Bolted joint calculation per VDI 2230

Formulas and fundamentals

Spring model and load factor

The mechanical model behind VDI 2230 is a preloaded spring system: the bolt acts as a tension spring with resilience δS, the clamped parts as a compression spring with resilience δP. An axial working load FA is shared via the load factor Φ:

Only the small portion FSA loads the bolt additionally, the remainder relieves the clamped plates. The load introduction factor n accounts for how close to the interface the working load is applied.

Quick sizing

The quick sizing mode answers the most common question in practice: what assembly preload and tightening torque can a bolt take? During tightening, tensile stress and torsional stress from the thread torque act simultaneously. Both are combined into an equivalent stress using the von Mises criterion, which may reach 90 percent of the minimum yield strength. This yields the permissible assembly preload FMzul and, with MA = FMzul·(0.16·P + 0.58·µG·d2 + µK·DKm/2), the tightening torque. These formulas reproduce the well-known VDI 2230 table values (Table A1) to within 1 percent.

Rule of thumb for friction: roughly 85 to 90 percent of the tightening torque is consumed by thread and head friction, only 10 to 15 percent generates preload. The friction coefficient is therefore the dominant uncertainty of any bolt assembly. That is why the tool uses separate friction coefficients for the thread (µG), the head bearing surface (µK) and the interface (µT).

Resiliences and preload

In verification mode the resiliences are computed explicitly: δS as a series connection of head, shank, free loaded thread, engaged thread and nut or tapped-hole region; δP via the equivalent deformation cone of the clamped parts, distinguishing whether the compression cone can develop fully or splits into cone and sleeve. From δS and δP follow the load factor ΦK = δP/(δS + δP) and the additional bolt load.

After assembly the joint loses preload through embedding, the plastic flattening of roughness peaks. The embedding amount fZ depends on surface roughness, load type and the number of contact surfaces and is converted into a force loss via FZ = fZ/(δS + δP). The required minimum assembly preload is:

The scatter of the tightening method is captured by the tightening factor αA: the bolt must withstand FMmax = αA·FMmin, and the verification reads:

Further verifications and model limitations

Four further verifications are performed: the static working-stress check using the equivalent stress with half the assembly torsion retained, the fatigue check with the permissible stress amplitude σASV = 0.85·(150/d + 45) for bolts heat-treated before thread rolling and a required safety SD ≥ 1.2, the surface pressure under the head bearing area against the limiting surface pressure pG of the softest clamped material, and the slip safety for shear transfer by friction grip.

Deliberate simplifications compared with the full guideline: the tool covers concentrically clamped and concentrically loaded single-bolt joints without one-sided opening. Eccentric clamping, temperature effects, multi-layer stacks of different materials, fine-pitch threads and reduced-shank bolts are not included. Tightening factors, embedding amounts and limiting surface pressures are tabulated guide values, and the chamfer diameter of the head bearing area is simplified to the clearance hole. For verifications that must be documented, the calculation per the full VDI 2230 Part 1 remains authoritative.

Calculating the tightening torque

The tightening torque MA is the torque applied at the wrench in order to generate a given assembly preload. It consists of three parts: the pitch component of the thread, the thread friction and the friction under the head bearing surface.

The VDI 2230 equation reads MA = FM·(0.16·P + 0.58·µG·d2 + µK·DKm/2), with the thread pitch P, the pitch diameter d2, the mean bearing diameter DKm and the friction coefficients µG in the thread and µK at the head. Substituting the permissible assembly preload FMzul for FM yields the tabulated tightening torque of the property class.

Friction is decisive: only about 10 to 15 percent of the torque generates preload, the rest is turned into heat in the thread and under the head. A tightening torque without the corresponding friction coefficient is therefore of little value. The calculator outputs MA in quick sizing mode for M4 to M36 and classes 8.8, 10.9 and 12.9, always for the friction coefficient set.

Calculating the preload

The preload FM is the tensile force in the bolt after assembly and thus the force that holds the interface together. It is the actual target quantity of any bolted joint; the tightening torque is merely the means of generating it indirectly.

The permissible assembly preload FMzul follows from the yield strength of the bolt. During tightening the tensile stress from FM and the torsional stress from the thread torque act simultaneously; both are combined into an equivalent stress using the von Mises criterion, which may utilise 90 percent of the minimum yield strength. The governing cross section is the smaller of the thread stress area and the shank area.

In verification mode the required preload is additionally determined from below: FMmin = FKerf + (1 − Φn)·FAmax + FZ covers the required clamp load, the preload drop caused by the working load and the embedding loss. With the tightening factor αA this gives FMmax = αA·FMmin, and the verification reads FMzul ≥ FMmax.

Worked example

Reference example, shear-loaded joint: a bracket transfers a static transverse force of 3 kN per bolt by friction grip (µT = 0.12, required slip safety 1.2). Selected is an M12 hexagon bolt as a tapped-hole joint in C45 steel, clamp length 30 mm (20 mm of which is shank), tightened with a torque wrench (αA = 1.6), µG = µK = 0.12.

The required clamp load is FKQerf = 1.2·3000/0.12 = 30,000 N. With the resiliences δS = 2.26·10⁻⁶ mm/N and δP = 0.33·10⁻⁶ mm/N and an embedding amount of 10 µm (shear, Rz 16), the embedding loss is about 3,870 N. This gives FMmin = 33,870 N and FMmax = 1.6·FMmin = 54,190 N.

An M12 of class 8.8 is not sufficient: its permissible assembly preload of 43.1 kN is below FMmax. Only class 10.9 with FMzul = 63.3 kN passes the verification; the corresponding tightening torque is 123 Nm. The case illustrates a typical eye-opener: transferring shear by friction grip consumes enormous preload and is a frequent cause of undersized joints.

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