### Two-Dimensional Finite Elements, Case Study Problems by Kotur Raghavan

General

In an earlier article features and applications of 2D continuum elements were discussed ( https://www.fembestpractices.com/2020/10/2d-finite-elements-overview-by-kotur.html ). The user has three options for non-axisymmetric problems. They are Plane Stress, Plane Strain and Generalized Plane Strain. In this brief article case studies are presented.

Problem Definition and Approach

The case study problem is shown I the figure below. The object is an extrusion of a square section with a circular cutout. The ratio of length (L) to breadth (B) is a parameter for the study. Analyses are carried out for two different load conditions – uniform internal unit pressure and linear thermal gradient in the radial direction.

Fig. 1

Fig. 2

The approach is to carry out studies using three-dimensional elements for different length ratios and compare the results obtained from two-dimensional models. Because of three planes of symmetry, it is sufficient to analyse only one-eighth of the component. Appropriate symmetry constraints are invoked. The analysis domain and the mesh pattern are shown in the figure above. In the model, the plane FF is either the free end or the axially constrained end. The plane designated as SY is the symmetry plane. Results are presented for the equivalent (von Mises) stress and the axial stress.

As per definition, this corresponds to the case of plane strain. As the ends are constrained, the axial deformation is zero all along the length and therefore the axial strain is zero. The component will experience axial stress due to Poisson effect.

Fig. 3

In the figure above, stress distributions are shown. As can be observed the state of stress is invariant along the length. These distributions are for a length ratio of unity. However, the results will be same for any length ratio.

Fig. 4

In the figure above are shown the results using two-dimensional elements with plane strain formulation. The results are exactly matching with those from the 3D model.

If the ends are free, the component will experience zero axial stress at the ends. As explained in the earlier article, the component will have a central core portion over which the axial stress remains constant. The variations of axial, von Mises and Y-normal (hoop) stresses along the line AB (Fig. 2) are presented in the graph below (Fig. 5).

Fig. 5

Fig. 6

Fig. 7

In the figures above, Stress patterns near the free end and the symmetry end are presented. A point to be noted here is that the axial stress is not making any significant contribution to the equivalent stress.

In the figure below a result of two-dimensional analysis with plane stress model is presented. The result pertains to equivalent stress. This matches well with the free end result in the 3D analysis.

Fig. 8

In the figure below results for the axial stress and equivalent stress from two-dimensional generalized plane stress model are presented. These results are seen to be matching well with the results from the 3D model near the symmetry plane.

Fig. 9

Thus the two simpler 2D models are predicting exactly the results at extreme ends from the 3D model.

The same model has been studied for thermal loading condition. Temperature distribution as shown in the figure below has been applied.

Fig. 10

In the figures below distributions of axial and equivalent stresses are presented,

Fig.11

Fig. 12

The above are to be compared with results from two-dimensional models. They are presented in the figures below.

Fig. 13

Fig. 14

The results from the GPS mode are exactly matching with 3D model results near the symmetry plane. However, at the free end, the 3D model is showing higher von Mises stress at the edge centre. The reason for the discrepancy is not clear. It will be investigated. Importantly, the usefulness of the GPS model has once again been established.

For the case of thermal load case the plane strain model has not been taken up. The reason is that the results will be qualitatively same as for the pressure loading case.

Summary

o   The study has established the applications of two-dimensional continuum models. They can be used where the necessary conditions needed for modelling 3D components with 2D models are satisfied. Plane stress and plane strain models are generally well known. Generalized plane strain model, on the other hand, is relatively less familiar.

o    The main difference between a plane stress model and a plane strain model is the presence of the axial stress component. But one stress component alone does not dictate design. Designs are normally based on failure criteria such as von Mises.

o   Based on the foregoing the main recommendation is that the users may carry out analyses using plane stress as well as GPS models. The higher of the values may be taken as the one governing design. This is valid irrespective of the length of the component if the ends are free.

o   If the ends are constrained there is no ambiguity at all and a plane strain model will give correct results.

Note

The results presented in this study are partially available in a NAFEMS publication authored by Angus Ramsay (NBC-03, The NAFEMS Benchmark Challenge, Volume 1, NAFEMS, 2018). The studies therein cover circular and elliptic cross sections for internal pressure load case only. That paper did not discuss the use of GPS formulation.