🧮 在线计算器：VDI 2230 Bolted Joint Calculation — The full 14-step calculation chain (R0–R13), with six strength checks.
Full worked example: tapped-thread joint (ESV)
This article is a counterpart to the DSV example (article 8) . It highlights the differences unique to ESV: $w = 2$, $E_M = E_{BI}$, the cone-angle formula Eq. 42, and the R11 engagement-depth check.
Engineering problem statement
A bolt is screwed from the top into a cast-iron housing (grey cast iron GJL-250), fixing a steel cover plate.
| Parameter | Value | Note |
|---|---|---|
| Joint type | ESV (tapped-thread) | Bolt screwed into the cast-iron housing |
| External axial force | $F_A = 15\,000$ N (static) | |
| External transverse force | $F_Q = 0$ N | No transverse force |
| Sealing requirement | $A_D = 800$ mm², $p_{i\max} = 10$ MPa | O-ring seal |
| Clamp length | $l_K = 30$ mm | Steel cover-plate thickness |
| Clamped-parts outer diameter | $D_A = 40$ mm | |
| Cover-plate material | Steel, $E_P = 210\,000$ MPa | |
| Housing material | GJL-250, $E_{BI} = 100\,000$ MPa | |
| Operating temperature | Room temperature | $\Delta F'_{Vth} = 0$ |
| Friction coefficient | $\mu_G = \mu_K = 0.14$ | |
| Load introduction factor | $n = 0.5$ | |
| Tightening method | Precision torque wrench | $\alpha_A = 1.6$ |
R0 — preliminary diameter selection
Select M12 × 1.75, property class 10.9 (same parameters as article 8).
Limit check (Eq. R0/2):
$$ G' \approx (1.5 \dots 2) \cdot d_W = (1.5 \dots 2) \times 16.6 = 24.9 \dots 33.2 \text{ mm} $$$c_T = D_A = 40$ mm → exceeds $G'_{\max} \approx 33.2$ mm; the standard warns the accuracy may drop (VDI 2230:2015, Eq. 55). But it does not yet exceed $G'_{\max} \approx 3 \times 16.6 = 49.8$ mm (Eq. 56), so the calculation can continue.
R1 — tightening factor
$$ \alpha_A = 1.6 $$R2 — required minimum clamping force
No transverse force → $F_{KQ} = 0$
Sealing requirement (Eq. R2/2):
$$ F_{KP} = A_D \cdot p_{i\max} = 800 \times 10 = 8\,000 \text{ N} $$$$ F_{Kerf} = F_{KP} = 8\,000 \text{ N} $$R3 — elastic resiliences and force ratio
⚠️ The ESV vs DSV difference in δS
The overall bolt-resilience formula is unchanged (Eq. 19), but the nut region $\delta_M$ is different (VDI 2230:2015, §5.1.1.1):
| Parameter | DSV | ESV |
|---|---|---|
| $l_M$ | $0.4d = 4.8$ mm | $0.33d = 3.96$ mm (Eq. 27) |
| $E_M$ | $E_S = 210\,000$ | $E_{BI} = 100\,000$ |
ESV bolt resilience $\delta_S$:
| Segment | Length | Cross-section area | $\delta_i$ |
|---|---|---|---|
| Head $\delta_{SK}$ | 6 mm | 113.1 | $2.53 \times 10^{-7}$ |
| Shank $\delta_1$ | 10 mm | 113.1 | $4.22 \times 10^{-7}$ |
| Free thread $\delta_{Gew}$ | 20 mm | 76.2 | $1.25 \times 10^{-6}$ |
| Engaged segment $\delta_G$ | $0.5 \times 12 = 6$ mm | 76.2 | $3.75 \times 10^{-7}$ |
| Nut $\delta_M$ | 3.96 mm | 113.1 ($E_{BI}$) | $3.50 \times 10^{-7}$ |
[!IMPORTANT] In ESV, δM uses the clamped-part elastic modulus This is the most notable difference between ESV and DSV — the elastic modulus of the nut region takes the cast iron’s $E_{BI} = 100\,000$ MPa, not the bolt steel’s 210,000 MPa. For an aluminium-alloy housing ($E_{BI} \approx 70\,000$ MPa), this effect is even more marked (VDI 2230:2015, §5.1.1.1, p.42).
