🧮 Online calculator: VDI 2230-1 Single-bolt Joint Calculation — The full 14-step calculation chain (R0–R13), with six strength checks. Use the inputs in the section “Re-run it in the calculator” to check this example step by step.
Full worked example: tapped thread joint (ESV)
This article works a tapped thread joint (ESV) example of our own, as a counterpart to the DSV example (article 8) . It highlights the differences unique to ESV: $w = 2$, $E_M = E_{BI}$, the cone-angle formula Eq. 42, the supplement $\delta_{PZu}$ to the clamped-part resilience in the load factor, and the R11 length of engagement check. It matches the VDI 2230-1 single-bolt joint calculator on this site. All inputs are listed in the section “Re-run it in the calculator” at the end, so you can check the numbers yourself.
Engineering problem statement
A bolt is screwed from the top into a cast-iron housing (grey cast iron GJL-250), fixing a steel cover plate.
| Parameter | Value | Note |
|---|---|---|
| Joint type | ESV (tapped thread) | Bolt screwed into the cast-iron housing |
| External axial force | $F_A = 15\,000$ N (static) | |
| External transverse force | $F_Q = 0$ N | No transverse force |
| Sealing requirement | $A_D = 800$ mm², $p_{i\max} = 10$ MPa | O-ring seal |
| Clamp length | $l_K = 30$ mm | Steel cover-plate thickness |
| Clamped-parts outer diameter | $D_A = 40$ mm | |
| Cover-plate material | S355 JO, $E_P = 205\,000$ MPa | $E_P$ and limiting surface pressure $p_G = 760$ MPa, both from VDI 2230:2015, Tabelle A9 |
| Housing material | GJL-250, $E_{BI} = 110\,000$ MPa | $R_m = 250$ MPa, $\tau_B = 290$ MPa (VDI 2230:2015, Tabelle A9) |
| Operating temperature | Room temperature | $\Delta F'_{Vth} = 0$ |
| Coefficient of friction | $\mu_G = \mu_K = 0.14$ | |
| Load introduction factor | $n = 0.5$ | |
| Tightening method | Torque wrench, target torque found by tests | $\alpha_A = 1.6$ |
R0 — preliminary diameter selection
Select M12 × 1.75, property class 10.9 (same thread data as article 8); hexagon-head bearing diameter $d_W = 17.0$ mm (the default in our calculator), through-hole diameter $d_h = 13.5$ mm, plain shank in the clamp length 10 mm.
Limiting size $G'$: this joint is concentrically clamped and concentrically loaded. The values $G' \approx (1.5 \dots 2)\,d_W$ of (R0/2) (Gl.(55), which assumes a steel tapped part) and $G'_{\max} \approx 3\,d_W$ (Gl.(56)) are validity limits only for eccentric cases (VDI 2230:2015, R0; §5.1.2.2). Here they are for reference only:
$$ G' \approx (1.5 \dots 2) \times 17.0 = 25.5 \dots 34 \text{ mm}, \qquad G'_{\max} \approx 3 \times 17.0 = 51 \text{ mm} $$$c_T = D_A = 40$ mm is above the upper value of $G'$ (34 mm) but below $G'_{\max} = 51$ mm. The tapped part here is cast iron, so the condition of Gl.(55) is not fully met either.
