Consider a hollow steel shaft with a 130 -mm outside...

Consider a hollow steel shaft with a 130 -mm outside diameter and a 70 -mm inside diameter. What is the maximum torque that this shaft can transmit if the shear yield stress is $1.4 \times 10^8 \mathrm{~Pa}$ and the behavior is elastic, perfectly-plastic?

Key Concepts

Shear Yield Stress

This key material property, often given for the specific material being used (like steel), defines the stress at which the material begins to yield or deform plastically under shear loading. It must not be exceeded in the design of shafts for safe torsional loading, particularly when the behavior is assumed to be elastic, perfectly-plastic.

Elastic, Perfectly-Plastic Behavior

This assumption about material behavior simplifies the analysis by considering that the material behaves elastically up to the yield point and then yields without further increase in stress. It is important for calculating the maximum torque transmitted without additional deformation, influencing the design and safety assessments of structural components.

Polar Moment of Inertia

This is a geometric property of a cross-section that quantifies its resistance to torsional deformation. For hollow shafts, the polar moment of inertia is computed by taking the difference between the moments of inertia of the outer and inner diameters. This value is necessary for determining how much torque the shaft can withstand before yielding.

Maximum Shear Stress

In torsion, the maximum shear stress occurs at the outer surface of the shaft. It is calculated using the relationship that ties together the torque, the polar moment of inertia of the shaft's cross-section, and the outer radius. Ensuring that this stress does not exceed the material's yield stress is crucial for safe operation.

Torsion in Circular Shafts

This concept deals with how circular shafts respond to twisting loads. The relationship between the applied torque, shear stress, and the geometry of the cross-section is central to understanding shaft design under torsion. It involves the torsion formula, which relates these quantities and is essential for determining the maximum torque a shaft can transmit before failure.

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