Post on 09-Apr-2018
International Journal of Engineering & Technology IJET-IJENS Vol:14 No:06 69
1410606-7373-IJET-IJENS © December 2014 IJENS I J E N S
Abstract-- A hybrid technique consisting of moving least
square differential quadrature and finite difference methods is
employed to examine bending problems of irregular composite
plates. Based on a transverse shear theory, the governing
equations of the problem are derived. The transverse deflection
and strain rotations of the plate are independently approximated
with moving least square approximations. The partial derivatives
of these quantities, at the interior nodes, are approximated using
finite difference approximations. The obtained results agreed
with the previous analytical and numerical ones. Further a
parametric study is introduced to investigate the effects of elastic
and geometric characteristics on behavior of the transverse
deflection and stress field quantities.
Index Term— Finite Difference;Moving Least Square
Differential Quadrature; Irregular Plates; Transverse Shear
Theory;Functionally Graded Material.
INTRODUCTION
The use of composite materials has gained much
popularity in recent years especially in extreme high
temperature environments. Functionally graded materials
(FGMs) are the new generation of composites in which the
micro-structural details are spatially varied through non-
uniform distribution of the reinforcement phase. This can be
achieved by using reinforcement with different properties, sizes
and shapes, as well as by interchanging the role of
reinforcement and matrix phase in a continuous manner. The
result is a macro-structure that produces continuous or smooth
change on thermal and mechanical properties at the
macroscopic level. Due to recent advances in material
processing capabilities, that aid in manufacturing wide variety
of functionally graded materials, its use in application
involving severe thermal environments is gaining acceptance in
composite community, the biomedical, electronics, aerospace
and aircraft industries [1].
Analysis of FG plate materials is one of the most
important problems in structural analysis. Due to the
mathematical complexity of such problems, only limited cases
can be solved analytically [2-4]. Finite element, Ritz method,
point collocation method, boundary element, and discrete
singular convolution methods have been widely applied to
solve numerically these plate problems [5-12]. The main
disadvantage of such techniques is to require a large number of
grid points as well as a large computre capacity to attain a
considerable accuracy [13-15].
Liew et al [16-22] have been introduced a new version
of differential quadrature (DQ) method termed by moving least
square differential quadrature method (MLSDQM). This
technique possessess the capability to deal with interfacial
composite plate problems as well as irregular shaped ones. But
withen the applications of MLSDQM, there were some
computational complications to determine the partial
derivatives of the field quantities [23,24]. Later, Wen and
Aliabadi reduced this difficulty through application of the finite
difference method (FDM) to find the partial derivatives of the
field quantities [25]. Finite difference method belongs to the
strong-form methods and the formulation procedure is
relatively simple and straight forward compared with meshless
techniques [26, 27].
The present work employes a hybird technique
consisting of MLSDQM and FDM to analyze bending
problems of irregular composite plates. The composite is made
of a FGM. The equilibrium equations are written according to
a transvers shear theory. The weighting coefficients, for the
approximated field quantities over the entire domain, are
obtained using MLSDQ technique. The partial derivatives of
these quantities, at the interior nodes, are approximated using
FDM. The obtained results are compared with the previous
analytical and numerical ones. Further a parametric study is
introduced to investigate the effects of elastic and geometric
characteristic of the problem on the obtained results.
2. FORMULATION OF THE PROBLEM
Consider a non-homogeneous composite consisting of an
isotropic plate bonded, (along x-axis), to another one made of a
FGM. The elastic characteristics of the composite vary such
that:
, , ,f y f y fG Ge E Ee (1)
Where G, E and v are shear modulus, Young’s modulus and
Poisson's ratio of the isotropic plate. Gf, E
f and v
f are shear
Quadrature Analysis of Functionally Graded
Materials
Ola Ragba, M.S. Matbuly
a,*, M. Nassar
b
a Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University,
P.O. 44519, Zagazig, Egypt. b Department of Engineering Mathematics and Physics, Faculty of Engineering, Cairo University, Giza, Egypt.
