SizingKitMechanical · Helical compression springs
Spring rate, and the fourth power on the wire
Wire diameter, coil diameter, active coils and the wire’s shear modulus give the rate of a helical compression spring in pounds per inch and newtons per millimeter, with the solid height, the travel left before the coils touch, the Wahl-corrected stress in the wire and a buckling check alongside it. Underneath the answer is a generated table of five wire sizes around yours, because rate goes with the fourth power of wire diameter and that is the one decision worth more than all the others together. Free, no signup, and nine ASTM spring wires with their moduli.
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- Wahl-corrected stress
- Index and buckling
The spring as it is wound
Four measurements and an end treatment. The end treatment is the one people leave out, and it decides how many of the coils you can see are actually doing any work.
A unit written after a number wins, so wire in millimeters and a free length in inches can go in together.
Calipers give the outside; the formula wants the mean, which is one wire diameter less. Putting the outside diameter straight into the equation makes the spring come out far too soft.
The term raised to the fourth power. Measure it on the straight part at the end of the spring, not across a coil.
Mean coil 0.67 in, spring index C = 8.38.
8 of them are active — 2 come off for the ends.
The commonest for a spring that must sit square under load. Solid height = total coils × wire diameter.
G = 11,500,000 psi (79.3 GPa). The default spring wire under about 1/8 in: high strength, tight tolerance, cheap. Not for use above about 250 °F.
Uncompressed length. It has no part in the rate at all, and it decides both the travel available and whether the spring buckles.
How far it is squashed in service. Load is the rate times this, and the wire stress follows the load.
Deliberately empty. The allowable depends on the wire diameter as well as the material and this site carries no table for it — take it from the wire specification for the exact size you are winding.
Separate from the end treatment above: that is how the wire was finished, this is what the spring sits against in the machine.
SPRING RATE
24.47 lbf/in
4.286 N/mm
LOAD AT WORKING DEFLECTION
24.47 lbf
109 N at 1 in
CORRECTED SHEAR STRESS
95.8 ksi
661 MPa, Wahl 1.175
| Mean coil diameter | 0.67 in 17.02 mm |
|---|---|
| Spring index C | 8.38 manufacturable band 4 to 12 |
| Active coils | 8 10 total, 2 inactive |
| Solid height | 0.8 in 20.32 mm |
| Travel to solid | 1.7 in 43.18 mm |
| Load at solid height | 41.6 lbf 185 N |
| Wahl correction factor | 1.175 adds 17.5% to the uncorrected stress |
| Slenderness, free length ÷ mean | 3.73 limit 5.26 at α = 0.5 |
RATE AGAINST WIRE SIZE, OUTSIDE DIAMETER HELD
| Wire | Index C | Rate | Change |
|---|---|---|---|
| 0.064 in | 10.72 | 9.34 lbf/in | -62% |
| 0.072 in | 9.42 | 15.49 lbf/in | -37% |
| 0.08 in | 8.38 | 24.47 lbf/in | — |
| 0.088 in | 7.52 | 37.14 lbf/in | +52% |
| 0.096 in | 6.81 | 54.56 lbf/in | +123% |
The table above is the whole argument for reading k = G·d⁴/(8·D³·n) carefully. Twenty percent more wire at the same outside diameter takes this spring from 24.47 lbf/in to 54.56 lbf/in, and twenty percent less takes it down to 9.34 lbf/in — a factor of 5.8 across a wire range you could hold in one hand. Part of that is the fourth power alone, which makes a 10% thicker wire 46% stiffer; the rest is the mean coil shrinking as the wire grows inside a held outside diameter, since the coil term is cubed and works the other way.
The stress figure carries the Wahl correction, which is why it is larger than the plain torsion result: the wire is curved, so the shear from the direct load adds to the torsional shear on the inside of the coil rather than subtracting. At C = 4 that correction is 40% and at C = 12 it is 12%, which is most of why a low index is avoided even when the coiler can manage it. There is no allowable stress on this page to compare it against, and that is deliberate: the allowable is wire-diameter-dependent as well as material-dependent, it varies far more between spring materials than the modulus does, and it differs again between static and fatigue service. Take it from the specification for the exact wire size.
Room temperature values. Shear modulus falls as temperature rises — a steel spring loses a few percent of its rate by 300 °F and considerably more beyond — so a spring that must hold a load hot has to be designed for it. The value also drifts slightly with the heavy cold work of coiling small-index springs. Allowable stress, which is what actually decides whether a spring survives, varies far more between these materials than modulus does and is wire-diameter-dependent; it is not tabulated here.
Two other things on this site are springs in disguise, and both are worth the comparison. A tightened bolt is a very stiff one — preload is the tension in it and the joint it clamps is a second spring in parallel. And the column of air behind an air cylinder’s piston has a rate in the same tens of pounds per inch as a modest steel spring, which is why a pneumatic clamp gives under a rising load. Nothing here is a stamped design: fatigue life, set, surge frequency and the allowable stress that decides all three belong to the spring maker and to a qualified engineer.
