Composite Structures 102 (2013) 306–314

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Composite Structures

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Corrigendum

Corrigendum to ‘‘Nonlinear dynamic response of imperfect eccentricallystiffened FGM double curved shallow shells on elastic foundation’’[Compos. Struct. 99 (2013) 88–96]

0263-8223/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compstruct.2013.03.009

DOI of original article: http://dx.doi.org/10.1016/j.compstruct.2012.11.017⇑ Tel.: +84 4 37547989; fax: + 84 4 37547724.

E-mail address: ducnd@vnu.edu.vn

Nguyen Dinh Duc ⇑University of Engineering and Technology, Vietnam National University, 144 Xuan Thuy, Cau Giay, Hanoi, Viet Nam

a r t i c l e i n f o

Article history:Available online 10 April 2013

Keywords:Nonlinear dynamicEccentrically stiffened FGM double curvedshallow shellsImperfectionElastic foundation

a b s t r a c t

This paper presents an analytical investigation on the nonlinear dynamic response of eccentrically stiff-ened functionally graded double curved shallow shells resting on elastic foundations and being subjectedto axial compressive load and transverse load. The formulations are based on the classical shell theorytaking into account geometrical nonlinearity, initial geometrical imperfection and the Lekhnitskysmeared stiffeners technique with Pasternak type elastic foundation. The non-linear equations are solvedby the Runge–Kutta and Bubnov–Galerkin methods. Obtained results show effects of material and geo-metrical properties, elastic foundation and imperfection on the dynamical response of reinforced FGMshallow shells. Some numerical results are given and compared with ones of other authors.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Functionally Graded Materials (FGMs), which are microscopi-cally composites and made from mixture of metal and ceramicconstituents, have received considerable attention in recent yearsdue to their high performance heat resistance capacity and excel-lent characteristics in comparison with conventional composites.By continuously and gradually varying the volume fraction of con-stituent materials through a specific direction, FGMs are capable ofwithstanding ultrahigh temperature environments and extremelylarge thermal gradients. Therefore, these novel materials are cho-sen to use in temperature shielding structure components of air-craft, aerospace vehicles, nuclear plants and engineeringstructures in various industries. As a result, in recent years impor-tant studies have been researched about the stability and vibrationof FGM plates and shells.

The research on FGM shells and plates under dynamic load isattractive to many researchers in the world.

Firstly we have to mention the research group of Reddy et al.The vibration of functionally graded cylindrical shells has beeninvestigated by Loy, Lam and Reddy [1]. Lam and Hua has takeninto account the influence of boundary conditions on the frequencycharacteristics of a rotating truncated circular conical shell [2]. In[3] Pradhan et al. studied vibration characteristics of FGM cylindri-cal shells under various boundary conditions. Ng et al. studied the

dynamic stability analysis of functionally graded cylindrical shellsunder periodic axial loading [4]. The group of Ng and Lam also pub-lished results on generalized differential quadrate for free vibrationof rotating composite laminated conical shell with various bound-ary conditions in 2003 [5]. In the same year, Yang and Shen [6]published the nonlinear analysis of FGM plates under transverseand in-plane loads.

In 2004, Zhao et al. studied the free vibration of two-side sim-ply-supported laminated cylindrical panel via the mesh-free kp-Ritz method [7]. About vibration of FGM plates Vel and Batra [8]gave three dimensional exact solution for the vibration of FGMrectangular plates. Also in this year, Sofiyev and Schnack investi-gated the stability of functionally graded cylindrical shells underlinearly increasing dynamic tensional loading in [9] and obtainedthe result for the stability of functionally graded truncated conicalshells subjected to a periodic impulsive loading [10], and they pub-lished the result of the stability of functionally graded ceramic–metal cylindrical shells under a periodic axial impulsive loadingin 2005 [11]. Ferreira et al. received natural frequencies of FGMplates by meshless method [12], 2006. In [13], Zhao et al. usedthe element-free kp-Ritz method for free vibration analysis of con-ical shell panels. Liew et al. studied the nonlinear vibration of acoating-FGM-substrate cylindrical panel subjected to a tempera-ture gradient [14] and dynamic stability of rotating cylindricalshells subjected to periodic axial loads [15]. Woo et al. investigatedthe non linear free vibration behavior of functionally graded plates[16]. Kadoli and Ganesan studied the buckling and free vibrationanalysis of functionally graded cylindrical shells subjected to atemperature-specified boundary condition [17]. Also in this year,

h

Rx

a b

z

y Ry

x

Fig. 1. Geometry and coordinate system of an eccentrically stiffened double curvedshallow FGM shell on elastic foundation.

