DIN 9250 Locking Torque Calculation: Tooth Engagement + Flat Friction Model

1. Formula Definition and Locking Mechanism

The ability of a DIN 9250 serrated lock washer to resist bolt loosening under vibration can be quantitatively described by the locking torque $M_{lock}$. When the bolt is subjected to an external loosening torque, two resistance torque components arise between the washer and the mating part, as well as the bolt head/nut:

• Tooth tip engagement mechanical locking torque: The tooth tips embed into the mating surface, forming a "micro-wedge" that requires overcoming shear/plowing forces to rotate.
• Flat friction sliding torque: The non-serrated portion of the washer (inter-tooth flat areas or the washer back face) still relies on conventional friction.

These two components act in parallel, and the total locking torque formula is:

Where: - $M_{lock}$ — Locking torque provided by the washer (N·mm) - $F_M$ — Current preload (N), typically taken as the minimum residual preload for a conservative assessment - $r_{tooth}$ — Equivalent diameter of tooth tip engagement (mm) - $\mu_{bite}$ — Equivalent friction coefficient of tooth tip engagement (dimensionless), combining mechanical interlocking and local friction - $r_{flat}$ — Equivalent diameter of the flat friction portion (mm) - $\mu_{flat}$ — Friction coefficient of the flat portion (non-serrated area), identical to that of a smooth surface

Formula meaning: The resistance torques from the two friction sources are superimposed, each resistance torque = force × friction coefficient × radius. Since $r$ is defined as a diameter, it must be divided by 2 to convert to a radius; if $r$ is directly taken as a radius, the denominator 2 is omitted from the formula. Engineering practice often uses diameter expression, so this form is retained.

2. Detailed Parameter Explanation

2.1 Tooth Tip Engagement Diameter $r_{tooth}$

The tooth tips engage the mating surface only within a portion of the washer's radial width. $r_{tooth}$ represents the average diameter of all tooth tip contact areas. It can be approximated as the average diameter of the tooth tip ring band:

• $D_{tooth,o}$ — Outer diameter of the tooth tips (typically equal to or slightly less than the washer outer diameter)
• $D_{tooth,i}$ — Inner diameter of the tooth tips (the termination diameter of the teeth near the washer inner hole)

If detailed geometric data is unavailable, a conservative approach is to take the arithmetic mean of the washer outer and inner diameters: $r_{tooth} \approx D_m = (D_e + D_i)/2$.

2.2 Flat Friction Diameter $r_{flat}$

This refers to the equivalent contact diameter of the flat annular area on the washer not covered by teeth (inter-tooth valleys or the washer back face). For a single-sided serrated washer, the other side is a smooth plane, and $r_{flat}$ can be taken as the average diameter of the washer's annular area; for double-sided serrated washers, the flat portion is very small and can be neglected.

Common treatment: When differentiation is not possible, $D_{km} = (D_e + D_i)/2$ can be uniformly used to represent the total effective friction diameter, but this confuses the two components. A more precise method: allocate the moment arm based on the tooth area ratio $A_{tooth}/A_{nom}$.

2.3 Tooth Tip Equivalent Friction Coefficient $\mu_{bite}$

is significantly higher than the conventional friction coefficient because it includes mechanical resistances such as plowing, cold welding, and shearing of the teeth. Measurements show that for steel‑steel connections with standard tooth profiles:

• $\mu_{bite}$ can reach 0.5 – 1.0, far exceeding $\mu_{flat}$ (0.1–0.2).
• It is a function of tooth count, tooth profile angle, and hardness matching, and can be estimated from tests or empirical formulas. If unavailable, $\mu_{bite} \approx 0.6 \sim 0.8$ can be used for conservative design.

2.4 Flat Friction Coefficient $\mu_{flat}$

This is the friction coefficient of a smooth plane under the same material and surface treatment conditions. Typical values are shown in the table below:

3. Formula Derivation Approach

The locking torque is obtained by integrating the tangential forces of each contact micro-region about the bolt axis. Simplified assumptions:

• The preload is uniformly distributed over the entire annular area of the washer, divided into the tooth zone and the non-tooth zone.
• The tangential resistance in the tooth zone is characterized by $\mu_{bite}$, with a moment arm of $r_{tooth}/2$.
• The tangential resistance in the non-tooth zone is characterized by $\mu_{flat}$, with a moment arm of $r_{flat}/2$.
• The load proportion carried by each part is based on area or contact pressure distribution. In the simplified formula, the direct product form with the preload implicitly assumes load distribution to both zones.

A more rigorous torque expression would be:

where $\alpha_{tooth}$ is the proportion of preload carried by the tooth zone. When the distribution difference is ignored and the total preload $F_M$ is directly multiplied by each moment arm, the given two-term sum form is obtained. This simplification is conservative (since it does not account for the tooth zone carrying only part of the load).

4. Calculation Example

Given: - Bolt M10, preload $F_M = 18\,000\ \text{N}$ - Serrated washer: outer diameter $D_e = 20$ mm, inner diameter $D_i = 10.5$ mm - Tooth zone outer diameter 19 mm, inner diameter 12 mm → $r_{tooth} = (19+12)/2 = 15.5$ mm - Flat portion diameter: conservatively take the entire washer average diameter $r_{flat} = (20+10.5)/2 = 15.25$ mm - $\mu_{bite} = 0.7$ (steel‑steel standard teeth) - $\mu_{flat} = 0.14$ (light oil film)

Calculation:

Comparison: For a smooth washer (only $\mu_{flat}=0.14$, $r_{flat}=15.25$ mm), the locking torque is:

The serrated washer increases the locking torque by approximately 6 times.

5. Locking Safety Factor Verification

The obtained $M_{lock}$ should be compared with the external loosening torque (e.g., the thread back-off torque $M_{thread\_backoff}$ caused by vibration) to calculate the safety factor:

If the safety factor is insufficient, the preload can be increased, the number of teeth can be increased (to raise $\mu_{bite}$), or a larger diameter washer can be selected.

6. Influencing Factors and Design Recommendations

1. Tooth profile and tooth count: Sharp teeth, more teeth → higher $\mu_{bite}$, $r_{tooth}$ closer to the outer diameter, larger locking torque.
2. Material hardness: Lower hardness of the mating part → deeper tooth penetration → higher $\mu_{bite}$, but surface crushing must be checked.
3. Preload decay: $F_M$ should be taken as the minimum residual preload; the locking torque decreases accordingly.
4. Repeated use: Tooth tip wear → reduced $\mu_{bite}$, potentially approaching $\mu_{flat}$.
5. Temperature: At high temperatures, material softening → reduced hardness → weakened engagement, while the flat friction coefficient may also change.
6. Coating: Soft, thick coatings (e.g., hot-dip galvanizing) hinder tooth tip penetration, reducing $\mu_{bite}$; these should be avoided or confirmed through testing.

Summary:
The DIN 9250 locking torque formula $M_{lock} = F_M (r_{tooth} \mu_{bite} + r_{flat} \mu_{flat}) / 2$ unifies the mechanical locking of teeth and flat friction into a linear superposition model, significantly increasing locking resistance through the high friction coefficient of the tooth zone. During design, geometric parameters and material characteristics must be reasonably selected, and the locking safety factor requirements must be satisfied.