19-10-2012, 03:50 PM
Finite Element Analysis
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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 be obtained 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 important problem 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 _nite element codes should plan their strategy toward this end,
supplementing the computer simulation with as much 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 _nd it e_ective to program their own code. A number of prewritten commercial
codes are available, representing a broad price range and compatible with machines from microcomputers
to supercomputers1. However, users with specialized needs should not necessarily
shy away from code development, and may _nd the code sources available in such texts as that
by Zienkiewicz2 to be a useful starting point. Most _nite element software is written in Fortran,
but some newer codes such as felt are in C or other more modern programming languages.
In practice, a _nite element analysis usually consists of three principal steps:
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 _xed displacements, and others
will have prescribed loads. These models can be extremely time consuming to prepare,
and commercial codes vie with one another to have the most user-friendly graphical \preprocessor"
Matrix analysis of trusses
Pin-jointed trusses, discussed more fully in Module 5, provide a good way to introduce FEA
concepts. The static analysis of trusses can be carried out exactly, and the equations of even
complicated trusses can be assembled in a matrix form amenable to numerical solution. This
approach, sometimes called \matrix analysis," provided the foundation of early FEA development.
Matrix analysis of trusses operates by considering the sti_ness of each truss element one
at a time, and then using these sti_nesses to determine the forces that are set up in the truss
elements by the displacements of the joints, usually called \nodes" in _nite element analysis.
Then noting that the sum of the forces contributed by each element to a node must equal the
force that is externally applied to that node, we can assemble a sequence of linear algebraic
equations in which the nodal displacements are the unknowns and the applied nodal forces are
known quantities.
General Stress Analysis
The element sti_ness matrix could be formed exactly for truss elements, but this is not the case
for general stress analysis situations. The relation between nodal forces and displacements are
not known in advance for general two- or three-dimensional stress analysis problems, and an
approximate relation must be used in order to write out an element sti_ness matrix.
In the usual \displacement formulation" of the _nite element method, the governing equations
are combined so as to have only displacements appearing as unknowns; this can be done by
using the Hookean constitutive equations to replace the stresses in the equilibrium equations by
the strains, and then using the kinematic equations to replace the strains by the displacements.
Stresses around a circular hole
We have considered the problem of a uniaxially loaded plate containing a circular hole in previous
modules, including the theoretical Kirsch solution (Module 16) and experimental determinations
using both photoelastic and moire methods (Module 17). This problem is of practical importance
|- for instance, we have noted the dangerous stress concentration that appears near rivet holes
| and it is also quite demanding in both theoretical and numerical analyses. Since the stresses
rise sharply near the hole, a _nite element grid must be re_ned there in order to produce
acceptable results.