Research ArticleBoundary Conditions in 2D Numerical and 3D Exact Models forCylindrical Bending Analysis of Functionally Graded Structures

F Tornabene1 S Brischetto2 N Fantuzzi1 and M Bacciocchi1

1DICAM Department University of Bologna Bologna Italy2Department of Mechanical and Aerospace Engineering Politecnico di Torino Torino Italy

Correspondence should be addressed to N Fantuzzi nicholasfantuzziuniboit

Received 8 August 2016 Revised 19 October 2016 Accepted 26 October 2016

Academic Editor Yuri S Karinski

Copyright copy 2016 F Tornabene et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The cylindrical bending condition for structural models is very common in the literature because it allows an incisive and simpleverification of the proposed plate and shell models In the present paper 2D numerical approaches (the Generalized DifferentialQuadrature (GDQ) and the finite element (FE) methods) are compared with an exact 3D shell solution in the case of free vibrationsof functionally graded material (FGM) plates and shells The first 18 vibration modes carried out through the 3D exact model arecompared with the frequencies obtained via the 2D numerical models All the 18 frequencies obtained via the 3D exact model arecomputed when the structures have simply supported boundary conditions for all the edges If the same boundary conditions areused in the 2D numerical models some modes are missed Some of these missed modes can be obtained modifying the boundaryconditions imposing free edges through the direction perpendicular to the direction of cylindrical bending However somemodescannot be calculated via the 2Dnumericalmodels evenwhen the boundary conditions aremodified because the cylindrical bendingrequirements cannot be imposed for numerical solutions in the curvilinear edges by definition These features are investigated inthe present paper for different geometries (plates cylinders and cylindrical shells) types of FGM law lamination sequences andthickness ratios

1 Introduction

The cylindrical bending conditions have a great diffusionin the open literature because they allow an immediate andsimple verification of several plate and shell models Someof these plate models which use the cylindrical bendinghypotheses for the verifications are discussed in the fol-lowing Chen et al [1] used the state-space method forthe investigation of a simply supported cross-ply laminatedplate embedding viscous interfaces in cylindrical bendingChen and Lee [2 3] also used the state-space method forthe cylindrical bending analysis of simply supported angle-ply laminated plates with interfacial damage and simplysupported angle-ply laminated plates subjected to a staticload An experimental three-roller cylindrical bending inves-tigation for plates was proposed by Gandhi and Raval [4]the comparison with analytical solutions was also conductedOral and Darendeliler [5] proposed a methodology for thedesign of plate-forming dies in cylindrical bending using

optimization techniques which allow the cost reductionsof die production Considering the large deflection of athin beam under certain conditions the solution of theplate problem is not unique [6] The bending-gradient platetheory seen as an extension of the Reissner-Mindlin platemodel was used in [7] for the cylindrical bending analysisof laminated plates Nimbolkar and Jain [8] used the sameReissner-Mindlin plate theory for the cylindrical bendinginvestigation of composite and elastic plates subjected to themechanical transverse load under plain strain conditionsSayyad and Ghugal [9] investigated laminated plates usinga 119899th order shear deformation theory under cylindricalbending requirements Some of the most important three-dimensional exact plate solutions were proposed by Pagano[10] and Pagano and Wang [11] for the cylindrical bendinganalysis of multilayered composite plates Saeedi et al [12]developed a 2D plate layer-wise model for the analysis ofdelamination growth in multilayered plates subjected tocylindrical bending loadings This method is alternative to

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2 Shock and Vibration

the finite element method Shu and Soldatos [13] developeda new stress analysis method to investigate the stress dis-tributions in angle-ply laminated plates when subjected tocylindrical bending Further cylindrical bending analyses forplates consider the inclusion of functionally graded material(FGM) layers typical examples are [14 15] In [16ndash21]the cylindrical bending analysis was proposed for multi-layered plates embedding piezoelectric andor FGM layersThe cylindrical bending analysis for multilayered compositeandor piezoelectric plates proposed in [22ndash24] includesthe application of thermal loads The cylindrical bendinganalysis of shells is not so diffused in the literature Fewcases can be found and they concern the cylindrical bendingconditions imposed to cylindrical shells Simply supportedangle-ply laminated cylindrical shells in cylindrical bendingwere investigated in [25] using the state-space formulation forthe bending and free vibration analyses A three-dimensionalpiezoelectric model based on the perturbation method forthe cylindrical bending electromechanical load applicationsto cylindrical shells was proposed in [26] Yan et al [27] ana-lyzed the cylindrical bending behavior of a simply supportedangle-ply laminated cylindrical shell embedding viscoelasticinterfaces

The authors proposed the free vibration analysis of simplysupported FGM plates and shells in [28 29] where the 3Dexact shell model was compared with several 2D numericalmodels such as the classical finite element (FE) one andthe classical and refined Generalized Differential Quadrature(GDQ) methods Similar comparisons were also proposedin [30 31] for one-layered and multilayered isotropic com-posite and sandwich plates and shells In such analyses theconsidered geometries are plates cylinders and cylindricaland spherical shell panels Low and high frequencies werecalculated for thick and thin simply supported plates andshells with constant radii of curvature The comparisonbetween the 2D numerical methods (FE and GDQ solutions)and the 3D exact solution is possible only if an exhaustivevibrationmode investigation is conductedThis investigationallows understanding how to make the comparison betweennumerical and analytical methods This comparison is noteasy because the exact 3D solution gives infinite vibrationmodes for all the possible combinations of half-wave numbers(119898 119899) and 2D numerical models propose a finite number ofvibration modes because they use a finite number of degreesof freedom in the plane and in the thickness directionIt should be underlined that the 3D exact model is ableto investigate any kind of geometric ratio (from extremelythick to very thin structures) nevertheless it can model onlysimply supported boundary conditions and it can investigatestructures wherein constant radii of curvature are presentThe 2D numerical approaches are more versatile (in termsof boundary conditions and type of structure analyzed)however they can be used only for moderately thick struc-tures It is obvious that thick structures can be analyzed onlywhen higher order displacement fields are set in such 2Dmodels Since commercial codes implement only through thethickness linear theories they can investigate accurately onlymoderately thick shells The method proposed in [28ndash31] tocompare the 3D exact shell model and the 2D numerical

models is based on the calculation of the frequencies viathe 2D numerical code and their appropriate visualizationof the relative vibration modes via the GDQ andor FEmethod After this visualization the evaluation of the 3Dexact frequencies is possible by means of the appropriateimposition of half-wave numbers obtained from the vibrationmode study From this procedure it is clear how the 3Danalysis could give some frequencies that cannot be calcu-lated by the 2D numerical codes This feature was not theaim of the papers [28ndash31] and for this reason these missedfrequencies are investigated in the present newwork for FGMstructures

This new analysis is conducted calculating the first 18frequencies for one-layered and sandwich FGM plates andshells by means of the 3D exact model The half-wavenumbers are combined along the two directions 120572 and 120573in the plane considering the simply supported boundaryconditions for the four edges In these first 18 frequenciesthere are some modes that cannot be calculated by meansof the 2D numerical models because they are not solutionsof the employed mathematical model with the four simplysupported edges In general these missed modes by the2D numerical models correspond to the cylindrical bendingconditions These conditions in the 3D exact solution meanthat one of the two half-wave numbers is equal to zerothrough an edge direction with a transverse displacementdifferent from zero along the perpendicular edge Theseparticular frequencies are not solutions of the 2D numericalmodels built with the four edges which are simply supportedbut a different mathematical model must be defined wheretwo edges are simply supported and the other two arefree These conditions allow the cylindrical bending withthe transverse displacement through two parallel edgesdifferent from zero in numerical models in order toobtain the missed frequencies and modes This condi-tion is possible only when the zero half-wave numberis imposed through a rectilinear side because the half-wave number equals zero in the curvilinear edge doesnot mean that all the derivatives in the that direction arezero because of the presence of the curvature This lastcondition is in conflict with the definition of cylindricalbending

The 3D exact shell model is described and validatedin Section 2 It uses the equilibrium equations writtenusing the generic curvilinear orthogonal coordinates Suchequations are solved in closed form by means of simplysupported boundary conditions The differential equationsalong 119911 written in layer-wise form are solved using theexponential matrix method (also known as transfer matrixmethod or the state-space approach) The 2D FE and GDQmodels are proposed and opportunely validated in Sec-tion 3 The 2D FE model makes use of a very commoncommercial finite element code The 2D GDQ models giveresults by means of an in-house academic software andthey are based on classical and refined shell models Theresults about the comparisons between 3D exact and 2Dnumerical models with the remark about the boundaryconditions are shown Section 4 Section 5 gives the mainconclusions

Shock and Vibration 3

x

y

z

a

b

(a)

z

b

R

a = 2R

(b)z

R

a

b

(c)

z

R

R

a ba b

(d)

Figure 1 Geometries of plates and shells with constant radii of curvature and their local reference system

2 Exact Solution of a 3D Shell Model

The three-dimensional exact shell model employed for thecomparisons proposed in Section 4 has been elaborated in[32ndash40] for the free frequency analyses of single-layered andmultilayered isotropic orthotropic composite FGM andsandwich plates and shells with constant radii of curvatureand for the free vibrations of single- and double-walledcarbon nanotubes including van der Waals interactions Theequilibrium equations proposed in the general orthogonalcurvilinear coordinate system (120572 120573 119911) are valid for bothplates and shells with constant radii of curvature For thesake of clarity the plates and shells geometries within theirlocal coordinate systems are depicted in Figure 1 The exactform of these equations is obtained using simply supportedboundary conditions while the differential equations in 119911are solved by means of the exponential matrix method(also known as transfer matrix method or the state-spaceapproach)

Considering spherical shells with constant radii of cur-vature 119877120572 and 119877120573 and embedding 119873119871 classical or FGM layers

the differential equations of equilibrium for the free vibrationanalysis written for a generic 119896 layer are

119867120573 120597120590119896120572120572120597120572 + 119867120572 120597120590119896120572120573120597120573 + 119867120572119867120573 120597120590119896120572119911120597119911+ (2119867120573119877120572 + 119867120572119877120573 ) 120590119896120572119911 = 120588119896119867120572119867120573119896

119867120573 120597120590119896120572120573120597120572 + 119867120572 120597120590119896120573120573120597120573 + 119867120572119867120573 120597120590119896120573119911120597119911+ (2119867120572119877120573 + 119867120573119877120572 ) 120590119896120573119911 = 120588119896119867120572119867120573V119896

119867120573 120597120590119896120572119911120597120572 + 119867120572 120597120590119896120573119911120597120573 + 119867120572119867120573 120597120590119896119911119911120597119911 minus 119867120573119877120572 120590119896120572120572 minus 119867120572119877120573 120590119896120573120573+ (119867120573119877120572 + 119867120572119877120573 ) 120590119896119911119911 = 120588119896119867120572119867120573119896

(1)

4 Shock and Vibration

It is remarked that this set of equations is valid onlyfor spherical shells wherein 119877120572 = 119877120573 = 119877 because fordoubly curved shells with variable radii of curvature suchequations take a different form The authors decided to keepsuch system with the two radii of curvature distinct becauseit will be much easier to understand the cylindrical bendingeffect for plates and shells starting from a general point ofview

In the case of shell geometries andor FGM layers eachphysical layer 119896 must be divided in an opportune numberof 119902 mathematical layers to approximate the shell curvatureandor the FGM law through the thickness direction 119911 Theindex 119895 indicates the total mathematical layers obtained fromthe subdivision of each 119896 layer in 119902 mathematical layers (119895 =119896 times 119902) The parametric coefficients for shells with constantradii of curvature can be written as

119867120572 = (1 + 119911119877120572) 119867120573 = (1 + 119911119877120573) 119867119911 = 1

(2)

A closed form solution for simply supported shells andplates can be developed by writing harmonic displacementsas

119906119895 (120572 120573 119911 119905) = 119880119895 (119911) 119890119894120596119905 cos (120572120572) sin (120573120573) V119895 (120572 120573 119911 119905) = 119881119895 (119911) 119890119894120596119905 sin (120572120572) cos (120573120573) 119908119895 (120572 120573 119911 119905) = 119882119895 (119911) 119890119894120596119905 sin (120572120572) sin (120573120573)

(3)

where 119880119895(119911) 119881119895(119911) and 119882119895(119911) indicate the displacementamplitudes in 120572 120573 and 119911 directions respectively 119894 indicatesthe imaginary unit 120596 = 2120587119891 is the circular frequencywhere 119891 is the frequency value in Hz 119905 is the symbol forthe time In the coefficients 120572 = 119898120587119886 and 120573 = 119899120587119887119898 and 119899 are the half-wave numbers and 119886 and 119887 are theshell dimensions in 120572 and 120573 directions respectively (all thesequantities are evaluated on the midsurface Ω0) 120588119895 is themass density (120590119895120572120572 120590119895120573120573 120590119895119911119911 120590119895120573119911 120590119895120572119911 120590119895120572120573) are the six stresscomponents and 119895 V119895 and 119895 are the second temporalderivative of the three displacement components 119877120572 and119877120573 indicate the radii of curvature defined with respect thereference surface Ω0 of the whole multilayered plate or shell119867120572 and 119867120573 proposed in (2) vary with continuity through thethickness direction of the multilayered structure Thereforethey depend on the thickness coordinate 119911 The referencesurface Ω0 of the structure is defined as the locus of pointswhich are positioned midway between the external surfacesDisplacement components are indicated as 119906 V and 119908 in 120572120573 and 119911 directions respectively

The final system in closed form is developed substituting(3) and constitutive and geometrical equations (which are not

here given for the sake of brevity but they can be found indetail in [32]) in (1)

(minus119862119895551198671198951205731198671198951205721198772120572 minus 11986211989555119877120572119877120573 minus 120572211986211989511119867119895120573119867119895120572 minus 120573211986211989566119867119895120572119867119895120573+ 1205881198951198671198951205721198671198951205731205962) 119880119895 + (minus12057212057311986211989512 minus 12057212057311986211989566) 119881119895

+ (12057211986211989511119867119895120573119867119895120572119877120572 + 12057211986211989512119877120573 + 12057211986211989555119867119895120573119867119895120572119877120572 + 12057211986211989555119877120573 ) 119882119895

+ (11986211989555119867119895120573119877120572 + 11986211989555119867119895120572119877120573 ) 119880119895119911 + (12057211986211989513119867119895120573 + 12057211986211989555119867119895120573)sdot 119882119895119911 + (11986211989555119867119895120572119867119895120573) 119880119895119911119911 = 0

(minus12057212057311986211989566 minus 12057212057311986211989512) 119880119895 + (minus119862119895441198671198951205721198671198951205731198772120573 minus 11986211989544119877120572119877120573minus 120572211986211989566119867119895120573119867119895120572 minus 120573211986211989522119867119895120572119867119895120573 + 1205881198951198671198951205721198671198951205731205962) 119881119895

+ (12057311986211989544119867119895120572119867119895120573119877120573 + 12057311986211989544119877120572 + 12057311986211989522119867119895120572119867119895120573119877120573 + 12057311986211989512119877120572 ) 119882119895

+ (11986211989544119867119895120572119877120573 + 11986211989544119867119895120573119877120572 ) 119881119895119911 + (12057311986211989544119867119895120572 + 12057311986211989523119867119895120572)sdot 119882119895119911 + (11986211989544119867119895120572119867119895120573) 119881119895119911119911 = 0

(12057211986211989555119867119895120573119867119895120572119877120572 minus 12057211986211989513119877120573 + 12057211986211989511119867119895120573119867119895120572119877120572 + 12057211986211989512119877120573 ) 119880119895

+ (12057311986211989544119867119895120572119867119895120573119877120573 minus 12057311986211989523119877120572 + 12057311986211989522119867119895120572119867119895120573119877120573 + 12057311986211989512119877120572 ) 119881119895

+ ( 11986211989513119877120572119877120573 + 11986211989523119877120572119877120573 minus 119862119895111198671198951205731198671198951205721198772120572 minus 211986211989512119877120572119877120573 minus 119862119895221198671198951205721198671198951205731198772120573minus 120572211986211989555119867119895120573119867119895120572 minus 120573211986211989544119867119895120572119867119895120573 + 1205881198951198671198951205721198671198951205731205962) 119882119895

+ (minus12057211986211989555119867119895120573 minus 12057211986211989513119867119895120573) 119880119895119911 + (minus12057311986211989544119867119895120572minus 12057311986211989523119867119895120572) 119881119895119911 + (11986211989533119867119895120573119877120572 + 11986211989533119867119895120572119877120573 ) 119882119895119911+ (11986211989533119867119895120572119867119895120573) 119882119895119911119911 = 0

(4)

Shock and Vibration 5

The cylindrical bending requirements for analytical methodsare obtained choosing one of the two half-wave numberswhich equals zero This condition means 120572 = 119898120587119886 =0 or 120573 = 119899120587119887 = 0 which allows the derivatives madein a particular direction (120572 or 120573) which equals zero tooThis mathematical condition means that each section in thecylindrical bending direction has always the same behavior

It is obvious that the cylindrical bending effect is properfor cylindrical structures only However the authors startedthe present treatise from the governing equations (1) ofspherical shell structures because one of the aims of thepresent contribution is to underline which are the extraterms (related to the curvature) that do not allow having thestructure to be in cylindrical bending conditions

The compact form of (4) has been elaborated in detail in[32] The final version is here proposed

D119895 120597U119895120597119911 = A119895U119895 (5)

where 120597U119895120597119911 = U1198951015840 andU119895 = [119880119895 119881119895 119882119895 1198801198951015840 1198811198951015840 1198821198951015840]Equation (5) is rewritten as

D119895U1198951015840 = A119895U119895 (6)

U1198951015840 = D119895minus1A119895U119895 (7)

U1198951015840 = A119895lowastU119895 (8)

with A119895lowast = D119895minus1A119895The exponential matrix method (also known as transfer

matrix method or the state-space approach) is employed forthe solution of (8) (details are given in [32])

U119895 (119911119895) = exp (A119895lowast119911119895)U119895 (0) with 119911119895 isin [0 ℎ119895] (9)

where 119911119895 is the thickness coordinate for the 119895 mathematicallayer with values from 0 at the bottom to ℎ119895 at the top Theexponential matrix is calculated using 119911119895 = ℎ119895 for each 119895mathematical layer

A119895lowastlowast = exp (A119895lowastℎ119895)= I + A119895lowastℎ119895 + A119895lowast22 ℎ1198952 + A119895lowast33 ℎ1198953 + sdot sdot sdot

+ A119895lowast119873119873 ℎ119895119873(10)

where I indicates the 6times6 identitymatrixThe expansion usedfor the exponentialmatrix in (10) has a fast convergence and itis not time consuming from the computational point of view119895 = 119872 mathematical layers are used to approximate theshell curvature andor the FGM law through the thickness119872 minus 1 transfer matrices must be defined by means ofthe interlaminar continuity conditions of displacements andtransverse shearnormal stresses at each interfaceThe closedform solution for plates or shells is obtained using simply

supported boundary conditions and free stress conditions atthe top and at the bottom surfaces The above conditionsallow the following final system

EU1 (0) = 0 (11)

and the 6 times 6 matrix E does not change independently of thenumber of layers 119872 even if a layer-wise approach is appliedU1(0) is the vector U defined at the bottom of the wholemultilayered shell or plate (first layer 1 with 1199111 = 0) All thesteps and details here omitted for the sake of brevity can befound in [32ndash40] where the proposed exact 3D shell modelhas been elaborated and tested for the first time for severalapplications Several validations for the 3D exact shell modelhave been shown in [32ndash40] For this reason the model isnow used with confidence for the results and comparisonsproposed in Section 4

The nontrivial solution ofU1(0) in (11) allows conductingthe free vibration analysis imposing the zero determinant ofmatrix E

det [E] = 0 (12)

Equation (12) is used to find the roots of an higher order poly-nomial written in 120582 = 1205962The imposition of a couple of (119898 119899)allows a certain number of circular frequencies 120596 = 2120587119891(from I to infin) to be calculated This number of frequenciesdepends on the order 119873 used in each exponential matrixA119895lowastlowast and the number 119872 of mathematical layers employedfor the shell curvature andor the FGM law approximationsThe results shown in [32ndash40] confirm that 119873 = 3 for theexponential matrix and 119872 = 100 for the mathematicallayers are appropriate to always calculate the correct freefrequencies for each structure lamination sequence numberof layers material FGM law and thickness value

3 2D Finite Element Analysis and 2DGeneralized Differential Quadrature Models

The 3D exact natural frequencies are compared with twonumerical 2D solutions given by a weak and a strongformulation The former is a classic 2D finite element (FE)model carried out through the commercial software Strand7The latter is a recent widespread technique known as Gener-alized Differential Quadrature (GDQ) method [41] Detailsrelated to convergence and accuracy of these techniquescan be found in the works [29ndash31] It is noted that theFE solution employs a classic linear theory known alsoas Reissner-Mindlin theory with a linear rotation of thefibers through the shell thickness The present GDQ solutionexploits higher order theories with a general expansionthrough the thickness of the structureThis general approachfalls within the framework of the unified theories becauseseveral structural models can be obtained via the repetitionof a fundamental nucleus Another difference between theGDQ applications and the FE ones is that the formerincludes the curvature effects and the rotary inertias in itsformulation

6 Shock and Vibration

The 2D kinematic hypothesis [42ndash50] considers the fol-lowing expansion of the degrees of freedom

U = 119873119888+1sum120591=0

F120591u(120591) (13)

where U represents the vector of the 3D displacementcomponents which is function of (120572 120573 119911) and u120591 is thevector of the displacement parameters function of (120572 120573)which depends on the free index 120591 Therefore u120591 lays on themidsurface of the shell [42] The matrix F120591(119894119895) = 120575119894119895119865120591 (for119894 119895 = 1 2 3) contains the so-called thickness functions where120575 is the Kronecker delta function The thickness functionsdescribe how the displacement parameters characterize thesolution through the thickness They can have any form ofany smooth function of 119911 however the classic form of thesefunctions is the power function Thus 119865120591 = 119911120591 can be 1(120591 = 0) 119911 (120591 = 1) 1199112 (120591 = 2) 1199113 (120591 = 3) and so onPlease note that according to the mathematical backgroundof the present Unified Formulation the index 119873119888 + 1 isreserved to the (optional) zigzag effect Since the aim of thepresent paper is to investigate the cylindrical bending effectfor functionally graded plates and cylindrical panels thefollowing numerical applications will consider a fixed valueof 119873119888 = 3 so that 120591 = 0 1 2 3 This choice is justified by thefact that third order theories are sufficient with a reasonablylimited number of degrees of freedom to obtain accuratesolutions when compared to 3D shell models [42ndash50] Formultilayered structures such as sandwich configurations thezigzag function must be added into the formulation Onceagain as shown in (13) the zigzag function is embedded atthe end of the expansion with the exponent 119873119888 + 1 Whensuch effect is neglected the sum of (13) stops at 119873119888

The same theory used for the FE solution can be carriedout in the GDQmethod setting 120591 = 0 1 and 119873119888 = 1 howeverit has been shown in other recent works that the strong formapproaches led tomore accurate results with respect to the FEones [46 48]

The relation between generalized strains 120576(120591) anddisplace-ments u(120591) can be set as

120576(120591) = DΩu

(120591) for 120591 = 0 1 2 119873119888 119873119888 + 1 (14)

where DΩ is a differential operator explicitly defined in [42]The present constitutive material model is linear and elastictherefore for a general 119896th laminaHookersquos law can be applied

120590(119896) = C(119896)120576(119896) (15)

where 120590(119896) is the vector containing the stress componentsand C(119896) represents the constitutive matrix for each 119896th ply[42] of the structure Each component of the constitutivematrix is indicated by the symbol 119862(119896)120585120578 and it is valid in thecurvilinear reference system 1198741015840120572120573119911 of the reference geome-tryThese coefficients are evaluatedwith the application of thetransformation matrix [42]

The stress resultants are the terms which correspond tothe punctual 3D stresses of the 3D doubly curved solidTheir definition can be achieved by developing the Hamiltonprinciple for the present structural theory [42]

S(120591) = 119873119871sum119896=1

int119911119896+1119911119896

(Z(120591))119879 120590(120591)119867120572119867120573119889119911for 120591 = 0 1 2 119873119888 119873119888 + 1

(16)

whereZ(120591) is a 6times9matrix containing the thickness functions119865120591 and the curvature coefficients119867120572 and119867120573 the definition ofsuchmatrix with some other details can be found in [42] S(120591)include the stress resultants as a function of the order 120591

The definition of (16) can be rewritten as

S(120591) = 119873119888+1sum119904=0

A(120591119904)120576(119904) for 120591 = 0 1 2 119873119888 119873119888 + 1 (17)

which is a function of generalized strains of order 119904 120576(119904)Note that the so-called elastic coefficients are defined as

A(120591119904) = 119873119871sum119896=1

int119911119896+1119911119896

(Z(120591))119879C(119896)Z(119904)119867120572119867120573119889119911 (18)

and they can be carried out by the following integral expres-sions

119860(120591119904)120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119911119904119911120591119867120572119867120573119867984858120572119867120577120573 119889119911

119860(119904)120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119911119904120591119911120591minus1119867120572119867120573119867984858120572119867120577120573 119889119911

119860(120591)120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119904119911119904minus1119911120591119867120572119867120573119867984858120572119867120577120573 119889119911

Shock and Vibration 7

119860( )120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119904119911119904minus1120591119911120591minus1119867120572119867120573119867984858120572119867120577120573 119889119911for 120591 119904 = 0 1 2 119873119888 119873119888 + 1 120585 120578 = 1 2 3 4 5 6 984858 120577 = 0 1 2

(19)

It is recalled that the exponent 119873119888 + 1 is reserved for theMurakami function otherwise this order is neglected Thesubscripts and indicate the derivatives of the thicknessfunctions 119865120591 and 119865119904 with respect to 119911 The subscripts 984858 and 120577are the exponents of the quantities119867120572 and 119867120573 whereas 120585 and120578 are the indices of the material constants 119862(119896)120585120578 for the 119896th ply

[42ndash49]Note that119862(119896)120585120578 119867120572 and119867120573 are quantities that dependon the thickness 119911 Therefore the integrals of (19) cannotbe computed analytically The present solution for solvingsuch integrals is to employ a numerical integration algorithmIn particular the so-called Generalized Integral Quadrature(GIQ) method is utilized For an accurate evaluation of theintegrals 51 points per layer are considered in the integration

The Hamilton principle supplies for the dynamic equilib-rium equations and the correspondent boundary conditionsIn detail the fundamental nucleus of three motion equationstakes the form

D⋆ΩS(120591) = 119873119888+1sum

119904=0

M(120591119904)u(119904)

for 120591 = 0 1 2 119873119888 119873119888 + 1(20)

whereD⋆Ω is the equilibrium operator andM(120591119904) is the inertialmatrix Their explicit expressions have been presented in aprevious authorsrsquo paper [42]

Combining the kinematic equation (14) constitutiveequation (17) and themotion equation (20) the fundamentalgoverning system in terms of displacement parameters iscarried out

119873119888+1sum119904=0

L(120591119904)u(119904) = 119873119888+1sum119904=0

M(120591119904)u(119904)

for 120591 = 0 1 2 119873119888 119873119888 + 1(21)

where L(120591119904) = D⋆ΩA(120591119904)DΩ is the fundamental operator [42]

Boundary conditions must be included for the solutionof the differential equations (21) The present paper dealsonly with free and simply supported boundary conditionsThe former is used to reproduce the cylindrical bendingconditions considering alternatively one free edge and onesimply supported edge When the four edges are all simplysupported the problem is the same as the ones investigated

in the previous papers [29ndash31] Simply supported conditionsare

119873(120591)120572 = 0119906(120591)120573 = 119906(120591)119911 = 0

120591 = 0 1 2 119873119888 119873119888 + 1 at 120572 = 1205720 or 120572 = 1205721 1205730 le 120573 le 1205731119906(120591)120572 = 0

119873(120591)120573 = 0119906(120591)119911 = 0

120591 = 0 1 2 119873119888 119873119888 + 1 at 120573 = 1205730 or 120573 = 1205731 1205720 le 120572 le 1205721

(22)

Free edge conditions are

119873(120591)120572 = 0119873(120591)120573 = 0119879(120591)120572 = 0120591 = 0 1 2 119873119888 119873119888 + 1 at 120572 = 1205720 or 120572 = 1205721 1205730 le 120573 le 1205731119873(120591)120572 = 0119873(120591)120573 = 0119879(120591)120573 = 0120591 = 0 1 2 119873119888 119873119888 + 1 at 120573 = 1205730 or 120573 = 1205731 1205720 le 120572 le 1205721

(23)

4 Results

The present section includes some numerical applicationsfor the comparison of the 3D exact shell vibrations withtwo numerical 2D models In particular classical simplysupported solutions and cylindrical bending (CB) behaviorof plates and shells are presentedThe following computationsfocus on plates circular cylinders and cylindrical panels dueto the fact that the CB effect occurs only when at least twoparallel straight edges are present in the structure

The first case is a square plate (119886 = 119887 = 1m) withsimply supported edges Two thickness ratios are analyzedthin 119886ℎ = 100 and moderately thick 119886ℎ = 20 structure Theplate is made of a single functionally graded layer (ℎ1 = ℎ)with volume fraction of ceramic (119888) phase

119881119888 (119911) = (05 + 119911ℎ)119901 with 119911 isin [minusℎ2 ℎ2 ] (24)

8 Shock and Vibration

Table 1 One-layered FGM plate with 119901 = 05 Analytical 3D versus 2D numerical models Frequency 119891 in Hz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119886ℎ = 1001 0 I 4019 mdash 3907 mdash 39210 1 I 4019 mdash 3907 mdash 39211 1 I 8037 8006 mdash 8036 mdash2 0 I 1607 mdash 1580 mdash 15840 2 I 1607 mdash 1580 mdash 15842 1 I 2008 2001 mdash 2008 mdash1 2 I 2008 2001 mdash 2008 mdash2 2 I 3212 3202 mdash 3211 mdash3 0 I 3613 mdash 3568 mdash 35770 3 I 3613 mdash 3568 mdash 35773 1 I 4014 4002 mdash 4013 mdash1 3 I 4014 4002 mdash 4013 mdash3 2 I 5215 5202 mdash 5215 mdash2 3 I 5215 5202 mdash 5215 mdash4 0 I 6416 mdash 6354 mdash 63650 4 I 6416 mdash 6354 mdash 63654 1 I 6816 6801 mdash 6815 mdash1 4 I 6816 6801 mdash 6815 mdash119886ℎ = 201 0 I 2002 mdash 1951 mdash 19510 1 I 2002 mdash 1951 mdash 19511 1 I 3989 3995 mdash 3988 mdash2 0 I 7916 mdash 7864 mdash 77970 2 I 7916 mdash 7864 mdash 77972 1 I 9858 9954 mdash 9857 mdash1 2 I 9858 9954 mdash 9857 mdash2 2 I 1560 1588 mdash 1560 mdash3 0 I 1749 mdash 1767 mdash 17290 3 I 1749 mdash 1767 mdash 17293 1 I 1936 1980 mdash 1936 mdash1 3 I 1936 1980 mdash 1936 mdash3 2 I 2490 2566 mdash 2490 mdash2 3 I 2490 2566 mdash 2490 mdash1 0 II (119908 = 0) 2775 2773 mdash 2775 mdash0 1 II (119908 = 0) 2775 2773 mdash 2775 mdash4 0 I 3034 mdash 3124 mdash 30070 4 I 3034 mdash 3124 mdash 3007

where 119901 is the power exponent which varies the volumefraction of the ceramic component through the thicknessTwo limit cases occur for (24) 119881119888 = 1 when 119901 = 0 and119881119888 rarr 0when119901 = infin In the following computations it is set as119901 = 05 Note that the volume fraction function in (24) givesa material which is ceramic (119888) rich at the top and metal (119898)rich at the bottomof the structureThemechanical properties(Youngmodulus andmass density) are given by the theory ofmixtures as

119864 (119911) = 119864119898 + (119864119888 minus 119864119898) 119881119888120588 (119911) = 120588119898 + (120588119888 minus 120588119898) 119881119888 (25)

where 119864119898 and 120588119898 are the elastic modulus and mass densityof the metallic phase respectively 119864119888 and 120588119888 are the corre-sponding properties of the ceramic phase Note that in allthe following computations the Poisson ratio is consideredconstant through the thickness thus ]119888 = ]119898 = ] = 03 Thetwo constituents have the following mechanical properties

119864119898 = 73GPa120588119898 = 2800 kgm3119864119888 = 380GPa120588119888 = 3800 kgm3

(26)

Shock and Vibration 9

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(a) Mode I (0 1) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

000501

01502

(b) Mode I (0 1) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(c) Mode II (119908 = 0) (0 1) 3D

002minus002

090908

08070706

06050504

04030302

020101

00minus01

1

(d) Mode II (119908 = 0) (0 1) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(e) Mode I (1 0) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

0minus005minus01

minus015minus02

(f) Mode I (1 0) 2D

Figure 2 Continued

10 Shock and Vibration

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(g) Mode II (119908 = 0) (1 0) 3D

002minus002

0908 08

0706 06

0504 04

0302 02

01 0 0

11

(h) Mode II (119908 = 0) (1 0) 2D

Figure 2 Differences between in-plane and cylindrical bending vibration modes for the 119886ℎ = 20 one-layered FGM plate with 119901 = 05 3Dmodes versus 2D modes

