The coil geometry turns an axial push into torsion along the wire, so rate uses the shear modulus G, not Young’s modulus E. k = Gd⁴/(8D³n).
Rate scales with wire diameter to the fourth power and falls with coil diameter cubed: doubling the wire multiplies the rate by sixteen.
The simple formula assumes straight wire; real coils concentrate stress on the inside. Kw = (4C−1)/(4C−4) + 0.615/C corrects for curvature and direct shear.
Free length minus wire diameter × total coils is all the travel there is. Once the coils touch, the stress check no longer applies.
Much lower and the spring is hard to coil with stress concentrating badly on the inside of each turn; much higher and it tends to buckle or tangle sideways. A spring much taller than its diameter can buckle like a slender column and needs a guide rod or sleeve.
The end coils are ground flat and squared off to sit against the mounting surfaces: they carry load but do not meaningfully twist, so they are excluded from the stiffness calculation though they add to the solid height. An extension spring uses the same rate formula but is usually wound with initial tension, so it does not start to extend until the load exceeds that preload, and its hooks have their own stress concentration.
A spring has a surge frequency of its own, ½·√(k/m) with its full active mass. Work it above about a thirteenth of that and the coils stop moving together, pass a wave along the spring, and clash, with stress far above the static calculation: why valve springs fail at a particular rpm rather than gradually, and why the fix is stiffer wire or fewer coils.
The stress calculation asks whether the wire survives; geometry asks whether there is room left to deflect, which usually bites first. Going solid is not a gentle limit: the component behind it suddenly sees whatever force is available.
The simple torsion formula assumes the wire is straight — real coiled wire has curvature that concentrates stress on the inside of each coil. The Wahl factor corrects for both that curvature and direct shear, and matters most for "tight" springs with a low spring index (D/d).
D/d between about 4 and 12 is the practical sweet spot — much lower and the spring becomes hard to coil and stress concentrates badly at the inside of each turn; much higher and the spring tends to buckle or tangle sideways under load.
The end coils of a compression spring are typically ground flat and squared off to sit properly against the mounting surfaces — they carry load but don't meaningfully twist, so they're excluded from the stiffness calculation even though they add to the spring's solid (fully compressed) length.
The same rate formula applies, but an extension spring is usually wound with initial tension between coils, so it doesn't start extending until the applied force exceeds that preload — and it needs hooks or loops engineered against their own separate stress concentration.
Releases in which this page changed, newest last. Derived from the archived copy of every release, not from notes written afterwards — so it reflects what actually shipped. Site-wide passes are left out; they are in the full changelog.