Nonlinear vibration of imperfect eccentrically stiffened functionallygraded double curved shallow shells resting on elastic foundationusing the first order shear deformation theory

Dao Huy Bich, Nguyen Dinh Duc, Tran Quoc Quan n

Vietnam National University, Hanoi, Vietnam 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam

a r t i c l e i n f o

Article history:Received 1 October 2013Received in revised form16 November 2013Accepted 19 December 2013Available online 30 December 2013

Keywords:Nonlinear dynamic responseNonlinear vibrationFGM double curved shallow shellsThe first order shear deformation theoryStress function

a b s t r a c t

This paper presents an analytical approach to investigate the nonlinear dynamic response and vibrationof imperfect eccentrically stiffened FGM thick double curved shallow shells on elastic foundation usingboth the first order shear deformation theory and stress function with full motion equations (not usingVolmir0s assumptions). The FGM shells are assumed to rest on elastic foundation and subjected tomechanical and damping loads. Numerical results for dynamic response of the FGM shells are obtainedby Runge–Kutta method. The results show the influences of geometrical parameters, the materialproperties, imperfections, the elastic foundations, eccentrically stiffeners and mechanical loads on thenonlinear dynamic response and nonlinear vibration of functionally graded double curved shallow shells.The numerical results in this paper are compared with results reported in other publications.

Crown Copyright & 2014 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Functionally Graded Materials (FGM) are homogeneous com-posite and microscopic scale materials with the mechanical andthermal properties varying smoothly and continuously from onesurface to the other. The properties of FGM shells are assumedto vary through the thickness of the structures. Due to the highheat resistance, FGMs have many practical applications suchas reactor vessels, aircrafts, space vehicles, defense industriesand other engineering structures.

Therefore, in the recent years, many investigations have beencarried out on the dynamic and vibration of FGM shells. Strozzi andPellicano [1] investigated the nonlinear vibrations of functionallygraded circular cylindrical shells by using the Sanders–Koiter theory.Sepiani et al. [2] studied the vibration and buckling analysis of two-layered functionally graded cylindrical shell, considering the effectsof transverse shear and rotary inertia. Nonlinear buckling analysis ofFGM shallow spherical shells under pressure loads was presented byGanapathi [3] by using finite element method, geometric nonlinear-ity is assumed only on the meridional direction in strain–displace-ment relations. The nonlinear vibration of heated bimetallic shallowshells of revolution is presented in work [4] of Wang and Song. Zhaoand Liew extended their previous works on isotropic conical panels

to analyze the free vibration of functionally graded conical shells byusing a meshless method [5]. Vibration analysis of ring-stiffenedconical-cylindrical spherical shells based on a modified variationalapproach is investigated by Qu et al. [6]. Sheng and Wang [7] haveconsidered the nonlinear vibration control of functionally gradedlaminated cylindrical shells based on Hamilton0s principle, VonKarman nonlinear theory and constant-gain negative velocity feed-back approach. Haddadpour et al. [8] obtained the free vibrationanalysis of functionally graded cylindrical shells including thermaleffects. Loy et al. [9] also focused the vibration of functionally gradedcylindrical shells. Xiang et al. [10] used Love0s first approximationtheory to analyze the natural frequencies of rotating functionallygraded cylindrical shells. An analysis on the nonlinear dynamics ofa clamped–clamped FGM circular cylindrical shell subjected to anexternal excitation and uniform temperature change is presentedin [11]. Hong [12] investigated the functionally graded materialshell with mounted magnetostrictive layer under thermal vibrationby using the generalized differential quadrature (GDQ) method.Chandrashekhar et al. [13] focused on nonlinear vibration analysisof composite laminated and sandwich plates with random materialproperties. Huang and Shen [14] studied nonlinear free and forcedvibration of simply supported shear deformable laminated plateswith piezoelectric actuators. Huang and Han [15] presented non-linear dynamic buckling problems of functionally graded cylindricalshell subjected to dependent axial load by using Budiansky–Rothdynamic buckling criterion [16]. Rafiee et al. [17] also published theresults on the nonlinear vibration and dynamic response of simply

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International Journal of Mechanical Sciences

0020-7403/$ - see front matter Crown Copyright & 2014 Published by Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijmecsci.2013.12.009

n Corresponding author.E-mail address: quantq_530@vnu.edu.vn (T. Quoc Quan).

International Journal of Mechanical Sciences 80 (2014) 16–28

supported piezoelectric functionally graded material shells undercombined electrical, thermal, mechanical and aerodynamic load-ing. Sofiyev [18,19] used the large deformation theory with vonKarman–Donnell-type of kinematic nonlinearity to study thenonlinear dynamics of FGM truncated conical shells. The vibrationof FGM thin cylindrical shells with exponential volume law hasbeen investigated by Shah et al. [20]. Amabili [21,22] studied thenonlinear vibrations of isotropic curved panels using the Donnell–Novozhilov shell theories. In these papers, the Lagrangianapproach was utilized to obtain the equations of motion includinggeometric imperfections and the in-plane inertia effects. Nonlinearvibrations of functionally graded shallow shells under a concen-trated force were studied by Alijani et al. [23]. Bich et al. [24]investigated the nonlinear vibration of functionally graded circularcylindrical shells based on improved Donnell equations. Basedon the first-order shear deformation theory, Tornabene [25] studiedthe dynamic behavior of moderately thick functionally gradedconical, cylindrical shells and annular plates. Additionally, Tornabeneet al. [26,27] investigated vibration of FGM and laminated doublycurved shells and panels. Based on physical neutral surface and highorder shear deformation theory, Zhang [28] analyzed the modelingand analysis of FGM rectangular plates. Patel et al. [29] consideredthe free vibration analysis of functionally graded elliptical cylindricalshells using higher-order theory. Up to date, a number of recentpublications focused on the dynamic of eccentrically stiffened doublycurved FGM shallow shells using the stress function and Volmir’shypothesis but they all use classical shell theory. Bich et al. studiednonlinear postbuckling and dynamic of eccentrically stiffened func-tionally graded shallow shells, panel and circular cylindrical shells[30–33]. Duc et al. [34,36] investigated nonlinear postbuckling andnonlinear dynamic response of imperfect eccentrically stiffened dou-bly curved FGM shallow shells on elastic foundations.

When using higher order shear deformation theories to studythe nonlinear dynamic analysis and vibrations of thick plates andshells, Volmir0s hypothesis is useless. Chorfi and Houmat [37]investigated nonlinear free vibrations of FGM doubly curved shellswith an elliptical plan-form using first order shear theory. Matsu-naga et al. [38] investigated free vibrations and stability of FGMdouble curved shallow shells according to a 2-D higher orderdeformation theory. Noted that in mentioned above references[37,38], all authors used displacements functions. There is no anypublication using simultaneous combination of the higher ordershear deformation theory and stress function to investigatedynamic for eccentrically stiffened FGM double curved shallowshells. This is the first paper presenting an analytical approach toinvestigate the nonlinear dynamic response and nonlinear vibra-tion of FGM shells using both the first order shear deformationtheory, smeared stiffeners Lekhnitsky0s technique and stress func-tion. Numerical results for dynamic response of the FGM shells areobtained by fourth-order Runge–Kutta method.

2. Double curved FGM shallow shell on elastic foundation

Consider a FGM double curved thin shallow shell of radii ofcurvature Rx,Ry length of edges a, b and uniform thickness h.A coordinate systemðx; y; zÞis established in which ðx; yÞ plane onthe middle surface of the shell and z on thickness directionð�h=2rzrh=2Þ as shown in Fig. 1.

