A New Sinusoidal Shear Deformation Theory for Bending, Buckling-1

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A new sinusoidal shear deformation theory for bending, buckling,and vibration of functionally graded plates

Huu-Tai Thai a , , Thuc P. Vo b ,ca Department of Civil and Environmental Engineering, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, Republic of Koreab School of Mechanical, Aeronautical and Electrical Engineering, Glyndwr University, Mold Road, Wrexham LL11 2AW, UK c Advanced Composite Training and Development Centre, Glyndwr University, Unit 5, Hawarden Industrial Park, Deeside, Flintshire CH5 3US, UK

a r t i c l e i n f o

Article history:Received 14 November 2011Received in revised form 6 July 2012Accepted 16 August 2012Available online 27 August 2012

Keywords:BendingBucklingVibrationFunctionally graded platePlate theory

a b s t r a c t

A new sinusoidal shear deformation theory is developed for bending, buckling, and vibra-tion of functionally graded plates. The theory accounts for sinusoidal distribution of trans-verse shear stress, and satises the free transverse shear stress conditions on the top andbottom surfaces of the plate without using shear correction factor. Unlike the conventionalsinusoidal shear deformation theory, the proposed sinusoidal shear deformation theorycontains only four unknowns and has strong similarities with classical plate theory inmany aspects such as equations of motion, boundary conditions, and stress resultantexpressions. The material properties of plate are assumed to vary according to powerlawdistribution of the volume fraction of the constituents. Equations of motion are derivedfrom the Hamiltons principle. The closed-form solutions of simply supported plates areobtained and the results are compared with those of rst-order shear deformation theoryand higher-order shear deformation theory. It canbe concluded that the proposed theory isaccurate and efcient in predicting the bending, buckling, and vibration responses of func-tionally graded plates.

2012 Elsevier Inc. All rights reserved.

1. Introduction

Functionally graded materials (FGMs) are a class of composites that have continuous variation of material properties fromone surface to another and thus eliminate the stress concentration found in laminated composites. FGMs are widely used inmany structural applications such as mechanics, civil engineering, aerospace, nuclear, and automotive. In company with theincrease in the application of FGM in engineering structures, many computational models have been developed for predict-ing the response of functionally graded (FG) plates. Thesemodels can either be developedusing displacement-based theories(when the principle of virtual work is used) or displacement-stress-based theories (when Reissners mixed variational the-

orem is used). In general, these theories can be classied into three main categories: classical plate theory (CPT); rst-ordershear deformation theory (FSDT); and higher-order shear deformation theory (HSDT).

The CPT, which neglects the transverse shear deformation effects, provides accurate results for thin plates [14] . For mod-erately thick plates, it underestimates deections and overestimates buckling loads and natural frequencies. The FSDT ac-counts for the transverse shear deformation effect, but requires a shear correction factor to satisfy the free transverseshear stress conditions on the top and bottomsurfaces of the plate [511] . Although the FSDT provides a sufciently accuratedescription of response for thin to moderately thick plates, it is not convenient to use due to difculty in determination of correct value of the shear correction factor. To avoid the use of shear correction factor, many HSDTs were developed based on

0307-904X/$ - see front matter 2012 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.apm.2012.08.008

Corresponding author. Tel.: + 82 2 2220 4154.E-mail addresses: [email protected] (H.-T. Thai), [email protected] (T.P. Vo).

Applied Mathematical Modelling 37 (2013) 32693281

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Applied Mathematical Modelling

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the assumption of quadratic, cubic or higher-order variations of in-plane displacements through the plate thickness, notableamong them are Reddy [12] , Karama et al. [13] , Zenkour [1416] , Xiao et al. [17] , Matsunaga [18] , Pradyumna and Bandyo-padhyay [19] , Fares et al. [20] , Talha and Singh [2122] , Benyoucef et al. [23] , Atmane et al. [24] , Meiche et al. [25] , Mantariet al. [26] , and Xiang et al. [27] . Among the aforementioned HSDTs, the well-known HSDTs with ve unknowns include: theReddys theory [12] , the sinusoidal shear deformation theory [1416] , the hyperbolic shear deformation theory [2324] , theexponential shear deformation theory [13,26] . Although the HSDTs with ve unknowns are sufciently accurate to predictresponse of thin to thick plate, their equations of motion are much more complicated than those of FSDT and CPT. Therefore,there is a scope to develop a HSDT which is simple to use.

