Compression Spring Rate and Solid Height Calculator

k = Gd⁴/(8D³n_a) with D the MEAN coil diameter, the four end types driving the active coil count and the solid height, and the published 20-to-80 per cent working window computed — so the page ends with a length you can put on a drawing.

Last updated 24 Sep 2026

Compression spring rate and solid height

Wire, OD, coils, end type → rate and solid height

Wire diameter d (mm)The single most powerful number on the page. The rate goes as d to the FOURTH power, so ten per cent more wire is forty-six per cent more rate — and the stress goes as 1/d³, so the same change takes a quarter off the stress. Nothing else here has that leverage.

Outside diameter OD (mm)The diameter you can measure with a caliper, and the one a bore has to clear. The rate formula does NOT use it: it uses the MEAN diameter D = OD − d, which this page computes for you. Putting OD into the formula where D belongs understates the rate by a third at an index of 7, and it is the commonest mistake in spring arithmetic.

TOTAL coils N_t (including the ends)Count every coil you can see, dead ends included. The rate uses the ACTIVE coils, which is this number minus the end coils, and how many that is depends entirely on the end type below.

End typeThis one dropdown changes the active coils, the solid height, the free length and the pitch all at once. At ten total coils, plain ends give 10 active coils and a solid height of 11d; squared and ground give 8 active and 10d. That is a 25 per cent difference in rate from a choice that looks cosmetic.

Wire materialThe rate is proportional to G, and G is not the same for every spring wire. 302 stainless is 69 GPa against carbon steel’s 79.3, so the same spring in stainless is 13 per cent softer; phosphor bronze is 41.4 GPa and is nearly half as stiff.

Free length L₀ (mm)The length with no load on it. If it is less than the solid height the spring does not exist, so this page floors it just above solid.

Working length you want (mm)The length the spring sits at when the machine is doing its job. The page gives the force there, the stress there and how much of the available travel it has used.

Not a circuit: the length budget of your own spring, drawn as a scale normalised to the free length so one figure serves every spring. The spring on the left is at its free length; the base of the scale is zero length and the top is L₀. Three markers move as you change the inputs: the SOLID HEIGHT, which is where the coils bind; the 80 PER CENT LIMIT, which is the shortest length published practice says you should work to; and the WORKING LENGTH you entered. The gap between the solid line and the 80 per cent line is the clearance you are supposed to keep, and a working line below it is a spring designed to coil bind. On the right the four end types are drawn as coil stacks and the one you selected is boxed. A boxed COIL is a dead coil, squared against its neighbour and taking no part in the deflection, and the line under the ground types is the ground face — so the twenty-five per cent rate difference between plain and squared ends is visible rather than described.

7.225N/mmExample

A 2 mm music wire spring, 16 mm outside diameter, ten total coils, squared and ground ends, 60 mm free length, working at 45 mm

The rate, the geometry, and where the mean diameter comes in

k = G·d⁴ / (8·D³·n_a) · D = OD − d · C = D/d · F = k·x · L_s = d·n_t or d(n_t + 1) · τ = K·8FD/(πd³)

k: spring rate: force per unit deflection, constant for a spring of uniform pitch and uniform wire
G: shear modulus of the WIRE, not of the steel in general — 79.3 GPa for carbon steel spring wire, 69 GPa for 302 stainless, 41.4 GPa for phosphor bronze
d: wire diameter. A fourth power, which is why it is the only lever that matters
D: MEAN coil diameter, OD − d. Not the outside diameter. This is the error the page exists to stop
n_a: ACTIVE coils: the total coils minus the end coils, which is 0, 1, 2 or 2 depending on the end type
C: spring index D/d. Below 4 the wire will not coil without cracking and the inner-fibre stress climbs; above 12 the springs tangle in the bin and buckle in the machine
L_s: solid height: the length at coil bind. d·n_t if the ends are ground, d(n_t + 1) if not
K: the stress correction factor — Wahl or Bergsträsser — which raises the nominal torsional stress at the inner fibre of the coil