For the FGM shell, the volume fractions of constituents areassumed to vary through the thickness according to the followingpower law distribution (P-FGM):

VcðzÞ ¼2zþh2h

� �N

; VmðzÞ ¼ 1�VcðzÞ; ð1Þ

where N is volume fraction index (0rNo1). Effective propertiesPref f of FGM shell are determined by linear rule of mixture as

Pref f ðzÞ ¼ PrcVcðzÞþPrmVmðzÞ; ð2Þin which Pr denotes a material property, and subscripts m andc stand for the metal and ceramic constituents, respectively.Specific expressions of elastic modulus E, Poisson ratio ν anddensity ρ are obtained by substituting Eq. (1) into Eq. (2) as

EðzÞ; vðzÞ; ρðzÞ½ � ¼ Em; νm; ρm� �þ Ecm; νcm; ρcm

� � 2zþh2h

� �N

; ð3Þ

where

Ecm ¼ Ec�Em; νcm ¼ νc�νm; ρcm ¼ ρc�ρm: ð4ÞIt is evident from Eqs. (3) and (4) that the upper surface of the

panel (z¼ �h=2) is metal-rich, while the lower surface (z¼ h=2) isceramic-rich, and the percentage of metal constituent in the panelis enhanced when N increases.

Assume that the shell is reinforced by eccentrically longitudinaland transversal homogeneous stiffeners with the elastic modulusE0. In order to provide the continuity between the shell andstiffeners, the full metal stiffeners are put at the metal-rich sideof the shell thus E0 ¼ Em and conversely full ceramic ones at theceramic-rich side, so that E0 ¼ Ec [24,30–35].

The shell–foundation interaction is represented by Pasternakmodel as

qe ¼ k1w�k2∇2w; ð5Þin which ∇2 ¼ ∂2=∂x2þ∂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 first-order shear deformation theory and theLekhnitsky smeared stiffeners are used to establish governingequations and to determine the nonlinear vibration of FGM thickshallow double curved shells.

The strain–displacement relations taking into account the vonKarman nonlinear terms are [39,40]

εx

εy

γxy

0B@

1CA¼

ε0xε0y

γ0xy

0BB@

1CCAþz

χxχyχxy

0B@

1CA;

γxzγyz

!¼

w;xþϕx

w;yþϕy

!; ð6Þ

with

ε0xε0y

γ0xy

0BB@

1CCA¼

u;x�w=Rxþw2;x=2

v;y�w=Ryþw2;y=2

u;yþv;xþw;xw;y

0BB@

1CCA;

χxχyχxy

0B@

1CA¼

ϕx;x

ϕy;y

ϕx;yþϕy;x

0B@

1CA; ð7Þ

h

Rx

a b

z

y Ry

x

Fig. 1. Geometry and coordinate system of P-FGM double curved shallow shell onelastic foundation.

D. Huy Bich et al. / International Journal of Mechanical Sciences 80 (2014) 16–28 17

in which ε0x and ε0y are normal strains and γ0xy is the shear strain inthe middle surface of the shell and γxz and γyz are the transverseshear strains components in the planes xz and yz, respectively.Rx;Ry are radii of curvatures; 1=Rx;1=Ry are principal curvatures ofthe shell. u; v;w are displacement components corresponding tothe coordinates ðx; y; zÞ, and ϕx;ϕy are the rotation angles of normalto the middle surface with respect to y and x axes, respectively.

Hooke0s law for a shell is defined as follows [24,30–35]:

sx ¼E

1�ν2ðεxþνεyÞ; sy ¼

E1�ν2

ðεyþνεxÞ;

sxy ¼E

2ð1þνÞγxy; sxz ¼E

2ð1þνÞγxz; syz ¼E

2ð1þνÞγyz; ð8Þ

and for stiffeners

ðsstx ;sst

y Þ ¼ E0ðεx; εyÞ; ð9Þ

The contribution of stiffeners can be accounted for using theLekhnitsky smeared stiffeners technique which is used in manyreferences [30–35,39]. Then integrating the stress–strain equa-tions and their moments through the thickness of the shell, theexpressions for force and moment resultants of an eccentricallystiffened FGM shallow shell are obtained

Nx ¼ I10þE0A1

s1

� �ε0x þ I20ε0yþðI11þC1Þχxþ I21χy;

Ny ¼ I20ε0x þ I10þE0A2

s2

� �ε0yþ I21χxþðI11þC2Þχy;

Nxy ¼ I30γ0xyþ2I31χxy;

Mx ¼ ðI11þC1Þε0x þ I21ε0yþ I12þE0I1s1

� �χxþ I22χy;

Mx ¼ I21ε0x þðI11þC2Þε0yþ I22χxþ I12þE0I2s2

� �χy;

Mxy ¼ I31γ0xyþ2I32χxy;

Qx ¼ KI30γxz;Qy ¼ KI30γyz; ð10Þ

with

I1j ¼Z h=2

�h=2

EðzÞ1�νðzÞ2

zj dz; j¼ 0;2

I2j ¼Z h=2

�h=2

EðzÞνðzÞ1�νðzÞ2

zj dz; j¼ 0;2

I3j ¼Z h=2

�h=2

EðzÞ2 ð1þνðzÞ½ �z

j dz¼ 12ðI1j� I2jÞ; j¼ 0;2

I1 ¼d1ðh1Þ3

12þA1ðz1Þ2; I2 ¼

d2ðh2Þ312

þA2ðz2Þ2;

C1 ¼E0A1z1

s1; C2 ¼

E0A2z2s2

;

z1 ¼h1þh2

; z2 ¼h2þh2

; ð11Þ

in which the width and thickness of longitudinal and transversalstiffeners are denoted by d1;h1 and d2;h2 respectively; s1; s2 arethe spacing of the longitudinal and transversal stiffeners. Thequantities A1;A2 are the cross-section areas of stiffeners andI1; I2; z1; z2 are the second moments of cross-section areas andthe eccentricities of stiffeners with respect to the middle surface ofshell respectively. K ¼ 5=6 is correction factors [39,40].

The nonlinear motion equation of FGM shallow shells based onLove0s theory [39,40]:

Nx;xþNxy;y ¼ I0∂2u∂t2

þ I1∂2ϕx

∂t2; ð12aÞ

Nxy;xþNy;y ¼ I0∂2v∂t2

þ I1∂2ϕy

∂t2; ð12bÞ

Qx;xþQy;yþNxw;xxþ2Nxyw;xy

þNyw;yyþq�k1wþk2∇2wþNx

RxþNy

Ry¼ I0

∂2w∂t2

þ2εI0∂w∂t

; ð12cÞ

Mx;xþMxy;y�Qx ¼ I2∂2ϕx

∂t2þ I1

∂2u∂t2

; ð12dÞ

Mxy;xþMy;y�Qy ¼ I2∂2ϕy

∂t2þ I1

∂2v∂t2

; ð12eÞ

where

Ii ¼Z h=2

�h=2ρðzÞzi dz; ði¼ 0;2Þ

I0 ¼ ρmhþρcmhNþ1

; I1 ¼ ρcmh2 1

Nþ2� 12ðNþ1Þ

� �;

I2 ¼ρmh

3

12þρcmh

3 1Nþ3

� 1Nþ2

þ 14ðNþ1Þ

� �; ð13Þ

in which q is an external pressure uniformly distributed on thesurface of the shell, ε is damping coefficient.

