DSpace at VNU: Nonlinear dynamic response and vibration of shear deformable imperfect eccentrically stiffened S-FGM circular cylindrical shells surrounded on elastic foundations

Accepted ManuscriptNonlinear dynamic response and vibration of shear deformable imperfect eccentrically stiffened S-FGM circular cylindrical shells surrounded on elastic foundations Nguy

Accepted Manuscript

Nonlinear dynamic response and vibration of shear deformable

imperfect eccentrically stiffened S-FGM circular cylindrical shells

surrounded on elastic foundations

Nguyen Dinh Duc, Pham Toan Thang

PII: S1270-9638(14)00230-2

Reference: AESCTE 3163

To appear in: Aerospace Science and Technology

Received date: 4 September 2014

Revised date: 1 November 2014

Accepted date: 9 November 2014

Please cite this article in press as: D.D Nguyen, T.T Pham, Nonlinear dynamic response and vibration of shear deformable imperfect eccentrically stiffened S-FGM circular cylindrical shells surrounded on elastic

foundations, Aerosp Sci Technol (2014), http://dx.doi.org/10.1016/j.ast.2014.11.005

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Nonlinear dynamic response and vibration of shear deformable imperfect

eccentrically stiffened S-FGM circular cylindrical shells

surrounded on elastic foundations

Nguyen Dinh Duc*, Pham Toan Thang Vietnam National University, Ha Noi, 144 XuanThuy – Cau Giay – Ha Noi – Viet Nam Email: ducnd@vnu.edu.vn; thangpt_55@vnu.edu.vn, Tel: +84-4-37547978; Fax: +84-4-37547424

Abstract: This paper presents an analytical approach to investigate the nonlinear dynamic response

and vibration of imperfect eccentrically stiffened functionally graded thick circular cylindrical shells surrounded on elastic foundations using both of the first order shear deformation theory and stress function with full motion equations (not using Volmir's assumptions) Material properties are graded in the thickness direction according to a Sigmoid power law distribution (S-FGM) in terms of the volume fractions of constituents with metal - ceramic - metal layers The S-FGM shells are subjected to mechanical and damping loads Numerical results for dynamic response of the shells are obtained by Runge-Kutta method The results show the influences of geometrical parameters, the volume fractions

of metal – ceramic – metal layers, imperfections, theelastic foundations, eccentrically stiffeners, pre–loaded axial compression and damping loads on the nonlinear dynamic response and nonlinear vibration of functionally graded cylindrical shells The proposed results are validated by comparing with other results reported in literature

Keywords: Nonlinear dynamic response, vibration, Sigmoid FGM thick circular cylindrical shells, the first order shear deformation theory, elastic foundations

1 Introduction

The idea of FGMs was first introduced in 1984 by a group of Japanese material scientists [1] Functionally graded materials (FGMs) are composite materials obtained by combining and mixing two or more different constituent materials, which are distributed along the thickness in

Corresponding author: Duc.N.D E-mail address: ducnd@vnu.edu.vn

accordance with a volume fraction law The FGM have received considerable attention in recent years due to their high performance heat resistance capacity and excellent characteristics

in comparison with conventional composites

Regarding to the dynamic and vibration of FGM plates and shells, Loy et al [2] analyzed the vibrations of the FGM cylindrical shells They found that the natural frequencies are affected by the constituent volume fractions and configurations of the constituent materials Pradhan et al [3] studied the vibration characteristics of FGM cylindrical shells made of stainless steel and zirconia under different boundary conditions Free vibration analysis of functionally graded cylindrical shells with holes was researched in [4] Ebrahimi and Najafizadeh [5] investigated the free vibration of a two-dimensional functionally graded circular cylindrical shell The equations of motion are based on the Love’s first approximation classical shell theory Shen [6] researched the large amplitude vibration behavior of a shear deformable FGM cylindrical shell of finite length embedded in a large outer elastic medium and in thermal environments Najafizadeh and Isvandzibaei [7,8] studied free vibration of FGM cylindrical shells with ring support by using Ritz method based on the first order and higher order shear deformation shell theories Haddadpour et al [9]considered free vibration of simply supported FGM cylindrical shells with four sets of in-plane boundary conditions by using Galerkin method based on the classical shell theory Alibeigloo et al [10] presented the numerical free vibration analysis for FGM cylindrical shell embedded thin piezoelectric layers Sofiyev and Kuruoglu [11] focused the torsional vibration and buckling of un-stiffened cylindrical shell with functionally graded coatings surrounded by an elastic medium Bich and Nguyen [12] used the displacement functions to investigate the nonlinear vibration of FGM un-stiffened cylindrical shells subjected to axial and transverse mechanical loads Their results shown that the Volmir’s assumption can be used for nonlinear dynamic analysis with an acceptable accuracy Shariyat [13] studied the dynamic buckling of imperfect FGM cylindrical shells with integrated surface-bonded sensor and actuator layers subjected to some complex combinations of thermo-electro-mechanical loads Shen [14-16] presented a postbuckling analysis of FGM cylindrical thin shells and FGM panels subjected to axial compression or external pressure in thermal environments Shen and Noda [17] obtained the postbuckling analysis for FGM cylindrical shells with piezoelectric actuators subjected to lateral pressure in

