Using the VDI 2230 load factor to budget preload loss

The most useful number in joint design

Ask an engineer for a joint's preload and you will get an answer. Ask what a micrometre of relaxation costs that joint and you usually will not, even though it is a two-line calculation and it is the number that decides whether the joint survives.

This article is that calculation, worked end to end.

Reference joint used throughout

Every figure in this article refers to the same reference joint, so numbers are comparable across articles and against your own calculations.

Stiffnesses are calculated to VDI 2230 Sheet 1 using the standard cone-of-compression method.

Step 1: bolt stiffness

The bolt is a series of cylindrical sections. Its stiffness is the series combination:

1/k_S = (1/E) · [ l_head/A_N + l_shank/A_shank + l_thread/A_3 + l_engaged/A_3 + l_nut/A_N ]

For M16 × 2,0 class 10.9, 48 mm clamp length, steel at E = 205 GPa:

k_S = 1/δ_S = 1,04 × 10⁹ N/m

Step 2: member stiffness

VDI 2230 models the compressed members as a truncated cone of material around the bolt. For a joint where the clamped diameter comfortably exceeds the cone, the compliance is:

δ_P = (2 / (π · E_P · D_h · tan φ)) · ln[ ((d_W + D_h)(d_W + D_h + 2·l_K·tan φ)) /
                                          ((d_W - D_h)(d_W - D_h + 2·l_K·tan φ)) ]

With d_W = 24,0 mm bearing diameter, D_h = 17,0 mm hole, l_K = 48 mm clamp length, and a cone half-angle φ giving tan φ ≈ 0,362 for this geometry:

δ_P = 1,75 × 10⁻¹⁰ m/N
k_P = 5,71 × 10⁹ N/m

The members are about 5,5 times stiffer than the bolt. That ratio is typical for a steel joint and it is the reason bolted joints work at all.

Step 3: the load factor

Φ = k_S / (k_S + k_P)
  = 1,04 / (1,04 + 5,71)
  = 0,154

Φ tells you what fraction of an external axial load appears as additional bolt tension. At Φ = 0,154, an external load of 10 kN raises bolt tension by only 1,54 kN. This is the classic and correct use of the load factor, and it is why preloaded joints are fatigue-tolerant.

Step 4: the number that actually matters

Now use the same stiffnesses in the other direction. When the stack gets shorter by δ, bolt and members both relax. The clamp force lost is:

ΔF_V = δ · (k_S · k_P) / (k_S + k_P)
     = δ · k_S · (1 − Φ)

For this joint:

ΔF_V/δ = 1,04 × 10⁹ × (1 − 0,154)
       = 0,884 × 10⁹ N/m
       = 0,884 kN per micrometre

0,884 kN per micrometre. From an assembly preload of 70,0 kN, that means 79 micrometres of total stack shortening takes the joint to zero.

Step 5: use it as a design gate

Then budget your relaxation sources and compare.

Any two of those together and the joint is below the DIN 25201-4 Annex B threshold on relaxation alone, before a single vibration cycle.

Making the joint less sensitive

The sensitivity is k_S · (1 − Φ). Reduce it and every micrometre costs less.

Longer grip. Doubling clamp length roughly halves k_S. This is the strongest lever available and it is why aerospace joints use long, thin bolts wherever geometry allows.

Reduced-shank bolts. Waisting the shank to the thread root diameter lowers k_S without changing grip. Standard practice in high-cycle joints.

Lower-modulus washers. Adds compliance in series. Effective, at the cost of stability.

Belleville stacks. The deliberate version of the above, adding a large, controlled compliance. Reduces sensitivity substantially and introduces its own force-versus-travel behaviour. See Belleville alternative.

Each of these makes the loss cheaper. None removes it. A joint at 30 micrometres of shortening with a halved k_S is at 86,6 percent instead of 73,1 percent. Whether that clears your required minimum clamp force is the question step 6 answers.

Removing the shortening instead

The other route is to hold clamp length constant by supplying the lost length back. Against a 0,884 kN per micrometre sensitivity, a 0,50 mm take-up reserve is worth 442 kN of cumulative clamp force recovery, which is six times the assembly preload of this joint. That is the arithmetic behind The self-tightening washer.

Spreadsheet checklist

1. Compute δ_S section by section. Do not use nominal shank area for the threaded length; use A_3, the minor diameter area.
2. Compute δ_P from the cone model, or take it from FE if your geometry is not cone-like.
3. Φ = k_S/(k_S + k_P).
4. Sensitivity = k_S · (1 − Φ), in kN per micrometre.
5. Sum your relaxation budget from surface finish, coatings, gaskets and thermal cycling.
6. Multiply, subtract from assembly preload, and compare against required minimum clamp force.
7. If step 6 fails, decide explicitly whether to reduce sensitivity, reduce the budget, or supply take-up.

Step 7 is a design decision. Most joints reach it by accident instead.

Full derivation and worked cases in The ISOKLAMP technical report.

Frequently asked questions

What is the VDI 2230 load factor?

The load factor Φ = k_S/(k_S + k_P) is the ratio of bolt stiffness to total joint stiffness. It expresses the fraction of an external axial load that appears as additional bolt tension. For an M16 × 2,0 class 10.9 joint at 48 mm grip, Φ = 0,154, so a 10 kN external load raises bolt tension by only 1,54 kN.

How do you calculate clamp force lost per micrometre?

Multiply bolt stiffness by one minus the load factor: ΔF_V/δ = k_S · (1 − Φ). For the reference joint that is 1,04 × 10⁹ N/m × 0,846 = 0,884 kN per micrometre. From a 70 kN assembly preload, 79 micrometres of total stack shortening opens the joint completely.

What bolt stiffness should I use for a threaded section?

Use A_3, the minor diameter stress area, not the nominal shank area. For M16 × 2,0 that is 144,1 mm² against 201,1 mm² nominal. Using nominal area for the threaded length underestimates bolt compliance by roughly 40 percent on that section and makes the joint look less sensitive to relaxation than it is.

How do I make a joint less sensitive to relaxation?

Reduce k_S · (1 − Φ). Longer grip is the strongest lever, since doubling clamp length roughly halves bolt stiffness. Reduced-shank or waisted bolts lower stiffness without changing grip. Lower-modulus or Belleville washers add compliance in series. All of these make each micrometre cheaper without removing the loss.

How much relaxation can a typical joint tolerate?

For the reference joint, 22 micrometres takes it to the DIN 25201-4 Annex B threshold of 80 percent residual. Six machined steel interfaces budget 20 to 35 micrometres of embedment alone. Add a zinc flake coating at 16 to 36 micrometres or a 3 mm PTFE gasket at 20 to 60 and the joint is below threshold on relaxation before any vibration is applied.

Take it further

• The ISOKLAMP technical report covers the full derivation, the geometry, and every dataset behind these figures.
• Full residual clamp force dataset gives the residual clamp force numbers for ten securing methods, with sources.
• Design partner programme is open. Eight slots, two per sector. Bring us a joint that keeps failing and we will run a VDI 2230 Sheet 1 analysis on it.

Engineering questions go to engineering@isoklamp.com. An engineer answers, not a form.

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