Equivalent Friction Diameter $D_{km}$: Calculation of Bearing Surface Friction Term

1. Definition and Function

In tightening torque calculations (VDI 2230 R13 or simplified K-factor method), the torque consumed by bearing surface friction accounts for a significant proportion.
The expression for bearing surface friction torque is:

Where $D_{km}$ is the equivalent friction diameter of the bearing surface. It simplifies the distributed friction force on the annular bearing surface into a concentrated force acting at a characteristic diameter, allowing the torque to be calculated directly in the form of "force × lever arm".

2. Simplified Calculation Formula

The most commonly used simplified calculation in engineering is the arithmetic mean of the inner and outer diameters:

• $d_w$ — Outer diameter of the bolt head (or nut) bearing surface (mm), i.e., the outer diameter of the annular contact area with the clamped part
• $d_h$ — Bolt hole diameter of the connected part (mm), or the inner diameter of the washer (if a washer is used, it is the inner hole diameter of the washer)

This formula assumes that the friction force is uniformly distributed over the annular surface and simplifies the resultant force application point to the midpoint between the inner and outer diameters. When the bearing surface is narrow (small $d_w/d_h$), this approximation has acceptable accuracy.

3. Exact Calculation Formula (Uniform Pressure Assumption)

Based on the integration of infinitesimal friction torque elements, assuming uniform contact pressure, the exact solution for the equivalent diameter is:

Brief derivation: Take an infinitesimal ring area $dA = 2\pi r dr$ at radius $r$, friction force $dF = \mu_K \cdot p \cdot dA$ ($p$ is uniform surface pressure), friction torque $dM = r \cdot dF$, integrate to obtain the total torque $M_K$, then compare with $M_K = \mu_K F_M D_{km}/2$ to derive this formula.

Comparison of the Two Algorithms

• When the bearing surface width is small ($d_w$ close to $d_h$), the results of both methods are similar.
• When the bearing surface width is large (e.g., flange bolts, large washers), the exact formula yields a $D_{km}$ value slightly larger than the arithmetic mean. This is because the outer area at a larger radius accounts for a greater proportion of the area and contributes more significantly to the friction torque. Using the exact value in design calculations is more conservative (estimates a larger bearing surface friction torque).

Example: Take $d_w = 30\text{ mm}, d_h = 20\text{ mm}$
- Simplified: $D_{km} = (30+20)/2 = 25\text{ mm}$
- Exact: $D_{km} = \frac{2}{3} \times \frac{30^3 - 20^3}{30^2 - 20^2} = \frac{2}{3} \times \frac{27000 - 8000}{900 - 400} = \frac{2}{3} \times \frac{19000}{500} = \frac{2}{3} \times 38 = 25.33\text{ mm}$
The difference is about 1.3%, negligible.

If the size difference is large, e.g., $d_w = 50, d_h = 30$:
- Simplified: $40\text{ mm}$
- Exact: $\frac{2}{3} \times \frac{125000 - 27000}{2500 - 900} = \frac{2}{3} \times \frac{98000}{1600} = \frac{2}{3} \times 61.25 = 40.83\text{ mm}$, difference about 2%. Still within engineering tolerance.

Therefore, in general bolted connections, the simplified formula is sufficiently accurate and more convenient for manual calculation.

4. Parameter Value Considerations

• $d_w$: Should take the actual outer diameter of the bearing surface. For standard hexagon head bolts, refer to the standard (e.g., ISO 4014) to obtain the width across flats $s$. The outer diameter of the bearing surface is usually slightly smaller than the width across flats (due to chamfering), commonly taken as $d_w \approx 0.95s$ or using the value given by the standard.
• $d_h$: Usually the bolt hole diameter. If the hole is a through hole and no washer is used, take the hole diameter directly; if a flat washer is used, $d_h$ takes the inner diameter of the washer. In this case, the bearing surface becomes between the washer and the clamped part, but friction usually occurs between the washer and the clamped part. The specific value depends on the actual location of the friction surface. In standard torque calculations, washers are often ignored, and the bearing surface dimensions of the bolt head/nut are used directly.
• When using DIN 25201 wedge-lock washers, the bearing surface friction may be transferred to the contact between the radial teeth of the washer and the workpiece. This contact surface is no longer a smooth annulus, but the concept of equivalent diameter can still be used approximately, taking the average diameter of the effective friction ring of the washer.

5. Application in the Torque Formula

Recall the VDI 2230 R13 total assembly torque formula:

The third term is the bearing surface friction term. The accuracy of $D_{km}$ directly affects the magnitude of this torque component. When the friction coefficient $\mu_K$ is taken as a fixed value, a slight change in $D_{km}$ will linearly affect this part of the torque, but the impact is usually much smaller than the variation caused by the dispersion of the friction coefficient.

6. Summary

• The equivalent friction diameter $D_{km}$ is a simplified representation of the annular friction surface.
• Commonly used simplified formula: $D_{km} = \frac{d_w + d_h}{2}$, easy to calculate and meets the accuracy requirements for most engineering applications.
• The exact formula $D_{km} = \frac{2}{3} \frac{d_w^3 - d_h^3}{d_w^2 - d_h^2}$ can provide a more conservative (larger) torque estimate in cases such as flange faces and large washers.
• During design, the sources of $d_w$ and $d_h$ should be clearly defined to ensure that the torque-preload relationship is based on correct geometric foundations.