thermal environments Loy et al [18] investigated the vibration of FGM cylindrical shells composed of stainless steel and nickel, considering the influence of the constituent volume fractions and the effects of the constituent materials on the frequencies Meiche et al [19] proposed a new hyperbolic shear deformation theory taking into account transverse shear deformation effects for the buckling and free vibration analysis of thick functionally graded sandwich plates Benachour et al [20] used the four variable refined plate theory for free vibration analysis of plates made of functionally graded materials with an arbitrary gradient Hebali et al [21] developed a new quasi-three-dimensional (3D) hyperbolic shear deformation theory for the bending and free vibration analysis of functionally graded plates Bessaim et al [22] studied a new higher-order shear and normal deformation theory for the bending and free vibration analysis of sandwich plates with functionally graded isotropic face sheets Larbi et al [23] investigated an efficient shear deformation beam theory based on neutral surface position

is developed for bending and frees vibration analysis of functionally graded beams Bouremanaet al [24] studied a new first-order shear deformation beam theory based on neutral surface position is developed for bending and free vibration analysis of functionally graded beams Meziane et al [25] proposed an efficient and simple refined shear deformation theory is presented for the vibration and buckling of exponentially graded material sandwich plate resting on elastic foundations under various boundaries Draiche et al [26] investigated the use

of trigonometric four variable plate theory for free vibration analysis of laminated rectangular plate supporting a localized patch mass

Today, functionally graded shells involving circular cylindrical shells are widely used in many important details of space vehicles, aircrafts, nuclear power plants and many other engineering applications For example, the strategic missiles using solid materials, they capable fly far beyond the continent with great velocity, so their hull could stand very high strength and high temperatures To satisfy it, the shell of the strategic missiles usually is made

of composite carbon-carbon or functionally graded materials (FGM) FGM circular cylindrical shell also could be used as the shell of a nuclear reactor or special engineering pipes, Regarding to the static and dynamic analysis of the FGM circular cylindrical shells, Duc and Thang [27] studied an analytical approach to investigate the nonlinear static buckling and postbuckling for imperfect eccentrically stiffened functionally graded thin circular

cylindrical shells surrounded on elastic foundation with ceramic–metal–ceramic layers and subjected to axial compression Duc and Thang [28] also investigated the nonlinear static buckling for imperfect functionally graded thin circular cylindrical shells reinforced by stiffeners in thermal environment

Some researchers have used the first-order and high-order shear deformation theories for buckling analysis of the perfect and imperfect thick composite cylindrical shells [29-31] Sheng and Wang [32] studied dynamic behavior for the functionally graded cylindrical shell with surface-bonded PZT piezoelectric layer under moving loads Shahsiah and Eslami [33] investigated the thermal buckling of FGM cylindrical shells under two types of thermal loads based on the first order shear deformation shell theory Shen [34] researched the large amplitude vibration behavior of a shear deformable FGM cylindrical shell of finite length embedded in a large outer elastic medium and in thermal environments Shahsiah and Eslami [35] presented the buckling temperature of simply supported FGM cylindrical shells under two cases of thermal loading using the first order shear deformation shell theory Bouderba et al [36] studied the thermomechanical bending response of functionally graded plates resting on Winkler-Pasternak elastic foundations.Tounsi et al [37] proposed a refined trigonometric shear deformation theory (RTSDT) taking into account transverse shear deformation effects is presented for the thermoelastic bending analysis of functionally graded sandwich plates Bourada et al [38] performed the use of a new four-variable refined plate theory for thermal buckling analysis of functionally graded material (FGM) sandwich plates Belabed et al [39] presented an efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates Bouiadjra [40] studied the nonlinear behavior of functionally graded material (FGM) plates under thermal loads using an efficient sinusoidal shear deformation theory Fekrar et al [41] developed a new sinusoidal higher-order plate theory for bending of exponential graded plates Bousahla et al [42] proposed a new trigonometric higher-order theory including the stretching effect for the static analysis of advanced composite plates such as functionally graded plates.Saidi et al [43] included an analytical solution to the thermo-mechanical bending analysis of functionally graded sandwich plates by using a new hyperbolic shear deformation theory Houari et al [44]developeda new higher order shear and normal deformation theory to simulate the thermoelastic bending of

e research

d functional

re symmetr

d on elastic Volmir's assudoesn’t eqnce of stifficated Thethe cylindumerical re

l and materavior of the

lations ffened S-F

on of an ecc

et al [45] prplates Noteshear defovibration ofthe nonlinlly graded t

ic through foundationumption is qual to Zerfeners and e

e Galerkin drical shellesult showsrial propertshells

ns using the not approp

ro [46] Fuelastic founmethod a

s to give the effects ties, elastic

circular cy

stiffened

S-new simple

e publicatioeory with d

M shells

ic and non

ar cylindricsurface byfirst order priate due tourthermore, ndations Tand Runge-expression

