🧮 Online calculator: VDI 2230-1 Single-bolt Joint Calculation — The full 14-step calculation chain (R0–R13) for concentric clamping and loading, with six strength checks; the eccentric cases in this article are not supported yet.
What the calculator covers: the calculator covers eccentric clamping and eccentric loading under a tensile working load (VDI 2230:2015, §5.3.2, Table 4, load cases I–III). The eccentric formulas in this article ($\delta_P^*$, $\delta_P^{**}$, $\Phi_{en}^*$, $F_{Kab}$ and the extra bending stress) are all in the calculator. An external moment $M_B$, a compressive working load, and eccentric joints whose clamped plates have different moduli of elasticity are not included yet; such joints still have to be calculated by hand following the standard.
Eccentric load and bending-moment effect — the “deep water” of VDI 2230
All the earlier worked examples assumed concentric symmetry ($s_{sym} = 0$, $a = 0$). But the standard states clearly: concentric symmetry is the minority case in engineering practice.
1. Why is eccentricity the norm?
The standard says (VDI 2230:2015, §5.1.2.3, p.57):
“The case of a concentrically clamped and concentrically loaded BJ is only rarely found in practice. In most cases, the line of action of the load $F_A$ does not lie in the bolt axis.”
Eccentricity comes from two independent sources, which must be handled separately:
| Type | Parameter | Physical meaning |
|---|---|---|
| Eccentric clamping (exzentrische Verspannung) | $s_{sym}$ | The bolt axis is offset from the symmetry axis of the clamped parts |
| Eccentric loading (exzentrischer Kraftangriff) | $a$ | The line of action of the external force is offset from the symmetry axis of the clamped parts |
2. Determining the eccentricity parameter $s_{sym}$
$s_{sym}$ describes the offset of the bolt axis relative to the geometric symmetry axis of the clamped parts (VDI 2230:2015, §5.1.2.2, Eq. 66):
$$ s_{sym} = \frac{c_T}{2} - e \leq \frac{G}{2} - e \tag{66} $$where $c_T$ is the interface dimension (in the plane of analysis) and $e$ is the shortest distance from the bolt axis to the interface edge.
When $s_{sym} = 0$ → the bolt lies on the symmetry axis of the clamped parts → concentric clamping When $s_{sym} \neq 0$ → the bolt is eccentric → it causes an additional bending moment
3. Determining the eccentric-loading distance $a$
$a$ is the distance from the substitutional line of action of the external force to the symmetry axis of the clamped parts (VDI 2230:2015, §5.2.1, p.61):
“The distance $a$ is the distance of the substitutional line of action of the axial working load from the axis of the laterally symmetrical deformation body, thus ultimately a lever arm.”
Determining $a$ is an elasto-mechanics problem — you need to find the point of zero bending moment nearest to the bolt. The standard gives the classic example of a connecting rod (Pleuel) (VDI 2230:2015, §5.2.1, p.62):
$$ a \approx 0.275 R $$This result is for a straight-split connecting rod with eccentric clamping, modelled as a circular ring of constant cross-section; $R$ is the radius of the centroid circle of the ring section (VDI 2230:2015, §5.2.1, Bild 15).
For a general structure, a statically indeterminate analysis that includes the bending deformation of the clamped parts is needed to find the point of zero moment nearest to the bolt (VDI 2230:2015, §5.2.1); in practice this is often done with finite element analysis.
4. Corrected resiliences $\delta_P^*$ and $\delta_P^{**}$
The eccentric effect increases the resilience of the clamped parts — because, on top of the axial compression, it adds a bending deformation.
4.1 Eccentric-clamping corrected resilience $\delta_P^*$
Used in the denominator of the force-ratio formula (VDI 2230:2015, §5.1.2.2, Eq. 67):
$$ \delta_P^* = \delta_P + \frac{s_{sym}^2 \cdot l_K}{E_P \cdot I_{Bers}} \tag{67} $$where $\delta_P$ is the symmetric resilience (Eq. 40 or 41) and $I_{Bers}$ is the substitutional moment of inertia of the deformation body.
