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Introduction to the Finite
Element Method
Spring 2010
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Course Objectives
• The student should be capable of writing
simple programs to solve different
problems using finite element method.
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Assessment
• 10% Assignments (1 per week)
• 20% Quizzes (best 2 out of 3)
– Week of 12/11/2006
– Week of 20/12/2006
– Week of 17/1/2006
• 20% Course Project
• 25% Midterm exam (Week of 2/12/2006)• 25% Final exam (starting 3/2/2007)
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Fundamental Course Agreement
• Homework is sent in electronic format (Nohardcopies are accepted)
• Computer programs have to written in
MATLAB or Mathematica script• No late homework is accepted
• No excuses are accepted for missing a
quiz• Best two out of three quizzes are counted
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References
• J.N. Reddy, “An Introduction to the FiniteElement Method” 3rd ed., McGraw Hill, ISBN007-124473-5
• D.V. Hutton, “Fundamentals of Finite Element
Analysis” 1st ed., McGraw Hill, ISBN 007-121857-2
• K. Bathe, “Finite Element Procedures,” PrenticeHall, 1996. (in library)
• T. Hughes, “The finite Element Method: LinearStatic and Dynamic Finite Element analysis,”Dover Publications, 2000. (in library)
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Numerical Solution of
Boundary Value Problems
Weighted Residual Methods
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Objectives
• In this section we will be introduced to thegeneral classification of approximatemethods
• Special attention will be paid for theweighted residual method
• Derivation of a system of linear equations
to approximate the solution of an ODE willbe presented using different techniques
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Why Approximate?
• Ignorance
• Readily Available Packages
• Need to Develop New Techniques• Good use of your computer!
• In general, the problem does not have an
analytical solution!
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Classification of Approximate
Solutions of D.E.’s
• Discrete Coordinate Method
– Finite difference Methods
– Stepwise integration methods
• Euler method
• Runge-Kutta methods
• Etc…
• Distributed Coordinate Method
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Distributed Coordinate Methods
• Weighted Residual Methods – Interior Residual
• Collocation
• Galrekin
• Finite Element
– Boundary Residual• Boundary Element Method
• Stationary Functional Methods
– Reyligh-Ritz methods – Finite Element method
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Basic Concepts
• A linear differential equation may be written in the form:
x g x f L
• Where L(.) is a linear differential operator.• An approximate solution maybe of the form:
n
i
ii xa x f 1
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Basic Concepts
• Applying the differential operator on the approximatesolution, you get:
01
1
x g x La
x g xa L x g x f L
n
i
ii
n
i
ii
x R x g x Lan
i
ii 1
Residue
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Handling the Residue
• The weighted residual methods are all
based on minimizing the value of the
residue.
• Since the residue can not be zero over the
whole domain, different techniques were
introduced.
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Collocation Method
• The idea behind the collocation method is
similar to that behind the buttons of your
shirt!
• Assume a solution, then force the residue
to be zero at the collocation points
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Collocation Method
0 j x R
01
j
n
i
jii
j
x F x La
x R
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Example Problem
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The bar tensile problem
02
2
x F
x
u EA
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Bar application
02
2
x F
x
u EA
n
i
ii xa xu1
x R x F
dx
xd a EA
n
i
i
i
12
2
Applying the collocation method
01
2
2
j
n
i
ji
i x F dx
xd a EA
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In Matrix Form
nnnnnn
n
n
x F
x F
x F
a
a
a
k k k
k k k
k k k
2
1
2
1
21
22212
12111
...
...
...
Solve the above system for the “generalized
coordinates” ai to get the solution for u(x)
j x x
iij
dx
xd EAk
2
2
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Notes on the trial functions
• They should be at least twice
differentiable!
• They should satisfy all boundary
conditions!
• Those are called the “ Admissibility
Conditions”.
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Using Admissible Functions
• For a constant forcing function, F(x)=f
• The strain at the free end of the bar should
be zero (slope of displacement is zero).
We may use:
l
xSin x
2
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Using the function into the DE:
• Since we only have one term in the series,we will select one collocation point!