Calculating from Eq. (10) yields

ε0x ¼ A22Nx�A12Ny�B11ϕx;x�B12ϕy;y;

ε0y ¼ A11Ny�A12Nx�B21ϕx;x�B22ϕy;y;

γ0xy ¼ A66Nxy�B66ðϕx;yþϕy;xÞ; ð14Þ

with

A11 ¼1Δ

I10þE0A1

s1

� �; A22 ¼

1Δ

I10þE0A2

s2

� �;

A12 ¼I20Δ; A66 ¼

1I30

;

Δ¼ I10þE0A1

s1

� �I10þ

E0A2

s2

� �� I220;

B11 ¼ A22ðI11þC1Þ�A12I21;

B22 ¼ A11ðI11þC2Þ�A12I21;

B12 ¼ A22I21�A12ðI11þC2Þ;B21 ¼ A11I21�A12ðI11þC1Þ;

B66 ¼I31I30

: ð15Þ

The stress function f ðx; y; tÞ is introduced as

Nx ¼ f ;yy; Ny ¼ f ;xx; Nxy ¼ � f ;xy: ð16Þ

Replacing Eq. (16) into Eqs. (12a) and (12b) yields

∂2u∂t2

¼ � I1I0

∂2ϕx

∂t2; ð17aÞ

∂2v∂t2

¼ � I1Io

∂2ϕy

∂t2: ð17bÞ

Substituting Eqs. (16) and (17) into Eqs. (12c–12e) leads to

Qx;xþQy;yþ f ;yyw;xx�2f ;xyw;xyþ f ;xxw;yy

þq�k1wþk2ðw;xxþw;yyÞþf ;xxRx

þ f ;yyRy

¼ I0∂2w∂t2

þ2εI0∂w∂t

; ð18aÞ

Mx;xþMxy;y�Qx ¼ I2�I21I0

!∂2ϕx

∂t2; ð18bÞ

Mxy;xþMy;y�Qy ¼ I2�I21Io

!∂2ϕy

∂t2: ð18cÞ

D. Huy Bich et al. / International Journal of Mechanical Sciences 80 (2014) 16–2818

By substituting Eq. (14) into Eq. (10) and then into Eqs. (18), thesystem of motion equations (18) is rewritten as follows:

L11ðwÞþL12ðϕxÞþL13ðϕyÞþP1ðw; f Þþq¼ Io∂2w∂t2

þ2εI0∂w∂t

;

L21ðwÞþL22ðϕxÞþL23ðϕyÞþP2ðf Þ ¼ I2�I21I0

!∂2ϕx

∂t2;

L31ðwÞþL32ðϕxÞþL33ðϕyÞþP3ðf Þ ¼ I2�I21Io

!∂2ϕy

∂t2; ð19Þ

where the linear operators Lijði¼ 1;3; j¼ 1;3Þ; P2; P3 and the non-linear operator P1 are defined as below:

L11ðwÞ ¼ KI30w;xxþKI30w;yy�k1wþk2ðw;xxþw;yyÞ;L12ðϕxÞ ¼ KI30ϕx;x;

L13ðϕyÞ ¼ KI30ϕy;y;

P1ðw; f Þ ¼ f ;yyw;xx�2f ;xyw;xyþ f ;xxw;yyþf ;yyRx

þ f ;xxRy

;

L21ðwÞ ¼ �KI30w;x;

L22ðϕxÞ ¼D11ϕx;xxþD66ϕx;yy�KI30ϕx;

L23ðϕyÞ ¼ ðD12þD66Þϕy;xy;

P2ðf Þ ¼ B21f ;xxxþðB11�B66Þf ;xyy;L31ðwÞ ¼ �KI30w;y;

L32ðϕxÞ ¼ ðD21þD66Þϕx;xy;

L33ðϕyÞ ¼D22ϕy;yyþD66ϕy;xx�KI30ϕy;

P3ðf Þ ¼ B12f ;yyyþðB22�B66Þf ;xxy;

D11 ¼ I12þE0I1s1

�B11ðI11þC1Þ� I21B21;

D22 ¼ I12þE0I2s2

�B22ðI11þC2Þ� I21B12;

D12 ¼ I22�B12ðI11þC1Þ� I21B22;

D21 ¼ I22�B21ðI11þC2Þ� I21B11;

D66 ¼ I32� I31B66: ð20ÞFor an imperfect FGM curved shell, Eq. (19) is modified into aform as

L11ðwÞþL12ðϕxÞþL13ðϕyÞþP1ðw; f ÞþP10ðwn; f Þþq

¼ Io∂2w∂t2

þ2εI0∂w∂t

;

L21ðwÞþL22ðϕxÞþL23ðϕyÞþP2ðf ÞþL21ðwnÞ ¼ I2�I21I0

!∂2ϕx

∂t2;

L31ðwÞþL32ðϕxÞþL33ðϕyÞþP3ðf ÞþL31ðwnÞ ¼ I2�I21Io

!∂2ϕy

∂t2; ð21Þ

in which wnðx; yÞ is a known function representing initial smallimperfection of the shell and

P10ðwn; f Þ ¼ KI30wn

;xxþKI30wn;yyþ f ;yyw

n;xx�2f ;xyw

n;xyþ f ;xxw

n;yy:

ð22ÞThe geometrical compatibility equation for an imperfect double

curved shallow shell is written as [34–36]

ε0x;yyþε0y;xx�γ0xy;xy ¼w2;xy�w;xxw;yyþ2w;xywn

;xy�w;xxwn

;yy

�w;yywn

;xx�w;yy

Rx�w;xx

Ryð23Þ

From the constitutive relations (14) in conjunction with Eq. (16)one can write

ε0x ¼ A22f ;yy�A12f ;xx�B11ϕx;x�B12ϕy;y;

ε0y ¼ A11f ;xx�A12f ;yy�B21ϕx;x�B22ϕy;y;

γ0xy ¼ �A66f ;xy�B66ðϕx;yþϕy;xÞ: ð24Þ

Setting Eq. (24) into Eq. (23) gives the compatibility equation ofan imperfect FGM double curved shell as

A11f ;xxxxþA22f ;yyyyþðA66�2A12Þf ;xxyy�B21ϕx;xxx�B12ϕy;yyy

þðB66�B11Þϕx;xyyþðB66�B22Þϕy;xxy� w2;xy�w;xxw;yy

�þ2w;xywn

;xy�w;xxwn

;yy�w;yywn

;xx�w;yy

Rx�w;xx

Ry

�¼ 0: ð25Þ

Eqs. (21) and (25) are basic nonlinear equations in terms ofvariables w and f and used to investigate the nonlinear vibrationof FGM thick double curved shells on elastic foundations.

4. Nonlinear dynamical analysis

An imperfect FGM shallow shell considered in this paper isassumed to be simply supported and subjected to uniformlydistributed pressure of intensity q and axial compression ofintensities Px and Py respectively at its cross section. Thus theboundary conditions are

w¼Nxy ¼ ϕy ¼Mx ¼ 0; Nx ¼ �Pxh at x¼ 0; a

w¼Nxy ¼ ϕx ¼My ¼ 0; Ny ¼ �Pyh at y¼ 0; b ð26ÞThe mentioned conditions (26) can be satisfied identically if thebuckling mode shape is chosen by

wðx; y; tÞ ¼WðtÞ sin λmx sin δny;ϕxðx; y; tÞ ¼ΦxðtÞ cos λmx sin δny;ϕyðx; y; tÞ ¼ΦyðtÞ sin λmx cos δny; ð27Þ

where λm ¼mπ=a; δn ¼ nπ=b, m;n¼ 1;2;… are the natural num-bers of half waves in the corresponding direction x; y; WðtÞ is thetime dependent total amplitude and Φx; Φy are the amplitudeswhich are functions dependent on time.

