Extension Spring Rate and Initial Tension Calculator

F = P_i + k·x, with the initial tension checked against the band that can actually be coiled at your spring index, and the hook computed separately in bending and in torsion — because the hook is what governs and what breaks.

Last updated 24 Sep 2026

Extension spring rate, initial tension and hook stress

Wire, OD, coils, hooks → force and the governing stress

Wire diameter d (mm)A fourth power in the rate and a cube in the stress, same as a compression spring — but on an extension spring it is also the thing that decides whether the HOOK survives, because the hook bending stress goes as 1/d³ too and starts from a much higher number.

Outside diameter OD (mm)Mean diameter D = OD − d, as always. On a close-wound extension spring the OD is easy to measure accurately because the coils are touching.

Body coils n (all of them are active)An extension spring has no dead end coils in the body: every coil deflects. The body length is d(n + 1) because the coils are wound touching.

Wire materialG sets the rate and the minimum tensile sets every allowable on the page. Note that initial tension is NOT achievable in every material: it is a cold-coiling effect, and a spring that has been stress-relieved hard or coiled hot may have little or none.

Initial tension P_iInitial tension is designed in by coiling the wire with a twist so that the coils press against each other. It is only achievable inside a band that narrows as the spring index rises — this page computes that band from Shigley’s fit to the Associated Spring chart and shows you where you are in it.

… or type your own initial tension (N)Only used when the dropdown above is set to the last option. Measure it by pulling the spring until the coils just separate.

Extension you want (mm)Measured from the free length, which this page computes from the body length plus two hooks.

Mean radius of the LOOP bend r₁ (mm)The radius of the bend where the hook leaves the body and turns over — the bend that carries the bending moment. A machine loop is typically half the mean coil diameter. This is where extension springs break.

Mean radius of the hook-body bend r₂ (mm)The side bend, where the hook twists out of the plane of the coil. It carries torsion, not bending. Shigley’s stated rule is that 2r₂/d should be greater than 4.

Not a circuit: two figures. On the left, the force–extension line in normalised coordinates — the horizontal axis is your own extension and the vertical axis your own force, so the line always ends at the top right corner whatever your numbers are. The arrow on the vertical axis marks the initial tension: the line starts THERE and not at the origin, and the height of that offset moves as you change the spring. The second line, drawn from the origin, is the answer a calculation that forgets initial tension gives — the gap between the two is the error, constant in newtons and therefore worst at small extensions. On the right, the hook in section with the two places it is stressed marked: A is the loop bend, which carries a BENDING moment against a section modulus half the torsional one, and B is the side bend, which carries torsion. The three stresses and the governing safety factor are printed beside it, and on most real springs A is the worst of them.

62.15NExample

A 1.6 mm music wire extension spring, 12.8 mm outside diameter, 30 body coils, machine loops with a 4.8 mm loop bend and a 4.0 mm side bend, pulled 30 mm

The rate, the offset, and the two hook stresses

F = P_i + k·x · k = Gd⁴/(8D³n) · P_i = τ_i·πd³/(8D) · σ_A = K_A·16FD/(πd³) + 4F/(πd²) · τ_B = K_B·8FD/(πd³)

P_i: initial tension: the force that must be applied before the coils separate at all. Put in by coiling, not by the designer’s choice
k: rate. The same formula as a compression spring, and every body coil is active
τ_i: the uncorrected torsional stress the initial tension corresponds to. Only achievable inside a published band that narrows as the index rises
σ_A: bending stress at the loop bend, from the moment FD/2 over the section modulus πd³/32, raised by the curved-beam factor K_A, plus the direct axial pull
τ_B: torsional stress at the side bend, from the torque FD/2 over πd³/16, raised by K_B — which is the Wahl factor with its direct-shear term removed
C₁, C₂: 2r₁/d and 2r₂/d, the bend indices. Shigley’s rule is C₂ > 4; both factors blow up as the bend tightens

