What spring rate is and what sets it
Spring rate is the force needed to compress a spring one unit of distance, and for a helical compression spring it is a constant: the load-deflection line is straight until the coils touch. A 300 lb/in spring takes 300 lb to compress the first inch, another 300 lb for the second, and so on. That linearity is why you can characterise the whole component with a single number.
The rate comes from treating each coil as a short torsion bar. The axial force acts at a radius of D/2, so it twists the wire, and the wire's resistance to twist is its torsional modulus G times its polar second moment of area, which goes as d4. Stack up the twist over the developed length of wire — proportional to D times N — and you arrive at k = Gd4/(8D3N).
The exponents are the practical message. Wire diameter enters to the fourth power, so a 5% thicker bar is a 21.6% stiffer spring — 1.054 = 1.2155. Mean coil diameter enters cubed and inverted, so a 5% larger coil is 13.6% softer, since 1.05−3 = 0.8638. Active coils enter linearly and inverted, so removing one coil from a ten-coil spring raises the rate by 11.1%: 10/9 = 1.111. Material barely moves at all across the steels, because every spring steel has a shear modulus within a few percent of 11.3 Mpsi.
Rate is not the same as load capacity. A spring's rate says nothing about how far it can travel before it takes a set or breaks — that is governed by wire stress, and stress depends on the same geometry in a different combination. Two springs can share a rate and have wildly different free lengths, travels and fatigue lives.
Measuring the four inputs correctly
Wire diameter is the easy one, but measure it with a micrometer on a straight section of a coil, not on a bend, and take two readings ninety degrees apart. Coiling can leave the section slightly oval, and because d is raised to the fourth power a 0.002 in reading error on a 0.437 in wire is a 1.8% rate error.
Mean coil diameter is the centre-to-centre diameter, which is not what a spring is usually sold by. A "2.5 inch coilover spring" is described by its inside diameter, because that is what has to clear the shock body. Mean diameter is ID + d: a 2.5 in ID spring wound from 0.4375 in wire has a mean diameter of 2.9375 in. Getting this wrong by one wire diameter is the single most common error on this calculation, and it is worth about a third of the rate.
Active coils are the coils free to deflect. Most compression springs have squared-and-ground ends: the last coil at each end is wound flat and ground so the spring sits square, and those two coils carry load without twisting appreciably. Count the total coils and subtract two. For squared-but-not-ground ends the convention is also to subtract two; for plain open ends, subtract nothing. Count carefully — half coils matter, and a spring with 8.5 total coils has 6.5 active.
Shear modulus G is the torsional modulus of rigidity, not Young's modulus. Music wire and hard-drawn carbon steel are taken at 11.5 Mpsi; oil-tempered, chrome silicon and chrome vanadium wires at 11.2 Mpsi; 302 stainless at 10.0 Mpsi; phosphor bronze at 6.0 Mpsi. Substituting stainless for carbon steel in an otherwise identical spring drops the rate by 13%, which is 10.0/11.5 − 1 = −0.1304.
Once you have the rate, deflection under load is straight Hooke's law, x = F/k — the same relationship the Hooke's law spring calculator works in the general case.
Worked example: a 2.5 in ID coilover spring
You have an unmarked coilover spring off a race car. It fits a 2.5 in ID perch, the wire micrometers at 0.4375 in, and you count 8.5 total coils with squared-and-ground ends, so 6.5 are active. It is a chrome silicon spring — but assume music wire at G = 11,500,000 psi and check the sensitivity afterwards.
- Mean coil diameter. D = ID + d = 2.5 + 0.4375 = 2.9375 in.
- Wire to the fourth. 0.43752 = 0.19140625, and squaring again gives d4 = 0.03663635 in4.
- Coil diameter cubed. 2.93752 = 8.62890625, times 2.9375 gives D3 = 25.347412 in3.
- Numerator. 11,500,000 × 0.03663635 = 421,318.05.
- Denominator. 8 × 25.347412 × 6.5 = 1,318.065.
- Rate. 421,318.05 ÷ 1,318.065 = 319.65 lb/in, which is 319.65 × 0.175127 = 55.98 N/mm.
- Spring index. C = 2.9375 ÷ 0.4375 = 6.71, comfortably inside the 4–12 coilable band.
- Deflection at 400 lb. 400 ÷ 319.65 = 1.251 in.
Sensitivity check: at chrome silicon's 11.2 Mpsi the rate becomes 319.65 × 11.2/11.5 = 311.3 lb/in — a 2.6% difference, smaller than the error you would introduce by mismeasuring the wire by 0.003 in. A catalogue spring would be sold as a 300 lb/in part, and the gap between 319.65 and the marked 300 is normal: manufacturers adjust free length and end grinding to hit a nominal rate, and rate tolerance on production springs is typically a few percent.
Reading the result: index, stacking and what a rate does not tell you
Check the spring index first. C = D/d below 4 means the wire is being bent around a very tight radius; the inside fibre sees a large stress concentration and the coiler needs special tooling. Above 12 the spring is slender enough to tangle in a bin and to buckle in service unless it runs over a rod or inside a tube. Between 5 and 9 is where standard production sits, and a result outside 4–12 usually means you mismeasured something.
Two springs in series always rate softer than either one. They see the same load and their deflections add, so 1/k = 1/k1 + 1/k2, and the result k1k2/(k1+k2) is smaller than the smaller of the two for any pair of positive rates. That is how a tender or helper spring works: a soft short spring stacked under a stiff main spring gives a low initial rate at droop and then goes solid, handing the full stiff rate to the wheel for the rest of the travel. Once the tender is coil-bound it stops contributing and the rate steps up to the main spring's value.
