Understanding Shear Stress in a Shaft Under Torque

When a shaft is subjected to an applied torque, it undergoes a twisting deformation. This twisting action causes internal shear stresses within the material of the shaft. The distribution of this shear stress across the cross-section of the shaft is not uniform; it varies depending on the distance from the center of the shaft.

Distribution of Shear Stress Due to Torque

For a solid circular shaft subjected to a torque (T), the shear stress (\(\tau\)) at any point at a distance 'r' from the center of the shaft is given by the torsion formula:

Where:

• \(\tau\) is the shear stress at distance 'r'.
• T is the applied torque.
• r is the radial distance from the center of the shaft to the point where shear stress is being calculated.
• J is the polar moment of inertia of the shaft's cross-section. For a solid circular shaft of radius R, \(J=\frac{\pi R^4}{2}\). J is a constant for a given shaft cross-section.

From the formula, \(\tau \propto r\), meaning the shear stress is directly proportional to the radial distance 'r' from the center.

Variation of Shear Stress from Center to Circumference

• At the center of the shaft, the radial distance \(r=0\). Substituting this into the torsion formula: \[ \tau = \frac{T \times 0}{J} = 0 \] So, the shear stress at the center is zero.
• At the circumference of the shaft, the radial distance \(r\) is equal to the radius of the shaft, \(r=R\). Substituting this into the formula: \[ \tau = \frac{TR}{J} = \tau_{\max} \] This value \(\frac{TR}{J}\) is the maximum shear stress induced in the shaft.

Therefore, the magnitude of shear stress induced in a shaft due to applied torque varies linearly from zero at the center to a maximum value at the circumference.