Torque Coefficient $K$ — Correction for Single-Side Serrated Conical Spring Washers

1. Review of the Standard K Factor

In rapid bolted joint calculations, the torque coefficient (also known as the nut factor) $K$ relates the tightening torque $M_A$, preload $F_M$, and nominal diameter $d$:

From the VDI 2230 R13 theoretical formula, $K$ can be expanded into a combination of geometric and frictional components:

The three terms correspond respectively to: useful work of the helix angle, thread friction, and bearing surface friction. With standard flat washers or direct head bearing, $\mu_K$ is the metal-to-metal friction coefficient, and $K$ typically ranges from 0.12 to 0.24.

2. Influence of Single-Side Serrated Conical Spring Washers on the Torque Coefficient

When using NFE 25-511 single-side serrated conical spring washers, the bearing surface is no longer a smooth plane but a radial serration interlocking pair. This alters the $K$ value in two ways:

1. Significant Increase in Equivalent Friction Coefficient
   The mechanical interlocking effect of the serrations causes the equivalent bearing surface friction coefficient $\mu_{eff}$ to be much higher than the smooth surface $\mu_K$ (reaching 0.3–0.5 or even higher). Consequently, the third term $\frac{D_{km}\,\mu_K}{2d}$ in the K factor is replaced by $\frac{D_{km}\,\mu_{eff}}{2d}$, leading to a substantial increase in the $K$ value.

2. Possible Change in Equivalent Friction Diameter
   Serration contact is not a continuous annular surface but consists of multiple radial tooth tip bands. If the classical equivalent diameter $D_{km}$ is still used, it is recommended to take the average diameter of the tooth tip bands (approximately the washer mean diameter $D_m = (D_e+D_i)/2$). For a conservative approach, the arithmetic mean of the outer and inner diameters of the original nut/washer bearing surface can also be retained.

Therefore, the torque coefficient for a single-side serrated washer $K_{dent}$ is:

Where: - $\mu_G$ — thread friction coefficient (same as for standard connections) - $\mu_{eff}$ — equivalent bearing surface friction coefficient provided by the serrated washer (including mechanical interlocking amplification)

3. Rapid Determination of $\mu_{eff}$

The calculation model for $\mu_{eff}$ has been detailed previously; its core is:

In engineering practice, for hardened spring steel serrated washers paired with standard steel mating parts, typical values are:

If measurement is not possible, a preliminary conservative value of $\mu_{eff} = 0.35$ can be used, followed by correction via ISO 16047 testing.

4. Calculation Example

Given:
- M10×1.5 bolt: $d=10$ mm, $P=1.5$ mm, $d_2=9.026$ mm
- Thread friction coefficient $\mu_G = 0.12$
- Single-side serrated washer under the nut: washer mean diameter $D_m = 15.5$ mm (used for equivalent diameter $D_{km}$)
- Serration equivalent friction coefficient $\mu_{eff} = 0.40$ (dry, oil-free)

Calculate K factor:

Comparison: With a standard flat washer ($\mu_K=0.12$), $K \approx 0.024 + 0.0628 + \frac{13.4 \times 0.12}{20} \approx 0.167$ (typical M10 value).
The serrated washer increases the torque coefficient from 0.167 to 0.397, an increase of approximately 140%.

5. Engineering Significance and Precautions

1. Higher Tightening Torque Required for the Same Preload
   With $K_{dent}$ as high as 0.3–0.5, the required tightening torque for a given target preload is significantly higher than for standard connections. The torque must be set according to the corrected $K$ value; otherwise, the preload will be severely insufficient.

2. Potential Reduction in Preload Scatter
   Although the $K$ value increases, the mechanical locking of the serrations reduces the random fluctuation component of the friction coefficient ($\mu_{eff}$ is less sensitive to lubrication). Consequently, the relative scatter of the preload may be better than with standard washers.

3. Series Use with DIN 25201 Wedge-Locking Washers
   If both wedge-locking washers and elastic serrated washers are used together, the total $K$ factor will combine the high friction from both interfaces: one from the wedge washer's serration interlock and the other from the elastic washer's serration interlock. In this case, $K$ can reach 0.5–0.7 and must be precisely determined through testing.

4. Surface Pressure Verification
   The increased torque leads to higher preload, and the local pressure at the serrations is extremely high. Strict surface pressure verification per R10 is mandatory.

5. Tool Capability
   The high $K$ value requires the tool to deliver higher torque. The capacity of the tightening tool must be checked to ensure it is sufficient.

Summary:
Single-side serrated conical spring washers significantly increase the torque coefficient $K$ through $\mu_{eff}$. In design, the corrected $K_{dent}$ should be used instead of the standard value. Accurate calculation requires measuring $\mu_{eff}$ and then substituting it into the formula $K_{dent} = \frac{0.16P}{d} + \frac{0.58 d_2 \mu_G}{d} + \frac{D_{km} \mu_{eff}}{2d}$ to ensure the target preload is achieved correctly.