The case we have used so far, where the torsional moment is a constant value, is somewhat a special case. In this example, we will use the solution for the general case. The angle of twist of two cross-sections separated by x is equal to:

\begin{aligned} &x_{1} \in<0 ; a> \\ &M\left(x_{1}\right)=m \cdot x_{1} \\ &M(0)=0 \\ &M(a)=m a \\ &x_{2} \in<0 ; a> \\ &M\left(x_{2}\right)=m \cdot a \\ &M(0)=m a \\ &M(a)=m a \\ &x_{3} \in<0 ; a> \\ &M\left(x_{3}\right)=m \cdot a+2 m \cdot x_{3} \\ &M(0)=m a \\ &M(a)=3 m a \end{aligned}

The graph of the torsional moment (not necessary)

The strength indicators for torsion and shear stresses:

\begin{aligned} W_{1} &=\frac{\pi \cdot\left(d^{4}-\left(\frac{3}{4} d\right)^{4}\right)}{16 d}=\frac{175 \pi d^{3}}{4096} \\ W_{2} &=W_{3}=\frac{\pi d^{3}}{16} \\ \tau_{1} &=\frac{4096 m a}{175 \pi d^{3}}=23,406 \frac{m a}{\pi d^{3}} \\ \tau_{2} &=\frac{16 m a}{\pi d^{3}} \\ \tau_{3} &=\frac{3 m a \cdot 16}{\pi d^{3}}=\frac{48 m a}{\pi d^{3}} \end{aligned}

Safety condition:

\(\tau_{\max }=\tau_{3} \leq k\) \(\frac{48 m a}{\pi d^{3}} \leq k \Rightarrow d \geq \sqrt[3]{\frac{48 m a}{\pi k}}\)

We calculate the moments of inertia for torsion (needed to calculate the angle of twist).

\begin{aligned} &I_{1}=\frac{\pi \cdot\left(d^{4}-\left(\frac{3}{4} d\right)^{4}\right)}{32}=\frac{175 \pi d^{4}}{8192} \\ &I_{2}=I_{3}=\frac{\pi d^{4}}{32} \end{aligned}

Total angle of twist:

\begin{aligned} &\varphi_{c}=\int_{0}^{a} \frac{m x_{1}}{G I_{1}} d x_{1}+\int_{0}^{a} \frac{m a}{G I_{2}} d x_{2}+\int_{0}^{a} \frac{m a+2 m \cdot x_{3}}{G I_{3}} d x_{3}= \\ &=\frac{m}{G I_{1}} \int_{0}^{a} x_{1} d x_{1}+\frac{m a}{G I_{2}} \int_{0}^{a} d x_{2}+\frac{m a}{G I_{3}} \int_{0}^{a} d x_{3}+\frac{2 m}{G I_{3}} \int_{0}^{a} x_{3} d x_{3}= \\ &=\frac{8192}{175 G \pi d^{4}} \frac{1}{2} a^{2}+\frac{32 m a}{G \pi d^{4}} a+\frac{32 m a}{G \pi d^{4}} a+\frac{32 \cdot 2 m a}{G \pi d^{4}} \frac{1}{2} a^{2}=\frac{20866 m a^{2}}{175 G \pi d^{4}}=119,2 \frac{m a^{2}}{G \pi d^{4}} \end{aligned}

Torsion

Strength of materials - Torsion - Example 9

Example 9