What
is Finite Element Analysis
Finite element analysis is a computerized
method for predicting how a real world object will react to forces, heat,
vibration, etc., in terms of whether it will break, wear out, or work the way
it was designed. It is called analysis, but in the product design cycle it is
used to predict what is going to happen when the product is used. The finite
element method works by breaking a real object down into a large number (1,000s
to 100,000s) of elements, such as little cubes. The behavior of each little
element, which is regular in shape, is readily predicted by set mathematical
equations. The computer then adds up all of the individual behaviors to predict
the behavior of the actual object. The finite in finite element analysis comes
from the idea that there are a finite number of elements in a finite element
model. Previously, engineers employed integral and differential calculus, which
breaks objects down into an infinite number of elements.
The finite element method is employed to
predict the behavior of things with respect to virtually all physical
phenomena:
Mechanical stress (stress analysis)
Mechanical vibration
Heat transfer (conduction, convection and
radiation)
Fluid flow (Both liquids and gaseous fluids)
Various electrical and magnetic phenomena
Acoustics
FEA Theory
In 1678, Robert Hooke set down the basis for
modern finite element stress analysis with Hooke's Law. Simply, an elastic body
stretches (strain) in proportion to the force (stress) on it. Mathematically:
F=kx where
F = force
k = proportional constant
x = distance of stretching
This is the only equation you needed to
understand finite element stress analysis. Hooke proved the equation by using
weights to stretch wires hanging from the ceiling. This experiment is repeated
every year in virtually every high school laboratory by students who study
physics.
Imagine that a coffee cup is sitting on a
table. It is broken down into 2,000 little brick elements. Each element has 8
corners, or nodes. All nodes on the bottom of the coffee cup are fixed (all
translations and rotations are constrained), so they cannot move. Now, let us
press down on just one node near the top of the cup.
That one node will move a little bit because
all materials have some elasticity. That movement would be described by F = kx
for that element except that other elements are in the way or are tending to
hold it back. In fact, as the force is transmitted through the first element,
it spreads out to other nodes. Without a computer, we would lose track of
events very quickly.
In the finite element method, a step occurs
called element stiffness formulation. What happens is that a stiffness, k, is
created for the relationship between every node on each element. Thus, every
node is connected to every other node on each element by a spring, which will
behave like F = kx. By so doing, we reduce the coffee cup to a large system of
springs. When the analysis is done a value for the translation, x, and force,
F, is determined for each node by the formula F = kx. Note: F and x are vectors
as each has a value and a direction. In the final step, results evaluation, the
stresses are determined by knowing the force at each node and the geometry of
each element.
Other physical phenomena such as heat
transfer, fluid flow, and electrical effects can be handled in a similar way by
using the pertinent governing equations.
Nodes and Elements
What is a Node
A node is a coordinate location in space
where the degrees of freedom (DOFs) are defined. The DOFs for this point
represent the possible movement of this point due to the loading of the
structure. The DOFs also represent which forces and moments are transferred
from one element to the next. The results of a finite element analysis,
(deflections and stresses), are usually given at the nodes.
In the real world, a point can move in 6
different directions, translation in X, Y, and Z, and rotation about X, Y, and
Z. In FEA, a node may be limited in the calculated motions for a variety of
reasons. For example, there is no need to calculate the out of plane
translation on a 2-D element; it would not be a 2-D element if its nodes were
allowed to move out of the plane.
The DOF of a node (which is based on the
element type) also relates what types of forces and restraints are transmitted
through the node to the element. A force (axial or shear) is equivalent to a
translation DOF. A moment is equivalent to a rotational DOF. Thus, to transfer
a moment about a certain axis, the node must have that DOF. If a node does not
have that rotational DOF, then applying a moment to the node will have no
effect on the analysis. Likewise, restraining that node with a rotational
boundary condition will have no effect; the node does not have the ability to
transmit the moment.
What is an Element
An element is the basic building block of
finite element analysis. There are several basic types of elements. Which type
of element for finite elements analysis that is used depends on the type of
object that is to be modeled for finite element analysis and the type of
analysis that is going to be performed.
An element is a mathematical relation that
defines how the degrees of freedom of a node relate to the next. These elements
can be lines (trusses or beams), areas (2-D or 3-D plates and membranes) or
solids (bricks or tetrahedrals). It also relates how the deflections create
.stresses.
Types of Finite Elements
1-D (Line) Element
2-D (Plane) Element
3-D (Solid) Element
-Typical Steps in FEA using ALGOR
In a typical stress analysis, there is a
basic set of steps that the analysis usually follows:
1. Create a mesh (a grid of nodes and
elements) that represents the model
2. Define a unit system
3. Define the model's analysis parameters
4. Define the element type and parameters
5. Apply the loads and the constraints
6. Assemble the element stiffness matrices
7. Solve the system of linear algebraic
equations
8. Calculate the results
9. Review the results
10. Generate a report of the analysis results
These steps are usually broken up into three
stages:
Setting up the model: Steps 1-5
Figure 1:
FEA model, including mesh, of a three-part assembly of a circular rod bonded to
two brackets, which are fixed using boundary conditions on the edges of the
holes located on their flat ends (red triangles on the underside). In the
Static Stress with Linear Material Models analysis, a surface force of 5.0×108
dynes is applied on one end of the rod (yellow arrows).
Analyzing
the model: Steps 6-8 (These steps are automatically performed by ALGOR)
Figure 2: von Mises
stress distributions obtained from the Static Stress with Linear Material
Models analysis of a three-partassembly of a circular rod bonded to two
brackets. The figure on the left is for the model without microholes, whereas
that on the right is for the model with micromoles. The latter predicts a
maximum stress 0.31% greater than the former
Figure 3:
Temperature distributions obtained from a Steady-State Heat Transfer analysis
for ring model with the base maintained at 100 °F, and a heat flux of 0.642 BTU
/ (sec·in2) applied to the inner surface. The figure on the left is for the
model without microholes, whereas that on the right is for the model with
microholes. The latter predicts a maximum temperature 0.349 °F greater than the
former.
Results evaluation:
Steps 9 and 10
A Brief History of the FEM
· 1943 ----- Courant (Variational methods)
· 1956 ----- Turner, Clough, Martin and
Topp (Stiffness)
· 1960 ----- Clough (“Finite Element”,
plane problems)
· 1970s ----- Applications on mainframe
computers
· 1980s ----- Microcomputers, pre- and
postprocessors
· 1990s ----- Analysis of large structural
systems
Available Commercial FEM Software
Packages
· ANSYS (General purpose, PC and
workstations)
· SDRC/I-DEAS (Complete
CAD/CAM/CAE package)
· NASTRAN (General purpose
FEA on mainframes)
· ABAQUS (Nonlinear and
dynamic analyses)
· COSMOS (General purpose
FEA)
· ALGOR (PC and workstations)
· PATRAN (Pre/Post
Processor)
· HyperMesh (Pre/Post
Processor)
· Dyna-3D (Crash/impact
analysis)
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