Understanding the following will be aided by a video course on Example 1.

Strength indicators and moments of inertia for basic plane figures |CLICK|.

Torsional moments and moments of inertia (for calculation convenience, we will check the relation of moments of inertia to each other).

\begin{aligned}\\ &M_{BC}=M_C\\ &M_{AB}=M_C+40\\ \\ &I_{BC}=\frac{\pi (2d)^4}{32}=\frac{\pi d^4}{2}\\ &I_{AB}=\frac{\pi d^4}{32}\\ \\ &\frac{I_{BC}}{I_{AB}}=16\ \Rightarrow I_{BC}=16I_{AB}\\ \end{aligned}

Geometrical condition - the total torsional angle is equal to 0: \(\varphi_c =0\)

\begin{aligned}\\ &\varphi=\frac{M_S\cdot l}{G\cdot I}\\ \\ &\varphi_c=\varphi_{BC}+\varphi_{AB}=0\\ &\frac{M_C\cdot 1.5}{G\cdot 16I_{AB}}+\frac{(M_C+40)\cdot 4}{G\cdot I_{AB}}=0\ |\cdot GI_{AB}\\ &0.09375M_c+4M_C+160=0\\ &M_C=-39.084\ kNm\\ \end{aligned}

Stresses in the rod

\begin{aligned}\\ &M_{BC}=-39.084\ kNm\\ &M_{AB}=0.916\ kNm\\ \\ &W_{BC}=\frac{\pi (2d)^3}{16}=\frac{\pi d^3}{2}\\ &W_{AB}=\frac{\pi d^3}{16}\\ \end{aligned}

Due to the significant difference in the torsional moment on characteristic intervals of the rod, maximum tangential stresses will occur in the BC section

\begin{aligned}\\ &\tau_{BC}=\frac{39.084\cdot 10^3}{\frac{\pi d^3}{2}}\le 70\cdot 10^6\\ &d\ge 0.07084\ m\\ &d=7.2\ cm\\ \\ \\ &\tau_{AB}=\frac{0.916\cdot 10^3}{\frac{\pi 0.072^3}{16}}=12.49\ MPa\\ &\tau_{BC}=\frac{39.084\cdot 10^3}{\frac{\pi 0.072^3}{2}}=-66.66\ MPa\\ \end{aligned}

Section torsional angle

\begin{aligned}\\ &\varphi_C=0\\ &\varphi_B=\varphi_C+\varphi_{BC}\\ \\ &\varphi=\frac{M_S\cdot l}{G\cdot I}\\ \\ &\varphi_B=\frac{-39.084\cdot 10^{3}\cdot 1.5}{80\cdot 10^{9}\cdot \frac{\pi 0.072^4}{2}}=-0.01736\ rad=-0.995\ [^o]\\ \end{aligned}

Charts

Torsion

Strength of materials - Torsion - Example 7

Example 7