Nonlinear response of shear deformable FGM curved panels resting on elastic foundations and...

Applied Mathematical Modelling xxx (2013) xxx–xxx

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Applied Mathematical Modelling

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Nonlinear response of shear deformable FGM curved panelsresting on elastic foundations and subjected to mechanicaland thermal loading conditions

0307-904X/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.apm.2013.11.015

⇑ Corresponding author. Tel.: +84 906193585.E-mail address: [email protected] (H.V. Tung).

Please cite this article in press as: H.V. Tung, N.D. Duc, Nonlinear response of shear deformable FGM curved panels resting on elastdations and subjected to mechanical and thermal loading conditions, Appl. Math. Modell. (2013), http://dx.doi.org/1j.apm.2013.11.015

Hoang Van Tung a,⇑, Nguyen Dinh Duc b

a Faculty of Civil Engineering, Hanoi Architectural University, Hanoi, Viet Namb University of Engineering and Technology, Vietnam National University, Hanoi, Viet Nam

a r t i c l e i n f o

Article history:Received 13 December 2011Received in revised form 19 September 2013Accepted 22 November 2013Available online xxxx

Keywords:Functionally graded materialsCurved panelsHigher order shear deformation shell theoryElastic foundationImperfection

a b s t r a c t

This paper presents an analytical investigation on the nonlinear response of thick function-ally graded doubly curved shallow panels resting on elastic foundations and subjected tosome conditions of mechanical, thermal, and thermomechanical loads. Material propertiesare assumed to be temperature independent, and graded in the thickness direction accord-ing to a simple power law distribution in terms of the volume fractions of constituents. Theformulations are based on higher order shear deformation shell theory taking into accountgeometrical nonlinearity, initial geometrical imperfection and Pasternak type elastic foun-dation. By applying Galerkin method, explicit relations of load-deflection curves for simplysupported curved panels are determined. Effects of material and geometrical properties, in-plane boundary restraint, foundation stiffness and imperfection on the buckling and post-buckling loading capacity of the panels are analyzed and discussed. The novelty of thisstudy results from accounting for higher order transverse shear deformation andpanel-foundation interaction in analyzing nonlinear stability of thick functionally gradedcylindrical and spherical panels.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

Functionally Graded Materials (FGMs), which are microscopically composites and made from mixture of metal and cera-mic constituents, have received considerable attention in recent years due to their high performance heat resistance capacityand excellent characteristics in comparison with conventional composites. By continuously and gradually varying the vol-ume fraction of constituent materials through a specific direction, FGMs are capable of withstanding ultrahigh temperatureenvironments and extremely large thermal gradients. Therefore, these novel materials are chosen to use in temperatureshielding structure components of aircraft, aerospace vehicles, nuclear plants, and engineering structures in various indus-tries. As a result, buckling and postbuckling behaviors of FGM plate and shell structures under different types of loading arepractically important problems as prerequisite conditions for safe and optimal design. Linear buckling behavior of simplysupported perfect and imperfect FGM plates under thermal loads has been reported in works [1–5] by using an analyticalapproach and classical and shear deformation plate theories. Zhao et al. [6] analyzed mechanical and thermal buckling ofFGM plates by using element-free kp-Ritz method. Postbuckling behavior of pure and hybrid FGM plates under various con-ditions of mechanical, thermal, and electric loadings have been investigated by Liew et al. [7,8] using differential quadraturemethod, Shen [9,10] making use of asymptotic perturbation technique and Lee et al. [11] employing the element-free kp-Ritz

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2 H.V. Tung, N.D. Duc / Applied Mathematical Modelling xxx (2013) xxx–xxx

method. Subsequently, the postbuckling and geometrically nonlinear response of FGM plates and cylindrical panels undervarious loading types has been treated in works [12–16] by similar numerical methods and different shell theories. By mak-ing use of meshless method and Sander’s first order shear deformation shell theory, Zhao et al. [17] analyzed the static re-sponse and free vibration of metal-ceramic functionally graded cylindrical panels under mechanical and thermal loads.Recently, Liew et al. [18] presented a review article of meshless methods and shear deformation theories for laminatedand functionally graded plates and shells. Also, Duc and Tung [19] and Woo et al. [20] reported analytical investigationson the nonlinear response of thin and moderately thick FGM cylindrical panels subjected to mechanical and thermomechan-ical loads.

The components of structures widely used in aircraft, reusable space transportation vehicles, and civil engineering areusually supported by an elastic foundation. Therefore, it is necessary to include effects of elastic foundation for a betterunderstanding of the buckling behavior and loading carrying capacity of plates and shells. Librescu and his co-workers haveextended their analytical works [21–23] to investigate the postbuckling behavior of flat and curved laminated compositepanels resting on Winkler elastic foundations [24,25]. The behavior of FGM plates and shells resting on an elastic foundationor surrounded by an elastic medium have been considered in works [26–33]. Bending behavior of FGM plates on Pasternaktype foundations has been studied by Huang et al. [26] and Zenkour [27] using analytical methods, and Shen and Wang [28]making use of asymptotic perturbation technique. Recently, Shen [29] and Shen et al. [30] investigated the postbucklingbehavior of FGM cylindrical shells subjected to axial compressive loads and internal pressure and surrounded by an elasticmedium modeled as a tensionless Pasternak type foundation. More recently, the authors have studied the postbucklingbehavior of thick FGM plates and thin FGM cylindrical panels resting on elastic foundations. Specifically, Duc and Tung[31] and Duc and Cong [32] used the higher order shear deformation plate theory to study the postbuckling of FGM andmid-plane symmetric S-FGM plates resting on elastic foundations and subjected to in-plane compressive and thermal loads.Also, Tung [33] made use of classical shell theory to analyze mechanical and thermomechanical postbuckling of thin FGMcylindrical panels on Pasternak type elastic foundations taking tangential restraint of boundary edges into consideration.

This paper extends previous works [19,31–33] to present an analytical investigation on the nonlinear response of thickFGM doubly curved panels resting on elastic foundations and subjected to mechanical, thermal, and thermomechanicalloads. The formulations are based on the higher order shear deformation shell theory taking geometrical nonlinearity, initialgeometrical imperfection, and elastic foundations into consideration. Pasternak model is used to represent the panel–foundation interaction. Explicit expressions of buckling loads and load-deflection curves for simply supported FGM curvedshallow panels are determined by Galerkin method. The effects of material and geometrical properties, in-plane restraint,foundation stiffness, and imperfection on the nonlinear response of the panels are analyzed and discussed.

2. FGM doubly curved panels on elastic foundations

Consider a ceramic–metal FGM doubly curved panel of radii of curvature Rx, Ry, length of edges a, b and uniform thicknessh resting on an elastic foundation.A coordinate system ðx; y; zÞ is established in which ðx; yÞ plane on the middle surface of thepanel and z is thickness direction ð�h=2 6 z 6 h=2Þ as shown in Fig. 1.

