Finite Element Primer for Engineers: Part 2
Mike Barton & S. D. Rajan
©, 2000, Barton & Rajan
2
Contents
•
Introduction to the Finite Element Method (FEM)
•
Steps in Using the FEM (an Example from Solid
Mechanics)
•
Examples
•
Commercial FEM Software
•
Competing Technologies
•
Future Trends
•
Internet Resources
•
References
©, 2000, Barton & Rajan
3
FEM Applied to Solid Mechanics Problems
Create elements
of the beam
d
xi 1
d
xi 2
d
yi 1
d
yi 2
1
2
4
3
Nodal displacement and forces
•
A
FEM
model
in
solid
mechanics
can
be
thought
of
as
a
system
of
assembled
springs
.
When
a
load
is
applied,
all
elements
deform
until
all
forces
balance
.
•
F = Kd
•
K is dependant upon Young’s
modulus and Poisson’s ratio,
as well as the geometry.
•
Equations from discrete elements
are assembled together to form
the global stiffness matrix.
•
Deflections are obtained by
solving the assembled set of
linear equations.
•
Stresses and strains are
calculated from the deflections.
©, 2000, Barton & Rajan
4
Classification of Solid

Mechanics Problems
Analysis of solids
Static
Dynamics
Behavior of Solids
Linear
Nonlinear
Material
Fracture
Geometric
Large Displacement
Instability
Plasticity
Viscoplasticity
Geometric
Classification of solids
Skeletal Systems
1D Elements
Plates and Shells
2D Elements
Solid Blocks
3D Elements
Trusses
Cables
Pipes
Plane Stress
Plane Strain
Axisymmetric
Plate Bending
Shells with flat elements
Shells with curved elements
Brick Elements
Tetrahedral Elements
General Elements
Elementary
Advanced
Stress Stiffening
©, 2000, Barton & Rajan
5
Governing Equation for Solid Mechanics Problems
[K]
{u}
=
{F
app
}
+
{F
th
}
+
{F
pr
}
+
{F
ma
}
+
{F
pl
}
+
{F
cr
}
+
{F
sw
}
+
{F
ld
}
[K]
=
total
stiffness
matrix
{u}
=
nodal
displacement
{F
app
}
=
applied
nodal
force
load
vector
{F
th
}
=
applied
element
thermal
load
vector
{F
pr
}
=
applied
element
pressure
load
vector
{F
ma
}
=
applied
element
body
force
vector
{F
pl
}
=
element
plastic
strain
load
vector
{F
cr
}
=
element
creep
strain
load
vector
{F
sw
}
=
element
swelling
strain
load
vector
{F
ld
}
=
element
large
deflection
load
vector
•
Basic equation for a static analysis is as follows:
©, 2000, Barton & Rajan
6
Six Steps in the Finite Element Method
•
Step 1

Discretization
: The problem domain is discretized
into a collection of simple shapes, or elements.
•
Step 2

Develop Element Equations
: Developed using the
physics of the problem, and typically Galerkin’s Method or
variational principles.
•
Step 3

Assembly
: The element equations for each element
in the FEM mesh are assembled into a set of global equations
that model the properties of the entire system.
•
Step 4

Application of Boundary Conditions
: Solution
cannot be obtained unless boundary conditions are applied.
They reflect the known values for certain primary unknowns.
Imposing the boundary conditions modifies the global
equations.
•
Step 5

Solve for Primary Unknowns
: The modified global
equations are solved for the primary unknowns at the nodes.
•
Step 6

Calculate Derived Variables
: Calculated using the
nodal values of the primary variables.
©, 2000, Barton & Rajan
7
Process Flow in a Typical FEM Analysis
Start
Problem
Definition
Pre

processor
•
Reads or generates
nodes and elements
(ex: ANSYS)
•
Reads or generates
material property data.
•
Reads or generates
boundary conditions
(loads and
constraints.)
Processor
•
Generates
element shape
functions
•
Calculates master
element equations
•
Calculates
transformation
matrices
•
Maps element
equations into
global system
•
Assembles
element equations
•
Introduces
boundary
conditions
•
Performs solution
procedures
Post

processor
•
Prints or plots
contours of stress
components.
•
Prints or plots
contours of
displacements.
•
Evaluates and
prints error
bounds.
Analysis and
design decisions
Stop
Step 1, Step 4
Step 6
Steps 2, 3, 5
©, 2000, Barton & Rajan
8
Step 1: Discretization

Mesh Generation
airfoil geometry
(from CAD program)
mesh
generator
surface model
ET,1,SOLID45
N, 1, 183.894081 ,

.770218637 , 5.30522740
N, 2, 183.893935 ,

.838009645 , 5.29452965
.
.
TYPE, 1
E, 1, 2, 80, 79, 4, 5, 83, 82
E, 2, 3, 81, 80, 5, 6, 84, 83
.
.
.
meshed model
©, 2000, Barton & Rajan
9
Step 4: Boundary Conditions for a Solid Mechanics Problem
•
Displacements
DOF constraints usually
specified at model boundaries to define rigid
supports.
•
Forces and Moments
Concentrated loads on
nodes usually specified on the model exterior.
•
Pressures
Surface loads usually specified on
the model exterior.
•
Temperatures
Input at nodes to study the
effect of thermal expansion or contraction.
•
Inertia Loads
Loads that affect the entire
structure (ex: acceleration, rotation).
©, 2000, Barton & Rajan
10
Step 4: Applying Boundary Conditions (Thermal Loads)
Temp
mapper
Nodes from
FE Modeler
Thermal
Soln Files
bf, 1,temp, 149.77
bf, 2,temp, 149.78
.
.
.
bf, 1637,temp, 303.64
bf, 1638,temp, 303.63
©, 2000, Barton & Rajan
11
Step 4: Applying Boundary Conditions (Other Loads)
•
Speed,
temperature
and
hub
fixity
applied
to
sample
problem
.
•
FE
Modeler
used
to
apply
speed
and
hub
constraint
.
antype,static
omega,10400*3.1416/30
d,1,all,0,0,57,1
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