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1A nth-order shear deformation theory for the bending analysis on the

functionally graded plates

Song Xianga*, Gui-wen Kanga

aLiaoning Key Laboratory of General Aviation, Shenyang Aerospace University, No. 37 Daoyi South Avenue,

Shenyang, Liaoning 110136, People's Republic of China

Abstract

This paper focus on the bending analysis of functionally graded plates by a nth-order shear deformation

theory and meshless global collocation method based on the thin plate spline radial basis function. Reddys

third-order theory can be considered as a special case of present nth -order theory (n=3). The governing

equations are derived by the principle of virtual work. The displacement and stress of a simply supported

functionally graded plate under sinusoidal load are calculated to verify the accuracy and efficiency of the

present theory.

Keywords: nth-order shear deformation theory; Bending analysis; Functionally graded plates; Meshless; Thin

plate spline;

1. Introduction

In recent years, functionally graded materials had been utilized in the aerospace and other industries

because of their superior heat-shielding properties. The functionally graded material for high-temperature

applications may be composed of ceramic and metal. The material properties of functionally graded material

vary continuously along certain dimension of the structure, but that of the fiber-reinforced laminated

composite materials are discontinuous across adjoining layers which result in the delaminating mode of

failure.

Many researchers have studied the behaviors of functionally graded plates. Vel and Batra [1] presented the

three-dimensional exact solution for the vibration of functionally graded rectangular plates. Ferreira et al. [2]

studied the static characteristics of functionally graded plates using third-order shear deformation theory and a

meshless method based on the multiquadrics radial basis function. Ferreira et al. [3] calculated the natural

frequencies of functionally graded plates by the multiquadrics radial basis function. Zenkour [4] proposed a

generalized shear deformation theory for bending analysis of functionally graded plates. Ferreira et al. [5]

Corresponding author. Tel.: +86 02489728667. Fax: +86 02489728690. E-mail address: xs74342@sina.com (Song Xiang).

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studied the static deformations of functionally graded plates using the radial basis function collocation method

and a higher-order shear deformation theory. They select the shape parameter in the radial basis functions by

an optimization procedure based on the cross-validation technique. Carrera et al. [6] presented the static

analysis of functionally graded material plates subjected to transverse mechanical loadings. The unified

formulation and principle of virtual displacements were employed to obtain both closed-form and finite

element solutions. Matsunaga [7] calculated the natural frequencies and buckling stresses of plates made of

functionally graded materials (FGMs) using a 2-D higher-order deformation theory. Carrera et al. [8]

evaluated the effect of thickness stretching in plate/shell structures made by materials which are functionally

graded (FGM) in the thickness directions. Xiang et al. [9] proposed a n-order shear deformation theory for

free vibration of functionally graded and composite sandwich plates.

In recent years, the various higher order shear deformation theories were proposed to analyze the plates.

Touratier [10] presented a standard plate theory which accounts for cosine shear stress distribution and free

boundary conditions for shear stress upon the top and bottom surfaces of the plate. Soldatos [11] presented a

general two-dimensional theory suitable for the static and/or dynamic analysis of a transverse shear

deformable plate, constructed of a homogeneous, monoclinic, linearly elastic material and subjected to any

type of shear tractions at its lateral plane. Karama et al. [12] presented a new multi-layer laminated composite

structure model to predict the mechanical behaviour of multi-layered laminated composite structures. They

introduced an exponential function as the shear stress function. Reddy [13] developed a higher-order shear

deformation theory which accounts for parabolic distribution of the transverse shear strains through the

thickness of the laminated plate. Aydogdu [14] proposed a new higher order laminated composite plate theory

in which a new shear stress function was used.

In this paper, a n-order shear deformation theory is used to analyze the static characteristics of functionally

graded plates. The present n-order shear deformation theory satisfies the zero transverse shear stress boundary

conditions on the top and bottom surface of the plate. The third-order theory of Reddy can be considered as a

special case of present n-order theory (n=3). Displacement and stress of the simply supported laminated plate

under sinusoidal load are computed by present n-order theory and a meshless global collocation method based

on the thin plate spline radial basis function. The results are compared with the available published results.