Two springs in parallel always rate stiffer than either one, because they share the load and deflect together, so the rates add. That is a nested-spring arrangement, or the pair of springs on one axle acting together on the chassis.
Rate at the spring is not rate at the wheel. Suspension geometry divides both force and travel, and the wheel rate falls with the square of the motion ratio, so a 320 lb/in spring on a 0.65 motion ratio delivers 320 × 0.652 = 135 lb/in at the tire. The wheel rate and motion ratio calculator handles that conversion, and the weight transfer calculator shows how much load the wheel actually has to absorb.
Rate of a 2.5 in ID coilover spring by wire diameter
| Wire diameter (in) | Mean coil dia (in) | Spring index C | Rate (lb/in) | Rate (N/mm) |
|---|---|---|---|---|
| 0.375 | 2.875 | 7.67 | 184.0 | 32.23 |
| 0.400 | 2.900 | 7.25 | 232.1 | 40.65 |
| 0.4375 | 2.9375 | 6.71 | 319.6 | 55.98 |
| 0.450 | 2.950 | 6.56 | 353.3 | 61.86 |
| 0.475 | 2.975 | 6.26 | 427.6 | 74.88 |
| 0.500 | 3.000 | 6.00 | 511.9 | 89.65 |
| 0.525 | 3.025 | 5.76 | 607.0 | 106.29 |
| 0.550 | 3.050 | 5.55 | 713.3 | 124.91 |
Catalogue springs hit round rates by varying active coil count as well as wire size, so a real 300 lb/in spring in this ID may use any of the top three wire sizes with a matching coil count.
This formula gives rate, not stress or travel
Nothing here tells you whether the spring survives the load you are applying. Torsional stress in the wire is τ = 8FD/(πd3) multiplied by a Wahl curvature factor that depends on the spring index, and it must stay below the material's allowable — typically around 45% of tensile strength for static service and well under that for fatigue. Nor does the rate tell you the usable travel: a spring goes solid at a height of roughly d × (total coils), and running a spring to coil bind under load will take a permanent set or break it. If you are designing a spring rather than identifying one, size the stress and the solid height as well, using a spring design manual or the manufacturer's data.
Where this calculation goes wrong
- Using inside or outside diameter as the mean. Mean is ID + d, or OD − d. On a 2.5 in ID spring with 0.4375 in wire, using 2.5 instead of 2.9375 overstates the rate by (2.9375/2.5)3 − 1 = 62%.
- Counting total coils as active coils. On a squared-and-ground spring, two coils are dead. Using 8.5 instead of 6.5 understates the rate by 6.5/8.5 − 1 = −24%.
- Confusing G with E. Young's modulus for steel is about 29–30 Mpsi; the shear modulus is about 11.5. Substituting E would nearly triple the answer.
- Applying it to a progressive spring. A variable-pitch or conical spring has no single rate. Coils bind progressively, active coil count falls as it compresses, and the rate rises through the stroke. This formula describes only the instantaneous rate of the coils still active.
- Ignoring end-coil effects on short springs. Below about three active coils the ends contribute meaningfully to deflection and the calculated rate runs high. Measure such springs on a tester.
- Forgetting the spring is only part of the ride rate. Bushing compliance, tire vertical stiffness and any bump stop in contact all sit in series or parallel with the spring at the wheel, which is why a car never rides at the rate you calculated here.
Measuring a rate instead of calculating it
Calculation is a check; measurement is the truth. To measure a rate, compress the spring on a press or a spring tester between flat parallel plates, record two load-deflection pairs well away from both ends of the travel — say at 20% and 60% of usable stroke — and divide the load difference by the deflection difference. Avoid the first fraction of an inch, where the ground ends are still seating, and stop short of coil bind, where the rate runs away.
Discrepancy between calculation and measurement is normal and informative. Measured rates on production springs typically fall a little below the formula, because the end coils are never perfectly dead and the helix angle is never zero. A measured rate far above the calculation usually means you counted too many active coils; far below usually means you used the ID as the mean diameter or the spring is partially coil-bound.
When you are picking a rate rather than checking one, work backwards from the wheel. Decide the ride frequency or the wheel rate you want, divide by the square of the motion ratio to get the spring rate, and only then choose the wire and coil geometry that produces it inside the space available. That order keeps you from designing a beautiful spring that does the wrong thing at the tire. From there, the tongue weight calculator and the towing capacity and payload calculator matter if the vehicle carries load, since static corner weights set where in its travel the spring is operating before you add any dynamic transfer.
Key terms
- Active coils
- Coils free to twist under load. On a squared-and-ground compression spring this is total coils minus two, the two ground end coils being effectively rigid.
- Spring index (C)
- Mean coil diameter divided by wire diameter. A shape measure that governs coilability and the stress concentration on the inner fibre of the wire.
- Shear modulus (G)
- Torsional modulus of rigidity — resistance to twisting. About 11.5 Mpsi for music wire, 11.2 for oil-tempered and chrome silicon alloys, 10.0 for 302 stainless.
- Coil bind
- The point at which all coils touch and the spring can compress no further. Solid height is approximately wire diameter times total coil count.
- Tender or helper spring
- A short soft spring stacked in series with the main spring to keep it captive at full droop. It goes solid early in the stroke, after which the assembly rates at the main spring alone.