While computers keep getting faster, and finite element software more efficient, there is still a lot of value in being able to simplify complex models. Finite elementbased stress analyses of axisymmetric structures range in size and materials from concrete containment buildings, to spinning steel engine parts, to elastomeric replacement blood vessels. My previous blogs 2D or not 2D? That is Often the FEA Modeling Question and Detailed Modeling of Threaded Connections have provided illustrations of axisymmetric geometry under axisymmetric loads. This post covers the ability to simulate nonaxisymmetric responses with an axisymmetric model, presenting two common approaches.
Three options for modeling the 3D geometric loading of axisymmetric geometry are:
1. Model in 3D
2. Use pure harmonic axisymmetric elements
3. Use general axisymmetric elements
Model in 3D:
While a 3D model is often the easiest method, since it avoids the need for simplifying assumptions, it is also by far the most computationally expensive. Consider a 360degree model created by sweeping a 2D mesh with each 3D element spanning 15 degrees. The number of degrees of freedom in the 3D model compared to the 2D axisymmetric model is either 72 or 90 times larger, depending upon the element order. Assuming that the CPU time scales roughly by the degrees of freedom ratio squared, which is a reasonable estimate if the 3D direct solver is forced to run out of core, the resulting increase in CPU time over the 2D model is between 5000 and 8000 times. Take the same axisymmetric model that ran in an hour, and it would run in 200 to 300 days in 3D!
Adding constraints to prevent torsional rigid body motion can be tricky in 3D modeling. The axisymmetric model inherently has the advantage of eliminating rotational and translational rigid body modes, simplifying these boundary condition requirements. The biggest advantage of 3D modeling is that there are no restrictions on the loading, materials and analysis type.
However, 3D loading does not always require 3D geometry. The approaches discussed below illustrate options for computing a 3D response with 2D elements. These pseudo 3D solutions require slightly more computational effort than the pure axisymmetric models, but still result in orders of magnitudes of CPU savings over the general 3D simulation.
Pure Harmonic Axisymmetric Elements:
Pure harmonic elements, such as the ANSYS PLANE25, allow for nonaxisymmetric loading and supports to be simulated via a series of harmonic functions (Fourier series), that are combined using superposition. These elements do not allow nonlinearities such as contact, nonlinear materials, or large deflection effects in the simulations. Analyses can be both static and dynamic, including even harmonic analyses (sinusoidally time varying loads) and thus resulting in a harmonic analysis of harmonic elements which would really test your analysis skills!
The most common use of these elements is the structural dynamics application to extract nonaxisymmetric natural frequencies and mode shapes. Figure 1, above, illustrates the nodal diameter=2 mode shape in an engine cover at several different PLANE25 element crosssections, along with the equivalent response captured with a 3D model.
In static analyses, any spatiallyvarying load can be applied to the geometry. This load is subsequently transformed into its Fourier components, each component is solved independently, and load combinations are used to combine the results. Figure 2 provides examples of modes 0 and 2 symmetric and antisymmetric force loading directions applicable to the ANSYS PLANE25 model. Theoretically, any arbitrary loading can be developed via superposition of the Fourier seriesbased loading. However, the input and validation can be challenging, making this application rarely used in current practice, since the general axisymmetric elements (discussed next) are much easier to use. Nevertheless, for modal analyses, PLANE25 elements are easy to define and provide a huge CPU savings over 3D modeling, and thus are still very popular.
Figure 2: Example Nodal Loads using Harmonic Elements, Corresponding Mode & Symmetry Input

General Axisymmetric Elements:
The second approach to a simplified 2D approach is the general axisymmetric element. An example of this element is ANSYS’ SOLID272, which utilizes nodal planes to define the harmonic response and loading, while providing a subset of nonlinear analysis capabilities. Figure 3 illustrates example applications.
Strengths of this methodology are:
 Computationally faster than 3D solids.
 No mesh discretization error in the circumferential direction, which can occur in 3D models, depending on mesh size relative to the mode shape of interest.
 3D visualization of model, loading and results.
 Support of large deflection.
 Support of nonlinear material properties (plasticity and hyperelasticity with UP formulation).
 Support of axisymmetric and nonaxisymmetric 3D contact (pointtopoint and pointtosurface elements for small deflections only).
 Nonaxisymmetric pressure and temperature loads are valid.
 Rotor Dynamics (Coriolis)  No need to calculate equivalent beam sections.
Limitations:
 Computationally much slower than the pure harmonic PLANE25.
 Limit of 12 nodal planes allows only up to 6 nodal diameters (05).
 Pointtosurface contact is not as robust as surfacetosurface modeling.
Comparative FEA solutions using the modal analysis model illustrated in Figure 1 are tabulated in Table 1, to provide a quantifiable comparison among the three methods described above. Conclusions from these simulations are:
 The first 15 frequencies are nearly identical for the three methods.
 The general harmonic method (ANSYS’s SOLID272 element) is not capable of extracting any modes with nodal diameters greater than 5.
 The pure harmonic method captures the first 25 modes with roughly 400 times less CPU time than the 3D model.
 The general harmonic method is 10 times slower than the pure harmonic method, but does provide nice 3D graphics as illustrated in the lower right image of Figure 3.
I am sure that there are many more applications of axisymmetric harmonic models that I have not touched on, and I welcome your input and feedback!