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Composite Structures 85 (2008) 95–104

Benchmark solutions for functionally graded thick plates restingon Winkler–Pasternak elastic foundations

Z.Y. Huang, C.F. Lu *, W.Q. Chen

Department of Civil Engineering, Zhejiang University, Hangzhou 310027, PR China

Available online 13 October 2007

Abstract

Exact solutions for functionally graded thick plates are presented based on the three-dimensional theory of elasticity. The plate isassumed isotropic at any point, while material properties to vary exponentially through the thickness. The system of governing partialdifferential equations is reduced to an ordinary one about the thickness coordinate by expanding the state variables into infinite dualseries of trigonometric functions. Interactions between the Winkler–Pasternak elastic foundation and the plate are treated as boundaryconditions. The problem is finally solved using the state space method. Effects of stiffness of the foundation, loading cases, and gradientindex on mechanical responses of the plates are discussed. It is established that elastic foundations affects significantly the mechanicalbehavior of functionally graded thick plates. Numerical results presented in the paper can serve as benchmarks for future analyses offunctionally graded thick plates on elastic foundations.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Exact solution; Functionally graded plates; Winkler–Pasternak elastic foundation; State space method

1. Introduction

Plates supported by elastic foundations have beenwidely adopted by many researchers to model various engi-neering problems during the past decades. To describe theinteractions of the plate and foundation as more appropri-ate as possible, scientists have proposed various kinds offoundation models, as documented well in Ref. [1]. Thesimplest model for the elastic foundation is the Winklermodel, which regards the foundation as a series of sepa-rated springs without coupling effects between each other,resulting in the disadvantage of discontinuous deflectionon the interacted surface of the plate. This was laterimproved by Pasternak [2] who took account of the inter-actions between the separated springs in the Winkler modelby introducing a new dependent parameter. From then on,the Pasternak model was widely used to describe the

0263-8223/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruct.2007.10.010

* Corresponding author. Tel.: +86 571 87952284; fax: +86 57187952165.

E-mail address: lucf@zju.edu.cn (C.F. Lu).

mechanical behavior of structure–foundation interactions[3–8].

On the other hand, functionally graded materials(FGMs) [9,10], a new generation of advanced inhomoge-neous composite materials first proposed for thermal barri-ers [11], have been increasingly applied for modernengineering structures in extremely higher temperatureenvironment. Many researches were conducted concerningthermal mechanical behavior of FGMs [12,13]. However,bending and vibration analyses of FGMs are quite limited,especially of those on elastic foundations. Cheng andKitipornchai [14] proposed a membrane analogy to derivean exact explicit eigenvalues for compression buckling,hydrothermal buckling, and vibration of FGM plates ona Winkler–Pasternak foundation based on the first-ordershear deformation theory. The same membrane analogywas later applied to the analyses of FGM plates and shellsbased on a third-order plate theory [15,16]. The free vibra-tion, transient response, large deflection and postbucklingresponses of FGM thin plates resting on Pasternak founda-tions were investigated by Yang and Shen [17,18] using themethod of differential quadrature and Galerkin procedure.

Shear layer ( pik )

Winkler springs ( wk )

Rigid substrate

z

y

x

FGM plate

Eh

E02l

1l

h

Fig. 1. A FGM rectangular plate resting on a Pasternak elasticfoundation.

96 Z.Y. Huang et al. / Composite Structures 85 (2008) 95–104

It should be mentioned that the classical thin plate the-ory [4,5,17,18] holds the Kirchhoff hypothesis that neglectsshear deformation in the plate, which is increasingly signif-icant when the plate becomes thicker. In comparison, theMindlin plate theory [3,14] accounts for the shear deforma-tions by introducing a shear correction factor, but it is lim-ited to moderately thick plates. Although the higher-orderplate theories [6,15] enhances the solution accuracy feasi-bly, only part of the elastic constants are considered, lead-ing to the fact that the results will remain the sameregardless of the variations of the elastic constants thatare not included in the theory. The most common featuresof these simplified theories lies in that the effect of trans-verse normal stress is ignored, due to which the resultsare bounded to inherent errors for extremely thick plates[19]. On the other hand, higher-order theories developedfor homogeneous materials are not always efficient formodeling inhomogeneous materials, especially for theoccasion where high gradient of material properties occurs.Furthermore, the assumption that the reaction forces bythe foundation acts on the mid-plane of the plate in thinplate theories will cause minor errors, but it is not the casefor thick plates. For a homogeneous thick plate, the effectsof elastic foundation on deformations of the top and bot-tom surfaces of the plate are absolutely different [8], indi-cating that, for a given FGM thick plate, the globalmechanical characteristics will be quite different if theharder or softer surface is subjected to the elasticfoundation.

