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Need for Computational Methods Solutions Using Either Strength of Materials or Theory ofElasticity Are Normally Accomplished for Regions and
Loadings With Relatively Simple Geometry
Many Applicaitons Involve Cases with Complex Shape,Boundary Conditions and Material Behavior
Therefore a Gap Exists Between What Is Needed in
Applications and What Can Be Solved by Analytical Closed-
form Methods
This Has Lead to the Development of SeveralNumerical/Computational Schemes Including: Finite
Difference, Finite Element and Boundary Element Methods
Introduction to Finite Element Methods
M.THIRUMALAIMUTHUKUMARAN AP/MECH Dr.NGPIT
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Advantages of Finite Element Analysis
- Models Bodies of Complex Shape
- Can Handle General Loading/Boundary Conditions
- Models Bodies Composed of Composite and Multiphase Materials
- Model is Easily Refined for Improved Accuracy by Varying
Element Size and Type (Approximation Scheme)
- Time Dependent and Dynamic Effects Can Be Included
- Can Handle a Variety Nonlinear Effects Including Material
Behavior, Large Deformations, Boundary Conditions, Etc.
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Basic Concept of the Finite Element Method
Any continuous solution field such as stress, displacement,
temperature, pressure, etc. can be approximated by a
discrete model composed of a set of piecewise continuous
functions defined over a finite number of subdomains.
Exact Analytical Solution
x
T
Approximate Piecewise
Linear Solution
x
T
One-Dimensional Temperature Distribution
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Two-Dimensional Discretization
-1-0.5
00.5
11.5
22.5
3
1
1.5
2
2.5
3
3.54
-3
-2
-1
0
1
2
xy
u(x,y)
Approximate Piecewise
Linear Representation
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Discretization Concepts
x
T
Exact T emperatu re Dis tributio n, T( x)
Fin ite E lem en t D isc retization
Li near In terpo lat ion Mo del (Four Elements)
Quadratic Interpolation M odel (Two Elements)
T1
T2 T2T3 T3
T4 T4T5
T1
T2
T3T4 T5
Piece wi se Linear Ap proxim ation
T
x
T1
T2T3 T3
T4 T5
T
T1
T2
T3T4 T5
Piece wi se Qua dra tic Approxim ati on
x
Temperature Continuous but with
Discon tinu ous T emp erature G radients
Temperature and Temp erature Gradients
Continuous
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Common Types of Elements
One-Dimensional Elements
Line
Rods, Beams, Trusses, Frames
Two-Dimensional ElementsTriangular, Quadrilateral
Plates, Shells, 2-D Continua
Three-Dimensional Elements
Tetrahedral, Rectangular Prism (Brick)
3-D Continua
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Discretization Examples
One-Dimensional
Frame Elements
Two-Dimensional
Triangular Elements
Three-Dimensional
Brick Elements
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Basic Steps in the Finite Element Method
Time Independent Problems- Domain Discretization
- Select Element Type (Shape and Approximation)
- Derive Element Equations (Variational and Energy Methods)
- Assemble Element Equations to Form Global System
[K]{U} = {F}
[K] = Stiffness or Property Matrix
{U} = Nodal Displacement Vector
{F} = Nodal Force Vector
- Incorporate Boundary and Initial Conditions
- Solve Assembled System of Equations for Unknown Nodal
Displacements and Secondary Unknowns of Stress and Strain Values
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Common Sources of Error in FEA