Since the FE commercial software does not have anembedded tool for the computation of functionally gradedproperties through the thickness the present FE applicationsare carried out building a fictitious laminated structure withisotropic plies In detail the thickness of the FGM lamina hasbeen divided into 100 fictitious layers and for each layer thecorresponding mechanical properties have been computedthrough (25) The numerical model has been made of auniform element division of 40 times 40 Quad8 elements for atotal number of 29280 degrees of freedom (dofs) The GDQsolution is computed with a not-uniform Chebyshev-Gauss-Lobatto grid [41] with 31 times 31 points for both thicknessratios It is underlined that since a single ply is set no zigzagfunction is needed in the computation It is recalled that thepresent higher order GDQ model considers a third orderexpansion with 120591 = 0 1 2 3 whereas the FE one has afirst order model (also known as Reissner-Mindlin model)Table 1 reports the results in terms of natural frequency(Hz) wherein the first column reports the number of half-waves (119898 119899) of the 3D exact solution Note that for a fixedset of half-wave numbers different mode shapes can occur(indicated by the second column ldquoModerdquo) In particular Iindicates the first mode and II indicates the second mode Inthe case of in-plane modes the caption (119908 = 0) is added tounderline that the vibration occurs in the plane and not outof plane The third column shows all the results for the 3Dexact solution both classical and cylindrical bending (CB)results It is noted that these frequencies are obtained witha single run of the 3D exact code keeping simply supportedboundary conditions for all the four edges of the plate Theother four columns of Table 1 refer to the numerical solutionsfor simply supported structures and for cylindrical bendingconditions (setting two free opposite edges) The cylindricalbending results are indicated by the acronym CB in the sametable

There is a very good agreement among the three solutionswhen simply supported conditions are considered On thecontrary a small error is observed in cylindrical bendingbecause the actual CB condition is not completely satisfieddue to the fact that the plate has a limited dimension in thesimply supported direction This feature can be detected inFigures 2 and 3 because the deformed shapes of the plate inthe CB configuration are not properly correct In fact thefibers parallel to the supported edges are not parallel amongthem but there is a curvature effect due to the free boundariesand to the limited length of the plate On the contrary thein-plane mode shapes (119908 = 0) when the plate is simplysupported on the four edges have all the fibers parallel amongthem also after the deformation These modes are the resultsobtained in the 3D solution with 119898 = 0 or 119899 = 0 combinedwith 119908 = 0 They are not cylindrical bending modes infact they can be found with 2D numerical models withoutmodifying the boundary conditions 3D frequencieswith119898 =0 or 119899 = 0 (with w different from 0) are CB modes which canbe obtained considering two simply supported edges and twofree edges in 2D numerical models

As it is well-known the ldquoperfectrdquo CB would occur whenthe simply supported direction has an ldquoinfiniterdquo lengthThis cannot be proven numerically but a trend can beobserved modeling different plates in cylindrical bendingconfiguration with different lengthsThese analyses are madekeeping one side of the plate constant and enlarging thelength of the other edge In the present computations 119886 iskept constant and 119887 = 119871 varies from 1 to 20 The results ofthe aforementioned analysis are listed in Table 2 and depictedin Figures 4 and 5 Table 2 includes the numerical valuesof the first mode shape in cylindrical bending for the 3Dexact solution the 2D FE and the 2D GDQmodels Figure 4shows that both GDQ and FE solutions tend to the 3D exactone increasing the length of the simply supported edge In

Shock and Vibration 11

ulowast

lowast

wlowast

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

(a) Mode I (1 0) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

000501

01502

(b) Mode I (1 0) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(c) Mode I (2 0) 3D

109

0807

060504

0302

01 0

109

0807

0605

0403

02010

0201501

0050

minus005minus01

minus015minus02

(d) Mode I (2 0) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(e) Mode I (3 0) 3D

10908

0706

050403

0201 0

109

0807

0605

0403

02010

0201501

0050

minus005minus01

minus015minus02

(f) Mode I (3 0) 2D

Figure 3 Continued

12 Shock and Vibration

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(g) Mode I (4 0) 3D

10908

0706050403

0201 0

10908

0706

0504

0302

010

0201501

0050

minus005minus01

minus015minus02

(h) Mode I (4 0) 2D

Figure 3 First four cylindrical bending vibration modes for the 119886ℎ = 20 one-layered FGM plate with 119901 = 05 3D modes versus 2D modes

3D exact2D GDQ2D FEM

f(H

z)

39

392

394

396

398

40

402

404

5 10 15 200La

Figure 4 First frequency 119891 in Hz for the 119886ℎ = 100 one-layeredFGM plate with 119901 = 05 convergence of 2D GDQ and 2D FEsolutions increasing the length of the simply supported sides

particular the GDQ solution is slightly more accurate thanthe FE one The 2D GDQ solution is carried out by enlargingthe length of one of the sides of the plate and the samenumberof dofs is kept On the contrary the FE model is generatedby 40 times 40 blocks of mesh In other words the plate with119871 = 2119886 has a mesh of 40 times 80 the one with 119871 = 3119886 has amesh 40 times 120 and so on When 119871 = 20119886 the error amongthe 3D exact and the 2D numerical solutions is very small andthis is proven also by the mode shapes of Figure 5 whereinthe boundary effect is negligible with respect the length of

the simply supported edges It must be underlined that it isimpossible to completely avoid the boundary effects due tothe free edges

The second example is related to a simply supportedcylinder with 119877120572 = 10m 119877120573 = infin 119886 = 2120587119877120572 and 119887 =20m and thickness ratios 119877120572ℎ = 100 and 119877120572ℎ = 10 Aone-layered FGM scheme is considered with ℎ1 = ℎ andthe same volume fraction function of (24) for the previousexample but setting 119901 = 20 The present material propertiesare the same as the flat plate The FE model has a 20 times80 Quad8 mesh with 27966 dofs and the GDQ solutionhas a 15 times 51 Chebyshev-Gauss-Lobatto grid Numericallyspeaking compatibility conditions must be enforced on theclosing meridian in order to obtain a closed revolution shell[42] To obtain the CB conditions numerically the cylinderis modeled completely free and the first 6 rigid mode shapesare neglected and removed from the numerical solution Itis noted that the GDQ model uses a third-order kinematicexpansion with no zigzag effect as the previous case sincea one-layer structure has been taken into account Table 3reports the results with the same meaning of the symbolsseen in Table 1 A very good agreement is observed when thecylinder is simply supported whereas when the cylinder iscompletely free some small errors can be observed but thenumerical solutions follow the 3D exact one

Results with 119899 = 0 and 119908 = 0 indicate the CB conditionsSuch results are obtained by means of 2D numerical modelsmodifying the boundary conditions (see 2D FE(CB) and 2DGDQ(CB)models)The cases 119899 = 0 combined with119908 = 0 arenot CB conditions and the frequencies can be found withoutmodifying the boundary conditions in the 2D numericalmodels

The final test is related to a cylindrical panel with 119877120572 =10m 119877120573 = infin 119886 = 1205873119877120572 and 119887 = 20m and thickness ratios119877120572ℎ = 100 and 119877120572ℎ = 10 The present shell is a sandwichstructure embedding a FGM core with ℎ2 = 07ℎ and two

Shock and Vibration 13

218

1614

121 1

08 0806 06

04 0402 02

0 0

0minus01minus02minus03minus04

(a) 119871119886 = 2

3

25

2

151

0806

040502

0 0

1

0minus01minus02minus03minus04minus05

(b) 119871119886 = 3

435

325

10505

0 0

15

0

minus02minus04minus06

1

2

(c) 119871119886 = 4

545

35

10505

0 0

0

minus04minus02

minus06minus08

1

25

4

3

215

(d) 119871119886 = 5

5

1

0 005

0

minus04minus02

minus06minus08

1

4

3

2

6

minus1

(e) 119871119886 = 6

5

10 0 051

43

2

67

89

100

minus05minus1

minus15

(f) 119871119886 = 10

Figure 5 First vibration mode for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 2D GDQ solution and edge effects versus the lengthof the simply supported sides

external skins with ℎ1 = ℎ3 = 015ℎ (ℎ1 + ℎ2 + ℎ3 = ℎ) Thebottom layer ismade of isotropicmetallicmaterial and the toplayer is made of ceramic material the mechanical propertiesare

119864119898 = 73GPa120588119898 = 2800 kgm3119864119888 = 200GPa120588119888 = 5700 kgm3

(27)

The FGM core has a volume fraction function defined as

119881119888 (119911) = (05 + 11991107ℎ)119901 with 119911 isin [minus035ℎ 035ℎ] (28)

where 119901 = 10 is employed in the present computations ThePoisson ratio is constant (]119888 = ]119898 = ] = 03) Equations (25)are still valid with 119881119888 defined as in (28)

The FEmodel is made of 40times40Quad8 elements for total29280 dofs and the GDQ model has a 31 times 31 Chebyshev-Gauss-Lobatto grid Since a sandwich structure is analyzedin the following a third-order theory with zigzag functionis considered in the computations for the GDQ method Thepresent natural frequencies are reported in Table 4 with the

14 Shock and Vibration

Table 2 First frequency 119891 in Hz for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 convergence of 2D GDQ and 2D FE solutionsincreasing the length of the simply supported sides

119871119886 1 2 3 4 5 6 8 10 15 20 3D exact2D GDQ 3921 3964 3980 3988 3993 3997 4001 4004 4007 4008 40192D FEM 3907 3949 3966 3974 3979 3983 3987 3989 3993 3994 4019

Table 3 One-layered FGM cylinder with 119901 = 20 Analytical 3D versus 2D numerical models Frequency 119891 in Hz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1004 0 I 0896 mdash 0897 mdash 08836 0 I 2532 mdash 2538 mdash 25118 0 I 4854 mdash 4867 mdash 481910 0 I 7848 mdash 7871 mdash 780112 0 I 1151 mdash 1155 mdash 114610 1 I 1356 1360 mdash 1357 mdash12 1 I 1441 1449 mdash 1445 mdash14 0 I 1584 mdash 1589 mdash 15778 1 I 1638 1640 mdash 1636 mdash14 1 I 1756 1767 mdash 1763 mdash16 0 I 2083 mdash 2090 mdash 207016 1 I 2207 2222 mdash 2216 mdash6 1 I 2441 2442 mdash 2438 mdash18 0 I 2649 mdash 2658 mdash 264014 2 I 2742 2752 mdash 2746 mdash18 1 I 2751 2772 mdash 2763 mdash16 2 I 2875 2890 mdash 2883 mdash12 2 I 2921 2926 mdash 2922 mdash119877120572ℎ = 104 0 I 8774 mdash 9085 mdash 86796 0 I 2456 mdash 2571 mdash 24306 1 I 3978 4089 mdash 3979 mdash4 1 I 4388 4357 mdash 4388 mdash8 0 I 4654 mdash 4920 mdash 46028 1 I 5575 5905 mdash 5576 mdash2 1 I 7202 7101 mdash 7202 mdash2 0 I (119908 = 0) 7322 7390 mdash 7319 mdash10 0 I 7419 mdash 7933 mdash 73476 2 I 8110 8261 mdash 8111 mdash10 1 I 8189 8811 mdash 8191 mdash8 2 I 8783 9173 mdash 8785 mdash4 2 I 9016 9084 mdash 9017 mdash12 0 I 1071 mdash 1159 mdash 106110 2 I 1081 1155 mdash 1081 mdash2 2 I 1089 1105 mdash 1089 mdash12 1 I 1142 1245 mdash 1143 mdash4 3 I 1263 1286 mdash 1263 mdash

same meaning of the columns seen in the previous casesIn order to achieve the cylindrical bending conditions thenumerical models have the straight edges simply supportedand the curved edges as free Due to the CB definitionit is not possible to achieve CB conditions when 119898 = 0

and 119899 = 1 2 3 because the cylindrical bending shouldoccur on the curved edge and this is not physically possiblebecause of the variation of displacement components due tothe curvature presence This aspect is indicated as NA (NotApplicable) solution in Table 4 Therefore this kind of 3D

Shock and Vibration 15

Table 4 Sandwich cylindrical shell panel embedding FGM core with 119901 = 10 Analytical 3D versus 2D numerical models Frequency 119891 inHz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1001 0 I 1948 mdash 1951 mdash 19412 0 I 8856 mdash 8872 mdash 88232 1 I 1104 1110 mdash 1104 mdash1 1 I 1850 1873 mdash 1850 mdash3 0 I 2038 mdash 2042 mdash 20303 1 I 2116 2128 mdash 2116 mdash2 2 I 2221 2242 mdash 2221 mdash3 2 I 2484 2496 mdash 2484 mdash3 3 I 3235 3254 mdash 3235 mdash4 0 I 3649 mdash 3660 mdash 36342 3 I 3692 3739 mdash 3692 mdash4 1 I 3714 3745 mdash 3714 mdash4 2 I 3940 3965 mdash 3941 mdash3 4 I 4226 4261 mdash 4226 mdash4 3 I 4378 4403 mdash 4378 mdash1 2 I 4568 4664 mdash 4568 mdash4 4 I 5035 5064 mdash 5035 mdash2 4 I 5040 5128 mdash 5040 mdash119877120572ℎ = 101 0 I 1905 mdash 1963 mdash 18841 1 I 3054 3145 mdash 3055 mdash1 2 I 6209 6409 mdash 6209 mdash2 0 I 8326 mdash 8857 mdash 82540 1 I 8710 mdash NA mdash NA0 1 II (119908 = 0) 8820 8956 mdash 8820 mdash2 1 I 8880 9531 mdash 8882 mdash0 2 I 9176 mdash NA mdash NA1 3 I 9576 9983 mdash 9567 mdash0 3 I 1036 mdash NA mdash NA2 2 I 1066 1147 mdash 1067 mdash0 4 I 1291 mdash NA mdash NA1 4 I 1335 1413 mdash 1335 mdash2 3 I 1364 1475 mdash 1364 mdash1 0 II (119908 = 0) 1665 1738 mdash 1665 mdash0 5 I 1683 mdash NA mdash NA2 4 I 1755 1918 mdash 1755 mdash0 2 II (119908 = 0) 1764 1791 mdash 1764 mdash

exact solutions cannot be achieved by any numerical modelOnce again solution with 119899 = 0 or 119898 = 0 combined with119908 = 0 (in-plane modes) can be found with 2D numericalmodels without modifying the simply supported boundaryconditions

5 Conclusions

Thepresent paper analyzes the cylindrical bending in the freefrequency analysis of functionally graded material (FGM)plates and cylindrical shells This condition is investigatedby means of a 3D exact shell model and two different 2D

numerical models a weak formulation based on classicalFinite Elements (FEs) and a strong formulation based onthe GDQ method The cylindrical bending (CB) behavior inthe 3D exact shell model can be analyzed using completelysimply supported structures and supposing one of the twohalf-wave numbers equals zeroThe numerical models obtainthe CB behavior using mixed simply supported and freeboundary conditions It has been demonstrated that theactual cylindrical bending (CB) is possible only with respectto straight edgesTheCBnatural frequencies of the numericalsolutions converge to the 3D exact CB ones if the length ofstraight simply supported edges is increased up to 20 times

16 Shock and Vibration

the value of the original edges The CB conditions are notpossible in 2D numericalmodels when the half-wave numberis zero in the curvilinear edge in this case the modificationof boundary conditions does not properly work In-planevibrationmodeswith zero transverse displacement combinedwith 119898 = 0 or 119899 = 0 cannot be defined as CB frequencies andthey can be calculated via 2D numerical models without themodification of the simply supported conditions for the fouredges All the considerations about the cylindrical bendingare valid for each geometry (plate cylinder and cylindricalshell) lamination scheme FGM law and thickness ratio

Competing Interests

The authors declare that they have no competing interests

References

[1] W Q Chen J B Cai and G R Ye ldquoResponses of cross-ply laminates with viscous interfaces in cylindrical bendingrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 823ndash835 2005

[2] W Q Chen and K Y Lee ldquoThree-dimensional exact analysisof angle-ply laminates in cylindrical bending with interfacialdamage via state-space methodrdquo Composite Structures vol 64no 3-4 pp 275ndash283 2004

[3] W Q Chen and K Y Lee ldquoTime-dependent behaviors of angle-ply laminates with viscous interfaces in cylindrical bendingrdquoEuropean Journal ofMechanicsmdashASolids vol 23 no 2 pp 235ndash245 2004

[4] A H Gandhi and H K Raval ldquoAnalytical and empiricalmodeling of top roller position for three-roller cylindricalbending of plates and its experimental verificationrdquo Journal ofMaterials Processing Technology vol 197 no 1ndash3 pp 268ndash2782008

[5] S Oral and H Darendeliler ldquoThe optimum die profile for thecylindrical bending of platesrdquo Journal of Materials ProcessingTechnology vol 70 no 1ndash3 pp 151ndash155 1997

[6] V A Jairazbhoy P Petukhov and J Qu ldquoLarge deflectionof thin plates in cylindrical bendingmdashnon-unique solutionsrdquoInternational Journal of Solids and Structures vol 45 no 11-12pp 3203ndash3218 2008

[7] A Lebee and K Sab ldquoA bending-gradient model for thickplates part II closed-form solutions for cylindrical bending oflaminatesrdquo International Journal of Solids and Structures vol 48no 20 pp 2889ndash2901 2011

[8] P V Nimbolkar and I M Jain ldquoCylindrical bending of elasticplatesrdquo Procedia Materials Science vol 10 pp 793ndash802 2015

[9] A S Sayyad and Y M Ghugal ldquoA nth-order shear deformationtheory for composite laminates in cylindrical bendingrdquo Curvedand Layered Structures vol 2 no 1 pp 290ndash300 2015

[10] N J Pagano ldquoExact solutions for composite laminates incylindrical bendingrdquo Journal of Composite Materials vol 3 no3 pp 398ndash411 1969

[11] N J Pagano and A S D Wang ldquoFurther study of compositelaminates under cylindrical bendingrdquo Journal of CompositeMaterials vol 5 pp 421ndash428 1971

[12] N Saeedi K Sab and J-F Caron ldquoCylindrical bending ofmultilayered plates with multi-delamination via a layerwisestress approachrdquo Composite Structures vol 95 pp 728ndash7392013

[13] X-P Shu and K P Soldatos ldquoCylindrical bending of angle-ply laminates subjected to different sets of edge boundaryconditionsrdquo International Journal of Solids and Structures vol37 no 31 pp 4289ndash4307 2000

[14] Z G Bian W Q Chen C W Lim and N Zhang ldquoAnalyticalsolutions for single- and multi-span functionally graded platesin cylindrical bendingrdquo International Journal of Solids andStructures vol 42 no 24-25 pp 6433ndash6456 2005

[15] H M Navazi and H Haddadpour ldquoNonlinear cylindricalbending analysis of shear deformable functionally graded platesunder different loadings using analytical methodsrdquo Interna-tional Journal of Mechanical Sciences vol 50 no 12 pp 1650ndash1657 2008

[16] W Q Chen J Ying J B Cai and G R Ye ldquoBenchmarksolution of imperfect angle-ply laminated rectangular plates incylindrical bending with surface piezoelectric layers as actuatorand sensorrdquo Computers and Structures vol 82 no 22 pp 1773ndash1784 2004

[17] W Q Chen and K Y Lee ldquoExact solution of angle-plypiezoelectric laminates in cylindrical bending with interfacialimperfectionsrdquo Composite Structures vol 65 no 3-4 pp 329ndash337 2004

[18] T Kant and SM Shiyekar ldquoCylindrical bending of piezoelectriclaminates with a higher order shear and normal deformationtheoryrdquo Computers and Structures vol 86 no 15-16 pp 1594ndash1603 2008

[19] P Lu H P Lee and C Lu ldquoAn exact solution for functionallygraded piezoelectric laminates in cylindrical bendingrdquo Interna-tional Journal of Mechanical Sciences vol 47 no 3 pp 437ndash4382005

[20] W Yan J Wang and W Q Chen ldquoCylindrical bendingresponses of angle-ply piezoelectric laminates with viscoelasticinterfacesrdquo Applied Mathematical Modelling vol 38 no 24 pp6018ndash6030 2014

[21] Y Y Zhou W Q Chen and C F Lu ldquoSemi-analytical solutionfor orthotropic piezoelectric laminates in cylindrical bendingwith interfacial imperfectionsrdquo Composite Structures vol 92no 4 pp 1009ndash1018 2010

[22] G P Dube S Kapuria and P C Dumir ldquoExact piezother-moelastic solution of a simply-supported orthotropic flat panelin cylindrical bendingrdquo International Journal of MechanicalSciences vol 38 pp 374ndash387 1996

[23] V N Pilipchuk V L Berdichevsky and R A IbrahimldquoThermo-mechanical coupling in cylindrical bending of sand-wich platesrdquoComposite Structures vol 92 no 11 pp 2632ndash26402010

[24] C Zhang S Di and N Zhang ldquoA new procedure for staticanalysis of thermo-electric laminated composite plates undercylindrical bendingrdquo Composite Structures vol 56 no 2 pp131ndash140 2002

[25] W Q Chen and K Y Lee ldquoState-space approach for statics anddynamics of angle-ply laminated cylindrical panels in cylindri-cal bendingrdquo International Journal of Mechanical Sciences vol47 no 3 pp 374ndash387 2005

[26] C-P Wu and Y-S Syu ldquoExact solutions of functionally gradedpiezoelectric shells under cylindrical bendingrdquo InternationalJournal of Solids and Structures vol 44 no 20 pp 6450ndash64722007

[27] W Yan J Ying and W Q Chen ldquoThe behavior of angle-ply laminated cylindrical shells with viscoelastic interfaces incylindrical bendingrdquo Composite Structures vol 78 no 4 pp551ndash559 2007

Shock and Vibration 17

[28] S Brischetto F Tornabene N Fantuzzi and E Viola ldquo3Dexact and 2D generalized differential quadraturemodels for freevibration analysis of functionally graded plates and cylindersrdquoMeccanica vol 51 no 9 pp 2059ndash2098 2016

[29] N Fantuzzi S Brischetto F Tornabene and E Viola ldquo2D and3D shell models for the free vibration investigation of func-tionally graded cylindrical and spherical panelsrdquo CompositeStructures vol 154 pp 573ndash590 2016

[30] S Brischetto and R Torre ldquoExact 3D solutions and finiteelement 2D models for free vibration analysis of plates andcylindersrdquo Curved and Layered Structures vol 1 no 1 pp 59ndash92 2014

[31] F Tornabene S Brischetto N Fantuzzi and E Viola ldquoNumer-ical and exact models for free vibration analysis of cylindricaland spherical shell panelsrdquo Composites Part B Engineering vol81 pp 231ndash250 2015

[32] S Brischetto ldquoExact elasticity solution for natural frequenciesof functionally graded simply-supported structuresrdquo ComputerModeling in Engineering amp Sciences vol 95 no 5 pp 391ndash4302013

[33] S Brischetto ldquoThree-dimensional exact free vibration analysisof spherical cylindrical and flat one-layered panelsrdquo Shock andVibration vol 2014 Article ID 479738 29 pages 2014

[34] S Brischetto ldquoA continuum elastic three-dimensional modelfor natural frequencies of single-walled carbon nanotubesrdquoComposites Part B Engineering vol 61 pp 222ndash228 2014

[35] S Brischetto ldquoAn exact 3d solution for free vibrations ofmultilayered cross-ply composite and sandwich plates andshellsrdquo International Journal of Applied Mechanics vol 6 no 6Article ID 1450076 42 pages 2014

[36] S Brischetto ldquoA continuumshellmodel including vanderWaalsinteraction for free vibrations of double-walled carbon nan-otubesrdquoCMES-ComputerModeling in Engineering and Sciencesvol 104 no 4 pp 305ndash327 2015

[37] S Brischetto F Tornabene N Fantuzzi and M BacciocchildquoRefined 2D and exact 3D shell models for the free vibrationanalysis of single- and double-walled carbon nanotubesrdquo Tech-nologies vol 3 no 4 pp 259ndash284 2015

[38] S Brischetto ldquoConvergence analysis of the exponential matrixmethod for the solution of 3D equilibrium equations for freevibration analysis of plates and shellsrdquo Composites Part BEngineering vol 98 pp 453ndash471 2016

[39] S Brischetto ldquoExact and approximate shell geometry in the freevibration analysis of one-layered and multilayered structuresrdquoInternational Journal of Mechanical Sciences vol 113 pp 81ndash932016

[40] S Brischetto ldquoCurvature approximation effects in the freevibration analysis of functionally graded shellsrdquo InternationalJournal of Applied Mechanics 2016

[41] F Tornabene N Fantuzzi F Ubertini and E Viola ldquoStrong for-mulation finite element method based on differential quadra-ture a surveyrdquo Applied Mechanics Reviews vol 67 no 2 ArticleID 020801 pp 1ndash55 2015

[42] F Tornabene E Viola and N Fantuzzi ldquoGeneral higher-orderequivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panelsrdquo CompositeStructures vol 104 pp 94ndash117 2013

[43] N Fantuzzi F Tornabene E Viola and A J Ferreira ldquoAstrong formulation finite element method (SFEM) based onRBF and GDQ techniques for the static and dynamic analysesof laminated plates of arbitrary shaperdquo Meccanica vol 49 no10 pp 2503ndash2542 2014

[44] F Tornabene N Fantuzzi and M Bacciocchi ldquoFree vibrationsof free-form doubly-curved shells made of functionally gradedmaterials using higher-order equivalent single layer theoriesrdquoComposites Part B Engineering vol 67 pp 490ndash509 2014

[45] F Tornabene and E Viola ldquoStatic analysis of functionallygraded doubly-curved shells and panels of revolutionrdquo Mecca-nica vol 48 no 4 pp 901ndash930 2013

[46] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoDynamic analysis of thick and thin elliptic shell structuresmade of laminated composite materialsrdquo Composite Structuresvol 133 pp 278ndash299 2015

[47] F TornabeneN FantuzziM Bacciocchi and EViola ldquoHigher-order theories for the free vibrations of doubly-curved lam-inated panels with curvilinear reinforcing fibers by meansof a local version of the GDQ methodrdquo Composites Part BEngineering vol 81 pp 196ndash230 2015

[48] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoFree vibrations of composite oval and elliptic cylinders bythe generalized differential quadrature methodrdquo Thin-WalledStructures vol 97 pp 114ndash129 2015

[49] F Tornabene and J N Reddy ldquoFGM and laminated doubly-curved and degenerate shells resting on nonlinear elastic foun-dations a GDQ solution for static analysis with a posterioristress and strain recoveryrdquo Journal of the Indian Institute ofScience vol 93 no 4 pp 635ndash688 2013

[50] N Fantuzzi F Tornabene and E Viola ldquoFour-parameter func-tionally graded cracked plates of arbitrary shape a GDQFEMsolution for free vibrationsrdquo Mechanics of Advanced Materialsand Structures vol 23 no 1 pp 89ndash107 2016

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Shock and Vibration

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2 Shock and Vibration

the finite element method Shu and Soldatos [13] developeda new stress analysis method to investigate the stress dis-tributions in angle-ply laminated plates when subjected tocylindrical bending Further cylindrical bending analyses forplates consider the inclusion of functionally graded material(FGM) layers typical examples are [14 15] In [16ndash21]the cylindrical bending analysis was proposed for multi-layered plates embedding piezoelectric andor FGM layersThe cylindrical bending analysis for multilayered compositeandor piezoelectric plates proposed in [22ndash24] includesthe application of thermal loads The cylindrical bendinganalysis of shells is not so diffused in the literature Fewcases can be found and they concern the cylindrical bendingconditions imposed to cylindrical shells Simply supportedangle-ply laminated cylindrical shells in cylindrical bendingwere investigated in [25] using the state-space formulation forthe bending and free vibration analyses A three-dimensionalpiezoelectric model based on the perturbation method forthe cylindrical bending electromechanical load applicationsto cylindrical shells was proposed in [26] Yan et al [27] ana-lyzed the cylindrical bending behavior of a simply supportedangle-ply laminated cylindrical shell embedding viscoelasticinterfaces

The authors proposed the free vibration analysis of simplysupported FGM plates and shells in [28 29] where the 3Dexact shell model was compared with several 2D numericalmodels such as the classical finite element (FE) one andthe classical and refined Generalized Differential Quadrature(GDQ) methods Similar comparisons were also proposedin [30 31] for one-layered and multilayered isotropic com-posite and sandwich plates and shells In such analyses theconsidered geometries are plates cylinders and cylindricaland spherical shell panels Low and high frequencies werecalculated for thick and thin simply supported plates andshells with constant radii of curvature The comparisonbetween the 2D numerical methods (FE and GDQ solutions)and the 3D exact solution is possible only if an exhaustivevibrationmode investigation is conductedThis investigationallows understanding how to make the comparison betweennumerical and analytical methods This comparison is noteasy because the exact 3D solution gives infinite vibrationmodes for all the possible combinations of half-wave numbers(119898 119899) and 2D numerical models propose a finite number ofvibration modes because they use a finite number of degreesof freedom in the plane and in the thickness directionIt should be underlined that the 3D exact model is ableto investigate any kind of geometric ratio (from extremelythick to very thin structures) nevertheless it can model onlysimply supported boundary conditions and it can investigatestructures wherein constant radii of curvature are presentThe 2D numerical approaches are more versatile (in termsof boundary conditions and type of structure analyzed)however they can be used only for moderately thick struc-tures It is obvious that thick structures can be analyzed onlywhen higher order displacement fields are set in such 2Dmodels Since commercial codes implement only through thethickness linear theories they can investigate accurately onlymoderately thick shells The method proposed in [28ndash31] tocompare the 3D exact shell model and the 2D numerical

models is based on the calculation of the frequencies viathe 2D numerical code and their appropriate visualizationof the relative vibration modes via the GDQ andor FEmethod After this visualization the evaluation of the 3Dexact frequencies is possible by means of the appropriateimposition of half-wave numbers obtained from the vibrationmode study From this procedure it is clear how the 3Danalysis could give some frequencies that cannot be calcu-lated by the 2D numerical codes This feature was not theaim of the papers [28ndash31] and for this reason these missedfrequencies are investigated in the present newwork for FGMstructures

This new analysis is conducted calculating the first 18frequencies for one-layered and sandwich FGM plates andshells by means of the 3D exact model The half-wavenumbers are combined along the two directions 120572 and 120573in the plane considering the simply supported boundaryconditions for the four edges In these first 18 frequenciesthere are some modes that cannot be calculated by meansof the 2D numerical models because they are not solutionsof the employed mathematical model with the four simplysupported edges In general these missed modes by the2D numerical models correspond to the cylindrical bendingconditions These conditions in the 3D exact solution meanthat one of the two half-wave numbers is equal to zerothrough an edge direction with a transverse displacementdifferent from zero along the perpendicular edge Theseparticular frequencies are not solutions of the 2D numericalmodels built with the four edges which are simply supportedbut a different mathematical model must be defined wheretwo edges are simply supported and the other two arefree These conditions allow the cylindrical bending withthe transverse displacement through two parallel edgesdifferent from zero in numerical models in order toobtain the missed frequencies and modes This condi-tion is possible only when the zero half-wave numberis imposed through a rectilinear side because the half-wave number equals zero in the curvilinear edge doesnot mean that all the derivatives in the that direction arezero because of the presence of the curvature This lastcondition is in conflict with the definition of cylindricalbending

The 3D exact shell model is described and validatedin Section 2 It uses the equilibrium equations writtenusing the generic curvilinear orthogonal coordinates Suchequations are solved in closed form by means of simplysupported boundary conditions The differential equationsalong 119911 written in layer-wise form are solved using theexponential matrix method (also known as transfer matrixmethod or the state-space approach) The 2D FE and GDQmodels are proposed and opportunely validated in Sec-tion 3 The 2D FE model makes use of a very commoncommercial finite element code The 2D GDQ models giveresults by means of an in-house academic software andthey are based on classical and refined shell models Theresults about the comparisons between 3D exact and 2Dnumerical models with the remark about the boundaryconditions are shown Section 4 Section 5 gives the mainconclusions

Shock and Vibration 3

x

y

z

a

b

(a)

z

b

R

a = 2R

(b)z

R

a

b

(c)

z

R

R

a ba b

(d)

Figure 1 Geometries of plates and shells with constant radii of curvature and their local reference system

2 Exact Solution of a 3D Shell Model

The three-dimensional exact shell model employed for thecomparisons proposed in Section 4 has been elaborated in[32ndash40] for the free frequency analyses of single-layered andmultilayered isotropic orthotropic composite FGM andsandwich plates and shells with constant radii of curvatureand for the free vibrations of single- and double-walledcarbon nanotubes including van der Waals interactions Theequilibrium equations proposed in the general orthogonalcurvilinear coordinate system (120572 120573 119911) are valid for bothplates and shells with constant radii of curvature For thesake of clarity the plates and shells geometries within theirlocal coordinate systems are depicted in Figure 1 The exactform of these equations is obtained using simply supportedboundary conditions while the differential equations in 119911are solved by means of the exponential matrix method(also known as transfer matrix method or the state-spaceapproach)

Considering spherical shells with constant radii of cur-vature 119877120572 and 119877120573 and embedding 119873119871 classical or FGM layers

the differential equations of equilibrium for the free vibrationanalysis written for a generic 119896 layer are

119867120573 120597120590119896120572120572120597120572 + 119867120572 120597120590119896120572120573120597120573 + 119867120572119867120573 120597120590119896120572119911120597119911+ (2119867120573119877120572 + 119867120572119877120573 ) 120590119896120572119911 = 120588119896119867120572119867120573119896

119867120573 120597120590119896120572120573120597120572 + 119867120572 120597120590119896120573120573120597120573 + 119867120572119867120573 120597120590119896120573119911120597119911+ (2119867120572119877120573 + 119867120573119877120572 ) 120590119896120573119911 = 120588119896119867120572119867120573V119896

119867120573 120597120590119896120572119911120597120572 + 119867120572 120597120590119896120573119911120597120573 + 119867120572119867120573 120597120590119896119911119911120597119911 minus 119867120573119877120572 120590119896120572120572 minus 119867120572119877120573 120590119896120573120573+ (119867120573119877120572 + 119867120572119877120573 ) 120590119896119911119911 = 120588119896119867120572119867120573119896

(1)

4 Shock and Vibration

It is remarked that this set of equations is valid onlyfor spherical shells wherein 119877120572 = 119877120573 = 119877 because fordoubly curved shells with variable radii of curvature suchequations take a different form The authors decided to keepsuch system with the two radii of curvature distinct becauseit will be much easier to understand the cylindrical bendingeffect for plates and shells starting from a general point ofview