The volume fractions of constituents are assumed to vary through the thickness according to the following power lawdistribution

Pleasedationj.apm.

VmðzÞ ¼2zþ h

2h

� �N

; VcðzÞ ¼ 1� VmðzÞ; ð1Þ

where N P 0 is volume fraction index. Effective properties Preff of FGM panel are determined by linear rule of mixture as

Preff ðzÞ ¼ PrmVmðzÞ þ PrcVcðzÞ; ð2Þ

where Pr denotes a temperature independent material property, and subscripts m and c stand for the metal and ceramic con-stituents, respectively.

Specific expressions of the modulus of elasticity E, the coefficient of thermal expansion a and the coefficient of thermalconduction K are obtained by substituting Eq. (1) into Eq. (2) as

x

y

zh

Rx Ryshear layer

a b

Fig. 1. Geometry and coordinate system of an FGM doubly curved panel on an elastic foundation.

cite this article in press as: H.V. Tung, N.D. Duc, Nonlinear response of shear deformable FGM curved panels resting on elastic foun-s and subjected to mechanical and thermal loading conditions, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/2013.11.015



H.V. Tung, N.D. Duc / Applied Mathematical Modelling xxx (2013) xxx–xxx 3

Pleasedationj.apm.

½EðzÞ;aðzÞ;KðzÞ� ¼ ½Ec;ac;Kc� þ ½Emc;amc;Kmc�2zþ h

2h

� �N

; ð3Þ

where

Emc ¼ Em � Ec; amc ¼ am � ac; Kmc ¼ Km � Kc; ð4Þ

and Poisson ratio m is assumed to be constant. It is evident from Eqs. (3), (4) that the upper surface of the panel (z = � h/2) isceramic-rich, while the lower surface (z = h/2) is metal-rich, and the percentage of ceramic constituent in the FGM panel isenhanced as N increases.

The FGM panel–foundation interaction is represented by Pasternak model as

qf ¼ k1w� k2r2w; ð5Þ

where r2 = o2/ox2 + o2/oy2, w is the deflection of the panel, k1 is Winkler foundation modulus and k2 is the shear layer foun-dation stiffness of Pasternak model.

3. Theoretical formulation

In this study, higher order shear deformation shell theory developed by Reddy and Liu [34] is used to establish governingequations and determine the nonlinear response of FGM curved panels. According to this theory, normal strains ex; ey,in-plane shear strain cxy and transverse shear deformations cxz; cyz are represented as

ex

ey

cxy

0B@

1CA ¼

e0x

e0y

c0xy

0B@

1CAþ z

k1x

k1y

k1xy

0BB@

1CCAþ z3

k3x

k3y

k3xy

0BB@

1CCA; ð6Þ

cxz

cyz

!¼

c0xz

c0yz

!þ z2 k2

xz

k2yz

!; ð7Þ

where

e0x

e0y

c0xy

0BB@

1CCA ¼

u;x �w=Rx þw2;x=2

v ;y �w=Ry þw2;y=2

u;y þ v ;x þw;xw;y

0B@

1CA;

k1x

k1y

k1xy

0BBB@

1CCCA ¼

/x;x

/y;y

/x;y þ /y;x

0B@

1CA;

k3x

k3y

k3xy

0BBB@

1CCCA ¼ �c1

/x;x þw;xx

/y;y þw;yy

/x;y þ phiy;x þ 2w;xy

0B@

1CA;

c0xz

c0yz

!¼

/x þw;x

/y þw;y

!;

k2xz

k2yz

!¼ �3c1

/x þw;x

/y þw;y

!;

ð8Þ

in which c1 = 4/3h2 and geometrical nonlinearity is incorporated. Also, u;v are the displacement components along the x, ydirections, respectively, and /x;/y are the rotations of normals to the midsurface with respect to y and x axes, respectively.

Hooke law for an FGM panel is defined as

ðrx;ryÞ ¼E

ð1� m2Þ ½ðex; eyÞ þ mðey; exÞ � ð1þ mÞaDTð1;1Þ�;

ðrxy;rxz;ryzÞ ¼E

2ð1þ mÞ ðcxy; cxz; cyzÞ;ð9Þ

where DT is temperature rise from stress free initial state. The force and moment resultants of the FGM panel are deter-mined as

ðNi;Mi; PiÞ ¼Z h=2

�h=2rjð1; z; z3Þdz; i ¼ x; y; xy;

ðQ i;KiÞ ¼Z h=2

�h=2rjð1; z2Þdz; i ¼ x; y; j ¼ xz; yz:

ð10Þ





Substitution of Eqs. (6), (7) into Eq. (9) and the result into Eqs. (10) give the constitutive relations in the form as

Pleasedationj.apm.

ðNx;Mx; PxÞ ¼1

1� m2 ðE1; E2; E4Þðe0x þ me0

yÞ þ ðE2; E3; E5Þðk1x þ mk1

yÞ þ ðE4; E5; E7Þðk3x þ mk3

yÞ � ð1þ mÞðU1;U2;U4Þh i

;

ðNy;My; PyÞ ¼1

1� m2 ðE1; E2; E4Þðe0y þ me0

x Þ þ ðE2; E3; E5Þðk1y þ mk1

x Þ þ ðE4; E5; E7Þðk3y þ mk3

x Þ � ð1þ mÞðU1;U2;U4Þh i

;

ðNxy;Mxy; PxyÞ ¼1

2ð1þ mÞ ½ðE1; E2; E4Þc0xy þ ðE2; E3; E5Þk1

xy þ ðE4; E5; E7Þk3xy�;

ðQ x;KxÞ ¼1

2ð1þ mÞ ½ðE1; E3Þc0xz þ ðE3; E5Þk2

xz�;

ðQ y;KyÞ ¼1

2ð1þ mÞ ½ðE1; E3Þc0yz þ ðE3; E5Þk2

yz�;

ð11Þ

where

ðE1; E2; E3; E4; E5; E7Þ ¼Z h=2

�h=2EðzÞð1; z; z2; z3; z4; z6Þdz;

ðU1;U2;U4Þ ¼Z h=2

�h=2EðzÞaðzÞDTðzÞð1; z; z3Þdz;

ð12Þ

and specific expressions of coefficients Ei ði ¼ 1� 7Þ are given in Eqs. (A1) in Appendix A. In the present study the temper-ature is assumed to be uniformly raised or steadily conducted only in the thickness direction.

