Spring Rate Calculator — Stiffness by Design
A coil spring’s stiffness is set by its geometry before it is ever wound: wire thickness to the fourth power fights coil diameter cubed, per active coil. Enter the spring’s dimensions, and this calculator returns its rate — plus the force at any deflection you specify.
A Bar Twisted Into a Helix
A coil spring is a torsion bar in disguise: compressing the spring twists the wire, coil by coil, and the stiffness sums that twisting into k = Gd⁴ ÷ (8D³N). Read the exponents as design levers: wire diameter d at the fourth power (the torsion bar's own stiffness scaling), coil diameter D cubed against you (a bigger coil is a longer lever arm twisting the wire), active coils N dividing the rate (each coil contributes its share of twist in series). The formula is exact for the ideal round-wire helix at modest deflections, and it is how every suspension spring, valve spring and ballpoint clicker was specified before it was wound: pick the rate the job needs, choose wire and coil to satisfy the equation within the space allowed, and check the wire's torsional stress at full travel. The calculator runs that specification loop forward — geometry in, rate and load out.
Worked Example: Five-Millimetre Wire, Six Coils
The defaults wind a plausible workshop spring: d = 5 mm, D = 40 mm, N = 6 active coils, steel (G = 79 GPa). k = 79×10⁹ × (0.005)⁴ ÷ (8 × (0.04)³ × 6) = 16.07 N/mm (16,073 N/m), and at the 20 mm deflection entered, the load is 321 N — about 33 kg of push, a suspension corner's small cousin. The levers' drama is worth a trial each: doubling the wire to 10 mm multiplies the rate sixteenfold (257 N/mm — undriveable in the same envelope), while doubling the coils to 12 merely halves it. The fourth power is the tyrant of spring design, which is why springs are specified to hundredths of a millimetre of wire and why a substituted "close enough" spring betrays itself instantly in the ride.
Rate, Preload and the Working Range
The rate is the slope; the spring's life is lived between two fences. Below, installation usually pre-compresses the spring (preload) so it never goes slack — the force at ride height is k × (installed compression), not zero, and the calculator's deflection field measures from free length, so installed forces need the installation's total compression entered. Above, the solid height (all coils touching, length = d × total coils) ends travel absolutely, and approaching it spikes both rate and stress; sane designs keep working deflection under about 75–80% of the distance to solid. Between the fences the spring is this page's linear citizen. The stress check — torsional stress ≈ 8FD ÷ (πd³), corrected by the Wahl factor for the wire's curvature — is the other half of the specification, named here because a rate without a stress audit is half a spring design.
The Suspension Tuner's Dialect
Chassis engineers speak of spring rate one step removed — as wheel rate, the stiffness measured at the tyre contact patch after the suspension linkage's motion ratio has had its say (the spring often moves a fraction of the wheel's travel, and the squares of ratios apply). A 16 N/mm coil like this page's worked example might present as 4 N/mm at the wheel through a 2:1 linkage — the same spring, a different dialect, converted by geometry. The calculator prices the spring on the bench; the wheel rate is that number after the car has translated it, and confusing the two dialects mis-specs a suspension by the ratio squared, an error this page prevents by defining its terms on its face.
The Mattress and the Trampoline
Domestic springs confess the same exponents: a mattress's pocket springs are soft because their wire is thin (the fourth power, forgiving at 1.5 mm) and their coils many; a trampoline's perimeter springs are the same formula preloaded hard, their rate chosen so the bed's centre deflection stores the jumper's energy and returns it on the beat. Even the humble retractable pen obeys — its spring's few newtons per millimetre are this equation at d = 0.35 mm, where the fourth power makes the rate exquisitely sensitive to the wire-drawing die's mood. The calculator scales from the pen to the lorry suspension without changing a symbol: that is what a derived formula, honestly exponented, buys its users. It also disciplines the modifier's itch: anyone tempted to swap springs by eye can first run both geometries through these five fields and learn whether the bargain is a tuning or a trap — the fourth power having ended more amateur suspensions than any pothole.
Related Tools
More from the Physics Calculators hub: the Young's Modulus Calculator — the tensile sibling of this page's shear modulus — and the Pulley Calculator, the other machine in this wave that trades force against distance.
Frequently Asked Questions
Why does cutting a spring make it stiffer?
Because N divides: chop two coils off a spring (a disreputable car-lowering method) and the remaining coils each twist more for the same travel — the rate rises in proportion to the coils removed, 6 → 4 active coils multiplying k by 1.5 — while the travel to solid shrinks doubly fast. The calculator prices the vandalism exactly: enter the reduced N and read the new rate. The same physics, used honestly, is how one spring blank is tuned into a product family: wind extra coils, close the ends to taste, and the rate is trimmed by the count that remains active.
What are active coils, precisely?
The coils free to flex: closed-and-ground ends sit flat against their seats and contribute almost no twist, so a spring with 8 total coils and two closed ends works as N = 6. Miscounting N is the commonest input error in this calculation (after measuring D to the outside instead of the centreline — the label insists on mean diameter for the same reason). When a measured spring disagrees with the calculator, audit N and D first; the formula has been right since the nineteenth century, and the discrepancy is nearly always in what was counted.
How do progressive springs work?
By making N a variable: coils are wound at varying pitch so that as load rises, the closest coils touch down (bind) progressively, removing themselves from the active count — N falls, k rises, and the spring stiffens with travel in a designed curve. Variable-diameter (barrel and conical) springs play the same trick geometrically. The calculator evaluates the constant-rate ideal at any stated N; the progressive spring is that evaluation staged along its travel, a staircase of this page's answers — which is how their designers in fact specify them.
Does the rate change with temperature?
Through G, gently: steel's shear modulus falls a few percent per hundred degrees, so a valve spring in a hot engine runs measurably softer than its bench figure (and creep-relaxation at temperature bleeds preload over time — the "sagged" spring of an old engine). At the cold end, G rises slightly and some alloys embrittle. The calculator's G input absorbs the correction when it matters: enter the hot modulus for the hot answer. For room-temperature machinery the standard 79 GPa of the label suffices, which is why it is the default.
What about extension and leaf springs?
Extension springs use the identical rate formula (the coils are wound pre-twisted shut, adding an initial tension the load must first overcome — a preload built into the winding); leaf springs are beams, not helices, and their rate comes from beam deflection (the EI of the Young's modulus page) rather than this page's torsion sum. Torsion springs — the clothespin's and the mousetrap's — rate in moment per radian by a bending formula of their own. Helical compression and extension springs are this calculator's stated jurisdiction, which covers the great majority of springs a reader will ever spec.
How do I measure a spring I already own?
Load it and measure: hang a known mass, note the deflection, and k = F ÷ x — two points (a second mass) verify the linearity this page assumes. A bathroom scale and a stack of books perform the audit respectably for soft springs; a press with a load cell does it for stiff ones. Compare the measurement with the calculator's geometry prediction: agreement within a few percent validates both the count of active coils and the wire measurement, and disagreement points at exactly the inputs the FAQ above warns about — the formula and the scale auditing each other, which is how spring catalogues maintain their honesty.
Who derived the spring formula?
The torsion-based analysis of helical springs was standardised in the mid-nineteenth century as wire-drawing made uniform spring wire an industrial commodity — the names attached are the era's engineer-analysts rather than a single inventor, with A. M. Wahl's twentieth-century stress-correction factor (for the wire's curvature) the last essential refinement. The spring is older than its formula by millennia (bow, trap, carriage leaf), but the wound coil at industrial scale waited for consistent wire — a reminder that calculators, like springs, are made of their materials' uniformity.