In the case of shell geometries andor FGM layers eachphysical layer 119896 must be divided in an opportune numberof 119902 mathematical layers to approximate the shell curvatureandor the FGM law through the thickness direction 119911 Theindex 119895 indicates the total mathematical layers obtained fromthe subdivision of each 119896 layer in 119902 mathematical layers (119895 =119896 times 119902) The parametric coefficients for shells with constantradii of curvature can be written as

119867120572 = (1 + 119911119877120572) 119867120573 = (1 + 119911119877120573) 119867119911 = 1

(2)

A closed form solution for simply supported shells andplates can be developed by writing harmonic displacementsas

119906119895 (120572 120573 119911 119905) = 119880119895 (119911) 119890119894120596119905 cos (120572120572) sin (120573120573) V119895 (120572 120573 119911 119905) = 119881119895 (119911) 119890119894120596119905 sin (120572120572) cos (120573120573) 119908119895 (120572 120573 119911 119905) = 119882119895 (119911) 119890119894120596119905 sin (120572120572) sin (120573120573)

(3)

where 119880119895(119911) 119881119895(119911) and 119882119895(119911) indicate the displacementamplitudes in 120572 120573 and 119911 directions respectively 119894 indicatesthe imaginary unit 120596 = 2120587119891 is the circular frequencywhere 119891 is the frequency value in Hz 119905 is the symbol forthe time In the coefficients 120572 = 119898120587119886 and 120573 = 119899120587119887119898 and 119899 are the half-wave numbers and 119886 and 119887 are theshell dimensions in 120572 and 120573 directions respectively (all thesequantities are evaluated on the midsurface Ω0) 120588119895 is themass density (120590119895120572120572 120590119895120573120573 120590119895119911119911 120590119895120573119911 120590119895120572119911 120590119895120572120573) are the six stresscomponents and 119895 V119895 and 119895 are the second temporalderivative of the three displacement components 119877120572 and119877120573 indicate the radii of curvature defined with respect thereference surface Ω0 of the whole multilayered plate or shell119867120572 and 119867120573 proposed in (2) vary with continuity through thethickness direction of the multilayered structure Thereforethey depend on the thickness coordinate 119911 The referencesurface Ω0 of the structure is defined as the locus of pointswhich are positioned midway between the external surfacesDisplacement components are indicated as 119906 V and 119908 in 120572120573 and 119911 directions respectively

The final system in closed form is developed substituting(3) and constitutive and geometrical equations (which are not

here given for the sake of brevity but they can be found indetail in [32]) in (1)

(minus119862119895551198671198951205731198671198951205721198772120572 minus 11986211989555119877120572119877120573 minus 120572211986211989511119867119895120573119867119895120572 minus 120573211986211989566119867119895120572119867119895120573+ 1205881198951198671198951205721198671198951205731205962) 119880119895 + (minus12057212057311986211989512 minus 12057212057311986211989566) 119881119895

+ (12057211986211989511119867119895120573119867119895120572119877120572 + 12057211986211989512119877120573 + 12057211986211989555119867119895120573119867119895120572119877120572 + 12057211986211989555119877120573 ) 119882119895

+ (11986211989555119867119895120573119877120572 + 11986211989555119867119895120572119877120573 ) 119880119895119911 + (12057211986211989513119867119895120573 + 12057211986211989555119867119895120573)sdot 119882119895119911 + (11986211989555119867119895120572119867119895120573) 119880119895119911119911 = 0

(minus12057212057311986211989566 minus 12057212057311986211989512) 119880119895 + (minus119862119895441198671198951205721198671198951205731198772120573 minus 11986211989544119877120572119877120573minus 120572211986211989566119867119895120573119867119895120572 minus 120573211986211989522119867119895120572119867119895120573 + 1205881198951198671198951205721198671198951205731205962) 119881119895

+ (12057311986211989544119867119895120572119867119895120573119877120573 + 12057311986211989544119877120572 + 12057311986211989522119867119895120572119867119895120573119877120573 + 12057311986211989512119877120572 ) 119882119895

+ (11986211989544119867119895120572119877120573 + 11986211989544119867119895120573119877120572 ) 119881119895119911 + (12057311986211989544119867119895120572 + 12057311986211989523119867119895120572)sdot 119882119895119911 + (11986211989544119867119895120572119867119895120573) 119881119895119911119911 = 0

(12057211986211989555119867119895120573119867119895120572119877120572 minus 12057211986211989513119877120573 + 12057211986211989511119867119895120573119867119895120572119877120572 + 12057211986211989512119877120573 ) 119880119895

+ (12057311986211989544119867119895120572119867119895120573119877120573 minus 12057311986211989523119877120572 + 12057311986211989522119867119895120572119867119895120573119877120573 + 12057311986211989512119877120572 ) 119881119895

+ ( 11986211989513119877120572119877120573 + 11986211989523119877120572119877120573 minus 119862119895111198671198951205731198671198951205721198772120572 minus 211986211989512119877120572119877120573 minus 119862119895221198671198951205721198671198951205731198772120573minus 120572211986211989555119867119895120573119867119895120572 minus 120573211986211989544119867119895120572119867119895120573 + 1205881198951198671198951205721198671198951205731205962) 119882119895

+ (minus12057211986211989555119867119895120573 minus 12057211986211989513119867119895120573) 119880119895119911 + (minus12057311986211989544119867119895120572minus 12057311986211989523119867119895120572) 119881119895119911 + (11986211989533119867119895120573119877120572 + 11986211989533119867119895120572119877120573 ) 119882119895119911+ (11986211989533119867119895120572119867119895120573) 119882119895119911119911 = 0

(4)

Shock and Vibration 5

The cylindrical bending requirements for analytical methodsare obtained choosing one of the two half-wave numberswhich equals zero This condition means 120572 = 119898120587119886 =0 or 120573 = 119899120587119887 = 0 which allows the derivatives madein a particular direction (120572 or 120573) which equals zero tooThis mathematical condition means that each section in thecylindrical bending direction has always the same behavior

It is obvious that the cylindrical bending effect is properfor cylindrical structures only However the authors startedthe present treatise from the governing equations (1) ofspherical shell structures because one of the aims of thepresent contribution is to underline which are the extraterms (related to the curvature) that do not allow having thestructure to be in cylindrical bending conditions

The compact form of (4) has been elaborated in detail in[32] The final version is here proposed

D119895 120597U119895120597119911 = A119895U119895 (5)

where 120597U119895120597119911 = U1198951015840 andU119895 = [119880119895 119881119895 119882119895 1198801198951015840 1198811198951015840 1198821198951015840]Equation (5) is rewritten as

D119895U1198951015840 = A119895U119895 (6)

U1198951015840 = D119895minus1A119895U119895 (7)

U1198951015840 = A119895lowastU119895 (8)

with A119895lowast = D119895minus1A119895The exponential matrix method (also known as transfer

matrix method or the state-space approach) is employed forthe solution of (8) (details are given in [32])

U119895 (119911119895) = exp (A119895lowast119911119895)U119895 (0) with 119911119895 isin [0 ℎ119895] (9)

where 119911119895 is the thickness coordinate for the 119895 mathematicallayer with values from 0 at the bottom to ℎ119895 at the top Theexponential matrix is calculated using 119911119895 = ℎ119895 for each 119895mathematical layer

A119895lowastlowast = exp (A119895lowastℎ119895)= I + A119895lowastℎ119895 + A119895lowast22 ℎ1198952 + A119895lowast33 ℎ1198953 + sdot sdot sdot

+ A119895lowast119873119873 ℎ119895119873(10)

where I indicates the 6times6 identitymatrixThe expansion usedfor the exponentialmatrix in (10) has a fast convergence and itis not time consuming from the computational point of view119895 = 119872 mathematical layers are used to approximate theshell curvature andor the FGM law through the thickness119872 minus 1 transfer matrices must be defined by means ofthe interlaminar continuity conditions of displacements andtransverse shearnormal stresses at each interfaceThe closedform solution for plates or shells is obtained using simply

supported boundary conditions and free stress conditions atthe top and at the bottom surfaces The above conditionsallow the following final system

EU1 (0) = 0 (11)

and the 6 times 6 matrix E does not change independently of thenumber of layers 119872 even if a layer-wise approach is appliedU1(0) is the vector U defined at the bottom of the wholemultilayered shell or plate (first layer 1 with 1199111 = 0) All thesteps and details here omitted for the sake of brevity can befound in [32ndash40] where the proposed exact 3D shell modelhas been elaborated and tested for the first time for severalapplications Several validations for the 3D exact shell modelhave been shown in [32ndash40] For this reason the model isnow used with confidence for the results and comparisonsproposed in Section 4

The nontrivial solution ofU1(0) in (11) allows conductingthe free vibration analysis imposing the zero determinant ofmatrix E

det [E] = 0 (12)

Equation (12) is used to find the roots of an higher order poly-nomial written in 120582 = 1205962The imposition of a couple of (119898 119899)allows a certain number of circular frequencies 120596 = 2120587119891(from I to infin) to be calculated This number of frequenciesdepends on the order 119873 used in each exponential matrixA119895lowastlowast and the number 119872 of mathematical layers employedfor the shell curvature andor the FGM law approximationsThe results shown in [32ndash40] confirm that 119873 = 3 for theexponential matrix and 119872 = 100 for the mathematicallayers are appropriate to always calculate the correct freefrequencies for each structure lamination sequence numberof layers material FGM law and thickness value

3 2D Finite Element Analysis and 2DGeneralized Differential Quadrature Models

The 3D exact natural frequencies are compared with twonumerical 2D solutions given by a weak and a strongformulation The former is a classic 2D finite element (FE)model carried out through the commercial software Strand7The latter is a recent widespread technique known as Gener-alized Differential Quadrature (GDQ) method [41] Detailsrelated to convergence and accuracy of these techniquescan be found in the works [29ndash31] It is noted that theFE solution employs a classic linear theory known alsoas Reissner-Mindlin theory with a linear rotation of thefibers through the shell thickness The present GDQ solutionexploits higher order theories with a general expansionthrough the thickness of the structureThis general approachfalls within the framework of the unified theories becauseseveral structural models can be obtained via the repetitionof a fundamental nucleus Another difference between theGDQ applications and the FE ones is that the formerincludes the curvature effects and the rotary inertias in itsformulation

6 Shock and Vibration

The 2D kinematic hypothesis [42ndash50] considers the fol-lowing expansion of the degrees of freedom

U = 119873119888+1sum120591=0

F120591u(120591) (13)

where U represents the vector of the 3D displacementcomponents which is function of (120572 120573 119911) and u120591 is thevector of the displacement parameters function of (120572 120573)which depends on the free index 120591 Therefore u120591 lays on themidsurface of the shell [42] The matrix F120591(119894119895) = 120575119894119895119865120591 (for119894 119895 = 1 2 3) contains the so-called thickness functions where120575 is the Kronecker delta function The thickness functionsdescribe how the displacement parameters characterize thesolution through the thickness They can have any form ofany smooth function of 119911 however the classic form of thesefunctions is the power function Thus 119865120591 = 119911120591 can be 1(120591 = 0) 119911 (120591 = 1) 1199112 (120591 = 2) 1199113 (120591 = 3) and so onPlease note that according to the mathematical backgroundof the present Unified Formulation the index 119873119888 + 1 isreserved to the (optional) zigzag effect Since the aim of thepresent paper is to investigate the cylindrical bending effectfor functionally graded plates and cylindrical panels thefollowing numerical applications will consider a fixed valueof 119873119888 = 3 so that 120591 = 0 1 2 3 This choice is justified by thefact that third order theories are sufficient with a reasonablylimited number of degrees of freedom to obtain accuratesolutions when compared to 3D shell models [42ndash50] Formultilayered structures such as sandwich configurations thezigzag function must be added into the formulation Onceagain as shown in (13) the zigzag function is embedded atthe end of the expansion with the exponent 119873119888 + 1 Whensuch effect is neglected the sum of (13) stops at 119873119888

The same theory used for the FE solution can be carriedout in the GDQmethod setting 120591 = 0 1 and 119873119888 = 1 howeverit has been shown in other recent works that the strong formapproaches led tomore accurate results with respect to the FEones [46 48]

The relation between generalized strains 120576(120591) anddisplace-ments u(120591) can be set as

120576(120591) = DΩu

(120591) for 120591 = 0 1 2 119873119888 119873119888 + 1 (14)

where DΩ is a differential operator explicitly defined in [42]The present constitutive material model is linear and elastictherefore for a general 119896th laminaHookersquos law can be applied

120590(119896) = C(119896)120576(119896) (15)

where 120590(119896) is the vector containing the stress componentsand C(119896) represents the constitutive matrix for each 119896th ply[42] of the structure Each component of the constitutivematrix is indicated by the symbol 119862(119896)120585120578 and it is valid in thecurvilinear reference system 1198741015840120572120573119911 of the reference geome-tryThese coefficients are evaluatedwith the application of thetransformation matrix [42]

The stress resultants are the terms which correspond tothe punctual 3D stresses of the 3D doubly curved solidTheir definition can be achieved by developing the Hamiltonprinciple for the present structural theory [42]

S(120591) = 119873119871sum119896=1

int119911119896+1119911119896

(Z(120591))119879 120590(120591)119867120572119867120573119889119911for 120591 = 0 1 2 119873119888 119873119888 + 1

(16)

whereZ(120591) is a 6times9matrix containing the thickness functions119865120591 and the curvature coefficients119867120572 and119867120573 the definition ofsuchmatrix with some other details can be found in [42] S(120591)include the stress resultants as a function of the order 120591

The definition of (16) can be rewritten as

S(120591) = 119873119888+1sum119904=0

A(120591119904)120576(119904) for 120591 = 0 1 2 119873119888 119873119888 + 1 (17)

which is a function of generalized strains of order 119904 120576(119904)Note that the so-called elastic coefficients are defined as

A(120591119904) = 119873119871sum119896=1

int119911119896+1119911119896

(Z(120591))119879C(119896)Z(119904)119867120572119867120573119889119911 (18)

and they can be carried out by the following integral expres-sions

119860(120591119904)120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119911119904119911120591119867120572119867120573119867984858120572119867120577120573 119889119911

119860(119904)120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119911119904120591119911120591minus1119867120572119867120573119867984858120572119867120577120573 119889119911

119860(120591)120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119904119911119904minus1119911120591119867120572119867120573119867984858120572119867120577120573 119889119911

Shock and Vibration 7

119860( )120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119904119911119904minus1120591119911120591minus1119867120572119867120573119867984858120572119867120577120573 119889119911for 120591 119904 = 0 1 2 119873119888 119873119888 + 1 120585 120578 = 1 2 3 4 5 6 984858 120577 = 0 1 2

(19)

It is recalled that the exponent 119873119888 + 1 is reserved for theMurakami function otherwise this order is neglected Thesubscripts and indicate the derivatives of the thicknessfunctions 119865120591 and 119865119904 with respect to 119911 The subscripts 984858 and 120577are the exponents of the quantities119867120572 and 119867120573 whereas 120585 and120578 are the indices of the material constants 119862(119896)120585120578 for the 119896th ply

[42ndash49]Note that119862(119896)120585120578 119867120572 and119867120573 are quantities that dependon the thickness 119911 Therefore the integrals of (19) cannotbe computed analytically The present solution for solvingsuch integrals is to employ a numerical integration algorithmIn particular the so-called Generalized Integral Quadrature(GIQ) method is utilized For an accurate evaluation of theintegrals 51 points per layer are considered in the integration

The Hamilton principle supplies for the dynamic equilib-rium equations and the correspondent boundary conditionsIn detail the fundamental nucleus of three motion equationstakes the form

D⋆ΩS(120591) = 119873119888+1sum

119904=0

M(120591119904)u(119904)

for 120591 = 0 1 2 119873119888 119873119888 + 1(20)

whereD⋆Ω is the equilibrium operator andM(120591119904) is the inertialmatrix Their explicit expressions have been presented in aprevious authorsrsquo paper [42]

Combining the kinematic equation (14) constitutiveequation (17) and themotion equation (20) the fundamentalgoverning system in terms of displacement parameters iscarried out

119873119888+1sum119904=0

L(120591119904)u(119904) = 119873119888+1sum119904=0

M(120591119904)u(119904)

for 120591 = 0 1 2 119873119888 119873119888 + 1(21)

where L(120591119904) = D⋆ΩA(120591119904)DΩ is the fundamental operator [42]

Boundary conditions must be included for the solutionof the differential equations (21) The present paper dealsonly with free and simply supported boundary conditionsThe former is used to reproduce the cylindrical bendingconditions considering alternatively one free edge and onesimply supported edge When the four edges are all simplysupported the problem is the same as the ones investigated

in the previous papers [29ndash31] Simply supported conditionsare

119873(120591)120572 = 0119906(120591)120573 = 119906(120591)119911 = 0

120591 = 0 1 2 119873119888 119873119888 + 1 at 120572 = 1205720 or 120572 = 1205721 1205730 le 120573 le 1205731119906(120591)120572 = 0

119873(120591)120573 = 0119906(120591)119911 = 0

120591 = 0 1 2 119873119888 119873119888 + 1 at 120573 = 1205730 or 120573 = 1205731 1205720 le 120572 le 1205721

(22)

Free edge conditions are

119873(120591)120572 = 0119873(120591)120573 = 0119879(120591)120572 = 0120591 = 0 1 2 119873119888 119873119888 + 1 at 120572 = 1205720 or 120572 = 1205721 1205730 le 120573 le 1205731119873(120591)120572 = 0119873(120591)120573 = 0119879(120591)120573 = 0120591 = 0 1 2 119873119888 119873119888 + 1 at 120573 = 1205730 or 120573 = 1205731 1205720 le 120572 le 1205721

(23)

4 Results

The present section includes some numerical applicationsfor the comparison of the 3D exact shell vibrations withtwo numerical 2D models In particular classical simplysupported solutions and cylindrical bending (CB) behaviorof plates and shells are presentedThe following computationsfocus on plates circular cylinders and cylindrical panels dueto the fact that the CB effect occurs only when at least twoparallel straight edges are present in the structure

The first case is a square plate (119886 = 119887 = 1m) withsimply supported edges Two thickness ratios are analyzedthin 119886ℎ = 100 and moderately thick 119886ℎ = 20 structure Theplate is made of a single functionally graded layer (ℎ1 = ℎ)with volume fraction of ceramic (119888) phase

119881119888 (119911) = (05 + 119911ℎ)119901 with 119911 isin [minusℎ2 ℎ2 ] (24)

8 Shock and Vibration

Table 1 One-layered FGM plate with 119901 = 05 Analytical 3D versus 2D numerical models Frequency 119891 in Hz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119886ℎ = 1001 0 I 4019 mdash 3907 mdash 39210 1 I 4019 mdash 3907 mdash 39211 1 I 8037 8006 mdash 8036 mdash2 0 I 1607 mdash 1580 mdash 15840 2 I 1607 mdash 1580 mdash 15842 1 I 2008 2001 mdash 2008 mdash1 2 I 2008 2001 mdash 2008 mdash2 2 I 3212 3202 mdash 3211 mdash3 0 I 3613 mdash 3568 mdash 35770 3 I 3613 mdash 3568 mdash 35773 1 I 4014 4002 mdash 4013 mdash1 3 I 4014 4002 mdash 4013 mdash3 2 I 5215 5202 mdash 5215 mdash2 3 I 5215 5202 mdash 5215 mdash4 0 I 6416 mdash 6354 mdash 63650 4 I 6416 mdash 6354 mdash 63654 1 I 6816 6801 mdash 6815 mdash1 4 I 6816 6801 mdash 6815 mdash119886ℎ = 201 0 I 2002 mdash 1951 mdash 19510 1 I 2002 mdash 1951 mdash 19511 1 I 3989 3995 mdash 3988 mdash2 0 I 7916 mdash 7864 mdash 77970 2 I 7916 mdash 7864 mdash 77972 1 I 9858 9954 mdash 9857 mdash1 2 I 9858 9954 mdash 9857 mdash2 2 I 1560 1588 mdash 1560 mdash3 0 I 1749 mdash 1767 mdash 17290 3 I 1749 mdash 1767 mdash 17293 1 I 1936 1980 mdash 1936 mdash1 3 I 1936 1980 mdash 1936 mdash3 2 I 2490 2566 mdash 2490 mdash2 3 I 2490 2566 mdash 2490 mdash1 0 II (119908 = 0) 2775 2773 mdash 2775 mdash0 1 II (119908 = 0) 2775 2773 mdash 2775 mdash4 0 I 3034 mdash 3124 mdash 30070 4 I 3034 mdash 3124 mdash 3007

where 119901 is the power exponent which varies the volumefraction of the ceramic component through the thicknessTwo limit cases occur for (24) 119881119888 = 1 when 119901 = 0 and119881119888 rarr 0when119901 = infin In the following computations it is set as119901 = 05 Note that the volume fraction function in (24) givesa material which is ceramic (119888) rich at the top and metal (119898)rich at the bottomof the structureThemechanical properties(Youngmodulus andmass density) are given by the theory ofmixtures as

119864 (119911) = 119864119898 + (119864119888 minus 119864119898) 119881119888120588 (119911) = 120588119898 + (120588119888 minus 120588119898) 119881119888 (25)

where 119864119898 and 120588119898 are the elastic modulus and mass densityof the metallic phase respectively 119864119888 and 120588119888 are the corre-sponding properties of the ceramic phase Note that in allthe following computations the Poisson ratio is consideredconstant through the thickness thus ]119888 = ]119898 = ] = 03 Thetwo constituents have the following mechanical properties

119864119898 = 73GPa120588119898 = 2800 kgm3119864119888 = 380GPa120588119888 = 3800 kgm3

(26)

Shock and Vibration 9

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(a) Mode I (0 1) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

000501

01502

(b) Mode I (0 1) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(c) Mode II (119908 = 0) (0 1) 3D

002minus002

090908

08070706

06050504

04030302

020101

00minus01

1

(d) Mode II (119908 = 0) (0 1) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(e) Mode I (1 0) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

0minus005minus01

minus015minus02

(f) Mode I (1 0) 2D

Figure 2 Continued

10 Shock and Vibration

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(g) Mode II (119908 = 0) (1 0) 3D

002minus002

0908 08

0706 06

0504 04

0302 02

01 0 0

11

(h) Mode II (119908 = 0) (1 0) 2D

Figure 2 Differences between in-plane and cylindrical bending vibration modes for the 119886ℎ = 20 one-layered FGM plate with 119901 = 05 3Dmodes versus 2D modes

Since the FE commercial software does not have anembedded tool for the computation of functionally gradedproperties through the thickness the present FE applicationsare carried out building a fictitious laminated structure withisotropic plies In detail the thickness of the FGM lamina hasbeen divided into 100 fictitious layers and for each layer thecorresponding mechanical properties have been computedthrough (25) The numerical model has been made of auniform element division of 40 times 40 Quad8 elements for atotal number of 29280 degrees of freedom (dofs) The GDQsolution is computed with a not-uniform Chebyshev-Gauss-Lobatto grid [41] with 31 times 31 points for both thicknessratios It is underlined that since a single ply is set no zigzagfunction is needed in the computation It is recalled that thepresent higher order GDQ model considers a third orderexpansion with 120591 = 0 1 2 3 whereas the FE one has afirst order model (also known as Reissner-Mindlin model)Table 1 reports the results in terms of natural frequency(Hz) wherein the first column reports the number of half-waves (119898 119899) of the 3D exact solution Note that for a fixedset of half-wave numbers different mode shapes can occur(indicated by the second column ldquoModerdquo) In particular Iindicates the first mode and II indicates the second mode Inthe case of in-plane modes the caption (119908 = 0) is added tounderline that the vibration occurs in the plane and not outof plane The third column shows all the results for the 3Dexact solution both classical and cylindrical bending (CB)results It is noted that these frequencies are obtained witha single run of the 3D exact code keeping simply supportedboundary conditions for all the four edges of the plate Theother four columns of Table 1 refer to the numerical solutionsfor simply supported structures and for cylindrical bendingconditions (setting two free opposite edges) The cylindricalbending results are indicated by the acronym CB in the sametable

There is a very good agreement among the three solutionswhen simply supported conditions are considered On thecontrary a small error is observed in cylindrical bendingbecause the actual CB condition is not completely satisfieddue to the fact that the plate has a limited dimension in thesimply supported direction This feature can be detected inFigures 2 and 3 because the deformed shapes of the plate inthe CB configuration are not properly correct In fact thefibers parallel to the supported edges are not parallel amongthem but there is a curvature effect due to the free boundariesand to the limited length of the plate On the contrary thein-plane mode shapes (119908 = 0) when the plate is simplysupported on the four edges have all the fibers parallel amongthem also after the deformation These modes are the resultsobtained in the 3D solution with 119898 = 0 or 119899 = 0 combinedwith 119908 = 0 They are not cylindrical bending modes infact they can be found with 2D numerical models withoutmodifying the boundary conditions 3D frequencieswith119898 =0 or 119899 = 0 (with w different from 0) are CB modes which canbe obtained considering two simply supported edges and twofree edges in 2D numerical models

As it is well-known the ldquoperfectrdquo CB would occur whenthe simply supported direction has an ldquoinfiniterdquo lengthThis cannot be proven numerically but a trend can beobserved modeling different plates in cylindrical bendingconfiguration with different lengthsThese analyses are madekeeping one side of the plate constant and enlarging thelength of the other edge In the present computations 119886 iskept constant and 119887 = 119871 varies from 1 to 20 The results ofthe aforementioned analysis are listed in Table 2 and depictedin Figures 4 and 5 Table 2 includes the numerical valuesof the first mode shape in cylindrical bending for the 3Dexact solution the 2D FE and the 2D GDQmodels Figure 4shows that both GDQ and FE solutions tend to the 3D exactone increasing the length of the simply supported edge In

Shock and Vibration 11

ulowast

lowast

wlowast

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

(a) Mode I (1 0) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

000501

01502

(b) Mode I (1 0) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(c) Mode I (2 0) 3D

109

0807

060504

0302

01 0

109

0807

0605

0403

02010

0201501

0050

minus005minus01

minus015minus02

(d) Mode I (2 0) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(e) Mode I (3 0) 3D

10908

0706

050403

0201 0

109

0807

0605

0403

02010

0201501

0050

minus005minus01

minus015minus02

(f) Mode I (3 0) 2D

Figure 3 Continued

12 Shock and Vibration

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(g) Mode I (4 0) 3D

10908

0706050403

0201 0

10908

0706

0504

0302

010

0201501

0050

minus005minus01

minus015minus02

(h) Mode I (4 0) 2D

Figure 3 First four cylindrical bending vibration modes for the 119886ℎ = 20 one-layered FGM plate with 119901 = 05 3D modes versus 2D modes

3D exact2D GDQ2D FEM

f(H

z)

39

392

394

396

398

40

402

404

5 10 15 200La

Figure 4 First frequency 119891 in Hz for the 119886ℎ = 100 one-layeredFGM plate with 119901 = 05 convergence of 2D GDQ and 2D FEsolutions increasing the length of the simply supported sides

particular the GDQ solution is slightly more accurate thanthe FE one The 2D GDQ solution is carried out by enlargingthe length of one of the sides of the plate and the samenumberof dofs is kept On the contrary the FE model is generatedby 40 times 40 blocks of mesh In other words the plate with119871 = 2119886 has a mesh of 40 times 80 the one with 119871 = 3119886 has amesh 40 times 120 and so on When 119871 = 20119886 the error amongthe 3D exact and the 2D numerical solutions is very small andthis is proven also by the mode shapes of Figure 5 whereinthe boundary effect is negligible with respect the length of

the simply supported edges It must be underlined that it isimpossible to completely avoid the boundary effects due tothe free edges

The second example is related to a simply supportedcylinder with 119877120572 = 10m 119877120573 = infin 119886 = 2120587119877120572 and 119887 =20m and thickness ratios 119877120572ℎ = 100 and 119877120572ℎ = 10 Aone-layered FGM scheme is considered with ℎ1 = ℎ andthe same volume fraction function of (24) for the previousexample but setting 119901 = 20 The present material propertiesare the same as the flat plate The FE model has a 20 times80 Quad8 mesh with 27966 dofs and the GDQ solutionhas a 15 times 51 Chebyshev-Gauss-Lobatto grid Numericallyspeaking compatibility conditions must be enforced on theclosing meridian in order to obtain a closed revolution shell[42] To obtain the CB conditions numerically the cylinderis modeled completely free and the first 6 rigid mode shapesare neglected and removed from the numerical solution Itis noted that the GDQ model uses a third-order kinematicexpansion with no zigzag effect as the previous case sincea one-layer structure has been taken into account Table 3reports the results with the same meaning of the symbolsseen in Table 1 A very good agreement is observed when thecylinder is simply supported whereas when the cylinder iscompletely free some small errors can be observed but thenumerical solutions follow the 3D exact one

Results with 119899 = 0 and 119908 = 0 indicate the CB conditionsSuch results are obtained by means of 2D numerical modelsmodifying the boundary conditions (see 2D FE(CB) and 2DGDQ(CB)models)The cases 119899 = 0 combined with119908 = 0 arenot CB conditions and the frequencies can be found withoutmodifying the boundary conditions in the 2D numericalmodels

The final test is related to a cylindrical panel with 119877120572 =10m 119877120573 = infin 119886 = 1205873119877120572 and 119887 = 20m and thickness ratios119877120572ℎ = 100 and 119877120572ℎ = 10 The present shell is a sandwichstructure embedding a FGM core with ℎ2 = 07ℎ and two

Shock and Vibration 13

218

1614

121 1

08 0806 06

04 0402 02

0 0

0minus01minus02minus03minus04

(a) 119871119886 = 2

3

25

2

151

0806

040502

0 0

1

0minus01minus02minus03minus04minus05

(b) 119871119886 = 3

435

325

10505

0 0

15

0

minus02minus04minus06

1

2

(c) 119871119886 = 4

545

35

10505

0 0

0

minus04minus02

minus06minus08

1

25

4

3

215

(d) 119871119886 = 5

5

1

0 005

0

minus04minus02

minus06minus08

1

4

3

2

6

minus1

(e) 119871119886 = 6

5

10 0 051

43

2

67

89

100

minus05minus1

minus15

(f) 119871119886 = 10

Figure 5 First vibration mode for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 2D GDQ solution and edge effects versus the lengthof the simply supported sides

external skins with ℎ1 = ℎ3 = 015ℎ (ℎ1 + ℎ2 + ℎ3 = ℎ) Thebottom layer ismade of isotropicmetallicmaterial and the toplayer is made of ceramic material the mechanical propertiesare

119864119898 = 73GPa120588119898 = 2800 kgm3119864119888 = 200GPa120588119888 = 5700 kgm3

(27)

The FGM core has a volume fraction function defined as

119881119888 (119911) = (05 + 11991107ℎ)119901 with 119911 isin [minus035ℎ 035ℎ] (28)

where 119901 = 10 is employed in the present computations ThePoisson ratio is constant (]119888 = ]119898 = ] = 03) Equations (25)are still valid with 119881119888 defined as in (28)

The FEmodel is made of 40times40Quad8 elements for total29280 dofs and the GDQ model has a 31 times 31 Chebyshev-Gauss-Lobatto grid Since a sandwich structure is analyzedin the following a third-order theory with zigzag functionis considered in the computations for the GDQ method Thepresent natural frequencies are reported in Table 4 with the

14 Shock and Vibration

Table 2 First frequency 119891 in Hz for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 convergence of 2D GDQ and 2D FE solutionsincreasing the length of the simply supported sides

119871119886 1 2 3 4 5 6 8 10 15 20 3D exact2D GDQ 3921 3964 3980 3988 3993 3997 4001 4004 4007 4008 40192D FEM 3907 3949 3966 3974 3979 3983 3987 3989 3993 3994 4019

Table 3 One-layered FGM cylinder with 119901 = 20 Analytical 3D versus 2D numerical models Frequency 119891 in Hz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1004 0 I 0896 mdash 0897 mdash 08836 0 I 2532 mdash 2538 mdash 25118 0 I 4854 mdash 4867 mdash 481910 0 I 7848 mdash 7871 mdash 780112 0 I 1151 mdash 1155 mdash 114610 1 I 1356 1360 mdash 1357 mdash12 1 I 1441 1449 mdash 1445 mdash14 0 I 1584 mdash 1589 mdash 15778 1 I 1638 1640 mdash 1636 mdash14 1 I 1756 1767 mdash 1763 mdash16 0 I 2083 mdash 2090 mdash 207016 1 I 2207 2222 mdash 2216 mdash6 1 I 2441 2442 mdash 2438 mdash18 0 I 2649 mdash 2658 mdash 264014 2 I 2742 2752 mdash 2746 mdash18 1 I 2751 2772 mdash 2763 mdash16 2 I 2875 2890 mdash 2883 mdash12 2 I 2921 2926 mdash 2922 mdash119877120572ℎ = 104 0 I 8774 mdash 9085 mdash 86796 0 I 2456 mdash 2571 mdash 24306 1 I 3978 4089 mdash 3979 mdash4 1 I 4388 4357 mdash 4388 mdash8 0 I 4654 mdash 4920 mdash 46028 1 I 5575 5905 mdash 5576 mdash2 1 I 7202 7101 mdash 7202 mdash2 0 I (119908 = 0) 7322 7390 mdash 7319 mdash10 0 I 7419 mdash 7933 mdash 73476 2 I 8110 8261 mdash 8111 mdash10 1 I 8189 8811 mdash 8191 mdash8 2 I 8783 9173 mdash 8785 mdash4 2 I 9016 9084 mdash 9017 mdash12 0 I 1071 mdash 1159 mdash 106110 2 I 1081 1155 mdash 1081 mdash2 2 I 1089 1105 mdash 1089 mdash12 1 I 1142 1245 mdash 1143 mdash4 3 I 1263 1286 mdash 1263 mdash

same meaning of the columns seen in the previous casesIn order to achieve the cylindrical bending conditions thenumerical models have the straight edges simply supportedand the curved edges as free Due to the CB definitionit is not possible to achieve CB conditions when 119898 = 0

and 119899 = 1 2 3 because the cylindrical bending shouldoccur on the curved edge and this is not physically possiblebecause of the variation of displacement components due tothe curvature presence This aspect is indicated as NA (NotApplicable) solution in Table 4 Therefore this kind of 3D

