The spring rate calculator above is a preliminary sizing and teaching tool, not a spring design. It derives the stiffness of a helical compression spring from its geometry and its material, rather than asking you for a rate you already know. That is the useful direction: a spring's rate is not something you choose directly, it is something that falls out of four decisions about wire, diameter, turns and material.
Arb Digital builds free physics calculators that expose the sensitivities rather than hiding them. This one prints the spring index and the Wahl stress-correction factor alongside the rate, because those two numbers tell you whether the spring you have just described can actually be made and whether the stress in it is anywhere near sensible.
What This Spring Rate Calculator Does
It computes the rate k in newtons per millimetre, and also in pounds per inch, from the wire diameter, the mean coil diameter, the number of active coils and the shear modulus of the wire. A helical compression spring works almost entirely in torsion — the wire twists rather than bends — which is why the shear modulus governs it and Young's modulus does not appear anywhere.
Alongside the rate, it runs a second calculation in whichever direction you need: a deflection from an applied load, or the load required to produce a given deflection. It then computes the shear stress in the wire at that condition, corrected by the Wahl factor for the stress concentration on the inside of the coil.
The spring index is the ratio of the coil diameter to the wire diameter, and it is the number a spring maker will look at first. Below about 4 the wire is very hard to coil without damaging it; above about 12 the spring becomes floppy, tangles in handling and is difficult to control dimensionally. Most manufacturable springs sit between 6 and 10.
How to Use It
- Measure the wire diameter carefully. The rate scales with the fourth power of it, so a two per cent measurement error becomes an eight per cent error in the answer.
- Use the mean coil diameter, not the outside diameter. Mean is measured to the centre of the wire, which is the outside diameter minus one wire diameter.
- Count active coils, not total coils. Closed and ground ends are not active. The usual convention is total coils minus two for squared and ground ends.
- Enter a shear modulus from your wire standard. Different steel wire grades differ, and stainless and non-ferrous alloys differ substantially more.
- Read the spring index before anything else. If it is outside roughly 4 to 12, the geometry may not be manufacturable regardless of what the rate says.
The Formula: How Spring Rate Is Calculated
For a helical compression spring of round wire, the rate is k = Gd4 ÷ (8D3n), where G is the shear modulus of the wire, d is the wire diameter, D is the mean coil diameter and n is the number of active coils. It comes from treating each element of the wire as a short torsion bar: the axial load applies a torque of FD/2 to the wire, and integrating the resulting twist along the whole wound length gives this expression. OpenStax University Physics Volume 1, section 12.3 on stress, strain and elastic modulus, sets out the shear relationship the derivation is built on.
The shear stress in the wire is τ = KW × 8FD ÷ (πd3), where KW is the Wahl correction factor KW = (4C − 1) ÷ (4C − 4) + 0.615 ÷ C, and C = D ÷ d is the spring index. The correction accounts for the fact that the inside of each coil is on a tighter radius than the outside, so the shear stress there is higher than the plain torsion formula predicts.
Work the defaults. With G = 79.3 GPa, d = 2 mm, D = 16 mm and n = 10, the rate is 79.3 × 109 × (0.002)4 ÷ (8 × (0.016)3 × 10) = 1.2688 ÷ 0.00032768 = 3,872 N/m, or 3.87 N/mm. The spring index is 16 ÷ 2 = 8, and the Wahl factor is 31/28 + 0.615/8 = 1.184.
Under a 20 N load the deflection is 20 ÷ 3,872 = 0.00517 m, or 5.17 mm. The corrected shear stress is 1.184 × 8 × 20 × 0.016 ÷ (π × 8 × 10−9) = 1.184 × 101.9 MPa = 120.6 MPa.
The Fourth Power Is the Whole Design
Look at the exponents. The rate goes with d to the fourth, with D to the minus third, and with n to the minus first. Those three sensitivities are wildly unequal, and knowing which lever to pull is most of what practical spring design consists of.
Increasing wire diameter by ten per cent multiplies the rate by 1.14 = 1.46. Increasing coil diameter by ten per cent multiplies it by 1.1−3 = 0.75. Adding one coil to ten reduces it by about nine per cent. If you need a large change in stiffness, the wire is the only realistic lever; if you need a fine adjustment, the coil count is the one to use.
The fourth power also means measurement precision matters far more than people expect. Wire is supplied to a tolerance, and the tolerance band on nominal 2 mm wire translates directly into a spread of several per cent in rate before anything else has gone wrong. That is why production springs are sorted by measured rate rather than trusted from a calculation.
Assumptions, and What Is Not Modelled
The formula assumes a helical compression spring of round wire, with a small helix angle, loaded axially, deflecting well within the elastic range, and with the ends free to rotate slightly as it compresses. The elastic requirement is the one that matters most: as OpenStax University Physics Volume 1, section 12.4 on elasticity and plasticity explains, once a material is loaded past its proportional limit it takes a permanent set, and a spring that has taken a set no longer has the rate it was made with.
