Torsion Shaft
Drive shafts, drill bits and turbine shafts are twisted rather than bent. Find how far a solid or hollow shaft twists under a torque, the shear stress in it and the power it carries — or solve for the torque, the length, the diameter or the material.
How Does a Shaft Twist?
A torque twists a shaft: each cross-section rotates a little relative to the next, and a straight line drawn along the surface turns into a helix. The resistance comes from the shear modulus G of the material and the polar moment of area J of the cross-section.
The shear stress is zero at the centre and largest at the outer surface. Material near the middle hardly works, so a hollow shaft is nearly as strong and stiff as a solid one at a fraction of the weight — that is why drive shafts and bicycle frames are tubes.
Live Diagram
twist exaggeratedTorsion Formulas
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Worked Techniques
For strength, the outer diameter grows only with the cube root of the torque: double the torque needs 26% more diameter.
The same power at a higher speed needs less torque, and so a thinner shaft. That is why motors run fast and a gearbox steps the speed down only where the high torque is needed.
Because of the fourth power, the inner core contributes little. A tube with d = 0.8 D keeps 59% of the stiffness of a solid shaft with only 36% of the material.
Sources & Verification
- R. C. Hibbeler, Mechanics of Materials in SI Units, 10th ed., Pearson, 2018 — Ch. 5 (Torsion).
- J. M. Gere & B. J. Goodno, Mechanics of Materials, 9th ed., Cengage Learning, 2018 — Ch. 3 (Torsion).
- R. G. Budynas & J. K. Nisbett, Shigley’s Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — Ch. 3 (Load and Stress Analysis) and Ch. 7 (Shafts and Shaft Components).