ME300H Introduction to Finite Element Methods Finite Element Analysis of Plane Elasticity.
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ME300H Introduction to Finite Element Methods Finite Element Analysis of Plane Elasticity
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Transcript of ME300H Introduction to Finite Element Methods Finite Element Analysis of Plane Elasticity.
ME300H Introduction to Finite Element Methods
Finite Element Analysis of Plane Elasticity
Review of Linear Elasticity
Linear Elasticity: A theory to predict mechanical response of an elastic body under a general loading condition.
Stress: measurement of force intensity
zzzyzx
yzyyyx
xzxyxx
zxxz
zyyz
yxxy
with
xx xy
yx yy
2-D
Review of Linear Elasticity
Traction (surface force) :
Equilibrium – Newton’s Law
0
0
Static
xyxxx
yx yyy
fx y
fx y
0F
x xx x xy y
y xy x yy y
t n n
t n n
t
Dynamic
xyxxx x
yx yyy y
f ux y
f ux y
Review of Linear Elasticity
Strain: measurement of intensity of deformation
1 1
2 2y yx x
xx xy xy yy
u uu u
x y x y
Generalized Hooke’s Law
yyxx zzxx
yyxx zzyy
yyxx zzzz
E E E
E E E
E E E
zxzxyzyzxyxy GGG
12
EG
zzyyxx
zzzz
yyyy
xxxx
e
Ge
Ge
Ge
2
2
2
1 1 2
E
Plane Stress and Plane Strain
Plane Stress - Thin Plate:
xy
y
x
22
22
xy
y
x
12
E00
01
E
1
E
01
E
1
E
xy
y
x
33
2212
1211
xy
y
x
C00
0CC
0CC
Plane Stress and Plane Strain
Plane Strain - Thick Plate:
xy
y
x
xy
y
x
12
E00
0211
E1
211
E
0211
E
211
E1
xy
y
x
33
2212
1211
xy
y
x
C00
0CC
0CC
Plane Stress: Plane Strain:
Replace E by and by21 E
1
Equations of Plane Elasticity
Governing Equations(Static Equilibrium)
Constitutive Relation (Linear Elasticity)
Strain-Deformation(Small Deformation)
xy
y
x
33
2212
1211
xy
y
x
C00
0CC
0CC
0yxxyx
0yx
yxy
x
ux
y
vy
y
u
x
vxy
0y
vC
x
uC
yx
vC
y
uC
x
0x
vC
y
uC
yy
vC
x
uC
x
22123333
33331211
Specification of Boundary Conditions
EBC: Specify u(x,y) and/or v(x,y) on
NBC: Specify tx and/or ty on
where
is the traction on the boundary at the segment ds.
yyyxyxyyxyxxxxyx nntnntjtitsT ; ;)(
Weak Formulation for Plane Elasticity
dxdyy
vC
x
uC
yx
vC
y
uC
xw0
dxdyx
vC
y
uC
yy
vC
x
uC
xw0
221233332
333312111
dstwdxdyy
vC
x
uC
y
w
x
v
y
uC
x
w0
dstwdxdyx
v
y
uC
y
w
y
vC
x
uC
x
w0
y222122
332
x1331
12111
where
y2212x33y
y33x1211x
ny
vC
x
uCn
x
v
y
uCt
nx
v
y
uCn
y
vC
x
uCt
are components of traction on the boundary
Finite Element Formulation for Plane Elasticity
n
1jj
22ij
n
1jj
21ij
2i
n
1jj
12ij
n
1jj
11ij
1i
vKuKF
vKuKF
Let
n
1jjj
n
1jjj
v)y,x()y,x(v
u)y,x()y,x(u
dxdyyy
Cxx
CK
Kdxdyxy
Cyx
CK
dxdyyy
Cxx
CK
ji22
ji33
22ij
21ji
ji33
ji12
12ij
ji33
ji11
11ij
dxdyfdstF
dxdyfdstF
yiyi2
i
xixi1
i
where
and
Constant-Strain Triangular (CST) Element for Plane Stress Analysis
Let1 2 3 1 1 2 2 3 3
5 6 7 1 1 2 2 3 3
( , )
( , )
u x y c c x c y u u u
v x y c c x c y v v v
1 1, xu F
1 1, yv F
2 2, xu F
3 3, xu F
2 2, yv F
3 3, yv F
2 3 3 2
1 2 3
3 2
1
2 e
x y x yx y
y yA
x x
3 1 1 3
2 3 1
1 3
1
2 e
x y x yx y
y yA
x x
1 2 2 1
3 1 2
2 1
1
2 e
x y x yx y
y yA
x x
Constant-Strain Triangular (CST) Element for Plane Stress Analysis
111 12 13 14 15 16 1
121 22 23 24 25 26 1
231 32 33 34 35 36 2
2241 42 43 44 45 46
3351 52 53 54 55 56
3361 62 63 64 65 66
1
4
x
y
x
ye
x
y
Fk k k k k k u
