Helical Spring Shear Stress (with the Wahl Factor)

If a spring coil is a twisted bar, then the stress in it should be the plain torsion result, $\tau = Tr/J$, which for our torque of $FD/2$ on a round wire works out to $8FD/\pi d^3$. That expression is correct for a straight bar and wrong for a coil, and the correction is what this page is really about.

Two things a straight bar does not have. First, the coil is CURVED, so the inside of the wire has a shorter path around the helix than the outside; the shear strain is crowded on the inner fibre, and the stress there runs higher than the torsion formula says. Second, the axial load is also carried as a direct transverse shear across the wire, which adds uniformly and is ignored entirely by the torsion result. A. M. Wahl folded both into a single multiplier in the 1940s, and it has carried his name ever since:

\[K_W = \frac{4C-1}{4C-4} + \frac{0.615}{C}, \qquad C = \frac{D}{d}\]

The first term is the curvature correction and the second is the direct shear. At a spring index of 8 the factor comes to 1.184; at an index of 4 it is 1.40; at an index of 12 it is 1.119. It is never below 1, and leaving it out does not merely lose a safety margin — it puts the peak stress in the wrong PLACE. The correction lands entirely on the inner surface of the coil, and that is exactly where a fatigue crack starts. Cut open any broken compression spring and the origin is almost always on the inside diameter, which is the physical signature of the term this factor represents.

What to compare the answer against depends on how the spring lives. A spring squeezed once and left there is a static problem, and the allowable shear is commonly taken near 45 per cent of the wire's tensile strength. A spring that cycles millions of times — a valve spring, a suspension spring, a relay contact — is a fatigue problem, with an allowable that comes from a fatigue diagram rather than a strength table, and that is far lower. It is also why shot peening is standard on cycling springs: putting a compressive residual stress into the inner surface pushes the crack-initiation site back into the metal and can raise fatigue life by a large multiple, for a process that costs almost nothing.

One further wrinkle the equation does not carry: extension springs. The body of an extension spring obeys everything above, but the failures happen in the END HOOKS, where the wire is bent through a tight radius and sees a bending stress on top of the torsion. A hook bent to a small radius will fail long before the body reaches its allowable, which is why extension-spring design pays as much attention to the ends as to the coils.