A torque T=750 kN ·m is applied to the hollow shaft shown...

A torque $T=750 \mathrm{kN} \cdot \mathrm{m}$ is applied to the hollow shaft shown that has a uniform 8 -mm wall thickness. Neglecting the effect of stress concentrations, determine the shearing stress at points $a$ and $b$.

Key Concepts

Polar Moment of Inertia

The polar moment of inertia is a geometric property of a cross-section that measures its resistance to torsion. For hollow shafts, this value is determined by the difference between the polar moments of inertia of the outer and inner circles. It is a crucial concept because it directly influences the magnitude of shear stresses developed under an applied torque, and is key in designing shafts to withstand torsional loads.

Hollow Shaft Geometry Considerations

When dealing with hollow shafts, factors such as wall thickness and the radii of the inner and outer surfaces significantly affect the structural behavior under torsion. Uniform wall thickness simplifies the analysis, but the distinction between inner and outer surfaces remains important since the maximum shear stress occurs at the outermost fiber while the inner surface experiences a different stress level. Understanding these geometric considerations is vital for accurate stress analysis and safe design.

Torsion in Shafts

This concept involves understanding how an applied twisting moment (torque) causes a shaft to experience angular deformation. In torsion, the material fibers experience shear stresses that vary linearly with the radial distance from the center, with maximum stress typically at the outer surface. Analyzing how the torque is resisted by the shaft is fundamental in ensuring structural integrity under twisting loads.

Shear Stress Distribution Under Torsion

When a torque is applied to a circular shaft, the resulting shear stress at any point in the cross-section is proportional to the product of the applied torque and the radial distance from the center, divided by the polar moment of inertia. This linear relationship implies that points farther from the center (such as the outer surface) experience higher shear stresses, making it critical to evaluate stress variations at different locations around the cross-section.

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