5  Torsion

5.1 Assumptions of Simple Torsion Theory

The theory applies to circular cross-sections (solid or hollow) and assumes:

• The material is homogeneous and isotropic
• Plane cross-sections remain plane after twisting
• Shear stress is proportional to shear strain (elastic behaviour)
• The shaft is straight and of uniform cross-section

5.2 The Torsion Formula

The fundamental relationship linking shear stress, torque, geometry, and angle of twist:

\[\frac{T}{J} = \frac{\tau}{r} = \frac{G\theta}{L}\]

where:

• \(T\) = applied torque (N·m)
• \(J\) = polar second moment of area (m⁴)
• \(\tau\) = shear stress at radius \(r\) (Pa)
• \(G\) = modulus of rigidity (Pa)
• \(\theta\) = angle of twist (radians)
• \(L\) = length of shaft (m)

5.3 Polar Second Moment of Area

For a solid circular shaft of diameter \(d\):

\[J = \frac{\pi d^4}{32}\]

For a hollow circular shaft with outer diameter \(d_o\) and inner diameter \(d_i\):

\[J = \frac{\pi (d_o^4 - d_i^4)}{32}\]

5.4 Shear Stress Distribution

Shear stress varies linearly from zero at the centre to a maximum at the outer surface:

\[\tau_{\max} = \frac{T \cdot r}{J} = \frac{16T}{\pi d^3} \quad \text{(solid shaft)}\]

A hollow shaft is more efficient — it carries almost the same torque for less material mass.

5.5 Angle of Twist

The total angle of twist along a shaft of length \(L\):

\[\theta = \frac{TL}{GJ}\]

For a stepped shaft (different diameters along its length), the total twist is the sum of twists in each segment:

\[\theta_{\text{total}} = \sum \frac{T_i L_i}{G J_i}\]

5.6 Power Transmission

The relationship between transmitted power, torque, and rotational speed:

\[P = T\omega = \frac{2\pi N T}{60}\]

where \(N\) is rotational speed in rpm and \(\omega\) is in rad/s. This is used to find the required shaft diameter for a given power and speed.

5.7 Composite (Compound) Shafts

5.7.1 Shafts in Series

The same torque passes through each section; total twist is the sum of individual twists:

\[\theta_{\text{total}} = \frac{TL_1}{GJ_1} + \frac{TL_2}{GJ_2}\]

5.7.2 Shafts in Parallel

Both shafts share the applied torque and twist by the same angle:

\[T = T_1 + T_2 \qquad \theta_1 = \theta_2\]

These two conditions give the equations needed to solve for the torque in each shaft.

5.8 Close-Coiled Helical Springs

A close-coiled helical spring under axial load \(W\) is primarily a torsion problem. The wire is subjected to torque \(T = WR\), where \(R\) is the mean coil radius.

Shear stress in the wire:

\[\tau = \frac{16WR}{\pi d^3}\]

Axial deflection:

\[\delta = \frac{64WR^3n}{Gd^4}\]

Stiffness of the spring:

\[k = \frac{W}{\delta} = \frac{Gd^4}{64R^3n}\]

where \(d\) is the wire diameter and \(n\) is the number of active coils.

5.9 Summary of Equations

Concept	Equation
Torsion formula	\(T/J = \tau/r = G\theta/L\)
Polar moment — solid shaft	\(J = \pi d^4 / 32\)
Polar moment — hollow shaft	\(J = \pi(d_o^4 - d_i^4)/32\)
Max shear stress (solid)	\(\tau_{\max} = 16T/\pi d^3\)
Angle of twist	\(\theta = TL/GJ\)
Power transmitted	\(P = 2\pi NT/60\)
Spring deflection	\(\delta = 64WR^3n/Gd^4\)
Spring stiffness	\(k = Gd^4/64R^3n\)

5.10 Summary

Torsion analysis always begins with the torsion formula \(T/J = \tau/r = G\theta/L\). For composite shaft problems, correctly identifying whether shafts are in series (same torque) or parallel (same twist) is the critical first step. The close-coiled helical spring is a direct application of torsion theory to a practical engineering component.