N.D. Duc / Composite Structures 102 (2013) 306–314 307

Wu et al. published their results of nonlinear static and dynamicanalysis of functionally graded plates [18]. Sofiyev has consideredthe buckling of functionally graded truncated conical shells underdynamic axial loading [19]. Prakash et al. studied the nonlinear axi-symmetric dynamic buckling behavior of clamped functionallygraded spherical caps [20]. In [21], Darabi et al. obtained the non-linear analysis of dynamic stability for functionally graded cylin-drical shells under periodic axial loading. Natural frequencies andbuckling stresses of FGM plates were analyzed by Matsunaga using2-D higher-order deformation theory [22]. In 2008, Shariyat alsoobtained the dynamic thermal buckling of suddenly heated tem-perature-dependent FGM cylindrical shells under combined axialcompression [23] and external pressure and dynamic buckling ofsuddenly loaded imperfect hybrid FGM cylindrical with tempera-ture-dependent material properties under thermo-electro-mechanical loads [24]. Allahverdizadeh et al. studied nonlinear freeand forced vibration analysis of thin circular functionally gradedplates [25]. Sofiyev investigated the vibration and stability behav-ior of freely supported FGM conical shells subjected to externalpressure [26], 2009. Shen published a valuable book ‘‘FunctionallyGraded materials, Non linear Analysis of plates and shells’’, in whichthe results about nonlinear vibration of shear deformable FGMplates are presented [27]. Last years, Zhang and Li published thedynamic buckling of FGM truncated conical shells subjected tonon-uniform normal impact load [28], Bich and Long studied thenon-linear dynamical analysis of functionally graded material shal-low shells subjected to some dynamic loads [29], Dung and Naminvestigated the nonlinear dynamic analysis of imperfect FGMshallow shells with simply supported and clamped boundary con-ditions [30]. Bich et al. has also considered the nonlinear vibrationof functionally graded shallow spherical shells [31].

In fact, the FGM plates and shells, as other composite struc-tures, ussually reinforced by stiffening member to provide thebenefit of added load-carrying static and dynamic capability witha relatively small additional weight penalty. Thus study on staticand dynamic problems of reinforced FGM plates and shells withgeometrical nonlinearity are of significant practical interest. How-ever, up to date, the investigation on static and dynamic of eccen-trically stiffened FGM structures has received comparatively littleattention. Recently, Bich et al. studied nonlinear dynamical analy-sis of eccentrically stiffened functionally graded cylindrical panels[32].

This paper presents an dynamic nonlinear response of doublecurved shallow eccentrically stiffened shells FGM resting on elasticfoundations and being subjected to axial compressive load andtransverse load. The formulations are based on the classical shelltheory taking into account geometrical nonlinearity, initial geo-metrical imperfection and the Lekhnitsky smeared stiffeners tech-nique with Pasternak type elastic foundation. The nonlineartransients response of doubly curved shallow shells subjected toexcited external forces obtained the dynamic critical bucklingloads are evaluated based on the displacement response usingthe criterion suggested by Budiansky–Roth. Obtained results showeffects of material, geometrical properties, eccentrically stiffened,elastic foundation and imperfection on the dynamical response ofFGM shallow shells.

2. Eccentrically stiffened double curved FGM shallow shell onelastic foundations

Consider a ceramic–metal stiffened FGM double curved shallowshell of radii of curvature Rx, Ry length of edges a, b and uniformthickness h resting on an elastic foundation.

A coordinate system (x, y, z) is established in which (x, y) planeon the middle surface of the panel and z is thickness direction (�h/2 6 z 6 h/2) as shown in Fig. 1.

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

VmðzÞ ¼2zþ h

2h

� �N

; VcðzÞ ¼ 1� VmðzÞ ð1Þ

where N is volume fraction index (0 6 N <1). Effective propertiesPreff of FGM panel are determined by linear rule of mixture as

Preff ðzÞ ¼ PrmVmðzÞ þ PrcVcðzÞ ð2Þ

where Pr denotes a material property, and subscripts m and c standfor the metal and ceramic constituents, respectively. Specificexpressions of modulus of elasticity E(z) and q(z) are obtained bysubstituting Eq. (1) into Eq. (2) as

½EðzÞ;qðzÞ� ¼ ðEm;qmÞ þ ðEcm;qcmÞ2zþ h

2h

� �N

ð3Þ

where

Ecm ¼ Ec � Em;qcm ¼ qc � qm; mðzÞ ¼ const ¼ m ð4Þ

It is evident from Eqs. (3), (4) that the upper surface of the panel(z = �h/2) is ceramic-rich, while the lower surface (z = h/2) is me-tal-rich, and the percentage of ceramic constituent in the panel isenhanced when N increases.

The panel–foundation interaction is represented by Pasternakmodel as

qe ¼ k1w� k2r2w ð5Þ

where r2 = @2/@x2 + @2/@ y2,w is the deflection of the panel, k1 isWinkler foundation modulus and k2 is the shear layer foundationstiffness of Pasternak model.

3. Theoretical formulation

In this study, the classical shell theory and the Lekhnitskysmeared stiffeners technique are used to obtain governing equa-tions and determine the nonlinear dynamical response of FGMcurved panels. The strain across the shell thickness at a distancez from the mid-surface are

ex

ey

cxy

0B@

1CA ¼

e0x

e0y

c0xy

0B@

1CA� z

kx

ky

2kxy

0B@

1CA ð6Þ

where e0x ; e0

x and c0xy are normal and shear strain at the middle sur-

face of the shell, and kx, ky,kxy are the curvatures. The nonlinearstrain–displacement relationship based upon the von Karman the-ory for moderately large deflection and small strain are:

x1

x2

ab

s2

hz2

z1

1

2

s2 s2 s2s1s1s1s1

b

O

z

Fig. 2. Configuration of an eccentrically stiffened shallow shells

308 N.D. Duc / Composite Structures 102 (2013) 306–314

e0x

e0y

c0xy

0B@

1CA 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;

kx

ky

kxy

0B@

1CA ¼

wx;x

wy;y

w;xy

0B@

1CA ð7Þ

In which u, v are the displacement components along the x, ydirections, respectively.