• The midpoint is a reasonable choice!
l
xSin
l EA
dx
xd EA
22
2
2
2
f aSinl
EA
1
2
42
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Solving:
• Then, the approximate
solution for this problem is:
• Which gives the maximum
displacement to be:
• And maximum strain to be:
EA f l
EA
f l
Sinl EA
f a2
2
2
21 57.0
24
42
l
xSin EA
f l xu 257.0
2
5.057.02
exact EA
f l l u
0.19.00 exact EA
lf u x
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The Subdomain Method (free
reading)
• The idea behind the
subdomain method is
to force the integral
of the residue to beequal to zero on an
subinterval of the
domain
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The Subdomain Method
01
j
j
x
x
dx x R
011
1
j
j
j
j
x
x
n
i
x
x
ii dx x g dx x La
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Bar application
02
2
x F
x
u EA
n
i
ii xa xu1
x R x F
dx
xd a EA
n
i
i
i
12
2
Applying the subdomain method
11
1
2
2 j
j
j
j
x
x
n
i
x
x
ii dx x F dx
dx
xd a EA
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In Matrix Form
11
2
2 j
j
j
j
x
x
i
x
x
i dx x F adxdx
xd EA
Solve the above system for the “generalized
coordinates” ai to get the solution for u(x)
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The Galerkin Method
• Galerkin suggested that the residue
should be multiplied by a weighting
function that is a part of the suggested
solution then the integration is performedover the whole domain!!!
• Actually, it turned out to be a VERY
GOOD idea
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The Galerkin Method
0 Domain
j dx x x R
01
Domain
j
n
i Domain
i ji dx x g xdx x L xa
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In Matrix Form
Domain
ji
Domain
i j dx x F xadx
dx
xd x EA
2
2
Solve the above system for the “generalized
coordinates” ai to get the solution for u(x)
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Same conditions on the functions
are applied
• They should be at least twice
differentiable!
• They should satisfy all boundary
conditions!
• Let’s use the same function as in the
collocation method:
l
x
Sin x 2
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Substituting with the approximate
solution:
Domain
j
n
i Domain
i ji dx x F xdx
dx
xd xa EA
1
2
2
l
l
fdxl
xSin
dxl
xSin
l
xSina
l EA
0
0
1
2
2
222
l l a
l EA
2
22 1
2
EA
fl l
EA
f a
2
3
2
1 52.016
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Substituting with the approximate
solution: (Int. by Parts)
Domain
j
n
i Domain
i ji dx x F xdx
dx
xd xa EA
1
2
2
l l a
l EA
2
22 1
2
EA
fl l
EA
f a
2
3
2
1 52.016
Domain
i j
l
i j
Domain
i j
dxdx
xd
dx
xd
dx
xd x
dx
dx
xd x
0
2
2
Zero!
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What did we gain?
• The functions are required to be less
differentiable
• Not all boundary conditions need to be
satisfied
• The matrix became symmetric!
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Summary
• We may solve differential equations using a
series of functions with different weights.
• When those functions are used, Residue
appears in the differential equation• The weights of the functions may be determined
to minimize the residue by different techniques
• One very important technique is the Galerkin
method.
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NOTE
• Next Sunday 5/11 (No lecture)
• Following week 12/11, Quiz #1 will be held
covering all the material up-to this lecture
• Homework #1 is due next week (Electronic
submission of report and code is
mandatory.
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Report Should Include …
• Cover page
• Introduction section indicating the
procedure you used with the equations as
implemented in your code
• Results section
• Observations and Conclusions if any
according to the output of your program.
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Homework #1
• Solve the beam bendingproblem, for beamdisplacement, for a simplysupported beam with a loadplaced at the center of the
beam using – Collocation Method
– Subdomain Method
– Galerkin Method
• Use three term Sin series thatsatisfies all BC’s
• Write a program that producesthe results for n-term solution.
)(4
4
x F dx
wd
0)()0(
0)()0(
2
2
2
2
dx
l wd
dx
wd
l ww
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Exact Solution
12/110
3
15
7
412
2/1060
13
12)(
23
3
x
x x x
x
x x
xw
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The Finite Element Method
2nd order DE’s in 1-D
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Objectives
• Understand the basic steps of the finite
element analysis
• Apply the finite element method to second
order differential equations in 1-D
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The Mathematical Model
• Solve:
• Subject to:
L x
f cudx
dua
dx
d
0
0
00 ,0 Qdxdu
auu L x
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Step #1: Discretization
• At this step, we divide
the domain into
elements.
