Science & Engineering
Compression spring rate.
Spring rate k = G d⁴ ÷ (8 N D³) from wire diameter, mean coil diameter, active coils and shear modulus, with the force at a given compression. Runs in your browser.
Spring geometry
Result
Fill the form and press Calculate.
Worked example
A steel spring, G = 79.3 GPa (79,300 N/mm²), made from 2 mm wire with a 16 mm mean coil diameter and 8 active coils: k = G d⁴ ÷ (8 N D³) = 79,300 × 2⁴ ÷ (8 × 8 × 16³) = 1,268,800 ÷ 262,144 = 4.84 N/mm (about 27.6 lbf/in). Compressed by 10 mm it pushes back with F = k x = 48.4 N. The spring index D ÷ d is 8.
How it works
For a round-wire helical compression spring the spring rate is k = G d⁴ ÷ (8 N D³), where G is the modulus of rigidity (shear modulus) of the wire, d the wire diameter, N the number of active coils and D the mean coil diameter (outside diameter minus the wire diameter). Active coils are those subject to flexure, always fewer than the total number of coils, so subtract the dead end coils. Hooke's law then gives the force F = k × compression.
The shear moduli offered are generic handbook values for the metal (steel 79.3, copper 44.7, titanium 41.4, aluminium 25.5 GPa); spring wire grades differ, so use the maker's value if you have it. The formula is an estimate: it ignores end effects and does not check stress, buckling, fatigue or solid height. The first and last 20% of the deflection range behave differently from the middle.
Not for safety-critical design
Planning and learning aid only. It is not for structural, pressure-system or other safety-critical design, and it is not professional engineering advice. Have a qualified engineer check anything that carries people or could fail dangerously.
Sources and basis: Compression Spring Constant Equation and Calculator (Engineers Edge); Helical Compression Spring Design Equations and Calculator (Engineers Edge); Shear modulus (Wikipedia). Checked 2026-10-08. Planning aid only; see the note above.
FAQ
What is the spring rate formula?
k = G d⁴ ÷ (8 N D³) for a round-wire helical compression spring: shear modulus times wire diameter to the fourth, over eight times the active coils times the mean coil diameter cubed.
What are active coils?
The coils that flex under load. It is always fewer than the total number of coils; the end coils that sit flat against the seats are not active.
Which diameter do I use for D?
The mean coil diameter: the outside diameter minus one wire diameter (equivalently the average of the inside and outside diameters).
What shear modulus should I use for spring steel?
The calculator offers 79.3 GPa for steel, a generic handbook value. Check the wire maker's data for the exact grade.
How accurate is the result?
It is an estimate. Spring rate is best determined by test, and it is only roughly constant across the middle of the deflection range.