Hong Life CalculatorsUnderstanding the torsion spring calculator — concept and principle
A torsion spring is a coiled wire that resists twisting, as in clothes pegs, mousetraps, hinges and garage doors. This calculator estimates the spring rate, the torque at a given angle and the bending stress in the wire from the wire diameter, mean coil diameter, number of active coils and the wire's Young's modulus.
How it is calculated
Design rate per turn: k′ = d⁴ × E ÷ (10.8 × D × N)
Torque at angle θ: M = k′ × θ ÷ 360°
Ideal rate per radian: d⁴ × E ÷ (64 × D × N)
Spring index C = D ÷ d; Ki = (4C² − C − 1) ÷ (4C(C − 1))
Bending stress σ = Ki × 32M ÷ (π d³)
- d wire diameter and D mean coil diameter in mm, N active body turns, E in GPa (×1000 to N/mm²), θ in degrees.
The principle
Twisting a torsion spring actually bends its wire around the coil, so the wire behaves like a long curved beam loaded in bending. Beam theory gives the ideal angle θ = 64MDN ÷ (d⁴E), so stiffness rises with the fourth power of wire diameter and falls with coil diameter and number of turns. Friction between coils and against the arbor makes real springs slightly softer, so Shigley's design equation uses 10.8 per turn instead of the ideal 64 ÷ 2π ≈ 10.2. Bending stress is highest on the inside of the coil, where the curved wire concentrates stress; Ki corrects for this.
Worked example
Default inputs: d = 2 mm, D = 20 mm, N = 6, E = 200 GPa, angle 90°.
- E = 200,000 N/mm²; d⁴ = 16 mm⁴.
- k′ = 16 × 200,000 ÷ (10.8 × 20 × 6) = 3,200,000 ÷ 1,296 = 2,469 N·mm per turn (= 6.859 N·mm per degree).
- M = 2,469 × 90 ÷ 360 = 617.3 N·mm = 0.6173 N·m.
- Ideal rate = 3,200,000 ÷ (64 × 120) = 416.7 N·mm per rad.
- C = 20 ÷ 2 = 10; Ki = (400 − 10 − 1) ÷ 360 = 1.081.
- σ = 1.081 × 32 × 617.3 ÷ (π × 8) = 849.3 MPa.
| Wire diameter d (mm) | Torque at this angle (N·mm) |
|---|---|
| 0.5 | 2.411 |
| 1 | 38.58 |
| 1.5 | 195.3 |
| 2 | 617.3 |
| 2.5 | 1,507 |
| 3 | 3,125 |
| 4 | 9,877 |
How to read the result
Compare the stress with the allowable bending stress for your wire grade and size from a spring-wire standard or supplier; high-strength music wire allows much more than ordinary steel. Spring index between about 4 and 12 is generally easy to manufacture. When the spring winds up, its coil diameter shrinks, so leave clearance around the arbor.
Common mistakes
- Entering the outer diameter instead of the mean coil diameter.
- Counting leg length as coils; long legs add extra deflection not included here.
- Using E for the wrong material; stainless and bronze wires are less stiff than carbon steel.
- Loading the spring in the unwinding direction, which raises stress.
Frequently asked questions
Why does wire diameter matter so much?
Stiffness depends on d⁴, so a 10 % thicker wire is about 46 % stiffer.
What is a typical Young's modulus for spring steel?
Around 200 GPa for carbon steel music wire.
Should a torsion spring wind tighter or looser under load?
It is normally loaded so that it winds tighter, which reduces stress.
Sources: Shigley & Mischke, Mechanical Engineering Design, torsion spring equations (Ki, stress, deflection and design rate k′ = d⁴E/(10.8DN)) · University of Idaho, Machine Design "Springs" poster (S. Scott, V. Schiweck, N. Cropper)