How to work out a compression spring's rate from the one in your hand
Four measurements, and one of them is not the number the equation wants.
Measure the wire and the coil, and say which coil diameter you measured
Calipers across a spring give the outside diameter; the rate equation wants the mean, which is center to center of the wire and one full wire diameter less. The selector above the field handles the conversion so the number you actually measured is the number you type. Take the wire diameter on the straightened tail at the end rather than across a coil, where the caliper jaws sit on two wires at a slight angle and read high.
Count every coil, then say how the ends are finished
Total coils is what you can see, half coils included. The end treatment then decides how many of them deflect: squared ends, ground or not, take two coils out of the count entirely, while plain ends take none. This is the commonest reason a calculated rate misses a measured one, and it is not a small effect — on a ten-coil spring, forgetting the two dead coils makes the calculated rate 20% too soft, and on a five-coil spring it is 40%.
Give it a free length and a working deflection, then read the wire table
Free length has no part in the rate at all, and it settles two other things: how much travel there is before the coils touch, and whether the spring is slender enough to buckle instead of compressing. Working deflection turns the rate into a load and the load into a corrected wire stress. Below all of that is the table that answers the question people actually arrive with — five wire diameters around yours, at the same outside diameter, with the rate each one gives.
Technical specifications
| Rate equation | k = G·d⁴/(8·D³·n) with d the wire diameter, D the mean coil diameter, n the active coils and G the shear modulus — a spring is a torsion bar wound into a helix, so the material term is shear and not Young's modulus |
|---|---|
| Wire materials | Nine shear moduli against their ASTM specifications — A228 music wire and A227 hard drawn at 11,500,000 psi, A229, A231 and A401 at 11,200,000, A313 stainless at 10,000,000, 17-7 PH at 11,000,000, B159 phosphor bronze at 6,000,000 and B197 beryllium copper at 7,000,000 |
| End treatments | Four, each with its own effect on both counts: closed and ground and closed-not-ground remove two coils from the active count, plain ends remove none, plain-ground removes one. Solid height adds a wire diameter for the two unground types |
| Stress correction | Wahl's factor, (4C − 1)/(4C − 4) + 0.615/C, applied to the torsional shear at the inside of the coil: 1.40 at an index of 4, 1.18 at 8, 1.12 at 12. The curvature of the wire is what makes the inside of the coil the highest-stressed point |
| Spring index | Reported and checked against 4 to 12 as the manufacturable band, with 6 to 9 the comfortable middle where most catalog compression springs sit. Outside it the page says so rather than refusing, because the arithmetic is still right and the coiling is the problem |
| Buckling check | Free length under 2.63·D/α, the absolute-stability criterion for steel: a slenderness limit of 5.26 with both ends squared and ground on flat parallel plates, 3.72 with one end pivoted, 2.63 with both pivoted and 1.32 with one end free |
| Worked example | 0.080 in music wire, 0.750 in outside diameter, 10 total coils closed and ground, 2.5 in free length: mean coil 0.670 in, index 8.38, 8 active coils, 24.47 lbf/in or 4.285 N/mm, solid at 0.800 in with 1.700 in of travel, and 95.8 ksi of corrected shear at 1 in of deflection |
| What the fourth power is worth | That same spring at 0.064 in wire is 9.34 lbf/in and at 0.096 in it is 54.56 lbf/in — a factor of 5.8 across a ±20% change in wire diameter, because the coil term is shrinking at the same time as the wire term grows |
Frequently asked questions
My spring measures softer than this says. What did I get wrong?
Almost always the active coil count. The end coils of a squared spring lie flat against the seat and do not twist, so they contribute nothing to the deflection, and the standard convention is to subtract two of them — but the transition from a dead end coil to a live body coil is gradual rather than a step, so the real count sits somewhere near the convention rather than on it. On a short spring where two coils are a third of the total, that approximation alone moves the rate by several percent. After that, check that you used the mean coil diameter rather than the outside, and that the wire is the size you think: a caliper reading 0.079 in on 0.080 in wire is a 5% error in rate all by itself.
Does swapping music wire for stainless change the rate?
Yes, by 13%, and it is the one material substitution that catches people out. Rate depends on the shear modulus, and among the steel spring wires that barely moves — music wire, hard drawn and oil tempered all sit between 11.2 and 11.5 million psi, so changing from one to another changes what the spring survives rather than what it feels like. Type 302 and 304 stainless is 10 million psi, which is 13% lower, so the identical geometry wound in stainless is 13% softer. Phosphor bronze at 6 million psi is nearly half, and a bronze spring is chosen for conductivity and corrosion resistance rather than for anything mechanical.
Can I cut coils off a spring to make it stiffer?
It works arithmetically and it ruins the ends. Rate is inversely proportional to active coils, so removing two of eight raises it by a third — but a cut leaves a sharp, unsquared, unground end that sits crooked in its seat, digs into whatever it bears on, and concentrates stress exactly where the spring is now shortest. It also loses free length faster than it gains rate, so the travel available drops. Where the spring is a cheap catalog part, ordering the right one is nearly always less trouble; where it is not, cutting it and having the end squared and ground is the honest version of the job.
