Allowable Assembly Stress Verification: VDI 2230 Step R7

1. Core Formula and Objective

After determining the maximum assembly preload $F_{Mmax}$, it is necessary to verify whether the maximum equivalent stress in the bolt during assembly exceeds the material's allowable range, to prevent yielding, necking, or fracture during tightening.

The verification condition given by VDI 2230‑1 (2015) Step R7 is:

Where: - $\sigma_{red,M}$ — Maximum equivalent stress in the assembly state (MPa), based on the von Mises yield criterion - $R_{p0.2}$ — 0.2% offset yield strength of the bolt material (MPa), taking the specified minimum value $R_{p0.2min}$ - $\nu$ — Utilization factor, taken as $\nu = 0.9$ for elastic tightening

Engineering significance: During elastic tightening, no macroscopic plastic deformation of the bolt is permitted. The utilization factor of 0.9 retains approximately 10% safety margin to account for uncertainties such as material batch scatter, simplifications in multiaxial stress calculations, and slight overtightening.

2. Composition and Calculation of Assembly Stress $\sigma_{red,M}$

During assembly, the bolt simultaneously experiences: - Tensile stress $\sigma_M$ — caused by the preload $F_{Mmax}$ - Torsional stress $\tau_M$ — caused by the thread friction torque $M_G$ (always present during tightening)

According to the von Mises criterion, the equivalent stress is:

2.1 Tensile Stress $\sigma_M$

• $A_S$ — Stress area of the bolt (mm²), calculated per thread standards (e.g., ISO 898‑1):
  $A_S = \frac{\pi}{4}\left(\frac{d_2 + d_3}{2}\right)^2$

,
where $d_2$ is the pitch diameter, and $d_3$ is the minor diameter ($d_3 = d_1 - \frac{H}{6}$, with $d_1$ as the basic minor diameter). - $F_{Mmax}$ — Maximum assembly preload (N) obtained from Step R6.

2.2 Torsional Stress $\tau_M$

The torsional stress is generated by the thread torque $M_G$ acting on the bolt shank:

• $M_G$ — Thread torque (N·mm), i.e., the portion of the total tightening torque used to overcome thread friction and the helix angle.
  According to the R13 formula:
  $$M_G = F_{Mmax} \left(0.16\,P + 0.58\,d_2\,\mu_G\right)$$

Note: To verify the worst-case scenario, $\mu_G$ should take the maximum possible value $\mu_{Gmax}$ (especially for torque-controlled methods), to obtain the largest $M_G$ and $\tau_M$.

• $W_p$ — Polar section modulus (mm³). For a circular cross-section:
  $$W_p = \frac{\pi}{16} d_S^3$$

where $d_S$ is typically taken as the equivalent diameter of the stress area: $d_S = \sqrt{\frac{4A_S}{\pi}}$. This simplification is conservative, as the actual torsional section of the threaded portion is slightly larger.

3. Values of the Utilization Factor $\nu$

Conservative design principle: When friction coefficients, preloads, etc., exhibit significant scatter, $\nu=0.9$ is a mandatory constraint for elastic tightening. If the calculated stress is slightly high, solutions include selecting a higher strength grade bolt, reducing the friction coefficient, or switching to an angle-controlled method.

4. Verification Procedure and Calculation Example

Step Review:

1. Obtain $F_{Mmin}$ from Step R5
2. Obtain $F_{Mmax}$ from Step R6
3. Determine bolt geometric parameters: $A_S, d_S, P, d_2$
4. Determine the upper limit of the friction coefficient $\mu_{Gmax}$ (if known)
5. Calculate tensile stress $\sigma_M = F_{Mmax}/A_S$
6. Calculate thread torque $M_G = F_{Mmax}(0.16P + 0.58d_2 \mu_{Gmax})$
7. Calculate torsional stress $\tau_M = M_G/W_p$, where $W_p = \frac{\pi}{16}d_S^3$
8. Calculate equivalent stress $\sigma_{red,M} = \sqrt{\sigma_M^2 + 3\tau_M^2}$
9. Verify $\sigma_{red,M} \le 0.9 \, R_{p0.2min}$

Numerical Example (M10 × 1.5, Grade 8.8 Bolt)

Given Data: - Bolt strength grade 8.8, $R_{p0.2min} = 640$ MPa - $A_S = 58.0$ mm², $d_S \approx 8.59$ mm, $W_p \approx \pi/16 \times (8.59)^3 = 124.3$ mm³ - Pitch $P = 1.5$ mm, pitch diameter $d_2 = 9.026$ mm - Assume from R6: $F_{Mmax} = 12880$ N - Upper limit of thread friction coefficient $\mu_{Gmax} = 0.15$

Calculation:

Allowable Stress:

Verification Result:

5. Key Considerations

1. Torsional stress cannot be ignored: For fine-pitch threads or high friction coefficients, $\tau_M$ can account for over 50% of the tensile stress, making the equivalent stress significantly higher than the pure tensile stress.
2. Friction coefficient selection: For torque-controlled methods, use the upper limit of $\mu_G$ to calculate $M_G$, covering the most severe torsional condition.
   For torque-angle or yield-controlled methods, when the bolt approaches yielding, the torsional stress decreases (due to stress redistribution). In such cases, the value of $\tau_M$ can be adjusted accordingly (VDI 2230 provides further guidance).
3. Stress area $A_S$: Should be calculated based on the minimum cross-section over the effective thread length, considering effects like thread runout.
4. Temperature correction: If the operating temperature is high, use the yield strength value corresponding to that temperature for $R_{p0.2}$.
5. Result handling: If $\sigma_{red,M} \le 0.9 R_{p0.2}$ is not satisfied, the following measures can be taken:
6. Increase the nominal bolt diameter (increasing $A_S, W_p$)
7. Select a higher strength grade (increasing $R_{p0.2}$)
8. Reduce the friction coefficient (decreasing $M_G$ and $\tau_M$)
9. Switch to a tightening method with lower scatter, reducing $\alpha_A$ and thus $F_{Mmax}$

Summary: Step R7 acts as a safety valve in connection design, ensuring the bolt remains within the elastic safety domain under the maximum assembly preload. It links preload analysis with bolt strength design and is a critical basis for material selection and tightening process determination.