Shock and Vibration 15

Table 4 Sandwich cylindrical shell panel embedding FGM core with 119901 = 10 Analytical 3D versus 2D numerical models Frequency 119891 inHz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1001 0 I 1948 mdash 1951 mdash 19412 0 I 8856 mdash 8872 mdash 88232 1 I 1104 1110 mdash 1104 mdash1 1 I 1850 1873 mdash 1850 mdash3 0 I 2038 mdash 2042 mdash 20303 1 I 2116 2128 mdash 2116 mdash2 2 I 2221 2242 mdash 2221 mdash3 2 I 2484 2496 mdash 2484 mdash3 3 I 3235 3254 mdash 3235 mdash4 0 I 3649 mdash 3660 mdash 36342 3 I 3692 3739 mdash 3692 mdash4 1 I 3714 3745 mdash 3714 mdash4 2 I 3940 3965 mdash 3941 mdash3 4 I 4226 4261 mdash 4226 mdash4 3 I 4378 4403 mdash 4378 mdash1 2 I 4568 4664 mdash 4568 mdash4 4 I 5035 5064 mdash 5035 mdash2 4 I 5040 5128 mdash 5040 mdash119877120572ℎ = 101 0 I 1905 mdash 1963 mdash 18841 1 I 3054 3145 mdash 3055 mdash1 2 I 6209 6409 mdash 6209 mdash2 0 I 8326 mdash 8857 mdash 82540 1 I 8710 mdash NA mdash NA0 1 II (119908 = 0) 8820 8956 mdash 8820 mdash2 1 I 8880 9531 mdash 8882 mdash0 2 I 9176 mdash NA mdash NA1 3 I 9576 9983 mdash 9567 mdash0 3 I 1036 mdash NA mdash NA2 2 I 1066 1147 mdash 1067 mdash0 4 I 1291 mdash NA mdash NA1 4 I 1335 1413 mdash 1335 mdash2 3 I 1364 1475 mdash 1364 mdash1 0 II (119908 = 0) 1665 1738 mdash 1665 mdash0 5 I 1683 mdash NA mdash NA2 4 I 1755 1918 mdash 1755 mdash0 2 II (119908 = 0) 1764 1791 mdash 1764 mdash

exact solutions cannot be achieved by any numerical modelOnce again solution with 119899 = 0 or 119898 = 0 combined with119908 = 0 (in-plane modes) can be found with 2D numericalmodels without modifying the simply supported boundaryconditions

5 Conclusions

Thepresent paper analyzes the cylindrical bending in the freefrequency analysis of functionally graded material (FGM)plates and cylindrical shells This condition is investigatedby means of a 3D exact shell model and two different 2D

numerical models a weak formulation based on classicalFinite Elements (FEs) and a strong formulation based onthe GDQ method The cylindrical bending (CB) behavior inthe 3D exact shell model can be analyzed using completelysimply supported structures and supposing one of the twohalf-wave numbers equals zeroThe numerical models obtainthe CB behavior using mixed simply supported and freeboundary conditions It has been demonstrated that theactual cylindrical bending (CB) is possible only with respectto straight edgesTheCBnatural frequencies of the numericalsolutions converge to the 3D exact CB ones if the length ofstraight simply supported edges is increased up to 20 times

16 Shock and Vibration

the value of the original edges The CB conditions are notpossible in 2D numericalmodels when the half-wave numberis zero in the curvilinear edge in this case the modificationof boundary conditions does not properly work In-planevibrationmodeswith zero transverse displacement combinedwith 119898 = 0 or 119899 = 0 cannot be defined as CB frequencies andthey can be calculated via 2D numerical models without themodification of the simply supported conditions for the fouredges All the considerations about the cylindrical bendingare valid for each geometry (plate cylinder and cylindricalshell) lamination scheme FGM law and thickness ratio

Competing Interests

The authors declare that they have no competing interests

References

[1] W Q Chen J B Cai and G R Ye ldquoResponses of cross-ply laminates with viscous interfaces in cylindrical bendingrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 823ndash835 2005

[2] W Q Chen and K Y Lee ldquoThree-dimensional exact analysisof angle-ply laminates in cylindrical bending with interfacialdamage via state-space methodrdquo Composite Structures vol 64no 3-4 pp 275ndash283 2004

[3] W Q Chen and K Y Lee ldquoTime-dependent behaviors of angle-ply laminates with viscous interfaces in cylindrical bendingrdquoEuropean Journal ofMechanicsmdashASolids vol 23 no 2 pp 235ndash245 2004

[4] A H Gandhi and H K Raval ldquoAnalytical and empiricalmodeling of top roller position for three-roller cylindricalbending of plates and its experimental verificationrdquo Journal ofMaterials Processing Technology vol 197 no 1ndash3 pp 268ndash2782008

[5] S Oral and H Darendeliler ldquoThe optimum die profile for thecylindrical bending of platesrdquo Journal of Materials ProcessingTechnology vol 70 no 1ndash3 pp 151ndash155 1997

[6] V A Jairazbhoy P Petukhov and J Qu ldquoLarge deflectionof thin plates in cylindrical bendingmdashnon-unique solutionsrdquoInternational Journal of Solids and Structures vol 45 no 11-12pp 3203ndash3218 2008

[7] A Lebee and K Sab ldquoA bending-gradient model for thickplates part II closed-form solutions for cylindrical bending oflaminatesrdquo International Journal of Solids and Structures vol 48no 20 pp 2889ndash2901 2011

[8] P V Nimbolkar and I M Jain ldquoCylindrical bending of elasticplatesrdquo Procedia Materials Science vol 10 pp 793ndash802 2015

[9] A S Sayyad and Y M Ghugal ldquoA nth-order shear deformationtheory for composite laminates in cylindrical bendingrdquo Curvedand Layered Structures vol 2 no 1 pp 290ndash300 2015

[10] N J Pagano ldquoExact solutions for composite laminates incylindrical bendingrdquo Journal of Composite Materials vol 3 no3 pp 398ndash411 1969

[11] N J Pagano and A S D Wang ldquoFurther study of compositelaminates under cylindrical bendingrdquo Journal of CompositeMaterials vol 5 pp 421ndash428 1971

[12] N Saeedi K Sab and J-F Caron ldquoCylindrical bending ofmultilayered plates with multi-delamination via a layerwisestress approachrdquo Composite Structures vol 95 pp 728ndash7392013

[13] X-P Shu and K P Soldatos ldquoCylindrical bending of angle-ply laminates subjected to different sets of edge boundaryconditionsrdquo International Journal of Solids and Structures vol37 no 31 pp 4289ndash4307 2000

[14] Z G Bian W Q Chen C W Lim and N Zhang ldquoAnalyticalsolutions for single- and multi-span functionally graded platesin cylindrical bendingrdquo International Journal of Solids andStructures vol 42 no 24-25 pp 6433ndash6456 2005

[15] H M Navazi and H Haddadpour ldquoNonlinear cylindricalbending analysis of shear deformable functionally graded platesunder different loadings using analytical methodsrdquo Interna-tional Journal of Mechanical Sciences vol 50 no 12 pp 1650ndash1657 2008

[16] W Q Chen J Ying J B Cai and G R Ye ldquoBenchmarksolution of imperfect angle-ply laminated rectangular plates incylindrical bending with surface piezoelectric layers as actuatorand sensorrdquo Computers and Structures vol 82 no 22 pp 1773ndash1784 2004

[17] W Q Chen and K Y Lee ldquoExact solution of angle-plypiezoelectric laminates in cylindrical bending with interfacialimperfectionsrdquo Composite Structures vol 65 no 3-4 pp 329ndash337 2004

[18] T Kant and SM Shiyekar ldquoCylindrical bending of piezoelectriclaminates with a higher order shear and normal deformationtheoryrdquo Computers and Structures vol 86 no 15-16 pp 1594ndash1603 2008

[19] P Lu H P Lee and C Lu ldquoAn exact solution for functionallygraded piezoelectric laminates in cylindrical bendingrdquo Interna-tional Journal of Mechanical Sciences vol 47 no 3 pp 437ndash4382005

[20] W Yan J Wang and W Q Chen ldquoCylindrical bendingresponses of angle-ply piezoelectric laminates with viscoelasticinterfacesrdquo Applied Mathematical Modelling vol 38 no 24 pp6018ndash6030 2014

[21] Y Y Zhou W Q Chen and C F Lu ldquoSemi-analytical solutionfor orthotropic piezoelectric laminates in cylindrical bendingwith interfacial imperfectionsrdquo Composite Structures vol 92no 4 pp 1009ndash1018 2010

[22] G P Dube S Kapuria and P C Dumir ldquoExact piezother-moelastic solution of a simply-supported orthotropic flat panelin cylindrical bendingrdquo International Journal of MechanicalSciences vol 38 pp 374ndash387 1996

[23] V N Pilipchuk V L Berdichevsky and R A IbrahimldquoThermo-mechanical coupling in cylindrical bending of sand-wich platesrdquoComposite Structures vol 92 no 11 pp 2632ndash26402010

[24] C Zhang S Di and N Zhang ldquoA new procedure for staticanalysis of thermo-electric laminated composite plates undercylindrical bendingrdquo Composite Structures vol 56 no 2 pp131ndash140 2002

[25] W Q Chen and K Y Lee ldquoState-space approach for statics anddynamics of angle-ply laminated cylindrical panels in cylindri-cal bendingrdquo International Journal of Mechanical Sciences vol47 no 3 pp 374ndash387 2005

[26] C-P Wu and Y-S Syu ldquoExact solutions of functionally gradedpiezoelectric shells under cylindrical bendingrdquo InternationalJournal of Solids and Structures vol 44 no 20 pp 6450ndash64722007

[27] W Yan J Ying and W Q Chen ldquoThe behavior of angle-ply laminated cylindrical shells with viscoelastic interfaces incylindrical bendingrdquo Composite Structures vol 78 no 4 pp551ndash559 2007

Shock and Vibration 17

[28] S Brischetto F Tornabene N Fantuzzi and E Viola ldquo3Dexact and 2D generalized differential quadraturemodels for freevibration analysis of functionally graded plates and cylindersrdquoMeccanica vol 51 no 9 pp 2059ndash2098 2016

[29] N Fantuzzi S Brischetto F Tornabene and E Viola ldquo2D and3D shell models for the free vibration investigation of func-tionally graded cylindrical and spherical panelsrdquo CompositeStructures vol 154 pp 573ndash590 2016

[30] S Brischetto and R Torre ldquoExact 3D solutions and finiteelement 2D models for free vibration analysis of plates andcylindersrdquo Curved and Layered Structures vol 1 no 1 pp 59ndash92 2014

[31] F Tornabene S Brischetto N Fantuzzi and E Viola ldquoNumer-ical and exact models for free vibration analysis of cylindricaland spherical shell panelsrdquo Composites Part B Engineering vol81 pp 231ndash250 2015

[32] S Brischetto ldquoExact elasticity solution for natural frequenciesof functionally graded simply-supported structuresrdquo ComputerModeling in Engineering amp Sciences vol 95 no 5 pp 391ndash4302013

[33] S Brischetto ldquoThree-dimensional exact free vibration analysisof spherical cylindrical and flat one-layered panelsrdquo Shock andVibration vol 2014 Article ID 479738 29 pages 2014

[34] S Brischetto ldquoA continuum elastic three-dimensional modelfor natural frequencies of single-walled carbon nanotubesrdquoComposites Part B Engineering vol 61 pp 222ndash228 2014

[35] S Brischetto ldquoAn exact 3d solution for free vibrations ofmultilayered cross-ply composite and sandwich plates andshellsrdquo International Journal of Applied Mechanics vol 6 no 6Article ID 1450076 42 pages 2014

[36] S Brischetto ldquoA continuumshellmodel including vanderWaalsinteraction for free vibrations of double-walled carbon nan-otubesrdquoCMES-ComputerModeling in Engineering and Sciencesvol 104 no 4 pp 305ndash327 2015

[37] S Brischetto F Tornabene N Fantuzzi and M BacciocchildquoRefined 2D and exact 3D shell models for the free vibrationanalysis of single- and double-walled carbon nanotubesrdquo Tech-nologies vol 3 no 4 pp 259ndash284 2015

[38] S Brischetto ldquoConvergence analysis of the exponential matrixmethod for the solution of 3D equilibrium equations for freevibration analysis of plates and shellsrdquo Composites Part BEngineering vol 98 pp 453ndash471 2016

[39] S Brischetto ldquoExact and approximate shell geometry in the freevibration analysis of one-layered and multilayered structuresrdquoInternational Journal of Mechanical Sciences vol 113 pp 81ndash932016

[40] S Brischetto ldquoCurvature approximation effects in the freevibration analysis of functionally graded shellsrdquo InternationalJournal of Applied Mechanics 2016

[41] F Tornabene N Fantuzzi F Ubertini and E Viola ldquoStrong for-mulation finite element method based on differential quadra-ture a surveyrdquo Applied Mechanics Reviews vol 67 no 2 ArticleID 020801 pp 1ndash55 2015

[42] F Tornabene E Viola and N Fantuzzi ldquoGeneral higher-orderequivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panelsrdquo CompositeStructures vol 104 pp 94ndash117 2013

[43] N Fantuzzi F Tornabene E Viola and A J Ferreira ldquoAstrong formulation finite element method (SFEM) based onRBF and GDQ techniques for the static and dynamic analysesof laminated plates of arbitrary shaperdquo Meccanica vol 49 no10 pp 2503ndash2542 2014

[44] F Tornabene N Fantuzzi and M Bacciocchi ldquoFree vibrationsof free-form doubly-curved shells made of functionally gradedmaterials using higher-order equivalent single layer theoriesrdquoComposites Part B Engineering vol 67 pp 490ndash509 2014

[45] F Tornabene and E Viola ldquoStatic analysis of functionallygraded doubly-curved shells and panels of revolutionrdquo Mecca-nica vol 48 no 4 pp 901ndash930 2013

[46] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoDynamic analysis of thick and thin elliptic shell structuresmade of laminated composite materialsrdquo Composite Structuresvol 133 pp 278ndash299 2015

[47] F TornabeneN FantuzziM Bacciocchi and EViola ldquoHigher-order theories for the free vibrations of doubly-curved lam-inated panels with curvilinear reinforcing fibers by meansof a local version of the GDQ methodrdquo Composites Part BEngineering vol 81 pp 196ndash230 2015

[48] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoFree vibrations of composite oval and elliptic cylinders bythe generalized differential quadrature methodrdquo Thin-WalledStructures vol 97 pp 114ndash129 2015

[49] F Tornabene and J N Reddy ldquoFGM and laminated doubly-curved and degenerate shells resting on nonlinear elastic foun-dations a GDQ solution for static analysis with a posterioristress and strain recoveryrdquo Journal of the Indian Institute ofScience vol 93 no 4 pp 635ndash688 2013

[50] N Fantuzzi F Tornabene and E Viola ldquoFour-parameter func-tionally graded cracked plates of arbitrary shape a GDQFEMsolution for free vibrationsrdquo Mechanics of Advanced Materialsand Structures vol 23 no 1 pp 89ndash107 2016

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Shock and Vibration

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DistributedSensor Networks

International Journal of

Shock and Vibration 3

x

y

z

a

b

(a)

z

b

R

a = 2R

(b)z

R

a

b

(c)

z

R

R

a ba b

(d)

Figure 1 Geometries of plates and shells with constant radii of curvature and their local reference system

2 Exact Solution of a 3D Shell Model

The three-dimensional exact shell model employed for thecomparisons proposed in Section 4 has been elaborated in[32ndash40] for the free frequency analyses of single-layered andmultilayered isotropic orthotropic composite FGM andsandwich plates and shells with constant radii of curvatureand for the free vibrations of single- and double-walledcarbon nanotubes including van der Waals interactions Theequilibrium equations proposed in the general orthogonalcurvilinear coordinate system (120572 120573 119911) are valid for bothplates and shells with constant radii of curvature For thesake of clarity the plates and shells geometries within theirlocal coordinate systems are depicted in Figure 1 The exactform of these equations is obtained using simply supportedboundary conditions while the differential equations in 119911are solved by means of the exponential matrix method(also known as transfer matrix method or the state-spaceapproach)

Considering spherical shells with constant radii of cur-vature 119877120572 and 119877120573 and embedding 119873119871 classical or FGM layers

the differential equations of equilibrium for the free vibrationanalysis written for a generic 119896 layer are

119867120573 120597120590119896120572120572120597120572 + 119867120572 120597120590119896120572120573120597120573 + 119867120572119867120573 120597120590119896120572119911120597119911+ (2119867120573119877120572 + 119867120572119877120573 ) 120590119896120572119911 = 120588119896119867120572119867120573119896

119867120573 120597120590119896120572120573120597120572 + 119867120572 120597120590119896120573120573120597120573 + 119867120572119867120573 120597120590119896120573119911120597119911+ (2119867120572119877120573 + 119867120573119877120572 ) 120590119896120573119911 = 120588119896119867120572119867120573V119896

119867120573 120597120590119896120572119911120597120572 + 119867120572 120597120590119896120573119911120597120573 + 119867120572119867120573 120597120590119896119911119911120597119911 minus 119867120573119877120572 120590119896120572120572 minus 119867120572119877120573 120590119896120573120573+ (119867120573119877120572 + 119867120572119877120573 ) 120590119896119911119911 = 120588119896119867120572119867120573119896

(1)

4 Shock and Vibration

It is remarked that this set of equations is valid onlyfor spherical shells wherein 119877120572 = 119877120573 = 119877 because fordoubly curved shells with variable radii of curvature suchequations take a different form The authors decided to keepsuch system with the two radii of curvature distinct becauseit will be much easier to understand the cylindrical bendingeffect for plates and shells starting from a general point ofview

In the case of shell geometries andor FGM layers eachphysical layer 119896 must be divided in an opportune numberof 119902 mathematical layers to approximate the shell curvatureandor the FGM law through the thickness direction 119911 Theindex 119895 indicates the total mathematical layers obtained fromthe subdivision of each 119896 layer in 119902 mathematical layers (119895 =119896 times 119902) The parametric coefficients for shells with constantradii of curvature can be written as

119867120572 = (1 + 119911119877120572) 119867120573 = (1 + 119911119877120573) 119867119911 = 1

(2)

A closed form solution for simply supported shells andplates can be developed by writing harmonic displacementsas

119906119895 (120572 120573 119911 119905) = 119880119895 (119911) 119890119894120596119905 cos (120572120572) sin (120573120573) V119895 (120572 120573 119911 119905) = 119881119895 (119911) 119890119894120596119905 sin (120572120572) cos (120573120573) 119908119895 (120572 120573 119911 119905) = 119882119895 (119911) 119890119894120596119905 sin (120572120572) sin (120573120573)

(3)

where 119880119895(119911) 119881119895(119911) and 119882119895(119911) indicate the displacementamplitudes in 120572 120573 and 119911 directions respectively 119894 indicatesthe imaginary unit 120596 = 2120587119891 is the circular frequencywhere 119891 is the frequency value in Hz 119905 is the symbol forthe time In the coefficients 120572 = 119898120587119886 and 120573 = 119899120587119887119898 and 119899 are the half-wave numbers and 119886 and 119887 are theshell dimensions in 120572 and 120573 directions respectively (all thesequantities are evaluated on the midsurface Ω0) 120588119895 is themass density (120590119895120572120572 120590119895120573120573 120590119895119911119911 120590119895120573119911 120590119895120572119911 120590119895120572120573) are the six stresscomponents and 119895 V119895 and 119895 are the second temporalderivative of the three displacement components 119877120572 and119877120573 indicate the radii of curvature defined with respect thereference surface Ω0 of the whole multilayered plate or shell119867120572 and 119867120573 proposed in (2) vary with continuity through thethickness direction of the multilayered structure Thereforethey depend on the thickness coordinate 119911 The referencesurface Ω0 of the structure is defined as the locus of pointswhich are positioned midway between the external surfacesDisplacement components are indicated as 119906 V and 119908 in 120572120573 and 119911 directions respectively

The final system in closed form is developed substituting(3) and constitutive and geometrical equations (which are not

here given for the sake of brevity but they can be found indetail in [32]) in (1)

(minus119862119895551198671198951205731198671198951205721198772120572 minus 11986211989555119877120572119877120573 minus 120572211986211989511119867119895120573119867119895120572 minus 120573211986211989566119867119895120572119867119895120573+ 1205881198951198671198951205721198671198951205731205962) 119880119895 + (minus12057212057311986211989512 minus 12057212057311986211989566) 119881119895

+ (12057211986211989511119867119895120573119867119895120572119877120572 + 12057211986211989512119877120573 + 12057211986211989555119867119895120573119867119895120572119877120572 + 12057211986211989555119877120573 ) 119882119895

+ (11986211989555119867119895120573119877120572 + 11986211989555119867119895120572119877120573 ) 119880119895119911 + (12057211986211989513119867119895120573 + 12057211986211989555119867119895120573)sdot 119882119895119911 + (11986211989555119867119895120572119867119895120573) 119880119895119911119911 = 0

(minus12057212057311986211989566 minus 12057212057311986211989512) 119880119895 + (minus119862119895441198671198951205721198671198951205731198772120573 minus 11986211989544119877120572119877120573minus 120572211986211989566119867119895120573119867119895120572 minus 120573211986211989522119867119895120572119867119895120573 + 1205881198951198671198951205721198671198951205731205962) 119881119895

+ (12057311986211989544119867119895120572119867119895120573119877120573 + 12057311986211989544119877120572 + 12057311986211989522119867119895120572119867119895120573119877120573 + 12057311986211989512119877120572 ) 119882119895

+ (11986211989544119867119895120572119877120573 + 11986211989544119867119895120573119877120572 ) 119881119895119911 + (12057311986211989544119867119895120572 + 12057311986211989523119867119895120572)sdot 119882119895119911 + (11986211989544119867119895120572119867119895120573) 119881119895119911119911 = 0

(12057211986211989555119867119895120573119867119895120572119877120572 minus 12057211986211989513119877120573 + 12057211986211989511119867119895120573119867119895120572119877120572 + 12057211986211989512119877120573 ) 119880119895

+ (12057311986211989544119867119895120572119867119895120573119877120573 minus 12057311986211989523119877120572 + 12057311986211989522119867119895120572119867119895120573119877120573 + 12057311986211989512119877120572 ) 119881119895

+ ( 11986211989513119877120572119877120573 + 11986211989523119877120572119877120573 minus 119862119895111198671198951205731198671198951205721198772120572 minus 211986211989512119877120572119877120573 minus 119862119895221198671198951205721198671198951205731198772120573minus 120572211986211989555119867119895120573119867119895120572 minus 120573211986211989544119867119895120572119867119895120573 + 1205881198951198671198951205721198671198951205731205962) 119882119895

+ (minus12057211986211989555119867119895120573 minus 12057211986211989513119867119895120573) 119880119895119911 + (minus12057311986211989544119867119895120572minus 12057311986211989523119867119895120572) 119881119895119911 + (11986211989533119867119895120573119877120572 + 11986211989533119867119895120572119877120573 ) 119882119895119911+ (11986211989533119867119895120572119867119895120573) 119882119895119911119911 = 0

(4)

Shock and Vibration 5

The cylindrical bending requirements for analytical methodsare obtained choosing one of the two half-wave numberswhich equals zero This condition means 120572 = 119898120587119886 =0 or 120573 = 119899120587119887 = 0 which allows the derivatives madein a particular direction (120572 or 120573) which equals zero tooThis mathematical condition means that each section in thecylindrical bending direction has always the same behavior

It is obvious that the cylindrical bending effect is properfor cylindrical structures only However the authors startedthe present treatise from the governing equations (1) ofspherical shell structures because one of the aims of thepresent contribution is to underline which are the extraterms (related to the curvature) that do not allow having thestructure to be in cylindrical bending conditions

The compact form of (4) has been elaborated in detail in[32] The final version is here proposed

D119895 120597U119895120597119911 = A119895U119895 (5)

where 120597U119895120597119911 = U1198951015840 andU119895 = [119880119895 119881119895 119882119895 1198801198951015840 1198811198951015840 1198821198951015840]Equation (5) is rewritten as

D119895U1198951015840 = A119895U119895 (6)

U1198951015840 = D119895minus1A119895U119895 (7)

U1198951015840 = A119895lowastU119895 (8)

with A119895lowast = D119895minus1A119895The exponential matrix method (also known as transfer

matrix method or the state-space approach) is employed forthe solution of (8) (details are given in [32])

U119895 (119911119895) = exp (A119895lowast119911119895)U119895 (0) with 119911119895 isin [0 ℎ119895] (9)

where 119911119895 is the thickness coordinate for the 119895 mathematicallayer with values from 0 at the bottom to ℎ119895 at the top Theexponential matrix is calculated using 119911119895 = ℎ119895 for each 119895mathematical layer

A119895lowastlowast = exp (A119895lowastℎ119895)= I + A119895lowastℎ119895 + A119895lowast22 ℎ1198952 + A119895lowast33 ℎ1198953 + sdot sdot sdot

+ A119895lowast119873119873 ℎ119895119873(10)

where I indicates the 6times6 identitymatrixThe expansion usedfor the exponentialmatrix in (10) has a fast convergence and itis not time consuming from the computational point of view119895 = 119872 mathematical layers are used to approximate theshell curvature andor the FGM law through the thickness119872 minus 1 transfer matrices must be defined by means ofthe interlaminar continuity conditions of displacements andtransverse shearnormal stresses at each interfaceThe closedform solution for plates or shells is obtained using simply

supported boundary conditions and free stress conditions atthe top and at the bottom surfaces The above conditionsallow the following final system

EU1 (0) = 0 (11)

and the 6 times 6 matrix E does not change independently of thenumber of layers 119872 even if a layer-wise approach is appliedU1(0) is the vector U defined at the bottom of the wholemultilayered shell or plate (first layer 1 with 1199111 = 0) All thesteps and details here omitted for the sake of brevity can befound in [32ndash40] where the proposed exact 3D shell modelhas been elaborated and tested for the first time for severalapplications Several validations for the 3D exact shell modelhave been shown in [32ndash40] For this reason the model isnow used with confidence for the results and comparisonsproposed in Section 4

The nontrivial solution ofU1(0) in (11) allows conductingthe free vibration analysis imposing the zero determinant ofmatrix E

det [E] = 0 (12)

Equation (12) is used to find the roots of an higher order poly-nomial written in 120582 = 1205962The imposition of a couple of (119898 119899)allows a certain number of circular frequencies 120596 = 2120587119891(from I to infin) to be calculated This number of frequenciesdepends on the order 119873 used in each exponential matrixA119895lowastlowast and the number 119872 of mathematical layers employedfor the shell curvature andor the FGM law approximationsThe results shown in [32ndash40] confirm that 119873 = 3 for theexponential matrix and 119872 = 100 for the mathematicallayers are appropriate to always calculate the correct freefrequencies for each structure lamination sequence numberof layers material FGM law and thickness value

3 2D Finite Element Analysis and 2DGeneralized Differential Quadrature Models

The 3D exact natural frequencies are compared with twonumerical 2D solutions given by a weak and a strongformulation The former is a classic 2D finite element (FE)model carried out through the commercial software Strand7The latter is a recent widespread technique known as Gener-alized Differential Quadrature (GDQ) method [41] Detailsrelated to convergence and accuracy of these techniquescan be found in the works [29ndash31] It is noted that theFE solution employs a classic linear theory known alsoas Reissner-Mindlin theory with a linear rotation of thefibers through the shell thickness The present GDQ solutionexploits higher order theories with a general expansionthrough the thickness of the structureThis general approachfalls within the framework of the unified theories becauseseveral structural models can be obtained via the repetitionof a fundamental nucleus Another difference between theGDQ applications and the FE ones is that the formerincludes the curvature effects and the rotary inertias in itsformulation

6 Shock and Vibration

The 2D kinematic hypothesis [42ndash50] considers the fol-lowing expansion of the degrees of freedom

U = 119873119888+1sum120591=0

F120591u(120591) (13)

where U represents the vector of the 3D displacementcomponents which is function of (120572 120573 119911) and u120591 is thevector of the displacement parameters function of (120572 120573)which depends on the free index 120591 Therefore u120591 lays on themidsurface of the shell [42] The matrix F120591(119894119895) = 120575119894119895119865120591 (for119894 119895 = 1 2 3) contains the so-called thickness functions where120575 is the Kronecker delta function The thickness functionsdescribe how the displacement parameters characterize thesolution through the thickness They can have any form ofany smooth function of 119911 however the classic form of thesefunctions is the power function Thus 119865120591 = 119911120591 can be 1(120591 = 0) 119911 (120591 = 1) 1199112 (120591 = 2) 1199113 (120591 = 3) and so onPlease note that according to the mathematical backgroundof the present Unified Formulation the index 119873119888 + 1 isreserved to the (optional) zigzag effect Since the aim of thepresent paper is to investigate the cylindrical bending effectfor functionally graded plates and cylindrical panels thefollowing numerical applications will consider a fixed valueof 119873119888 = 3 so that 120591 = 0 1 2 3 This choice is justified by thefact that third order theories are sufficient with a reasonablylimited number of degrees of freedom to obtain accuratesolutions when compared to 3D shell models [42ndash50] Formultilayered structures such as sandwich configurations thezigzag function must be added into the formulation Onceagain as shown in (13) the zigzag function is embedded atthe end of the expansion with the exponent 119873119888 + 1 Whensuch effect is neglected the sum of (13) stops at 119873119888

The same theory used for the FE solution can be carriedout in the GDQmethod setting 120591 = 0 1 and 119873119888 = 1 howeverit has been shown in other recent works that the strong formapproaches led tomore accurate results with respect to the FEones [46 48]

The relation between generalized strains 120576(120591) anddisplace-ments u(120591) can be set as

120576(120591) = DΩu

(120591) for 120591 = 0 1 2 119873119888 119873119888 + 1 (14)

where DΩ is a differential operator explicitly defined in [42]The present constitutive material model is linear and elastictherefore for a general 119896th laminaHookersquos law can be applied

120590(119896) = C(119896)120576(119896) (15)

where 120590(119896) is the vector containing the stress componentsand C(119896) represents the constitutive matrix for each 119896th ply[42] of the structure Each component of the constitutivematrix is indicated by the symbol 119862(119896)120585120578 and it is valid in thecurvilinear reference system 1198741015840120572120573119911 of the reference geome-tryThese coefficients are evaluatedwith the application of thetransformation matrix [42]

The stress resultants are the terms which correspond tothe punctual 3D stresses of the 3D doubly curved solidTheir definition can be achieved by developing the Hamiltonprinciple for the present structural theory [42]

S(120591) = 119873119871sum119896=1

int119911119896+1119911119896

(Z(120591))119879 120590(120591)119867120572119867120573119889119911for 120591 = 0 1 2 119873119888 119873119888 + 1

(16)

whereZ(120591) is a 6times9matrix containing the thickness functions119865120591 and the curvature coefficients119867120572 and119867120573 the definition ofsuchmatrix with some other details can be found in [42] S(120591)include the stress resultants as a function of the order 120591

The definition of (16) can be rewritten as

S(120591) = 119873119888+1sum119904=0

A(120591119904)120576(119904) for 120591 = 0 1 2 119873119888 119873119888 + 1 (17)

which is a function of generalized strains of order 119904 120576(119904)Note that the so-called elastic coefficients are defined as

A(120591119904) = 119873119871sum119896=1

int119911119896+1119911119896

(Z(120591))119879C(119896)Z(119904)119867120572119867120573119889119911 (18)

and they can be carried out by the following integral expres-sions

119860(120591119904)120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119911119904119911120591119867120572119867120573119867984858120572119867120577120573 119889119911

119860(119904)120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119911119904120591119911120591minus1119867120572119867120573119867984858120572119867120577120573 119889119911

119860(120591)120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119904119911119904minus1119911120591119867120572119867120573119867984858120572119867120577120573 119889119911

Shock and Vibration 7

119860( )120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119904119911119904minus1120591119911120591minus1119867120572119867120573119867984858120572119867120577120573 119889119911for 120591 119904 = 0 1 2 119873119888 119873119888 + 1 120585 120578 = 1 2 3 4 5 6 984858 120577 = 0 1 2

(19)

It is recalled that the exponent 119873119888 + 1 is reserved for theMurakami function otherwise this order is neglected Thesubscripts and indicate the derivatives of the thicknessfunctions 119865120591 and 119865119904 with respect to 119911 The subscripts 984858 and 120577are the exponents of the quantities119867120572 and 119867120573 whereas 120585 and120578 are the indices of the material constants 119862(119896)120585120578 for the 119896th ply

[42ndash49]Note that119862(119896)120585120578 119867120572 and119867120573 are quantities that dependon the thickness 119911 Therefore the integrals of (19) cannotbe computed analytically The present solution for solvingsuch integrals is to employ a numerical integration algorithmIn particular the so-called Generalized Integral Quadrature(GIQ) method is utilized For an accurate evaluation of theintegrals 51 points per layer are considered in the integration

The Hamilton principle supplies for the dynamic equilib-rium equations and the correspondent boundary conditionsIn detail the fundamental nucleus of three motion equationstakes the form

D⋆ΩS(120591) = 119873119888+1sum

119904=0

M(120591119904)u(119904)

for 120591 = 0 1 2 119873119888 119873119888 + 1(20)

whereD⋆Ω is the equilibrium operator andM(120591119904) is the inertialmatrix Their explicit expressions have been presented in aprevious authorsrsquo paper [42]

Combining the kinematic equation (14) constitutiveequation (17) and themotion equation (20) the fundamentalgoverning system in terms of displacement parameters iscarried out

