Effective Friction Coefficient
Effective Friction Coefficient
Formula Expression
Parameters
| Symbol | Name | Unit |
|---|---|---|
| mu_flat | mu_flat | — |
| size_key | size_key | — |
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Equivalent Friction Coefficient $\mu_{eff}$: Single-Side Serrated Conical Spring Washer
1. Background and Definition
In bolted joint torque‑preload calculations, the bearing surface friction coefficient $\mu_K$ is a key parameter determining torque consumption and preload scatter.
The bearing surface of a single-side serrated conical spring washer (NFE 25-511) is not a smooth plane but features radial serrations. During tightening, the serration tips bite into the mating part and the bolt head/nut surface, creating mechanical interlocking whose slip resistance far exceeds that of ordinary plane friction.
To continue using the classic Coulomb friction model in VDI 2230 or standard torque formulas, the equivalent friction coefficient $\mu_{eff}$ is introduced, converting the mechanical locking effect of the serrations into a virtual friction coefficient:
- $\mu_{base}$ — Base friction coefficient, the friction coefficient of a smooth surface under the same material and lubrication conditions
- $\Delta\mu_{dent}$ — Serration additional coefficient, determined by serration geometry, number of serrations, embedment depth, and material hardness
Physical meaning: Even with good lubrication, serration engagement can raise the equivalent friction coefficient to 0.3–0.5 or higher, significantly improving slip resistance and anti-loosening capability.
2. Mechanism of Serration Amplification Effect
Under preload $F_M$, the single-side serrations embed into the mating surface, producing the following effects: - Cold welding/micro-interlocking: Local pressure at serration tips is extremely high, penetrating the surface oxide layer and creating local adhesion, requiring a large tangential force to shear. - Plowing effect: During tangential sliding, serrations plow grooves into the mating surface, greatly increasing resistance. - Self-locking wedge effect: The inclined flanks of the serrations generate a wedging force under tangential load, preventing slip.
These effects combine so that the tangential resistance at the interface far exceeds $F_M \cdot \mu_{base}$ and is not solely dependent on surface condition — this is the fundamental reason for the outstanding anti-slip capability of single-side serrated washers.
3. Theoretical Calculation Model for $\mu_{eff}$
Based on serration geometry and mechanical analysis, $\Delta\mu_{dent}$ can be approximated as:
where: - $\eta$ — Engagement efficiency factor (0.5–1.0), depending on serration geometry, embedment depth, and hardness ratio of mating materials - $\sum A_{dent}$ — Sum of actual bearing areas of all serrations (mm²), i.e., the effective contact area between serration tips and the mating surface - $A_{nom}$ — Nominal bearing surface area (mm²), $A_{nom} = \frac{\pi}{4}(D_e^2 - D_i^2)$ - $\tau_{joint}$ — Shear strength at the serration joint (MPa), typically taken as the shear strength of the softer material - $p_{nom}$ — Nominal bearing surface pressure (MPa), $p_{nom} = F_M / A_{nom}$
Since $p_{nom} = F_M / A_{nom}$ and the total shear force of all serrations $\sum F_{shear} = \tau_{joint} \cdot \sum A_{dent}$, $\Delta\mu_{dent}$ can be understood as the ratio of additional tangential resistance provided by the serrations to the axial force.
Simplified engineering formula (when serrations are fully embedded):
Assuming full serration engagement and shear strength equal to $0.6 R_m$ of the softer material, and given the serration area ratio $R_a = \sum A_{dent} / A_{nom}$:
4. Key Parameters Influencing $\mu_{eff}$
| Parameter | Influence Trend | Engineering Value/Note |
|---|---|---|
| Number of serrations $z$ | $z$ ↑ → $R_a$ ↑ → $\mu_{eff}$ ↑ | But individual serration area is limited; too many serrations reduce embedment depth |
| Serration height and sharpness | Tall, sharp serrations embed deeper, plowing effect stronger | Standard serration geometry defined by NFE 25-511 |
| Washer hardness | Washer must be harder than the mating part to ensure serrations bite rather than crush | Recommended ≥ 400 HV |
| Mating part hardness and strength | Softer mating part → deeper serration embedment → higher $\mu_{eff}$, but risk of crushing | Must check R10 surface pressure |
| Surface coating | Coating reduces $\mu_{base}$, but serrations can still penetrate coating for mechanical engagement | Zinc-nickel, Dacromet coatings have minimal effect |
| Nominal pressure $p_{nom}$ | $p_{nom}$ ↑ → $\Delta\mu_{dent}$ term decreases (denominator increases), but $R_a$ increases with deeper embedment; overall effect complex | $\mu_{eff}$ is relatively stable within the working preload range |
5. Typical Numerical Reference
For conventional steel-steel connections with a single-side serrated conical spring washer (hardened spring steel, 10–12 serrations), the equivalent friction coefficient:
| Lubrication Condition | Smooth Surface $\mu_{base}$ | Serrated Washer $\mu_{eff}$ |
|---|---|---|
| Dry, no oil | 0.18 – 0.25 | 0.35 – 0.50 |
| Light oil film | 0.12 – 0.16 | 0.30 – 0.45 |
| Good lubrication (grease) | 0.08 – 0.12 | 0.25 – 0.35 |
Practical recommendation: Due to variations in serration geometry, materials, and assembly conditions, it is recommended to determine $\mu_{eff}$ for the actual combination via ISO 16047 torque‑clamp force testing during the design phase, obtaining the $K$ factor or direct $\mu_G, \mu_K$ values.
6. Application in Torque Calculation
Once $\mu_{eff}$ is obtained, it can be used as the bearing surface friction coefficient $\mu_K$ in the VDI 2230 R13 formula:
where $D_{km}$ is the equivalent friction diameter of the washer bearing surface.
Notes: - Due to the larger $\mu_{eff}$, the bearing surface torque share will increase significantly (can exceed 60% of total torque). - If the bolt head/nut contact surface is also serrated (i.e., double-side serrated washer), $\mu_{eff}$ applies to both friction surfaces. - The thread friction coefficient $\mu_G$ is unaffected by the washer and is still determined by the lubrication condition of the bolt and nut.
7. Example Calculation
Given: M10 connection, single-side serrated washer under the nut, washer outer diameter 20 mm, inner diameter 11 mm, 12 serrations, serration area ratio $R_a \approx 0.15$.
- Mating part: S235 steel ($R_m = 360$ MPa, softer), $\tau_{joint} \approx 0.6 \times 360 = 216$ MPa.
- Preload $F_M = 18\,000$ N.
- Smooth surface base friction coefficient (with oil) $\mu_{base} = 0.12$.
Nominal area and pressure:
Serration additional term:
Equivalent friction coefficient:
Interpretation: Even with a base oil friction of only 0.12, the serration effect raises the equivalent friction coefficient to about 0.52, greatly enhancing the bearing surface's slip resistance. Correspondingly, the bearing surface friction term in the tightening torque will increase, and the target torque must be adjusted accordingly during design.
Summary:
$\mu_{eff}$ converts the mechanical interlocking effect of a single-side serrated conical spring washer into an increment in the friction coefficient, facilitating integration into standard torque calculation systems. In design, experimental determination should be the basis, with theoretical estimation serving as a reference during the concept phase. Correctly evaluating $\mu_{eff}$ allows full utilization of the anti-loosening and anti-slip potential of serrated washers, while avoiding risks from insufficient torque or over-tightening.