The initial imperfection wn is assumed to have the same formof the shell deflection w, i.e.

wnðx; yÞ ¼W0 sin λmx sin δny; ð28Þin which W0 ¼ const is a known initial amplitude.

Substituting Eqs. (27) and (28) into the compatibility equa-tion (25), we define the stress function as

f ðx; y; tÞ ¼ A1ðtÞ cos 2λmxþA2ðtÞ cos 2δny

þA3ðtÞ sin λmx sin δny�12 Pxhy2�1

2 Pyhx2; ð29Þ

with

A1 ¼δ2n

32A11λ2m

WðWþ2W0Þ; A2 ¼λ2m

32A22δ2n

WðWþ2W0Þ;

A3 ¼1

ðA11λ4mþA22δ

4nþðA66�2A12Þλ2mδ2nÞ

δ2nRx

þλ2mRy

!W

þðB21λ3mΦxðtÞþB12δ

3nΦyðtÞþðB11�B66Þλmδ2nΦxðtÞþðB22�B66Þλ2mδnΦyðtÞÞðA11λ

4mþA22δ

4nþðA66�2A12Þλ2mδ2nÞ

:

ð30ÞReplacing Eqs. (27) and (29) into the equations of motion (21) andthen applying Galerkin method we obtain

l11Wþ l12Φxþ l13Φyþ l14ðWþW0ÞΦxþ l15ðWþW0ÞΦy

þn1ðWþW0Þþn2WðWþW0Þþn3WðWþ2W0Þþn4WðWþW0ÞðWþ2W0Þþn5qþn6

¼ Iod2Wdt2

þ2εI0dWdt

;

l21Wþ l22Φxþ l23Φyþn7ðWþW0Þþn8WðWþ2W0Þ ¼ ρ1 €Φx;

l31Wþ l32Φxþ l33Φyþn9ðWþW0Þþn10WðWþ2W0Þ ¼ ρ1 €Φy; ð31Þin which specific expressions of coefficient l1iði¼ 1;5Þ;

ljkðj¼ 2;3; k¼ 1;3Þ;nmðm¼ 1;10Þ; ρ1 are give in Appendix A.

D. Huy Bich et al. / International Journal of Mechanical Sciences 80 (2014) 16–28 19

4.1. Natural frequencies

In the case of q¼ 0, the natural frequencies of the perfect shellcan be determined by solving the following equation:

l11þn1þ I0ω2 l12 l13l21þn7 l22þρ1ω

2 l23l31þn9 l32 l33þρ1ω

2

¼ 0: ð32Þ

Three angular frequencies of the FGM shallow shell in the axial,circumferential and radial directions are determined by solvingEq. (32) and the smallest one is being considered.

4.2. Nonlinear dynamic responses

Consider a double curved FGM shallow shell acted on by anuniformly distributed transverse load q¼Q sin Ωt (Q is theamplitude of uniformly excited load, Ω is the frequency of theload). The system equation (31) has the form

Iod2Wdt2

þ2εI0dWdt

� l11W� l12Φx� l13Φy� l14ðWþW0ÞΦx

� l15ðWþW0ÞΦy�n1ðWþW0Þ�n2WðWþW0Þ�n3WðWþ2W0Þ�n4WðWþW0ÞðWþ2W0Þ ¼ n5Q sin Ωtþn6;

l21Wþ l22Φxþ l23Φyþn7ðWþW0Þþn8WðWþ2W0Þ ¼ ρ1 €Φx;

l31Wþ l32Φxþ l33Φyþn9ðWþW0Þþn10WðWþ2W0Þ ¼ ρ1 €Φy: ð33ÞBy using Eq. (33), three aspects are taken into consideration:

fundamental frequencies of natural vibration of the FGM shallowshell, frequency–amplitude relation of nonlinear free vibration andnonlinear response of FGM shell. The nonlinear dynamicalresponses of the FGM shallow shells can be obtained by solvingthis equation combined with initial conditions to be assumed asWð0Þ ¼ 0; ðdW=dtÞð0Þ ¼ 0 by using the fourth-order Runge–Kuttamethod.

4.3. Simplified assumption

For further research, we next consider the hypothetical case ofrotations Φx; Φy exist, but the inertial forces caused by the rotationangles Φx; Φy are small so they can be ignored. The system Eq. (33)can be written as follows:

Iod2Wdt2

þ2εI0dWdt

� l11W� l12Φx� l13Φy� l14ðWþW0ÞΦx

� l15ðWþW0ÞΦy�n1ðWþW0Þ�n2WðWþW0Þ�n3WðWþ2W0Þ

�n4WðWþW0ÞðWþ2W0Þ ¼ n5Q sin Ωtþn6;

l21Wþ l22Φxþ l23Φyþn7ðWþW0Þþn8WðWþ2W0Þ ¼ 0;

l31Wþ l32Φxþ l33Φyþn9ðWþW0Þþn10WðWþ2W0Þ ¼ 0: ð34ÞSolving the second and third obtained equations with respect

to Φx and Φy then substituting the results into the first equationyields

Iod2Wdt2

þ2εI0dWdt

�a1W�a2ðWþW0Þ�a3WðWþW0Þ�a4WðWþ2W0Þ

�a5WðWþW0ÞðWþ2W0Þ�a6ðWþW0Þ2 ¼ n5Q sin Ωtþn6;

ð35Þwhere

a1 ¼ l11þ l12l23l31� l21l33l22l33� l32l23

þ l13l32l21� l22l31l22l33� l32l23

;

a2 ¼ n1þ l12n9l23�n7l33l22l33� l32l23

þ l13n9l22�n7l32l32l23� l22l33

;

a3 ¼ n2þ l14l23l31� l21l33l22l33� l32l23

þ l15l32l21� l22l31l22l33� l32l23

;

a4 ¼ n3þ l12n10l23�n8l33l22l33� l32l23

þ l13n10l22�n8l32l32l23� l22l33

;

a5 ¼ l14n10l23�n8l33l22l33� l32l23

þ l15n10l22�n8l32l32l23� l22l33

;

a6 ¼ l14n9l23�n7l33l22l33� l32l23

þ l15n9l22�n7l32l32l23� l22l33

: ð36Þ

In other hand, from Eq. (35) the fundamental frequencies of a perfectshell can be determined approximately by an explicit expression:

ωmn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ða1þa2Þ

I0

s: ð37Þ

Consider nonlinear vibration of a perfect shell, Eq. (35) has ofthe form

Iod2Wdt2

þ2εI0dWdt

�ða1þa2ÞW�ða3þa4þa6ÞW2�a5W3 ¼ n5Q sin Ωtþn6:

ð38ÞEq. (38) can be rewritten as

d2Wdt2

þ2εdWdt

þω2mnðW�MW2þNW3Þ�F sin Ωt�H ¼ 0; ð39Þ

with ω2mn ¼ �ða1þa2Þ=I0 is the fundamental frequency of linear

vibration of the FGM shell and

M¼ �ða3þa4þa6Þða1þa2Þ

; N¼ a5ða1þa2Þ

; F ¼ n5QIo

; H¼ n6

I0

Seeking solution as W ¼ A sin Ωt and applying Galerkin pro-cedure to Eq. (39), the amplitude–frequency relation of nonlinearforced vibration is obtained:

Ω2�4επΩ�ω2

mn 1� 83π

MAþ3N4A2

� �¼ �F

A�4HAπ

: ð40Þ

By denoting α¼Ω=ωmn Eq. (40) is rewritten as

α2� 4επωmn

α� 1� 83π

MAþ3N4A2

� �¼ � F

Aω2mn

� 4HAπω2

mn: ð41Þ

For the nonlinear vibration of the shell without damping, thisrelation has of the form

α2� 1� 83π

MAþ3N4A2

� �¼ � F

Aω2mn

� 4HAπω2

mn: ð42Þ

Note that, in the case of FGM shell without axial compressionPx; Py, the term including H in Eqs. (39)–(42) is neglected.