Worked example

A 1.6 mm music wire extension spring, 12.8 mm outside diameter, 30 body coils, machine loops with a 4.8 mm loop bend and a 4.0 mm side bend, pulled 30 mm
Mean diameter D = 12.8 − 1.6 = 11.2 mm, index C = 11.2/1.6 = 7.0. Same arithmetic as a compression spring, and the same trap: the outside diameter is not D
Rate: every body coil is active on an extension spring, so n = 30 and k = Gd⁴/(8D³n) = 79,300 × 1.6⁴ / (8 × 11.2³ × 30) = 1.5413 N/mm
NOW THE PART A COMPRESSION SPRING DOES NOT HAVE. This spring was coiled with a twist, so the coils press on each other and the force does not start at zero. At C = 7 the published achievable band for the initial tension stress is 87.4 to 134.1 MPa; take the middle, 110.8 MPa, and turn it back into a force with P_i = τ_iπd³/(8D) = 15.91 N
So the force at 30 mm of extension is F = P_i + kx = 15.91 + 1.5413 × 30 = 62.15 N. A page that ignored P_i would say 46.24 N — 26 per cent low. At 2 mm of extension it would be 84 per cent low
THE HOOK, WHICH IS THE POINT OF THIS PAGE. The loop bend carries a bending moment FD/2 over a section modulus πd³/32, raised by the curved-beam factor: C₁ = 2 × 4.8/1.6 = 6.0, K_A = 1.1417, and adding the direct axial pull gives σ_A = 1,019 MPa
The side bend carries torsion instead: C₂ = 2 × 4.0/1.6 = 5.0, K_B = 1.1875, and τ_B = K_B·8FD/(πd³) = 514 MPa. The body itself, Wahl corrected, is only 525 MPa
WHICH GOVERNS. Music wire at 1.6 mm has a minimum tensile of 2,065 MPa. The published allowables are 45 per cent of it for the body in torsion (929 MPa), 75 per cent for the hook in bending (1,549 MPa) and 40 per cent for the hook in torsion (826 MPa). That gives safety factors of 1.77 on the body, 1.52 on the hook bend and 1.61 on the hook body. The hook bend governs, by a wide margin, which is why extension springs break at the hook and almost never in the middle
Both bend indices here are above Shigley's stated minimum of 4, and the hook still governs. Tighten the loop bend to 1.6 mm (C₁ = 2) and the hook bending stress rises to 1,437 MPa and the safety factor falls to 1.08 — from a bend radius the drawing never mentioned. And note what this spring does NOT have: a solid stop. A compression spring cannot be pushed past coil bind; an extension spring pulls until something yields, so no geometry protects it from an overload

How much initial tension you can actually have, against spring index

Initial tension is not a free parameter: it is put into the spring by coiling the wire with a twist, and how much twist the coiler can hold depends on how tight the coil is. A low-index spring can be wound hard; a high-index one cannot, and above about C = 16 there is very little initial tension to be had at all. The band here is Shigley’s published fit to the Associated Spring preferred-range chart, τ_i = 33,500/exp(0.105 C) ± 1,000[4 − (C−3)/6.5] psi, converted to MPa. It was checked against Shigley’s own published solutions to two problems and reproduces both to four significant figures: 4.251 lbf at C = 7.738 and 8.55 lbf at C = 4.91, each at the low edge of the band. A different published form gives the band as (0.4 to 0.8)·S_ut/C, which at C = 10 in music wire is 12.8 to 25.7 ksi against this fit’s 8.8 to 14.6; where they disagree, ask the spring maker what they can wind, because this is a MANUFACTURING limit and not a strength one. The same designation can mean different dimensions in different standards families — ANSI against ISO, inch against metric, one national standard against another. The family used here is named beside every figure; check which one your part was made to.
Spring index C	Lowest τ_i (MPa)	Middle (MPa)	Highest τ_i (MPa)	The same band in ksi	P_i for a 1.6 mm wire at that index (N)
4	125.2	151.8	178.3	18.2–25.9	38.14
5	111.2	136.6	162.1	16.1–23.5	27.47
6	98.6	123.0	147.4	14.3–21.4	20.61
7	87.4	110.8	134.1	12.7–19.4	15.91
8	77.4	99.7	122.0	11.2–17.7	12.53
10	60.7	80.8	101.0	8.8–14.6	8.13
12	47.5	65.5	83.5	6.9–12.1	5.49
14	37.2	53.1	69.0	5.4–10.0	3.81
16	29.3	43.0	56.8	4.2–8.2	2.70