This paper aims to provide an exact three-dimensionalelasticity solution for the bending behavior of FGM thickplates on a Winkler–Pasternak foundation. The plate isassumed isotropic at any point in the plate volume, withthe Young’s modulus varying exponentially through thethickness [20] while the Poisson’s ratio remaining constant.The state space method (SSM) [21,22] is used to express thebasic equations of elasticity into a condensed first-ordersimultaneous partial differential equation. The stress anddisplacement components are expanded into an infinite dualseries of trigonometric functions about the in-plane coordi-nates for a fully simply supported plate, leading to a systemof ordinary differential equation about the thickness coordi-nate, for which an exact solution is possible. The reactions bythe elastic foundation acting on the bottom surface of theplate are treated as boundary conditions. Effects of the foun-dation stiffness, loading cases, and gradient index of theYoung’s modulus on the mechanical behavior of FGMplates are investigated. Numerical results are presented andexpected to serve as benchmarks for future analyses ofFGM thick plates on elastic foundations.

2. Theoretical formulations

Consider a rectangular FGM plate having the thicknessh, length l1, and width l2, as depicted in Fig. 1. It is assumedto be rested on a Winkler–Pasternak type elastic founda-tion with the Winkler stiffness of kw and shear stiffness of

kpi (i = x,y). The Cartesian coordinate system is estab-lished so that 0 6 x 6 l1, 0 6 y 6 l2, and 0 6 z 6 h.

The plate is assumed isotropic at any point in the vol-ume with constant Poisson’s ratio m, while the Young’smodulus E varies exponentially through thickness accord-ing to the following form,

E � E0ekz; ð1Þwhere k is the gradient index, and defined by

k ¼ 1

hln

Eh

E0

� �: ð2Þ

The subscript ‘0’ and ‘h’ here and in the following text rep-resent the values of the quantity that is followed at the low-er and top surfaces of the plate, respectively.

2.1. Basic equations of three-dimensional elasticity

In the absence of body forces, the equilibrium equationsfor the elastic body due to infinitesimal deformation areexpressed as

oxrx þ oysxy þ ozsxz ¼ 0;

oxsxy þ oyry þ ozsyz ¼ 0;

oxsxz þ oysyz þ ozrz ¼ 0;

ð3Þ

where the operator o denotes the derivatives about the suf-fix coordinate of the variable that is followed. For isotropicmaterials, the constitutive equations are

rx

ry

rz

8>><>>:9>>=>>; ¼

lþ 2G l l

l lþ 2G l

l l lþ 2G

26643775

oxu

oyv

ozw

8>><>>:9>>=>>;; ð4aÞ

syz

sxz

sxy

8><>:9>=>; ¼

G 0 0

0 G 0

0 0 G

264375 oywþ ozv

ozuþ oxw

oxvþ oyu

8><>:9>=>;; ð4bÞ

where l and G are the Lame constants with the same var-iation scheme of Young’s modulus E.

Z.Y. Huang et al. / Composite Structures 85 (2008) 95–104 97

Further supposing that the plate is simply supported atthe four edges with the following typical boundaryconditions,

w ¼ v ¼ rx ¼ 0 at x ¼ 0; a; ð5aÞw ¼ u ¼ ry ¼ 0 at y ¼ 0; b: ð5bÞ

To capture such supporting conditions, the displacementand stress components, according to the method of super-position, can be expanded into infinite dual trigonometricseries of

rz

u

v

w

sxz

syz

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;¼X1n¼0

X1m¼0

G0ekhfRfðfÞ sinðnpnÞ sinðmpgÞhUðfÞ cosðnpnÞ sinðmpgÞhV ðfÞ sinðnpnÞ cosðmpgÞhW ðfÞ sinðnpnÞ sinðmpgÞG0ekhfC1ðfÞ cosðnpnÞ sinðmpgÞG0ekhfC3ðfÞ sinðnpnÞ cosðmpgÞ

