DIN 6796 Force‑Deflection Equation: Load $F(s)$ at Any Compression

1. Washer Type and Mechanical Model

DIN 6796 specifies disc spring washers (conical spring washers), typically with a smooth conical surface and no serrations, used in bolted connections to provide axial elastic force, compensating for preload loss due to embedding, thermal expansion, etc. Their mechanical behavior fully follows the Almen‑Laszlo disc spring theory, allowing precise description of the load $F(s)$ at any axial compression $s$.

2. Core Formula

The axial force of a disc washer at axial compression $s$ ($0 \le s \le h_0$) is:

Where: - $F(s)$ — axial force of the washer at compression $s$ (N) - $E$ — material elastic modulus (steel ≈ 206 000 MPa) - $\nu$ — Poisson's ratio (steel ≈ 0.3) - $t$ — washer material thickness (mm) - $h_0$ — free cone height (mm), i.e., the height difference between inner and outer edges in the free state - $D_e$ — washer outer diameter (mm) - $K_1$ — shape factor, determined solely by the outer-to-inner diameter ratio $c = D_e/D_i$

2.1 Shape Factor $K_1$

Physical meaning of $K_1$: Characterizes the relationship between the bending stiffness of the disc cross-section and the radial width ratio. The larger $c$ (wider washer), the larger $K_1$, and the "softer" the washer.

3. Dimensionless Form (Convenient for Programming and Tables)

Define dimensionless quantities:

The force formula then becomes:

In this form, the value in brackets reflects the nonlinear variation of the disc effect with compression. When $\delta = \eta$ (i.e., $s=h_0$), the bracket equals 1, and the force reaches the flat force $F_{flat}$.

4. Characteristics of the Force‑Deflection Curve

Typical DIN 6796 washers ($\eta \approx 0.4 \sim 1.3$) exhibit the following curve shapes:

• $\eta < \sqrt{2}$ (common case): Stiffness decreases with increasing compression, the curve gradually flattens.
• $\eta \approx \sqrt{2}$: Approximately constant stiffness (linear spring) in the working range.
• $\eta > \sqrt{2}$: Stiffness first increases then decreases, with an inflection point; a negative stiffness region may appear (not used in conventional bolted connections).

DIN 6796 washers for bolts are typically designed with $\eta \approx 0.6 \sim 0.9$ to ensure approximately linear elastic behavior within the normal preload range.

Important limitation: To avoid washer flipping or excessive stress, the recommended maximum working compression is:

Within this range, the formula provides good accuracy, and the washer remains in the elastic safe zone.

5. Integration with Bolt Design Systems

5.1 Washer Stiffness

The tangent stiffness of the washer at a given working point $s_0$ is:

This can be obtained by differentiating the force-deflection equation or directly from standard curves. This stiffness is used in VDI 2230 for compliance superposition:

Total system compliance $\delta_{total} = \delta_S + \delta_P + \delta_W$, which is then used to calculate preload loss and load distribution.

5.2 Compression Corresponding to Preload

To find the washer compression for a target preload $F_{M}$, the nonlinear equation $F(s) = F_M$ must be solved. This can be done using numerical methods or by consulting manufacturer-provided $F-s$ tables. Ensure that $s$ lies within the range $0.1h_0 \le s \le 0.75h_0$.

5.3 Flat Force Check

When $s = h_0$, the formula simplifies to the flat force:

It must be ensured that $F_{Mmax} \le F_{flat}/S_{flat}$ ($S_{flat} \ge 1.3$), as detailed in the "Flat Force Calculation" section.

6. Calculation Example

Washer parameters (similar to M10 DIN 6796): - $D_e = 20$ mm, $D_i = 10.2$ mm → $c = 1.961$, $\ln c \approx 0.673$ - $t = 1.5$ mm, $h_0 = 1.0$ mm → $\eta = 0.667$ - $E = 206\,000$ MPa, $\nu = 0.3$

Step 1: Calculate $K_1$

Step 2: Find load at $s = 0.5$ mm

Bracket term: $(\eta - \delta)(\eta - \delta/2) + 1 = (0.667-0.333)(0.667-0.167)+1 = 0.334 \times 0.5 + 1 = 1.167$
Base constant: $\dfrac{4E}{1-\nu^2} \approx 905\,495$ MPa
Shape factor: $\dfrac{t^4}{K_1 D_e^2} = \dfrac{1.5^4}{0.694 \times 20^2} = \dfrac{5.0625}{0.694 \times 400} = \dfrac{5.0625}{277.6} \approx 0.01824$

Step 3: Verification
The flat force $F_{flat}$ of this washer is (bracket = 1 when $\delta=\eta=0.667$):

At a working compression of 0.5 mm, the force is 6.4 kN, approximately 58% of the flat force, within the safe elastic range.

7. Application Notes

1. Serration Effect
   If the washer surface has serrations (e.g., NFE 25-511 type), the force-deflection relationship must be multiplied by a serration correction factor $\beta_{strié}$ (0.7～0.9). Standard DIN 6796 washers are typically unserrated, so this formula applies directly.

2. Material and Temperature
   The elastic modulus $E$ decreases with increasing temperature, reducing the force at elevated temperatures. Correction based on the working temperature is required.

3. Manufacturing Tolerances
   Tolerances on thickness $t$ and cone height $h_0$ significantly affect the force (due to the $t^3$ relationship). For design, use the combination of minimum thickness and minimum cone height to calculate the minimum force, ensuring that elastic compensation requirements are met under the most unfavorable conditions.

4. Multiple Washer Combinations

5. In series (stacked): Total deflection = single washer deflection × number of washers, total load = single washer load.
6. In parallel (nested): Total load = single washer load × number of washers, total deflection = single washer deflection.
   The combined force-deflection relationship is obtained by superposition according to these rules.

Summary:
The force-deflection equation for DIN 6796 disc washers, based on the Almen‑Laszlo theory, fully describes the elastic load variation from the free state to complete flattening. Using this equation, the washer compression, stiffness, and flat force safety margin can be accurately obtained for any preload, forming the basis for elastic compensation design in bolted connections. Design should be integrated with the VDI 2230 system to ensure the washer's working point remains in the elastic safe zone.