• The elements areconnected at nodes.
• All properties of the
domain are defined at
those nodes.
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Step #2: Element Equations
• Let’s concentrate ourattention to a singleelement.
• The same DE applieson the element level,hence, we may followthe procedure forweighted residual
methods on theelement level!
21
0
x x x
f cu
dx
dua
dx
d
21
2211
21
,
,,
QdxduaQ
dxdua
u xuu xu
x x x x
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Polynomial Approximation
• Now, we may propose an approximate
solution for the primary variable, u(x),
within that element.
• The simplest proposition would be a
polynomial!
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Polynomial Approximation
• Interpolating the values
of displacement
knowing the nodal
displacements, we maywrite:
01 b xb xu
01111 b xbu xu
212
11
12
2 u x x x xu
x x x x xu
02122 b xbu xu
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Polynomial Approximation
eu xu
uuu
u x x
x x
u x x
x x
xu
2
1
212211
2
12
1
1
12
2
St #2 El t E ti
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Step #2: Element Equations
(cont’d)
• Assuming constant
domain properties:
• Applying the
Galerkin method:
21
2
2
0
x x x
f cudxud a
022
Domain
jii jii
j dx f xu x xcudx
xd xa
St #2 El t E ti
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Step #2: Element Equations
(cont’d)
• Note that:
• And:
ee hdx
xd
hdx
xd 1
,
1 21
Domain
i j
x
x
i
j
Domain
i j
dxdx
xd
dx
xd a
dx
xd xa
dxdx
xd xa
2
1
2
2
St #2 El t E ti
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Step #2: Element Equations
(cont’d)
• For i=j=1: (and ignoring boundary terms)
• Which gives:
01
2
1
2
1
2
2
2
x
x eee
dxh
x x f u
h
x xc
h
a
023
1
ee
e
fhuchha
St #2 El t E ti
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Step #2: Element Equations
(cont’d)
• Repeating for all terms:
• The above equation is called the element
equat ion .
1
1
221
12
611
11
2
1 ee
e
fh
u
uch
h
a
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What happens for adjacent
elements?
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Homework #2
• Derive the element equation without
ignoring the boundary terms.
• What are differences in the element
equation.
• The solution should be handed using the
same report format (use equation editor to
write your report).
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Finite Element Procedure
1. Connecting Elements
2. Boundary Conditions3. Solving Equations
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Objectives
• Learn how the finite element model for the
whole domain is assembled
• Learn how to apply boundary conditions
• Solving the system of linear equations
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Recall
• In the previous lecture, we obtained the
element equation that relates the element
degrees of freedom to the externally
applied f ields
• Which maybe written:
1
1
221
12
611
11
2
1 ee
e
fh
u
uch
h
a
2
1
2
1
43
21
f f
uu
k k k k
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Two –Element example
1
2
1
1
1
2
1
1
1
4
1
3
1
2
1
1
f f
uu
k k k k
2
2
2
1
2
2
2
1
2
4
2
3
2
2
2
1
f
f
u
u
k k
k k
3
2
1
3
2
1
3
2
1
2
4
2
3
2
2
2
1
1
4
1
3
1
2
1
1
0
0
Q
f f
f
uu
u
k k k k k k
k k
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Illustration: Bar application
1. Discretization: Divide the bar into N number of
elements. The length of each element will be(L/N)
2. Derive the element equation from the differentialequation for constant properties an externally
applied force:
02
2
x F x
u EA
02
1
2
x
x
i ji j
e
dx f udx
d
dx
d
h
EA
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Performing Integration:
1
1
211
11
2
1 e
e
e
e
fh
u
u
h
EA
Note that if the integration is evaluated from 0 to he,where he is the element length, the same results
will be obtained.
