🧮 Online calculator: VDI 2230-1 Single-bolt Joint Calculation — The full 14-step calculation chain (R0–R13), with six strength checks.
Bolt elastic resilience δS — breaking the bolt into a chain of cylinders
The previous article set up the framework of the spring model; this article provides the first concrete number for the numerator and denominator of the force ratio $\Phi$ — how “soft” is the bolt itself?
1. Basic principle: series cylindrical segments
VDI 2230 treats the bolt as a tension spring made of several series cylindrical segments of different cross-sections (VDI 2230:2015, §5.1.1, Bild 6).
The elastic resilience of each cylindrical segment is (VDI 2230:2015, §5.1.1.1, Eq. 18):
$$ \delta_i = \frac{l_i}{E_S \cdot A_i} \tag{18} $$where $l_i$ is the segment length, $A_i$ the segment cross-section area, and $E_S$ the elastic modulus of the bolt material.
Because the segments are in series, the total resilience is the sum of the segment resiliences (VDI 2230:2015, §5.1.1.1, Eq. 19):
$$ \delta_S = \delta_{SK} + \delta_1 + \delta_2 + \dots + \delta_{Gew} + \delta_{GM} \tag{19} $$In series = add directly
The standard explains: “In the bolt, the cylindrical elements are arranged in a row, so that the total elastic resilience $\delta_S$ is determined by adding the resiliences of the individual cylindrical elements within the clamp length and the further deformation regions.” (VDI 2230:2015, §5.1.1.1, p.40)
2. Substitutional lengths and formulas of each segment
The standard computes not only the segments inside the clamp length, but also the regions affected by force outside the clamp length — the head and the nut / blind-hole region. Each is described below:
2.1 Bolt head $\delta_{SK}$
The standard uses a “substitutional extension length” (Ersatzdehnlänge) to model the elastic deformation of the head (VDI 2230:2015, §5.1.1.1, Eq. 29):
$$ \delta_{SK} = \frac{l_{SK}}{E_S \cdot A_N} \tag{29} $$where $A_N = \frac{\pi}{4} d^2$ (nominal-diameter cross-section area, Eq. 25), and the substitutional length is:
- Hexagon-head bolt: $l_{SK} = 0.5 \cdot d$ (Eq. 30, for a medium through-hole diameter $d_h$)
- Socket-head cap screw: $l_{SK} = 0.4 \cdot d$ (Eq. 31)
Source: (VDI 2230:2015, §5.1.1.1, Eq. 29-31)
2.2 Plain-shank segments inside the clamp length $\delta_i$
Each plain-shank segment (diameter may differ) is computed directly by Eq. (18). For a standard bolt, the clamp length usually contains:
- Unthreaded shank segment: cross-section area $A_i = \frac{\pi}{4} d_i^2$ ($d_i$ is the actual diameter of that segment)
- Reduced-shank bolt (Dehnschaft): cross-section area computed from the waist diameter $d_T$
2.3 Free thread segment $\delta_{Gew}$
For the thread segment inside the clamp length that is not engaged, use the thread minor-diameter cross-section $A_{d_3}$ (VDI 2230:2015, §5.1.1.1, Eq. 28):
$$ \delta_{Gew} = \frac{l_{Gew}}{E_S \cdot A_{d_3}} \tag{28} $$where $A_{d_3} = \frac{\pi}{4} d_3^2$ (Eq. 23).