R1 — tightening factor
$$ \alpha_A = 1.6 $$R2 — required minimum clamp load
No transverse force → $F_{KQ} = 0$
Sealing requirement (Eq. R2/2):
$$ F_{KP} = A_D \cdot p_{i\max} = 800 \times 10 = 8\,000 \text{ N} $$$$ F_{Kerf} = F_{KP} = 8\,000 \text{ N} $$R3 — elastic resiliences and load factor
⚠️ The ESV vs DSV difference in δS
The overall bolt-resilience formula is unchanged (Eq. 19), but the nut region $\delta_M$ is different (VDI 2230:2015, §5.1.1.1):
| Parameter | DSV | ESV |
|---|---|---|
| $l_M$ | $0.4d = 4.8$ mm | $0.33d = 3.96$ mm (Eq. 27) |
| $E_M$ | $E_S = 210\,000$ | $E_{BI} = 110\,000$ |
ESV bolt resilience $\delta_S$:
| Segment | Length | Cross-section area | $\delta_i$ |
|---|---|---|---|
| Head $\delta_{SK}$ | 6 mm | 113.1 | $2.53 \times 10^{-7}$ |
| Shank $\delta_1$ | 10 mm | 113.1 | $4.21 \times 10^{-7}$ |
| Free thread $\delta_{Gew}$ | 20 mm | 76.2 | $1.249 \times 10^{-6}$ |
| Engaged thread $\delta_G$ | $0.5 \times 12 = 6$ mm | 76.2 | $3.75 \times 10^{-7}$ |
| Nut $\delta_M$ | 3.96 mm | 113.1 ($E_{BI}$) | $3.18 \times 10^{-7}$ |
[!IMPORTANT] In ESV, δM uses the elastic modulus of the tapped part This is the most notable difference between ESV and DSV — the elastic modulus of the nut region takes the cast iron’s $E_{BI} = 110\,000$ MPa, not the bolt steel’s 210,000 MPa. For an aluminium-alloy housing ($E_{BI} \approx 70\,000$ MPa), this effect is even more marked (VDI 2230:2015, §5.1.1.1, p.42). And $\delta_M$ appears twice in an ESV: once in the denominator, as part of $\delta_S$, and once in the numerator, as the supplement $\delta_{PZu}$ to the clamped-part resilience (see the load factor below).
ESV clamped-parts resilience
For ESV the joint coefficient $w = 2$ and the cone angle uses Eq. 42. No separate outer diameter of the base body is given, so $D_A' = D_A$:
$$ \beta_L = l_K / d_W = 30 / 17.0 = 1.765 $$$$ y = D_A' / d_W = 40 / 17.0 = 2.353 $$$$ \tan\varphi_E = 0.348 + 0.013\ln(1.765) + 0.193\ln(2.353) = 0.348 + 0.0074 + 0.1651 = 0.5205 $$Limiting diameter (Eq. 39, $w = 2$):
$$ D_{A,Gr} = 17.0 + 2 \times 30 \times 0.5205 = 48.2 \text{ mm} $$$D_A = 40 < D_{A,Gr} = 48.2$ → use Eq. 41 (cone + sleeve):
$$ \delta_P = \frac{\frac{2}{2 \times 13.5 \times 0.5205}\ln\left[\frac{(17.0+13.5)(40-13.5)}{(17.0-13.5)(40+13.5)}\right] + \frac{4}{40^2-13.5^2}\left[30-\frac{40-17.0}{2 \times 0.5205}\right]}{205\,000 \times \pi} $$Logarithmic term: $\ln\left[\frac{30.5 \times 26.5}{3.5 \times 53.5}\right] = \ln(4.316) = 1.462$
Sleeve length: $l_H = l_K - \frac{D_A - d_W}{w \cdot \tan\varphi} = 30 - \frac{23.0}{1.041} = 30 - 22.09 = 7.91$ mm
$$ \delta_P = \frac{\frac{2 \times 1.462}{14.05} + \frac{4 \times 7.91}{1417.8}}{644\,026} = \frac{0.208 + 0.022}{644\,026} = 3.58 \times 10^{-7} \text{ mm/N} $$Load factor Φ
For ESV $w = 2$, and by Gl.(81) $\delta_{PZu} = (w-1)\delta_M = \delta_M = 3.18 \times 10^{-7}$ mm/N. In an ESV a large part of the force flows through the screwed-in thread, and this term corrects for it (VDI 2230:2015, §5.3.1). With (R3/3):
$$ \Phi_n = n \cdot \frac{\delta_P + \delta_{PZu}}{\delta_S + \delta_P} = 0.5 \times \frac{3.58 + 3.18}{26.16 + 3.58} = 0.5 \times \frac{6.76}{29.74} = 0.114 $$The exact value from the calculator is $\Phi = 0.1137$. Without $\delta_{PZu}$ you would get only 0.060 and underestimate the bolt’s share of the load by almost half. This is the key difference between ESV and DSV in the load factor.
