Compression spring design: spring rate formula, spring index, free length and solid height (with worked example)

The spring rate formula: k = G·d⁴ / (8·D³·n)

When a helical compression spring is squeezed, the wire itself is loaded in torsion. Treating each coil as a twisted bar gives the spring rate (stiffness) k — the force needed per millimetre of deflection:

k = F / δ = G·d⁴ / (8·D³·n)　　δ = 8·F·D³·n / (G·d⁴)

d is the wire diameter, D the mean coil diameter (through the wire centre), n the number of active coils and G the shear modulus of the material (N/mm²). Catalogues usually list the outer diameter OD or inner diameter ID: D = OD − d = ID + d. Only active coils deflect; the closed dead coils at the ends are not counted in n but are included in the total coils Nt.

Wire diameter acts to the fourth power and coil diameter to the third, so change d or D to move the rate a lot and fine-tune with the coil count. A music-wire spring with d = 1.5 mm, D = 13.5 mm and n = 8 has k = 78500 × 1.5⁴ / (8 × 13.5³ × 8) ≈ 2.52 N/mm.

TIPWire tolerance matters more than people expect: ±0.02 mm on a 1 mm wire changes k by about ±8%. When the force must be accurate, specify a load at a given length with a tolerance and let the spring maker adjust the coils, rather than specifying k alone.

Spring index C = D/d and the Wahl correction

The spring index C describes how tightly the spring is wound. A small C bends the wire severely during coiling, concentrates stress on the inside and makes the spring hard to manufacture; a large C gives a soft spring that tangles, buckles easily and is hard to hold to tolerance on OD and free length. Design for 4–12, ideally 6–10. The spring generator warns when C is below 4 or above 16.

C = D / d　　Kw = (4C − 1) / (4C − 4) + 0.615 / C

Once wire is coiled, the shear stress on the inside of the coil is higher than in a straight bar in torsion. The Wahl factor Kw corrects for both curvature and direct shear — the smaller C, the larger the correction:

TIPWhen space is tight, fix the OD (or the ID and the guide rod) first, then choose the wire for C = 6–10. For OD 15 mm, d = 1.5 mm gives D = 13.5 mm and C = 9 — right in the ideal range.

End types, free length and solid height

Free length L0 is the unloaded length; solid height Ls is the length when every coil touches its neighbour and the spring cannot compress further. L0 − Ls is the maximum available deflection. The end type sets how many coils are inactive (end coils Ne) and how the lengths are calculated:

The formulas follow Shigley's Mechanical Engineering Design; the closed-end rules match the spring generator, which adds one dead coil with pitch equal to the wire diameter at each end, while grinding removes about half a wire from each end. Closed ends let the spring stand square; ground ends also make the seat perpendicular to the axis with minimum wobble — the choice for precision mechanisms.

• Keep the shortest working length at least 10–15% of the stroke above Ls. Never run a spring solid: the coils clash and the stress jumps.
• Work between about 20% and 80% of the maximum deflection: too little and the end coils seat unevenly and the rate is non-linear; too much and you are close to solid.
• Grinding plain ends leaves a tiny seat, so the generator recommends closed ends instead.

Materials and shear modulus G: SWP, SUS304, phosphor bronze

G is the shear modulus (modulus of rigidity), related to Young's modulus by G = E / (2(1 + ν)). The table lists the design values used by JIS B 2704. Tensile strength varies strongly with wire size (thinner wire is stronger); the column gives indicative values around ⌀1 mm — always check the wire standard.

τ = Kw · 8·F·D / (π·d³)

For static loads (few cycles), keep the Wahl-corrected shear stress below about 45% of the tensile strength for music and hard-drawn wire, or 35% for stainless — Shigley's guideline for springs without set removal. Fatigue applications with hundreds of thousands of cycles need lower stresses; use the spring maker's fatigue diagrams.

Buckling check

Stable if L0 < 5.26·D (flat parallel ends, guided)　　L0 < 2.63·D (pivoted ends)

A slender spring bows sideways like a column when compressed. Beyond these ratios, add a guide rod inside or a sleeve outside, with enough clearance to avoid rubbing.

TIPStainless has about 13% lower G than music wire, so swapping materials at the same size drops k by about 13%; remove a coil (e.g. 8 → 7) to keep the rate. The generator uses G = 79.3 GPa (generic steel), about 1% above the JIS value of 78.5 GPa.

Worked example: from load requirement to wire, coils and free length

Requirement: the spring sits on a ⌀8 mm guide rod inside a ⌀16 mm bore. It must give 10 N preload at an installed length of 25 mm and 25 N at 19 mm (6 mm stroke). Material: music wire SWP-B, closed ends.

1. Spring rate: k = (25 − 10) / (25 − 19) = 15 / 6 = 2.5 N/mm.
2. Size: OD 15 mm (1 mm clearance to the ⌀16 bore) with 1.5 mm wire gives D = 13.5 mm, ID 12 mm (ample clearance to the ⌀8 rod) and C = 9.
3. Active coils: n = G·d⁴ / (8·D³·k) = 78500 × 5.0625 / (8 × 2460.4 × 2.5) ≈ 8.08, so n = 8 and the actual k ≈ 2.52 N/mm.
4. Free length: L0 = 25 + 10 / 2.52 ≈ 29.0 mm; loads at 25 mm and 19 mm are about 10.1 N and 25.2 N.
5. Solid height: closed ends give Nt = 10 and Ls = (10 + 1) × 1.5 = 16.5 mm. The shortest working length of 19 mm is 2.5 mm above Ls (about 42% of the stroke), and the working points sit at 32–80% of the 12.5 mm maximum deflection.
6. Pitch: p = (L0 − 3d) / n = (29 − 4.5) / 8 ≈ 3.06 mm; with ground ends p = (29 − 3) / 8 = 3.25 mm and Ls = 15 mm.
7. Stress: Kw = 1.162, so at F = 25 N τ = 1.162 × 8 × 25 × 13.5 / (π × 1.5³) ≈ 296 MPa; at solid (F ≈ 31.5 N) τ ≈ 373 MPa — far below 0.45 × about 2000 MPa ≈ 900 MPa.
8. Buckling: L0 / D = 29 / 13.5 ≈ 2.15 < 2.63, so it is stable even with unguided ends.

Enter wire 1.5, OD 15, 8 active coils, pitch 3.06 and closed ends in the spring generator to download a STEP / STL for interference checks. Its spec card shows k ≈ 2.55 N/mm because it uses G = 79.3 GPa. The OD grows only about 0.03 mm when compressed, so 1 mm of bore clearance is plenty.

TIPWhen ordering from a spring maker, state the material, wire d, OD or ID (and the rod or bore it works on), free length L0, active or total coils, end type, and a load at a length with tolerance (e.g. 25 N ± 10% at 19 mm). Specifying the load is easier to make than fixing L0, n and k all at once.