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modulus, Young’s modulus and Poisson's ratio of the FG plate.
is a constant characterizing the composite gradation.
Assume that the composite is subjected to a pure bending due
to a laterally distributed load q(x,y). Based on a first-order
shear deformation theory, the equilibrium equations for such
composite thin plate can be written as [28]:
,xyxx
x
MMQ
x y
(2-a)
,xy yy
y
M MQ
x y
(2-b)
( , )yx
QQq x y
x y
(2-c)
Where ,( , , ),ijM i j x y are the bending and twisting
moment resultants.
,( , ),iQ i x y are the shearing force resultants.
The transverse deflection ( , )w x y and the normal
rotations ( , ), ( , )x yx y x y are related to the moment
and shear resultants through the following constitutive relations
[29].
,yx
xxM Dx y
(3-a)
,y x
yyM Dy x
(3-b)
1,
2
yxxyM D
y x
(3-c)
,x x
wQ kG h
x
(4-a)
,y y
wQ kG h
y
(4-b)
Where 3
212(1 )
E hD
is the flexural rigidity of the plate.
h is thickness of the plate.
k is the shear correction factor [29-30], which is to be taken 5/6.
On suitable substitution from Eqs.(3) and (4) into (2), the equilibrium equations can be reduced to:
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22 2
2 2
(1 ) (1 ) (1 )0,
2 2 2
y yx x xx
wD kGh
x y y x xx y
(5-a)
2 2 2
2 2
(1 ) (1 )0,
2 2
y y yx x wD kGh y
x y y x yy x
(5-b)
2 2
2 20
y yxy
w w wkGh qe
y x yx y
(5-c)
Where 0 for FG part while =0 for isotropic one.
According to the type of supporting the boundary conditions can be described as:
(a)Simply supported:
SS1
0, 0, 0, n nsw M M
(6)
SS2
0, 0, 0, s nw M
(7)
(b) Clamped:
0, 0, 0, s nw
(8)
(c) Free: 0, 0, 0, n n nsQ M M
(9)
Where the subscripts n and s represent the normal and tangent directions to the boundary edge, respectively; Mn, Mns
and Qn denote the normal bending moment, twisting moment and shear force on the plate edge; n and s are the
normal and tangent rotations about the plate edge.
The continuity conditions, (along the interface), must be also satisfied such that:
( ,0 ) ( ,0 ), ( ,0 ) ( ,0 ), ( ,0 ) ( ,0 ),f f f
xx xx xy xyw x w x M x M x M x M x
(10)
Which means that the deflection and moments, (along the interface), of isotropic and FG plates must be equaled.
Further, the force resultants and the rotations on the edge can be expressed in terms of the basic unknowns
x yand as follows [29-30]:
2 22 ,nn x xx x y xy y yyM n M n n M n M
(11-a)
2 2( ) ( ),ns x y xy x y yy xxM n n M n n M M
(11-b)
,n x x y yQ n Q n Q
(11-c)
,n x x y yn n
(11-d)
s x y y xn n
(11-e)
Where, x yn and n are the direction cosines at a point on the boundary edge.
3. SOLUTION OF THE PROBLEM
A hybrid technique consisting of MLSDQM and FDM is employed to solve the problem as follows:
Descretize the domain of the problem, into a finite number of nodes: {Xi=(xi, yi), i=1,N}. Each node is
associated with three nodal unknowns (w, and x y ). The influence domain, for each node, is determined as shown
in Fig.(1). Over each influence domain, (, i=1,N), the nodal unknowns can be approximated as [16-18]:
1
( ) ( , ) ( , ) ( , ) , ( , , ),( 1, ).n
h ji i i i i j i i x y
j
x y x y x y w i N
x (12)
Fig. 1. Domain descretization for moving least squares differential quadrature method.
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Where n is the number of nodes within the influence domain,. , , h h h hx yw are approximate values for
nodal unknowns w, and x y , respectively. ( , )j i ix y is defined as the shape function of MLS approximation
over the influence domain,, i=1,N). The nodal parameters: , ,i i ix yw are always not equal to the physical
values ( , ), ( , ), ( , )i i x i i y i iw x y x y x y , since the MLS shape functions ( , )j i ix y do not satisfy the Kronecker
delta condition generally.