If F ¼ 0, i.e. no excitation acting on the shell, the frequency–amplitude relation of the free nonlinear vibration is obtained:

ω2NL ¼ ω2

mn 1� 83π

MAþ3N4A2

� ��4HAπ

: ð43Þ

5. Numerical results and discussion

In order to illustrate the present approach, we consider aceramic–metal FGM shell that consists of aluminum (metal) andalumina (ceramic) with the following properties [34–36]

Ec ¼ 380� 109 N=m2; ρc ¼ 3800 kg=m3;

Em ¼ 70� 109 N=m2; ρm ¼ 2702 kg=m3;

vm ¼ 0:3177; νc ¼ 0:24:

The parameters for the stiffeners are

z1 ¼ z2 ¼ 0:0225ðmÞ; s1 ¼ s2 ¼ 0:025ðmÞ;h1 ¼ h2 ¼ 0:0125ðmÞ; d1 ¼ d2 ¼ 0:004ðmÞ:

D. Huy Bich et al. / International Journal of Mechanical Sciences 80 (2014) 16–2820

5.1. Validation of the present formulation

In order to validate the present study, the fundamental fre-quency parameter of un-stiffened FGM shallow shells is comparedwith other studies.

Table 1 compares the values of the fundamental frequencyparameter Ψ ¼ωmnh

ffiffiffiffiffiffiffiffiffiffiffiffiρc=Ec

pof the perfect un-stiffened FGM shal-

low shell a=b¼ 1; b=h¼ 0:1� �

in this paper with the results withthose presented by Alijani et al. [23] used Donnell0s nonlinearshallow shell theory, Chorfi and Houmat [37] accorded to the first-order shear deformation theory and Matsunaga [38] based on thetwo-dimensional (2D) higher-order theory. Poisson ratio is chosento be 0.3. From Table 1, it is seen that the present values are notsignificantly different in this comparison study.

Next, the frequency of natural vibration ωmn of spherical panelsðRx ¼ Ry ¼ 5 ðmÞ;N¼ 1Þ is compared with the result of Bich et al. [32]

based on classical shell theory. As shown in Table 2, a goodagreement can be witnessed.

Table 3 compares the values of the fundamental frequencies ofthe spherical FGM shallow shell with Duc [34]: m¼ n¼ 1;

Table 1Comparison among the fundamental natural parameter Ψ ¼ωmnh

ffiffiffiffiffiffiffiffiffiffiffiffiρc=Ec

pfor a Al/Al2O3 FGM shallow shell without elastic foundations ða=b¼ 1; b=h¼ 0:1Þ.

Structures a/Rx b/Ry N Ψ ¼ωmnhffiffiffiffiffiffiffiffiffiffiffiffiρc=Ec

pPresent

Ref. [23] Ref. [37] Ref. [38]

Plate 0 0 0 0.0597 0.0577 0.0588 0.05810.5 0.0506 0.049 0.0492 0.05021 0.0456 0.0442 0.043 0.04464 0.0380 0.0383 0.0381 0.0387

Spherical shell 0.5 0.5 0 0.0779 0.0762 0.0751 0.07670.5 0.0676 0.0664 0.0657 0.06681 0.0617 0.0607 0.0601 0.06114 0.0519 0.0509 0.0503 0.0513

Cylindrical shell 0 0.5 0 0.0648 0.0629 0.0622 0.06320.5 0.0553 0.054 0.0535 0.05431 0.0501 0.049 0.0485 0.05014 0.0430 0.0419 0.0413 0.0422

Hyperbolic paraboloidal shell �0.5 0.5 0 0.0597 0.058 0.0563 0.05920.5 0.0506 0.0493 0.0479 0.04961 0.0456 0.0445 0.0432 0.04484 0.0396 0.0385 0.0372 0.0389

Table 2Comparison of natural frequencies ðrad=sÞ of the FGM spherical shallow shell witha¼ b¼ 0:8 ðmÞ; h¼ 0:01ðmÞ.

(m,n) Un-stiffened Stiffened

Present Ref. [32] Present Ref. [32]

(1,1) 1775.96 1809.95 2517.87 2560.43(1,2) and (2,1) 2407.32 2437.29 6710.93 6743.12(2,2) 3282.12 3299.81 9275.76 9309.30(1,3) and (3,1) 3897.84 3931.47 14352.33 14576.20(2,3) and (3,2) 4877.61 4920.50 15860.25 16063.76

Table 3Effects of volume ratio N on fundamental frequencies of the spherical FGM doublecurved shallow shell in comparison with Duc0s results [34].

N ωL ðrad=sÞ

Stiffened Un-stiffened

Present Ref. [34] Present Ref. [34]

0 55:726� 105 56:130� 105 54:957� 105 55:667� 105

1 38:642� 105 39:034� 105 38:064� 105 38:515� 105

2 31:125� 105 31:982� 105 30:892� 105 31:441� 105

5 23:653� 105 24:047� 105 22:931� 105 23:477� 105

Fig. 2. Dynamic response of imperfect eccentrically stiffened FGM double curvedshallow shells on elastic foundation. b=a¼ 1; b=h¼ 30; N ¼ 1; m¼ n¼ 1; Rx ¼ Ry ¼6 ðmÞ; k1 ¼ k2 ¼ 0; W0 ¼ 0; ε¼ 0; q¼ 5000 sin ð500 tÞ. (—, present; - - -, Duc [34]).

Table 4Comparison of frequencies (s�1) calculated by Eq. (32) and Eq. (37) of the FGM shellwith a=b¼ 1; b=h¼ 20; m;nð Þ ¼ ð1;1Þ.

N Natural frequencies obtained fromEq. (32) with full motion equations

Natural frequencies obtained fromEq. (37) with simplified assumption

Stiffened Un-stiffened Stiffened Un-stiffened

0 2264.8 1926.5 2267.0 1930.50.5 2035.6 1585.4 2039.2 1588.71 1722.4 1393.8 1725.5 1398.62 1532.8 1024.5 1538.9 1031.53 1265.9 884.4 1269.2 889.65 1034.3 647.3 1037.4 650.01 876.5 417.9 878.3 418.3

D. Huy Bich et al. / International Journal of Mechanical Sciences 80 (2014) 16–28 21

a¼ b¼ 2 ðmÞ; h¼ 0:01ðmÞ; k1 ¼ 200; k2 ¼ 10; ν¼ 0:3; Rx ¼ Ry ¼3 ðmÞ. From Table 3, as can be seen that the present values are notsignificantly different from the results of Duc [34].

