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

Preliminary sizing of a bolted joint: from load to size, fast

Before getting into complex system-level bolted-joint calculations, the first challenge an engineer faces is often: “for a given working load, how large should the bolt be? Is M8 enough, or M12?” A sound preliminary design / pre-selection (German: Vorauslegung) quickly fixes a suitable nominal diameter and does an early “sweep” for common failure modes (such as fatigue fracture, or loss of preload from crushing of the bearing surface). This article shows how to do a quick pre-selection from the working load, and how to check the feasibility of the design with simplified formulas.

1. Quick pre-selection list

If you only need a quick estimate with no calculation, go straight to the standalone article:

📋 A minimal preliminary estimation table for bolt size — read the recommended nominal diameter and property class directly from the load range, a result in seconds.

Section 2 onward covers the more precise Kübler-equation method.

2. Strength-area estimation: the Kübler equation (German: Gleichung von Kübler)

The Kübler equation is used mainly for a more precise preliminary design / pre-selection (German: Vorauslegung) of a bolted joint. By making reasonable assumptions on the safe side, it computes the minimum required thread stress cross-section (German: Spannungsquerschnitt des Gewindes $A_{\mathrm{s}}$) or waist cross-section (German: Taillenquerschnitt $A_{\mathrm{T}}$) of the bolt, so that the joint will not be pulled apart and the preload will not be lost, helping the designer quickly fix a reasonable bolt size.

The Kübler equation is written as:

$$ A_{\mathrm{s}} \text{ or } A_{\mathrm{T}} \geq \dfrac{F_{\mathrm{B}} + F_{\mathrm{Kl}}}{\dfrac{R_{\mathrm{p}0,2}}{\kappa \cdot k_{\mathrm{A}}} - \beta \cdot E_{\mathrm{S}} \cdot \dfrac{f_{\mathrm{Z}}}{l_{\mathrm{k}}}} $$

The physical meaning, definition and value guidance of each parameter:

  • $A_{\mathrm{s}}$ or $A_{\mathrm{T}}$: the target value the preliminary design / pre-selection must find, used to back out the minimum required bolt size.

    1. $A_{\mathrm{s}}$ (thread stress cross-section): used to evaluate an ordinary bolt or a fully threaded bolt. The actual load-bearing cross-section of the thread lies between the thread pitch diameter ($d_2$) (German: Flankendurchmesser) and the thread minor diameter ($d_3$) (German: Kerndurchmesser). It is the equivalent area of the weakest failure section of the thread, with the standard formula: $$ A_s = \frac{\pi}{4} \left( \frac{d_2 + d_3}{2} \right)^2 $$ Note: in real engineering you usually do not compute this value by hand. After finding $A_{\mathrm{s}}$, the designer looks up a standard thread table directly and picks the nearest larger standard thread size.
    2. $A_{\mathrm{T}}$ (waist cross-section, German: Taillenquerschnitt): for a reduced-shank bolt (German: Dehnschrauben) used for fatigue resistance or thermal-expansion compensation, the diameter of the reduced shank $d_{\mathrm{T}}$ is deliberately machined smaller than the thread minor diameter (usually $d_{\mathrm{T}} \approx 0.9 d_3$); the weakest section of the whole connection is then no longer the thread but the shank. The formula is: $$ A_{\mathrm{T}} = \frac{\pi}{4} d_{\mathrm{T}}^2 $$

  • $F_{\mathrm{B}}$: the axial working load the bolt carries (German: axiale Betriebskraft). This value is usually given directly by the overall structural design requirements.

  • $F_{\mathrm{Kl}}$: the required clamping force of the flange or clamped parts (German: geforderte Klemmkraft). The minimum clamping force the structure needs to prevent the interface from separating, slipping or leaking — also set by the overall structure.

  • $R_{\mathrm{p}0,2}$ (0.2% yield strength): its value is set directly by the bolt property class (German: Festigkeitsklasse) the designer selects (per DIN EN ISO 898-1).

    • Class 8.8: $R_{p0,2}$ is usually taken as $640 \text{ N/mm}^2$.
    • Class 10.9: $R_{p0,2} = 10 \times 9 \times 10 = 900 \text{ N/mm}^2$.
    • Class 12.9: $R_{p0,2} = 12 \times 9 \times 10 = 1080 \text{ N/mm}^2$.

