satish second presentation final

8/8/2019 Satish Second Presentation Final

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College of Engineering Pune (COEP)

Forerunners in Technical Education

ByBy

Gaikwad Satish B.Gaikwad Satish B.(M0910S16)(M0910S16)

(STO2(STO2--14)14)

Under the supervision ofUnder the supervision of

Dr. S.D. KULKARNIDr. S.D. KULKARNI

1

STATIC ANALYSIS OF THICK ISOTROPIC RECTANGULAR AND SKEW

PLATES USING DISCRETE KIRCHHOFF QUADRILATERAL ELEMENTS

BASED ON REDDY`S THIRD ORDER SHEAR DEFORMATION THEORY


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Introduction

Literature review

Displacement field approximation

Finite element Formulation

References

CONTENTS

2


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INTRODUCTION

3

Consider isotropic thick plate of thickness h` as shown in Fig..


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LITERATURE REVEIW:

4

Reddy J.N(1984) (TSDT) A simple higher order Theory for plates.

Batoz and Tahar (1982)(DKQ)

Evaluation of new quadrilateral thin plate element.

Modhave (2009) (MDKQ) Static analysis of thin rectangular & Skew plates using

MDKQ.

K T S RIyengar(1974)-MIF Analysis of Thick Plate.

Tarun Kant(1980)-RHSDT Numerical analysis of Thick Plate.


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Cont

5

J N Reddy and Phan(1983)-

HSDT

Stability and vibration analysis of plates using HSDT.

Ine-Wei Liu (1994) 32-DOF based energy orthogonality.

J. Kong & Y. K(1994) Generalized spline finite strip,formulation based third order

theory.


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Cont

6

The displacement field approximation is:-

u1(x, y, z) = u0(x, y) z w0,x+ z=0x(x, y) + z2x(x, y) + z3^x(x, y)

u2(x, y, z) = v0(x, y) z w0,y+ z=0y(x, y) + z2y (x, y) + z

3^y(x, y)

u3(x, y, z) = w0(x, y)

The functions x, x, y, y are determined using the conditions that transverse

shear strains, K xz=0 and Kyz=0 vanish on the plate top and bottom.

The displacement field then becomes;2

1 0 0 ,

2

2 0 0 ,

3 0

4

3

4

3

x ox ox

y ox ox

z wu u zw z

h x

z wu v zw z

h x

u w

] ]

] ]

x ! x -

x ! x -

!


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Cont

7

The generalized strains , K can be expressed as

The generalized strains , K can be expressed as:-

where

0 0 2( )zk R z k I I! , 0zK ]!

0x,x

00y,y

0x ,y 0 y,x

u

u

u u

I ! -

0,xx

00,yy

0,xy

k

2

! -

0x

20y

0x 0y

k

! ] ] ]-

3

24( )3zR z zh

!


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Cont

8


Stress-strain relationship for isotropic thick plates:-

where

? Ax

y

xy

W

W W

X

! -

? Azx

yz

XX

X

!

- ? A

x

y

xy

I

I I

K

! -

? Azx

yz

KK

K

!

-

? A ? AQW I ! - ? A ? AQX K ! -

2

10

2

11 02

EQ

R

RR

!

- - 2

1 0

1 011

0 02

E

Q

R

RRR

! - -


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Cont

9


Stress Resultants:-

Substitution of the constitutive equations into the expressions of stress resultants , following

generalized plate constitutive relations,

xx

y y

xyxy

d zN

N d z

Nd z

W

W

X

! -

-

.

.

.

xx

y y

xyxy

zdzM

M zdz

Mzdz

W

W

X

! -

-

. ( )

. ( )

. ( )

xx

y y

xyxy

R z dzP

P R z dz

PR z dz

W

W

X

! -

-

. ,

. ,

xz Z x

yyz Z

R dzQ

Q R dz

X

X

! - -

? A

0

0

02

0 0

0 0 ,

0 0

N A

M D Q A

P H

I

O ]

O

! !

- - -


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Cont

10


Where

? A 2

3 3

1 0

1 01

10 0

2 X

EhA

R

RR

R

!

-

? A3

2

3 3

1 0

1 012(1 )

10 0

2 X

EhD

R

RR

R

!

-

? A3

2

3 3

1 017

1 0315(1 )

10 0

2 X

EhR

RR

R

! -

2

2 2

10

2 2

13(1 )0 2 X

EhA

R

RR

!