Apply the MLS technique to approximate uh(x) to u(x), for any x , such as [31-32]:
(13)
Where 1 2( ) ( ), ( ), , ( )T
ma a a ax x x x is a vector of unknown
coefficients. 1 2( ) ( ), ( ), , ( )x x x xTmP p p p is a complete set of monomial basis. m is the number of
basis terms. For example, in two dimensional problems: m=3 for linear basis ( ) 1, ,TP x yx while m=6 for
quadratic basis: 2 2( ) 1, , , , ,TP x y x xy yx .
The coefficients ( ),( 1, )ja j mx , can be obtained at any point x by minimizing the following weighted quadratic
form:
2 2
1 1
( ) ( - )( ( ) ) ( - )( ( ) ( ) )n n
h Ti i i i i i
i i
u u u
a x x x x x P x a x (14)
Where n is the number of nodes in the neighborhood of x and ui is the nodal parameter of u(x) at point xi. ϖi(x) =ϖ(x-
xi) is a positive weight function which decreases as x xi increases. It always takes unit value at the sampling
point x and vanishes outside the domain of influence for x.
The stationary value of Π(a) with respect to a(x) leads to a linear relation between the coefficient vector a(x)
and the vector of fictitious nodal values u, such as:
( ) ( ) ( )A x a x B x u (15-a)
from which 1( ) ( ) ( )a x A x B x u , (15-b)
where 1 2
1
( ) ( ) ( ) ( ) ( ) ( ) ( ), ,n
TT Ti i i i i i n
i
u u u
A x P x x P x x P x P x u=
1 1 2 2( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )n n B x =P x x x P x x P x x P x
.
On suitable substitution from Eq. (15-b) into (13), uh(x) can then be expressed in terms of the shape functions as:
1
1
( ) ( ) ( ) ( ) ( ) ( ) ( )n
h T T
i i i
i
u u
x P x a x P x A x B x u= x (16)
Where the nodal shape function: ϕi(x) =PT(x) A
-1(x) Bi(x) (17)
It should be noted that the MLS shape function and its derivatives are dependent on the weight function and the radius
of influence domain. It's also required that n ≥ m in the domain of influence so that the matrix A(x) in Eqs.(15) can be
inverted[16-18].
The shape functions ( , )j i ix y can be determined as follows [33]:
Equation (17) can be rewritten as:
Ti T 1
i ix P x A x B x x B x (18)
Since A(x) is a symmetric matrix, then Eq.(18) yields
A x x P x (19)
Therefore, the problem of determination of the shape function is reduced to solution of Eq.(19). This Equation can be
solved using LU decomposition and back-substitution, which requires fewer computations than the inversion of A(x).
For the partial derivatives of ( , )j i ix y , one can employ a squared finite difference grid with mesh side hm as
shown in Fig. 2. Each node is associated with three nodal unknowns (w, and x y ), such that the partial derivatives of
( , )j i ix y can be approximated as:
1
( ) ( ) ( ) ( ) ( ),m
h Ti i
i
u P a P a
x x x x x
International Journal of Engineering & Technology IJET-IJENS Vol:14 No:06 73
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1 1 1 1
, ,2 2
1 1 1 1 1 1 1 1
, 2
1 1( , ) ( 2 ), ( , ) ( 2 ),
1( , ) ( ),
4
( , , ),( , , )and ( , 1, )
m m
m
i j ij i j ij ij ij
LL i j KK i j
i j i j i j i j
LK i j
x y
x y x yh h
x yh
w L K x y i j N
(20)
A suitable linear interpolation must be applied to modify Eqs. (20) for the intermediate points, (resulting from domain
irregularities), as shown in Fig. 2.