Worked example

A 2 mm music wire spring, 16 mm outside diameter, ten total coils, squared and ground ends, 60 mm free length, working at 45 mm
Get the MEAN diameter first, because every other number depends on it and this is where the mistake is made: D = OD − d = 16 − 2 = 14 mm. Not 16. Using 16 would give a rate of 4.840 N/mm, a third low
Spring index C = D/d = 14/2 = 7.0, comfortably inside the preferred 4 to 12. That index is also what sets the stress correction, on the index and Wahl factor page
Active coils. Squared and ground ends take two coils out of the count, so n_a = n_t − 2 = 10 − 2 = 8. This is the step the end-type table above is for, and choosing plain ends instead would leave all ten active
Now the rate: k = Gd⁴/(8D³n_a) = 79,300 × 2⁴ / (8 × 14³ × 8) = 1,268,800 / 175,616 = 7.225 N/mm. Music wire's G is 79,300 N/mm²; in 302 stainless the same spring would be 6.286 N/mm, thirteen per cent softer for nothing but the material
Solid height, which is the dimension the housing has to allow for: ground ends give L_s = d·n_t = 2 × 10 = 20 mm. The available deflection is therefore 60 − 20 = 40 mm, and that 40 mm is the whole budget
At the 45 mm working length the deflection is 15 mm, so F = k·x = 7.225 × 15 = 108.4 N. That is 37.5 per cent of the available travel — inside the published 20 to 80 per cent window, with room at both ends
AND HERE IS THE LIMIT NOBODY COMPUTES. Eighty per cent of the available deflection is 32.0 mm, so the shortest length this spring should ever be worked to is 28.0 mm, at 231 N. Below that you are into the clearance that published practice reserves for end-coil contact, pitch variation and free-length tolerance. Solid height is 20 mm and 289 N; that is a number to know and never a number to design to
The stress check, for completeness: K_W at C = 7 is 1.2129, the uncorrected stress at the working load is 483 MPa and the corrected stress is 586 MPa. Music wire at 2 mm has a minimum tensile of 2,000 MPa and Shigley's static allowable is 45 per cent of it, 900 MPa. So the spring is at 65 per cent of allowable at the working length and 174 per cent at solid — which is another reason not to go there

The four end types, and what each one does to your spring

The first six columns are the published table; the last four are what it does to THIS page’s example — a 2 mm wire, ten total coils, 14 mm mean diameter, 60 mm free length. Read the last four and the point is unmissable: the same ten coils of the same wire on the same diameter give a rate of 5.780 N/mm with plain ends and 7.225 N/mm squared and ground, a difference of 25 per cent, because two of the ten coils have stopped being springs and become seats. Grounding is worth exactly one wire diameter of solid height and nothing else; squaring is worth two active coils and nothing else. Squared and ground is the normal choice for anything that has to sit square and not wander, and it is the most expensive of the four to make. Plain ends on a spring that has to push on a flat face will cock and buckle, which is the buckling page‘s subject. These dimensions come from a published standard’s table, not from a formula. The standard itself is cited below and the printed values are attributed to the catalogue they were taken from; a different publisher may round differently in the last digit.
End type	End coils n_e	Total coils n_t	Free length L₀	Solid length L_s	Pitch p	n_a here	L_s here	Rate here	Pitch here
Plain (open, not ground)	0	n_a	p·n_a + d	d(n_t + 1)	(L₀ − d)/n_a	10	22.0 mm	5.780 N/mm	5.80 mm
Plain and ground (open, ends ground flat)	1	n_a + 1	p(n_a + 1)	d·n_t	L₀/(n_a + 1)	9	20.0 mm	6.422 N/mm	6.00 mm
Squared / closed (not ground)	2	n_a + 2	p·n_a + 3d	d(n_t + 1)	(L₀ − 3d)/n_a	8	22.0 mm	7.225 N/mm	6.75 mm
Squared / closed and ground	2	n_a + 2	p·n_a + 2d	d·n_t	(L₀ − 2d)/n_a	8	20.0 mm	7.225 N/mm	7.00 mm

