Looking at the load on the shaft, we can draw a graph of the torsional moments:

We calculate the unit change in angle for the individual intervals.

Condition for the maximum angle of twist:

\(\varphi_{\max } \leq \varphi_{\text {dop }}\) $$ \begin{aligned} &\varphi_{\max }=\frac{30 \cdot 10^{3} \cdot 2 \cdot 32}{80 \cdot 10^{9} \cdot \pi D^{4}}+\frac{30 \cdot 10^{3} \cdot 2 \cdot 32}{80 \cdot 10^{9} \cdot \pi \cdot\left(D^{4}-\left(\frac{1}{2} D\right)^{4}\right)}= \\ &=\frac{7,64 \cdot 10^{-6}}{D^{4}}+\frac{8,15 \cdot 10^{-6}}{D^{4}}=\frac{15,79 \cdot 10^{-6}}{D^{4}} \end{aligned} $$ $$ \frac{15,79 \cdot 10^{-6}}{D^{4}} \leq 0,4 \cdot \frac{\pi}{180}=6,98 \cdot 10^{-3} $$ \(D \geq 0,218 m\)

Strength condition:

\(\tau_{\max } \leq \tau_{d o p}\) \begin{aligned} &\tau_{\max }=\tau_{B C} \\ &\tau_{B C}=\frac{M_{B C}}{W_{B C}} \end{aligned} \(W_{B C}=\frac{\pi\left(D^{4}-(0,5 D)^{4}\right)}{16 D}=0,184 D^{3}\) \begin{aligned} &\frac{30 \cdot 10^{3}}{0,184 D^{3}} \leq 60 \cdot 10^{6} \\ &D \geq 0,1395 m \end{aligned}

I dimension the diameter: D=0,22 m

\begin{aligned} &W_{A B}=\frac{\pi 0,22^{3}}{16}=2,09 \cdot 10^{-3} \mathrm{~m}^{3} \\ &W_{B C}=0,184 \cdot 0,22^{3}=1,96 \cdot 10^{-3} \mathrm{~m}^{3} \end{aligned}

Shear stresses \(\tau=\frac{M_s}{W_s}\)

\begin{aligned} &W_{A B}=\frac{\pi 0,22^{3}}{16}=2,09 \cdot 10^{-3} \mathrm{~m}^{3} \\ &W_{B C}=0,184 \cdot 0,22^{3}=1,96 \cdot 10^{-3} \mathrm{~m}^{3} \\ &\tau_{A B}=\frac{30 \cdot 10^{3}}{2,09 \cdot 10^{-3}}=14,35 \mathrm{MPa} \\ &\tau_{B C}=\frac{30 \cdot 10^{3}}{1,96 \cdot 10^{-3}}=15,31 \mathrm{MPa} \end{aligned}

Angle of twist of the shaft:

\begin{aligned} &\varphi_{A}=0 \\ &\varphi_{B}=\varphi_{A}+\varphi_{A B}=\frac{30 \cdot 10^{3} \cdot 2 \cdot 32}{80 \cdot 10^{9} \cdot \pi 0,22^{4}}=3,26 \cdot 10^{-3} \mathrm{rad}=0,187^{\circ} \\ &\varphi_{C}=\varphi_{B}+\varphi_{B C}=3,26 \cdot 10^{-3}+\frac{30 \cdot 10^{3} \cdot 2 \cdot 32}{80 \cdot 10^{9} \cdot \pi \cdot\left(0,22^{4}-0,11^{4}\right)}=3,26 \cdot 10^{-3}+3,48 \cdot 10^{-3}=6,74 \cdot 10^{-3} \mathrm{rad}=0,386^{\circ} \end{aligned}

Graphs:

Torsion

Strength of materials - Torsion - Example 6

Example 6