  • $k_{\mathrm{A}}$ (tightening factor, German: Anziehfaktor): defined as the ratio of the maximum to the minimum preload that can occur at assembly ($F_{Mmax} / F_{Mmin}$), covering friction-coefficient scatter, tool error and operator variation. Its value depends on the tightening method:

    • Impact wrench (Schlagschrauber): very large scatter, $k_A = 2.5 \sim 4.0$, usually not recommended for demanding bolted joints.
    • Torque-controlled tightening (drehmomentgesteuertes Anziehen): with an ordinary torque wrench, the range is generally $1.6 \sim 2.0$. With a precision torque wrench that signals, it can be narrowed to $1.4 \sim 1.8$. Take the smaller value (such as 1.6) when the parts are relatively rigid (small tightening angle); take the larger value when the parts are softer (large tightening angle) or when a blind hole has higher hardness.
    • Yield-point-controlled tightening (streckgrenzgesteuertes Anziehen): simply take $k_A = 1.0$. Because this method tightens the bolt into the over-elastic (yield) region, the material’s own yield point acts as a “safety valve”, almost removing the effect of friction-coefficient variation, so the overload risk is very low.

  • $\kappa$ (reduction factor, German: Reduktionsfaktor): when the bolt is tightened, the applied torque $M_A$ is transmitted through the thread and the bearing surface, producing two stresses in the bolt shank at the same time:

    1. Axial tensile stress $\sigma_M$ (Montagezugspannung) — this is exactly the preload we want;
    2. Torsional shear stress $\tau_t$ (Torsionsspannung) — a “by-product” of thread friction that contributes nothing to load carrying.

    By the von Mises equivalent-stress criterion (Gestaltänderungsenergie-Hypothese), the equivalent stress of these two combined is $\sigma_{\mathrm{red}} = \sqrt{\sigma_M^2 + 3\,\tau_t^2}$. This is the total load the bolt material actually “feels”. In other words: even if the axial tensile stress $\sigma_M$ is still well below $R_{p0.2}$, once the torsional component is added, $\sigma_{\mathrm{red}}$ may already be close to yield.

    $\kappa$ is a simplified expression of this effect — it tells us: for a given friction condition, the axial usable stress in the denominator is really only $R_{p0.2} / \kappa$, not $R_{p0.2}$ itself. The larger the friction (higher $\mu_G$), the greater the fraction of torque “wasted” on friction, the higher the corresponding $\tau_t$, the larger $\kappa$, and the less margin left for axial load.

    • The value depends on the total friction coefficient ($\mu_{ges}$) of the thread surface and on the bolt type (plain-shank or reduced-shank); see the table below:
    $\mu_G$ 0.08 0.10 0.12 0.14 0.20
    Plain-shank bolt $\kappa$ 1.11 1.15 1.19 1.24 1.41
    Reduced-shank bolt $\kappa$ 1.15 1.20 1.25 1.32 1.52

  • $\beta$ (compliance factor, German: Nachgiebigkeitsfaktor): reflects the difference in elastic deformability from bolt to bolt.

    • Ordinary plain-shank bolt (Schaftschrauben, e.g. per DIN EN ISO 4014): thicker shank, less flexible, $\beta \approx 1.1$.
    • Fully threaded bolt (Ganzgewindeschrauben, e.g. per DIN EN ISO 4017): threaded throughout, somewhat more flexible, $\beta \approx 0.8$.
    • Reduced-shank bolt (Dehnschrauben): excellent elastic deformability, with the waist about 90% of the thread minor diameter, $\beta \approx 0.6$.

  • $E_{\mathrm{S}}$ (elastic modulus, German: E-Modul): for most ordinary steel bolts at room temperature, the elastic modulus is taken as a constant $E_S \approx 210000 \text{ N/mm}^2$.

  • $f_{\mathrm{Z}}$ (embedding, German: Setzbetrag): the irreversible deformation from the microscopic roughness of the interfaces (under the bolt head, on the thread flanks, between the parts) being plastically flattened after tightening and loading, which causes a preload loss ($F_{\mathrm{Z}}$).

    • Note: in the Kübler equation, if there is no precise data, an average value of $0.011 \text{ mm}$ is usually taken. The actual embedding depends strongly on the number and roughness of the interfaces; moreover, for a transversely loaded (querbeansprucht) bolted joint, the embedding rises markedly.

  • $l_{\mathrm{k}}$ (clamp length, German: Klemmlänge): the total thickness of all the compressed clamped parts in the direction of the bolt force.

