🧮 Online calculator: VDI 2230-1 Single-bolt Joint Calculation — The full 14-step calculation chain (R0–R13), with six strength checks.
Six strength checks and the tightening torque: R7–R13
R1–R6 answered “how much preload to apply”; R7–R13 answer “can the bolt take it”.
1. R7 — assembly stress check (§5.5.1)
This is the first gate: the bolt does not exceed yield strength during assembly.
The standard allows the use of a fraction of the yield strength (usually $\nu = 0.9$, i.e. 90%); the allowable assembly comparison stress is (VDI 2230 Blatt 1:2015, §4.2 R7):
$$ \sigma_{red,Mzul} = \nu \cdot R_{p0.2\min} \tag{R7/1} $$The allowable assembly preload $F_{Mzul}$ accounts for the combined stress of tension and torsion that the bolt carries at the same time during tightening (VDI 2230 Blatt 1:2015, §4.2 R7):
$$ F_{Mzul} = A_0 \cdot \frac{\nu \cdot R_{p0.2\min}}{\sqrt{1 + 3 \left[\frac{3}{2} \frac{d_2}{d_0}\left(\frac{P}{\pi \cdot d_2} + 1.155 \mu_{G\min}\right)\right]^2}} \tag{R7/2} $$$A_0$ is the weakest cross-section and $d_0$ its diameter: normally $A_S$ with $d_0 = d_S = (d_2 + d_3)/2$; $d_0 = d_{i\min}$ if the shank is thinner than $d_S$; $d_0 = d_T$ for reduced-shank bolts. (R7/2) is Eq. (143) in §5.5.1 multiplied by $\nu \cdot A_0$ (Eq. (144)) (VDI 2230 Blatt 1:2015, §5.5.1).
Check condition:
$$ F_{Mzul} \geq F_{M\max} \tag{R7/3} $$If it is not met, the standard requires explicitly (VDI 2230 Blatt 1:2015, §4.2 R7):
“If the requirement is not met, a larger bolt nominal diameter is to be selected and the calculation repeated from R2.”
2. R8 — service stress check (§5.5.2)
The total bolt force under service contains the preload and the additional bolt force (VDI 2230 Blatt 1:2015, §4.2 R8):
$$ F_{S\max} = F_{Mzul} + \Phi_{en}^{*} \cdot F_{A\max} - \Delta F_{Vth} \tag{R8/1} $$[!IMPORTANT] Sign convention for the thermal effect Note: here, if $\Delta F_{Vth} > 0$ (the thermal effect reduces the preload), set $\Delta F_{Vth} = 0$ — this is the conservative direction, making sure the check is based on the maximum possible bolt force (VDI 2230 Blatt 1:2015, §4.2 R8).
The maximum tensile stress and torsional stress:
$$ \sigma_{z\max} = F_{S\max} / A_0 \tag{R8/2} $$$$ \tau_{\max} = M_G / W_P \tag{R8/3} $$where $M_G = F_{Mzul}\dfrac{d_2}{2}\left(\dfrac{P}{\pi d_2} + 1.155\,\mu_{G\min}\right)$ and $W_P = \dfrac{\pi}{16} d_0^3$ (VDI 2230 Blatt 1:2015, §4.2 R8).
The standard recommends a reduction factor $k_\tau = 0.5$: in service the torsional stress in the bolt is always lower than at assembly. In elastically tightened joints it drops to about half after some years under static load, and under alternating load it disappears completely over time (VDI 2230 Blatt 1:2015, §4.2 R8, §5.5.2.1):
$$ \sigma_{red,B} = \sqrt{\sigma_{z\max}^2 + 3(k_\tau \cdot \tau_{\max})^2} \tag{R8/4} $$Check condition and safety factor:
$$ S_F = \frac{R_{p0.2\min}}{\sigma_{red,B}} \geq 1.0 \tag{R8/5-2} $$3. R9 — fatigue stress check (§5.5.3)
If the joint carries a dynamic load (the external force alternates between $F_{A\max}$ and $F_{A\min}$), fatigue must be checked.
