1. Solid Shaft Stress: [cite: 23, 25]
For a solid shaft of diameter $D$, the polar moment of inertia is $J_s = \frac{\pi D^4}{32}$. [cite: 23, 25]
The maximum shear stress $\tau$ at the outer surface ($r = D/2$) is:
$$\tau = \frac{T \cdot (D/2)}{\pi D^4 / 32} = \frac{16T}{\pi D^3}$$ [cite: 23, 25]
2. Hollow Shaft Stress: [cite: 23, 25]
For the hollow shaft with $D_o = D$ and $D_i = D/2$:
$$J_h = \frac{\pi}{32} (D^4 - (D/2)^4) = \frac{\pi}{32} (D^4 - \frac{D^4}{16}) = \frac{\pi D^4}{32} (\frac{15}{16})$$ [cite: 23, 25]
The new maximum shear stress $\tau'$ at the same outer radius $r = D/2$ is:
$$\tau' = \frac{T \cdot (D/2)}{J_h} = \frac{T \cdot (D/2)}{\frac{\pi D^4}{32} \cdot \frac{15}{16}} = \frac{16T}{\pi D^3} \cdot \frac{16}{15}$$ [cite: 23, 25]
3. Final Comparison: [cite: 23, 25]
Substituting the original $\tau$ value:
$$\tau' = \frac{16}{15} \tau$$ [cite: 23, 25]