A shaft is made of an aluminum alloy having an allowable...

A shaft is made of an aluminum alloy having an allowable shear stress of $\tau_{\text {allow }}=100$ MPa. If the diameter of the shaft is $100 \mathrm{mm}$, determine the maximum torque $\mathbf{T}$ that can be transmitted. What would be the maximum torque $\mathbf{T}^{\prime}$ if a 75 -mm-diameter hole were bored through the shaft? Sketch the shear-stress distribution along a radial line in each case.

Key Concepts

Allowable Shear Stress

Allowable shear stress is a material property indicating the maximum shear stress a material can handle without failure. In design, this value is compared to the calculated shear stress from applied loads to ensure safe operation within the material's strength limits.

Solid versus Hollow Shafts

Understanding the differences between solid and hollow shafts is crucial in design because the presence of a hole alters the cross-sectional geometry, thereby affecting the polar moment of inertia and the distribution of shear stress. Hollow shafts or shafts with bored holes can offer weight savings while potentially reducing the maximum torque that can be safely transmitted.

Shear Stress Distribution

The shear stress distribution along a radial line in a shaft under torsion reveals the variation from the center to the edge. For a solid shaft, the shear stress increases linearly from zero at the center to a maximum at the outer surface, while for a hollow shaft the distribution is non-zero only in the material region, with discontinuities at the inner and outer radii. Sketching these distributions helps visualize how the stress is spread across the cross-section.

Polar Moment of Inertia

The polar moment of inertia quantifies a cross-section’s resistance to torsional deformation. For circular sections, its value depends on the radius (or inner and outer radii for hollow shafts) and is essential in calculating shear stress due to a given applied torque using the torsion formula.

Shear Stress in Torsion

This concept involves understanding how a torque applied to a shaft produces a shear stress distribution across its cross-section. The shear stress is highest at the outer surface and varies linearly from the center to the edge in a solid circular shaft, which is critical in designing shafts to ensure they do not exceed material limits.

Torsion Formula

The torsion formula relates the applied torque to the resulting shear stress through the expression ? = T*c/J, where ? is the shear stress, T is the applied torque, c is the distance from the center to the outer fiber, and J is the polar moment of inertia. This formula is fundamental in evaluating whether a shaft will remain within allowable stress limits under load.

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