A torque of 3300 N -m is transmitted through a 50-mm-diameter solid shaft having a yield stress

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A torque of 3300 N -m is transmitted through a...

A torque of 3300 N -m is transmitted through a $50-\mathrm{mm}$-diameter solid shaft having a yield stress in shear of $1.1 \times 10^4 \mathrm{~Pa}$. Is there plastic deformation for this loading? If so, how far has the plastic action penetrated from the surface assuming elastic, perfectly plastic behavior? What is the maximum possible static moment?

Key Concepts

Yielding in Torsion

Yielding in the context of torsion refers to the point at which the induced shear stress in the material exceeds its yield strength, causing plastic deformation. When the shear stress at the outer surface of a shaft surpasses the yield stress of the material, plastic deformation begins there and may progress inward. Recognizing the onset and progression of yielding is important for ensuring that structural components function safely and effectively under load.

Torsional Loading

This concept involves applying a twisting load or torque to a circular shaft. In torsion, the applied torque creates shear stresses across the cross-section, with the stresses varying linearly from the center (zero stress) to the surface (maximum stress). Understanding torsional loading is essential for predicting how shafts and beams will behave under twist, which is critical in many mechanical and structural engineering applications.

Shear Stress Distribution

In a circular shaft under torsion, the resulting shear stress is not uniformly distributed; it increases linearly from the center to the outer surface. This distribution is given by the relation ? = T·r/J, where ? is the shear stress at a radial distance r, T is the applied torque, and J is the polar moment of inertia. This concept is key in determining the points in the material that are most likely to yield or fail under loading.

Elastic-Perfectly Plastic Behavior

This is an idealized material model where the material behaves elastically up to the yield point and then deforms plastically without any strain hardening. In torsional analysis, assuming elastic-perfectly plastic behavior simplifies the calculation of how far plastic deformation penetrates from the outer surface into the material. This concept is widely used in the design and analysis of shafts and other components where plastic deformation is a concern.

Polar Moment of Inertia and Torsion Formula

The polar moment of inertia, denoted as J, is a geometric property of a cross-section that quantifies its resistance to torsion. For a solid circular shaft, J is calculated using the formula J = (?·d^4)/32, where d is the diameter of the shaft. This property is central to the torsion formula ? = T·r/J, which links the applied torque and radial position to the induced shear stress, forming the foundation for torsional analysis in engineering.

Maximum Static Torque

The maximum static torque, often referred to as the limit torque, is the highest torque that can be applied to a shaft before it fully plastically deforms or fails. It is determined by integrating the material’s yield condition over the entire cross-sectional area. This concept is critical in design as it sets the upper bound for the safe operational torque that a shaft can transmit without catastrophic failure.

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