What spring rate means
Spring rate (k) is the force required to compress a spring by one millimetre, expressed in N/mm. A spring with k = 12 N/mm produces 120 N when compressed 10 mm.
Compression springs behave linearly through their working range: compress twice as far, get twice the force. That linearity breaks down as the spring approaches solid height, and it never applies at all to conical, barrel or variable-pitch geometries — those cannot be described by a single value of k.
The formula
For a cylindrical compression spring:
For spring steel G is typically taken as 79,500 N/mm²; for stainless, around 70,000 N/mm².
D is not the outside diameter. It is the mean coil diameter: outside diameter minus wire diameter. This is the single most common error in practice, and because D enters the formula cubed, it distorts the result badly.
Worked example
Take a spring with 4 mm wire, 34 mm outside diameter, 10 total coils, ends closed and ground.
1 · Find the mean coil diameter — D = 34 − 4 = 30 mm
2 · Find the active coils — With closed and ground ends, the two end coils carry no load: n = 10 − 2 = 8
3 · Substitute
Compressed 20 mm, this spring produces roughly 236 N.
Our calculator runs the same formula if you want to try your own dimensions.
Talk to our engineersWhat moves the result
The exponents are not equal. Wire diameter enters to the fourth power and coil diameter to the third, so a small dimensional shift becomes a large force shift. Using the example above:
| Change | New value | Effect on k |
|---|---|---|
| Wire diameter up 5% | d: 4.00 → 4.20 mm | k rises 21.6% |
| Coil diameter up 10% | D: 30 → 33 mm | k falls 24.9% |
| One coil added | n: 8 → 9 | k falls 11.1% |
| Switch to stainless | G: 79,500 → 70,000 | k falls 11.9% |
The finest way to tune spring rate is not wire diameter but coil count. Moving the wire by 0.1 mm shifts the force by roughly 10%; adding a coil is a more controlled intervention and can usually be done with existing tooling.
Five common mistakes
- 1Using outside diameter as D. In the example above, entering 34 instead of 30 understates k by 31%.
- 2Treating total coils as active coils. Closed and ground ends remove two coils; closed ends remove roughly two as well.
- 3Using E instead of G. Compression and torsion bar springs use the shear modulus; the modulus of elasticity belongs to torsion spring moment calculations. The two differ by a factor of about 2.6.
- 4Ignoring buckling. Springs with a free length to mean diameter ratio above roughly 4 will move laterally even when the calculation is correct. Check this ratio wherever the spring is unguided.
- 5Forgetting solid height. Linearity ends as the spring approaches solid. The working stroke should finish at least 10–15% above solid height.
When the maths is right but the spring isn't
The formula gives you geometry, not performance. Two springs with identical k values will behave completely differently in service if their material, heat treatment or surface treatment differ. When a spring holds its force but fails early, the problem is almost never in the calculation — look at surface quality, shot peening and heat treatment instead.
- k = G·d⁴ / (8·D³·n), valid for cylindrical compression springs.
- D is the mean coil diameter — outside diameter minus wire diameter.
- Active coils are total coils minus the end coils.
- Wire diameter is the most sensitive variable; fine tuning is done with coil count.
- A correct calculation does not guarantee service life — that is a materials and surface question.
Editorial note — This article is general information. Formulas, standards and values given here describe established engineering practice and are not a design specification. Any final spring design must be verified against the load case, operating conditions and applicable standards of the individual application. Tenvor Springs accepts no liability for designs derived from this text without such verification.