02
1
2
x
x
i ji j
e
dx f udx
d
dx
d
h
EA
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Two –Element bar example
1
2
1
1
1
2
1
1
11
11
f f
uu
h EA
e
2
2
2
1
2
2
2
1
11
11
f
f
u
u
h
EA
e
0
0
1
2
1
2110
121
011
3
2
1 R
fh
uu
u
h
EA e
e
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Applying Boundary Conditions
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Applying BC’s
• For the bar with fixed left side and free
right side, we may force the value of the
left-displacement to be equal to zero:
0
0
1
2
1
2
0
110
121
011
3
2
R fh
u
uh
EA e
e
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Solving
• Removing the first row and column of the
system of equations:
• Solving:
1
2
211
12
3
2 e
e
fh
u
u
h
EA
4
3
2
2
3
2
EA
fh
u
ue
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Secondary Variables
• Using the values of the displacements
obtained, we may get the value of the
reaction force:
0
0
1
2
1
2
2
4
2
3
0
110
121
011 R fh
fh
fh e
e
e
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Secondary Variables
• Using the first equation, we get:
• Which is the exact value of the reactionforce.
R fh fh ee
22
3
e fh R 2
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Summary
• In this lecture, we learned how to
assemble the global matrices of the finite
element model; how to apply the boundary
conditions, and solve the system ofequations obtained.
• And finally, how to obtain the secondary
variables.
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Homework #3
• Problems #3.9 & 3.13 from the text book
• Write down a computer code that solves
the problem for N elements.
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Bars and Trusses
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Objectives
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Bar Example (Ex. 4.5.2, p. 187)
• Consider the bar shown in the above figure.
• It is composed of two different parts. One steel tapered
part, and uniform Aluminum part.• Calculate the displacement field using finite elementmethod.
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Bar Example
• The bar may be represented by two
elements.
• The stiffness matrices of the two elements
may be obtained using the followingintegration:
2
1
2
122
22
21
2
1
11
11 x
x
ee
ee
x
x
e dx
hh
hh
x EAdxdx
d
dx
d
dx
d dx
d
x EA K
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Bar Example
• For the Aluminum bar: E=107 psi, and A=1
in2. we get:
• For the Steel bar: E=38107 psi, and
A=(1.5-0.5x/96) in2. we get:
1111
120
10
11
11
120
10 7
2
7 2
1
x
x Al
dx K
11
11
96
10.75.4
11
11
96
5.05.1
96
10.3 7
2
7 2
1
x
x
Fe dx x
K
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Bar Example
• Assembling the Stiffness matrix and
utilizing the external forces, we get:
• The boundary conditions may be applied
and the system of equations solved.
0
0
10
10.2
0
33.833.80
33.88.575.49
05.495.49
10
5
5
3
2
1
4
R
u
u
u
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Bar Example
• Solving, we get:
• For the secondary
variables:
inu
u
181.0
061.0
3
2
lb R 30000
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Reading Task
• Please read and understand examples,4.5.1 & 4.5.3.
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Trusses
• A truss is a set of bars that are connectedat frictionless joints.
• The Truss bars are generally oriented in
the plain.
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Trusses
• Now, the problem lies in thetransformation of the local displacements
of the bar, which are always in the
direction of the bar, to the global degreesof freedom that are generally oriented in
the plain.
E i f M i
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Equation of Motion
0
0
0000
01010000
0101
2
1
2
2
1
1
F
F
v
u
v
u
h
EA
T f ti M t i
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Transformation Matrix
DOF d Transforme DOF Local v
u
v
u
CosSin
SinCos
CosSin
SinCos
v
u
v
u
2
2
1
1
2
2
1
1
00
00
00
00
DOF
d Transforme DOF Local T
Th E i f M i B
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The Equation of Motion Becomes
• Substituting into theFEM:
• Transforming the
forces:
• Finally:
F T K
F T T K T T T
F K
R ll
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Recall
T K T K T
CosSin
SinCos
CosSin
SinCos
T
00
00
00
00
Where:
0000
0101
0000
0101
h
EA K
Element Stiffness Matrix in Global
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Element Stiffness Matrix in Global
Coordinates
CosSinSinCos
CosSin
SinCos
CosSin
SinCos
CosSin
SinCos
h
EA K
00
00
00
00
0000
0101
0000
0101
00
00
00
00
Element Stiffness Matrix in Global
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Element Stiffness Matrix in Global
Coordinates
22
22
22
22
22
12
2
1
22
12
2
1
22
1
22
1
22
12
2
1
SinSinSinSin
SinCosSinCos
SinSinSinSin
SinCosSinCos
h
EA K
E l 4 6 1 196 201
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Example: 4.6.1 pp. 196-201
• Use the finite element analysis to find thedisplacements of node C.
