Compression Spring Rate Formula (with worked example)

The formula

k = G · d⁴ / (8 · D³ · Na)

• k — spring rate (N/mm)
• G — shear modulus of the wire (steel is about 79,300 MPa; check your material data)
• d — wire diameter (mm)
• D — mean coil diameter = outer diameter − d (mm)
• Na — active coils (total coils minus the end coils that do not deflect)

Worked example

Steel spring, wire d = 2 mm, outer diameter 20 mm, so D = 18 mm, with 10 active coils:

k = 79300 × 2⁴ / (8 × 18³ × 10) ≈ 2.72 N/mm. Force at 10 mm deflection ≈ 27.19 N.

Spring index C = D/d = 9. Springs with an index of roughly 4 to 12 are generally easier to manufacture consistently. With closed and ground ends, total coils = Na + 2 = 12, so solid height ≈ 12 × 2 = 24 mm.

These are theoretical values; end conditions, tolerances and set removal affect real springs. For extension and torsion springs or exact stress checks, use the free Spring Rate Calculator.