Embedded Loss Calculator

Embedded Loss in Bolted Joints

Mechanism

Embedded loss occurs when surface asperities at contact interfaces in a bolted joint undergo plastic deformation under pressure. When a bolt is tightened, the clamping force causes high local stresses at the microscopic high points where surfaces meet. These stresses exceed the material’s yield strength, causing permanent flattening of the asperities.

This localized plastic deformation effectively reduces the grip length (distance between bolt head and nut), causing a decrease in bolt stretch and corresponding reduction in preload. The phenomenon takes place at all interfaces: under the bolt head, under the nut, at the thread contacts, and between any plates in the joint.

Quantitative Data

Research shows that each interface in a joint typically experiences:

• 1-3 μm of embedding for smooth ground surfaces (Rz < 10 μm)
• 2-4 μm for medium roughness surfaces (Rz 10-40 μm)
• 3-7 μm for rough surfaces (Rz 40-160 μm)

The total embedding in a joint is the sum of all interface contributions. A typical steel joint with three interfaces (bolt head, nut face, one plate-to-plate) may experience 5-15 μm total embedding.

This small dimensional change can cause significant preload loss. In standard joints, embedding typically results in 5-10% preload reduction. However, in small-diameter fasteners, this loss can reach 38% or more of initial preload.

Calculation Methodology

The direct mathematical relationship between embedding and preload loss is:

Preload Loss (kN) = Bolt Stiffness (kN/mm) × Total Embedding (mm)

For practical analysis, we can expand this into a complete calculation process:

1. Calculate bolt stiffness (Kb):

   Kb = (E × A) / L

   Where E is elastic modulus (N/mm²), A is bolt stress area (mm²), and L is grip length (mm).

2. Determine total embedding (δe):

   δe = n × δi

   Where n is number of interfaces and δi is embedding per interface (μm).

3. Calculate absolute preload loss:

   ΔF = Kb × (δe / 1000)

   Note: δe must be converted from μm to mm for this calculation.

4. Calculate percentage loss:

   % Loss = (ΔF / Fi) × 100%

   Where Fi is initial preload.

Alternatively, percentage loss can be calculated directly by comparing embedding to initial bolt elongation:

% Loss = (δe / δb) × 100%

Where δb is initial elastic bolt elongation under preload (μm), calculated as:

δb = (Fi × L × 1000) / (E × A)

Critical Factors

Bolt Size Effect

The percentage of preload lost increases dramatically as bolt diameter decreases. Research demonstrates that while an M12 bolt might lose 12% of preload, an M4 bolt with identical interfaces will lose approximately 38%.

This occurs because the absolute embedding (in microns) is similar regardless of bolt size, but smaller bolts have less elastic elongation. Mathematically, bolt elongation scales with L/d², so for any given embedding distance, percentage loss increases as diameter decreases.

Grip Length/Diameter Ratio

The ratio of grip length to bolt diameter (L/d) is critical. VDI 2230 recommends maintaining a minimum L/d ratio of 3 to limit embedding losses. Short bolts (L/d < 3) are particularly vulnerable because:

% Loss ∝ δe/(L/d²)

Research indicates that with L/d = 1, percentage losses can be 3-4 times higher than with L/d = 5 for the same absolute embedding.

Material Effects

Material properties significantly influence embedding:

• Soft materials embed more than hard materials. Aluminum joints exhibit approximately 50% more embedding than steel joints under identical conditions.
• VDI 2230 recommends doubling the expected embedding values for aluminum compared to steel.
• Stainless steel fasteners typically show slightly higher embedding than carbon steel fasteners due to lower hardness.

Surface Finish and Coatings

Surface roughness directly impacts embedding magnitude:

• Rougher surfaces produce more embedding, with approximately 10-20% of peak-to-valley roughness height translating to embedding distance.
• Coatings affect embedding based on their thickness and hardness:
  • PTFE coatings (20-50 μm thick) may add 2-5 μm of embedding through compression
  • Thin electroplated coatings (5-15 μm) have minimal additional effect
  • Hard ceramic coatings (TiN, etc.) do not significantly compress but may alter surface roughness
  • Thick soft coatings like alkyd (80 μm) can increase embedding substantially

Loading Type

The type of service loading affects embedding severity:

• Shear (transverse) loads produce 20-60% more embedding than pure axial loads
• Vibration accelerates embedding through micro-slip at interfaces
• Most embedding (approximately 80%) occurs in the first loading cycle

Time-Dependent Behavior

Embedding occurs in distinct phases:

1. Initial tightening: Some asperity flattening occurs during tightening and is automatically compensated by continued torque application.
2. Post-tightening settlement (minutes to hours): Additional embedding occurs as compressed asperities yield, causing immediate preload drop.
3. First load application (hours to days): The majority of remaining embedding occurs during the first few service load cycles or thermal cycles.
4. Long-term behavior: After initial settling, embedding typically stabilizes unless other factors (creep, corrosion, etc.) are present.

