Symbol	Name	Unit
bolt_grade	bolt_grade	—
cam_angle_deg	cam_angle_deg	°
mu_cam	mu_cam	—
nominal_dia	nominal_dia	—
preload	preload	N

Locking Torque Calculation: Core Indicator of DIN 25201 Wedge-Locking Performance

1. Definition and Basic Formula

In the DIN 25201 wedge-locking washer system, the locking torque $M_{lock}$ is the torque generated by wedge surface friction and geometric ramp angle that resists loosening rotation when the bolt tends to rotate loose. It is the core design indicator for evaluating the locking capability of wedge washers.

The basic calculation formula is:

\boxed{M_{lock} = F_M \cdot \tan(\alpha + \rho) \cdot \frac{D_{cam}}{2}}

Where:

$M_{lock}$ — Locking torque (N·mm or N·m), acting opposite to the loosening rotation direction
$$F_M$$ — Actual preload in the current connection (typically the residual preload after assembly) (N)
$\alpha$ — Wedge ramp angle (cam angle of the wedge washer), typically 6°～8°
$\rho$ — Wedge surface friction angle, $\rho = \arctan(\mu_{cam})$ , where $\mu_{cam}$ is the friction coefficient between wedge surfaces
$D_{cam}$ — Equivalent friction diameter of the wedge surface (mm), taken as the mean contact diameter of the wedge surface

This formula is identical in form to the horizontal force required to push a load up an inclined plane, with a clear physical meaning: loosening the bolt is equivalent to "lifting" the preload up a wedge surface with an inclination angle $\alpha$ while overcoming wedge surface friction.

2. Formula Derivation and Physical Meaning

2.1 Wedge Surface Force Analysis

For a pair of assembled wedge washers, the wedge surface contact between the inner and outer washers can be simplified as an inclined plane with an inclination angle $\alpha$ . When the bolt attempts to loosen counterclockwise, the outer washer (engaged with the nut/bolt head) must slide upward relative to the inner washer (engaged with the clamped part) along the wedge surface.

Enlarging the local wedge surface:

Outer washer
/|
F_M / | Normal force N
/ | Friction force μN (along the slope)
/ |
/ |
/α |
+------+------→ Tangential force F_t (rotation direction)
Inner washer

The axial load is the preload $$F_M$$ , acting vertically downward (clamping direction).
The outer washer moving upward along the wedge surface is equivalent to pushing $$F_M$$ upward along the inclined plane.
The tangential force $$F_t$$ required in the circumferential direction originates from the loosening torque at the radius $D_{cam}/2$ .

According to the inclined plane principle, the tangential force required to push the slider upward (overcoming the axial force $$F_M$$ and simultaneously overcoming friction) is:

F_t = F_M \cdot \tan(\alpha + \rho)

Where the friction angle $\rho$ satisfies $\tan\rho = \mu_{cam}$ . $\alpha + \rho$ means that not only the lifting component due to the wedge inclination must be overcome, but also the additional friction force.

This tangential force $$F_t$$ acts at the radius $D_{cam}/2$ , so the corresponding torque is:

M_{lock} = F_t \cdot \frac{D_{cam}}{2} = F_M \cdot \tan(\alpha + \rho) \cdot \frac{D_{cam}}{2}

This completes the derivation of the locking torque.

2.2 Why $\tan(\alpha+\rho)$ Instead of $\tan(\alpha-\rho)$

If the wedge surface is in a self-locking state (i.e., $\alpha < \rho$ ), the loosening process corresponds to "downward sliding," and the required active torque would be $F_M \cdot \tan(\rho - \alpha) \cdot D_{cam}/2$ , a smaller value. If $\alpha > \rho$ , self-locking is not possible.
However, DIN 25201 design does not rely on wedge surface friction self-locking. Instead, it uses a large wedge angle (above 6°) such that $\alpha$ is typically greater than $\rho$ (for steel-on-steel dry friction $\mu \approx 0.15, \rho \approx 8.5°$ , but washers often have coatings, so $\mu$ may be lower). The wedge surface itself is not self-locking in static mechanics.
During loosening, bolt rotation forces the outer washer to "climb up" the wedge surface, which further stretches the bolt, increasing the preload $$F_M$$ . The system is in a dynamic forced locking state. Therefore, to complete one full loosening rotation, continuous energy input is required to overcome the resistance torque corresponding to the $\tan(\alpha+\rho)$ term.
Consequently, the loosening torque must exceed $M_{lock}$ for the connection to loosen.