-


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Cont

11


Boundary Conditions:-

The boundary conditions for various edge conditions are given by,

Simply-supported:

At x = 0 and x = a : w0 = 0 and U0y = 0

At y = 0 and y = b : w0 = 0 and U0x = 0

Clamped:

At x = 0 and x = a : w0=0 andU0x = 0 andU0y = 0

At y = 0 and y = b : w0 = 0 and U0x = 0 and U0y = 0


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Cont

12


Finite Element formulation:-

Consider the 4-node quadrilateral element with 7-D.O.F. per node..

For discrete Kirchhoff quadrilateral element , kirchhoff hypothesis is imposed

only at the nodes on the element boundary.

0x, 0y are interpolated independently but are subsequently related by imposing

the constraints at discretepoints.


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Cont

13


The in plane displacement u0x, u0y and the shear strains 0x, 0y, are interpolated usingbilinear Lagrange interpolation functions.

Forx and y, interpolation functions as proposed by Batoz and Tahar.

Bilinear Interpolation Functions-

Where

The bilinear interpolation functionsNi are

0

e

x oxu Nu! 0e

y oy

u Nu!0

e

x oxN]

] ! 0e

y oy

N]] !

1 2 3 4 Te

ox ox ox ox oxu u u u u !- 1 2 3 4 Te

oy oy oy oy oyu u u u u !- 1 2 3 4 Te

ox ox ox ox ox= = = = = !- 1 2 3 4

Te

oy oy oy oy oy= = = = = !-

11

1 14

N s t ! 21

1 14

N s t ! 31

1 14

N s t ! 41

1 14

N s t!


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Cont

14


The original shape functions are of the form

The smoothing shape function is chosen as a complete quadratic fit, one order less than the

actual functions

2 2 2 21 2 3 4 5 6 7 8iN a a a a a a a a\ L \ \L L \ L \L!

m i n i

ia \ L!

2 21 2 3 4 5 6i N b b b b b b\ \ \!


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Cont

15


Subsitute smooth Shape functions:

The unknown coefficients bi are obtained in terms of the known coefficients ai

The problem thus reduces to solving a set of six linear simultaneous equations defined by

For i= 1 to 6,

Using above relationUox & Uoy interpolated as

Where The expressions of

2i iA

N N d d T \ L!

/ 0id daT !

1 2 8 N N N N ! - K iN


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Cont

16


2 2

1

2 2

2

2 2

3

2 2

4

2

5

2

6

2

7

2

8

1 1 1 1 1 1

4 12 12 4 4 41 1 1 1 1 1

4 12 12 4 4 4

1 1 1 1 1 1

4 12 12 4 4 4

1 1 1 1 1 1

4 12 12 4 4 4

1 1 1

2 3 2

1 1 1

2 3 2

1 1 1

2 3 2

1 1 12 3 2

N

N

N

N

N

N

N

N

\ \ \

\ \ \

\ \ \

\ \ \

\

\

\

\

!

!

!

!

!

!

!

!


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Cont

17


Modified Interpolation Functions-

Where

0 0

e

x xNU U!

0 0

e

y yNU U!

1 2 8

0 0 0 0

Te

x x x xU U U U ! - L

1 2 8

0 0 0 0

Te

y y y yU U U U ! - L

1 2 8N N N N ! - L


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Cont

18


Sixteen nodal rotation variables i0x, i0y for the eight nodes (i= 1 8) are related to

the twelve degrees of freedom deflection vectorwe0 of the four corner nodes .

Batoz and Tahar Conditions:

1) Impose at mid side nodes iof sidejljoining corner nodesj and l(i= 5 to 8) the

constraints is = wi0,s where w

i0,s is obtained by considering cubic variation ofw along

the side of length ajlas

2) Approximate the variation ofn to be linear along each side, i.e.

( ) ( )

( ) ( ) ( )

, , ,

, , , , ,

3 1

2 43 1

2 4

i j l j l

s o o o s o s

jl

j li j l

s o o o xsx o ysy o xsx o ysy

jl

w w w w w

a

w w w w w w wa

= - - - +

= - - - + + +

_ a, , , , , ,

1 1

2 2

i j li j l

n n n o xnx o yny o xnx o yny o xnx o ynyU U U! !


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Cont

19


By the above procedure 0x and 0y can be finally expressed in terms ofwe0.

