Key takeaways
- Rate equals shear modulus times wire diameter to the fourth power, divided by eight times mean diameter cubed times active coils.
- Wire diameter is the most sensitive term, so the wire tolerance class belongs in the calculation.
- The formula assumes a straight, uniform helix and a small deflection; it is a design estimate, not a substitute for a load test.
- Count active coils from the end type, not from the total coil count on the drawing.
The formula
For a compression spring made from round wire, the rate is calculated as k equals G times d to the fourth power, divided by eight times D cubed times n. In the formula, k is the rate in newtons per millimetre, G is the shear modulus of the material, d is the wire diameter, D is the mean coil diameter and n is the number of active coils.
Use consistent units, usually newtons and millimetres, with G in N/mm2. Music wire has a shear modulus of roughly 79,000 to 81,000 N/mm2. Stainless 302 is close to 69,000 N/mm2. Use the value for the actual material and condition, not a generic steel figure.
The formula is an elastic calculation. It assumes a uniform helix, a circular wire section, a small deflection and no coil contact. Those assumptions are reasonable for many springs and poor for some. Knowing where they break is as important as being able to enter the numbers.
A worked example
Take an illustrative spring in music wire with a wire diameter of 1.2 mm, a mean coil diameter of 9.6 mm and 8 active coils. Using G equals 79,300 N/mm2, the wire term d to the fourth is 2.0736 mm4 and the mean diameter term D cubed is 884.74 mm3.
The numerator is 79,300 times 2.0736, which is 164,436. The denominator is 8 times 884.74 times 8, which is 56,623. Dividing gives a rate of about 2.90 N/mm. At a deflection of 5 mm, the load is about 14.5 N.
The spring index for this example is 9.6 divided by 1.2, which is 8. That is a comfortable index, so the basic formula is likely to be close to a measured result. A spring with an index of 3.5 would need a curvature correction.
These numbers are illustrative. The material modulus, the wire tolerance and the end coils all move the real result.
| Term | Symbol | Value | Effect of a 10 per cent increase |
|---|---|---|---|
| Wire diameter | d | 1.20 mm | Rate rises by about 46 per cent |
| Mean coil diameter | D | 9.60 mm | Rate falls by about 25 per cent |
| Active coils | n | 8 | Rate falls by about 9 per cent |
| Shear modulus | G | 79,300 N/mm2 | Rate rises by about 10 per cent |
| Rate | k | About 2.90 N/mm | Result of the terms above |
Illustrative calculation only. Use the modulus for the actual material and condition, and confirm the rate with a load test.
What each term does
The exponents in the formula explain why spring design is sensitive to small changes. Wire diameter enters to the fourth power, so a 10 per cent increase in wire size raises the rate by about 46 per cent. Mean coil diameter enters cubed in the denominator, so a 10 per cent increase lowers the rate by about 25 per cent. Active coils enter linearly, so a 10 per cent increase lowers the rate by about 9 per cent.
This is why a wire tolerance class matters. A nominal 1.20 mm wire may be supplied within a tolerance band that is a few hundredths of a millimetre wide. That band is small in absolute terms and significant in rate terms. A spring that has to meet a tight load band needs a wire tolerance that supports it, or a 100 per cent load check.
The modulus term is often treated as a constant. It is not. It changes with temperature, so use the modulus at the service temperature for a hot or cold spring.
Where the formula stops matching test data
The first limit is the spring index. The simple formula treats the wire as a straight bar in torsion. In a tightly coiled spring, the inside of the coil carries higher stress than the outside, and the curvature correction becomes significant. Below an index of about 4, the rate estimate may still be usable, but the stress estimate needs the correction.
The second limit is deflection. The formula assumes the coils remain separate and the helix angle stays small. A spring compressed close to solid height behaves differently, because the coils interact and the load rises sharply. A spring with a large pitch angle also behaves differently from the ideal helix.
The third limit is the end condition. A closed and ground end has dead coils that do not deflect. If the calculation uses the total coil count instead of the active count, the rate will be too low. The error can be large on a short spring with few coils.
The fourth limit is the material condition. Residual stress from coiling, a set operation or a stress relief changes the load at a given length. The formula gives the elastic rate of the finished geometry, not the load history of the part.
Counting active coils
Active coils are the coils that are free to deflect. For a closed and ground end, the two end coils are effectively dead, so the active count is the total count minus two. For an open end, the end coils contribute partly, and the count depends on how the spring seats. For a plain end, the seating is less controlled and the effective count is harder to state.
The drawing should give the total coils and the end type, and the calculation should derive the active count from them. If the drawing gives only the active count, the supplier still needs the total count to make the part. Give both, and say which one the load call is based on.
A spring with fewer than three active coils is difficult to calculate and to test. The end effects dominate, and the load at length becomes sensitive to seating. A disc spring may be a better answer than a helix with two active coils.
How to check a calculated rate
The most direct check is a load test at two lengths. Measure the load at each, calculate the slope, and compare it with the predicted rate. A single load point does not separate a rate error from a free length error.
Ask for the test fixture to state the lengths, the load cell, the platen arrangement and the speed. A spring loaded quickly can read differently from one loaded slowly, especially if the coils are seating. Let the spring settle before taking the reading.
If the measured rate is outside the band, check the wire diameter and the active coil count before changing the design. Record the actual wire size with the test result so the next batch has a baseline.
- Measure the wire diameter on the sample, not from the certificate alone
- Count the active coils on the sample and compare with the drawing
- Test at two lengths so the rate can be separated from the free length
- State the test speed and whether the spring was settled first
- Record the load at the installed length and at the working length
- Keep the result with the material certificate and the first article
Frequently asked questions
What is the formula for spring rate?
For a compression spring, rate equals G d to the fourth divided by 8 D cubed n, where G is the shear modulus, d is the wire diameter, D is the mean coil diameter and n is the number of active coils. Use consistent units and the modulus for the actual material.
Why does my calculated rate differ from the test result?
The usual causes are the wire diameter being outside the nominal, the active coil count being different from the assumption, the mean diameter being measured incorrectly, or the spring not being settled before the test. Check those before changing the formula.
Does the rate formula work for extension springs?
The same elastic form applies, but the initial tension in an extension spring changes the load at the start of the travel. The rate is the slope after the coils separate. State the initial tension separately and test it as well as the rate.
How accurate is the spring rate formula?
For a spring with a comfortable index, a uniform helix and a small deflection, the estimate is usually close enough for design. It is not a substitute for a load test, especially near solid height, at a low index, or after a set.
Sources and further reading
- EN 13906-1, Cylindrical helical springs made from round wire and bar, compression springs
- EN 10270-1, Steel wire for mechanical springs, patented cold drawn unalloyed spring steel wire
- Qumek spring rate worksheets and load test records
Qumek Engineering Desk
Engineering team
The engineering team behind Qumek drawings, DFM notes, tolerances and inspection standards. Every guide is reviewed against the drawings and processes used on live programmes.
Check a calculated spring rate against a test
Send the wire size, the mean diameter, the active coils, the material and the target load. The engineering desk will check the arithmetic and the assumptions behind it.
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