Domain Approximation
Element Interpolation/Approximation
Numerical Integration Errors
(Including Spatial and Time Integration)
Computer Errors (Round-Off, Etc., )
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Measures of Accuracy in FEA
Accuracy
Error = |(Exact Solution)-(FEM Solution)|
Convergence
Limit of Error as:
Number of Elements (h-convergence)
or
Approximation Order (p-convergence)
IncreasesIdeally, Error 0 as Number of Elements or
Approximation Order
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Two-Dimensional Discretization Refinement
(Discretization with 228 Elements)
(Discretization with 912 Elements)
(Triangular Element)
(Node)
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One Dimensional Examples
Static Case
1 2
u1 u2
Bar Element
Uniaxial Deformation of Bars
Using Strength of Materials Theory
Beam Element
Deflection of Elastic Beams
Using Euler-Bernouli Theory
1 2
w1 w2
q2
q1
dx
duau
qcuaudx
d
,
:ionSpecificatCondtionsBoundary
0)(
:EquationalDifferenti
)(,,,
:ionSpecificatCondtionsBoundary
)()(
:EquationalDifferenti
2
2
2
2
2
2
2
2
dx
wdb
dx
d
dx
wdb
dx
dww
xf
dx
wdb
dx
d
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Two Dimensional Examples
u1
u2
1
2
3 u3
v1
v2
v3
1
2
3
f1
f2
f3
Triangular Element
Scalar-Valued, Two-DimensionalField Problems
Triangular Element
Vector/Tensor-Valued, Two-Dimensional Field Problems
yx ny
nxdn
d
yxfyx
,
:ionSpecificatCondtionsBoundary
),(
:EquationialDifferentExample
2
2
2
2
yxy
yxx
y
x
n
y
vC
x
uCn
x
v
y
uCT
nx
v
y
uCn
y
vC
x
uCT
Fy
v
x
u
y
Ev
Fy
v
x
u
x
Eu
221266
661211
2
2
ConditonsBoundary
0)1(2
0)1(2
entsDisplacemofTermsinEquationsFieldElasticity
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Development of Finite Element Equation The Finite Element Equation Must Incorporate the Appropriate Physicsof the Problem
For Problems in Structural Solid Mechanics, the Appropriate Physics
Comes from Either Strength of Materials or Theory of Elasticity
FEM Equations are Commonly Developed UsingDirect, Variational-
Virtual Work or Weighted ResidualMethods
Variational-Virtual Work Method
Based on the concept of virtual displacements, leads to relations between internal and
external virtual work and to minimization of system potential energy for equilibrium
Weighted Residual Method
Starting with the governing differential equation, special mathematical operations
develop the weak form that can be incorporated into a FEM equation. This
method is particularly suited for problems that have no variational statement. For
stress analysis problems, a Ritz-Galerkin WRM will yield a result identical to that
found by variational methods.
Direct Method
Based on physical reasoning and limited to simple cases, this method is
worth studying because it enhances physical understanding of the process
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Simple Element Equation Example
Direct Stiffness Derivation
1 2k
u1 u2
F1 F2
}{}]{[
rmMatrix Foinor
2NodeatmEquilibriu
1NodeatmEquilibriu
2
1
2
1
212
211
FuK
F
F
u
u
kk
kk
kukuF
kukuF
Stiffness Matrix Nodal Force Vector
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Common Approximation Schemes
One-Dimensional Examples
Linear Quadratic Cubic
Polynomial Approximation
Most often polynomials are used to construct approximation
functions for each element. Depending on the order of
approximation, different numbers of element parameters are
needed to construct the appropriate function.
