Transcript FEA Short Course

Procedures of Finite Element Analysis
Two-Dimensional Elasticity Problems
Professor M. H. Sadd
Two Dimensional Elasticity Element Equation
Orthotropic Plane Strain/Stress Derivation
Using Weak Form – Ritz/Galerin Scheme
Displacement Formulation Orthotropic Case
Hooke's Law
 
u
v     u v 
 C11
 C12   C66     Fx  0
x 
x
y  y   y x 
 x  C11ex  C12e y
   u v   
u
v 



  Fy  0
C


C

C
 66

 12
22
x   y x  y 
x
y 
 xy  C66exy
 y  C12ex  C22e y
Isotropic Material
 E
 1   2  plane stress
C11  C22   E(1 - )
 plane strain

 (1  )(1 - 2)
 E
 1   2  plane stress
E
C12  
,
C



 plane stress and plane strain
66
E
2
(
1


)
 plane strain

 (1  )(1 - 2)
Two Dimensional Elasticity Weak Form
Mulitply Each Field Equation by Test Function & Integrate Over Element
  

u
v     u v 
he  w1   C11
 C12   C66     Fx dxdy  0
e
x
y  y   y x 
 x 

    u v   

u
v 
he  w2  C66      C12
 C22   Fy dxdy  0
e
x
y 
 x   y x  y 

he  element th ickness (constant)
Use Divergence Theorem to Trade Differentiation On To Test Function
 w 
u
v  w   u v  
he   1  C11
 C12   1 C66    dxdy  he  w1 Fx dxdy  he  w1Tx ds
e
e
x
y  y   y x  
 x 
e
 w
he   2
e
 x
  u v  w2 
u
v 



dxdy  he  w2 Fy dxdy  he  w2Ty ds
C


C

C
 66 
22
 y  12 x
e

y

x

y



e
 

 u v 
 u v 

u
v 
u
v 
Tx   C11
 C12 nx  C66   n y , Ty  C66   nx   C11
 C22 n y
x
y 
x
y 

 y x 
 y x 

Two Dimensional Elasticity Ritz-Galerkin Method
N
N
j 1
j 1
Let w1  w2  i , u   u j  j , v   v j  j in weak forms 
 [ K 11 ]
 12 T
[ K ]
where
[ K 12 ] {u} {F 1}
   2   [ K ]{U }  {F }
22  
[ K ] {v} {F }

  j
 i  j 
dxdy
K ij11  he   C11 i
 C66
e
x x
y y 


 i  j
 i  j 
12
21

dxdy
K ij  K ij  he   C12
 C66
e
x y
y x 


 i  j
 i  j 
22
dxdy
K ij  he   C66
 C22
e

x

x

y

y


Fi1  he   i Fx dxdy  he   iTx ds , Fi 2  he   i Fy dxdy  he   iTy ds
e
e
e
e
Two Dimensional Elasticity Element Equation
Triangular Element N = 3
(x3,y3)

3
 1
(x1,y1)
 [ K 11 ]
 12 T
[ K ]
e
11
 K11

 
 
[ K 12 ] {u} {F 1}
 2  
22  
}
v
{
[ K ]   {F }
 
 

 
2

(x2,y2)
(e)
 K11

 
 

 
 

 
12
K11
11
K12
12
K12
11
K13
22
K11
21
K12
22
K12
21
K13

11
K 22
12
K 22
11
K 23


22
K 22
21
K 23



11
K 33




(e)
K12
(e)
K13
(e)
K14
(e)
K15
(e)
K 22
(e)
K 23
(e)
K 24
(e)
K 25

(e)
K 33
(e)
K 34
(e)
K 35


(e)
K 44
(e)
K 45



(e)
K 55




12
  u1   F11 
K13
  2
22  
v
K13
  1   F1 
1
12  
K 23
u 2   F2 

   2
22 
v
K 23   2   F2 
12  
u3   F31 
K 33
   2 
22
K 33   v3   F3 
(e)
  u1   F11 
K16
  2
(e)  
v
K 26
1
    F1 
1
(e)  
K 36
u 2   F2 

   2
(e) 
v
K 46   2   F2 
(e)  
u3   F31 
K 56
   2 
(e)
K 66   v3   F3 
Two Dimensional Elasticity Element Equation
Plane Strain/Stress Derivation Using Virtual Work Statement
y
(x3,y3)