It does not model buckling, which becomes a risk when the free length exceeds roughly four times the mean coil diameter for a spring with unguided ends. It does not model solid height, where the coils touch and the spring stops being a spring. It does not model fatigue, which for a cyclically loaded spring is usually the governing design consideration rather than static stress. It does not model set, relaxation, surging at resonance, or the effect of shot peening on fatigue life.
Extension springs are a different case again, because they carry an initial tension that has to be overcome before any deflection happens at all. Their rate follows the same formula, but their force-deflection line does not pass through the origin, so treating them with this page's output alone will misestimate the force at small deflections.
Why This Page Publishes No Allowable Stress Values
The calculator reports the shear stress in the wire and stops there. There is no table of allowable stresses, and that is deliberate: it follows the same rule as the breaker size calculator, which publishes no ampacity table. Permissible spring stress depends on the wire specification, the diameter, whether the spring has been shot peened, whether it is set-removed, whether the loading is static or cyclic, and the operating temperature. A single generic number covering all of that would be wrong far more often than it was right.
Those values live in the wire standard and the spring design standard that apply to your material and jurisdiction, and the margin against them is set through a safety factor or a fatigue criterion. The factor of safety calculator handles the ratio; the choice of allowable belongs to a qualified engineer working from the governing document.
Where This Sits Next to the Other Elasticity Tools
The Hooke's law calculator starts from a rate you already know and solves F = kx for force, stiffness or extension. This page produces that rate in the first place, from geometry and material. Used together, one designs the spring and the other predicts what it will do.
The shear modulus calculator is where the G value in the formula comes from, either measured from a test or derived from Young's modulus and Poisson's ratio. The shear stress calculator covers direct and transverse shear in structural members, which is a different geometry from the torsional shear in a spring wire. For the energy the compressed spring is holding, the elastic potential energy calculator takes the rate and the deflection and returns it.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using the outside diameter as the coil diameter — mean diameter is the outside diameter minus one wire diameter, and because the rate goes with D cubed the error is large.
- Counting total coils instead of active coils — closed and ground ends carry load but do not deflect, and the usual convention subtracts two from the total.
- Using Young's modulus instead of the shear modulus — a helical spring works in torsion, so G governs. For steel, E is roughly 2.6 times G, and the error is that big.
- Ignoring the spring index — a geometry with an index below 4 or above 12 may be impractical to manufacture whatever the calculated rate says.
- Treating the static stress as the design criterion for a cycling spring — fatigue usually governs a spring that moves in service, and it is a completely separate calculation.
Related Free Tools From Arb Digital
Take the rate into the Hooke's law calculator for force and extension, and the elastic potential energy calculator for the energy stored. Get the material constant from the shear modulus calculator, and compare against the structural cases in the shear stress calculator and the stress and strain calculator. The factor of safety calculator covers the margin, and the force converter moves loads between newtons, pounds-force and kilograms-force. Everything Arb Digital publishes is on the free online tools hub.
Frequently Asked Questions
It is the force needed to deflect a spring by one unit of length, usually quoted in newtons per millimetre or pounds per inch. For a linear spring it is constant throughout the working range, which is what makes a single number a useful description of the spring at all.
Because the rate depends on the fourth power of it. The wire is loaded in torsion, and its torsional stiffness goes with the fourth power of diameter, so a ten per cent increase in wire size makes the spring 46 per cent stiffer. No other input on the page has comparable leverage.
A coil that is free to deflect. Closed and ground end coils sit flat against the seat and do not contribute to the deflection, so for squared and ground ends the usual convention is that active coils equal total coils minus two.
A correction that accounts for the higher shear stress on the inside of a coil, where the wire is on a tighter radius, plus the direct shear from the load itself. It depends only on the spring index, and it is close to 1.18 at an index of 8 and rises sharply for tighter indices.
Most manufacturable springs sit between about 6 and 10. Below 4 the wire is very hard to coil without damage and the stress concentration becomes severe. Above 12 the spring becomes unstable in handling and difficult to hold to dimensional tolerance.
The rate formula is the same, but extension springs are usually wound with an initial tension that must be overcome before any deflection occurs. Their force-deflection line is therefore offset from the origin, so the load at a given deflection will be higher than this page predicts.
No. It reports a calculated stress at the condition you entered. Whether that stress is acceptable depends on the wire specification, the surface treatment, the temperature and whether the loading is static or cyclic, and that judgement belongs to a qualified engineer working from the applicable standard.
This tool is provided for educational and preliminary sizing use. It is not a spring design, it publishes no allowable stress values of its own, and it does not model buckling, solid height, fatigue, set or surge. Any load-bearing or safety-critical spring must be designed and verified by a qualified engineer.