The nonlinear equilibrium equations of a perfect FGM doubly curved shallow panel based on the higher order shear defor-mation theory are [34]

Nx;x þ Nxy;y ¼ 0; ð13aÞ

Nxy;x þ Ny;y ¼ 0; ð13bÞ

Qx;x þ Q y;y � 3c1ðKx;x þ Ky;yÞ þ c1ðPx;xx þ 2Pxy;xy þ Py;yyÞ þNx

Rxþ Ny

Ryþ Nxw;xx

þ 2Nxyw;xy þ Nyw;yy þ q� k1wþ k2r2w ¼ 0; ð13cÞ

Mx;x þMxy;y � Q x þ 3c1Kx � c1ðPx;x þ Pxy;yÞ ¼ 0; ð13dÞ

Mxy;x þMy;y � Q y þ 3c1Ky � c1ðPxy;x þ Py;yÞ ¼ 0; ð13eÞ

where q is external pressure uniformly distributed on the upper surface of the panel and the panel–foundation interactionhas been included. The last three equations of Eqs. (13) may be rewritten into two equations in terms of variables w and/x,x + /y,y by substituting Eqs. (11) into Eqs. (13c)–(13e). Subsequently, elimination of the variable /x,x + /y,y from two theresulting equations leads to the following system of equilibrium equations

Nx;x þ Nxy;y ¼ 0;

Nxy;x þ Ny;y ¼ 0;

c21ðD2D5=D4 � D3Þr6wþ ðc1D2=D4 þ 1ÞD6r4w

þ ð1� c1D5=D4Þr2 Nxw;xx þ 2Nxyw;xy þ Nyw;yy þNx

Rxþ Ny

Ryþ q� k1wþ k2r2w

� �

� D6=D4 Nxw;xx þ 2Nxyw;xy þ Nyw;yy þNx

Rxþ Ny


� �¼ 0;

ð14Þ

where

D1 ¼E1E3 � E2

2

E1ð1� m2Þ ; D2 ¼E1E5 � E2E4

E1ð1� m2Þ ; D3 ¼E1E7 � E2

4

E1ð1� m2Þ ;

D4 ¼ D1 � c1D2; D5 ¼ D2 � c1D3; D6 ¼1

2ð1þ mÞ ðE1 � 6c1E3 þ 9c21E5Þ:

ð15Þ

For an imperfect FGM curved panel, Eqs. (14) are modified into the form as





Pleasedationj.apm.

c21ðD2D5=D4 � D3Þr6wþ ðc1D2=D4 þ 1ÞD6r4wþ ð1

� c1D5=D4Þr2 f;yyðw;xx þw�;xxÞ � 2f ;xyðw;xy þw�;xyÞ þ f;xxðw;yy þw�;yyÞ þf;yy

Rxþ f;xx


� �

� D6=D4 f;yyðw;xx þw�;xxÞ � 2f ;xyðw;xy þw�;xyÞ þ f;xxðw;yy þw�;yyÞ þf;yy

Rxþ f;xx


� �¼ 0; ð16Þ

in which w⁄(x, y) is a known function representing initial small imperfection of the panel. Note that the termsr6w andr4ware unchanged because these terms are obtained from the expressions for bending moments Mij and higher order momentsPij and these moments depend not on the total curvature but only on the change in curvature of the panel [4]. Also, f(x, y) is astress function defined as

Nx ¼ f;yy; Ny ¼ f;xx; Nxy ¼ �f;xy: ð17Þ

The geometrical compatibility equation for an imperfect doubly curved shallow panel is written as

e0x;yy þ e0

y;xx � c0xy;xy ¼ w2

;xy �w;xxw;yy þ 2w;xyw�;xy �w;xxw�;yy �w;yyw�;xx �w;yy

Rx�w;xx

Ry: ð18Þ

From the constitutive relations (11) in conjunction with Eq. (17) one can write

ðe0x ; e

0yÞ ¼

1E1ðf;yy; f;xxÞ � mðf;xx; f;yyÞ � E2ðk1

x ; k1yÞ � E4ðk3

x ; k3yÞ þU1ð1;1Þ

h i;

c0xy ¼ �

1E1

2ð1þ mÞf;xy þ E2k1xy þ E4k3

xy

h i:

ð19Þ

Setting Eqs. (19) into Eq. (18) gives the compatibility equation of an imperfect FGM doubly curved panel as

r4f � E1 w2;xy �w;xxw;yy þ 2w;xyw�;xy �w;xxw�;yy �w;yyw�;xx �

w;yy

Rx�w;xx

Ry

� �¼ 0: ð20Þ

Eqs. (16) and (20) are nonlinear equations in terms of variables w and f and used to investigate the stability of thick FGMdoubly curved panels resting on elastic foundations and subjected to mechanical, thermal, and thermomechanical loads. Asthe curvature radii of FGM doubly curved panel tend to infinity, i.e. Rx ?1 and Ry ?1, Eqs. (16) and (20) reduce to equi-librium and compatibility equations obtained in [31] for thick FGM rectangular plates on elastic foundations. Also, special-ization of Eqs. (16) and (20) for case of Rx ?1 and neglecting transverse shear deformation, gives corresponding equationsderived in work [33] for thin FGM cylindrical panels on elastic foundations.

In the present study, the edges of curved panels are assumed to be simply supported. Depending on the in-plane restraintat the edges, three cases of boundary conditions, labelled as Cases 1, 2, and 3 will be considered [12,20–22].

Case 1. Four edges of the panel are simply supported and freely movable (FM). The associated boundary conditions are

w ¼ Nxy ¼ /y ¼ Mx ¼ Px ¼ 0; Nx ¼ Nx0 at x ¼ 0; a;

w ¼ Nxy ¼ /x ¼ My ¼ Py ¼ 0; Ny ¼ Ny0 at y ¼ 0; b:ð21Þ

Case 2. Four edges of the panel are simply supported and immovable (IM). In this case, boundary conditions are

w ¼ u ¼ /y ¼ Mx ¼ Px ¼ 0; Nx ¼ Nx0 at x ¼ 0; a;

w ¼ v ¼ /x ¼ My ¼ Py ¼ 0; Ny ¼ Ny0 at y ¼ 0; b:ð22Þ

Case 3. All edges are simply supported. Two edges x ¼ 0; a are freely movable, whereas the remaining two edges y ¼ 0; b areimmovable. For this case, the boundary conditions are defined as

w ¼ Nxy ¼ /y ¼ Mx ¼ Px ¼ 0; Nx ¼ Nx0 at x ¼ 0; a;

w ¼ v ¼ /x ¼ My ¼ Py ¼ 0; Ny ¼ Ny0 at y ¼ 0; b;ð23Þ

where Nx0;Ny0 are in-plane compressive loads at movable edges (i.e. Case 1 and the first of Case 3) or are fictitious compres-sive edge loads at immovable edges (i.e. Case 2 and the second of Case 3).