$$ F_{SA} = 0.1137 \times 15\,000 = 1\,705 \text{ N}, \qquad F_{PA} = (1 - 0.1137) \times 15\,000 = 13\,295 \text{ N} $$R4 — preload change
Take $f_Z$ from Tabelle 5, using the roughest class ($R_z = 40 \dots 160\ \mu$m, tension/compression). An ESV has only one bearing surface (under the bolt head) and 1 inner interface (between cover plate and housing):
$f_Z$ = thread 3 μm + head bearing surface 4 μm + 1 inner interface 3 μm = 10 μm (VDI 2230:2015, §5.4.2.1, Tabelle 5)
$$ F_Z = \frac{0.010}{2.974 \times 10^{-6}} = 3\,363 \text{ N} $$Note: the values in Tabelle 5 were measured on steel parts. A cast-iron housing is outside that range, so we take the roughest class to stay on the safe side.
R5 — minimum assembly preload
$$ F_{M\min} = 8\,000 + (1-0.1137) \times 15\,000 + 3\,363 = 8\,000 + 13\,295 + 3\,363 = 24\,658 \text{ N} $$R6 — maximum assembly preload
$$ F_{M\max} = 1.6 \times 24\,658 = 39\,452 \text{ N} $$R7 — assembly stress check
The permissible assembly preload follows from (R7/2) with $\mu_G = 0.14$; the other values are the same as for M12 in article 8:
$$ F_{Mzul} = 84.26 \times \frac{846}{\sqrt{1 + 3 \times [1.573 \times (0.0513 + 0.1617)]^2}} = 84.26 \times \frac{846}{\sqrt{1.337}} = 84.26 \times 731.7 = 61\,657 \text{ N} $$$$ 61\,657 \geq 39\,452 \;\Rightarrow\; F_{Mzul} \geq F_{M\max} $$✅ M12-10.9 is enough! (In contrast to the DSV example — for ESV the transverse force is zero and the clamp load needed for sealing is smaller, so M12 passes.)
R8–R12 check summary
| Step | Check item | Computed value | Permissible value | Safety factor | Result |
|---|---|---|---|---|---|
| R8 | Service stress | $\sigma_{red,B} = 803.5$ N/mm² | $R_{p0.2} = 940$ N/mm² | $S_F = 1.17$ | ✅ |
| R9 | Fatigue | Static load, $\sigma_a = 0$ | — | — | not needed |
| R10 | Surface pressure (steel cover plate) | $p_{B\max} = 756$ N/mm² | $p_G = 760$ N/mm² | $S_P = 1.03$, $S_{pB} = 1.01$ | ✅ (small margin) |
| R11 | Length of engagement | $m_{eff,\min} = 13.6$ mm | — | — | see below |
| R12 | Slip | $F_Q = 0$ → not needed | — | — | — |
R8 — service stress
Torque-controlled tightening, so the torsional stress is included by (R8/4) with $k_\tau = 0.5$ (VDI 2230:2015, R8):
$$ F_{S\max} = F_{Mzul} + \Phi F_A = 61\,657 + 0.1137 \times 15\,000 = 63\,363 \text{ N} $$$$ \sigma_{z\max} = \frac{63\,363}{84.26} = 752.0 \text{ N/mm}^2, \qquad \tau_{\max} = \frac{M_G}{W_P} = \frac{71\,324}{218.2} = 326.9 \text{ N/mm}^2 $$$$ \sigma_{red,B} = \sqrt{752.0^2 + 3 \times (0.5 \times 326.9)^2} = 803.5 \text{ N/mm}^2, \qquad S_F = \frac{940}{803.5} = 1.17 \geq 1.0 $$R10 — surface pressure
The bolt head presses on the steel cover plate, so R10 checks the cover plate (S355 JO, $p_G = 760$ N/mm²). The GJL-250 housing ($p_G = 850$ N/mm² in Tabelle A9) is not under the head, so R10 does not involve it.
