Bolt Torque from Preload⚠ unverified

Understanding the Parameters

• Target preload $F_i$ — the clamp force you want in the bolt, typically $0.75\,A_t\sigma_p$ (Bolt Preload). Torque scales linearly with it.

• Nominal diameter $d$ — the lever-arm scale of the thread and bearing surfaces; torque scales with $d$ because both the thread helix radius and the collar friction radius grow with the bolt.

• Nut factor $K$ — a single empirical coefficient (~0.20 for as-received steel, ~0.15 lubricated, up to ~0.30 dry/rusty) that lumps together thread friction, collar (under-head) friction, and the thread-lead geometry. It is the dominant source of preload uncertainty.

• Why torque control scatters — because $K$ varies with lubrication, plating, surface finish, and speed, torque control delivers preload only to about $\pm 25\%$. Critical joints use turn-of-nut or direct bolt-stretch measurement instead.

Derivation (Approaching a Proof)

The applied torque does work against three resistances as the bolt advances: the thread-lead component (the useful part that stretches the bolt), the thread friction, and the collar (under-head/nut-face) friction. A detailed power-screw analysis (see Bolt Torque Preload) gives the torque to produce tension $F_i$ as

$$T = F_i\left[\underbrace{\frac{d_m}{2}\!\left(\frac{\ell + \pi\mu d_m\sec\alpha}{\pi d_m - \mu\ell\sec\alpha}\right)}_{\text{thread}} + \underbrace{\frac{\mu_c\,d_c}{2}}_{\text{collar}}\right],$$

with $d_m$ the mean thread diameter, $\ell$ the lead, $\alpha$ the thread half-angle, and $\mu,\mu_c$ the thread and collar friction coefficients. All the geometry and friction collapse into one dimensionless group when normalised by the nominal diameter $d$:

$$T = K\,F_i\,d, \qquad K \equiv \frac{1}{d}\left[\frac{d_m}{2}\!\left(\frac{\ell + \pi\mu d_m\sec\alpha}{\pi d_m - \mu\ell\sec\alpha}\right) + \frac{\mu_c d_c}{2}\right].$$

For standard ISO/UN threads this evaluates to $K \approx 0.20$ for unlubricated steel — the origin of the familiar rule of thumb.

Dimensional check. $T = K\,F_i\,d = (\text{–})\cdot\text{N}\cdot\text{m} = \text{N}\cdot\text{m}$ — a torque, as required.

History and Development

The nut-factor form $T = K F_i d$ is a deliberate simplification of the full power-screw torque equation, made so that assembly torque could be tabulated with a single coefficient per lubrication condition. It appears in every fastener torque chart and in VDI 2230. Its weakness — that $K$ hides a $\pm 25\%$ preload scatter — drove the development of more repeatable methods (torque-angle, yield-point, ultrasonic bolt-stretch) for joints where preload accuracy is critical.

Related Concepts: Bolt Preload from Torque, Bolt Torque Preload, Bolt Preload, Bolt Proof Load, Bolt Yield Torque, Joint Stiffness Ratio

Notes: Inverse of Bolt Preload from Torque; see Bolt Torque Preload for the full power-screw expansion of $K$. $K$ is the lumped nut factor (~0.2 steel, ~0.15 lubricated), not a bare $\mu$. Torque control → preload scatter ±25%.