The working window, for this page’s example spring

Two published rules bound the useful travel and they do not quite agree, so both are printed. MW Components and *The Engineer*‘s “Top Tips for Compression Spring Design” independently give the same window: use the centre 20 to 80 per cent of the available deflection, reserving the first and last 15 to 20 per cent. Shigley states it differently, as a fractional overrun to closure ξ ≥ 0.15 — the force at solid must be at least 1.15 times the maximum working force, which permits 87 per cent of the available travel rather than 80. The page computes both limits. The reason for either is the same and it is not stress: it is that the LAST few per cent of travel is where the end coils touch their neighbours, where the rate stops being linear, where the coil pitch is no longer uniform and where a tolerance on free length or coil count turns into a solid stack. A spring designed to its own solid height has no margin for any of that. This page sizes a part; it does not certify one. Where the answer carries a consequence — a load path, a lifting duty, a pressure boundary, a fastener holding something that can fall — confirm it against the design code that governs the application, and against the manufacturer’s own rating, before relying on it.
Percentage of available deflection	Length	Deflection	Force	Corrected stress	Per cent of allowable	Verdict
10 %	56.00 mm	4.00 mm	28.9 N	156 MPa	17 %	too little travel to be worth a spring
20 %	52.00 mm	8.00 mm	57.8 N	312 MPa	35 %	the published working window
30 %	48.00 mm	12.00 mm	86.7 N	469 MPa	52 %	the published working window
40 %	44.00 mm	16.00 mm	115.6 N	625 MPa	69 %	the published working window
50 %	40.00 mm	20.00 mm	144.5 N	781 MPa	87 %	the published working window
60 %	36.00 mm	24.00 mm	173.4 N	937 MPa	104 %	the published working window
70 %	32.00 mm	28.00 mm	202.3 N	1,093 MPa	122 %	the published working window
80 %	28.00 mm	32.00 mm	231.2 N	1,250 MPa	139 %	the published working window
90 %	24.00 mm	36.00 mm	260.1 N	1,406 MPa	156 %	beyond the published limit
100 %	20.00 mm	40.00 mm	289.0 N	1,562 MPa	174 %	beyond the published limit

Wire diameter is a fourth power, and it is the only real lever on this page

Everything else on this page is a first or third power; the wire is a fourth. Ten per cent more wire is 46 per cent more rate, twenty per cent is 107 per cent — the spring more than doubles. And it cuts both ways at once, which is the part that surprises people: the stress at a GIVEN DEFLECTION goes up with the thicker wire, because the force rose faster than the section modulus did, but the stress at a GIVEN FORCE falls as 1/d³. So thicker wire is the answer to “this spring is too soft” and also, at constant load, the answer to “this spring is overstressed”. It is almost never the answer to both at once. If you need more rate AND less stress, the move is a larger mean diameter with thicker wire, which is to say a bigger spring, or fewer active coils, which costs you travel. This is a first-pass calculation on an idealised part — uniform section, static load, no stress raisers beyond those stated, and room-temperature material properties. Real parts have fillets, keyways, surface finish and duty cycles that a closed-form answer cannot see.
Change	Wire d	Rate multiplier d⁴	Rate	Force at 15 mm deflection	Corrected stress	Per cent of allowable
20 % thinner	1.600 mm	0.410×	2.959 N/mm	44.4 N	451 MPa	50 %
10 % thinner	1.800 mm	0.656×	4.740 N/mm	71.1 N	517 MPa	57 %
as entered	2.000 mm	1.000×	7.225 N/mm	108.4 N	586 MPa	65 %
10 % thicker	2.200 mm	1.464×	10.578 N/mm	158.7 N	657 MPa	73 %
20 % thicker	2.400 mm	2.074×	14.981 N/mm	224.7 N	731 MPa	81 %

Mean diameter, active coils, and why solid height is a fact and not a design point

D is the MEAN diameter and it is not the diameter you measured. The rate formula is k = Gd⁴/(8D³n_a), and D is the diameter of the circle the centre of the wire travels on — the outside diameter minus one wire diameter. Feed the outside diameter into that formula instead and, at a spring index of 7, you get a rate a third too low. The error is invisible because the answer still looks like a spring rate; it only shows up when the spring arrives and is stiffer than the drawing said. Every dimension on this page is computed from OD and d so the question cannot arise.

The end type is not cosmetic: it decides how many of your coils are springs. A coil that is squared against its neighbour is a seat, not a spring, and it takes no part in the deflection. Plain ends leave every coil active; plain and ground take one out; squared and squared-and-ground take two. On ten total coils that is the difference between a rate of 5.780 and 7.225 N/mm from an identical-looking spring. The same choice moves the solid height by one wire diameter — grinding buys you exactly that and nothing more — and changes the pitch formula, because the free length has to account for the dead coils. The table above gives all four, and gives them for your own numbers.