The force and moment resultants of the FGM panel are deter-mined by

ðNi;MiÞ ¼Z h=2

�h=2rið1; zÞdz i ¼ x; y; xy ð8Þ

The constitutive stress–strain equations by Hooke law for theshell material are omitted here for brevity. The shell reinforcedby eccentrically longitudinal and transversal stiffeners is shownin Fig. 1. The shallow shell is assumed to have a relative small riseas compared with its span. The contribution of stiffeners can be ac-counted for using the Lekhnitsky smeared stiffeners technique.Then intergrading the stress–strain equations and their momentsthrough the thickness of the shell, the expressions for force andmoment resultants of an eccentrically stiffened FGM shallow shellare obtained [32]:

Nx ¼E1

1� m2 þEA1

s1

� �e0

x þE1m

1� m2 e0y �

E2

1� m2 þ C1

� �kx �

E2m1� m2 ky

Ny ¼E1m

1� m2 e0x þ

E1

1� m2 þEA2

s2

� �e0

y �E2m

1� m2 kx �E2

1� m2 þ C2

� �ky

Mx ¼E2

1� m2 þ C1

� �e0

x þE2m

1� m2 e0y �

E3

1� m2 þEI1

s1

� �kx �

E3m1� m2 ky

My ¼E2m

1� m2 e0x þ

E2

1� m2 þ C2

� �e0

y �E3m

1� m2 kx �E3

1� m2 þEI2

s2

� �ky

Nxy ¼1

2ð1þ mÞ E1c0xy � 2E2kxy

� �

Mxy ¼1

2ð1þ mÞ E2c0xy � 2E3kxy

� �ð9Þ

where

E1 ¼ Em þEcm

N þ 1

� �h

E2 ¼EcmNh2

2ðN þ 1ÞðN þ 2Þ

E3 ¼Em

12þ Ecm

1N þ 3

� 1N þ 2

þ 14N þ 4

� �� �h3

C1 ¼EA1z1

s1; C2 ¼

EA2z2

s2

ð10Þ

In above relations (9) and (10), the quantities A1, A2 are the crosssection areas of the stiffeners and I1, I2, z1, z2 are the second mo-ments of cross section areas and eccentricities of the stiffeners withrespect to the middle surface of the shell respectively, E is elasticitymodulus in the axial direction of the corresponding stiffener witchis assumed identical for both types of stiffeners (Fig. 2). In order toprovide continuity between the shell and stiffeners, suppose thatstiffeners are made of full metal (E = Em) if putting them at the me-tal-rich side of the shell, and conversely full ceramic stiffeners(E = Ec) at the ceramic-rich side of the shell [32].

The nonlinear dynamic equations of a FGM shallow shells basedon the classical shell theory are [33]

Nx;x þ Nxy;y ¼ q@2u@t2

Nxy;x þ Ny;y ¼ q@2v@t2

Mx;xx þ 2Mxy;xy þMy;yy þNx

Rxþ Ny

Ryþ Nxw;xx þ 2Nxyw;xy þ Nyw;yy þ q

� k1wþ k2r2w ¼ q@2w@t2 ð11Þ

where

q ¼Z h

2

�h2

qðzÞdzþ q0A1

s1þ A2

s2

� �¼ qm þ

qcm

N þ 1

� �hþ q0

A1

s1þ A2

s2

� �

ð12Þ

in which q0 is the mass density of stiffeners; q @2u@t2 ! 0 and q @2v

@t2 ! 0into consideration because of u� w,v� w the Eq. (11) can berewritten as:

Mx;xx þ 2Mxy;xy þMy;yy þNx

Rxþ Ny

Ryþ Nxw;xx þ 2Nxyw;xy þ Nyw;yy þ q

� k1wþ k2r2w ¼ q@2w@t2 ð13Þ

Calculating from Eq. (9), obtained:

e0x ¼ A22Nx � A12Ny þ B11kx þ B12ky

e0y ¼ A11Ny � A12Nx þ B21kx þ B22ky

c0xy ¼ A66Nxy þ 2A66kxy

ð14Þ

where

A11 ¼1D

EA1

s1þ E1

1� m2

� �; A22 ¼

1D

EA2

s2þ E1

1� m2

� �

A12 ¼1D

E1m1� m2 ; A66 ¼

2ð1þ mÞE1

D ¼ EA1

s1þ E1

1� m2

� �EA2

s2þ E1

1� m2

� �� E1m

1� m2

� �2

B11 ¼ A22 C1 þE2

1� m2

� �� A12

E2m1� m2 ; B22 ¼ A11 C2 þ

E2

1� m2

� �� A12

E2m1� m2

B12 ¼ A22E2m

1� m2 � A12E2

1� m2 þ C2

� �; B21 ¼ A11

E2m1� m2 � A12

E2

1� m2 þ C1

� �

B66 ¼E2

E1

ð15Þ

Substituting once again Eq. (14) into the expression of Mij in (9)leads to:

