DIN 2093 Four-Point (OM, I, II, III) Full Stress Calculation

1. Definition of Characteristic Points

When a disc spring is axially compressed, the stress distribution across the cross-section is non-uniform. To comprehensively evaluate static and fatigue strength, DIN 2093 specifies that stresses must be calculated at the following four key positions (sometimes referred to as five points, but typically four):

For static applications, the primary control is that the compressive stress at point OM does not exceed the material yield limit; for fatigue applications, both the compressive stress amplitude at point OM and the tensile stress amplitude at point I must be verified.

2. Stress Calculation Formulas for Each Point (Almen‑Laszlo Theory)

2.1 Common Parameters

• Outer diameter $D_e$, inner diameter $D_i$, thickness $t$, free cone height $h_0$
• Modulus of elasticity $E$, Poisson's ratio $\nu$
• Dimensionless compression $\delta = s/t$, dimensionless cone height $\eta = h_0/t$
• Outer-to-inner diameter ratio $c = D_e/D_i$

Geometric coefficients:

Stress common factor:

2.2 Stress at Point OM (Upper Inner Edge, Compressive)

• The negative sign indicates compressive stress; the absolute value is often used for verification.
• This is the point with the largest absolute stress, typically controlling plastic deformation and fatigue.

2.3 Stress at Point I (Lower Outer Edge, Tensile)

• If the result is positive, it is tensile stress, representing the most critical tensile stress point;
• If $\eta$ is small and deformation is limited, this point may be compressive, but for most disc springs in the working range, it exhibits tensile stress.

2.4 Stress at Point II (Upper Outer Edge)

• The absolute stress at this point is typically smaller, often compressive or a small tensile stress.

2.5 Stress at Point III (Lower Inner Edge)

• This point is similar to point OM, but the sign of the second term in the brackets is opposite; it often appears as compressive or a small tensile stress, depending on the geometry.

3. Strength Evaluation Criteria

3.1 Static Application

• The compressive stress (absolute value) at point OM must be less than the material yield limit $R_{p0.2}$ divided by the safety factor:

  $$|\sigma_{OM}| \le \frac{R_{p0.2}}{S_F}, \quad S_F \ge 1.2$$

• If tensile stress occurs at other points, they should also be verified according to this criterion.

3.2 Fatigue Application

• Point OM: Calculate the compressive stress amplitude $\sigma_{a,OM} = (|\sigma_{OM,max}| - |\sigma_{OM,min}|)/2$ and compare it with the material compressive fatigue limit.
• Point I: If tensile stress is present, calculate the tensile stress amplitude and verify it. The tensile stress amplitude at point I is often a dominant factor in fatigue failure.

4. Calculation Example

Disc Spring Specification (DIN 2093 Series A, 20×10.2×1.5):

• $D_e = 20\ \text{mm}, D_i = 10.2\ \text{mm} \Rightarrow c = 1.9608$
• $t = 1.5\ \text{mm}, h_0 = 1.0\ \text{mm} \Rightarrow \eta = 0.6667$
• Material spring steel, $E = 206\,000\ \text{MPa}, \nu = 0.3$

Step 1: Calculate Constants

Step 2: Take Working Compression $s = 0.6\ \text{mm}$ ($\delta = 0.4$)

Calculate common terms:

Step 3: Calculate Stresses at Each Point

• $\sigma_{OM} = -7\,332 \times 0.4 \times 0.9128 \approx -2\,677\ \text{MPa}$
• $\sigma_{I} = 7\,332 \times (0.4 / 1.9608) \times 0.4460 \approx 7\,332 \times 0.2041 \times 0.4460 \approx 667\ \text{MPa}$
• $\sigma_{II} = 7\,332 \times 0.2041 \times (-0.9810) \approx -1\,467\ \text{MPa}$
• $\sigma_{III} = 7\,332 \times 0.4 \times (-0.5142) \approx -1\,508\ \text{MPa}$

Interpretation: - The compressive stress at point OM is the largest (2 677 MPa), controlling static strength. - Point I exhibits tensile stress (667 MPa), which has a significant impact on fatigue. - The absolute stresses at points II and III are also relatively large, but the failure risk at these locations is slightly lower; verification is still required.

5. Conclusion

Using the four formulas above, the complete stress state at the four key points of a disc spring can be obtained at any compression level. The compressive stress at point OM and the tensile stress at point I are the primary focus for design and fatigue verification, while points II and III provide a comprehensive understanding of the cross-sectional stress distribution to prevent local failure. Based on the stress calculation results, appropriate disc spring specifications, materials, and surface treatments can be selected to ensure safety and reliability.