DIN 2093 Linear Static FEA: Small Deformation Verification and Analytical Comparison

1. Basic Assumptions of Linear Static FEA

Linear static finite element analysis (Linear Static FEA) solves for the displacement, strain, and stress of a disc spring under external loads, based on the following assumptions:

• Geometric Linearity: Small deformation assumption, i.e., deformations are much smaller than structural dimensions, and geometric stiffness changes (such as stress stiffening, large rotations) are not considered.
• Material Linearity: The stress-strain relationship satisfies Hooke's law, with no plasticity.
• Constant Contact State: If contact is included in the model, it is assumed that the contact area and state do not change with deformation (or there is no contact).

For disc springs, when the dimensionless deflection $\delta = s/t$ is small, generally limited to $s/h_0 < 0.3$, the deformation of the disc spring is in an approximately linear stage, the force-deflection relationship is nearly linear, and the above assumptions are essentially valid. Under these conditions, linear FEA shows good agreement with the analytical solution (Almen‑Laszlo) and can be used to verify model correctness.

2. Basic Equations of Linear Static FEA

In the global coordinate system, linear static finite elements reduce to solving a system of linear algebraic equations:

where: - $\mathbf{K}$ — System stiffness matrix, assembled from element stiffness matrices, which is a constant matrix in linear analysis. - $\mathbf{U}$ — Nodal displacement vector (unknown). - $\mathbf{F}$ — Nodal load vector (known external loads, such as prescribed displacements or force boundary conditions).

For small deformation elastic problems, the element stiffness matrix is given by the following integral (using one element as an example):

• $\mathbf{B}$ — Element strain-displacement matrix.
• $\mathbf{D}$ — Material elasticity matrix (for isotropic materials, containing $E, \nu$).

After solving the above system for $\mathbf{U}$, the strain $\boldsymbol{\varepsilon} = \mathbf{B} \mathbf{U}$ and stress $\boldsymbol{\sigma} = \mathbf{D} \boldsymbol{\varepsilon}$ can be calculated.

3. Analysis Procedure and Key Modeling Points

When performing linear static FEA of disc springs, note the following:

1. Geometric Model: Create based on the nominal dimensions of the disc spring, ignoring small chamfers, but retaining the accurate diameters of the inner and outer edges.
2. Material Properties: Input the elastic modulus $E$ and Poisson's ratio $\nu$, consistent with the values used in the analytical formula.
3. Boundary Conditions:
4. Typically, a fixed constraint (limiting axial displacement) or a symmetry constraint is applied to the outer edge of the lower surface of the disc spring.
5. An axial prescribed displacement $s$ (or axial concentrated force) is applied to the outer or inner edge of the upper surface to simulate compression.
6. Mesh Density: Refer to mesh convergence criteria, with ≥ 4 layers in the thickness direction (linear hexahedral elements) and ≥ 48 elements around the circumference to ensure stress accuracy.
7. Solution: The solver computes the displacement field and outputs reaction forces and stress distribution.

4. Comparison and Verification with the Almen‑Laszlo Analytical Solution

The Almen‑Laszlo analytical formula is an accurate theoretical solution for disc springs under small deformation. For comparison and verification, select the same compression $s$ and calculate:

• Analytical load $F_{AL}(s)$ (see formula in previous sections);
• FEA reaction force $F_{FEA}$: Extract the total axial reaction force at the nodes where the prescribed displacement is applied.

The relative error between the two is defined as:

Simultaneously, the OM point stress can be compared:

• Analytical OM point stress $\sigma_{OM,AL}$;
• Maximum compressive stress at the inner edge of the upper surface from FEA $\sigma_{OM,FEA}$.

5. Error Assessment and Model Credibility

Within the small deflection range satisfying $s/h_0 < 0.3$, the load error between linear FEA and the analytical solution should be < 5%, and the OM point stress error should be < 5%~8% (stress is more sensitive to mesh and edge constraints). If the errors are within the allowable range, it indicates that:

• The FEA model geometry, material, and boundary conditions are set correctly;
• The mesh density has converged;
• It can serve as a benchmark for subsequent nonlinear, large deflection, or combined loading analyses.

If the errors exceed the limits, check: - Whether the mesh meets convergence requirements (refine in the thickness direction); - Whether the boundary conditions are consistent with the analytical assumptions (the analytical solution assumes uniform axial compression without radial constraint); - Whether the geometric nonlinearity switch has been incorrectly activated (large deformation option must be turned off for linear static analysis); - Whether the elastic modulus, Poisson's ratio, etc., are input correctly.

6. Calculation Example

Disc Spring: $D_e = 40\ \text{mm}, D_i = 20.4\ \text{mm}, t = 2.0\ \text{mm}, h_0 = 0.9\ \text{mm}$, $E = 206\,000\ \text{MPa}, \nu = 0.3$.
Compression: $s = 0.2\ \text{mm}$, giving $s/h_0 \approx 0.222 < 0.3$, satisfying the small deformation condition.

Analytical Calculation: - Calculate $K_1, C_1, C_2$ (omitted), substitute into the load formula to obtain $F_{AL} \approx 2\,360\ \text{N}$. - Absolute OM point stress $\sigma_{OM,AL} \approx 745\ \text{MPa}$.

FEA Model: Linear hexahedral elements, 4 layers in the thickness direction, 48 elements around the circumference. An axial displacement of 0.2 mm is applied to the inner edge of the upper surface, and the axial displacement of the outer edge of the lower surface is fixed.

FEA Results: - Reaction force $F_{FEA} = 2\,410\ \text{N}$ - OM point compressive stress $\sigma_{OM,FEA} = 768\ \text{MPa}$

Errors:

Both load and stress errors are within the allowable range, the FEA model is verified, and it can be used for subsequent analyses.

7. Applicability and Limitations of Linear Static FEA

• Valid Range: $s/h_0 < 0.3$, the force-deflection relationship is linear or weakly nonlinear, and linear analysis provides sufficient accuracy.
• Out-of-Range Effects: When $s/h_0 > 0.3$, geometric nonlinear effects of the disc spring (such as stiffness change, inner/outer diameter expansion) become significant. Linear analysis will overestimate stiffness and underestimate force. In this case, geometric nonlinearity (large deformation) must be activated, and a finer mesh may be required.
• Contact Nonlinearity: If the disc spring stack involves contact issues such as friction or separation, linear static analysis is not applicable; nonlinear contact analysis is required.

8. Conclusion

Linear static FEA is the first step in verifying the accuracy of a disc spring finite element model. Under small deformation conditions, by comparing the load and stress with the Almen‑Laszlo analytical solution and ensuring the errors are within acceptable limits (load < 5%, stress < 8%), a credible numerical model can be established, providing a solid foundation for subsequent nonlinear analysis and simulation of disc spring stacks.