3
 1
(x1,y1)
Virtual Work Statement
e = 12 + 23 + 31
Ve  he  e
S e  he e
e

Ve
2
(Element Geometry)
Se
Ve
he  ( x e x   y e y  2 xy e xy )dxdy
 he = thickness
(x2,y2)
x
ij eij dV   Ti n ui dS   Fi ui dV
e
 he  (Txn u  T yn v)ds  he  ( Fx u  Fy v)dxdy  0
e
e
  e T    
x
x 

 u T Txn  
 u T  Fx  




he    e y   y  dxdy  he      n  ds  he       dxdy  0
e 
e  v
 e  v
F 

T 
    y  
    y 
 2e xy   xy  


Two Dimensional Elasticity Element Equation
Interpolation Scheme
u( x, y )   ui  i ( x, y )
u 
    []{d }
v 
v ( x , y )   vi  i ( x , y )
i
i
 e x   / x
0 
0 
 / x
u



 
{e}   e y   0
 / y      0
 / y []{d }  [ B]{d }




2e   / y  / x  v   / y  / x 



 xy  
{}  [C ]{e}  [C ][ B]{d }
General Orthotropi c Material
C11 C12
[C ]  C12 C22

0
 0
0 
0 

C66 
Isotropic Material
 E
 1   2  plane stress
C11  C22   E(1 - )
 plane strain

(1


)(1
2

)

 E
 1   2  plane stress
C12  
E
 plane strain

(1


)(1
2

)

E
C66   
 plane stress and plane strain
2(1  )
Two Dimensional Elasticity Element Equation
Virtual Work Statement
 n
 Fx 
T
T Tx
he  {δd } ([ B] [C ][ B]){d }dxdy  he  {δd } [ψ ]  dxdy  he  {δd } [ψ ]  n ds
e
e
e
 Fy 
T y 
T
T
T
T
Element Equation
[ K ]{d }  {F }  {Q}
[ K ]  he  [ B]T [C ][ B]dxdy  Stiffness Matrix
e
 Fx 
{F }  he  [ψ ]  dxdy  Body Force Vector
e
 Fy 
T
n


T
x
T
{Q}  he  [ψ ]  n ds  Loading Vector
e
T y 
Triangular Element With Linear Approximation
v3
u ( x1 , y1 )  u1  c1  c 2 x1  c3 y1
u3
(x3,y3)
u( x, y )  c1  c2 x  c3 y  u ( x2 , y 2 )  u 2  c1  c 2 x 2  c3 y 2
u ( x3 , y 3 )  u 3  c1  c 2 x3  c3 y3
v2
3
y
3
2
v1
(x2,y2)
u( x, y )  u11 ( x, y )  u2  2 ( x, y )  u3 3 ( x, y )   ui  i ( x, y )
u2
i 1
 i ( x, y ) 
1
(x1,y1)
 i  x j y k  xk y j
u1
x
1
( i   i x   i y )
2 Ae
 i  xk  x j
i  y j  yk
 i ( x j , y j )   ij ,
3

i 1
i
1
Lagrange Interpolation Functions
3
2
1
3
3
3
1
1

1
1

2
1

2
1

2
Triangular Element With Linear Approximation
u  1
 
v   0
 e x   / x
0 
u 

 
{e}   e y   0
 / y   


2e   / y  / x  v 

 xy  
0 
 / x
 0
 / y [ψ ]{d }  [ B]{d }


 / y  / x 
0
2
0
3
1
0
2
0
 1

 x
[ B]   0

 1

 y
0
1
y
1
x
 u1 
v 
 1
0  u 2 
   [ψ ]{d }
 3   v 2 
u3 
 
 v3 
 2
x
0
 2
y
0
 2
y
 2
x
 3
x
0
 3
y

0 
1 0  2

 3 
1 

0 1 0
y  2 Ae 
  1 1  2
 3 

x 
0
3
2
0
2
3
Stiffness Matrix
[ K ]  he Ae [ B]T [C ][ B]
12C11   12C66 1  1C12  1  1C66 1 2C11   1  2 C66 1  2 C12   2  1C66


 12 C22  12 C66
 2  1C12  1  2C66  1  2 C22  1 2C66



 22C11   22C66
 2  2 C12   2  2C66
he 
[K ] 

