🧮 Online calculator: VDI 2230-1 Single-bolt 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

R0 roughly sizes the nominal diameter $d$ with the four-step procedure of Table A7 (VDI 2230 Blatt 1:2015, §4.2 R0, Table A7):

A. Pick the table force: in the first column, take the nearest value that is not smaller than the load. If axial and transverse loads act together and $F_{A\max} < F_{Q\max}/\mu_{T\min}$, use only $F_{Q\max}$.
B. Step up for the load type: 4 steps when designing for a transverse load; when designing for an axial load, 2 steps for dynamic and eccentric, 1 step for dynamic and concentric or static and eccentric, 0 steps for static and concentric. This gives $F_{M\min}$.
C. Step up for the tightening method: 2 steps for a simple power screwdriver; 1 step for a torque wrench or precision screwdriver; 0 steps for angle-controlled or yield-controlled tightening. This gives $F_{M\max}$.
D. Read the diameter in that row for the property class.

R0 is only a pre-selection. R7 ($F_{Mzul} \geq F_{M\max}$) confirms whether the diameter is large enough.

For eccentrically clamped or eccentrically loaded joints, also check that the calculation is valid: the interface size $c_T$ in the plane of the bolt axis and the line of action of the axial working load must not exceed these limiting dimensions (VDI 2230 Blatt 1:2015, §4.2 R0). Use (R0/1) for a through-bolt joint (DSV/TBJ) and (R0/2) for a tapped-thread joint (ESV/TTJ):

$$ \mathrm{DSV:}\quad G = h_{\min} + d_W \tag{R0/1} $$$$ \mathrm{ESV:}\quad G' \approx (1.5 \dots 2)\, d_W \tag{R0/2} $$

This limit applies to eccentric cases only. If $c_T$ exceeds $G$ (or $G'$), the calculation error becomes relatively large (VDI 2230 Blatt 1:2015, §4.2 R0).

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 Blatt 1:2015, §4.2 R1, §5.4.3):

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

The standard gives reference values for the tightening methods in Table A8 of the annex. The table below lists the most common rows (VDI 2230 Blatt 1:2015, Table A8):

Tightening method	$\alpha_A$ (Table A8)
Yield-controlled / angle-controlled tightening	table 1.2–1.4; in design use $\alpha_A = 1$ (the bolt is sized for $F_{M\min}$; the scatter is still there)
Hydraulic tensioning, no friction and no torsion	1.1–1.4
Torque-controlled, target torque from tests on the original joint	1.4–1.6
Torque-controlled, target torque from an estimated friction coefficient (friction class B / A)	1.6–2.0 / 1.7–2.5
Impact wrench, stall wrench or pulse wrench; tightening by hand feel	2.5–4

[!warning] $\alpha_A = 1$ does not mean zero scatter For yield-controlled and angle-controlled tightening the design uses $\alpha_A = 1$ only because the bolt is sized for $F_{M\min}$ and needs no separate design for $F_{M\max}$. The real scatter is still there (table 1.2–1.4); it now comes mainly from the scatter of the yield strength within the bolt batch (VDI 2230 Blatt 1:2015, §4.2 R1, 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 Blatt 1:2015, §4.2 R2):

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 $M_{Y\max}$ is the torque about the bolt axis (not a bending moment), $r_a$ the friction radius from the clamped-part geometry, $q_F$ / $q_M$ the numbers of internal interfaces transmitting the transverse force / the torque, and $\mu_{T\min}$ the minimum static friction coefficient at the interface (Table A6). This is Eq. (108) in §5.4.1 (VDI 2230 Blatt 1:2015, §4.2 R2, §5.4.1).

