vibration and buckling analysis of functionally graded ......lar plate subjected to a transverse...

456

Abstract In this paper a new eight-unknown higher order shear deformation theory is proposed to study the buckling and free vibration of func-tionally graded (FG) material plates. The theory bases on full twelve-unknown higher order shear deformation theory, simultane-ously satisfies zero transverse shear stress at the top and bottom surfaces of FG plates. Equations of motion are derived from Hamil-ton’s principle. The critical buckling load and the vibration natural frequency are analyzed. The accuracy of present analytical solution is confirmed by comparing the present results with those available in existing literature. The effect of power law index of functionally graded material, side-to-thickness ratio on buckling and free vibra-tion responses of FG plates is investigated. Keywords Buckling, vibration analysis, functionally graded materials, higher order shear deformation theory, closed-form solution.

Vibration and Buckling Analysis of Functionally Graded Plates Using New Eight-Unknown Higher Order Shear Deformation Theory

NOMENCLATURE

x, y, z coordinates a, b, h length, width, and thickness of the plate p volume fraction index u, v, w in-plane and transverse displacement at mid-plane of the plate u v w0 0 0, , displacements of the mid-plane in the x; y; z directions

Tran Ich Thinh a Tran Minh Tu b Tran Huu Quoc c Nguyen Van Long d

a Ha NoiUniversity of Science and Technol-ogy, 1Dai Co Viet Road, Ha Noi, Viet Nam E-mail address: [email protected] b University of Civil Engineering, 55 GiaiPhong Road, Ha Noi, Viet Nam E-mail address: [email protected] c University of Civil Engineering, 55 GiaiPhong Road, Ha Noi, Viet Nam E-mail address: [email protected] d Construction Technical College No.1, Trung Van, TuLiem, Ha Noi, Viet Nam E-mail address: [email protected] http://dx.doi.org/10.1590/1679-78252522

Received 09.10.2015 In revised form 17.12.2015 Accepted 22.12.2015 Available online 05.01.2016

T.I. Thinh et al. / Vibration and Buckling Analysis of Functionally Graded Plates Using New Eight-Unknown Higher Order… 457

Latin American Journal of Solids and Structures 13 (2016) 456-477

NOMENCLATURE (continuation)

x y zu v w0 0 0, , , , ,q q q* * * * * *

higher-order terms of displacements in the Taylor series expansion

w , ŵ natural frequency and non-dimensional natural frequency cr crN N,

critical buckling load and non-dimensional critical buckling load

ijQ stiffness coefficients of FG plates

x y xyN0 0 0,N ,N

in-plane pre-buckling loads

1 2,g g in-plane load parameters

c c cE , ,n r , Young’s modulus, Poisson coefficient, mass density of the ceramic

m m mE , ,n r Young’s modulus, Poisson coefficient, mass density of the metal U, V, K strain energy, external work, kinetic energy

1 INTRODUCTION

Ever since invented by Japanese scientists in the 80s of the last century, Functionally Graded Mate-rials (FGMs) has been more and more widely applied in many fields such as aircraft industry, nuclear industry, civil engineering, automotive, biomechanics, optics… Typical FGMs are composed of ce-ramic and metal materials. Ceramic provides high temperature resistance while metals have high toughness; thus FGMs are usually used in the manufacture of heat-resistance structural components such as airplane fuselages or walls in nuclear reaction plants.…The understanding of the behavior of FGM, therefore, is very much desired. Studying the static and dynamic behavior of FGM structures has become an interesting topic for researchers around the world.

There have been many computational models and methods of calculation proposed for FGM plates. The classical plate theory (CPT) that bases on Kirchhoff-Love’s assumption, is only suitable for thin plates since it ignores the effects of transverse shear deformation. For moderately thick plates, numerical results calculated using CPT yield lower deflection, higher natural frequency and buckling load in comparison with experimental results. In order to correct this inaptitude, the first-order shear deformation theories (FSDT) have been initially proposed by Reissner and further developed by Mindlin. Although FSDT describes more realistic behavior of thin to moderately thick plates, the parabolic distribution of transverse shear stress through the thickness of the plate is not properly reflected, thus the shear correction factor is introduced. The determination of this factor is not simple as it depends on the loading, boundary condition, materials etc…

To avoid using shear correction factor, higher order shear deformation theories (HSDTs) are proposed. Based on third order shear deformation theory with five displacement unknowns, Reddy (2000) developed analytical and finite element solutions for static and dynamic analysis of functionally graded rectangular plates. The formulation accounts for the thermo-mechanical coupling, time de-pendency, and the von Kárman-type geometric non-linearity. Bodaghi et al. (2010) used Reddy’s third order shear deformation theory and Levy-type solution for buckling analysis of thick functionally graded rectangular plates. Also with five displacement unknowns, Zenkour (2006) used his generalized shear deformation theory to study static behaviors of simply supported functionally graded rectangu-lar plate subjected to a transverse uniform load. Employing finite element method based on nine

458 T.I. Thinh et al. / Vibration and Buckling Analysis of Functionally Graded Plates Using New Eight-Unknown Higher Order…


unknowns higher order shear deformation theory, Pandya and Kant (1988) investigated deflections, in-plane and inter-laminar stresses of thick laminated composite plates. This displacement model assumed non-linear and constant variation of in-plane and transverse displacement, respectively, through the plate thickness. Using eleven-unknown displacement field and finite element method, Talha and Singh (2010) analyzed static response and natural frequency of functionally graded plates. Higher order terms of the displacement field are determined by vanishing the transverse shear stresses on the top and bottom surfaces of the plate.Kim and Reddy (2013) also used acouple of stress-based third-ordertheory with elevenunknowns to analyze the bending, vibration and buckling behaviors of FG plates by analytical method. Based on the higher-order refined theories, Jha et al. (2012) presented analytical solutions for free vibration analysis of simply supported rectangular functionally graded plates. This HSDT introduces twelve displacement unknowns, and correctly describes the quadratic distribution of transverse normal strain across the thickness although the values at the top and bottom are non-zero. A comprehensive review of the various methods employed to study the static, dynamic and stability behaviors of functionally graded plates can be found in the work of Swaminathan et al. (2015). The review focuses on comparing the stress, vibration and buckling characteristics of FGM plates using different theories. It is observed that most of the above mentioned HSDTs require addi-tional computation efforts due to the additional unknowns introduced to them (usually nine, eleven or thirteen unknowns depending on each particular theory).

In this paper, a new higher order displacement field based on twelve-unknown higher order shear deformation theory is developed to analyze the free vibration and buckling of functionally graded plates. The new form is dictated by the satisfaction of vanishing transverse shear stress at the top and bottom surfaces of the plate. With this proposed higher order displacement field, the number of displacement unknowns reduces from twelve to eight, thus savingcomputational time and optimizing the storage capacity of computers. The accuracy of the present theory is verified by comparisonwith previous studies. 2 KINEMATICS

The displacement components u(x,y,z), v(x,y,z) and w(x,y,z) at any point in the plate can be expanded in Taylor's series in terms of the thickness coordinate as (Jha – 2012):

x x

y y

z z

u x y z t u x y t z x y t z u x y t z x y t

v x y z t v x y t z x y t z v x y t z x y t

w x y z t w x y t z x y t z w x y t z x y t

2 * 3 *0 0

2 * 3 *0 0

2 * 3 *0 0

( , , , ) ( , , ) ( , , ) ( , , ) ( , , );

( , , , ) ( , , ) ( , , ) ( , , ) ( , , );

( , , , ) ( , , ) ( , , ) ( , , ) ( , , ).

