Teaching
Finite Element Analysis to Second Year Students

Marshall F. Coyle

1031 Edgecomb Ave.

York, PA 17403-3398

Email: mfc5@psu

Christal
G. Keel

Pennsylvania State University-York

Email: christal@keel.org

ABSTRACT

Finite element analysis (FEA) is a powerful analytical tool used to evaluate
structural, dynamic, thermal, fluid, and electrical engineering problems.
In the past, only specialists with access to mainframes conducted finite element
analyses due to the massive processing power required. However, the recent
advances in microcomputer technology allow this processing capability to be
available to virtually anyone. Engineering students can now solve complex
problems that would not be feasible or practical to solve by hand at a much
earlier point in the curriculum. Still, a person using FEA software who does
not have a clear understanding of the basic engineering concepts could obtain
erroneous solutions, leading to a detrimental outcome. This paper discusses
the justification for offering FEA to second year students, as opposed to
the current fourth-year placement we see most often. Included are examples
of exercises that will attempt to reinforce to the student the importance
of rigorous attention to the fundamental engineering concepts crucial to any
analysis.

**INTRODUCTION**

Finite element analysis software allows us to simulate the performance
of a structure or component. Analyses can be conducted on structures of practically
any geometry, with countless degrees of freedom.^{1, 2, 3} Because
of the recent advances in the user interface of many FEA packages, it is now
practical for students with only a basic knowledge of FEA and the principles
of mechanics to conduct complex analyses.^{2} The availability of
tutorials, workbooks, and online support contribute to a user-friendly environment
for the student.^{5, 6}

Until now, most introductory courses in finite
elements have been offered to engineering students in the fourth year of study,
with more advanced theory or application being reserved for graduate programs.
However, some engineering professors believe that the time has come to introduce
this technology much earlier in the curriculum. In the beginning, care must
be taken to ensure that the student has a firm understanding of Statics and
Strength of Materials. Using finite element analysis as a tool, teachers can
enhance the comprehension of these courses, as well as stress proper planning
of a problem and setting reasonable expectations as to the outcome.^{2}
Seeing the physical representation of these basic theories can lead to an
increased appreciation for the student
of forces, boundary conditions, and the significance of material properties.^{7}
Applying a finite element analysis to a common truss problem, for example,
can help the student visualize the analysis in a way previously not possible,
as well as demonstrate the importance of boundary conditions and how they
can affect the solution.

**PRESENT TREND**

In an effort to get a general feel for when (and if) FEA is being taught
in engineering programs, an informal survey was conducted. This survey was
administered using an engineering technology list server containing roughly
2,000 people, representing over 500 educational institutions, along with a
list of people recognized as experts in the finite element analysis field.^{4}
The survey asked educators 1) whether they currently offered finite element
analysis in their curriculum, 2) when it was presented, 3) whether it was
an elective, and 4) which discipline they taught.

Even though the response was small (18 responses), we feel that it reflects
the current trend toward FEA being offered in engineering programs. The results
indicate that roughly 78% of the universities currently offer the FEA course
in the junior or senior level of a four-year program. In roughly half of these
colleges, FEA was offered as an elective. A finite element analysis course
was offered in virtually all of the mechanical and mechanical technology programs,
and most of the civil engineering curriculums. Curiously, only a few responses
related to structural engineering programs offered finite element analysis
courses. The prerequisites varied slightly from one discipline to the other:
mechanical programs required Strength of Materials and/or Statics in 85% of
the cases. Other fields emphasized more engineering math and some computer
aided design (CAD) exposure.

**WHY INTRODUCE FEA EARLY IN
THE CURRICULUM?**

It is typical for Mechanical Engineering disciplines to teach Statics
and Strength of Materials early in the curriculum. For many students, visualizing
what is actually happening within a structure is a common problem, particularly
in 3-D situations. Using finite element analysis not only allows the student
to *see* the resulting stresses and deformations, but also serves to
emphasize the critical nature of applying proper boundary conditions when
solving a problem.

The assumptions made in Strength of Materials’ beam problems regarding
stress and strain distributions can now be seen in a visually dramatic way.
A typical Statics example is a truss beam problem utilizing the assumption
that all members are two force members joined by frictionless pins. By reproducing
the truss beam using finite element analysis software, the student can prove
that this assumption is valid. Finite element analysis can also be used to
verify the hand calculations the student has performed on a problem. Until
now, the perception has been that the FEA was solely the province of specialists.^{2}
However, the rapid expansion in the microprocessor technology, as well as
the straightforward user interface of many finite element analysis packages,
is causing a gradual change in this opinion. Home computers are powerful enough
to rival the mainframes of just a few years ago. Many software manufacturers
offer real-time online support, context-sensitive help, tutorials, and better
documentation than was ever available before. Now, novice FEA users who understand
the basic theories, methodologies, and pitfalls of finite element analysis
can perform a useful and acceptable model. Engineers just entering the job
market are being exposed to basic forms of FEA embedded in CAD packages in
the form of behavioral modeling features. With the job market clamoring for
entry-level engineers, it becomes even more imperative that students be exposed
to finite element analysis as early as is practical.