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

$$ F_{KA} = F_{Kab} = F_{A\max}\frac{A_D\,(a\,u - s_{sym}\,u)}{I_{BT} + s_{sym}\,u\,A_D} + M_{B\max}\frac{u\,A_D}{I_{BT} + s_{sym}\,u\,A_D} \tag{R2/3} $$

For the geometric meaning of $s_{sym}$, $a$ (always taken as positive) and $u$, see §5.3.2 and the eccentric-load article (VDI 2230 Blatt 1:2015, §4.2 R2, §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 , the bolt-resilience article and the clamped-parts resilience article . A recap of the key formulas:

Concentric clamping and loading ($s_{sym} = 0$, $a = 0$) (VDI 2230 Blatt 1:2015, §4.2 R3, Eq. (83)):

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

Eccentric clamping and loading (VDI 2230 Blatt 1:2015, §4.2 R3):

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

The output of this step also includes the additional bolt force and the unloading force of the clamped parts (VDI 2230 Blatt 1:2015, §4.2 R3):

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

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 Blatt 1:2015, §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.”

Examples from Table 5 (steel, uncoated): for $R_z < 10\ \mu\text{m}$ under axial tension/compression, 3 µm in the thread, 2.5 µm per head/nut bearing surface and 1.5 µm per inner interface; under shear 3 / 3 / 2 µm. Rougher surfaces give larger values at the bearing surfaces and interfaces (up to 4–6.5 µm / 3–3.5 µm for $R_z$ 40–160 µm); the thread value stays at 3 µm. $f_Z$ is the sum of all contributions (VDI 2230 Blatt 1:2015, Table 5).

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.

(R4/2) is Eq. (118) in §5.4.2.3, a design simplification for the case where the change of the elastic modulus can be ignored. Otherwise use the full Eq. (117) and also consider the preload loss from relaxation (VDI 2230 Blatt 1:2015, §4.2 R4, §5.4.2.3).

Sign convention: $\Delta F'_{Vth} > 0$ means the thermal effect causes a preload loss (same sign as $F_Z$); $\Delta F'_{Vth} < 0$ means the preload rises. The zero rules in R5, R8, R10 and R12 all build on this convention.

[!IMPORTANT] A key point to note The standard states in R5: if it cannot be fully ensured that the load is always applied only after the working (steady-state) temperature is reached, then for $\Delta F'_{Vth} < 0$ (the thermal effect raises the preload) set $\Delta F'_{Vth} = 0$ when computing $F_{M\min}$. Do not rely on the thermal effect to supply clamping force (VDI 2230 Blatt 1:2015, §4.2 R5).

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 VDI 2230 main dimensioning formula, Eq. (16), with the $\alpha_A$ factor removed. Its logic is (VDI 2230 Blatt 1:2015, §4.2 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 Blatt 1:2015, §4.2 R6):

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

$F_{M\max}$ is the largest preload that can occur during assembly. R7 compares it with the permissible assembly preload of the bolt ($F_{Mzul} \geq F_{M\max}$); R8 and the later checks all use $F_{Mzul}$, and the peak in service is $F_{S\max} = F_{Mzul} + \Phi_{en}^{*} F_{A\max} - \Delta F_{Vth}$. The four forces increase in this order: $F_{M\min} < F_{M\max} \leq F_{Mzul} < F_{S\max}$ (VDI 2230 Blatt 1:2015, §4.2 Figure 5, R7, R8).

8. The design logic chain of R0–R6

R0 → preliminary d (Table A7)
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: R0 first pre-selects the diameter with Table A7. R2–R6 then work backward from the functional minimum at the interface, $F_{Kerf}$, to the required range of assembly preload, and R7 uses it to check whether the pre-selected diameter is large enough. If not, choose a larger nominal diameter and repeat from R2 (VDI 2230 Blatt 1:2015, §4.2 R0, R7). The backward chain sits between pre-selection and verification; the two are not opposites.

Data basis and accuracy statement

All formulas and reference values in this article are from VDI 2230 Blatt 1:2015-11: the §4.2 calculation summary, §5.3–§5.4, Table 5 and Tables A7 and A8 of the annex.

Disclaimer: This article is for engineering teaching reference only.

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

1. R0 — preliminary diameter selection#

2. R1 — tightening factor αA#

3. R2 — required minimum clamping force FKerf#

3.1 Transmitting transverse force and/or torque#

3.2 Sealing function#

3.3 Preventing opening#

3.4 Total requirement#

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

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

5.1 Embedding loss#

5.2 Thermal expansion difference#

6. R5 — minimum assembly preload#

7. R6 — maximum assembly preload#

8. The design logic chain of R0–R6#

Data basis and accuracy statement#