Finite element analysis (FEA) involves breaking a model down into small pieces called finite elements. FEA was first developed in 1943 and involved numerical analysis techniques. By the 1970s, FEA was used primarily by aerospace, automotive, and defense industries due to the high cost of computers. Modern FEA involves preprocessing like meshing a model, applying properties and boundary conditions, solving the model using software, and postprocessing to analyze results like stresses and displacements.

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History of Finite Element Analysis
Finite Element Analysis (FEA) was first developed in 1943 by R.
Courant, who utilized the Ritz method of numerical analysis and
minimization of variational calculus.
A paper published in 1956 by M. J. Turner, R. W. Clough, H. C.
Martin, and L. J. Topp established a broader definition of
numerical analysis. The paper centered on the "stiffness and
deflection of complex structures".
By the early 70's, FEA was limited to expensive mainframe
computers generally owned by the aeronautics, automotive,
defense, and nuclear industries. Since the rapid decline in the cost
of computers and the phenomenal increase in computing power,
FEA has been developed to an incredible precision.

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Basics of Finite Element Analysis
Why FEM ?
• Modern mechanical design involves
complicated shapes, sometimes made of
different materials.
• Engineers need to use FEM to evaluate their
designs.

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Basics of Finite Element Analysis
FEA Applications
• Evaluate the stress or temperature
distribution in a mechanical component.
• Perform deflection analysis.
• Analyze the kinematics or dynamic response.
• Perform vibration analysis.

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Finite element analysis starts with an approximation of the region of
interest into a number of meshes (triangular elements). Each mesh is
connected to associated nodes (black dots) and thus becomes a finite
element.
Basics of Finite Element Analysis
Consider a cantilever beam shown.

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Basics of Finite Element Analysis
• After approximating the object by finite elements,
each node is associated with the unknowns to be
solved.
• For the cantilever beam the displacements in x and
y would be the unknowns.
• This implies that every node has two degrees of
freedom and the solution process has to solve 2n
degrees of freedom.
• Once the displacements have been computed, the
strains are derived by partial derivatives of the
displacement function and then the stresses are
computed from the strains.

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Formulation of the Finite Element Method
• The classical finite element analysis code (h version)
The system equations for solid and structural
mechanics problems are derived using the principle of
virtual displacement and work (Bathe, 1982).
• The method of weighted residuals (Galerkin Method)
weighted residuals are used as one method of finite
element formulation starting from the governing differential
equation.
• Potential Energy and Equilibrium; The Rayleigh-Ritz
Method.
Involves the construction of assumed displacement field.
Uses the total potential energy for an elastic body

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Finite Element Analysis
• Pre-Processing
• Solving Matrix (solver)
• Post-Processing
FEA requires three steps
FEA is a mathematical representation of a physical system
and the solution of that mathematical representation

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FEA Pre-Processing
Mesh
Mesh is your way of communicating geometry to
the solver, the accuracy of the solution is primarily
dependent on the quality of the mesh.
The better the mesh looks, the more accurate the
solution is.
A good-looking mesh should have well-shaped
elements, and the transition between densities
should be smooth and gradual without skinny,
distorted elements.

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FEA Pre-Processing - meshing
The mesh transition from .05 to .5 element size without control of transition
(a) creates irregular mesh around the hole which will yield disappointing
results.

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FEA Pre-Processing
Finite elements supported by most finite-element codes:

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FEA Pre-Processing – Elements
Beam Elements
Beam elements typically fall into two categories; able to
transmit moments or not able to transmit moments.
Rod (bar or truss) elements cannot carry moments.
Entire length of a modeled component can be captured with a
single element. This member can transmit axial loads only and
can be defined simply by a material and cross sectional area.

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FEA Pre-Processing – Elements
The most general line element is a beam.
(a) and (b) are higher order line elements.

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FEA Pre-Processing – Elements
Plate and Shell Modeling
Plate and shell are used interchangeably and refer to surface-
like elements used to represent thin-walled structures.
A quadrilateral mesh is usually more accurate than a mesh of
similar density based on triangles. Triangles are acceptable in
regions of gradual transitions.