Spring index C = D/d is the number a spring maker asks for first. It is the ratio of the coil to the wire, and it decides three things at once: whether the spring can be coiled at all, how much the stress at the inner fibre exceeds the nominal value, and how badly the spring tangles and buckles. Below about 4 the wire is being wrapped round a radius tight enough to crack it and the tooling loads are severe. Above about 12 the springs knit together in the bin and wander in the machine. In between, the Wahl and Bergsträsser factors tell you what the curvature of the wire is doing to the stress — about 40 per cent at C = 4, about 11 per cent at C = 12.

Never design to solid height. Solid height is a fact about the spring, not a place to work it. Published practice leaves clearance: MW Components and The Engineer‘s design tips both put the usable travel at the centre 20 to 80 per cent of the available deflection, reserving 15 to 20 per cent at each end; Shigley states the same idea as a fractional overrun to closure of at least 0.15. The reason is not stress. It is that the last few per cent of travel is where the end coils begin touching their neighbours and the rate stops being linear, and it is where the free-length and coil-count tolerances live. A spring designed to 95 per cent of its travel goes solid on some units and not on others, and a spring at coil bind is a solid steel tube in the load path with no compliance at all. This page computes both published limits and the length each one corresponds to.

What this calculation cannot see. It is a linear elastic model of a uniform helix. It does not see a spring that has been pre-set (deliberately compressed to solid in manufacture to raise the usable stress), which most quality compression springs have been. It does not see fatigue — a static allowable says nothing about a spring cycling ten million times, and shot peening changes that answer more than any dimension does. It does not see the spring buckling, which the buckling page computes and which is a geometry problem rather than a stress one. It does not see temperature, which takes modulus out of every spring material above a stated limit and is on the wire material page. And it does not see surging — a spring has natural frequencies of its own, and a valve spring or a clutch spring driven near one of them will fail from a stress the static calculation never predicted.

Frequently asked questions

Is D the outside diameter or the mean diameter?

The mean, always: D = OD − d, the diameter of the circle the centre of the wire follows. This is the single commonest error in spring arithmetic and it is worth being blunt about the size of it. Because the rate goes as 1/D³, using the outside diameter at a spring index of 7 gives a rate 33 per cent too low; at an index of 4 it is 42 per cent too low. This page takes the outside diameter as the input because that is what you can measure, and does the subtraction for you.

How do I know how many coils are active?

Count the total coils and subtract the end coils, and the end coils depend entirely on how the ends are finished. Plain (open) ends: zero — every coil is active. Plain and ground: one. Squared (closed), ground or not: two. So a ten-coil spring has 10, 9, 8 or 8 active coils depending on a detail that costs nothing to specify and changes the rate by 25 per cent. The table on this page gives all four end types with the free length, solid height and pitch formula for each.

Why must I not design a spring to its solid height?

Because the last part of the travel is not spring. As the coils close the end coils begin touching their neighbours, the active coil count effectively falls, and the rate stops being linear — the spring gets stiffer than the calculation says, at exactly the moment you least want it to. On top of that, free length and coil count both carry manufacturing tolerances, so a design at 95 per cent of travel will bottom out on some units and not on others. Published practice reserves 15 to 20 per cent at each end; Shigley’s version is a fractional overrun to closure of at least 0.15. A spring at coil bind transmits the full load straight through with no compliance, which is usually the failure the spring was fitted to prevent.

What is a good spring index, and what happens outside the range?

Between 4 and 12 is the range Shigley states and spring makers prefer. Below 4 the wire is bent round a radius tight enough that the outer fibre can crack in coiling, the tooling forces rise sharply, and the stress concentration at the inner fibre climbs — the Wahl factor is 1.40 at C = 4 against 1.12 at C = 12. Above 12 nothing is overstressed, but the springs tangle with each other badly enough to need individual packing, and they are slender enough to buckle in service. If your index is outside the range, changing the mean diameter is usually cheaper than changing the wire.

Does grinding the ends change the rate?

No — grinding changes the solid height by exactly one wire diameter and changes nothing else about the rate. What changes the rate is SQUARING (closing) the ends, which takes two coils out of the active count. The two are usually specified together, which is why they get confused. A squared but unground spring has the same rate as a squared and ground one of the same total coils, and a solid height one wire diameter greater. What grinding really buys is squareness: an unground end does not sit flat, so the load goes in off-centre.