Fig. 2 shows the comparison of dynamic responses of the shellin this paper based on the first order shear theory and the resultsin Duc [34] based on the classic theory. As can be seen that there isa little difference.

5.2. Nonlinear dynamic responses

In this section, we will consider the effects of the characteristicsof FGM shell, the pre-loaded axial compression, the elasticfoundation and the dimensional ratios on the nonlinear dynamicresponses of the FGM shallow shell. First, we examine thefrequencies in the case of non-ignored inertial forces caused bytwo rotations Φx; Φy in Eq. (32) and compare with the frequenciesin the case of ignored due to the assumption inertial forces causedby too small rotations Φx; Φy (Eq. 37). From Table 4, we can seethat the two values of the natural frequencies from Eqs. (32) and(37) quite close together. The obtained results also show that thenatural frequencies of stiffened shells are greater than ones of un-stiffened shells. The natural frequencies of stiffened and un-stiffened FGM shallow shells observed to be dependent on theconstituent volume fractions, they decreases when increasing thepower index N, furthermore with greater value N the effects ofstiffeners is observed to be stronger.

The effects of elastic foundations on the natural oscillationfrequency of the FGM shell are shown in Table 5. The value of thenatural oscillation frequency increases when the values k1 and k2increase. Furthermore, the Pasternak elastic foundation influenceson the natural oscillation frequency larger than the Winklerfoundation. Table 5 also shows that the natural oscillation fre-quency increases when m;nð Þ are increased.

Comparison of dynamic responses established by the full orderof motion Eq. (33) and by the simplified Eq. (35) is shown in Fig. 3when the frequency of excited pressure load is far away from thenatural frequency of the shell and Fig. 4 when the frequency ofexcitation is near to the natural frequency of the shell. It can beseen that there is not much difference between the nonlineardynamic responses of two cases also.

Figs. 5–7 represent the nonlinear dynamic responses of the shellswith different values of the volume ratio N when the frequency of

Table 5Effects of elastic foundations on natural frequencies (s�1) of the FGM shell witha=b¼ 1; a=h¼ 20; N¼ 1, Px ¼ 0; Py ¼ 0.

(m,n) k1,k2

(0, 0) (0.3, 0) (0.3, 0.02) (0, 0.04)

1;1ð Þ 1722.4 1824.71 2071.87 21742.781;2ð Þ and 2;1ð Þ 4735.2 6357.49 7584.40 8013.032;2ð Þ 7543.92 8751.92 10636.45 11028.721;3ð Þ and 3;1ð Þ 10266.27 12287.02 14047.12 15648.412;3ð Þ and 3;2ð Þ 14292.85 15642.97 17254.21 19234.24

Fig. 3. Comparison of dynamic responses in the two cases established by Eqs. (33)and (35). b=a¼ 1; b=h¼ 30; N ¼ 1; m¼ n¼ 1; Rx ¼ Ry ¼ 6 ðmÞ; k1 ¼ k2 ¼ 0; W0 ¼ 0;ε¼ 0:1; q¼ 5000 sin ð500 tÞ. (—, using Eq. (33); - - -, using the full order Eq. (35)).

Fig. 4. Comparison of dynamic responses in the two cases established by Eqs. (33) and (35). b=a¼ 1; b=h¼ 30; N ¼ 2; m¼ n¼ 1; Rx ¼ Ry ¼ 6 ðmÞ; k1 ¼ k2 ¼ 0;W0 ¼ 0; q¼ 4500 sin ð1530 tÞ.

D. Huy Bich et al. / International Journal of Mechanical Sciences 80 (2014) 16–2822

the exciting force is near to the natural frequency of the shell. Thenonlinear dynamic responses with volume ratio N¼ 0, naturalfrequency of the shell ω¼ 2264: 8 ðs�1Þ, frequency of the externalload Ω¼ 2270:5 ðs�1Þ are shown in Fig. 5. The nonlinear dynamicresponses with volume ratio N¼ 1, natural frequency of the shellω¼ 1722:4 ðs�1Þ, frequency of the external load Ω¼ 1725:6 ðs�1Þare shown in Fig. 6. The nonlinear dynamic responses with N¼ 5,natural frequency of the shell ω¼ 1034:3 ðs�1Þ, frequency of theexternal load Ω¼ 1040:2 ðs�1Þare shown in Fig. 7. We can see thatwhen the frequencies of the external forces close to the value of thenatural frequencies of the shell, the nonlinear vibration of the shelloccurs the phenomenon like the harmonic beat phenomenon of thelinear vibration.

Fig. 8 gives the effect of the power law index N on the nonlineardynamic response of FGM shells with a=b¼ 1; a=h¼ 20;Px ¼ 0; Py ¼ 0; ε¼ 0:1 when the frequency of external force q isfar away from the natural oscillation frequency of the FGM shell

with N¼ 0; 1; 5. It can be seen that the amplitude of the non-linear dynamic response of FGM shell increases when increasingthe power law index N.

Figs. 9 and 10 illustrate the effect of geometric factors of theFGM shell on nonlinear dynamic response with N¼ 1; Px ¼ 0;Py ¼ 0; ε¼ 0:1. From Fig. 9 the amplitude of the FGM shellincreases when increasing the ratio a=b. Fig. 10 describes the effectof the ratio b=h on the nonlinear dynamic response of FGM shells,the shell fluctuates stronger when the ratio b=h is increased.

Figs. 11 and 12 show the effect of elastic foundations on thenonlinear dynamic response of the FGM shell with a=b¼ 1;a=h¼ 20; N¼ 1; Px ¼ 0; Py ¼ 0. Fig. 11 shows the effect of theWinkler foundation. It is clear that the shell fluctuation amplitudedecreases when the module k1 of Winkler foundation increases.The parameter k2 of the Pasternak foundation also has a similarbehavior. The graphs in Figs. 11 and 12 show the beneficial effectsof elastic foundations on the nonlinear dynamic response of FGM

Fig. 5. Nonlinear response of the FGM shell with N¼ 0. b=a¼ 1; b=h¼ 20;N ¼ 1; m¼ n¼ 1; Rx ¼ Ry ¼ 6 ðmÞ; k1 ¼ k2 ¼ 0; W0 ¼ 0; ε¼ 0:1; Q ¼ 5000 ðN=m2Þ.

Fig. 6. Nonlinear response of the FGM shell with N ¼ 1. b=a¼ 1; b=h¼ 20;N ¼ 1; m¼ n¼ 1; Rx ¼ Ry ¼ 6 ðmÞ; k1 ¼ k2 ¼ 0; W0 ¼ 0; ε¼ 0:1; Q ¼ 5000 ðN=m2Þ.

Fig. 7. Nonlinear response of the FGM shell with N ¼ 5. b=a¼ 1; b=h¼ 20;N ¼ 1; m¼ n¼ 1; Rx ¼ Ry ¼ 6 ðmÞ; k1 ¼ k2 ¼ 0; W0 ¼ 0; ε¼ 0:1; Q ¼ 5000 ðN=m2Þ.

Fig. 8. Effect of power law index N on nonlinear dynamical response of the FGMshells. b=a¼ 1; b=h¼ 20; m¼ n¼ 1; Rx ¼ Ry ¼ 6 ðmÞ; k1 ¼ k2 ¼ 0; W0 ¼ 0; ε¼ 0:1;q¼ 5000 sin ð500 t Þ. (—, N ¼ 0; …., N ¼ 1; - - -, N ¼ 5).