2. The governing equations based on the nth-order shear deformation theory

The displacement field of the n-order shear deformation theory is

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1

1

1 2 ( , )( , ) ( , ) ( , )

1 2 ( , )( , ) ( , ) ( , ) , 3,5,7,9...

( , )

n

n

x x

n

n

y y

w x yU u x y z x y z x y

n h x

w x yV v x y z x y z x y n

n h y

W w x y

= + +

= + + = =

(1)

where u , v , w , x and y are the unknown displacement functions. h is the thickness of the plate.

The strain can be expressed in the form of

1 2

2

1 2

2

1 2

1

1 2

1 2

1 22

21

n

nx xx

n

y yn

y

n

y ynx xxy

n

yz y

u wz z

x x n h x x

v wz z

y y n h y y

u v wz z

y x y x n h y x x y

z

h

= + +

= + +

= + + + + +

= + 1

21

n

xz x

w

y

z w

h x

= +

(2)

We obtain the following Euler-Lagrange equations using the dynamic version of the principle of virtual

displacements

2 22

2 2 1 2 2

1 1 2

1 1 2

0

0

2

0

0

xyx

xy y

y y xy yx x x

xy xyx xx x

xy xy y y

y y

NN

x y

N N

x y

Q R P PQ R PC C C q

x x y y x x y y

M PM PC C Q C R

x x y y

M P M PC C Q C R

x x y y

+ =

+ =

+ + + + =

+ + =

+ + =

(3)

where

1

1

1 2n

Cn h

=

,

1

2

2n

Ch

=

.

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4

/ 2

/2

h

hN dz

= , / 2

/2

h

hM zdz

= , / 2

/2

hn

hP z dz

= / 2

/2

h

zh

Q dz

= ,/ 2

1

/2

hn

zh

R z dz

= , , = ,x y (4) The stress-strain relationships of the functionally graded plate in the global x-y-z coordinate system can be

written as

11 12

12 11

66

66

66

( ) ( ) 0 0 0

( ) ( ) 0 0 0

0 0 ( ) 0 0

0 0 0 ( ) 0

0 0 0 0 ( )

xx x

yy y

xy xy

yz yz

zx zx

Q z Q z

Q z Q z

Q z

Q z

Q z

=

(5)

where

11 2

( )( )

1

E zQ z

=

, 12 2

( )( )

1

E zQ z

=

, 66( )

( )2(1 )

E zQ z

=

+ (6)

In the Eq. (6), is the Poissons ratio, the variation of Youngs modulus E is given as:

1( ) ( )

2

p

c m m

zE z E E E

h

= + +

(7)

where cE and mE denote the elasticity modulus of the ceramic and metal, respectively. p is power law index. z

is the distance from mid-plane. h is the thickness of the plate. As can be seen Eq. (7), ( ) cE z E= at the top

surface / 0.5z h = , and ( ) mE z E= at the bottom surface / 0.5z h = . Top surface of functionally graded

plate is pure ceramic, and bottom surface is pure metal.

Substituting Eq. (2) and Eq. (5) into Eq. (4), the resultants of functionally graded plate can be expressed in terms

of displacement as follows

2 2

11 11 1 11 12 12 1 122 2

y yx xx

u w v wN A B C E A B C E

x x x x y y y y

= + + + + +

2 2

12 12 1 12 11 11 1 112 2

y yx xy

u w v wN A B C E A B C E

x x x x y y y y

= + + + + +

2

66 66 1 66 2y yx x

xy

u v wN A B C E

y x y x y x x y

= + + + + +

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2 2

11 11 1 11 12 12 1 122 2

y yx xx

u w v wM B D C EE B D C EE

x x x x y y y y

= + + + + +

2 2

12 12 1 12 11 11 1 112 2

y yx xy

u w v wM B D C EE B D C EE

x x x x y y y y

= + + + + +

2

66 66 1 66 2y yx x

xy

u v wM B D C EE

y x y x y x x y

= + + + + +

2 2

11 11 1 11 12 12 1 122 2

y yx xx

u w v wP E EE C H E EE C H

x x x x y y y y

= + + + + +

2 2

12 12 1 12 11 11 1 112 2

y yx xy

u w v wP E EE C H E EE C H

x x x x y y y y

= + + + + +

2

66 66 1 66 2y yx x

xy

u v wP E EE C H

y x y x y x x y

= + + + + +

( )66 2 66x x wQ A C DDx

= +

( )66 2 66y y wQ A C DDy

= +

( )66 2 66x x wR DD C Fx

= +

( )66 2 66y y wR DD C Fy

= + 8 / 2

/ 2( )