The fact is that FEA may not be effortless, but it is presented in a more
straightforward manner than it has ever been before. However, we cannot allow
this fact to lead us to underestimate the complexity of the technology. Obtaining
a good looking model with applied loads in no way decreases the need to *understand*
the problem being approached or the impact of the assumptions used to build
it. Obtaining an attractive looking mesh may be relatively trouble-free, but
in no way guarantees us an accurate analysis. Obtaining a solution with the
press of a few buttons may be tremendously satisfying, but cannot deliver
a true understanding of the workings of the technology. Finally, obtaining
a result is only half the battle: interpreting the result and having the skill
to know whether it makes *sense* will be the real benchmark of a finite
element analysis.

**METHODS USED TO PRESENT FEA
TO STUDENTS**

Most schools place Statics in the first or second year of a mechanical
engineering program. It is assumed that a fundamental math and physics background
has been acquired to understand the basic principles of Newtonian mechanics.
The concepts of free body diagrams, forces, and reactions must be well understood
at this stage. Ideally, the student will have had some introduction into the
analysis of structures and the forces in beams. It would be best if these
principals were reviewed to ensure that the class, as a whole, is up to speed
and can proceed at the same level. This is where the FEA will begin to enhance
the classroom activities. Quite often, the student has had some introductory
courses in CAD, so a short primer on the basic navigation of your specific
FEA software should be sufficient. This kind of “hands-on” approach to the
software allows the student to become comfortable with the software.

During this introduction, call attention to the three phases of conducting a finite element analysis: preprocessing, solution, and post-processing. Also, at this time, the building blocks of finite element models (nodes and elements) should be explained. The next step is to begin with a basic problem, model it in the finite element software, and compare the FEA solution with an analytical solution. The 2-D axial loaded rod shown in Figure 1 illustrates a good initial finite element analysis exercise. This exercise demonstrates Strength

Figure 1: 2-D Axially Loaded Rod.

In this example, Figure 1, the rod is broken up into three segments (elements)
having various lengths (l_{i}) and cross-sectional areas (A_{i}).
The boundary conditions at node 4 can be altered by either applying a load (P)
or displacement (u_{y}) along with applying a change in temperature
(DT) to the model. The materials properties, such as Young’s
Modulus (E_{i}) and coefficient of thermal expansion (a_{i}), may also
be varied. The student is required to validate his or her modeling techniques
by comparing the FEA results with the following Strength of Material equations:

s_{i} = F_{i}/A_{i
}

d_{i} = (F_{i}
l_{i})/(A_{i} E_{i}) + a_{i} DT l_{i}

µ_{i} = d_{i} / l_{i}

s_{i} = E_{i}(µ_{i }- a_{i} DT)

The variables should be initially set to yield solutions that are relatively computationally easy; i.e., A

Another useful exercise can be to perform a static analysis of a universal truss structure, such as the Warren Truss illustrated in Figure 2.

Figure 2: Warren Truss.

Students typically encounter some manner of truss problem in their first
Statics class, accompanied by the standard truss assumption; i.e., that all
the joints are smooth, frictionless pins. However, anyone can tell by looking
at an *actual* truss structure that this is not the case and this discrepancy
often confuses many students and shrouds the process with suspicion. Have the
students’ first model the truss using link elements and then model the same
structure as a frame using beam elements. It can clearly be shown that the difference
between the two results is negligible. This method serves to validate the general
truss assumptions as well as to alleviate the tendency of some students to question
every supposition.

Engineering students are usually exposed to concepts such as Saint-Venants’ Principle, stress/strain distributions, and stress concentration factors. Contour plots from FEA of solid models can vividly demonstrate these concepts

**CONCLUSION**

Graduating engineers who stay in the engineering field will eventually
find themselves involved with FEA. The impressive graphical and animation
features associated with FEA programs can enhance a students’ physical feel
for structural problems. The student can visually see the effects of varying
the boundary conditions. FEA can also demonstrate or validate common assumptions
engineers use to simplify calculations. FEA can be used as a primer or virtual
lab for a Strength of Materials class; however, using finite element software
without really appreciating the fundamental engineering tenets can result
in profoundly misleading or inaccurate solutions. Therefore, it is important
that the student has a keen awareness of the power and potential misuse
of FEA.

**REFERENCE**

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