Embedding Settlement Preload Loss: VDI 2230 Step R4

1. Definition and Physical Background

After assembly of a bolted joint, due to the micro-roughness of contact surfaces and thread flank clearances, plastic flattening occurs under the preload pressure. This causes a slight reduction in both the effective thickness of the clamped parts and the effective length of the bolt, a phenomenon known as embedding relaxation.
The macroscopic consequence is: the bolt tensile length decreases → preload drops. VDI 2230 designates this preload loss as $F_Z$, calculates it in Step R4, and compensates for it in the minimum assembly preload in Step R5.

Core Formula:

Where: - $F_Z$ — Preload loss due to embedding (N) - $f_{Z,total}$ — Total embedding amount of all contact surfaces in the joint (mm) - $\delta_S$ — Bolt elastic compliance (mm/N) - $\delta_P$ — Clamped part elastic compliance (mm/N)

Physical Meaning: Converts embedding displacement into force loss. The stiffer the system (smaller $\delta_S+\delta_P$), the more severe the preload drop for the same embedding amount.

2. Derivation Approach

The basic relationship between preload, elongation, and system compliance is:

When embedding settlement $f_Z$ occurs, the effective clamping length of the bolt decreases, equivalent to applying a reverse displacement, causing a preload drop $\Delta F = F_Z$. Within the elastic range, the relationship between displacement and force increment is:

3. Determination of Total Embedding Amount $f_{Z,total}$

is the sum of the unit contact surface embedding amounts for all surfaces in the joint that come into contact and may experience embedding. VDI 2230 provides reference embedding amounts (μm) for each pair of contact surfaces, selected based on surface condition, material, and machining process.

Reference Embedding Amounts per Contact Surface (VDI 2230 Table A6, etc.):

MoS₂ Coating: Due to its lubricating and friction-reducing effects, the surface conforms more easily under pressure, resulting in minimal micro-embedding, typically in the 1–3 μm range; a design value of 2 μm can be used.

Total Embedding Amount Calculation:

Multiple embedding contact surfaces exist in a joint, including: - Contact surface between bolt head/nut and washer (or directly with clamped part) - Contact surface between washer and clamped part (if washers are used) - Interface between clamped parts - Thread flank contact surface (usually counted as one contact surface)

Where $n$ is the total number of embedding contact surfaces. VDI 2230 provides typical contact surface counts for common joint types:

Conservative Design: Each contact surface is taken with the larger $f_Z$ value for the respective surface.

4. Calculation of Compliances $\delta_S$ and $\delta_P$

These two values are calculated in detail in VDI 2230 Step R3.

• Bolt Compliance $\delta_S$:
  $$\delta_S = \sum \frac{l_i}{E_S \cdot A_i}$$

Considers the series compliance of each bolt segment (shank, threaded section, head influence, etc.).

• Clamped Part Compliance $\delta_P$: For simple cylindrical structures, the cone-sleeve model can be used to calculate the equivalent area and length, and thus the compliance. Precise values may require finite element analysis.

Engineering Trends: - Long bolts, soft materials → large $\delta_S + \delta_P$ → small $F_Z$ for the same embedding amount, small preload loss. - Short bolts, rigid connections (e.g., thick flanges) → small $\delta_S + \delta_P$ → large preload loss, embedding must be strictly controlled.

5. Calculation Example

Given: - M10 bolt, clamping length 50 mm, bolt compliance $\delta_S = 1.2 \times 10^{-6}$ mm/N - Clamped part compliance $\delta_P = 0.8 \times 10^{-6}$ mm/N - Joint includes: bolt head directly on clamped part 1, interface between clamped part 1 and clamped part 2, nut on clamped part 2. Total of 3 contact surfaces. - Surface condition: general machining, uncoated, take $f_Z = 5$ μm/surface. - Thread pair embedding is included in the above count, no additional.

Total Embedding Amount:

Total Compliance:

Preload Loss:

Interpretation: If the initial assembly preload is 25 000 N, after embedding loss it will drop to approximately 17 500 N (a 30% decrease). This loss must be compensated for in Step R5 by increasing $F_{Mmin}$.

6. Influence of Surface Treatment on $f_Z$ and $F_Z$

• MoS₂ Coating: Minimal embedding amount (1–3 μm), smallest preload loss, suitable for high-precision, high-reliability joints.
• Phosphating + Oil: Embedding amount approximately 2–4 μm, economical, controllable loss.
• General Uncoated: Embedding amount 3–6 μm, moderate loss.
• Hot-Dip Galvanizing: Rough surface, embedding amount as high as 8–14 μm, significant preload loss, requiring special compensation or multiple tightening cycles to eliminate embedding.
• Cadmium/Zinc Plating (electroplated): Thin layer, embedding amount 2–4 μm, but hydrogen embrittlement must be considered.

Engineering Measures to Reduce Embedding Loss: - Improve machining accuracy and surface finish of mating surfaces. - Use lubricants or MoS₂ coatings. - Employ multiple tightening cycles (assemble → loosen → re-tighten) to pre-flatten micro-roughness. - Use lower surface pressure where permissible.

7. Integration into the VDI 2230 Design Process

The $F_Z$ calculated in Step R4 is passed to the minimum assembly preload formula in Step R5:

Thus, the target preload is increased by the amount $F_Z$ in advance to compensate for embedding loss, ensuring the joint maintains the required minimum clamping force throughout its service life.

Summary: Embedding settlement is an inherent physical phenomenon causing preload decay. It can be quantitatively assessed using $F_Z = f_{Z,total}/(\delta_S + \delta_P)$. Appropriate selection of surface treatment (especially using low-embedding coatings like MoS₂) can significantly reduce this loss, allowing for smaller bolt sizes or improved joint reliability.