Angle of Twist (φ = TL/JG)

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.