The Wahl factor: the real stress inside a spring

Every helical compression or extension spring looks, at first glance, like a simple torsion problem: apply an axial force, the wire twists about its own axis, and the spring deflects. But the wire is not straight — it is wound into a helix, and that curvature distorts the stress distribution across the section. The real shear stress on the inner face of the coil is always higher than the classic torsion formula predicts. The Wahl factor is the correction that closes exactly this gap.

Ignoring this factor is one of the most common — and most expensive — mistakes in spring design. It leads the engineer to underestimate the peak stress precisely where the spring always cracks first: the inner surface of the wire. This guide explains the physics of the phenomenon, presents the formulas, works a full numeric example with units, and shows how to use the Wahl factor to size springs that survive fatigue.

What the Wahl factor is and how it works

When an axial force F compresses or stretches a spring, each section of the wire is loaded mainly in torsion, producing torsional shear stress. If the wire were straight and the force acted with no eccentricity, that stress would be distributed symmetrically around the section. In a real spring, however, two additional effects add to the pure torsion.

The first is direct (transverse) shear: the force F itself, besides generating the torque, passes through the wire section and produces a uniform shear stress. This component adds to the torsional shear on the inner side of the coil and subtracts from it on the outer side. The second effect is curvature: because the wire is coiled, the inner fibres have a smaller bend radius and a shorter path than the outer ones, which concentrates the strain — and therefore the stress — on the face pointing toward the spring axis.

Added together, these two effects make the maximum stress always appear on the inner fibre of the coil rather than on the outer surface. The Wahl factor (Kw) is a dimensionless multiplier that captures this increase: it takes the nominal torsional stress and corrects it to the true peak value the material actually feels. The tighter the spring is wound, the larger the correction.

Spring index: the parameter that controls everything

The Wahl factor depends neither on the material nor on the load — it depends only on geometry, summarised in a single number: the spring index, C. It is the ratio of the mean coil diameter (D, measured between the wire centres) to the wire diameter (d).

The index tells you how tightly the spring is wound. A low index (C near 4) means tight coils around a thick wire — lots of curvature, lots of stress concentration. A high index (C above 10) means an open spring with gentle curvature and a small correction. The usual manufacturable range runs from 4 to 12; below 4 the coiling is difficult and the wire may crack during forming, above 12 the spring becomes fragile to handle and prone to tangling.

The Wahl factor formula

In the 1940s A. M. Wahl derived an expression that combines the direct-shear correction and the curvature correction into a single coefficient, written purely as a function of the index C. This is the formula used in the stress check of most helical springs.

The first term, (4C − 1)/(4C − 4), represents mainly the curvature effect; the second, 0.615/C, adds the direct-shear contribution. Notice that both grow as C decreases — which is why low-index springs are penalised the most. A few reference values help internalise the behaviour:

In other words, a spring of index 4 sees roughly 40% more stress on the inner fibre than the nominal torsion suggests, while one of index 12 sees only about 12% more. That is a huge difference, and it completely changes the fatigue safety margin.

• C = 4 → Kw ≈ 1.40 (correction of +40%)
• C = 6 → Kw ≈ 1.25 (correction of +25%)
• C = 8 → Kw ≈ 1.18 (correction of +18%)
• C = 12 → Kw ≈ 1.12 (correction of +12%)

From nominal to corrected stress

The nominal torsional shear stress in a compression or extension spring is given by the classic expression 8·F·D/(π·d³). It represents the theoretical average, without accounting for the asymmetry of the distribution. To obtain the true peak stress on the inner fibre, you simply multiply it by the Wahl factor.

In this formula, F is the axial force (N), D is the mean diameter (mm), d is the wire diameter (mm), and the result τ comes out in MPa (equivalent to N/mm²). It is this corrected value, not the nominal one, that must be compared against the material's allowable stress — whether in static sizing or in a fatigue analysis using a Goodman diagram or similar.

A worked numeric example

Let us apply everything to a concrete spring — the same one used in this guide's preset. Take a working force F = 100 N, mean diameter D = 18 mm and wire diameter d = 3 mm.

First, the index: C = D / d = 18 / 3 = 6. With C = 6, the Wahl factor becomes Kw = (4·6 − 1)/(4·6 − 4) + 0.615/6 = 23/20 + 0.1025 = 1.15 + 0.1025 ≈ 1.25.

Now the nominal stress: τ_nom = 8 · 100 · 18 / (π · 3³) = 14400 / (π · 27) = 14400 / 84.8 ≈ 170 MPa. Applying the correction: τ = 1.25 × 170 ≈ 213 MPa.

The difference is about 25%. Had the designer sized the spring on the nominal 170 MPa alone, they would believe they had a comfortable margin against the material limit — yet the inner fibre of the wire would be working at 213 MPa. Under cyclic loading, that is exactly where — on the inner surface of the first active coil — the fatigue crack nucleates and grows. Ignoring the Wahl factor does not change where the spring fails; it only changes whether or not you predicted that failure.

The Bergsträsser factor: the standards' alternative

The Wahl factor is the most traditional, but not the only one. Many modern standards and handbooks prefer the Bergsträsser factor (KB), which gives practically identical results with a more compact and slightly more accurate expression near the usual range of indices.

For our example with C = 6, KB = (4·6 + 2)/(4·6 − 3) = 26/21 ≈ 1.24 — versus Wahl's 1.25, a negligible difference for practical purposes. The choice between them rarely changes a design decision; what matters is applying some correction factor and being consistent throughout the calculation, including the fatigue analysis.

How to use the Wahl factor in design

In practice, the Wahl factor should enter every stress check of a helical compression or extension spring. A few guidelines help you take advantage of it rather than merely suffer its consequences:

Note also how the factor talks to the other spring parameters. Increasing D (a wider spring) raises the nominal stress, but it also raises C and lowers Kw — the effects oppose each other. Increasing d strongly lowers the nominal stress (the d³ in the denominator) while at the same time lowering C and raising Kw. That is why the index is the most powerful design variable: it simultaneously governs stiffness, manufacturability and stress concentration.

• Prefer indices between 5 and 9 whenever possible: they keep Kw moderate and the spring manufacturable.
• If the application demands a low index, budget the stress with the corresponding Kw from the outset — never with the nominal stress.
• Focus attention on the inner fibre of the first active coil: that is where fatigue begins.
• Good surface finish and no decarburisation on the inner face greatly extend fatigue life.
• Always use the corrected stress when building the fatigue (Goodman) diagram, never the nominal one.

How molas.app.br computes this automatically

In the molas.app.br 3D designer, the index, the Wahl factor and the corrected stress are recomputed in real time with every adjustment to wire diameter, outer diameter or number of coils — so you see the effect of geometry on the peak inner-fibre stress instantly, without redoing the arithmetic by hand.