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Clamped-parts resilience δP and the Rötscher cone model
The previous article settled how “soft” the bolt is ($\delta_S$); this one answers how “soft” the clamped parts are ($\delta_P$) — the other half of the force ratio $\Phi$ calculation.
1. Why is δP harder to compute than δS?
The bolt resilience $\delta_S$ can be broken simply into series cylindrical segments that add up. But the clamped parts are a completely different case — the standard states (VDI 2230:2015, §5.1.2, p.45):
“The calculation of the elastic resilience $\delta_P$ of the parts preloaded by the bolt […] proves to be difficult on account of the three-dimensional stress and deformation state which forms when preload is applied.”
The preload spreads outward from the bearing surface under the bolt head; the compression zone widens gradually from the bearing surface toward the interface, in a shape close to a paraboloid of revolution (Rotationsparaboloid) (VDI 2230:2015, §5.1.2, p.45, [7; 9; 10]).
VDI 2230’s solution is: use an equivalent deformation cone (Ersatzverformungskegel) to replace the real paraboloid compression zone, so that it has the same elastic resilience — this is the so-called Rötscher cone model.
2. Core formula: computing δP
The general integral form of the clamped-parts resilience is (VDI 2230:2015, §5.1.2, Eq. 38):
$$ \delta_P = \int_{y=0}^{y=l_K} \frac{\mathrm{d}y}{E(y) \cdot A(y)} \tag{38} $$Integrating over the cone, the standard gives a closed-form analytical solution.
2.1 Case one: $D_A \geq D_{A,Gr}$ (cone does not reach the outer wall)
The cone can fully develop, unconstrained by the outer wall (VDI 2230:2015, §5.1.2.1, Eq. 40):
$$ \delta_P = \frac{2 \ln \left[ \frac{(d_W + d_h)(d_W + w \cdot l_K \cdot \tan\varphi - d_h)}{(d_W - d_h)(d_W + w \cdot l_K \cdot \tan\varphi + d_h)} \right]}{w \cdot E_P \cdot \pi \cdot d_h \cdot \tan\varphi} \tag{40} $$2.2 Case two: $d_W < D_A < D_{A,Gr}$ (cone + sleeve combination)
After the cone develops to the outer wall, the remaining part becomes a hollow sleeve of constant wall thickness (VDI 2230:2015, §5.1.2.1, Eq. 41):
$$ \delta_P = \frac{\frac{2}{w \cdot d_h \cdot \tan\varphi} \ln\left[\frac{(d_W + d_h)(D_A - d_h)}{(d_W - d_h)(D_A + d_h)}\right] + \frac{4}{D_A^2 - d_h^2}\left[l_K - \frac{D_A - d_W}{w \cdot \tan\varphi}\right]}{E_P \cdot \pi} \tag{41} $$2.3 The decision boundary: limiting diameter $D_{A,Gr}$
The standard uses a limiting diameter to decide which formula to use (VDI 2230:2015, §5.1.2, Eq. 39):
$$ D_{A,Gr} = d_W + w \cdot l_K \cdot \tan\varphi \tag{39} $$where the joint coefficient is:
| Joint type | $w$ | Source |
|---|---|---|
| ESV (tapped-thread) | 2 | §5.1.2 |
| DSV (through-bolt) | 1 | §5.1.2 |
- $D_A \geq D_{A,Gr}$: use Eq. 40 (pure cone)
- $d_W < D_A < D_{A,Gr}$: use Eq. 41 (cone + sleeve)
- $d_W \geq D_A$: sleeve only (hollow cylinder)
3. Cone angle φ — not a constant!
This is the most subtle part of the VDI 2230 resilience model. The cone angle $\varphi$ is not a fixed value but a variable determined by the geometry of the clamped parts (VDI 2230:2015, §5.1.2.1, Eq. 42-45):
Tapped-thread joint ESV / TTJ:
$$ \tan\varphi_E = 0.348 + 0.013 \ln\beta_L + 0.193 \ln y \tag{42} $$Through-bolt joint DSV / TBJ:
$$ \tan\varphi_D = 0.362 + 0.032 \ln(\beta_L / 2) + 0.153 \ln y \tag{43} $$where:
$$ \beta_L = l_K / d_W \tag{44} $$$$ y = D_A' / d_W \tag{45} $$The standard notes that, as an approximation, $\tan\varphi = 0.6$ can be taken within the following dimension-ratio ranges (VDI 2230:2015, §5.1.2.1, p.51):
- ESV: $\beta_L = 4 \dots 6$, $y = 2.5 \dots 4$
- DSV: $\beta_L = 0.5 \dots 4$, $y = 4 \dots 6$
“In this case, the maximum error when calculating the plate resilience is about 5%.” (VDI 2230:2015, §5.1.2.1, p.51)
4. Handling multi-layer plates
When the clamped parts are made of multiple layers of different materials (different $E_P$), the standard requires the cone and sleeve to be further split into sub-segments matching each layer (VDI 2230:2015, §5.1.2.1, Eq. 52-53):
Recursion of the bearing diameter of each cone sub-segment (Eq. 52):
$$ d_{W,i} = d_W + 2 \cdot \tan\varphi \cdot \sum_{i=1}^{j} l_{i-1} \tag{52} $$The total resilience is the sum of the sub-segments (Eq. 53):
$$ \delta_P = \sum_{i=1}^{j} \delta_{Pi}^V + \sum_{i=j+1}^{m} \delta_{Pi}^H \tag{53} $$5. Summary of DSV vs ESV differences in the δP calculation
| Parameter | DSV (through-bolt) | ESV (tapped-thread) | Source |
|---|---|---|---|
| Joint coefficient $w$ | 1 | 2 | §5.1.2, Eq. 39 |
| Cone-angle formula | Eq. 43 | Eq. 42 | §5.1.2.1 |
| Number of deformation cones | 2 (one at head + one at nut) | 1 (one at head) | Bild 8, 9 |
| Nut region $E_M$ | $E_S$ (bolt material) | $E_{BI}$ (clamped-part material) | §5.1.1.1 |
The ESV $w = 2$ means only one full cone extends from the bolt head to the interface, while the DSV $w = 1$ means the head and the nut each contribute one cone, meeting in the middle of the clamp length.
6. Numerical sensitivity note
The logarithmic term in Eq. 40 has $(d_W - d_h)$ in the denominator. When $d_W \approx d_h$ (the bearing-surface outer diameter is close to the through-hole diameter), this difference approaches zero and can cause numerical instability. In engineering practice this means: when the bearing surface is too small, the clamped-parts resilience rises sharply — which is exactly the expected physical behaviour (the compression zone degenerates into an extremely thin ring).
Data basis and accuracy statement
All formulas in this article are from VDI 2230 Blatt 1:2015-11, §5.1.2. The accuracy of the cone-angle regression formulas (Eq. 42/43) is about ±5% (§5.1.2.1).
Disclaimer: This article is for engineering teaching reference only. The final responsibility for engineering safety verification rests with the user.
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