Note: further verification steps after preliminary design

Since the Kübler equation is only a “preliminary sizing” in the pre-design stage, after looking up the actual bolt size (diameter $d$) from a table, the designer should — without doing the very complex full VDI 2230 verification — still perform at least the following two quick checks:

  1. Fatigue-limit check (for strong dynamic loads): If the bolt carries a dynamic pulsating load in service, its alternating stress amplitude $\pm \sigma_a$ must be estimated and verified not to exceed the bolt’s fatigue limit $\pm \sigma_A$, to prevent fatigue fracture.
  2. Bearing-pressure check (Flächenpressung): To prevent the base material from creeping and crushing under the bolt head or nut because the local contact area is too small (which would further increase the embedding), the bearing pressure must be checked to be less than the material’s limiting pressure $p_G$. The formula: $p \approx \frac{F_{sp} / 0.9}{A_p} \leq p_G$.

3. Simplified fatigue-strength check

In the early design stage of a bolted joint (preliminary design / pre-selection), the engineer usually has not yet fixed the specific flange dimensions and assembly details, so cannot do a precise joint diagram (German: Verspannungsschaubild) calculation based on VDI 2230. However, for a joint under strong dynamic load, fatigue fracture (German: Dauerbruch) is the most fatal failure mode. To avoid the risk early, a simplified fatigue-strength check formula on the safe side can be used:

$$ \pm \sigma_a \approx \pm k \cdot \frac{F_{Bo} - F_{Bu}}{A_s} \le \pm \sigma_A $$

The left side $\pm \sigma_a$ (alternating stress amplitude, German: Ausschlagspannung) is the dynamic stress the bolt sees in the actual case; the right side $\pm \sigma_A$ (fatigue limit, German: Ausschlagfestigkeit) is the bolt’s own maximum resistance to fatigue fracture. As long as the left side does not exceed the right side, the bolt will not suffer fatigue fracture.

1. Estimate the alternating stress amplitude $\pm \sigma_a$

  • $F_{Bo}$: the upper limit of the axial working load (German: oberer Grenzwert der axialen Betriebskraft).
  • $F_{Bu}$: the lower limit of the axial working load (German: unterer Grenzwert der axialen Betriebskraft). Their difference $(F_{Bo} - F_{Bu})$ is the total dynamic load pulsation the joint sees in service.
  • $A_s$: the thread stress cross-section (German: Spannungsquerschnitt des Gewindes), obtainable directly from standard thread tables (see DIN 13-1 etc.) or computed with the formula in section 2.
  • $k$: the clamped-part material influence factor. In a precise VDI 2230 calculation, the external dynamic load does not act entirely on the bolt but is shared by the stiffness ratio (the ratio of resiliences $\delta_S$ and $\delta_P$) of the bolt and the clamped parts; the bolt carries only part of it — the “additional bolt force” ($F_{SA}$). The parameter $k$ is a minimal summary of this stiffness-sharing mechanism, and its value depends entirely on the base material:
    • Steel (Stahl): $k = 0.1$. High elastic modulus, very stiff steel flange, absorbing most of the deformation, so the dynamic additional bolt force passed to the bolt is very small (about 10%).
    • Cast iron (Gusseisen): $k = 0.125$. Lower elastic modulus than steel, slightly less stiff, so the bolt shares more of the dynamic load (about 12.5%).
    • Aluminium (Aluminium): $k = 0.15$. Elastic modulus only about one-third of steel, a “softer” base with larger elastic recovery, so a larger fraction of the alternating load falls on the bolt (about 15%). Conclusion: the softer the clamped part (base), the lower its stiffness, the weaker its ability to “shelter” the bolt, and the greater the fatigue stress amplitude the bolt must bear.

2. Estimate the fatigue limit $\pm \sigma_A$

The fatigue limit (German: Ausschlagfestigkeit) depends on the order of thread manufacturing and heat treatment:

Rolled-before-heat-treated thread (SV) (Schlussvergütetes Gewinde / SV)

That is, the thread is cut or rolled first and the bolt is heat treated as a whole last — the normal manufacturing state for class 8.8, 10.9 and 12.9 bolts. Extensive testing shows that the fatigue limit of an SV bolt is almost independent of the mean bolt force (i.e. the preload level) and is mainly controlled by the diameter, with the estimate:

$$ \pm \sigma_{A(SV)} \approx \pm 0.85 \cdot \left( \frac{150}{d} + 45 \right) $$
  • The size effect (Größeneinfluss) shown by the parameter $d$ (thread nominal diameter, in mm): with $d$ in the denominator, the formula reveals a very important rule: as the bolt diameter grows, its fatigue limit gradually drops. A large bolt is more likely to have internal microscopic metallurgical defects, and its surface stress gradient is relatively flat, so its fatigue sensitivity to the notch (thread) is far higher than a small bolt’s.
  • Safety reduction factor $0.85$: fatigue life in testing often shows a wide scatter band (Streuung). Multiplying by 0.85 lowers the theoretical value to the lower bound of the scatter band, ensuring enough margin (on the safe side) already at the preliminary design / pre-selection stage.