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;ð6Þ

and

rx

ry

sxy

8><>:9>=>; ¼ G0ekhf

X1n¼0

X1m¼0

RnðfÞ sinðnpnÞ sinðmpgÞRgðfÞ sinðnpnÞ sinðmpgÞC2ðfÞ cosðnpnÞ cosðmpgÞ

8><>:9>=>;;ð7Þ

where n = x/l1, g = y/l2, and f = z/h are the non-dimen-sional coordinates, n and m are the half-wave numbers inthe x and y directions, respectively. Substituting Eqs. (6)and (7) into the equilibrium Eq. (3) leads to the followinguncoupled equations of motion for each pair of (n,m),

knRn � kmC2 þ ofC1 þ khC1 ¼ 0

� knC2 þ kmRg þ ofC3 þ khC3 ¼ 0

� knC1 � kmC3 þ ofRf þ khRf ¼ 0

ð8Þ

where kn = nph/l1 and km = mph/l2. Similarly, theconstitutive equations for a given pair of (n,m) areobtained as

Rn

Rg

Rf

8><>:9>=>; ¼ 1

G0

�C0kn �l0km l0of

�l0kn �C0km l0of

�l0kn �l0km C0of

264375 U

V

W

8><>:9>=>;; ð9aÞ

C3

C1

C2

8><>:9>=>; ¼

0 of km

of 0 kn

km kn 0

264375 U

V

W

8><>:9>=>;: ð9bÞ

where C0 = l0 + 2G0.

2.2. State space method (SSM)

It is noted that with the aid of Eqs. (6) and (7), the prob-lem is reduced to a system of ordinary differential equa-tions about f. According to the routine procedure ofSSM [21,22], the governing Eqs. (8) and (9) can be reducedto the following set of simultaneous first-order ordinarydifferential equation

d

dfdðfÞ ¼ AnmdðfÞ ð10Þ

Here, d ¼ Rf U V W C1 C3½ �T is called the statevector, and the coefficient matrix Anm is obtained as

Anm

¼

�kh 0 0 0 kn km

0 0 0 �kn 1 0

0 0 0 �km 0 1

c0 c1kn c1km 0 0 0

�c1kn c2k2n þ k2

m ð1þ 2c1Þknkm 0 �kh 0

�c1km ð1þ 2c1Þknkm c2k2m þ k2

n 0 0 �kh

2666666664

3777777775;

where

c0 ¼G0

C0

; c1 ¼l0

C0

; c2 ¼1

G0

C0 �l2

0

C0

� �:

Accompanied with the state Eq. (1), the induced variablesare determined by

Rn ¼ c1Rf � c2knU � 2c1kmV ;

Rg ¼ c1Rf � 2c1knU � c2kmV ;

C2 ¼ knV þ kmU :

ð11Þ

According to matrix operation, the general solution to Eq.(10) is obtained as

dðfÞ ¼ efAnmd0; ð12Þwhere d0 is the state vector at the bottom surface of theplate. It is noted that the state vector d(f) at an arbitrarycoordinate f is obtained by transforming d0 through atransfer matrix exp(fAnm). Setting f = 1 in Eq. (12) resultsin

dh ¼ Td0; ð13Þwhere T = exp(Anm) is the global transfer matrix, dh is thestate vector at the top surface of the plate. Eq. (13) formsthe global transfer relation of the state vectors at the bot-tom and top surfaces of the plates, from which the equa-tion governing bending behavior of the plate can beobtained by incorporating the lateral traction boundaryconditions and the effect of elastic foundations.

2.3. Coupled effect of elastic foundation and solution

Since the bottom surface of the plate is assumed sub-jected to Winkler–Pasternak elastic foundation (seeFig. 1), the reaction–deflection relation at the bottom sur-face of the model is expressed by

rz0 ¼ kww0 � kpxo

2w0

ox2� kpy

o2w0

oy2ð14Þ

at z = 0, where rz0 is the density of reaction force on theplate bottom, w0 the deflection of that surface. If the foun-dation is isotropic, it is obvious that kpx = kpy = kp.

