VDI 2230 Compliance of Clamped Parts $\delta_P$

1. Definition and Physical Significance

The compliance of clamped parts $\delta_P$ represents the axial elastic compression (mm/N) of the clamped components under a unit bolt preload. In VDI 2230-1, $\delta_P$ together with the bolt compliance $\delta_S$ determines:

• Load distribution factor $\Phi^* = \dfrac{\delta_P}{\delta_S + \delta_P}$
• Preload loss due to embedding $F_Z = \dfrac{f_Z}{\delta_S + \delta_P}$
• Additional bolt force under working load $F_{SA} = \Phi^* \cdot F_A$

The compliance calculation for clamped parts is significantly more complex than for bolts because compressive stress is not uniformly distributed over the entire cross-section of the clamped parts, but spreads conically from the bolt head/nut.

2. Deformation Cone Model (Basic Assumptions)

VDI 2230 employs the Deformation Cone model to estimate the equivalent compliance of clamped parts. It assumes that compressive stress is uniformly distributed within a virtual cone with an apex angle of $2\varphi$, while material outside the cone does not participate in load bearing.

Standard recommendations: - For rigid materials such as steel and cast iron, the half-apex angle of the deformation cone $\varphi \approx 30^\circ$, i.e., $\tan\varphi \approx 0.577$ - For softer materials such as aluminum alloys, $\varphi$ is appropriately reduced

3. Basic Compliance Formula (Standard Case)

For the ideal case of a circular cross-section, bolt axis centered, constant clamping length $l_K$, the compliance of clamped parts is:

Where: - $E_P$: Elastic modulus of the clamped part material (MPa); for multiple layers of different materials, calculate segmentally - $d_h$: Bolt hole diameter (mm), typically nominal bolt diameter + 1 mm (clearance hole) - $d_w$: Outer diameter of the bolt head/nut bearing surface (mm); for hexagon heads, $d_w \approx 0.95 \times s$ ($s$ = width across flats) - $l_K$: Clamping length (mm), i.e., total thickness of clamped parts - $\varphi$: Half-apex angle of deformation cone; for steel, $30^\circ$ ($\tan30^\circ \approx 0.577$)

4. Simplified Substitute Cone Method ($\omega$ Method)

When the logarithmic formula above is inconvenient, VDI 2230 also recommends using an equivalent substitute cylinder to approximate compliance. Define the equivalent cross-sectional area $A_{ers}$ such that:

The substitute cross-sectional area $A_{ers}$ is determined by:

(Applicable under the "infinite" flat plate assumption where the deformation cone does not reach the component edge)

If the actual component width $D_{limit}$ restricts cone expansion, a correction using the actual component boundary with a reduction factor should be applied.

5. Correction for Eccentric Clamping and Eccentric Loading

When the bolt is not at the symmetry center of the clamped parts, or when the load acts eccentrically, the compliance of the clamped parts increases. VDI 2230 introduces the eccentric substitute compliance $\delta_P^*$ to replace the basic compliance:

Where: - $s_{sym}$: Load eccentricity (mm), i.e., distance from the working load line of action to the bolt axis - $I_{Bers}$: Area moment of inertia of the substitute cross-section of clamped parts (mm⁴), calculated based on the equivalent cylinder diameter

This correction implies that under eccentric loading, the clamped parts undergo not only uniform compression but also bending, increasing compliance. This causes $\Phi^*$ to rise, increasing the additional force on the bolt, which is detrimental to fatigue.

6. Multi-Layer Material Connections

When clamped parts consist of multiple layers of different materials/thicknesses, compliance must be calculated layer by layer and then summed. For each layer, if the deformation cone does not exceed the layer thickness:

Total compliance:

If the deformation cone penetrates through one layer into the next, integration must be limited to the actual $l_i$ of that layer.

7. Calculation Example

Given: - M10 bolt, hexagon head width across flats $s = 16\ \text{mm}$, $d_w \approx 0.95 \times 16 = 15.2\ \text{mm}$ - Bolt hole $d_h = 11\ \text{mm}$ - Clamped parts: two steel plates, total clamping length $l_K = 40\ \text{mm}$, material steel $E_P = 206\,000\ \text{MPa}$ - $\varphi = 30^\circ$, $\tan\varphi = 0.577$

Step 1: Substitute into compliance formula

Step 2: Calculate term by term

• Denominator constant: $206\,000 \cdot \pi \cdot 11 \cdot 0.577 \approx 206\,000 \cdot 19.95 \approx 4.11 \times 10^6$
• Deformation cone outer diameter: $d_w + 2 l_K \tan\varphi = 15.2 + 2 \times 40 \times 0.577 = 15.2 + 46.16 = 61.36\ \text{mm}$
• First term in numerator: $(61.36 - 11)(15.2 + 11) = 50.36 \times 26.2 \approx 1319.4$
• First term in denominator: $(61.36 + 11)(15.2 - 11) = 72.36 \times 4.2 \approx 303.9$
• Ratio: $1319.4 / 303.9 \approx 4.341$
• $\ln(4.341) \approx 1.468$

Step 3: Final compliance

Interpretation: The compliance of these clamped parts is approximately $3.57 \times 10^{-7}\ \text{mm/N}$, meaning that under 10 000 N preload, the clamped parts compress by about 0.0036 mm. This is one order of magnitude smaller than the bolt compliance $4.37 \times 10^{-6}$, indicating that the clamped parts stiffness is much greater than the bolt stiffness—a typical characteristic of most designs.

8. Position in the VDI 2230 Calculation Chain

The compliance of clamped parts $\delta_P$ is primarily used in: - R3: Together with bolt compliance $\delta_S$ to calculate the load distribution factor $\Phi^*$ - R4: To calculate preload loss due to embedding settlement $F_Z$ - R5: To determine the minimum assembly preload $F_{Mmin}$ - R8/R9: To evaluate working stress and fatigue safety

Accurate $\delta_P$ is fundamental to the reliability of bolted joint design. When the geometry of clamped parts is complex (e.g., thin plates, ribs, non-axisymmetric), it is recommended to use finite element analysis to obtain more precise compliance values.

Summary: The compliance of clamped parts $\delta_P$ in VDI 2230 is calculated based on the deformation cone model, with the core being the logarithmic expression or the substitute cylinder method. The constant $\tan\varphi \approx 0.577$ is key in the formula. Eccentric loading increases compliance and requires correction via $\delta_P^*$. Correct compliance values are prerequisites for accurately calculating preload loss and load distribution.