---
title: "Derivation of the torsion equation"
subject: "Strength of Materials"
chapter: 4
key: "semester_4/strength_of_materials_mechanical:torsion-equation-derivation-circular-shaft"
source: https://www.bctengineer.com/q/semester_4/strength_of_materials_mechanical/torsion-equation-derivation-circular-shaft
site: BCT Engineering
---

# Derivation of the torsion equation

Strength of Materials · Chapter 4: Torsion

- Asked in: year not given, 8 marks
- Video: [TORSION EQUATION : DERIVATION](https://www.youtube.com/watch?v=vtybG1Or620) — TECHNICAL CLASSES
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/strength_of_materials_mechanical/torsion-equation-derivation-circular-shaft
- Whole chapter: https://www.bctengineer.com/md/semester_4/strength_of_materials_mechanical/chapter-4

## Question

State the assumptions of the theory of torsion and derive the torsion equation T/J = tau/r = G*theta/L for a solid circular shaft. Define polar modulus and torsional rigidity.

## Short answer

Assumptions: straight circular shaft, homogeneous, isotropic, Hooke's law, plane sections remain plane, uniform twist. Shear strain γ = rθ/L, stress τ = Gγ, so τ/r = Gθ/L. Torque T = (Gθ/L)J, giving T/J = τ/r = Gθ/L. Polar modulus Z_p = J/r; torsional rigidity = GJ.

## Answer

#### Assumptions
1. The shaft is straight, of uniform circular cross-section, and the material is homogeneous and isotropic.
2. The material obeys Hooke's law; stresses stay within the elastic (proportional) limit.
3. Plane transverse sections remain plane and circular after twisting; radii remain straight.
4. The twist is uniform along the length and the torque acts in a plane perpendicular to the axis.
5. Shear strain varies linearly with the distance from the axis.

#### Derivation
```text
  fixed end                free end
     |  A ------------- B     line AB before twisting
     |  A ------------- B'    after twisting
     |<------- L ------->|    arc BB' = r*theta
```

A line AB on the surface of the shaft of radius $r$ and length $L$ moves to AB' when the free end twists through angle $\theta$ (radians). The arc BB' $= r\theta$, and the shear strain (small angle) is

$$
\gamma = \frac{BB'}{L} = \frac{r\theta}{L}
$$

By Hooke's law in shear, $\tau = G\gamma$, so at the surface

$$
\frac{\tau}{r} = \frac{G\theta}{L}
$$

At a general radius $\rho$, $\tau_\rho = G\rho\theta/L$, hence shear stress varies **linearly** with radius: zero at the axis and maximum at the surface.

The torque on a ring of area $dA = 2\pi\rho\,d\rho$ is $dT = \tau_\rho\,\rho\,dA$.

$$
T = \int \tau_\rho\,\rho\,dA = \frac{G\theta}{L}\int \rho^2\,dA = \frac{G\theta}{L}J
$$

where $J = \int\rho^2\,dA$ is the polar second moment of area. Combining:

$$
\boxed{\frac{T}{J} = \frac{\tau}{r} = \frac{G\theta}{L}}
$$

#### Section properties
| Section | $J$ |
|---|---|
| Solid, diameter $d$ | $\pi d^4/32$ |
| Hollow, $d_o,\ d_i$ | $\pi(d_o^4 - d_i^4)/32$ |

- **Polar modulus** $Z_p = J/r$ (so $T = \tau_{max}Z_p$). For a solid shaft $Z_p = \pi d^3/16$.
- **Torsional rigidity** $= GJ$, the torque needed to produce a unit angle of twist per unit length ($T/\theta = GJ/L$).

Power transmitted: $P = T\omega = \dfrac{2\pi N T}{60}$ with $N$ in rpm.

## Check yourself

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

1. In the derivation, what is the shear strain at the surface of a shaft of radius r, length L, twist theta?
   A. gamma = L theta / r
   B. gamma = r theta / L
   C. gamma = theta / (r L)
   D. gamma = G r theta / L
   Answer: B. Arc BB' = r theta, so gamma = BB'/L = r theta / L. (Key point: Shear strain at surface)

2. How does shear stress vary across a solid circular shaft section?
   A. Uniform
   B. Linearly: zero at the axis and maximum at the surface
   C. Maximum at the axis
   D. Parabolic, zero at the surface
   Answer: B. tau_rho = G rho theta / L, directly proportional to radius. (Key point: Variation of shear stress)

3. Which pair is correct?
   A. Polar modulus = J/r; torsional rigidity = GJ
   B. Polar modulus = G J; torsional rigidity = J/r
   C. Polar modulus = J r; torsional rigidity = G/J
   D. Polar modulus = J/G; torsional rigidity = T/J
   Answer: A. Z_p = J/r gives T = tau_max Z_p, and GJ is the torque needed per unit angle of twist per unit length. (Key point: Polar modulus and torsional rigidity)

4. How is T = G theta J / L obtained?
   A. Integrating tau x rho x dA over the section, with J = integral rho^2 dA
   B. Integrating tau dA over the section
   C. Differentiating tau
   D. Using the average stress times r
   Answer: A. dT = tau_rho rho dA, and T = (G theta / L) integral rho^2 dA = G theta J / L. (Key point: Torque from ring integration)

## Related in Strength of Materials

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## Related in other subjects

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- [Torsional pendulum, modulus of rigidity](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Abar-torsional-pendulum-rigidity) — Engineering Physics, chapter 1, asked 2×
- [Elastic-plastic torsion of a circular shaft](https://www.bctengineer.com/question.md?key=semester_5%2Fmechanics_of_solids%3Aplastic-torsion-circular-shaft) — Mechanics of Solids, chapter 12