= + + +

= + + +

= + + +

(1)

where u v w, , denote the displacements of a point along the (x,y, z) coordinates. u v w0 0 0, , are corresponding displacements of a point on the mid-plane. xq , yq and zq are rotations of transverse

normal to the mid-plane about the y-axis, x-axis and z-axis, respectively. x yu v w0 0 0, , , ,q q* * * * * and zq

* are the higher-order terms in the Taylor series expansion and they represent higher-order transverse cross-sectionaldeformation modes.



For plates under bending, the transverse shear stresses xzs , yzs must be vanished at the top and bottom surfaces. These conditions lead to the requirement that the corresponding transverse strains

on these surfaces to be zero. From xz yzh h

x y x y, , , , 02 2

g gæ ö æ ö÷ ÷ç ç÷ ÷ = =ç ç÷ ÷ç ç÷ ÷ç çè ø è ø

, we obtain:

( )

( )

; ;

; .

z x z x x x x x

z y z y y y y y

hu w w

hh

v w wh

2* * * *0 , , 0, 0,2

2* * * *0 , , 0, 0,2

1 4 12 8 331 4 12 8 33

q q q q

q q q q

=- - =- + -

=- - =- + - (2)

The displacement field (1) becomes:

( ) ( )( ) ( )

.

x x x

y z y z y y y y

z z

x z x z x

w w z z w z

z zu u z c c w w

z zv v z c c w w

*1 2 0, 0,

2 3* *

0 , 1 , 2 0, 0,

2 * 3 *0 0

2 3*

0 , , ;

;

2 3

2 3

é ùê úë û

é ù+ ê úë û= + + +

= + - + - + +

= + - - + +

(3-a)

with:

hc c

h

2

1 2 2

4; .

4= = (3-b)

Using the strain-displacement relations of the theory of elasticity, the linear strains are obtained:

x x x x x y y y y y

z z z z xy xy y xy xy

xz xz xz xz xz yz yz yz yz yz

z z z z z z

z z z z z

z z z z z z

0 0 2 * 3 * 0 0 2 * 3 *

0 0 2 * 0 0 2 * 3 *

0 0 2 * 3 * 0 0 2 * 3 *

; ;

; ;

; .

e e k e k e e k e k

e e k e g e k e k

g g k g k g g k g k

= + + + = + + +

= + + = + + +

= + + + = + + +

(4-a)

where:

{ } { } { } { }{ } ( ) ( ) ( )

{ } ( )

x y z xy x y z y x x y z xy x x y y x y y x

x y z xy z xx z xx z yy z yy z z xy z xy

x y xy x x xx

u v u v w

c c c

c w w

0 0 0 0 0 0 0 0 *0, 0, 0, 0, , , 0 , ,

* * * * * * * *, 1 , , 1 , , 1 ,

* * *2 , 0,

, , , , , , ; , , , , ,2 , ;

1 1, , , , , 3 , ;

2 21

, ,3

e e e g q k k k k q q q q

e e e g q q q q q q

k k k q

= + = +ì üï ïï ï= - + - + - +í ýï ïï ïî þ

= - + +

( ) ( )( )

( )( ){ } { } { } { }{ } ( ) ( ){ } { } { }

xx y y yy yy

x y y x xy xy

xz yz x x y y xz yz z x z y

xz yz x x y y xz yz z x z y

c w w

c w w

w w c c

c w c w

* *0, 2 , 0, 0,

*2 , , 0, 0,

0 0 0 0 * *0, 0, 1 , 1 ,

* * * * * *2 0, 2 0, , ,

1, ,

31

2 2 ;3

, , ; , , ;

, , ; , , .

q

q q

g g q q k k q q

g g q q k k q q

ìïï - + +íïïîüïï- + + + ýïïþ

= + + = - -

= - + - + =

(4-b)

In the above formulas, a comma followed by x or y denotes differentiation with respect to the coordinates x or yrespectively.



3 CONSTITUTIVE EQUATION

Consider a simply supported linearly elastic rectangular FG plate of uniform thickness h as shown in Figure 1. The Poisson’s ratio n is assumed to be constant across the plate thickness. The Young’s modulus, the mass densityof the FG plate is assumed to follow the power law distribution alongthe thickness, and expressed as (Reddy – 2000):

( ) ( )p

m c m

zE z E E E

h

1

2

æ ö÷ç ÷= + - +ç ÷ç ÷çè ø (5-a)

( ) ( )p

m c m

zz

h

1

2r r r r

æ ö÷ç ÷= + - +ç ÷ç ÷çè ø (5-b)

In the above formula, subscript c refers to ceramic material and subscript m refers to metal material of the FG plate. It is clear from the expression that the top surface (z h / 2= ) of the FG plate is ceramic-rich and the bottom ( z h / 2=- ) is metal-rich in constituents.

Figure 1: Geometry of FG plate with positive set of reference axes.

The stress-strain relationship for the FG plate can be written as:

x x

y y

z z

xy xy

xz xz

yz yz

Q Q Q

Q Q Q

Q Q Q

Q

Q

Q

11 12 13

21 22 23

31 32 33

44

55

66

0 0 0

0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

s es es es gs gs g

é ùì ü ì üï ï ï ïï ï ï ïê úï ï ï ïê úï ï ï ïï ï ï ïê úï ï ï ïï ï ï ïê úï ï ï ïê úï ï ï ïï ï ï ï=í ý í ýê úï ï ï ïê úï ï ïï ï ïê úï ï ïê úï ï ïï ï ïê úï ï ïï ï ïê úï ï ïï ï ïî þ î þë û

ïïïïïïïïï

(6-a)

in which ( )x y z xz yz xy, , , , ,s s s s s s are the stresses, and ( )x y z xz yz xy, , , , ,e e e g g g are the strains with re-spect to axes x, y, z. The elements of stiffness matrix ijQ are defined as follows:



( )( )( )

( )( )

( )

;

;

.

EQ Q Q

EQ Q Q Q Q Q

EQ Q Q

11 22 33

12 23 13 21 32 31

44 55 66

1

1 1 2

1 1 2

2 1

n

n nn

n n

n

-= = =

+ -

= = = = = =+ -

= = =+

(6-b)

4 EQUILIBRIUM EQUATIONS

Hamilton’s principle is used to derive the equations of motion. The principle can be stated in analyt-ical form as:

( )T

U V K dt0

0 .d d d= + -ò (7-a)

where δU is the variation of strain energy; δV is the variation of external work; and δKis the variation of kinetic energy.

The variation of strain energy of the plate can be calculated by:

( )h

x x y y z z xy xy xz xz yz yzA h

U dAdz/2

/2

d s de s de s de s dg s dg s dg-

= + + + + +ò ò (7-b)

The variation of work done by in-planeand transverse loads is given by:

zA A

V N w dA q w dA0d d d+ +=- -ò ò

(7-c)

where x xy yw w w

N N N Nx xx y

2 2 20 0 00 0 0

2 22

¶ ¶ ¶= + +

¶ ¶¶ ¶

and zq+ is the transverse load at the top surface of the plate, z z

h h hw w w

2 3

0 02 2 8q q+ * *= + + + is the

transverse displacement of any point on the top surface of the plate; x y xyN0 0 0,N ,N are in-plane pre-

buckling applied loads. The variation of kinetic energy of the plate can be written in the form:

( ) ( )h

A h

K u u v v w w z dAdz/2

/2

d d d d r-

= + +ò ò (7-d)

where the dot-superscript indicates the differentiation with respect to the time variable t; ( )zr is the mass density.



