Transcript Document

ECIV 720 A
Advanced Structural Mechanics
and Analysis
Lecture 12:
Isoparametric CST
Area Coordinates
Shape Functions
Strain-Displacement Matrix
Rayleigh-Ritz Formulation
Galerkin Formulation
FEM Solution: Area Triangulation
Area is Discretized into Triangular Shapes
FEM Solution: Area Triangulation
One Source of Approximation
FEM Solution: Nodes and Elements
Points where corners of triangles meet
Define NODES
Non-Acceptable Triangulation
…
Nodes should be defined on corners of
ALL adjacent triangles
FEM Solution: Nodes and Elements
vi
ui
Y
X
Each node translates in X and Y
FEM Solution: Objective
q6
q5
3 (x3,y3)
v
q1
q2
u
1 (x1,y1)
• Use Finite Elements to Compute
Approximate Solution At Nodes
• Interpolate u and v at any point
from Nodal values q1,q2,…q6
q4
q3
2 (x2,y2)
Intrinsic Coordinate System
3 (x3,y3)
h
x
2 (x2,y2)
2 (0,1)
1 (x1,y1)
h
x
Map Element
Define Transformation
3 (0,0)
1 (1,0)
Parent
Area Coordinates
Y
Location of P can be
defined uniquely
2
A1
L1 
A
A2
L2 
A
A3
L3 
A
X
A1
3
P
1
Area Coordinates
Area Coordinates and Shape Functions
L1  x L2  h L3  1  x  h
3 (x3,y3)
x
2 (x2,y2)
1 (x1,y1)
h
Area Coordinates are linear
functions of x and h
h
Are equal to 1 at nodes they
correspond to
2 (0,1)
Are equal to 0 at all other nodes
x
3 (0,0)
1 (1,0)
Natural Choice for Shape Functions
Shape Functions
N1  L1  x
N 2  L2  h
0 x 1
N3  L3  1  x  h
0 h 1
3 (x3,y3)
Y
X
x
1 (x1,y1)
2 (x2,y2)
h
Geometry from Nodal Values
x  N1 x1  N 2 x2  N3 x3
y  N1 y2  N 2 y2  N3 y3
N1  L1  x
N 2  L2  h
N3  L3  1  x  h
x  x1  x3 x  x2  x3 h  x3
 x13x  x23h  x3
y   y1  y3 x   y2  y3 h  y3
 y13x  y23h  y3
Intrinsic Coordinate System
3 (x3,y3)
x  x13x  x23h  x3
y  y13x  y23h  y3
Map Element
x
2 (x2,y2)
h
1 (x1,y1)
Transformation
2 (0,1)
h
x
3 (0,0)
1 (1,0)
Parent
Displacement Field from Nodal Values
3
q6
u  N1q1  N 2 q3  N3q5
q5
x
1
v  N1q2  N 2 q4  N3q6
v
q2
u
q4
q1
2
q3
h
 N1
u
0
0
N1
N2
0
0
N2
N3
0
 q1 
q 
 2
0  q3 
   Nq

N 3  q 4 
q5 
 
q6 
Strain Tensor from Nodal Values of
Displacements
Strain Tensor
 u 


 x 
 v 
ε

 y 
 u  v 
 y x 
Need Derivatives
u
x
v
y
u
y
v
x
u  N1q1  N 2 q3  N3q5
v  N1q2  N 2 q4  N3q6
u and v functions of x and h
Jacobian of Transformation
u u x u y