4 Ae 



 22C 22   22C66











1 3C11   1  3C66
 3  1C12  1  3C66
 2 3C11   2  3C66
 3  2C12   2  3C66
 32C11   32C66

1  3C12   3  1C66 

 1  3C22  1 3C66 
 2  3C12   3  2C66 

 2  3C22   2 3C66 
 3  3C12   3  3C66 

 32 C22   32C66 
0
3 

 3 
Loading Terms for Triangular Element
With Uniform Distribution
y
(x3,y3)

3
 1
(x1,y1)
e = 12 + 23 + 31
{F} 
Txn 
{Q}  he  [ψ ]  n ds

T y 
Ve  he  e
S e  he e
e
2
 he = thickness
(x2,y2)
x
he Ae
{Fx Fy Fx Fy Fx Fy }T
3
T
Txn 
Txn 
Txn 
T
T
 he  [ψ ]  n ds  he  [ψ ]  n ds  he  [ψ ]  n ds
12
23
31
T y 
T y 
T y 
T
(Element Geometry)
 1Txn 

n
 1T y 
 T n 
T n 
hL
he  [ψ ]T  xn ds  he   2 xn ds  e 12
12
12  T
2
T y 
 2 y
n
 3Tx 

n
 3T y 
Txn 
 n
T y 
Txn 
 n
T y 
0
 
 0 12
0
0
 
n
T n 


 n
T
h
L
hL
x
T
T Tx
e 23  x 
he  [ψ ]  n ds 
, he  [ψ ]  n ds  e 31

n
23
31
2 T y 
2
T y 
T y 
n
Tx 
 n
T y  23
Txn 
 n
T y 
 0 
 
0
Txn 
 n
T y  31
Rectangular Element Interpolation
y
4
3
Bilinear A pproximati on
u ( x, y )  c1  c2 x  c3 y  c4 xy
b
1
2
x
 1 ( x, y )u1   2 ( x, y )u2   3 ( x, y )u3   4 ( x, y )u4
a
Interpolat ion Functions
y
 x 
1  1  1  
 a  b 
x
y
 2  1  
a b
x y
3 
ab
 x y
 4  1  
 ab
Two Dimensional Elasticity Element Equation
Rectangular Element N = 4
(x4,y4)

4
(x3,y3)

3
 [ K 11 ]
 12 T
[ K ]
e
2
1

(x1,y1)
11
 K11

 
 

 
 

 
 

 
[ K 12 ] {u} {F 1}

   
[ K 22 ] {v} {F 2 }

(x2,y2)
12
K11
11
K12
12
K12
11
K13
12
K13
11
K14
K1122
K1221
K1222
K1321
K1322
K1421

11
K 22
12
K 22
11
K 23
12
K 23
11
K 24


22
K 22
21
K 23
22
K 23
21
K 24



11
K 33
12
K 33
11
K 34




22
K 33
21
K 34





11
K 44






12
  u1   F11 
K14
   
K1422   v1   F12 
12 
u 2   F21 
K 24
  2
22  
v
K 24
  2    F2 
  1
12  
u
K 34
3
   F3 

22 
2
K 34
 v3   F3 
   1
12 
u
K 44
F
 4   4 
22 
 v4   F42 
K 44
Rectangular Element With BiLinear Approximation
u  1 0  2
 
 v   0 1 0
 1

0 
 / x
 x
[ B ]   0
 / y [ψ ]   0

 / y  / x 
 
 1
 y
 1 y
0
  a 1  b 



1 x

0
 1  

b a
 1
x
1
y
  1    1  
a b
 b  a 
0
3
2
0
0
1
y
1
x
 2
x
0
 2
y
3 0
0
 2
y
 2
x
1 y
1  
a b
0

x
ab
4
0
0
x
ab
1 y
1  
a b

u1 
v 
 1
u2 
0  v2 
   [ψ ]{d }
 4  u3 
v3 
 
u4 
v4 
 3
x
0
 3
y
y
ab
0
x
ab
 4
x
0
 3
y
 3
x
0
 4
y
 4
x
0
 4
y
0
x
ab
y
ab

y
ab
0
1 x
1  
b a











1 x
1  
b a
y 


ab 
0
Two Dimensional Elasticity
Rectangular Element Equation - Orthotropic Case
(x4,y4)

4
(x3,y3)