h

ij ijh

A Q z dz

= , / 2

/ 2( )

h

ij ijh

B zQ z dz

= , / 2

2

/2( )

h

ij ijh

D z Q z dz

= / 2

1

/ 2( )

hn

ij ijh

DD z Q z dz

= , / 2

/2( )

hn

ij ijh

E z Q z dz

= , / 2

1

/ 2( )

hn

ij ijh

EE z Q z dz+

= / 2

2 2

/2( )

hn

ij ijh

F z Q z dz

= , / 2

2

/2( )

hn

ij ijh

H z Q z dz

= (9)

By substituting Eq. (8) into Eq. (3), the governing equations in terms of displacements can be obtained as

follows:

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2 22 22 2 3 3

11 12 11 12 1 11 1 122 2 2 3 2

2 22 22 2 3

66 66 1 662 2 2 2

A +A B B

A B 2 0

y yx x

y yx x

u v w wC E C E

x x y x x y x x x y x y

u v wC E

y x y y x y y x y x y

+ + + +

+ + + + + + =

(10)

2 22 22 2 3 2 2

66 66 1 66 12 112 2 2 2 2

2 22 2 3 3

12 11 1 12 1 112 2 2 3

A B 2 A A

B B 0

y yx x

y yx x

u v w u vC E

x y x x y x x y x x y x y y

w wC E C E

x y y x y x y y y

+ + + + + + +

+ + + + =

(11)

2 2 2 2

66 2 66 66 2 662 2 2 2

2 2 2 2

2 66 2 66 66 2 662 2 2 2

1 1

A DD A DDy yx x

y yx x

w w w wC C

x x x x y y y y

w w w wC DD C F DD C F

x x x x y y y y

C E

+ + + + +

+ + + + + +

3 33 33 3 4 4

1 12 11 12 1 11 1 123 2 3 2 3 4 2 2 2

3 33 33 3 4

12 11 12 11 1 12 1 112 3 2 3 2 2 2 3

y yx x

y yx x

u v w wE EE EE C H C H

x x y x x y x x x y x y

u v wE E EE EE C H C H

x y y x y y x y x y y

+ + + + + +

+ + + + +

4

4

3 33 33 3 4

66 66 1 662 2 2 2 2 2 2 22 2 2 2

y yx x

w

y

u v wE EE C H

x y x y x y x y x y x y x y

q

+ + + + + + =

(12)

2 22 22 2 3 3 2 2

11 12 11 12 1 11 1 12 662 2 2 3 2 2

2 22 2 3

66 1 66 662 2 2

B +B D D B

D 2

y yx x

y yx x

u v w w u vC EE C EE

x x y x x y x x x y x y y x y

wC EE A

y x y y x y x y

+ + + + + +

+ + + +

2

2 66 2 66

2 22 22 2 3 3 2 2

1 11 12 11 12 1 11 1 12 662 2 2 3 2 2

22

66 2

2

E E EE EE E

EE

x x x

y yx x

x

w w wC DD C F

x x x

u v w w u vC C H C H

x x y x x y x x x y x y y x y

y

+ + + +

+ + + + + + +

+ +

22 3

1 66 2 22 0

y yx wC Hx y y x y x y

+ + =

(13)

