DIN 9250 Torque Coefficient K Value: Tooth Surface Friction Correction and Regression Range

1. Definition of Torque Coefficient K

In bolted connections, the torque coefficient $K$ (nut factor) relates the tightening torque $M_A$ to the resulting preload $F_M$ and the nominal bolt diameter $d$:

Its engineering form is:

Through the $K$ value, the required tightening torque can be quickly calculated from the target preload, or the preload can be estimated from a known torque. $K$ combines the effects of thread friction, bearing surface friction, and the helix angle, and is a dimensionless empirical coefficient.

2. Theoretical Composition of the K Coefficient

The theoretical expression for $K$ can be derived from the precise torque formula of VDI 2230 (Kellermann & Klein):

Therefore:

The three terms correspond to: - Useful work of the helix angle (very small contribution, approx. 1%–4%) - Thread friction contribution - Bearing surface friction contribution (usually the largest, 40%–60%)

For DIN 9250 serrated lock washers, the bearing surface is no longer a smooth plane; its friction behavior is dominated by the mechanical interlocking of the teeth. The standard friction coefficient $\mu_K$ must be replaced by the equivalent tooth surface friction coefficient $\mu_{serr}$:

3. Determination of the Equivalent Tooth Surface Friction Coefficient $\mu_{serr}$

The equivalent friction coefficient $\mu_{serr}$ for serrated washers is related to the smooth surface friction coefficient $\mu_{flat}$ by an amplification factor $k_{serr}$:

• $\mu_{flat}$ — Friction coefficient of the same material and coating on a smooth surface (typically 0.08–0.25)
• $k_{serr}$ — Tooth surface amplification factor, dependent on tooth count, tooth profile, and material hardness matching (typically 1.3–2.5)

For steel‑steel connections with standard tooth profiles (10–16 teeth), $k_{serr}$ commonly ranges from 1.5 to 2.0. The resulting $\mu_{serr}$ is typically between 0.20 and 0.45, significantly higher than the $\mu_K$ of standard smooth washers.

4. Typical K Value Range (DIN 9250 Serrated Washer + Steel Bolt)

The table below provides reference ranges for the torque coefficient $K_{serr}$ when using DIN 9250 washers under common lubrication and surface treatment conditions. Assumes bolts M6–M20, with thread friction coefficient $\mu_G$ and washer bearing surface friction $\mu_{serr}$ within the same lubrication system.

Notes: - The thread friction coefficient $\mu_G$ is typically taken as the value for the same lubrication state as the washer side $\mu_{flat}$ (e.g., both phosphated + oil, $\mu_G\approx0.12$). - $K_{serr}$ is 30%–70% higher than the K value for a smooth washer under the same conditions (e.g., 0.14–0.22). - If no measured values are available for design, select the "Design Recommended Value" from the table and perform a preload scatter analysis of ±15%.

5. Calculation Example

Given: - Bolt M10×1.5, grade 8.8, $d = 10$ mm, $P = 1.5$ mm, $d_2 = 9.026$ mm - Thread friction coefficient $\mu_G = 0.12$ - Serrated washer DIN 9250: $D_e = 20$ mm, $D_i = 10.5$ mm → $D_{km} \approx 15.25$ mm - Smooth surface friction coefficient $\mu_{flat} = 0.14$ (phosphated + light oil) - Tooth surface amplification factor $k_{serr} = 1.7$

Calculation:

Comparison: Using a smooth washer ($\mu_K = 0.14$):

The serrated washer increases the K value from 0.194 to 0.268, an increase of approximately 38%.

Tightening torque for target preload $F_M = 20\,000$ N: - Smooth washer: $M_A = 0.194 \times 20\,000 \times 10 = 38.8$ N·m - Serrated washer: $M_A = 0.268 \times 20\,000 \times 10 = 53.6$ N·m

6. Main Factors Influencing the K Value

• Tooth profile and tooth count: Denser and deeper teeth increase $\mu_{serr}$, leading to a higher $K$ value.
• Material hardness matching: The washer must be harder than the mating part; a greater hardness difference leads to deeper penetration and a higher $K$ value.
• Surface coating: Soft coatings (e.g., Dacromet) may be penetrated by the teeth, with the $K$ value still primarily determined by base material interlocking; thick coatings (e.g., hot-dip galvanizing) hinder penetration, reducing the $K$ value and increasing scatter.
• Reuse: Tooth wear reduces $k_{serr}$, causing the $K$ value to regress towards that of a smooth washer. Recalibration is required before each reuse.
• Preload level: If the preload is too high and the teeth are fully flattened, the $K$ value tends to saturate and no longer increases linearly with preload.

7. Experimental Regression and Standard Determination of K Value

Due to the complexity of tooth surface friction, theoretical calculations can only provide initial estimates. Accurate K values must be determined through experimental regression, with the standard method being ISO 16047 (Fasteners – Torque/clamp force testing). The specific procedure:

1. Tighten the actual bolt‑washer‑joint assembly on a test machine, simultaneously recording torque $M_A$ and preload $F_M$.
2. Calculate $K = M_A/(F_M d)$ for each test.
3. Calculate the mean $\bar{K}$ and standard deviation from multiple data points to assess scatter.
4. Design values $K_{min}$ and $K_{max}$ are determined from statistical upper and lower limits (e.g., ±3σ) for calculating preload extremes.

For mass production, it is recommended to perform spot checks of the $K$ value for each batch of supplied washers to control tightening quality.

8. Integration with the VDI 2230 System

• R13 Precise torque calculation: Directly use $\mu_{serr}$ in the bearing surface friction term, bypassing the K factor.
• R6 Maximum preload: When using the torque control method, preload scatter can be determined by combining the coefficient of variation of the $K$ value and $\alpha_A$.
• R10 Surface pressure: Due to the higher $K$ value, the preload at a given torque may be lower than expected, but the local pressure at the tooth tips is extremely high, requiring separate verification of the mating part surface against crushing.

9. Precautions

• K value is not a constant: It is sensitive to surface condition, temperature, and tightening speed. Generic values from different batches cannot be directly applied.
• Washer orientation: The serrated washer must be installed with the teeth facing the mating part (or bolt head). Do not install it backwards.
• Single use: The interlocking characteristics of serrated washers degrade after disassembly. The K value may decrease by 10%–20% upon reuse; the washer should be replaced.
• Tool setting: Since $K_{serr}$ is higher than for standard washers, the tightening tool must deliver a higher torque. Verify that the tool's capacity is sufficient.

Summary:
The torque coefficient $K$ for DIN 9250 serrated washers is significantly higher than for smooth washers due to the equivalent tooth surface friction coefficient $\mu_{serr}$, with a typical range of 0.18 to 0.38. The $K$ value can be estimated by substituting $\mu_{serr}$ into the theoretical formula, but final determination requires testing per ISO 16047. An accurate $K$ value is essential for ensuring the correct preload is achieved at the specified tightening torque.