The spring I received is stiffer than this page predicts. Why?

Three usual causes, in order of likelihood. First, the active coil count is not what you assumed — check the end type, and count the total coils on the actual part rather than on the drawing. Second, the mean diameter is smaller than specified: the rate goes as 1/D³, so a two per cent diameter error is a six per cent rate error, and coiling tolerances of that size are normal. Third, you are measuring near solid, where the end coils have begun to touch and the rate is genuinely higher than the linear value. Measure the rate between 30 and 60 per cent of available deflection, which is where the linear model is honest.

Can I just use 79.3 GPa for G whatever the wire is?

No, and it is a bigger error than it looks. 79.3 GPa is right for carbon steel spring wire — music wire, hard drawn, oil tempered. 302 stainless is 69 GPa, so a stainless spring of identical geometry is 13 per cent softer. Phosphor bronze is 41.4 GPa, nearly half. Substituting stainless for carbon steel “for corrosion resistance” without recalculating gives a spring that misses its force by more than any manufacturing tolerance ever will. The wire material page carries the moduli, the strengths and the temperature limits together.

Related calculators

References

R. G. Budynas and J. K. Nisbett, Shigley’s Mechanical Engineering Design, chapter 10 “Mechanical Springs”. The source for the four-way end-condition table (Table 10-1, itself attributed there to Associated Spring’s Design Handbook), for the Bergsträsser factor, for the buckling constants C₁′ and C₂′, for the torsion-spring constant 10.8, and for the S_ut = A/d^m constants. Two of its published worked solutions (problems 10-35 and 10-37) were reproduced here to four significant figures as a check on the initial-tension band, and the end-condition table was checked cell for cell against RoyMech’s independent printing.
RoyMech, Helical Spring Design — Compression and Extension Springs and Spring Materials. Used as the independent check on the four-way end-condition table (which agrees with Shigley’s Table 10-1 cell for cell), on the SI A and m constants, and on the shear moduli. RoyMech’s own published position is worth quoting because it is the honest one: “do not use a fixed percentage of tensile strength as a universal spring allowable”.
Compression Spring Store / The Spring Store, Helical Spring Formulas and Equations. The independent printing of the four pitch formulas — (L−d)/n_a open, (L−3d)/n_a closed, (L−2d)/n_a closed and ground — and of the two solid-height rules, d(N+1) unground and dN ground, which is what confirmed the end table independently of both textbook printings.
MW Components, How to Select a Compression Spring and How to Select an Extension Spring. The source for the working-deflection window: “reserve at least the first and last 15 to 20 per cent of the range”, and “if critical force-versus-deflection linearity is required, only the centre 20 to 80 per cent of the available deflection range should be employed”. The Engineer‘s published “Top Tips for Compression Spring Design” gives the same 20 to 80 per cent window independently.
Spring Manufacturers Institute, Handbook of Spring Design, and Associated Spring / Barnes Group, Design Handbook. The origin of the end-condition table, of the preferred initial-tension band plotted against spring index, and of the stability curves. Both are cited by name and NOT reproduced; the values used here were taken from the named catalogue and textbook printings below, which is this site’s standing policy on copyrighted design data.
EN 13906-1, -2 and -3, Cylindrical helical springs made from round wire and bar — Calculation and design (Part 1 compression, Part 2 extension, Part 3 torsion). Cited by number, not reproduced. It is the European counterpart to the treatment on these pages and it is the document a European drawing will name; the closed forms used here (k = Gd⁴/8D³n, the Wahl correction, the bending rate of a torsion spring) are the same ones, and every figure this site prints is either computed from a definition or attributed to a named catalogue.
ISO 2162-1, Technical product documentation — Springs — Part 1: Simplified representation, and ISO 26909, Springs — Vocabulary. Cited by number for the terms used here — free length, solid length, active coils, spring index — which are not this site’s coinages.
A. M. Wahl, Mechanical Springs (2nd edition, McGraw-Hill). The origin of the correction factor that carries his name and of the curved-beam factors used for extension-spring hooks and torsion-spring coils. Cited by name; the factors themselves are printed here because they are algebraic expressions in the spring index, not tabulated data.