Substituting the expressions for δU, δV, and δK from Eqs. (7b)–(7d) into Eq.(7a) and integrating by parts, then collecting x y z zu v w w

* *0 0 0 0d d d dq dq dq d dq, , , , , , , , the following equations of motion of the

plate are obtained:

( ) ( ): x x xy y x z x z x x xu N N I u J I c I c w w* *0 , , 0 0 1 2 , 1 , 3 2 0, 0,1 12 3d q q q+ = + - + - + (8-a)

( ) ( ): xy x y y y z y z y y yv N N I v J I c I c w w* *0 , , 0 0 1 2 , 1 , 3 2 0, 0,1 12 3d q q q+ = + - + - + (8-b)

( ) ( ) ( )( ) ( )

: x xx xy xy y yy x x y y x x y y z

z z x y x x y y z

z

cw M M M c Q Q Q Q q N

I w I I w I I c u v c J I c

I I c w I c w

* * * * *20 , , , 2 , , , , 0

* * 20 0 1 2 0 3 3 2 0, 0, 2 4 , , 5 2

2 * 2 2 2 *5 6 2 0 6 2 0

23

1 1 1

3 3 61 1 1

6 9 9

d

q q q q q

q

++ + - + + + + + =

+ + + + + + + - -

- - -

(8-c)

( ) ( ):x x x xy y x x xy y x xx z x z x x x

cM M M M c Q Q

J u K J c J c J w J w

* * *2, , , , 2

* *1 0 2 3 , 1 3 , 2 4 0, 4 0,

31 1 1 1

2 2 3 3

dq

q q q

+ - + - +

=- - + + + + (8-d)

( ) ( ):y xy x y y xy x y y y yy z y z y y y

cM M M M c Q Q

J v K J c J c J w J w

* * *2, , , , 2

* *1 0 2 3 , 1 3 , 2 4 0, 4 0,

31 1 1 1

2 2 3 3

dq

q q q

+ - + - +

=- - + + + + (8-e)

( ) ( )( )

:z x xx xy xy y yy z z z x y

x x y y z z z

hN N N N q N I w I I u v I w

J I I I c I c w I w

* * * *, , , 1 1 0 2 2 0, 0, 3 0

* 2 2 * 2 2 *3 , , 4 4 4 1 5 2 0 5 0

1 12

2 2 21 1 1 1 1

2 4 4 6 6

dq q

q q q q q

++ + - + + = + + + +

+ + + - - - -

(8-f)

( ): x xx xy xy y yy z z zyx

z z z

u vhw M M M M q N I w I I I w

x y

J I I I c I c w I wx y

2* * * * *0 00 , , , 2 2 0 3 3 4 0

* 2 2 * 2 2 *4 5 5 5 1 6 2 0 6 0

1 12 2

3 4 3

1 1 1 1 1

3 6 6 9 9

d q

qqq q q

+æ ö¶ ¶ ÷ç ÷ç+ + - + + = + + + +÷ç ÷÷ç ¶ ¶è ø

æ ö¶¶ ÷ç ÷ç+ + + - - - - ÷ç ÷ç ÷¶ ¶ ÷çè ø

(8-g)

( ) ( ) ( )( ) ( )

:z x xx xy xy y yy z x x y y x x y y z

x y x x y y z z z

z

c hN N N N S S c S S q N

I c u v c J I w I I c I c

I w I c w I w I

3* * * * * * *1

, , , , , 1 , , 3

2 2 2 *2 1 0, 0, 1 3 , , 3 0 4 4 1 4 1

2 2 * * *5 0 5 1 0 5 0 6

2 32 8

1 1 1 1

2 2 4 41 1

6 6

dq

q q q q q

q

++ + - + + - + + + =

+ + + + + - - -

- - + +

(8-h)



where, x y2 2 2 2 2/ / = ¶ ¶ + ¶ ¶ is Laplacian operator in two-dimensional Cartesian coordinate sys-tem, and the stress resultants are defined by:

{ }x x x xx

h h xy y yy y

yz z z zz h h

xyxy xy xy xyxy

N MNM

MN Nz dz zdz M

N MNM

N MN

**

/2 /2*2 *

**/2 /2

*

1 ; ;

s ss ss ss s

- -

é ù ì ü ì ü ì üï ï ï ï ï ïê ú ìï ï ï ï ï ï ïï ï ï ï ï ïê ú ï ï ï ï ï ïï ï ï ï ï ïê ú ï ï ï ï ï ï= =ê ú í ý í ý í ý íï ï ï ï ï ïê ú ï ï ï ï ï ïê ú ï ï ï ï ï ïï ï ï ï ï ïê ú ï ï ï ï ï ïï ï ï ï ï ïî þ î þ î þê úë û

ò ò

{ } { }

h x

yh

xy

h hx x xz x x xz

yz yzy y y yh h

z dz

Q Q S Sz dz z z dz

Q Q S S

/23

/2

/2 /2* *2 3

* */2 /2

;

1 ; .

sss

s ss s

-

- -

üï ì üï ïï ï ï ïï ï ï ïï ïï ï ï ï=ý í ýï ï ï ïï ï ï ïï ï ï ïï ï ï ïî þï ïî þ

é ù é ùì ü ì üï ï ï ïê ú ê úï ï ï ïï ï ï ï= =í ý í ýê ú ê úï ï ï ïê ú ê úï ï ï ïï ï ï ïî þ î þë û ë û

ò

ò ò

(9)

and ( )i iI J K2, , are mass inertias defined as:

{ } ( ){ }h

h

I I I I I I z z z z z z z dz/2

2 3 4 5 61 2 3 4 5 6

/2

, , , , , , , , , ,r-

= ò

i i iJ I I c i K I I c I c2

2 2 2 2 4 2 6 2

1 2 1, 1,3,4;

3 3 9+= - = = - +

(10)

i i i i i i iz z zi x x x x y y y

i i i i iz z zy xy x xy xy

w w w wN N N N N N N N

x x x x y y yw w

N N N N N ix y x y x y x yy

2 2 2 * 2 * 2 2 2 *1 2 3 1 20 0 0 0

2 2 2 2 2 2 2

2 * 2 2 2 * 2 *3 1 2 30 0

22 2 2 2 ; 0,1,2,3.

q q

q

+ + + + +

+ + + +

¶ ¶ ¶ ¶ ¶ ¶ ¶= + + + + + +

¶ ¶ ¶ ¶ ¶ ¶ ¶¶ ¶ ¶ ¶ ¶

+ + + + + =¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶¶

(11-a)

x y xy

x y xy

x y xy

x y xy

x y xy

x y xy

x y xy

N N NzN N NzN N N

zN N Nh

zN N N

zN N N

zN N N

0 0 0

1 1 1

22 2 2

33 3 3

44 4 4

55 5 5

66 6 6

1

1

ì ü ì üï ï ï ïï ï ï ïï ï ï ïï ï ï ïï ï ï ïï ï ï ïï ï ï ïï ï ï ïï ï ï ïï ï ï ïï ï ï ïï ï ï=í ý í ýï ï ïï ï ïï ï ïï ï ïï ï ïï ï ïï ï ïï ï ïï ï ïï ï ïï ï ïï ï ïîî þ

