In 1960, the first documented application named “the finite element method” was a 2D simulation of a gravity dam (Clough, 1960). This civil engineering application was part of the beginning of a new era in structural analysis and design. Yet in 2017, 2D stress analyses appear to have become a lost art. Increases in computational resources and software efficiencies have made 3D simulations the norm. But, many lessons can still be learned via 2D analyses where high-fidelity nonlinear structural response or automated design optimization can be captured in a reasonable time.

Most finite element analysis (FEA) simulations start from CAD drawings (see the author’s blog post, Practical FEA Simulations, at https://caeai.com/blog/practical-fea-simulations). Since the drawing file is often a 3D rendering, the analyst will often start by creating a 3D finite element model. Yet for many problems, a 2D simulation will provide quicker, more accurate results, allowing for additional design iterations and even design optimization in the same time required to perform a single 3D analysis. At a minimum, when the required simulations are highly nonlinear, lessons learned from 2D simulations can help streamline the input assumptions and convergence efficiency of the future 3D run. Two-dimensional loading can be applied in many forms, including in-plane forces, moments, pressures, accelerations, and temperatures. Special-purpose harmonic elements can also be used to represent non-axisymmetric loading on an axisymmetric FEA model using a Fourier series-based load case superposition approach.

Two-dimensional analyses can be used to model thin-walled structures using a plane stress assumption, very long construction systems using a plane strain assumption, and buildings of revolution using an axisymmetric solution. All three simulation types use the same 2D FEA mesh but different element stiffness formulations to simulate the physical differences, as summarized below.

**Plane stress modeling**

Plane stress is applicable for thin- to moderately thick-walled geometry, where zero normal and shear stress perpendicular to the structural model is assumed. The 2D plane stress FEA model has to lie in a single plane at Z=0 in most codes, but the geometry it represents does not have to be of constant thickness. Modeling the life of a prestressed concrete beam, simulating the long-term effects of concrete creep and shrinkage along with thermal-induced cracking, would be an example where a 2D simulation is of great value since this type of detailed analysis requires a time-dependent solution that requires many equilibrium iterations (see Figure 1 for a 2D model of a standard four-point bend test).

Plane stress simulations allow for variable thickness inputs that can be combined with plane strain and/or axisymmetric models. When combining plane stress and axisymmetric elements, the plane stress element thickness must be consistent with the axisymmetric stiffness formulation, which is typically the full 360-degree stiffness. Thus, when representing a planer structure such as ribs or buttresses in a domed or cylindrical structure, the thickness defined for these non-axisymmetric components should be the combined thicknesses of all the ribs in the model.

An explicit example would be modeling an axisymmetric hyperbolic cooling tower that includes a base beam foundation superstructure. The photo on the left of Figure 2 illustrates the axisymmetric superstructure and the supporting beam framed base, which allows for a clear passage of intake air. The images on the right of Figure 2 illustrate the 2D finite element model used in preliminary design. In this case, 2D beam elements are used to represent the combined stiffness of all the support beams where increased thickness is used to represent the equivalent stiffness of the entire array of support beams.

This 2D analysis model would be ideal for computing displacements and nominal design stresses under construction sequence loading where concrete curing is accounted for while minimizing the computing resources needed to simulate the step-by-step nonlinear “birth analysis” process.

**Plane strain modeling**

Plane strain assumes the out-of-plane geometry is large and/or constrained and that loading does not vary in the out-of-plane direction (Z) such that Z displacements are neglected. Common applications of plane strain include the analysis of dams and tunnels. The out-of-plane strain is either prescribed to be zero or held at a constant value in the special case of “generalized plane strain.” The physical geometry does not have to be rectangular, but it must be defined by a zero or constant rate of out-of-plane response.

The generalized plane strain (constant out-of-plane strain) is typically required for temperature loads that would create near infinite normal stresses with the zero Z plane strain modeling assumption. Figure 3 illustrates a generalized plane strain model of a bridge deck cross-section. This simplified model is ideal for developing displacements and nominal stresses under cyclic thermal loading where nonlinear material response such as concrete creep and shrinkage might be simulated. While the model is predominately constructed from plane strain elements, the lateral braces are simulated with plane stress elements with reduced stiffness.

**Axisymmetric modeling**

For structures of revolution, including domes, pipes, piles, silos, or tanks subjected to axisymmetric loading, the 2D axisymmetric element formulation can save considerable computational time with increased stress accuracy. A parametric axisymmetric model of a concrete silo is used to illustrate the value of this 2D simulation. Figure 4 illustrates the design analysis performed, along with an example response surface interaction plot illustrating the sensitivity of adjusted design parameters on the structural response of the silo. Highlights from these design simulations include:

- A fully parametric model of the silo cross section is created as illustrated in the left image of Figure 4 to allow for variation in any design dimension.
- Loading is defined via surface effect elements on the free internal model edges (representing the surfaces) utilizing a linearly increasing pressure representing the silo payload, such as grain.
- Material properties can be either linear or nonlinear depending on the analysis objective.
- Since the model solution time is very fast, a series of simulations were performed utilizing an automated design of experiments technique where multiple input and output variables can be monitored.
- In the bottom center images of Figure 4, the impact on the hopper height and ring beam to wall radii are defined as the input variables for this example.
- The output response monitored in this case is the maximum wall displacement. The right image shows the variation of displacement in the cross-section where the image is expanded to a 3D sector for better visualization.
- The response surface graph in the top center image illustrates the highly nonlinear interaction diagram documenting the variation in maximum displacement versus the associated changes in the parametric input variables.
- Utilizing a simple 2D model as shown allows the designer to quickly evaluate the impact of many input design variables that will be used to develop a more stable design. Once the stable design is developed, the 2D model can be easily revolved into a 90-degree or 180-degree sector upon which non-axisymmetric controlling wind loading could be simulated.

These are just a few of the many options in 2D finite element modeling. I welcome comments and/or examples that illustrate the value on performing 2D FEA simulations.

Cloud, R.W., 1960, The Finite Element Method in Plane Stress Analysis, Proc. 2nd ASCE Conf. On Electronic Computation, Pittsburg.

**Peter R. Barrett, P.E., **vice president of CAE Associates (https://caeai.com), manages consulting engineering services and software training. With more than 30 years of experience in thermal-structural nonlinear and dynamic analysis applications using the finite element method, his structural engineering applications include Nuclear, Aerospace, Biomedical, and Offshore structures.