🧮 在线计算器：Bolt Pre-selection Calculator — Fast sizing with the Kübler equation; supports a design mode and a check mode.
The reduction factor $\kappa$ explained: the “invisible tax” of tightening a bolt
📎 Prerequisite reading: this article is a deeper supplement to Bolt Pre-selection (001) — From Load to Size , focusing on the physical nature and derivation of the $\kappa$ parameter in the denominator of the Kübler equation.
In the design and installation of a bolted joint, the reduction factor $\kappa$ (Reduktionsfaktor) is a very central parameter. It answers a key question: while the bolt is being tightened, how much strength is still “available” for axial load carrying? Below we break down its definition, derivation and formula, based on the public technical standard VDI 2230.
1. First, the physical background: the “double burden” during tightening
While a bolt is being tightened, the bolt shank carries two loads at the same time, and is therefore in a two-way (2D) stress state:
- Axial tensile stress ($\sigma_{M}$, Montagezugspannung): produced as the bolt is stretched while being screwed down along the thread — this is exactly the preload we want.
- Torsional shear stress ($\tau_t$, Torsionsspannung): produced as the bolt shank is twisted to overcome the thread-surface friction — a “by-product” of thread friction that contributes nothing to load carrying.
Now the problem: the bolt material’s strength parameters (such as the yield strength $R_{p0,2}$) are all measured by a uniaxial tension test. But the bolt is really in a multi-axial combined stress state of “both pulled and twisted”. We need a way to convert this combined stress into an equivalent uniaxial stress, so it can be compared with $R_{p0,2}$.
2. Theoretical basis: the fourth strength theory (von Mises yield criterion)
For the ductile steels commonly used in high-strength bolts, tests show the von Mises yield criterion (German: Gestaltänderungsenergiehypothese, GEH for short) gives the most accurate prediction, because it compares the energy needed to cause a change of shape (not a change of volume) in the material.
From the 3D general form to the bolt’s 2D special case
By the von Mises criterion, the general equivalent-stress formula in a three-way (spatial) stress state is:
$$ \sigma_{red} = \frac{1}{\sqrt{2}} \sqrt{(\sigma_1 - \sigma_2)^2 + (\sigma_1 - \sigma_3)^2 + (\sigma_2 - \sigma_3)^2} $$where $\sigma_1, \sigma_2, \sigma_3$ are the three principal stresses on the element.
For a surface element of the bolt shank, it carries tensile stress only along the axis and shear stress on the cross-section, with no other normal stress in the radial or circumferential direction. So it is in a plane stress state, and one of the three principal stresses must be zero (let $\sigma_2 = 0$). Substituting and simplifying:
$$ \sigma_{red} = \sqrt{\sigma_1^2 - \sigma_1\sigma_3 + \sigma_3^2} $$To compute directly from the normal and shear stresses known in engineering, introduce the plane-stress von Mises formula in Cartesian coordinates:
$$ \sigma_{v} = \sqrt{\sigma_x^2 + \sigma_y^2 - \sigma_x\sigma_y + 3\tau^2} $$In the specific case of bolt tightening:
- Axial tensile stress $\sigma_x = \sigma_{M}$
- Transverse normal stress $\sigma_y = 0$
- Shear stress $\tau = \tau_t$
After substitution, all the $\sigma_y$ terms cancel, giving the equivalent-stress formula for bolt assembly:
$$\boxed{\sigma_{red} = \sqrt{\sigma_{M}^2 + 3\cdot\tau_t^2}}$$This is the total load the bolt material really “feels”. Even if the axial tensile stress $\sigma_M$ has not yet reached $R_{p0,2}$, once the torsional component is added, $\sigma_{red}$ may already be close to yield.
3. The strict definition of $\kappa$
The theoretical definition of the reduction factor $\kappa$ is: the ratio of the combined equivalent stress to the pure axial tensile stress:
$$ \kappa = \frac{\sigma_{red}}{\sigma_{M}} $$Substituting the equivalent-stress formula into the definition:
$$ \kappa = \frac{\sqrt{\sigma_{M}^2 + 3 \cdot \tau_t^2}}{\sigma_{M}} = \sqrt{1 + 3 \left(\frac{\tau_t}{\sigma_{M}}\right)^2} $$The physical meaning is clear: because a torque is always present ($\tau_t > 0$), $\kappa$ must be greater than 1. $\kappa$ reflects the “degree of reduction” of the bolt’s tensile load capacity by the torsional shear stress — like an “invisible tax”: the greater the thread friction, the higher this tax rate.