Fk k k k k k v
Fk k k k k k u
Fvk k k k k kA
Fuk k k k k k
Fvk k k k k k
2 2 2 2 2
11 11 2 3 33 3 2 21 12 2 3 3 2 33 2 3 22 22 3 2 33 2 3
2 2
31 11 3 1 2 3 33 1 3 3 2 32 12 3 1 3 2 33 1 3 3 2 33 11 3 1 33 1 3
41 12 2 3 33 1 3
; ;
; ;
k c y y c x x k c y y x x c y y k c x x c y y
k c y y y y c x x x x k c y y x x c x x x x k c y y c x x
k c y y c x x x
2
3 2 42 22 1 3 3 2 33 2 3 3 1 43 12 1 3 3 1 33 1 3
2 2
44 22 1 3 33 3 1 51 11 1 2 2 3 33 2 1 3 2 52 12 1 2 33 2 1 3 2
53 11 1 2 3 1 33 2 1 1 3 54
; ;
; ;
;
x k c x x x x c y y y y k c x x y y c x x
k c x x c y y k c y y y y c x x x x k c y y c x x x x
k c y y y y c x x x x k c
2 2
12 1 2 1 3 33 2 1 1 3 55 11 1 2 33 2 1
61 12 2 3 33 2 1 3 2 62 22 2 1 3 2 33 1 2 2 3 63 12 3 1 33 2 1 1 3
64 22 1 3 2 1 33 1 2 3 1 65 12 1 2 2 1
;
y y x x c x x x x k c y y c x x
k c y y c x x x x k c x x x x c y y y y k c y y c x x x x
k c x x x x c y y y y k c y y x x c
2 2 2
33 2 1 66 22 2 1 33 1 2 x x k c x x c y y
4-Node Rectangular Element for Plane Stress Analysis
Let
443322118765
443322114321
vvvvxycycxcc)y,x(v
uuuuxycycxcc)y,x(u
b
y
a
x1
b
y
a
xb
y1
a
x
b
y1
a
x1
43
21
4-Node Rectangular Element for Plane Stress Analysis
For Plane Strain Analysis:21
EE
1
and
Loading Conditions for Plane Stress Analysis
n
1jj
22ij
n
1jj
21ij
2i
n
1jj
12ij
n
1jj
11ij
1i
vKuKF
vKuKF
dxdyfdstF
dxdyfdstF
yiyi2
i
xixi1
i
Evaluation of Applied Nodal Forces
dstF xi1
i
tdy16
y1
b
y1
a
xdstFF
b
0
2
ox2
)A(12
)A(x2
3.383dy168
y
16
y
8
y1100dy1.0
16
y11000
8
y1
8
8F
8
0 2
3
2
28
0
2)A(
x2
tdy16
y1
b
y
a
xdstFF
b
0
2
ox3
)A(13
)A(x3
350dy168
y
8
y100dy1.0
16
y11000
8
y
8
8F
8
0 2
38
0
2)A(
x3
Evaluation of Applied Nodal Forces
tdy16
8y1
b
y1
a
xdstFF
b
0
2
ox2
)B(12
)B(x2
7.216dy168
y
16
y
32
y5
4
3100dy1.0
16
8y11000
8
y1
8
8F
8
0 2
3
2
28
0
2)B(
x2
tdy16
8y1
b
y
a
xdstFF
b
0
2
ox3
)B(13
)B(x3
7.116dy168
y
16
y2
32
y3100dy1.0
16
8y11000
8
y
8
8F
8
0 2
3
2
28
0
2)B(
x3
Element Assembly for Plane Elasticity
4
4
3
3
2
2
1
1
)A()A(
y
x
y
x
y
x
y
x
v
u
v
u
v
u
v
u
F
F
F
F
F
F
F
F
3
3
4
4
2
2
1
1
��������A
B
1 2
3 4
34
65
6
6
5
5
4
4
3
3
)B()B(
y
x
y
x
y
x
y
x
v
u
v
u
v
u
v
u
F
F
F
F
F
F
F
F
3
3
4
4
2
2
1
1
��������
Element Assembly for Plane Elasticity
1 2
3 4
65
A
B
6
6
5
5
4
4
3
3
2
2
1
1
)B(y
)B(x
)B(y
)B(x
)B(y
)A(y
)B(x
)A(x
)B(y
)A(y
)B(x
)A(x
)A(y
)A(x
)A(y
)A(x
v
u
v
u
v
u
v
u
v
u
v
u
0000
0000
0000
0000
0000
0000
0000
0000
F
F
F
F
FF
FF
FF
FF
F
F
F
F
3
3
4
4
23
23
14
14
2
2
1
1
Comparison of Applied Nodal Forces
Discussion on Boundary Conditions
•Must have sufficient EBCs to suppress rigid body translation and rotation
• For higher order elements, the mid side nodes cannot be skipped while applying EBCs/NBCs
Plane Stress – Example 2
Plane Stress – Example 3
Evaluation of Strains
44332211
44332211
vvvv)y,x(v
uuuu)y,x(u
b
y
a
x1
b
y
a
xb
y1
a
x
b
y1
a
x1
43
21
4
1jj
jj
j
4
1jj
j
4
1jj
j
xy
y
x
vx
uy
vy
ux
x
v
y
uy
vx
u
Evaluation of Stresses
4
4
3
3
2
2
1
1
xy
y
x
v
u
v
u
v
u
v
u
ab
y
a
x1
b
1
ab
y
ab
x
b
y1
a
1
ab
x
b
y1
a
1
a
x1
b
1a
x1
b
10
ab
x0
ab
x0
a
x1
b
10
0ab
y0
ab
y0
b
y1
a
10
b
y1
a
1
Plane Stress Analysis Plane Strain Analysis
xy
y
x
22
22
xy
y
x
12
E00
01
E
1
E
01
E
1
E
xy
y
x
xy
y
x
12
E00
0211
E1
211
E
0211
E
211
E1
