DIN 9250 FEM Verification of Tooth Root Stress Concentration Factor $k_t$

1. Verification Objectives

The tooth root stress concentration factor $k_t$ is a critical parameter for the strength verification of DIN 9250 washers. It converts the nominal stress at the OM point of a smooth disc spring into the actual peak stress at the tooth root. Due to the complex tooth geometry, analytical solutions provide only approximations, and the recommended values in standards or handbook tables are based on statistical experience.
The purpose of FEM numerical verification is to:

• Accurately calculate $k_t$ for a given tooth geometry;
• Evaluate the accuracy of analytical formulas (e.g., empirical formulas based on notch radius and tooth depth);
• Analyze the influence of loading conditions (preload magnitude, tooth tip friction) on $k_t$;
• Provide validated $k_t$ values for design, avoiding excessive conservatism or unsafe assumptions.

2. Definition and Analytical Reference of $k_t$

In the washer body, the theoretical stress concentration factor $k_t$ is defined as:

• $\sigma_{max}^{el}$ — Maximum principal stress at the tooth root in the elastic solution (typically the absolute value of radial compressive stress)
• $\sigma_{nom}$ — Nominal stress in the same cross-section without a notch, i.e., the value calculated from the smooth disc spring formula at the OM point $\sigma_{OM}$

Analytical approximation formula (for bending stress concentration at the tooth root) can refer to the "notched beam under bending" model, for example:

Where $d$ is the tooth depth and $r$ is the tooth root fillet radius. However, this formula does not account for the helix angle of the teeth or the disc effect, resulting in limited accuracy.

DIN 9250 standards or manufacturers often provide empirical $k_t$ values in tabular form (e.g., 1.8 to 2.5). These values originate from extensive testing but lack theoretical support for specific dimensions.

3. FEM Modeling Methodology

3.1 Geometry and Mesh

• A two-dimensional plane strain cross-section of a single tooth is taken (assuming the tooth is infinitely long, neglecting circumferential curvature).
• The tooth geometry is accurately reproduced: tooth tip angle, tooth root fillet radius $r$, tooth depth, and unit thickness in the tooth width direction.
• Mesh refinement is applied in the tooth root region, typically using second-order triangular or quadrilateral elements, ensuring at least 8 to 10 elements along the fillet.
• Model extent: includes a sufficiently large surrounding area to avoid boundary effects, with symmetry conditions or fixed constraints applied at the radial boundaries.

3.2 Material Properties

• Linear elastic: Washer material is spring steel, $E = 206\,000$ MPa, $\nu = 0.3$.
• Analysis type: Static linear static analysis.

3.3 Loads and Boundary Conditions

• Simulate disc compression: Apply a bending moment/force at the outer or inner edge of the washer consistent with the nominal stress at the OM point.
• Common method: Apply a uniform pressure or forced displacement on the tooth tip (contact surface) to induce the same bending and membrane stresses in the cross-section as under actual working conditions.
• Extract the stress distribution at the OM point cross-section (through the tooth root).

3.4 Solution and Post-Processing

• Calculate nodal von Mises stress or maximum principal stress.
• Identify the absolute maximum compressive stress $\sigma_{max}^{FEM}$ at the tooth root fillet.
• Simultaneously, remove the tooth groove (smooth conical surface) in the same model to calculate the nominal stress $\sigma_{nom}^{FEM}$ at the OM point (or directly use the smooth disc spring formula).
• Calculate the FEM stress concentration factor:
  $$k_t^{FEM} = \frac{\sigma_{max}^{FEM}}{\sigma_{nom}^{FEM}}$$

4. Analytical Comparison and Deviation Analysis

Compare the FEM-derived $k_t^{FEM}$ with the analytical formula or standard table value $k_t^{ref}$, and calculate the relative deviation:

Common Sources of Deviation:

Typical Deviation Range: - For standard tooth geometries, the deviation between FEM and standard table values is generally within ±15%. - If the analytical formula originates from a simplified model, the deviation can reach +30% (conservative) or -20% (unsafe). - It is recommended to use the FEM-calibrated $k_t$ as the design input.

5. Verification Report Output Content

Upon completion of the FEM verification, the design documentation should include:

1. Geometric parameters and material
2. FEM model description (element type, mesh density, boundary conditions)
3. Nominal stress and peak stress contour plots
4. Numerical $k_t$ and its deviation from the reference value
5. Corrective conclusion: If the deviation exceeds ±10%, the FEM value should be used in the design, or the safety factor $S_F$ should be increased to above 1.5.

6. Example

For an M10 DIN 9250 washer, tooth root fillet $r = 0.2$ mm, tooth depth $d = 0.45$ mm, tooth tip angle 60°.
Analytical prediction: $k_t \approx 2.3$ (from table).
FEM result: OM point nominal stress $\sigma_{nom} = 1\,850$ MPa, maximum compressive stress at tooth root $\sigma_{max} = 4\,200$ MPa, therefore $k_t^{FEM} = 2.27$.
Deviation $\Delta k_t \approx -1.3\%$, validating the reliability of the table value, which can be directly adopted.

Summary:
FEM is the most reliable tool for determining the tooth root stress concentration factor. By comparing with analytical solutions or standard values, the design margin can be quantified and conservatism can be adjusted. For critical safety components, it is recommended to replace empirical coefficients with FEM-calibrated $k_t$ to achieve a balance between lightweight design and reliability.