The approximate solutions of w and f satisfying boundary conditions (21)–(23) are assumed to be [19–22]

ðw;w�Þ ¼ ðW;lhÞ sin kmx sin dny; ð24aÞ

f ¼ A1 cos 2kmxþ A2 cos 2dnyþ A3 sin kmx sin dnyþ 12

Nx0y2 þ 12

Ny0x2; ð24bÞ

/x ¼ B1 cos kmx sin dny;/y ¼ B2 sin kmx cos dny; ð24cÞ





where km ¼ mp=a, dn = np/b, W is amplitude of the deflection and l is imperfection parameter. The coefficients Ai ði ¼ 1� 3Þare determined by substitution of Eqs. (24a,b) into Eq. (20) as

Pleasedationj.apm.

A1 ¼E1d

2n

32k2m

WðW þ 2lhÞ; A2 ¼E1k

2m

32d2n

WðW þ 2lhÞ; A3 ¼E1

ðk2m þ d2

nÞ2

d2n

Rxþ k2

m

Ry

!W: ð25Þ

Similarly, setting Eqs. (8) and (11) into Eqs. (13d,e) and introduction of Eqs. (24a,c) into the resulting equations, the coef-ficients B1;B2 are obtained as

B1 ¼a12a23 � a22a13

a212 � a11a22

W; B2 ¼a12a13 � a11a23

a212 � a11a22

W; ð26Þ

where

ða11; a22; a12Þ ¼ ðc21D3 þ D1 � 2c1D2Þðk2

m; d2n; mkmdnÞ þ

1� m2ðc2

1D3 þ D1 � 2c1D2Þðd2n; k

2m; kmdnÞ þ D6ð1;1;0Þ;

ða13; a23Þ ¼ c1D5ðk3m þ kmd2

n; d3n þ dnk

2mÞ � D6ðkm; dnÞ:

ð27Þ

Subsequently, Eqs. (24a,b) are substituted into Eq. (16) and applying the Galerkin method for the resulting equation asprocedure developed in [21–25]. Specifically, introduction of Eqs. (24a,b) into Eq. (16) and then multiplicating the obtainedequation by sin kmx sin dny and integration the result over the panel area yield

�c21ðD2D5 � D3D4Þ

D6ðk2

m þ d2nÞ

3 þ ðc1D2 þ D4Þðk2m þ d2

nÞ2 þ D4 � c1D5

D6ðk2

m þ d2nÞ þ 1

� �½k1 þ k2ðk2

m þ d2nÞ�

�

þ E1

ðk2m þ d2

nÞ2

d2n

Rxþ k2

m

Ry

!2ðD4 � c1D5Þ

D6ðk2

m þ d2nÞ þ 1

� �9=;W

� 32E1k2md2

n

3mnp2ðk2m þ d2

nÞ2

d2n

Rxþ k2

m

Ry

!2ðD4 � c1D5Þ

D6ðk2

m þ d2nÞ þ 1

� �WðW þ lhÞ

� 2E1

3mnp2

4k2md2

n D4 � c1D5ð ÞD6

1Rxþ 1

Ry

� �þ k2

m

Rxþ d2

n

Ry

" #WðW þ 2lhÞ

þ E1

16ðk4

m þ d4nÞðD4 � c1D5Þ

D6ðk2

m þ d2nÞ þ 1

� �WðW þ lhÞðW þ 2lhÞ

þ ðD4 � c1D5ÞD6

ðk2m þ d2

nÞ þ 1� �

ðNx0k2m þ Ny0d

2nÞðW þ lhÞ � 16

mnp2

Nx0

Rxþ Ny0

Ry

� �� 16q

mnp2 ¼ 0; ð28Þ

where m, n are odd numbers. This is basic equation governing the nonlinear response of thick FGM doubly curved shallowpanels under mechanical, thermal, and thermomechanical loading conditions. In what follows, some common loading con-ditions will be considered.

3.1. Mechanical nonlinear response

Consider a simply supported FGM curved panel with all movable edges and resting on an elastic foundation. Two cases ofmechanical loads will be analyzed.

3.1.1. FGM doubly curved panel under uniform external pressureConsider a simply supported FGM doubly curved panel with movable edges and only subjected to external pressure q

uniformly distributed on the upper surface of the panel. In this case, Nx0 = Ny0 = 0, and Eq. (28) is reduced to

q ¼ b11�W þ b21

�Wð �W þ lÞ þ b31�Wð �W þ 2lÞ þ b41

�Wð �W þ lÞð �W þ 2lÞ ð29Þ

where

b11 ¼mnp8ð�D3

�D4 � �D2�D5Þ

9B6h�D6

ðm2B2a þ n2Þ3 þmnp6

48B4h

ð4�D2 þ 3�D4Þðm2B2a þ n2Þ2 þmnp2B2

a�D1n

16B4h

½K1B2a þ p2K2ðm2B2

a þ n2Þ�

þ mnp2�E1B2an

16B2hðm2B2

a þ n2Þ2ðn2Rax þm2BaRbyÞ

2; ð30Þ

b21 ¼�2m2n2p2B3

a�E1ðn2Rax þm2BaRbyÞ

9B5h�D6ðm2B2

a þ n2Þ22p2ð3�D4 � 4�D5Þðm2B2

a þ n2Þ þ 3B2h�D6

h i;

b31 ¼��E1

72B5h�D6

m2n2p4B2að3�D4 � 4�D5ÞðBaRax þ RbyÞ þ 3p2B2

h�D6ðm2B3

aRax þ n2RbyÞh i

;

b41 ¼mnp6�E1n

256B4h

ðm4B4a þ n4Þ;





and

Pleasedationj.apm.

Bh ¼ b=h; Ba ¼ b=a; �W ¼W=h; Rax ¼ a=Rx; Rby ¼ b=Ry;

K1 ¼k1a4

D1; K2 ¼

k2a2

D1; �Ei ¼ Ei=hi ði ¼ 1� 7Þ;

�D1 ¼�E1

�E3 � �E22

�E1ð1� m2Þ; �D2 ¼

�E1�E5 � �E2

�E4

�E1ð1� m2Þ; �D3 ¼

�E1�E7 � �E2

4�E1ð1� m2Þ

;

�D4 ¼ �D1 �43

�D2; �D5 ¼ �D2 �43

�D3; �D6 ¼1

2ð1þ mÞ ð�E1 � 8�E3 þ 16�E5Þ;

n ¼ p2ð3�D4 � 4�D5Þ3B2

h�D6

ðm2B2a þ n2Þ þ 1:

ð31Þ

Eq. (29) is closed-form relation of pressure-deflection curves for movable edges FGM doubly curved panels under uniformexternal pressure. It is evident from Eq. (29) that pressure-loaded curved panels is not imperfection sensitivity. In otherwords, there is no dramatic difference between pressure-deflection curves of perfect and imperfect cylindrical panels.

