Finite Element Analysis (FEA)
Finite Element Analysis is a powerful numerical procedure than enables engineers to
acquire information about their designs that would be difficult, if not impossible, to
determine analytically. Finite Element Analysis (FEA) is used in virtually every industry
you can think of, however it is particularly valuable to engineers in the automotive and
aircraft industries.
To perform finite element analysis the engineer must provide the following information:
1. The geometry of the part to be analyzed
2. The material properties of the part to be analyzed
Strength Properties (Structural and Dynamic Analsysis--stresses, strains, mode
shapes, etc.)
Elastic Modulus, E
Shear Modulus, G
Poisson's ratio, ν
Thermal Properties (Heat transfer problems, thermal analysis)
Is the material homogenous? Is the material isotropic, anisotropic, orthotropic?
Is the material a composite?
3. The loads on the part.
4. How the part is constrained to resist loading.
Other things that the engineer must be able to determine include:
Is the loading static or time-variant?
Is it safe to assume that loads and stresses and/or deflections are related
linearly?
Do material properties change with deflection?
Do boundary conditions remain constant with loading?
The Art of Finite Element Analysis
Modeling for FEA requires a thorough understanding and accurate representation of the
part to be analyzed--otherwise the analysis is just a pretty picture of stress and/or strain
fields that are likely to be extremely inaccurate. However, accurate modeling is often
easier said than done, particularly where loading and boundary conditions are concerned.
The best modelers are those people that understand both the strengths and weaknesses
of FEA, have confidence and convincing evidence that the loads and boundary conditions
they have applied accurately model the circumstances, and… .that have a number of
years of experience in using FEA.
While we are beginning to understand and use FEA, we will analyze models that are
relatively easy to model and for which we can acquire analytical solutions for stresses
and strains. Modelers in industry don't generally have this luxury--otherwise they would
not need FEA software. However, it is more likely than not that the engineer modeling
with FEA will simplify the model to the degree possible before running the analysis.
How does FEA work?
A part's "stiffness" is related to a) its geometry and b) its material properties
For example, take a simple geometry like the one shown below:
F
y
:
x
z
This loading will cause this part to bend about the z-axis. It is intuitive that if the beam is
made of steel that it would be stiffer than a beam made of rubber. It is also clear that the
resistance to bending (stiffness) would be greater if the load were applied parallel to the
z-axis, causing bending about the y-axis.
y
x
z
F
In the equation shown below, the matrix K, represents a combination of geometry and
material properties resulting in stiffness.

K 
{x}= {F}
The x term is a vector of "displacement values" that are unknown before the FEA
program is run--these terms may be thought of as deflections. And the F vector contains
information about the loads on the part.
This "system" of equations is exactly like the scalar equation you learned about in
physics that says: a spring with a spring rate of K, will deflect an amount x, due to the
force, F, stretching (or compressing) the spring..
The way that FEA works, in general, is like this:
The geometry of the part is divided into thousands of little pieces called "elements". The
vertex of every element is called a node. Inside the software, there are equations, called
shape functions, that tell the software how to vary the values of x across the element.
Average values of x are determined at the nodes. In fact the only place that the engineer
can access values of stress and/or deflections are at the nodes. The finer the "mesh" of
elements, the more accurate the nodal values will be.
In addition to telling the software what kinds of loads are imposed on the part anad what
type of material the part is made of, the engineer must also tell the software how the part
resists the loads imposed on it. We recall well that every load acting on an object has an
equal but opposite load acting on it. For example, a round cantilevered beam that is
subjected to twisting, will resist the external twisting moment with equal but opposite
twisting at the wall. However, the way in which the FEA modeler would communicate this
information to the FEA software would be through the use of "boundary conditions."
Boundary conditions tell the FEA software how loading is resisted by constraining
displacements and rotations of certain nodes. In the case of the cantilevered round
beam, the engineer would constrain the nodes at the beam-wall interface by instructing
the software to not allow translation of the nodes at the wall in x, y, or z. Also, depending
on the orientation of the co-ordinate system, the engineer would need to instruct the
software not to allow twisting of nodes around two of the axis.
The following picture shows you a "discretized" shaft. Notice the triangular shaped
elements and where they meet (the nodes)
The more elements a model contains the more accurate the average value of the stress
or strain at the nodes. The trade off for more accuracy is processing time--the more
elements and nodes, the longer it takes to generate results.
Mesh size, loadings, and boundary conditions play a critical role in producing
accurate and reliable finite element models.
The next picture shows the loading (a torque around the z axis) and the boundary
conditions (no translations in x, y, or z; no rotations around y or x).