A 2 -in.-diameter solid shaft is made of a mild steel that...

A 2 -in.-diameter solid shaft is made of a mild steel that is assumed to be elastoplastic with $\tau_{Y}=20$ ksi. Determine the maximum shearing stress and the radius of the elastic core caused by the application of a torque of magnitude $(a) 30$ kip $\cdot$ in. $(b)$ 40 kip -in.

Key Concepts

Torsion of Circular Shafts

This concept involves analyzing how a solid circular shaft responds to an applied torque. The fundamental relation used is ? = T * r / J, where ? is the shear stress, T is the applied torque, r is the radial distance from the center, and J is the polar moment of inertia. This analysis helps determine the distribution of shear stress over the cross?section and predicts the maximum shear stress occurring at the outer surface of the shaft.

Elastoplastic Behavior

Elastoplastic behavior refers to the material response when it exhibits elastic deformation up to a certain yield point (in this case, a yield shear stress of ?Y = 20 ksi) followed by plastic deformation beyond that limit. In the context of torsion, the shaft will first respond elastically at lower applied torques until the shear stress at the outer fibers reaches the yield stress, after which the outer region begins to yield while a smaller elastic core remains.

Elastic Core in Plastic Deformation

When a shaft under torsion begins to yield, the material does not yield uniformly across the entire cross-section. Instead, there is an elastic core that remains below the yield stress, surrounded by a plastically deformed annulus. Determining the radius of this elastic core involves equating the shear stress (which varies linearly with the radius in the elastic region) to the yield shear stress, thus defining the boundary between elastic and plastic regions.

Yield Criterion in Torsion

The yield criterion, applied in torsion, identifies the maximum shear stress that the material can sustain in the elastic regime before plastic deformation begins. For the case of elastoplastic materials like mild steel, the yield shear stress (?Y) is a critical threshold. When the applied torque generates a shear stress equal to ?Y at a certain radial distance, that distance marks the boundary between the elastic and plastic zones, guiding the evaluation of the maximum stresses and the extent of the elastic core.

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