3.1.2. FGM cylindrical panels under axial compressive loadsConsider a movable edges FGM cylindrical panel with radius of curvature R, axial direction along x axis and y axis is in

circumferential direction. The panel is supported by an elastic foundation and subjected to axial compressive loads Fx uni-formly distributed at two curved edges x = 0, a in the absence of external pressure and thermal loads. In this case, q = 0,Ny0 = 0, Nx0 = � Fxh, Rx ?1, Ry = R, and Eq. (28) leads to

Fx ¼ b12

�W�W þ l

þ b22�W þ b32

�Wð �W þ 2lÞ�W þ l

þ b42�Wð �W þ 2lÞ; ð32Þ

where

b12 ¼16p4ð�D3

�D4 � �D2�D5Þðm2B2

a þ n2Þ3

3m2B2aB2

h½p2ð3�D4 � 4�D5Þðm2B2a þ n2Þ þ 3B2

h�D6�þ p2 �D6ð4�D2 þ 3�D4Þðm2B2

a þ n2Þ2

m2B2a ½p2ð3�D4 � 4�D5Þðm2B2

a þ n2Þ þ 3B2h�D6�

þ�D1

m2p2B2h


a þ n2Þ� þ�E1m2B4

aða=RÞ2

p2ðm2B2a þ n2Þ2

;

b22 ¼ �32mn�E1B3

aða=RÞ½2p2ð3�D4 � 4�D5Þðm2B2a þ n2Þ þ 3B2

h�D6�

3p2Bhðm2B2a þ n2Þ2½p2ð3�D4 � 4�D5Þðm2B2


;

b32 ¼ �2n�E1Baða=RÞ½4m2p2B2

að3�D4 � 4�D5Þ þ 3B2h�D6�

3m3p2B2aBh½p2ð3�D4 � 4�D5Þðm2B2


;

b42 ¼p2�E1ðm4B4

a þ n4Þ16m2B2

aB2h

:

ð33Þ

Eq. (32) indicates the initial imperfection sensitivity of axially compressed cylindrical panels. Specifically, the imperfectpanels (l – 0) will be deflected at the onset of compression. In contrast, geometrically perfect FGM cylindrical panels (l = 0)exhibit a bifurcation type buckling behavior and buckling compressive loads can be predicted from Eq. (32) as Fxb = b12. Incase of transverse shear deformation is omitted and Rx ?1, Eqs. (29) and (32) are reduced to results reported in [33] forthin FGM cylindrical panels resting on elastic foundations and subjected to mechanical loads.

3.2. Thermal and thermomechanical nonlinear response

A simply supported FGM curved panel resting on an elastic foundation and with all immovable edges is considered. Thepanel is subjected to uniform external pressure q and simultaneously exposed to uniform temperature field or subjected tothrough the thickness temperature gradient. The in-plane condition on immovability at all edges, i.e. u = 0 at x ¼ 0; a andv = 0 at y ¼ 0; b, is fulfilled in an average sense as [21,22]
Z b
0

Z a

0

@u@x

dxdy ¼ 0;Z a

0

Z b

0

@v@y

dydx ¼ 0: ð34Þ

From Eqs. (8) and (11) one can obtain the following expressions in which Eq. (17) and imperfection have been included

@u@x¼ 1

E1ðf;yy � mf;xxÞ �

E2

E1/x;x þ

c1E4

E1ð/x;x þw;xxÞ �

12

w2;x �w;xw�;x þ

U1

E1þ w

Rx;

@v@y¼ 1

E1ðf;xx � mf;yyÞ �

E2

E1/y;y þ

c1E4

E1ð/y;y þw;yyÞ �

12

w2;y �w;yw�;y þ

U1

E1þ w

Ry:

ð35Þ





Substitution of Eqs. (24) into Eqs. (35) and then the result into Eqs. (34) give fictitious edge compressive loads as

Pleasedationj.apm.

Nx0 ¼�U1

1� m� 4

mnp2ð1� m2Þ ðE2 � c1E4ÞðkmB1 þ mdnB2Þ � c1E4ðk2m þ md2

nÞþ E11Rxþ m

Ry

� �� ð1� m2ÞE1d

2n

ðk2m þ d2

nÞ2

d2n

Rxþ k2

m

Ry

!" #W

þ E1

8ð1� m2Þ ðk2m þ md2

nÞWðW þ2lhÞ;

Ny0 ¼�U1

1� m� 4

mnp2ð1� m2Þ ðE2 � c1E4ÞðmkmB1 þ dnB2Þ � c1E4ðmk2m þ d2

nÞþ E1mRxþ 1

Ry

� �� ð1� m2ÞE1k

2m

ðk2m þ d2

nÞ2

d2n

Rxþ k2

m

Ry

!" #W

þ E1

8ð1� m2Þ ðmk2m þ d2

nÞWðW þ2lhÞ: ð36Þ

Specific expressions of parameter U1 in two cases of thermal loading will be determined.

3.2.1. Uniform temperature riseThe FGM curved panel is entirely exposed to thermal environments uniformly raised from stress free initial state Ti to

final value Tf and temperature difference DT = Tf � Ti is considered to be independent of thickness variable z. The thermalparameter is obtained from Eqs. (12) as

U1 ¼ LhDT; L ¼ Ecac þEcamc þ Emcac

N þ 1þ Emcamc

2N þ 1: ð37Þ

3.2.2. Through the thickness temperature gradientThe metal-rich surface temperature Tm is maintained at reference value while ceramic-rich surface temperature Tc is

elevated and nonlinear steady temperature conduction is governed by one-dimensional Fourier equation

ddz

KðzÞdTdz

� �¼ 0; Tðz ¼ �h=2Þ ¼ Tc; Tðz ¼ h=2Þ ¼ Tm: ð38Þ

Using K(z) in Eqs. (3), the solution of Eq. (38) may be found in terms of polynomial series, and the first eight terms of thisseries gives the following approximation

TðzÞ ¼ Tm þ DT � DTrP5

j¼0ð�rN Kmc=KcÞ

j

jNþ1P5j¼0ð�Kmc=KcÞj

jNþ1

ð39Þ

where r = (2z + h)/2h and DT = Tc � Tm is defined as the temperature change between two surfaces of the FGM panel.Introduction of Eq. (39) into Eq. (12) gives the thermal parameter as

U1 ¼ ðL� HÞhDT; ð40Þ

where

H ¼P5

j¼0ð�Kmc=KcÞj

jNþ1EcacjNþ2þ

EcamcþEmcacðjþ1ÞNþ2 þ

Emcamcðjþ2ÞNþ2

h iP5

j¼0ð�Kmc=KcÞj

jNþ1

: ð41Þ

Subsequently, setting Eq. (37) into Eq. (36) and then the result into Eq. (28) give

q ¼ b13�W þ b23

�Wð �W þ lÞ þ b33�Wð �W þ 2lÞ þ b43

�Wð �W þ lÞð �W þ 2lÞ � b53LDT

1� mð42Þ

where coefficients bj3 ðj ¼ 1� 5Þ are can be found in Eqs. (A2)–(A4) in Appendix A.Eq. (42) expresses explicit relation of pressure-deflection curves for FGM curved panels supported by elastic foundations

and under combined action of uniformly raised temperature field and uniform external pressure. A similar expression forFGM curved panels simultaneously subjected to uniform external pressure and temperature gradient across the thicknessmay be obtained as Eq. (42), provided L is replaced by (L � H).