Special Approximation
For some cases (e.g. infinite elements, crack or other singular
elements) the approximation function is chosen to have special
properties as determined from theoretical considerations
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One-Dimensional Bar Element
udVfuPuPedV jjii
}]{[:LawStrain-Stress
}]{[}{][
)(:Strain
}{][)(:ionApproximat
dB
dBdN
dN
EEe
dx
dux
dx
d
dx
due
uxu
k
kk
k
kk
L
TT
j
iTL
TTfdxA
P
PdxEA
00][}{}{}{][][}{ NdddBBd
L
TL
TfdxAdxEA
00][}{}{][][ NPdBB
VectorentDisplacemNodal}{
VectorLoading][}{
MatrixStiffness][][][
0
0
j
i
LT
j
i
LT
u
u
fdxAP
P
dxEAK
d
NF
BB
}{}]{[ FdK
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One-Dimensional Bar Element
A = Cross-sectional Area
E = Elastic Modulusf(x) = Distributed Loading
dVuFdSuTdVe iV
iS
i
n
iijV
ijt
Virtual Strain Energy = Virtual Work Done by Surface and Body Forces
For One-Dimensional Case
udVfuPuPedV jjii
W
(i) (j)
Axial Deformation of an Elastic Bar
Typical Bar Element
dx
duAEP ii
dx
duAEP
j
j iu ju
L
x
(Two Degrees of Freedom)
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Linear Approximation Scheme
VectorentDisplacemNodal}{
MatrixFunctionionApproximat][
}]{[1
)()(
1
2
1
2
1
21
2211
2112
1
212
11
21
d
N
dN
ntDisplacemelasticeApproximat
u
u
L
x
L
x
u
u
u
uxux
uL
xu
L
xx
L
uuuu
Laau
au
xaau
x (local coordinate system)(1) (2)
iu ju
L
x(1) (2)
u(x)
x(1) (2)
y 1(x) y 2(x)
1
y k(x) Lagrange Interpolation Functions
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Element EquationLinear Approximation Scheme, Constant Properties
VectorentDisplacemNodal}{
1
1
2][}{
11
1111
1
1
][][][][][
2
1
2
1
02
1
02
1
00
u
u
LAf
P
Pdx
L
xL
x
AfP
PfdxAP
P
L
AEL
LL
L
LAEdxAEdxEAK
oL
o
L T
LT
LT
d
NF
BBBB
1
1
211
11}{}]{[
2
1
2
1 LAf
P
P
u
u
L
AE oFdK
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Quadratic Approximation Scheme
}]{[
)()()(
42
3
2
1
321
332211
2
3213
2
3212
11
2321
dN
ntDisplacemeElasticeApproximat
u
u
u
u
uxuxuxu
LaLaau
LaLaau
au
xaxaau
x(1) (3)
1u 3u
(2)
2u
L
u(x)
x(1) (3)(2)
x(1) (3)(2)
1
y 1(x) y 3(x)
y 2(x)
3
2
1
3
2
1
781
8168
187
3F
F
F
u
u
u
L
AE
quationElement
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Lagrange Interpolation FunctionsUsing Natural or Normalized Coordinates
11
(1) (2))1(
2
1
)1(2
1
2
1
)1(2
1
)1)(1(
)1(21
3
2
1
)1)(3
1)(
3
1(
16
9
)3
1)(1)(1(
16
27
)3
1)(1)(1(
16
27
)
3
1)(
3
1)(1(
16
9
4
3
2
1
(1) (2) (3)
(1) (2) (3) (4)
ji
jiji
,0,1)(
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Simple ExampleP
A1,E1,L1 A2,E2,L2
(1) (3)(2)
1 2
0000
011
011
1ElementEquationGlobal
)1(
2
)1(1
3
2
1
1
11 P
P
U
U
U
L
EA
)2(
2
)2(
1
3
2
1
2
22
0
110
110
000
2ElementEquationGlobal
P
P
U
U
U
L
EA
3
2
1
)2(
2
)2(
1
)1(
2
)1(1
3
2
1
2
22
2
22
2
22
2
22
1
11
1
11
1
11
1
11
0
0
EquationSystemGlobalAssembled
P
P
P
P
PP
P
U
U
U
L
EA
L
EA
