Master De Ingenieria Mecanica Y Material

Finite Element Analysis
David Roylance
Department of Materials Science and Engineering
Massachusetts Institute of Technology
Cambridge, MA 02139
February 28, 2001

Introduction
Finite element analysis (FEA) has become commonplace in recent years, and is now the basis
of a multibillion dollar per year industry. Numerical solutions to even very complicated stress
problems can now beobtained routinely using FEA, and the method is so important that even
introductory treatments of Mechanics of Materials – such as these modules – should outline its
principal features.
In spite of the great power of FEA, the disadvantages of computer solutions must be kept in
mind when using this and similar methods: they do not necessarily reveal how the stresses are
influenced by importantproblem variables such as materials properties and geometrical features,
and errors in input data can produce wildly incorrect results that may be overlooked by the
analyst. Perhaps the most important function of theoretical modeling is that of sharpening the
designer’s intuition; users of finite element codes should plan their strategy toward this end,
supplementing the computer simulation with asmuch closed-form and experimental analysis as
possible.
Finite element codes are less complicated than many of the word processing and spreadsheet
packages found on modern microcomputers. Nevertheless, they are complex enough that most
users do not find it effective to program their own code. A number of prewritten commercial
codes are available, representing a broad price range and compatiblewith machines from microcomputers to supercomputers1. However, users with specialized needs should not necessarily
shy away from code development, and may find the code sources available in such texts as that
by Zienkiewicz2 to be a useful starting point. Most finite element software is written in Fortran,
but some newer codes such as felt are in C or other more modern programming languages.
Inpractice, a finite element analysis usually consists of three principal steps:
1. Preprocessing: The user constructs a model of the part to be analyzed in which the geometry is divided into a number of discrete subregions, or “elements,” connected at discrete
points called “nodes.” Certain of these nodes will have fixed displacements, and others
will have prescribed loads. These models can beextremely time consuming to prepare,
and commercial codes vie with one another to have the most user-friendly graphical “preprocessor” to assist in this rather tedious chore. Some of these preprocessors can overlay
a mesh on a preexisting CAD file, so that finite element analysis can be done conveniently
as part of the computerized drafting-and-design process.
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2

C.A. Brebbia, ed., FiniteElement Systems, A Handbook, Springer-Verlag, Berlin, 1982.
O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method, McGraw-Hill Co., London, 1989.

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2. Analysis: The dataset prepared by the preprocessor is used as input to the finite element
code itself, which constructs and solves a system of linear or nonlinear algebraic equations
Kij uj = fi
where u and f are the displacements andexternally applied forces at the nodal points. The
formation of the K matrix is dependent on the type of problem being attacked, and this
module will outline the approach for truss and linear elastic stress analyses. Commercial
codes may have very large element libraries, with elements appropriate to a wide range
of problem types. One of FEA’s principal advantages is that many problem types canbe
addressed with the same code, merely by specifying the appropriate element types from
the library.
3. Postprocessing: In the earlier days of finite element analysis, the user would pore through
reams of numbers generated by the code, listing displacements and stresses at discrete
positions within the model. It is easy to miss important trends and hot spots this way,
and modern codes use...