D. Huy Bich et al. / International Journal of Mechanical Sciences 80 (2014) 16–28 23

shells, namely the amplitude of the shell decreases when it isrested on elastic foundations, and the beneficial effect of thePasternak foundation is better than the Winkler one.

Fig. 13 indicates the effect of exciting force amplitude onnonlinear dynamic response in the case of Q ¼ ð5000;7500;10 000 ðN=m2ÞÞ and Px ¼ 0; Py ¼ 0, the FGM shell fluctuationamplitudes increase when Q increases. Fig. 14 shows the effect ofpre-loaded axial compression Px on the nonlinear dynamicresponse (with Py ¼ 0). This figure also indicates that the nonlineardynamic response amplitude of the FGM shell increases when thevalue of the pre-loaded axial compressive force Px increases.

Fig. 15 shows us that there is no significant change in thedynamic response of the FGM spherical shallow shell in case ofν¼ νðzÞ and ν¼ const.

Fig. 16 describes the nonlinear vibration of eccentrically stif-fened FGM shallow shell depending on initial imperfection of theshell. Obviously, the amplitude of vibration will increase and losethe stability if the initial imperfection increases. We can see that

the imperfect coefficient has a significant effect on the dynamicresponse of the FGM shell.

Fig. 17 shows the effects of stiffeners on the nonlinear dynamicresponse of the spherical FGM shallow shells. The result showsthat the stiffeners strongly decrease vibration amplitude ofthe shell.

The deflection–velocity relation has the closed curve form as inFig. 18. Deflection Wand velocity dW=dt are equal to 0 at initial timeand final time of beat and the contour of this relation corresponds tothe period which has the greatest amplitude of beat.

5.3. Frequency–amplitude curves

Fig. 19 presents the effects of the external forces and stiffenerson frequency–amplitude in the case of forced vibration. We see

Fig. 10. Effect of ratio b=h on nonlinear dynamical response of the FGM shells.b=a¼ 1; N ¼ 1; m¼ n¼ 1; Rx ¼ Ry ¼ 6 ðmÞ; k1 ¼ k2 ¼ 0; W0 ¼ 0; ε¼ 0:1; q¼ 5000sin ð500 tÞ: ( —, b=h¼ 20; …, b=h¼ 30; - - -, b=h¼ 40).

Fig. 9. Effect of ratio a=b on nonlinear dynamical response of the FGM shells.b=h¼ 20; N ¼ 1; m¼ n¼ 1; Rx ¼ Ry ¼ 6 ðmÞ; k1 ¼ k2 ¼ 0; W0 ¼ 0; ε¼ 0:1;q¼ 5000 sin ð500 tÞ: (—, a=b¼ 0:5; …, a=b¼ 1; - - -, a=b¼ 2).

Fig. 11. Effect of the linear Winkler foundation on nonlinear dynamical response ofthe FGM shell. b=a¼ 1; b=h¼ 20; N ¼ 1; m¼ n¼ 1; Rx ¼ Ry ¼ 6 ðmÞ; k2 ¼ 0;W0 ¼ 0; ε¼ 0:1; q¼ 5000 sin ð500 tÞ: (—, k1 ¼ 0:5 ðGPa=mÞ; …., k1 ¼ 0:3 ðGPa=mÞ;- - -, k1 ¼ 0).

Fig. 12. Effect of the Pasternak foundation on nonlinear dynamical response of theFGM shell. b=a¼ 1; b=h¼ 20; N ¼ 1; m¼ n¼ 1; Rx ¼ Ry ¼ 6 ðmÞ; k1 ¼ 0; W0 ¼ 0;ε¼ 0:1; q¼ 5000 sin ð500 tÞ: (—, k2 ¼ 0:04 ðGPa=mÞ; ….., k2 ¼ 0:02 ðGPa=mÞ; - - -,k2 ¼ 0).

D. Huy Bich et al. / International Journal of Mechanical Sciences 80 (2014) 16–2824

that with the same frequency, the amplitude of stiffened shells issmaller than one of un-stiffened shells.

Fig. 20 shows the effects of elastic foundations on the fre-quency–amplitude relations of the FGM shells. One can see thatwith the same frequency, FGM shells on elastic foundations havesmaller amplitude than the shells without elastic foundations, andthe Pasternak foundation influences more sensitively than Winklerfoundation on the frequency–amplitude relations of the shells.

Fig. 21 shows the effect of excitation force Q on the frequency–amplitude of nonlinear vibration of FGM shallow shells. As can beseen, when the excitation force decreases, the curves of forcedvibration are closer to the curve of free vibration.

Effect of pre-loaded axial compression Px on the frequency–amplitude of nonlinear vibration of FGM shallow shells with Py ¼ 0without damping is shown in Fig. 22. Clearly, the extreme points offrequency–amplitude curve decrease when pre-loaded axial com-pression decreases.

6. Conclusions

This is the first paper presenting an analytical approach toinvestigate the nonlinear dynamic response and vibration ofimperfect eccentrically stiffened FGM double curved thick shallowshells using both the first order shear deformation theory andstress function. The FGM shell is assumed to rest on elasticfoundation and subjected to mechanical and damping loads.Numerical results for dynamic response of the FGM shell areobtained by Runge–Kutta method.

From the obtained results in this paper we can conclude that:

� The initial imperfection has a significant influence on thenonlinear dynamic response and nonlinear vibration of theFGM shells.

� The stiffener system strongly enhances the load-carrying capa-city of the FGM shallow shells.

Fig. 13. Nonlinear dynamic responses of the FGM shell with different loads.b=a¼ 1; b=h¼ 20; N ¼ 1; m¼ n¼ 1; Rx ¼ Ry ¼ 6 ðmÞ; k1 ¼ k2 ¼ 0; W0 ¼ 0; ε¼0:1; Ω¼ 500 ðs�1Þ: (—, Q ¼ 5000 ðN=m2Þ; …, Q ¼ 7500 ðN=m2Þ; - - -, Q ¼10 000 ðN=m2Þ).

Fig. 14. Effect of pre-loaded axial Px compression on nonlinear response of the FGMshell. b=a¼ 1; b=h¼ 20; N ¼ 1; m¼ n¼ 1; Rx ¼ Ry ¼ 6 ðmÞ; k1 ¼ k2 ¼ 0; W0 ¼ 0;ε¼ 0:1; q¼ 5000 sin ð500 tÞ: (—, Px ¼ 0; …, Px ¼ 300 ðMPaÞ; - - -, Px ¼ 500 ðMPaÞ).

Fig. 15. Effect of Poisson ratio on dynamic response of the FGM spherical shallowshell. b=a¼ 1; b=h¼ 20; N¼ 1; m¼ n¼ 1; Rx ¼ Ry ¼ 6 ðmÞ; W0 ¼ 0; ε¼ 0:1;q¼ 5000 sin ð500 tÞ: (—, ν¼ νðzÞ; - - -,ν¼ const).

Fig. 16. Effect of imperfection W0 on nonlinear dynamic response of FGM shell.b=a¼ 1; b=h¼ 30; N¼ 1; m¼ n¼ 1; Rx ¼ Ry ¼ 6 ðmÞ; k1 ¼ k2 ¼ 0; ε¼ 0:1; q¼5000 sin ð500 tÞ (—, W0 ¼ 0; …, W0 ¼ 10�5; - - -, W0 ¼ 3� 10�5).

