e e For element equilibrium, - - PowerPoint PPT Presentation

34

Discretization

e e

  



     

4 1 4 1 4 1

, , ,

• x

y

n w

• i
• i

i n x i xi i n y i yi i

w N x y w N x y N x y

 

   

     

  

  

4 1 n i i i  

  u N u

i

N

• x

y

w i i i

N N N

 

          

 

e e e e e

U W U W          

Concisely, For element equilibrium, 1 2 3 4

            

• x

y

w u

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Finite Element Discretization

If we define 1 1 1 2 2 2

( , , , , , ,......., , , )

• x

y

• x

y

• NN

xNN yNN

w w w       

t

d

and

1 2 1 2 1 2

| | | | | | | | |

• x

x x y y y

w w w NN NN NN

N N N N N N N N N

     

               N

 u Nd

Then we can also write,

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 

1 1  



 

b b i i bi i bi b i b

ε L u L N u B u B =L N B d ; ; Q ;

b NN i NN i

 

1 



 

s s NN s i i i=1 si i si s i s

ε L u L N u B u B =L N B d ; ; Q ;

NN i

 

1 NN i



i i i i

ε=Lu B u B =LN Bd Q ; Q ;

Discrete Strains

37

y x y x

i i bi i i

N x N y N N x y

   

                             B

• y
• x

w i i si w i i

N N x N N y

 

                  B

Finite Element Discretization

38

Finite Element Discretization

1 1 1

1 2 1 2

  

 

 

e t e t e

u K u u f d K d - d f

ij

NN NN NN e t t i j i i i j j where the element stiffness matrix linking node i and j is given by

e t ij i j A

dA  K B DB

39

t

e t t i i i A S

dA dS  

 

f N b N t

e t A

dA   K B DB

and the consistent nodal forces for node I are given by The complete element matrices may be written as

t

e A S

dA dS  

 

t t

f N b N t

Finite Element Discretization

41

Boundary Conditions

Simply Supported

• n

n

• n

n nt t t nt n n

Clamped Free Symmetric Skew Symmetric w Q Q w M M M M                

1 2

• nt

t n n

SS SS w w M M M       

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Effective Design Of Reissner-Mindlin Plate Bending Element

One must consider and study the following points on which considerable body of knowledge is now available.

• Locking : shear constraints

: Kirchhoff mode criteria

• Uniform reduced/selective integrations

(Equivalence with mixed methods)

• Spurious zero energy modes/rank deficiency

43

P h L P

44

TIP DISPLACEMENT (DEEP CANT. BEAM)

N 1 Point Quadrature 2 Points Quadrature 1 0.762 0.416 x 10-1 2 0.940 0.445 4 0.985 0.762 8 0.996 0.927 16 0.999 0.981

45

TIP DISPLACEMENT (THIN CANT. BEAM)

N 1 Point Quadrature 2 Points Quadrature 1 0.750 0.200 x 10-4 2 0.938 0.800 x 10-4 4 0.984 0.320 x 10-3 8 0.996 0.128 x 10-3 16 0.999 0.512 x 10-3

46

1 1 1 1                Kb l l EI l l 1 1 2 1 1 2 1 2 3 1 2 6 1 1 2 1 1 2 1 2 6 1 2 3                    K s l l l l Gh l l l l

 

2 s b

O l h  K K

Discrete Bending Energy Discrete Shear Energy

1 l h 

 

1 1 2 2

   d

t

w w

1 1 2 2   d K d d d

t t e b s

U K

 

2 s b e e

U U O l h 

b e

U

s e

U

l h  



s e

U



b e e

U U

2-Node Timoshenko Beam Element

… Contradiction Physics

47

REMEDIAL MEASURES

Use of Discrete Kirchhoff Procedure Element matrix equation is stabilized by typing the two independent dofs. at discrete point according to Kirchhoff mode criterion. Method is effective but implementation is complicated NOT WIDELY USED. Use Of Selective Reduced Integration Total strain energy is split into bending and to shear energies. A one order lower quadrature formula than the normal one is used for evaluating constrained shear energy component.

48

* Not communicable in mesh of two or more elements.

1 2

, 3 3 3 3 2 2 2 2 2 2

• 1*

1 mod   w Serendipity Serendipity Lagrange Serendipity Lagrange Lagrange X b X b Integration X X S X S Spurious zero energy es

Serendipity Heterosis Lagrange Hughes and Cohen(1978), Computers and Structures, 9, 445-450.