VDI 2230 Deformation Diagram Data

1. Definition and Function of the Deformation Diagram

The deformation diagram (force-deformation diagram) is the core visualization tool of the systematic bolted joint calculation according to VDI 2230. It is based on the elastic compliance curves of the bolt and the clamped parts, and presents key physical quantities such as assembly preload, working load, residual clamping force, and preload loss in a single two-dimensional diagram.

Using the deformation diagram, the designer can intuitively assess: - Whether the joint interface separates under working load - Whether the bolt is overloaded - Whether sealing/anti-slip requirements are still met after preload loss - The influence of tightening method and friction scatter on joint reliability

2. Basic Elements of the Deformation Diagram

The deformation diagram uses force $F$ as the ordinate and axial displacement $f$ as the abscissa. The diagram contains two characteristic lines:

• Bolt characteristic line: slope $k_S = 1/\delta_S$, representing the force-elongation relationship of the bolt under tension
• Clamped parts characteristic line: slope $k_P = 1/\delta_P$, representing the force-compression relationship of the clamped parts under compression

The two lines intersect at the assembly point, forming a closed force-displacement triangle.

3. Calculation of Key Point Coordinates in the Deformation Diagram

3.1 Bolt Characteristic Line

Bolt compliance $\delta_S$, its force-deformation relationship is:

Usually taking the assembly preload $F_M$ as the reference point:

The bolt characteristic line passes through the origin $(0, 0)$ and the point $(\delta_S F_M, F_M)$.

3.2 Clamped Parts Characteristic Line

Clamped parts compliance $\delta_P$, the compression under assembly preload $F_M$ is:

The clamped parts characteristic line passes through the origin $(0, 0)$ and the point $(-\delta_P F_M, F_M)$ (when displacement in the compression direction is taken as positive, the clamped parts displacement is negative or the diagram is reversed).

In the deformation diagram, the two lines are usually drawn symmetrically on either side of the vertical axis, forming a "butterfly-shaped" force-displacement diagram.

3.3 Assembly Point A

Under assembly preload $F_M$, the bolt elongation is $f_{SM} = \delta_S F_M$, and the clamped parts compression is $f_{PM} = \delta_P F_M$.

3.4 Point After Application of Working Load $F_A$

After the working load is applied, the bolt force increases to $F_S = F_M + \Phi^* F_A$, and the residual clamping force on the clamped parts decreases to $F_{res} = F_M - (1 - \Phi^*)F_A$.

The bolt displacement increases by $\Delta f_S = \delta_S \cdot \Phi^* F_A$, and the clamped parts compression decreases by $\Delta f_P = \delta_P \cdot (1 - \Phi^*)F_A$.

Due to displacement compatibility, $\Delta f_S = \Delta f_P$, which is the geometric origin for deriving $\Phi^* = \delta_P/(\delta_S + \delta_P)$.

3.5 Point After Preload Loss $F_Z$

Due to embedding settlement, thermal effects, etc., after a preload loss $F_Z$, the effective preload decreases to $F_{Meff} = F_M - F_Z$. Correspondingly, the bolt elongation decreases and the clamped parts compression also decreases.

4. Complete Data Point Calculation for the Deformation Diagram

4.1 Given Conditions

• Bolt compliance $\delta_S$ (mm/N)
• Clamped parts compliance $\delta_P$ (mm/N)
• Assembly preload $F_M$ (N)
• External load $F_A$ (N)
• Load factor $\Phi^*$
• Preload loss $F_Z$ (N)

4.2 Key Point Coordinates

4.3 Safety Margin Calculation

• Separation safety margin: $F_{res} - F_{Kerf}$, where $F_{Kerf}$ is the required minimum residual clamping force
• Bolt strength margin: ratio of $F_{S,max}$ to the bolt yield load $F_{0.2} = R_{p0.2} \cdot A_S$
• Critical joint separation load: $F_{A,crit} = \dfrac{F_M}{(1 - \Phi^*)}$, when $F_A$ exceeds this value, the joint interface opens

5. Calculation Example

Given: - $\delta_S = 4.37 \times 10^{-6}\ \text{mm/N}$ - $\delta_P = 3.57 \times 10^{-7}\ \text{mm/N}$ - $F_M = 20,000\ \text{N}$ - $F_A = 10,000\ \text{N}$ - $\Phi^* = 0.0755$ - $F_Z = 2,000\ \text{N}$

Point Calculations:

• Assembly point A: $f_S = 4.37\times10^{-6} \times 20,000 = 0.0874\ \text{mm}$, $F = 20,000\ \text{N}$

• Bolt working point B: $F_S = 20,000 + 0.0755 \times 10,000 = 20,755\ \text{N}$ $f_S = 4.37\times10^{-6} \times 20,755 \approx 0.0907\ \text{mm}$

• Clamped parts working point C: $F_{res} = 20,000 - (1 - 0.0755) \times 10,000 = 20,000 - 9,245 = 10,755\ \text{N}$ $f_P = 3.57\times10^{-7} \times 10,755 \approx 0.00384\ \text{mm}$

• Point after loss A': $F_{Meff} = 20,000 - 2,000 = 18,000\ \text{N}$ $f_S = 4.37\times10^{-6} \times 18,000 \approx 0.0787\ \text{mm}$

Deformation Diagram Interpretation: - Critical joint separation load $F_{A,crit} = 20,000 / (1 - 0.0755) \approx 21,630\ \text{N}$, current $F_A = 10,000$, safety margin is sufficient. - Maximum bolt force $20,755\ \text{N}$, well below the proof load of 37,400 N for M10 grade 8.8, strength margin is large. - Residual clamping force after relaxation is 18,000 N, if $F_{Kerf} = 15,000\ \text{N}$, it is still acceptable.

6. Position of the Deformation Diagram in the VDI 2230 Process

• R3: Calculate $\delta_S, \delta_P$, establish the deformation diagram framework
• R4: Calculate $F_Z$, update the deformation diagram after loss
• R5: Back-calculate the required minimum $F_{Mmin}$ from $F_{Kerf}$
• R6: Considering tightening scatter $\alpha_A$, determine the maximum $F_{Mmax}$
• R7/R8: Check bolt stress on the deformation diagram

The deformation diagram runs through the entire VDI 2230 calculation process and is the core tool for understanding the mechanical behavior of the joint and verifying reliability.

Summary: The VDI 2230 deformation diagram, based on the compliance of the bolt and clamped parts, visualizes preload, working load, load factor, and preload loss in a unified manner. Key point coordinates are calculated from $\delta_S F$ and $\delta_P F$, and the diagram allows intuitive assessment of joint separation, bolt overload, and safety status after relaxation.