[!NOTE] SV vs SG process distinction SV (Schlussvergütet), i.e. “rolled-before-heat-treated thread (SV)”: the thread is machined first and heat treated last. The heat treatment relieves the machining stresses, so the fatigue limit is independent of the preload — the standard normal state. SG (Schlussgewalzt), i.e. “rolled-after-heat-treated thread (SG)”: heat treated first and the thread rolled last. This process leaves a residual compressive stress at the thread root, giving a higher fatigue limit, but one more affected by the preload (see next section).

Rolled-after-heat-treated thread (SG) (Schlussgewalztes Gewinde / SG)

The thread is rolled after heat treatment (higher manufacturing cost). The residual compressive stress kept at the thread root markedly raises the fatigue life, but the strengthening effect depends on the mean bolt force, so the fatigue limit must be corrected to:

$$ \pm \sigma_{A(SG)} \approx \left( 2 - \frac{F_m}{F_{0,2}} \right) \cdot \sigma_{A(SV)} $$

where $F_m$ is the mean bolt force (German: Schrauben-Mittelkraft) and $F_{0,2}$ is the force at which the bolt material reaches the 0.2% yield strength.

3. Fatigue safety-factor check

Once the actual alternating stress amplitude $\sigma_a$ and the bolt fatigue limit $\sigma_A$ are computed, the fatigue safety factor (German: dynamische Sicherheit) is:

$$ S_D = \frac{\sigma_A}{\sigma_a} \ge 1.2 $$

$S_D \ge 1.2$ is the lower bound required by the standard, ensuring the bolted joint does not suffer fatigue fracture under long-term dynamic service.

Summary: with this simplified check formula, the engineer needs only the range of external load pulsation $(F_{Bo} - F_{Bu})$, the base material type (which sets $k$) and a preliminary bolt diameter $d$ to quickly assess the fatigue-fracture risk in tens of seconds, fixing a reasonable starting bolt size for the later detailed VDI 2230 calculation (covering preload scatter, embedding loss, tightening factor, etc.).

4. Preliminary bearing-pressure check (German: Flächenpressung)

Exceeding the bearing pressure causes the base material to crush microscopically under the bolt head or nut, leading to an irreversible loss of preload (Vorspannkraftverlust). The mechanism: if the bearing pressure exceeds the crushing limit of the base material (Quetschgrenze), the contact region yields plastically (Setzen), the bolt springs back elastically, and the preload drops; when the loss is too large, the joint loosens or fails.

Check formula:

$$ p \approx \frac{F_{sp} / 0.9}{A_p} \le p_G $$

Definition and value guidance of each parameter:

  • $F_{sp} / 0.9$ (estimated maximum working tension): $F_{sp}$ is the assembly clamping force at 90% yield utilization. In real service the bolt must also add the shared external axial working load ($\Phi \cdot F_B$), so approximating $F_{Smax}$ by $F_{sp} / 0.9$ gives a safe estimate of the maximum total bolt tension without a full VDI 2230 calculation.

  • $A_p$ (effective bearing area): the net area of actual contact between the bolt head or nut and the base surface, i.e. the total underside area of the head/nut minus the through-hole area.

  • $p_G$ (limiting pressure): the allowable pressure limit of the clamped-part (not the bolt) material. $p_G$ should take the value for the base material; the allowable pressure of aluminium alloy is far below that of steel, so it especially needs checking when a high-strength bolt is used on a soft base.

Measures when it exceeds the limit (when $p > p_G$):

  1. Use a flanged bolt or nut: enlarge the bearing underside area to directly lower the pressure.
  2. Add a hardened (heat-treated) washer: insert a high-hardness steel washer between the head/nut and the soft base to spread the concentrated pressure over a larger area.
  3. Change the base material: choose a material with a higher allowable pressure to meet the strength requirement.

Data basis and accuracy statement

The calculation method in this article is a pre-selection estimate (Vorauslegung), with an accuracy of ±15–20% for engineering pre-selection. A precise check should use the full VDI 2230-1 calculation method.

Data sources:

  • DIN 13-1 (ISO 261) — metric ISO thread geometry
  • DIN EN ISO 898-1 — bolt mechanical properties (property class and Rp0.2)
  • DIN EN ISO 4014 / 4017 — hexagon-head bolt dimensions
  • DIN EN 20273 — through-hole diameter
  • VDI 2230-1 — systematic calculation method for high-strength bolted joints

Disclaimer: This article is for engineering estimation and teaching reference only. The results do not replace a professional engineer’s design judgement and final sign-off. The final responsibility for engineering safety verification rests with the user.

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