of charactefoundation

ylindrical s

-FGM thick

e first-orderons mentiondisplaceme

nlinear vibrcal shells w

y Sigmoid-lashear defor

o the fact th

in this paTherefore, t-Kutta met

n of naturaeristics of f

ns and ecce

shells surr

k circular cy

r shear defoned above [

nt function

ration of imwith metal-c

aw distriburmation thehat the rightaper, we tothe calculatthod are u

al frequencfunctionallyentrically st

ounded on

ylindrical sh

ormation [29-45],

ns while

mperfect ceramic-ution (S-eory and

t side of ook into ting has used for cies and

y graded tiffeners

n elastic

hell

For an S-FGM cylindrical shell made of two different constituent materials with ceramic-metal layers, the volume fractions V z c and V z m can be written in the Sigmoid power law distribution as [27-28,47]

metal-2

22

21,

V z

z h

with volume fraction index N dictates the material variation profile through the S-FGM shell

thickness, the subscriptsm andc are metal and ceramic constituents respectively

The shell-foundation interaction is represented by Pasternak model as

( , )( , )( , ),

, ,, ,

e displacem

y

G denote t

SDT and verately large

te system of

fo

rder shear dligible and displaceme

ment compothe mid-pla

von-Karma

er deflection

f a circular undations

deformationnormal doent field fo

onent alongane rotation

, z coordinaransverse no

displacemenobtain

ounded on e

ch assumes dicular to thwithin the

(4)

ate directioormal abou

nt relation

elastic

that the

he shell is

mid-ons, and

ut the y

n which

y x

G G

E I I

h

h

j j

0 0

( )( )

where the coupling parameters C C x, y are negative for outside stiffeners and positive for inside one; I I x, y are the second moments of cross-section areas; s s x, y are the spacing of the

longitudinal and circumferential stiffeners; ,z z x yare the eccentricities of stiffeners with respect

to the middle surface of shell; and the width and thickness of longitudinal and circumferential stiffeners are denoted by ,d h x x andd h respectively y, y A A are the cross-section areas of x, y

stiffeners And K is correction factors, 4

y x

y x

y x

F

G G

F

G G

H

ss

ss

ss

2 2

h



 ¨ (14) ,

f x y is stress function defined by

, , , , ,

N  f N  f N  f (15) Substituting Eq (15) into Eqs (13a) and (13b) yields

2 2

0 2 2 1

0

,,

,

x x

y y

G G

s

y y

Assume thatw x y , is a known function representing initial small imperfection, Eqs (21) for

an imperfect S-FGM circular cylindrical shell can be modified as the following form [19-20]:

' * 2

2 2

0 2 2

y x

x

y x

y

y x

F

G G

F

G G

H

ss

ss

ss

elastic foundations using the first order shear deformation theory

3 Nonlinear dynamic analysis

An imperfect S-FGM circularcylindrical shell considered is assumed to be simply supported and subjected to uniformly distributed pressure of intensity q and axial compression P x.Thus the boundary conditions are

The initial imperfection w*is assumed to have the same form of the shell deflectionw, i.e

in which W0 is known initial amplitude

Substituting Eqs (29) and (30) into the compatibility equation (27), we define the stress function as

(33)

in which specific expressions of coefficients H i1i( 1,5),H jk(j2,3,k 1,3),n m m( 1,10),S1

are given in Appendix A

3.1 Natural frequencies

Taking linear parts of Eqs (33) and putting q 0, the natural frequencies of the perfect shell

can be determined directly by solving determinant

3.2 Nonlinear dynamic responses

Consider a functionally graded circular cylindrical shell acted on by an uniformly distributed transverse load q t Qsin8 ( Q is the amplitude of uniformly excited load, t 8 is the frequency of the load) The system Eqs (33) have the form

dt

For further research, we next consider the hypothetical case of rotations ,' ' exist, but the x yinertial forces caused by the rotation angles ,' ' are small so they can be ignored The x ysystem Eqs (35) can be written as follows

in which specific expressions of coefficients b i  i 1,6 is give in Appendix B

In other hand, from Eq (37) the fundamental frequencies of a perfect shell can be determined approximately by an explicit expression

Seeking solution as W Asin8 and applying Galerkin procedure to Eq (40), the t

amplitude-frequency relation of nonlinear forced vibration is obtained

4 Numerical results and discussions

In order to illustrate the present approach, we consider an FGM shell that consists of aluminum (metal) and alumina (ceramic) with the following properties [46]

To validate the analysis, result for the natural frequencies of the isotropic circular cylindrical shell obtained from expression Eq (34) with N 0 (in this case, the S-FGM shell and the P-FGM shell are same and they are made of full metal) is compared with Loy et al [18] In Table

1, the comparison shows that the present result is agreed well with those in the literature

Fig 2 and Table 2 show the comparison of dynamic response of the S-FGM shells in this paper based on the first order shear theory and the results in Bich et al [12] for the P-FGM shell based on the classic theory with the same geometrical parameters without elastic foundations Clearly, in Table 2 we can see that there is very a little difference between natural frequencies, but in Fig 3, it is easy to recognize that the obtained amplitudes of frequency in this paper of the S-FGM shell are smaller than the one of the P-FGM shell in [12]