This paper aims to develop a simple sinusoidal shear deformation theory for bending, buckling, and vibration analyses of FG plates. This theory is based on assumption that the in-plane and transverse displacements consist of bending and shearparts. Unlike the conventional sinusoidal shear deformation theory [1416] , the proposed sinusoidal shear deformation the-ory contains four unknowns and has strong similarities with CPT in many aspects such as equations of motion, boundaryconditions, and stress resultant expressions. Material properties of FG plate are assumed to vary according to power law dis-tribution of the volume fraction of the constituents. Equations of motion are derived from the Hamiltons principle. Theclosed-form solutions are obtained for simply supported plates. Numerical examples are presented to verify the accuracyof the proposed theory in predicting the bending, buckling, and vibration responses of FG plates. It should be pointed outthat Merdaci et al. [28] , Ameur et al. [29] , and Tounsi et al. [30] recently developed a theory that is similar with the presentone. However, their works are limited to only bending problems.

2. Theoretical formulations

2.1. Basic assumptions

The assumptions of the present theory are as follows:

i. The displacements are small in comparison with the plate thickness and, therefore, strains involved are innitesimal.ii. The transverse normal stress r z is negligible in comparison with in-plane stresses r x and r y.

iii. The transverse displacement u3 includes two components of bending wb and shear w s . These components are func-tions of coordinates x, y, and time t only.

u3 x; y; z ; t wb x; y; t ws x; y; t : 1

iv. The in-plane displacements u1 and u2 consist of extension, bending, and shear components.u1 u ub us and u2 v v b v s: 2

- The bending components ub and v b are assumed to be similar to the displacements given by the classical plate theory.Therefore, the expressions for ub and v b are

ub z @ wb@ x

and v b z @ wb@ y

: 3a

- The shear components us and v s give rise, in conjunction with w s , to the sinusoidal variations of shear strains c xz , c yz andhence to shear stresses r xz , r yz through the thickness h of the plate in such a way that shear stresses r xz , r yz are zero at thetop and bottom surfaces of the plate. Consequently, the expression for us and v s can be given as

us z hp sin

p z h @ ws@ x and v s z hp sin p z h @ ws@ y : 3b

2.2. Kinematics

Based on the assumptions made in the preceding section, the displacement eld can be obtained using Eqs. (1)-(3) as

u1 x; y; z ; t u x; y; t z @ wb@ x

f @ ws

@ x ;

u2 x; y; z ; t v x; y; t z @ wb@ y

f @ ws

@ y ;

u3 x; y; z ; t wb x; y; t ws x; y; t ;

4

where

f z hp sin

p z h

: 5

3270 H.-T. Thai, T.P. Vo / Applied Mathematical Modelling 37 (2013) 32693281


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The kinematic relations can be obtained as follows:

e xe yc xy

8:9>=>;

z j b xj b yj b xy

8>:9>=>;

f j s xj s yj s xy

8:9>=>;

@ u@ x@ v @ y

@ u@ y

@ v @ x

8>:9>=>;

;j b xj b yj b xy

8>:9>=>;

@ 2 wb@ x2

@ 2 wb@ y2

2 @ 2 wb

@ x@ y

8>>>:9>>=>>;

;j s xj s yj s xy

8>>:9>>=>>;

; cs yz

cs xz @ ws@ y

@ ws@ x( ); g 1 df dz cos p z h :

7

2.3. Constitutive equations

The material properties of FG plate are assumed to vary continuously through the thickness of the plate in accordancewith a power law distribution as

P z P m P c P m 12

z h

p

; 8

where P represents the effective material property such as Youngs modulus E and mass density q subscripts m and c rep-resent the metallic and ceramic constituents, respectively; and p is thevolume fraction exponent. The value of p equal to zerorepresents a fully ceramic plate, whereas innite p indicates a fully metallic plate. Since the effects of the variation of Pois-sons ratio m on the response of FG plates are very small [3132] , the Poissons ratio m is usually assumed to be constant. Thelinear constitutive relations of a FG plate can be written as

r xr yr xyr yz r xz

8>>>>>>>:

9>>>>=>>>>;

E z 1 m2

1 m 0 0 0m 1 0 0 00 0 1 m2 0 00 0 0 1 m2 00 0 0 0 1 m2

26666643777775

e xe yc xyc yz c xz

8>>>>>>>:

9>>>>=>>>>;: 9

2.4. Equations of motion

Hamiltons principle is used herein to derive the equations of motion. The principle canbe stated in analytical form as [33]