L
EA
L
EA
L
EA
L
EAL
EA
L
EA
0
LoadingedDistributZeroTake
f
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Simple Example Continued
P
A1,E1,L1 A2,E2,L2
(1) (3)(2)
1 2
0
0ConditionsBoundary
)2(
1
)1(
2
)2(
2
1
PP
PP
U
P
P
U
U
L
EA
L
EA
L
EA
L
EA
L
EA
L
EA
LEA
LEA
0
0
0
0
EquationSystemGlobal Reduced
)1(
1
3
2
2
22
2
22
2
22
2
22
1
11
1
11
1
11
1
11
PU
U
L
EA
L
EA
LEA
LEA
LEA
0
3
2
2
22
2
22
2
22
2
22
1
11
LEA ,,Properties
mFor Unifor
PU
U
L
AE 0
11
12
3
2
PPAE
PLU
AE
PLU )1(132 ,
2,Solving
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One-Dimensional Beam ElementDeflection of an Elastic Beam
2
2423
1
1211
2
2
2
4
2
2
2
3
12
2
21
2
2
1
,,,
,
,
dx
dwuwu
dx
dwuwu
dx
wdEIQ
dx
wdEI
dx
dQ
dx
wd
EIQdx
wd
EIdx
d
Q
I = Section Moment of Inertia
E = Elastic Modulus
f(x) = Distributed Loading
W
(1) (2)
Typical Beam Element
1w
L
2w1 2
1M2M
1V 2V
x
Virtual Strain Energy = Virtual Work Done by Surface and Body Forces
wdVfwQuQuQuQedV 44332211
L
TL
dVfwQuQuQuQdxEI0
443322110
][}{][][ NdBB T
(Four Degrees of Freedom)
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Beam Approximation FunctionsTo approximate deflection and slope at each
node requires approximation of the form3
4
2
321)( xcxcxccxw
Evaluating deflection and slope at each node
allows the determination ofci thus leading to
FunctionsionApproximatCubicHermitethearewhere
,)()()()()( 44332211
i
uxuxuxuxxw
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Beam Element Equation
LT
L
dVfwQuQuQuQdxEI0443322110
][}{][][ NdBB T
4
3
2
1
}{
u
u
u
u
d ][][
][ 4321
dx
d
dx
d
dx
d
dx
d
dx
d
NB
22
22
30
233
3636
323
3636
2][][][
LLLL
LL
LLLL
LL
L
EIdxEI
L
BBK T
L
LfL
Q
Q
Q
Q
u
u
u
u
LLLL
LL
LLLL
LL
L
EI
6
6
12
233
3636
323
3636
2
4
3
2
1
4
3
2
1
22
22
3
L
LfLdxfdxf
LLT
6
6
12][
0
4
3
2
1
0N
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FEA Beam Problem
f
a b
UniformEI
0
0
0
0
6
6
12
000000
000000
00/2/3/1/3
00/3/6/3/6
00/1/3/2/3
00/3/6/3/6
2)1(
4
)1(
3
)1(
2
)1(
1
6
5
4
3
2
1
22
2323
22
2323
Q
Q
Q
Q
a
a
fa
U
U
U
U
U
U
aaaa
aaaa
aaaa
aaaa
EI
1Element
)2(
4
)2(
3
)2(
2
)2(
1
6
5
4
3
2
1
22
2323
22
2323
0
0
/2/3/1/300
/3/6/3/600
/1/3/2/300
/3/6/3/600
000000
000000
2
Q
Q
Q
Q
U
U
U
U
U
U
bbbb
bbbb
bbbb
bbbbEI
2Element
(1) (3)(2)
1 2
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FEA Beam Problem
)2(
4
)2(
3
)2(
2
)1(
4
)2(
1
)1(
3
)1(
2
)1(
1
6
5
4
3
2
1
23
2
232233
2
2323
0
0
6
6
12
/2
/3/6
/1/3/2/2
/3/6/3/3/6/6
00/1/3/2
00/3/6/3/6
2
Q
Q
Q
Q
a
a
fa
U
U
U
U
U
U
a
aa
aaba
aababa
aaa
aaaa
EI
SystemAssembledGlobal
0,0,0 )2(4)2(
3
)1(
12
)1(
11 QQUwU
ConditionsBoundary
0,0 )2(2)1(
4
)2(
1
)1(
3 QQQQ
ConditionsMatching
0
0
0
0
0
0
6
12
/2
/3/6
/1/3/2/2
/3/6/3/3/6/6
2
4
3
2
1
23
2
332233
afa
U
U
U
U
a
aa
aaba
aababa
EI
SystemReduced
Solve System for Primary Unknowns U1,U2,U3,U4
Nodal Forces Q1and Q2Can Then Be Determined