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2 22 22 2 3 2

66 66 1 66 122 2 2 2

2 22 22 3 3

11 12 11 1 12 1 112 2 2 2 3

66

B D 2 B

B D D

y yx x

y yx x

u v w uC EE

x y x x y x x y x x y x y

v w wC EE C EE

y x y y x y x y y y

A

+ + + + + + +

+ + + +

2 22

2 66 2 66 1 66 2

2 22 2 3 2 2

66 1 66 12 112 2 2 2

22

12 11 12

2 E

EE 2 E E

EE EE

y y y

y yx x

yx

w w w u vC DD C F C

y y y x y x

w u vC H

x y x x y x x y x y y

Cx y y

+ + + + + +

+ + + + + +

+

22 3 3

12 1 112 2 30

yx w wH C Hx y x y y y

+ + =

(14)

For the simply supported plates, the boundary conditions are as follows:

10, : 0, 0, 0, 0, 0y x x xx a v w M C P N= = = = = =

10, : 0, 0, 0, 0, 0x y y yy b u w M C P N= = = = = = 15 For the clamped supported plates, the boundary conditions are as follows:

0, 0, 0, 0, 0x yu v w = = = = = 16 3. Solution methods

According to the meshless global collocation method, the solutions of Eqs. (10-14) can be approximated in the

form of

1

( )N

u i t

j ij

j

u g r e =

=

1

( )N

v i t

j ij

j

v g r e =

=

1

( )N

w i t

j ij

j

w g r e =

=

1

( )xN

i t

x j ij

j

g r e =

=

1

( )yN

i t

y j ij

j

g r e

=

= (17)

where N is the total number of nodes, u

j , v

j , w

j , x

j

and yj are 5N unknown coefficients, is the

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natural frequency of free vibration, ( )ijg r is radial basis function, ijr is the distance between the node Xi and

node Xj, The multiquadric, inverse multiquadric, Gaussian and thin plate spline radial basis functions include a

shape parameter which have important effect on the accuracy. The choice of the shape parameter in thin plate spline

radial basis function is easier than other radial basis functions. Radial basis function used in this paper is the thin

plate spline radial basis function as follows:

6

10logijr

ijg r= (18)

Thin plate spline radial basis function has the disadvantage of singularity when the distance between two

nodes is zero. In order to eliminate the singularity of thin plate spline radial basis function, when the distance

between two nodes is zero, 2 2

ij ijr r = + .

where is infinitesimal. The infinitesimal value of this paper is 110-30.

Substituting Eq. (17) into Eqs. (10-14) and boundary conditions. The discretized governing equations and

boundary conditions can be expressed as

{ }0

Lg q

Bg

=

19

Then

1

0

Lg q

Bg

=

20

The deflection can be calculated by subsitituting the Eq. (20) into the third formulation of Eq. (17), the stress can

be obtained by the constitutive equations.

4. Numerical examples

In the present paper, a simply supported square functionally graded plate of side a and thickness h under

sinusoidal load q is considered. The available published results for the static analysis of the clamped

functionally graded plates are very sparse, so only simply supported functionally graded plate is considered.

The form of the q is 0 sin( / )sin( / )q q x a y api pi= . The functionally graded plate comprised of metal and

ceramic. The material properties of the metal and ceramic are as follows:

Metal: Em =70GPa, m =0.3

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Ceramic: Ec =380GPa, c =0.3

The meshless global radial point collocation method based on the thin plate spline radial basis function is

used to solve the governing differential equations. The node distribution is 21h21. Because the 21x21 node distribution can produce the results with good accuracy and is moderate in its computing time.

The transverse displacements, the normal stresses and the in-plane and transverse shear stresses are

presented in normalized form as:

3

4

0

10 ( / 2, / 2)c

h E w a aw

q a= ,

0

( / 2, / 2, / 2)xx

a a h h

q a

= ,

0

( / 2, / 2, / 3)y

y

a a h h

q a

=

0

(0,0, / 3)xy

xy

h h

q a

= , 0

( / 2,0, / 6)yz

yz

a h h

q a

= ,

0

(0, / 2,0)xzxz

a h

q a

=

Table 1 lists the dimensionless stresses and displacements of a simply supported functionally graded square

plate of a/h=10. It can be found that the present theory produces the close results to those of Zenkour [4] and

Carrera et al. [6], the accuracy of displacements is higher than that of stress, and the accuracy of plane shear

stress is higher than that of normal stress. Figs. 1-4 show the dimensionless displacement and stress of a

functionally graded square plate (a/h=10, p=1). It can be found from the Figs. 1-4 that present theory with n=5

or 7 produce the more close results to those of Zenkour [4]. The differences with respect to the results of

Zenkour [4] are due to the meshless global collocation method which has been used to solve the governing

equations. If there is derivative boundary condition, the solution accuracy of the meshless global collocation

method will deteriorate.