0 Z T

0dU dV dK dt ; 10

where dU is the variation of strain energy; dV is the variation of potential energy; and dK is the variation of kinetic energy.The variation of strain energy of the plate is calculated by

dU Z V r xde x r yde y r xydc xy r yz dc yz r xz dc xz dAdz

Z A

N x@ du

@ x M b x

@ 2 dwb

@ x2 M

s x

@ 2 dws

@ x2 N y

@ dv

@ y M b y

@ 2 dwb

@ y2 M

s y

@ 2 dws

@ y2

( N xy

@ du

@ y

@ dv

@ x 2M b xy @ 2 dwb@ x@ y 2M s xy @ 2 dw s@ x@ y Q yz @ dws@ y Q xz @ dws@ x )dA; 11 where N , M , and Q are the stress resultants dened as

N i; M bi ;M si Z

h=2

h=21; z ; f r idz ; i x; y; xy and Q i Z

h=2

h=2 g r idz ; i xz ; yz : 12

The variation of potential energy of the applied loads can be expressed as

dV Z A qdu3 dA Z A N 0 x @ u3@ x @ du3@ x N 0 y @ u3@ y @ du3@ y 2N 0 xy @ u3@ x @ du3@ y dA; 13 where q and N

0

x; N 0

y; N 0

xy are transverse and in-plane applied loads, respectively.

H.-T. Thai, T.P. Vo / Applied Mathematical Modelling 37 (2013) 32693281 3271


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The variation of kinetic energy of the plate can be written as

dK Z V _u1 d _u1 _u2 d _u2 _u3 d _u3q z dAdz Z A I 0_ud _u _v d _v _wb _wsd _wb _wsf I 1 _u @ d

_wb@ x

@ _wb

@ x d _u _v

@ d _wb@ y

@ _wb

@ y d _v

J 1 _u@ d _ws

@ x

@ _ws@ x

d _u _v @ d _ws

@ y

@ _ws@ y

d _v

I 2

@ _wb@ x

@ d _wb@ x

@ _wb

@ y@ d _wb

@ y

K 2 @ _w s@ x @ d _w s@ x @ _w s@ y @ d _ws@ y J 2 @ _wb@ x

@ d _ws@ x

@ _w s

@ x@ d _wb

@ x

@ _wb@ y

@ d _ws@ y

@ _ws

@ y@ d _wb

@ y dA; 14 where dot-superscript convention indicates the differentiation with respect to the time variable t ; q z is the mass density;and I 0 ; I 1 ; J 1 ; I 2 ; J 2 ; K 2 are mass inertias dened as

I 0 ; I 1 ; J 1 ; I 2 ; J 2 ; K 2 Z h=2

h=21; z ; f ; z 2 ; zf ; f 2q z dz 15

Substituting the expressions for dU , dV , and dK from Eqs. (11)(14) into Eq. (10) and integrating by parts, and collectingthe coefcients of du , dv , dw b , and dw s , the following equations of motion of the plate are obtained

du : @ N x

@ x

@ N xy@ y

I 0 u I 1@ wb@ x

J 1@ w s@ x

; 16a

dv : @ N xy

@ x

@ N y@ y

I 0 v I 1@ wb@ y

J 1@ ws@ y

; 16b

dwb : @ 2M b x

@ x2 2

@ 2M b xy@ x@ y

@ 2 M b y@ y2

q ~N I 0wb w s I 1@ u@ x

@ v @ y I 2r 2 wb J 2r 2 w s; 16c

dw s : @ 2 M s x

@ x2 2

@ 2 M s xy@ x@ y

@ 2M s y@ y2

@ Q xz

@ x

@ Q yz @ y

q ~N I 0wb w s J 1@ u@ x

@ v @ y J 2r 2 wb K 2 r 2 ws; 16d

where

~N N 0 x@ 2wb ws

@ x2 N 0 y

@ 2wb ws@ y2

2N 0 xy@ 2wb ws

@ x@ y : 17

The boundary conditions of present theory involve specifying the six following pairs:

un or N nn ;us or N ns ;

@ wb=@ n or M bnn ;

@ w s=@ n or M snn ;

wb or V bn ;

w s or V sn ;

18

where

un un x v n y; us un y v n xN nn N xn2

x N yn2

y 2N xyn xn y; N ns N y N xn xn y N xyn2

x n2

y;

M bnn M b xn

2 x M

b yn

2 y 2M

b xyn xn y; M

snn M

s xn

2 x M

s yn

2 y 2M

s xyn xn y;

V bn Q b xn x Q

b yn y

@ M bns@ s

; V sn Q s xn x Q

s yn y

@ M sns@ s

;

Q b x @ M b x

@ x

@ M b xy@ y

N 0 x@ wb ws

@ x N 0 xy

@ wb ws@ y

I 1 u I 2@ wb@ x

J 2@ ws@ x

;

Q b y @ M b y@ y

@ M b xy

@ x N 0 xy

@ wb ws@ x

N 0 y@ wb ws

@ y I 1 v I 2

@ wb@ y

J 2@ w s@ y

;

