Quadrature Analysis of Functionally Graded Materials … graded materials (FGMs) are the new...

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.



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:



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.



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



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.



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


(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


(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.



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


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


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


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


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


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


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


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


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


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).



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



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



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



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).



REFERENCES [1] Ichikawa K., editor, Functionally graded materials in the 21st

Century: a work shop on trends and forecasts, Japan. Kluwer

Academic Publishers, (2000).

[2] Wang C.M., Lim G.T., Reddy J.N., Lee K.H.;'' Relationships between

bending solutions of Reissner and Mindlin plate theories'', Eng.

Struct.,23: 838–849, (2001).

[3] Ashraf M. Zenkour;'' Exact mixed-classical solutions for the bending

analysis of shear deformable rectangular plates'', Appl. Math. Model,

27: 515–534, (2003).

[4] Timoshenko S.P., Woinowsky-Krieger S., Theory of plates and

shells, 2nd Edition, McGraw-Hill, New York, (1959).

[5] Tran Minh Tu , Le Ngoc Thach, Tran Huu Quoc ; ''Finite element

modeling for bending and vibration analysis of laminated and

sandwich composite plates based on higher-order theory'',

Computational Materials Science ,49: S390–S394,(2010).

[6] Hrabok M.M., Hrudey T.M.;" A review and catalogue of plate

bending finite elements ", Computers & Structures,19: 479-

495,(1984).

[7] Liew K.M., Wang C.M., Xiang Y., Kitipornchai S., Vibration of

Mindlin Plates—Programming the p-Version Ritz Method, Elsevier

Science, Oxford, (1998).

[8] Gupta U.S., Jain S.K., Jain D.;'' Method of collocation by derivatives

in the study of axisymmetric vibration of circular plates'', Computers

and Structures, 57: 841–845, (1995).

[9] Masa. Tanaka, Matsumoto T., Shiozaki A.;" Application of the

boundary-domain-element method to the harmonic-bending-vibration

problem of thin elastic plates ", Engineering Analysis with Boundary

Elements ,13: 239-247,(1994).

[10] Masa. Tanaka, Bercin A.N.;"Static bending analysis of stiffened

plates using the boundary element method ", Engineering Analysis

with Boundary Elements, 21: 147-154, (1998).

[11] Civalek Ö;"Three-dimensional vibration, buckling and bending

analyses of thick rectangular plates based on discrete singular

convolution method ", International Journal of Mechanical Sciences,

49: 752-765, (2007).

[12] Civalek Ö;" A four-node discrete singular convolution for geometric

transformation and its application to numerical solution of vibration

problem of arbitrary straight-sided quadrilateral plates ", Applied

Mathematical Modelling, 33: 300-314, (2009).

[13] Belytschko T., Krongauz Y., Organ D., Fleming M., Krysl P.;''

Meshless methods: an overview and recent developments'', Computer

Methods in Applied Mechanics and Engineering ,139: 3–47, (1996).

[14] Jafari A.A., Eftekhari S.A.; " An efficient mixed methodology for

free vibration and buckling analysis of orthotropic rectangular plates"

, Applied Mathematics and Computation, 218: 2670-2692, (2011).

[15] Xinwei Wang, Zhe Wu; " Differential quadrature analysis of free

vibration of rhombic plates with free edges", Applied Mathematics

and Computation, 225: 171-183, (2013).

[16] Liew K.M., Huang Y.Q., Reddy J.N.; '' A hybrid moving least

squares and differential quadrature (MLSDQ) meshfree method'',

Int.J. Comput. Engrg. Sci. , 3: 1–12, (2002) .

[17] Liew K.M., Huang Y.Q., Reddy J.N.; "Analysis of general shaped

thin plates by the moving least squares differential quadrature

method", Finite Elements in Analysis and Design, 40: 1453-1474,

(2004).

[18] Liew K. M., Haung Y. Q., Reddy J. N.; "Moving least squares

differential quadrature method and its application to the analysis of

shear deformable plates", Int. J. Numer. Meth. Engng. , 56: 2331-

2351, (2003).