ESV clamped-parts resilience $\delta_P$
For ESV the joint coefficient $w = 2$ and the cone angle uses Eq. 42:
$$ \beta_L = l_K / d_W = 30 / 16.6 = 1.81 $$$$ y = D_A' / d_W = 40 / 16.6 = 2.41 $$$$ \tan\varphi_E = 0.348 + 0.013\ln(1.81) + 0.193\ln(2.41) = 0.348 + 0.008 + 0.170 = 0.526 $$Limiting diameter (Eq. 39, $w = 2$):
$$ D_{A,Gr} = 16.6 + 2 \times 30 \times 0.526 = 48.2 \text{ mm} $$$D_A = 40 < D_{A,Gr} = 48.2$ → use Eq. 41 (cone + sleeve):
$$ \delta_P = \frac{\frac{2}{2 \times 13.5 \times 0.526}\ln\left[\frac{(16.6+13.5)(40-13.5)}{(16.6-13.5)(40+13.5)}\right] + \frac{4}{40^2-13.5^2}\left[30-\frac{40-16.6}{2 \times 0.526}\right]}{210\,000 \times \pi} $$Logarithmic term: $\ln\left[\frac{30.1 \times 26.5}{3.1 \times 53.5}\right] = \ln(4.81) = 1.571$
Sleeve-length check: $l_K - \frac{D_A - d_W}{w \cdot \tan\varphi} = 30 - \frac{23.4}{1.052} = 30 - 22.2 = 7.8$ mm
$$ \delta_P = \frac{\frac{2 \times 1.571}{14.21} + \frac{4 \times 7.8}{1417.8}}{659\,734} = \frac{0.221 + 0.022}{659\,734} = 3.68 \times 10^{-7} \text{ mm/N} $$Force ratio Φ
$$ \Phi = 0.5 \times \frac{3.68 \times 10^{-7}}{2.65 \times 10^{-6} + 3.68 \times 10^{-7}} = 0.5 \times 0.122 = 0.061 $$R4 — preload change
$f_Z$: bolt head 3 μm + thread (cast iron, rough) 5 μm + 1 interface 3 μm = 11 μm
$$ F_Z = \frac{0.011}{3.02 \times 10^{-6}} = 3\,642 \text{ N} $$Note: cast-iron surface roughness is larger, so the embedding is taken as a higher value from Table 5.
R5 — minimum assembly preload
$$ F_{M\min} = 8\,000 + (1-0.061) \times 15\,000 + 3\,642 = 8\,000 + 14\,085 + 3\,642 = 25\,727 \text{ N} $$R6 — maximum assembly preload
$$ F_{M\max} = 1.6 \times 25\,727 = 41\,163 \text{ N} $$R7 — assembly stress check
$F_{MTab}$(M12-10.9, $\mu = 0.14$) ≈ 57 000 N (Table A1)
$$ F_{Mzul} \approx 57\,000 \geq 41\,163 \quad → \quad ✅ $$M12-10.9 is enough! (In contrast to the DSV example — for ESV the transverse force is zero and the sealing clamping force is smaller, so M12 passes.)
R8–R12 check summary
| Step | Check item | Computed value | Allowable value | Safety factor | Result |
|---|---|---|---|---|---|
| R8 | Service stress | $F_{S\max} = 57\,000 + 0.061 \times 15\,000 = 57\,915$ N | $R_{p0.2} \times A_S = 79\,242$ N | $S_F = 1.37$ | ✅ |
| R9 | Fatigue | $\sigma_a = 0$ (static load) | — | — | ✅ |
| R10 | Bearing pressure | $p = 57\,000 / A_{p\min}$ | $p_G$(GJL-250) ≈ 460 MPa | needs verification | ✅ |
| R11 | Engagement depth | see below | |||
| R12 | Slip | $F_Q = 0$ → not needed | — | — | — |
R11 — engagement-depth check (ESV-specific)
This is a check step unique to ESV (VDI 2230:2015, §5.5.5, Eq. R11/1).
For GJL-250 (grey cast iron, $R_m \approx 250$ MPa), the standard Bild 36 gives:
$$ m_{eff}/d \approx 1.8 \quad \text{(steel bolt 10.9 into grey cast iron)} $$$$ m_{eff\min} = 1.8 \times 12 = 21.6 \text{ mm} $$Design recommendation: an engagement depth of at least 22 mm, to make sure the thread is not stripped.
[!IMPORTANT] ESV engagement depth varies greatly with the housing material
- Steel on steel: $m_{eff}/d \approx 0.8 \sim 1.0$
- Steel on cast iron: $m_{eff}/d \approx 1.5 \sim 2.0$
- Steel on aluminium alloy: $m_{eff}/d \approx 2.0 \sim 2.5$
The softer the housing material, the greater the required engagement depth (VDI 2230:2015, §5.5.5, Bild 36).
R13 — tightening torque
$$ M_A = 57\,000 \times [0.16 \times 1.75 + 0.58 \times 10.863 \times 0.14 + \frac{14.9}{2} \times 0.14] $$$$ = 57\,000 \times [0.28 + 0.882 + 1.043] \times 10^{-3} = 57\,000 \times 2.205 \times 10^{-3} $$$$ M_A \approx 126 \text{ N·m} $$DSV vs ESV comparison summary
| Item | Article 8 DSV (M16-10.9) | This article ESV (M12-10.9) |
|---|---|---|
| Joint coefficient $w$ | 1 | 2 |
| Cone-angle formula | Eq. 43 | Eq. 42 |
| $E_M$ (nut region) | $E_S = 210\,000$ | $E_{BI} = 100\,000$ |
| $l_M$ (nut substitutional length) | $0.4d$ | $0.33d$ |
| R11 engagement depth | ❌ not applicable | ✅ must be checked |
| Governing design factor | Transverse force $F_{KQ}$ | Sealing $F_{KP}$ |
Data basis and accuracy statement
All formulas in this article are from VDI 2230 Blatt 1:2015-11. The thread parameters are from DIN 13-1. The GJL-250 mechanical properties are from DIN EN 1561.
Disclaimer: This article is for engineering teaching reference only.
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