3
e
1

(x1,y1)
2

(x2,y2)
General Orthotropi c Material
C11 C12
[C ]  C12 C22

0
 0
0 
0 

C66 
Stiffness Matrix
[ K ]  he Ae [ B]T [C ][ B]
1  b

a  1
1  2b
a  1
1b
1
1b
2a  1

1
 3  a C11  b C66  4 C12  C66  6   a C11  b C66  4 C12  C66   6  a C11  C66   4 C12  C66  6  a C11  b C66  4  C12  C66  u1   F1 







 
  v1   F12 
22
21
22
21
22
21
22

K
K
K
K
K
K
K

   1 
11
12
12
13
13
14
14
11
12
11
12
11
12

 u2   F2 


K 22
K 22
K 23
K 23
K 24
K 24

 v   F 2 
22
21
22
21
22



K 22
K 23
K 23
K 24
K 24

  2    21 

 u3   F3 
11
12
11
12




K 33
K 33
K 34
K 34

 v  F 2 

 3   3 





K 3322
K 3421
K 3422

 u4   F41 
11
12






K
K
44
44

   2 

 v4   F4 







K 4422
FEA of Elastic 1x1 Plate Under Uniform Tension
Element 1: 1 = -1, 2 = 1, 3 = 0, 1 = 0, 2 = -1, 3 = 1, A1 = ½.
y

0
1
 1 

2



E

2(1   2 )  








3
4
2
3

3
2
T
1
1

1
1
2

2
x
1
1 
2
3
2

1 

2
1 

2
3
2





0
1 

2
1 

2
1 
2
1 
2

 
(1)
 (1)
0  u1  T1x 
   (1) 
 v1(1)  T1 y 
   (1)  T (1) 
 u 2    2 x 
(1)
(1)
 1  v 2  T2 y 
  (1)   (1) 
 u 3  T3 x 
0  v (1)  T (1) 
3y 
 3  
1 
Element 2: 1 = 0, 2 = 1, 3 = -1, 1 = -1, 2 = 0, 3 = 1, A1 = ½
1  
 2
 

 
E

2  
2(1   )

 

 

0
0
1 

1 
2
0

1


0
1 
2







1 
2

1
1 
2
3
2

1 
2
1

 ( 2)
( 2)
 u1  T1x 
 v1( 2 )  T1(y2 ) 
   ( 2) 
( 2)
1    u 2  T2 x 

 ( 2)    ( 2) 
2  v 2  T2 y 
1     ( 2 )  T ( 2 ) 

 u3
3x
2   ( 2)   ( 2) 
3    v3  T3 y 
2 
FEA of Elastic Plate
Assembled Global System
3
4

2
3

3
2
T
1
1

1
1
2

2
 K11(1)











 K11( 2 )
K12(1)  K12( 2 )
K13(1)
K14(1)
K15(1)  K13( 2 )
K16(1)  K14( 2 )
K15(1)

(1)
(2)
K 22
 K 22
(1)
K 23
(1)
K 24
(1)
(2)
K 25
 K 23
(1)
(2)
K 26
 K 24
(1)
K 25


(1)
K 33
(1)
K 34
(1)
K 35
(1)
K 36
0



K
(1)
44
(1)
45
(1)
46
0




(1)
(2)
K 55
 K 33
(1)
(2)
K 56
 K 34
(2)
K 35





(1)
(2)
K 66
 K 44
(2)
K 45






(2)
K 55







K
Loading Condtions
Boundary Conditions
U1 = V1 = U4 = V4 = 0
K
T2(x1)  T / 2 , T2(1y)  0 , T3(x1)  T2(x2 )  T / 2 , T3(y1)  T2(y2 )  0
Reduced System
(1)
 K 33

 
 

 
(1)
(2)
K16(1)  U 1  T1 x  T1 x 
(2) 
  (1)
(1)  
K 26
  V1  T1 y  T1 y 
(1)

0  U 2   T2 x

  
(1)
0  V2   T2 y

   (1)
(2) 
(2)  
K 36 U 3  T3 x  T2 x 
(1)
(2)
(2)  
K 46
 V3  T3 y  T2 y 
 

(2)
(2)  
U
T
K 56
4
3
x

  
(2)
(2) 



V
T
K 66   4  
3y

(1)
K 34
(1)
K 35
(1)
K 44
(1)
K 45

(1)
( 2)
K 55
 K 33


 U 2  T / 2 
  

(1)
V
0
K 46




2



(1)
( 2) 
K 56  K 34  U 3  T / 2 
(1)
( 2)  
K 66
 K 44
  V3   0 
(1)
K 36
Solution of Elastic Plate Problem
Choose Material Properties:
E = 207GPa and v = 0.25
3
4

