Fatigue Strength Verification Under Alternating Stress: VDI 2230 Step R9

1. Verification Purpose and Standard

When a bolted joint is subjected to periodically varying service loads, the internal bolt stress fluctuates accordingly. Even if the maximum stress is well below the yield strength, fatigue fracture can occur. VDI 2230‑1 Step R9 specifically verifies this risk, with the core criterion:

• $\sigma_a$ — Bolt stress amplitude (MPa)
• $\sigma_A$ — Bolt fatigue limit (stress amplitude, MPa), standard reference value approx. 60 MPa (rolled thread)
• $S_D$ — Fatigue safety factor, requirement $S_D \ge 1.2$

If this condition is satisfied, the bolt has sufficient fatigue life under cyclic loading (theoretically infinite life or a specified finite life).

2. Detailed Formulas

2.1 Stress Amplitude $\sigma_a$

Under service load, the axial force in the bolt fluctuates between a maximum value $F_{SAo}$ and a minimum value $F_{SAu}$. The stress amplitude is taken as the force amplitude divided by the stress area:

Physical meaning: Only the alternating component is considered; the constant tensile stress from preload does not contribute to fatigue damage (the mean stress effect is already accounted for in the fatigue limit or corrected separately).

2.2 Bolt Force Limits $F_{SAo}$ and $F_{SAu}$

These values depend on the range of external load variation, the load factor $\Phi^*$, and the assembly preload:

• When the external load varies between $0$ and $F_A$:
  $$F_{SAo} = F_M + \Phi^* \cdot F_A \quad (\text{at maximum external load})$$

Hence:

The stress amplitude then simplifies to:

• If the external load does not start from zero, use the actual minimum external load to calculate $F_{SAu} = F_M + \Phi^* \cdot F_{A,min}$; the difference remains $\Phi^* (F_{Amax} - F_{Amin})$.

Note: To obtain the most unfavorable fatigue condition, the external load $F_A$ should be taken as the maximum variation amplitude of the service load, and the preload $F_M$ should be taken as the minimum possible value $F_{Mmin}$ (since a lower preload may increase the load factor $\Phi^*$ and make the bolt stress amplitude more critical for fatigue).

3. Fatigue Limit $\sigma_A$ Values

VDI 2230 provides reference values for the fatigue limit of threaded connections under standard test conditions, in units of MPa (stress amplitude). These values already include the stress concentration effect of the thread notch, so they can be used directly in calculations without an additional notch factor.

Influencing factors: - Rolled thread: Surface compressive residual stress improves fatigue strength, increasing it by 30%–60% compared to cut threads. - Thread root radius: Increasing the root radius reduces stress concentration. - Surface treatment: Zinc plating and phosphating have a minor effect on the fatigue limit, but hot-dip galvanizing may reduce fatigue strength (note hydrogen embrittlement). - Bolt size: The values above are relatively stable for M8–M20; for very large diameters, the fatigue limit decreases slightly (size effect).

Design recommendation: In the absence of specific test data, taking $\sigma_A = 60$ MPa for conventional rolled threads is a common safe reference in VDI 2230.

4. Fatigue Safety Factor $S_D$

accounts for the scatter in the fatigue limit, uncertainties in load assumptions, and the consequences of failure. VDI 2230 specifies a minimum requirement:

More specific recommended values:

5. Calculation Example

Connection parameters: - M10×1.5, grade 8.8 rolled thread, $A_S = 58.0$ mm² - Pulsating service load: $F_A = 5\,000$ N (0 to 5000 N) - Load factor $\Phi^* = 0.25$ - Fatigue limit $\sigma_A = 60$ MPa, safety factor $S_D = 1.5$ (significant dynamic loading)

Calculated stress amplitude:

Allowable stress amplitude:

Verification: $10.8\ \text{MPa} < 40\ \text{MPa}$ → Fatigue safe, ample margin.

6. Key Factors Affecting Fatigue Strength and Improvement Measures

1. Reduce stress amplitude
2. Increase bolt diameter (increase $A_S$)
3. Reduce load factor $\Phi^*$: use a more elastic bolt (e.g., slender shank) or stiffer clamped parts

4. Reduce the amplitude of the external load

5. Increase fatigue limit

6. Use rolled threads instead of cut threads
7. Increase bolt strength grade (fatigue limit increases approximately proportionally; e.g., grade 12.9 rolled can be taken as 70 MPa)

8. Thread root rolling or shot peening

9. Optimize preload

10. High preload reduces $\Phi^*$, but also increases mean stress; VDI 2230 fatigue verification primarily considers stress amplitude, so high preload is beneficial.

11. However, ensure the maximum stress does not exceed the limit from R8.

12. Avoid additional bending

13. Ensure bolt alignment to avoid eccentric loading, which induces additional bending stress; bending significantly increases the fatigue stress amplitude.

7. Position in the Overall VDI 2230 Process

• R5: Determine minimum preload $F_{Mmin}$
• R6: Determine maximum preload $F_{Mmax}$
• R7: Assembly stress verification (static strength)
• R8: Service stress verification (static strength, maximum load)
• R9: Fatigue strength verification (alternating load)

R9 is the final strength verification step in the design process, ensuring the joint does not fail due to fatigue under long-term cyclic loading. If the verification fails, the bolt size, geometry, or material must be revised.

Summary: The fatigue strength verification formula under alternating load $\sigma_a = (F_{SAo} - F_{SAu})/(2A_S) \le \sigma_A / S_D$ centers on the stress amplitude, using the fatigue limit $\sigma_A$ (approx. 60 MPa for rolled threads) based on threaded connection fatigue tests for safety assessment. Accurately calculating the bolt force variation amplitude and appropriately selecting the fatigue limit and safety factor are key to ensuring joint durability.