Torsion Spring Stress⚠ unverified

Understanding the Parameters

• Applied moment $M$ — the winding moment (the "load" on a torsion spring is a torque about the coil axis). Stress is proportional to $M$; the maximum operating moment sets the peak stress.

• Wire diameter $d$ — enters as $1/d^3$ via the section modulus of the round wire. As always, wire diameter dominates, and it trades off against the rate requirement.

• Correction factor $K$ — accounts for the fact that the wire is curved: the inner fibre of the coil is shorter and carries higher bending stress than a straight beam would. For round wire it is a function of the spring index, e.g. $K_i = \dfrac{4C^2 - C - 1}{4C(C-1)}$ (inner fibre). As $C \to \infty$, $K \to 1$ (straight-wire bending).

• Coil diameter $D$ — only relevant for computing $C$ and hence $K$; it is not in the bare formula.

Derivation (Approaching a Proof)

The wire of a torsion spring carries the load as a bending moment $M$ about its own cross-section (winding the coil bends the wire). The peak bending stress in a straight beam is given by the flexure formula $\sigma = Mc/I$ (see Beam Bending Stress). For round wire, the extreme-fibre distance is $c = d/2$ and the second moment of area is $I = \pi d^4/64$, so:

$$\sigma = \frac{M(d/2)}{\pi d^4/64} = \frac{32 M}{\pi d^3}.$$

This is the straight-beam result. Because the spring wire is curved into a coil, the inner fibre experiences a higher stress than the flexure formula predicts — the same crowding effect that gives helical compression springs their Wahl factor. Introducing the curvature correction factor $K$ (a function of the spring index $C$):

$$\sigma = K\,\frac{32 M}{\pi d^3} = \frac{32 M K}{\pi d^3}.$$

As the coil becomes large relative to the wire ($C$ large), $K \to 1$ and the stress reduces to ordinary straight-wire bending.

Dimensional check. $[\sigma] = \dfrac{\text{N}\cdot\text{m}}{\text{m}^3} = \dfrac{\text{N}}{\text{m}^2} = \text{Pa}$ (K dimensionless). ✓

History and Development

The bending analysis of torsion-spring wire, with the inner-fibre curvature correction, is standard in Wahl's Mechanical Springs and Shigley. Because torsion springs are loaded in bending, they use the tensile/bending allowables and Goodman fatigue criteria (Goodman Line, Fatigue Endurance Limit) rather than the shear allowables of compression springs — an important distinction in design.

Related Concepts: Torsion Spring Rate, Beam Bending Stress, Spring Index, Helical Spring Stress, Goodman Line, Fatigue Endurance Limit

Notes: Bending stress (uses the section modulus of round wire), not torsional shear. $K$ from the spring index (inner-fibre correction); coil diameter $D$ enters only through $K$ (see registry note). Compare against the wire's allowable bending/fatigue stress.