119873119888+1sum119904=0

L(120591119904)u(119904) = 119873119888+1sum119904=0

M(120591119904)u(119904)

for 120591 = 0 1 2 119873119888 119873119888 + 1(21)

where L(120591119904) = D⋆ΩA(120591119904)DΩ is the fundamental operator [42]

Boundary conditions must be included for the solutionof the differential equations (21) The present paper dealsonly with free and simply supported boundary conditionsThe former is used to reproduce the cylindrical bendingconditions considering alternatively one free edge and onesimply supported edge When the four edges are all simplysupported the problem is the same as the ones investigated

in the previous papers [29ndash31] Simply supported conditionsare

119873(120591)120572 = 0119906(120591)120573 = 119906(120591)119911 = 0

120591 = 0 1 2 119873119888 119873119888 + 1 at 120572 = 1205720 or 120572 = 1205721 1205730 le 120573 le 1205731119906(120591)120572 = 0

119873(120591)120573 = 0119906(120591)119911 = 0

120591 = 0 1 2 119873119888 119873119888 + 1 at 120573 = 1205730 or 120573 = 1205731 1205720 le 120572 le 1205721

(22)

Free edge conditions are

119873(120591)120572 = 0119873(120591)120573 = 0119879(120591)120572 = 0120591 = 0 1 2 119873119888 119873119888 + 1 at 120572 = 1205720 or 120572 = 1205721 1205730 le 120573 le 1205731119873(120591)120572 = 0119873(120591)120573 = 0119879(120591)120573 = 0120591 = 0 1 2 119873119888 119873119888 + 1 at 120573 = 1205730 or 120573 = 1205731 1205720 le 120572 le 1205721

(23)

4 Results

The present section includes some numerical applicationsfor the comparison of the 3D exact shell vibrations withtwo numerical 2D models In particular classical simplysupported solutions and cylindrical bending (CB) behaviorof plates and shells are presentedThe following computationsfocus on plates circular cylinders and cylindrical panels dueto the fact that the CB effect occurs only when at least twoparallel straight edges are present in the structure

The first case is a square plate (119886 = 119887 = 1m) withsimply supported edges Two thickness ratios are analyzedthin 119886ℎ = 100 and moderately thick 119886ℎ = 20 structure Theplate is made of a single functionally graded layer (ℎ1 = ℎ)with volume fraction of ceramic (119888) phase

119881119888 (119911) = (05 + 119911ℎ)119901 with 119911 isin [minusℎ2 ℎ2 ] (24)

8 Shock and Vibration

Table 1 One-layered FGM plate with 119901 = 05 Analytical 3D versus 2D numerical models Frequency 119891 in Hz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119886ℎ = 1001 0 I 4019 mdash 3907 mdash 39210 1 I 4019 mdash 3907 mdash 39211 1 I 8037 8006 mdash 8036 mdash2 0 I 1607 mdash 1580 mdash 15840 2 I 1607 mdash 1580 mdash 15842 1 I 2008 2001 mdash 2008 mdash1 2 I 2008 2001 mdash 2008 mdash2 2 I 3212 3202 mdash 3211 mdash3 0 I 3613 mdash 3568 mdash 35770 3 I 3613 mdash 3568 mdash 35773 1 I 4014 4002 mdash 4013 mdash1 3 I 4014 4002 mdash 4013 mdash3 2 I 5215 5202 mdash 5215 mdash2 3 I 5215 5202 mdash 5215 mdash4 0 I 6416 mdash 6354 mdash 63650 4 I 6416 mdash 6354 mdash 63654 1 I 6816 6801 mdash 6815 mdash1 4 I 6816 6801 mdash 6815 mdash119886ℎ = 201 0 I 2002 mdash 1951 mdash 19510 1 I 2002 mdash 1951 mdash 19511 1 I 3989 3995 mdash 3988 mdash2 0 I 7916 mdash 7864 mdash 77970 2 I 7916 mdash 7864 mdash 77972 1 I 9858 9954 mdash 9857 mdash1 2 I 9858 9954 mdash 9857 mdash2 2 I 1560 1588 mdash 1560 mdash3 0 I 1749 mdash 1767 mdash 17290 3 I 1749 mdash 1767 mdash 17293 1 I 1936 1980 mdash 1936 mdash1 3 I 1936 1980 mdash 1936 mdash3 2 I 2490 2566 mdash 2490 mdash2 3 I 2490 2566 mdash 2490 mdash1 0 II (119908 = 0) 2775 2773 mdash 2775 mdash0 1 II (119908 = 0) 2775 2773 mdash 2775 mdash4 0 I 3034 mdash 3124 mdash 30070 4 I 3034 mdash 3124 mdash 3007

where 119901 is the power exponent which varies the volumefraction of the ceramic component through the thicknessTwo limit cases occur for (24) 119881119888 = 1 when 119901 = 0 and119881119888 rarr 0when119901 = infin In the following computations it is set as119901 = 05 Note that the volume fraction function in (24) givesa material which is ceramic (119888) rich at the top and metal (119898)rich at the bottomof the structureThemechanical properties(Youngmodulus andmass density) are given by the theory ofmixtures as

119864 (119911) = 119864119898 + (119864119888 minus 119864119898) 119881119888120588 (119911) = 120588119898 + (120588119888 minus 120588119898) 119881119888 (25)

where 119864119898 and 120588119898 are the elastic modulus and mass densityof the metallic phase respectively 119864119888 and 120588119888 are the corre-sponding properties of the ceramic phase Note that in allthe following computations the Poisson ratio is consideredconstant through the thickness thus ]119888 = ]119898 = ] = 03 Thetwo constituents have the following mechanical properties

119864119898 = 73GPa120588119898 = 2800 kgm3119864119888 = 380GPa120588119888 = 3800 kgm3

(26)

Shock and Vibration 9

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(a) Mode I (0 1) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

000501

01502

(b) Mode I (0 1) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(c) Mode II (119908 = 0) (0 1) 3D

002minus002

090908

08070706

06050504

04030302

020101

00minus01

1

(d) Mode II (119908 = 0) (0 1) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(e) Mode I (1 0) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

0minus005minus01

minus015minus02

(f) Mode I (1 0) 2D

Figure 2 Continued

10 Shock and Vibration

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(g) Mode II (119908 = 0) (1 0) 3D

002minus002

0908 08

0706 06

0504 04

0302 02

01 0 0

11

(h) Mode II (119908 = 0) (1 0) 2D

Figure 2 Differences between in-plane and cylindrical bending vibration modes for the 119886ℎ = 20 one-layered FGM plate with 119901 = 05 3Dmodes versus 2D modes

Since the FE commercial software does not have anembedded tool for the computation of functionally gradedproperties through the thickness the present FE applicationsare carried out building a fictitious laminated structure withisotropic plies In detail the thickness of the FGM lamina hasbeen divided into 100 fictitious layers and for each layer thecorresponding mechanical properties have been computedthrough (25) The numerical model has been made of auniform element division of 40 times 40 Quad8 elements for atotal number of 29280 degrees of freedom (dofs) The GDQsolution is computed with a not-uniform Chebyshev-Gauss-Lobatto grid [41] with 31 times 31 points for both thicknessratios It is underlined that since a single ply is set no zigzagfunction is needed in the computation It is recalled that thepresent higher order GDQ model considers a third orderexpansion with 120591 = 0 1 2 3 whereas the FE one has afirst order model (also known as Reissner-Mindlin model)Table 1 reports the results in terms of natural frequency(Hz) wherein the first column reports the number of half-waves (119898 119899) of the 3D exact solution Note that for a fixedset of half-wave numbers different mode shapes can occur(indicated by the second column ldquoModerdquo) In particular Iindicates the first mode and II indicates the second mode Inthe case of in-plane modes the caption (119908 = 0) is added tounderline that the vibration occurs in the plane and not outof plane The third column shows all the results for the 3Dexact solution both classical and cylindrical bending (CB)results It is noted that these frequencies are obtained witha single run of the 3D exact code keeping simply supportedboundary conditions for all the four edges of the plate Theother four columns of Table 1 refer to the numerical solutionsfor simply supported structures and for cylindrical bendingconditions (setting two free opposite edges) The cylindricalbending results are indicated by the acronym CB in the sametable

There is a very good agreement among the three solutionswhen simply supported conditions are considered On thecontrary a small error is observed in cylindrical bendingbecause the actual CB condition is not completely satisfieddue to the fact that the plate has a limited dimension in thesimply supported direction This feature can be detected inFigures 2 and 3 because the deformed shapes of the plate inthe CB configuration are not properly correct In fact thefibers parallel to the supported edges are not parallel amongthem but there is a curvature effect due to the free boundariesand to the limited length of the plate On the contrary thein-plane mode shapes (119908 = 0) when the plate is simplysupported on the four edges have all the fibers parallel amongthem also after the deformation These modes are the resultsobtained in the 3D solution with 119898 = 0 or 119899 = 0 combinedwith 119908 = 0 They are not cylindrical bending modes infact they can be found with 2D numerical models withoutmodifying the boundary conditions 3D frequencieswith119898 =0 or 119899 = 0 (with w different from 0) are CB modes which canbe obtained considering two simply supported edges and twofree edges in 2D numerical models

As it is well-known the ldquoperfectrdquo CB would occur whenthe simply supported direction has an ldquoinfiniterdquo lengthThis cannot be proven numerically but a trend can beobserved modeling different plates in cylindrical bendingconfiguration with different lengthsThese analyses are madekeeping one side of the plate constant and enlarging thelength of the other edge In the present computations 119886 iskept constant and 119887 = 119871 varies from 1 to 20 The results ofthe aforementioned analysis are listed in Table 2 and depictedin Figures 4 and 5 Table 2 includes the numerical valuesof the first mode shape in cylindrical bending for the 3Dexact solution the 2D FE and the 2D GDQmodels Figure 4shows that both GDQ and FE solutions tend to the 3D exactone increasing the length of the simply supported edge In

Shock and Vibration 11

ulowast

lowast

wlowast

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

(a) Mode I (1 0) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

000501

01502

(b) Mode I (1 0) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(c) Mode I (2 0) 3D

109

0807

060504

0302

01 0

109

0807

0605

0403

02010

0201501

0050

minus005minus01

minus015minus02

(d) Mode I (2 0) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(e) Mode I (3 0) 3D

10908

0706

050403

0201 0

109

0807

0605

0403

02010

0201501

0050

minus005minus01

minus015minus02

(f) Mode I (3 0) 2D

Figure 3 Continued

12 Shock and Vibration

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(g) Mode I (4 0) 3D

10908

0706050403

0201 0

10908

0706

0504

0302

010

0201501

0050

minus005minus01

minus015minus02

(h) Mode I (4 0) 2D

Figure 3 First four cylindrical bending vibration modes for the 119886ℎ = 20 one-layered FGM plate with 119901 = 05 3D modes versus 2D modes

3D exact2D GDQ2D FEM

f(H

z)

39

392

394

396

398

40

402

404

5 10 15 200La

Figure 4 First frequency 119891 in Hz for the 119886ℎ = 100 one-layeredFGM plate with 119901 = 05 convergence of 2D GDQ and 2D FEsolutions increasing the length of the simply supported sides

particular the GDQ solution is slightly more accurate thanthe FE one The 2D GDQ solution is carried out by enlargingthe length of one of the sides of the plate and the samenumberof dofs is kept On the contrary the FE model is generatedby 40 times 40 blocks of mesh In other words the plate with119871 = 2119886 has a mesh of 40 times 80 the one with 119871 = 3119886 has amesh 40 times 120 and so on When 119871 = 20119886 the error amongthe 3D exact and the 2D numerical solutions is very small andthis is proven also by the mode shapes of Figure 5 whereinthe boundary effect is negligible with respect the length of

the simply supported edges It must be underlined that it isimpossible to completely avoid the boundary effects due tothe free edges

The second example is related to a simply supportedcylinder with 119877120572 = 10m 119877120573 = infin 119886 = 2120587119877120572 and 119887 =20m and thickness ratios 119877120572ℎ = 100 and 119877120572ℎ = 10 Aone-layered FGM scheme is considered with ℎ1 = ℎ andthe same volume fraction function of (24) for the previousexample but setting 119901 = 20 The present material propertiesare the same as the flat plate The FE model has a 20 times80 Quad8 mesh with 27966 dofs and the GDQ solutionhas a 15 times 51 Chebyshev-Gauss-Lobatto grid Numericallyspeaking compatibility conditions must be enforced on theclosing meridian in order to obtain a closed revolution shell[42] To obtain the CB conditions numerically the cylinderis modeled completely free and the first 6 rigid mode shapesare neglected and removed from the numerical solution Itis noted that the GDQ model uses a third-order kinematicexpansion with no zigzag effect as the previous case sincea one-layer structure has been taken into account Table 3reports the results with the same meaning of the symbolsseen in Table 1 A very good agreement is observed when thecylinder is simply supported whereas when the cylinder iscompletely free some small errors can be observed but thenumerical solutions follow the 3D exact one

Results with 119899 = 0 and 119908 = 0 indicate the CB conditionsSuch results are obtained by means of 2D numerical modelsmodifying the boundary conditions (see 2D FE(CB) and 2DGDQ(CB)models)The cases 119899 = 0 combined with119908 = 0 arenot CB conditions and the frequencies can be found withoutmodifying the boundary conditions in the 2D numericalmodels

The final test is related to a cylindrical panel with 119877120572 =10m 119877120573 = infin 119886 = 1205873119877120572 and 119887 = 20m and thickness ratios119877120572ℎ = 100 and 119877120572ℎ = 10 The present shell is a sandwichstructure embedding a FGM core with ℎ2 = 07ℎ and two

Shock and Vibration 13

218

1614

121 1

08 0806 06

04 0402 02

0 0

0minus01minus02minus03minus04

(a) 119871119886 = 2

3

25

2

151

0806

040502

0 0

1

0minus01minus02minus03minus04minus05

(b) 119871119886 = 3

435

325

10505

0 0

15

0

minus02minus04minus06

1

2

(c) 119871119886 = 4

545

35

10505

0 0

0

minus04minus02

minus06minus08

1

25

4

3

215

(d) 119871119886 = 5

5

1

0 005

0

minus04minus02

minus06minus08

1

4

3

2

6

minus1

(e) 119871119886 = 6

5

10 0 051

43

2

67

89

100

minus05minus1

minus15

(f) 119871119886 = 10

Figure 5 First vibration mode for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 2D GDQ solution and edge effects versus the lengthof the simply supported sides

external skins with ℎ1 = ℎ3 = 015ℎ (ℎ1 + ℎ2 + ℎ3 = ℎ) Thebottom layer ismade of isotropicmetallicmaterial and the toplayer is made of ceramic material the mechanical propertiesare

119864119898 = 73GPa120588119898 = 2800 kgm3119864119888 = 200GPa120588119888 = 5700 kgm3

(27)

The FGM core has a volume fraction function defined as

119881119888 (119911) = (05 + 11991107ℎ)119901 with 119911 isin [minus035ℎ 035ℎ] (28)

where 119901 = 10 is employed in the present computations ThePoisson ratio is constant (]119888 = ]119898 = ] = 03) Equations (25)are still valid with 119881119888 defined as in (28)

The FEmodel is made of 40times40Quad8 elements for total29280 dofs and the GDQ model has a 31 times 31 Chebyshev-Gauss-Lobatto grid Since a sandwich structure is analyzedin the following a third-order theory with zigzag functionis considered in the computations for the GDQ method Thepresent natural frequencies are reported in Table 4 with the

14 Shock and Vibration

Table 2 First frequency 119891 in Hz for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 convergence of 2D GDQ and 2D FE solutionsincreasing the length of the simply supported sides

119871119886 1 2 3 4 5 6 8 10 15 20 3D exact2D GDQ 3921 3964 3980 3988 3993 3997 4001 4004 4007 4008 40192D FEM 3907 3949 3966 3974 3979 3983 3987 3989 3993 3994 4019

Table 3 One-layered FGM cylinder with 119901 = 20 Analytical 3D versus 2D numerical models Frequency 119891 in Hz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1004 0 I 0896 mdash 0897 mdash 08836 0 I 2532 mdash 2538 mdash 25118 0 I 4854 mdash 4867 mdash 481910 0 I 7848 mdash 7871 mdash 780112 0 I 1151 mdash 1155 mdash 114610 1 I 1356 1360 mdash 1357 mdash12 1 I 1441 1449 mdash 1445 mdash14 0 I 1584 mdash 1589 mdash 15778 1 I 1638 1640 mdash 1636 mdash14 1 I 1756 1767 mdash 1763 mdash16 0 I 2083 mdash 2090 mdash 207016 1 I 2207 2222 mdash 2216 mdash6 1 I 2441 2442 mdash 2438 mdash18 0 I 2649 mdash 2658 mdash 264014 2 I 2742 2752 mdash 2746 mdash18 1 I 2751 2772 mdash 2763 mdash16 2 I 2875 2890 mdash 2883 mdash12 2 I 2921 2926 mdash 2922 mdash119877120572ℎ = 104 0 I 8774 mdash 9085 mdash 86796 0 I 2456 mdash 2571 mdash 24306 1 I 3978 4089 mdash 3979 mdash4 1 I 4388 4357 mdash 4388 mdash8 0 I 4654 mdash 4920 mdash 46028 1 I 5575 5905 mdash 5576 mdash2 1 I 7202 7101 mdash 7202 mdash2 0 I (119908 = 0) 7322 7390 mdash 7319 mdash10 0 I 7419 mdash 7933 mdash 73476 2 I 8110 8261 mdash 8111 mdash10 1 I 8189 8811 mdash 8191 mdash8 2 I 8783 9173 mdash 8785 mdash4 2 I 9016 9084 mdash 9017 mdash12 0 I 1071 mdash 1159 mdash 106110 2 I 1081 1155 mdash 1081 mdash2 2 I 1089 1105 mdash 1089 mdash12 1 I 1142 1245 mdash 1143 mdash4 3 I 1263 1286 mdash 1263 mdash

same meaning of the columns seen in the previous casesIn order to achieve the cylindrical bending conditions thenumerical models have the straight edges simply supportedand the curved edges as free Due to the CB definitionit is not possible to achieve CB conditions when 119898 = 0

and 119899 = 1 2 3 because the cylindrical bending shouldoccur on the curved edge and this is not physically possiblebecause of the variation of displacement components due tothe curvature presence This aspect is indicated as NA (NotApplicable) solution in Table 4 Therefore this kind of 3D

Shock and Vibration 15

Table 4 Sandwich cylindrical shell panel embedding FGM core with 119901 = 10 Analytical 3D versus 2D numerical models Frequency 119891 inHz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1001 0 I 1948 mdash 1951 mdash 19412 0 I 8856 mdash 8872 mdash 88232 1 I 1104 1110 mdash 1104 mdash1 1 I 1850 1873 mdash 1850 mdash3 0 I 2038 mdash 2042 mdash 20303 1 I 2116 2128 mdash 2116 mdash2 2 I 2221 2242 mdash 2221 mdash3 2 I 2484 2496 mdash 2484 mdash3 3 I 3235 3254 mdash 3235 mdash4 0 I 3649 mdash 3660 mdash 36342 3 I 3692 3739 mdash 3692 mdash4 1 I 3714 3745 mdash 3714 mdash4 2 I 3940 3965 mdash 3941 mdash3 4 I 4226 4261 mdash 4226 mdash4 3 I 4378 4403 mdash 4378 mdash1 2 I 4568 4664 mdash 4568 mdash4 4 I 5035 5064 mdash 5035 mdash2 4 I 5040 5128 mdash 5040 mdash119877120572ℎ = 101 0 I 1905 mdash 1963 mdash 18841 1 I 3054 3145 mdash 3055 mdash1 2 I 6209 6409 mdash 6209 mdash2 0 I 8326 mdash 8857 mdash 82540 1 I 8710 mdash NA mdash NA0 1 II (119908 = 0) 8820 8956 mdash 8820 mdash2 1 I 8880 9531 mdash 8882 mdash0 2 I 9176 mdash NA mdash NA1 3 I 9576 9983 mdash 9567 mdash0 3 I 1036 mdash NA mdash NA2 2 I 1066 1147 mdash 1067 mdash0 4 I 1291 mdash NA mdash NA1 4 I 1335 1413 mdash 1335 mdash2 3 I 1364 1475 mdash 1364 mdash1 0 II (119908 = 0) 1665 1738 mdash 1665 mdash0 5 I 1683 mdash NA mdash NA2 4 I 1755 1918 mdash 1755 mdash0 2 II (119908 = 0) 1764 1791 mdash 1764 mdash

exact solutions cannot be achieved by any numerical modelOnce again solution with 119899 = 0 or 119898 = 0 combined with119908 = 0 (in-plane modes) can be found with 2D numericalmodels without modifying the simply supported boundaryconditions

5 Conclusions

Thepresent paper analyzes the cylindrical bending in the freefrequency analysis of functionally graded material (FGM)plates and cylindrical shells This condition is investigatedby means of a 3D exact shell model and two different 2D

numerical models a weak formulation based on classicalFinite Elements (FEs) and a strong formulation based onthe GDQ method The cylindrical bending (CB) behavior inthe 3D exact shell model can be analyzed using completelysimply supported structures and supposing one of the twohalf-wave numbers equals zeroThe numerical models obtainthe CB behavior using mixed simply supported and freeboundary conditions It has been demonstrated that theactual cylindrical bending (CB) is possible only with respectto straight edgesTheCBnatural frequencies of the numericalsolutions converge to the 3D exact CB ones if the length ofstraight simply supported edges is increased up to 20 times

16 Shock and Vibration

the value of the original edges The CB conditions are notpossible in 2D numericalmodels when the half-wave numberis zero in the curvilinear edge in this case the modificationof boundary conditions does not properly work In-planevibrationmodeswith zero transverse displacement combinedwith 119898 = 0 or 119899 = 0 cannot be defined as CB frequencies andthey can be calculated via 2D numerical models without themodification of the simply supported conditions for the fouredges All the considerations about the cylindrical bendingare valid for each geometry (plate cylinder and cylindricalshell) lamination scheme FGM law and thickness ratio

Competing Interests

The authors declare that they have no competing interests

References

[1] W Q Chen J B Cai and G R Ye ldquoResponses of cross-ply laminates with viscous interfaces in cylindrical bendingrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 823ndash835 2005

[2] W Q Chen and K Y Lee ldquoThree-dimensional exact analysisof angle-ply laminates in cylindrical bending with interfacialdamage via state-space methodrdquo Composite Structures vol 64no 3-4 pp 275ndash283 2004

[3] W Q Chen and K Y Lee ldquoTime-dependent behaviors of angle-ply laminates with viscous interfaces in cylindrical bendingrdquoEuropean Journal ofMechanicsmdashASolids vol 23 no 2 pp 235ndash245 2004

[4] A H Gandhi and H K Raval ldquoAnalytical and empiricalmodeling of top roller position for three-roller cylindricalbending of plates and its experimental verificationrdquo Journal ofMaterials Processing Technology vol 197 no 1ndash3 pp 268ndash2782008

[5] S Oral and H Darendeliler ldquoThe optimum die profile for thecylindrical bending of platesrdquo Journal of Materials ProcessingTechnology vol 70 no 1ndash3 pp 151ndash155 1997

[6] V A Jairazbhoy P Petukhov and J Qu ldquoLarge deflectionof thin plates in cylindrical bendingmdashnon-unique solutionsrdquoInternational Journal of Solids and Structures vol 45 no 11-12pp 3203ndash3218 2008

[7] A Lebee and K Sab ldquoA bending-gradient model for thickplates part II closed-form solutions for cylindrical bending oflaminatesrdquo International Journal of Solids and Structures vol 48no 20 pp 2889ndash2901 2011

[8] P V Nimbolkar and I M Jain ldquoCylindrical bending of elasticplatesrdquo Procedia Materials Science vol 10 pp 793ndash802 2015

[9] A S Sayyad and Y M Ghugal ldquoA nth-order shear deformationtheory for composite laminates in cylindrical bendingrdquo Curvedand Layered Structures vol 2 no 1 pp 290ndash300 2015

[10] N J Pagano ldquoExact solutions for composite laminates incylindrical bendingrdquo Journal of Composite Materials vol 3 no3 pp 398ndash411 1969

[11] N J Pagano and A S D Wang ldquoFurther study of compositelaminates under cylindrical bendingrdquo Journal of CompositeMaterials vol 5 pp 421ndash428 1971

[12] N Saeedi K Sab and J-F Caron ldquoCylindrical bending ofmultilayered plates with multi-delamination via a layerwisestress approachrdquo Composite Structures vol 95 pp 728ndash7392013

[13] X-P Shu and K P Soldatos ldquoCylindrical bending of angle-ply laminates subjected to different sets of edge boundaryconditionsrdquo International Journal of Solids and Structures vol37 no 31 pp 4289ndash4307 2000

[14] Z G Bian W Q Chen C W Lim and N Zhang ldquoAnalyticalsolutions for single- and multi-span functionally graded platesin cylindrical bendingrdquo International Journal of Solids andStructures vol 42 no 24-25 pp 6433ndash6456 2005

[15] H M Navazi and H Haddadpour ldquoNonlinear cylindricalbending analysis of shear deformable functionally graded platesunder different loadings using analytical methodsrdquo Interna-tional Journal of Mechanical Sciences vol 50 no 12 pp 1650ndash1657 2008

[16] W Q Chen J Ying J B Cai and G R Ye ldquoBenchmarksolution of imperfect angle-ply laminated rectangular plates incylindrical bending with surface piezoelectric layers as actuatorand sensorrdquo Computers and Structures vol 82 no 22 pp 1773ndash1784 2004

[17] W Q Chen and K Y Lee ldquoExact solution of angle-plypiezoelectric laminates in cylindrical bending with interfacialimperfectionsrdquo Composite Structures vol 65 no 3-4 pp 329ndash337 2004

[18] T Kant and SM Shiyekar ldquoCylindrical bending of piezoelectriclaminates with a higher order shear and normal deformationtheoryrdquo Computers and Structures vol 86 no 15-16 pp 1594ndash1603 2008

[19] P Lu H P Lee and C Lu ldquoAn exact solution for functionallygraded piezoelectric laminates in cylindrical bendingrdquo Interna-tional Journal of Mechanical Sciences vol 47 no 3 pp 437ndash4382005

[20] W Yan J Wang and W Q Chen ldquoCylindrical bendingresponses of angle-ply piezoelectric laminates with viscoelasticinterfacesrdquo Applied Mathematical Modelling vol 38 no 24 pp6018ndash6030 2014

[21] Y Y Zhou W Q Chen and C F Lu ldquoSemi-analytical solutionfor orthotropic piezoelectric laminates in cylindrical bendingwith interfacial imperfectionsrdquo Composite Structures vol 92no 4 pp 1009ndash1018 2010

[22] G P Dube S Kapuria and P C Dumir ldquoExact piezother-moelastic solution of a simply-supported orthotropic flat panelin cylindrical bendingrdquo International Journal of MechanicalSciences vol 38 pp 374ndash387 1996

[23] V N Pilipchuk V L Berdichevsky and R A IbrahimldquoThermo-mechanical coupling in cylindrical bending of sand-wich platesrdquoComposite Structures vol 92 no 11 pp 2632ndash26402010

[24] C Zhang S Di and N Zhang ldquoA new procedure for staticanalysis of thermo-electric laminated composite plates undercylindrical bendingrdquo Composite Structures vol 56 no 2 pp131ndash140 2002

[25] W Q Chen and K Y Lee ldquoState-space approach for statics anddynamics of angle-ply laminated cylindrical panels in cylindri-cal bendingrdquo International Journal of Mechanical Sciences vol47 no 3 pp 374ndash387 2005

[26] C-P Wu and Y-S Syu ldquoExact solutions of functionally gradedpiezoelectric shells under cylindrical bendingrdquo InternationalJournal of Solids and Structures vol 44 no 20 pp 6450ndash64722007

[27] W Yan J Ying and W Q Chen ldquoThe behavior of angle-ply laminated cylindrical shells with viscoelastic interfaces incylindrical bendingrdquo Composite Structures vol 78 no 4 pp551ndash559 2007

Shock and Vibration 17

[28] S Brischetto F Tornabene N Fantuzzi and E Viola ldquo3Dexact and 2D generalized differential quadraturemodels for freevibration analysis of functionally graded plates and cylindersrdquoMeccanica vol 51 no 9 pp 2059ndash2098 2016

[29] N Fantuzzi S Brischetto F Tornabene and E Viola ldquo2D and3D shell models for the free vibration investigation of func-tionally graded cylindrical and spherical panelsrdquo CompositeStructures vol 154 pp 573ndash590 2016

[30] S Brischetto and R Torre ldquoExact 3D solutions and finiteelement 2D models for free vibration analysis of plates andcylindersrdquo Curved and Layered Structures vol 1 no 1 pp 59ndash92 2014

[31] F Tornabene S Brischetto N Fantuzzi and E Viola ldquoNumer-ical and exact models for free vibration analysis of cylindricaland spherical shell panelsrdquo Composites Part B Engineering vol81 pp 231ndash250 2015

[32] S Brischetto ldquoExact elasticity solution for natural frequenciesof functionally graded simply-supported structuresrdquo ComputerModeling in Engineering amp Sciences vol 95 no 5 pp 391ndash4302013

[33] S Brischetto ldquoThree-dimensional exact free vibration analysisof spherical cylindrical and flat one-layered panelsrdquo Shock andVibration vol 2014 Article ID 479738 29 pages 2014

[34] S Brischetto ldquoA continuum elastic three-dimensional modelfor natural frequencies of single-walled carbon nanotubesrdquoComposites Part B Engineering vol 61 pp 222ndash228 2014

[35] S Brischetto ldquoAn exact 3d solution for free vibrations ofmultilayered cross-ply composite and sandwich plates andshellsrdquo International Journal of Applied Mechanics vol 6 no 6Article ID 1450076 42 pages 2014

[36] S Brischetto ldquoA continuumshellmodel including vanderWaalsinteraction for free vibrations of double-walled carbon nan-otubesrdquoCMES-ComputerModeling in Engineering and Sciencesvol 104 no 4 pp 305ndash327 2015

[37] S Brischetto F Tornabene N Fantuzzi and M BacciocchildquoRefined 2D and exact 3D shell models for the free vibrationanalysis of single- and double-walled carbon nanotubesrdquo Tech-nologies vol 3 no 4 pp 259ndash284 2015

[38] S Brischetto ldquoConvergence analysis of the exponential matrixmethod for the solution of 3D equilibrium equations for freevibration analysis of plates and shellsrdquo Composites Part BEngineering vol 98 pp 453ndash471 2016

[39] S Brischetto ldquoExact and approximate shell geometry in the freevibration analysis of one-layered and multilayered structuresrdquoInternational Journal of Mechanical Sciences vol 113 pp 81ndash932016

[40] S Brischetto ldquoCurvature approximation effects in the freevibration analysis of functionally graded shellsrdquo InternationalJournal of Applied Mechanics 2016

[41] F Tornabene N Fantuzzi F Ubertini and E Viola ldquoStrong for-mulation finite element method based on differential quadra-ture a surveyrdquo Applied Mechanics Reviews vol 67 no 2 ArticleID 020801 pp 1ndash55 2015

[42] F Tornabene E Viola and N Fantuzzi ldquoGeneral higher-orderequivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panelsrdquo CompositeStructures vol 104 pp 94ndash117 2013

[43] N Fantuzzi F Tornabene E Viola and A J Ferreira ldquoAstrong formulation finite element method (SFEM) based onRBF and GDQ techniques for the static and dynamic analysesof laminated plates of arbitrary shaperdquo Meccanica vol 49 no10 pp 2503ndash2542 2014

[44] F Tornabene N Fantuzzi and M Bacciocchi ldquoFree vibrationsof free-form doubly-curved shells made of functionally gradedmaterials using higher-order equivalent single layer theoriesrdquoComposites Part B Engineering vol 67 pp 490ndash509 2014

[45] F Tornabene and E Viola ldquoStatic analysis of functionallygraded doubly-curved shells and panels of revolutionrdquo Mecca-nica vol 48 no 4 pp 901ndash930 2013

[46] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoDynamic analysis of thick and thin elliptic shell structuresmade of laminated composite materialsrdquo Composite Structuresvol 133 pp 278ndash299 2015

[47] F TornabeneN FantuzziM Bacciocchi and EViola ldquoHigher-order theories for the free vibrations of doubly-curved lam-inated panels with curvilinear reinforcing fibers by meansof a local version of the GDQ methodrdquo Composites Part BEngineering vol 81 pp 196ndash230 2015

[48] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoFree vibrations of composite oval and elliptic cylinders bythe generalized differential quadrature methodrdquo Thin-WalledStructures vol 97 pp 114ndash129 2015

[49] F Tornabene and J N Reddy ldquoFGM and laminated doubly-curved and degenerate shells resting on nonlinear elastic foun-dations a GDQ solution for static analysis with a posterioristress and strain recoveryrdquo Journal of the Indian Institute ofScience vol 93 no 4 pp 635ndash688 2013

[50] N Fantuzzi F Tornabene and E Viola ldquoFour-parameter func-tionally graded cracked plates of arbitrary shape a GDQFEMsolution for free vibrationsrdquo Mechanics of Advanced Materialsand Structures vol 23 no 1 pp 89ndash107 2016

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International Journal of

4 Shock and Vibration

It is remarked that this set of equations is valid onlyfor spherical shells wherein 119877120572 = 119877120573 = 119877 because fordoubly curved shells with variable radii of curvature suchequations take a different form The authors decided to keepsuch system with the two radii of curvature distinct becauseit will be much easier to understand the cylindrical bendingeffect for plates and shells starting from a general point ofview

In the case of shell geometries andor FGM layers eachphysical layer 119896 must be divided in an opportune numberof 119902 mathematical layers to approximate the shell curvatureandor the FGM law through the thickness direction 119911 Theindex 119895 indicates the total mathematical layers obtained fromthe subdivision of each 119896 layer in 119902 mathematical layers (119895 =119896 times 119902) The parametric coefficients for shells with constantradii of curvature can be written as