On suitable substitution from Eqs. (12) and (20) into (5), the problem can be reduced to the following system of
linear algebraic equations:
2 1 1 1 1
1
1 1 1 1 1 1 1 1
1 1 1 14 4 2 1
2 2 2 2
10, ( 1, )
2
m
nx j y j x j i j i j ij ij ijij ij ij x ij y x x x x x
j
i j i j i j i jy y y y
h kGhc w D c kGh D c D
D i N
(21-a)
2 1 1 1 1 1 1 1 1
1
1 1 1 1
14
2
1 14 2 1 0, ( 1, )
2 2
m
ny j x j y j i j i j i j i jij ij x ij ij y x x x x
j
i j i j ij ij ijy y y y y
h kGhc w D c D c kGh D
D i N
(21-b)
2 ( )
1 1 1 1
1
.4 , ( , 1, )
jyni j i j ij ij ij y j x j y j m
ij ij x ij ij y
j
h q ew w w w w c w c c c i j N
kGh
(21-c)
To satisfy the boundary conditions, the first order partial derivative of the shape function can be approximated using
MLSDQM as follows [18]:
Differentiate Eq.(19) with respect to I, (I=x,y) such as:
, , ,( ) ( ) ( ) ( ) ( ), ( , )I I I I x y A x x P x A x x (22)
The first order partial derivative of the shape function can be described as:
, , ,( ) ( ) ( )B ( ) ( )B ( ), ( , ).L T T
j L i j i j L i j i j i j L ic L x y x x x x x x (23)
Further
,
1
( , ) , ( , , ), ( , ), ( 1, ).n
L j
L i i j x y
j
x y c w L x y i N
(24)
Therefore, the boundary conditions can be reduced to the following linear algebraic equations:
(a)Simply supported:
SS1 2 2 2 2
1
0, ( ) (1 ) ( ) (1 ) 0, n
i x y j y x j
x y ij x y ij x x y ij x y ij y
j
w n n c n n c n n c n n c
2 2 2 2
1
( ) 2 ( ) 2 0, ( 1, )n
y x j x y jx y ij x y ij x x y ij x y ij y
j
n n c n n c n n c n n c i N
(25)
Fig. 2. Grid descretization for the hybrid technique consisting of FDM and MLSDQM.
International Journal of Engineering & Technology IJET-IJENS Vol:14 No:06 74
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SS2
1
0, 0, n
i j j
ij x y y x
j
w n n
2 2 2 2
1
( ) (1 ) ( ) (1 ) 0, ( 1, )n
x y j y x jx y ij x y ij x x y ij x y ij y
j
n n c n n c n n c n n c i N
(26)
(b) Clamped: 1 1
0, 0, 0, ( 1, )n n
i j j j j
ij x y y x ij x x y y
j j
w n n n n i N
(27)
(c)Free: 1
0,n
jx y j jx ij y ij x ij x y ij y
j
n c n c w n n
2 2 2 2
1
( ) (1 ) ( ) (1 ) 0,n
x y j y x jx y ij x y ij x x y ij x y ij y
j
n n c n n c n n c n n c
2 2 2 2
1
( ) 2 ( ) 2 0,( 1, )n
y x j x y jx y ij x y ij x x y ij x y ij y
j
n n c n n c n n c n n c i N
(28)
Along the interface, the following algebraic equation must also be considered:
1( , ) ( , ), ( , ) ( , ), ( , ) ( , ), ( 1, )f f f
i i i i xx i i xx i i xy i i xy i iw x y w x y M x y M x y M x y M x y i N
(29)
Where N1 is the number of nodes along the interface.
4. NUMERICAL RESULTS
For the present results, Gaussian weight function with a circular influence domain is adopted for the MLS
approximation because of its partial derivatives with respect to x and y coordinates exist to any desired order. It takes
the form of [16-18]: 2 2
2
exp( ( / ) ) exp( ( / ) )
( ) 1 exp( ( / ) )
0
ii
i
i
d c r cd r
w x r c
d r
(30)
Where2 2( ) ( )i i id x x y y , is the distance from a nodal xi to a field x in the influence domain of xi. r is the
radius of support domain and c is the dilation parameter In the present work, the dilation parameter is selected such
as: c=r/4.
