Spring Rate Calculator
Compute the spring rate (stiffness) of a helical compression spring from its wire and coil geometry. Use this during spring selection or custom spring design to match a required force-deflection characteristic.
About this calculator
The spring rate k of a helical compression spring is determined by the formula k = (G·d⁴) / (8·D³·n), where G is the shear modulus of the wire material, d is the wire diameter, D is the mean coil diameter, and n is the number of active coils. This equation comes from the theory of torsion in curved beams: the wire is primarily subjected to torsional shear as the spring deflects. Because d appears to the fourth power and D to the third power, small changes in wire diameter have a very strong effect on stiffness. For steel springs, G ≈ 79,000 MPa; for stainless steel, G ≈ 69,000 MPa. Once k is known, the deflection under a given load F is simply δ = F / k, following Hooke's Law.
How to use
Design a steel spring (G = 79,000 MPa) with wire diameter d = 3 mm, mean coil diameter D = 25 mm, and n = 10 active coils. k = (G·d⁴) / (8·D³·n) = (79,000 × 3⁴) / (8 × 25³ × 10). Numerator: 79,000 × 81 = 6,399,000. Denominator: 8 × 15,625 × 10 = 1,250,000. k = 6,399,000 / 1,250,000 ≈ 5.12 N/mm. If a 50 N load is applied, the spring deflects δ = 50 / 5.12 ≈ 9.8 mm.
Frequently asked questions
What is the formula for calculating helical compression spring rate?
The spring rate formula is k = (G·d⁴) / (8·D³·n), where G is the shear modulus of the spring wire, d is the wire diameter, D is the mean coil diameter (center-to-center of the wire), and n is the number of active coils. Active coils are those that actually deflect under load — closed end coils that are ground flat are typically not counted as active. This formula assumes the wire behaves purely in torsion, which is valid for springs with a spring index (D/d) greater than about 4.
How does changing wire diameter affect spring rate?
Wire diameter has the most powerful influence on spring rate because it appears as d⁴ in the numerator. Doubling the wire diameter increases the spring rate by a factor of 2⁴ = 16, all else being equal. This makes wire diameter the primary design lever when a large change in stiffness is needed. Conversely, the mean coil diameter D appears as D³ in the denominator, so increasing D reduces stiffness significantly — a larger coil diameter makes for a softer spring.
What value of shear modulus should I use for different spring materials?
For standard carbon steel music wire (the most common spring material), use G ≈ 79,000–81,000 MPa. For 302/304 stainless steel springs, use G ≈ 68,000–69,000 MPa, which results in a noticeably softer spring for the same geometry. Phosphor bronze springs have G ≈ 41,000 MPa, making them considerably more flexible. Always use the shear modulus, not the tensile (Young's) modulus, in the spring rate formula because the wire deforms primarily in torsion.