Bolt Fatigue Factor⚠ unverified
Mechanical / Joints · Compute the Goodman fatigue factor of safety for a bolted joint
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| Fi | Fi | N | 1.0 | Bolt preload |
| C | C | — | 1.0 | Joint stiffness ratio (dimensionless) |
| Fe | Fe | N | 1.0 | External load applied to the joint |
| At | At | m**2 | 1.0 | Tensile-stress area of the bolt |
| Se | Se | Pa | 1.0 | Endurance limit of the bolt material |
| Sut | Sut | Pa | 1.0 | Ultimate tensile strength of the bolt material |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | σa | — | Fatigue factor of safety (dimensionless). Returns 0.0 when ``At``, ``Se``, or ``Sut`` is not positive |
The science & history
Understanding the Parameters
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Joint constant $C$ — the fatigue lever. The external load fluctuates by $F_e$, but the bolt only sees $C F_e$ of it (Bolt Load External); halving that gives the alternating force $C F_e/2$. A small $C$ (soft bolt, stiff members) is the primary fatigue defence.
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Alternating stress $\sigma_a = C F_e/(2 A_t)$ — the cyclic stress amplitude at the thread root region. It is what Goodman limits against; everything else sets the allowable.
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Preload $F_i$ → mean stress $\sigma_i = F_i/A_t$ — preload raises the mean stress but, crucially, does not add to the alternating stress; on a Goodman diagram the operating point rides up the line from the preload point, so higher preload actually reduces the room for $\sigma_a$ but stabilises the joint.
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$S_e$ and $S_{ut}$ — the endurance limit and ultimate strength anchor the Goodman line. For rolled threads $S_e$ already includes the severe thread stress concentration; bolts are notoriously fatigue-sensitive at the first engaged thread.
Derivation (Approaching a Proof)
The external load cycles between $0$ and $F_e$, so the bolt-load increment cycles between $0$ and $C F_e$. Its alternating and mean parts (as stresses over $A_t$) are
$$\sigma_a = \frac{C F_e}{2 A_t}, \qquad \sigma_m = \sigma_i + \frac{C F_e}{2 A_t} = \frac{F_i}{A_t} + \sigma_a,$$
where the preload $\sigma_i = F_i/A_t$ shifts the mean but not the amplitude. The Goodman line for a bolt preloaded to $\sigma_i$ (the load line starts at $\sigma_i$, not the origin) gives the factor of safety guarding the first-cycle margin to the Goodman line as
$$n_f = \frac{S_e\,(S_{ut} - \sigma_i)}{\sigma_a\,(S_{ut} + S_e)}.$$
$n_f > 1$ places the operating point safely below the Goodman line for infinite life; $n_f < 1$ predicts fatigue failure. This is the same Goodman construction as Shaft Fatigue Factor and Weld Fatigue Factor, specialised to the preloaded-bolt load line.
Dimensional check. $\sigma_a = \dfrac{C F_e/2}{A_t} = \dfrac{\text{N}}{\text{m}^2} = \text{Pa}$; and $n_f = \dfrac{S_e(S_{ut}-\sigma_i)}{\sigma_a(S_{ut}+S_e)}$ is $\dfrac{\text{Pa}\cdot\text{Pa}} {\text{Pa}\cdot\text{Pa}}$ = dimensionless — a factor of safety, as required.
History and Development
Bolt fatigue analysis is where the elastic joint model earns its keep. The insight that preload converts a fully-reversed external load into a small-amplitude, high-mean bolt stress (amplitude $C F_e/2$, not $F_e/2$) explains why properly preloaded bolts survive and loose bolts fail quickly. The preloaded-bolt Goodman line (Shigley) and the German VDI 2230 method formalise it; rolled (not cut) threads and adequate preload are the two practical levers that keep $n_f$ above 1.
Related Concepts: Bolt Load External, Joint Stiffness Ratio, Bolt Preload, Goodman Line, Shaft Fatigue Factor, Weld Fatigue Factor, Bolt Tensile Stress Area
Notes: Displayed equation is the intermediate $\sigma_a$; the tool returns the Goodman factor $n_f$ (output should be symbol $n_f$, not $\sigma_a$; $\sigma_a$ itself is in Pa). Preload raises mean, not amplitude. Uses the preloaded-bolt load line. Rolled threads improve $S_e$.