Also, a scaling factor dmax is defined as: max
m
rd
g (31)
Where gm is the grid size, which can be regarded as the distance between the nodal point xi and the 2nd
nearest
neighboring field nodes. For regular FD grid spaced by hm, gm can be taken as: gm= .
For practical purposes the numerical results are normalized such as [18]:
For squared plates: 4 2100 / ( ), 10 / ( ), / , ( , , ).s
ij ij ij ijw wD qa M M qa E i j x y (32)
For skew rhombic plate shown in Fig. (3): 4 21600 / ( ), 40 / ( ), / , ( , , ).s
ij ij ij ijw wD qa M M qa E i j x y (33)
Where , ,ij ijw M are the normalized deflection, moments, and stresses, respectively. a is a composite width, (a=
a1+a2 ). sD is the flexural rigidity of the isotropic plate.
A numerical scheme is designed to investigate the influence of computational characteristics,( radius of
support domain r, completeness order of the basis functions Nc, and scaling factors dmax ) on accuracy of the obtained
results. The boundary conditions (25-29) are directly substituted into equilibrium ones (21). The reduced system is
a
θ x
y
b
Fig. 3. Skew rhombic plate.
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solved using MATLAB. The problem is solved over a regular grid with N1= (7, 9,.., 25). For the concerned boundary
conditions, Table (1) shows that the results for N1 = 11 are nearly the same as those corresponding to N1=
(13,15,..,25). Therefore, a grid 11X11 nodes is considered to implement the parametric study. For different scaling
factors dmax (ranging from 2 to 10) and completeness order Nc, (ranging from 2 to 7), accurate results can be attained
when dmax ≤ Nc + 0.5, as in Figs. (4) and (5). This was previously recorded in [17]. Table I
Comparison between the obtained deflection and the previous analytical ones at the centre of a squared plate: Nc=5, dmax=8.
Boundary conditions
Number
of grid nodes
SS1 SS2 CC
Exact [4, 29] Obtained results Exact [4, 29] Obtained results Exact [4, 29] Obtained results
7x7 0.4273 0.1094 0.4273 0.7236 0.1496 0.1691
9x9 0.4273 0.2858 0.4273 0.4204 0.1496 0.1534
11x11 0.4273 0.42708 0.4273 0.427311 0.1496 0.1505
13x13 0.4273 0.427295 0.4273 0.4273 0.1496 0.1510
15x15 0.4273 0.4273 0.4273 0.4273 0.1496 0.1507
9 10 11 12 13 14 15
Number of grid points N1
Nc=2
0.6
0.7
0.8
0.9
1.0
No
rmal
ized
cen
ter
def
lect
ion
Present results
h=1.5
Previous results[36]
Isotropic
(2,18)(12,16)
(0,0)
(16,14)
x
y
Present results
h=0.5dmax=6
dmax=7
dmax=8
dmax=6
dmax=7
dmax=8
9 10 11 12 13 14 15
Number of grid points N1
Nc=3
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
No
rmal
ized
cen
ter
def
lect
ion
Present results
h=1.5
Previous results[36]
Isotropic
(2,18)(12,16)
(0,0)
(16,14)
x
y
Present results
h=0.5dmax=6
dmax=7
dmax=8
dmax=6
dmax=7
dmax=8
9 10 11 12 13 14 15
Number of grid points N1
Nc=4
0.900
0.925
0.950
0.975
1.000
1.025
1.050
No
rmal
ized
cen
ter
def
lect
ion
Present results
h=1.5
Previous results[36]Isotropic
(2,18)(12,16)
(0,0)
(16,14)
x
y Present results
h=0.5
dmax=6
dmax=7
dmax=8
dmax=6
dmax=7
dmax=8
9 10 11 12 13 14 15
Number of grid points N1
Nc=5
1.00
1.25
1.50
1.75
2.00
2.25
2.50
No
rmal
ized
cen
ter
def
lect
ion
Present results
h=1.5
Previous results[36]
Isotropic
(2,18)(12,16)
(0,0)
(16,14)
x
y
Present results
h=0.5
dmax=6
dmax=7
dmax=8dmax=6
dmax=7
dmax=8
Fig. 4. Variation of the results with the completeness order Nc , scaling factor dmax and the number of grid points N1 for an irregular quadrilateral
plate with various thickness h.
