Compression spring rate calculator
Enter the wire diameter, the outer diameter of the coils, the number of active coils and the shear modulus of the wire. The calculator gives the spring rate in N/mm and lbf/in, the mean coil diameter and the spring index.
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How to use it
- Choose the units: millimetres with G in GPa, or inches with G in millions of psi.
- Enter the wire diameter, the outer diameter of the coils and the number of active coils.
- Enter the shear modulus G of the wire material.
- Read the spring rate in N/mm and lbf/in, the mean coil diameter and the spring index.
k = G·d⁴ / (8·D³·n)
- d — wire diameter;
- D — mean coil diameter = outer diameter − d;
- n — active coils;
- G — shear modulus (1 GPa = 1,000 N/mm²).
The spring index C = D/d describes how tightly the wire is wound; springs with a very low or very high index are harder to make.
Worked example
Wire 2 mm, outer diameter 22 mm, 10 active coils, G = 79.3 GPa: D = 22 − 2 = 20 mm, and
k = 79,300 × 2⁴ ÷ (8 × 20³ × 10) = 1,268,800 ÷ 640,000 = 1.9825 N/mm, or 11.32 lbf/in. The index is 20 ÷ 2 = 10.
In inches: d = 0.1 in, OD = 1.1 in, 10 coils, G = 11.5 Mpsi: k = 11,500,000 × 0.0001 ÷ (8 × 1 × 10) = 14.375 lbf/in.
Tips
- Load at a given deflection: F = k × x. The spring above pushes back with 1.9825 × 10 = 19.8 N when compressed 10 mm.
- Measure the wire with a micrometer; an error of 0.05 mm on a 2 mm wire changes the rate by 10 %.
- This is the rate only. Stress, solid height and buckling need separate checks.
Frequently asked questions
What is the spring rate formula?
k = G × d⁴ / (8 × D³ × n), where G is the shear modulus of the wire, d the wire diameter, D the mean coil diameter (outer diameter minus d) and n the number of active coils. In N, mm and MPa the result is in N/mm.
How many active coils does a compression spring have?
Coils that are free to deflect. With squared and ground ends, a common style, the two end coils are inactive, so active coils ≈ total coils − 2. With plain open ends nearly all coils are active.
What shear modulus should I use?
About 79 GPa (11.5 million psi) is the usual figure for carbon spring steels such as music wire; stainless steels are lower. Take the value for your wire from the supplier, since it varies with alloy and treatment.
Why does wire diameter matter so much?
The rate grows with the fourth power of d. A wire 10 % thicker gives a spring about 46 % stiffer, while a coil diameter 10 % larger makes it about 25 % softer.