19.12 : Thin-Walled Hollow Shafts

In analyzing a thin-walled hollow shaft subjected to torsional loading, a segment with width dx is isolated for examination. Despite its equilibrium state, this segment faces torsional shearing forces at its ends. These forces are quantitatively described by the product of the longitudinal shearing stress on the segment's minor surface and the area of this surface, leading to the concept of shear flow. This shear flow is consistent throughout the structure, indicating a uniform distribution of torsional stress.

Further analysis involves examining a small section of the shaft's wall, identified by ds. The force exerted on this section by the shear flow is calculated by multiplying the shear flow by the length of the section. The torsional impact on the shaft is determined by calculating the moment this force generates about a specific point, O, within the shaft's hollow. This calculation involves multiplying the force by the distance from O to the line of action of the force, perpendicular to the force's direction.

Integrating this moment across the entire shaft yields the expression for the total torque affecting the hollow cylinder. This integral calculation offers a deeper understanding of how torsional forces influence the structural behavior and integrity of thin-walled hollow shafts, providing key insights into their design and analysis under torsional loading.