El t E ti
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Element Equations
0000
0101
0000
0101
1
L
EA K
1010
0000
1010
0000
2
L
EA K
3536.03536.03536.03536.03536.03536.03536.03536.0
3536.03536.03536.03536.0
3536.03536.03536.03536.0
3
L
EA K
A bl P d
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Assembly Procedure
3536.13536.0103536.03536.0
3536.03536.0003536.03536.0
101000
000101
3536.03536.0003536.03536.0
3536.03536.0013536.03536.1
L EA K
Gl b l F V t
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Global Force Vector
P
P
F F
F
F
F
F
F
F
F F
F y
x
y
x
y
x
y
x
y
x
2
2
2
1
1
3
3
2
2
1
1
Remember!
NO distributed loadis applied to a truss
B d C diti
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Boundary Conditions
02211
V U V U
Remove the corresponding rows and columns
P
P
V
U
L
EA
23536.13536.0
3536.03536.0
3
3
Continue! (as before)
R lt
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Results
EA
PLV
EA
PLU
3 ,828.5
33
P F F
P F P F
y x
y x
3 ,0
, ,
22
11
Postcomp tation
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Postcomputation
e
e
e
ee
A
P
A
P 21
e
e
ee
e
e
u
u
L
E A
P
P
2
1
2
1
11
11
2
2
1
1
2
2
1
1
00
00
00
00
v
u
vu
CosSin
SinCos
CosSinSinCos
v
u
vu
Postcomputation
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Postcomputation
A
P
A
P 2 ,
3 ,0 )3()2()1(
Summary
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Summary
• In this lecture we learned how to apply thefinite element modeling technique to bar
problems with general orientation in a
plain.
Homework #5
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Homework #5
• Problem 4.27, – Due 13/12/2006 before 9:00am
• Problem 4.44,
– Due 20/12/2006 before 9:00am
Announcements
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Announcements
• Compensation Tutorial for E15: – Next Sunday 17/12/2006 3rd Period in H6
• Next Lecture:
– Wednesday 20/12/2006 3rd Period in H6
• Next Quiz:
– Wednesday 20/12/2006 3rd Period in H6
– (This Lecture is included)
Term Projects
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Term Projects
• A problem has got to be solved using thefinite element method
• A report is going to be presented by each
group presenting the problem and itssolution
The Report should contain:
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The Report should contain:
• Cover page – Project Title
– Names of team members
• Table of contents
• Introduction and literature survey – Introduction to the problem
– Historical background and relevance of the problem
– Papers and books that presented the problem
– Latest achievements in the problem
The Report should contain:
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The Report should contain:
• The finite element derivation – Governing equation
– Derivation of the element matrices
• Using Glerkin method• Application of Symbolic manipulator to derive the
matrix equations will be appreciated
– Solution procedure
The Report should contain:
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The Report should contain:
• The numerical results and verification – Program results
– Verification of results compared to published results
– Parametric study
• Discussion – Observations of the results
– Further work that may be performed with the problem
– Future developments of the model
• References
Evaluation
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Evaluation
• Report (50%)
• Code (30%)
– Structured: Functions built, easily modified
– Readability: Organization, remarks
– Length: The shorter the better
• Results (20%)
Projects
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Projects
• Heat transfer in a 2-D heat sink
• 2-D flow around a blunt body in a wind
tunnel
• Vibration characteristics of a pipe withinternal fluid flow
• Panel flutter of a beam
• Rotating Timoshenko beam/blade
Heat transfer in a 2 D heat sink
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Heat transfer in a 2-D heat sink
• The heat sink will have heat flowing fromone side
• Convection transfer on the surfaces
• Different boundary conditions on the otherthree sides
• Plot contours of temperature distribution
with different boundary conditions
2-D flow around a blunt body in a
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wind tunnel
• Potential flow in a duct
• Rectangular body with different
Dimensions
• Study the effect of the body size on theflow speed on both sides
• Plot contours of potential function,
pressure, and velocity potential
Vibration characteristics of a pipe
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with internal fluid flow
• Study the change of the naturalfrequencies with the flow speed under
different boundary conditions and fluid
density• Indicate the flow speeds at which
instabilities occur
Panel flutter of a beam
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Panel flutter of a beam
• A fixed-fixed beam is subjected to flowover its surface
• Plot the effect of the flow speed on the
natural frequencies of the beam• Indicate the speed at which instability
occurs
Rotating Timoshenko beam/blade
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Rotating Timoshenko beam/blade
• Rotating beams undergo centrifugaltension that results in the change of its
natural frequencies
• Study the effect of rotation speed on thebeam natural frequencies and frequency
response to excitations at the root
Teams
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Teams
• 2-3 Students teams
• Names and selected projects should be
submitted before 4PM on Thursday
21/12/2006
Work Progress
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Work Progress
• A report should be submitted By 4PM every Wednesday• 27/12/2006
– The report should contain a preliminary literature survey
– Problem statement
– Governing equations
• 10/1/2007
– The report should contain a deeper literature survey – The preliminary derivations of the finite element model
• 17/1/2007 – A more mature version of the report should be presented
– Preliminary results of the code
– List of the program script should be included
• 24/1/2007 – Final version of the report should be presented together with the code
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Beams and Frames
Beams and Frames
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Beams and Frames
• Beams are the most-used structuralelements.