[19] Liew K.M., Xin Zhao, Ferreira A.J.M; " A review of meshless

methods for laminated and functionally graded plates and shells",

Composite Structures, 93:2031-2041, (2011).

[20] Ping Zhu, Zhang L.W., Liew K.M.;" Geometrically nonlinear thermo

mechanical analysis of moderately thick functionally graded plates

using a local Petrov-Galerkin approach with moving Kriging

interpolation", Composite Structures, 107:298-314, (2014).

[21] Lanhe W., Hongjun W, Daobin W.; '' Dynamic stability analysis of

FGM plates by the moving least squares differential quadrature

method'', Composite Structures, 77:383-394, (2007).

[22] Zhang L.W., Zhu P., Liew K.M.;" Thermal buckling of functionally

graded plates using a local Kriging meshless method", Composite

Structures, 108:472-492, (2014).

[23] Bui T.Q., Nguyen M.N., Zhang Ch.; "Buckling analysis of Reissner-

Mindlin plates subjected to in plane edge loads using a shear locking

free and mesh free method ", Engineering Analysis with Boundary

Elements , 35: 1038-1053, (2011).

[24] Ola Ragb , Matbuly M.S., Nassar M.; " Analysis of Composite Plates

Using Moving Least Squares Differential Quadrature Method",

Applied Mathematics and Computation, 238: 225–236, (2014).

[25] Wen P.H.,Ali abadi M.H.,'' A hybrid finite difference and moving

least square method for elasticity problems'', Engineering Analysis

with Boundary Elements, 36: 600–605,(2012).

[26] Jensen P.S.,''Finite difference technique for variable grids'', Comput

Struct ,2:17–29, (1972).

[27] Liszka T.,Orkisz j.,''The finite difference method at arbitrary irregular

grids and its application in applied mechanics'', Comput Struct,

11:83–95,( 1980).

[28] Panc. V., Theory of elastic plates, Leydon : Noordhoff International

Publishing, 13-41, (1975).

[29] Reddy J.N., Theory and analysis of elastic plates, Taylor& Francis:

Philadelphia, PA, (1999).

[30] Reddy J.N., Mechanics of laminated composite plates. Theory and

Analysis, CRC Press, Boca Raton, Fl, (1997).

[31] Lancaster P., Salkauskas K.; "Surfaces generated by moving least

squares methods", Mathematics of computation, 37: 141-158,

(1981).

[32] Breitkopf P., Rassineux A., Touzot G., Villon P.; ''Explicit form and

efficient computation of MLS shape functions and their derivatives'',

Int. J. Numer. Methods Engrg. 48: 451–466, (2000).

[33] Belytschko T., Krongauz Y., Fleming M., Organ D., Liu W.K.; "

Smoothing and accelerated computations in the element free Galerkin

method", Journal of Computational and Applied Mathematics ,74:

111-126, (1996).

[34] Liew K. M., Han J. B.;"Bending analysis of simply supported shear

deformable skew plates", Journal of Engineering Mechanics (ASCE),

123: 214-221, (1997).

[35] Sengupta D.; "Performance study of a simple finite element in the

analysis of skew rhombic plates ", Computers and Structures,

54:1173-1182, (1995).

[36] Liew K. M., Han J. B.;"Bending solution for thick plates with

quadrangular boundary", Journal of Engineering Mechanics (ASCE) ,

124:9-17, (1998).

http://www.sciencedirect.com.ezproxy.zu.edu.eg:81/science/article/pii/0045794984900555





http://www.sciencedirect.com.ezproxy.zu.edu.eg:81/science/article/pii/S0955799798000022








http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TWP-4MVVST7-1&_user=10&_coverDate=02%2F29%2F2008&_alid=845672680&_rdoc=115&_fmt=high&_orig=search&_cdi=5568&_st=13&_docanchor=&view=c&_ct=663&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=f49601191172c7c87ab5c5a5a500b23b

Quadrature Analysis of Functionally Graded Materials … graded materials (FGMs) are the new...

Documents

Transcript of Quadrature Analysis of Functionally Graded Materials … graded materials (FGMs) are the new...