The stress amplitude (VDI 2230 Blatt 1:2015, §4.2 R9):
$$ \sigma_a = \frac{F_{SAo} - F_{SAu}}{2 A_S} \tag{R9/1} $$where $F_{SAo}$ and $F_{SAu}$ are the maximum and minimum additional bolt loads. For eccentric loading, use the bending stress amplitude instead (VDI 2230 Blatt 1:2015, §4.2 R9):
$$ \sigma_{ab} = \frac{\sigma_{SAbo} - \sigma_{SAbu}}{2} \tag{R9/2} $$with $\sigma_{SAb}$ from Eq. (188) in §5.5.3.
The standard gives reference values for the bolt fatigue limit (related to the stress cross-section $A_S$, $N_D \geq 2 \times 10^6$):
Rolled before heat treatment (SV, schlussvergütet) (VDI 2230 Blatt 1:2015, §4.2 R9):
$$ \sigma_{ASV} = 0.85 \cdot (150/d + 45) \tag{R9/5-1} $$Rolled after heat treatment (SG, schlussgewalzt) (VDI 2230 Blatt 1:2015, §4.2 R9):
$$ \sigma_{ASG} = (2 - F_{Sm}/F_{0.2\min}) \cdot \sigma_{ASV} \tag{R9/5-2} $$The gain of SG shrinks as the mean bolt force $F_{Sm}$ rises; as $F_{Sm} \to F_{0.2\min}$ it tends to the SV value.
Check condition ($\sigma_{a/ab}$ means $\sigma_a$ for concentric and $\sigma_{ab}$ for eccentric loading):
$$ \sigma_{a/ab} \leq \sigma_{AS} \tag{R9/3} $$$$ S_D = \frac{\sigma_{AS}}{\sigma_{a/ab}} \geq 1.0 \tag{R9/4} $$The user sets the safety factor; the standard cites reference [4], which recommends $S_D \geq 1.2$ (VDI 2230 Blatt 1:2015, §4.2 R9).
4. R10 — bearing pressure check (§5.5.4)
The bearing pressure under the bolt head / nut must not exceed the limiting surface pressure $p_G$ of the clamped-part material, otherwise it causes creep and preload loss. Under the head and under the nut of a through-bolt joint, use the $p_G$ of the part in contact at that side; if the two parts are different materials, check each side separately (VDI 2230 Blatt 1:2015, §5.5.4).
The standard explains (VDI 2230 Blatt 1:2015, §4.2 R10):
“Surface pressures which cause creep (time-dependent plastic flowing) in conjunction with a loss of preload should not become effective.”
Assembly state:
$$ p_{M\max} = F_{Mzul} / A_{p\min} \leq p_G \tag{R10/1} $$Service state:
$$ p_{B\max} = (F_{V\max} + F_{SA\max} - \Delta F_{Vth}) / A_{p\min} \leq p_G \tag{R10/2} $$Note: if $\Delta F_{Vth} > 0$, set $\Delta F_{Vth} = 0$ here (VDI 2230 Blatt 1:2015, §4.2 R10).
For yield-controlled or angle-controlled tightening, use the following; the factor 1.4 covers the scatter of the yield point (VDI 2230 Blatt 1:2015, §4.2 R10):
$$ p_{\max} = \frac{F_{MTab}}{A_{p\min}} \cdot 1.4 \tag{R10/3} $$Safety factor:
$$ S_p = p_G / p_{M/B\max} \geq 1.0 \tag{R10/4} $$5. R11 — minimum engagement depth (§5.5.5)
To prevent thread stripping, the bolt must be covered well enough by the nut thread or the internal thread. A tapped-thread joint (ESV) is the most common critical case; a standard nut (except thin types) carries the full load as long as its property class is not lower than that of the bolt (VDI 2230 Blatt 1:2015, §4.2 R11, §5.5.5):
$$ F_{mS} \leq \min(F_{mGM},\; F_{mGS}) \tag{R11/1} $$That is, the maximum tensile force of the bolt must be less than the critical stripping force of the internal thread or the bolt thread. (At R11 the standard prints $F_{MGS}$, which does not match its §2 symbol list; we follow the symbol list.)