(1) (3)(2)
1 2
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Special Features of Beam FEA
Analytical Solution Gives
Cubic Deflection Curve
Analytical Solution Gives
Quartic Deflection Curve
FEA Using Hermit Cubic InterpolationWill Yield Results That Match Exactly
With Cubic Analytical Solutions
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Truss ElementGeneralization of Bar Element With Arbitrary Orientation
x
y
k=AE/L
cos,sin cs
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Frame ElementGeneralization of Bar and Beam Element with Arbitrary Orientation
W
(1) (2)
1w
L
2w1 2
1M2M
1V 2V
2P1P1u 2u
4
3
2
2
1
1
2
2
2
1
1
1
22
2323
22
2323
460
260
6120
6120
0000
260
460
6120
6120
0000
P
Q
Q
P
w
u
w
u
L
EI
L
EI
L
EI
L
EIL
EI
L
EI
L
EI
L
EI L
AE
L
AEL
EI
L
EI
L
EI
L
EIL
EI
L
EI
L
EI
L
EIL
AE
L
AE
Element Equation Can Then Be Rotated to Accommodate Arbitrary Orientation
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Some Standard FEA References
Bathe, K.J., Finite Element Procedures in Engineering Analysis, Prentice-Hall, 1982, 1995.
Beer, G. and Watson, J.O., Introduction to Finite and Boundary Element Methods for Engineers,John Wiley, 1993
Bickford, W.B., A First Course in the Finite Element Method, Irwin, 1990.
Burnett, D.S., Finite Element Analysis, Addison-Wesley, 1987.
Chandrupatla, T.R. and Belegundu, A.D., Introduction to Finite Elements in Engineering, Prentice-Hall, 2002.
Cook, R.D., Malkus, D.S. and Plesha, M.E., Concepts and Applications of Finite Element Analysis, 3rdEd., John Wiley,
1989.
Desai, C.S., Elementary Finite Element Method, Prentice-Hall, 1979.
Fung, Y.C. and Tong, P., Classical and Computational Solid Mechanics, World Scientific, 2001.
Grandin, H., Fundamentals of the Finite Element Method, Macmillan, 1986.
Huebner, K.H., Thorton, E.A. and Byrom, T.G., The Finite Element Method for Engineers, 3rdEd., John Wiley, 1994.
Knight, C.E., The Finite Element Method in Mechanical Design, PWS-KENT, 1993.
Logan, D.L., A First Course in the Finite Element Method, 2ndEd., PWS Engineering, 1992.
Moaveni, S., Finite Element Analysis Theory and Application with ANSYS,2ndEd., Pearson Education, 2003.
Pepper, D.W. and Heinrich, J.C., The Finite Element Method: Basic Concepts and Applications, Hemisphere, 1992.
Pao, Y.C., A First Course in Finite Element Analysis, Allyn and Bacon, 1986.
Rao, S.S., Finite Element Method in Engineering, 3rdEd., Butterworth-Heinemann, 1998.
Reddy, J.N., An Introduction to the Finite Element Method, McGraw-Hill, 1993.Ross, C.T.F., Finite Element Methods in Engineering Science,Prentice-Hall, 1993.
Stasa, F.L., Applied Finite Element Analysis for Engineers, Holt, Rinehart and Winston, 1985.
Zienkiewicz, O.C. and Taylor, R.L., The Finite Element Method, Fourth Edition, McGraw-Hill, 1977, 1989.