What the bend radius does to the hook — at this page’s own working load

The hook is where extension springs fail, and the bend radius is the reason. Both Wahl curved-beam factors climb steeply as the bend tightens and both are singular at a radius of half a wire diameter, which is a bend the wire could not survive anyway. Shigley’s stated rule is that C₂ = 2r₂/d should exceed 4, and the same reasoning applies to r₁. Read the last column: at a generous 5d bend the hook is comfortable at this load, and at a tight 0.75d bend it is not. The practical consequence is a drawing note. “Machine loop” on a drawing leaves the bend radius to the coiler, and coilers make it as tight as the tooling allows because that is fastest. If the hook stress matters, dimension the bend radius. 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.
Bend radius r₁	C₁ = 2r₁/d	Bending factor K_A	Torsion factor K_B	Hook bending stress (MPa)	Per cent of allowable
0.75d	1.50	2.1667	2.5000	1,906	123 %
1.00d	2.00	1.6250	1.7500	1,437	93 %
1.50d	3.00	1.3333	1.3750	1,185	76 %
2.00d	4.00	1.2292	1.2500	1,095	71 %
3.00d	6.00	1.1417	1.1500	1,019	66 %
5.00d	10.00	1.0806	1.0833	966	62 %

What a page that forgets initial tension gets wrong

The offset is constant in newtons and therefore enormous in percentage terms wherever the extension is small — which is where most extension springs actually work. At 2 mm of extension this spring is carrying 19.0 N and the zero-intercept answer is 3.1 N, an error of 516 per cent. At 25 mm the same 15.9 N is a 41 per cent error. That is why an extension spring’s rate must be measured between two extensions rather than from a single force and length, and why a catalogue that quotes only a rate for an extension spring has not told you what you need. 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.
	This spring	The same rate through the origin	Difference	Error
Force at zero extension	15.91 N	0.00 N	15.91 N	∞
Force at 2 mm	18.99 N	3.08 N	15.91 N	516 %
Force at 6 mm	25.15 N	9.25 N	15.91 N	172 %
Force at 12 mm	34.40 N	18.50 N	15.91 N	86 %
Force at 25 mm	54.44 N	38.53 N	15.91 N	41 %

The offset that is not in the textbook formula, and the hook that is the real spring

An extension spring’s force does not start at zero. It is coiled with a twist so the coils press against each other, and that preload — initial tension, P_i — has to be overcome before the spring moves at all. The characteristic is F = P_i + kx: a straight line with the right slope and the wrong intercept if you forget it. The error is constant in newtons and therefore worst where the extension is small, which is where most extension springs live. On this page’s example a calculation that omits P_i is 84 per cent low at 2 mm of extension. It also means the rate of an extension spring cannot be measured from one force and one length: you need two points and the slope between them.

Initial tension is not a design variable, it is a manufacturing one. How hard the coiler can twist the wire depends on how tight the coil is, so the achievable initial tension is a BAND that narrows as the spring index rises. This page computes that band from Shigley’s published fit to the Associated Spring chart, and shows whether the value you want sits inside it, above it or below it. Above the band the coiler cannot hold the coils together that hard; below it the coils do not stay closed and the spring rattles in the assembly. At a spring index above about 16 there is very little initial tension available at all — which is a real constraint on any mechanism that relies on the spring holding itself shut.

The hook is a different calculation, and it is usually what governs. The body of the spring is in torsion, exactly like a compression spring. The hook is not. Where the hook turns over, the wire is a curved beam in BENDING, carrying a moment FD/2 against a section modulus of πd³/32 — half the torsional one — and raised further by a curved-beam stress concentration that climbs steeply as the bend tightens. Where the hook twists out of the plane of the coil it is in torsion, with its own factor. This page computes both, compares each against its own published allowable, and names which governs. On the defaults the hook bend is the answer with a safety factor of 1.52 against the body’s 1.77. That is why extension springs break at the hook.

And an extension spring has no solid stop. A compression spring cannot be pushed past coil bind: the geometry itself is an overload protection, and a spring that is safe at solid cannot be damaged by any force at all. An extension spring has nothing of the kind. Pull it further and it simply goes on taking stress until the hook opens or the wire yields, and the first sign is usually a spring that has grown permanently longer and lost its preload. If the mechanism can overtravel, the stop has to be designed in somewhere else — a hard stop in the linkage, a slot, a second spring in series. This is the single most important difference between the two spring types and it is a design responsibility, not a calculation.

