In Fig. P.14.21 is shown a shaft composed of an inside solid...

In Fig. P.14.21 is shown a shaft composed of an inside solid rod baving $G=7 \times 10^{10} \mathrm{~Pa}$ and an outer sleeve having $G=1.4 \times 10^{11} \mathrm{~Pa}$. What is the maximum stress wben shafts are connected at the right end only?

Key Concepts

Torsion in Shafts

This concept deals with how a circular shaft twists under an applied torque. It involves the relationship between the applied torque, the material properties, and the geometric characteristics of the shaft, which together determine the angular deformation and resulting stress distribution.

Shear Stress Distribution

Shear stress in a twisting shaft is not uniform and typically varies linearly from the center to the outer surface. Calculating the maximum shear stress usually involves identifying the location where the stress is highest, which is typically at the outermost fibers of the shaft.

Composite Shaft Analysis

When a shaft is composed of different materials or parts, such as an inner core and an outer sleeve, each with its own material properties, the overall behavior under torsion must be analyzed by considering the interaction between these two components. Compatibility of deformation and equilibrium of torque transfer between the different parts are central to this analysis.

Shear Modulus

The shear modulus is a material property that quantifies its rigidity under shear loading. In torsion problems, when different parts of a composite structure have different shear moduli, these differences must be accounted for in the stress analysis, as they directly affect the distribution of shear strains and stresses within each component.

Polar Moment of Inertia

The polar moment of inertia is a geometric property that reflects how an area is distributed about an axis and is crucial in torsion analysis. It influences the magnitude of shear stress induced by a given torque. In composite shafts, the individual polar moments of inertia of the different components contribute to the overall torsional stiffness and stress distribution.

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