Joint Stiffness⚠ unverified
Mechanical / Joints · Compute the combined stiffness of a bolted joint
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 | k | N/m | Combined series stiffness, in newtons per metre (N/m). Returns 0.0 when the total stiffness ``kb + km`` is not positive |
The science & history
Understanding the Parameters
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Bolt stiffness $k_b$ — the softer of the two usually dominates a series combination: $k$ can never exceed the smaller of $k_b$ and $k_m$. A slender bolt therefore controls the series rate.
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Member stiffness $k_m$ — from the cone-frustum estimate Bolted Joint Stiffness Member. Members are typically several times stiffer than the bolt, so $k \approx k_b$ in many joints.
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Series vs parallel — the same two springs combine two ways depending on the question: series (this page) for the compliance of the bolt-and-member stack seen end-to-end; parallel ($k_b + k_m$) for how they share an external load. Keeping the two straight is the crux of bolted-joint analysis.
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Result $k$ — always smaller than either input, and smaller than the parallel sum.
Derivation (Approaching a Proof)
Two linear springs of rates $k_b$ and $k_m$ combine in series when the same force $F$ passes through both and their deflections add. Each stretches by $F/k_b$ and $F/k_m$ respectively, so the total relative displacement is
$$\delta = \frac{F}{k_b} + \frac{F}{k_m} = F\left(\frac{1}{k_b} + \frac{1}{k_m}\right).$$
The effective series stiffness is force over total displacement:
$$k = \frac{F}{\delta} = \frac{1}{\dfrac{1}{k_b} + \dfrac{1}{k_m}} = \frac{k_b\,k_m}{k_b + k_m}.$$
(Contrast the parallel case, where both springs share a common displacement and their forces add, giving $k_b + k_m$ — the combination that produces the joint constant $C$.)
Dimensional check. $\dfrac{k_b\,k_m}{k_b+k_m} = \dfrac{(\text{N}/\text{m})^2}{\text{N}/\text{m}} = \text{N}/\text{m}$ — a stiffness, as required.
History and Development
Series/parallel spring combination is elementary mechanics, but its careful application to bolted joints — recognising that the same bolt and members combine one way for stack compliance and another for external load sharing — is what the Shigley/VDI 2230 joint model formalises. The reciprocal-sum rate here underlies the joint-diagram construction (the triangular force–deflection diagram) from which preload, external load, and separation are all read off geometrically.
Related Concepts: Joint Stiffness Ratio, Bolted Joint Stiffness Member, Bolt Shank Stiffness, Bolt Thread Stiffness, Bolt Load External, Member Load External
Notes: Series rate (reciprocal sum) — always smaller than either spring. For external-load sharing use the parallel sum $k_b+k_m$ via Joint Stiffness Ratio instead. Bolt stiffness $k_b$ from Bolt Shank Stiffness + Bolt Thread Stiffness in series; $k_m$ from Bolted Joint Stiffness Member.