What this page leaves out. Hook geometry beyond the two bend radii — a full crossover loop, an extended hook, a swivel end or a threaded insert each change the load path in ways no closed form covers. Fatigue: these are static allowables, and an extension spring cycling in the millions is a different problem in which the hook is even more dominant. The tolerance on initial tension, which is wide. Hydrogen embrittlement from plating, which kills high-strength spring wire and which is why plated springs get baked. And buckling, which extension springs do not do — that is the compression spring’s problem.

Frequently asked questions

What is initial tension and why does it matter?

It is the force that has to be applied to an extension spring before the coils separate at all — built in by coiling the wire with a twist so the coils press against one another. It matters because the force–extension line starts at P_i rather than at the origin: F = P_i + kx. A calculation that uses F = kx is wrong by P_i at every extension, and since P_i is a constant, the percentage error is worst at small extensions — which is where most extension springs work. It also means you cannot get the rate from a single force measurement.

Why do extension springs always break at the hook?

Because the hook is a different stress problem and a worse one. The body is in torsion with a section modulus of πd³/16; the hook bend is in bending with πd³/32, half as much, carrying the same FD/2 moment — so it starts at twice the stress before any correction. Then the curved-beam factor adds more, and it climbs steeply as the bend radius tightens. On this page’s own defaults the hook bend is at 66 per cent of its allowable while the body is at 56 per cent. The single most effective thing you can do about it is dimension the bend radius on the drawing.

How much initial tension can I ask for?

Only what the coil can hold, and that depends on the spring index. The published band runs from roughly 125 to 178 MPa of uncorrected torsional stress at C = 4, down to about 48 to 84 MPa at C = 12, and to very little above C = 16. This page computes the band for your index and tells you whether your figure is inside it. It is a manufacturing limit, not a strength limit, so no material change moves it — the only lever is the spring index itself.

Is there a way to make an extension spring safe against overload?

Not in the spring. A compression spring has coil bind, which is a geometric hard stop that cannot be exceeded; an extension spring has nothing equivalent, so it goes on taking load until something yields. The answer is always outside the spring: a mechanical stop in the linkage that reaches before the spring does, a slot that runs out, or a second stiffer spring in series that takes over. If you cannot fit one, size the spring so that the maximum travel the mechanism physically permits is still inside the allowable, which is the extension figure this page computes.

What are the different hook types for?

A machine loop is the cheapest — the wire is simply bent over into a loop in line with the body — and it has the tightest bends, so it is the weakest. A crossover centre loop brings the wire across the middle of the coil, which gives a gentler side bend. An extended loop moves the bend away from the body altogether and is the strongest of the three, at the cost of length and money. Where the hook stress governs and cannot be fixed by bend radius alone, a swivel hook or a threaded insert takes the bending out of the wire completely.

Why is the body allowable lower for an extension spring than for a compression spring of the same wire?

Because an extension spring is loaded for as long as it is fitted. A compression spring at its free length carries nothing; an extension spring carries its own initial tension against the coils from the moment it is made, and in service it is almost always fitted with some preload. That is a sustained stress rather than an occasional one, and it makes relaxation and set more likely. Century Spring publishes 30 to 45 per cent of minimum tensile for an extension body where a compression spring of the same wire gets 45; Shigley’s Table 10-7 is less conservative. Both readings are on the wire material page.

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.
Stock Springs / Stock Drive Products, Initial Tension in Extension Springs. The trade form of the initial-tension relation, P = S·d³/(2.55·D), whose 2.55 is 8/π — that identity is how this page confirmed the relation rather than taking it on trust.
Newcomb Spring Corp., Extension Spring Initial Tension and Torsion Spring Static Operating Stress. Cited for the qualitative relation — “the higher the spring index the lower the initial tension will be” — which the fitted band on this page reproduces, and for its published position that maximum operating stresses for torsion springs should be confirmed with the maker rather than taken from a general table.
Century Spring Corp., Extension Springs design guide. The published allowable stresses for extension springs: body wire “between 30 and 45 per cent of the material’s minimum tensile strength”, hook in torsion 30–45 % depending on material, hook in bending 75 %. More conservative than Shigley’s Table 10-7 for the body, and both are printed on the wire page.
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.
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.
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”.