Transcript Finite Element Analysis MEEN 5330 Dustin Grant
Finite Element Analysis
MEEN 5330 Dustin Grant Kamlesh Borgaonkar Varsha Maddela Rupakkumar Patel Sandeep Yarlagadda
Introduction What is finite element analysis, FEA?
What is FEA used for?
1D Rod Elements, 2D Trusses
Basic Concepts
f
T
Loads
P i
ji
,
j
f
~
i
0 Equilibrium Boundary conditions
Development of Theory Rayleigh-Ritz Method Total potential energy equation Galerkin’s Method
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
1D Rod Elements Loading consists of three types : body force f , traction force T, point load P i 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
1D Rod Elements Element strain energy Element -1 Element-2
U e
1 2
q T
[
k e
]
q
Element stiffness matrix [
k e
]
E e A e l e
1 1 1 1 Load vectors Element body load vector Element traction-force vector
f
e
A e l e f
2
T
e
Tl e
2 1 1 1 1
Example 1D Rod Elements Example 1 Problem 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 Given:
E
6
psi A e
1.2
in
2
q
1 0.02
in q
2 0.025
in
Solution: 1) Displacement (q) at point P We have (
x
2 2
x
1 ) (
x x
1 2 Now linear shape functions N 1 ( ) and N 2 ( ) are given by
N
1 1 2 0.375
And
N
2 1 2 0.625
2D Truss 2 DOF Transformations Modified Stiffness Matrix Methods of Solving
2D Truss Transformation Matrix Direction Cosines
l e
x
2
x
1 [
L
]
l
0
m
0
l
0 0
m
y
2
y
1 2
l
cos
x
2
x
1
l e m
sin
y
2
y
1
l e
2D Truss Element Stiffness Matrix [
k e
]
E e A e l e
l
2
lm l
2
lm lm m
2
lm
m
2
l
2
lm l
2
lm
m lm m
2
lm
2
Methods of Solving Elimination Approach Eliminate Constraints Penalty Approach Will not discuss Today
Elimination Method Set defection at the constraint to equal zero
Elimination Method Modified Equation DOF’s 1,2,4,7,8 equal to zero
2D Truss Element Stresses
E e
l l e
m l m
q
Element Reaction Forces
R
Q
2D Truss Development of Tables Coordinate Table Connectivity Table Direction Cosines Table
2D Truss Coordinate Table
2D Truss Connectivity Table
2D Truss Direction Cosines Table
l e
x
2
x
1 2
y
2
y
1 2
l
cos
x
2
l e
x
1
m
sin
y
2
l e
y
1
Example 2D Truss
MATLAB Program TRUSS2D.M
3D Truss Stiffness Matrix 3D Transformation Matrix Direction Cosines [
L
]
l
0
m
0
n
0
l
0 0
m
0
n
l e
x
2
x
1
y
2
y
1
z
2
z
1 2
l
cos
x
2
x
1
l e m
cos
y
2
y
1
l e n
cos
z
2
z
1
l e
3D Truss Stiffness Matrix 3D Stiffness Matrix [
k e
]
E e A e l e
l ln
2
lm l ln
2
lm lm m
2
mn
lm
m
2
mn ln mn n
2
ln
mn
n
2
l
2
lm
ln l
2
lm ln
lm
m
2
mn lm m
2
mn
ln mn
n n ln mn
2 2
Conclusion Good at Hand Calculations, Powerful when applied to computers Only limitations are the computer limitations
References
Homework