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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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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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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
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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
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The generalized strains , K can be expressed as
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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The generalized strains , K can be expressed as
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
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The generalized strains , K can be expressed as
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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The generalized strains , K can be expressed as
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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The generalized strains , K can be expressed as
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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The generalized strains , K can be expressed as
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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The generalized strains , K can be expressed as
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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The generalized strains , K can be expressed as
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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The generalized strains , K can be expressed as
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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The generalized strains , K can be expressed as
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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The generalized strains , K can be expressed as
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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The generalized strains , K can be expressed as
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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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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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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