Knowing that the stepped shaft shown transmits a torque of...

Knowing that the stepped shaft shown transmits a torque of magnitude $T=2.50$ kip $\cdot$ in. $,$ determine the maximum shearing stress in the shaft when the radius of the fillet is $(a) r=\frac{1}{8}$ in. (b) $r=\frac{3}{16}$ in.

Key Concepts

Torque

Torque is the rotational force applied to a member, such as a shaft, creating a twisting action. It is critical in mechanical engineering because it induces shear stresses within the material, and the magnitude of the torque directly influences the level of stress experienced.

Shear Stress in Torsion

Shear stress in a shaft under torsion can be determined using the relationship ? = T*r/J, where ? is the shear stress, T is the applied torque, r is the radial position from the center (usually taken as the outer radius for maximum stress), and J is the polar moment of inertia of the shaft's cross-section. This equation is fundamental for evaluating the material's response to twisting loads.

Polar Moment of Inertia

The polar moment of inertia is a geometric parameter that quantifies a cross-section's resistance to torsional deformation. For circular cross-sections, it is typically calculated as J = (?*d^4)/32, and its value greatly influences the distribution of shear stress across the shaft.

Stress Concentration

Stress concentration occurs when there is an abrupt change in a member's geometry, such as at a fillet or a notch, resulting in a localized increase in stress. In the context of a stepped shaft, regions around the fillet experience higher stresses than predicted by simple torsion theory, making it an essential factor in design and analysis to prevent failure.

Fillet Radius in Stepped Shafts

The fillet radius in a stepped shaft smooths the transition between different shaft diameters. Its size affects the distribution of stress; a smaller fillet radius typically leads to a higher stress concentration, whereas a larger fillet radius can reduce the peak stress. Understanding its influence is key for optimizing design to ensure durability under torsional loads.

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