Q s x @ M s x

@ x

@ M s xy@ y

Q xz N 0 x

@ wb ws@ x

N 0 xy@ wb ws

@ y J 1 u J 2

@ wb@ x

K 2@ ws@ x

;

Q s y @ M s y@ y

@ M s xy

@ x Q yz N

0 xy

@ wb ws@ x

N 0 y@ wb ws

@ y J 1 v J 2

@ wb@ y

K 2@ ws@ y

;

M bns M b y M b xn xn y M b xyn2 x n2 y; M sns M s y M s xn xn y M s xyn2 x n2 y:

19



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By substituting Eq. (6) into Eq. (9) and the subsequent results into Eq. (12) , the stress resultants are obtained as

f N gf M bgf M sg

8>:9>=>;

A B Bs

B D Ds

Bs Ds H s264

375f e0gf j bgf j sg

8>:9>=>;

and fQ g As f csg; 20

where

A ; B ; Bs ; D ; Ds ; H s 1 m 0m 1 00 0 1 m=2264 375

A; B; Bs; D; Ds; H s; 21a

A; B; Bs; D; Ds; H s Z h=2

h=21; z ; f ; z 2 ; zf ; f 2

E z 1 m2 dz ; 21b

As 1 0

0 1 As; As Z h=2

h=2

E z 21 m

g 2 dz : 21c

By substituting Eq. (20) into Eq. (16) , the equations of motion can be expressed in terms of displacements u; v ; wb ; w sas

A @ 2

u@ x2 1 m2 @ 2

u@ y2 1 m2 @ 2

v

@ x@ y ! Br2 @ wb@ x B

sr 2 @ ws@ x I 0u I 1 @

wb@ x J 1 @

ws@ x ; 22a

A @ 2 v

@ y2

1 m2

@ 2 v @ x2

1 m

2@ 2 u

@ x@ y ! Br 2 @ wb@ y Bsr 2 @ ws@ y I 0 v I 1 @ wb@ y J 1 @ ws@ y ; 22b Br 2 @

u@ x

@ v @ y Dr 4wb Dsr 4w s q ~N I 0wb ws I 1 @

u@ x

@ v @ y I 2 r 2 wb J 2r 2 ws; 22c

Bsr 2 @ u

@ x

@ v @ y Dsr 4 wb H sr 4ws Asr 2w s q ~N I 0 wb ws J 1 @

u@ x

@ v @ y J 2 r 2 wb K 2r 2 ws: 22d

Clearly, when the effect of transverse shear deformation is neglected ( w s

0), Eq. (22) yields the equations of motion of

FG plate based on the classical plate theory.

3. Closed-form solution for the simply supported rectangular plate

Consider a simply supported rectangular plate with length a and width b under transverse load q and in-plane forces intwo directions ( N 0 x c1 N cr ; N

0 y c2 N cr ; N

0 xy 0). Based on the Navier approach, the following expansions of displacements are

chosen to automatically satisfy the simply supported boundary conditions of plate

u x; y; t X1

m1X1

n1

U mn eix t cos a xsin b y;

v x; y; t X1

m1X1

n1

V mn eix t sin a xcos b y;

wb x; y; t X1

m1X1

n1W bmn eix t sin a xsin b y;

w s x; y; t X1

m1X1

n1

W smn eix t sin a xsin b y;

23

where i ffiffiffiffiffiffiffi

1p , a mp=a , b np=b, U mn ; V mn ; W bmn ; W smn are coefcients, and x is the frequency of vibration. The trans-verse load q is also expanded in the double-Fourier sine series as

q x; y X1

m1X1

n1

Q mn sin a xsin b y; 24

where

Q mn 4

ab Z a

0

Z b

0q x; ysin a xsin b ydxdy

q0 for sinusoidally distributed load ;16 q

0mnp2 for uniformly distributed load( 25



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Substituting Eq. (23) into Eq. (22) , the closed-form solutions can be obtained from

s11 s12 s13 s14s12 s22 s23 s24s13 s23 s33 k s34 ks14 s24 s34 k s44 k

2666437775

x 2

m11 0 m13 m140 m22 m23 m24

m13 m23 m33 m34m14 m24 m34 m44

2666437775

0BBB@1CCCA

U mnV mn

W bmnW smn

8>>>>>:9>>>=>>>;

00

Q mnQ mn

8>>>>>:9>>>=>>>;

