Solid shaft in torsion: torque, shear stress and angle of twist

A 30 mm solid steel shaft (G = 80 GPa) transmits 15 kW at 1450 rpm. Over a length of 0.8 m, find the torque, the maximum shear stress and the angle of twist.

Given

Solution

1. 1.Torque from power and speed
   $$T=\frac{P}{\omega }$$
   $$T=98.79\mathrm{N}\cdot \mathrm{m}$$
2. 2.Polar moment of a solid shaft
   $$J=\frac{\pi d^4}{32}$$
   $$J=79522\mathrm{m}{\mathrm{m}}^4$$
3. 3.Maximum shear stress at the surface
   $${\tau }_{max}=\frac{T(d/2)}{J}$$
   $${\tau }_{max}=18.63\mathrm{MPa}$$
4. 4.Angle of twist
   $$\varphi =\frac{TL}{GJ}$$
   $$\varphi =0.712^{\circ }$$

Worked in ProveCalc

Power over rpm only gives N·m once rpm is read as revolutions per minute in radians per second; the engine does that conversion itself, and the angle of twist comes out in degrees on request.

The results above were evaluated by ProveCalc's compute engine (SymPy and Pint), the same engine the desktop app runs on your machine.

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