3. Detailed Parameter Explanation

Parameter	Symbol	Unit	Meaning and Value
Preload	$$F_M$$	N	Current axial preload of the connection. In locking analysis, the minimum possible preload (considering embedding and thermal losses) is typically used to obtain a conservative locking torque.
Wedge ramp angle	$\alpha$	°	Characteristic angle of DIN 25201 washers. Standard series typically use 6°; heavy-duty series may use 7°～8°. Increasing $\alpha$ significantly raises the locking torque, but washer strength must be considered.
Wedge surface friction coefficient	$\mu_{cam}$	—	Friction coefficient between inner and outer washer wedge surfaces, depending on material, surface treatment, and lubrication (typically intentionally unlubricated to maintain high friction). Typical design value 0.10～0.20.
Wedge surface friction angle	$\rho$	°	$\rho = \arctan(\mu_{cam})$ . For example, $\mu_{cam}=0.15 \rightarrow \rho \approx 8.53°$ .
Wedge equivalent diameter	$D_{cam}$	mm	Mean diameter of the wedge surface contact ring, determined by washer geometry. Can be approximated as $D_{cam} \approx (D_{cam,o} + D_{cam,i})/2$ , or calculated using precise formulas.

4. Role of Locking Torque in the Locking System

4.1 Relationship with Thread Loosening Torque

The torque of the thread itself in the loosening direction (when self-locking is absent or insufficient) is:

M_{G,off} = F_M \cdot \frac{d_2}{2} \cdot \tan(\rho' - \varphi)

When vibration causes a significant reduction in the thread friction coefficient $\mu_G$ , $\rho'$ decreases, potentially making $\rho' < \varphi$ . The thread loses its self-locking capability, and $M_{G,off}$ becomes negative (i.e., the thread will spontaneously loosen).

In the DIN 25201 system, because the outer teeth of the washer lock the nut and the clamped part respectively, any loosening rotation must simultaneously drive wedge surface sliding. Therefore, the total resistance torque against loosening is:

M_{resist} = M_{lock} + M_{G,off} \quad (\text{when } M_{G,off} \text{ is positive})

M_{resist} = M_{lock} - |M_{G,off}| \quad (\text{when the thread becomes unstable})

Since $\alpha$ is 6°～8°, much larger than the thread helix angle $\varphi$ (approximately 2°～4°), and $\tan(\alpha+\rho)$ is much larger than $\tan(\rho'-\varphi)$ , $M_{lock}$ is numerically an order of magnitude greater than the thread loosening torque. Even if thread friction completely disappears, $M_{lock}$ remains very high, ensuring the connection will never loosen.

4.2 Locking Condition

The design must satisfy:

M_{lock} > M_{disturb,max}

Where $M_{disturb,max}$ is the maximum loosening disturbance torque from external vibration, impact, etc. This is typically verified through Junker transverse vibration testing, where standards (e.g., ISO 16130) require the washer to maintain preload retention above a specific threshold under vibration.

5. Calculation Example

Given Conditions

Bolt M10×1.5, strength grade 8.8
Assembly preload $F_M = 25\,000 \text{ N}$
Wedge washer DIN 25201, wedge angle $\alpha = 6°$
Wedge surface friction coefficient $\mu_{cam} = 0.15$ , corresponding to $\rho = \arctan(0.15) \approx 8.53°$
Washer wedge mean diameter $D_{cam} = 13 \text{ mm}$