DIN 2093 Limit Load Analysis: Plastic Failure and Safety Factor

1. Definition of Limit Load

As the axial compression of a disc spring increases, the material transitions from elastic to plastic behavior, ultimately reaching the limit load—the load at which the structure undergoes unacceptable plastic deformation or loses its load-bearing capacity. For disc springs, the limit state is typically determined by the following criteria:

• Equivalent plastic strain criterion: The equivalent plastic strain $\varepsilon_{eq}^{p}$ at the critical point (usually the OM point) reaches a certain threshold (e.g., 0.2% or 2%), indicating local failure.
• Global yielding: The plastic zone penetrates the entire cross-section, causing the load‑displacement curve to exhibit a plateau or an extremum point.

2. Control Point for Plastic Failure — OM Point

The stress distribution in a disc spring is non-uniform. The OM point (inner edge of the upper surface) experiences the maximum compressive stress (in absolute value) and is the first location to yield. As compression increases, the plastic zone expands inward from the OM point.

Methods for determining the limit load: - Numerical method: Using elastic-plastic finite element analysis (FEA), gradually increase the compression and monitor the equivalent plastic strain at the OM point. When $\varepsilon_{eq}^{p}$ reaches the specified threshold, the corresponding external load (or compression) defines the limit state. - Analytical approximation: Correlate the elastic stress formula at the OM point with the material yield strength. When the elastic nominal OM stress reaches $R_{p0.2}$, yielding begins at the OM point surface. However, due to stress redistribution, the disc spring can still carry additional load until the plastic zone expands sufficiently to reach the true limit.

Therefore, the limit load is typically higher than the elastic limit load.

3. Safety Factor Requirements

To prevent plastic collapse or excessive permanent deformation during service, the design load must be less than the limit load with an adequate safety margin. DIN 2093 recommends:

Here, $SF$ is defined as:

• $F_{limit}$ — Limit load (N)
• $F_{work}$ — Maximum working load (N)
• $s_{limit}$ — Limit compression (mm), typically corresponding to the compression at which the equivalent plastic strain reaches the critical value
• $s_{work}$ — Maximum working compression (mm)

4. Procedure for Determining the Limit Load

1. Establish an elastic-plastic FEA model: Include disc spring geometry, the material's true stress‑strain curve (including the hardening region after yield), and boundary conditions.
2. Apply a gradually increasing axial displacement and perform a nonlinear static analysis.
3. Extract the equivalent plastic strain $\varepsilon_{eq}^{p}$ at the OM point and plot the load‑displacement curve.
4. Identify the state point satisfying the failure criterion:
5. The load/compression corresponding to $\varepsilon_{eq}^{p} = 0.2\%$ or 2%.
6. Or the starting point of the plateau on the load‑displacement curve.
7. Take the load at this state as $F_{limit}$.

5. Calculation Example (Conceptual)

A disc spring specification: $D_e=40\ \text{mm}, D_i=20.4\ \text{mm}, t=2.0\ \text{mm}, h_0=1.2\ \text{mm}$, material 50CrV4, $R_{p0.2}=1500\ \text{MPa}$.

Elastic analysis: At $s = 0.8\ \text{mm}$, the elastic stress at the OM point ≈ 1600 MPa (slightly exceeding yield), indicating that yielding begins near this compression.

Elastic-plastic FEA with stepwise loading: - At $s \approx 1.0\ \text{mm}$, the equivalent plastic strain at the OM point reaches 0.2%, corresponding to a load $F_{limit} \approx 9\,500\ \text{N}$. - If the maximum working load $F_{work} \leq 9\,500 / 1.5 \approx 6\,330\ \text{N}$ (static), or $9\,500 / 2.0 = 4\,750\ \text{N}$ (cyclic), the design is safe.

If the actual working load is 5 000 N under cyclic conditions, $SF = 9\,500/5\,000 = 1.9$, which is close to but slightly below 2.0. Consideration is needed whether to accept this or reduce the load.

6. Design Recommendations

• Prioritize elastic design: For conventional applications, the elastic stress at the OM point should be $\le R_{p0.2}/1.2$ to avoid any plasticity.
• Evaluate limit load: Only utilize the plastic reserve when unavoidable high loads or compact designs are required. However, the limit load must be verified by FEA, and the safety factor must be satisfied.
• Strictly limit cyclic loads: Entering the plastic regime under cyclic loading can induce low-cycle fatigue. The $SF$ must be ≥ 2.0, and strain‑fatigue analysis is recommended.
• Pre‑setting (strong pressing): Controlled flattening during manufacturing is allowed to produce a predictable set loss, but the spring must not re‑enter the plastic regime during service.

Summary: The limit load of a disc spring is governed by the equivalent plastic strain at the OM point and is determined via elastic-plastic FEA. The design safety factor is ≥ 1.5 for static loads and ≥ 2.0 for cyclic loads, ensuring that the disc spring does not suffer plastic failure or excessive permanent deformation during its service life.