Substituting the basic assumption of state variables ofEqs. (6) and (7) into Eq. (14), one gets

Table 1Comparisons of central deflections 103Dw(0.5l, 0.5l, 0.5h)/ql4 of a uni-formly loaded homogeneous square plate (l/h = 100, m = 0.3) on Win-kler–Pasternak foundations

Kw Kp Results

Present Lam et al. [28]

1 1 3.8546 3.85334 0.7630 0.76354 0.1153 0.115

34 1 3.2105 3.21034 0.7317 0.73254 0.1145 0.115

54 1 1.4765 1.47634 0.5704 0.57054 0.1095 0.109

-2 -1 0 10

0.2

0.4

0.6

0.8

1

104u(0,l/2,z)/h

z/h

-1 -0.8 -0.6 -0.4 -0.2 00

0.2

0.4

0.6

0.8

1

σz(l/2,l/2,z)/q

0

z/h

-60 -40 -20 0 200

0.2

0.4

0.6

0.8

1

σx(l/2,l/2,z)/q

0

z/h

0w pK K= = 100,w pK K=

200,w pK K=

Fig. 2. Through-thickness distributions of displacements and stresses of a mode

98 Z.Y. Huang et al. / Composite Structures 85 (2008) 95–104

Rf0 ¼1

G0

kwhþ kpx

hk2

n þkpy

hk2

m

� �W 0 ¼ bW 0: ð15Þ

For a practical problem, the following lateral tractionboundary conditions

Rfh ¼ Rfh; C1h ¼ C1h; C3h ¼ C3h; ð16aÞC10 ¼ C10; C30 ¼ C30; ð16bÞ

together with Eq. (15) should be incorporated into Eq. (13)for unique solution. In Eq. (16), the bar over a variable de-notes that the stress is prescribed.

For the convenience of incorporating boundary condi-tions at the top and bottom surfaces, Eq. (13) is re-expressed into the form of

-10 -9 -8 -7 -6 -50

0.2

0.4

0.6

0.8

1

104w(l/2,l/2,z)/h

z/h

-3 -2 -1 00

0.2

0.4

0.6

0.8

1

τxz(0,l/2,z)/q

0

z/h

-10 0 10 20 300

0.2

0.4

0.6

0.8

1

τxy(0,0,z)/q

0

z/h

0= 200, 0w pK K=

10= 200, 25w pK K

=

= =

rately thick square FGM plate on an elastic foundation (Case 1, h/l = 0.1).

Z.Y. Huang et al. / Composite Structures 85 (2008) 95–104 99

Rf

UVWC1

C3

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;h

¼

t11 t12 t13 t14 t15 t16

t21 t22 t23 t24 t25 t26

t31 t32 t33 t34 t35 t36

t41 t42 t43 t44 t45 t46

t51 t52 t53 t54 t55 t56

t61 t62 t63 t64 t65 t66

26666664

37777775Rf

UVWC1

C3

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;0

; ð17Þ

where tij are the elements of the transfer matrix T. Substi-tuting Eqs. (15) and (16) into the above expression, the fol-lowing unique solution algebraic equation is extracted,

t12 t13 bt11 þ t14

t52 t53 bt51 þ t54

t62 t63 bt61 þ t64

24 35 UVW

8<:9=;

0

¼Rfh

C1h

C3h

8<:9=;�

t15 t16

t55 t56

t65 t66

24 35 C10

C30

� �; ð18Þ

-1 0 1 20

0.2

0.4

0.6

0.8

1

104u(0,l/2,z)/h

z/h

-1.2 -0.9 -0.6 -0.3 00

0.2

0.4

0.6

0.8

1

σz(l/2,l/2,z)/q

0

z/h

-20 0 20 40 600

0.2

0.4

0.6

0.8

1

σ x(l/2,l/2,z)/q0

z/h

0w pK K= = 100,w pK K=

200,w pK K=

Fig. 3. Through-thickness distributions of displacements and stresses of a mode

from which, a unique solution is obtained as

U

V

W

8><>:9>=>;

0

¼t12 t13 bt11þ t14

t52 t53 bt51þ t54

t62 t63 bt61þ t64

264375�1

Rfh

C1h

C3h

8><>:9>=>;�

t15 t16

t55 t56

t65 t66

264375 C10

C30

( )0B@1CA:ð19Þ

By combining the displacements at the bottom surface withthe traction conditions and substituting them into the gen-eral solution Eq. (13), the state vector at an arbitrary coor-dinate f is thus attained.

