§1The torsion formula
Twist shears the shaft in rings: zero at the axis, maximum at the skin. One formula for stress, one for wind-up.
G is the shear modulus (steel ≈ 80 GPa). The d³ in stress and d⁴ in stiffness are the levers of the whole page: a small diameter increase buys a lot of both.
Contents§2Power to torque
The running example: 15 kW at 1440 rev/min is T = 9549 × 15/1440 = 99.5 N·m. The same power at 240 rev/min — after a 6:1 reduction — is 597 N·m: gearboxes multiply the shaft-sizing problem exactly as fast as they multiply torque, which is why the slow shaft is always the fat one.
Contents§3Design for strength
99.5 N·m with τ_allow = 40 MPa (a modest figure that quietly covers keyways and shock): d³ = 16T/πτ = 16 × 99 500/(π × 40) = 12 670 mm³, d = 23.3 → Ø25 mm, running at 32.4 MPa. Strength alone says 25. §4 says otherwise.
§4Design for stiffness
A shaft that winds up ruins timing, chatters gears and stores spring energy for the worst moment. Working guidance for machine drives: hold twist to about ¼–1° per metre.
At Ø25: J = 38 300 mm⁴, so θ = TL/GJ = 99 500 × 1000/(80 000 × 38 300) = 0.0324 rad/m = 1.86°/m — double the 1°/m guideline. Solving J for 1°/m needs 71 200 mm⁴ → d = 29.2 → Ø30 mm. Stiffness added 5 mm the stress calculation never asked for — on transmission shafts it usually does.
§5Hollow shafts
Torsion barely uses the core, so remove it: a bore costs little stiffness and saves real mass.
Keep the same J with dᵢ/d₀ = 0.6: d₀ = 30/(1 − 0.6⁴)^¼ = 31.1 → Ø32 × Ø19 bore. Cross-section area falls to about 0.69 of the solid’s — roughly 31 % lighter for one millimetre more outside diameter. Where inertia matters (indexing drives, robotics) the saving compounds, since the removed metal was also the slowest to accelerate.
§6Keyways and combined loading
Real shafts carry bending from belt pulls and gear forces on top of torque, and they are slotted for keys exactly where both peak.
A standard profiled keyway costs a shaft on the order of a quarter of its torsional strength and adds a stress-raiser at its ends — sled-runner ends and generous fillets recover much of it. Combined bending-and-torsion design replaces T with an equivalent torque T_e = √(M² + T²) (and M with an equivalent moment ½(M + T_e)) — the classical shortcut for ductile shafts; a full treatment belongs with the Machine Elements pages. The habit that matters here: place keyways and shoulders away from the bending peak when the layout allows, and radius everything.
Contents§7Quick reference
The working core of the page on one card rack.
Torsion
τ = 16T/πd³
θ = TL/GJ · J = πd⁴/32
Torque
T = 9549 P(kW)/n
Stiffness guide
¼–1° per metre
usually governs drives
Hollow
J = π(d₀⁴−dᵢ⁴)/32
bore 0.6d₀ ≈ −31 % mass
Combined
T_e = √(M² + T²)
radius every keyway end