Torsion Spring Rate and Deflection Calculator

k = Ed⁴/(10.8·D·n) — E, because the wire is in bending — printed per turn, per degree and per radian, with the legs counted, the inner-fibre stress at the leg transition, and the wound-down inside diameter that actually sizes the arbor.

Last updated 24 Sep 2026

Torsion spring rate, torque and wound-down diameter

Wire, coils, legs, angle → rate, torque and arbor size

Wire diameter d (mm)A fourth power in the rate, exactly as for a compression spring, and a cube in the stress. The difference is that the wire here is in BENDING, so the modulus that matters is E and not G.

Outside diameter OD (mm)At free length. The coil gets SMALLER as the spring is wound up, and this page computes by how much — that number, not this one, is what sizes the arbor.

Body coils nThe coils in the body, before the legs are counted. The legs add a fraction of a coil of their own, which this page works out below.

Wire materialE, not G. A torsion spring’s wire is bent, not twisted, so the rate is proportional to Young’s modulus. The ratio E/G is about 2.6 for steel and 2.5 for phosphor bronze, so a material swap moves a torsion spring and a compression spring by different amounts.

Length of leg 1 (mm)Measured from the body to the point the load acts. A straight leg bends too, and it is worth (l₁ + l₂)/(3πD) of an extra coil — which on short springs with long legs can be a large fraction of the total.

Length of leg 2 (mm)

Angular deflection you want (°)In degrees, in the winding-up direction. 360° is one full turn; the page converts to turns and radians for you because published torsion-spring rates use all three.

Which way the load actsA torsion spring must be loaded in the direction that WINDS IT TIGHTER. Loaded the other way the residual stresses from coiling work against you and the coil grows off its arbor instead of onto it.

Print the rate asTorsion spring rate is published per turn, per degree and per radian by different sources, and the constant in the formula changes with it. All three are shown below; this picks which one is the headline.

Not a circuit: the coil seen end on, to scale, with the free circle drawn once and the WOUND-DOWN circle drawn as a family that shrinks as you increase the angular deflection. The inner circle is the largest arbor that still clears at ten per cent, and it shrinks with the coil — which is the whole message of the figure. The wire length πDn cannot change, so winding the spring tighter adds turns and the only way to do that is to make the circle smaller; an arbor chosen from the free inside diameter is an arbor the spring grips at full deflection. The two legs are drawn with the closing direction arrowed: a torsion spring must be loaded that way, because loaded the other way the coil grows instead and comes off its arbor. On the right, the torque rising in a straight line while the inside diameter falls, which are the two things the designer has to satisfy at once.

3,390.3205N·mm/unitExample

A 2 mm music wire torsion spring, 16 mm outside diameter, six body coils, two 30 mm straight legs, wound up 90° in the closing direction

Bending, not torsion — which is why it is E

k = E·d⁴/(10.8·D·n) per revolution · M = k·θ · n = n_body + (l₁+l₂)/(3πD) · D′ = D·n_b/(n_b + θ_rev) · σ = K_i·32M/(πd³)

E: Young’s modulus, NOT the shear modulus. The wire of a torsion spring is bent round its own axis of curvature; it is not twisted. This is the whole reason the formula looks like a beam formula
10.8: the published constant for a rate per REVOLUTION. The theoretical value from the bending strain energy is 32/π = 10.186; 10.8 is empirical and about six per cent softer, the difference being friction between coils
n: effective coils: the body coils plus the compliance of the two straight legs, worth (l₁+l₂)/(3πD) coils between them
D′: the mean diameter after winding up θ revolutions. Follows from the wire length πDn being constant, so it is exact rather than fitted
K_i: the inner-fibre stress factor for a curved beam, (4C²−C−1)/(4C(C−1)). The inner fibre is the one that governs, and the outer fibre’s factor is below 1
σ: bending stress, so the allowable is the BENDING one — about 75 per cent of minimum tensile for steel, against 45 per cent in torsion for a compression spring