{ }h

x y xyh

N N N dz/2

0 0 0

/2

;-

ïïïïïïïïïïïïïþ

ò (11-b)

5 NAVIER’S SOLUTION

Consider a simply supported rectangular FG plate with length a, width b under in-plane loads in two directions ( x cr y cr xyN N N N N

0 0 01 2, , 0g g= = = ). The associated simply supported boundary conditions

are as follows:

At edge x = 0 and x = a: y z z x xv w w M M

* * *0 0 00, 0, 0, 0, 0, 0, 0, 0.q q q= = = = = = = =

At edge y = 0 and y = b: x z z y yu w w M M* * *

0 0 00, 0, 0, 0, 0, 0, 0, 0.q q q= = = = = = = =



Following Navier’s solution procedure, the displacement variables are chosen to satisfy the above simply supported boundary condition with the form(for the buckling and vibration problems, the transverse load is set to be zero):

( ) ( )

( ) ( )

( )

i t i tmn mn

m n m n

i t i tmn x xmn

m n m n

i ty ymn z

m n

u x y t u e x y v x y t v e x y

w x y t w e x y t e x y

x y t e x y x y

0 0 0 01 1 1 1

0 01 1 1 1

1 1

, , cos sin ; , , sin cos ,

, , sin x sin y; , , cos sin ,

, , sin cos ; , ,

w w

w w

w

a b a b

a b q q a b

q q a b q

¥ ¥ ¥ ¥

= = = =¥ ¥ ¥ ¥

= = = =¥ ¥

= =

= =

= =

=

åå åå

åå åå

åå ( )

( ) ( )

i tzmn

m n

i t i tmn z zmn

m n m n

z

t e x y

w x y t w e x y x y t e x y

1 1

* * * *0 0

1 1 1 1

sin sin ,

, , sin sin ; , , sin sin .

p 0

w

w w

q a b

a b q q a b

¥ ¥

= =¥ ¥ ¥ ¥

= = = =+

=

= =

=

åå

åå åå

(12)

where: i 1= - is the imaginary unit. mn mn mn xmn ymn zmn mn zmnu v w w* *

0 0 0 0, , , , , , ,q q q q are coefficients, and

ω is the natural frequency; m n, 1,3,5,7,...= . Substituting Eq. (12) into Eq. (8a-h), the closed-form solutions can be obtained from:

s s s s s s s s

s s s s s s s s

s s s k s s s s k s

s s s s s s s s

s s s s s s s s

s s s s s s k s s k

s s s k s s s s k s

s s s

11 12 13 14 15 16 17 18

21 22 23 24 25 26 27 28

31 32 33 33 34 35 36 37 37 38

41 42 43 44 45 46 47 48

51 52 53 54 55 56 57 58

61 62 63 64 65 66 66 67 68 68

71 72 73 73 74 75 76 77 77 78

81 82 8

+ +

+ ++ +

x

y

z

z

u

v

w

M

w

s s s k s s k

0

0

0

2

8 8

0

3 84 85 86 86 78 88 88

qw q

q

q

´

*

*

ì üæ öé ù ï ï÷ç ï ïê ú ÷ç ï ï÷ç ï ïê ú ÷ç ï ï÷çê ú ÷ï ïç ÷ï ïê ú ÷ç ï ï÷ç ï ïê ú ÷ç ï ï÷çê ú ï ï÷ç ÷ï ïê úç ÷ï ïé ùç ÷-ê ú í ý÷ç ê úë û ÷ç ï ïê ú ÷ç ï ï÷çê ú ï÷ç ï÷ê úç ÷ïç ÷ïê ú ÷ç ï÷çê ú ï÷ç ï÷çê ú ï÷ç ÷ïê úç + + ÷ï÷çè øê ú ïë û î þ

0=ïïïïïïïïïïï

(13-a)

where the elements of matrix [S], [M] are defined in the Appendix, and:

( ) ( )( ) ( )

cr cr

cr cr

hk N k k k N

h hk k k N k N

22 2 2 2

33 1 2 37 73 66 1 24 6

2 2 2 268 86 77 1 2 88 1 2

; ;12

; .80 448

g a g b g a g b

g a g b g a g b

= + = = = +

= = = + = + (13-b)

The system of Eq. (13a) maybe used to obtain the solutions of the buckling problem soft he FG plates by dropping all the inertia terms ( 0w = ), and the solutions of the free vibration problems of the plates by removing in-plane loads ( crN 0= ).



6 RESULTS AND DISCUSSION

With self-developed Matlab’s code, various numerical examples are presented and discussed for veri-fying the accuracy and efficiency of the present theory in predicting the buckling and vibration re-sponses of simply supported FG plates. The considered FG plate composesof aluminum (as metal) and alumina (as ceramic) with the following material properties:

Al:Em = 70 GPa, νm = 0.3, ρm = 2702 kg/m3; Al2O3: Ec = 380 GPa, νc= 0.3, ρc= 3800 kg/m3.

For convenience, the following non-dimensional forms are used:

( )c cr c

crcc

E h N aD N h

ED

3 2

22ˆ; ; ˆ .

12 1

rw w

pn= = =

-

In order to emphasize the efficiency of present eight-unknown HSDT, the calculated results are compared with other shear deformation theories. The following models of shear deformation theories are used in this section:

HSDT-12:

xxu u z z u z2 * *

03

0= + + + ;

yyv v z z v z2 * *

03

0= + + + ;

zzw w z z w z2 * *

03

0= + + + ;

HSDT-5:

xx

wzu u z

xh0

2

3

04

3

æ ö¶ ÷ç ÷ç= + - + ÷ç ÷÷ç ¶è ø ;

yy

wzv v z

yh0

2

3

04

3

æ ö¶ ÷ç ÷ç= + - + ÷ç ÷÷ç ¶è ø ;

w w0= .

HSDT-4: b sw wz fx x

u u0¶ ¶

- -¶ ¶

= ;

b sw wz fy y

v v0¶ ¶

- -¶ ¶

= ;

b sw w= + w .

where h zf zh

.p

p

æ ö÷ç ÷= - ç ÷ç ÷çè øsin

6.1 Buckling Analysis

Example 1. Functionally gradedAl/Al2O3 square (b/a=1) and rectangular (b/a=2) plates subjected to biaxial compression (γ1 = -1, γ2 = -1) are considered. Table 1 gives some numerical results showing the accuracy of the present non-dimensional buckling loads with various values of side-to-thickness ratio. The obtained results based on proposed HSDT are compared with the results of Thai and Choi (2012) whichwere based on an efficient and simple refined theory. The theory, which Thai and Choi used is similar with the classical plate theory in many aspects; it accounts for a quadratic variation of the transverse shear strains across the thickness and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate. A good agreement between the results is observed.



