🧮 在线计算器：VDI 2230 Bolted 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):
$$ \delta_S = \delta_{SK} + \delta_1 + \delta_2 + \dots + \delta_{Gew} + \delta_{GM} \tag{19} $$[!NOTE] “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)
- 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) | $E_{BI}$ (clamped-part material) | §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 takes the clamped-part material (usually 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 |
Computing the resilience of each segment with $E_S = 210\,000$ N/mm² and summing gives $\delta_S$.
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):
$$ \beta_i = \frac{l_i}{E \cdot I_i} \tag{33} $$The bending resilience does not take part directly in V1 (concentric symmetric loading), but it is required in the eccentric-load mode.
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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