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MachineCalcs

Spring rate formula explained

Skip the math — use the compression spring calculator .

The spring rate k is the force a helical compression spring produces per unit of deflection — its stiffness, in N/mm or lbf/in. For a round-wire helical spring it is:

k = G · d⁴ / (8 · D³ · Nₐ)

where:

  • G — the material’s shear modulus (≈ 79.3 GPa / 11.5 Mpsi for music wire).
  • d — the wire diameter.
  • D — the mean coil diameter (outside diameter minus one wire diameter).
  • Nₐ — the number of active coils (total coils minus the inactive end coils).

Why wire diameter dominates

Wire diameter appears to the fourth power, while coil diameter appears to the third power in the denominator. So a small change in wire size has an outsized effect: going from 1.0 mm to 1.2 mm wire (a 20% increase) multiplies the rate by 1.2⁴ ≈ 2.07 — it more than doubles the stiffness. If a spring is far too soft or too stiff, the wire diameter is the first lever to reach for.

Worked example

Music wire, d = 1.0 mm, mean coil D = 10 mm, Nₐ = 8 active coils, with G = 79.3 GPa:

k = 79 300 · 1⁴ / (8 · 10³ · 8) = 79 300 / 64 000 ≈ 1.24 N/mm

So every millimetre of compression takes about 1.24 N. At 10 mm of deflection the force is about 12.4 N. (This is exactly what the calculator returns for these inputs.)

Springs in series and parallel

Two springs in parallel (side by side) add their rates: k = k₁ + k₂. Two springs in series (end to end) combine like resistors in parallel: 1/k = 1/k₁ + 1/k₂, so the stack is softer than either spring alone.

The rate is only half the story — you also need to check that the corrected shear stress stays under the material’s allowable and that the spring won’t buckle. The calculator does both alongside the rate.

Frequently asked questions

What is the formula for spring rate?
For a helical compression spring, k = G·d⁴/(8·D³·Nₐ), where G is the shear modulus, d the wire diameter, D the mean coil diameter and Nₐ the number of active coils.
Why does wire diameter matter so much?
Wire diameter enters to the fourth power, so doubling the wire diameter makes the spring 16× stiffer. It is the single most powerful way to change spring rate.
How do I make a spring softer without changing the wire?
Increase the mean coil diameter (rate falls with the cube of D) or add active coils (rate is inversely proportional to Nₐ).

use the compression spring calculator.

Last reviewed: 2026-05-29.