4.2 Eccentric-loading corrected resilience $\delta_P^{**}$
Used in the numerator of the force-ratio formula (VDI 2230:2015, §5.1.2.3, Eq. 71):
$$ \delta_P^{**} = \delta_P + \frac{a \cdot s_{sym} \cdot l_K}{E_P \cdot I_{Bers}} \tag{71} $$$\delta_P^{**}$ can be smaller than, equal to, or larger than $\delta_P^*$, depending on the relative magnitude and direction of $a$ and $s_{sym}$.
4.3 Eccentric force ratio $\Phi_{en}^*$
The most general force-ratio formula (VDI 2230:2015, Eq. R3/4):
$$ \Phi_{en}^* = n \cdot \frac{\delta_P^{**} + \delta_{PZu}}{\delta_S + \delta_P^*} \tag{R3/4} $$Simplified forms for different eccentric cases (VDI 2230:2015, Eq. 73-75):
| Case | Calculation of $F_{SA}$ | Source |
|---|---|---|
| $s_{sym} \neq 0$, $a > 0$ | $n \cdot \frac{\delta_P^{**}}{\delta_S + \delta_P^*} \cdot F_A$ | Eq. 73 |
| $s_{sym} \neq 0$, $a = 0$ | $n \cdot \frac{\delta_P}{\delta_S + \delta_P^*} \cdot F_A$ | Eq. 74 |
| $a = s_{sym} \neq 0$ | $n \cdot \frac{\delta_P^*}{\delta_S + \delta_P^*} \cdot F_A$ | Eq. 75 |
These three forms are for DSV. For ESV, the extra load-introduction resilience $\delta_{PZu}$ must be added as described in §5.3.1 (VDI 2230:2015, §5.1.2.3).
[!CAUTION] The risk when $a$ and $s_{sym}$ are on different sides The standard warns (VDI 2230:2015, §5.1.2.3, p.58): “If $a$ and $s_{sym}$ do not lie on the same side of the axis of symmetry, the additional bolt load $F_{SA}$ […] may become larger than calculated.” This case should be avoided by design.
5. Substitutional moment of inertia $I_{Bers}$
The $I_{Bers}$ in the eccentricity-correction formulas is the equivalent section moment of inertia of the deformation body (cone + sleeve).
Cone part (Eq. 59):
$$ I_{Bers}^V = 0.147 \cdot \frac{(D_A - d_W) \cdot d_W^3 \cdot D_A^3}{D_A^3 - d_W^3} \tag{59} $$Eccentricity correction (Steiner shift term, Eq. 60):
$$ I_{Bers}^{Ve} = I_{Bers}^V + s_{sym}^2 \cdot \frac{\pi}{4} D_A^2 \tag{60} $$If $D_A \geq D_{A,Gr}$ (no deformation sleeve), use $D_{A,Gr}$ in place of $D_A$ in (59) and (60) (VDI 2230:2015, §5.1.2.2).
Sleeve part (Eq. 61):
$$ I_{Bers}^H = \frac{b \cdot c_T^3}{12} \tag{61} $$with $b \leq D_{A,Gr}$.
Combined body (Eq. 62):
$$ I_{Bers} = \frac{l_K}{\frac{2}{w}(l_V / I_{Bers}^{Ve}) + l_H / I_{Bers}^H} \tag{62} $$This $I_{Bers}$ does not subtract the hole. For the extra bending stress $\sigma_{SAb}$ the bolt cross-section is not counted, and the value with the hole taken out is used instead (VDI 2230:2015, §5.1.2.2, Eq. 64):
$$ \bar I_{Bers} = I_{Bers} - \frac{\pi}{64} d_h^4 \tag{64} $$6. Opening check (Klaffen)
Eccentric loading causes the interface to begin losing contact pressure on the bending-tension side — that is, the interface opens (VDI 2230:2015, §5.3.2).
