Determination of Shear Stress and Angle of Twist of Shafts

Definition

Torsion refers to the twisting of a structural member, such as a shaft, when it is subjected to an external torque (twisting moment) that tends to produce rotation about the longitudinal axis of the member. The shear stress is the internal resistance developed within the material to oppose this twisting, while the angle of twist represents the angular displacement of one cross-section relative to another along the length of the shaft.

Main Content

1. The Torsion Equation

The torsion equation relates the applied torque ($T$), the maximum shear stress ($\tau$), the polar moment of inertia ($J$), the modulus of rigidity ($G$), the angle of twist ($\theta$), and the length of the shaft ($L$).
The formula is expressed as: $\frac{T}{J} = \frac{\tau}{r} = \frac{G\theta}{L}$

2. Shear Stress Distribution

Shear stress varies linearly across the cross-section of a circular shaft.
It is zero at the center (neutral axis) and reaches its maximum value at the outermost surface of the shaft.

       Shear Stress Profile
      /------------------\
     /    /      |      \    \
    |    |       |       |    |
    |----|-------+-------|----|  <-- Maximum at Surface
    |    |       |       |    |
     \    \      |      /    /
      \------------------/

3. Angle of Twist

The angle of twist is the measure of how much a shaft rotates under a torque. It is directly proportional to the torque and length, and inversely proportional to the stiffness of the material ($G$) and the geometry ($J$).
If the torque or cross-section changes along the length, the total twist is the sum of twists for each individual segment.

Working / Process

1. Identify Geometric and Material Properties

Determine the radius ($r$) and the polar moment of inertia ($J$) based on the shaft geometry. For a solid shaft, $J = \frac{\pi d^4}{32}$, and for a hollow shaft, $J = \frac{\pi (D^4 - d^4)}{32}$.
Identify the material property $G$ (modulus of rigidity) which defines how resistant the material is to shearing deformation.

2. Calculate Maximum Shear Stress

Use the rearranged torsion formula: $\tau_{max} = \frac{T \cdot r}{J}$.
Ensure the units are consistent (usually Newtons, millimeters, and Pascals/MPa) to avoid calculation errors. For example, if a torque of 1000 Nm is applied to a 50mm diameter shaft, calculate $J$ first, then solve for $\tau$.

3. Calculate Angle of Twist

Use the rearranged torsion formula: $\theta = \frac{T \cdot L}{G \cdot J}$.
Note that $\theta$ must be calculated in radians. If the final answer is required in degrees, multiply the result by $\frac{180}{\pi}$.
Always verify if the shaft is uniform; if the torque $T$ or diameter changes, calculate $\theta$ for each part and sum them up: $\theta_{total} = \sum \frac{T_i L_i}{G_i J_i}$.

Advantages / Applications

Used extensively in mechanical engineering for the design of transmission shafts in engines, wind turbines, and industrial machinery.
Essential for determining the "factor of safety" to prevent structural failure or permanent deformation of machine parts under operating loads.
Helps engineers select appropriate materials and diameters to minimize weight while maintaining the required strength and stiffness.

Summary

The determination of shear stress and angle of twist is a fundamental analysis process that ensures shafts can transmit power without exceeding material strength or functional deformation limits. By using the torsion formula, engineers calculate the peak internal shear stress to prevent yielding and the total angular displacement to ensure precise machine operation. Important terms to remember include Polar Moment of Inertia (J), Modulus of Rigidity (G), Torque (T), and Shear Stress ($\tau$).