Several shafts are shrink-fitted over a solid shaft (see...

Several shafts are shrink-fitted over a solid shaft (see Fig. P.14.62) so as to form a combined member which can transmit torque with no slippage between the members. Explain why the assumption of cross sections rotating as platelets still applies. If each material is linear and elastic, show using equilibrium, Hooke's law, and the assumption above that $$ \begin{aligned} T & =\alpha\left[\left(\int_0^{R_1} r^2 d A\right) G_1+\left(\int_{R_2}^{R_2} r^2 d A\right) G_2+\left(\int_{2_3}^{E_3} r^2 d A\right) G_3\right] \\ & =\alpha\left[G_2 J_1+G_2 J_2+G_3 J_3\right] \end{aligned} $$ where $\alpha$ is the rate of twist.

Several shafts are shrink-fitted over a solid shaft (see Fig. P.14.62) so as to form a combined member which can transmit torque with no slippage between the members. Explain why the assumption of cross sections rotating as platelets still applies. If each material is linear and elastic, show using equilibrium, Hooke's law, and the assumption above that
$$
\begin{aligned}
T & =\alpha\left[\left(\int_0^{R_1} r^2 d A\right) G_1+\left(\int_{R_2}^{R_2} r^2 d A\right) G_2+\left(\int_{2_3}^{E_3} r^2 d A\right) G_3\right] \\
& =\alpha\left[G_2 J_1+G_2 J_2+G_3 J_3\right]
\end{aligned}
$$
where $\alpha$ is the rate of twist.

Key Concepts

Torsion in Circular Shafts

This concept involves understanding how a circular shaft twists under an applied torque. The twisting causes shear stresses and strains that vary linearly with the radial distance from the center. Analyzing these stresses and strains forms the basis for calculating the shaft’s deformation and the distribution of torque along its cross-section.

Saint?Venant's Assumption (Cross?Section Rotating as Platelets)

Saint?Venant's assumption states that plane cross?sections of a shaft remain plane after deformation and rotate as rigid plates, which implies that warping is negligible. This assumption simplifies the analysis by allowing the strain to be considered uniformly across the cross-section, and it remains valid even for composite shafts with different materials as long as bonding prevents relative displacements.

Hooke's Law in Torsion

In the context of torsion, Hooke's law relates shear stress to shear strain by the shear modulus of the material. For linear elastic materials, this linear relationship allows for the direct calculation of stress within each material layer, and it provides the foundation for summing the contributions of different materials in a composite shaft to determine the overall torsional response.

Polar Moment of Inertia

The polar moment of inertia is an integral measure of the cross-section’s resistance to twisting. It is calculated as the integral of r² dA over the cross-sectional area and is crucial in determining how the distribution of material about the axis of rotation contributes to the shaft’s ability to resist torsion.

Equilibrium of Torsional Moments in Composite Members

When multiple shafts or materials are combined into one composite member, equilibrium of torsional moments must be maintained across the entire cross-section. This concept uses the idea that each material contributes to the overall torque according to its polar moment of inertia and shear modulus, ensuring that the composite behaves as a unified structure with no relative slippage between the layers.

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