The standard’s approach: as long as the interface size does not exceed the limit $G$ or $G'$, the pressure from the preload is taken as evenly spread over the contact area $A_D$, and the bending stress from the working load as linear. The interface pressure along the coordinate $x$ is then (VDI 2230:2015, §5.3.2, Eq. 99):
$$ p(x) = -\frac{F_K}{A_D} + \frac{M_{Kl}}{I_{BT}}\,x \tag{99} $$Setting the pressure to zero at the edge on the opening side, $x = u$, and solving for the clamp load at which opening just starts gives (VDI 2230:2015, §5.3.2, Eq. 107):
$$ F_{Kab} = F_A \frac{(a - s_{sym})\,u\,A_D}{I_{BT} + s_{sym}\,u\,A_D} + M_B \frac{u\,A_D}{I_{BT} + s_{sym}\,u\,A_D} \tag{107} $$Here $u$ is the distance from the edge on the opening side to the centroid line of the interface, and $I_{BT}$ is the moment of inertia of the interface area; $a$ is always positive, and the signs of $s_{sym}$ and $u$ follow the sign rules of Tabelle 4 in the standard.
This $F_{Kab}$ is the clamp load $F_{KA}$ needed to prevent opening in R2 (Eq. R2/3). It is added to the sealing requirement $F_{KP}$, and the sum is compared with the slip requirement $F_{KQ}$; the larger value is $F_{Kerf}$ (Eq. R2/4).
If the clamp load is too low and the interface has opened, the resilience of the clamped parts increases nonlinearly (VDI 2230:2015, §5.3.3), and the standard’s linear spring model no longer applies. This is also why the standard stresses that $F_{Kerf}$ must be large enough to prevent opening.
7. Practical advice for eccentric design
Based on the standard’s requirements, the design of an eccentric joint should follow these principles:
Constructional level
- Minimize $s_{sym}$ — place the bolt near the symmetry axis of the clamped parts
- Keep $a$ and $s_{sym}$ on the same side — avoid an unfavourable lever effect (Eq. 73–75)
- Control the interface dimension $c_T \leq G$ — ensure the linear spring model is valid (Eq. 54, 55)
Calculation level
- Do not use the concentric-symmetric formulas — unless it is verified that $s_{sym} \approx 0$ and $a \approx 0$
- Do not subtract the through-hole area when computing $I_{Bers}$ — because the bolt takes part in bending through the head and the nut; only the extra bending stress uses $\bar I_{Bers}$ with the hole taken out (Eq. 64)
- Check critical joints with finite element analysis — finding the eccentricity parameters $a$ and $n$ often needs finite element support
The standard sums up (VDI 2230:2015, §5.1.2.3, p.57):
“It is to be taken into account that, under unfavorable cross-sectional conditions, even relatively small eccentricities of the load introduction may have a considerable effect on the deformation behaviour of the clamped parts.”
Data basis
All formulas in this article are from VDI 2230 Blatt 1:2015-11, §5.1.2.2, §5.1.2.3, §5.2.1 and §5.3.2.
Disclaimer: This article is for engineering teaching reference only. The accuracy of the eccentric calculation depends heavily on the precise determination of $a$ and $s_{sym}$; checking critical joints with finite element analysis is advised.
📚 Series navigation
← Previous: Full Worked Example (ESV) ｜ Next: Annex B, Example B1 →
Full-series review
| # | Title | Core |
|---|---|---|
| 001 | VDI 2230 Overview | necessity, main equation |
| 002 | Scope and Standard Positioning | boundaries, standard family |
| 003 | Spring Model and Force Distribution | force ratio Φ |
| 004 | Bolt Elastic Resilience δS | Eq. 17-31 |
| 005 | Clamped-Parts Resilience δP | Rötscher cone |
| 006 | Preload Design R1-R6 | design part |
| 007 | Strength Checks R7-R13 | verification part |
| 008 | Worked Example DSV | M12→M16 iteration |
| 009 | Worked Example ESV | cast-iron housing |
| 010 | Eccentric Load (this article) | $\delta_P^*$, $\delta_P^{**}$ |
| 011 | Annex B, Example B1 | central bolt of a hydraulic cylinder |
| 012 | Annex B, Example B2 | transverse load and slip of a disc coupling |
| 013 | Annex B, Example B3 | torque transfer by a hollow flywheel centre bolt |