Question 5.2: A solid aluminum alloy (GAl = 28 GPa) shaft 60 mm in diamete......
A solid aluminum alloy (G_{Al} = 28 GPa) shaft 60 mm in diameter and 1000 mm long is to be replaced by a tubular steel shaft (G_{St} = 84 GPa) of the same outer diameter such that the new shaft will exceed neither (1) twice the maximum shear stress nor (2) the angle of twist of the aluminum shaft. What should be the inner radius of the tubular steel shaft? Which of the two criteria, (1) strength or (2) stiffness, governs?
Given: Dimensions of the aluminum shaft.
Find: Dimensions of the steel shaft (same length, same outer diameter) that will meet strength and stiffness requirements.
Assume: Hooke’s law applies.
We will design first for strength, then for stiffness
1. Designing for strength:
\begin{aligned} \tau_{\max , \mathrm{St}} & \leq 2 \tau_{\max , \mathrm{Al}}, \\ \left(\frac{T c}{J}\right)_{\mathrm{St}} & \leq 2\left(\frac{T c}{J}\right)_{\mathrm{Al}}. \end{aligned}Since we are told that the outer diameters of both shafts are equal, and since the applied torque T does not change, we are simply requiring that
\begin{aligned} J_{\mathrm{St}} & \geq \frac{1}{2} J_{\mathrm{Al}}, \\ \frac{\pi}{2}\left[(0.03 \mathrm{~m})^4-r_i^4\right] & \geq \frac{\pi}{4}\left[(0.03 \mathrm{~m})^4\right], \\ r_i^4 & \leq \frac{2}{\pi}\left[\frac{\pi}{2}(0.03 \mathrm{~m})^4-\frac{\pi}{4}(0.03 \mathrm{~m})^4\right], \\ r_i^4 & \leq \frac{1}{2}(0.03 \mathrm{~m})^4=405 \times 10^{-9} \mathrm{~m}^4, \\ r_i & \leq 25.2 \mathrm{~mm} . \end{aligned}2. Designing for stiffness:
\begin{aligned} \phi_{\mathrm{St}} & \leq \phi_{\mathrm{Al}}, \\ \frac{T L}{J_{\mathrm{St}} G_{\mathrm{St}}} & \leq \frac{T L}{J_{\mathrm{Al}} G_{\mathrm{Al}}}, \\ J_{\mathrm{St}} G_{\mathrm{St}} & \geq J_{\mathrm{Al}} G_{\mathrm{Al}}, \\ \frac{\pi}{2}\left[(0.03 \mathrm{~m})^4-r_i^4\right]\left(84 \times 10^9 \mathrm{~Pa}\right) & \geq \frac{\pi}{2}\left[(0.03 \mathrm{~m})^4\right]\left(28 \times 10^9 \mathrm{~Pa}\right), \end{aligned}
Solve for r_i \leq 27.1 \mathrm{~mm} .
The inner radius of the steel shaft must be r_i ≤ 25.2 mm, as strength govern