The expression of the interpolation function xi andy

i for x, y

0x e

x owU ! 0y e

y owU ! 1 2 12.....x x x x

H H H H !- 1 2 12.....

y y y y

! -

1 5 5 8 8

2 1 5 5 8 8

3 5 5 8 8

4 6 6 5 5

5 2 6 6 5 5

6 6 6 5 5

7 7 7 6 6

8 3 7 7 6 6

9 7 7 6 6

1 0 8 8 7 7

1 1 4 8

1 .5 [ ] ,

[ ] ,

[ ] ,

1 .5 [ ] ,[ ] ,

[ ] ,

1 .5 [ ] ,

[ ] ,

[ ] ,

1 .5 [ ] ,[

x

x

x

x

x

x

x

x

x

x

x

H a N a N

H N c N c N

H b N b N

H a N a N H N c N c N

H b N b N

H a N a N

H N c N c N

H b N b N

H a N a N H N c

!

!

!

! !

!

!

!

!

! !

8 7 7

1 2 8 8 7 7

] ,

[ ] ,x

N c N

H b N b N

!

1 5 5 8 8

2 3

3 1 5 5 8 8

4 6 6 5 5

5 6

6 2 6 6 5 5

7 7 7 6 6

8 9

9 3 7 7 6 6

1 0 8 8 7 7

1 1 1 2

1 2 4 8 8 7 7

1 .5[ ] ,

,

[ ] ,

1 .5[ ] ,,

[ ] ,

1 .5[ ] ,

,

[ ] ,

1 .5[ ] ,,

[ ] ,

y

y x

y

y

y x

y

y

y x

y

y

y x

y

d N d N

H H

H N e N e N

H d N d N H H

H N e N e N

H d N d N

H H

H N e N e N

H d N d N H H

H N e N e N

!

!

!

! !

!

!

!

!

! !

!


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Cont

20

Generalized Strains:-

The generalized displacement vectorUe is defined in terms of the nodaldisplacement variables Uei as

The generalized strains 0

, k0

, k2

, 0 can be expressed in terms ofUe

as

0 0 0

e i i i i i i i

i x x x x y x yU u u U U ] ] !-

1 2 3 4

eT eT eT eT eTU U U U U !-

0

0

eB UII !0

0

e

kB UO !

2

2 e

kB UO ! 20

eB U]] !

1 2 3 4

0 0 0 0 0 B B B B B

I I I I I !-

1 2 3 4

0 0 0 0 0 B B B B BO O O O O !-

1 2 3 42 2 2 2 2 B B B B BO O O O O !-

1 2 3 4

2 0 0 0 0 B B B B B] ] ] ] ] !-


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Cont

21

The element stiffness matrix obtained as

The Load vector obtained as

Summing up the contributions for all elements for an arbitrary value of virtual

displacements

0 0 0 00

[ ]e T T T

ko kK B AB B DB B B d xdyH

O O

!

[ ]e T

z N p d xdy! %

KU P!


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1. Reddy J.N.A simple higher order Theory for laminated Composite plates. Journal for

applied echanics,vol-51/745,1984

2. Batoz JL, Tahar MB. Evaluation of a new quadrilateral thin plate bending element. In-

ternational journal fornu erical ethods inengineering, 18, 1655-1677, 1982

3. Modhave NS. Static analysis of thin isotropic rectangular and skew plates using

modified discrete Kirchhoff quadrilateral element.M

.T

ech.T

hesis, College ofEngineeringPune, 2009

4. K T Sundara raja Iyengar, K Chandrashekra. On the analysis of thick plate, Journal

ofaeronautical science, 28(34), 1961.

5. Tarun Kant. Numerical analysis of thick plates, Depart ent of CivilEngineering,

IndianInstitute ofTechnology,Bo bay 400 076, India

REFERANCES

22


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6 J. N. Reddy and N. D. Phan. Semi-analytical shape functions in the finite Element.analysis of rectangular plates. Journal of Soundandvibration, 242(3), 427-443, 2001

7. J N Reddy. Relation between classical & higher order. Journalacta mechanica, 1998

plates. Computers andStructures, 54(6), 1173-1182, 2000.

8. Alon Yair, Static analysis of thick laminate plates using higher order three dimensionalfinite element, Thesis Californiamonterey school, -1990

9. Ine-Wei Liu,An element for static,vibration & buckling analysis of thick

plates..dimensional finite element, Journal of computers & structures, vol 59-1994.

8. J. Kong & Y. K.,A generalized spline finite strip for the analysis of plates. Thin walled

andstructures, vol 22-1994

REFERANCES

23


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THANK YOU

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satish second presentation final

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