Angle of Twist (φ = TL/JG)
Also known as shaft twist · TL/JG
Worked example: 1 kN·m over 2 m of 50 mm steel shaft → 2.334 deg — press Try an example to run it live, then adjust anything.
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Angle of Twist (φ = TL/JG) explained
This is the torsional counterpart of δ = PL/AE: the product JG is the torsional rigidity, and dividing it by L gives the shaft's torsional spring rate. A 2 m length of 50 mm steel shaft (J = 6.136 × 10⁻⁷ m⁴, G = 80 GPa) under 1 kN·m twists φ = 1000 × 2 ÷ (6.136 × 10⁻⁷ × 80 × 10⁹) = 0.0407 rad = 2.33°. The formula returns radians internally; this calculator displays degrees, so watch which one you are reading.
Long shafts are usually governed by twist rather than by stress. The old machine-design rule of thumb is to hold a power transmission shaft to about one degree of twist per twenty diameters of length, because more than that and the driven end lags the driver enough to upset timing, chatter a cutting tool, or set up torsional vibration. Torsional resonance is a real killer in reciprocating engine and compressor drivelines — the crankshaft, flywheel and driven mass form a torsional oscillator whose stiffness is exactly JG/L, and running near its natural frequency has snapped many crankshafts. Enter J in m⁴ as a plain number, and note this holds for circular sections only.
Angle of Twist (φ = TL/JG) formula
- = Angle of twist (°)
- = Applied torque (N·m)
- = Shaft length (m)
- = Polar moment of inertia (mm⁴)
- = Shear modulus (kPa)
Missing one of these? Work it out first, then come back
- Angle of twist — Torque with a Lever Arm (τ = rF sin θ), Angular Velocity (ω = θ/t)
- Applied torque — Torsional Shear Stress (τ = Tr/J), Torque
- Shaft length — Normal Strain (ε = δ/L), Thermal Linear Expansion
- Polar moment of inertia — Torsional Shear Stress (τ = Tr/J), Polar Moment of Inertia — Solid Shaft
- Shear modulus — Shear Modulus (G = τ/γ), Relation Between E, G and ν