119867120572 = (1 + 119911119877120572) 119867120573 = (1 + 119911119877120573) 119867119911 = 1

(2)

A closed form solution for simply supported shells andplates can be developed by writing harmonic displacementsas

119906119895 (120572 120573 119911 119905) = 119880119895 (119911) 119890119894120596119905 cos (120572120572) sin (120573120573) V119895 (120572 120573 119911 119905) = 119881119895 (119911) 119890119894120596119905 sin (120572120572) cos (120573120573) 119908119895 (120572 120573 119911 119905) = 119882119895 (119911) 119890119894120596119905 sin (120572120572) sin (120573120573)

(3)

where 119880119895(119911) 119881119895(119911) and 119882119895(119911) indicate the displacementamplitudes in 120572 120573 and 119911 directions respectively 119894 indicatesthe imaginary unit 120596 = 2120587119891 is the circular frequencywhere 119891 is the frequency value in Hz 119905 is the symbol forthe time In the coefficients 120572 = 119898120587119886 and 120573 = 119899120587119887119898 and 119899 are the half-wave numbers and 119886 and 119887 are theshell dimensions in 120572 and 120573 directions respectively (all thesequantities are evaluated on the midsurface Ω0) 120588119895 is themass density (120590119895120572120572 120590119895120573120573 120590119895119911119911 120590119895120573119911 120590119895120572119911 120590119895120572120573) are the six stresscomponents and 119895 V119895 and 119895 are the second temporalderivative of the three displacement components 119877120572 and119877120573 indicate the radii of curvature defined with respect thereference surface Ω0 of the whole multilayered plate or shell119867120572 and 119867120573 proposed in (2) vary with continuity through thethickness direction of the multilayered structure Thereforethey depend on the thickness coordinate 119911 The referencesurface Ω0 of the structure is defined as the locus of pointswhich are positioned midway between the external surfacesDisplacement components are indicated as 119906 V and 119908 in 120572120573 and 119911 directions respectively

The final system in closed form is developed substituting(3) and constitutive and geometrical equations (which are not

here given for the sake of brevity but they can be found indetail in [32]) in (1)

(minus119862119895551198671198951205731198671198951205721198772120572 minus 11986211989555119877120572119877120573 minus 120572211986211989511119867119895120573119867119895120572 minus 120573211986211989566119867119895120572119867119895120573+ 1205881198951198671198951205721198671198951205731205962) 119880119895 + (minus12057212057311986211989512 minus 12057212057311986211989566) 119881119895

+ (12057211986211989511119867119895120573119867119895120572119877120572 + 12057211986211989512119877120573 + 12057211986211989555119867119895120573119867119895120572119877120572 + 12057211986211989555119877120573 ) 119882119895

+ (11986211989555119867119895120573119877120572 + 11986211989555119867119895120572119877120573 ) 119880119895119911 + (12057211986211989513119867119895120573 + 12057211986211989555119867119895120573)sdot 119882119895119911 + (11986211989555119867119895120572119867119895120573) 119880119895119911119911 = 0

(minus12057212057311986211989566 minus 12057212057311986211989512) 119880119895 + (minus119862119895441198671198951205721198671198951205731198772120573 minus 11986211989544119877120572119877120573minus 120572211986211989566119867119895120573119867119895120572 minus 120573211986211989522119867119895120572119867119895120573 + 1205881198951198671198951205721198671198951205731205962) 119881119895

+ (12057311986211989544119867119895120572119867119895120573119877120573 + 12057311986211989544119877120572 + 12057311986211989522119867119895120572119867119895120573119877120573 + 12057311986211989512119877120572 ) 119882119895

+ (11986211989544119867119895120572119877120573 + 11986211989544119867119895120573119877120572 ) 119881119895119911 + (12057311986211989544119867119895120572 + 12057311986211989523119867119895120572)sdot 119882119895119911 + (11986211989544119867119895120572119867119895120573) 119881119895119911119911 = 0

(12057211986211989555119867119895120573119867119895120572119877120572 minus 12057211986211989513119877120573 + 12057211986211989511119867119895120573119867119895120572119877120572 + 12057211986211989512119877120573 ) 119880119895

+ (12057311986211989544119867119895120572119867119895120573119877120573 minus 12057311986211989523119877120572 + 12057311986211989522119867119895120572119867119895120573119877120573 + 12057311986211989512119877120572 ) 119881119895

+ ( 11986211989513119877120572119877120573 + 11986211989523119877120572119877120573 minus 119862119895111198671198951205731198671198951205721198772120572 minus 211986211989512119877120572119877120573 minus 119862119895221198671198951205721198671198951205731198772120573minus 120572211986211989555119867119895120573119867119895120572 minus 120573211986211989544119867119895120572119867119895120573 + 1205881198951198671198951205721198671198951205731205962) 119882119895

+ (minus12057211986211989555119867119895120573 minus 12057211986211989513119867119895120573) 119880119895119911 + (minus12057311986211989544119867119895120572minus 12057311986211989523119867119895120572) 119881119895119911 + (11986211989533119867119895120573119877120572 + 11986211989533119867119895120572119877120573 ) 119882119895119911+ (11986211989533119867119895120572119867119895120573) 119882119895119911119911 = 0

(4)

Shock and Vibration 5

The cylindrical bending requirements for analytical methodsare obtained choosing one of the two half-wave numberswhich equals zero This condition means 120572 = 119898120587119886 =0 or 120573 = 119899120587119887 = 0 which allows the derivatives madein a particular direction (120572 or 120573) which equals zero tooThis mathematical condition means that each section in thecylindrical bending direction has always the same behavior

It is obvious that the cylindrical bending effect is properfor cylindrical structures only However the authors startedthe present treatise from the governing equations (1) ofspherical shell structures because one of the aims of thepresent contribution is to underline which are the extraterms (related to the curvature) that do not allow having thestructure to be in cylindrical bending conditions

The compact form of (4) has been elaborated in detail in[32] The final version is here proposed

D119895 120597U119895120597119911 = A119895U119895 (5)

where 120597U119895120597119911 = U1198951015840 andU119895 = [119880119895 119881119895 119882119895 1198801198951015840 1198811198951015840 1198821198951015840]Equation (5) is rewritten as

D119895U1198951015840 = A119895U119895 (6)

U1198951015840 = D119895minus1A119895U119895 (7)

U1198951015840 = A119895lowastU119895 (8)

with A119895lowast = D119895minus1A119895The exponential matrix method (also known as transfer

matrix method or the state-space approach) is employed forthe solution of (8) (details are given in [32])

U119895 (119911119895) = exp (A119895lowast119911119895)U119895 (0) with 119911119895 isin [0 ℎ119895] (9)

where 119911119895 is the thickness coordinate for the 119895 mathematicallayer with values from 0 at the bottom to ℎ119895 at the top Theexponential matrix is calculated using 119911119895 = ℎ119895 for each 119895mathematical layer

A119895lowastlowast = exp (A119895lowastℎ119895)= I + A119895lowastℎ119895 + A119895lowast22 ℎ1198952 + A119895lowast33 ℎ1198953 + sdot sdot sdot

+ A119895lowast119873119873 ℎ119895119873(10)

where I indicates the 6times6 identitymatrixThe expansion usedfor the exponentialmatrix in (10) has a fast convergence and itis not time consuming from the computational point of view119895 = 119872 mathematical layers are used to approximate theshell curvature andor the FGM law through the thickness119872 minus 1 transfer matrices must be defined by means ofthe interlaminar continuity conditions of displacements andtransverse shearnormal stresses at each interfaceThe closedform solution for plates or shells is obtained using simply

supported boundary conditions and free stress conditions atthe top and at the bottom surfaces The above conditionsallow the following final system

EU1 (0) = 0 (11)

and the 6 times 6 matrix E does not change independently of thenumber of layers 119872 even if a layer-wise approach is appliedU1(0) is the vector U defined at the bottom of the wholemultilayered shell or plate (first layer 1 with 1199111 = 0) All thesteps and details here omitted for the sake of brevity can befound in [32ndash40] where the proposed exact 3D shell modelhas been elaborated and tested for the first time for severalapplications Several validations for the 3D exact shell modelhave been shown in [32ndash40] For this reason the model isnow used with confidence for the results and comparisonsproposed in Section 4

The nontrivial solution ofU1(0) in (11) allows conductingthe free vibration analysis imposing the zero determinant ofmatrix E

det [E] = 0 (12)

Equation (12) is used to find the roots of an higher order poly-nomial written in 120582 = 1205962The imposition of a couple of (119898 119899)allows a certain number of circular frequencies 120596 = 2120587119891(from I to infin) to be calculated This number of frequenciesdepends on the order 119873 used in each exponential matrixA119895lowastlowast and the number 119872 of mathematical layers employedfor the shell curvature andor the FGM law approximationsThe results shown in [32ndash40] confirm that 119873 = 3 for theexponential matrix and 119872 = 100 for the mathematicallayers are appropriate to always calculate the correct freefrequencies for each structure lamination sequence numberof layers material FGM law and thickness value

3 2D Finite Element Analysis and 2DGeneralized Differential Quadrature Models

The 3D exact natural frequencies are compared with twonumerical 2D solutions given by a weak and a strongformulation The former is a classic 2D finite element (FE)model carried out through the commercial software Strand7The latter is a recent widespread technique known as Gener-alized Differential Quadrature (GDQ) method [41] Detailsrelated to convergence and accuracy of these techniquescan be found in the works [29ndash31] It is noted that theFE solution employs a classic linear theory known alsoas Reissner-Mindlin theory with a linear rotation of thefibers through the shell thickness The present GDQ solutionexploits higher order theories with a general expansionthrough the thickness of the structureThis general approachfalls within the framework of the unified theories becauseseveral structural models can be obtained via the repetitionof a fundamental nucleus Another difference between theGDQ applications and the FE ones is that the formerincludes the curvature effects and the rotary inertias in itsformulation

6 Shock and Vibration

The 2D kinematic hypothesis [42ndash50] considers the fol-lowing expansion of the degrees of freedom

U = 119873119888+1sum120591=0

F120591u(120591) (13)

where U represents the vector of the 3D displacementcomponents which is function of (120572 120573 119911) and u120591 is thevector of the displacement parameters function of (120572 120573)which depends on the free index 120591 Therefore u120591 lays on themidsurface of the shell [42] The matrix F120591(119894119895) = 120575119894119895119865120591 (for119894 119895 = 1 2 3) contains the so-called thickness functions where120575 is the Kronecker delta function The thickness functionsdescribe how the displacement parameters characterize thesolution through the thickness They can have any form ofany smooth function of 119911 however the classic form of thesefunctions is the power function Thus 119865120591 = 119911120591 can be 1(120591 = 0) 119911 (120591 = 1) 1199112 (120591 = 2) 1199113 (120591 = 3) and so onPlease note that according to the mathematical backgroundof the present Unified Formulation the index 119873119888 + 1 isreserved to the (optional) zigzag effect Since the aim of thepresent paper is to investigate the cylindrical bending effectfor functionally graded plates and cylindrical panels thefollowing numerical applications will consider a fixed valueof 119873119888 = 3 so that 120591 = 0 1 2 3 This choice is justified by thefact that third order theories are sufficient with a reasonablylimited number of degrees of freedom to obtain accuratesolutions when compared to 3D shell models [42ndash50] Formultilayered structures such as sandwich configurations thezigzag function must be added into the formulation Onceagain as shown in (13) the zigzag function is embedded atthe end of the expansion with the exponent 119873119888 + 1 Whensuch effect is neglected the sum of (13) stops at 119873119888

The same theory used for the FE solution can be carriedout in the GDQmethod setting 120591 = 0 1 and 119873119888 = 1 howeverit has been shown in other recent works that the strong formapproaches led tomore accurate results with respect to the FEones [46 48]

The relation between generalized strains 120576(120591) anddisplace-ments u(120591) can be set as

120576(120591) = DΩu

(120591) for 120591 = 0 1 2 119873119888 119873119888 + 1 (14)

where DΩ is a differential operator explicitly defined in [42]The present constitutive material model is linear and elastictherefore for a general 119896th laminaHookersquos law can be applied

120590(119896) = C(119896)120576(119896) (15)

where 120590(119896) is the vector containing the stress componentsand C(119896) represents the constitutive matrix for each 119896th ply[42] of the structure Each component of the constitutivematrix is indicated by the symbol 119862(119896)120585120578 and it is valid in thecurvilinear reference system 1198741015840120572120573119911 of the reference geome-tryThese coefficients are evaluatedwith the application of thetransformation matrix [42]

The stress resultants are the terms which correspond tothe punctual 3D stresses of the 3D doubly curved solidTheir definition can be achieved by developing the Hamiltonprinciple for the present structural theory [42]

S(120591) = 119873119871sum119896=1

int119911119896+1119911119896

(Z(120591))119879 120590(120591)119867120572119867120573119889119911for 120591 = 0 1 2 119873119888 119873119888 + 1

(16)

whereZ(120591) is a 6times9matrix containing the thickness functions119865120591 and the curvature coefficients119867120572 and119867120573 the definition ofsuchmatrix with some other details can be found in [42] S(120591)include the stress resultants as a function of the order 120591

The definition of (16) can be rewritten as

S(120591) = 119873119888+1sum119904=0

A(120591119904)120576(119904) for 120591 = 0 1 2 119873119888 119873119888 + 1 (17)

which is a function of generalized strains of order 119904 120576(119904)Note that the so-called elastic coefficients are defined as

A(120591119904) = 119873119871sum119896=1

int119911119896+1119911119896

(Z(120591))119879C(119896)Z(119904)119867120572119867120573119889119911 (18)

and they can be carried out by the following integral expres-sions

119860(120591119904)120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119911119904119911120591119867120572119867120573119867984858120572119867120577120573 119889119911

119860(119904)120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119911119904120591119911120591minus1119867120572119867120573119867984858120572119867120577120573 119889119911

119860(120591)120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119904119911119904minus1119911120591119867120572119867120573119867984858120572119867120577120573 119889119911

Shock and Vibration 7

119860( )120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119904119911119904minus1120591119911120591minus1119867120572119867120573119867984858120572119867120577120573 119889119911for 120591 119904 = 0 1 2 119873119888 119873119888 + 1 120585 120578 = 1 2 3 4 5 6 984858 120577 = 0 1 2

(19)

It is recalled that the exponent 119873119888 + 1 is reserved for theMurakami function otherwise this order is neglected Thesubscripts and indicate the derivatives of the thicknessfunctions 119865120591 and 119865119904 with respect to 119911 The subscripts 984858 and 120577are the exponents of the quantities119867120572 and 119867120573 whereas 120585 and120578 are the indices of the material constants 119862(119896)120585120578 for the 119896th ply

[42ndash49]Note that119862(119896)120585120578 119867120572 and119867120573 are quantities that dependon the thickness 119911 Therefore the integrals of (19) cannotbe computed analytically The present solution for solvingsuch integrals is to employ a numerical integration algorithmIn particular the so-called Generalized Integral Quadrature(GIQ) method is utilized For an accurate evaluation of theintegrals 51 points per layer are considered in the integration

The Hamilton principle supplies for the dynamic equilib-rium equations and the correspondent boundary conditionsIn detail the fundamental nucleus of three motion equationstakes the form

D⋆ΩS(120591) = 119873119888+1sum

119904=0

M(120591119904)u(119904)

for 120591 = 0 1 2 119873119888 119873119888 + 1(20)

whereD⋆Ω is the equilibrium operator andM(120591119904) is the inertialmatrix Their explicit expressions have been presented in aprevious authorsrsquo paper [42]

Combining the kinematic equation (14) constitutiveequation (17) and themotion equation (20) the fundamentalgoverning system in terms of displacement parameters iscarried out

119873119888+1sum119904=0

L(120591119904)u(119904) = 119873119888+1sum119904=0

M(120591119904)u(119904)

for 120591 = 0 1 2 119873119888 119873119888 + 1(21)

where L(120591119904) = D⋆ΩA(120591119904)DΩ is the fundamental operator [42]

Boundary conditions must be included for the solutionof the differential equations (21) The present paper dealsonly with free and simply supported boundary conditionsThe former is used to reproduce the cylindrical bendingconditions considering alternatively one free edge and onesimply supported edge When the four edges are all simplysupported the problem is the same as the ones investigated

in the previous papers [29ndash31] Simply supported conditionsare

119873(120591)120572 = 0119906(120591)120573 = 119906(120591)119911 = 0

120591 = 0 1 2 119873119888 119873119888 + 1 at 120572 = 1205720 or 120572 = 1205721 1205730 le 120573 le 1205731119906(120591)120572 = 0

119873(120591)120573 = 0119906(120591)119911 = 0

120591 = 0 1 2 119873119888 119873119888 + 1 at 120573 = 1205730 or 120573 = 1205731 1205720 le 120572 le 1205721

(22)

Free edge conditions are

119873(120591)120572 = 0119873(120591)120573 = 0119879(120591)120572 = 0120591 = 0 1 2 119873119888 119873119888 + 1 at 120572 = 1205720 or 120572 = 1205721 1205730 le 120573 le 1205731119873(120591)120572 = 0119873(120591)120573 = 0119879(120591)120573 = 0120591 = 0 1 2 119873119888 119873119888 + 1 at 120573 = 1205730 or 120573 = 1205731 1205720 le 120572 le 1205721

(23)

4 Results

The present section includes some numerical applicationsfor the comparison of the 3D exact shell vibrations withtwo numerical 2D models In particular classical simplysupported solutions and cylindrical bending (CB) behaviorof plates and shells are presentedThe following computationsfocus on plates circular cylinders and cylindrical panels dueto the fact that the CB effect occurs only when at least twoparallel straight edges are present in the structure

The first case is a square plate (119886 = 119887 = 1m) withsimply supported edges Two thickness ratios are analyzedthin 119886ℎ = 100 and moderately thick 119886ℎ = 20 structure Theplate is made of a single functionally graded layer (ℎ1 = ℎ)with volume fraction of ceramic (119888) phase

119881119888 (119911) = (05 + 119911ℎ)119901 with 119911 isin [minusℎ2 ℎ2 ] (24)

8 Shock and Vibration

Table 1 One-layered FGM plate with 119901 = 05 Analytical 3D versus 2D numerical models Frequency 119891 in Hz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119886ℎ = 1001 0 I 4019 mdash 3907 mdash 39210 1 I 4019 mdash 3907 mdash 39211 1 I 8037 8006 mdash 8036 mdash2 0 I 1607 mdash 1580 mdash 15840 2 I 1607 mdash 1580 mdash 15842 1 I 2008 2001 mdash 2008 mdash1 2 I 2008 2001 mdash 2008 mdash2 2 I 3212 3202 mdash 3211 mdash3 0 I 3613 mdash 3568 mdash 35770 3 I 3613 mdash 3568 mdash 35773 1 I 4014 4002 mdash 4013 mdash1 3 I 4014 4002 mdash 4013 mdash3 2 I 5215 5202 mdash 5215 mdash2 3 I 5215 5202 mdash 5215 mdash4 0 I 6416 mdash 6354 mdash 63650 4 I 6416 mdash 6354 mdash 63654 1 I 6816 6801 mdash 6815 mdash1 4 I 6816 6801 mdash 6815 mdash119886ℎ = 201 0 I 2002 mdash 1951 mdash 19510 1 I 2002 mdash 1951 mdash 19511 1 I 3989 3995 mdash 3988 mdash2 0 I 7916 mdash 7864 mdash 77970 2 I 7916 mdash 7864 mdash 77972 1 I 9858 9954 mdash 9857 mdash1 2 I 9858 9954 mdash 9857 mdash2 2 I 1560 1588 mdash 1560 mdash3 0 I 1749 mdash 1767 mdash 17290 3 I 1749 mdash 1767 mdash 17293 1 I 1936 1980 mdash 1936 mdash1 3 I 1936 1980 mdash 1936 mdash3 2 I 2490 2566 mdash 2490 mdash2 3 I 2490 2566 mdash 2490 mdash1 0 II (119908 = 0) 2775 2773 mdash 2775 mdash0 1 II (119908 = 0) 2775 2773 mdash 2775 mdash4 0 I 3034 mdash 3124 mdash 30070 4 I 3034 mdash 3124 mdash 3007

where 119901 is the power exponent which varies the volumefraction of the ceramic component through the thicknessTwo limit cases occur for (24) 119881119888 = 1 when 119901 = 0 and119881119888 rarr 0when119901 = infin In the following computations it is set as119901 = 05 Note that the volume fraction function in (24) givesa material which is ceramic (119888) rich at the top and metal (119898)rich at the bottomof the structureThemechanical properties(Youngmodulus andmass density) are given by the theory ofmixtures as

119864 (119911) = 119864119898 + (119864119888 minus 119864119898) 119881119888120588 (119911) = 120588119898 + (120588119888 minus 120588119898) 119881119888 (25)

where 119864119898 and 120588119898 are the elastic modulus and mass densityof the metallic phase respectively 119864119888 and 120588119888 are the corre-sponding properties of the ceramic phase Note that in allthe following computations the Poisson ratio is consideredconstant through the thickness thus ]119888 = ]119898 = ] = 03 Thetwo constituents have the following mechanical properties

119864119898 = 73GPa120588119898 = 2800 kgm3119864119888 = 380GPa120588119888 = 3800 kgm3

(26)

Shock and Vibration 9

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(a) Mode I (0 1) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

000501

01502

(b) Mode I (0 1) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(c) Mode II (119908 = 0) (0 1) 3D

002minus002

090908

08070706

06050504

04030302

020101

00minus01

1

(d) Mode II (119908 = 0) (0 1) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(e) Mode I (1 0) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

0minus005minus01

minus015minus02

(f) Mode I (1 0) 2D

Figure 2 Continued

10 Shock and Vibration

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(g) Mode II (119908 = 0) (1 0) 3D

002minus002

0908 08

0706 06

0504 04

0302 02

01 0 0

11

(h) Mode II (119908 = 0) (1 0) 2D

Figure 2 Differences between in-plane and cylindrical bending vibration modes for the 119886ℎ = 20 one-layered FGM plate with 119901 = 05 3Dmodes versus 2D modes

Since the FE commercial software does not have anembedded tool for the computation of functionally gradedproperties through the thickness the present FE applicationsare carried out building a fictitious laminated structure withisotropic plies In detail the thickness of the FGM lamina hasbeen divided into 100 fictitious layers and for each layer thecorresponding mechanical properties have been computedthrough (25) The numerical model has been made of auniform element division of 40 times 40 Quad8 elements for atotal number of 29280 degrees of freedom (dofs) The GDQsolution is computed with a not-uniform Chebyshev-Gauss-Lobatto grid [41] with 31 times 31 points for both thicknessratios It is underlined that since a single ply is set no zigzagfunction is needed in the computation It is recalled that thepresent higher order GDQ model considers a third orderexpansion with 120591 = 0 1 2 3 whereas the FE one has afirst order model (also known as Reissner-Mindlin model)Table 1 reports the results in terms of natural frequency(Hz) wherein the first column reports the number of half-waves (119898 119899) of the 3D exact solution Note that for a fixedset of half-wave numbers different mode shapes can occur(indicated by the second column ldquoModerdquo) In particular Iindicates the first mode and II indicates the second mode Inthe case of in-plane modes the caption (119908 = 0) is added tounderline that the vibration occurs in the plane and not outof plane The third column shows all the results for the 3Dexact solution both classical and cylindrical bending (CB)results It is noted that these frequencies are obtained witha single run of the 3D exact code keeping simply supportedboundary conditions for all the four edges of the plate Theother four columns of Table 1 refer to the numerical solutionsfor simply supported structures and for cylindrical bendingconditions (setting two free opposite edges) The cylindricalbending results are indicated by the acronym CB in the sametable

There is a very good agreement among the three solutionswhen simply supported conditions are considered On thecontrary a small error is observed in cylindrical bendingbecause the actual CB condition is not completely satisfieddue to the fact that the plate has a limited dimension in thesimply supported direction This feature can be detected inFigures 2 and 3 because the deformed shapes of the plate inthe CB configuration are not properly correct In fact thefibers parallel to the supported edges are not parallel amongthem but there is a curvature effect due to the free boundariesand to the limited length of the plate On the contrary thein-plane mode shapes (119908 = 0) when the plate is simplysupported on the four edges have all the fibers parallel amongthem also after the deformation These modes are the resultsobtained in the 3D solution with 119898 = 0 or 119899 = 0 combinedwith 119908 = 0 They are not cylindrical bending modes infact they can be found with 2D numerical models withoutmodifying the boundary conditions 3D frequencieswith119898 =0 or 119899 = 0 (with w different from 0) are CB modes which canbe obtained considering two simply supported edges and twofree edges in 2D numerical models

As it is well-known the ldquoperfectrdquo CB would occur whenthe simply supported direction has an ldquoinfiniterdquo lengthThis cannot be proven numerically but a trend can beobserved modeling different plates in cylindrical bendingconfiguration with different lengthsThese analyses are madekeeping one side of the plate constant and enlarging thelength of the other edge In the present computations 119886 iskept constant and 119887 = 119871 varies from 1 to 20 The results ofthe aforementioned analysis are listed in Table 2 and depictedin Figures 4 and 5 Table 2 includes the numerical valuesof the first mode shape in cylindrical bending for the 3Dexact solution the 2D FE and the 2D GDQmodels Figure 4shows that both GDQ and FE solutions tend to the 3D exactone increasing the length of the simply supported edge In

Shock and Vibration 11

ulowast

lowast

wlowast

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

(a) Mode I (1 0) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

000501

01502

(b) Mode I (1 0) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(c) Mode I (2 0) 3D

109

0807

060504

0302

01 0

109

0807

0605

0403

02010

0201501

0050

minus005minus01

minus015minus02

(d) Mode I (2 0) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(e) Mode I (3 0) 3D

10908

0706

050403

0201 0

109

0807

0605

0403

02010

0201501

0050

minus005minus01

minus015minus02

(f) Mode I (3 0) 2D

Figure 3 Continued

12 Shock and Vibration

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(g) Mode I (4 0) 3D

10908

0706050403

0201 0

10908

0706

0504

0302

010

0201501

0050

minus005minus01

minus015minus02

(h) Mode I (4 0) 2D

Figure 3 First four cylindrical bending vibration modes for the 119886ℎ = 20 one-layered FGM plate with 119901 = 05 3D modes versus 2D modes

3D exact2D GDQ2D FEM

f(H

z)

39

392

394

396

398

40

402

404

5 10 15 200La

Figure 4 First frequency 119891 in Hz for the 119886ℎ = 100 one-layeredFGM plate with 119901 = 05 convergence of 2D GDQ and 2D FEsolutions increasing the length of the simply supported sides

particular the GDQ solution is slightly more accurate thanthe FE one The 2D GDQ solution is carried out by enlargingthe length of one of the sides of the plate and the samenumberof dofs is kept On the contrary the FE model is generatedby 40 times 40 blocks of mesh In other words the plate with119871 = 2119886 has a mesh of 40 times 80 the one with 119871 = 3119886 has amesh 40 times 120 and so on When 119871 = 20119886 the error amongthe 3D exact and the 2D numerical solutions is very small andthis is proven also by the mode shapes of Figure 5 whereinthe boundary effect is negligible with respect the length of

the simply supported edges It must be underlined that it isimpossible to completely avoid the boundary effects due tothe free edges

The second example is related to a simply supportedcylinder with 119877120572 = 10m 119877120573 = infin 119886 = 2120587119877120572 and 119887 =20m and thickness ratios 119877120572ℎ = 100 and 119877120572ℎ = 10 Aone-layered FGM scheme is considered with ℎ1 = ℎ andthe same volume fraction function of (24) for the previousexample but setting 119901 = 20 The present material propertiesare the same as the flat plate The FE model has a 20 times80 Quad8 mesh with 27966 dofs and the GDQ solutionhas a 15 times 51 Chebyshev-Gauss-Lobatto grid Numericallyspeaking compatibility conditions must be enforced on theclosing meridian in order to obtain a closed revolution shell[42] To obtain the CB conditions numerically the cylinderis modeled completely free and the first 6 rigid mode shapesare neglected and removed from the numerical solution Itis noted that the GDQ model uses a third-order kinematicexpansion with no zigzag effect as the previous case sincea one-layer structure has been taken into account Table 3reports the results with the same meaning of the symbolsseen in Table 1 A very good agreement is observed when thecylinder is simply supported whereas when the cylinder iscompletely free some small errors can be observed but thenumerical solutions follow the 3D exact one

Results with 119899 = 0 and 119908 = 0 indicate the CB conditionsSuch results are obtained by means of 2D numerical modelsmodifying the boundary conditions (see 2D FE(CB) and 2DGDQ(CB)models)The cases 119899 = 0 combined with119908 = 0 arenot CB conditions and the frequencies can be found withoutmodifying the boundary conditions in the 2D numericalmodels

The final test is related to a cylindrical panel with 119877120572 =10m 119877120573 = infin 119886 = 1205873119877120572 and 119887 = 20m and thickness ratios119877120572ℎ = 100 and 119877120572ℎ = 10 The present shell is a sandwichstructure embedding a FGM core with ℎ2 = 07ℎ and two

Shock and Vibration 13

218

1614

121 1

08 0806 06

04 0402 02

0 0

0minus01minus02minus03minus04

(a) 119871119886 = 2

3

25

2

151

0806

040502

0 0

1

0minus01minus02minus03minus04minus05

(b) 119871119886 = 3

435

325

10505

0 0

15

0

minus02minus04minus06

1

2

(c) 119871119886 = 4

545

35

10505

0 0

0

minus04minus02

minus06minus08

1

25

4

3

215

(d) 119871119886 = 5

5

1

0 005

0

minus04minus02

minus06minus08

1

4

3

2

6

minus1

(e) 119871119886 = 6

5

10 0 051

43

2

67

89

100

minus05minus1

minus15

(f) 119871119886 = 10

Figure 5 First vibration mode for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 2D GDQ solution and edge effects versus the lengthof the simply supported sides

external skins with ℎ1 = ℎ3 = 015ℎ (ℎ1 + ℎ2 + ℎ3 = ℎ) Thebottom layer ismade of isotropicmetallicmaterial and the toplayer is made of ceramic material the mechanical propertiesare

119864119898 = 73GPa120588119898 = 2800 kgm3119864119888 = 200GPa120588119888 = 5700 kgm3

(27)

The FGM core has a volume fraction function defined as

119881119888 (119911) = (05 + 11991107ℎ)119901 with 119911 isin [minus035ℎ 035ℎ] (28)

where 119901 = 10 is employed in the present computations ThePoisson ratio is constant (]119888 = ]119898 = ] = 03) Equations (25)are still valid with 119881119888 defined as in (28)

The FEmodel is made of 40times40Quad8 elements for total29280 dofs and the GDQ model has a 31 times 31 Chebyshev-Gauss-Lobatto grid Since a sandwich structure is analyzedin the following a third-order theory with zigzag functionis considered in the computations for the GDQ method Thepresent natural frequencies are reported in Table 4 with the

14 Shock and Vibration

Table 2 First frequency 119891 in Hz for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 convergence of 2D GDQ and 2D FE solutionsincreasing the length of the simply supported sides

119871119886 1 2 3 4 5 6 8 10 15 20 3D exact2D GDQ 3921 3964 3980 3988 3993 3997 4001 4004 4007 4008 40192D FEM 3907 3949 3966 3974 3979 3983 3987 3989 3993 3994 4019

Table 3 One-layered FGM cylinder with 119901 = 20 Analytical 3D versus 2D numerical models Frequency 119891 in Hz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1004 0 I 0896 mdash 0897 mdash 08836 0 I 2532 mdash 2538 mdash 25118 0 I 4854 mdash 4867 mdash 481910 0 I 7848 mdash 7871 mdash 780112 0 I 1151 mdash 1155 mdash 114610 1 I 1356 1360 mdash 1357 mdash12 1 I 1441 1449 mdash 1445 mdash14 0 I 1584 mdash 1589 mdash 15778 1 I 1638 1640 mdash 1636 mdash14 1 I 1756 1767 mdash 1763 mdash16 0 I 2083 mdash 2090 mdash 207016 1 I 2207 2222 mdash 2216 mdash6 1 I 2441 2442 mdash 2438 mdash18 0 I 2649 mdash 2658 mdash 264014 2 I 2742 2752 mdash 2746 mdash18 1 I 2751 2772 mdash 2763 mdash16 2 I 2875 2890 mdash 2883 mdash12 2 I 2921 2926 mdash 2922 mdash119877120572ℎ = 104 0 I 8774 mdash 9085 mdash 86796 0 I 2456 mdash 2571 mdash 24306 1 I 3978 4089 mdash 3979 mdash4 1 I 4388 4357 mdash 4388 mdash8 0 I 4654 mdash 4920 mdash 46028 1 I 5575 5905 mdash 5576 mdash2 1 I 7202 7101 mdash 7202 mdash2 0 I (119908 = 0) 7322 7390 mdash 7319 mdash10 0 I 7419 mdash 7933 mdash 73476 2 I 8110 8261 mdash 8111 mdash10 1 I 8189 8811 mdash 8191 mdash8 2 I 8783 9173 mdash 8785 mdash4 2 I 9016 9084 mdash 9017 mdash12 0 I 1071 mdash 1159 mdash 106110 2 I 1081 1155 mdash 1081 mdash2 2 I 1089 1105 mdash 1089 mdash12 1 I 1142 1245 mdash 1143 mdash4 3 I 1263 1286 mdash 1263 mdash

same meaning of the columns seen in the previous casesIn order to achieve the cylindrical bending conditions thenumerical models have the straight edges simply supportedand the curved edges as free Due to the CB definitionit is not possible to achieve CB conditions when 119898 = 0

and 119899 = 1 2 3 because the cylindrical bending shouldoccur on the curved edge and this is not physically possiblebecause of the variation of displacement components due tothe curvature presence This aspect is indicated as NA (NotApplicable) solution in Table 4 Therefore this kind of 3D

