NFE 25-511 Installation Verification: Compression Check

1. Verification Objective

In bolted connections using NFE 25-511 single-sided toothed conical spring washers, the actual compression $s$ of the washer after assembly must simultaneously satisfy two conditions:

1. Elastic range condition: The compression must not exceed the maximum allowable elastic compression $s_{adm}$ of the washer, to prevent flattening, collapse, or yielding.
2. Functional condition: The compression must not be less than a certain minimum recommended value $s_{min}$, to ensure the washer provides sufficient elastic restoring force to compensate for embedding and thermal losses.

The core of installation verification is: Determine the washer compression $s$ from the target preload $F_{M,target}$, and verify $s_{min} \le s \le s_{adm}$, while also confirming that the OM point stress meets strength requirements.

2. Washer Force‑Deflection Relationship (Review)

According to NFE 25-511 and the Almen‑Laszlo modified formula, the relationship between washer axial force $F$ and compression $s$ is:

For symbol meanings and units, refer to the previous "Force‑Deflection Characteristics" section. For simplicity, define dimensionless variables:

and coefficients:

The force‑compression equation simplifies to:

3. Determining Compression from Target Preload

The axial force on the washer after assembly equals the bolt preload $F_M$. During installation, the nominal target preload $F_{M,target}$ (which can be $F_{Mmax}$ or an intermediate value) is typically used as the washer load. Since the force‑deflection relationship is nonlinear, solving for $\delta$ from $F$ requires solving a cubic equation:

This equation can be solved via numerical iteration (Newton's method, bisection method) or using Excel's goal seek. For conventional design, engineering approximations can also be used:

• Small deformation linearization: When $\delta \ll \eta$, $F \approx C_F \cdot \delta \cdot (\eta^2+1)$, so $\delta \approx \dfrac{F_{M,target}}{C_F (\eta^2+1)}$. However, NFE 25-511 washers typically operate in the higher utilization range of $\delta = 0.3 \sim 0.7\eta$, where this approximation has significant deviation and is only suitable as an initial value.

• Chart method: Directly read from the standard $F-s$ curve provided by the manufacturer.

Recommended method: Use a solver to find the root under the constraint $0 < \delta < \eta$. During the design phase, it is common to calculate multiple points in the working range and use interpolation.

4. Allowable Compression Range

4.1 Maximum Allowable Compression $s_{adm}$

The maximum compression is limited by two factors:

• Flattening limit: The compression must not exceed the cone height, i.e., $s < h$. However, to prevent drastic changes in characteristics, it is recommended:
  $$s \le 0.75\,h \quad \left( \delta \le 0.75\,\eta \right)$$

When $\delta > 0.75\eta$, the OM point stress increases rapidly, the force‑deflection curve tends to flatten or even decrease, and the washer is prone to inversion deformation.

• Strength limit: At this compression, the OM point stress must satisfy (see "Washer Body Combined Stress" section):
  $$\sigma_{OM}(\delta) \le \sigma_{adm} \quad (\text{e.g., } 0.9\,R_{p0.2} \text{ or fatigue allowable})$$

is calculated using the same Almen‑Laszlo stress formula, multiplied by the tooth root notch factor . The strength condition may be more stringent than 0.75h, so must be determined comprehensively.

4.2 Minimum Recommended Compression $s_{min}$

The washer must have a certain initial compression to ensure it can still maintain a positive elastic force after embedding and thermal losses occur. The minimum recommended compression is determined by the required minimum elastic restoring force, but generally not less than:

Too small a compression places the washer in the low-stiffness region of the force‑deflection curve, resulting in poor elastic compensation and susceptibility to gaps due to vibration.

5. Installation Verification Procedure

1. Obtain washer parameters: $D_e, D_i, t, h, \beta_{strié}, \alpha_k$, material $E, \nu, R_{p0.2}$.
2. Determine target preload $F_{M,target}$ (typically $F_{Mmax}$ or $F_{Mnom}$; conservatively use $F_{Mmax}$).
3. Calculate washer force constant: $C_F$ and $\eta$.
4. Solve for compression $s_{target}$: Solve $\delta$ from $F_{M,target} = F(s)$, then $s = \delta \cdot t$.

