The allowable shearing stress is 15 ksi in the steel rod A B...

The allowable shearing stress is 15 ksi in the steel rod $A B$ and 8 ksi in the brass rod $B C .$ Knowing that a torque of magnitude $T=10$ kip.in. is applied at $A,$ determine the required diameter of $(a) \operatorname{rod} A B$ $(b) \operatorname{rod} B C$

Key Concepts

Torsion in Circular Shafts

This concept involves the analysis of how circular shafts behave under torsional loads. When a torque is applied to a circular shaft, it induces shear stresses that vary linearly with the radial distance from the center of the shaft. Understanding torsion is essential for determining how the applied torque is resisted by the shaft and for ensuring that the design meets strength and safety requirements.

Polar Moment of Inertia

The polar moment of inertia is a geometric property of a cross-section that indicates its resistance to torsion, similar to how the moment of inertia relates to bending. For circular shafts, it is calculated using the formula J = ?d?/32, where d is the diameter. This property is crucial because it directly influences the stress distribution when the shaft is subjected to a twisting load.

Shear Stress Distribution in Torsion

Shear stress in a shaft under torsion is not uniformly distributed; it depends on the distance from the center. The maximum shear stress occurs at the outer surface and is computed using the formula ? = T·c/J, where T is the applied torque, c is the outer radius, and J is the polar moment of inertia. This relationship forms the basis for designing shafts that can safely carry applied torques without exceeding the material's strength limits.

Allowable Shear Stress

The allowable shear stress is the maximum shear stress that a material can sustain without failure. It is a critical design parameter that accounts for the material properties and safety factors. In torsion problems, the allowable shear stress is used to determine the minimum required cross-sectional dimensions to ensure that the shaft will perform reliably under the applied loads.

Design of Composite Shafts

Composite shafts are those constructed from segments made of different materials, each of which may have different mechanical properties such as allowable shear stress. The design process involves ensuring that each segment is sized appropriately to handle the specific portion of the applied load based on its material’s strength. This requires applying torsion formulas separately for each material while maintaining the integrity and functionality of the overall structure.

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