Mx ¼ B11Nþx B21N�y D11kx � D12ky

Mx ¼ B12Nþx B22N�y D21kx � D22ky

Mxy ¼ B66N�xy2D66kxy

ð16Þ

N.D. Duc / Composite Structures 102 (2013) 306–314 309

where

D11 ¼EI1

s1þ E3

1� m2 � C1 þE2

1� m2

� �B11 �

E2m1� m2 B21

D22 ¼EI2

s2þ E3

1� m2 � C2 þE2

1� m2

� �B22 �

E2m1� m2 B12

D12 ¼E3m

1� m2 � C1 þE2

1� m2

� �B12 �

E2m1� m2 B22

D21 ¼E3m

1� m2 � C2 þE2

1� m2

� �B21 �

E2m1� m2 B11

D66 ¼E3

2ð1þ mÞ �E2

2ð1þ mÞB66

ð17Þ

Then Mij into Eq. (13) and f(x, y) is stress function defined by

Nx ¼ f;yy; Ny ¼ f;xx; Nxy ¼ �f;xy ð18Þ

For an imperfect FGM curved panel, Eq. (13) are modified intoform

B21f;xxxx þ B12f;yyyy þ ðB11 þ B22 � 2B66Þf;xxyy � D11w;xxxx

� D22w;yyyy � ðD12 þ D21 þ 4D66Þw;xxyy þ D11w�;xxxx

þ D22w�;yyyy þ ðD12 þ D21 þ 4D66Þw�;xxyy þ f;yyw;xx � 2f ;xyw;xy

þ f;xxw;yy þf;yy

Rxþ f;xx

Ryþ q� k1wþ k2r2w ¼ q

@2w@t2 ð19Þ

in which w⁄(x, y) is a known function representing initial smallimperfection of the eccentrically stiffened shallow shells. The geo-metrical compatibility equation for an imperfect shallow shells iswritten

e0x;yy þ e0

y;xx � c0xy;xy ¼ w2

;xy �w;xxw;yy �w�2;xy þw�;xxw�;yy

�w;yy �w�;yy

Rx�

w;xx �w�;xx

Ry: ð20Þ

From the constitutive relations (18) in conjunction with Eq. (14)one can write

e0x ¼ A22f;yy � A12f;xx þ B11w;xx þ B12w;yy

e0y ¼ A11f;xx � A12f;yy þ B21w;xx þ B22w;yy

c0xy ¼ �A66f;xy þ 2A66w;xyÞ

ð21Þ

Setting Eq. (21) into Eq. (20) gives the compatibility equation ofan imperfect eccentrically stiffened shallow FGM shells as

A11f;xxxx þ ðA66 � 2A12Þf;xxyy þ A22f;yyyy þ B21w;xxxx

þ ðB11 þ B22 � 2B66Þw;xxyy þ B12w;yyyy ¼ w2;xy �w;xxw;yy

�w�2;xy þw�;xxw�;yy �w;yy �w�;yy

Rx�

w;xx �w�;xx

Ryð22Þ

Eqs. (19) and (22) are nonlinear equations in terms of variablesw and f and used to investigate the nonlinear dynamic and nonlin-ear stability of thick imperfect stiffened FGM double curved panelson elastic foundations subjected to mechanical, thermal and ther-mo mechanical loads.

4. Nonlinear dynamic analysis

In the present study, suppose that the stiffened FGM shallowshell is simply supported at its all edges and subjected to a trans-verse load q(t), compressive edge loads r0(t) and p0(t). The bound-ary conditions are

w ¼ Nxy ¼ Mx ¼ 0; Nx ¼ �r0h at x ¼ 0; aw ¼ Nxy ¼ My ¼ 0; Ny ¼ p0h at y ¼ 0; b:

ð23Þ

where a and b are the lengths of in-plane edges of the shallow shell.

The approximate solutions of w, w⁄ and f satisfying boundaryconditions (23) are assumed to be [27–31]

w ¼WðtÞ sin kmx sin dny ð24aÞw� ¼W0 sin kmx sin dny ð24bÞf ¼ gðtÞ½sin kmx sin dny� hðxÞ �xðyÞ� ð24cÞ

where km = mp/a, dn = np/b and W is the maximum deflection; W0 isa constant; h(x) and x(y) chosen such that:

gh00ðxÞ ¼ p0h gx00ðyÞ ¼ r0h ð25Þ

Subsequently, substitution of Eqs. (24a and b) into Eqs. (22) and(24c) into Eq. (19) and applying the Galerkin procedure for theresulting equation yield leads to:

