🧮 在线计算器：VDI 2230 Bolted Joint Calculation — The full 14-step calculation chain (R0–R13), with six strength checks.
The “spring philosophy” of a bolted joint — the core physical model of VDI 2230
1. The spring model: the physical basis of VDI 2230
The starting point of all VDI 2230 calculations is to model the bolted joint as two sets of springs (VDI 2230:2015, §3.2, p.20):
“In this model, the bolt and the clamped parts are considered as tension and compression springs with the elastic resiliences $\delta_S$ and $\delta_P$.”
- The bolt = a spring in tension, with elastic resilience (elastische Nachgiebigkeit) $\delta_S$
- The clamped parts (flanges, backing plates, etc.) = a spring in compression, with elastic resilience $\delta_P$
[!NOTE] Resilience vs stiffness Resilience $\delta$ is the reciprocal of stiffness $k$: $\delta = f/F = 1/k$. VDI 2230 uses resilience rather than stiffness because the resiliences of the bolt’s segments add directly in a series system (VDI 2230:2015, §5.1.1.1, Eq. 19):
$$\delta_S = \delta_{SK} + \delta_1 + \delta_2 + \dots + \delta_{Gew} + \delta_{GM}$$This is more intuitive than summing the reciprocals of stiffnesses.
When the bolt is tightened, it stretches (storing elastic energy) and the clamped parts are compressed (also storing elastic energy). The two constrain each other through the contact surfaces of the bolt head and the nut, forming a force-balanced system.
2. The joint diagram (Verspannungsschaubild)
This is the most classic visualization tool in VDI 2230 (VDI 2230:2015, §3.2, Bild 1 / Bild 2). The standard represents the force–deformation relationship of the bolt and the clamped parts with two sloped lines:
Assembled state (no external force):
- Under the preload $F_V$, the bolt stretches by $f_{SM}$, with slope $1/\delta_S$
- Under the same $F_V$, the clamped parts are compressed by $f_{PM}$, with slope $1/\delta_P$
- The two lines meet at the preload $F_V$ — this is the assembly equilibrium point
After applying an axial external force $F_A$:
- The bolt force increases by $F_{SA}$ (additional bolt force, Schraubenzusatzkraft)
- The compression force of the clamped parts decreases by $F_{PA}$ (plate unloading force)
- The standard gives the key force-balance relations (VDI 2230:2015, §3.2.1, Eq. 6, 7):
3. The force ratio Φ — the core parameter of external-force distribution
The force ratio $\Phi$ (Kraftverhältnis) is the most important intermediate quantity in VDI 2230. It answers a fundamental question: what fraction of the external force is transmitted to the bolt?
The standard gives different formulas for different clamping and loading cases:
Concentric symmetric loading and clamping ($s_{sym} = 0$, $a = 0$), from Eq. (83) (VDI 2230:2015, §4.2, Eq. R3/3):
$$ \Phi_n = n \cdot \frac{\delta_P + \delta_{PZu}}{\delta_S + \delta_P} \tag{R3/3} $$Eccentric clamping and loading (the most common case, $s_{sym} \neq 0$, $a > 0$) (VDI 2230:2015, Eq. R3/4):
$$ \Phi_{en}^{*} = n \cdot \frac{\delta_P^{**} + \delta_{PZu}}{\delta_S + \delta_P^{*}} \tag{R3/4} $$where $\delta_P^{*}$ is from Eq. (67) and $\delta_P^{**}$ from Eq. (71) (VDI 2230:2015, §5.3.1).
Key parameters
| Parameter | Meaning | Source |
|---|---|---|
| $\delta_S$ | bolt elastic resilience | §5.1.1, Eq. 19 |
| $\delta_P$ | clamped-parts elastic resilience | §5.1.2, Eq. 40-53 |
| $n$ | load introduction factor (Krafteinleitungsfaktor) | §5.2.2 |
| $\delta_{PZu}$ | additional resilience (load-introduction position correction) | §5.3.1 |
The physical meaning of the load introduction factor $n$: the standard §5.2.2 states that $n$ describes where the external force acts inside the clamped parts. When the load introduction point is in the plane of the bolt head / nut, $n = 1$ (the most unfavourable case); when the load introduction point is near the centre of the interface, $n$ can be as small as about 0.3 (VDI 2230:2015, §5.2.2). Determining $n$ often needs engineering judgement or FEM support.
4. Residual clamping force — the functional bottom line
After applying the external force $F_A$, the residual clamping force (Restklemmkraft) at the interface is the preload minus the unloading effect. The standard’s constraint on the minimum required clamping force $F_{Kerf}$ is (VDI 2230:2015, §5.4.1, Eq. R2/4):
$$ F_{Kerf} \geq \max\left(F_{KQ};\; F_{KP} + F_{KA}\right) \tag{R2/4} $$This $F_{Kerf}$ comes from three functional requirements (VDI 2230:2015, §4.2, Eq. R2/1–R2/3):
- Friction force transfer: $F_{KQ} = \dfrac{F_{Q\max}}{q_F \cdot \mu_{T\min}} + \dfrac{M_{Y\max}}{q_M \cdot r_a \cdot \mu_{T\min}}$ (transmit transverse force and/or torque, R2/1)
- Sealing function: $F_{KP} = A_D \cdot p_{i\max}$ (resist medium pressure, R2/2)
- Prevent opening: $F_{KA}$ (prevent one-sided opening of the interface, R2/3)
The core design constraint is: under all operating conditions, the residual clamping force must always stay above $F_{Kerf}$.
5. The main equation: everything converges here
All of the analysis above finally converges to the main equation of VDI 2230 (VDI 2230:2015, §4.2, Eq. 16):
$$ F_{M\max} = \alpha_A \cdot \left[ F_{Kerf} + (1 - \Phi) \cdot F_A + F_Z + \Delta F'_{Vth} \right] \tag{16} $$The standard positions this formula:
“All of these factors (Figure 5) are an integral part of the main dimensioning formula, which is the basis for the bolt calculation.”
(VDI 2230:2015, §4.2, p.30)
The physical meaning of each term:
| Term | Meaning | Source step |
|---|---|---|
| $F_{Kerf}$ | minimum clamping force required for function | R2, Eq. R2/4 |
| $(1-\Phi) \cdot F_A$ | unloading effect of the external force on the interface | R3, Eq. R3/2 |
| $F_Z$ | preload loss due to embedding | R4, Eq. R4/1 |
| $\Delta F'_{Vth}$ | preload change due to thermal expansion difference | R4, Eq. R4/2 |
| $\alpha_A$ | scatter amplification factor of the tightening method | R1, Eq. R1/1 |
The design logic: start from the interface’s functional requirement $F_{Kerf}$, add the external-force effect and the various preload losses term by term, then multiply by the tightening scatter, to back out the maximum preload that must be applied at assembly — then verify through the six checks R7–R12 whether the bolt can carry it.
6. Next step: into the resilience calculation
The framework of the spring model is now set up. To compute the force ratio $\Phi$, we need the specific values of $\delta_S$ and $\delta_P$. The next article goes into the elastic resilience of the bolt $\delta_S$ — how does VDI 2230 break a real bolt down into series cylindrical segments to compute its resilience?
Data basis and accuracy statement
All quotations in this article are from VDI 2230 Blatt 1:2015-11. Citation format: (VDI 2230:2015, section, formula/figure/table/page).
Disclaimer: This article is for engineering teaching reference only. The final responsibility for engineering safety verification rests with the user.
📚 Series navigation
← Previous: Scope and Standard Positioning ｜ Next: Bolt Elastic Resilience δS →