Thermal Strength Utilization Rate η_hot: High-Temperature Load Capacity Evaluation

1. Definition and Purpose

The thermal strength utilization rate $\eta_{hot}$ is a core indicator for evaluating the residual load capacity of disc springs at high temperatures, defined as:

• $F_{cold}$: Maximum allowable working load (permissible load, N) at room temperature (20 °C)
• $F_{hot}$: Maximum allowable working load (N) at operating temperature $T$

This coefficient comprehensively reflects the dual impact of elastic modulus reduction and material yield strength degradation on the load capacity of disc springs.

2. Determination of Room Temperature and High-Temperature Permissible Loads

2.1 Room Temperature Permissible Load $F_{cold}$

The permissible load of a disc spring at room temperature is limited by both yield strength and flattening force:

where $F_{flat,20}$ is the flattening force at room temperature, $S_F$ is the safety factor (≥ 1.3), $A_S$ is the characteristic area, and $k_{stress}$ is the stress conversion coefficient.

2.2 High-Temperature Permissible Load $F_{hot}$

The load capacity at high temperature is reduced due to two factors: 1. Elastic modulus reduction: Under the same deflection, the force decreases as $F_{flat}(T) = F_{flat,20} \cdot E(T)/E_{20}$ 2. Yield strength reduction: The maximum allowable stress at the OM point drops to $\sigma_{y}(T)/S_F$

The smaller of these two values is taken as the high-temperature permissible load:

Therefore, the thermal strength utilization rate can be simplified as:

Typically, the yield strength degrades faster than the elastic modulus, so $\eta_{hot}$ is primarily determined by the high-temperature yield strength retention rate.

3. Evaluation Criteria and Countermeasures

4. Comparison of Thermal Strength Utilization Rates for Typical Materials

Note: Estimated based on yield strength retention rate.

5. Calculation Example

Given: Disc spring material 51CrV4, room temperature permissible load $F_{cold} = 10,000\ \text{N}$, operating temperature 300 °C.
For 51CrV4 at 300 °C, elastic modulus retention rate ≈ 92%, yield strength retention rate ≈ 80%.

Since the yield strength retention rate is lower, $\eta_{hot} \approx 80\%$.
Then $F_{hot} = 10,000 \times 0.80 = 8,000\ \text{N}$.

Assessment: $\eta_{hot}$ is exactly at the 60%~80% boundary, falling into the "acceptable" lower limit. If the design still uses 10,000 N, the disc spring will undergo plastic deformation (Set loss exceeding limits). The working load must be reduced to below 8,000 N, or switch to H13 (at 300 °C, $\eta_{hot} \approx 85\%$, $F_{hot} \approx 8,500\ \text{N}$) for a better margin.

6. Engineering Application Recommendations

• Prioritize $\eta_{hot}$ for material selection: For high-temperature springs, do not only consider room temperature strength; the $\eta_{hot}$ at the target temperature must be calculated to ensure design margin.
• Load derating: When $60\% \le \eta_{hot} < 80\%$, the maximum allowable deflection calculated at room temperature must be multiplied by $\eta_{hot}$ for derating use.
• Avoid critical applications: Do not design disc springs for long-term operation under conditions where $\eta_{hot}$ is close to 60%, as manufacturing tolerances and temperature fluctuations can easily lead to insufficient load capacity.

Summary: The thermal strength utilization rate $\eta_{hot}$ unifies the temperature-induced degradation of elastic modulus and yield strength into an intuitive residual capacity percentage, serving as a rapid criterion for high-temperature disc spring derating design and material selection.