DIN 9250 Tooth Tip Contact Pressure: Hertz Line Contact Model

1. Formula and Physical Significance

The locking ability and collapse resistance of serrated washers are directly related to the tooth tip contact pressure. Under preload, the tooth tip and the mating surface form a narrow strip contact (approximate line contact), and the maximum contact pressure can be calculated using the classical Hertz line contact theory:

Where: - $p_{max}$ — Maximum tooth tip contact pressure (MPa) - $F_M$ — Bolt preload (N), typically the maximum assembly preload $F_{Mmax}$ is used for verification - $E^*$ — Equivalent elastic modulus (MPa), dependent on the elastic constants of the washer and the mating part - $n$ — Number of teeth on the washer (assuming uniform load distribution) - $r_{tip}$ — Tooth tip radius of curvature (mm), describing the degree of rounding at the tooth tip - $w_{tooth}$ — Tooth load-bearing width (mm), i.e., the contact length of the tooth in the radial direction

This formula simplifies the complex elastic-plastic contact of the tooth tip to a line contact problem between a cylinder (tooth tip) and a plane (mating part), used to evaluate whether macroscopic plastic collapse will occur at the tooth tip and to provide a basis for tooth profile optimization.

2. Equivalent Elastic Modulus $E^*$

The tooth tip and the mating surface are typically made of different materials. The equivalent modulus is calculated by the series combination of the elastic constants of the two contacting bodies:

• Washer is generally made of spring steel: $E_1 \approx 206\,000\ \text{MPa},\ \nu_1 = 0.3$
• Mating part is structural steel: same parameters; if aluminum alloy, $E_2 \approx 70\,000\ \text{MPa},\ \nu_2 = 0.33$

For steel‑steel contact:

3. Formula Derivation (Hertz Line Contact)

In classical Hertz theory, for a cylinder of radius $R$ in contact with a plane under a load per unit length $F'$, the maximum contact pressure is:

Parameter definitions: - $F'$ — Normal force per unit contact line length (N/mm) - $R$ — Equivalent radius of curvature of the tooth tip (mm) - $E^*$ — Equivalent elastic modulus (MPa)

For a DIN 9250 washer, the normal force on each tooth is $F_{tooth} = F_M / n$. This force is distributed along the tooth width $w_{tooth}$, so the load per unit length is:

Taking the tooth tip radius of curvature $r_{tip}$ as the equivalent radius $R$ and substituting yields the given formula:

4. Relationship Between Contact Pressure and Material Strength

Hertz theory provides the elastic contact pressure. When $p_{max}$ exceeds the material hardness (or approximately $1.6\sigma_y$), local plastic yielding will occur in the tooth tip contact zone, the contact area will expand, and the pressure will redistribute until the average pressure drops to the constrained hardness of the material (approximately $3\sigma_y$).

Design control should consider: - If elastic behavior at the tooth tip is desired (rare), require $p_{max} \le 1.6\,\sigma_y$. - Micro-plasticity is generally allowed to form a reliable mechanical interlock, but macroscopic collapse must be avoided. A conservative criterion is:

where $H_V$ is the Vickers hardness of the mating part (MPa), and $\sigma_y$ is the yield strength (MPa).

5. Calculation Example

Given: - M10 bolt, maximum preload $F_{Mmax} = 20\,000\ \text{N}$ - Washer tooth count $n = 12$, tooth width $w_{tooth} = 1.5\ \text{mm}$ - Tooth tip radius of curvature $r_{tip} = 0.15\ \text{mm}$ - Steel‑steel contact: $E^* \approx 113\,200\ \text{MPa}$ - Mating part: S355 steel, yield strength $\sigma_y \approx 355\ \text{MPa}$, hardness approximately 1500 MPa

Calculation:

This value is far higher than the material hardness (1500 MPa), indicating that significant plastic deformation will inevitably occur at the tooth tip, and the contact area will increase substantially until the average pressure drops to approximately $3\sigma_y \approx 1065\ \text{MPa}$. This is the physical basis for the tooth's ability to "bite" into the mating part.

6. Application and Limitations of the Formula

Applications: - Compare the influence of different tooth profile designs (varying $r_{tip}$, tooth count, tooth width) on contact pressure, optimizing the tooth profile to avoid excessive penetration or yielding of the tooth tip itself. - Evaluate the pressure level after changing the mating part material (e.g., replacing steel with aluminum), predicting the risk of excessive collapse. - Provide an elastic pressure reference value for analytical formulas of plastic penetration depth.

Limitations: - The formula assumes perfectly elastic behavior. In reality, tooth tip pressures are extremely high and always enter the plastic regime. The actual maximum pressure will not exceed the material hardness, so this value is only for theoretical comparison and cannot be directly used as the true contact pressure. - Does not account for tooth tip friction, coatings, or material work hardening. - Assumes uniform load distribution; in practice, due to manufacturing tolerances, only some teeth may bear the load.

Advanced Treatment: To obtain a more realistic contact pressure distribution, elastic-plastic finite element (FEM) analysis is required, as described in previous sections.

Summary:
The DIN 9250 tooth tip contact pressure formula $p_{max} = \sqrt{ F_M E^* / (\pi n r_{tip} w_{tooth}) }$ is a direct application of Hertz line contact theory. It provides an upper bound estimate of the elastic tooth tip pressure. In design, it can be compared with material hardness to prevent excessive penetration and serve as a reference index for tooth profile optimization. In practice, the tooth tip inevitably enters the plastic regime, and this formula should be used in conjunction with a plastic model.