Joint Stiffness Ratio⚠ unverified
Mechanical / Joints · Compute the joint stiffness ratio
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| kb | kb | N/m | 1.0 | Bolt stiffness |
| km | km | N/m | 1.0 | Member (clamped material) stiffness |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | C | — | Stiffness ratio (dimensionless). Returns 0.0 when the total stiffness ``kb + km`` is not positive |
The science & history
Understanding the Parameters
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Bolt stiffness $k_b$ — a slender bolt (long grip, small area) is soft, giving a small $C$ so the bolt sees little of the external load. This is the design goal: soft bolt, stiff members.
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Member stiffness $k_m$ — stiff clamped members (short grip, large area, high modulus) also lower $C$. Adding a soft gasket raises member compliance, lowering $k_m$ and pushing $C$ up — worse for fatigue.
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Ratio $C$ — bounded between 0 and 1. $C \to 0$ means the members carry the load and the bolt is fatigue-safe; $C \to 1$ (e.g. a soft joint with a very stiff bolt) throws the whole load onto the bolt.
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Complement $1-C$ — the fraction carried by the members, used directly in Member Load External.
Derivation (Approaching a Proof)
Preload $F_i$ stretches the bolt and compresses the members equally. Now apply an external tensile load $P$ across the joint. The bolt and members deform together by a common additional displacement $\delta$ (they stay in contact while the joint is clamped), so they act as parallel springs.
The extra bolt load and the reduction in member compression are
$$\Delta F_b = k_b\,\delta, \qquad \Delta F_m = k_m\,\delta.$$
Equilibrium requires the two increments to sum to the applied load, $\Delta F_b + \Delta F_m = P$, so
$$\delta = \frac{P}{k_b + k_m}, \qquad \Delta F_b = \frac{k_b}{k_b + k_m}\,P = C P.$$
Thus $C = k_b/(k_b+k_m)$ is exactly the share of $P$ that increases the bolt load. The members lose $(1-C)P$ of their compression.
Dimensional check. $C = \dfrac{k_b}{k_b+k_m} = \dfrac{\text{N}/\text{m}}{\text{N}/\text{m}}$ = dimensionless — a pure ratio, as required.
History and Development
The stiffness ratio is the heart of the elastic bolted-joint model developed through the 20th century (Shigley, VDI 2230). Its power is counter-intuitive: a properly preloaded joint puts most of a fluctuating external load into unloading the members rather than loading the bolt, so the bolt's alternating stress is only $C$ times what a naïve "bolt carries all the load" analysis would predict. Getting $C$ right — through the member-stiffness estimate Bolted Joint Stiffness Member — is what makes bolted joints survive millions of load cycles.
Related Concepts: Joint Stiffness, Bolted Joint Stiffness Member, Bolt Load External, Member Load External, Bolt Fatigue Factor, Bolt Shank Stiffness
Notes: $0 < C < 1$; typical $0.2$–$0.5$. Design goal: soft bolt + stiff members → small $C$ → bolt fatigue-safe. Soft gaskets raise $C$ (worse). $1-C$ is the member share (Member Load External). Bolt and members act as parallel springs under external load.