### 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.

**Pressure Load, Constrained Ends**

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.

**Pressure Load, Free Ends.**

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.

**Thermal Loading**

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.

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