g A11m4 þ ðA66 � 2A12Þm2n2k2 þ A22n4k4� � a2

p2

n2k2

Rxþm2

Ry

!ðW �W0Þ

þW B21m4 þ ðB11 þ B22 � 2B66Þm2n2k2 þ B12n4k4� þ 16

3mnk2

p2 W2 �W20

� �¼ 0 ð26Þ

gp4

a4 B21m4 þ ðB11 þ B22 � B66Þn2m2k2 þ B12n4k4� � ðW

�W0Þp4

a4 D11m4 þ ðD12 þ D21 þ 4D66Þn2m2k2 þ D22n4k4� þ 32

3Wgmnp2 k2

a4 þp2hW

a2 ðm2r0 þ n2p0k2Þ

� p2

a2 gm2

Ryþ n2k2

Rx

!� 16h

mnp2

r0

Rxþ p0

Ry

� �þ 16q

mnp2 � k1W

� k2Wp2

a2 ðm2 þ k2n2Þ ¼ q

@2W@t2 ð27Þ

where m,n are odd numbers, and k ¼ ab.

Eliminating g from two obtained equations leads to non-linearsecond-order ordinary differential equation for f(t):

Wp2ha2 ðm

2r0 þ n2p0k2Þ � k1 � k2

p2

a2 ðm2 þ k2n2Þ � p4

a4

P2

P1þ p2

a2

m2

Ryþ n2k2

Rx

!P2

P1

" #

þ ðW �W0Þp2

a2

m2

Ryþ n2k2

Rx

!P2

P1� p4

a4 P3 �m2

Ryþ n2k2

Rx

!21P1

24

35

þ ðW2 �W20Þ

1a2

m2

Ryþ n2k2

Rx

!16mnk2

3P1� 16mnp2k2

a4

P2

P1

" #

þ�W2 32mnp2k2

3a4

P2

P1þWðW �W0Þ

32mnk2

3a2

m2

Ryþ n2k2

Rx

!1P1

�WðW2 �W20Þ

512m2n2k9a4

1P1� 16h

mnp2

r0

Rxþ p0

Ry

� �þ 16q

mnp2 ¼ q@2W@t2 ð28Þ

where

P1 ¼ A11m4 þ ðA66 � 2A12Þm2n2k2 þ A22n4k4

P2 ¼ B21m4 þ ðB11 þ B22 � 2B66Þm2n2k2 þ B12n4k4

P3 ¼ D11m4 þ ðD12 þ D21 þ 4D66Þm2n2k2 þ D22n4k4

ð29Þ

The obtained Eq. (28) is a governing equation for dynamicimperfect stiffened FGM doubly-curved shallow shells in general.The initial conditions are assumed as Wð0Þ ¼W0; _Wð0Þ ¼ 0. Thenonlinear Eq. (28) can be solved by the Newmark’s numerical inte-gration method or Runge–Kutta method.

4.1. Nonlinear vibration of eccentrically stiffened FGM shallow shell

Consider an imperfect stiffened FGM shallow shell acted on byuniformly distributed excited transverse q(t) = Qsin Xt, i.e.p0 = r0 = 0, from (28) we have

310 N.D. Duc / Composite Structures 102 (2013) 306–314

Q 1W þ Q 2ðW �W0Þ þ Q3 W2 �W20

� �� Q 4W2 þ Q 5WðW

�W0Þ � Q 6W W2 �W20

� �þ Q 7 sin Xt ¼ q

@2W@t2 ð30Þ

where

Q 1 ¼ k1 þ k2p2

a2 ðm2 þ k2n2Þ þ p4

a4

P2

P1� p2

a2

m2

Ryþ n2k2

Rx

!P2

P1

Q 2 ¼ �p2

a2

m2

Ryþ n2k2

Rx

!P2

P1þ p4

a4 P3 þm2

Ryþ n2k2

Rx

!21P1

Q 3 ¼1a2

m2

Ryþ n2k2

Rx

!16mnk2

3P1� 16mnp2k2

a4

P2

P1

Q 4 ¼32mnp2k2

3a4

P2

P1

Q 5 ¼32mnk2

3a2

m2

Ryþ n2k2

Rx

!1P1

Q 6 ¼512m2n2k

9a4

1P1

Q 7 ¼16Q0

mnp2

ð31Þ

From Eq. (30) the fundamental frequencies of natural vibrationof the imperfect stiffened FGM shell can be determined by therelation:

xL ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1qðQ 1 þ Q 2Þ

sð32Þ

The equation of nonlinear free vibration of a perfect FGM shal-low panel can be obtained from:

€W þ H1W þ H2W2 þ H3W3 ¼ 0 ð33Þ

where denoting

H1 ¼ x2L ¼

1qðQ1 þ Q 2Þ

H2 ¼Q4 � Q3 � Q5

q

H3 ¼Q6

q

ð34Þ

Seeking solution as W(t) = scos xt and applying procedure likeGarlerkin method to Eq (33), the frequency-amplitude relation ofnonlinear free vibration is obtained

xNL ¼ xL 1þ 8H2

3px2L

sþ 3H3

4x2L

s2

� �12

ð35Þ

where xNL is the nonlinear vibration frequency and s is the ampli-tude of nonlinear vibration.