Once surfaces have conformed, embedding does not continue indefinitely under normal conditions. This explains why re-tightening fasteners after initial service (e.g., wheel lug nuts after 100 miles) is effective—it restores preload after most embedding has occurred.

Standardized Calculation Approach (VDI 2230)

The VDI 2230 standard provides a systematic method for embedding calculations:

1. Determine embedding per interface (fZ) from tables based on surface roughness, loading type
2. Calculate total embedding based on number of interfaces
3. Compute bolt stiffness and resulting preload loss
4. Add a preload allowance (FZ) to the required minimum preload (FM) to compensate: FA = FM + FZ

VDI 2230 recommends increasing the embedding values by factors of:

• 2.0× for aluminum or other soft materials
• 1.5× for mixed loading conditions

Practical Design Solutions

Design Phase Mitigation

• Compensatory preload: Increase initial preload by the calculated percentage loss (typically 5-15%)
• Grip length optimization: Maintain L/d ratio ≥ 3 to minimize percentage losses
• Surface specifications: Specify smoother surface finishes (Rz < 10 μm) at critical interfaces
• Hardware selection: Use hardened washers to reduce embedding in softer materials
• Material specifications: Specify appropriate hardness at contact surfaces

Assembly Phase Mitigation

• Tightening method: Angle-controlled tightening (torque + angle) can compensate for embedding during installation
• Bolt tensioning: Hydraulic tensioners can achieve higher initial preload to offset expected losses
• Re-tightening procedures: Specify re-tightening after 24 hours or after first thermal/load cycle
• Spring elements: Use Belleville washers or disc springs for critical joints to maintain force despite embedding

Calculator Application

The embedded loss calculator implements these calculation principles to predict preload loss for specific joint configurations. For accurate results:

1. Enter precise bolt dimensions and material properties
2. Specify the correct number of interfaces (including thread interface)
3. Select appropriate surface roughness or enter measured values
4. Consider the loading type (axial/shear/mixed)
5. Examine both absolute and percentage loss results

The calculator outputs provide essential design data:

• The total embedding distance (μm)
• Absolute preload loss (kN)
• Percentage of initial preload lost
• Bolt and joint stiffness values
• Tailored design recommendations

This information allows for precise design adjustments to ensure adequate preload retention throughout the joint’s service life. For critical applications, the percentage loss should be incorporated into tightening specifications as a minimum preload margin.

Industry Practices

Industry approaches to embedding vary by sector:

Automotive

Typically allows 5-10% preload loss in design, uses torque-angle tightening and re-tightening for critical fasteners (cylinder head bolts, wheel lugs), standardizes to VDI 2230.

Aerospace

Uses a fixed 5% preload allowance in design calculations, employs surface treatment specifications, rarely permits re-tightening, relies on high-precision fasteners with controlled surface finish.

Pressure Vessels

ASME PCC-1 requires bolt stress margins, specifies re-tightening after thermal cycling, uses hydraulic tensioning to maximize initial bolt stretch, employs hardened washers and controlled surface preparation.

Structural Engineering

Accounts for slip in connections, requires minimum preload verification, specifies hardened washers for all structural bolts, follows standard installation procedures to manage embedding effects.

Experimental Validation

Research studies have validated embedding calculations through various methodologies:

• Ultrasonic bolt elongation measurement shows 5-10% preload reduction in standard joints after tightening
• Direct tension indicator (DTI) washers demonstrate embedding through reduced compression
• Torque-angle signature analysis identifies embedding as a non-linear region following initial tightening
• Microscopic examination confirms asperity flattening at interfaces after loading

Experiments have also confirmed the disproportionate effect on small-diameter fasteners, with M4 bolts showing three times the percentage loss of M12 bolts under identical conditions.

Additional Considerations

Combined Effects

Embedding often combines with other preload loss mechanisms:

• Stress relaxation: High-temperature applications can see additional 2-5% loss from stress relaxation
• Thermal effects: Differential thermal expansion can exacerbate embedding through cyclic loading
• Creep: Polymer coatings or soft materials may continue to deform over time
• Corrosion: Surface oxidation can accelerate embedding or create oxide layers that subsequently crush

Environmental Effects

Certain environments modify embedding behavior:

• High temperature: Reduces material yield strength, potentially increasing embedding
• Cryogenic conditions: Increases material hardness, potentially reducing embedding
• Corrosive environments: May increase interface roughness through pitting
• Vibration: Accelerates embedding through micro-movement at interfaces

References

• VDI 2230 Part 1 (2015) – “Systematic calculation of high duty bolted joints”
• ISO 898-1/2 – “Mechanical properties of fasteners”
• Bickford, J. – “Introduction to the Design and Behavior of Bolted Joints”
• ASME PCC-1 – “Guidelines for Pressure Boundary Bolted Flange Joint Assembly”
• Friede, R. & Lange, J. (2010) – “Loss of Preload in Bolted Connections Due to Embedding and Self Loosening”
• Bolt Science Technical Articles – “Embedding Loss in Bolted Joints”