DIN 6796 von Mises Equivalent Stress: Multiaxial Plastic Yield Assessment

1. Application Background

DIN 6796 disc springs experience a multiaxial stress state in their cross-section during operation, particularly at the inner edge (OM, uM points), where radial stress, tangential stress, and shear stress coexist. Although failure in the compressive stress region is often controlled by empirical allowable compressive stress, for tensile stress regions (e.g., uM point) and cases requiring precise evaluation of plastic safety margins, the von Mises yield criterion must be applied to reduce multiaxial stresses to an equivalent uniaxial stress, which is then compared with the material's tensile yield strength $R_{p0.2}$.

2. von Mises Equivalent Stress Formula

For the general case with known principal stresses $\sigma_1, \sigma_2, \sigma_3$:

In disc springs, most regions can be approximated as a plane stress state (thickness-direction stress $\sigma_z \approx 0$), and radial stress $\sigma_r$ and tangential stress $\sigma_t$ are typically treated as principal stresses, with shear stress $\tau_{rz}$ superimposed separately.

In practice, when the three principal stress directions are clearly defined, the following formula can be used:

For the free surfaces at the inner and outer edges of the spring, shear stress $\tau_{rz}=0$, and tangential stress $\sigma_t$ also varies depending on edge conditions. Therefore, the equivalent stress at each characteristic point can be simplified.

3. Theoretical Calculation of Stress Components

Based on an extension of the Almen‑Laszlo theory, the stress components within the disc spring cross-section can be approximated by the following formulas (applicable to smooth conical surfaces without teeth):

Common parameters:

3.1 Inner Edge Points (OM / uM)

At the inner diameter edge ($r = D_i/2$): - Radial stress: $\sigma_{OM}$ on the upper surface (OM), $\sigma_{uM}$ on the lower surface (uM) (formulas as in previous chapter) - Tangential stress: Due to the free edge, $\sigma_t = 0$ - Shear stress: At the edge, $\tau_{rz} = 0$ (free surface)

Therefore, the stress state at the inner edge points is uniaxial stress, and the equivalent stress is directly equal to the absolute value of the radial stress:

3.2 Outer Edge Points (OU / uU)

At the outer diameter edge ($r = D_e/2$): - Radial stress: $\sigma_{OU}$ on the upper surface (OU), $\sigma_{uU}$ on the lower surface (uU) - Tangential stress: Similarly, $\sigma_t = 0$ - Shear stress: $\tau = 0$

Hence, the outer edge points are also in a uniaxial stress state, and the equivalent stress is the absolute value of the radial stress. Consequently, the stresses at the traditional five characteristic points are directly the principal stresses, and the equivalent stress equals their absolute value. This is precisely why stress verification for disc springs often directly uses $\sigma_{OM}$, $\sigma_{uM}$, etc.

3.3 Internal Cross-Section Points (if verification is needed)

If verification of non-edge points is required, tangential stress must be considered. At any radius $r$, the tangential stress can be approximated by:

However, for springs of conventional dimensions, the absolute stress at internal points is always smaller than at the inner edge, so verification is typically unnecessary.

4. Plastic Yield Criterion

For tensile stress-controlled points (e.g., uM point when $\sigma_{uM} > 0$), the following is used:

If the uM point is in compression, it is evaluated using the allowable compressive stress, and the von Mises criterion is not mandatory.

For compressive stress-controlled points (e.g., OM point), although the stress is uniaxial compression, the material's compressive strength is significantly higher than its tensile yield due to the triaxial compressive stress background. Using $0.9R_{p0.2}$ would be overly conservative. DIN 2093 allows evaluation using the allowable compressive stress $\sigma_{zul,c} \approx (1.4\sim1.6)R_{p0.2}$. In this case, no equivalent stress conversion is needed; a direct comparison $|\sigma_{OM}| \le \sigma_{zul,c}$ is sufficient.

5. Calculation Example

Using the M10 DIN 6796 spring: $c=1.961$, $K_1=0.6947$, $C_1=0.427$, $C_2=0.7135$, $C_\sigma=7332$ MPa, $R_{p0.2}=1500$ MPa.

Assume a working deflection $s=0.6$ mm, $\delta=0.4$.

• OM point:
  $$\sigma_{OM} = -7332 \times 0.4 \times [0.427(0.6667-0.2) + 0.7135] = -7332 \times 0.4 \times [0.199 + 0.7135] = -7332 \times 0.4 \times 0.9125 \approx -2676\ \text{MPa}$$

This is compressive stress; the equivalent stress is 2676 MPa. Compared with the allowable compressive stress of 2200~2400 MPa, it is slightly high, requiring a reduction in deflection.

• uM point:
  $$\sigma_{uM} = 7332 \times 0.4 \times [0.427\times 0.4667 - 0.7135] = 7332 \times 0.4 \times [0.199 - 0.7135] = 7332 \times 0.4 \times (-0.5145) \approx -1509\ \text{MPa}$$

This is compressive stress; tensile yield verification is not required. It can be evaluated using the allowable compressive stress.

• If a tensile stress state exists (e.g., when $\eta$ is large or deflection is small, making uM positive), the equivalent stress is calculated and compared with $0.9\times1500=1350$ MPa.

6. Advantages of Low Cone Height in DIN 6796

A low $h_0/t$ ratio reduces bending stress components, therefore: - The uM point is less likely to develop high tensile stress; in most operating conditions, uM is in compression or at a very low tensile stress level, eliminating concerns about tensile yield. - Although the absolute compressive stress at the OM point remains relatively high, it is significantly lower than in high-cone springs of the same size, allowing for greater deflection. - Equivalent stress verification in DIN 6796 springs can often be simplified to controlling the compressive stress at the OM point, greatly streamlining the design process.

7. Design Procedure Summary

1. Determine the maximum deflection $s_{max}$ based on the preload force.
2. Calculate the radial stresses at the five characteristic points.
3. Tensile stress points (typically only uM may be tensile): If $\sigma > 0$, then $\sigma_v = \sigma$, requiring $\sigma_v \le 0.9R_{p0.2}$.
4. Compressive stress points (e.g., OM): Use the allowable compressive stress $\sigma_{zul,c}$ directly for verification.
5. If a more rigorous multiaxial assessment is needed, additional calculation of $\sigma_t$ and $\tau$ for internal high-stress points can be performed for von Mises superposition, but this is not required in conventional design.

Conclusion:
The von Mises equivalent stress in DIN 6796 springs is crucial for verifying plastic yield caused by tensile stress. Due to the uniaxial stress characteristic at edge points, the equivalent stress equals the absolute value of the radial stress at that point, simplifying the judgment to a direct comparison between stress and the material's allowable value. The low cone height design minimizes the risk in the tensile stress region, enhancing the elastic safety margin of the spring.