The required effective engagement depth can be read approximately from Figure 36 (Bild 36): the horizontal axis is the shear strength $\tau_{BM}$ of the internal-thread material ($\tau_B/R_m$ from Table 6 times the $R_m$ of that material); pick the curve for bolt class 8.8 / 10.9 / 12.9 and read $m_{eff}/d$. The softer the material, the deeper the engagement: about $0.8d$ for steel, and much more than $1d$ for aluminium and magnesium alloys; how much depends on $\tau_{BM}$ and the bolt class, and the standard gives only the chart, not a number for these metals (VDI 2230 Blatt 1:2015, §4.2 R11, §5.5.5.1 Figure 36; shear strengths of aluminium and magnesium parts in Annex A, Table A9).
You can also calculate it with the analytic chain of §5.5.5, Eqs. (195)–(213): the strength ratio $R_S$ tells which side is critical, then Eq. (210) (internal thread critical) or Eq. (213) (bolt thread critical) with the thread tolerance limits gives the required depth. The total engagement depth is $m_{ges} = m_{eff} + m_{Zu}$ with $m_{Zu} = 2P$ (Eqs. (208), (209)) (VDI 2230 Blatt 1:2015, §5.5.5).
How $R_S$ is found depends on the material of the internal thread (VDI 2230 Blatt 1:2015, §5.5.5):
- Both threads in steel: both sides have the same $\tau_B/R_m$, so the tensile-strength ratio $R_{mM}/R_{mS}$ can replace the shear-strength ratio. This is Eq. (199), derived from Eqs. (195)–(198) under the steel premise of Eq. (196).
- Internal thread not in steel (cast iron, aluminium, magnesium and so on): that condition does not hold, so use the definition, Eq. (195), with the shear-strength ratio $\tau_{BM}/\tau_{BS}$. Here $\tau_{BS}$ is $\tau_B/R_m$ from Table 7 times the bolt $R_m$.
For a tapped-thread joint, the calculator has a “Tapped part is steel” switch. Yes uses Eq. (199) and needs the tensile strength $R_{mM}$ of the tapped part; No uses Eq. (195) and needs its shear strength $\tau_{BM}$. When you pick the tapped-part material from Table A9, the switch is set from the material group, and $R_{mM}$ and $\tau_{BM}$ are filled in too. With the thread tolerance limits added, the calculator follows this chain; without them it uses a rough $1.0d$ estimate and warns that for low shear-strength materials such as aluminium and magnesium, the length of engagement read from Figure 36 can be much more than $1.0d$.
6. R12 — slip safety (§5.5.6)
Both the transverse force and the torque about the bolt axis $M_Y$ must be carried by friction at the interface. The minimum residual clamping force (VDI 2230 Blatt 1:2015, §4.2 R12):
$$ F_{KR\min} = \frac{F_{Mzul}}{\alpha_A} - (1 - \Phi_{en}^{*}) F_{A\max} - F_Z - \Delta F_{Vth} \tag{R12/1} $$[!warning] The thermal zero rule here is the opposite of R8 and R10 Here, if $\Delta F_{Vth} < 0$ (heating raises the preload), set $\Delta F_{Vth} = 0$ as a rule (VDI 2230 Blatt 1:2015, §4.2 R12); it may be counted only if the transverse load always acts after the steady-state temperature is reached (VDI 2230 Blatt 1:2015, §5.5.6.3). In R8 and R10 the rule is to set $\Delta F_{Vth} = 0$ when it is $> 0$. The reason is that the safe direction differs: R8 and R10 want a large bolt force or surface pressure, R12 wants a small residual clamping force, so each keeps only the unfavourable side.
The clamping force required to carry the transverse force and the torque (VDI 2230 Blatt 1:2015, §4.2 R12):
$$ F_{KQerf} = \frac{F_{Q\max}}{q_F\,\mu_{T\min}} + \frac{M_{Y\max}}{q_M\,r_a\,\mu_{T\min}} \tag{R12/2} $$This is the same formula as (R2/1) in R2 (Eq. (108)); $M_Y$ is the torque about the bolt axis, not a bending moment. If you keep only the transverse-force term you lose the torque: for a pure-torque case $F_{KQerf}$ becomes 0 and the slip check means nothing.