x x x y x
u u x u y


h x h y h
 u   x
 x   x
 u    x
  
 h   h
J
y   u 
x   x 
  u 
y 
 
h   y 
Jacobian of Transformation
v v x v y


x x x y x
v v x v y


h x h y h
 v   x
 x   x
 v    x
  
 h   h
J
y   v 
x   x 
  v 
y 
 
h   y 
Jacobian of Transformation – Physical Significance
 x
 x
J
 x
 h
y 
x 

y 
h 
 x13
J
 x23
y13 

y23 
x  x13x  x23h  x3
y  y13x  y23h  y3
x
 x13
x
y
 y13
x
x
 x23
h
y
 y23
h
Jacobian of Transformation – Physical Significance
3 (x3,y3)
r1
x
r1  x1  x3 i   y1  y3 j
r2
r2  x2  x3 i   y2  y3 j
2 (x2,y2)
1 (x1,y1)
h
i
r1  r2  x13
x23
j k
x13
y13 0 
x23
y23 0
y13
k
y23
Jacobian of Transformation – Physical Significance
k
r1
x
3 (x3,y3)
r2
r1  r2 
2 (x2,y2)
y13
x23
y23
k
2 Aelem
1 (x1,y1)
h
x13
Compare to Jacobian
 x13
J
 x23
det J  2 Aelem
y13 

y23 
Jacobian of Transformation
 u 
 u 
 x 
 x 
 u   J  u 
 
 
 y 
 h 
 v 
 v 
 x 
 x 
 v   J  v 
 
 
 y 
 h 
Solve
Solve
 u 
 u 
 
 x 
1  x 
 u   J  u 
 
 
 y 
 h 
 v 
 v 
 
 x 
1  x 
 v   J  v 
 
 
 y 
 h 
1  y23  y13 
1
J 



det J  x23 x13  2 Aelem
1
 y23  y13 
 x

x
13 
 23
Strain Tensor from Nodal Values of Displacements
N1  x
N2  h
u


N 1 0 N 2
x x
u


N 1 0 N 2
h h
N3  1  x h
0 N3
 q1 
q 
 2
0   q1  q5 

q6 
0 N3
 q1 
q 
 2
0   q3  q5 

q6 
Strain Tensor from Nodal Values of Displacements
v


0 N1 0 N 2
x x
 q1 
q 
 2
0 N 3    q2  q6 

q6 
v


0 N1 0 N 2
h h
 q1 
q 
 2
0 N 3    q4  q6 

q6 
Strain Tensor from Nodal Values of
Displacements


 x
ε  0



 y

0

  u 
1



y  v  2 Aelem


x 
 y23
 0

 x32
0
y31
0
y12
x32
0
x13
0
y23
x13
y31
x21
B
e = B q
Looks Familiar?
 q1 
0  
q2 

x21  
 
y12   
q6 
q
Strain-Displacement Matrix
xij  xi  x j
3 (x3,y3)
yij  yi  y j
x
1 (x1,y1)
ex 
1
 
ε  e y  
  2 Aelem
 xy 
2 (x2,y2)
 y23
 0

 x32
h
0
x32
y23
y31
0
x13
0
x13
y31
y12
0
x21
 q1 
0  
q2 

x21  
 
y12   
q6 
Is constant within each element - CST
Stresses
 x 
E
 
 y  
2
  1 
 xy 

 
0  ex
1 


 1
0  e y 

1    
0 0
  xy 
2 

σ  Dε
e = B q
σ  DBq
Element Stiffness Matrix ke
3 (x3,y3)
x
1 (x1,y1)
2 (x2,y2)
h
1 T
U e   ε σDdV
2 Ve
dV  tdA
e = B q
 = D B qe
Ue 
1 T T
q e  B DBtdA q e
le
2

1 T T
 q e B DBt  dAq e
2 
A

ke
Formulation of Stiffness Equations
T (force/area)
y
x
z
Assume
Plane
Stress
t
Tt (force/length)
P
y
x
P
Total Potential Approach
Tt (force/length)
Total Potential
P
1 T
T
T
T
   σ εdV   u fdV   u TdS   u i Pi
V
S
2 V
i
Total Potential Approach
1 T
T
T
T
   σ εdV   u fdV   u TdS   u i Pi
V
S
2 V
i
Tt (force/length)
1 T
    ε DεtdA
Ae
e 2
P
   uT ftdA    uT Ttdl   uTi Pi
e
Ae
e
le
i
Total Potential Approach
1 T
    ε DεtdA
Ae
2
e
   u ftdA
T
e
Ae
   u Ttdl
T
e
le
  u Pi
T
i
i
Ue 
1 T T
q e  B DBtdA q e
le
2

