Torsion Spring Rate Calculator (Torque per Degree)

How it works

k (N·mm/turn) = E d⁴ ÷ (10.8 D N_a), N_a = body turns + (L₁ + L₂) ÷ (3π D) · M = k × θ · σ = 32 M K_i ÷ (π d³), K_i = (4C² − C − 1) ÷ (4C(C − 1)) · wound diameter D′ = N D ÷ (N + θ/360)

A torsion spring stores energy by bending its wire rather than twisting it, so the elastic modulus replaces the shear modulus and the wire behaves as a curved beam whose angle of rotation per unit torque is proportional to its length — the body turns plus a share of the legs — and inversely to the fourth power of the diameter. The 10.8 in the denominator is Shigley's empirical constant that accounts for curvature and friction between coils, and the inside-fibre factor raises the calculated bending stress where the wire curves most tightly. Winding the spring tighter reduces its diameter and lengthens the body, which is why arbors are sized below the loaded inside diameter and housings leave room, and why springs are always loaded in the winding direction.

Worked example

Music wire 2 mm coiled at a 16 mm mean diameter with 6 body turns and two 30 mm legs has about 6.4 effective turns and a rate of 207,000 × 2⁴ ÷ (10.8 × 16 × 6.4) ≈ 3,000 N·mm per turn — 8.3 N·mm per degree. At 90° it delivers 750 N·mm (25 N at a leg tip), with a bending stress near 1,050 MPa against roughly 1,560 MPa allowable; the inside diameter closes from 14 to about 13.4 mm, so a 12 mm arbor fits.

Frequently asked questions