Coiled Spring
Compute spring index c, Wahl factor k, torsional stress τ, deflection δ and spring rate κ for helical coil springs.
Inputs
Formula
c=D/d; k=(4c−1)/(4c−4)+0.615/c; τ=k·8WD/(πd³); δ=8NeD³W/(Gd⁴); κ=Gd⁴/(8NeD³)
G≈80 GPa for steel; N = Ne+2 for ground-end compression springs.
Fundamentals
Helical coil springs store energy and damp shock via torsion of the wire — the most common elastic element.
Under axial load $F$ the wire is mainly in torsion; stiffness $k=G d^4/(8 D^3 n)$, $G$ shear modulus, $d$ wire dia, $D$ mean dia, $n$ active coils.
History
Helical springs obey Hooke's law (1676, ut tensio sic vis); hot-coiling in the 19th c. brought them to vehicle suspension.
Engineering applications
Used in suspension, valves, clutches and as buffers or force-measuring elements.
Glossary
| Stiffness $k$ | Force per deflection, $k=F/\delta$. |
| Active coils $n$ | Coils that deform; more = softer. |
| Shear modulus $G$ | Material's resistance to shear distortion. |
How to use
- Fill in Applied load W, Mean coil dia. D, Wire dia. d, Active coils Ne, Shear mod. G in the Inputs section (watch the unit on each field).
- Click Calculate; the tool evaluates the formula shown above.
- Read Spring index c, Wahl factor k, Torsional stress τ, Deflection δ, Spring rate κ in the results area.
Formula notesHelical spring: wire dia d = 1.6·∛(F·C/τ); spring index C = D/d; Wahl K = (4C−1)/(4C−4) + 0.615/C.
Formula · Worked Example · Knowledge
Formula
Helical coil spring: spring index $c=D/d$, Wahl factor $k=(4c-1)/(4c-4)+0.615/c$, stress $\tau=k\,8WD/(\pi d^3)$, deflection $\delta=8N_e D^3 W/(G d^4)$.
Worked Example
W=100 N, D=20, d=2, Ne=8, G=80 GPa → c=10, τ≈142 MPa, δ≈20 mm.
Key Points
- Very small or large c hurts winding/stability.
- Rate $\kappa=G d^4/(8N_e D^3)$ depends only on geometry & material.
- Ground-end compression springs: total coils $N=N_e+2$.
Parameters
Inputs
| Parameter | Symbol | Unit | Default |
|---|---|---|---|
| Applied load W (N) | W | N | 100 |
| Mean coil dia. D (mm) | D | mm | 20 |
| Wire dia. d (mm) | d | mm | 2 |
| Active coils Ne | Ne | 8 | |
| Shear mod. G (GPa) | G | GPa | 80 |
Outputs
| Result | Symbol | Unit |
|---|---|---|
| Spring index c | c | |
| Wahl factor k | k | |
| Torsional stress τ | tau | MPa |
| Deflection δ | delta | mm |
| Spring rate κ | kappa | N/mm |
Applications
- Common engineering use cases
FAQ
What formula does this tool use?
This tool computes per ISO / AGMA / ASME standard formulas: c=D/d; k=(4c−1)/(4c−4)+0.615/c; τ=k·8WD/(πd³); δ=8NeD³W/(Gd⁴); κ=Gd⁴/(8NeD³)
How accurate are the results?
Results match input precision, based on SI units and common engineering approximations; for critical duty re-check with a safety factor.
Where is it used?
Common engineering use cases