Spring Calculator
Spring rate k = Gd⁴/8D³n, deflection, Wahl-corrected shear stress, solid height, force to solid and a buckling check for helical compression springs.
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About this toolWhat is the Spring Calculator?
A helical compression spring is a torsion bar wound into a coil. Push on it and the wire twists, which is why its stiffness depends on the shear modulus of the wire rather than Young's modulus, and why the stress that matters is shear. Four dimensions decide almost everything: wire diameter, coil diameter, number of active coils and free length. This calculator turns those into spring rate, deflection under load, shear stress, solid height and the force at which the spring goes solid.
Choose music wire, 302 stainless, chrome-silicon or chrome-vanadium and the typical shear modulus fills in (you can overwrite it). Give the coil diameter as outside, mean or inside diameter, set the total coils and end type, and enter a working load. The page default, a 2 mm music-wire spring with an 18 mm outside diameter and 10 coils squared and ground, has a rate of 4.99 N/mm and deflects 10.0 mm under 50 N.
A second panel checks strength and stability: the wire's minimum tensile strength from its diameter, the allowable shear stress for a static spring, the safety factor if the spring is pressed solid, and the free length beyond which it may buckle.
- Spring rate, deflection, compressed length and Wahl-corrected shear stress
- Four end types with active coils, solid height and pitch from Shigley Table 10-1
- Typical G for music wire, 302 stainless, chrome-silicon and chrome-vanadium, editable
- Wire tensile strength, allowable shear, safety factor at solid and buckling length
- Metric or inch units with in-place conversion; copy and share
How to use the Spring Calculator
- Choose metric (mm, N, GPa) or inch units (in, lbf, Mpsi). Switching converts what you have entered.
- Pick the wire material. The shear modulus fills in with a typical value you can edit.
- Enter the wire diameter, the coil diameter and whether it is the outside, mean or inside diameter.
- Enter total coils, the free length and the end type, then the working load.
- Read rate, deflection, compressed length and stress, then check the safety factor at solid and the buckling length.
Worked examples
Music-wire spring (page default)
Input
d 2 mm, OD 18 mm, 10 coils squared and ground, free length 44 mm, 50 N
Output
k = 4.99 N/mm; deflection 10.03 mm; τ = 301.5 MPa; solid height 20 mm; force to solid 119.7 N
Mean diameter is 16 mm, so C = 8 and the Wahl factor is 1.184. At solid the stress is 722 MPa against an allowable 900 MPa (45% of 2,000 MPa), a safety factor of 1.25, so the spring is solid-safe.
Same spring in 302 stainless
Input
Material 302 stainless (G = 69.0 GPa), other inputs unchanged
Output
k = 4.21 N/mm; deflection at 50 N 11.9 mm; allowable shear 591 MPa
Stainless is 16% less stiff and has a lower allowable stress, so the same geometry would take a set if pressed solid. Use thicker wire or fewer coils pressed less far.
Inch-unit spring
Input
d 0.080 in, OD 0.75 in, 12 coils squared and ground, free length 2.5 in, music wire G = 11.85 Mpsi, 10 lbf
Output
k = 20.2 lbf/in; deflection 0.50 in; τ = 39.2 ksi; solid height 0.96 in
Force to solid is 31.1 lbf. Index 8.4 is in the comfortable range for coiling.
Background and insight
The rate formula k = Gd⁴ / (8D³Na) comes from treating each active coil as a straight bar under torsion: the axial load F acts at the coil radius, giving a torque FD/2, and the twist of a length πDNa of wire turns into axial deflection. The fourth power on d and the cube on D make the rate very sensitive to both. A 10% thicker wire is 46% stiffer; a 10% larger coil is 25% softer.