Worked example

A 2 mm music wire torsion spring, 16 mm outside diameter, six body coils, two 30 mm straight legs, wound up 90° in the closing direction
The name is misleading and the first step is to say so: the wire in a torsion spring is in BENDING, not torsion. The coil is being wound tighter, which bends the wire about its own axis of curvature. That is why the rate uses E, 206,800 N/mm² for music wire, and not G
Mean diameter D = 16 − 2 = 14 mm, index C = 7.0 — the same arithmetic as every other helical spring
The legs bend too, and the published allowance is worth (l₁+l₂)/(3πD) = 60/(3π × 14) = 0.4547 coils. So the effective coil count is 6 + 0.4547 = 6.4547, not 6. That is 7.6 per cent of the body, from two legs of entirely ordinary length
Rate: k = Ed⁴/(10.8·D·n) = 206,800 × 2⁴ / (10.8 × 14 × 6.4547) = 3,390.3 N·mm per revolution, which is 9.4176 N·mm per degree and 539.6 N·mm per radian. All three are the same spring; the constant changes with the unit and 10.8 belongs to the revolution
A note on that 10.8, because it is not the number the mathematics gives. Integrate the bending strain energy of the coil and the constant is exactly 32/π = 10.186 per revolution, which would make this spring 3,595 N·mm/rev. The published 10.8 is empirical and 6.0 per cent softer. The difference is friction between the coils as they slide on one another
Torque at 90°: M = 9.4176 × 90 = 847.6 N·mm, or 0.848 N·m
NOW THE DIMENSION THAT SEIZES SPRINGS. The wire length πDn is fixed, so winding the spring up by a quarter of a turn gives it 6.25 coils of the same wire and the diameter must shrink: D′ = D·n/(n + θ) = 14 × 6/6.25 = 13.440 mm, so the inside diameter falls from 12.00 mm to 11.440 mm, a shrinkage of 4.7 per cent. Allow ten per cent clearance and the largest arbor this spring will run on is 10.30 mm — not the 10.80 mm the free diameter suggests
The stress, at the leg-to-body transition where the inner fibre is worst: the nominal bending stress is 32M/(πd³) = 1,079 MPa, the inner-fibre factor at C = 7 is K_i = 1.1190, so σ = 1,208 MPa. Music wire at 2 mm has a minimum tensile of 2,000 MPa and the static BENDING allowable is about 75 per cent of it, 1,500 MPa — note that this is a much higher percentage than the 45 per cent a compression spring gets, because bending strength exceeds shear strength. This spring is at 81 per cent of it and it would reach the allowable at about 112°

The same spring, in the three units its rate gets published in

This is the most common way to get a torsion spring wrong, and it is not a physics mistake — it is a units mistake. The familiar constant 10.8 belongs to a rate PER REVOLUTION. Per degree the constant is 3,888 and per radian it is 67.86, and at least one manufacturer’s published design guide prints R = Ed⁴/(10.8 D N) with units of “N·mm per degree”, which is out by a factor of 360. The fourth column is the other half of the story: the theoretical constant, from integrating the bending strain energy of the coil, is exactly 32/π = 10.186 per revolution. The 10.8 everybody uses is an EMPIRICAL value about six per cent softer, and the six per cent is friction between the coils sliding on one another as the spring winds. Use 10.186 and you will predict a spring stiffer than the one you get. 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.
Convention	Formula	This spring (N·mm per unit)	The frictionless constant	Rate that constant would give
Per revolution (per turn, per 360°)	E·d⁴/(10.80·D·n)	3,390.3205	10.186	3,594.7145
Per degree	E·d⁴/(3,888.00·D·n)	9.4176	3,666.930	9.9853
Per radian	E·d⁴/(67.86·D·n)	539.5863	64.000	572.1166

The coil winding down onto its arbor

The wire in a torsion spring does not stretch, so the length πDn is a constant. Wind the spring up by θ turns and it now has n + θ coils of that same wire, so the diameter must fall: D′ = D·n/(n + θ). That is not a fit or an approximation, it is conservation of wire length, and it is why an arbor sized to the FREE inside diameter is an arbor the spring grips at full deflection. On this six-coil example a single full turn takes 17 per cent off the inside diameter. Note also that the body gets LONGER as it winds — the extra turns have to go somewhere — so a torsion spring in a pocket needs axial clearance as well as radial. 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.
Deflection	Turns	Inside diameter	Shrinkage	Largest arbor at 10 % clearance	Torque
0°	0.000	12.000 mm	0.00 %	10.800 mm	0.0 N·mm
45°	0.125	11.714 mm	2.38 %	10.543 mm	423.8 N·mm
90°	0.250	11.440 mm	4.67 %	10.296 mm	847.6 N·mm
180°	0.500	10.923 mm	8.97 %	9.831 mm	1,695.2 N·mm
270°	0.750	10.444 mm	12.96 %	9.400 mm	2,542.7 N·mm
360°	1.000	10.000 mm	16.67 %	9.000 mm	3,390.3 N·mm

