Torsion of shafts, explained

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Where the stress is

Twist a round shaft and each cross-section rotates a little relative to the next. Plane sections stay plane and radii stay straight, so the shear strain grows linearly from zero at the axis to its maximum at the outer surface. In the elastic range the stress does the same.

J is the polar moment of area, the torsion equivalent of the second moment of area in bending. For a circle it is exactly twice I. Because material near the axis carries almost no stress, a hollow shaft gives up very little strength for a large saving in weight, which is why drive shafts are tubes.

From power to diameter

A motor is specified by power and speed, not torque. Torque is power divided by angular speed in radians per second: T = P/ω, with ω = 2πn/60 when n is in rpm. The same power at a lower speed means more torque, which is why the output shaft of a gearbox is thicker than its input.

Two checks size a shaft. Strength: the maximum shear stress must stay below an allowable value. Stiffness: the twist per unit length must stay small enough for whatever the shaft drives. Long shafts are often limited by stiffness, not strength.

A worked number

A solid steel shaft, 25 mm in diameter and 1 m long, carries 10 kW at 1450 rpm. ω = 2π × 1450 / 60 ≈ 151.8 rad/s, so T = 10,000 / 151.8 ≈ 65.9 N·m.

J = π × 0.025⁴ / 32 ≈ 3.83 × 10⁻⁸ m⁴. The surface stress is 65.9 × 0.0125 / 3.83 × 10⁻⁸ ≈ 21.5 MPa, and with G = 80 GPa the twist is 65.9 / (80 × 10⁹ × 3.83 × 10⁻⁸) ≈ 0.0215 rad, about 1.23°.

Common mistakes

• Dividing power by speed in rpm instead of rad/s, which is off by a factor of about ten.
• Using the bending I instead of the polar J. For a circular section J = 2I.
• Sizing the plain shaft and forgetting the keyway or shoulder, where stress concentration multiplies the peak.