Two shafts are made of the same material. The cross section...

Two shafts are made of the same material. The cross section of shaft $A$ is a square of side $b$ and that of shaft $B$ is a circle of diameter $b$ Knowing that the shafts are subjected to the same torque, determine the ratio $\tau_{A} / \tau_{B}$ of maximum shearing stresses occurring in the shafts.

Key Concepts

Shear Stress Under Torsion

This concept refers to the phenomenon whereby a shaft under a twisting force (torque) develops internal shear stresses, with the maximum stress occurring at the outermost fibers. The calculation is usually based on the formula ? = T·r/J, where T is the applied torque, r is the distance from the center of the shaft to its outer edge (or extreme fiber), and J is the polar moment of inertia of the cross-section.

Polar Moment of Inertia

The polar moment of inertia is a geometric property of a cross-section that measures its resistance to torsional deformation. Its value depends on the shape of the cross-section and is used in calculating the shear stress in a shaft. For example, circular and square cross-sections have different formulas for J, meaning they will distribute shear stresses differently under the same torsional load.

Effect of Cross-Sectional Geometry on Torsion

The geometry of a shaft’s cross-section significantly influences its torsional performance. Different shapes, such as circles and squares, have different distributions of material relative to their center and thus different polar moments of inertia. This leads to variations in maximum shear stress for a given torque even when the overall cross-sectional area may be similar.

Transcript

Please give Ace some feedback