It should be pointed out that numerical instabilities willalways occur using the global analysis of Eq. (13), when the

-10 -9 -8 -7 -6 -50

0.2

0.4

0.6

0.8

1

104w(l/2,l/2,z)/h

z/h

-3 -2 -1 00

0.2

0.4

0.6

0.8

1

τxz

(0,l/2,z)/q0

z/h

-30 -20 -10 0 100

0.2

0.4

0.6

0.8

1

τxy(0,0,z)/q

0

z/h

0= 200, 0w pK =

10= 200, 25w pK K= =

=K

rately thick square FGM plate on an elastic foundation (Case 2, h/l = 0.1).

100 Z.Y. Huang et al. / Composite Structures 85 (2008) 95–104

plate is extremely thick, or the foundation is highly stiff, orthe half-wave number n (or m) is too large [23]. For this sit-uation, the authors introduced the method of sub-struc-tures and the joint coupling matrices [24] to enhance thestabilities of the numerical calculation of the state spacemethod. The joint coupling matrices will be applied inthe present work for stable calculation for highly thickplates. However, detailed procedure of the method was welldescribed by Lu et al. [25–27], and hence is not repeatedhere for brevity.

3. Numerical results

For numerical illustrations, only square FGM plates onisotropic elastic foundations are considered, that is,

-1.5 -1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

106u(0,l/2,z)/h

z/h

-1 -0.8 -0.6 -0.4 -0.2 00

0.2

0.4

0.6

0.8

1

σz(l/2,l/2,z)/q

0

z/h

-3 -2 -1 0 10

0.2

0.4

0.6

0.8

1

σx (l/2,l/2,z)/q0

z/h

0w pK K= = 100,w pK K=

200,wK K=

Fig. 4. Through-thickness distributions of displacements and stresses of a stro

l1 = l2 = l and kpx = kpy = kp. In the calculations, theYoung’s moduli at the two surfaces of the plate, exceptfor the case concerning effects of gradient index, areassumed of eE and 10eE, respectively, while the Poisson’sratio is taken as m = 0.3. In all examples, the foundationparameters are presented in the non-dimensional form of

Kw ¼kwl4

D; Kp ¼

kpl2

D;

where D ¼ eEh3=12ð1� m2Þ is a reference bending rigidity ofthe plate.

In order to validate the present formulations, numericalresults for bending of a homogeneous thin plate (k = 0, l/h = 100, m = 0.3) are compared to that obtained by Lamet al. [28] using Green’s functions. The plate is assumedsubjected to uniform pressure q0 (E/105) on the top surface

-4 -3 -2 -1 00

0.2

0.4

0.6

0.8

1

106w(l/2,l/2,z)/h

z/h

-0.6 -0.45 -0.3 -0.15 00

0.2

0.4

0.6

0.8

1

τxz(0,l/2,z)/q

0

z/h

-0.5 0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

τxy (0,0,z)/q0

z/h

0=

10p =

200, 0w pK =

200, 25w pK K= =

=K

ngly thick square FGM plate on an elastic foundation (Case 1, h/l = 0.5).

Z.Y. Huang et al. / Composite Structures 85 (2008) 95–104 101

and the results for the central deflection of the plate aregiven in Table 1. The present results are obtained by takingtwenty-one terms in Eqs. (6) and (7) and found to agreewell with that reported by Lam et al. [28].