4.2. Nonlinear dynamic buckling analysis of imperfect eccentricallystiffened FGM shallow shell

The aim of considered problems is to search the critical dynamicbuckling loads. They can be evaluated based on the displacementresponses obtained from the motion Eq. (28). This criterion sug-gested by Budiansky and Roth is employed here as it is widely ac-cepted. This criterion is based on that, for large values of loadingspeed the amplitude-time curve of obtained displacement re-sponse increases sharply depending on time and this curve ob-tained a maximum by passing from the slope point, and at thetime t = tcr a stability loss occurs, and here t = tcr is called criticaltime and the load corresponding to this critical time is called dy-namic critical buckling load.

4.2.1. Imperfect eccentrically stiffened FGM cylindrical panel acted onby axial compressive load

Eq. (28) in this case Rx ?1, Ry = R, p0 = q = 0;r0 – 0 can berewritten as:

Wp2ha2 m2r0 � k1 � k2

p2

a2 ðm2 þ k2n2Þ � p4P2

a4P1þ p2m2P2

a2RP1

� �

þ ðW �W0Þp2m2P2

a2RP1� p4

a4 P3 �m4

R2P1

� �

þ W2 �W20

� � 1a2

16m3nk2

3P1R� 16mnp2k2

a4

P2

P1

" #þ�W2

� 32mnp2k2

3a4

P2

P1þWðW �W0Þ

32mnk2

3a2

m2

R1P1

�W W2 �W20

� �512m2n2k9a4

1P1¼ q

@2W@t2 ð36Þ

The static critical load can be determined by the equation to bereduced from Eq. (36) by putting €W ¼ 0; W0 ¼ 0

Wp2ha2 m2r0 ¼W k1 þ k2

p2

a2 ðm2 þ k2n2Þ þ p4P2

a4P1

�

�p2m2P2

a2RP1� p2m2P2

a2RP1þ p4

a4 P3 þm4

R2P1

�

�W2 1a2

16m3nk2

3P1R� 16mnp2k2

a4

P2

P1

"

�32mnp2k2

3a4

P2

P1þ 32mnk2

3a2

m2

R1P1

#

þW3 512m2n2k9a4

1P1

ð37Þ

Taking of W – 0, i.e. considering the shell after the loss of stabil-ity we obtain

p2ha2 m2r0 ¼ k1 þ k2

p2

a2 ðm2 þ k2n2Þ þ p4P2

a4P1� p2m2P2

a2RP1� p2m2P2

a2RP1

þ p4

a4 P3 þm4

R2P1�W

1a2

16m3nk2

3P1R� 16mnp2k2

a4

P2

P1

"

�32mnp2k2

3a4

P2

P1þ 32mnk2

3a2

m2

R1P1

#þW2 512m2n2k

9a4

1P1

ð38Þ

From Eq. (38) the upper buckling load can be determined byW = 0

rupper ¼a2

m2hp2 k1þk2p2

a2 ðm2þk2n2Þþp4P2

a4P1�p2m2P2

a2RP1�p2m2P2

a2RP1þp4

a4 P3þm4

R2P1

� �ð39Þ

And the lower buckling load is found using the condition dr0dW ¼ 0,

it follows:

rlower ¼a2

p2hm2 k1 þ k2p2

a2 ðm2 þ k2n2Þ þp4P2

a4P1�p2m2P2

a2RP1

�

�p2m2P2

a2RP1þp4

a4 P3 þm4

R2P1� 9a4P1

1024m2n2k

1a2

16m3nk2

3P1R� 16mnp2k2

a4

P2

P1� 32mnp2k2

3a4

P2

P1þ 32mnk2

3a2

m2

R1P1

" #2

þþ 4512m2n2k

9a4

1P1

� �2 1a2

16m3nk2

3P1R� 16mnp2k2

a4

P2

P1

"

�32mnp2k2

3a4

P2

P1þ 32mnk2

3a2

m2

R1P1

##ð40Þ

Table 1The dependence of the fundamental frequencies of nature vibration of spherical FGMdouble curved shallow shell on volume ratio N.

N xL(rad/s)

Reinforced Unreinforced

0 56.130 � 105 55.667 � 105

1 39.034 � 105 38.515 � 105

2 31.982 � 105 31.441 � 105

5 24.047 � 105 23.477 � 105

Table 2The dependence of the fundamental frequencies of nature vibration of spherical FGMdouble curved shallow shell on elastic foundations.

K1, K2 xL (rad/s)

Reinforced Unreinforced

K1 = 200, K2 = 0 33.574 � 10 5 32.865 � 105

K1 = 200, K2 = 10 39.034 � 10 5 38.515 � 105

K1 = 200, K2 = 20 44.079 � 10 5 43.273 � 105

K1 = 200, K2 = 30 48.535 � 10 5 46.371 � 105

K1 = 0, K2 = 10 26.734 � 105 25.646 � 105

K1 = 100, K2 = 10 31.534 � 10 5 30.078 � 105

K1 = 150, K2 = 10 35.585 � 10 5 35.033 � 105

K1 = 200, K2 = 10 39.034 � 10 5 38.515 � 105

Table 3Comparison of - with result reported by Bich et. al. [32], Alijani et. al. [34], Chorfi andHoumat [35] and Matsunaga [36].

