Introduction
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Transcript Introduction
EMA 405
Introduction
Syllabus
Textbook: none
Prerequisites: EMA 214; 303, 304, or 306;
EMA 202 or 221
Room: 2261 Engineering Hall
Time: TR 11-12:15
Course Materials: ecow2.engr.wisc.edu
Instructors
Jake Blanchard, Room 143 ERB,
phone: 263-0391
e-mail: [email protected]
office hours: TBD
Grading
Homeworks – 40%
Quiz – 20%
Design Problem – 20%
Final Project – 20%
Schedule
Topics
Introduction
FEA Theory
Intro to ANSYS
Trusses
Plane Stress/Strain
Axisymmetric
3-D Problems
Beams
Plates
Heat Transfer
Multiple Load
Steps
Plasticity
The finite element method
Began in 1940’s to help solve problems in
elasticity and structures
It has evolved to solve nonlinear, thermal,
structural, and electromagnetic problems
Key commercial codes are ANSYS,
ABAQUS, Nastran, etc.
We’ll use ANSYS, but other codes are as
good or better (…a “religious” question)
The Process
Build a model
◦
◦
◦
◦
◦
Geometry
Material Properties
Discretization/mesh
Boundary conditions
Load
Solve
Postprocessing
Structural Elements
Truss
Beams
Planar
3-D
Plate
Elements
Truss
Beam
Planar
Shell
Brick
Finite Element Fundamentals
The building block of FEM is the element
stiffness matrix
3
a
1
2
a
f1x
f
1 y k11
f 2 x k 21
f2 y
f 3 x k61
f 3 y
k12
k 22
k62
u1
k16 v1
k 26 u 2
v2
k66 u3
v3
Now Put Several Together
7
8
9
5
4
6
1
2
3
F K U
Global Stiffness
[K] is a composite of the
element stiffness elements
Once K is known, we can
choose forces and calculate
displacements, or choose
displacements and calculate
forces
Boundary conditions are
needed to allow solution
f 1x
f
1y
f 2x
f
2y
f 3x
f
3y
f
4x
f 4y
f
F 5 x
f 5y
f 6x
f
6y
f 7x
f 7y
f
8x
f 8y
f
9x
f 9 y
u1
v
1
u 2
v 2
u 3
v3
u
4
v 4
u
U 5
v 5
u 6
v 6
u
7
v 7
u
8
v8
u 9
v 9
Element Stiffness
f3y
f3x
3
a
f2y
1
2
f1y
a
f1x
v3
f2x
u3
y
x
v2
v1
u1
u2
How Do We Get Element Stiffness?
u [ A]c
assum e
u ( x, y ) c1 c2 x c3 y
v( x, y ) c4 c5 x c6 y
x1 0;
y1 0
x2 a;
y2 0
x3 0;
y3 a
Rewrite as
matrix
equation
c A1u
Coordinates
of element
corners
u1 c1 ; v1 c4
u2 c1 c2 a; v2 c4 c5 a
u3 c1 c3 a; v3 c4 c6 a
u1 1 0 0 c1
c
u
1
a
0
2
2
u 1 0 a c
3
3
Substitute
coordinates
into assumed
functions
a
A1 1 1
a
1
c1
a
1
c2 1
c a 1
3
0 0
1 0
0 1
0 0 u1
1 0 u2
0 1 u3
Continued…
c1
a 0 0 u1
1
u
c
1
1
0
2
2
c a 1 0 1 u
3
3
c1
Rewrite
u 1 x y c2
assumed
functions
c
3
u 1 1
a
x
a 0 0 u1
y 1 1 0 u 2
1 0 1 u3
u 1 1
a
x
au1
y u1 u2
u u
1 3
Multiply
1
u ( x, y ) au1 u1 u2 x u1 u3 y
a
Substitute
Continued
1
u ( x, y ) a x y u1 xu2 yu3
a
u ( x, y ) N1u1 N 2u 2 N 3u3
x y
N1 1
a a
x
N2
a
y
N3
a
Sim ilarly
v( x, y ) N1v1 N 2 v2 N 3v3
Collect
terms
Stress-Strain
x y
N 3
u N1
N 2
N1 1
x
u1
u2
u3
a a
x x
x
x
x
u
u1 u 2
N2
x
a
x
a a
y
v
v1 v3
N3
y
a
y
a a
u ( x, y ) N1u1 N 2u 2 N 3u3
u v
u1 u3 v1 v2
xy
v ( x, y ) N1v1 N 2 v2 N 3v3
y x
a a a a
Stress-Strain
u1
v
1
x