b/a a/h Method p

0 0.5 1 2 5 10 20 100

1

5 Thai

(2012) 8.0105 5.3127 4.1122 3.1716 2.5265 2.2403 2.0035 1.6293

Present 8.0826 5.3716 4.1643 3.2132 2.5549 2.2621 2.0205 1.6426

10 Thai

(2012) 9.2893 6.0615 4.6696 3.6315 3.0177 2.7264 2.4173 1.9099

Present 9.3139 6.0810 4.6867 3.6455 3.0280 2.7346 2.4236 1.9146

20 Thai

(2012) 9.6764 6.2834 4.8337 3.7686 3.1724 2.8834 2.5494 1.9961

Present 9.6831 6.2887 4.8384 3.7723 3.1753 2.8857 2.5512 1.9974

50 Thai

(2012) 9.7907 6.3485 4.8818 3.8088 3.2186 2.9307 2.5891 2.0217

Present 9.7918 6.3494 4.8826 3.8095 3.2191 2.9311 2.5894 2.0219

100 Thai

(2012) 9.8073 6.3579 4.8888 3.8147 3.2254 2.9376 2.5948 2.0254

Present 9.8075 6.3581 4.8890 3.8148 3.2255 2.9377 2.5949 2.0255

2

5 Thai

(2012) 5.3762 3.5388 2.7331 2.1161 1.7187 1.5370 1.3692 1.0990

Present 5.4090 3.5652 2.7563 2.1348 1.7320 1.5474 1.3772 1.1051

10 Thai

(2012) 5.9243 3.8565 2.9689 2.3117 1.9332 1.7517 1.5510 1.2200

Present 5.9343 3.8644 2.9758 2.3174 1.9374 1.7551 1.5536 1.2219

20 Thai

(2012) 6.0794 3.8565 3.0344 2.3665 1.9955 1.8152 1.6044 1.2547

Present 6.0821 3.9473 3.0363 2.3680 1.9967 1.8161 1.6051 1.2552

50 Thai

(2012) 6.1244 3.9708 3.0533 2.3823 2.0137 1.8338 1.6200 1.2647

Present 6.1248 3.9711 3.0536 2.3826 2.0139 1.8340 1.6201 1.2648

100 Thai

(2012) 6.1308 3.9744 3.0560 2.3846 2.0164 1.8365 1.6222 1.2662

Present 6.1309 3.9745 3.0561 2.3847 2.0164 1.8366 1.6223 1.2662

Table 1: Non-dimensional critical buckling load crN of simply supported Al/Al2O3 plate subjected to biaxial compression (γ1 = -1, γ2 = -1).

Example 2. In this example, a moderately thick (a/h = 10) rectangular (b/a = 2) FG plate with different values of power law index p is examined. Table 2 contains the non-dimensional buckling loads calculated by present and various shear deformation theories: first-order shear deformation theory with 5 unknowns (FSDT), third-order shear deformation theory with 5 unknowns (HSDT-5), simple higher-order shear deformation theory with 4 unknowns (HSDT-4), and full higher-order shear deformation theory with 12 unknowns (HSDT-12). Fig. 2 exhibits a variation of non-dimensional buckling loads crN̂ versus power law index p of rectangular FG plates (b/a = 2, a/h = 10)with various types of loading. It is observed that the non-dimensional critical buckling load decreases as p increases,



the variation of the non-dimensional buckling load is considerable when p is small, and the fully-ceramic plate gives the largest critical buckling load. An excellent agreement between the results predicted by present HSDT and full HSDT with 12 displacement unknowns for all values of power law indexis shown.

p Method Loading type (γ1, γ2)

(-1, 0) (0, -1) (-1, -1)

0

FSDT 1.5093 6.0372 1.2074 HSDT-4 1.5094 6.0376 1.2075 HSDT-5 1.5093 6.0373 1.2075 HSDT-12 1.5119 6.0475 1.2095

Present 1.5119 6.0475 1.2095

1


Present 0.7582 3.0326 0.6065

2


Present 0.5904 2.3616 0.4723 3 FSDT 0.5359 2.1436 0.4287 HSDT-4 0.5337 2.1350 0.4270 HSDT-5 0.5338 2.1353 0.4271 HSDT-12 0.5348 2.1390 0.4278 Present 0.5351 2.1404 0.4281

5


Present 0.4936 1.9744 0.3949

10


Present 0.4471 1.7886 0.3577

Table 2: Comparison of non-dimensional critical buckling load crN̂ of plates under different loading types

with different values of power-law index p (b/a = 2, a/h = 10).



Figure 2: The variation of non-dimensional critical buckling load

crN̂ of rectangular plate versus power law

index p (b/a = 2, a/h = 10) with various shear deformation theories.

Example 3. Thick and thin rectangular FG plates (p = 5)with side-to-thickness ratio varies from 5 to 100 are analyzed using present HSDT and various shear deformation theories. The non-dimensional buckling loads

crN̂ under uniaxial and bi-axial compression are presented in Table 3. Fig. 3 shows a

variation of non-dimensional buckling loads crN̂ with respect to side-to-thickness ratio a/hof rectan-gular FG plates (b/a = 2, p = 5). It can be seen that the non-dimensional buckling load increases by the increase of thickness ratioa/h, and the variation of the non-dimensional buckling loadbecomes significant for thick plate.The difference in the results obtained using proposed HSDT and the rest of HSDT increases with a decreases in the value of the side-to-thickness ratioa/h.It is emphasized that the proposed HSDT model contains a fewer number of unknowns than those associated with the full HSDT theory. However, an excellent agreement between the results predicted by present HSDT and full HSDT with 12 displacement unknowns also can be observed, and FSDT overestimates the buck-ling loads of FG thick plate as it neglects the thickness stretching effect.

Figure 3: The variation of non-dimensional critical buckling load N̂ of rectangular plate versus

thickness ratio a/h (b/a = 2, p = 5) with various shear deformation theories.



a/h Method Loading type (γ1, γ2)

(-1, 0) (0, -1) (-1, -1)

5


Present 0.4413 1.7650 0.3530

10


Present 0.4936 1.9744 0.3949

20


Present 0.5087 2.0348 0.4070 30 FSDT 0.5119 2.0475 0.4095 HSDT-4 0.5114 2.0458 0.4092 HSDT-5 0.5115 2.0458 0.4092 HSDT-12 0.5116 2.0462 0.4092 Present 0.5116 2.0464 0.4093

50


Present 0.5131 2.0524 0.4105

100


Present 0.5137 2.0549 0.4110

Table 3: Comparison of non-dimensional critical buckling load N̂ of plates under different loading types with various values of side-to-thickness ratio a/h (b/a = 2, p = 5).

6.2 Free Vibration Analysis

Example 4. The next verification is performed for moderately thick and thick FG square plates. Different values of power law index are considered. The non-dimensional fundamental frequencies are given in Table 4. Obtained results are compared with solutions using first-order and higher-order shear deformation theories provided by Hosseini-Hashemi (2011), and sinusoidal shear deformation theory provided by Thai (2013). It can be seen that the difference between the results is very small.



a/h Method Power law index (p)

0 0.5 1 4 10

5

FSDT (Hosseini-2011) 0.2112 0.1805 0.1631 0.1397 0.1324 HSDT (Hosseini-2011) 0.2113 0.1805 0.1631 0.1398 0.1301 HSDT-4 (Thai-2013) 0.2113 0.1807 0.1631 0.1377 0.1300

Present 0.2122 0.1816 0.1640 0.1386 0.1307

10

FSDT (Hosseini-2011) 0.0577 0.0490 0.0442 0.0382 0.0366 HSDT (Hosseini-2011) 0.0577 0.0490 0.0442 0.0381 0.0364 HSDT-4 (Thai-2013) 0.0577 0.0490 0.0442 0.0381 0.0364

Present 0.0578 0.0491 0.0443 0.0381 0.0364

Table 4: Comparison of non-dimensional fundamental frequency ŵ of square plate. Example 5. Non-dimensional frequencies ŵ of moderately thick rectangular FG plates (b/a = 2, a/h = 10) for different values of power law index p and various modes of vibration are presented in Table 5. Figure 4 illustrates the variation of non-dimensional fundamental frequency (m=n=1) with respect to power law index p. As can be seen from the presented results, the non-dimensional natural frequencydecreases with increasing value of power law index p. It is basically due to the fact that Young’s modulus of ceramic is higher than metal.For the same value of p, the non-dimensional natural frequency increases for higher modes. Figure 4 also shows that the non-dimensional natural-frequency decreases significantly when p is small.