$D_{Ki}$ is not given and is taken as the through-hole diameter $d_h = 13.5$ mm (the area comes out a little too large, so the result is on the unsafe side):
$$ A_{p\min} = \frac{\pi}{4}(17.0^2 - 13.5^2) = 83.8 \text{ mm}^2 $$- Assembly state: $p_{M\max} = F_{Mzul}/A_{p\min} = 61\,657/83.8 = 735$ N/mm², $S_P = 1.03$;
- Working state: $p_{B\max} = (61\,657 + 1\,705)/83.8 = 756$ N/mm², $S_{pB} = 1.01$.
Both pass, but with a small margin. A chamfer at the hole would make the real area even smaller, so a washer under the head is worth considering (outer diameter $d_{Wa} = d_W + 1.6\,h_S$, Gl.(194)).
R11 — length of engagement check (ESV-specific)
This check step is unique to ESV (VDI 2230:2015, §5.5.5, R11). First, the strength ratio $R_S$ tells which thread strips first. It is defined with the shear strengths and the thread shear areas (VDI 2230:2015, §5.5.5, Gl.(195)):
$$ R_S = \frac{\tau_{BM} \cdot A_{SGM}}{\tau_{BS} \cdot A_{SGS}} $$The shear areas of the internal and external thread, $A_{SGM}$ and $A_{SGS}$, come from Gl.(197) and (198). In their ratio the length of engagement cancels, so the ratio depends only on the thread geometry; for M12 coarse it is 1.386. GJL-250 has $\tau_{BM} = 290$ N/mm² (Tabelle A9). A 10.9 bolt has $\tau_B/R_m = 0.62$ (Tabelle 7), so $\tau_{BS} = 0.62 \times 1040 = 644.8$ N/mm²:
$$ R_S = 1.386 \times \frac{290}{644.8} = 0.623 < 1 $$The internal thread is critical: under overload the thread in the housing strips first.
[!note] Why not Gl.(199) Gl.(199) is derived from Gl.(195)–(198) under the steel premise of Gl.(196) and replaces the shear-strength ratio with the tensile-strength ratio $R_{mM}/R_{mS}$. This assumes that both thread materials have the same $\tau_B/R_m$, so it is valid only for steel on steel (VDI 2230:2015, §5.5.5, Gl.(196), (199)). GJL-250 is grey cast iron with $\tau_B/R_m = 290/250 \approx 1.16$, while a 10.9 bolt has only 0.62. The condition does not hold, so this example uses Gl.(195). Steel-formula comparison: using Gl.(199) by mistake would give $R_S \approx 0.33$, and $m_{eff,\min}$ below would come out about 3.4 % too small, which is on the unsafe side.
When the internal thread is critical, the required length of engagement depends on the shear strength $\tau_{BM}$ of the internal-thread material (290 N/mm² here). If the actual thread sizes are not known, the worst tolerance case is used: bolt major diameter at its lower limit, internal pitch diameter at its upper limit, and $R_{m,\max} \approx 1.2\,R_{m,\min}$ (VDI 2230:2015, §5.5.5.1, Gl.(210)). For M12 6g/6H the limit sizes are $d_{\min} = 11.701$ mm and $D_{2\max} = 11.063$ mm (ISO 965):
$$ m_{ges,\min} = \frac{R_{m,\max} A_S P}{C_1 C_3 \tau_{BM,\min}\left[\frac{P}{2} + (d_{\min} - D_{2\max})\tan 30^\circ\right]\pi d_{\min}} + 2P $$$$ = \frac{1.2 \times 1040 \times 84.3 \times 1.75}{1 \times 1.019 \times 290 \times [0.875 + 0.638 \times 0.577] \times \pi \times 11.701} + 3.5 = 13.63 + 3.5 = 17.13 \text{ mm} $$Here $C_1 = 1$ (no size of the material around the internal thread is given, so the usual ESV case $s/d > 1.9$ is assumed) and $C_3 = 0.728 + 1.769 R_S - 2.896 R_S^2 + 1.296 R_S^3 = 1.019$ (Gl.(202), valid for $0.4 < R_S < 1$). Taking off $2P$ for the thread run-out and chamfer gives the minimum effective length of engagement:
$$ m_{eff,\min} = 17.13 - 2 \times 1.75 = 13.63 \text{ mm}, \qquad m_{eff}/d \approx 1.1 $$Design recommendation: use an effective length of engagement of 14 mm; the total length of engagement is then $m_{ges} = 14 + 2 \times 1.75 = 17.5$ mm. Both are above the required 13.63 mm and 17.13 mm, so the check passes, but the margin is only about 0.4 mm.
