Stepped shaft, three torques: twist at free end

Reading of the figure: fixed at the left end; three segments each 500 mm long with diameters 80, 60 and 40 mm. Torques of 10 kN·m (at the end of segment 1), 4 kN·m (end of segment 2) and 2 kN·m (free end) act in the same sense. $G = 80$ GPa $= 8\times10^4\ \text{N/mm}^2$.

Torque in each segment (section method from the free end)

Twist of each segment

$$\varphi_i = \frac{T_iL_i}{GJ_i}, \qquad J_i = \frac{\pi}{32}d_i^4$$ $$\varphi_{free} = 0.02487 + 0.02947 + 0.04974 = 0.10408\ \text{rad} = 5.963^\circ$$

Maximum shear stress

$$\tau = \frac{16T}{\pi d^3}: \qquad \tau_1 = 159.2,\quad \tau_2 = 141.5,\quad \tau_3 = 159.2\ \text{N/mm}^2$$

The maximum, 159.2 N/mm², occurs in segments 1 and 3 (the 80 mm and 40 mm parts); segment 2 has the lowest stress.

Answer: angle of twist at the free end $= 5.96^\circ$ (0.1041 rad); $\tau_{max} = 159.2$ MPa, in segments 1 and 3.