1 T T
 q e B DBt  dAq e
2 
A

1 T
 q e k eq e
2
Total Potential Approach
1 T
    ε DεtdA
Ae
2
e
   u ftdA
T
e
Ae
   u Ttdl
T
e
le
  u Pi
T
i
i
Work Potential
of Body Forces
WP of Body Forces
2
Element e
fx
fy
3
u
WP
  u ftdA
T
f
Ae
 te  uf x  vf y dA
Ae
v
1
WP of Body Forces
WP
f
 te  uf x  vf y dA
Ae
u  N1q1  N 2 q3  N3q5
v  N1q2  N 2 q4  N3q6
3
q6
q5
v
q2
x
1
u
q4
q1
2
q3
h
3
q6
WP of Body Forces
q5
v
q2
x
1
u
q4
q1
2
1
Ae Ni dA  3 Ae ,
i  1,2,3
q3
h
WP
f
 q1  te f x  N1dA   q2  te f y  N1dA 
Ae
Ae




q3  te f x  N 2 dA   q4  te f y  N 2 dA 
Ae
Ae




q5  te f x  N 3dA   q6  te f y  N 3dA 
Ae
Ae




WP of Body Forces
WP
  u ftdA  q f e
T
f
T
Ae
 q1
q2
 fx 
f 
te Ae  y 
 q6 

3 f 
x
 
 f y 
Nodal Equivalent
Body Force Vector
Total Potential Approach
1 T
    ε DεtdA
Ae
2
e
   u ftdA
T
e
Ae
   u Ttdl
T
e
le
  u Pi
T
i
i
Work Potential
of Tractions
WP of Traction
Tt (force/length)
2
3
1
Distributed Load acting on EDGE of element
WP of Traction
Components
Tx1,Ty1
Known Distribution
Tx2,Ty2
Normal Pressure
p1, p2
Known Distribution
WP of Traction
Directional cos
Ty2
l12  x  y
2
21
Tx2
c  cos  y21 / l12
s  sin   x12 / l12
Ty1
Tx1
Components
Tx1  cp1
Normal Pressure
p 1 , p2
2
21
Known Distribution
Tx 2  cp2
Ty1   sp1
Ty 2   sp2
WP of Traction
2
WP T   u Ttdl
T
v
l12
Ty
Tx u
3
1
 te 
l12
uT
x
 vTy dA
WP of Traction
Ty2
2
WP T   u Ttdl
T
Tx2
l12
 te 
l12
3
uT
x
 vTy dA
Ty1
Tx1
1
u  N1q1  N 2 q3  N3q5
Tx  N1Tx1  N 2Tx 2
v  N1q2  N 2 q4  N3q6
Ty  N1Ty1  N 2Ty 2
WP of Traction
WP T   u Ttdl
T
l12
 te 
l12
uT
x
 vTy dA  qTe
 2Tx1  Tx 2 
2T  T 
tel1 2  y1 y 2 
 q1 q2 q3 q4 