What the legs are worth, on this page’s six-coil example

A straight leg is a cantilever and it bends under the same moment the coils do, so it adds compliance. The published allowance is (l₁+l₂)/(3πD) extra coils, and on a short spring with long legs it is not a rounding error: at 80 mm legs on a six-coil body it is 1.21 coils, which is 20 per cent of the body. Ignore it and the spring comes out softer than the calculation said, which is the direction that costs you force rather than breaking anything. Two cautions on the allowance itself. It assumes STRAIGHT legs loaded at their ends; a leg with a bend, a hook or a roller on it is a different cantilever. And it assumes the load acts perpendicular to the leg, which is only true at one angle of a spring that moves. 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.
Each leg	Extra coils (l₁+l₂)/(3πD)	Effective coils	Rate per degree	Change against the 30 mm case
0 mm	0.0000	6.0000	10.1313 N·mm/°	7.58 %
10 mm	0.1516	6.1516	9.8817 N·mm/°	4.93 %
20 mm	0.3032	6.3032	9.6440 N·mm/°	2.40 %
30 mm	0.4547	6.4547	9.4176 N·mm/°	0.00 %
50 mm	0.7579	6.7579	8.9951 N·mm/°	-4.49 %
80 mm	1.2126	7.2126	8.4280 N·mm/°	-10.51 %

Bending not torsion, an empirical constant, and the diameter that shrinks onto the arbor

The wire is in bending, not torsion, despite the name. A torsion spring is called that because the SPRING carries a torque, not because the wire is twisted. Winding the coil tighter bends each element of wire about its own axis of curvature, exactly as a curved beam bends, and the rate is therefore proportional to Young’s modulus E rather than to the shear modulus G. That single fact changes three things: the rate formula, the stress formula (32M/πd³, a bending stress, against 8FD/πd³ in torsion), and the allowable, which is about 75 per cent of minimum tensile in bending against 45 per cent in shear. A page that uses G here is out by a factor of E/G, which for steel is about 2.6.

10.8 is not the number the mathematics gives. Integrate the bending strain energy of a coil of wire and the rate comes out as Ed⁴/(10.186·D·n) per revolution, the constant being exactly 32/π. Every spring maker publishes 10.8, which is six per cent softer, and the six per cent is friction: the coils of a torsion spring slide on one another as it winds, and the measured rate is consistently below the frictionless one. Both figures are on this page, and the honest reading is that the 10.8 is the one to design with and the 10.186 is the one that tells you why your measurement will not repeat exactly.

A torsion spring must be wound so the load closes it. Every quantity on this page goes the wrong way if the load opens the coil. The residual stresses from coiling are favourable in the closing direction and unfavourable in the opening one, so the usable stress is lower. The coil GROWS as it unwinds instead of shrinking, so it comes off the arbor rather than onto it. The body shortens instead of lengthening. And the spring tends to work its way off the end of the arbor. If a mechanism genuinely has to work the other way, the answer is a spring wound the opposite hand — not a derating and not a bigger spring.

The coil diameter shrinks as it deflects, and this is what seizes springs. The wire does not stretch, so its length πDn is constant. Wind the spring up by θ turns and it has n + θ coils of the same wire, which it can only do by getting smaller: D′ = D·n/(n + θ). That is conservation of wire length, not an empirical correction, and it is exact. On this page’s example a quarter turn takes 4.7 per cent off the inside diameter. An arbor sized to the free inside diameter is an arbor the spring grips at full deflection, and a gripped torsion spring is a spring whose rate is suddenly whatever the friction is. Size the arbor from the WOUND-DOWN diameter with clearance — ten per cent is the published rule — and remember that the body lengthens at the same time, so the pocket needs axial room too.

