Spring rate (spring constant): formula and calculation (Hooke's law)

The spring rate k (also called stiffness) measures how much force a spring needs to deflect one unit of length. It's the single most important property of a spring: it defines whether it is 'stiff' or 'soft'.

Hooke's law

Hooke's law states that, within the elastic range, force is proportional to deflection. The constant of proportionality is k. In a compression spring, compressing 10 mm with k = 5 N/mm takes 50 N.

Helical spring rate formula

For a round-wire helical spring, k depends on the material's shear modulus G, the wire diameter d, the mean diameter D and the number of active coils Na. Note the power: wire diameter enters to the fourth — doubling d multiplies stiffness by 16.

What raises or lowers the rate

• Thicker wire (larger d): raises k strongly (d⁴ effect).
• Larger diameter (larger D): lowers k (1/D³ effect).
• More active coils (larger Na): lowers k proportionally.
• Material: G varies little across steels (~79.3 GPa), more for stainless (~69 GPa).

Worked example

A steel spring (G = 79.3 GPa) with d = 2 mm, D = 18 mm and 6 active coils has k ≈ 79300 · 2⁴ / (8 · 18³ · 6) ≈ 4.5 N/mm. To make it stiffer without changing size, the most efficient path is increasing the wire diameter.

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