Elemental Stress Continuity in Finite Element Analysis (FEA)

Finite Element Analysis (FEA) is a widely-used numerical technique for solving complex engineering problems. After obtaining nodal displacements by solving FEM equations, the question arises: how do FEA solvers calculate stress values that are continuous at element boundaries? This article addresses this topic by explaining the challenges in ensuring stress continuity and the approaches used in FEA solvers to achieve approximate, yet effective stress distributions.

Overview of FEA and Stress Analysis

Finite Element Analysis (FEA) is a powerful tool used to analyze the behavior of complex structures under various loads and conditions. Key equations involved in FEA include the strain-displacement relation (kinematic equations) and the stress-strain relation (constitutive equations). The primary unknowns in FEA are nodal displacements, typically denoted as u, v, w for the x, y, and z directions, respectively.

Calculating Nodal Displacements

The nodal displacements are determined by solving the weak form of the governing equations. A common method involves minimizing the potential energy of the system, as described in the equation below:

(2)

Once the nodal displacements are known, the strain values for each element can be calculated using the strain-displacement relation.

Stress Calculation and Discontinuities

Using the displacements and strain-displacement relation, the elements' stresses can be calculated. However, a direct application of stress-strain formulas often leads to discontinuities at element interfaces. This is because stress discontinuities can arise due to the interpolation functions used to approximate the stresses. Even though the equilibrium equations may not be satisfied internally at element interfaces, they are generally satisfied globally.

Visualizing Strain Continuity

To better understand this concept, consider a two-rod system as shown in the figures. The nodal displacements and element stresses for the system are as follows:

Hint: Consider the following parameters:

AEL 15×103 in-2 A 2.25 in2 L 20 in

When using an FEA solver, the strains are calculated at each element. However, the values at the inter-element boundaries are not necessarily continuous. This is illustrated in Fig. 1, where the strain values at each element are plotted. On the other hand, Fig. 2 shows the strains interpolated at the nodal points, which appears more continuous but is a visual trick for the eye.

Global vs Local Equilibrium

While FEA solutions are approximate and may not satisfy local equilibrium at element boundaries, they generally satisfy global equilibrium conditions. This is why FEA is a robust and widely used method for representing actual solutions. Refining the mesh can lead to more accurate and continuous stress distributions, as the approximation errors are reduced.

Conclusion and Future Improvements

In conclusion, FEA provides a reliable and efficient method to analyze complex structures, but it is important to be cautious about the approximations made. While stress continuity is not always guaranteed at element boundaries, the global accuracy of the solution is typically reassuring for practical engineering applications.

References:

Hallquist, J.O. 2006. LS-DYNA Theory Manual. Rao, S. S. 2011. The Finite Element Method in Engineering, 5th ed. Florida, USA: Elsevier Inc.