Bolt Torque Preload⚠ unverified
Mechanical / Fasteners · Compute the tightening torque required to achieve a target preload
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| Fi | Fi | N | 1.0 | Target bolt preload |
| d | d | m | 1.0 | Nominal bolt diameter |
| mu | μ | — | 1.0 | Coefficient of friction at the threads |
| mu_t | μt | — | 1.0 | Coefficient of friction under the bolt head (collar). Default is None, in which case it is set equal to ``mu`` |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | T | N.m | Tightening torque, in newton-metres (N.m) |
The science & history
Understanding the Parameters
-
Target preload $F_i$ — the clamp force you want; torque is directly proportional to it. Choose it as a fraction of the proof load (see Bolt Proof Load).
-
Nominal diameter $d$ — the moment-arm scale; larger bolts need more torque for the same preload.
-
Thread friction $\mu$ — resists the nut sliding up the thread flanks; the dominant loss along with collar friction. Lubrication lowers it and thus the required torque.
-
Collar friction $\mu_t$ — friction between the turning bolt head (or nut) and the joint surface, acting at roughly the same diameter scale. It typically consumes more torque than thread friction because it acts at a larger radius. Defaults to $\mu$ when not specified.
Registry note: this is a simplified form. It captures the two friction contributions ($\approx \mu\,F_i d/2$ each) but omits the thread-lead (pitch) term that actually stretches the bolt, and it lumps the collar radius into $d/2$. The full power-screw torque (see Bolt Preload from Torque) includes $\tan\lambda$ and separate mean thread/collar diameters. Also, the default $\mu = \mu_t = 1.0$ is non-physical (real values ≈0.12–0.2). Flagged in Known Issues.
Understanding the Simplification
Comparing with the rigorous power-screw torque, $T = F_i\big[\tfrac{d_m}{2}(\tan\lambda + \mu\sec\alpha)/(\dots) + \mu_c d_c/2\big]$, this formula keeps only the friction pieces and approximates both mean diameters by the nominal $d$:
$$T \approx F_i\left(\mu\,\frac{d}{2} + \mu_t\,\frac{d}{2}\right) = \frac{F_i d}{2}(\mu + \mu_t).$$
Derivation (Approaching a Proof)
The tightening torque balances the resisting friction moments as the bolt is turned to develop preload $F_i$. Two frictional interfaces resist:
-
Thread friction. The clamp force $F_i$ presses the thread flanks together; the friction force is $\mu F_i$, acting at approximately the thread radius $d/2$, giving a moment $\mu F_i (d/2)$.
-
Collar friction. The same clamp force presses the rotating bearing face against the joint; its friction $\mu_t F_i$ acts at approximately $d/2$, giving $\mu_t F_i (d/2)$.
Summing the two friction moments:
$$T = \mu F_i \frac{d}{2} + \mu_t F_i \frac{d}{2} = \frac{F_i d}{2}(\mu + \mu_t).$$
The (small) additional torque that actually raises the nut along the thread lead — the part that does useful work stretching the bolt — is neglected here; it adds a $\tan\lambda$ term in the full analysis and accounts for only ~10 % of the total. Equating this to the nut-factor form $T = K F_i d$ shows $K \approx (\mu + \mu_t)/2$.
Dimensional check. $[T] = \text{N} \cdot \text{m} \cdot (\text{dimensionless}) = \text{N}\cdot\text{m}$. ✓
History and Development
Torque–preload analysis descends from the power-screw mechanics of the 19th century (the bolt thread is an inclined plane wrapped into a helix). Shigley, Machinery's Handbook, and VDI 2230 give the full and simplified forms; the recognition that ~90 % of tightening torque is friction — and only ~10 % stretches the bolt — is the key practical insight and the reason torque is a poor proxy for preload.
Related Concepts: Bolt Preload from Torque, Bolt Torque from Preload, Bolt Yield Torque, Bolt Proof Load, Bolt Shank Stiffness
Notes: Simplified friction-only form (omits the thread-lead term; lumps collar radius into $d/2$). Use realistic friction (≈0.12–0.2), not the 1.0 defaults. Equivalent nut factor $K \approx (\mu+\mu_t)/2$.