• Many real structures may be approximated
as beam elements• Two main beam theories:
– Euler-Bernoulli beam theory
– Timoshenko beam theory
Euler-Bernoulli Beam Theory
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Euler Bernoulli Beam Theory
• The main assumption in the Euler-Bernoulli beam theory is that the beam’s
thickness is too small compared to the
beam length• That assumption resulted in that the sheer
deformation of the beam may be
neglected without much error in theanalysis
Governing Equation
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Governing Equation
• The equation governing the deformation ofand E-B beam under transverse loading
may be written in the form:
)(22
2
2
x F dxwd x EI
dxd
The Thin-Beam Elements
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The Thin Beam Elements
• The thin beam element has a special feature,namely, the two degrees of freedom at each
node are related.
Beam Interpolation Function
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Beam Interpolation Function
a x H xw
34
2321)( xa xa xaa xw
a x x x xw 321
a x xa x H adx
xdH
dx
xdw
x
2
3210
Beam Interpolation Function
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Beam Interpolation Function
a H ww 001
aT
a
a
aa
l H
l H
H H
w
w
ww
x
x
4
3
2
1
2
2
1
1
0
0
'
'
a H ww x 0'0' 1 al H wl w
2
al H wl w x 2''
Beam Interpolation Function
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Beam Interpolation Function
4
3
2
1
2
32
2
2
1
1
3210
1
0010
0001
'
'
a
a
a
a
l l
l l l
w
w
w
w
2
2
1
1
2323
22
4
3
2
1
'
'
1212
1323
0010
0001
w
w
w
w
l l l l
l l l l
a
a
a
a
Beam Interpolation Function
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Beam Interpolation Function
ee w x N wT x H a x H xw 1
ewT a 1
4
1i
ii w x N xw
Beam Interpolation Function
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Beam Interpolation Function
2
32
3
3
2
2
3
32
3
3
2
2
23
2
231
l
x
l
x
l
x
l
x l
x
l
x x
l
x
l
x
x N x N T
Interpolation Functions
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Interpolation Functions
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
X
N ( x
)
N1
N2
N3
N4
Beam Stiffness Matrix
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Beam Stiffness Matrix
• The governing equation is:
• Using the series solution
)(2
2
2
2
x F dx
wd x EI
dx
d
4
1i
ii w x N xw
Beam Stiffness Matrix
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Beam Stiffness Matrix
• The governing equation becomes
• Applying Galerkin method:
)()(4
1
2
2
2
2
x R x F wdx
N d x EI
dx
d
i
i
i
ee l
j
i
ii
l
j dx N x F wdx N d x EI
dxd dx N x R
0
4
12
2
2
2
0
)()(
Beam Stiffness Matrix
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Beam Stiffness Matrix
• Using integration by parts, twice, andignoring the boundary terms, we get:
• In matrix form: 0)(0
4
12
2
2
2
el
ji
i
ji
dx N x F wdx
N d
dx
N d
x EI
ee l
xx
e
l
xx xx dx N x F wdx N N x EI 00
)(
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Use of Symbolic Manipulator
Beam Example
Optional Homework #6
http://localhost/var/www/apps/conversion/tmp/scratch_4/BeamProgram.pdfhttp://localhost/var/www/apps/conversion/tmp/scratch_4/BeamProgram.pdf
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Optional Homework #6
• Derive the expression for the interpolationfunction for a beam in terms of nodal
displacements and slopes.