6 Tx1  2Tx 2 
Nodal Equivalent
Ty1  2Ty 2 
Traction Vector
Total Potential Approach
1 T
    ε DεtdA
Ae
2
e
   u ftdA
T
e
Ae
   u Ttdl
T
e
le
  u Pi
T
i
i
Work Potential
of Concentrated Loads
WP of Concentrated Loads
P
WP
Pi
 u Pi  ui Pxi  vi Pyi
T
i
Indicates that at location of point loads
a node must be defined
In Summary
1 T
    ε DεtdA
Ae
2
e
1 T
   q e k eq e
e 2
   u ftdA
 q f
   u Ttdl
  q Te
  u Pi
  Q Pi
T
e
Ae
T
e
le
T
i
i
T
e e
e
T
e
e
T
i
i
After Superposition
1 T
   q e k eq e
e 2
 q f
T
e e
e
  q Te
T
e
e
  Q Pi
T
i
i
1 T
T
  Q KQ  Q F
2
where
F   f e  Te   P
e
Minimizing wrt Q
0  KQ  F
Galerkin Approach
Tt (force/length)
P
Galerkin
0
 σ εφdV   φ fdV   φ TdS   φ P
T
V
T
V
T
S
T
i
i
i
Galerkin Approach
0
 σ εφdV   φ fdV   φ TdS   φ P
T
T
V
T
V
T
i
S
i
i
Tt (force/length)
0

Ae
e
ε Dεφ tdA
T
P
   φ ftdA    φ Ttdl   φ Pi
T
e
Ae
T
e
le
T
i
i
Galerkin Approach
Introduce Virtual Displacement Field f
φ  Nψ
y6
3
y5
fy
y2
1
x
fx
εφ  Bψ
y4
y1
2
y3
h
y 1 
y 
2

ψ
 
 
y 6 
Galerkin Approach
0

Ve
e
σ εφ dV
T
   φ fdV
T
e
Ve
   φ TdA
T
e
Ae
  φ Pi
T
i
i
U e
Virtual Strain Energy
of element e
Element Stiffness Matrix ke
3 (x3,y3)
x
1 (x1,y1)
U e 
2 (x2,y2)
h

Ae
σ εφ dV
T
dV  tdA
φ  Nψ e
e = B ye
 = D B qe
U e 
ψ
T
e
B
DB
tdA
q
e

T
le
 T

 ψ B DBt  dAq e
A


T
e
ke
U e  ψ k eq e
T
e
Galerkin Approach
0

Ve
e
σ εφ dV
T
   φ fdV
T
e
Ve
   φ TdA
T
e
Ae
  φ Pi
T
i
i
Virtual
Work Potential
of Body Forces
WP of Body Forces
fx
fy
3
fy
WP f   φ ftdA
T
2
Element e
fx
Ae
 te  f x f x  f y f y dA
Ae
φ  Nψ
1
As we have already seen
WP of Body Forces
WP
  ψ ftdA  ψ f e
T
f
T
Ae
 y 1 y 2
 fx 
f 
te Ae  y 
 y6

3 f 
x
 
 f y 
Nodal Equivalent
Body Force Vector
Galerkin Approach
0

Ve
e
σ εφ dV
T
   φ fdV
T
e
Ve
   φ TdA
T
e
Ae
  φ Pi
T
i
i
Virtual
Work Potential
of Traction
WP of Traction
2
WP T   φ Ttdl
fy
T
l12
Ty
Tx fx
3
 te 
1
f x  N1y 1  N 2y 3  N3y 5
f y  N1y 2  N 2y 4  N 3y 6
l12
f T
x x
 f yTy dA
Tx  N1Tx1  N 2Tx 2
Ty  N1Ty1  N 2Ty 2
WP of Traction
WP T   φ Ttdl
T
l12
 te 
l12
f T
x x
 f yTy dA  ψ Te
T
e
 2Tx1  Tx 2 
2T  T 
tel12  y1 y 2 
 y 1 y 2 y 3 y 4 


6 Tx1  2Tx 2 
Nodal Equivalent
Ty1  2Ty 2 
Traction Vector
Galerkin Approach
0

Ve
e
σ εφ dV
T
   φ fdV
T
e
Ve
   φ TdA
T
e
Ae
  φ Pi
T
i
i
Virtual
Work Potential
of Point Loads
WP of Concentrated Loads
P
WP P  φ Pi  f xi Pxi  f yi Pyi
i
T
i
Indicates that at location of point loads
a node must be defined
In Summary
0

