Circular Drive Shaft Torsional Shear Stress (Mechanical Engineering)

A mechanical design engineer at the UNLV Howard R. Hughes College of Engineering is analyzing the torsional load capacity of a solid alloy steel drive shaft powering an industrial centrifugal water chiller.

The drive shaft has a circular cross-section of radius $R = 3\text{ cm}$. When transmitted engine torque twists the shaft, internal elastic torsional shear stress $\tau$ develops. According to the mechanics of materials elastic torsion formula, shear stress varies directly with radial distance from the central rotational axis, modeled in Cartesian coordinates as:

To evaluate the total internal resisting torque $T$ developed across the shaft cross-section, the engineer must integrate the moment of shear stress ($d T = r \cdot \tau \, dA$) over the circular domain $D$:

1. Define the circular domain of integration $D$ in polar coordinates with radial bounds $r$ and angular limits $\theta$.
2. Transform the torque integrand into polar coordinates, incorporating the Jacobian area differential $dA = r \, dr \, d\theta$.
3. Evaluate the double integral to determine the total internal resisting torque $T$ in Newton-centimeters ($\text{N}\cdot\text{cm}$) and convert to Newton-meters ($\text{N}\cdot\text{m}$).
4. Explain how this polar integral relates to the polar moment of inertia $J$ and prevents torsional shearing yield failure in rotating machinery drivetrains.