; 26

where

s11 Aa2 1 m

2 Ab2 ; s12

1 m2

Aab; s22 1 m

2 Aa2 Ab2

s13 Baa 2 b2; s14 Bsaa2 b2 ; s23 Bba 2 b2s24 Bsba 2 b2; s33 Da 2 b22 ; s34 Dsa2 b22

s44 H sa 2 b22 Asa2 b2; k N cr c1a 2 c2 b2

m11 m22 I 0 ; m13 a I 1 ; m14 a J 1 ; m23 bI 1 ; m24 b J 1m33 I 0 I 2a 2 b2; m34 I 0 J 2a2 b2; m44 I 0 K 2 a 2 b2:

27

4. Results and discussion

In this section, various numerical examples are presented and discussed to verify the accuracy of present theory in pre-dicting the bending, buckling, and vibration responses of simply supported FG plates. For numerical results, an Al/Al 2O3 platecomposed of aluminum (as metal) and alumina (as ceramic) is considered. The Youngs modulus and density of aluminumare E m 70 GPa and q m 2702 kg/m 3 , respectively, and those of alumina are E c 380 GPa and q c 3800 kg/m 3 , respec-tively. For verication purpose, the obtained results are compared with those predicted using various plate theories. Thedescription of various displacement models is given in Table 1 . In all examples, a shear correction factor of 5/6 is used forFSDT and the rotary inertias are included in all theories. The Poissons ratio of the plate is assumed to be constant throughthe thickness and equal to 0.3. For convenience, the following nondimensionalizations are used in presenting the numericalresults in graphical and tabular form:

w 10E c h

3

q0a4 w

a2

;b2 ; r x hq0a r x a2 ; b2 ; h2 ; r y hq0a r y a2 ; b2 ; h3 ; D Eh

3

12 1 m2 ;

r xy hq0ar xy 0; 0; h3 ; N N

cr b2

p 2D ; N Na

2

E mh3 ; x x h ffiffiffiffiffiffiffiffiffiffiqc =E c q ; x x a

2

h ffiffiffiffiffiffiffiqc =E c q :28

4.1. Bending problem

Example 1: The rst example is carried out for square plate subjected to uniformly distributed load ( a 10 h). Table 2shows the comparison of nondimensional deections and stresses obtained by present theory with those given by Zenkour[16] based on sinusoidal shear deformation theory (SSDT). It can be seen that the proposed new SSDT and conventional SSDT[16] give identical results of deections as well as stresses for all values of power law index p. It should be noted that theproposed new SSDT involves four unknowns as against ve in case of conventional SSDT [16] . It is observed that the stressesfor a fully ceramic plate are the same as those for a fully metal plate. This is due to the fact that the plate for these two casesis fully homogenous and the nondimensional stresses do not depend on the value of the elastic modulus.

Example 2: Table 3 shows the comparison of nondimensional deections and stresses of square plate subjected to sinu-soidally distributed load ( a 10 h). The obtained results are compared with those given by Benyoucef et al. [23] based on thehyperbolic shear deformation theory (HSDT). It can be seen that an excellent agreement is obtained for all values of powerlaw index p.

Table 1

Displacement models.

Model Theory Unknowns

CPT Classical plate theory 3FSDT First-order shear deformation theory 5TSDT Third-order shear deformation theory 5HSDT Hyperbolic shear deformation theory 5SSDT Sinusoidal shear deformation theory 5

Present New sinusoidal shear deformation theory 4



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To illustrate the accuracy of present theory for wide range of power law index p and thickness ratio a=h , the variations of nondimensional deection

w with respect to power law index p and thickness ratio a=h are illustrated in Fig. 1 and Fig. 2 ,

Table 2

Comparison of nondimensional deection and stresses of square plate under uniformly distributed load ( m, n = 100 term series, a = 10h).

p Method

w r x

r y r xy

r yz r xz

Ceramic SSDT [16] 0.4665 2.8932 1.9103 1.2850 0.4429 0.5114Present 0.4665 2.8932 1.9103 1.2850 0.4429 0.5114

