---
title: "Stepped shaft: maximum torque at C"
subject: "Strength of Materials"
chapter: 6
key: "semester_3/strength_of_materials:stepped-shaft-fixed-max-torque-at"
source: https://www.bctengineer.com/q/semester_3/strength_of_materials/stepped-shaft-fixed-max-torque-at
site: BCT Engineering
---

# Stepped shaft: maximum torque at C

Strength of Materials · Chapter 6: Torsion

- Asked in: 2076 Chaitra, 8 marks
- Video: [Exam #3 2015 Fall Problem #1 Indeterminate Torque](https://www.youtube.com/watch?v=qYXmebMC2ys) — Emerson Engineering
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/strength_of_materials/stepped-shaft-fixed-max-torque-at
- Whole chapter: https://www.bctengineer.com/md/semester_3/strength_of_materials/chapter-6

## Question

A steel shaft is connected to fixed supports as shown in figure. Limiting shear stress in the material is 50 MPa. Determine the maximum torque that can be applied at joint C. What is the shear stress at A? [Figure: shaft fixed at A (left) and B (right); segment AC diameter 20 mm and length 0.7 m; segment CB diameter 30 mm and length 0.5 m; torque T kNm applied at C.]

## Short answer

Maximum torque at C is 302.5 N·m, limited by segment CB reaching 50 MPa. Shear stress at A (segment AC) is 23.8 MPa.

## Answer

**Data:** $\tau_{allow} = 50\ \text{MPa}$; segment AC: $d_1 = 20$ mm, $L_1 = 0.7$ m; segment CB: $d_2 = 30$ mm, $L_2 = 0.5$ m; same material; torque $T$ applied at C; both ends fixed.

#### Distribution of torque
Let $T_A$ and $T_B$ be the end reactions: $T_A + T_B = T$ (equilibrium). The rotation of C calculated from either end must be equal:

$$
\frac{T_AL_1}{GJ_1} = \frac{T_BL_2}{GJ_2} \Rightarrow \frac{T_A}{T_B} = \frac{L_2/J_2}{L_1/J_1} = \frac{0.5/30^4}{0.7/20^4} = 0.1411
$$

So $T_A = 0.1236\,T$ and $T_B = 0.8764\,T$.

#### Limiting torque
Stress in each segment is $\tau = \dfrac{16T_i}{\pi d^3}$.

- AC reaches 50 MPa when $T_A = \dfrac{\pi(50)(20)^3}{16} = 78.54$ N·m. Then $T = T_A/0.1236 = 635.2$ N·m.
- CB reaches 50 MPa when $T_B = \dfrac{\pi(50)(30)^3}{16} = 265.07$ N·m. Then $T = T_B/0.8764 = 302.5$ N·m.

The smaller value governs (segment CB is stressed to the limit first):

$$
T_{max} = 302.5\ \text{N·m}
$$

#### Shear stress at A
With this torque, $T_A = 37399.9$ N·mm and $T_B = 265071.9$ N·mm, so

$$
\tau_A = \frac{16T_A}{\pi(20)^3} = 23.81\ \text{MPa}, \qquad \tau_B = \frac{16T_B}{\pi(30)^3} = 50.00\ \text{MPa}
$$

**Answer:** maximum torque at C $= 302.5$ N·m ($\approx 0.302$ kN·m); shear stress at A (segment AC) $= 23.8$ MPa.

## Check yourself

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

1. What principle splits torque?
   A. T_A = T_B
   B. Rotation of C is same from either side: T_A L1/GJ1 = T_B L2/GJ2
   C. Stress equal
   D. Length equal
   Answer: B. Both segments twist the same. (Key point: Torque split)

2. Which segment takes more torque?
   A. AC
   B. Equal
   C. CB, 87.6% of T, being thicker and shorter
   D. Neither
   Answer: C. Torsional stiffness GJ/L. (Key point: Share)

3. Which segment reaches 50 MPa first?
   A. AC, 635 N.m
   B. AC, 302 N.m
   C. Both together
   D. CB, so T_max = 302.5 N.m
   Answer: D. Smaller T governs. (Key point: Governing)

4. What is tau at A?
   A. 23.8 MPa
   B. 50 MPa
   C. 100 MPa
   D. 12 MPa
   Answer: A. 16 T_A/(pi 20^3). (Key point: Stress at A)

## Related in Strength of Materials

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- [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: torsional reactions, rotations, max shear](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Ashaft-torsional-reactions-copper-steel) — 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: 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
- [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

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