The solid circular shaft shown is made of a steel that is...

The solid circular shaft shown is made of a steel that is assumed to be elastoplastic with $\tau_{Y}=145 \mathrm{MPa}$. Determine the magnitude $T$ of the applied torques when the plastic zone is $(a) 16 \mathrm{mm}$ deep, (b) $24 \mathrm{mm}$ deep.

Key Concepts

Yield Criterion in Torsion

The yield criterion, such as reaching a critical shear stress value (yield shear stress), indicates when parts of a shaft begin to plastically deform under applied torsional loads. In torsional problems, this is typically examined by comparing the maximum shear stress in the material with the known yield stress, which is instrumental for predicting the onset and extent of plasticity.

Elastoplastic Material Behavior

Elastoplasticity refers to materials that exhibit linear elastic behavior up to a certain stress threshold (the yield stress), beyond which plastic deformations occur without significant additional stress increase. This concept is critical when determining the load levels that cause parts of a structure to yield and form plastic zones.

Plastic Zone Analysis

Plastic zone analysis involves determining the region within the cross-section where the material has yielded and behaves plastically while the remainder might still behave elastically. This mixed state requires integrating the contributions of both zones to calculate overall response parameters, such as the applied torque, and is essential for understanding the performance and safety of the structure under high loads.

Torsion in Circular Shafts

This concept involves analyzing the behavior of circular shafts subjected to an applied torque, which induces a distribution of shear stresses across the cross-section. The relationship ? = (T*r)/J (where T is the torque, r the radial distance, and J the polar moment of inertia) underpins the analysis and is fundamental for designing members that resist twisting loads.

Shear Stress Distribution in Circular Cross-Sections

In a circular shaft under torque, the shear stress is not uniform; it varies linearly from zero at the center to a maximum at the outer surface. Understanding this distribution is essential since it determines where the material first reaches its yield point, and it is critical for both elastic and inelastic analysis.

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