The effects of foundation stiffness, loading cases, andgradient index on the mechanical behavior of a FGM platewill be intensively discussed in the following text. Here, thephase ‘‘loading case” is referred to the occasion that theplate is arranged with the softer or harder surface attachedto the elastic foundation, whereas the other (harder orsofter surface) is subjected to the applied load. For the con-venience of citing, Case 1 and Case 2 are designated to theplates in the above two occasions, respectively. For simplic-ity, but without loss of generality, the plate is assumed sub-jected to a sinusoidal distributed pressure q(n,g) =q0 sin(npn)sin (mpg) on the top surface with q0 ¼ eE=105.

-1 -0.5 0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

106u(0,l/2,z)/h

z/h

-1.5 -1 -0.5 00

0.2

0.4

0.6

0.8

1

σz(l/2,l/2,z)/q

0

z/h

-1 0 1 20

0.2

0.4

0.6

0.8

1

σx(l/2,l/2,z)/q

0

z/h

0w pK K= = 100,wK K=

200,wK K=

Fig. 5. Through-thickness distributions of displacements and stresses of a stro

Firstly, the effects of foundation stiffness and loadingcase are considered. The through-thickness distributionsof displacements and stresses of moderately thick squareFGM plates (h/l = 0.1) on different elastic foundationsare plotted in Figs. 2 (Case 1) and 3 (Case 2), that ofstrongly thick plates (h/l = 0.5) given in Figs. 4 (Case 1)and 5 (Case 2).

It is seen from Figs. 2 and 3 that all displacements andstresses except for the transverse normal stress decreasegradually as either Kw or Kp increases. Decreases of dis-placements indicate that increasing the foundation stiffnesswill certainly enhance the deformation rigidity of the plate.Comparisons of Figs. 2 and 3 show that the magnitudes ofdisplacements and stresses (except rz) at the physical levelin the thickness direction of a given moderately thickFGM plate are almost the same for Case 1 and Case 2, that

-5 -4 -3 -2 -1 00

0.2

0.4

0.6

0.8

1

106w(l/2,l/2,z)/h

z/h

-0.5 -0.4 -0.3 -0.2 -0.1 00

0.2

0.4

0.6

0.8

1

τxz

(0,l/2,z)/q0

z/h

-1 -0.5 0 0.50

0.2

0.4

0.6

0.8

1

τxy

(0,0,z)/q0

z/h

0p = 200, 0w pK K= =

10p = 200, 25w pK K= =

ngly thick square FGM plate on an elastic foundation (Case 2, h/l = 0.5).

102 Z.Y. Huang et al. / Composite Structures 85 (2008) 95–104

is, exchanging the two surfaces applied by the externalloads and elastic foundations affects little on the stress stateand deformation of the moderately thick plate.

However, influences of the foundation stiffness onbehavior of a strongly thick FGM plate are comparativelymore complicated, as shown in Figs. 4 and 5. When Kw orKp increases, the positive in-plane displacement u and stresssxy decreases monotonically, but the negative values ofthese two quantities do not necessarily undertake mono-tonic increase or decrease. The figures for the in-plane dis-placement u indicate that the horizontal plane undergoingzero in-plane deformations moves increasingly away fromthe foundation supported surface as the foundation stiff-ness increases. This is rather different from the phenomenaobserved for the moderately thick plate. However, the vari-ations of the remaining given stresses and displacementversus Kw and Kp are almost the same as that in Figs. 2and 3. Comparisons of the results in Figs. 4 and 5 showthat the magnitudes or variation schemes of the displace-ments and stresses relative to the physical location in theplate of Case 1 andCase 2 are considerably different. Non-

wKpK0

250

500

0

250

500-10

-8

-6

-4

-2

0

104 w

(l/2

,l/2,

h/2)

/h

Fig. 6. Variation of deflection w(l/2, l/2,h/2) and in-plane normal stress rx(l/2, l

Kp (Case 1, h/l = 0.1).

wKpK

0

250

500

0

250

500-3.5

-3

-2.5

-2

-1.5

106 w

(l/2

,l/2,

h/2)

/h

Fig. 7. Variation of deflection w(l/2, l/2,h/2) and in-plane normal stress rx(l/2, l

Kp (Case 1, h/l = 0.5).

linearities of the displacements in the plate of Case 2 aremuch more obvious than that in Case 1.