What this page does not cover. Fatigue: these are static allowables and a torsion spring cycling in the millions is a different calculation in which the leg-to-body transition dominates. Legs that are not straight cantilevers, which the (l₁+l₂)/(3πD) allowance does not describe. Double torsion springs and springs with a centre bend, which are two springs in parallel about a common axis. Friction against the arbor, which adds hysteresis to the torque–angle line that this page draws as a straight one. And the free angle itself, which carries a manufacturing tolerance wide enough that anything positional should be specified as a torque at an angle rather than as a rate and a free position.

Frequently asked questions

Why does a torsion spring use E and not G?

Because the wire is bent, not twisted. The name refers to the torque the spring as a whole carries; the wire itself is a curved beam being bent about its own axis of curvature as the coil winds tighter. Bending means Young’s modulus, so the rate is Ed⁴/(10.8·D·n) rather than Gd⁴/(8D³n). It also means the stress is a bending stress, 32M/πd³, and the allowable is the bending one — about 75 per cent of minimum tensile for steel rather than the 45 per cent a compression spring gets in shear.

Is the rate per turn, per degree or per radian?

All three get published, and the constant changes with the unit, which is where the mistakes come from. Ed⁴/(10.8·D·n) is a rate PER REVOLUTION. Divide by 360 for per degree, which makes the constant 3,888; divide by 2π for per radian, which makes it 67.86. At least one published manufacturer’s guide prints the 10.8 form and labels it N·mm per degree, which is wrong by a factor of 360. This page prints all three side by side.

How do I size the arbor?

From the WOUND-DOWN inside diameter, not the free one. The wire length is constant, so winding the spring up by θ turns makes the mean diameter D·n/(n + θ) and the inside diameter shrinks with it. Take the value at your maximum deflection, subtract ten per cent for clearance, and that is the largest arbor. Getting this wrong does not usually break anything — it produces a spring whose torque is unpredictable because part of the load is going into friction against the arbor, and which is noticeably worse going back than coming.

Why does 10.8 appear rather than 10.2?

Because 10.2 is the frictionless answer and springs have friction. Integrating the bending strain energy of the coil gives a constant of exactly 32/π = 10.186 per revolution. Measured torsion springs come out about six per cent softer, because the coils slide against one another as the spring winds, and 10.8 is the empirical constant that matches. Both are computed on this page. Design with 10.8; expect the measured rate to scatter, because friction is what the difference is made of.

Do the legs affect the rate?

Yes, and more than people expect. A straight leg is a cantilever carrying the same moment the coils do, so it adds compliance, and the published allowance is (l₁+l₂)/(3πD) extra coils. On this page’s six-coil example two 30 mm legs are worth 0.455 of a coil, about 8 per cent of the body; at 80 mm legs it is 1.21 coils. Leave it out and the spring is softer than calculated. The allowance assumes straight legs loaded perpendicular at their ends, so a bent or hooked leg needs its own treatment.

What happens if the spring is loaded the wrong way round?

Everything goes the wrong direction at once. The coiling residual stresses, which help in the closing direction, hurt in the opening one, so the usable stress drops. The coil diameter grows instead of shrinking, so the spring lifts off its arbor and wanders. The body shortens instead of lengthening. And the spring tends to walk off the end of the arbor. There is no correction factor for this; the answer is a spring wound the other hand.

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.
WNJ Spring Machine, Torsion Spring Design: Equations, Materials & Machine Guide. The source for the mandrel rule — at least ten per cent clearance between the deflected inside diameter and the mandrel — and for the leg allowance L/(3πD). A WARNING about it: it prints R = Ed⁴/(10.8 D N) with units of “N·mm/°”. With the constant 10.8 that rate is per REVOLUTION, not per degree; per degree the constant is 3,888. This page prints all three unit conventions side by side for exactly that reason.
Master Spring & Wire Form Co., Material Data and Torsion Spring Design. The second published modulus and maximum-temperature table, carried because it DISAGREES with Suhm’s on two rows — 300 °F against 250 °F for oil-tempered wire and 600 °F against 500 °F for type 302 stainless — and on the shear modulus of phosphor bronze (6.25 against 6.0 Mpsi). Both readings are printed on the wire page.
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.
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”.