Torsional Spring Calculator

The helical torsion spring rate is k = E*d^4/(64*D*Na), giving torque per radian. E is Young's modulus, d the wire diameter, D the mean coil diameter and Na the number of active turns. Example: a music-wire spring (E = 207 GPa) with d = 2 mm, D = 20 mm and Na = 8 gives k = 0.3234 N*m/rad, which is 5.6450 N*mm/deg or 2.0322 N*m per revolution.

Use the definition k = T/theta: divide the applied torque T by the angular deflection theta. Both must use consistent angle units, for example N*m per radian or lbf*in per degree. Rearranged, it also gives torque T = k*theta and deflection theta = T/k.

The Wahl factor corrects the straight-wire bending stress for curvature: Ki = (4C-1)/(4C-4) + 0.615/C, where C = D/d is the spring index. For C = 10, Ki = 1.1448, meaning the real peak stress is about 14.5% higher than the uncorrected value.

Bending stress is S = Ki*32*M/(pi*d^3), with Ki the Wahl factor, M the applied torque and d the wire diameter. Example: M = 1 N*m on a d = 2 mm, D = 20 mm spring (C = 10, Ki = 1.1448) gives S = 1457.65 MPa, so check it against the wire material's tensile strength.

Spring index C = D/d should generally stay between 4 and 12. Below 4 the wire is hard to coil and the Wahl correction grows sharply; above 12 the spring gets floppy, wastes space and is prone to buckling or tangling.

Winding the spring up tightens the coils, so the mean diameter shrinks to Dc = D*Na/(Na+n), where n is the deflection in revolutions. The inside diameter shrinks with it, so the mounting mandrel must be sized for the loaded ID with clearance, typically about 10%, to avoid binding.

Always load a torsion spring in the wind-up direction (the direction that tightens the coils). Loading it to unwind raises the bending stress on the wire and can permanently set or break the spring.