Question 4.5: A solid aluminum alloy shaft 60 mm in diameter and 1000 mm l...

We design first for strength and then for stiffness. From Table 2.1

or another source, we find the appropriate material properties: G_{Al} = 28 GPa and G_{St} = 84 GPa.

1. Designing for Strength

\underset{Steel}{\tau _{\max }} \leq 2\underset{Al}{\tau _{\max }} .

\left\lgroup\frac{Tc}{J}\right\rgroup _{St}\leq 2\left\lgroup\frac{Tc}{J}\right\rgroup _{Al} .

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

J_{St} ≥ \frac{1}{2}J_{Al} .

\frac{\pi}{2} \left[(0.03  \textrm{m})^4-r^{4}_{i}\right]\geq \frac{\pi}{4}\left[(0.03  \textrm{m})^4\right]w .

r^{4}_{i}\leq \frac{2}{\pi}\left[ \frac{\pi}{2}(0.03  \textrm{m})^4-\frac{\pi}{4}(0.03  \textrm{m})^4\right] .

r^{4}_{i}\leq \frac{2}{\sout{\pi}}\frac{\sout{\pi}}{4}(0.03  \textrm{m})^4=405×10^{-9}  \textrm{m}^4 .

r_i \leq 25.2 mm.

2. Designing for Stiffness

\phi _{_{St}}\leq \phi _{_{Al}}.

\frac{TL}{J_{St}G_{St}} \leq \frac{TL}{J_{Al}G_{Al}} .

J_{St}G_{St} \geq J_{Al}G_{Al} .

\frac{\pi}{2} \left[(0.03  \textrm{m})^4-r^{4}_{i} \right](84\times 10^9  \textrm{Pa})\geq \frac{\pi}{2} \left[(0.03  \textrm{m})^4\right](28\times 10^9  \textrm{Pa}) .

Solve for r_i ≤ 27.1 mm. The inner radius of the steel shaft must be r_i ≤ 25.2 mm, as strength governs.