Spring Rate Calculator
Calculate spring rate (spring constant k), deflection, and force using Hooke's Law.
Find coil spring rate from wire diameter, coil diameter, and number of coils.
What Is Spring Rate? Spring rate (k) is the force required to deflect a spring by one unit of length. k = F / x (force / deflection). Units: N/mm, N/m, lb/in, lb/ft. A stiffer spring has a higher k. A softer spring has a lower k. Named after Robert Hooke, English physicist, who stated this law in 1678.
Hooke’s Law F = k × x F = force applied to spring (N or lb) x = deflection (mm or inches) k = spring rate (N/mm or lb/in) Valid only in the elastic region — springs have a maximum load (solid height or yield point).
Coil Spring Rate Formula For a cylindrical coil spring: k = (G × d⁴) / (8 × D³ × N) G = shear modulus of spring wire material d = wire diameter D = mean coil diameter (outer diameter minus wire diameter) N = number of active coils This formula is used by spring manufacturers worldwide.
Shear Modulus (G) for Common Spring Materials Steel (music wire, ASTM A228): G = 81.5 GPa (11.8 × 10⁶ psi) Stainless steel (17-7 PH): G = 69 GPa (10.0 × 10⁶ psi) Chrome-silicon steel (ASTM A401): G = 80.7 GPa (11.7 × 10⁶ psi) Phosphor bronze: G = 41.4 GPa (6.0 × 10⁶ psi) Titanium alloy: G = 40 GPa (5.8 × 10⁶ psi) Beryllium copper: G = 48 GPa (7.0 × 10⁶ psi)
Active Coils Total coils = active coils + inactive (closed) end coils. Compression spring with closed ends: 2 inactive coils → N_active = N_total − 2. Extension springs: all coils are usually active.
Spring Index Spring index C = D/d (coil diameter / wire diameter). Typical range: C = 4 to 12. Low C (<4): very stiff, difficult to manufacture, high stress. High C (>12): prone to buckling, less stable.
Series and Parallel Springs Springs in series (end-to-end): 1/k_total = 1/k₁ + 1/k₂ + … → k_total < smallest individual spring rate Springs in parallel (side-by-side): k_total = k₁ + k₂ + … → k_total > largest individual spring rate