Finite Element Formulation Tarun Kant Department of Civil - - PowerPoint PPT Presentation

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Reissner-Mindlin Plate Bending Finite Element Formulation

Tarun Kant

Department of Civil Engineering Indian Institute of Technology Bombay

E-mail: tkant@civil.iitb.ac.in Website: www.civil.iitb.ac.in/~tkant

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BEAM

w u x z Δx x τxz σx p(x)

z

dz

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Euler-Bernoulli Beam

( , ) ( ,0) ( , ) ( ,0)

z

u u x z u x z z w x z w x



    

• r more concisely

( ) ( ) ( )

• u

u x z x u z w w x w        

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S-D Relations

setting 0 gives us

• x
• xz

xz

• u

u z x x x w u w z x x w x                            

• Causes occurrence of second derivative of wo in

PE functional “π”

• Involves C1 continuous interpolation functions in

finite element formulation

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,

x xz x

  

Non-vanishing stresses and strains

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Before deformation

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After deformation

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Timoshenko Beam

Same as that of Euler-Bernoulli Beam

,

x xz

 

Displacement Model Non-Vanishing Stresses/Strains

 

, ( )   

x xz

x

S-D Relations

• x

u z x x        

• xz

w x        

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• Π contains only first derivatives of wo and θ
• FE Formulation uses simple Co shape functions

Timoshenko Beam

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Euler-Bernoulli Beam, Poisson-Kirchhoff Plate and Love Shell Theories

• Thin beam, plate, shell, i.e., (h/L) << 1 or (h/a) << 1 or

(h/R) << 1

• Tangential displacements vary linearly through the

thickness.

• Transverse shear deformation is neglected, γ1z= γ2z=0
• Transverse normal strain is zero, εz = 0
• Transverse normal stress is small (neglected), σz ≈ 0

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After deformation

Reissner-Mindlin theory γ or φ θ

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Kirchhoff Hypothesis

• Transverse shear deformability of structural elements

(especially of shear deformable FRC laminates) is not accounted in the formulation

• Transverse normal rotations of the cross sections

becomes first derivatives of transverse displacement components,

• Transverse displacement field turns C1 continuous

Limitations

• i

i

w x     

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Remarks

• Popular displacement – based formulations include

–transverse shear deformation/s. ?

• This is not only due to the significance of shear

deformation effects in moderately thick/multilayered/FR composite elements.

• But mainly because the resulting formulation is Co

which enables simple interpolations.

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• The classical Poisson-Kirchhoff plate theory requires C1

continuity.

• Co– continuous finite element interpolations are easily

constructed.

• Multi-dimensional C1 -interpolations are difficult/tedious to

construct.

• Considerable ingenuity was required to develop compatible

C1- continuous plate/shell elements; resulting schemes have always been extremely complicated in one way or another.

Remarks (Contd.)

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• Since mid-seventies, there is turning away from

Poisson-Kirchhoff theory based elements to those based on Reissner-Mindlin theory which not only requires Co-continuity but also accommodates transverse shear deformations. However, these too, initially, were not without its own inherent difficulties.

• Locking in ‘thin’ regime
• Spurious ‘zero-energy’ modes

Remarks (Contd.)

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Positive set of stress components

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Positive set of displacement components

z y x

w

• w
• v

v

u

• u

x



y



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Kinematics

( , , ) ( , ) ( , , ) ( , ) ( , , ) ( , )

y x

• u x y z

z x y v x y z z x y w x y z w x y      

( , , ) ( , , ) ( , , )

x y

• x

y

u v w w

• r

w      

t t

u u

z y x

w

• w
• v

v

u

• u

x



y



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Strains

y x x x y y y x xy xy

• xz

y x

• yz

x y

u z z x x v z z y y u v z z y x y x w u w z x x w v w z y y                                                                     

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Middle Surface Strains

1 1 ε Lu                                                           

• x

y

x y w x y x y

b b s s

ε L u ε L u  

b

x L y x y                             

1 1

s

x L y                  

( , , ) ( , ) ( , )

t b t s t

ε ε ε

x y xy y x t t b s

         

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Total Potential Energy 1 2

t

t t t V V S

U W dV dV dS     

  

= ε σ u b u t

int u : ε : σ : b : t : : vector of displacement components of a po in plate space vector of strain components vector of stress components vector of body forces vector of boundary tractions V sol :

t

ution domain S part of boundary on which boundary tractions are prescribed

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Strain Energy

       

1 2 1 2 1 ( ( ) ( ) ( ) 2 1 2 1 . 2

b s b x x y y xy xy V x x y y xy xy V x x y y xy xy A z z z x x y y xy xy A t b b A

U U U U dV z z z dV zdz zdz zdz dA M M M dA dA                                      

       

= = = = ε σ

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( , , ) ( , , )

t b x y xy t b x y xy

in which M M M      σ ε

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Shear Strain Energy

       

1 2 1 2 1 ( ) ( ) 2 1 2 1 . 2

s xz xz yz yz V xz x yz y V x xz y yz A z z x x y y A t s s A

U dV dAdz dz dz dA Q Q dA dA                         

      

= = = = ε σ

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Strain Energy

       

1 1 . . 2 2 1 1 . . 2 2

b s A A A A

U U U dA dA dA dA      

   

t t b b s s t t b b b s s s

ε σ ε σ ε D ε ε D ε

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Work Done

• z

A A

W w p dxdy dxdy  

 

t

u p

( , , ) ( ,0,0)

t

• x

y t z

w p     u p

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Equilibrium

 

U W U W U W

• r

U W                

Minimum Potential Energy Principle Virtual Work Principle

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Constitutive Relations - 3D

     

, , ,

x y xy x y

       

t b t s t t t b s

ε ε ε ε ,ε

     

, , , ,

x y xy xz yz t t b s

       

t b t s t

σ σ σ σ σ

b s

 

b b s s

σ E ε σ E ε σ = Eε

Define,

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Constitutive Relations - 3D (Contd.)

11 12 22 33 1 2 x x y y xy xy xz xz S yz yz S

E E E sym E E E                                                                        

b b b s s s

σ E ε σ E ε

1 2

; ; 2(1 )

S S

E E G E G      σ = Eε  

11 22 2 12 11 33 11

; 1 ; 1 2 E E E E E E E G          

For isotropic material,

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Constitutive Relations - 2D

11 12 22 33 x x b y y xy xy

M D D M D sym D M                                    

b b

σ D ε

1 2 x x S s s y y S

Q D Q D                               D ε

 

3 3 11 22 12 11 33 11 2 2 1 2

1 ; ; ; 2 12 12 1 5 ; ; 2(1 ) 6 12

S S

Eh h D D D D D D G E D D kGh G k

• r

                  σ Dε

For isotropic material,

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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

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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

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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

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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

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Mindlin-Reissner Plate

Convergence Criteria

• 1. All three rigid body modes must be exactly representable
• 2. The following five constant strain states must be exactly representable

, , ,     

x y xy x y

curvatures transverse shear strains

• These conditions are satisfied for isoparametric elements
• Recent developments have also provided these condition for

non-standard isoparametric elements

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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