Table 6 shows non-dimensional frequencies ŵ of thin to thick rectangular FG plates (b/a = 2, p = 5) for different values of side-to-thickness ratio a/hand various modes of vibration. Figure 5 depict the variation of non-dimensional fundamental frequency (m=n=1) with respect to side-thickness-ratio a/h.

Similarly it is observed that the non-dimensional frequency decreases as the side-to-thickness ratio decreases. The fall in non-dimensional frequency is observed up to around a/h =20, beyond this no changes in non-dimensional frequency are distinguished.

Figure 4: Comparison of the variation of

non-dimensional fundamental frequency ŵ of square plate versus power law index p.

Figure 5: Comparison of the variation of non-dimensional fundamental frequency ŵ of square plate versus thickness ratio a/h.



p Method Mode (m,n)

1 (1, 1) 2 (1, 2) 3 (2, 1) 3 (2,2)

0

FSDT 0.0365 0.0577 0.1183 0.1376 HSDT-4 0.0365 0.0577 0.1183 0.1377 HSDT-5 0.0365 0.0577 0.1183 0.1376 HSDT-12 0.0365 0.0578 0.1186 0.1381

Present 0.0365 0.0578 0.1186 0.1381

1


Present 0.0280 0.0443 0.0912 0.1063

2


Present 0.0254 0.0402 0.0826 0.0962 3 FSDT 0.0246 0.0388 0.0796 0.0927 HSDT-4 0.0245 0.0387 0.0792 0.0921 HSDT-5 0.0245 0.0387 0.0792 0.0921 HSDT-12 0.0245 0.0387 0.0794 0.0923 Present 0.0245 0.0388 0.0794 0.0924

5


Present 0.0239 0.0377 0.0770 0.0894

10


Present 0.0231 0.0364 0.0740 0.0859

Table 5: Non-dimensional frequency ŵ of plates with different values of power-law index p (b/a = 2, a/h = 10).

All above obtained results are studied using different plate theories. From table 5 and 6, it is apparent that the present proposed HSDT and full HSDT with 12 displacement unknowns give almost identical results for all values of power law index p and side-to-thickness ratio a/h. This emphasizes again, the benefits of the proposed HSDT in comparison with the full HSDT, as the proposed HSDT uses fewer displacement unknowns but requires less computational effort.



a/h Method Mode (m,n)

1 (1, 1) 2 (1, 2) 3 (2, 1) 3 (2,2)

5


Present 0.0894 0.1365 0.2586 0.2946

10


Present 0.0239 0.0377 0.0770 0.0894

20


Present 0.0061 0.0097 0.0204 0.0239 30 FSDT 0.0027 0.0043 0.0092 0.0108 HSDT-4 0.0027 0.0043 0.0092 0.0108 HSDT-5 0.0027 0.0043 0.0092 0.0108 HSDT-12 0.0027 0.0043 0.0092 0.0108 Present 0.0027 0.0043 0.0092 0.0108

50


Present 0.0010 0.0016 0.0033 0.0039

100


Present 0.0002 0.0004 0.0008 0.0010

Table 6: Non-dimensional frequency ŵ of plates with different values of side-to-thickness a/h (b/a = 2, p = 5). 7 CONCLUSIONS

The new eight-unknown HSDT is proposed based on full twelve-unknown HSDT and satisfies van-ishing transverse stresses at the top and bottom surface of FG plates. The accuracy of numerical solutions has been validated against existing results in available literatures.The effects of the side-to-thickness ratio and the power law index of constituent volume fraction on the buckling loads and on the natural frequencies are also discussed. The results show that the buckling loads increase, and the natural frequencies decrease significantly with increasing power law index. It can be observed by the presented results that the gradation of the constitutive components is an important parameter



for buckling and free vibration analysis of FG plates.The present formulation for FG plates involves less computation compared to full twelve-unknown higher-order shear deformation theory,while gives identical results as full twelve-unknown higher-order shear deformation theory.The numerical results of critical buckling loads and natural frequencies should serve as a reference for any other analyti-cal/computational model of FG plates. Acknowledgements

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number: 107.02-2013.25 .

References

Bodaghi, M., Saidi, A, R. (2010). Levy-type solution for buckling analysis of thick functionally graded rectangular plates based on the higher-order shear deformation plate theory. Applied Mathematical Modelling 34, 3659–3673. Hosseini-Hashemi, S., Fadaee, M., Atashipour, S.R. (2011). Study on the free vibration of thick functionally graded rectangular plates according to a new exact closed-form procedure. Composite Structures 93(2), 722–735. Hosseini-Hashemi, S., Fadaee, M., Atashipour, S.R. (2011). A new exact analytical approach for free vibration of Reissner-Mindlin functionally graded rectangular plates. International Journal of Mechanical Science. 53(1), 11–22. Jha, D.K., Kant, T., Singh, R.K. (2012). Higher order shear and normal deformation theory for natural frequency of functionally graded rectangular plates. Nuclear Engineering and Design. 250, 8–13. Kim, J., Reddy, J.N. (2013). Analytical solutions for bending, vibration, and buckling of FGM plates using a couple stress-based third-order theory. Composite Structures 103(2013) 86–98 Pandya, B.N., Kant, T. (1988). Finite element stress analysis of laminated composites using higher order displacement model. Composites Science and Technology. 32, 137-155. Reddy, J.N. (2000). Analysis of functionally graded plates. International Journalfor Numerical Methods in Engineering 47(1-3),663-84. Shufrin, I., Eisenberger, M. (2005). Stability and vibration of shear deformable plates-first order and higher order analyses. International Journal of Solids and Structures. 42(3–4), 1225-1251. Swaminathan, K., Naveenkumar, D.T., Zenkour, A.M., Carrera, E. (2015). Stress, vibration and buckling analyses of FGM plates - A state-of-the-art review. Composite Structures. 120, 10–31. Talha, M., Singh, B.N. (2010). Static response and free vibration analysis of FGM plates using higher order shear deformation theory. Applied Mathematical Modelling. 34, 3991–4011. Thai, H. T., Choi, D. H. (2012). An efficient and simple refined theory for buckling analysis of functionally graded plates. Applied Mathematical Modelling 36, 1008–1022 Thai, H.T., Vo, T. P. (2013). A new sinusoidal shear deformation theory for bending, buckling, and vibration of functionally graded plates. Applied Mathematical Modelling. 37, 3269–3281. Zenkour, A. M. (2006). Generalized shear deformation theory for bending analysis of functionally graded plates. Applied Mathematical Modelling. 30, 67–84.



Appendix A. Elements of [D1], [D2], [D3], [D4] matrices.