What is the spring index for, if the rate equation does not use it?
It is the manufacturability check and the stress correction, and the rate equation is exactly why it has to be looked at separately. Index is the mean coil diameter divided by the wire diameter: below about 4 the coiler cannot bend the wire that tightly without cracking it, and above about 12 the spring is floppy enough to tangle in a bin, to wander off center under load and to defy any rate tolerance. It also sets the Wahl correction, which is 1.40 at an index of 4 and 1.12 at 12 — so a tightly wound spring carries a 40% stress penalty on the inside of the coil for geometry alone, before it has been loaded any harder than its lazy cousin.
What happens when a compression spring goes solid?
It stops being a spring and becomes a stack of steel washers, and whatever was pushing on it now applies its full force to the assembly through a rigid column. That is not automatically damaging — many designs deliberately allow a spring to close, and a spring that has been stress-relieved and pre-set will take it — but nothing about the load is controlled any more, and any part sized on the spring force is now sized on nothing. The travel available is the free length minus the solid height, and the solid height depends on the ends: closed and ground is total coils times wire diameter, and the unground versions add one more wire diameter on top.
How do I pick a wire size to hit a target rate?
Change the wire before you change anything else, because rate goes with the fourth power of it and with only the first power of the coil count. Ten percent thicker wire is 46% stiffer at a fixed mean coil diameter, and rather more than that at a fixed outside diameter, because the mean coil shrinks as the wire grows and that term is cubed working the other way. The table under the result on this page prints five sizes around yours for exactly this, with the spring index of each so you can see when a heavier wire has taken the design below index 4 and out of the manufacturable range.
Why does the page not tell me whether my spring is overstressed?
Because the allowable stress it would have to compare against is not a number this site can honestly supply. It depends on the wire diameter as well as the material — thin wire is drawn harder and is stronger — it varies far more between spring materials than the modulus does, and it differs again between static service and fatigue service, where the governing figure is a stress range rather than a peak. There is a box to enter your own figure from the wire specification, and when it is filled the page reports the ratio. Left empty, it reports the stress and stops, which is the honest end of what a table of moduli can tell you.
About k = G·d⁴/(8·D³·n), term by term
A helical compression spring is a torsion bar wound into a helix. Pushing on the ends does not bend the wire in any useful sense — it twists it — which is why the material property in the rate equation is the shear modulus rather than Young’s modulus, and why the stress that eventually breaks a spring is a shear stress at the inside surface of the coil. Once that is clear the equation reads straightforwardly: stiffness rises with the wire’s resistance to twist, falls as the lever arm of the coil grows, and falls as more coils are added in series. What is not obvious is how unequal the four terms are. The shear modulus barely moves across the steel spring wires, spanning 11.2 to 11.5 million psi, so choosing chrome silicon over music wire changes what the spring endures and leaves what it feels like almost untouched. Active coils are linear. The coil diameter is cubed. And the wire diameter is raised to the fourth power, so a wire one tenth thicker is a spring 46% stiffer.
The two inputs that go wrong in practice are both counting problems rather than measurement problems. The first is the coil diameter: a caliper gives the outside, and the equation wants the mean, which is one whole wire diameter smaller. On the worked example above that is the difference between 0.750 and 0.670 in, and since the term is cubed, using the outside figure makes the spring come out 29% softer than it is. The second is the active coil count. Squared end coils lie flat against their seats and do not twist, so convention subtracts two of them from the total — and the convention is an approximation, because the transition from a dead end coil to a live body coil is gradual. Manufacturers’ own figures for the same end style differ by a fraction of a coil, which is why a short spring is harder to predict than a long one: on five total coils that fraction is several percent of the rate, and on twenty it is noise.
What this page will not do is tell you the spring is safe. It reports the corrected shear stress, with Wahl’s factor applied because the curvature of the wire piles the direct shear on top of the torsional shear at the inside of the coil, and it reports the spring index that sets both that correction and whether the thing can be coiled at all. The allowable stress to compare against is genuinely not available as a table: it depends on wire diameter as much as on material, because thin wire is drawn harder and is stronger, and it changes again between static and fatigue service. So there is a field to enter the figure from your own wire specification, and when it is empty the page stops where the data stops. That is the same reason a preloaded bolt, which is a very stiff spring in series with the joint it clamps, is designed against a proof stress the fastener standard actually publishes rather than against a rule of thumb. Fatigue life, set, surge frequency and the allowable that decides all three belong to the spring maker and to a qualified engineer; this page is the arithmetic that tells you which questions to put to them.
What happens to the spring you measure
Every number on this page is worked out by JavaScript running in the tab you are reading it in. Nothing you type — loads, lengths, nameplate ratings, the rates your utility charges you — is uploaded, logged or kept, which is also why the calculators carry on working in a mechanical room with no signal.
Wire sizes, coil counts, materials and the allowable stress you take off your own wire specification stay in this tab and are gone when it closes — nothing is saved between visits, so a design worked out here leaves no record anywhere but your notebook.