In case of FGM curved panels with two movable edges x ¼ 0; a and the remaining edges y ¼ 0; b are immovable undercombination of uniform external pressure q and thermal loadings DT in the absence of edge compressive loads at movableedges x ¼ 0; a, explicit expressions of pressure-deflection curves take the form as Eq. (42) in which the coefficients bj3 arereplaced by coefficients bj4 ðj ¼ 1� 5Þ, respectively, displayed in detail in Eqs. (A5) in Appendix A.

From Eq. (42) and its temperature gradient counterpart, closed-form relations of temperature-deflection curves with andwithout the presence of pre-existent external pressure for FGM curved panels on elastic foundations with immovable edgesmay be obtained. However, such specific expressions are omitted here for sake of brevity.





4. Results and discussion

To validate the present formulation, a perfect simply supported FGM square plate without foundation interaction that iscomposed of Ti–6Al–4V and aluminum oxide and subjected to a uniform load q is considered. The elasticity moduli of Ti–6Al–4V and aluminum oxide as given in references [15–17] are Em = 105.7 � 109(N/m2) and Ec = 320.2 � 109(N/m2), respec-tively. Fig. 2 depicts the variation in the non-dimensional central deflection �W ¼W=h with the load parameter qa4/(Emh4) forthree various values of volume fraction index and the results are compared with those obtained by Zhao and Liew [15] usingthe element-free kp-Ritz method. To consist with the results reported in Ref. [15] and for sake of comparison, solid curves inFig. 2 are plotted from specialization of Eq. (29) for mentioned case of FGM plate in which Em and Ec have been interchangedand N⁄ is volume fraction index as the definition in Eq. (1) is given for Vc, i.e. VcðzÞ ¼ ð1=2þ z=hÞN

�. As can be seen, a very good

agreement is obtained in this comparison.As second example for verification of the present method, a perfect simply supported FGM cylindrical panel without foun-

dation interaction and made of silicon nitride (Si3N4) and stainless steel (SUS304) and subjected to uniform axial compres-sion is considered. The elasticity moduli of silicon nitride and stainless steel at room temperature (T0 = 300K), respectively,are Ec = 322.27 � 109 (N/m2) and Em = 207.79 � 109 (N/m2) [12,17], whereas Poisson ratio of both constituents is m = 0.28. Thecritical buckling loads are given in Table 1 and compared with those reported by Shen [12] using a singular perturbationtechnique. Again, a good agreement is observed in this comparison. The present predictions for the critical buckling loadsare slightly higher than Shen’s results (about 4%) because Ref. [12] used a boundary layer theory of shell buckling, whichincludes the effects of nonlinear prebuckling deformations and more precisely predict the buckling loads of cylindrical shells.

Fig. 2. Comparisons of nonlinear response for simply supported (Ti-6Al-4V/aluminum oxide) FGM plates.

Table 1Comparisons of critical buckling loads Fxcrbh (in MN) for perfect Si3N4/SUS304 cylindrical panels subjected to axial compression at room temperature (a/b = 1.2,b = 0.3m, a/R = 0.5, b/h = 30, m = 0.28, m = 1, n = 1, T0 = 300K).

N 0 0.2 0.5 1.0 2.0 5.0 8.0 10

Shen [12] 4.9565 5.4489 5.8836 6.2758 6.6488 7.0594 7.2280 7.2968Present 5.1946 5.7122 6.1670 6.5755 6.9632 7.3917 7.5691 7.6417

Table 2Critical buckling loads Fxcr (in GPa) of perfect FGM cylindrical panels without elastic foundations subjected to axialcompression (a/b = 1, K1 = 0, K2 = 0 movable edges).

N a/R = 0.3, b/h = 20 a/R = 0.5, b/h = 20 a/R = 0.5, b/h = 40

0 0.7835 (1,1)a 1.0672 (1,1) 0.5913 (3,1)0.5 1.7831 2.4856 1.35591.0 2.2047 3.1166 1.68445.0 3.2414 4.5721 2.474810 3.6110 5.0368 2.7468

a The number in brackets indicate the buckling mode (m, n).

Please cite this article in press as: H.V. Tung, N.D. Duc, Nonlinear response of shear deformable FGM curved panels resting on elastic foun-dations and subjected to mechanical and thermal loading conditions, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.11.015



Table 3Effects of foundation stiffness on the critical buckling loads Fxcr (in GPa) of perfect FGM cylindrical panels subjected to axial compression (a/b = 1.0, a/R = 0.5,b/h = 30, m = 1, n = 1 movable edges).

(K1, K2) N = 0 N = 0.2 N = 0.5 N = 1.0 N = 5.0 ceramic

(0,0) 0.7227 1.2487 1.7190 2.1817 3.1942 3.9233(50,0) 0.7588 1.3104 1.7992 2.2793 3.3380 4.1192

(100,0) 0.7949 1.3722 1.8794 2.3769 3.4818 4.3151(50,10) 0.9012 1.5542 2.1158 2.6648 3.9057 4.8925

(200,10) 1.0095 1.7395 2.3563 2.9577 4.3371 5.4801(50,20) 1.0437 1.7980 2.4323 3.0502 4.4734 5.6658

Fig. 3. Effects of the N index on the pressure-deflection curves of FGM spherical panels (immovable edges).

Fig. 4. Comparisons of pressure-deflection curves of FGM spherical panels for various cases of in-plane restraint.






The remainder of this section presents illustrative results for ceramic–metal curved panels made of aluminum andalumina with the following properties [2–5]

Pleasedationj.apm.

Em ¼ 70GPa; am ¼ 23� 10�6 oC�1; Km ¼ 204W=mK;

Ec ¼ 380GPa; ac ¼ 7:4� 10�6 oC�1; Kc ¼ 10:4W=mK;ð43Þ

Poisson ratio m = 0.3 and have a square planform (a = b).The effects of material distribution and geometrical parameters on the buckling behavior of FGM cylindrical panels under

axial compressive load are shown in Table 2. As can be observed, the critical buckling loads are increased as the volume frac-tion index N and length to curvature radius ratio a/R increase.

Fig. 5. Effects of the N index on the postbuckling of FGM cylindrical panels under axial compression (movable edges).

Fig. 6. Effects of the N index on the nonlinear response of FGM spherical panels under uniform temperature rise (immovable edges).





Table 3 illustrates the effects of foundation stiffness on the buckling loads of FGM cylindrical panels subjected to axialcompression. It is evident that the presence of elastic foundations enhances the buckling loads and dimensionless stiffnessK2 of Pasternak model has very beneficial and sensitive influence on the buckling resistance capacity of the panels.