In Figs. 59 we present the evolution of the displacement and stresses through the thickness direction for

various p values. According to the Figs. 59, the p values have more influence on the transverse shear stress

than they do on the normal stress and plane shear stress.

The present results are also compared with the three-dimensional solutions of Kashtalyan [15] and the

refined two-dimensional solutions of Carrera et al. [6]. Kashtalyan [15] considers a simply supported square

plate of thickness h and side-to-thickness ratio a/h = 3. The load and boundary conditions are the same as the

above examples. The shear modulus is assumed to vary exponentially through the thickness (Poissons ratio is

considered to be constant) according to

( / 0.5)

1( )z hG z G e = , 1

2(1 )

EG

=

+, / 2 / 2h z h 21

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The transverse displacement is presented in normalized form as:

1

0

( / 2, / 2)G w a aw

q h= 22

The dimensionless displacements of a simply supported square plate with a/h = 3 are listed in Table 2.

According to the Table 2, with the increase of n value, the present results are gradually close to the

three-dimensional solutions of Kashtalyan [15] and the refined two-dimensional solutions of Carrera et al. [6].

5. Conclusions

In this paper, a n-order shear deformation theory is used to analyze the static characteristics of functionally

graded plates. The third-order theory of Reddy can be considered as a special case of present n-order theory

(n=3). Displacement and stress of the simply supported functionally graded plate under sinusoidal load are

computed by present n-order theory and a meshless global collocation method based on the thin plate spline

radial basis function. The results are compared with available published results and the agreement is found to

be good. Through numerical experiments, it is found that present theory does not require any shear correction

factor and allows the user to experiment the best order number to approximate the structural problem under

investigation. If there is derivative boundary condition, the solution accuracy of the meshless global collocation

method will deteriorate. The authors will study the improvement of solution accuracy in the future article.

References

[1] Vel, S.S., Batra, R.C.: Three-dimensional exact solution for the vibration of functionally graded

rectangular plates. J Sound Vib 272, 703730 (2004)

[2] Ferreira, A.J.M., Batra, R.C., Roque, C.M.C., et al.: Static analysis of functionally graded plates using

third-order shear deformation theory and a meshless method. Composite Structures 69(4), 449457

(2005)

[3] Ferreira, A.J.M., Batra, R.C., Roque, C.M.C., et al.: Natural frequencies of functionally graded plates by a

meshless method. Composite Structures 75 (14), 593600 (2006)

[4] Zenkour, A.M.: Generalized shear deformation theory for bending analysis of functionally graded plates.

Applied Mathematical Modeling 30, 6784 (2006)

[5] Ferreira, A.J.M., Roque, C.M.C., Jorge, R.M.N., et al.: Analysis of functionally graded plates by a robust

meshless method. Mechanics of Advanced Materials and Structures 14 (8), 577587 (2007)

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[6] Carrera, E., Brischetto, S., Robaldo, A.: Variable kinematic model for the analysis of functionally graded

material plates. AIAA Journal 46, 194203 (2008)

[7] Matsunaga, H.: Free vibration and stability of functionally graded plates according to a 2-D higher-order

deformation theory. Composite Structures 82, 499512 (2008)

[8] Carrera, E., Brischetto, S., Cinefra, M., Soave, M.: Effects of thickness stretching in functionally graded

plates and shells. Composites Part B: Engineering 42, 123133 (2011)

[9] Xiang, S., Jin, Y.X., Bi, Z.Y., et al.: A n-order shear deformation theory for free vibration of functionally

graded and composite sandwich plates. Composite Structures 93, 28262832 (2011)

[10] Touratier, M.: An efficient standard plate theory. Int J Eng Sci 29(8), 901916 (1991)

[11] Soldatos, K.P.: A transverse shear deformation theory for homogeneous monoclinic plates. Acta Mech 94,