[!note] How to read Bild 36 Bild 36 has the shear strength $\tau_{BM}$ of the internal-thread material on the horizontal axis and three curves for bolt property classes 8.8 / 10.9 / 12.9: the lower $\tau_{BM}$, the larger the required $m_{eff}/d$ (VDI 2230:2015, §5.5.5.1, Bild 36). The curves are calculated with the tolerance limits of Gl.(210) above and take M4 as the worst size (upper envelope), so for M12 the chart gives a slightly more conservative value than the formula. The horizontal axis is the shear strength, not the tensile strength $R_m$; reading the chart with $R_m$ gives a wrong answer.
R13 — tightening torque
Friction diameter of the bearing surface $D_{Km} = (d_W + D_{Ki})/2 = (17.0 + 13.5)/2 = 15.25$ mm:
$$ M_A = 61\,657 \times \left[0.16 \times 1.75 + 0.58 \times 10.863 \times 0.14 + \frac{15.25}{2} \times 0.14\right] $$$$ = 61\,657 \times [0.280 + 0.882 + 1.068] = 61\,657 \times 2.230 = 137\,470 \text{ N·mm} $$$$ M_A \approx 137.5 \text{ N·m} $$DSV vs ESV comparison summary
| Item | Article 8 DSV (M16-10.9) | This article ESV (M12-10.9) |
|---|---|---|
| Joint coefficient $w$ | 1 | 2 |
| Cone-angle formula | Eq. 43 | Eq. 42 |
| $E_M$ (nut region) | $E_S = 210\,000$ | $E_{BI} = 110\,000$ |
| $l_M$ (nut substitutional length) | $0.4d$ | $0.33d$ |
| $\delta_{PZu}$ (supplement to the clamped-part resilience) | 0 | $\delta_M$ |
| R11 length of engagement | not applicable | must be checked |
| Governing design factor | Transverse force $F_{KQ}$ | Sealing $F_{KP}$ |
Re-run it in the calculator
Open the VDI 2230-1 single-bolt joint calculator , fill in the inputs below, keep all other inputs at their defaults, and click Run Calculation. This is our own example, so the table gives every input.
