🧮 在线计算器：Bolt Pre-selection Calculator — Fast sizing with the Kübler equation; supports a design mode and a check mode.

Introduction: the rubber-band-and-sponge model

To fully understand the physical logic behind the “bearing-pressure check”, we can picture the bolted-joint system as “a stretched rubber band (the bolt) clamping a sponge (the clamped part)”.

The heart of this part is to reveal a key question: why, when a part’s surface is crushed just a little bit, does the whole bolted joint face the risk of complete failure?

1. From microscopic crushing to macroscopic “embedding” (Setzen)

1. How does bearing pressure (Flächenpressung) arise?

When a bolt is tightened, it is stretched, producing a large axial tension (preload). This tension must be transmitted to the clamped part through the underside of the bolt head or nut (the bearing surface).

By the pressure formula $p = \frac{F}{A_p}$ (pressure = force ÷ contact area), because the effective bearing area under the bolt head ($A_p$) is usually small, the part surface in this annular region carries an extremely concentrated, large pressure.

2. The physical mechanism of microscopic crushing and plastic embedding

Any machined part surface, however smooth it looks, is made up of uneven “peaks and valleys” (surface roughness) at the microscopic level.

  • Microscopic crushing: when the bearing pressure exceeds the “crushing limit” (Quetschgrenze), or limiting surface pressure (Grenzflächenpressung $p_G$), of the base material, these tiny metal “peaks” cannot bear the heavy load and are plastically flattened (plastisches Einebnen von Oberflächenrauhigkeiten).
  • Macroscopic creep (Kriechen): if the pressure is badly over the limit, not only is the microscopic roughness flattened, but even the macroscopic metal base under the bolt head undergoes time-dependent yield flow (material creep), pressing a visible shallow dent into the part surface.

In engineering, the permanent thinning of a part caused by this crushing and flattening of the surface is collectively called “embedding” (Setzen).

2. The fatal chain reaction: why does embedding cause the system to collapse?

Understanding this step is the key to understanding bolt failure. We need to bring in a core mechanics concept we have stressed repeatedly in earlier tutorials: the bolt itself is an extremely stiff spring.

1. Why does “embedding” cause an irreversible loss of preload (Vorspannkraftverlust)?

  • Elastic recoil: when the stiff bolt is tightened, it is stretched by even just a few tenths of a millimetre. If the part beneath it thins due to “embedding” (say the embedding amount is $f_Z$), the originally stretched bolt, like a spring, immediately recoils elastically by the same distance to fill the gap.
  • Sudden drop in tension: by Hooke’s law, when the spring’s stretch shrinks, its tension must drop with it. So the microscopic crushing of the part surface immediately turns into a geometric recoil of the bolt, finally appearing as an irreversible loss of preload (i.e. clamping force) ($F_Z$).
  • Quantified calculation: this mechanical relation can be computed precisely by the formula $F_Z = \frac{f_Z}{\delta_S + \delta_P}$, where $\delta_S$ and $\delta_P$ are the resiliences of the bolt and the part.

2. The final consequence: loosening and fatigue fracture (Lockern und Versagen)

Under a dynamic alternating load (such as high-frequency vibration or alternating tension-compression), the bolted joint must always keep enough residual clamping force to lock the parts tightly.

  • Loosening (Lockern): if the preload loss from embedding is too large, the residual clamping force will not be able to press the two parts firmly together.
  • Failure progression: once the clamping force is badly insufficient, the parts slide relative to each other under a transverse external force (fretting). This not only causes the thread to self-loosen (selbsttätiges Losdrehen), but also makes the bolt itself bear fatal alternating bending and shear stresses. In the end, this easily triggers the most dreaded failure mode in mechanical engineering — bolt fatigue fracture (Dauerbruch).

As we can see, “a dent pressed into the surface → the bolt stress recoils → the preload disappears → the bolt breaks by vibration” is an extremely vicious domino effect.

3. Engineering response: how to run the bearing-pressure check

To break the vicious cycle above, a preliminary bearing-pressure check (Überprüfung der Flächenpressung) must be done in the design-draft stage (Vorauslegung) of the bolted joint or during an operation assessment.

In engineering practice, a simplified approximate formula is usually used for the preliminary assessment, with a precise check done later in the detailed VDI 2230 verification (the standard’s step R10). Below are the detailed steps and the underlying mechanical logic:

1. The core preliminary check formula

In the preliminary design stage, the following calculation-engine approximate formula can be used to check whether the actual pressure $p$ is less than or equal to the material’s limiting surface pressure $p_G$:

$$p \approx \frac{F_{sp} / 0,9}{A_p} \le p_G$$

The derivation logic of the formula: In the actual working state, the maximum total tension $F_{Smax}$ the bolt carries equals the assembly preload $F_{sp}$ plus the additional bolt tension caused by the external working load (i.e. $\Phi \cdot F_B$). To simplify the calculation in the early sizing stage, engineers usually use $F_{sp} / 0,9$ to roughly and safely estimate the maximum total axial tension the bolt may bear under extreme conditions (the implied assumption: the additional working load takes up at most this remaining 10% of the load space).