Shock and Vibration 15

Table 4 Sandwich cylindrical shell panel embedding FGM core with 119901 = 10 Analytical 3D versus 2D numerical models Frequency 119891 inHz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1001 0 I 1948 mdash 1951 mdash 19412 0 I 8856 mdash 8872 mdash 88232 1 I 1104 1110 mdash 1104 mdash1 1 I 1850 1873 mdash 1850 mdash3 0 I 2038 mdash 2042 mdash 20303 1 I 2116 2128 mdash 2116 mdash2 2 I 2221 2242 mdash 2221 mdash3 2 I 2484 2496 mdash 2484 mdash3 3 I 3235 3254 mdash 3235 mdash4 0 I 3649 mdash 3660 mdash 36342 3 I 3692 3739 mdash 3692 mdash4 1 I 3714 3745 mdash 3714 mdash4 2 I 3940 3965 mdash 3941 mdash3 4 I 4226 4261 mdash 4226 mdash4 3 I 4378 4403 mdash 4378 mdash1 2 I 4568 4664 mdash 4568 mdash4 4 I 5035 5064 mdash 5035 mdash2 4 I 5040 5128 mdash 5040 mdash119877120572ℎ = 101 0 I 1905 mdash 1963 mdash 18841 1 I 3054 3145 mdash 3055 mdash1 2 I 6209 6409 mdash 6209 mdash2 0 I 8326 mdash 8857 mdash 82540 1 I 8710 mdash NA mdash NA0 1 II (119908 = 0) 8820 8956 mdash 8820 mdash2 1 I 8880 9531 mdash 8882 mdash0 2 I 9176 mdash NA mdash NA1 3 I 9576 9983 mdash 9567 mdash0 3 I 1036 mdash NA mdash NA2 2 I 1066 1147 mdash 1067 mdash0 4 I 1291 mdash NA mdash NA1 4 I 1335 1413 mdash 1335 mdash2 3 I 1364 1475 mdash 1364 mdash1 0 II (119908 = 0) 1665 1738 mdash 1665 mdash0 5 I 1683 mdash NA mdash NA2 4 I 1755 1918 mdash 1755 mdash0 2 II (119908 = 0) 1764 1791 mdash 1764 mdash

exact solutions cannot be achieved by any numerical modelOnce again solution with 119899 = 0 or 119898 = 0 combined with119908 = 0 (in-plane modes) can be found with 2D numericalmodels without modifying the simply supported boundaryconditions

5 Conclusions

Thepresent paper analyzes the cylindrical bending in the freefrequency analysis of functionally graded material (FGM)plates and cylindrical shells This condition is investigatedby means of a 3D exact shell model and two different 2D

numerical models a weak formulation based on classicalFinite Elements (FEs) and a strong formulation based onthe GDQ method The cylindrical bending (CB) behavior inthe 3D exact shell model can be analyzed using completelysimply supported structures and supposing one of the twohalf-wave numbers equals zeroThe numerical models obtainthe CB behavior using mixed simply supported and freeboundary conditions It has been demonstrated that theactual cylindrical bending (CB) is possible only with respectto straight edgesTheCBnatural frequencies of the numericalsolutions converge to the 3D exact CB ones if the length ofstraight simply supported edges is increased up to 20 times

16 Shock and Vibration

the value of the original edges The CB conditions are notpossible in 2D numericalmodels when the half-wave numberis zero in the curvilinear edge in this case the modificationof boundary conditions does not properly work In-planevibrationmodeswith zero transverse displacement combinedwith 119898 = 0 or 119899 = 0 cannot be defined as CB frequencies andthey can be calculated via 2D numerical models without themodification of the simply supported conditions for the fouredges All the considerations about the cylindrical bendingare valid for each geometry (plate cylinder and cylindricalshell) lamination scheme FGM law and thickness ratio

Competing Interests

The authors declare that they have no competing interests

References

[1] W Q Chen J B Cai and G R Ye ldquoResponses of cross-ply laminates with viscous interfaces in cylindrical bendingrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 823ndash835 2005

[2] W Q Chen and K Y Lee ldquoThree-dimensional exact analysisof angle-ply laminates in cylindrical bending with interfacialdamage via state-space methodrdquo Composite Structures vol 64no 3-4 pp 275ndash283 2004

[3] W Q Chen and K Y Lee ldquoTime-dependent behaviors of angle-ply laminates with viscous interfaces in cylindrical bendingrdquoEuropean Journal ofMechanicsmdashASolids vol 23 no 2 pp 235ndash245 2004

[4] A H Gandhi and H K Raval ldquoAnalytical and empiricalmodeling of top roller position for three-roller cylindricalbending of plates and its experimental verificationrdquo Journal ofMaterials Processing Technology vol 197 no 1ndash3 pp 268ndash2782008

[5] S Oral and H Darendeliler ldquoThe optimum die profile for thecylindrical bending of platesrdquo Journal of Materials ProcessingTechnology vol 70 no 1ndash3 pp 151ndash155 1997

[6] V A Jairazbhoy P Petukhov and J Qu ldquoLarge deflectionof thin plates in cylindrical bendingmdashnon-unique solutionsrdquoInternational Journal of Solids and Structures vol 45 no 11-12pp 3203ndash3218 2008

[7] A Lebee and K Sab ldquoA bending-gradient model for thickplates part II closed-form solutions for cylindrical bending oflaminatesrdquo International Journal of Solids and Structures vol 48no 20 pp 2889ndash2901 2011

[8] P V Nimbolkar and I M Jain ldquoCylindrical bending of elasticplatesrdquo Procedia Materials Science vol 10 pp 793ndash802 2015

[9] A S Sayyad and Y M Ghugal ldquoA nth-order shear deformationtheory for composite laminates in cylindrical bendingrdquo Curvedand Layered Structures vol 2 no 1 pp 290ndash300 2015

[10] N J Pagano ldquoExact solutions for composite laminates incylindrical bendingrdquo Journal of Composite Materials vol 3 no3 pp 398ndash411 1969

[11] N J Pagano and A S D Wang ldquoFurther study of compositelaminates under cylindrical bendingrdquo Journal of CompositeMaterials vol 5 pp 421ndash428 1971

[12] N Saeedi K Sab and J-F Caron ldquoCylindrical bending ofmultilayered plates with multi-delamination via a layerwisestress approachrdquo Composite Structures vol 95 pp 728ndash7392013

[13] X-P Shu and K P Soldatos ldquoCylindrical bending of angle-ply laminates subjected to different sets of edge boundaryconditionsrdquo International Journal of Solids and Structures vol37 no 31 pp 4289ndash4307 2000

[14] Z G Bian W Q Chen C W Lim and N Zhang ldquoAnalyticalsolutions for single- and multi-span functionally graded platesin cylindrical bendingrdquo International Journal of Solids andStructures vol 42 no 24-25 pp 6433ndash6456 2005

[15] H M Navazi and H Haddadpour ldquoNonlinear cylindricalbending analysis of shear deformable functionally graded platesunder different loadings using analytical methodsrdquo Interna-tional Journal of Mechanical Sciences vol 50 no 12 pp 1650ndash1657 2008

[16] W Q Chen J Ying J B Cai and G R Ye ldquoBenchmarksolution of imperfect angle-ply laminated rectangular plates incylindrical bending with surface piezoelectric layers as actuatorand sensorrdquo Computers and Structures vol 82 no 22 pp 1773ndash1784 2004

[17] W Q Chen and K Y Lee ldquoExact solution of angle-plypiezoelectric laminates in cylindrical bending with interfacialimperfectionsrdquo Composite Structures vol 65 no 3-4 pp 329ndash337 2004

[18] T Kant and SM Shiyekar ldquoCylindrical bending of piezoelectriclaminates with a higher order shear and normal deformationtheoryrdquo Computers and Structures vol 86 no 15-16 pp 1594ndash1603 2008

[19] P Lu H P Lee and C Lu ldquoAn exact solution for functionallygraded piezoelectric laminates in cylindrical bendingrdquo Interna-tional Journal of Mechanical Sciences vol 47 no 3 pp 437ndash4382005

[20] W Yan J Wang and W Q Chen ldquoCylindrical bendingresponses of angle-ply piezoelectric laminates with viscoelasticinterfacesrdquo Applied Mathematical Modelling vol 38 no 24 pp6018ndash6030 2014

[21] Y Y Zhou W Q Chen and C F Lu ldquoSemi-analytical solutionfor orthotropic piezoelectric laminates in cylindrical bendingwith interfacial imperfectionsrdquo Composite Structures vol 92no 4 pp 1009ndash1018 2010

[22] G P Dube S Kapuria and P C Dumir ldquoExact piezother-moelastic solution of a simply-supported orthotropic flat panelin cylindrical bendingrdquo International Journal of MechanicalSciences vol 38 pp 374ndash387 1996

[23] V N Pilipchuk V L Berdichevsky and R A IbrahimldquoThermo-mechanical coupling in cylindrical bending of sand-wich platesrdquoComposite Structures vol 92 no 11 pp 2632ndash26402010

[24] C Zhang S Di and N Zhang ldquoA new procedure for staticanalysis of thermo-electric laminated composite plates undercylindrical bendingrdquo Composite Structures vol 56 no 2 pp131ndash140 2002

[25] W Q Chen and K Y Lee ldquoState-space approach for statics anddynamics of angle-ply laminated cylindrical panels in cylindri-cal bendingrdquo International Journal of Mechanical Sciences vol47 no 3 pp 374ndash387 2005

[26] C-P Wu and Y-S Syu ldquoExact solutions of functionally gradedpiezoelectric shells under cylindrical bendingrdquo InternationalJournal of Solids and Structures vol 44 no 20 pp 6450ndash64722007

[27] W Yan J Ying and W Q Chen ldquoThe behavior of angle-ply laminated cylindrical shells with viscoelastic interfaces incylindrical bendingrdquo Composite Structures vol 78 no 4 pp551ndash559 2007

Shock and Vibration 17

[28] S Brischetto F Tornabene N Fantuzzi and E Viola ldquo3Dexact and 2D generalized differential quadraturemodels for freevibration analysis of functionally graded plates and cylindersrdquoMeccanica vol 51 no 9 pp 2059ndash2098 2016

[29] N Fantuzzi S Brischetto F Tornabene and E Viola ldquo2D and3D shell models for the free vibration investigation of func-tionally graded cylindrical and spherical panelsrdquo CompositeStructures vol 154 pp 573ndash590 2016

[30] S Brischetto and R Torre ldquoExact 3D solutions and finiteelement 2D models for free vibration analysis of plates andcylindersrdquo Curved and Layered Structures vol 1 no 1 pp 59ndash92 2014

[31] F Tornabene S Brischetto N Fantuzzi and E Viola ldquoNumer-ical and exact models for free vibration analysis of cylindricaland spherical shell panelsrdquo Composites Part B Engineering vol81 pp 231ndash250 2015

[32] S Brischetto ldquoExact elasticity solution for natural frequenciesof functionally graded simply-supported structuresrdquo ComputerModeling in Engineering amp Sciences vol 95 no 5 pp 391ndash4302013

[33] S Brischetto ldquoThree-dimensional exact free vibration analysisof spherical cylindrical and flat one-layered panelsrdquo Shock andVibration vol 2014 Article ID 479738 29 pages 2014

[34] S Brischetto ldquoA continuum elastic three-dimensional modelfor natural frequencies of single-walled carbon nanotubesrdquoComposites Part B Engineering vol 61 pp 222ndash228 2014

[35] S Brischetto ldquoAn exact 3d solution for free vibrations ofmultilayered cross-ply composite and sandwich plates andshellsrdquo International Journal of Applied Mechanics vol 6 no 6Article ID 1450076 42 pages 2014

[36] S Brischetto ldquoA continuumshellmodel including vanderWaalsinteraction for free vibrations of double-walled carbon nan-otubesrdquoCMES-ComputerModeling in Engineering and Sciencesvol 104 no 4 pp 305ndash327 2015

[37] S Brischetto F Tornabene N Fantuzzi and M BacciocchildquoRefined 2D and exact 3D shell models for the free vibrationanalysis of single- and double-walled carbon nanotubesrdquo Tech-nologies vol 3 no 4 pp 259ndash284 2015

[38] S Brischetto ldquoConvergence analysis of the exponential matrixmethod for the solution of 3D equilibrium equations for freevibration analysis of plates and shellsrdquo Composites Part BEngineering vol 98 pp 453ndash471 2016

[39] S Brischetto ldquoExact and approximate shell geometry in the freevibration analysis of one-layered and multilayered structuresrdquoInternational Journal of Mechanical Sciences vol 113 pp 81ndash932016

[40] S Brischetto ldquoCurvature approximation effects in the freevibration analysis of functionally graded shellsrdquo InternationalJournal of Applied Mechanics 2016

[41] F Tornabene N Fantuzzi F Ubertini and E Viola ldquoStrong for-mulation finite element method based on differential quadra-ture a surveyrdquo Applied Mechanics Reviews vol 67 no 2 ArticleID 020801 pp 1ndash55 2015

[42] F Tornabene E Viola and N Fantuzzi ldquoGeneral higher-orderequivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panelsrdquo CompositeStructures vol 104 pp 94ndash117 2013

[43] N Fantuzzi F Tornabene E Viola and A J Ferreira ldquoAstrong formulation finite element method (SFEM) based onRBF and GDQ techniques for the static and dynamic analysesof laminated plates of arbitrary shaperdquo Meccanica vol 49 no10 pp 2503ndash2542 2014

[44] F Tornabene N Fantuzzi and M Bacciocchi ldquoFree vibrationsof free-form doubly-curved shells made of functionally gradedmaterials using higher-order equivalent single layer theoriesrdquoComposites Part B Engineering vol 67 pp 490ndash509 2014

[45] F Tornabene and E Viola ldquoStatic analysis of functionallygraded doubly-curved shells and panels of revolutionrdquo Mecca-nica vol 48 no 4 pp 901ndash930 2013

[46] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoDynamic analysis of thick and thin elliptic shell structuresmade of laminated composite materialsrdquo Composite Structuresvol 133 pp 278ndash299 2015

[47] F TornabeneN FantuzziM Bacciocchi and EViola ldquoHigher-order theories for the free vibrations of doubly-curved lam-inated panels with curvilinear reinforcing fibers by meansof a local version of the GDQ methodrdquo Composites Part BEngineering vol 81 pp 196ndash230 2015

[48] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoFree vibrations of composite oval and elliptic cylinders bythe generalized differential quadrature methodrdquo Thin-WalledStructures vol 97 pp 114ndash129 2015

[49] F Tornabene and J N Reddy ldquoFGM and laminated doubly-curved and degenerate shells resting on nonlinear elastic foun-dations a GDQ solution for static analysis with a posterioristress and strain recoveryrdquo Journal of the Indian Institute ofScience vol 93 no 4 pp 635ndash688 2013

[50] N Fantuzzi F Tornabene and E Viola ldquoFour-parameter func-tionally graded cracked plates of arbitrary shape a GDQFEMsolution for free vibrationsrdquo Mechanics of Advanced Materialsand Structures vol 23 no 1 pp 89ndash107 2016

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Shock and Vibration

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International Journal of

Shock and Vibration 5

The cylindrical bending requirements for analytical methodsare obtained choosing one of the two half-wave numberswhich equals zero This condition means 120572 = 119898120587119886 =0 or 120573 = 119899120587119887 = 0 which allows the derivatives madein a particular direction (120572 or 120573) which equals zero tooThis mathematical condition means that each section in thecylindrical bending direction has always the same behavior

It is obvious that the cylindrical bending effect is properfor cylindrical structures only However the authors startedthe present treatise from the governing equations (1) ofspherical shell structures because one of the aims of thepresent contribution is to underline which are the extraterms (related to the curvature) that do not allow having thestructure to be in cylindrical bending conditions

The compact form of (4) has been elaborated in detail in[32] The final version is here proposed

D119895 120597U119895120597119911 = A119895U119895 (5)

where 120597U119895120597119911 = U1198951015840 andU119895 = [119880119895 119881119895 119882119895 1198801198951015840 1198811198951015840 1198821198951015840]Equation (5) is rewritten as

D119895U1198951015840 = A119895U119895 (6)

U1198951015840 = D119895minus1A119895U119895 (7)

U1198951015840 = A119895lowastU119895 (8)

with A119895lowast = D119895minus1A119895The exponential matrix method (also known as transfer

matrix method or the state-space approach) is employed forthe solution of (8) (details are given in [32])

U119895 (119911119895) = exp (A119895lowast119911119895)U119895 (0) with 119911119895 isin [0 ℎ119895] (9)

where 119911119895 is the thickness coordinate for the 119895 mathematicallayer with values from 0 at the bottom to ℎ119895 at the top Theexponential matrix is calculated using 119911119895 = ℎ119895 for each 119895mathematical layer

A119895lowastlowast = exp (A119895lowastℎ119895)= I + A119895lowastℎ119895 + A119895lowast22 ℎ1198952 + A119895lowast33 ℎ1198953 + sdot sdot sdot

+ A119895lowast119873119873 ℎ119895119873(10)

where I indicates the 6times6 identitymatrixThe expansion usedfor the exponentialmatrix in (10) has a fast convergence and itis not time consuming from the computational point of view119895 = 119872 mathematical layers are used to approximate theshell curvature andor the FGM law through the thickness119872 minus 1 transfer matrices must be defined by means ofthe interlaminar continuity conditions of displacements andtransverse shearnormal stresses at each interfaceThe closedform solution for plates or shells is obtained using simply

supported boundary conditions and free stress conditions atthe top and at the bottom surfaces The above conditionsallow the following final system

EU1 (0) = 0 (11)

and the 6 times 6 matrix E does not change independently of thenumber of layers 119872 even if a layer-wise approach is appliedU1(0) is the vector U defined at the bottom of the wholemultilayered shell or plate (first layer 1 with 1199111 = 0) All thesteps and details here omitted for the sake of brevity can befound in [32ndash40] where the proposed exact 3D shell modelhas been elaborated and tested for the first time for severalapplications Several validations for the 3D exact shell modelhave been shown in [32ndash40] For this reason the model isnow used with confidence for the results and comparisonsproposed in Section 4

The nontrivial solution ofU1(0) in (11) allows conductingthe free vibration analysis imposing the zero determinant ofmatrix E

det [E] = 0 (12)

Equation (12) is used to find the roots of an higher order poly-nomial written in 120582 = 1205962The imposition of a couple of (119898 119899)allows a certain number of circular frequencies 120596 = 2120587119891(from I to infin) to be calculated This number of frequenciesdepends on the order 119873 used in each exponential matrixA119895lowastlowast and the number 119872 of mathematical layers employedfor the shell curvature andor the FGM law approximationsThe results shown in [32ndash40] confirm that 119873 = 3 for theexponential matrix and 119872 = 100 for the mathematicallayers are appropriate to always calculate the correct freefrequencies for each structure lamination sequence numberof layers material FGM law and thickness value

3 2D Finite Element Analysis and 2DGeneralized Differential Quadrature Models

The 3D exact natural frequencies are compared with twonumerical 2D solutions given by a weak and a strongformulation The former is a classic 2D finite element (FE)model carried out through the commercial software Strand7The latter is a recent widespread technique known as Gener-alized Differential Quadrature (GDQ) method [41] Detailsrelated to convergence and accuracy of these techniquescan be found in the works [29ndash31] It is noted that theFE solution employs a classic linear theory known alsoas Reissner-Mindlin theory with a linear rotation of thefibers through the shell thickness The present GDQ solutionexploits higher order theories with a general expansionthrough the thickness of the structureThis general approachfalls within the framework of the unified theories becauseseveral structural models can be obtained via the repetitionof a fundamental nucleus Another difference between theGDQ applications and the FE ones is that the formerincludes the curvature effects and the rotary inertias in itsformulation

6 Shock and Vibration

The 2D kinematic hypothesis [42ndash50] considers the fol-lowing expansion of the degrees of freedom

U = 119873119888+1sum120591=0

F120591u(120591) (13)

where U represents the vector of the 3D displacementcomponents which is function of (120572 120573 119911) and u120591 is thevector of the displacement parameters function of (120572 120573)which depends on the free index 120591 Therefore u120591 lays on themidsurface of the shell [42] The matrix F120591(119894119895) = 120575119894119895119865120591 (for119894 119895 = 1 2 3) contains the so-called thickness functions where120575 is the Kronecker delta function The thickness functionsdescribe how the displacement parameters characterize thesolution through the thickness They can have any form ofany smooth function of 119911 however the classic form of thesefunctions is the power function Thus 119865120591 = 119911120591 can be 1(120591 = 0) 119911 (120591 = 1) 1199112 (120591 = 2) 1199113 (120591 = 3) and so onPlease note that according to the mathematical backgroundof the present Unified Formulation the index 119873119888 + 1 isreserved to the (optional) zigzag effect Since the aim of thepresent paper is to investigate the cylindrical bending effectfor functionally graded plates and cylindrical panels thefollowing numerical applications will consider a fixed valueof 119873119888 = 3 so that 120591 = 0 1 2 3 This choice is justified by thefact that third order theories are sufficient with a reasonablylimited number of degrees of freedom to obtain accuratesolutions when compared to 3D shell models [42ndash50] Formultilayered structures such as sandwich configurations thezigzag function must be added into the formulation Onceagain as shown in (13) the zigzag function is embedded atthe end of the expansion with the exponent 119873119888 + 1 Whensuch effect is neglected the sum of (13) stops at 119873119888

The same theory used for the FE solution can be carriedout in the GDQmethod setting 120591 = 0 1 and 119873119888 = 1 howeverit has been shown in other recent works that the strong formapproaches led tomore accurate results with respect to the FEones [46 48]

The relation between generalized strains 120576(120591) anddisplace-ments u(120591) can be set as

120576(120591) = DΩu

(120591) for 120591 = 0 1 2 119873119888 119873119888 + 1 (14)

where DΩ is a differential operator explicitly defined in [42]The present constitutive material model is linear and elastictherefore for a general 119896th laminaHookersquos law can be applied

120590(119896) = C(119896)120576(119896) (15)

where 120590(119896) is the vector containing the stress componentsand C(119896) represents the constitutive matrix for each 119896th ply[42] of the structure Each component of the constitutivematrix is indicated by the symbol 119862(119896)120585120578 and it is valid in thecurvilinear reference system 1198741015840120572120573119911 of the reference geome-tryThese coefficients are evaluatedwith the application of thetransformation matrix [42]

The stress resultants are the terms which correspond tothe punctual 3D stresses of the 3D doubly curved solidTheir definition can be achieved by developing the Hamiltonprinciple for the present structural theory [42]

S(120591) = 119873119871sum119896=1

int119911119896+1119911119896

(Z(120591))119879 120590(120591)119867120572119867120573119889119911for 120591 = 0 1 2 119873119888 119873119888 + 1

(16)

whereZ(120591) is a 6times9matrix containing the thickness functions119865120591 and the curvature coefficients119867120572 and119867120573 the definition ofsuchmatrix with some other details can be found in [42] S(120591)include the stress resultants as a function of the order 120591

The definition of (16) can be rewritten as

S(120591) = 119873119888+1sum119904=0

A(120591119904)120576(119904) for 120591 = 0 1 2 119873119888 119873119888 + 1 (17)

which is a function of generalized strains of order 119904 120576(119904)Note that the so-called elastic coefficients are defined as

A(120591119904) = 119873119871sum119896=1

int119911119896+1119911119896

(Z(120591))119879C(119896)Z(119904)119867120572119867120573119889119911 (18)

and they can be carried out by the following integral expres-sions

119860(120591119904)120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119911119904119911120591119867120572119867120573119867984858120572119867120577120573 119889119911

119860(119904)120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119911119904120591119911120591minus1119867120572119867120573119867984858120572119867120577120573 119889119911

119860(120591)120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119904119911119904minus1119911120591119867120572119867120573119867984858120572119867120577120573 119889119911

Shock and Vibration 7

119860( )120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119904119911119904minus1120591119911120591minus1119867120572119867120573119867984858120572119867120577120573 119889119911for 120591 119904 = 0 1 2 119873119888 119873119888 + 1 120585 120578 = 1 2 3 4 5 6 984858 120577 = 0 1 2

(19)

It is recalled that the exponent 119873119888 + 1 is reserved for theMurakami function otherwise this order is neglected Thesubscripts and indicate the derivatives of the thicknessfunctions 119865120591 and 119865119904 with respect to 119911 The subscripts 984858 and 120577are the exponents of the quantities119867120572 and 119867120573 whereas 120585 and120578 are the indices of the material constants 119862(119896)120585120578 for the 119896th ply

[42ndash49]Note that119862(119896)120585120578 119867120572 and119867120573 are quantities that dependon the thickness 119911 Therefore the integrals of (19) cannotbe computed analytically The present solution for solvingsuch integrals is to employ a numerical integration algorithmIn particular the so-called Generalized Integral Quadrature(GIQ) method is utilized For an accurate evaluation of theintegrals 51 points per layer are considered in the integration

The Hamilton principle supplies for the dynamic equilib-rium equations and the correspondent boundary conditionsIn detail the fundamental nucleus of three motion equationstakes the form

D⋆ΩS(120591) = 119873119888+1sum

119904=0

M(120591119904)u(119904)

for 120591 = 0 1 2 119873119888 119873119888 + 1(20)

whereD⋆Ω is the equilibrium operator andM(120591119904) is the inertialmatrix Their explicit expressions have been presented in aprevious authorsrsquo paper [42]

Combining the kinematic equation (14) constitutiveequation (17) and themotion equation (20) the fundamentalgoverning system in terms of displacement parameters iscarried out

119873119888+1sum119904=0

L(120591119904)u(119904) = 119873119888+1sum119904=0

M(120591119904)u(119904)

for 120591 = 0 1 2 119873119888 119873119888 + 1(21)

where L(120591119904) = D⋆ΩA(120591119904)DΩ is the fundamental operator [42]

Boundary conditions must be included for the solutionof the differential equations (21) The present paper dealsonly with free and simply supported boundary conditionsThe former is used to reproduce the cylindrical bendingconditions considering alternatively one free edge and onesimply supported edge When the four edges are all simplysupported the problem is the same as the ones investigated

in the previous papers [29ndash31] Simply supported conditionsare

119873(120591)120572 = 0119906(120591)120573 = 119906(120591)119911 = 0

120591 = 0 1 2 119873119888 119873119888 + 1 at 120572 = 1205720 or 120572 = 1205721 1205730 le 120573 le 1205731119906(120591)120572 = 0

119873(120591)120573 = 0119906(120591)119911 = 0

120591 = 0 1 2 119873119888 119873119888 + 1 at 120573 = 1205730 or 120573 = 1205731 1205720 le 120572 le 1205721

(22)

Free edge conditions are

119873(120591)120572 = 0119873(120591)120573 = 0119879(120591)120572 = 0120591 = 0 1 2 119873119888 119873119888 + 1 at 120572 = 1205720 or 120572 = 1205721 1205730 le 120573 le 1205731119873(120591)120572 = 0119873(120591)120573 = 0119879(120591)120573 = 0120591 = 0 1 2 119873119888 119873119888 + 1 at 120573 = 1205730 or 120573 = 1205731 1205720 le 120572 le 1205721

(23)

4 Results

The present section includes some numerical applicationsfor the comparison of the 3D exact shell vibrations withtwo numerical 2D models In particular classical simplysupported solutions and cylindrical bending (CB) behaviorof plates and shells are presentedThe following computationsfocus on plates circular cylinders and cylindrical panels dueto the fact that the CB effect occurs only when at least twoparallel straight edges are present in the structure

The first case is a square plate (119886 = 119887 = 1m) withsimply supported edges Two thickness ratios are analyzedthin 119886ℎ = 100 and moderately thick 119886ℎ = 20 structure Theplate is made of a single functionally graded layer (ℎ1 = ℎ)with volume fraction of ceramic (119888) phase

119881119888 (119911) = (05 + 119911ℎ)119901 with 119911 isin [minusℎ2 ℎ2 ] (24)

8 Shock and Vibration

Table 1 One-layered FGM plate with 119901 = 05 Analytical 3D versus 2D numerical models Frequency 119891 in Hz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119886ℎ = 1001 0 I 4019 mdash 3907 mdash 39210 1 I 4019 mdash 3907 mdash 39211 1 I 8037 8006 mdash 8036 mdash2 0 I 1607 mdash 1580 mdash 15840 2 I 1607 mdash 1580 mdash 15842 1 I 2008 2001 mdash 2008 mdash1 2 I 2008 2001 mdash 2008 mdash2 2 I 3212 3202 mdash 3211 mdash3 0 I 3613 mdash 3568 mdash 35770 3 I 3613 mdash 3568 mdash 35773 1 I 4014 4002 mdash 4013 mdash1 3 I 4014 4002 mdash 4013 mdash3 2 I 5215 5202 mdash 5215 mdash2 3 I 5215 5202 mdash 5215 mdash4 0 I 6416 mdash 6354 mdash 63650 4 I 6416 mdash 6354 mdash 63654 1 I 6816 6801 mdash 6815 mdash1 4 I 6816 6801 mdash 6815 mdash119886ℎ = 201 0 I 2002 mdash 1951 mdash 19510 1 I 2002 mdash 1951 mdash 19511 1 I 3989 3995 mdash 3988 mdash2 0 I 7916 mdash 7864 mdash 77970 2 I 7916 mdash 7864 mdash 77972 1 I 9858 9954 mdash 9857 mdash1 2 I 9858 9954 mdash 9857 mdash2 2 I 1560 1588 mdash 1560 mdash3 0 I 1749 mdash 1767 mdash 17290 3 I 1749 mdash 1767 mdash 17293 1 I 1936 1980 mdash 1936 mdash1 3 I 1936 1980 mdash 1936 mdash3 2 I 2490 2566 mdash 2490 mdash2 3 I 2490 2566 mdash 2490 mdash1 0 II (119908 = 0) 2775 2773 mdash 2775 mdash0 1 II (119908 = 0) 2775 2773 mdash 2775 mdash4 0 I 3034 mdash 3124 mdash 30070 4 I 3034 mdash 3124 mdash 3007

where 119901 is the power exponent which varies the volumefraction of the ceramic component through the thicknessTwo limit cases occur for (24) 119881119888 = 1 when 119901 = 0 and119881119888 rarr 0when119901 = infin In the following computations it is set as119901 = 05 Note that the volume fraction function in (24) givesa material which is ceramic (119888) rich at the top and metal (119898)rich at the bottomof the structureThemechanical properties(Youngmodulus andmass density) are given by the theory ofmixtures as

119864 (119911) = 119864119898 + (119864119888 minus 119864119898) 119881119888120588 (119911) = 120588119898 + (120588119888 minus 120588119898) 119881119888 (25)

where 119864119898 and 120588119898 are the elastic modulus and mass densityof the metallic phase respectively 119864119888 and 120588119888 are the corre-sponding properties of the ceramic phase Note that in allthe following computations the Poisson ratio is consideredconstant through the thickness thus ]119888 = ]119898 = ] = 03 Thetwo constituents have the following mechanical properties

119864119898 = 73GPa120588119898 = 2800 kgm3119864119888 = 380GPa120588119888 = 3800 kgm3

(26)

Shock and Vibration 9

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(a) Mode I (0 1) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

000501

01502

(b) Mode I (0 1) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(c) Mode II (119908 = 0) (0 1) 3D

002minus002

090908

08070706

06050504

04030302

020101

00minus01

1

(d) Mode II (119908 = 0) (0 1) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(e) Mode I (1 0) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

0minus005minus01

minus015minus02

(f) Mode I (1 0) 2D

Figure 2 Continued

10 Shock and Vibration

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(g) Mode II (119908 = 0) (1 0) 3D

002minus002

0908 08

0706 06

0504 04

0302 02

01 0 0

11

(h) Mode II (119908 = 0) (1 0) 2D

Figure 2 Differences between in-plane and cylindrical bending vibration modes for the 119886ℎ = 20 one-layered FGM plate with 119901 = 05 3Dmodes versus 2D modes

Since the FE commercial software does not have anembedded tool for the computation of functionally gradedproperties through the thickness the present FE applicationsare carried out building a fictitious laminated structure withisotropic plies In detail the thickness of the FGM lamina hasbeen divided into 100 fictitious layers and for each layer thecorresponding mechanical properties have been computedthrough (25) The numerical model has been made of auniform element division of 40 times 40 Quad8 elements for atotal number of 29280 degrees of freedom (dofs) The GDQsolution is computed with a not-uniform Chebyshev-Gauss-Lobatto grid [41] with 31 times 31 points for both thicknessratios It is underlined that since a single ply is set no zigzagfunction is needed in the computation It is recalled that thepresent higher order GDQ model considers a third orderexpansion with 120591 = 0 1 2 3 whereas the FE one has afirst order model (also known as Reissner-Mindlin model)Table 1 reports the results in terms of natural frequency(Hz) wherein the first column reports the number of half-waves (119898 119899) of the 3D exact solution Note that for a fixedset of half-wave numbers different mode shapes can occur(indicated by the second column ldquoModerdquo) In particular Iindicates the first mode and II indicates the second mode Inthe case of in-plane modes the caption (119908 = 0) is added tounderline that the vibration occurs in the plane and not outof plane The third column shows all the results for the 3Dexact solution both classical and cylindrical bending (CB)results It is noted that these frequencies are obtained witha single run of the 3D exact code keeping simply supportedboundary conditions for all the four edges of the plate Theother four columns of Table 1 refer to the numerical solutionsfor simply supported structures and for cylindrical bendingconditions (setting two free opposite edges) The cylindricalbending results are indicated by the acronym CB in the sametable

