A torsion spring is the one spring type that almost everybody calculates with the wrong modulus. It looks like a compression spring, it is wound like a compression spring, and it is named after torsion — but the wire inside it is not being twisted. When you wind the legs of a torsion spring towards each other, the wire is loaded as a long curved beam in bending. That single fact changes the stiffness equation from one that uses the shear modulus G to one that uses Young's modulus E, and it changes the stress from a shear stress to a bending stress on the inside surface of the coil. Get the modulus wrong and the rate comes out roughly two and a half times too low.
This calculator from Arb Digital takes the geometry and the material figures you supply and returns the angular rate, the torque at a stated deflection, the corrected peak bending stress and how much of your allowable stress that consumes. It deliberately overlaps with nothing else on the site: the spring rate calculator handles helical compression and extension springs, where the wire really is in torsion and the rate does depend on G; the Hooke's law calculator handles the linear force-displacement case in one dimension; and the torque calculator resolves a force and a lever arm rather than deriving a spring's stiffness. Sections and shafts loaded in real torsion belong to the polar moment of inertia calculator, which gives J, the torsion constant and the angle of twist — that is a different job from this one, and the boundary is worth holding on to.
What This Torsion Spring Calculator Does
You give it four geometric and material inputs — wire diameter, mean coil diameter, active coil count and Young's modulus — plus a deflection angle and your own allowable bending stress. From those it computes the spring index, the angular rate in newton-millimetres per degree and per revolution, the torque developed at your deflection, the Wahl-corrected bending stress at the inner fibre of the coil, the percentage of your allowable stress that represents, and the strain energy stored.
It also reports the numbers people forget. Winding a torsion spring in the closing direction makes the body shrink: the coil diameter reduces and the body length grows, so a spring that fits its arbor freely at rest can bind on it under load. The tool computes the wound-down inside diameter from the published relation so you can compare it against your arbor. It shows the rate on the pure beam-theory basis alongside the design basis with the friction allowance, because those two numbers differ by about six per cent and the difference is often mistaken for a manufacturing fault.
How to Use It
- Measure or specify the wire and coil. Wire diameter is straightforward. Mean coil diameter is the outside diameter minus one wire diameter, or the inside diameter plus one wire diameter — not the outside diameter itself, which is the single commonest input error on this page.
- Count the active coils. For a torsion spring this is the number of coils in the body. If the straight arms are significant, add their combined length in the arms box and the tool converts it to an equivalent coil count so the rate reflects the whole bent length.
- Enter Young's modulus for your wire. Not the shear modulus. Take it from the wire specification for the exact alloy and temper you are buying; drawn spring wire, stainless and phosphor bronze differ, and cold work changes it.
- Set the deflection and your allowable stress. Deflection is in the wind-up direction from free. The allowable stress must come from your own reading of the wire specification, your duty cycle and your fatigue requirement.
- Read the rate, the torque and the stress share together. A rate that suits the mechanism is useless if the stress share is over 100 per cent, and a comfortable stress share is useless if the torque is wrong at working deflection.
The Formula: How a Torsion Spring Rate Is Calculated
Treat the coiled wire as a beam of length L = πDN carrying a constant bending moment M. Standard beam theory gives the end rotation as θ = ML/(EI), and for round wire the second moment of area is I = πd4/64. Substituting and rearranging gives the rate per radian:
k = M/θ = Ed4/(64DN) in newton-millimetres per radian, with E in N/mm2 and every length in millimetres. Multiply by 2π to work per revolution, which gives Ed4/(10.186DN). Spring design practice replaces 10.186 with 10.8, an empirical allowance for friction between adjacent coils and between the spring body and its arbor, so the working design form is krev = Ed4/(10.8DN). Both are on this page because both are in circulation and a six per cent discrepancy between a calculation and a test is usually just this.