u 2
y B
v2
xy
u3
v3
1 0 1 0 0 0
1
B 0 1 0 0 0 1
a
1 1 0 1 1 0
k B DB dV tAB T D B
T
V
Comes from
minimizing total
potential energy
(variational
principles)
Material Properties
[D] comes from
the stress-strain
equations
For a linear,
elastic, isotropic
material
Strain Energy
x
x
y [ D ] y
xy
xy
1
0
E
1
[ D]
0
2
1
1
0 0
2
T
1
U D dV
2 V
Final Result for Our Case
[k ]
A
AEt
2a 2 1 2
1
2
3
1
3
2
2
2
2
1 1 0
1 1 0
2
2
2
1 2
a
2
[k ]
Et
4 1 2
1
2
3
1
3
2
2
2
2
1 1 0
1 1 0
2
2
2
1 1 2
1 1 2
0
0
2
1
1
0
1
1
0
0
0
2
1 1 2
1 1 2
0
0
2
1
1
0
1
1
0
0
0
2
or
f1x
f
1y
f 2 x
Et
2
f
4
1
2
y
f3x
f 3 y
1
2
3
1
3
2
2
2
2
1 1 0
1 1 0
2
2
2
1 1 2 u1
1 1 2 v1
0
0
2 u2
1
1
0 v2
1
1
0 u3
0
0
2 v3
Examples
f1x
f
1y
f 2 x
Et
2
f
2 y 4 1
f3x
f 3 y
u1
3
1
2
u1
1
1
2
Examples
f1x
f
1y
f 2 x
Et
2
f
2 y 4 1
f3x
f 3 y
v1
1
3
2
v1
1
1
2
Prescribe forces
F
Process
What do we know? – v1=v2=0; f3y=F; all
horizontal forces are 0
Remove rigid body motion – arbitrarily set
u1=0 to remove horizontal translation;
hence, f1x is a reaction
Reduce matrix to essential elements for
calculating unknown displacements – cross
out rows with unknown reactions and
columns with displacements that are 0
Solve for displacements
Back-solve for reaction forces
Equations
1
2
3
1
3
2
2
2
2
1 1 0
1 1 0
2
2
2
f1x
f
1y
0
Et
2
f
4
1
2y
0
F
or
0
Et
0
2
4
1
F
0
2
0 1
2
0
2 u2
0 u3
2 v3
1 1 2 0
1 1 2 0
0
0
2 u2
1
1
0 0
1
1
0 u3
0
0
2 v3
Solution
u 2
2F
0
u3
Et
v
1
3
Putting 2 Together
3
4
2
a
1
1
2
a
Element 2 Stiffness Matrix
2
(3)
3
1
T=
1 (4)
Rotate 180o
2
c s 0 0 0 0
-s c 0 0 0 0
0 0 c s 0 0
0 0 -s c 0 0
0 0 0 0 c s
0 0 0 0 -s c
3 (2)
K’ = TTKT
For 180o rotation
K’=K
Just rearrange the rows and columns top correspond to
global numbering scheme (in red).
Element Matrices
f 1x
f
1y
f 2x
Et
f2y
2
f 3x 4 1
f 3y
f
4
x
f4y
f 1x
f
1y
f 2x
Et
f2y
2
f 3x 4 1
f 3y
f 4x
f4y
1
2
3
1
3
2
2
2
2
0
1 1
1 1
0
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
2
0
0
2
0
1
0
0 1 2
0 1 2
1 1 2
1 1 2
0
0
2
1
1
0
1
0
0
0
1
0
0
0
0
0
0
2
0
0
1
0
2
0
2
2
0
1
1
1
0
2
0
0
0
0
0
0
0
0
0
0
0 u1
0 v1
0 u 2
0 v 2
0 u 3
0 v 3
0 u 4
0 v 4
0
0 u1
0
0 v1
1 1 u 2
2
2 v 2
2
2 u 3
1 1 v 3
3
1 u 4
1
3 v 4
Add the element matrices
f 1x
f
1y
f 2x
Et
f2y
2
f 3x 4 1
f 3y
f
4
x
f4y
1
2
1 1
2
0
0 u1
3
1
3
2
1 1
2
0
0 v1
2
2
3
0
0
1
1 1 u 2
1
1
0
3
1
0
2
2
v 2
1 1
0
1
3
0
2
2 u 3
2
1
0
0
3
1 1 v 3
2
0
0
1
2
2
1 3
1 u 4
0
1
2
2
1 1
3 v 4
0
What if triangles have midside nodes?
u ( x, y) c1 c2 x c3 y c4 x c5 xy c6 y
2
3
2
v( x, y) c7 c8 x c9 y c10 x c11 xy c12 y 2
2
4
5
1
2
6
What about a quadrilateral element?
u( x, y) c1 c2 x c3 y c4 xy
v( x, y) c5 c6 x c7 y c8 xy
3
4
1
2
What about arbitrary shapes?
For most problems, the element shapes
are arbitrary, material properties are more
general, etc.
Typical solution is to integrate stiffness
solution numerically
Typically gaussian quadrature, 4 points
k B DBdV
T
V