Ve
e
σ εφ dV
T
0
ψ
T
e
k eq e
e
   φ fdV
ψ f
   φ TdA
  ψ Te
  φ Pi
  φ Pi
T
e
Ve
T
e
Ae
T
i
i
T
e e
e
T
e
e
T
i
i
After Superposition
0
 ψ k eq e
T
e
e
ψ f
T
e e
e
  ψ Te
T
e
e
  φ Pi
T
i
i
0  Ψ KQ  Ψ F
T
T
where
F   fe  Te   P
e
Y is arbitrary and thus
0  KQ  F
Stress Calculations
BC
0  KQ  F
3
x
1
Q K F
q6
q5
For Each Element
ee = Be qe
v
q2
1
u
q4
q1
2
q3
h
 y23
1 
Be 
0

2A
 x32
0
x32
y23
y31
0
x13
0
x13
y31
y12
0
x21
0
x21 

y12 
Stress Calculations
3
ee = Be qe
q6
q5
v
q2
x
1
u
q4
q1
2
q3
h
 x 
E
 
 y  
2
1


 
 xy 

 
0  ex
1 
 
 1

0 e y 

1    
0 0
  xy 
2 

σ e  De ε e
σ e  De B e q e
Constant
Summary of Procedure
Tt (force/length)
Discretize domain in CST
Nodes should be placed at
- point loads
- start & end of distributed loads
Summary of Procedure
For Every Element Compute
•Strain-Displacement Matrix B
1
B
2 Aelem
3
q6
 y23
 0

 x32
q5
v
q2
x
1
u
q1
q4
2
q3
h
0
x32
y23
y31
0
x13
0
x13
y31
y12
0
x21
0
x21 

y12 
Summary of Procedure
•Element Stiffness Matrix
 T

k e  B DBt  dA
A


•Node Equivalent Body Force Vector
 fx 
f 
te Ae  y 
fe 

3 f 
x
 
 f y 
Summary of Procedure
•Node Equivalent Traction Vector
1 2
e
T
 2Tx1  Tx 2 
2T  T 
tel1 2  y1 y 2 



6 Tx1  2Tx 2 
Ty1  2Ty 2 
For ALL loaded sides
Summary of Procedure
Collect ALL Point Loads
in Nodal Load Vector
Py1
1
Px1
PyN
N
PxN
 Px1 
P 
 y1 
 Px 2 
 
P   Py 2 
  
 
 PxN 
P 
 yN 
Summary of Procedure
Form Stiffness Equations
K
  k e
e
F
  f e  Te   P
e
Q  q1 q2  qn 
T
F  F1 F2  Fn 
T
F  KQ
Summary of Procedure
F  KQ
Solve
Apply Boundary Conditions
1
Q K F
For Every Element Compute Stress
σ e  De B e q e
Example
Tt=200 lb/in
(0,2)
(3,2)
fx=0
fy=60 lb/in2
(0,0)
(3,0)
ANSYS Solution – Coarse Mesh
2-D Constant Stress Triangle
Comments
• First Element for Stress Analysis
• Does not work very well
• When in Bending under-predicts displacements
– Slow convergence for fine mesh
• For in plane strain conditions – Mesh “Locks”
– No Deformations
Element Defects
Element Defects
Constant Stress Triangles
Exact
Y-Deflection &
X-Stress about
¼ of actual
Element Defects
x1=0, y1=0
x2=a, y2=0
x3=0, y3=a
 1
ex   a
  
e y    0
  
 xy    1
a

0
1
a
1
a
1
a
0
0
0
0
0
0
1
a
1
a
0

0  0 
 
1  u 2 
 
a  v2 

0  0 
  0 
 
u2
ex 
a
ey  0
 xy
u2

a
?
Spurious Shear Strain Absorbs Energy – Larger Force Required
Element Defects
Rubber Like Material ~0.5
Mesh Lock