To characterize the behavior of the panels, deformations in which the central region of a panel moves towards the con-cave side of the panel are referred to as inward deflections (or positive deflections). Deformations in the opposite directionare defined to as outward deflections (or negative deflections) [23]. In addition, the results presented in the following figurescorrespond to deformation mode with half-wave numbers m = n = 1, and unless otherwise stated, the FGM panel–foundationinteraction is ignored.

Fig. 3 shows the effects of the volume fraction index N on the nonlinear response of FGM spherical panels with immovableedges under uniform external pressure. As can be seen, although the pressure-deflection curves become higher when N in-creases, the snap-through behavior also to be more severe. Therefore, in spite of excellent temperature resistance capacity ofceramic, this constituent should be restricted in pressure-loaded panels to avoid brittle fracture.

Fig. 7. Effects of the N index on the nonlinear response of FGM spherical panels under temperature gradient (immovable edges).

Fig. 8. Effects of N on the nonlinear response of FGM cylindrical panels under uniform temperature rise (immovable edges).





A comparison of the nonlinear response of FGM spherical panels under external pressure and three various cases ofin-plane restraint is given in Fig. 4. It is evident that both pressure-deflection curves and the severity of snap-through insta-bility are enhanced as number of edges under in-plane restraint is increased.

Fig. 5 shows the effects of the volume fraction index N on the postbuckling behavior of FGM cylindrical panels withmovable edges under axial compressive loads. It is similar to situation in Fig. 3, the increase of buckling loads andload-deflection curves is paid by an unstable postbuckling behavior, i.e. a very intense snap-through phenomenon, whenN is increased.

Figs. 6 and 7 show the effects of the N index on the nonlinear response of FGM spherical panels with immovable edgesunder uniform temperature rise and through the thickness temperature gradient, respectively, whereas Fig. 8 gives the load-deflection curves of FGM cylindrical panels with immovable edges and various values of N under uniform temperature rise.

As can be observed, the panels deflect outwards under thermal loads and the response of the panels are relatively benign,i.e. there is no snap-through phenomenon and bifurcation type buckling.

Fig. 9. Effects of elastic foundations on the pressure-deflection curves of FGM cylindrical panels (immovable edges).

Fig. 10. Effects of elastic foundations on the pressure-deflection curves of FGM spherical panels (immovable edges).





In addition, FGM cylindrical panels exhibit a better capacity of temperature resistance with stable load-deflection curvesin comparison with that of FGM spherical panels. In fact, compressive forces are developed due to thermal loads andimmovability at edges and these forces make the panels to deflect towards convex side at the onset of thermal loadingby virtue of curvature of the panels. Furthermore, it is similar to case of pressure-loaded panels, FGM curved panels subjectedto thermal loads are not sensitive to initial imperfection. Specifically, there is no dramatic difference in the nonlinearresponse of perfect and imperfect FGM curved panels exposed to pressure or thermal loadings.

The effects of elastic foundations on the nonlinear response of pressure-loaded perfect FGM cylindrical and spherical pan-els with immovable edges are considered in Figs. 9 and 10, respectively. These figures show a beneficial influence of elasticmedia on the nonlinear response of panels. Specifically, the capability of load carrying is enhanced and the intensity of snap-through behavior is reduced due to the presence of elastic foundations. In addition, the shear layer stiffness K2 of Pasternak

Fig. 11. Effects of elastic foundations on the postbuckling curves of FGM cylindrical panels under axial compression (movable edges).

Fig. 12. Effects of elastic foundations on the load-deflection curves of FGM cylindrical panels under uniform temperature rise (immovable edges).





foundation model has more pronounced influence than the modulus K1 of Winkler model on the pressure withstanding andstabilizing of FGM curved panels.

Fig. 11 analyzes the effects of elastic foundations on the postbuckling behavior of FGM cylindrical panels withmovable edges and subjected to axial compression. Obviously, both buckling loads and postbuckling equilibriumpaths of FGM panels become considerably higher due to the support of elastic media, especially Pasternak type foun-dations. However, the severity of snap-through instability is almost unchanged with different values of foundationparameters.

Elastic foundations also have benefit to the loading bearing capability of FGM cylindrical panels with immovable edgesand exposed to thermal environments as shown in Fig. 12. Due to the tension of foundation spring (i.e. mutual interactionbetween the panel and foundation) the negative deflection of FGM panels is dropped.

Fig. 13 illustrates interactive effects of elastic foundations and imperfections on the pressure-deflection curves of FGMspherical panels with movable edges. As mentioned in [23], negative or positive imperfections produce perturbations inthe panel geometry that move the central region of a panel outward or inward, respectively. It seems, from this figure, that

Fig. 13. Interactive effects of elastic foundations and imperfections on the pressure-deflection curves of FGM spherical panels (movable edges).

Fig. 14. Effects of imperfection on the nonlinear response of FGM spherical panels under uniform temperature rise (immovable edges).




Fig. 15. Interactive effects of temperature environments and elastic foundations on the nonlinear response of FGM cylindrical panels under uniformexternal pressure (immovable edges).


effective curvature of panels is increased by negative imperfections and, as a result, the panels experience a more intensesnap-through response for the same foundation parameters. Nevertheless, the enhancement of foundation stiffnessconsiderably improves the loading bearing capacity and stability of pressure-deflection curves for both FGM panels withpositive and negative imperfections.

Fig. 14 assesses the effects of initial imperfection on the nonlinear response of FGM spherical panels withimmovable edges and subjected to uniform temperature rise. It is inverse to case of pressure-loaded FGM panels,negative imperfections have a beneficial and stabilizing influence on the behavior of heated FGM panels in deep regionof deflection.

Finally, Fig. 15 shows the interactive effects of temperature environments and elastic foundations on the nonlinearresponse of FGM cylindrical panels with immovable edges under uniform external pressure. It is seen that the panelsexhibit a bifurcation buckling behavior due to the presence of pre-existent thermal loads. Both buckling pressure andthe intensity of snap-through phenomenon are enhanced when the temperature field is elevated. In contrast, the severityof snapping is reduced and postbuckling curves become more stable by virtue of elastic foundations, especially Pasternaktype foundations.

5. Concluding remarks

The paper presents an analytical investigation on the nonlinear response of thick FGM doubly curved panels resting onelastic foundations and subjected to mechanical and thermal loading conditions. The formulations are based on the higherorder shear deformation shell theory taking into account geometrical nonlinearity, initial imperfection and elastic founda-tions. Galerkin method is used to obtain explicit expressions of load-deflection curves. The results show that elastic founda-tions, especially Pasternak type foundations have a very beneficial influence on the buckling loads and postbuckling loadcarrying capacity of FGM curved panels. Specifically, buckling loads and postbuckling strength are enhanced, and intensityof snap-through phenomenon of curved panels becomes more benign due to presence of elastic foundations. The study alsoshows the pronounced effects of volume fraction index, in-plane restraint and temperature conditions, and imperfection onthe nonlinear response of FGM curved panels.

