Concept of Pure Torsion and the Torsion Equation

Definition

Pure torsion occurs when a structural member, such as a circular shaft, is subjected to equal and opposite couples (torques) acting in planes perpendicular to the axis of the member. In this state, the cross-sections of the member remain plane, and radii remain straight, though they rotate by a certain angle known as the angle of twist.

Main Content

1. Pure Torsion

A member is in pure torsion when it is acted upon only by twisting moments (torques) at its ends, with no bending or axial loads present.
Imagine a circular shaft fixed at one end; when a wrench applies a twisting force at the other end, the entire length of the shaft experiences the same internal torque.

2. The Torsion Equation

The torsion equation relates the applied torque to the internal shear stress developed within the material of the shaft.
The fundamental formula is: $\frac{T}{J} = \frac{\tau}{r} = \frac{G\theta}{L}$
Where $T$ is Torque, $J$ is the Polar Moment of Inertia, $\tau$ is Shear Stress, $r$ is the radial distance, $G$ is the Shear Modulus, $\theta$ is the angle of twist, and $L$ is length.

3. Assumptions for the Theory

The material is homogeneous (same properties throughout) and isotropic (same properties in all directions).
The shaft is circular in cross-section, and plane sections before twisting remain plane after twisting.
The stress developed does not exceed the limit of proportionality (Hooke’s Law applies).

Working / Process

1. Identifying Torque and Geometry

Determine the external torque ($T$) being applied to the shaft (usually in N-m).
Calculate the Polar Moment of Inertia ($J$) based on the geometry: For a solid shaft, $J = \frac{\pi d^4}{32}$; for a hollow shaft, $J = \frac{\pi (D^4 - d^4)}{32}$.

2. Visualizing Stress Distribution

The shear stress ($\tau$) varies linearly from zero at the center (neutral axis) to a maximum at the outer surface of the shaft.

       Shear Stress Distribution:

       (Center)    (Surface)
          0 --------> Tau_max
          |           |
          |           |
          |           |
          0 --------> Tau_max

3. Calculating Angle of Twist

Use the rearranged torsion equation: $\theta = \frac{TL}{GJ}$.
Ensure units are consistent (e.g., Torque in N-mm, Length in mm, Modulus in N/mm², and $J$ in mm⁴) to obtain the angle of twist in radians.

Advantages / Applications

Transmission of power: Used extensively in automobile drive shafts and machine tool spindles to transmit rotational energy.
Material Testing: Torsion tests are used to determine the shear modulus ($G$) and shear strength of metallic materials.
Design Optimization: The torsion equation allows engineers to select the minimum shaft diameter required to prevent failure due to excessive shear stress.

Summary

Pure torsion refers to the state where a member is twisted by moments about its longitudinal axis. The torsion equation serves as the governing mathematical relationship to determine shear stress and deformation in circular shafts. Key terms include Polar Moment of Inertia ($J$), which represents the shaft's resistance to torsion, and the Angle of Twist ($\theta$), which measures the angular deformation of the material.