What is the twist at A for the shaft showe in Fig. P.14.66?...

What is the twist at $A$ for the shaft showe in Fig. P.14.66? The diameter varies linearly as indicated. Assume that the simple theory for uniform shatis is valid locally here. (Take $G=20 \times 10^6 \mathrm{psi}$.)

Key Concepts

Torsion in Shafts

This concept deals with how circular shafts twist when subjected to a torque. The fundamental relation connects the applied torque to the resulting shear stress and angular deformation (twist) through the geometry of the shaft and the material properties.

Polar Moment of Inertia

The polar moment of inertia (J) quantifies the distribution of a cross?section's area relative to an axis and is critical for understanding torsion. For circular cross sections, J is proportional to the fourth power of the diameter, making variations in diameter highly influential on the torsional stiffness.

Variable Cross-Section Analysis

When a shaft’s diameter changes along its length, its torsional stiffness also varies. This concept involves treating the shaft as a series of infinitesimal segments, each with its own uniform cross-sectional properties, and then integrating the local torsion equations along the length.

Shear Modulus in Torsion

The shear modulus (G) is a material property that measures the stiffness of a material in response to shear deformations. In torsion problems, G appears in the relationship between applied torque and the resulting twist, influencing how much angular deformation occurs for a given load.

Integration of Differential Elements

Since the shaft has a non-uniform geometry, the total twist is computed by integrating the local twist per unit length over the span of the shaft. This approach, which stems from considering differential segments of the shaft as locally uniform, allows one to sum the contributions of each segment to the overall twist.

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