Acknowledgment

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) underGrant number 107.01-2012.02.

Appendix A

The coefficients Ei in Eqs. (12)





Pleasedationj.apm.

E1 ¼ Echþ EmchN þ 1

; E2 ¼EmcNh2

2ðN þ 1ÞðN þ 2Þ ;

E3 ¼Ech3

12þ Emch3 1

4ðN þ 1Þ �1

ðN þ 2ÞðN þ 3Þ

� �;

E4 ¼Emch4

N þ 118� 3

4ðN þ 2Þ þ3

ðN þ 3ÞðN þ 4Þ

� �;

E5 ¼Ech5

80þ Emch5

N þ 11

16� 1

2ðN þ 2Þ þ3

ðN þ 2ÞðN þ 3Þ �12

ðN þ 2ÞðN þ 4ÞðN þ 5Þ

� �;

E7 ¼Ech7

448þ Emch7

N þ 11

64� 6

32ðN þ 2Þ þ30

16ðN þ 2ÞðN þ 3Þ �15

ðN þ 2ÞðN þ 3ÞðN þ 4Þ þ90

ðN þ 2ÞðN þ 3ÞðN þ 4ÞðN þ 5Þ

�

� 360ðN þ 2ÞðN þ 3ÞðN þ 4ÞðN þ 6ÞðN þ 7Þ

�:

ðA1Þ

The coefficients bj3 ðj ¼ 1� 5Þ in Eq. (42)

b13 ¼mnp8ð�D3

�D4 � �D2�D5Þ

9B6h�D6

ðm2B2a þ n2Þ3 þmnp6ð4�D2 þ 3�D4Þ

48B4h

ðm2B2a þ n2Þ2 þ

�E1mnp2B2anðn2Rax þm2BaRbyÞ

2

16B2hðm2B2

a þ n2Þ2

þmnp2 �D1B2an

16B4h


a þ n2Þ�

þ 43mnp2ð1� m2ÞB3

h

pBhð3�E2 � 4�E4Þ mB2aRax

�B1 þ mmBaRby�B1 þ mnBaRax

�B2 þ nRby�B2

� �h�4p2�E4ðm2B3

aRax þ mm2B2aRby þ mn2BaRax þ n2RbyÞ þ 3Bh

�E1ðB2aR2

ax þ 2mBaRaxRby þ R2byÞ

�3Bhð1� m2Þ�E1B2aðn2Rax þm2BaRbyÞ

2

ðm2B2a þ n2Þ2

#;

b23 ¼ �2�E1m2n2p2B2

að2n� 1Þ3B3

hðm2B2a þ n2Þ2

ðn2BaRax þm2B2aRbyÞ

þ p2n

12ð1� m2ÞB4h

4p2�E4ðm4B4a þ 2mm2n2B2

a þ n4Þ � pBhð3�E2 � 4�E4Þðm3B3a�B1 þ mm2nB2

a�B2 þ n3�B2 þ mmn2Ba

�B1Þh

�3Bh�E1ðm2B3

aRax þ mm2B2aRby þ mn2BaRax þ n2RbyÞ þ

6ð1� m2ÞBh�E1m2n2B3

a

ðm2B2a þ n2Þ2

ðn2Rax þm2BaRbyÞ#;

b33 ¼ �p4�E1m2n2B2

a

18B5h�D6

ð3�D4 � 4�D5ÞðBaRax þ RbyÞ �p2�E1

24ð1� m2ÞB3h

½ð4� m2Þðm2B3aRax þ n2RbyÞ þ 3mBaðm2BaRby þ n2RaxÞ�;

b43 ¼mnp6�E1n

256ð1� m2ÞB4h

½ð3� m2Þðm4B4a þ n4Þ þ 4mm2n2B2

a �;

b53 ¼mnp4nðm2B2

a þ n2Þ16B2

h

ð �W þ lÞ � 1BhðBaRax þ RbyÞ;

ðA2Þ

where

�B1 ¼�a12�a23 � �a22�a13

�a212 � �a11�a22

; �B2 ¼�a12�a13 � �a11�a23

�a212 � �a11�a22

ðA3Þ

and

ð�a11; �a22; �a12Þ ¼p2

B2h

169

�D3 þ �D1 �83

�D2

� �ðm2B2

a ;n2;mmnBaÞ þ

ð1� mÞp2

2B2h

169

�D3 þ �D1 �83

�D2

� �ðn2;m2B2

a ;mnBaÞ þ �D6ð1;1;0Þ;

ð�a13; �a23Þ ¼4p3 �D5

3B3h

ðm3B3a þmn2Ba;n3 þm2nB2

aÞ �p�D6

BhðmBa;nÞ:

ðA4Þ

The coefficients bj4 ðj ¼ 1� 5Þ in case of FGM curved panels with movable edges x ¼ 0; a and two edges y ¼ 0; b areimmovable.





Pleasedationj.apm.

b14 ¼mnp8ð�D3

�D4 � �D2�D5Þ

9B6h�D6

ðm2B2a þ n2Þ3 þmnp6ð4�D2 þ 3�D4Þ

48B4h

ðm2B2a þ n2Þ2 þ mnp2�E1B2

an

16B2hðm2B2

a þ n2Þ2ðn2Rax þm2BaRbyÞ

2

þmnp2 �D1B2an

16B4h

� ½K1B2a þ p2K2ðm2B2

a þ n2Þ� � 4�E1ðm2B2a � mn2ÞBa

mnp2B2hðm2B2

a þ n2Þ2ðn2RaxRby þm2BaR2

byÞ

þ 4�E2�B2Rby

mpB2h

� 16�E4Rby

3mpB3h

ðBh�B2 þ npÞ þ

4�E1R2by

mnp2B2h

;

b24 ¼ �2�E1m2n2p2B3

að2n� 1Þ3B3

hðm2B2a þ n2Þ2

ðn2Rax þm2BaRbyÞ

þ nn2p2�E1Baðm2B2

a � mn2Þ4B3

hðm2B2a þ n2Þ2

ðn2Rax þm2BaRbyÞ �n3p3�E2

�B2

4B3h

þ n3p3�E4

3B4h

ðBh�B2 þ npÞ � n2p2�E1Rby

4B3h

" #;

b34 ¼ ��E1

24B3h

4m2n2p4B2að3�D4 � 4�D5Þ

3B2h�D6

ðBaRax þ RbyÞ þ p2ðm2B3aRax þ n2RbyÞ

" #� n2p2�E1Rby

8B3h

;

b44 ¼mnp6�E1n

256B4h

ðm4B4a þ 3n4Þ; b54 ¼ ð1� mÞ mn3p4n

16B2h

ð �W þ lÞ � Rby

Bh

" #:

ðA5Þ

References

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