Check condition:
$$ F_{KR\min} > F_{KQerf} \tag{R12/3} $$Slip safety factor:
$$ S_G = \frac{F_{KR\min}}{F_{KQerf}} > 1.0 \tag{R12/4} $$The user sets the safety factor; usual values (VDI 2230 Blatt 1:2015, §4.2 R12):
- Static load: $S_G \geq 1.2$
- Alternating $F_Q$ and/or $M_Y$: $S_G \geq 1.8$
If an overload overcomes the static friction at the interface (or with fitted bolts), you must also rule out shearing of the bolt: $\tau_{Q\max} = F_{Q\max}/A_\tau$ (R12/5), $S_A = \tau_B/\tau_{Q\max} \geq 1.1$ (R12/7), with $\tau_B/R_m$ from Table 7 (VDI 2230 Blatt 1:2015, §4.2 R12). This is the second line of defence after friction fails; it cannot replace the slip check.
7. R13 — tightening torque (§5.4.3)
The final step turns the design result into installation parameters the workshop can carry out (VDI 2230 Blatt 1:2015, §4.2 R13):
$$ M_A = F_{Mzul} \left[0.16 \cdot P + 0.58 \cdot d_2 \cdot \mu_{G\min} + \frac{D_{Km}}{2} \cdot \mu_{K\min}\right] \tag{R13/1} $$(R13/1) is Eq. (127) in §5.4.3.1 with $F_M = F_{Mzul}$, and $D_{Km} = (d_W + D_{Ki})/2$ (Eq. (128)). Always use the minimum friction coefficients $\mu_{G\min}$ and $\mu_{K\min}$ to calculate the tightening torque (VDI 2230 Blatt 1:2015, §5.4.3.1). With locking elements, add the prevailing (overbolting) torque $M_U$ and the additional head torque $M_{KZu}$ (VDI 2230 Blatt 1:2015, §4.2 R13):
$$ M_{A,S} = M_A + M_U + M_{KZu} \tag{R13/2} $$The three terms in the brackets of (R13/1) represent, respectively:
| Term | Physical meaning |
|---|---|
| $0.16 \cdot P$ | helix-angle effect (pitch contribution) |
| $0.58 \cdot d_2 \cdot \mu_{G\min}$ | thread-flank friction torque |
| $\frac{D_{Km}}{2} \cdot \mu_{K\min}$ | bolt-head / nut bearing-surface friction torque |
[!NOTE] The make-up of the torque In a typical case, 80–90% of the total tightening torque is spent on thread and bearing-surface friction, and only about 10–20% (the $0.16P$ term) is turned into preload. For the calculator’s default example (M14, $\mu_G = \mu_K = 0.10$): bearing-surface friction about 46%, thread friction about 38%, pitch term about 16%. This is why the friction coefficient is so critical to preload design.
8. The logical hierarchy of the six checks
R7 assembly stress check → bolt does not exceed yield at assembly (static, one-off)
R8 service stress check → does not exceed yield in service (static, continuous)
R9 fatigue stress check → no fatigue fracture under alternating load (dynamic, long-term)
R10 bearing pressure check → no crushing under the head / nut (material, creep)
R11 engagement depth check → thread not stripped (geometry, mainly ESV)
R12 slip safety check → interface does not slip (friction, transverse force / torque)
R13 tightening torque → turned into workshop parameters (output)
If R7 is not met → select a larger bolt and recompute from R2 If R8–R12 are not met → adjust bolt size / property class / tightening method / friction coefficient / external force
Data basis and accuracy statement
All formulas in this article are from VDI 2230 Blatt 1:2015-11: the §4.2 calculation summary, §5.4.3 and §5.5. The recommended safety-factor values are from the text of the individual R steps in §4.2. The torque shares in R13 are example values from the MechCalc calculator’s default case, not data from the standard.
Disclaimer: This article is for engineering teaching reference only.
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