2. Calculation and values of the key parameters

  • $F_{sp}$ (peak assembly force): This is the maximum assembly clamping force of the bolt material at 90% yield-strength utilization. In actual engineering calculations, this value can be read directly from a standard data table (such as Table TB 8-14) by the “bolt property class”, “thread size” and “estimated thread friction coefficient”.
  • $A_p$ (effective bearing area): The effective bearing area is the annular area of real contact between the bolt head or nut and the pressed part surface. Its formula: $$A_p \approx \frac{\pi}{4}(d_w^2 - d_h^2)$$ where $d_w$ is the outer contact-circle diameter under the bolt head or nut (such as the across-flats size of a hexagon head, or the minimum of the flange-face outer diameter), and $d_h$ is the through-hole diameter in the flange (usually taking the medium fit-hole diameter per DIN EN 20273).
  • $p_G$ (limiting pressure / Grenzflächenpressung): This is set by the clamped-part material — the allowable pressure limit the material can bear without severe creep or crushing. This value usually needs a table lookup (such as Table TB 8-10b). For example, the limiting surface pressure of steel is usually far above that of cast iron, which in turn is far above that of light aluminium alloy.

3. A special correction for high-precision tightening methods

If you use a modern high-precision yield-point-controlled (streckgrenzgesteuert) or angle-controlled (drehwinkelgesteuert) method, the bolt is deliberately tightened at assembly to 100% of the material yield strength, or even into the plastic region; the conservative “divide by 0.9” formula can no longer be used.

In this advanced case, because the maximum assembly force rises markedly, the preliminary check formula must be corrected as follows:

$$p = 1,4 \cdot \frac{F_{sp}}{A_p} \le p_G$$

The deep origin of the factor 1.4: this is not an arbitrary safety factor but the combined product of three physical variables:

  1. The statistical error of the ratio of the material’s actual maximum yield strength to the nominal minimum yield strength ($\approx 1.2$)
  2. The effect of the reciprocal of the yield-utilization bound ($1 / 0.9 \approx 1.11$)
  3. The strengthening from work hardening of the material in the plastic region ($\approx 1.05$)

4. The precise check in VDI 2230 (step R10)

If your preliminary estimate passes and you move into the rigorous, systematic VDI 2230 calculation (the standard’s step R10), the pressure check is further split strictly into two extreme cases, “assembly state” and “service state”, for precise verification:

  1. Assembly state (Montagezustand): Here only the extreme point of the assembly stage is considered: the allowable maximum assembly preload $F_{Mzul}$ and the worst-case minimum possible contact area $A_{p\min}$. $$p_{M\max} = \frac{F_{Mzul}}{A_{p\min}} \le p_G$$
  2. Service state (Betriebszustand): Here all the adverse superpositions of the later working environment are included: the maximum residual preload $F_{V\max}$, the maximum external additional bolt working load $F_{SA\max}$, and the thermal-deformation force $\Delta F_{Vth}$ from possible temperature-load environmental decay are precisely combined. $$p_{B\max} = \frac{F_{V\max} + F_{SA\max} - \Delta F_{Vth}}{A_{p\min}} \le p_G$$

4. What to do if the pressure is over the limit?

If the actual pressure computed in the design stage is greater than the limiting pressure of the base material ($p > p_G$), it means the flange surface is at high risk of crushing and will most likely cause a preload loss.

As an engineer, you must never force ahead in this situation. Starting from the mechanical essence, take a combination of the following three optimization measures:

  1. Enlarge the contact area (i.e. enlarge the denominator, increase $A_p$): Use a flanged hexagon bolt/nut, or switch to a special bolt type with no undercut groove under the head to increase the contact area.
  2. Add a hard, thick washer (Unterlegscheiben): We cannot press directly onto a soft base. Add a specially heat-treated, very hard, thick washer between the bolt head and the soft base to “spread the pressure” (for a washer of thickness $h_S$, the equivalent spread outer diameter on its back can usually be estimated by the mechanical-cone approximation as $d_{Wa} = d_W + 1,6 \cdot h_S$).
  3. Change the base material (i.e. enlarge the numerator, increase $p_G$): This is also the most thorough way: switch to a clamped-part material with a higher yield strength and a harder compressive capacity (such as from aluminium alloy to reinforced carbon-steel plate).