Lecture 8 Notes
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Transcript Lecture 8 Notes
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
xx xy xz
yx yy yz
zx zy zz
2-D
xx
yx
xy yx
with
yz zy
xz zx
xy
yy
Review of Linear Elasticity
Traction (surface force) :
t
t x xx nx xy n y
t y xy nx yy n y
Equilibrium – Newton’s Law
F 0
xx xy
fx 0
x
y
yx yy
fy 0
x
y
Static
xx xy
f x ux
x
y
yx yy
f y uy
x
y
Dynamic
Review of Linear Elasticity
Strain: measurement of intensity of deformation
1
1 ux u y
xy xy
2
2 y
x
u
xx x
x
yy
u y
y
Generalized Hooke’s Law
xx
xx
E
yy
zz
xx
E
xx
E
yy
E
yy
E
E
yy
E
zz
zz
E
zz
E
xx e 2G xx
yy e 2G yy
zz e 2G zz
e xx yy zz
xy G xy yz G yz zx G zx
E
G
21
E
1 1 2
Plane Stress and Plane Strain
Plane Stress - Thin Plate:
x C 11
y C 12
0
xy
C 12
C 22
0
0 x
0 y
C 33 xy
E
x 1 2
E
y
2
1
xy 0
E
1 2
E
1 2
0
x
0 y
E xy
21
0
Plane Stress and Plane Strain
Plane Strain - Thick Plate:
x C 11
y C 12
0
xy
C 12
C 22
0
1 E
x 1 1 2
E
y
1 1 2
xy
0
0 x
0 y
C 33 xy
Plane Strain:
Plane Stress:
Replace E by
E
1 2
and by
1
E
1 1 2
1 E
1 1 2
0
x
0 y
E xy
2 1
0
Equations of Plane Elasticity
Governing Equations
(Static Equilibrium)
Constitutive Relation
(Linear Elasticity)
x C 11
y C 12
0
xy
C 12
C 22
0
0 x
0 y
C 33 xy
x xy
0
x
y
xy y
0
x
y
Strain-Deformation
(Small Deformation)
v
y
y
v u
x y
u
x
x
xy
u
v
u
v
0
C 12 C 33
C 33
C 11
x
x
y y
y
x
u
v
u
v
0
C
C
C
C
33
33
12
22
y
x y
x
y
x
Specification of Boundary Conditions
EBC: Specify u(x,y) and/or v(x,y) on G
NBC: Specify tx and/or ty on G
where T (s) t x i t y j ; t x xxnx xyny ; t y yxnx yyny
is the traction on the boundary G at the segment ds.
Weak Formulation for Plane Elasticity
v
u
v
u
dxdy
C 33
C 12 C 33
0 w1 C 11
x
y
y y
x
x
v
u
v
u
C 12
C 22 dxdy
C 33
0 w 2 C 33
y
x
x y
y
x
w 1
u v
u
v w 1
C 11
C 12
C 33
dxdy w1 t x ds
0
x
y y
y x
x
G
w 2
u v w 2
u
v
C 12
C 33
C 22 dxdy w 2 t y ds
0
x
y
y x y
x
G
where
u v
u
v
C 12 n x C 33 n y
t x C 11
x
y
y x
t C u v n C u C v n
33
22
x 12 x
y
y
y
x
y
are components of
traction on the
boundary G
Finite Element Formulation for Plane Elasticity
Let
n
u( x , y ) j ( x , y )u j
j 1
n
v ( x , y ) j ( x , y )v j
j 1
where
and
n
1 n 11
12
F
K
u
K
vj
i
ij
j
ij
j 1
j 1
n
n
2
21
Fi K ij u j K ij22 v j
j 1
j 1
11
i j
i
C 33
K ij C 11
x x
y
12
i j
i
K
C
C
ij
12 x y 33 y
22
i j
i
C 22
K ij C 33
x x
y
1
Fi i t x ds i f x dxdy
G
F 2 t ds f dxdy
i G i y i y
j
dxdy
y
j
dxdy K 21
ji
x
j
dxdy
y
Constant-Strain Triangular (CST) Element for Plane
Stress Analysis
v2 , F2 y
u2 , F2 x
v3 , F3 y
u3 , F3 x
v1, F1 y
u1, F1x
Let
u( x, y ) c1 c2 x c3 y 1u1 2u2 3u3
v( x, y ) c5 c6 x c7 y 1v1 2v2 3v3
x2 y3 x3 y2
1 x y
1
y 2 y3
2 Ae
x x
2
3
x3 y1 x1 y3
1 x y y y
2
3 1
2 Ae
x x
1 3
3
1
x1 y2 x2 y1
y
y1 y2
2 Ae
x x
2 1
x
Constant-Strain Triangular (CST) Element for Plane
Stress Analysis
k11
k
21
1 k31
4 Ae k41
k51
k61
k12
k22
k13 k14
k23 k24
k15
k25
k32
k33
k34
k35
k42
k43 k44
k45
k52
k53
k54
k55
k62
k63
k64
k65
k16 u1 F1 x
k26 v1 F1 y
k36 u2 F2 x
F