D. Huy Bich et al. / International Journal of Mechanical Sciences 80 (2014) 16–28 25

Fig. 17. Effect of stiffeners on nonlinear dynamic response of the shallow sphericalFGM shell. b=a¼ 1; b=h¼ 20; N ¼ 1; m¼ n¼ 1; Rx ¼ Ry ¼ 6 ðmÞ; ε¼ 0:1; q¼ 5000sin ð500 tÞ; W0 ¼ 0 (—, eccentrically stiffened FGM shell; - - -, FGM shell withouteccentrically stiffened).

Fig. 18. Deflection–velocity relation of FGM shallow shell. b=a¼ 1; b=h¼ 30; N¼ 1;m¼ n¼ 1; Rx ¼ Ry ¼ 6 ðmÞ; k1 ¼ k2 ¼ 0; ε¼ 0:1; q¼ 5000 sin ð500 tÞ.

Fig. 19. The frequency–amplitude curve of nonlinear vibration of FGM shells.b=a¼ 1; b=h¼ 30; N¼ 1; m¼ n¼ 1; Rx ¼ Ry ¼ 6 ðmÞ; k1 ¼ k2 ¼ 0; ε¼ 0; q¼ 5000sin ð500 tÞ (—, free vibration; - - -, forced vibration).

Fig. 20. Effect of the elastic foundations on frequency–amplitude curve of FGMshells in the case of free vibration and no damping. b=a¼ 1; b=h¼ 20; m¼ n¼ 1;Rx ¼ Ry ¼ 6 ðmÞ; W0 ¼ 0; ε¼ 0; q¼ 5000 sin ð500 tÞ.

Fig. 21. Effect of excitation force Q on the frequency–amplitude curve of FGMshell. b=a¼ 1; b=h¼ 30; N ¼ 1; m¼ n¼ 1; Rx ¼ Ry ¼ 6 ðmÞ; k1 ¼ k2 ¼ 0; ε¼ 0;Ω¼ 500 ðs�1Þ.

Fig. 22. Effect of pre-loaded axial compression Px on the frequency–amplitude ofthe FGM shell (—, free vibration; - - -, forced vibration).b=a¼ 1; b=h¼ 30; N ¼ 1;m¼ n¼ 1; Rx ¼ Ry ¼ 6 ðmÞ; k1 ¼ k2 ¼ 0; ε¼ 0:1; Q ¼ 105ðN=m2Þ.

D. Huy Bich et al. / International Journal of Mechanical Sciences 80 (2014) 16–2826

� The pre-loaded axial compressions strongly influence on thecritical dynamic response of the stiffened shallow shell.

� The elastic foundations have a strong effect on the nonlineardynamic response of the FGM shells and the beneficial effect ofthe Pasternak foundation is better than the Winkler one.

Especially, the comparisons of the fundamental frequencies andthe nonlinear responses between using the full order equation andthe approximate equation (when the inertial forces caused byrotations Φx; Φy in motion equations are ignored) are done. Thetested results show that the assumption ignoring the inertialforces caused by two rotations Φx; Φy is reasonable and can beused, for simplicity, in the approximate cases.

The present results comparing with results received by othermethods indicate the reliability of the calculations.

Acknowledgement

This work was supported by the Grant in Mechanics coded107.02-2013.06 of the National Foundation for Science and Tech-nology Development of Vietnam – NAFOSTED. The authors aregrateful for this support.

Appendix A

l11 ¼ �m2π2

a2Ryþn2π2

b2Rx

� 2A11

m4π4

a4 þA22n4π4

b4þðA66�2A12Þm2n2π4

a2b2

� þk1þk2m2π2

a2þn2π2

b2

� �264

375;

l12 ¼ �KI30mπ

aþ m2π2

a2Ryþn2π2

b2Rx

! ðB66�B11Þmn2π3

ab2�B21

m3π3

a3

h iA11

m4π4

a4 þA22n4π4

b4þðA66�2A12Þm2n2π4

a2b2

� ;

l13 ¼ �KI30nπbþ m2π2

a2Ryþn2π2

b2Rx

! ðB66�B22Þm2nπ3a2b �B12

n3π3

b3

h iA11

m4π4

a4 þA22n4π4

b4þðA66�2A12Þm2n2π4

a2b2

� ;

l14 ¼8mnπ2

ab

ðB66�B11Þmn2π3

ab2�B21

m3π3

a3

h iA11

m4π4

a4 þA22n4π4

b4þðA66�2A12Þm2n2π4

a2b2

� ;

l15 ¼8mnπ2

ab

ðB66�B22Þm2nπ3a2b �B12

n3π3

b3

h iA11

m4π4

a4 þA22n4π4

b4þðA66�2A12Þm2n2π4

a2b2

� ;n1 ¼ �KI30

m2π2

a2þn2π2

b2

� ��h Px

m2π2

a2þPy

n2π2

b2

� �;

n2 ¼8mnπ2

3ab

m2π2

a2Ryþn2π2

b2Rx

� A11

m4π4

a4 þA22n4π4

b4þðA66�2A12Þm2n2π4

a2b2

� ;n3 ¼ �1

6mb

A22naRxþ naA11mbRy

� �;

n4 ¼ �18

m2π2

a2 þn2π2

b2

� m2b2

A22n2a2þ n2a2

A11m2b2

� ;

n5 ¼16

mnπ2; n6 ¼

�16hmnπ2

Px

RxþPy

Ry

� �;

l21 ¼ ðB11�B66Þmn2π3

ab2

m2π2

a2Ryþn2π2

b2Rx

� A11

m4π4

a4 þA22n4π4

b4þðA66�2A12Þm2n2π4

a2b2

� þ

þB21

m2π2

a2Ryþn2π2

b2Rx

� A11

m4π4

a4 þA22n4π4

b4þðA66�2A12Þm2n2π4

a2b2

� m3π3

a3;

l22 ¼ D11mπ

a

� 2þD66

nπb

� 2þ I30þ

B21m3π3

a3 þðB11�B66Þmn2π3

ab2

h i2A11

m4π4

a4 þA22n4π4

b4þðA66�2A12Þm2n2π4

a2b2

� ;

l23 ¼ D12þD66ð Þmnπ2

abþ

B21B12m3n3π6

a3b3þB21ðB22�B66Þm5nπ6

a5b

h iA11

m4π4

a4 þA22n4π4

b4þðA66�2A12Þm2n2π4

a2b2

� ;n7 ¼ KI30

mπ

a;

n8 ¼ �43B21

A11

nπ

ab2;

l31 ¼ ðB22�B66Þm2nπ3

a2b

m2π2

a2Ryþn2π2

b2Rx

� A11

m4π4

a4 þA22n4π4

b4þðA66�2A12Þm2n2π4

a2b2

�

þB12

m2π2

a2Ryþn2π2

b2Rx

� A11

m4π4

a4 þA22n4π4

b4þðA66�2A12Þm2n2π4

a2b2

� n3π3

b3;

l32 ¼ ðD21þD66Þmnπ2

abþ

B212

n6π6

b6þB12ðB22�B66Þm2n4π6

a2b4

A11m4π4

a4 þA22n4π4

b4þðA66�2A12Þm2n2π4

a2b2

� ;

l33 ¼D22nπb

� 2þD66

mπ

a

� 2þ I30þ

B12n3π3

b3þðB22�B66Þm2nπ3

a2b

h i2A11

m4π4

a4 þA22n4π4

b4þðA66�2A12Þm2n2π4

a2b2

� ;n9 ¼ KI30

nπb;

n10 ¼ �43B12

A22

mπ

a2b;

ρ1 ¼I21I0� I2:

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