# Compression, extension and torsion springs: sizes, rate formula, printed envelopes

> How do I calculate the rate of a compression spring?

- Category: Hardware › Motion
- Kind: Hardware
- Basis: Music-wire (ASTM A228) spring design: k = G·d⁴ / (8·D³·n) with G = 79.3 GPa, D = mean diameter, n = active coils; envelopes in common hobby sizes
- Page: https://xemstudio.com/objects/springs/ · JSON: https://xemstudio.com/api/geometry?object=springs · Plain text: https://xemstudio.com/objects/springs/sheet.txt
- License: dimension sheet CC BY 4.0 (Xem Studio); linked files under their publishers' licenses

## Notes

- Rate k (N/mm) = G d⁴ / (8 D³ n): wire 1.0 mm, OD 10 (D = 9), 10 active coils → 79300 × 1 / (8 × 729 × 10) = 1.36 N/mm. Halve the coils, double the rate. Spring index D/d between 4 and 12 is makeable; 6–9 is comfortable.
- Compression: free length L, solid height ≈ (n + 2) × d, max deflection about 0.8 × (L − solid). Extension: same k, plus initial tension; hooks need D/2 of extra length each end. Torsion: rate in N·mm/deg = E d⁴ / (3667 D n) with E = 207 GPa.
- Printed TPU springs work for small forces (ratchet pawls, detents); PLA/PETG helices break. The files here are envelopes for fit and to check pocket clearance (pocket ID = OD + 0.5; guide rod = ID − 0.5).
- Where they go on the LES rigs: pawl return (torsion 0.8 × 8), diaphragm return and displacer-rod bias (compression 0.5 × 6 × 20), belt tensioner (extension 1.0 × 10 × 50).

## Sizes

| type | wire_mm | od_mm | length_mm | coils | rate_n_per_mm | index_d_over_d |
|---|---|---|---|---|---|---|
| compression | 0.5 | 6 | 20 | 8 | 0.47 | 11.0 |
| compression | 0.8 | 8 | 25 | 10 | 1.09 | 9.0 |
| compression | 1.0 | 10 | 30 | 10 | 1.36 | 9.0 |
| compression | 1.2 | 12 | 40 | 12 | 1.36 | 9.0 |
| compression | 1.6 | 16 | 50 | 10 | 2.18 | 9.0 |
| extension | 0.6 | 6 | 30 | 20 | 0.41 | 9.0 |
| extension | 1.0 | 10 | 50 | 25 | 0.54 | 9.0 |
| torsion | 0.8 | 8 | 10 | 6 | — | 9.0 |

## Where the files are

- Wikipedia: [Spring (device)](https://en.wikipedia.org/wiki/Spring_(device)) — web — CC BY-SA 4.0 (text)
- McMaster-Carr: [Compression springs (free STEP per part, for your own design use; no redistribution)](https://www.mcmaster.com/products/compression-springs/) — STEP, PDF — McMaster-Carr site terms (verify at link)
- Lee Spring: [Spring design formulas](https://www.leespring.com/learn-about-springs) — web — Lee Spring site (verify at link)
- Long Earth Society: [Engine and rig builds](https://longearthsociety.com/engineering/) — web, video — LES site

## Free STEP/STL (8 parts, CC BY 4.0, Xem Studio)

Index: https://xemstudio.com/files/hardware/springs/index.json

- compression-d0.5-od6-l20: [STEP](https://xemstudio.com/files/hardware/springs/compression-d0.5-od6-l20.step) · [STL](https://xemstudio.com/files/hardware/springs/compression-d0.5-od6-l20.stl) — compression spring envelope: wire 0.5, OD 6, 20 mm, 8 coils
- compression-d0.8-od8-l25: [STEP](https://xemstudio.com/files/hardware/springs/compression-d0.8-od8-l25.step) · [STL](https://xemstudio.com/files/hardware/springs/compression-d0.8-od8-l25.stl) — compression spring envelope: wire 0.8, OD 8, 25 mm, 10 coils
- compression-d1-od10-l30: [STEP](https://xemstudio.com/files/hardware/springs/compression-d1-od10-l30.step) · [STL](https://xemstudio.com/files/hardware/springs/compression-d1-od10-l30.stl) — compression spring envelope: wire 1.0, OD 10, 30 mm, 10 coils
- compression-d1.2-od12-l40: [STEP](https://xemstudio.com/files/hardware/springs/compression-d1.2-od12-l40.step) · [STL](https://xemstudio.com/files/hardware/springs/compression-d1.2-od12-l40.stl) — compression spring envelope: wire 1.2, OD 12, 40 mm, 12 coils
- compression-d1.6-od16-l50: [STEP](https://xemstudio.com/files/hardware/springs/compression-d1.6-od16-l50.step) · [STL](https://xemstudio.com/files/hardware/springs/compression-d1.6-od16-l50.stl) — compression spring envelope: wire 1.6, OD 16, 50 mm, 10 coils
- extension-d0.6-od6-l30: [STEP](https://xemstudio.com/files/hardware/springs/extension-d0.6-od6-l30.step) · [STL](https://xemstudio.com/files/hardware/springs/extension-d0.6-od6-l30.stl) — extension spring envelope: wire 0.6, OD 6, 30 mm, 20 coils
- extension-d1-od10-l50: [STEP](https://xemstudio.com/files/hardware/springs/extension-d1-od10-l50.step) · [STL](https://xemstudio.com/files/hardware/springs/extension-d1-od10-l50.stl) — extension spring envelope: wire 1.0, OD 10, 50 mm, 25 coils
- torsion-d0.8-od8-l10: [STEP](https://xemstudio.com/files/hardware/springs/torsion-d0.8-od8-l10.step) · [STL](https://xemstudio.com/files/hardware/springs/torsion-d0.8-od8-l10.stl) — torsion spring envelope: wire 0.8, OD 8, 10 mm, 6 coils

## Questions

**How do I calculate a compression spring rate?**

k = G·d⁴ / (8·D³·n): shear modulus 79.3 GPa for music wire, d wire diameter, D mean coil diameter (OD − d), n active coils. 1 mm wire, 10 mm OD, 10 coils → 1.36 N/mm.

**What is the solid height of a compression spring?**

About (total coils) × wire diameter: a 10-coil, 1 mm wire spring with closed ends is 12 mm solid; usable travel is about 80% of free length minus solid height.

**Can I 3D print springs?**

Only in TPU for light duty (pawl returns, detents, compliant mechanisms); rigid filaments fatigue in a few cycles. Use the printed envelope to check the pocket, then buy the steel spring.

## Fits with

- [Ratchet wheels and pawl (anti-reverse / one-way drive)](https://xemstudio.com/objects/ratchet-and-pawl/)
- [Clevis pins, R-clips, cotter pins, clevises, universal joint, rod ends (heim joints)](https://xemstudio.com/objects/pins-clevises-and-joints/)
- [Balloons, gloves, inner tubes and silicone sheet as diaphragms](https://xemstudio.com/objects/balloons-and-diaphragms/)
- [Shafts, D-flats, set-screw collars, E-clips (DIN 6799) and circlips (DIN 471)](https://xemstudio.com/objects/shaft-collars-clips-and-shafts/)