### 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
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
21  

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 
21    
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
ab

4-Node Rectangular Element for Plane Stress Analysis
For Plane Strain Analysis:
E 
E
1  2
and
 

1 
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  10001     0.1dy  1000  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
10001     0.1dy  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   y8 
 1   o 1  
 tdy
a
b    16  
  y  8 2 
8 3
8
y
5 y y2
y3 
dy  216.7
 2
 1  10001  
  0.1dy  1000  
2 
8
8
16
4
32
16
8

16
 


 
1 (B)
3
8
b
b
0
2
x y   y8 
 o 1  
 tdy
a b   16  
  y  8 2 
8 3y
8 y
2 y2
y3 
dy  116.7
10001  
 2 
  0.1dy  1000 
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)
u3  
v 
 3  
u4  
 
 v 4  
 
u5  
 v5  
 
u6  
 v6  
 
 Fx 1 
F 
 y1 
 Fx 2 
 
 F y2 
 
 Fx 4 
 F y4 
 
 Fx 3 
 Fy 
 3
( A)
 
   
     



     















 
    


 
    


     



     
( A)
 u1 
v 
 1 
u2  
 
 v 2  
 
u3  
 v 3  
 
u4  
 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   u1  
 
0   v1  
0  u2  
 
0   v 2  
  u3  
 
   v 3  
 
  u4  

   v 4  
 
  u5  

   v5  
 u 
   6  

  
 v6  
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  
ab

 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 
21    
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
```