A A A B B B C C C D D



B B B C C C D D D E E

B B B C C C D D D E E

B B B C C C D D DD

11 12 13 11 12 13 11 12 13 11 12

21 22 23 21 22 23 21 22 23 21 22

31 32 33 31 32 33 31 32 33 31 32

11 12 13 11 12 13 11 12 13 11 12

21 22 23 21 22 23 21 22 23 21 22

31 32 33 31 32 33 31 32 31é ù =ê úë û E E

C C C D D D E E E F F



D D D E E E F F F G G

D D D E E E F F F G G

3 31 32

11 12 13 11 12 13 11 12 13 11 12

21 22 23 21 22 23 21 22 23 21 22

31 32 33 31 32 33 31 32 33 31 32

11 12 13 11 12 13 11 12 13 11 12

21 22 23 21 22 23 21 22 23 21 22

é ùê úê úêêêêêêêêêêêêêêêêêêêêë û

;

úúúúúúúúúúúúúúúúúúúú

( ) ( )h

ij ij ij ij ij ij ij ijh

A B C D E F G Q z z z z z z dz/2

2 3 4 5 6

/2

, , , , , , 1, , , , , , .-

= ò

A B C D

B C D ED

C D E F

D E F G

44 44 44 44

44 44 44 442

44 44 44 44

44 44 44 44

;

é ùê úê úê úé ù = ê úê úë û ê úê úê úë û

A B C D

B B D ED

C D E F

D E F G

55 55 55 55

55 55 55 553

55 55 55 55

55 55 55 55

;


A B C D

B C D ED

C D E F

D E F G

66 66 66 66

66 66 66 664

66 66 66 66

66 66 66 66

;


Appendix B. The global linear stiffness matrix [K], global mass matrix [M] and global displacement [Q].

s s s s s s s s

s s s s s s s s

s s s k s s s s k s

s s s s s s s s

s s s s s s s s

s s s s s s k s s k

s s s k s s s s k s

s s

11 12 13 14 15 16 17 18

21 22 23 24 25 26 27 28

31 32 33 33 34 35 36 37 37 38

41 42 43 44 45 46 47 48

51 52 53 54 55 56 57 58

61 62 63 64 65 66 66 67 68 68

71 72 73 73 74 75 76 77 77 78

81 82

+ +

=

+ ++ +

K

s s s s k s s k83 84 85 86 86 78 88 88

;

é ùê úê úê úê úê úê úê úê úê úê úê úê úê úê úê ú+ +ê úë û

;

m m m m m m

m m m m m m

m m m m m m m m

m m m m m m

m m m m m m

m m m m m m m m

m m m m m m m m

m m m m m m m m

11 13 14 16 17 18

22 23 25 26 27 28

31 32 33 34 35 36 37 38

41 43 44 46 47 48

52 53 55 56 57 58

61 62 63 64 65 66 67 68

71 72 73 74 75 76 77 78

81 82 83 84 85 86 78 88

0 0

0 0

0 0

0 0

é ùê úê úêêêêê

= êêêêêêêêêë û

M

úúúúúúúúúúúúúú

{ }Tmn mn mn xmn ymn zmn mn zmnu v w w* *0 0 0 0 .q q q q=Q



Coefficients of matrix [S].

( ) D c D c D cs A A s s A A s s2 2 3 211 2 12 2 44 211 11 44 12 21 12 44 13 312

; ; ;3 3 3

a b ab a abæ ö÷ç ÷ç= + = = + = =- - + ÷ç ÷÷çè ø

D c D cs s B B2 211 2 44 214 41 11 44 ;3 3

a bæ ö æ ö÷ ÷ç ç÷ ÷ç ç= = - + -÷ ÷ç ç÷ ÷÷ ÷ç çè ø è ø

D c D cs s B B 12 2 44 215 51 12 44 ;3 3

abæ ö÷ç ÷ç= = + - - ÷ç ÷÷çè ø

C C D D Ds s A C s s B3 2 3 211 12 11 12 4416 61 13 44 17 71 13

2; 2 ;

2 2 3 3 3a a ab a a ab

æ ö æ ö÷ ÷ç ç÷ ÷ç ç= =- - - + = =- - - +÷ ÷ç ç÷ ÷÷ ÷ç çè ø è ø

C c C cs s C C c3 211 1 12 118 81 13 44 13 ;2 2

a a abæ ö÷ç ÷ç= =- - - + ÷ç ÷÷çè ø

s A A2 222 44 22 ;a b= +

D c D c D cs s 2 321 2 44 2 22 223 32

2;

3 3 3a b b

æ ö÷ç ÷ç= =- + -÷ç ÷÷çè ø

D c D cs s B B 21 2 44 224 42 21 44 ;3 3

abæ ö÷ç ÷ç= = + - - ÷ç ÷÷çè ø

D c D c C Cs s B B s s C A2 2 2 344 2 22 2 21 2225 52 44 22 26 62 44 23; ;3 3 2 2

a b a b b bæ ö æ ö æ ö÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç= = - + - = =- + - +÷ ÷ ÷ç ç ç÷ ÷ ÷÷ ÷ ÷ç ç çè ø è ø è ø

D D D C c C cs s B s s C c C2 3 2 321 44 22 21 1 22 127 72 23 28 82 44 1 23

22 ; 3 ;

3 3 3 2 2a b b b a b b b

æ ö æ ö÷ ÷ç ç÷ ÷ç ç= =- + - - = =- + - -÷ ÷ç ç÷ ÷÷ ÷ç çè ø è ø

( ) ( )

( )

G G G cG cs A E c C c

G cA E c C c

2212 21 44 24 2 2 2 211 2

33 55 55 2 55 2

22 2 422 2

66 66 2 66 2

42

9 9

2 ;9

a a a b

b b

+ += + + - + +

+ + - +

( )G c E c E c E c G c G cs s A C c E c2 2 2

3 2 211 2 11 2 12 2 44 2 12 2 44 234 43 55 55 2 55 2

2 22 ;

9 3 3 3 9 9a a ab

æ ö æ ö÷ ÷ç ç÷ ÷ç ç= = - + - + - + - -÷ ÷ç ç÷ ÷ç ç÷ ÷÷ ÷ç çè ø è ø

( )E c E c G c G c G c E cs s A C c E c2 2 2

2 2 312 2 44 2 12 2 44 2 22 2 22 235 53 66 66 2 66 2

2 22 ;

3 3 9 9 9 3a b b b

æ ö æ ö÷ ÷ç ç÷ ÷ç ç= =- + - - + - + + -÷ ÷ç ç÷ ÷ç ç÷ ÷÷ ÷ç çè ø è ø

F c D c F c F c F c D c F cs s 4 2 2 2 2 411 2 13 2 12 2 21 2 44 2 23 2 22 236 63

2;

6 3 6 6 3 3 6a a a b b b

æ ö÷ç ÷ç= = + + + - + +÷ç ÷÷çè ø

G c E c G c G c G c E c G cs s 4 2 2 2 2 411 2 13 2 12 2 21 2 44 2 23 2 22 237 73

2 4 2;

9 3 9 9 9 3 9a a a b b b

æ ö÷ç ÷ç= = + + + + + +÷ç ÷÷çè ø

( )

( )

F c c F c c F c c F c cs s D B c F c F c D c c

F c cD B c F c F c D c c

4 2 2 211 1 2 12 1 2 21 1 2 44 1 238 83 55 55 1 13 2 55 2 55 1 2

2 422 1 266 66 1 23 2 66 2 66 1 2

2

6 6 6 3

;6

a a a b

b b

æ ö÷ç ÷ç= = + - + - + + + + ÷ç ÷÷çè ø

+ - + - + +

G c E c G c E cs C C E c C c A

2 22 2 211 2 11 2 44 2 44 2

44 11 44 55 2 55 2 55

2 22 ;