(a/Rx, b/Ry) N Present Ref. [32] Ref. [34] Ref. [35] Ref. [36]

FGM plate(0, 0) 0 0.0562 0.0597 0.0597 0.0577 0.0588

0.5 0.0502 0.0506 0.0506 0.0490 0.04921 0.0449 0.0456 0.0456 0.0442 0.04034 0.0385 0.0396 0.0396 0.0383 0.038110 0.0304 0.0381 0.0380 0.0366 0.0364

FGM cylindrical panel(0, 0.5) 0 0.0624 0.0648 0.0648 0.0629 0.0622

0.5 0.0528 0.0553 0.0553 0.0540 0.05351 0.0494 0.0501 0.0501 0.0490 0.04854 0.0407 0.0430 0.0430 0.0419 0.041310 0.0379 0.0409 0.0408 0.0395 0.0390

0 0.05 0.1 0.155.7413

5.7414

5.7415

5.7416

5.7417

5.7418

5.7419x 104

τ

ωNL

(rad

/s)

Reinforced, Rx=Ry=3(m), N=5

Reinforced, Rx=R(y)=3(m), N=0

Unreinforced, Rx=Ry=3(m), N=5

Unreinforced, Rx=Ry=3(m), N=0

Fig. 3. Frequency-amplitude relation.

N.D. Duc / Composite Structures 102 (2013) 306–314 311

4.2.2. Imperfect eccentrically stiffened shallow FGM cylindrical panelsubjected to transverse load

Eq. (28) in this case Rx ?1,Ry = R, p0 = r0 = 0 can be rewritten as:

W �k1 � k2p2

a2 ðm2 þ k2n2Þ � p4

a4

P2

P1þ p2n2k2P2

a2RP1

" #

þ ðW �W0Þp2m2P2

a2RP1� p4

a4 P3 �m4

RP1

� �

þ W2 �W20

� � 1a2

16m3nk2

3RP1� 16mnp2k2

a4

P2

P1

" #

þ�W2 32mnp2k2

3a4

P2

P1þWðW �W0Þ

32m3nk2

3Ra2

1P1

�W W2 �W20

� �512m2n2k9a4

1P1þ 16q

mnp2 ¼ q@2W@t2 ð41Þ

The static critical load can be determined by the equation to be re-duced from Eq. (41) by putting €W ¼ 0;W0 ¼ 0 and using conditiondq

dW ¼ 0.

4.2.3. Imperfect eccentrically stiffened FGM shallow spherical panelunder transverse load

Eq. (28) in this case Rx = Ry = R, p0 = r0 = 0 can be rewritten as:

W �k1 � k2p2

a2 ðm2 þ k2n2Þ � p4

a4

P2

P1þ p2

a2

m2 þ n2k2

R

!P2

P1

" #

þ ðW �W0Þp2

a2

m2 þ n2k2

R

!P2

P1� p4

a4 P3 �m2 þ n2k2

R

!21P1

24

35

þ W2 �W20

� � 1a2

m2 þ n2k2

R

!16mnk2

3P1� 16mnp2k2

a4

P2

P1

" #

þ�W2 32mnp2k2

3a4

P2

P1þWðW �W0Þ

32mnk2

3a2

m2 þ n2k2

R

!1P1

�W W2 �W20

� �512m2n2k9a4

1P1þ 16q

mnp2 ¼ q@2W@t2 ð42Þ

The static critical load can be determined by the equation to bereduced from Eq. (42) by putting €W ¼ 0; W0 ¼ 0 and using condi-tion dq

dW ¼ 0.

5. Numerical results and discussion

The eccentrically stiffened FGM shells considered here are shal-low shell with in-plane edges:

a ¼ b ¼ 2 m; h ¼ 0:01 m;

Em ¼ 70� 109 N=m2; Ec ¼ 380� 109 N=m2;

qm ¼ 2702 kg=m3; qc ¼ 3800 kg=m3;

s1 ¼ s2 ¼ 0:4; z1 ¼ z2 ¼ 0:0225 ðmÞ; m ¼ 0:3

ð43Þ

Table 1 presents the dependence of the fundamental frequenciesof nature vibration of spherical FGM shallow shell on volume ratio Nin which m ¼ n ¼ 1; a ¼ b ¼ 2 ðmÞ, h ¼ 0:01 ðmÞ; K1 ¼ 200; K2 ¼10, R¼x Ry ¼ 3 ðmÞ; W0 ¼ 1e� 5

From the results of Table 1, it can be seen that the increase ofvolume ration N will lead to the decrease of frequencies of naturevibration of spherical FGM shallow shell.

Table 2 presents the frequencies of nature vibration of sphericaldouble curved FGM shallow shell depending on elastic founda-tions. These results show that the increase of the coefficients ofelastic foundations will lead to the increase of the frequencies ofnature vibration. Moreover, the Pasternak type elastic foundationhas the greater influence on the frequencies of nature vibrationof FGM shell than Winkler model does.

Based on (28) the nonlinear vibration of imperfect eccentricallystiffened shells under various loading cases can be performed.

