How to use the compression spring rate calculator
- Pick mm, N, MPa or inch, lbf, psi.
- Enter the wire diameter, the mean coil diameter and the number of active coils.
- Choose the material or enter your own shear modulus, then add a deflection and a free length if you know them.
- Read the spring rate, the force and stress at that deflection and the deflection to coil contact.
Formula
Rate k = G d^4 / (8 D^3 Na)
Force F = k x deflection
Spring index C = D / d; Wahl factor Kw = (4C - 1) / (4C - 4) + 0.615 / C
Shear stress = Kw x 8 F D / (pi d^3); solid height = d x total coils
G is the shear modulus of the wire, d the wire diameter, D the mean coil diameter (center to center of wire) and Na the active coils. Total coils default to the active coils plus 2.
A metric and an inch spring
Steel wire 2 mm thick wound to a mean diameter of 20 mm with 10 active coils and G = 79,300 MPa has a rate of 79,300 x 16 / (8 x 8000 x 10) = 1.9825 N/mm. Compressed 5 mm it pushes with 9.9125 N, at a shear stress of 72.245 MPa with a spring index of 10 and a Wahl factor of 1.1448. With 12 total coils the solid height is 24 mm, so a 60 mm spring can compress 36 mm, at which the force is 71.37 N.
An inch spring with 0.08 in wire, 0.75 in mean diameter and 8 active coils has a rate of 17.448 lbf/in, and a force of 8.724 lbf at 0.5 in. Its stress is about 37,592 psi.
What sets the stiffness
The wire diameter counts to the fourth power, so a slight increase in thickness makes the spring much stiffer. The coil diameter counts to the third power in the other direction, so a wider coil is much softer. More active coils make a softer spring in direct proportion. Dead coils at the ends do not flex and are left out of the active count.
Related tools include the spring oscillation calculator, the metal weight calculator, the sheet metal bend allowance calculator and the torque wrench extension calculator.
Checking the stress
The Wahl factor corrects the simple shear stress for the extra stress on the inside of a coil, which grows as the spring index falls. The stress shown is a static check at one deflection. It does not include fatigue, set or stress relief, which matter for springs that cycle or stay compressed. Always compare with the allowable stress for your wire from the spring maker.
Assumptions and limits
- Round wire, a straight cylindrical coil and an elastic range with no permanent set.
- Shear modulus values are textbook examples: spring steel 79.3 GPa, stainless 69 GPa, phosphor bronze 41.4 GPa. Use supplier data for real parts.
- Total coils default to the active coils plus 2. You can enter another count.
- A spring index below 4 or above 12 gets a warning, since such springs are hard to wind.
Frequently asked questions
How do I calculate spring rate from dimensions?
Use k = G d^4 / (8 D^3 Na). For 2 mm steel wire, 20 mm mean coil diameter and 10 active coils that is 1.98 N/mm.
What is the mean coil diameter?
It is measured from the center of the wire on one side of the coil to the center of the wire on the other, which equals the outer diameter minus one wire diameter.
What are active coils?
They are the coils that flex when the spring is loaded. The end coils that rest against the seat or are ground flat are not active, so total coils are usually about two more.
Why does the page warn about the spring index?
A spring index C = D / d between about 4 and 12 is easy to coil and behaves predictably. Outside that range the spring is hard to make, and the simple formulas are less reliable.