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FEA Pre-Processing – Elements
Solid Element Modeling
Tetrahedral (tet) mesh is the only generally
accepted means to fill a volume, used as auto-
mesh by many FEA codes.
10-node Quadratic

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CAD Modeling for FEA
• CAD models prepared by the design group for
eventual FEA.
• CAD models prepared without consideration of
FEA needs.
• CAD models unsuitable for use in analysis due to
the amount of rework required.
• Analytical geometry developed by or for analyst
for sole purpose of FEA.
CAD and FEA activities should be coordinated at the early stages
of the design process to minimize the duplication of effort.

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CAD Modeling for FEA
• Solid chunky parts (thick-walled, low aspect ratio)
parts mesh cleanly directly off CAD models.
• Clean geometry
geometrical features must not prevent the mesh from
being created. The model should not include buried
features.
• Parent-child relationships
parametric modeling allows defining features off other
CAD features.

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CAD Modeling for FEA
Short edges and Sliver surfaces
Short edges and sliver surfaces usually accompany each other and
on large faces can cause highly distorted elements or a failed mesh.

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CAD Modeling for FEA – Sliver Surfaces
The rounded rib on the
inside of the piston has a
thickness of .30 and a
radius of .145, as a result
a flat surface of .01 by 2.5
is created. A mesh size
of .05 is required to avoid
distorted elements. This
results in a 290,000
nodes. If the radius is
increased to .15, a mesh
size of .12 is sufficient
which results in 33,500
nodes.
Flat surface

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CAD Modeling for FEA
Fillet across shallow angle
Sliver surface caused by a slightly
undersized fillet
Sliver surface caused by
misaligned features.

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Guidelines for Geometry Planning
• Delay inclusion of fillets and chamfers as long as
possible.
• Try to use permanent datums as references where
possible to minimize dependencies.
• Avoid using fillet or draft edges as references for
other features (parent-child relationship)
• Never bury a feature in your model. Delete or
redefine unwanted or incorrect features.

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Guidelines for Part Simplification
• Outside corner breaks or rounds.
• Small inside fillets far from areas of interest.
• Screw threads or spline features unless they are
specifically being studied.
• Small holes outside the load path.
• Decorative or identification features.
• Large sections of geometry that are essentially
decoupled from the behavior of interested section.
In general, features listed below could be considered for
suppression. But, consider the impact before suppression.

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Guidelines for Part Simplification
Fillet added
to the rib
Holes removed
Fillet
removed
Ribs needed
for casting
removed

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CAD Modeling for FEA
Model Conversion
• Try to use the same CAD system for all
components in design.
• When the above is not possible, translate
geometry through kernel based tools such as
ACIS or Parasolids. Using standards based
(IGES, DXF, or VDA) translations may lead to
problem.
• Visually inspect the quality of imported
geometry.
• Avoid modification of the imported geometry in
a second CAD system.
• Use the original geometry for analysis. If not
possible, use a translation directly from the
original model.

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Example of a solid model corrupted by
IGES transfer

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FEA Pre-Processing
Material Properties
The only material properties that are generally required
by an isotropic, linear static FEA are: Young’s modulus
(E), Poisson’s ratio (v), and shear modulus (G).
G = E / 2(1+v)
Provide only two of the three properties.
Thermal expansion and simulation analysis require
coefficient of thermal expansion, conductivity and
specific heat values.

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FEA Pre-Processing
Nonlinear Material Properties
A multi-linear model requires the input of stress-strain
data pairs to essentially communicate the stress-strain
curve from testing to the FE model
Highly deformable, low stiffness, incompressible materials,
such as rubber and other synthetic elastomers require
distortional and volumetric constants or a more complete set
of tensile, compressive, and shear force versus stretch curve.
A creep analysis requires time and temperature dependent
creep properties. Plastic parts are extremely sensitive to this
phenomenon

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FEA Pre-Processing
• Their properties hold constant throughout the assigned entity.
• Average values are used (variation could be up to 15%).
• Localized changes due to heat or other processing effects are
not accounted for.
• Any impurities present in the parent material are neglected.
Comments
If possible, obtain material property values specific to the
application under analysis.
If you are selecting the property set from the code’s
library, be aware of the assumptions made with this
selection.

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FEA Pre-Processing
Boundary Conditions
In FEA, the name of the game is “boundary
condition”, that is calculating the load and figuring
out constraints that each component experiences in
its working environment.
“Garbage in, garbage out”
The results of FEA should include a complete
discussion of the boundary conditions.