4. From the stress ratio to a computable algebraic formula
To turn the stress ratio into specific geometry and force parameters, we expand further:
- Tensile stress: $\sigma_{M} = F_{VM} / A_0$, where $A_0 = \frac{\pi}{4} d_0^2$
- Shear stress: $\tau_t = M_G / W_P$
- Thread friction torque: $M_G \approx F_{VM} \cdot \left(0{,}159 \cdot P + 0{,}577 \cdot \mu_G \cdot d_2\right)$
A key detail: the fully plastic section modulus
When computing the shear stress, consider that at normal tightening (such as using 90% of the yield strength) the section edge of the bolt is often already close to or into the plastic state. To use the material’s plastic support effect (Tragreserven), the torsional section modulus $W_P$ in engineering standards usually does not use the elastic formula $\frac{\pi}{16} d_0^3$, but the fully plastic torsional section modulus:
$$ W_{P,pl} = \frac{\pi}{12} d_0^3 $$Substituting $M_G$, $A_0$ and $W_{P,pl}$ into the ratio $\tau_t / \sigma_{M}$ gives the final algebraic formula for $\kappa$.
5. The specific formula
By VDI 2230 and public engineering textbooks, the specific formula for $\kappa$ can be written in the following two mathematically equivalent algebraic forms:
Form one (simplified engineering expression)
$$ \kappa = \sqrt{1 + 3 \cdot \left[ \frac{3}{d_0} \cdot \left(0{,}159 \cdot P + 0{,}577 \cdot \mu_G \cdot d_2\right) \right]^2} $$Form two (VDI 2230 standard expression)
$$ \kappa = \sqrt{1 + 3 \cdot \left[ \frac{3}{2} \cdot \frac{d_2}{d_0} \left(\frac{P}{\pi \cdot d_2} + 1{,}155 \cdot \mu_G\right) \right]^2} $$Formula parameters
| Parameter | Meaning | Value source |
|---|---|---|
| $P$ | thread pitch (Gewindesteigung) | DIN 13-1 standard thread table |
| $d_2$ | thread pitch diameter (Flankendurchmesser) | DIN 13-1 standard thread table |
| $d_0$ | section reference diameter | ordinary bolt: $(d_2+d_3)/2$; reduced-shank bolt: $d_T$ |
| $\mu_G$ | thread friction coefficient (Reibungszahl im Gewinde) | VDI 2230-1, Tab.A2 |
6. Engineering use: the role of $\kappa$ in the Kübler equation
The formula shows clearly: the thread friction coefficient $\mu_G$ is the most central variable that decides the size of $\kappa$.
The larger $\mu_G$, the greater the torque used to overcome friction during tightening, the stronger the torsional shear stress produced, and thus the marked increase in $\kappa$. Since the material’s total load capacity is fixed, this greatly cuts the axial tension we actually need.
In engineering practice (for example, in a bolt pre-selection with the Kübler equation), the engineer divides the material’s yield strength by the reduction factor (i.e. $R_{p0,2} / \kappa$) to assess the effective axial load capacity the bolt can really provide under a given lubrication state.
Typical $\kappa$ value reference
| $\mu_G$ | 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 |
Engineering insight: take a plain-shank bolt with $\mu_G = 0.14$, $\kappa = 1.24$, which means the axial usable stress is only $R_{p0,2} / 1.24 \approx 81\%$. About 19% of the material’s load capacity is “eaten” by thread friction. This is why good lubrication (lowering $\mu_G$) is critical to fully using the bolt’s strength.
7. The safety criterion in VDI 2230
In the most authoritative bolt-calculation standard, VDI 2230, to make sure the bolt does not suffer permanent plastic yield at assembly — even under the combined tension and torsion load — it must strictly hold that:
$$ \sigma_{red} \le 0.9 \cdot R_{p0,2} $$That is, the computed combined equivalent stress must not exceed 90% of the material yield strength. This 10% safety margin is exactly the load-carrying room reserved for the external working load that may be added during operation.
Data basis and accuracy statement
The derivation in this article is based on the following public technical standards:
- VDI 2230-1 — systematic calculation method for high-strength bolted joints
- DIN 13-1 (ISO 261) — metric ISO thread geometry
- DIN EN ISO 898-1 — bolt mechanical properties
Disclaimer: This article is for engineering 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.
📚 Series navigation
← Back: Bolt Pre-selection (001) — From Load to Size ｜ Next: A Minimal Pre-selection Table →