1 SSDT [16] 0.9287 4.4745 2.1962 1.1143 0.5446 0.5114Present 0.9287 4.4745 2.1692 1.1143 0.5446 0.5114

2 SSDT [16] 1.1940 5.2296 2.0338 0.9907 0.5734 0.4700Present 1.1940 5.2296 2.0338 0.9907 0.5734 0.4700

3 SSDT [16] 1.3200 5.6108 1.8593 1.0047 0.5629 0.4367Present 1.3200 5.6108 1.8593 1.0047 0.5629 0.4367

4 SSDT [16] 1.3890 5.8915 1.7197 1.0298 0.5346 0.4204Present 1.3890 5.8915 1.7197 1.0298 0.5346 0.4204

5 SSDT [16] 1.4356 6.1504 1.6104 1.0451 0.5031 0.4177Present 1.4356 6.1504 1.6104 1.0451 0.5031 0.4177

6 SSDT [16] 1.4727 6.4043 1.5214 1.0536 0.4755 0.4227Present 1.4727 6.4043 1.5214 1.0536 0.4755 0.4227

7 SSDT [16] 1.5049 6.6547 1.4467 1.0589 0.4543 0.4310Present 1.5049 6.6547 1.4467 1.0589 0.4543 0.4310

8 SSDT [16] 1.5343 6.8999 1.3829 1.0628 0.4392 0.4399

Present 1.5343 6.8999 1.3829 1.0628 0.4392 0.43999 SSDT [16] 1.5617 7.1383 1.3283 1.0620 0.4291 0.4481

Present 1.5617 7.1383 1.3283 1.0662 0.4291 0.4481

10 SSDT [16] 1.5876 7.3689 1.2820 1.0694 0.4227 0.4552Present 1.5876 7.3689 1.2820 1.0694 0.4227 0.4552

Metal SSDT [16] 2.5327 2.8932 1.9103 1.2850 0.4429 0.5114Present 2.5327 2.8932 1.9103 1.2850 0.4429 0.5114

Table 3

Comparison of nondimensional deection and stresses of square plate under sinusoidally distributed load ( a=10h).

p Method

w

r x

r y

r xy

r yz

r xz

Ceramic HSDT [23] 0.2960 1.9955 1.3121 0.7065 0.2132 0.2462Present 0.2960 1.9955 1.3121 0.7065 0.2132 0.2462

1 HSDT [23] 0.5889 3.0870 1.4894 0.6110 0.2622 0.2462Present 0.5889 3.0870 1.4894 0.6110 0.2622 0.2462

2 HSDT [23] 0.7572 3.6094 1.3954 0.5441 0.2763 0.2265Present 0.7573 3.6094 1.3954 0.5441 0.2763 0.2265

3 HSDT [23] 0.8372 3.8742 1.2748 0.5525 0.2715 0.2107Present 0.8377 3.8742 1.2748 0.5525 0.2715 0.2107

4 HSDT [23] 0.8810 4.0693 1.1783 0.5667 0.2580 0.2029Present 0.8819 4.0693 1.1783 0.5667 0.2580 0.2029

5 HSDT [23] 0.9108 4.2488 1.1029 0.5755 0.2429 0.2017Present 0.9118 4.2488 1.1029 0.5755 0.2429 0.2017

6 HSDT [23] 0.9345 4.4244 1.0417 0.5803 0.2296 0.2041Present 0.9356 4.4244 1.0417 0.5803 0.2296 0.2041

7 HSDT [23] 0.9552 4.5971 0.9903 0.5834 0.2194 0.2081Present 0.9562 4.5971 0.9903 0.5834 0.2194 0.2081

8 HSDT [23] 0.9741 4.7661 0.9466 0.5856 0.2121 0.2124Present 0.9750 4.7661 0.9466 0.5856 0.2121 0.2124

9 HSDT [23] 0.9917 4.9303 0.9092 0.5875 0.2072 0.2164Present 0.9925 4.9303 0.9092 0.5875 0.2072 0.2164

10 HSDT [23] 1.0083 5.0890 0.8775 0.5894 0.2041 0.2198Present 1.0089 5.0890 0.8775 0.5894 0.2041 0.2198

Metal HSDT [23] 1.6071 1.9955 1.3121 0.7065 0.2132 0.2462Present 1.6070 1.9955 1.3121 0.7065 0.2132 0.2462



8/13

respectively, for square plate subjected to sinusoidally distributed load. The obtained results are compared with those pre-dicted by CPT and TSDT [12] . It can be seen that the results of present theory and TSDT are almost identical, and the CPTunderestimates the deection of plate. Since the transverse shear deformation effects are not considered in CPT, the valuesof nondimensional deection

w predicted by CPT are independent of thickness ratio a=h (see Fig. 2 ).

4.2. Buckling problem

Due to the variation of material properties through the thickness, the stretching-bending coupling exists in FG plate. Thiscoupling produces deection and bending moments when plate is subjected to in-plane compressive loads. Hence, bifurca-tion-type buckling will not occur [34,35] . The conditions for bifurcation-type buckling to occur under the action of in-planeloads are examined by Aydogdu [36] and Naderi and Saidi [37] . It is observed that the bifurcation-type buckling occurs whenthe plate is fully clamped. For movable-edge plate, the bifurcation-type buckling occurs when the in-plane loads are appliedat the neutral surface. Therefore, the buckling analysis is presented herein for FG plate subjected to in-plane loads acting onthe neutral surface.

