19.12:
Thin-Walled Hollow Shafts
Consider a small portion of width 'dx' from a thin-walled hollow shaft of thickness 't' subjected to torsional loading.
Since the portion is in equilibrium, the only forces acting on it are the shearing forces exerted on the ends of the portion.
Expressing the shearing forces as the product of the longitudinal shearing stress on the small face and of the area of that face, an expression for shear flow can be derived, which remains constant throughout the member.
Now, consider a small element of the hollow shaft wall section of length 'ds'. The magnitude of the shearing force exerted on the element is expressed as the product of the shear flow and length of the element.
The moment of this force about an arbitrary point O within the cavity of the hollow shaft is obtained by multiplying the force by the perpendicular distance from point O to the line of action of the force.
Integrating the moment equation gives the expression for applied torque on the entire thin-walled hollow cylinder.
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.
