Formula Expression
Parameters
| Symbol | Name | Unit |
|---|---|---|
| De | De | mm |
| Di | Di | mm |
| h0 | h0 | mm |
| s0 | s0 | mm |
| t | t | mm |
| temp_C | temp_C | °C |
| time_hours | time_hours | h |
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DIN 2093 Residual Stress (Load Retention) and Replacement Criteria
1. Definition
After long-term operation of a disc spring under constant deflection, its load capacity gradually decreases due to stress relaxation.
The ratio of the residual load $F(t)$ at a given time $t$ to the initial load $F_0$ is defined as the relaxation ratio (load retention rate):
Therefore, the residual load can be expressed as:
If stress (e.g., OM point stress) is used in the design, the following also applies:
Here, "residual stress" essentially refers to the effective stress (or load) that the disc spring can provide after long-term relaxation, not the residual internal stress within the stress field.
2. Determination of Relaxation Ratio $R(t)$
The relaxation ratio can be obtained from the relaxation model in DIN 2093 (see the "Relaxation Ratio" section):
- $R_{\infty}$ — Relaxation limit at infinite time (e.g., 0.90 indicates that 90% of the load can be retained ultimately)
- $\tau$ — Relaxation time constant, strongly dependent on temperature (Arrhenius relationship)
- $\beta$ — Shape factor (0.3~0.8)
When the operating temperature $T$ differs from the test temperature, $\tau(T)$ must be corrected using the Arrhenius formula.
3. Residual Load Capacity Assessment
In design, the minimum residual load during service life $F_{res,min}$ is typically used as the basis for verification:
- $t_{service}$ — Expected total service time (or inspection interval)
This residual load must still satisfy the functional requirements of the connection (e.g., preventing slippage or separation, i.e., not less than $F_{Kerf}$ or $F_{Mmin}$).
4. Replacement Threshold
When the residual load (or residual stress) of the disc spring drops below a certain critical percentage of the initial design requirement, its elastic compensation capacity is severely insufficient, and replacement or retightening is necessary. Engineering experience suggests:
| Application Type | Replacement Threshold (Residual Load / Initial Required Load) | Description |
|---|---|---|
| General Industrial Connections | ≤ 70% | Replacement recommended when residual force falls below 70% of the design minimum requirement |
| Critical Safety Applications | ≤ 80% | e.g., pressure vessels, aerospace components, elevator brakes; replace immediately when below 80% |
Here, the "initial required load" typically refers to the minimum preload $F_{Mmin}$ determined during design (already including compensation for embedding and thermal loss).
That is, when:
the disc spring must be replaced or retightened to the initial preload.
5. Example
Given:
- Initial preload of disc spring $F_0 = 8\,000\ \text{N}$
- Design minimum preload requirement $F_{Mmin} = 7\,000\ \text{N}$
- Relaxation model parameters: $R_{\infty}=0.88$, $\tau = 3\,000\ \text{h}$, $\beta = 0.5$
- Inspection after an expected operating time of 10 000 hours
Step 1: Calculate the relaxation ratio
Step 2: Residual load
Step 3: Assessment
- Residual load / Minimum required load = $7\,194 / 7\,000 \approx 1.028$, i.e., 102.8%, above the 70% and 80% thresholds, no replacement needed.
If high temperature reduces $\tau$ to 800 h, then
$R(10\,000) \approx 0.88 + 0.12 \cdot \exp(-\sqrt{12.5}) \approx 0.88 + 0.12 \times 0.029 \approx 0.8835$ $F_{res} \approx 7\,068\ \text{N}$, still slightly above 7 000 N, but with very little safety margin; close monitoring or early replacement is recommended.
6. Design Recommendations
- Long-term service design: Select disc spring materials and heat treatment processes (e.g., special tempering, shot peening with thermal stabilization) that yield a high $R_{\infty}$ (>0.92).
- High-temperature applications: The Arrhenius acceleration effect must be considered to predict the long-term residual load at the maximum operating temperature and to determine the replacement interval accordingly.
- Safety factor: For enclosed structures where periodic replacement is impossible, the design allowable residual load should be set at least 1.2 times the threshold value.
- Recording and monitoring: Record the initial load and service time of the disc spring in the equipment history, and assess the residual load capacity in conjunction with the temperature history.
Summary: The residual load of a disc spring is the product of the initial load and the relaxation ratio. When the residual load drops below 70% (general) or 80% (critical) of the design minimum requirement, replacement or retightening is mandatory. Reasonably predicting the relaxation ratio and setting the replacement threshold are key to ensuring the long-term reliable operation of disc spring connections.