DIN 2093 Goodman Fatigue Verification

1. Basic Principles of the Goodman Diagram

The fatigue failure of disc springs under cyclic loading is determined by both the stress amplitude $\sigma_a$ and the mean stress $\sigma_m$. The Goodman diagram is one of the most commonly used mean stress correction methods in engineering, expressing the allowable stress amplitude under different mean stresses as a linear boundary line.

For high-cycle fatigue, the Goodman relationship can be written as:

If a safety factor $n_s$ is introduced, the allowable criterion becomes:

Where:

• $\sigma_a$ — Working stress amplitude (MPa), taken as the absolute value of the compressive stress amplitude at the OM point for disc springs
• $\sigma_m$ — Working mean stress (MPa), taken as the absolute value of the mean compressive stress at the OM point (conservative approach)
• $\sigma_{-1}$ — Material fatigue limit under fully reversed loading ($R=-1$) (MPa), approximately $0.35 \sim 0.5\,\sigma_b$ for spring steel
• $\sigma_b$ — Material tensile strength (MPa)
• $n_s$ — Fatigue safety factor, typically required to be $n_s \ge 1.2 \sim 1.5$

When the inequality is satisfied, the design is acceptable under the given safety factor. This criterion applies to high-cycle fatigue ($N > 10^4$ cycles).

2. Determination of Parameters

2.1 Working Stress Amplitude $\sigma_a$ and Mean Stress $\sigma_m$

From the minimum deflection $s_{min}$ and maximum deflection $s_{max}$ of the disc spring working cycle, calculate the radial stress (compressive stress, take absolute value) at the OM point (inner edge of the upper surface):

Then:

Note: For disc springs, the OM point is always under compression, therefore $\sigma_m$ is a compressive stress. The Goodman diagram was originally designed for tensile stress; when used for compressive stress, the absolute value can be substituted, yielding a conservative result. The fatigue limit diagram (Haigh diagram) given in DIN 2093 is itself intended for the compressive stress state at the OM point, where the allowable stress amplitude decreases with increasing mean compressive stress, a trend consistent with the Goodman line.

2.2 Fully Reversed Fatigue Limit $\sigma_{-1}$

For quenched and tempered spring steel (e.g., 50CrV4), the fully reversed bending fatigue limit $\sigma_{-1}$ is typically in the range of $0.35 \sim 0.5\,\sigma_b$. If measured data is unavailable, a conservative estimate can be taken as:

• For spring steel with $\sigma_b = 1500 \sim 1800\ \text{MPa}$, $\sigma_{-1} \approx 600 \sim 900\ \text{MPa}$.
• This value is based on fully reversed loading of smooth specimens. The actual fatigue limit of disc springs may be slightly lower due to factors such as cross-sectional shape and surface quality. DIN 2093 recommends using experimentally determined group fatigue limits (Group 1/2/3), which can be directly substituted into the $\sigma_{-1}$ position in the Goodman equation.

If the pulsating fatigue limit $\sigma_{A0}$ (corresponding to $R=0$) is used instead of $\sigma_{-1}$, the Goodman relationship still holds; it is only necessary to determine the corresponding point on the diagram.

2.3 Tensile Strength $\sigma_b$

Take the minimum specified tensile strength $R_m$ from the material standard. Common disc spring materials:

3. Safety Factor $n_s$

The selection of the fatigue safety factor $n_s$ should comprehensively consider:

• Accuracy and variability of the load
• Reliability of material data
• Surface treatment (shot peening can improve, should be reflected in $\sigma_{-1}$)
• Severity of failure consequences

Recommended values:

For disc springs, the standard fatigue diagram in DIN 2093 already implies a certain safety factor, but for manual calculations, a safety factor of no less than 1.2 should still be applied.

4. Calculation Example

Given: - Disc spring: 50CrV4, $\sigma_b = 1600\ \text{MPa}$, take $\sigma_{-1} = 0.4 \times 1600 = 640\ \text{MPa}$ - Working cycle: at $s_{min}=0.2\ \text{mm}$, $\sigma_{OM,min}=1042\ \text{MPa}$; at $s_{max}=0.6\ \text{mm}$, $\sigma_{OM,max}=1896\ \text{MPa}$ - Calculation:

• Substitute into the Goodman criterion (left side):

  $$\frac{427}{640} + \frac{1469}{1600} = 0.667 + 0.918 = 1.585$$

• Requirement: $1.585 \le 1/n_s$, i.e., $n_s \le 0.63$, far less than 1, not passed.

This indicates that the mean stress at this operating point is too high. The compression amount must be reduced, the disc spring specification changed, or shot peening applied to increase $\sigma_{-1}$.

Adjustment: If the maximum deflection is reduced to $s_{max}=0.4\ \text{mm}$, resulting in $\sigma_{OM,max} \approx 1370\ \text{MPa}$, then:

Here, $n_s \le 0.99$, still not satisfied. Continue reducing deflection or switch to a Group 1 thin disc spring. If shot peening raises $\sigma_{-1}$ to 800 MPa:

Slightly insufficient; further fine-tuning is possible.

5. Integration with the DIN 2093 Standard Fatigue Diagram

The appendix of DIN 2093 provides fatigue limit diagrams (Haigh diagrams) for each group of disc springs, with the horizontal axis representing mean stress (absolute value at the OM point) and the vertical axis representing stress amplitude. During design, the allowable stress amplitude for the corresponding point can be directly read from the diagram without needing to calculate the Goodman line. However, when the operating point falls outside the range of the standard diagram, the Goodman method described above can be used for extrapolation, noting its conservative nature.

6. Conclusion

• The Goodman fatigue verification provides an intuitive linear mean stress correction method for disc springs.
• By calculating $\sigma_a$ and $\sigma_m$, combined with the material properties $\sigma_{-1}$ and $\sigma_b$, the fatigue safety can be quickly assessed.
• The design should aim for a safety factor of no less than 1.2; if this is not met, priority should be given to optimizing the operating point, selecting a thinner disc spring, or implementing shot peening treatment.