🧮 在线计算器：VDI 2230 Bolted Joint Calculation — The full 14-step calculation chain (R0–R13), with six strength checks.

Preload design: from functional requirement to assembly preload (R1–R6)

The 14-step VDI 2230 calculation chain splits into a design part (R0–R6) and a verification part (R7–R13). This article explains the core steps of the design part.

1. R0 — preliminary diameter selection

The task of R0 is to preliminarily fix the nominal diameter $d$ based on experience or a simplified method (such as the Kübler equation), and to check the interface limiting dimension (VDI 2230:2015, §4.2, Eq. R0/1, R0/2):

$$ G = h_{\min} + d_W \tag{R0/1} $$

If the interface dimension $c_T$ exceeds $G$, the standard’s cone-resilience formula produces a notable error and FEM verification should be considered.

2. R1 — tightening factor αA

The tightening factor $\alpha_A$ describes the ratio of the maximum to the minimum preload that the same tightening method can produce under the same conditions (VDI 2230:2015, §5.4.3, Eq. R1/1):

$$ \alpha_A = \frac{F_{M\max}}{F_{M\min}} \tag{R1/1} $$

The standard gives reference values for different tightening methods in Table A8:

Tightening method $\alpha_A$ Source
Manual torque wrench 2.5–4.0 Table A8
Precision torque wrench (μ known) 1.4–1.6 Table A8
Yield-point-controlled tightening 1.0 Table A8
Angle-controlled tightening 1.0 Table A8

Physical meaning: the larger $\alpha_A$, the greater the uncertainty of the preload. The design must check strength against the maximum preload $F_{M\max}$ and function against the minimum preload $F_{M\min}$ — both ends must be safe.

3. R2 — required minimum clamping force FKerf

$F_{Kerf}$ is the starting point of the whole design calculation — it answers “at least how much residual clamping force does the interface need to meet the functional requirement”. The standard gives three functional sources (VDI 2230:2015, §4.2, Eq. R2/1–R2/4):

3.1 Transmitting transverse force and/or torque

$$ F_{KQ} = \frac{F_{Q\max}}{q_F \cdot \mu_{T\min}} + \frac{M_{Y\max}}{q_M \cdot r_a \cdot \mu_{T\min}} \tag{R2/1} $$

where $q_F$ and $q_M$ are the numbers of interfaces taking part in slipping (single shear $q = 1$, double shear $q = 2$), and $\mu_{T\min}$ is the minimum interface friction coefficient.

3.2 Sealing function

$$ F_{KP} = A_D \cdot p_{i\max} \tag{R2/2} $$

where $A_D$ is the sealing area and $p_{i\max}$ is the maximum medium pressure.

3.3 Preventing opening

$$ F_{KA} \tag{R2/3} $$

The specific value of $F_{KA}$ depends on the eccentric geometry and the bending-moment effect, see §5.3.2.

3.4 Total requirement

$$ F_{Kerf} \geq \max\left(F_{KQ};\; F_{KP} + F_{KA}\right) \tag{R2/4} $$

[!NOTE] Choosing the design starting point Many people get stuck at R2 — because they do not know what value $F_{Kerf}$ should take. The standard’s logic is: start from the function. If your joint must both transmit transverse force and seal, take the larger of the two. If none of the three items above applies (pure axial tension with no sealing), then $F_{Kerf} = 0$.