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Boundary Conditions
Loads
Loads are used to represent inputs to the system.
They can be in the forms of forces, moments,
pressures, temperature, or accelerations.
Constraints
Constraints are used as reactions to the applied
loads. Constraints can resist translational or
rotational deformation induced by applied loads.

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Boundary Conditions
Linear Static Analysis
Boundary conditions are assumed constant from
application to final deformation of system and all loads
are applied gradually to their full magnitude.
Dynamic Analysis
The boundary conditions vary with time.
Non-linear Analysis
The orientation and distribution of the boundary
conditions vary as displacement of the structure is
calculated.

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Boundary Conditions
Degrees of Freedom
Spatial DOFs refer to the three translational and three rotational
modes of displacement that are possible for any part in 3D
space. A constraint scheme must remove all six DOFs for the
analysis to run.
Elemental DOFs refer to the ability of each element to transmit
or react to a load. The boundary condition cannot load or
constrain a DOF that is not supported by the element to which
it is applied.

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Boundary Conditions
Constraints and their geometric equivalent in classic
beam calculation.
Fixed support
Pin support
Roller support

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Boundary Conditions
A solid face should always have at least three points in
contact with the rest of the structure. A solid element
should never be constrained by less than three points and
only translational DOFs must be fixed.
Accuracy
The choice of boundary conditions has a direct impact
on the overall accuracy of the model.
Over-constrained model – an overly stiff model due
to poorly applied constraints.

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Boundary Conditions -Example
Excessive Constraints
Model of the chair seat with patches representing the tops of
the legs.
Patch 3
Patch 1
Patch 2
Patch 4

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Patch 3
Patch 1
Patch 2
Patch 4
Boundary Conditions -Example
It may appear to be acceptable to constrain each circular patch
in vertical translation while leaving the rotational DOFs
unconstraint. This causes the seat to behave as if the leg-to-
seat interfaces were completely fixed.
A more realistic constraint scheme would be to pin the
center point of each circular patch (translational),
allowing the patch to rotate. Each point should be fixed
vertically, and horizontal constraints should be selectively
applied so that in-plane spatial rotation and rigid body
translation is removed without causing excessive
constraints.

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Boundary Conditions -Example
• Constraining the center point of patch 1 in all 3
translational DOFs.
• Constraining x and y translations of the center point of
patch 2.
• Constraining z and y translation of the center point of
patch 3.
• Constraining just the y translation of the center point of
patch 4.
This scheme allows in-
plane translation induced
by bending of the seat
without rigid body
translation or rotation.
Patch 3
Patch 1
Patch 2
Patch 4

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Summary of Pre-Processing
• Build the geometry
• Make the finite-element mesh
• Add boundary conditions; loads and
constraints
• Provide properties of material
• Specify analysis type (static or dynamic,
linear or non-linear, plane stress, etc.)
These activities are called finite element modeling.

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Solving the Model - Solver
Once the mesh is complete, and the properties and
boundary conditions have been applied, it is time to solve
the model. In most cases, this will be the point where you
can take a deep breath, push a button and relax while the
computer does the work for a change.
Multiple Load and Constraint Cases
In most cases submitting a run with multiple load cases will
be faster than running sequential, complete solutions for
each load case.
Final Model Check

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Unexpectedly high or low displacements (by order of magnitude)
could be caused by an improper definition of load and/or
elemental properties.
Post-Processing, Displacement Magnitude

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Post-Processing, Displacement Animation
Animation of the model displacements serves as the best means of
visualizing the response of the model to its boundary conditions.

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Post-Processing, FEA of a connecting rod

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Second Mode (Twisting)
The magnitude of the stresses should not be entirely unexpected.
First Mode (Bending)
Post-Processing, Stress Results

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Deformation of a duct under thermal load
Post-Processing, thermal analysis

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Post-ProcessingView Animated
Displacements
Does the shape of deformations make sense?
View Displacement
Fringe Plot
Yes
Review Boundary
Conditions
No
Are magnitudes in line with your expectations?
View Stress
Fringe Plot
Yes
Is the quality and mag. Of stresses acceptable?
Review Load Magnitudes
and Units
No
Review Mesh Density
and Quality of Elements
No
View Results Specific
To the Analysis
Yes

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FEA - Flow Chart