9 10 11 12 13 14 15
Number of grid points N1
h=0.5
0.98
0.99
1.00
1.01
1.02
1.03
1.04
1.05
1.06
1.07
1.08
1.09
1.10N
orm
aliz
ed c
ente
r def
lect
ion
Present results
Previous results[36]
Isotropic
(2,18)(12,16)
(0,0)
(16,14)
x
y
dmax=6
dmax=5
dmax=7
dmax=8
9 10 11 12 13 14 15
Number of grid points N1
h=1.5
1.00
1.25
1.50
1.75
2.00
2.25
2.50
Norm
aliz
ed c
ente
r def
lect
ion
Present results
Previous results[36]
Isotropic
(2,18)(12,16)
(0,0)
(16,14)
x
y
dmax=6
dmax=5
dmax=7
dmax=8
Fig. 5. Variation of the results with the scaling factor dmax and the number of grid points N1 for an irregular quadrilateral plate with
various thickness h, ( Nc=5).
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To examine the validity of the obtained results, the bending problem of regular and irregular isotropic plates
is solved and compared with the previous analytical ones in [4, 29, 34, and 35]. Tables (2) to (5) show a very good
agreement between the obtained results and the previous analytical solutions. For simply supported plates, the error
between obtained results and the previous exact ones in [4, 29, 34, and 35] is . While this error is
for clamped plates. Figures (4) and (5), show also that the accuracy of the obtained results increases with
increasing both of the completeness order Nc and the radius of support domain r for different values of plate thickness
h. These results exactly agrees with that recorded in [36].
Table II
Comparison between the obtained deflection and the previous analytical ones [4, 29] for clamped circular plate at the centre.
Table III
Comparison between the obtained deflection and the previous analytical ones for clamped rectangular plates at the centre.
Support
size Exact [4, 29]
Obtained results
Nc=4 Nc=5 Nc=6 Nc=7
r=.7a 1.6339 1.6339 1.3924 1.5590 3.0671
r=.8a 1.6339 1.6339 1.6339 1.6339 1.6501
r=.9a 1.6339 1.6339 1.6339 1.6339 1.6288
r=a 1.6339 1.6339 1.6339 1.6339 1.6339
r=1.1a 1.6339 1.6339 1.6339 1.6339 1.6339
b/a
W(0,0) 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
Exact
[4,29] 0.00126 0.00150 0.00172 0.00191 0.00207 0.00220 0.00230 0.00238 0.00245 0.00249 0.00254
Obtained
results 0.00126054 0.00150028 0.0017196 0.0019097 0.0020702 0.0022003 0.0022301 0.0023803 0.0024495 0.0024897 0.00254078
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Table IV
Comparison between the obtained deflection and the previous analytical ones [4, 29] for simply supported square plates.
Table V
Comparison between the obtained deflection and the previous analytical ones [34, 35] for simply supported skew rhombic plates at the centre.