2.4 Engaged thread segment + nut / blind-hole region $\delta_{GM}$
This is the most complex part. It contains two sub-terms (VDI 2230:2015, §5.1.1.1, Eq. 20):
$$ \delta_{GM} = \delta_G + \delta_M \tag{20} $$Engaged segment $\delta_G$ — the part of the bolt engaged in the nut or blind hole (Eq. 21, 22):
$$ \delta_G = \frac{l_G}{E_S \cdot A_{d_3}}, \qquad l_G = 0.5 \cdot d \tag{21, 22} $$Nut / blind-hole region $\delta_M$ — caused by the bending and compressive deformation of the thread teeth (Eq. 24):
$$ \delta_M = \frac{l_M}{E_M \cdot A_N} \tag{24} $$The standard describes the physical origin of $\delta_M$:
“$\delta_M$ results from the axial relative movement between bolt and nut or internal thread as a result of the elastic bending and compressive deformation of the teeth of the bolt and nut threads and of the arching and compressive deformation of the nut.” (VDI 2230:2015, §5.1.1.1, p.41)
Key difference: DSV vs ESV
| Parameter | Through-bolt joint DSV | Tapped-thread joint ESV | Source |
|---|---|---|---|
| $l_M$ | $0.4 \cdot d$ | $0.33 \cdot d$ | Eq. 26, 27 |
| $E_M$ | $E_S$ (bolt material) | $E_{BI}$ (material of the part with the internal thread) | §5.1.1.1, p.42 |
This is the first notable difference between DSV and ESV in the resilience calculation — for ESV, the elastic modulus of the nut region is that of the part with the internal thread (the part the bolt is screwed into, often cast iron or aluminium), not the bolt steel.
3. Full example: M12 × 1.75 standard hexagon-head bolt
Assume an M12 × 1.75 hexagon-head bolt, through-bolt joint (DSV), clamp length $l_K = 40$ mm, of which the plain shank is 25 mm and the free thread is 15 mm.
The substitutional lengths and resiliences of each segment:
| Segment | Length | Cross-section area | Formula source |
|---|---|---|---|
| Head $\delta_{SK}$ | $l_{SK} = 0.5 \times 12 = 6$ mm | $A_N = \frac{\pi}{4} \times 12^2 = 113.1$ mm² | Eq. 29, 30 |
| Shank $\delta_1$ | $l_1 = 25$ mm | $A_1 = 113.1$ mm² | Eq. 18 |
| Free thread $\delta_{Gew}$ | $l_{Gew} = 15$ mm | $A_{d_3} = \frac{\pi}{4} \times 9.853^2 = 76.2$ mm² | Eq. 28 |
| Engaged segment $\delta_G$ | $l_G = 0.5 \times 12 = 6$ mm | $A_{d_3} = 76.2$ mm² | Eq. 21, 22 |
| Nut $\delta_M$ | $l_M = 0.4 \times 12 = 4.8$ mm | $A_N = 113.1$ mm² | Eq. 24, 26 |
With $E_S = 210\,000$ N/mm² (values in mm/N): $\delta_{SK}=2.53\times10^{-7}$, $\delta_1=1.053\times10^{-6}$, $\delta_{Gew}=9.37\times10^{-7}$, $\delta_G=3.75\times10^{-7}$, $\delta_M=2.02\times10^{-7}$; in total $\delta_S \approx 2.82\times10^{-6}$ mm/N (our calculator gives the same result for the same input).
4. Bending resilience $\beta_S$ (brief note)
In §5.1.1.2 the standard also defines the bolt’s bending resilience $\beta_S$ (Biegenachgiebigkeit), used to compute the additional bending-moment effect under eccentric loading (VDI 2230:2015, §5.1.1.2, Eq. 34):
$$ \beta_S = \beta_{SK} + \beta_1 + \beta_2 + \dots + \beta_{Gew} + \beta_M + \beta_G \tag{34} $$The structure of the bending-resilience calculation is exactly like that of the axial resilience, only replacing the cross-section area $A_i$ with the section moment of inertia $I_i$ (Eq. 33):
$$ \beta_i = \frac{l_i}{E \cdot I_i} \tag{33} $$With concentric clamping and concentric loading, the bending resilience is not used (this is the case our calculator covers). With eccentric clamping or eccentric loading, $\beta_S$ is used to calculate the additional bending stress in the bolt (§5.1.1.2, §5.5.3); eccentric cases are covered in part 010 of this series .
Data basis and accuracy statement
All formulas in this article are from VDI 2230 Blatt 1:2015-11, §5.1.1. The thread geometry parameter in the worked example ($d_3 = 9.853$ mm for M12×1.75) is from DIN 13-1.
Disclaimer: This article is for engineering teaching reference only. The final responsibility for engineering safety verification rests with the user.
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