Next, a global observation concerning the effects of thefoundation stiffness on the deflections and stresses of FGMplate of Case 1 is presented in Fig. 6 (h/l = 0.1) and Fig. 7(h/l = 0.5). Figs. 6a and 7a are for the transverse displace-ment at the central point of the mid-plane, while Figs. 6band 7b for the maximal in-plane normal stress at the cen-tral point of the top surface. The curves in Figs. 2 and 4convince the monotonic decrease of the displacement andin-plane normal stress versus the increasing of Kw or Kp.It is also informed from Figs. 6 and 7 that the incrementsof these two quantities becomes smaller as Kw or Kp

increases. Such effects of Kw is more obvious than that ofKp. This phenomenon for the strongly thick plate (Fig. 7)is much more distinguishable than that for the moderatelythick plate (Fig. 6).

Finally, the effects of gradient index on the behavior of astrongly thick (h/l = 0.5) FGM plate on a Pasternak foun-dation (Kw = 100 and Kp = 10) are presented in Fig. 8. Theloading case is taken as Case 1, in which the Young’s mod-

wKpK0

250

500

0

250

500-50

-40

-30

-20

-10

0

σ x(l/2

,l/2,

h)/q

0

/2,h) of a strongly thick FGM plate versus the foundation stiffness Kw and

wKpK0

250

500

0

250

500-2.5

-2

-1.5

σ x(l/2

,l/2,

h)/q

0

/2,h) of a strongly thick FGM plate versus the foundation stiffness Kw and

-3 -2 -1 0 1 20

0.2

0.4

0.6

0.8

1

106u(0,l/2,z)/h

z/h

-10 -8 -6 -4 -2 00

0.2

0.4

0.6

0.8

1

106w(l/2,l/2,z)/h

z/h

-1 -0.8 -0.6 -0.4 -0.2 00

0.2

0.4

0.6

0.8

1

σz (l/2,l/2,z)/q0

z/h

-0.8 -0.6 -0.4 -0.2 00

0.2

0.4

0.6

0.8

1

τxz(0,l/2,z)/q

0

z/h

-4 -3 -2 -1 0 10

0.2

0.4

0.6

0.8

1

σx(l/2,l/2,z)/q

0

z/h

-0.5 0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

τxy

(0,0,z)/q0

z/h

0hE E= 010hE E= 050hE E=

Fig. 8. Through-thickness distributions of displacements and stresses of strongly thick FGM plates with different values of gradient index or Eh/E0 on aPasternak foundation. (Case 1, h/l = 0.5, Kw = 100, Kp = 10).

Z.Y. Huang et al. / Composite Structures 85 (2008) 95–104 103

ulus at the upper surface (Eh) is taken to be eE, 10eE, and50eE, respectively, while that at the lower surface remainingunchanged (E0 ¼ eEÞ. It is observed that, with the increas-ing gradient index, i.e., the variation of Young’s modulusbecomes increasingly abrupt through the thickness, thevariations of displacements along the thickness directionbecome more gentle, whereas the stresses vary in anincreasing abrupt manner. Meanwhile, the plane whereno in-plane normal stress occurs moves increasinglytoward the harder surface.

4. Conclusions

Exact three-dimensional elasticity solutions are pre-sented for FGM thick plates resting on a Winkler–Paster-nak elastic foundation, using the state space method.Illustrating examples are carried out, with the most impor-

tant conclusions that the effects of foundation stiffness onmechanical responses of the plate are considerably differ-ent, especially for the strongly thick plates, and that, fora given FGM thick plate, the mechanical behavior of theplate with the softer surface supported by elastic founda-tion differ significantly from that of the plate with theharder surface subjected to the same foundation. Numeri-cal results given in the present paper render a benchmarkfor the analyses of FGM thick plates on elastic foundationsin the future. Effects of elastic foundations on thermo-mechanical behavior of FGM thick plates will fall intothe scope of subsequent investigations.

Acknowledgements

This work was supported by the National Natural Sci-ence Foundation of China (Nos. 10432030 and 10702061)

104 Z.Y. Huang et al. / Composite Structures 85 (2008) 95–104

and the China Postdoctoral Science Foundation (No.20060401071).

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