finite element analysis meen 5330 dustin grant kamlesh borgaonkar varsha maddela rupakkumar patel...
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Finite Element Analysis
MEEN 5330Dustin GrantKamlesh BorgaonkarVarsha MaddelaRupakkumar PatelSandeep Yarlagadda
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Introduction
What is finite element analysis, FEA?
What is FEA used for?
1D Rod Elements, 2D Trusses
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Basic Concepts
Loads
Equilibrium
Boundary conditions
fT
iP
0~
, ijji f
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Development of Theory
Rayleigh-Ritz MethodTotal potential energy equation
Galerkin’s Method
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1D Rod Elements To understand and solve 2D and 3D
problems we must understand basic of 1D problems.
Analysis of 1D rod elements can be done using Rayleigh-Ritz and Galerkin’s method
To solve FEA problems same are modified in the Potential-Energy approach and Galerkin’s approach
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1D Rod Elements Loading consists of three types : body
force f , traction force T, point load Pi
Body force: distributed force , acting on every elemental volume of body i.e. self weight of body.
Traction force: distributed force , acting on surface of body i.e. frictional resistance, viscous drag and surface shear
Point load: a force acting on any single point of element
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1D Rod Elements
Element strain energy
Element stiffness matrix
Load vectorsElement body load vectorElement traction-force vector
qkqU eTe
][
2
1
11
11][
e
eee
l
AEk
1
1
2
flAf eee
1
1
2ee Tl
T
Element -1 Element-2
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Example 1D Rod ElementsExample 1Problem statement: (Problem 3.1 from Chandrupatla and Belegunda’s book)Consider the bar in Fig.1, determine the following by hand calculation: 1) Displacement at point P 2) Strain and stress
3) Element stiffness matrix 4) strain energy in element
21.2eA in
630 10E psi 1 0.02q in
2 0.025q in
Given:
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Solution:
1) Displacement (q) at point P
We have
12 1
2( ) 1
( )
2(20 15) 1 0.25
(23 15)
x xx x
Now linear shape functions N1( ) and N2( ) are given by
1
1( ) 0.375
2N
And 2
1( ) 0.625
2N
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2D Truss
2 DOF
Transformations
Modified Stiffness Matrix
Methods of Solving
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2D Truss
Transformation MatrixDirection Cosines
ml
mlL
00
00][
212
212 yyxxle
el
xxl 12cos
el
yym 12sin
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2D Truss
Element Stiffness Matrix
22
22
22
22
][
mlmmlm
lmllml
mlmmlm
lmllml
l
AEk
e
eee
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Methods of Solving
Elimination ApproachEliminate Constraints
Penalty ApproachWill not discuss Today
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Elimination Method
Set defection at the constraint to equal zero
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Elimination Method
Modified Equation DOF’s 1,2,4,7,8 equal to zero
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2D Truss
Element Stresses
Element Reaction Forces
qmlmll
E
e
e
QKR
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2D Truss
Development of Tables
Coordinate TableConnectivity TableDirection Cosines Table
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2D Truss
Coordinate Table
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2D Truss
Connectivity Table
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2D Truss
Direction Cosines Table
2
1 22
1 2y y x x le
el
x xl
1 2cos
el
y ym
1 2sin
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Example 2D Truss
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MATLAB Program TRUSS2D.M
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3D Truss Stiffness Matrix
3D Transformation MatrixDirection Cosines
nml
nmlL
000
000][
212
212
212 zzyyxxle
el
xxl 12cos
el
yym 12cos
el
zzn 12cos
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3D Truss Stiffness Matrix
3D Stiffness Matrix
22
22
22
22
22
22
][
nmnlnnmnln
mnmlmmnmlm
lnlmllnlml
nmnlnnmnln
mnmlmmnmlm
lnlmllnlml
l
AEk
e
eee
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Conclusion
Good at Hand Calculations, Powerful when applied to computers
Only limitations are the computer limitations
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References
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Homework