Example 3: Since the buckling results of FG plate are not available in the literature, only isotropic plate is used herein forverication. Table 4 shows the nondimensional buckling loads

^

N of isotropic plate ( p 0) subjected to different loadingtypes for different values of aspect ratio a=b and thickness ratio h=b. The obtained results are compared with those of Shufrinand Eisenberger [38] based on FSDT and TSDT. A good agreement is found for all cases ranging from moderately thick to verythick plates.

0

0.3

0.6

0.9

1.2

1.5

0 2 4 6 8 10p

CPTTSDTPresent

w

0

0.3

0.6

0.9

1.2

1.5

CPTTSDTPresent

Fig. 1. Comparison of the variation of nondimensional deection

w of square plate under sinusoidally distributed load versus power law index p (a = 5h).

0

0.3

0.6

0.9

1.2

1.5

0 10 20 30 40 50

a/h

CPTTSDTPresent

w

p=10

p=1

p=0

0

0.3

0.6

0.9

1.2

1.5CPTTSDTPresent

w

p=10

p=1

p=0

Fig. 2. Comparison of the variation of nondimensional deection

w of square plate under sinusoidally distributed load versus thickness ratio a /h.



9/13

Table 4

Comparison of nondimensional critical buckling load ^

N of isotropic plate under different loading types ( p =0).

a /b h/b Method Loading type ( c 1 , c 2)

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

1 0.1 FSDT [38] 3.7865 3.7865 1.8932TSDT [38] 3.7866 3.7866 1.8933Present 3.7869 3.7869 1.8935

0.2 FSDT [38] 3.2637 3.2637 1.6319TSDT [38] 3.2653 3.2653 1.6327Present 3.2666 3.2666 1.6333



1.5 0.1 FSDT [38] 4.0250 2.0048 1.3879TSDT [38] 4.0253 2.0048 1.3879Present 4.0258 2.0049 1.3880


0.3 FSDT [38] 2.5457 1.5267 1.0570

TSDT [38] 2.5545 1.5285 1.0582Present 2.5580 1.5295 1.0589


2 0.1 FSDT [38] 3.7865 1.5093 1.2074TSDT [38] 3.7866 1.5093 1.2075Present 3.7869 1.5094 1.2075



0.4 FSDT [38] 1.9034 0.9991 0.7992

TSDT [38] 1.9230 1.0015 0.8012Present 1.9292 1.0025 0.8020

0

2

4

6

8

10

0 2 4 6 8 10p

CPTTSDTPresent

N

0

2

4

6

8

10

CPTTSDTPresent

Fig. 3. Comparison of the variation of nondimensional critical buckling load

N of square plate under biaxial compression versus power law index p (a = 5h).



10/13

The variations of nondimensional critical buckling load

N with respect to power law index p and thickness ratio a=h areillustrated in Figs. 3 and 4 , respectively, for square plate under biaxial compression. It is observed that the proposed theoryand TSDT give almost identical results, and CPT overestimates the buckling loads of plate due to ignoring transverse sheardeformation effects. The difference between CPT and shear deformation theories (TSDT and present theory) decreases whenthe thickness ratio a=h increases (see Fig. 4 ).

4.3. Free vibration problem

Example 4: Nondimensional fundamental frequencies ^

x of square plate for different values of thickness ratio h=a andpower law index p are presented in Table 5 . The obtained results are compared with those reported by and Hosseini-Has-hemi et al. [11] based on FSDT and and Hosseini-Hashemi et al. [39] based on TSDT. It is observed that there is an excellentagreement between the results predicted by present theory, FSDT [11] , and TSDT [39] .

Example 5: To verify the higher order modes of vibration, the rst four nondimensional frequencies

x are compared inTable 6 for rectangular plate ( b 2awith thickness ratio varied from 5 to 20 and power law index varied from 0 to 10. Thenondimensional frequencies obtained by using proposed theory and TSDT [12] are compared with those given by Hosseini-Hashemi et al. [11] based on FSDT. It can be seen that the results predicted by proposed theory and TSDT are almost identicalfor all modes of vibration of thin to thick plates. Also, the proposed theory gives more accurate prediction of natural fre-quency compared to FSDT. The maximum difference between present theory and TSDT is 0.1% at the fourth mode(a=h 5, p 5), while the maximum difference between FSDT and TSDT is 5.64% at the fourth mode ( a=h 5, p 10). How-ever, this difference decreases at lower modes of vibration. For example, the differences between FSDT and TSDT are 1.26%,1.78%, 2.44%, and 5.64% at the rst, second, third, and fourth modes ( a=h 5, p 10), respectively.

