Post on 22-Dec-2015
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