Torsion vs bending; hollow shaft replaced by solid

Torsion and bending

Replacing the hollow shaft by a solid shaft

Equal polar modulus means $Z_p$ (hollow) $= Z_p$ (solid). Hollow shaft: $D = 30$ cm, $d = 20$ cm. Solid alloy shaft: diameter $D_s$.

$$Z_p = \frac{\pi}{16}\,\frac{D^4-d^4}{D} = \frac{\pi}{16}\,\frac{30^4-20^4}{30} = 4254.2\ \text{cm}^3, \qquad Z_p = \frac{\pi}{16}D_s^3$$ $$D_s^3 = \frac{30^4-20^4}{30} = 21666.67 \Rightarrow D_s = 27.88\ \text{cm}$$

Ratio of torsional rigidities

Torsional rigidity is $GJ$. Hollow (steel) $J_h = \dfrac{\pi}{32}(30^4-20^4) = 63813.6\ \text{cm}^4$; solid (alloy) $J_s = \dfrac{\pi}{32}D_s^4 = 59300.2\ \text{cm}^4$, and $G_{steel} = 2.5\,G_{alloy}$:

$$\frac{(GJ)_{hollow}}{(GJ)_{solid}} = \frac{2.5\,G_{alloy}\,J_h}{G_{alloy}\,J_s} = 2.5\times\frac{63813.6}{59300.2} = 2.690$$

Answer: solid shaft diameter $= 27.88$ cm; ratio of torsional rigidities (hollow : solid) $= 2.69$.