195200 (1992)

[12] Karama, M., Afaq, K.S., Mistou, S.: Mechanical behaviour of laminated composite beam by new

multi-layered laminated composite structures model with transverse shear stress continuity. Int J Solids

Struct 40, 15251546 (2003)

[13] Reddy, J.N.: A simple higher-order theory for laminated composite plates. J Appl Mech 51, 745752

(1984)

[14] Aydogdu, M.: A new shear deformation theory for laminated composite plates. Compos Struct 89(1),

94101 (2009)

[15] Kashtalyan, M.: Three-Dimensional Elasticity Solution for Bending of Functionally Graded Rectangular

Plates. European Journal of Mechanics A/Solids 23(5), 853864 (2004).

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Table 1 Dimensionless stresses and displacements of a functionally graded square plate (a/h=10)

p Method wx y xy

1 Zenkour [4] 0.5889 3.0870 1.4894 0.6110

Carrera et al. [6] 0.5875 _ 1.5062 0.6081

Present n=3 0.5895 3.2480 1.5287 0.6295

Present n=5 0.5886 3.1018 1.5007 0.6205

Present n=7 0.5878 3.0203 1.4830 0.6114

Present n=9 0.5855 3.1006 1.5007 0.6198

2 Zenkour [4] 0.7573 3.6094 1.3954 0.5441

Carrera et al. [6] 0.7570 _ 1.4147 0.5421

Present n=3 0.7581 3.7062 1.4222 0.5559

Present n=5 0.7572 3.5609 1.3960 0.5503

Present n=7 0.7549 3.5946 1.4065 0.5513

Present n=9 0.7541 3.4942 1.3828 0.5463

4 Zenkour [4] 0.8819 4.0693 1.1783 0.5667

Carrera et al. [6] 0.8823 _ 1.1985 0.5666

Present n=3 0.8824 3.9371 1.1474 0.5620

Present n=5 0.8809 4.0079 1.1796 0.5709

Present n=7 0.8746 4.1122 1.1939 0.5787

Present n=9 0.8742 4.1312 1.2037 0.5726

7 Zenkour [4] 0.9562 4.5971 0.9903 0.5834

Carrera et al. [6] 0.9554 _ 1.0117 0.5852

Present n=3 0.9563 4.6568 1.0026 0.5868

Present n=5 0.9523 4.6667 1.0026 0.5957

Present n=7 0.9477 4.5561 1.0020 0.5922

Present n=9 0.9469 4.5469 0.9985 0.5877

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Table 2 Dimensionless displacements of a simply supported square plate with a/h = 3

N =10-1 =10-6

3 -1.4861 -1.4137

5 -1.4781 -1.4061

7 -1.4649 -1.3936

9 -1.4544 - 1.3835

11 -1.4463 -1.3759

13 - 1.4401 -1.3700

15 -1.4352 -1.3653

17 -1.4312 -1.3615

19 - 1.4279 - 1.3584

21 -1.4252 -1.3558

Kashtalyan [15] -1.4146 -1.3426

Carrera et al. [6] -1.4145 -1.3424

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Fig. 1. Dimensionless displacement of a functionally graded square plate (a/h=10, p=1)

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Fig. 2. Dimensionless stress x of a functionally graded square plate (a/h=10, p=1)

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Fig. 3. Dimensionless stress y of a functionally graded square plate (a/h=10, p=1)

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Fig. 4. Dimensionless stress xy of a functionally graded square plate (a/h=10, p=1)

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Fig. 5. Nondimensional stress x through the thickness direction of square functionally graded plate under

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Fig. 6. Nondimensional stress y through the thickness direction of square functionally graded plate under

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Fig. 7. Nondimensional stress xy through the thickness direction of square functionally graded plate under

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Fig. 8. Nondimensional stress xz through the thickness direction of square functionally graded plate under

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Fig. 9. Nondimensional stress yz through the thickness direction of square functionally graded plate under

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> A nth-order theory for bending analysis of the functionally graded plates.

> Present theory does not require any shear correction factor

> Present theory allows the user to experiment the best order number.