Composite Structures 99 (2013) 88–96

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Composite Structures

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Nonlinear dynamic response of imperfect eccentrically stiffened FGM doublecurved shallow shells on elastic foundation

Nguyen Dinh DucUniversity 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 1 December 2012

Keywords:Nonlinear dynamicEccentrically stiffened FGM double curvedshallow shellsImperfectionElastic foundation

0263-8223/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.compstruct.2012.11.017

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

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.

� 2012 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 arechosen to use in temperature shielding structure components ofaircraft, aerospace vehicles, nuclear plants and engineering struc-tures in various industries. As a result, in recent years importantstudies have been researched about the stability and vibration ofFGM 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 Lam and Reddy (1999) in [1]. Lam and Li 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 thedynamic 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 vibration

ll rights reserved.

of 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 nonlinear 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,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 nonlinearaxisymmetric dynamic buckling behavior of clamped functionally

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 99 (2013) 88–96 89

graded 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 heatedtemperature-dependent FGM cylindrical shells under combinedaxial compression [23] and external pressure and dynamicbuckling of suddenly loaded imperfect hybrid FGM cylindricalwith temperature–dependent material properties under thermo-electro-mechanical loads [24]. Allahverdizadeh et al. studiednonlinear free and forced vibration analysis of thin circularfunctionally graded plates [25]. Sofiyev investigated the vibrationand stability behavior of freely supported FGM conical shells sub-jected to external pressure [26], 2009. Shen published a valuablebook ‘‘Functionally Graded materials, Nonlinear Analysis of platesand shells’’, in which the results about nonlinear vibration of sheardeformable FGM plates are presented [27]. Last years, Zhang and Lipublished the dynamic buckling of FGM truncated conical shellssubjected to non-uniform normal impact load [28], Bich and Long(2010) studied the non-linear dynamical analysis of functionallygraded material shallow shells subjected to some dynamic loads[29], Dung and Nam investigated the nonlinear dynamic analysisof imperfect FGM shallow shells with simply supported andclamped boundary conditions [30]. Bich et al. has also consideredthe nonlinear vibration of functionally graded shallow sphericalshells [31].

In fact, the FGM plates and shells, as other composite struc-tures, usually reinforced by stiffening member to provide thebenefit of added load-carrying static and dynamic capabilitywith a relatively small additional weight penalty. Thus studyon static and dynamic problems of reinforced FGM plates andshells with geometrical nonlinearity are of significant practicalinterest. However, up to date, the investigation on static anddynamic of eccentrically stiffened FGM structures has receivedcomparatively little attention. Recently, Bich et al. studied non-linear dynamical analysis of eccentrically stiffened functionallygraded 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/oy2, 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:

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.

x1

x2

ab

s2

hz2

z1

1

2

s2 s2 s2s1s1s1s1

b

O

z

Fig. 2. Configuration of an eccentrically stiffened shallow shells.

90 N.D. Duc / Composite Structures 99 (2013) 88–96

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

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 C1

� �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 þEA1

s1

� �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

are made of full metal (E = Em) if putting them at the metal-rich sideof the shell, and conversely full ceramic stiffeners (E = Ec) at theceramic-rich side of the shell. In above relations (9) and (10), thequantities A1, A2 are the cross section areas of the stiffeners andI1, I2, z1, z2 are the second moments of cross section areas and eccen-tricities of the stiffeners with respect to the middle surface of theshell respectively, E is elasticity modulus in the axial direction ofthe corresponding stiffener witch is assumed identical for bothtypes of stiffeners (Fig. 2). In order to provide continuity betweenthe shell and stiffeners, suppose that stiffeners

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

qðzÞdz ¼ qm þqcm

N þ 1

� �h ð12Þ

in which q @2u@t2 ! 0 and q @2v

@t2 ! 0 into consideration because ofu� w, v� w the Eq. (11) can be rewritten 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 þE2

1� m2

� �� A12

E2m1� m2

B12 ¼ A22E2m

1� m2 � A12E2

1� m2 þ C2

� �;

B21 ¼ A11E2m

1� m2 � A12E2

1� m2 þ C1

� �

B66 ¼E2

E1

ð15Þ

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

Mx ¼ B11Nx þ B21Ny � D11kx � D12ky

Mx ¼ B12Nx þ B22Ny � D21kx � D22ky

Mxy ¼ B66Nxy � 2D66kxy

ð16Þ

N.D. Duc / Composite Structures 99 (2013) 88–96 91

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 the 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 into form

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

þ f;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) onecan 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;xy

ð21Þ

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

A11f;xxxxþðA66�2A12Þf;xxyyþA22f;yyyyþB21w;xxxx

þðB11þB22�2B66Þw;xxyyþB12w;yyyy

¼w2;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 variables wand f and used to investigate the nonlinear dynamic and nonlinearstability of thick imperfect stiffened FGM double curved panels onelastic foundations subjected to mechanical, thermal and thermomechanical 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 ð24aÞ

w� ¼W0 sin kmx sin dny ð24bÞ

f ¼ gðtÞ sin kmx sin dny� hðxÞ �xðyÞ½ � 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 Eq. (24a,b) into Eq. (22), (24c) into Eq.(19) and applying the Galerkin procedure for the resulting equationyield 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 163

mnk2

p2 ðW2 �W2

0Þ ¼ 0 ð26Þ

gp4

a4 B21m4þðB11þB22�B66Þn2m2k2þB12n4k4h i

�ðW�W0Þp4

a4 D11m4þðD12þD21þ4D66Þn2m2k2þD22n4k4h i

þ323

Wgmnp2 k2

a4þp2hW

a2 ðm2r0þn2p0k2Þ�p2

a2 gm2

Ryþn2k2

Rx

!

