VDI 2230 Bolt Compliance $\delta_S$

1. Definition and Significance

In the systematic calculation of bolted joints (VDI 2230-1), bolt compliance $\delta_S$ represents the elastic elongation of the bolt under a unit axial force (mm/N). It determines the bolt elongation under working load and the load distribution factor $\Phi^*$, forming the basis for subsequent calculations of preload changes, fatigue, and joint reliability.

Physical relationship:

where $\Delta l$ is the axial elongation (mm), and $F$ is the axial tensile force (N).

The reciprocal of compliance is stiffness $k_S = 1/\delta_S$.

2. Composition of Bolt Compliance

VDI 2230 divides the bolt into several characteristic segments, each with different cross-sectional areas and lengths. Compliance can be considered as the series superposition of each segment:

• $l_i$: Length of the $i$-th bolt segment (mm)
• $A_i$: Cross-sectional area of the $i$-th segment (mm²)
• $E_S$: Elastic modulus of the bolt material (MPa), typically $E_S = 206\,000\ \text{MPa}$ for steel bolts

Typical bolt segments include: bolt head, unthreaded shank, threaded engagement portion, free threaded portion, etc. Standard calculations usually simplify these into the following parts.

3. Compliance Calculation for Each Segment (VDI 2230 Standard Method)

3.1 Bolt Head Compliance $\delta_{SK}$

The bolt head also undergoes compressive deformation under load, and its compliance can be approximated as:

• $d$: Nominal bolt diameter (mm)
• $A_N$: Nominal stress area of the bolt (mm²), $A_N = \frac{\pi}{4} d^2$
• The factor 0.4 is an empirical value accounting for head deformation effects.

3.2 Unthreaded Shank Compliance $\delta_1$

For a smooth shank segment of length $l_1$ and diameter $d$:

3.3 Unengaged Threaded Shank Compliance $\delta_2$

For the threaded segment not engaged in a nut or tapped hole, its compliance is calculated based on the thread minor diameter $d_3$, but considering the elastic effect of the threads, the effective area can be taken as the stress area $A_S$ or the minor diameter area. VDI 2230 recommends:

where $d_3$ is the thread minor diameter (mm), obtainable from thread standards.

3.4 Engaged Thread Segment Compliance $\delta_3$

For the engaged portion within a nut or tapped hole, due to the elastic deformation of the thread flanks, its compliance is greater than that of an equivalent length of smooth shank. VDI 2230 provides:

Additionally, the compliance of the thread flanks $\delta_P$ should be superimposed, but $\delta_3$ often already partially includes this effect. Simplified engineering calculations typically compute the engaged length using the thread minor diameter area.

3.5 Total Bolt Compliance (Typical Joint)

In common bolted joints, the total compliance is the sum of the above segments. Furthermore, VDI 2230 also recommends considering the local compliance of the thread flanks (approximately 5% to 10% of total compliance for standard nuts), which can be added as an additional term:

4. Simplified Algorithm for Equivalent Bolt Compliance (for Preliminary Design)

When detailed segmentation is not possible, a simplified formula can be used:

• $l_K$: Total clamping length of the bolt (mm), i.e., total thickness of the clamped parts
• $A_S$: Bolt stress area (mm²), according to ISO 898-1 formula:
  $$A_S = \frac{\pi}{4} \left( \frac{d_2 + d_3}{2} \right)^2$$

where $d_2$ is the thread pitch diameter, and $d_3$ is the thread minor diameter. - The second term $0.4/(E_S d)$ is an approximation of the head compliance.

This formula is frequently used in preliminary selection (VDI 2230 R0).

5. Calculation Example

Given: M10 × 1.5 bolt, grade 8.8, clamping length $l_K = 50\ \text{mm}$, bolt material steel $E_S = 206\,000\ \text{MPa}$. - M10 coarse thread: $d_2 = 9.026\ \text{mm}$, $d_3 = 8.160\ \text{mm}$ - Stress area:

Simplified calculation:

The compliance is approximately $4.37 \times 10^{-6}\ \text{mm/N}$, meaning the bolt elongates by about 0.044 mm under an axial force of 10 000 N.

6. Important Notes

• Multi-material joints: If the bolt material is not steel (e.g., titanium alloy, superalloy), the elastic modulus $E_S$ must use the corresponding value.
• Temperature effects: At high temperatures, the elastic modulus decreases, increasing compliance. $E_S(T)$ should be used in calculations.
• Fine threads: The stress area $A_S$ for fine threads is slightly larger than for coarse threads, resulting in slightly lower compliance.
• Very long bolts: When the length-to-diameter ratio is large, the simplified compliance of the threaded segment may introduce significant errors; detailed segmentation is recommended.

7. Position in the VDI 2230 Calculation Chain

Bolt compliance $\delta_S$ is primarily used for: - R3: Together with joint compliance $\delta_P$, to calculate the load distribution factor $\Phi^* = \delta_P / (\delta_S + \delta_P)$ - R4/R5: To calculate embedding loss $\Delta F_Z = f_Z / (\delta_S + \delta_P)$ - R8: When evaluating working stress, in conjunction with preload and external load variations

Accurate compliance values are the cornerstone of the entire bolted joint design calculation.

Summary: VDI 2230 bolt compliance $\delta_S$ is calculated using the segment superposition method, with the core formula $\delta_S = \sum l_i/(E_S A_i)$, where the bolt head and threaded portions use empirical corrections. In engineering practice, the simplified formula $\delta_S \approx l_K/(E_S A_S) + 0.4/(E_S d)$ is commonly used for rapid assessment.