VDI 2230 Load Factor $\Phi^*$

1. Definition and Physical Meaning

In bolted joints, the external axial working load $F_A$ is not entirely borne by the bolt. Due to the different elastic compliances of the bolt and the clamped parts, the load is distributed between them in a certain proportion. The load factor $\Phi^*$ (also called force ratio or load distribution factor) is the key parameter describing this distribution:

• $F_{SA}$: Additional bolt force caused by external load (N)
• $F_{PA}$: Reduction in clamping force of the clamped parts caused by external load (N)
• $F_A$: External axial working load (N)

Physical picture: After assembly, the bolt is in tension and the clamped parts are in compression. When the external load $F_A$ is applied, the bolt elongates further, and the compression of the clamped parts decreases. The deformation compatibility of the system determines that the bolt only bears a small portion of the external load, while most of the external load serves to "unload" the compression of the clamped parts.

The smaller $\Phi^*$ is, the smaller the additional force on the bolt, which is beneficial for fatigue; the larger $\Phi^*$ is, the larger the additional force on the bolt, increasing the bolt stress amplitude.

2. Basic Formula (Concentric Loading, Load Introduction at the Interface)

When the working load acts directly at the interface (e.g., a gasket installed at the interface) and the load axis coincides with the bolt axis, the load factor is:

Where: - $\delta_S$: Elastic compliance of the bolt (mm/N), calculated according to the bolt compliance section of VDI 2230 - $\delta_P$: Elastic compliance of the clamped parts (mm/N), calculated based on the deformation cone model

Properties: - $0 < \Phi^* < 1$ - The more flexible the bolt (larger $\delta_S$), the smaller $\Phi^*$ → smaller bolt force - The stiffer the clamped parts (smaller $\delta_P$), the smaller $\Phi^*$ → smaller bolt force - Typical values: For steel-steel joints, $\Phi^*$ is usually between 0.15 and 0.35

3. General Formula with Load Introduction Factor $n$

In actual structures, the external load rarely acts exactly at the center of the interface. The axial position of the load application affects the load distribution. VDI 2230 introduces the load introduction factor $n$ for correction:

The range of $n$ depends on the load application position:

More precise $n$ values can be determined according to the charts or formulas given in VDI 2230, related to the distance $a$ from the load introduction point to the interface and the dimensions of the substitute cone.

4. Load Factor for Eccentric Loading

When the working load has an eccentricity $a$ (relative to the bolt axis), the bolt and clamped parts are subjected not only to axial force but also to bending moment. In this case, the load factor increases to:

Or in abbreviated form:

Where: - $a$: Eccentricity of the working load relative to the bolt axis (mm) - $s_{sym}$: Distance from the bolt axis to the bending neutral axis of the clamped parts (mm) - $I_{Bers}$: Moment of inertia of the substitute cross-section of the clamped parts (mm⁴) - $I_S$: Moment of inertia of the bolt cross-section (mm⁴), $I_S = \frac{\pi}{64} d^4$ (approximate)

Effect: The larger the eccentricity, the larger $\Phi_{en}^*$, the larger the additional bolt force, and the bending stress also increases, reducing fatigue life. Therefore, eccentric loading should be avoided in design.

5. Load Factor for Multi-Bolt Joints

For a joint consisting of $m$ bolts, if each bolt shares the external load uniformly, the load factor for a single bolt is still calculated using the formulas above, but the external load $F_A$ should be taken as the share borne by a single bolt.

If the load distribution is uneven, finite element analysis or detailed calculations are required to determine $F_A$ for each bolt.

6. Calculation Example

Given: - M10 bolt, compliance $\delta_S = 4.37 \times 10^{-6}\ \text{mm/N}$ - Steel clamped parts, compliance $\delta_P = 3.57 \times 10^{-7}\ \text{mm/N}$ - External load $F_A = 10,000\ \text{N}$, concentric loading, load introduction at the interface

Basic load factor:

Additional bolt force:

It can be seen that out of the 10 kN external load, the bolt only bears an additional approximately 755 N, while the remaining 9,245 N serves to release the compression of the clamped parts. This is the reason why "the bolt force is small."

If the load eccentricity $a = 20\ \text{mm}$ and $s_{sym} = 10\ \text{mm}$, a simplified estimation of the additional term would increase $\Phi_{en}^*$ to around 0.12~0.18, potentially doubling the additional bolt force.

7. Position in the VDI 2230 Procedure

• R3: Calculate $\delta_S, \delta_P$, obtain $\Phi^*$
• R4: Calculate the preload loss $F_Z$ due to embedding, using $\delta_S + \delta_P$ in the denominator
• R5: Determine the minimum assembly preload $F_{Mmin} = F_{Kerf} + (1 - \Phi^*)F_A + F_Z$
• R8: Calculate the maximum bolt force under operating conditions $F_{Smax} = F_{Mmax} + \Phi^* F_A$, perform stress verification
• R9: Calculate the stress amplitude $\sigma_a = \frac{\Phi^* F_A}{2 A_S}$, perform fatigue verification

An accurate $\Phi^*$ is a central pillar of the entire bolted joint design and analysis.

8. Important Notes

• Compliance unit consistency: $\delta_S$ and $\delta_P$ must use the same units (e.g., mm/N).
• Selection of load introduction factor $n$ significantly affects the result and should be strictly determined according to the charts or formulas in VDI 2230, avoiding arbitrary values.
• Nonlinear contact: When there are gaps or soft gaskets between the clamped parts, $\delta_P$ increases significantly, raising $\Phi^*$, which requires separate handling.
• High-temperature effects: Changes in the elastic modulus at high temperatures affect both $\delta_S$ and $\delta_P$, potentially changing $\Phi^*$, requiring re-evaluation.
• Experimental verification: For critical joints, load distribution can be measured using photoelasticity or strain gauges to verify the theoretical calculation.

Summary: The load factor $\Phi^*$ is a key parameter determining the additional bolt force and fatigue strength in bolted joint design. The basic formula is $\Phi^* = \delta_P/(\delta_S + \delta_P)$, and the general formula introduces the load introduction factor $n$ and eccentricity correction. Correct calculation of $\Phi^*$ requires accurate compliance data and reasonable assumptions about load introduction, making it an essential part of the systematic VDI 2230 design process.