• Try to use a symbolic manipulator togenerate the expressions.
)(4
4
2
2
x F dxwd EI
dt wd A
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Two Dimensional Elements
2-D Elements
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• In this section, we will be introduced to twodimensional elements with single degree
of freedom per node.
• Detailed attention will be paid torectangular elements.
For the 2-D BV Problem
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• Let’s consider a problem with a singledependent variable
• We may set one degree of freedom to
each node; say f i.• Further, let’s only consider a rectangular
element that is aligned with the physical
coordinates
A Rectangular Element
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g
• For the approximationof a general function
f(x,y) over the element
you need a 2-D
interpolation function
xya ya xaa y x f 4321,
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Let’s follow the same
procedure!
2-D Interpolation Function
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p
a y x H y x f ,, xya ya xaa y xf 4321),(
a H f f 0,00,0 1
aT
aa
a
a
b H ba H
a H
H
f f
f
f
4
3
2
1
4
3
2
1
0,
0,
0,0
aa H f a f ,00, 2 aba H f ba f ,, 3 ab H f b f ,0,0 4
2-D Interpolation Function
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p
4
3
2
1
4
3
2
1
001
1
001
0001
a
a
a
a
b
abba
a
f
f
f
f
4
3
2
1
4
3
2
1
1111
100
1
00
11
0001
f
f
f f
abababab
bb
aa
a
a
aa
2-D Interpolation Function
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p
e f y x N a y x H y x f ,,,
ab xy
b y
ab
xyab
xy
a
xab
xy
b
y
a
x
y x N y x N T
1
,,
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How does this look like?
2-D Interpolation Functions
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p
0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1
0
0.3
0.6
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
N1
x
y
0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1
0
0.3
0.6
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
N2
x
y
2-D Interpolation Functions
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p
0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1
0
0.3
0.6
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
N3
x
y
0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1
0
0.3
0.6
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
N4
x
y
Example: Laplace Equation
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p p q
02
02
2
2
2
y x
ei
ii y x N y x N ,,4
1
Example: Laplace Equation
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p p q
ei
ii y x N y x N ,,4
1
0 e
Area
y y x x dA N N N N
Applying the Galerkin method and integrating by parts,
the element equation becomes
The Element Equaiton
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0222
222
222
222
6
1
22222222
22222222
22222222
22222222
e
babababa
babababa
babababa
babababa
ab
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The Logistic Problem!
The Logistic Problem
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• In the 2-D problems, the numberingscheme, usually, is not as straight forward
as the 1-D problem
1-D Example
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• Element #1 is associated with nodes 1&2• Element #2 is associated with nodes 2&3, etc…
2-D Example
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2-D Example
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For Element #5
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Global Node Number Local Node Number
51
62
93
84
Contribution of element #5 to global
matrix
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
matrix121110987654321
1
2
3
4
1,31,41,21,15
2,32,42,22,16
7
4,34,44,24,18
3,33,43,23,19
10
11
12
A Solution for the Logistics’
Problem
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Problem
• One solution of the logistic problem is tokeep a record of elements and the
mapping of the local numbering scheme to
the global numbering scheme in a table!
Elements Register: Global
Numbering
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Numbering
Node Number ElementNumber 4321
45211
785421011873
56324
896551112986
Algorithm for Assembling Global
Matrix
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Matrix
1. Create a square matrix “ A”;N*N (N=Number of nodes)
2. For the ith element
3. Get the element matrix “B”
4. For the jth node
5. Get its global number k6. For the mth node
7. Get its global number n
8. Let Akn=Akn+B jm9. Repeat for all m
10. Repeat for all j11. Repeat for all i
Node Number Element
Number 4321
45211
78542
1011873
56324
89655
1112986
121110987654321
1
2
3
4
5
6
7
8
9
10
11
12