| Input (label in the calculator) | Value | Note |
|---|---|---|
| Calculation Mode | Verification Mode (specified) | |
| Joint Type | TTJ — Tapped hole/ESV | |
| Bolt size / Head Type | M12 / Hexagon head | $d_W$ uses the default 17.0 mm |
| Clamped layers | One layer, 30 mm | Click the material button, pick S355 JO from Table A9 (fills in $E_P = 205\,000$ and $p_G = 760$) |
| Equivalent OD at the interface $D_A$ | 40 mm | Leave $D_A'$ empty |
| Calculate the clamped parts as a TBJ | Automatic | No $D_A'$ given, so no switch to DSV |
| Through-hole Diameter $d_h$ / Unthreaded shank length in the clamp $l_1$ | 13.5 mm / 10 mm | |
| Tapped-body modulus $E_{BI}$ | Click “Pick material” next to it and pick GJL-250 from Table A9 | This also fills in $E_{BI} = 110\,000$, $R_{mM} = 250$ and $\tau_{BM,min} = 290$, and sets “Tapped part is steel” to No |
| Tapped part is steel | No (shear strengths, Eq.(195)) | Set by the material pick above; if you type the values in by hand, choose No yourself |
| Nut/internal-thread shear strength $\tau_{BM,min}$ | 290 N/mm² | Filled in by the material pick; needed when the tapped part is not steel, because $R_S$ is found from it |
| Bolt major diameter, lower limit $d_{min}$ | 11.701 mm | M12 6g |
| Internal pitch diameter, upper limit $D_{2max}$ | 11.063 mm | M12 6H |
| Available length of engagement $m_{eff,vorh}$ | 14 mm | Optional; with it R11 gives a pass, without it R11 is marked “not checked” |
| Axial Load $F_A$ / Transverse Load $F_Q$ | 15 000 N / 0 | |
| Internal Pressure $p_i$ / Sealing Area $A_D$ | 10 N/mm² / 800 mm² | |
| Load Introduction Factor $n$ | 0.5 | |
| Strength Class | 10.9 | |
| Thread / Head Friction | 0.14 / 0.14 | Keep the interface friction at its default; with no transverse load it does not change the result |
| Tightening Method / Tightening Factor $\alpha_A$ | Torque controlled / 1.6 | |
| Utilisation $\nu$ | 0.9 | |
| Embedding $f_Z$ | 0.010 mm | The 10 μm embedding amount from R4 |
| Include Thermal Effects / Fatigue Check | No / No |
How the results map to this article:
| Calculator output | Calculator value | Where in this article |
|---|---|---|
| Bolt resilience $\delta_S$ / clamped-part resilience $\delta_P$ | $2.616 \times 10^{-6}$ / $3.578 \times 10^{-7}$ mm/N | R3 |
| Load factor $\Phi$ | 0.1137 | R3 (the “Steps” tab shows $\delta_{PZu} = 3.1831 \times 10^{-7}$) |
| Required clamp load $F_{Kerf}$ | 8 000 N | R2 |
| Preload loss due to embedding $F_Z$ | 3 363 N | R4 |
| $F_{M\min}$ / $F_{M\max}$ / $F_{Mzul}$ | 24 658 / 39 452 / 61 657 N | R5–R7 |
| Maximum bolt load $F_{S\max}$ / comparative stress $\sigma_{red,B}$ | 63 363 N / 803.5 N/mm² | R8 |
| Assembly-state / working-state pressure | 735.4 / 755.7 N/mm² | R10 |
| Strength ratio $R_S$ / R11 critical thread | 0.6232 / internal | R11 (the “Steps” tab shows Eq.(195)) |
| Factor $C_3$ | 1.0194 | R11 |
| Required total length of engagement $m_{ges,min}$ / minimum effective length of engagement $m_{eff,min}$ | 17.127 / 13.63 mm | R11 |
| Tightening torque $M_A$ | 137 470 N·mm | R13 |
| $S_F$ / $S_P$ / $S_{pB}$ | 1.17 / 1.03 / 1.01 | Summary |
When the tapped part is not steel, $\tau_{BM,min}$ is needed to find $R_S$; without it the calculator reports an input error. Fill in both $d_{min}$ and $D_{2max}$, or neither; giving only one of them is also an input error. With neither, the calculator still finds that the internal thread is critical, but it does not use the tolerance version of Gl.(210); it falls back to a rough estimate $m_{eff} = 1.0d = 12$ mm, which is lower than the 13.63 mm found here, and says so in the “Steps” tab.
Data basis and accuracy statement
All formulas in this article are from VDI 2230 Blatt 1:2015-11. The thread parameters are from DIN 13-1, and the M12 6g/6H limit sizes follow ISO 965. The material values of GJL-250 and S355 JO are from VDI 2230 Tabelle A9. The bearing diameter $d_W$ of the hexagon head is the default value in our calculator.
Disclaimer: This article is for engineering teaching reference only.
📚 Series navigation
← Previous: Full Worked Example (DSV) ｜ Next: Eccentric Load and Bending-Moment Effect →