Stress in the wire is not just torque over section modulus. On the inside of the coil the wire is curved and the direct shear from F adds to the torsional shear, so the peak is higher. A. M. Wahl's 1929 correction factor Kw = (4C − 1)/(4C − 4) + 0.615/C, with spring index C = D/d, covers both effects. At C = 8 it adds 18%; at C = 4 it adds 40%, which is one reason springs with an index under 4 are hard to make. Shigley's current editions use the similar Bergsträsser factor; the two differ by about 1% at C = 8 and under 2% at C = 4, with Wahl slightly higher.
Active coils and solid height depend on how the ends are finished. Following Shigley Table 10-1: plain ends have all coils active and solid height d(Nt + 1); plain and ground ends lose one coil; squared (closed) ends lose two coils with solid height d(Nt + 1); squared and ground ends lose two coils with solid height d·Nt. Material data used here are typical values from Shigley Tables 10-4 to 10-6: G = 81.7 GPa for music wire (ASTM A228), 69.0 GPa for 302 stainless (A313) and 77.2 GPa for chrome-silicon (A401) and chrome-vanadium (A231); minimum tensile strength Sut = A / d^m; and allowable static shear of 45% of Sut for music wire, 35% for stainless and 50% for the alloy steels.
Common use cases
Specifying a replacement spring
Measure a spring you have, enter its dimensions, and get the rate and solid force to quote to a spring maker.
Designing a return or preload spring
Iterate wire size and coils until the rate, travel and stress at solid all work.
Checking a catalogue part
Confirm a stock spring's stated rate and see the stress it will see at your working deflection.
Teaching and coursework
Work Shigley-style spring problems with the formula steps visible.
Tips and limitations
- Keep the spring index between about 4 and 12. Below 4 the wire is hard to coil; above 12 springs tangle and buckle.
- Aim for 3 to 15 active coils. With very few coils the end coils make the rate unpredictable.
- Design so the spring is solid-safe: the stress at solid height should stay under the allowable, so an accidental full compression does not set it.
- Shot peening and presetting (removing set) raise the usable stress, but the allowable values here assume neither.
- Springs that cycle millions of times need a fatigue check with the Goodman or Sines criterion; static allowables are not enough.
Frequently asked questions
What is the formula for spring rate?
For a helical compression spring k = G d⁴ / (8 D³ Na), with G the shear modulus, d the wire diameter, D the mean coil diameter and Na the number of active coils. In N, mm and MPa it gives N/mm.
What is the Wahl correction factor?
Kw = (4C − 1)/(4C − 4) + 0.615/C, with C = D/d. It multiplies the nominal torsional stress 8FD/πd³ to account for coil curvature and direct shear, giving the peak stress on the inside of the coil.
How do I calculate the solid height of a spring?
For squared and ground or plain and ground ends, solid height = d x total coils. For plain or squared unground ends it is d x (total coils + 1). These are the Shigley Table 10-1 values; real solid height is often a little more.
How many active coils does a spring with squared and ground ends have?
Two fewer than the total. Squared ends are closed against the next coil, so one coil at each end carries no deflection. A 10-coil squared and ground spring has 8 active coils.
What shear modulus should I use for music wire?
About 81.7 GPa (11.85 Mpsi) for small diameters, falling to about 80 GPa for wire over 3 mm, per Shigley Table 10-5. Use 69.0 GPa for 302 stainless and 77.2 GPa for chrome-silicon or chrome-vanadium.
Should I use mean or outside diameter in the spring formula?
Mean diameter, which is the outside diameter minus one wire diameter. Using the outside diameter makes the calculated rate too low; for a 2 mm wire and 18 mm OD the error is 30%.
When will a compression spring buckle?
With both ends squared and ground on parallel flat plates, Shigley gives a stable free length up to about 5.26 times the mean diameter. Longer springs need a guide rod or bore, and less well constrained ends buckle sooner.
Is there an in-depth guide to the Spring Calculator?
Yes. “Compression Spring Design: Rate, Wahl Stress and Solid Height” covers the background, how it works, common pitfalls and worked examples. Read the full guide.
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