k46 v2 2 y
k56 u3 F3 x
k66 v3 F3 y
k11 c11 y2 y3 c33 x3 x2 ; k21 c12 y2 y3 x3 x2 c33 y2 y3 ; k22 c22 x3 x2 c33 y2 y3
2
2
2
2
2
k31 c11 y3 y1 y2 y3 c33 x1 x3 x3 x2 ; k32 c12 y3 y1 x3 x2 c33 x1 x3 x3 x2 ; k33 c11 y3 y1 c33 x1 x3
2
2
k41 c12 y2 y3 c33 x1 x3 x3 x2 ; k42 c22 x1 x3 x3 x2 c33 y2 y3 y3 y1 ; k43 c12 x1 x3 y3 y1 c33 x1 x3
2
k44 c22 x1 x3 c33 y3 y1 ; k51 c11 y1 y2 y2 y3 c33 x2 x1 x3 x2 ; k52 c12 y1 y2 c33 x2 x1 x3 x2
2
2
k53 c11 y1 y2 y3 y1 c33 x2 x1 x1 x3 ; k54 c12 y1 y2 x1 x3 c33 x2 x1 x1 x3 ; k55 c11 y1 y2 c33 x2 x1
2
2
k61 c12 y2 y3 c33 x2 x1 x3 x2 k62 c22 x2 x1 x3 x2 c33 y1 y2 y2 y3 k63 c12 y3 y1 c33 x2 x1 x1 x3
k64 c22 x1 x3 x2 x1 c33 y1 y2 y3 y1 k65 c12 y1 y2 x2 x1 c33 x2 x1
2
k66 c22 x2 x1 c33 y1 y2
2
2
4-Node Rectangular Element for Plane Stress Analysis
u( x , y ) c1 c2 x c3 y c4 xy 1 u1 2 u2 3 u3 4 u4
Let v( x , y ) c c x c y c xy v v v v
5
6
7
8
1 1
2 2
3 3
4 4
x
y
x
y
1 1 1 2 1
a
b
a
b
x y
x y
3
4 1
a b
ab
4-Node Rectangular Element for Plane Stress Analysis
For Plane Strain Analysis:
E
E
1 2
and
1
Loading Conditions for Plane Stress Analysis
n
1 n 11
12
F
K
u
K
i ij j ij v j
j 1
j 1
n
n
Fi 2 K ij21 u j K ij22 v j
j 1
j 1
1
Fi i t x ds i f x dxdy
G
F 2 t ds f dxdy
i G i y i y
Evaluation of Applied Nodal Forces
Fi1 i t x ds
G
1 ( A)
2
Fx(2A ) F
Fx(2A )
8
0
( A)
x3
F
8
0
G
0
2
x
y y
1 o 1 tdy
a
b 16
y 2
8
8
y
y y2
y3
dy 383.3
1 10001 0.1dy 1000 1 2
2
8
8
16
8
16
8
16
1 ( A)
3
Fx(3A ) F
2 t x ds
b
3 t x ds
G
b
0
2
x y y
o 1 tdy
a b 16
y 2
8 y
8 y
y3
dy 350
10001 0.1dy 100
2
0
88
16
8
8
16
Evaluation of Applied Nodal Forces
1 (B)
2
Fx(2B ) F
Fx(2B )
8
0
(B)
x3
F
2 t x ds
0
G
F
3 t x ds
G
Fx(3B )
0
2
x
y y8
1 o 1
tdy
a
b 16
y 8 2
8 3
8
y
5 y y2
y3
dy 216.7
2
1 10001
0.1dy 1000
2
8
8
16
4
32
16
8
16
1 (B)
3
8
b
b
0
2
x y y8
o 1
tdy
a b 16
y 8 2
8 3y
8 y
2 y2
y3
dy 116.7
10001
2
0.1dy 1000
2
88
8 16
16
32 16
Element Assembly for Plane Elasticity
5
6
B
3
4
4
3
A
1
2
Fx 1
F
y1
Fx 2
F y2
Fx 4
F y4
Fx 3
Fy
3
(B)
(B)
u3
v
3
u4
v 4
u5
v5
u6
v6
Fx 1
F
y1
Fx 2
F y2
Fx 4
F y4
Fx 3
Fy
3
( A)
( A)
u1
v
1
u2
v 2
u3
v 3
u4
v 4
Element Assembly for Plane Elasticity
5
6
B
4
3
A
2
1
Fx(1A )
F y(1A )
Fx(2A )
( A)
F
y2
F ( A ) F ( B )
x1
x(4A )
(B)
F y 4 F y1
( A)
( B )
F
F
x2
x3
F y( A ) F y( B )
3 (B) 2
Fx 4
0
0
F y(4B )
(B)
Fx 3
0
0
F y(3B )
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 u1
0 v1
0 u2
0 v 2
u3
v 3
u4
v 4
u5
v5
u
6
v6
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
x
a
x y
3
a b
y
b
x
y
1
a
b
x y
4 1
ab
1 1 1 2
u( x , y ) 1 u1 2 u2 3 u3 4 u4
v ( x , y ) 1v1 2 v 2 3 v 3 4 v 4
4
j
u
uj
j 1 x
x x
4
j
v
v
y
j
y
y
j 1
xy u v 4 j
j
u
v
y x
j 1 y j x j
Evaluation of Stresses
u1
v
1
1
y
1
y
y
y
0
0
0
0
1
a 1 b
u2
a
b
ab
ab
x
1 x
x
x
1 x v 2
0
1
0
0
0
1
y
b a
ab
ab
b a u3
xy
1
x
1
y
x
1
y
x
y
1
x
y
1 1
v3
1
1
b a
a b
ab
a b ab ab b a
ab u
4
v
4
Plane Stress Analysis
E
x 1 2
E
y
2
1
xy 0
E
1 2
E
1 2
0
x
0 y
E xy
21
0
Plane Strain Analysis
1 E
x 1 1 2
E
y
1 1 2
xy
0
E
1 1 2
1 E
1 1 2
0
x
0 y
E xy
2 1
0