5. Check range:
   

   $$s_{min} \le s_{target} \le s_{adm} \quad (s_{adm} = \min[0.75h,\ s_{\sigma\_adm}])$$

6. Check OM point stress (if not already included in $s_{adm}$): Calculate $\sigma_{OM}$ at this compression and multiply by $\alpha_k$, verify $\le \sigma_{adm}$.

7. If not acceptable:
8. $s_{target} > s_{adm}$: Washer load capacity insufficient; select a larger size (from Z→M→L) or increase the number of washers (stacked combination), or reduce preload (requires bolt redesign).
9. $s_{target} < s_{min}$: Preload too low; washer cannot provide elastic function; consider reducing washer stiffness (choose a smaller model or thinner), or increase preload.
10. Stress exceeds limit: Reduce compression (i.e., reduce preload) or select a higher-strength material.

6. Calculation Example

Given: M10 matching L-type washer (parameters as in previous example):

mm, mm, mm, mm, , ,
Material 50CrV4, $E=206000$ MPa, $\nu=0.3$, $R_{p0.2}=1500$ MPa.
Target preload $F_{M,target}=11\,000$ N (taken from the upper limit of the L-type recommended working load).

Step 1: Calculate constants

,

(calculated previously)

(Exact value varies due to rounding; simplified here for demonstration.)

Step 2: Solve for $\delta$
Equation: $15000 \cdot \delta [ (0.667-\delta)(0.667-0.5\delta) + 1 ] = 11000$, i.e.,

By trial, $\delta \approx 0.41$ gives left side ≈ 0.73. Take $\delta=0.41$ → $s = 0.41 \times 1.8 = 0.738$ mm.

Step 3: Check range

mm. taken as 0.1h=0.12 mm.

→ Range satisfied.

Step 4: OM point stress check (refer to previous formula)
Calculate smooth OM stress:

(as before)

Coefficient $\frac{4E}{1-\nu^2} \frac{t^2}{K_1 D_e^2} = 905495 \times \frac{3.24}{475} \approx 6173$ (unit stress equivalent)

Bracket term = $0.592\times0.462 + 0.796 = 1.069$

MPa
Including notch factor $\alpha_k=1.8$, $\sigma_{OM} = 1.8 \times 2706 = 4871$ MPa, far exceeding $R_{p0.2}=1500$ MPa, not acceptable!

This indicates that under these parameters, a preload of 11 000 N cannot be sustained (stress too high). In reality, the L-type recommended working load of 9 000 – 11 200 N from the previous example may be based on a smaller compression or different washer data. This discrepancy arises because the washer geometry used in this example may correspond to an older model, and the coefficient selection for the force‑deflection formula differs from actual washers. In engineering practice, directly use the manufacturer's force‑deflection curve and allowable compression table; this calculation serves only as a methodological demonstration.

If the stress check passes, verification is complete; if it fails, reduce the target preload or replace the washer.

7. Notes

• Multiple washer combinations: When a single washer cannot meet compression or strength requirements, washers can be used in groups.
• Stacked (series): Multiple washers with the same cone orientation stacked; total compression stroke equals the sum of individual compressions, total load capacity unchanged (equivalent to lengthening the spring).

• Nested (parallel): Multiple washers with alternating cone orientations; total load capacity equals the sum of individual capacities, but compression stroke is the same as a single washer.
  Combined use must be approved by NFE 25-511 or verified by testing.

• Actual friction effects: The force‑deflection formula assumes smooth loading; actual tooth friction causes hysteresis, with the unloading curve slightly lower than the loading curve. This has a minor effect on compression verification but must be considered in preload‑torque conversion.

• Tolerance considerations: Washer thickness, cone height, etc., have manufacturing tolerances. The most unfavorable combination (maximum thickness, minimum cone height, etc.) should be used to check the limit compression, ensuring safety within the tolerance range.

Conclusion:
The installation verification of NFE 25-511 washers involves back-calculating the washer compression from the target preload and confirming that it lies within the elastic safe range and that stress does not exceed limits. This method ensures that the washer can reliably provide elastic force under design loads, preventing flattening or overload, and is an important guarantee of connection reliability. For accurate execution, rely on manufacturer-provided characteristic data or measured force‑deflection curves.