� 16hmnp2

r0

Rxþp0

Ry

� �þ 16q

mnp2�k1W�k2Wp2

a2 ðm2þk2n2Þ¼q

@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 þn2p0k2Þ � k1 � k2

p2

a2 ðm2 þ k2n2Þ �p4

a4

P2

P1þp2

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

" #

�W2 32mnp2k2

3a4

P2

P1þWðW �W0Þ

32mnk2

3a2

m2

Ryþn2k2

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 dynamic imperfectstiffened FGM doubly-curved shallow shells in general. The initialconditions are assumed as Wð0Þ ¼W0; _Wð0Þ ¼ 0. The nonlinearEq. (28) can be solved by the Newmark’s numerical integrationmethod or Runge–Kutta method.

92 N.D. Duc / Composite Structures 99 (2013) 88–96

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) = QsinXt, i.e.p0 = r0 = 0, from (28) we have

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) = scosxt and applying procedure likeGalerkin method to Eq. (33), the frequency–amplitude relation ofnonlinear free vibration is obtained

xNL ¼ xL 1þ 8H2

3px2L

sþ 3H3

4x2L

s2� �1

2

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 displacement

responses 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

The 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

" #

�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

P1

"

þ32mnk2

3a2

m2

R1P1

#þW3 512m2n2k

9a4

1P1

ð37Þ

Taking of W – 0, i.e. considering the shell after the loss of stabilitywe 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

�W1a2

16m3nk2

3P1R� 16mnp2k2

a4

P2

P1� 32mnp2k2

3a4

P2

P1

"

þ32mnk2

3a2

m2

R1P1

#þW2 512m2n2k

9a4

1P1

ð38Þ

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

rupper ¼a2

m2hp2 k1 þ k2p2

a2 ðm2 þ k2n2Þ þ p4P2

a4P1� p2m2P2

a2RP1

�

�p2m2P2

a2RP1þ p4

a4 P3 þm4

R2P1

�ð39Þ

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

follows

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]

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 � 105 32.865 � 105

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

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

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

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

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

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

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

rlower ¼a2

p2hm2 k1 þ k2p2

a2 ðm2 þ k2n2Þ þ p4 P2a4 P1� p2 m2 P2

a2 RP1� p2 m2 P2

a2 RP1þ p4

a4 P3 þ m4

R2 P1� 9a4 P1

1024m2 n2k1

a216m3 nk2

3P1 R � 16mnp2k2

a4P2P1� 32mnp2k2

3a4P2P1þ 32mnk2

3a2m2

R1

P1

h i2þ 4 512m2 n2k

9a41

P1

� �21

a216m3 nk2

3P1 R � 16mnp2k2

a4P2P1� 32mnp2k2

3a4P2P1þ 32mnk2

3a2m2

R1

P1

h i� �

ð40Þ

0 0.05 0.1 0.155.7413

5.7414

5.7415

5.7416

5.7417

5.7418

5.7419x 104

τ

ω NL (r

ad/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 99 (2013) 88–96 93

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

The Eq. (28) in this case Rx ?1, Ry = R, p0 = r0 = 0 can be rewrit-ten 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

" #

�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 bereduced from Eq. (41) by putting €W ¼ 0;W0 ¼ 0 and using condi-tion dq

dW ¼ 0.

4.2.3. Imperfect eccentrically stiffened FGM shallow spherical panelunder transverse load

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

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

" #

�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 discussions

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

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

a ¼ b ¼ 2m; h ¼ 0:01m;

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 0:0225ðmÞ; m ¼ 0:3

ð43Þ

The Table 1 presents the dependence of the fundamental fre-quencies of nature vibration of spherical FGM shallow shell on vol-ume ratio N in which m ¼ n ¼ 1; a ¼ b ¼ 2ðmÞ; h ¼ 0:01ðmÞ; K1 ¼200; K2 ¼ 10; Rx ¼ Ry ¼ 3ðmÞ; W0 ¼ 1e� 5.

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

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

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.

94 N.D. Duc / Composite Structures 99 (2013) 88–96

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. Par-ticularly for spherical panel we put 1

Rx¼ 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 frequencyparameter - ¼ xLh

ffiffiffiffiqcEc

q(In the Table 1–3, xL is calculated from Eq.

(32)) given by the present analysis with the result of Alijani et al.[34] based on the Donnell’s nonlinear shallow shell theory, Chorfiand Haumat [35] based on the first-order shear deformation theoryand Matsunaga [36] based on the two-dimensional (2D) higher or-der theory for the perfect unreinforced FGM cylindrical panel. Theresults 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 chosen materialproperties in (43). As in Table 3, we can observe a very good agree-ment 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)) withm ¼ n ¼ 1; a ¼ b ¼ 2ðmÞ; h ¼ 0:01ðmÞ; K1 ¼ 200; K2 ¼ 10; Rx ¼Ry ¼ 3ðmÞ; W0 ¼ 1e� 5. As expected the nonlinear vibration fre-quencies of reinforced spherical shallow FGM shells are greaterthan ones of unreinforced 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 FGMshell 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 foundations

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

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

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

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

N.D. Duc / Composite Structures 99 (2013) 88–96 95

played negative role on dynamic response of the shell: the larger k1

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 FGM

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

96 N.D. Duc / Composite Structures 99 (2013) 88–96

panels. 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 andbeing subjected to axial compressive load and transverse load.The formulations are based on the classical shell theory taking intoaccount 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 theNational Foundation for Science and Technology Developmentof Vietnam-NAFOSTED. The author is grateful for this financialsupport.

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