asteel-shaft-200hp-150rpm) — Strength of Materials, chapter 6
- [Fixed shaft, two torques: couples, diameter](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Ashaft-fixed-ends-torques-30-375) — Strength of Materials, chapter 6

## Related in other subjects

- [Statically indeterminate shaft with torque](https://www.bctengineer.com/question.md?key=semester_4%2Fstrength_of_materials_mechanical%3Ashaft-fixed-both-ends-torque) — Strength of Materials, chapter 4
- [Castigliano: fixed-end moment with 50 kNm at overhang](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acastigliano-fixed-moment-50knm-overhang) — Theory of Structures II, chapter 2