DIN 2093 Stress Amplitude and Mean Stress: Fatigue Assessment Fundamentals

1. Basic Definitions

When a disc spring operates under cyclic loading, the stress at the critical point (typically the OM point) varies periodically between the stress $\sigma_{min}$ corresponding to the minimum working deflection $s_{min}$ and the stress $\sigma_{max}$ corresponding to the maximum working deflection $s_{max}$. The two fundamental parameters for fatigue assessment are:

Stress amplitude $\sigma_a$:

Mean stress $\sigma_m$:

Where $\sigma_{max}$ and $\sigma_{min}$ are the calculated radial stresses at the OM point (compressive stresses, typically negative; for convenience, their absolute values are often used in fatigue analysis).

Additionally, the stress ratio $R$ can assist in determining the load type:

• Pulsating compression cycle: $R = 0$ (from zero compressive stress to maximum compressive stress)
• Fully reversed cycle: $R = -1$ (rarely occurs for disc springs)

2. OM Point Stress Calculation

The radial stress at the OM point (inner edge of the upper surface) is given by the Almen‑Laszlo theory:

Where: - $E$ — Elastic modulus, $\nu$ — Poisson's ratio - $t$ — Thickness, $h_0$ — Free cone height, $D_e$ — Outer diameter - $K_1$ — Shape factor, $C_1, C_2$ — Geometric constants, all determined solely by the outer-to-inner diameter ratio $c = D_e/D_i$ (calculation formulas are given in the previous section on OM point stress)

For a given deflection $s$, $\sigma_{OM}$ is uniquely determined. Substituting $s_{min}$ and $s_{max}$ yields $\sigma_{OM,min}$ and $\sigma_{OM,max}$, from which $\sigma_a$ and $\sigma_m$ are calculated (both taken as absolute values).

3. Fatigue Verification Criterion (DIN 2093 Method)

DIN 2093 provides a fatigue limit diagram (Haigh diagram) for disc springs, based on extensive fatigue testing. The design must satisfy:

Where $\sigma_{A,OM}(\sigma_m)$ is the maximum allowable stress amplitude at a given mean stress $\sigma_m$, obtainable from the fatigue limit curve in the standard.

For commonly used quenched and tempered spring steels (e.g., 50CrV4), DIN 2093 provides a simplified allowable stress amplitude curve, which can be fitted as:

Where: - $\sigma_{A0}$ — Pulsating fatigue limit (allowable stress amplitude at $R=0$), approximately 600 ~ 800 MPa for shot-peened disc springs - $R_m$ — Material tensile strength (≈ 1500 ~ 1800 MPa)

Safety factor: To account for the scatter in fatigue strength, a fatigue safety factor $S_D \ge 1.2$ should be introduced:

4. Calculation Example

Disc spring specification (DIN 2093 Series A, 20×10.2×1.5): - $D_e = 20\ \text{mm}$, $D_i = 10.2\ \text{mm}$, $t = 1.5\ \text{mm}$, $h_0 = 1.0\ \text{mm}$ - Material: 50CrV4, $R_m \approx 1600\ \text{MPa}$, pulsating fatigue limit $\sigma_{A0} \approx 700\ \text{MPa}$

Working cycle: - Minimum deflection $s_{min} = 0.2\ \text{mm}$ ($\delta = 0.1333$) - Maximum deflection $s_{max} = 0.8\ \text{mm}$ ($\delta = 0.5333$)

Step 1: Calculate corresponding stresses

Using the OM stress formula (with the previously calculated $C_\sigma \approx 7332\ \text{MPa}$ and geometric coefficients): - $s_{min}=0.2$: $\sigma_{OM,min} \approx -1042\ \text{MPa}$ (absolute value 1042 MPa) - $s_{max}=0.8$: $\sigma_{OM,max} \approx -2926\ \text{MPa}$ (absolute value 2926 MPa)

Step 2: Calculate stress amplitude and mean stress

Step 3: Fatigue verification

Using the Goodman correction:

This indicates that the material cannot withstand fatigue loading at this mean stress level; the maximum deflection must be reduced, or the disc spring dimensions must be changed.

If the working cycle is adjusted to $s_{min}=0.2\ \text{mm}$, $s_{max}=0.5\ \text{mm}$: - $\sigma_{OM,max} \approx 1896\ \text{MPa}$, $\sigma_{OM,min} = 1042\ \text{MPa}$ - $\sigma_a = (1896-1042)/2 = 427\ \text{MPa}$, $\sigma_m = 1469\ \text{MPa}$ - Allowable stress amplitude $\sigma_{A,OM} = 700 \times (1 - 1469/1600) \approx 57\ \text{MPa}$, still far below 427 MPa, indicating that this disc spring is unsuitable for operation at this mean stress level. A different specification is required (e.g., increasing $t$, reducing $h_0/t$).

Conclusion: In disc spring fatigue verification, the mean stress at the OM point is often very high, leading to a significant reduction in the allowable stress amplitude. Design should prioritize controlling the maximum deflection to keep $\sigma_m$ within the material's fatigue allowable range, or use materials with higher fatigue limits and surface strengthening measures.

Summary: The stress amplitude $\sigma_a$ and mean stress $\sigma_m$ are calculated from the OM point stresses at the minimum/maximum working deflections and form the basis for fatigue assessment. DIN 2093 provides an allowable stress amplitude curve based on the Goodman relationship; the design must ensure $\sigma_a \le \sigma_{A,OM}/S_D$. Due to the prevalent high mean compressive stress in disc springs, fatigue design often becomes the limiting factor.