19.12 : 薄肉中空軸（薄肉中空シャフト）

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.