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 dAq 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 dAq 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
l12 x y
2
21
Tx2
c cos y21 / l12
s sin x12 / l12
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
l12
Ty
Tx u
3
1
te
l12
uT
x
vTy dA
WP of Traction
Ty2
2
WP T u Ttdl
T
Tx2
l12
te
l12
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
l12
te
l12
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 εφdV φ fdV φ TdS φ P
T
V
T
V
T
S
T
i
i
i
Galerkin Approach
0
σ εφdV φ fdV φ TdS φ P
T
T
V
T
V
T
i
S
i
i
Tt (force/length)
0
Ae
e
ε Dεφ 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
εφ Bψ
y4
y1
2
y3
h
y 1
y
2
ψ
y 6
Galerkin Approach
0
Ve
e
σ εφ 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 εφ 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 dAq e
A
T
e
ke
U e ψ k eq e
T
e
Galerkin Approach
0
Ve
e
σ εφ 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 εφ 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
l12
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
l12
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
l12
te
l12
f T
x x
f yTy dA ψ Te
T
e
2Tx1 Tx 2
2T T
tel12 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 εφ 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 εφ 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