x
y
.5- -.4 -.3 .2- .1- 0 .1 .2 .3 .4 .5
Exact[4,29] Obtained results Exact[4,29] Obtained results Exact[4,29] Obtained results Exact[4,29] Obtained results Exact[4,29] Obtained results Exact[4,29] Obtained results Exact[4,29] Obtained results Exact[4,29] Obtained results Exact[4,29] Obtained results Exact[4,29] Obtained results Exact[4,29] Obtained results
-.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-.4 0 0 .040691 .0406914 .076366 .076365 .103675 .10367497 .12069 .1206897 .126456 .1264556 .12069 .1206897 .103675 .10367497 .076366 .076365 .040691 .0406914 0 0
-.3 0 0 .078388 .0783882 .146964 .1469640 .199333 .1993326 .231903 .2319025 .24293 .2429299 .231903 .2319025 .199333 .1993326 .146964 .1469640 .078388 .0783882 0 0
-.2 0 0 .109917 .1099169 .205700 .2057002 .278562 .2785617 .32376 .3237596 .339044 .3390438 .32376 .3237596 .278562 .2785617 .205700 .2057002 .109917 .1099169 0 0
-.1 0 0 .131702 .13170197 .245875 .2458755 .33235 .3323497 .385855 .3858547 .403936 .403928 .385855 .3858547 .33235 .3323497 .245875 .2458755 .131702 .13170197 0 0
0 0 0 .139800 .1398003 .260577 .2605766 .351854 .3518542 .408268 .4082679 .427314 .42731396 .408268 .4082679 .351854 .3518542 .260577 .2605766 .139800 .1398003 0 0
.1 0 0 .131702 .13170197 .245875 .2458755 .33235 .3323497 .385855 .3858547 .403936 .403928 .385855 .3858547 .33235 .3323497 .245875 .2458755 .131702 .13170197 0 0
.2 0 0 .109917 .1099169 .205700 .2057002 .278562 .2785617 .32376 .3237596 .339044 .3390438 .32376 .3237596 .278562 .2785617 .205700 .2057002 .109917 .1099169 0 0
.3 0 0 .078388 .0783882 .146964 .1469640 .199333 .1993326 .231903 .2319025 .24293 .2429299 .231903 .2319025 .199333 .1993326 .146964 .1469640 .078388 .0783882 0 0
.4 0 0 .040691 .0406914 .076366 .076365 .103675 .10367497 .12069 .1206897 .126456 .1264556 .12069 .1206897 .103675 .10367497 .076366 .076365 .040691 .0406914 0 0
.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
θ
Support
size
150
300
450
600
750
Exact Obtained results Exact Obtained results Exact Obtained results Exact Obtained results Exact Obtained results
[34] [35] Nc=4 Nc=5 [34] [35] Nc=4 Nc=5 [34] [35] Nc=4 Nc=5 [34] [35] Nc=4 Nc=5 [34] [35] Nc=4 Nc=5
r=.5a .0640 .0605 0.0651 0.0626 .6609 .6587 0.6429 0.6248 2.1193 2.1285 1.4257 1.7483 4.1026 4.1079 0.7151 1.2063 5.8231 5.8172 1.574 3.4005
r=.6a .0640 .0605 0.0641 0.0651 .6609 .6587 0.651 0.6466 2.1193 2.1285 1.9384 1.9394 4.1026 4.1079 3.0779 3.1346 5.8231 5.8172 4.7888 5.1277
r=.7a .0640 .0605 0.0632 0.0647 .6609 .6587 0.655 0.6452 2.1193 2.1285 2.0541 2.0109 4.1026 4.1079 3.8776 3.7838 5.8231 5.8172 5.5824 5.6014
r=.8a .0640 .0605 0.0626 0.0635 .6609 .6587 0.6574 0.6401 2.1193 2.1285 2.0888 2.0354 4.1026 4.1079 4.0293 3.9534 5.8231 5.8172 5.7072 5.6952
International Journal of Engineering & Technology IJET-IJENS Vol:14 No:06 78
1410606-7373-IJET-IJENS © December 2014 IJENS I J E N S
Further, a parametric study is introduced to investigate the effect of geometric and elastic characteristics of
the plate on behavior of the obtained results. Figures (6-9) show that the values of normalized deflection decrease with
increasing both of the graduation parameter γ and the interface location (a1/a2). While, these values increase with
increasing of the aspect ratio (a/b), as shown in Figs. (9), where a and b are the width and length of the rectangular
plate. Figures (10) and (11) show normalized stress distribution, yy , through different locations of the composite.
Figures (6-11) insist the advantage of FG composites treating the material discontinuity problems.