9 3 9 3a b

æ ö æ ö÷ ÷ç ç÷ ÷ç ç= - + + - + + - +÷ ÷ç ç÷ ÷ç ç÷ ÷÷ ÷ç çè ø è ø



D F c D c D F c F cs s B D3 211 11 2 13 2 12 12 2 44 246 64 13 44 ;2 6 3 2 6 3

a a abæ ö æ ö æ ö÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç= =- - + - - + - -÷ ÷ ÷ç ç ç÷ ÷ ÷÷ ÷ ÷ç ç çè ø è ø è ø

E G c E c E E G c G cs s C3 211 11 2 13 2 12 44 12 2 44 247 74 13

2 2 22 ;

3 9 3 3 3 9 9a a ab

æ ö æ ö æ ö÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç= =- - + - - + - -÷ ÷ ÷ç ç ç÷ ÷ ÷÷ ÷ ÷ç ç çè ø è ø è ø

( )D c F c cs s D D B c F c F c D c c

D c F c c F c cD c

311 1 11 1 248 84 55 13 55 1 13 2 55 2 55 1 2

212 1 12 1 2 44 1 244 1

32 6

;2 6 3

a a

ab

æ ö÷ç ÷ç= =- - + - - + - +÷ç ÷÷çè øæ ö÷ç ÷ç- + - - ÷ç ÷÷çè ø

G c E c G c E cs C C E c C c A

2 22 2 244 2 44 2 22 2 22 2

55 44 22 66 2 66 2 66

2 22 ;

9 3 9 3a b

æ ö æ ö÷ ÷ç ç÷ ÷ç ç= - + + - + + - +÷ ÷ç ç÷ ÷ç ç÷ ÷÷ ÷ç çè ø è ø

D F c F c D c D F cs s D B2 312 12 2 44 2 32 2 22 22 256 65 44 32 ;2 6 3 3 2 6

a b b bæ ö æ ö æ ö÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç= =- + - - + - - -÷ ÷ ÷ç ç ç÷ ÷ ÷÷ ÷ ÷ç ç çè ø è ø è ø

E E G c G c E c E G cs s C2 321 44 21 2 44 2 23 2 22 22 257 75 23

2 2 22 ;

3 3 9 9 3 3 9a b b b

æ ö æ ö æ ö÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç= =- + - - + - - -÷ ÷ ÷ç ç ç÷ ÷ ÷÷ ÷ ÷ç ç çè ø è ø è ø

( )D c F c c F c cs s D c D D B c F c F c D c c

D c F c c

221 1 21 1 2 44 1 258 85 44 1 66 23 66 1 23 2 66 2 66 1 2

322 1 22 1 2

32 6 3

;2 6

a b b

b

æ ö÷ç ÷ç= =- + - - + - - + - +÷ç ÷÷çè øæ ö÷ç ÷ç- - ÷ç ÷÷çè ø

E C C E E C C Es E A4 2 2 2 2 411 13 31 12 21 23 32 2266 44 33;4 2 2 4 4 2 2 4

a a a b b bæ ö æ ö æ ö÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç= + + + + + + + + +÷ ÷ ÷ç ç ç÷ ÷ ÷÷ ÷ ÷ç ç çè ø è ø è ø

( ) ( )F F F F Fs s D D D D B4 2 2 2 2 411 12 21 44 2267 76 13 31 23 32 332

2 ;6 6 6 3 6

a a a b b bæ ö÷ç ÷ç= = + + + + + + + + +÷ç ÷÷çè ø

E c E C c E c E c E C cs s E c

E cC

4 2 2 2 211 1 13 31 1 12 1 21 1 23 32 168 86 44 1

422 133

3 3

4 2 2 4 4 2 2

3 ;4

a a a b b

b

æ ö æ ö æ ö÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç= = + + + + + + +÷ ÷ ÷ç ç ç÷ ÷ ÷÷ ÷ ÷ç ç çè ø è ø è ø

+ +

G E E G G G E E Gs C4 2 2 2 2 411 13 31 12 21 44 23 32 2277 33

2 2 4 2 24 ;

9 3 3 9 9 9 3 3 9a a a b b b

æ ö æ ö æ ö÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç= + + + + + + + + +÷ ÷ ÷ç ç ç÷ ÷ ÷÷ ÷ ÷ç ç çè ø è ø è ø



( ) ( )F c F c F c F c F cs s F D c F D cD

4 2 2 2 2 411 1 12 1 21 1 44 1 22 178 87 31 13 1 32 23 1

33

2

6 6 6 3 66 ;

a a a b b bæ ö÷ç ÷ç= = + + + + + + + +÷ç ÷÷çè ø

+

E c E c E c E c E cs G C c E c E c

E c E c E cG C c E c E

2 2 24 2 2 2 2 211 1 13 1 31 1 12 1 21 1

88 55 55 1 55 1 44 1

22 2 423 1 32 1 22 1

66 66 1 66 1 33

3 32

4 2 2 4 4

3 32 9 ;

2 2 4

a a a b

b b

æ öæ ö ÷ç÷ç ÷÷ çç= + + + + - + + + ÷÷ çç ÷÷ ç÷ç ÷÷çè ø è øæ ö÷ç ÷ç+ + + + - + +÷ç ÷÷çè ø

( ) ( ) ( )cr cr crh hk N k k k N k k k N2 4

2 2 2 2 2 233 1 2 37 73 66 1 2 68 86 77 1 2; ; ;12 80

g a g b g a g b g a g b= + = = = + = = = +

( )cr hk N6

2 288 1 2 .448

g a g b= +

Coefficients of matrix [M].

cm m I m m I m m J m m I211 22 0 13 31 3 14 41 1 16 61 2; ; ; ;3 2

a a= = = =- = = = =-

cm m I m m I117 71 3 18 81 2; ;3 2

aa= =- = =-

cm m I m m J223 32 3 25 52 1; ;3

b= =- = =

( )ccm m I m m I m m I m I I

2 2 221

26 62 2 27 72 3 28 82 2 33 0 6; ; ; ;2 3 2 9

a bbb b += =- = =- = =- = +

( )cc cm m J m m J m m I I

2 222 2

34 43 4 35 53 4 36 63 1 5; ; ;3 3 6

a ba b += =- = =- = = +

( )m m I I m m K

2 2

38 83 3 5 44 55 2; ;6

a b+= = + = = m m J m m J46 64 3 47 74 4; ;2 3

a a= =- = =-

c cm m J m m J m m J m m J1 148 84 3 56 65 3 57 75 4 58 85 3; ; ; ;2 2 3 2

a bb b= =- = =- = =- = =-

( ) ( ) ( )cm I I m m I I m m I

2 2 2 2 2 21

66 2 4 67 76 3 5 68 86 4; ; 1 ;4 6 4

a b a b a bæ ö+ + + ÷ç ÷ç ÷ç= + = = + = = + ÷ç ÷ç ÷ç ÷÷çè ø

( )m I I

2 2

77 4 6;9

a b+= +

( ) ( )c cm m I m I I

2 2 2 2 21 1

78 87 5 88 4 61 ; .6 4

a b a bæ ö+ +÷ç ÷ç ÷ç= = + = +÷ç ÷ç ÷ç ÷÷çè ø

vibration and buckling analysis of functionally graded ......lar plate subjected to a transverse...

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