2
3

3
2
T
1
1
1

1
2
U 2   0.492 
V   0.081 
 2 

11

  
T  10 m
U 3   0.441 
 V3   0.030 

2
Note the lack of symmetry in the displacement solution
Axisymmetric Formulation
z
 constant
plane
4
3
1
2
Quad Element
ur  N1u1  N 2 u2  N 3u3  N 4 u4
u z  N1v1  N 2 v2  N 3v3  N 4 v4
r
Strain - Displaceme nt


 er   1r
e  


{ e}       r
 ez   0


2erz 
 

 z


0
 r

1
0  u  
 r    r
  u z  
0

z 




r 
 z

0

0 N
 1
  0
z 


r 
0
N2
0
N3
0
N4
N1
0
N2
0
N3
0
 u1 
v 
 1
u2 
 
0  v2 
   [ B]{d }
N 4  u3 
 v3 
 
u4 

v4 

Axisymmetric Formulation
Strain - Displaceme nt
 N 1
 r
 er 
N
e 
 1
  
r
{ e}  
  [ B]{d }  
 0
 ez 


2erz 

 N 1

 z
0
0
N 1
z
N 1
r
N 2
r
N2
r
0
N 2
z
0
0
N 2
z
N 2
r
N 3
r
N3
r
0
0
N 3
z
N 3
r
0
N 3
z
N 4
r
N4
r
0
N 4
z
 u1 
 
0   v1 
 u2 
0  v 
2


N 4  u3 
 
z   v 
N 4   3 
 u
r   4 

v4 

Stress - Strain Relations


0  e 
1  
r 
r
 

 

1 

0 e 
E
 

  
{ σ}    

1 
0   e   [C ]{e}  [C ][ B]{d }
 

(
1


)(
1

2

)
z
 
1  2   z 

0
0

2 e 
 0
 rz 

2   rz 
Element Equation
Stiffness Matrix
[ K ]{d }  {F }  {Q}
[ K ]   [ B]T [C ][ B] rdrdz
e
Two-Dimensional FEA Code
MATLAB PDE Toolbox
- Simple Application Package
For Two-Dimensional Analysis
Initiated by Typing “pdetool”
in Main MATLAB Window
- Includes a Graphical User
Interface (GUI) to:
- Select Problem Type
- Select Material Constants
- Draw Geometry
- Input Boundary Conditions
- Mesh Domain Under Study
- Solve Problem
- Output Selected Results
Two-Dimensional FEA Example
Using MATLAB PDE Toolbox
Cantilever Beam Problem
L/2c = 5
g1=0
2c = 0.4
g2=100
L=2
Mesh: 4864 Elements, 2537 Nodes
FEA MATLAB PDE Toolbox Example
Cantilever Beam Problem
Stress Results
E = 10x106 , v = 0.3
Contours of sx
g1=0
2c = 0.4
g2=100
L=2
 max 
Mc (40)( 2)(0.2)

 3000
I
(1)(0.4)3 / 12
FEA Result: smax = 3200
FEA MATLAB PDE Toolbox Example
Cantilever Beam Problem
Displacement Results
Contours of Vertical
Displacement v
E = 10x106 , v = 0.3
g1=0
2c = 0.4
g2=100
L=2
vmax
PL3
(40)( 2)3


 0.002
3EI (3)(107 )(1)(0.4)3 / 12
FEA Result: vmax = 0.00204
Two-Dimensional FEA Example
Using MATLAB PDE Toolbox
Plate With Circular Hole
Contours of Horizontal Stress x
Stress Concentration Factor: K  2.7
Theoretical Value: K = 3
Two-Dimensional FEA Example
Using MATLAB PDE Toolbox
Plate With Circular Hole
Contours of Horizontal Stress x
Stress Concentration Factor: K  3.5
Theoretical Value: K = 4
FEA MATLAB Example
Plate with Elliptical Hole
(Finite Element Mesh: 3488 Elements, 1832 Nodes)
Aspect Ratio b/a = 2
(Contours of Horizontal Stress x)
Stress Concentration Factor K  3.3
Theoretical Value: K = 5
FEA Example
Diametrical Compression of Circular Disk
(FEM Mesh: 1112 Elements, 539 Nodes)
(FEM Mesh: 4448 Elements, 2297 Nodes)
(Contours of Max Shear Stress)
(Contours of Max Shear Stress)
Theoretical Contours of
Maximum Shear Stress
Experimental Photoelasticity
Isochromatic Contours