A 90-N ·m torque is applied to a hollow shaft having the...

A $90-\mathrm{N} \cdot \mathrm{m}$ torque is applied to a hollow shaft having the cross section shown. Neglecting the effect of stress concentrations, determine the shearing stress at points $a$ and $b$

Key Concepts

Torsion

Torsion refers to the twisting of an object due to an applied torque. It is a fundamental concept in mechanics that describes how a force applied at a distance from an axis causes rotation and internal shear stresses within the material. In engineering, understanding torsion is crucial for designing shafts and other components to ensure they can safely transmit torque without failing.

Shear Stress in Torsion

Shear stress in a torsioned object is the internal stress that results from the applied torque. The stress is distributed across the cross section and is often highest at points furthest from the center. The relation ? = T?/J connects the applied torque (T), the radial distance (?), and the polar moment of inertia (J), allowing for the calculation of shear stress at any point in the section.

Polar Moment of Inertia

The polar moment of inertia is a geometric property of the cross section that measures its resistance to torsional deformation. It plays a critical role in determining how torque is distributed across a cross section, directly influencing the shear stress experienced by the material. For hollow and composite sections, the calculation of the polar moment of inertia accounts for the distribution of material around the center, which is essential for accurate stress analysis.

Neglecting Stress Concentrations

Neglecting stress concentrations involves simplifying the analysis by assuming that local increases in stress due to geometric discontinuities or abrupt changes in cross section are not present. This assumption is made to focus on the overall stress distribution in the component, simplifying calculations and providing an idealized understanding of how shear stress develops under torsion.

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