Bolt Preload from Torque⚠ unverified

Understanding the Parameters

• Tightening torque $T$ — the moment applied at the wrench. Preload is directly proportional to it, which is why torque wrenches are the everyday preload tool despite their imprecision.

• Nominal diameter $d$ — sets the length scale over which the friction and thread forces act. Larger bolts need proportionally more torque for the same preload.

• Nut factor $K$ — the crucial fudge factor that bundles thread friction, under-head (collar) friction, and thread geometry into one number. Typical values: $K \approx 0.20$ dry/as-received, $\approx 0.15$ lightly lubricated, down to $\approx 0.10$–$0.12$ with wax or moly. Because $K$ appears in the denominator, halving friction nearly doubles the preload for the same torque — the source of torque-method scatter (±25–30 % is typical).

Derivation (Approaching a Proof)

The nut factor is a compact wrapper for the full power-screw torque analysis. The tightening torque must overcome three moments: raising the load up the thread helix, thread friction, and collar (under-head or under-nut) friction. The long-form result is

$$T = F_i\left[\underbrace{\frac{d_m}{2}\left(\frac{\tan\lambda + \mu\sec\alpha}{1 - \mu\tan\lambda\sec\alpha}\right)}_{\text{thread: lead + friction}} + \underbrace{\mu_c\,\frac{d_c}{2}}_{\text{collar friction}}\right],$$

where $d_m$ is the mean thread diameter, $\lambda$ the lead angle, $\alpha$ the thread half-angle, and $\mu$, $\mu_c$ the thread and collar friction coefficients. Every geometric and frictional term inside the brackets is proportional to the bolt size, so the whole bracket can be written as a single dimensionless constant times the nominal diameter:

$$T = K\,F_i\,d \;\Longrightarrow\; F_i = \frac{T}{K d}.$$

That is the definition of the nut factor: $K = \dfrac{1}{d}\big[\text{bracket above}\big]$. For standard 60° threads with typical friction ($\mu \approx \mu_c \approx 0.15$), the bracket evaluates to $K \approx 0.20$. The lead (bolt-stretching) term contributes only ~10 % of the torque; the rest is friction — which is why $K$, and hence preload, is so sensitive to lubrication.

Dimensional check. $[F_i] = \dfrac{\text{N}\cdot\text{m}}{(\text{dimensionless})\,\text{m}} = \text{N}$. ✓

History and Development

The nut-factor method dates to early-20th-century fastener practice and was consolidated in Machinery's Handbook, Shigley, and manufacturer torque charts. Its known imprecision drove the development of better preload-control methods — turn-of-the-nut, torque-to-yield (angle control), bolt elongation measurement, and ultrasonic and strain-gauged bolts — for critical joints (engine heads, aerospace, structural steel). The full power-screw derivation behind $K$ is the same one used for Bolt Torque Preload and lead screws.

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

Notes: $K$ is empirical and lubrication-sensitive (≈0.2 dry, ≈0.15 lubed) — expect ±25 % preload scatter. Assumes no prevailing torque (locknuts add a term). For precision use angle control or direct elongation measurement.