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

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: DOI: Reference: S1270-9638(14)00230-2 10.1016/j.ast.2014.11.005 AESCTE 3163 To appear in: Aerospace Science and Technology Received date: September 2014 Revised date: November 2014 Accepted date: 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 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain 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 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 unstiffened 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 F FGM sandw wich plates Sadoune et e al [45] prresented a nnew simplee first-orderr shear defoormation ttheory for laminated l composite c plates p Notee that all thee publicatioons mentionned above [[29-45], tthe authorss used the first order shear deformation theeory with ddisplacement functionns while studying no onlinear dyn namic and vibration v off thick FGM M shells In thiss paper, wee research the nonlinear dynamiic and nonnlinear vibrration of im mperfect eccentricallly stiffened d functionallly graded thick t circulaar cylindriccal shells w with metal-cceramicm metal layerrs, which arre symmetric through the middle surface byy Sigmoid-laaw distribuution (SF FGM) and surrounded d on elastic foundation ns using the first order shear deforrmation theeory and V assu umption is not approppriate due too the fact thhat the rightt side of stress functtion The Volmir's equations of o motion doesn’t eq qual to Zerro [46] Fuurthermore, in this paaper, we toook into account off the presen nce of stifffeners and elastic e founndations T Therefore, tthe calculatting has bbecome more m compliicated Thee Galerkin method aand Runge Kutta metthod are uused for dynamic analysis a of the cylind drical shells to give expressionn of naturaal frequenccies and nnonlinear response Nu umerical reesult shows the effects of characteeristics of ffunctionallyy graded m materials, geometrical g l and materrial propertties, elastic foundationns and ecceentrically sttiffeners on the dynaamical behaavior of the shells Theoretiical formulations 2.1 Eccenttrically stifffened S-FGM thick circular cyylindrical sshells surrounded on n elastic foundation ns Fig.1 Configuratio C on of an ecccentrically stiffened S FGM thickk circular cyylindrical shhell For an S-FGM cylindrical shell made of two different constituent materials with metalceramic-metal layers, the volume fractions Vc z and Vm z can be written in the Sigmoid power law distribution as [27-28,47] £¦ z h ơN ƯƯ ưư , Ư đ h Ư Vc z  Ô ƯƯ 2 z h ơN , ƯƯ h ưưđ Ư ƯƠ Vm z Vc z  1, h N p 0,  b z b 0, h 0b z b , (1) where the Young’s modulus E , the Poisson’s ratio O are expressed as: £¦ z sy sx sx sy (6) Hooke’s law for the cylindrical shell is defined as [27-28, 49-50] E T x , T y   O  ¡¢ Fx , F y O F y , Fx ¯°± , E E T xy  H xy , T xz  H xz , 2(1 O ) 2(1 O ) E T yz  H yz 2(1 O ) (7) and for stiffeners T xst , T yst  E0 Fx , Fy , (8) The constitutive stress-strain equations by Hook law for the shell material are omitted here for brevity The shell reinforced by eccentrically longitudinal and circumferential stiffeners is shown in Fig The contribution of stiffeners can be accounted for using the Lekhnitsky smeared stiffeners technique and the force and moment resultants of an eccentrically stiffened S-FGM shell are shown as [27-28, 49-50]:  E A¬ N x  žž I10 x ­­­ F0 x I 20F0 y I11 C x Dx I 21D y , žŸ sx ưđ E0 Ay ơư ưư F0 y I 21Dx I11 C y D y , N y  I 20F0 x žž I10 .. .Nonlinear dynamic response and vibration of shear deformable imperfect eccentrically stiffened S-FGM circular cylindrical shells surrounded on elastic foundations Nguyen Dinh... investigate the nonlinear dynamic response and vibration of imperfect eccentrically stiffened functionally graded thick circular cylindrical shells surrounded on elastic foundations using both of. .. theelastic foundations, eccentrically stiffeners, pre– loaded axial compression and damping loads on the nonlinear dynamic response and nonlinear vibration of functionally graded cylindrical shells