The stress is a bending stress, σ = Mc/I = 32M/(πd3) for a straight beam, but the wire is curved, and curvature concentrates stress on the inner fibre. The Wahl bending correction for round wire at the inside of the coil is Kbi = (4C2 − C − 1)/(4C(C − 1)), where C = D/d is the spring index. The bending relation itself is the standard one set out in the OpenStax treatment of stress, strain and elastic modulus, and the curved-beam behaviour that the correction factor captures is developed in the Cambridge teaching material on bending and torsion of beams.
Worked example, matching the defaults. With d = 2 mm, D = 16 mm, N = 8 and E = 200 GPa = 200,000 N/mm2: d4 = 16, so Ed4 = 3,200,000. The design rate per revolution is 3,200,000/(10.8 × 16 × 8) = 3,200,000/1,382.4 = 2,314.81 N·mm per turn, which is 6.4300 N·mm per degree. At 90° the torque is 578.70 N·mm. The index is C = 8, so Kbi = (256 − 8 − 1)/(4 × 8 × 7) = 247/224 = 1.10268. The uncorrected stress is 32 × 578.70/(π × 8) = 736.83 MPa, and the corrected inner-fibre stress is 812.5 MPa — 67.7 per cent of a 1,200 MPa allowable. Strain energy is ½Mθ = ½ × 578.70 × 1.5708 = 454.5 mJ. Those figures were worked out by hand before the code was written and the page reproduces them exactly.
Wire Diameter Is the Only Input That Really Matters
The rate goes as d4. Nothing else in the equation has anything like that leverage. Increase the wire from 2.0 mm to 2.2 mm — a ten per cent change you can barely see — and the rate rises by 46 per cent. Drop to 1.8 mm and it falls by 34 per cent. This is why torsion springs are tuned by coil count and coil diameter in production rather than by wire size: N and D are both first-power terms, so they give a designer proportional, controllable adjustment, while wire diameter is a coarse step.
Spring Index: Why 4 to 12 Is the Practical Band
The spring index C = D/d decides how manufacturable the spring is and how hard the inner fibre is working. At low index — a fat wire on a small coil, say C = 3 — the Wahl bending factor climbs above 1.3, so the inner surface carries thirty per cent more stress than the straight-beam formula suggests, and the wire is difficult to wind without cracking or excessive residual stress. At high index — thin wire on a big coil, C above about 15 — the correction factor falls close to 1.0, but the spring becomes floppy, tangles in handling, and its coils vary more from piece to piece.
Direction of Wind, and Why It Is Not Optional
A torsion spring must be loaded in the direction that closes the coils. Wind it that way and the body diameter reduces, the coils press together, and the residual stresses left by the coiling process work in your favour. Load it the other way, opening the coils, and the residual stress from forming now adds to the applied stress instead of subtracting from it, so the usable stress range drops sharply and fatigue life falls with it.
The dimensional consequence is the one that bites in assembly. As the spring winds down, the inside diameter shrinks according to the published relation IDloaded = N × IDfree/(N + θrev), where θrev is the deflection in revolutions. Wind a spring with eight coils through 90 degrees, a quarter turn, and the inside diameter drops by about three per cent. The tool reports the loaded inside diameter in the note, and an arbor sized to the free inside diameter will be gripped at working deflection. Body length grows at the same time, because the same wire length occupies a smaller diameter.
Fatigue, Set and the Limits of a Static Calculation
Everything on this page is a static calculation. It tells you the stress at one deflection. It says nothing at all about how many cycles the spring will survive, and for a spring that cycles, fatigue is normally the governing criterion rather than static strength. A cycling spring is assessed on the stress range between its installed and working deflections, not on the peak alone, and the allowable range for a given life comes from published fatigue data for that wire in that surface condition.
Where This Calculation Stops
The arms are the other simplification. Straight arms bend as cantilevers under load, adding compliance the coil-body equation does not include, and a long arm can contribute more deflection than several coils. The equivalent-length input here handles the first-order case where the arms are short and reasonably stiff. Arms that are long, tapered, or loaded away from their tips need to be modelled as separate beams and combined in series with the body — our beam deflection calculator covers that side, and the Young's modulus calculator and stress strain calculator deal with the material properties both models need.