4. R3 — force ratio Φ (external-force distribution)

This step was discussed in detail in the spring model article and the bolt-resilience article . A recap of the key formulas:

Concentric symmetric (VDI 2230:2015, Eq. R3/3):

$$ \Phi_n = n \cdot \frac{\delta_P + \delta_{PZu}}{\delta_S + \delta_P} \tag{R3/3} $$

Eccentric (VDI 2230:2015, Eq. R3/4):

$$ \Phi_{en}^{*} = n \cdot \frac{\delta_P^{**} + \delta_{PZu}}{\delta_S + \delta_P^{*}} \tag{R3/4} $$

The output of this step also includes the additional bolt force and the plate unloading force (Eq. R3/1, R3/2):

$$ F_{SA} = \Phi \cdot F_A \tag{R3/1} $$$$ F_{PA} = (1 - \Phi) \cdot F_A \tag{R3/2} $$

5. R4 — preload change (the “preload thieves”)

5.1 Embedding loss

$$ F_Z = \frac{f_Z}{\delta_S + \delta_P} \tag{R4/1} $$

The standard explains (VDI 2230:2015, §5.4.2, R4):

“The guide values for the amounts of embedding $f_Z$ in the case of bolts, nuts and compact clamped parts made of steel can be taken from Table 5.”

Typical values from Table 5: under the bolt head ≈ 3 μm, nut / internal thread ≈ 3 μm, each internal interface ≈ 3 μm.

5.2 Thermal expansion difference

$$ \Delta F'_{Vth} = \frac{l_K \cdot (\alpha_S \cdot \Delta T_S - \alpha_P \cdot \Delta T_P)}{\delta_S \frac{E_{SRT}}{E_{ST}} + \delta_P \frac{E_{PRT}}{E_{PT}}} \tag{R4/2} $$

Symbols: $\alpha_S, \alpha_P$ are the thermal expansion coefficients, $\Delta T$ is the temperature rise, and the subscripts RT/T denote the elastic-modulus correction at room / operating temperature.

[!IMPORTANT] A key point to note The standard states clearly in R5: if $\Delta F'_{Vth} < 0$ (i.e. the thermal effect increases the preload), then when computing $F_{M\min}$ set $\Delta F'_{Vth} = 0$. This is a conservative treatment — do not rely on the thermal effect to keep the clamping force (VDI 2230:2015, R5, p.25).

6. R5 — minimum assembly preload

$$ F_{M\min} = F_{Kerf} + (1 - \Phi_{en}^{*}) \cdot F_{A\max} + F_Z + \Delta F'_{Vth} \tag{R5/1} $$

This is the version of the VDI 2230 main equation (Eq. 16) with the $\alpha_A$ factor removed. Its logic is (VDI 2230:2015, §5.4.3, R5):

“The required minimum assembly preload is obtained while taking into account preload changes and assuming the greatest possible relief of the joint.”

Read it right to left: the preload must be large enough that, after subtracting the external-force unloading $(1-\Phi)F_A$, the embedding loss $F_Z$ and the thermal change $\Delta F'_{Vth}$, a clamping force of $F_{Kerf}$ still remains.

7. R6 — maximum assembly preload

$$ F_{M\max} = \alpha_A \cdot F_{M\min} \tag{R6/1} $$

Source (VDI 2230:2015, §5.4.3, R6):

“Taking into account (R1/1), the possible maximum assembly preload is calculated.”

$F_{M\max}$ is the maximum preload that can occur at assembly (because $\alpha_A > 1$); the later strength checks (R7–R8) are all based on this worst case.

8. The design logic chain of R0–R6

R0 → preliminary d
R2 → FKerf (functional requirement)
R3 → Φ (resilience ratio → force distribution)    needs δS(R3→§5.1.1), δP(R3→§5.1.2)
R4 → FZ, ΔF'Vth (preload loss)  needs δS, δP, fZ(Table 5), thermal parameters
R5 → FMmin = FKerf + (1-Φ)FA + FZ + ΔF'Vth
R1 → αA (tightening scatter)
R6 → FMmax = αA · FMmin
↓
enter the checks R7–R13

The key insight: R1–R6 is a from-requirement-to-preload derivation chain. Its core idea is “work backward from the functional bottom line of the interface” — the exact opposite of the traditional “select the bolt first, then check” approach.

Data basis and accuracy statement

All formulas in this article are from VDI 2230 Blatt 1:2015-11, §4.2 calculation summary and §5.4.

Disclaimer: This article is for engineering teaching reference only.

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