Fig. 4. Effect of eccentrically stiffeners on nonlinear dynamic response of theshallow spherical FGM shell.

Fig. 5. Deflection-velocity relation of the eccentrically stiffened shallow sphericalFGM shell.

Fig. 6. Influence of elastic foundations on nonlinear dynamic response of theeccentrically stiffened shallow spherical FGM shell.

Fig. 7. Effect of volume metal-ceramic on nonlinear response of the eccentricallystiffened shallow spherical FGM shell.

312 N.D. Duc / Composite Structures 102 (2013) 306–314

Particularly for spherical panel we put 1Rx¼ 1

Ryin (28), for cylindrical

shell 1Rx¼ 0 and for a plate 1

Rx¼ 1

Ry¼ 0.

Table 3 presents the comparison on the fundamental frequency

parameter - ¼ xLhffiffiffiffiqcEc

q(In the Tables 1–3, xL is calculated from

Eq. (32)) given by the present analysis with the result of Alijaniet al. [34] based on the Donnell’s nonlinear shallow shell theory,Chorfi and Haumat [35] based on the first–order shear deformationtheory and Matsunaga [36] based on the two-dimensional (2D)higher order theory for the perfect unreinforced FGM cylindricalpanel. The results in Table 3 were obtained with m = n = 1,a = b = 2 (m), h = 0.02 (m), K1 = 0, K2 = 0; W⁄ = 0 and with the cho-sen material properties in (43). As in Table 3, we can observe a verygood agreement in this comparison study.

Fig. 3 shows the relation frequency-amplitude of nonlinear freevibration of reinforced and unreinforced spherical shallow FGMshell on elastic foundation (calculated from Eq. (35)) with m ¼n ¼ 1; a ¼ b ¼ 2 ðmÞ, h ¼ 0:01 ðmÞ; K1 ¼ 200, K2 ¼ 10; R¼x Ry ¼ 3ðmÞ; W0 ¼ 1e� 5. As expected the nonlinear vibration frequenciesof reinforced spherical shallow FGM shells are greater than ones ofunreinforced shells.

The nonlinear Eq. (28) is solved by Runge–Kutta method. Thebelow figures, except Fig. 6, are calculated basing on k1 = 100;k2 = 10.

Fig. 4 shows the effect of eccentrically stiffeners on nonlinearrespond of the FGM shallow shell on elastic foundation. The FGM

Fig. 8. Effect of dynamic loads on nonlinear response.

Fig. 9. Effect of Rx on nonlinear dynamic response.

Fig. 10. Influence of initial imperfection on nonlinear dynamic response of theeccentrically stiffened spherical panel.

Fig. 11. Nonlinear dynamic response of eccentrically stiffened spherical andcylindrical FGM panel.

N.D. Duc / Composite Structures 102 (2013) 306–314 313

shell considered here is spherical panel Rx = Ry = 5 m. Clearly, thestiffeners played positive role in reducing amplitude of maximumdeflection. Relation of maximum deflection and velocity for spher-ical shallow shell is expressed in Fig. 5.

Fig. 6 shows influence of elastic foundations on nonlinear dy-namic response of spherical panel. Obviously, elastic foundationsplayed negative role on dynamic response of the shell: the largerk1 and k2 coefficients are, the larger amplitude of deflections is.

Fig. 7 shows effect of volume metal-ceramic on nonlinear dy-namic response of the eccentrically stiffened shallow sphericalFGM shell.

Figs. 8 and 9 show effect of dynamic loads and Rx on nonlineardynamic response of the eccentrically stiffened shallow sphericalFGM shell.

Fig. 10 shows influence of initial imperfection on nonlinear dy-namic response of the eccentrically stiffened spherical panel. Theincrease in imperfection will lead to the increase of the amplitudeof maximum deflection.

Fig. 11 shows nonlinear dynamic response of shallow eccentri-cally stiffened spherical and eccentrically stiffened cylindrical FGMpanels. For eccentrically stiffened cylindrical FGM panel, in thiscase, the obtained results is identical to the result of Bich in [32].

6. Concluding remarks

This paper presents an analytical investigation on the nonlineardynamic response of eccentrically stiffened functionally gradeddouble curved shallow shells resting on elastic foundations and

314 N.D. Duc / Composite Structures 102 (2013) 306–314

being subjected to axial compressive load and transverse load. Theformulations are based on the classical shell theory taking into ac-count geometrical nonlinearity, initial geometrical imperfectionand the Lekhnitsky smeared stiffeners technique with Pasternaktype elastic foundation. The nonlinear equations are solved bythe Runge–Kutta and Bubnov-Galerkin methods. Some resultswere compared with the ones of the other authors.

Obtained results show effects of material, geometrical proper-ties, eccentrically stiffened, elastic foundation and imperfectionon the dynamical response of reinforced FGM double curved shal-low shells. Hence, when we change these parameters, we can con-trol the dynamic response and vibration of the FGM shallow shellsactively.

Acknowledgments

This work was supported by Project in Mechanics of the NationalFoundation for Science and Technology Development of Vietnam-NAFOSTED. The author is grateful for this financial support.

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