(a) For the 3 -in.-diameter solid cylinder and loading...

(a) For the 3 -in.-diameter solid cylinder and loading shown, determine the maximum shearing stress. ( $b$ ) Determine the inner diameter of the 4 -in.-diameter hollow cylinder shown, for which the maximum stress is the same as in part $a$.

Key Concepts

Equating Maximum Stress Conditions

In engineering design, it is common to set the maximum stresses in different configurations equal in order to achieve similar performance or safety criteria. This involves solving for unknown parameters, such as inner dimensions, by using established stress formulas and ensuring that the stress levels stay within acceptable limits across different designs.

Comparison of Solid and Hollow Cylinders in Torsion

This concept involves understanding how different geometrical configurations affect a cylinder's resistance to twisting. A hollow cylinder, with less material in the core, will have a different polar moment of inertia compared to a solid cylinder, leading to different maximum shear stresses under the same loading conditions. Equating these stresses can be used to design cylinders with equivalent performance.

Shear Stress in Torsion

This concept explains how a twisting moment or torque applied to a cylindrical object induces a distribution of shear stresses. The maximum shear stress occurs at the outer surface of the cylinder and is calculated using the relationship ? = T·r/J, where T is the applied torque, r is the radial distance from the center, and J is the polar moment of inertia.

Polar Moment of Inertia

The polar moment of inertia is a measure of an object's resistance to torsional deformation. For circular cross-sections, it is computed differently for solid and hollow cylinders. In a solid cylinder, it is proportional to the fourth power of the diameter, whereas in a hollow cylinder, it is the difference between the fourth powers of the outer and inner diameters, reflecting the effect of material distribution on torsional performance.

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