DIN 9250 Washer Body Combined Stress Verification: Tooth Root Notch Factor Method

1. Verification Purpose and Applicable Objects

DIN 9250 toothed lock washers typically feature a disc spring body with radial fine teeth on one or both sides. Under preload, the washer body experiences combined bending and compressive stress, with the maximum compressive stress still occurring at the OM point on the inner edge of the upper surface. However, due to the presence of tooth grooves, the cross-section changes abruptly in this area, causing severe stress concentration. To ensure the washer does not yield or fail due to fatigue during assembly and operation, the nominal stress at the OM point must be multiplied by the tooth root notch factor $k_t$ to obtain the actual peak stress for verification.

The strength condition is:

• $\sigma_{red}$ — Maximum local stress at the tooth root (MPa), treated here as uniaxial compressive stress; the equivalent stress is its absolute value
• $\sigma_{OM}$ — Nominal compressive stress at the OM point (MPa) calculated using the smooth disc spring formula, taken as absolute value
• $k_t$ — Tooth root stress concentration factor (≥1), comprehensively reflecting the stress increase caused by the tooth groove geometry
• $R_{p0.2}$ — 0.2% offset yield strength of the washer material (MPa)
• $S_F$ — Static strength safety factor, typically ≥ 1.2

2. Calculation of Nominal OM Point Stress $\sigma_{OM}$

The conical elastic body of DIN 9250 washers follows the Almen‑Laszlo disc spring theory. For a smooth conical surface without teeth, the compressive stress at the OM point is (negative value taken as absolute):

The meaning of parameters in this formula is identical to that for DIN 6796 (see relevant section). The calculation must use the actual washer geometry $D_e, D_i, t, h_0$ and material properties $E, \nu$.

Note: This calculation yields the nominal stress for a smooth disc spring without considering tooth groove weakening; therefore, $\sigma_{OM}$ is a nominal value. The actual stress is higher at the tooth root.

3. Tooth Root Notch Factor $k_t$

is the theoretical stress concentration factor at the tooth root, defined as the ratio of the local maximum elastic stress to the nominal stress. It primarily depends on:

• Tooth geometry parameters: tooth depth, tooth root fillet radius $r$, tooth pitch
• Loading type: dominated by bending normal stress

Since DIN 9250 tooth profiles are standardized, $k_t$ is typically provided by the manufacturer or standard, denoted as _TOOTH_NOTCH_FACTOR. In the absence of precise data, the following empirical ranges can be referenced:

Conservative Design: If an exact value cannot be obtained, $k_t = 2.5$ can be used for preliminary verification, to be corrected after experimental validation.

Determination Method: _TOOTH_NOTCH_FACTOR may be a variable name in design tables. It can be obtained from standards based on the number of teeth, tooth root fillet radius, and the ratio of tooth height to thickness. It is recommended to use the notch factor provided by the manufacturer.

4. Allowable Stress and Safety Factor

Although the OM point is under a triaxial compressive stress state, and the material's compressive capacity is higher than its tensile yield strength, for a unified conservative calculation, the tensile yield strength $R_{p0.2}$ is still used as the baseline, with a safety factor $S_F \ge 1.2$. For high-hardness spring steel (e.g., 50CrV4, $R_{p0.2} \approx 1500\ \text{MPa}$), the allowable stress is:

This may seem low, as the allowable compressive stress can typically reach $1.4R_{p0.2}$. However, at the tooth root notch, the stress state is complex and may include tensile stress components. Therefore, VDI 2230 and DIN 9250 design practice typically uses the tensile yield strength divided by the safety factor as a general criterion, which is conservative. If the washer material has a clearly defined allowable compressive stress, it can be specified separately.

For dynamic loading, fatigue strength (stress amplitude) verification is also required, which is not covered here.

5. Complete Verification Procedure

1. Determine washer geometry and material: $D_e, D_i, t, h_0$, $E, R_{p0.2}$, tooth geometry parameters.
2. Calculate the working deflection $s$: Back-calculate from the bolt preload $F_{Mmax}$ using the force-deflection formula for a smooth washer (or a formula adjusted for tooth cross-section reduction, but the OM point stress calculation still uses the smooth formula, with the notch factor applied separately).
3. Calculate the nominal OM stress $\sigma_{OM}$ (take absolute value).
4. Determine $k_t$: Look up the _TOOTH_NOTCH_FACTOR table or use a recommended value.
5. Calculate the local stress: $\sigma_{red} = k_t \cdot |\sigma_{OM}|$.
6. Verify: $\sigma_{red} \le R_{p0.2} / S_F$ ($S_F \ge 1.2$).
7. If not satisfied, reduce the preload, select a thicker or larger outer diameter washer, or increase the tooth root fillet radius to lower $k_t$.

6. Calculation Example

Given: - DIN 9250 washer for M10: $D_e = 20\ \text{mm}$, $D_i = 10.2\ \text{mm}$, $t = 1.5\ \text{mm}$, $h_0 = 1.0\ \text{mm}$ - Material 50CrV4, $E=206\,000\ \text{MPa}$, $R_{p0.2}=1500\ \text{MPa}$ - Working preload $F_{Mmax}=12\,000\ \text{N}$, back-calculated deflection from force-deflection curve $s \approx 0.6\ \text{mm}$ ($\delta=0.4$) - Smooth washer stress factor $C_\sigma \approx 7332\ \text{MPa}$, $C_1=0.427$, $C_2=0.7135$, $\eta=0.6667$

Nominal OM stress (absolute value):

Determine $k_t$: Standard tooth profile, take $k_t = 2.2$.

Local stress:

Allowable stress:

, verification fails! This indicates the preload is too high or the washer specification is unsuitable.

Adjustment: Reduce $F_{Mmax}$ to 8 000 N, corresponding $s \approx 0.45\ \text{mm}$, $\sigma_{OM}$ drops to approximately 1800 MPa, then $\sigma_{red}=1800 \times 2.2 = 3960\ \text{MPa}$, still exceeds the limit. This means that simply reducing the load is insufficient; the washer size must be increased (e.g., using a thicker washer or larger outer diameter), a higher strength material must be selected, or it must be confirmed through actual testing that the allowable value can be increased under compressive stress (if the local compressive stress is permissible up to 3000 MPa and the actual notch factor is smaller, it might pass). This example highlights the significant impact of the tooth root notch factor on strength, which must be carefully considered in design.

7. Discussion and Recommendations

• Tooth root optimization: Increasing the tooth root fillet radius can effectively reduce $k_t$ and should be considered during the mold design phase.
• Experimental validation: For critical washers, the actual stress concentration factor should be determined through strain gauge measurements or photoelastic testing.
• Multiaxial effects: Tensile stress components may exist at the tooth root. If the washer is subjected to repeated loading, fatigue verification is more critical and should not rely solely on static strength.
• Conservative strategy: If $k_t$ is not precisely known, use a larger value and limit the working deflection to $\le 0.5h_0$ to maintain a safety margin.

Summary:
The combined stress verification for DIN 9250 washers introduces the tooth root notch factor $k_t$ to convert the nominal OM point stress from the smooth disc spring formula into the actual peak stress, which is then compared with the material's allowable stress. Correctly determining $k_t$ is the key to the strength design of this type of washer and must be based on standards or experimental data.