The variations of nondimensional fundamental frequency

x of square plate with respect to power law index p and thick-ness ratio a

=h are compared in Figs. 5 and 6, respectively. It is observed that the nondimensional frequencies

xpredicted by

present theory and TSDT are almost identical, and the CPT overestimates the frequency of thick plate.

0

2

4

6

8

10

a/h

CPTTSDTPresent

N

p=0

p=1

p=10

0

2

6

8

10

0 10 20 30 40 50a/h

CPTTSDTPresent

N

p=0

p=1

p=10

Fig. 4. Comparison of the variation of nondimensional critical buckling load

N of square plate under biaxial compression versus thickness ratio a /h .

Table 5

Comparison of nondimensional fundamental frequency ^

x of square plate.

a /h Method Power law index (p)

0 0.5 1 4 10

5 FSDT [11] 0.2112 0.1805 0.1631 0.1397 0.1324HSDT [39] 0.2113 0.1807 0.1631 0.1378 0.1301Present 0.2113 0.1807 0.1631 0.1377 0.1300

10 FSDT [11] 0.0577 0.0490 0.0442 0.0382 0.0366HSDT [39] 0.0577 0.0490 0.0442 0.0381 0.0364Present 0.0577 0.0490 0.0442 0.0381 0.0364

20 FSDT [11] 0.0148 0.0125 0.0113 0.0098 0.0094HSDT [39] 0.0148 0.0125 0.0113 0.0098 0.0094

Present 0.0148 0.0125 0.0113 0.0098 0.0094



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Table 6

Comparison of the rst four nondimensional frequencies

x of rectangular plate ( b = 2a).

a /h Mode( m ,n) Method Power law index (p)

0 0.5 1 2 5 8 10

5 1 (1,1) FSDT [11] 3.4409 2.9322 2.6473 2.4017 2.2528 2.1985 2.1677TSDT 3.4412 2.9347 2.6475 2.3949 2.2272 2.1697 2.1407Present 3.4416 2.9350 2.6478 2.3948 2.2260 2.1688 2.1403

2 (1,2) FSDT [11] 5.2802 4.5122 4.0773 3.6953 3.4492 3.3587 3.3094TSDT 5.2813 4.5180 4.0781 3.6805 3.3938 3.2964 3.2514Present 5.2822 4.5187 4.0787 3.6804 3.3914 3.2947 3.2506





3 (1,3) FSDT [11] 9.1876 7.8145 7.0512 6.4015 6.0247 5.8887 5.8086

TSDT 9.1880 7.8189 7.0515 6.3886 5.9765 5.8341 5.7575Present 9.1887 7.8194 7.0519 6.3885 5.9742 5.8324 5.7566





4 (2,1) FSDT [11] 12.4560 10.5660 9.5261 8.6572 8.1875 8.0207 7.9166

TSDT 12.4562 10.5677 9.5261 8.6509 8.1636 7.9934 7.8909Present 12.4565 10.5680 9.5263 8.6508 8.1624 7.9925 7.8905

2

3

4

5

6

0 2 4 6 8 10p

CPTTSDTPresent

2

3

4

5

6

0 2 4 6 8 10p

CPTTSDTPresent

Fig. 5. Comparison of the variation of nondimensional fundamental frequency

x of square plate versus power law index p (a = 5h).



12/13

5. Conclusions

A newsinusoidal shear deformation theory is developed for bending, buckling, and vibration of FG plates. Unlike the con-ventional sinusoidal shear deformation theory, the proposed sinusoidal shear deformation theory contains only four un-knowns and has strong similarities with the CPT in many aspects such as equations of motion, boundary conditions, andstress resultant expressions. Equations of motion are derived from the Hamiltons principle. Closed-form solutions are ob-tained for simply supported plates. The accuracy of the proposed theory has been veried for the bending, buckling, and freevibration analyses of FG plates. All comparison studies show that the deection, stress, buckling load, and natural frequencyobtained by the proposed theory with four unknowns are almost identical with those predicted by other shear deformationtheories containing ve unknowns. The practical utilities of this theory are: (1) there is no need to use a shear correctionfactor; (2) the nite element model based on this theory will be free from shear locking since the CPT comes out as a specialcase of the present theory; and (3) the theory is simple and time efcient due to involving only four unknowns. In conclusion,

it can be said that the proposed theory is accurate and efcient in predicting the bending, buckling, and vibration responsesof FG plates.

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CPTTSDTPresent

p=0

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2

3

4

5

6

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CPTTSDTPresent

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Fig. 6. Comparison of the variation of nondimensional fundamental frequency

x of square plate versus thickness ratio a /h.



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A New Sinusoidal Shear Deformation Theory for Bending, Buckling-1

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