DIN 9250 Spring Force‑Deflection Characteristics: Disc Spring Elasticity + Tooth Section Reduction

1. Washer Characteristics

The toothed lock washers specified in DIN 9250 typically have a disc (conical) elastic body and are machined with radial fine teeth on one or both sides. They simultaneously possess: - Elastic compensation capability: Relying on the disc geometry to provide spring force, compensating for preload relaxation; - Mechanical locking capability: Teeth bite into the mating surface, increasing anti-rotation torque.

Due to the presence of tooth grooves, the bending stiffness of the washer cross-section is lower than that of a smooth disc spring of the same size. Its force‑deflection relationship must be modified based on the classic Almen‑Laszlo equation by introducing a tooth section reduction factor $β_{serr}$.

2. Basic Formula for Smooth Disc Springs (Almen‑Laszlo)

The axial force $F_{Almen}(s)$ for a smooth conical washer without teeth is:

Where: - $E$ — Material elastic modulus - $\nu$ — Poisson's ratio - $t$ — Washer thickness - $h_0$ — Free cone height - $D_e$ — Outer diameter - $K_1$ — Shape factor, depending only on the outer-to-inner diameter ratio $c = D_e/D_i$

3. Tooth Section Reduction Factor $β_{serr}$

Tooth grooves remove part of the cross-section, reducing the bending stiffness of the washer. This results in lower axial force at the same compression compared to a smooth washer. The reduction factor is introduced:

• $β_{serr} \le 1$, dimensionless
• When toothless (smooth), $β_{serr}=1$

Physical meaning of $β_{serr}$: It comprehensively reflects the degree of reduction in the section moment of inertia caused by the number of teeth, tooth width, and tooth depth. The denser and deeper the teeth, the smaller $β_{serr}$, and the "softer" the washer.

4. Methods for Determining $β_{serr}$

4.1 Theoretical Estimation Based on Section Moment of Inertia

If the radial cross-section of the washer is considered a rectangular beam, the ratio of the net section moment of inertia $I_{net}$ at the tooth groove to the gross moment of inertia $I_{gross}$ can be approximated as the reduction factor:

However, this geometric simplification has significant error; engineering practice relies more on empirical data.

4.2 Empirical Values (Common Tooth Profiles for DIN 9250)

Design Recommendation: If actual measurement is not possible, a preliminary design can use $β_{serr}=0.80$, and then correct it through compression testing during the prototype phase.

5. Force‑Deflection Curve Characteristics

After introducing $β_{serr}$, the overall shape of the washer's force‑deflection curve remains unchanged (still characteristic of a disc spring), but the ordinate is proportionally scaled down. This means: - Reduced flattening force: $F_{flat,serr} = β_{serr} \cdot F_{flat,Almen}$ - Reduced stiffness: $k_{serr} = β_{serr} \cdot k_{Almen}$ - Reduced energy storage: The elastic energy absorbed at the same compression decreases

Due to the mechanical interlocking effect of the teeth, the actual working point of the washer is usually located on the rising section of the force‑deflection curve. Lower stiffness is beneficial for compensating for settlement, but it must be ensured that the flattening force is still greater than the maximum working preload.

6. Calculation Example

Given DIN 9250 washer (for M10): - $D_e = 20$ mm, $D_i = 10.2$ mm → $c=1.961$, $K_1 \approx 0.6947$ - $t = 1.5$ mm, $h_0 = 1.0$ mm - Material steel, $E=206\,000$ MPa, $\nu=0.3$ - Tooth profile results in $β_{serr}=0.80$

Flattening force for smooth washer (calculated previously):

Flattening force for toothed washer:

Force at an arbitrary compression: Taking $s=0.5$ mm as an example, the force for the smooth washer is approximately $6\,420$ N, so the force for the toothed washer is $0.80 \times 6\,420 \approx 5\,136$ N.

Design Note: Due to the reduced flattening force, the upper limit of the permissible preload for this washer decreases correspondingly. It must be ensured that $F_{Mmax} \le F_{flat,serr} / 1.3$.

7. Integration with Design Systems

1. VDI 2230 Compliance Calculation: Washer compliance $\delta_W = 1/k_{serr}$. Since stiffness decreases, compliance increases, raising the total system compliance, which is beneficial for reducing preload loss due to embedding.
2. Torque Coefficient K: The friction effect of the tooth surface requires a separate correction of $\mu_K$ using $k_{serr}$; see the chapter on torque‑preload relationships in DIN 9250.
3. Stress Verification: When calculating stress at points like OM, if the tooth groove is located at the point of maximum stress (inner edge), an additional notch stress concentration factor must be considered. This can be evaluated in conjunction with $β_{serr}$, but is generally not used directly for stress calculation.

Summary:
The force‑deflection characteristics of DIN 9250 toothed conical washers are based on the Almen‑Laszlo equation for smooth disc springs, modified by multiplying by the tooth section reduction factor $β_{serr}$. $β_{serr}$ quantifies the reduction in stiffness caused by the tooth grooves and is an important correction parameter for washer selection and preload matching. In practical applications, priority should be given to obtaining accurate force‑deflection curves through compression testing.