A sleeve is welded to a shaft at C in Fig. P14.18. What is...

A sleeve is welded to a shaft at $C$ in Fig. P14.18. What is the twist angle at $D$ from the $500-\mathrm{N}-\mathrm{m}$ torque on sleeve CD? Use $G=10 \times 10^{11} \mathrm{~Pa}$ as the shear modulas for both sleeve and shaft. Neglect twisting deformation of that part of $D$ which is perpendicular to $A B$ at $C$.

Key Concepts

Torsional Deformation

Torsional deformation refers to the twisting of an object due to an applied torque. When a cylindrical member such as a shaft or a sleeve is subjected to a torque, its cross sections rotate relative to each other, causing shear stresses and strains within the material. This concept forms the basis for analyzing how much angular displacement occurs along the length of the member when it is loaded in torsion.

Angle of Twist Calculation

The angle of twist is calculated by relating the applied torque to the material’s resistance to twisting. For a member with a uniform cross-section, it is given by the formula ? = TL/(JG), where T is the applied torque, L is the length over which the torque is applied, J is the polar moment of inertia of the cross-sectional area, and G is the shear modulus of the material. This relationship is fundamental in predicting how much rotation occurs under a given load.

Polar Moment of Inertia

The polar moment of inertia is a geometric property of a cross section that measures its resistance to torsion. It depends strongly on the shape and dimensions of the section. For circular cross sections, the polar moment of inertia plays a critical role in the torsion formula, directly influencing the stiffness of the component against twisting loads.

Shear Modulus

The shear modulus is a material property that quantifies the material’s rigidity under shear stress. In the context of torsion, it represents how resistant the material is to deformation when a shear stress is applied. A higher shear modulus indicates that the material will undergo less angular deformation for a given applied torque, making it a key parameter in torsion problems.

Welded Connections in Torsion

Welded connections, such as a sleeve welded to a shaft, are critical in ensuring continuity between different components. In torsion problems, a good welded connection means that the applied torque is effectively transferred from one component to the other without significant slip. This allows the combined system to be analyzed as a continuous element, simplifying the calculation of the overall twist and ensuring that deformations in the welded joint are appropriately considered or neglected based on the problem assumptions.

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