-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5
y
0.00
0.04
0.08
0.12
0.16
Norm
aliz
ed d
efle
ctio
n
Isotropic
FGM
x=0
x=0.2
x=0.3
x=0.4
-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5
y
0.00
0.04
0.08
0.12
0.16
Norm
aliz
ed d
efle
ctio
n
Isotropic
FGM
x=0
x=0.2
x=0.3
x=0.4
(a) (b)
Fig. 6. Normalized deflection distribution through different locations for FG clamped rectangular plate: (a) γ=1 (b) γ=10
-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5
y
0.0
0.5
1.0
1.5
2.0
2.5
Norm
aliz
ed d
efle
ctio
n
x=0
x=0.2
x=0.3
x=0.4
FGM
Isotropic
-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5
y
0.0
0.5
1.0
1.5
2.0
2.5
No
rmal
ized
def
lect
ion
x=0
x=0.2
x=0.3
x=0.4
FGM
Isotropic
(a) (b)
Fig. 7. Normalized deflection distribution through different locations for FG simply supported skew plate: (a) γ=1 (b) γ=10
-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5
y
0
1
2
3
4
5
6
7
8
No
rmal
ized
def
lect
ion
x=0
x=0.2
x=0.3
x=0.4
FGM
Isotropic
-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5
y
0
1
2
3
4
5
6
7
8
No
rmal
ized
def
lect
ion x=0
x=0.2
x=0.3
x=0.4
FGM
Isotropic
(a) (b)
Fig. 8. Normalized deflection distribution through different locations for FG simply supported circular plate: (a) γ=1 (b) γ=10
International Journal of Engineering & Technology IJET-IJENS Vol:14 No:06 79
1410606-7373-IJET-IJENS © December 2014 IJENS I J E N S
5. CONCLUSION
This work extends the applications of quadrature
techniques for solving bending problems of regular and
irregular FG composites. The method of solution consists of
MLSDQM and FDM. This overcame the computational
difficulties arising with the application of MLSDQ
approximations. The shape function and their partial
derivatives are simply determined. As well as, derivatives of
the field quantities are simply approximated over a finite
difference grid. The obtained results agreed with the previous
analytical and numerical ones. Further a parametric study is
introduced to investigate the effects of computational,
geometric, and elastic characteristics of the problem on values
of the obtained results.
-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5
Location along the interface
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Norm
aliz
ed d
efle
ctio
n
FGM
a1a2
b
a=a1+a2
a/b=2,a1/a2=1
a/b=1,a1/a2=2
a/b=1,a1/a2=4
a/b=1,a1/a2=1
Isotropic
-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5
Location along the interface
0.0
0.1
0.2
0.3
0.4
0.5
Norm
aliz
ed d
efle
ctio
n
Isotropic FGM
a1a2
b
a=a1+a2
a/b=1.2,a1/a2=1
a/b=1,a1/a2=2
a/b=1,a1/a2=4
a/b=1,a1/a2=1
(a) (b)
Fig. 9. Variation of the normalized deflection with the aspect ratio for FG plate:
(a) Simply supported (b) Clamped-clamped, (γ=10).
-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5
y
-0.015
-0.010
-0.005
0.000
0.005
0.010
0.015
Norm
aliz
ed
yy
x=0
x=0.1
x=0.2
x=0.3
FGM
Isotropic
-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5
y
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Norm
aliz
ed
yy
x=0
x=0.1
x=0.2
x=0.3
FGM
Isotropic
(a) (b)
Fig. 10. Normalized stress distribution through different locations for FG
(a) Clamped-clamped (b) Simply supported, (γ=10, h=0.1).
-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5
y
-1.6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Norm
aliz
ed
yy
x=0
x=0.1
x=0.2
x=0.3
FGM
Isotropic
-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5
y
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Norm
aliz
ed
yy
x=0
x=0.1
x=0.2
x=0.3
FGM
Isotropic
(a) (b)
Fig. 11. Normalized stress distribution through different locations for FG circular plate
(a) Clamped-clamped (b) Simply supported, (γ=10, h=0.1).
International Journal of Engineering & Technology IJET-IJENS Vol:14 No:06 80
1410606-7373-IJET-IJENS © December 2014 IJENS I J E N S
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