Spring Index⚠ unverified
Mechanical / Springs · Spring index (coil-to-wire diameter ratio)
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| D | D | m | 0.04 | Mean coil diameter |
| d | d | m | 0.005 | Wire diameter |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| C | C | — | Spring index |
The science & history
Understanding the Parameters
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Mean coil diameter $D$ — measured to the centre of the wire (not the outside or inside of the coil), so $D = D_{\text{outer}} - d$. It sets the moment arm $D/2$ through which the axial force twists the wire, and therefore scales the stress and the softness of the spring.
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Wire diameter $d$ — the thickness of the wire itself. Because the wire carries the load in torsion, its diameter enters spring stress as $d^3$ and spring rate as $d^4$ — small changes in wire size have an outsized effect, which is why $d$ is chosen from standard gauge tables.
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The index $C$ — being a pure ratio, it is scale-independent: a tiny instrument spring and a truck valve spring with the same $C$ share the same relative curvature and stress-concentration behaviour. This is what makes $C$ the master design parameter.
Derivation (Approaching a Proof)
The spring index is a definition, not a derived law — but its significance comes from how it governs the curvature stress correction. A helical spring loaded axially twists its wire; the inner fibre of the curved wire is shorter than the outer, so the shear stress there is higher than simple torsion predicts. A. M. Wahl quantified this with a correction factor that depends only on $C$:
$$K_W = \frac{4C - 1}{4C - 4} + \frac{0.615}{C}.$$
As $C \to \infty$ (a nearly straight wire) $K_W \to 1$ (no curvature effect); as $C$ falls toward 4, $K_W$ climbs above 1.4, sharply raising the peak stress (Helical Spring Stress). This is exactly why $C$ is bounded below in practice. The index also appears implicitly in the rate and deflection formulas through the $D^3/d^4$ and $D^4/d^4$ groupings, which are powers of $C$.
Dimensional check. $[C] = \dfrac{\text{m}}{\text{m}} = 1$ (dimensionless). ✓
History and Development
The spring index and its role in curvature stress date to the early 20th century and were made rigorous by A. M. Wahl (Mechanical Springs, 1944), whose curvature-plus-shear correction factor is written purely in terms of $C$. It remains the first quantity computed in any spring design (Shigley, Associated Spring / SPEC handbooks), with the $4 \le C \le 12$ guideline universal across the industry.
Related Concepts: Helical Spring Rate, Helical Spring Stress, Wahl Correction Factor, Spring Buckling, Helical Spring Deflection
Notes: Use the mean coil diameter ($D = D_o - d$). Recommended range $C = 4$–$12$; low $C$ raises the Wahl factor and coiling difficulty, high $C$ invites buckling and tangling.