Arb Digital builds technical calculators and reference pages for engineering and manufacturing sites — the kind that earn links because the maths is correct and the assumptions are stated.
Browse the free tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using the shear modulus G. A torsion spring's wire is in bending. The rate uses Young's modulus E. Substituting G, which is roughly 0.385E for steel, makes the spring look two and a half times softer than it is.
- Entering the outside diameter as the mean diameter. Mean coil diameter is the outside diameter minus one wire diameter. On a small spring that error alone can shift the rate by twenty per cent or more.
- Skipping the Wahl bending correction. The straight-beam stress understates the inner-fibre stress by ten per cent at index 8 and by over thirty per cent at index 3. That gap is exactly where a spring fails.
- Loading the spring in the opening direction. It must be wound in the closing direction. Opening it adds the forming residual stress to the applied stress and cuts fatigue life sharply.
- Sizing the arbor to the free inside diameter. The body shrinks as it winds down. An arbor that is a neat fit at rest becomes an interference fit at working deflection, and the friction that follows destroys rate repeatability.
Related Free Tools From Arb Digital
For helical compression and extension springs, where the wire genuinely is in torsion and the rate depends on the shear modulus, use the spring rate calculator, with the shear modulus calculator for G itself. For a linear force-per-millimetre relation with no geometry involved, the Hooke's law calculator and the elastic potential energy calculator are the right pair. On the section side, the section modulus calculator gives the elastic and plastic section moduli for standard shapes, the polar moment of inertia calculator handles shafts in real torsion, and the shear stress calculator covers the shear side of a loaded section. Browse the full free online tools hub for the rest.
Frequently Asked Questions
Young's modulus E. Despite the name, the wire in a torsion spring is loaded in bending as a curved beam, not in torsion. Compression and extension springs are the ones that twist their wire and therefore depend on the shear modulus G.
Per radian the rate is E d to the fourth divided by 64 D N, with E in newtons per square millimetre and lengths in millimetres. Per revolution that becomes E d to the fourth over 10.186 D N in theory, and spring design practice uses 10.8 in the denominator to allow for coil and arbor friction.
The spring index C is the mean coil diameter divided by the wire diameter. It sets the Wahl bending correction factor, which raises the inner-fibre stress above the straight-beam value, and it governs how easily the spring can be wound. Most production springs sit between index 4 and 12.
Because winding a torsion spring in its working direction closes the coils onto a smaller diameter. The loaded inside diameter is the free inside diameter multiplied by N divided by N plus the deflection in revolutions, so an arbor sized to the free dimension will be gripped under load.
It should be loaded only in the direction that closes the coils. Loading it in the opening direction adds the residual stress left by the coiling process to the applied stress instead of subtracting it, which reduces the usable stress range and shortens fatigue life.
No. It is a static calculation that reports the stress at one deflection. A cycling spring is governed by the stress range between installed and working positions and by published fatigue data for that wire and surface condition, none of which is modelled here.
One is pure beam theory with 10.186 in the denominator; the other is the design value with 10.8, which includes an empirical allowance for friction between adjacent coils and against the arbor. The roughly six per cent gap between them is normal and often explains a mismatch between calculation and test.
Arms bend as cantilevers and add compliance the coil-body equation does not include. Entering their combined length converts them to an equivalent extra bent length, which handles short, stiff arms. Long or tapered arms should be modelled as separate beams and combined in series with the body.
This tool is provided for education and engineering estimation only. It evaluates published spring relations from figures you enter, and it does not select a material, verify a design, or confirm that any spring is fit for service. Young's modulus and the allowable bending stress must come from your wire specification, and any load-bearing or safety-related spring should be designed and signed off by a qualified mechanical engineer.