There is a very good agreement among the three solutionswhen simply supported conditions are considered On thecontrary a small error is observed in cylindrical bendingbecause the actual CB condition is not completely satisfieddue to the fact that the plate has a limited dimension in thesimply supported direction This feature can be detected inFigures 2 and 3 because the deformed shapes of the plate inthe CB configuration are not properly correct In fact thefibers parallel to the supported edges are not parallel amongthem but there is a curvature effect due to the free boundariesand to the limited length of the plate On the contrary thein-plane mode shapes (119908 = 0) when the plate is simplysupported on the four edges have all the fibers parallel amongthem also after the deformation These modes are the resultsobtained in the 3D solution with 119898 = 0 or 119899 = 0 combinedwith 119908 = 0 They are not cylindrical bending modes infact they can be found with 2D numerical models withoutmodifying the boundary conditions 3D frequencieswith119898 =0 or 119899 = 0 (with w different from 0) are CB modes which canbe obtained considering two simply supported edges and twofree edges in 2D numerical models

As it is well-known the ldquoperfectrdquo CB would occur whenthe simply supported direction has an ldquoinfiniterdquo lengthThis cannot be proven numerically but a trend can beobserved modeling different plates in cylindrical bendingconfiguration with different lengthsThese analyses are madekeeping one side of the plate constant and enlarging thelength of the other edge In the present computations 119886 iskept constant and 119887 = 119871 varies from 1 to 20 The results ofthe aforementioned analysis are listed in Table 2 and depictedin Figures 4 and 5 Table 2 includes the numerical valuesof the first mode shape in cylindrical bending for the 3Dexact solution the 2D FE and the 2D GDQmodels Figure 4shows that both GDQ and FE solutions tend to the 3D exactone increasing the length of the simply supported edge In

Shock and Vibration 11

ulowast

lowast

wlowast

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

(a) Mode I (1 0) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

000501

01502

(b) Mode I (1 0) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(c) Mode I (2 0) 3D

109

0807

060504

0302

01 0

109

0807

0605

0403

02010

0201501

0050

minus005minus01

minus015minus02

(d) Mode I (2 0) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(e) Mode I (3 0) 3D

10908

0706

050403

0201 0

109

0807

0605

0403

02010

0201501

0050

minus005minus01

minus015minus02

(f) Mode I (3 0) 2D

Figure 3 Continued

12 Shock and Vibration

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(g) Mode I (4 0) 3D

10908

0706050403

0201 0

10908

0706

0504

0302

010

0201501

0050

minus005minus01

minus015minus02

(h) Mode I (4 0) 2D

Figure 3 First four cylindrical bending vibration modes for the 119886ℎ = 20 one-layered FGM plate with 119901 = 05 3D modes versus 2D modes

3D exact2D GDQ2D FEM

f(H

z)

39

392

394

396

398

40

402

404

5 10 15 200La

Figure 4 First frequency 119891 in Hz for the 119886ℎ = 100 one-layeredFGM plate with 119901 = 05 convergence of 2D GDQ and 2D FEsolutions increasing the length of the simply supported sides

particular the GDQ solution is slightly more accurate thanthe FE one The 2D GDQ solution is carried out by enlargingthe length of one of the sides of the plate and the samenumberof dofs is kept On the contrary the FE model is generatedby 40 times 40 blocks of mesh In other words the plate with119871 = 2119886 has a mesh of 40 times 80 the one with 119871 = 3119886 has amesh 40 times 120 and so on When 119871 = 20119886 the error amongthe 3D exact and the 2D numerical solutions is very small andthis is proven also by the mode shapes of Figure 5 whereinthe boundary effect is negligible with respect the length of

the simply supported edges It must be underlined that it isimpossible to completely avoid the boundary effects due tothe free edges

The second example is related to a simply supportedcylinder with 119877120572 = 10m 119877120573 = infin 119886 = 2120587119877120572 and 119887 =20m and thickness ratios 119877120572ℎ = 100 and 119877120572ℎ = 10 Aone-layered FGM scheme is considered with ℎ1 = ℎ andthe same volume fraction function of (24) for the previousexample but setting 119901 = 20 The present material propertiesare the same as the flat plate The FE model has a 20 times80 Quad8 mesh with 27966 dofs and the GDQ solutionhas a 15 times 51 Chebyshev-Gauss-Lobatto grid Numericallyspeaking compatibility conditions must be enforced on theclosing meridian in order to obtain a closed revolution shell[42] To obtain the CB conditions numerically the cylinderis modeled completely free and the first 6 rigid mode shapesare neglected and removed from the numerical solution Itis noted that the GDQ model uses a third-order kinematicexpansion with no zigzag effect as the previous case sincea one-layer structure has been taken into account Table 3reports the results with the same meaning of the symbolsseen in Table 1 A very good agreement is observed when thecylinder is simply supported whereas when the cylinder iscompletely free some small errors can be observed but thenumerical solutions follow the 3D exact one

Results with 119899 = 0 and 119908 = 0 indicate the CB conditionsSuch results are obtained by means of 2D numerical modelsmodifying the boundary conditions (see 2D FE(CB) and 2DGDQ(CB)models)The cases 119899 = 0 combined with119908 = 0 arenot CB conditions and the frequencies can be found withoutmodifying the boundary conditions in the 2D numericalmodels

The final test is related to a cylindrical panel with 119877120572 =10m 119877120573 = infin 119886 = 1205873119877120572 and 119887 = 20m and thickness ratios119877120572ℎ = 100 and 119877120572ℎ = 10 The present shell is a sandwichstructure embedding a FGM core with ℎ2 = 07ℎ and two

Shock and Vibration 13

218

1614

121 1

08 0806 06

04 0402 02

0 0

0minus01minus02minus03minus04

(a) 119871119886 = 2

3

25

2

151

0806

040502

0 0

1

0minus01minus02minus03minus04minus05

(b) 119871119886 = 3

435

325

10505

0 0

15

0

minus02minus04minus06

1

2

(c) 119871119886 = 4

545

35

10505

0 0

0

minus04minus02

minus06minus08

1

25

4

3

215

(d) 119871119886 = 5

5

1

0 005

0

minus04minus02

minus06minus08

1

4

3

2

6

minus1

(e) 119871119886 = 6

5

10 0 051

43

2

67

89

100

minus05minus1

minus15

(f) 119871119886 = 10

Figure 5 First vibration mode for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 2D GDQ solution and edge effects versus the lengthof the simply supported sides

external skins with ℎ1 = ℎ3 = 015ℎ (ℎ1 + ℎ2 + ℎ3 = ℎ) Thebottom layer ismade of isotropicmetallicmaterial and the toplayer is made of ceramic material the mechanical propertiesare

119864119898 = 73GPa120588119898 = 2800 kgm3119864119888 = 200GPa120588119888 = 5700 kgm3

(27)

The FGM core has a volume fraction function defined as

119881119888 (119911) = (05 + 11991107ℎ)119901 with 119911 isin [minus035ℎ 035ℎ] (28)

where 119901 = 10 is employed in the present computations ThePoisson ratio is constant (]119888 = ]119898 = ] = 03) Equations (25)are still valid with 119881119888 defined as in (28)

The FEmodel is made of 40times40Quad8 elements for total29280 dofs and the GDQ model has a 31 times 31 Chebyshev-Gauss-Lobatto grid Since a sandwich structure is analyzedin the following a third-order theory with zigzag functionis considered in the computations for the GDQ method Thepresent natural frequencies are reported in Table 4 with the

14 Shock and Vibration

Table 2 First frequency 119891 in Hz for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 convergence of 2D GDQ and 2D FE solutionsincreasing the length of the simply supported sides

119871119886 1 2 3 4 5 6 8 10 15 20 3D exact2D GDQ 3921 3964 3980 3988 3993 3997 4001 4004 4007 4008 40192D FEM 3907 3949 3966 3974 3979 3983 3987 3989 3993 3994 4019

Table 3 One-layered FGM cylinder with 119901 = 20 Analytical 3D versus 2D numerical models Frequency 119891 in Hz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1004 0 I 0896 mdash 0897 mdash 08836 0 I 2532 mdash 2538 mdash 25118 0 I 4854 mdash 4867 mdash 481910 0 I 7848 mdash 7871 mdash 780112 0 I 1151 mdash 1155 mdash 114610 1 I 1356 1360 mdash 1357 mdash12 1 I 1441 1449 mdash 1445 mdash14 0 I 1584 mdash 1589 mdash 15778 1 I 1638 1640 mdash 1636 mdash14 1 I 1756 1767 mdash 1763 mdash16 0 I 2083 mdash 2090 mdash 207016 1 I 2207 2222 mdash 2216 mdash6 1 I 2441 2442 mdash 2438 mdash18 0 I 2649 mdash 2658 mdash 264014 2 I 2742 2752 mdash 2746 mdash18 1 I 2751 2772 mdash 2763 mdash16 2 I 2875 2890 mdash 2883 mdash12 2 I 2921 2926 mdash 2922 mdash119877120572ℎ = 104 0 I 8774 mdash 9085 mdash 86796 0 I 2456 mdash 2571 mdash 24306 1 I 3978 4089 mdash 3979 mdash4 1 I 4388 4357 mdash 4388 mdash8 0 I 4654 mdash 4920 mdash 46028 1 I 5575 5905 mdash 5576 mdash2 1 I 7202 7101 mdash 7202 mdash2 0 I (119908 = 0) 7322 7390 mdash 7319 mdash10 0 I 7419 mdash 7933 mdash 73476 2 I 8110 8261 mdash 8111 mdash10 1 I 8189 8811 mdash 8191 mdash8 2 I 8783 9173 mdash 8785 mdash4 2 I 9016 9084 mdash 9017 mdash12 0 I 1071 mdash 1159 mdash 106110 2 I 1081 1155 mdash 1081 mdash2 2 I 1089 1105 mdash 1089 mdash12 1 I 1142 1245 mdash 1143 mdash4 3 I 1263 1286 mdash 1263 mdash

same meaning of the columns seen in the previous casesIn order to achieve the cylindrical bending conditions thenumerical models have the straight edges simply supportedand the curved edges as free Due to the CB definitionit is not possible to achieve CB conditions when 119898 = 0

and 119899 = 1 2 3 because the cylindrical bending shouldoccur on the curved edge and this is not physically possiblebecause of the variation of displacement components due tothe curvature presence This aspect is indicated as NA (NotApplicable) solution in Table 4 Therefore this kind of 3D

Shock and Vibration 15

Table 4 Sandwich cylindrical shell panel embedding FGM core with 119901 = 10 Analytical 3D versus 2D numerical models Frequency 119891 inHz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1001 0 I 1948 mdash 1951 mdash 19412 0 I 8856 mdash 8872 mdash 88232 1 I 1104 1110 mdash 1104 mdash1 1 I 1850 1873 mdash 1850 mdash3 0 I 2038 mdash 2042 mdash 20303 1 I 2116 2128 mdash 2116 mdash2 2 I 2221 2242 mdash 2221 mdash3 2 I 2484 2496 mdash 2484 mdash3 3 I 3235 3254 mdash 3235 mdash4 0 I 3649 mdash 3660 mdash 36342 3 I 3692 3739 mdash 3692 mdash4 1 I 3714 3745 mdash 3714 mdash4 2 I 3940 3965 mdash 3941 mdash3 4 I 4226 4261 mdash 4226 mdash4 3 I 4378 4403 mdash 4378 mdash1 2 I 4568 4664 mdash 4568 mdash4 4 I 5035 5064 mdash 5035 mdash2 4 I 5040 5128 mdash 5040 mdash119877120572ℎ = 101 0 I 1905 mdash 1963 mdash 18841 1 I 3054 3145 mdash 3055 mdash1 2 I 6209 6409 mdash 6209 mdash2 0 I 8326 mdash 8857 mdash 82540 1 I 8710 mdash NA mdash NA0 1 II (119908 = 0) 8820 8956 mdash 8820 mdash2 1 I 8880 9531 mdash 8882 mdash0 2 I 9176 mdash NA mdash NA1 3 I 9576 9983 mdash 9567 mdash0 3 I 1036 mdash NA mdash NA2 2 I 1066 1147 mdash 1067 mdash0 4 I 1291 mdash NA mdash NA1 4 I 1335 1413 mdash 1335 mdash2 3 I 1364 1475 mdash 1364 mdash1 0 II (119908 = 0) 1665 1738 mdash 1665 mdash0 5 I 1683 mdash NA mdash NA2 4 I 1755 1918 mdash 1755 mdash0 2 II (119908 = 0) 1764 1791 mdash 1764 mdash

exact solutions cannot be achieved by any numerical modelOnce again solution with 119899 = 0 or 119898 = 0 combined with119908 = 0 (in-plane modes) can be found with 2D numericalmodels without modifying the simply supported boundaryconditions

5 Conclusions

Thepresent paper analyzes the cylindrical bending in the freefrequency analysis of functionally graded material (FGM)plates and cylindrical shells This condition is investigatedby means of a 3D exact shell model and two different 2D

numerical models a weak formulation based on classicalFinite Elements (FEs) and a strong formulation based onthe GDQ method The cylindrical bending (CB) behavior inthe 3D exact shell model can be analyzed using completelysimply supported structures and supposing one of the twohalf-wave numbers equals zeroThe numerical models obtainthe CB behavior using mixed simply supported and freeboundary conditions It has been demonstrated that theactual cylindrical bending (CB) is possible only with respectto straight edgesTheCBnatural frequencies of the numericalsolutions converge to the 3D exact CB ones if the length ofstraight simply supported edges is increased up to 20 times

16 Shock and Vibration

the value of the original edges The CB conditions are notpossible in 2D numericalmodels when the half-wave numberis zero in the curvilinear edge in this case the modificationof boundary conditions does not properly work In-planevibrationmodeswith zero transverse displacement combinedwith 119898 = 0 or 119899 = 0 cannot be defined as CB frequencies andthey can be calculated via 2D numerical models without themodification of the simply supported conditions for the fouredges All the considerations about the cylindrical bendingare valid for each geometry (plate cylinder and cylindricalshell) lamination scheme FGM law and thickness ratio

Competing Interests

The authors declare that they have no competing interests

References

[1] W Q Chen J B Cai and G R Ye ldquoResponses of cross-ply laminates with viscous interfaces in cylindrical bendingrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 823ndash835 2005

[2] W Q Chen and K Y Lee ldquoThree-dimensional exact analysisof angle-ply laminates in cylindrical bending with interfacialdamage via state-space methodrdquo Composite Structures vol 64no 3-4 pp 275ndash283 2004

[3] W Q Chen and K Y Lee ldquoTime-dependent behaviors of angle-ply laminates with viscous interfaces in cylindrical bendingrdquoEuropean Journal ofMechanicsmdashASolids vol 23 no 2 pp 235ndash245 2004

[4] A H Gandhi and H K Raval ldquoAnalytical and empiricalmodeling of top roller position for three-roller cylindricalbending of plates and its experimental verificationrdquo Journal ofMaterials Processing Technology vol 197 no 1ndash3 pp 268ndash2782008

[5] S Oral and H Darendeliler ldquoThe optimum die profile for thecylindrical bending of platesrdquo Journal of Materials ProcessingTechnology vol 70 no 1ndash3 pp 151ndash155 1997

[6] V A Jairazbhoy P Petukhov and J Qu ldquoLarge deflectionof thin plates in cylindrical bendingmdashnon-unique solutionsrdquoInternational Journal of Solids and Structures vol 45 no 11-12pp 3203ndash3218 2008

[7] A Lebee and K Sab ldquoA bending-gradient model for thickplates part II closed-form solutions for cylindrical bending oflaminatesrdquo International Journal of Solids and Structures vol 48no 20 pp 2889ndash2901 2011

[8] P V Nimbolkar and I M Jain ldquoCylindrical bending of elasticplatesrdquo Procedia Materials Science vol 10 pp 793ndash802 2015

[9] A S Sayyad and Y M Ghugal ldquoA nth-order shear deformationtheory for composite laminates in cylindrical bendingrdquo Curvedand Layered Structures vol 2 no 1 pp 290ndash300 2015

[10] N J Pagano ldquoExact solutions for composite laminates incylindrical bendingrdquo Journal of Composite Materials vol 3 no3 pp 398ndash411 1969

[11] N J Pagano and A S D Wang ldquoFurther study of compositelaminates under cylindrical bendingrdquo Journal of CompositeMaterials vol 5 pp 421ndash428 1971

[12] N Saeedi K Sab and J-F Caron ldquoCylindrical bending ofmultilayered plates with multi-delamination via a layerwisestress approachrdquo Composite Structures vol 95 pp 728ndash7392013

[13] X-P Shu and K P Soldatos ldquoCylindrical bending of angle-ply laminates subjected to different sets of edge boundaryconditionsrdquo International Journal of Solids and Structures vol37 no 31 pp 4289ndash4307 2000

[14] Z G Bian W Q Chen C W Lim and N Zhang ldquoAnalyticalsolutions for single- and multi-span functionally graded platesin cylindrical bendingrdquo International Journal of Solids andStructures vol 42 no 24-25 pp 6433ndash6456 2005

[15] H M Navazi and H Haddadpour ldquoNonlinear cylindricalbending analysis of shear deformable functionally graded platesunder different loadings using analytical methodsrdquo Interna-tional Journal of Mechanical Sciences vol 50 no 12 pp 1650ndash1657 2008

[16] W Q Chen J Ying J B Cai and G R Ye ldquoBenchmarksolution of imperfect angle-ply laminated rectangular plates incylindrical bending with surface piezoelectric layers as actuatorand sensorrdquo Computers and Structures vol 82 no 22 pp 1773ndash1784 2004

[17] W Q Chen and K Y Lee ldquoExact solution of angle-plypiezoelectric laminates in cylindrical bending with interfacialimperfectionsrdquo Composite Structures vol 65 no 3-4 pp 329ndash337 2004

[18] T Kant and SM Shiyekar ldquoCylindrical bending of piezoelectriclaminates with a higher order shear and normal deformationtheoryrdquo Computers and Structures vol 86 no 15-16 pp 1594ndash1603 2008

[19] P Lu H P Lee and C Lu ldquoAn exact solution for functionallygraded piezoelectric laminates in cylindrical bendingrdquo Interna-tional Journal of Mechanical Sciences vol 47 no 3 pp 437ndash4382005

[20] W Yan J Wang and W Q Chen ldquoCylindrical bendingresponses of angle-ply piezoelectric laminates with viscoelasticinterfacesrdquo Applied Mathematical Modelling vol 38 no 24 pp6018ndash6030 2014

[21] Y Y Zhou W Q Chen and C F Lu ldquoSemi-analytical solutionfor orthotropic piezoelectric laminates in cylindrical bendingwith interfacial imperfectionsrdquo Composite Structures vol 92no 4 pp 1009ndash1018 2010

[22] G P Dube S Kapuria and P C Dumir ldquoExact piezother-moelastic solution of a simply-supported orthotropic flat panelin cylindrical bendingrdquo International Journal of MechanicalSciences vol 38 pp 374ndash387 1996

[23] V N Pilipchuk V L Berdichevsky and R A IbrahimldquoThermo-mechanical coupling in cylindrical bending of sand-wich platesrdquoComposite Structures vol 92 no 11 pp 2632ndash26402010

[24] C Zhang S Di and N Zhang ldquoA new procedure for staticanalysis of thermo-electric laminated composite plates undercylindrical bendingrdquo Composite Structures vol 56 no 2 pp131ndash140 2002

[25] W Q Chen and K Y Lee ldquoState-space approach for statics anddynamics of angle-ply laminated cylindrical panels in cylindri-cal bendingrdquo International Journal of Mechanical Sciences vol47 no 3 pp 374ndash387 2005

[26] C-P Wu and Y-S Syu ldquoExact solutions of functionally gradedpiezoelectric shells under cylindrical bendingrdquo InternationalJournal of Solids and Structures vol 44 no 20 pp 6450ndash64722007

[27] W Yan J Ying and W Q Chen ldquoThe behavior of angle-ply laminated cylindrical shells with viscoelastic interfaces incylindrical bendingrdquo Composite Structures vol 78 no 4 pp551ndash559 2007

Shock and Vibration 17

[28] S Brischetto F Tornabene N Fantuzzi and E Viola ldquo3Dexact and 2D generalized differential quadraturemodels for freevibration analysis of functionally graded plates and cylindersrdquoMeccanica vol 51 no 9 pp 2059ndash2098 2016

[29] N Fantuzzi S Brischetto F Tornabene and E Viola ldquo2D and3D shell models for the free vibration investigation of func-tionally graded cylindrical and spherical panelsrdquo CompositeStructures vol 154 pp 573ndash590 2016

[30] S Brischetto and R Torre ldquoExact 3D solutions and finiteelement 2D models for free vibration analysis of plates andcylindersrdquo Curved and Layered Structures vol 1 no 1 pp 59ndash92 2014

[31] F Tornabene S Brischetto N Fantuzzi and E Viola ldquoNumer-ical and exact models for free vibration analysis of cylindricaland spherical shell panelsrdquo Composites Part B Engineering vol81 pp 231ndash250 2015

[32] S Brischetto ldquoExact elasticity solution for natural frequenciesof functionally graded simply-supported structuresrdquo ComputerModeling in Engineering amp Sciences vol 95 no 5 pp 391ndash4302013

[33] S Brischetto ldquoThree-dimensional exact free vibration analysisof spherical cylindrical and flat one-layered panelsrdquo Shock andVibration vol 2014 Article ID 479738 29 pages 2014

[34] S Brischetto ldquoA continuum elastic three-dimensional modelfor natural frequencies of single-walled carbon nanotubesrdquoComposites Part B Engineering vol 61 pp 222ndash228 2014

[35] S Brischetto ldquoAn exact 3d solution for free vibrations ofmultilayered cross-ply composite and sandwich plates andshellsrdquo International Journal of Applied Mechanics vol 6 no 6Article ID 1450076 42 pages 2014

[36] S Brischetto ldquoA continuumshellmodel including vanderWaalsinteraction for free vibrations of double-walled carbon nan-otubesrdquoCMES-ComputerModeling in Engineering and Sciencesvol 104 no 4 pp 305ndash327 2015

[37] S Brischetto F Tornabene N Fantuzzi and M BacciocchildquoRefined 2D and exact 3D shell models for the free vibrationanalysis of single- and double-walled carbon nanotubesrdquo Tech-nologies vol 3 no 4 pp 259ndash284 2015

[38] S Brischetto ldquoConvergence analysis of the exponential matrixmethod for the solution of 3D equilibrium equations for freevibration analysis of plates and shellsrdquo Composites Part BEngineering vol 98 pp 453ndash471 2016

[39] S Brischetto ldquoExact and approximate shell geometry in the freevibration analysis of one-layered and multilayered structuresrdquoInternational Journal of Mechanical Sciences vol 113 pp 81ndash932016

[40] S Brischetto ldquoCurvature approximation effects in the freevibration analysis of functionally graded shellsrdquo InternationalJournal of Applied Mechanics 2016

[41] F Tornabene N Fantuzzi F Ubertini and E Viola ldquoStrong for-mulation finite element method based on differential quadra-ture a surveyrdquo Applied Mechanics Reviews vol 67 no 2 ArticleID 020801 pp 1ndash55 2015

[42] F Tornabene E Viola and N Fantuzzi ldquoGeneral higher-orderequivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panelsrdquo CompositeStructures vol 104 pp 94ndash117 2013

[43] N Fantuzzi F Tornabene E Viola and A J Ferreira ldquoAstrong formulation finite element method (SFEM) based onRBF and GDQ techniques for the static and dynamic analysesof laminated plates of arbitrary shaperdquo Meccanica vol 49 no10 pp 2503ndash2542 2014

[44] F Tornabene N Fantuzzi and M Bacciocchi ldquoFree vibrationsof free-form doubly-curved shells made of functionally gradedmaterials using higher-order equivalent single layer theoriesrdquoComposites Part B Engineering vol 67 pp 490ndash509 2014

[45] F Tornabene and E Viola ldquoStatic analysis of functionallygraded doubly-curved shells and panels of revolutionrdquo Mecca-nica vol 48 no 4 pp 901ndash930 2013

[46] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoDynamic analysis of thick and thin elliptic shell structuresmade of laminated composite materialsrdquo Composite Structuresvol 133 pp 278ndash299 2015

[47] F TornabeneN FantuzziM Bacciocchi and EViola ldquoHigher-order theories for the free vibrations of doubly-curved lam-inated panels with curvilinear reinforcing fibers by meansof a local version of the GDQ methodrdquo Composites Part BEngineering vol 81 pp 196ndash230 2015

[48] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoFree vibrations of composite oval and elliptic cylinders bythe generalized differential quadrature methodrdquo Thin-WalledStructures vol 97 pp 114ndash129 2015

[49] F Tornabene and J N Reddy ldquoFGM and laminated doubly-curved and degenerate shells resting on nonlinear elastic foun-dations a GDQ solution for static analysis with a posterioristress and strain recoveryrdquo Journal of the Indian Institute ofScience vol 93 no 4 pp 635ndash688 2013

[50] N Fantuzzi F Tornabene and E Viola ldquoFour-parameter func-tionally graded cracked plates of arbitrary shape a GDQFEMsolution for free vibrationsrdquo Mechanics of Advanced Materialsand Structures vol 23 no 1 pp 89ndash107 2016

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6 Shock and Vibration

The 2D kinematic hypothesis [42ndash50] considers the fol-lowing expansion of the degrees of freedom

U = 119873119888+1sum120591=0

F120591u(120591) (13)

where U represents the vector of the 3D displacementcomponents which is function of (120572 120573 119911) and u120591 is thevector of the displacement parameters function of (120572 120573)which depends on the free index 120591 Therefore u120591 lays on themidsurface of the shell [42] The matrix F120591(119894119895) = 120575119894119895119865120591 (for119894 119895 = 1 2 3) contains the so-called thickness functions where120575 is the Kronecker delta function The thickness functionsdescribe how the displacement parameters characterize thesolution through the thickness They can have any form ofany smooth function of 119911 however the classic form of thesefunctions is the power function Thus 119865120591 = 119911120591 can be 1(120591 = 0) 119911 (120591 = 1) 1199112 (120591 = 2) 1199113 (120591 = 3) and so onPlease note that according to the mathematical backgroundof the present Unified Formulation the index 119873119888 + 1 isreserved to the (optional) zigzag effect Since the aim of thepresent paper is to investigate the cylindrical bending effectfor functionally graded plates and cylindrical panels thefollowing numerical applications will consider a fixed valueof 119873119888 = 3 so that 120591 = 0 1 2 3 This choice is justified by thefact that third order theories are sufficient with a reasonablylimited number of degrees of freedom to obtain accuratesolutions when compared to 3D shell models [42ndash50] Formultilayered structures such as sandwich configurations thezigzag function must be added into the formulation Onceagain as shown in (13) the zigzag function is embedded atthe end of the expansion with the exponent 119873119888 + 1 Whensuch effect is neglected the sum of (13) stops at 119873119888

The same theory used for the FE solution can be carriedout in the GDQmethod setting 120591 = 0 1 and 119873119888 = 1 howeverit has been shown in other recent works that the strong formapproaches led tomore accurate results with respect to the FEones [46 48]

The relation between generalized strains 120576(120591) anddisplace-ments u(120591) can be set as

120576(120591) = DΩu

(120591) for 120591 = 0 1 2 119873119888 119873119888 + 1 (14)

where DΩ is a differential operator explicitly defined in [42]The present constitutive material model is linear and elastictherefore for a general 119896th laminaHookersquos law can be applied

120590(119896) = C(119896)120576(119896) (15)

where 120590(119896) is the vector containing the stress componentsand C(119896) represents the constitutive matrix for each 119896th ply[42] of the structure Each component of the constitutivematrix is indicated by the symbol 119862(119896)120585120578 and it is valid in thecurvilinear reference system 1198741015840120572120573119911 of the reference geome-tryThese coefficients are evaluatedwith the application of thetransformation matrix [42]

The stress resultants are the terms which correspond tothe punctual 3D stresses of the 3D doubly curved solidTheir definition can be achieved by developing the Hamiltonprinciple for the present structural theory [42]

S(120591) = 119873119871sum119896=1

int119911119896+1119911119896

(Z(120591))119879 120590(120591)119867120572119867120573119889119911for 120591 = 0 1 2 119873119888 119873119888 + 1

(16)

whereZ(120591) is a 6times9matrix containing the thickness functions119865120591 and the curvature coefficients119867120572 and119867120573 the definition ofsuchmatrix with some other details can be found in [42] S(120591)include the stress resultants as a function of the order 120591

The definition of (16) can be rewritten as

S(120591) = 119873119888+1sum119904=0

A(120591119904)120576(119904) for 120591 = 0 1 2 119873119888 119873119888 + 1 (17)

which is a function of generalized strains of order 119904 120576(119904)Note that the so-called elastic coefficients are defined as

A(120591119904) = 119873119871sum119896=1

int119911119896+1119911119896

(Z(120591))119879C(119896)Z(119904)119867120572119867120573119889119911 (18)

and they can be carried out by the following integral expres-sions

119860(120591119904)120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119911119904119911120591119867120572119867120573119867984858120572119867120577120573 119889119911

119860(119904)120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119911119904120591119911120591minus1119867120572119867120573119867984858120572119867120577120573 119889119911

119860(120591)120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119904119911119904minus1119911120591119867120572119867120573119867984858120572119867120577120573 119889119911

Shock and Vibration 7

119860( )120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119904119911119904minus1120591119911120591minus1119867120572119867120573119867984858120572119867120577120573 119889119911for 120591 119904 = 0 1 2 119873119888 119873119888 + 1 120585 120578 = 1 2 3 4 5 6 984858 120577 = 0 1 2

(19)

It is recalled that the exponent 119873119888 + 1 is reserved for theMurakami function otherwise this order is neglected Thesubscripts and indicate the derivatives of the thicknessfunctions 119865120591 and 119865119904 with respect to 119911 The subscripts 984858 and 120577are the exponents of the quantities119867120572 and 119867120573 whereas 120585 and120578 are the indices of the material constants 119862(119896)120585120578 for the 119896th ply

[42ndash49]Note that119862(119896)120585120578 119867120572 and119867120573 are quantities that dependon the thickness 119911 Therefore the integrals of (19) cannotbe computed analytically The present solution for solvingsuch integrals is to employ a numerical integration algorithmIn particular the so-called Generalized Integral Quadrature(GIQ) method is utilized For an accurate evaluation of theintegrals 51 points per layer are considered in the integration

The Hamilton principle supplies for the dynamic equilib-rium equations and the correspondent boundary conditionsIn detail the fundamental nucleus of three motion equationstakes the form

D⋆ΩS(120591) = 119873119888+1sum

119904=0

M(120591119904)u(119904)

for 120591 = 0 1 2 119873119888 119873119888 + 1(20)

whereD⋆Ω is the equilibrium operator andM(120591119904) is the inertialmatrix Their explicit expressions have been presented in aprevious authorsrsquo paper [42]

Combining the kinematic equation (14) constitutiveequation (17) and themotion equation (20) the fundamentalgoverning system in terms of displacement parameters iscarried out

119873119888+1sum119904=0

L(120591119904)u(119904) = 119873119888+1sum119904=0

M(120591119904)u(119904)

for 120591 = 0 1 2 119873119888 119873119888 + 1(21)

where L(120591119904) = D⋆ΩA(120591119904)DΩ is the fundamental operator [42]

Boundary conditions must be included for the solutionof the differential equations (21) The present paper dealsonly with free and simply supported boundary conditionsThe former is used to reproduce the cylindrical bendingconditions considering alternatively one free edge and onesimply supported edge When the four edges are all simplysupported the problem is the same as the ones investigated

in the previous papers [29ndash31] Simply supported conditionsare

119873(120591)120572 = 0119906(120591)120573 = 119906(120591)119911 = 0

120591 = 0 1 2 119873119888 119873119888 + 1 at 120572 = 1205720 or 120572 = 1205721 1205730 le 120573 le 1205731119906(120591)120572 = 0

119873(120591)120573 = 0119906(120591)119911 = 0

120591 = 0 1 2 119873119888 119873119888 + 1 at 120573 = 1205730 or 120573 = 1205731 1205720 le 120572 le 1205721

(22)

Free edge conditions are

119873(120591)120572 = 0119873(120591)120573 = 0119879(120591)120572 = 0120591 = 0 1 2 119873119888 119873119888 + 1 at 120572 = 1205720 or 120572 = 1205721 1205730 le 120573 le 1205731119873(120591)120572 = 0119873(120591)120573 = 0119879(120591)120573 = 0120591 = 0 1 2 119873119888 119873119888 + 1 at 120573 = 1205730 or 120573 = 1205731 1205720 le 120572 le 1205721

(23)

4 Results

The present section includes some numerical applicationsfor the comparison of the 3D exact shell vibrations withtwo numerical 2D models In particular classical simplysupported solutions and cylindrical bending (CB) behaviorof plates and shells are presentedThe following computationsfocus on plates circular cylinders and cylindrical panels dueto the fact that the CB effect occurs only when at least twoparallel straight edges are present in the structure

The first case is a square plate (119886 = 119887 = 1m) withsimply supported edges Two thickness ratios are analyzedthin 119886ℎ = 100 and moderately thick 119886ℎ = 20 structure Theplate is made of a single functionally graded layer (ℎ1 = ℎ)with volume fraction of ceramic (119888) phase

119881119888 (119911) = (05 + 119911ℎ)119901 with 119911 isin [minusℎ2 ℎ2 ] (24)

8 Shock and Vibration

Table 1 One-layered FGM plate with 119901 = 05 Analytical 3D versus 2D numerical models Frequency 119891 in Hz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119886ℎ = 1001 0 I 4019 mdash 3907 mdash 39210 1 I 4019 mdash 3907 mdash 39211 1 I 8037 8006 mdash 8036 mdash2 0 I 1607 mdash 1580 mdash 15840 2 I 1607 mdash 1580 mdash 15842 1 I 2008 2001 mdash 2008 mdash1 2 I 2008 2001 mdash 2008 mdash2 2 I 3212 3202 mdash 3211 mdash3 0 I 3613 mdash 3568 mdash 35770 3 I 3613 mdash 3568 mdash 35773 1 I 4014 4002 mdash 4013 mdash1 3 I 4014 4002 mdash 4013 mdash3 2 I 5215 5202 mdash 5215 mdash2 3 I 5215 5202 mdash 5215 mdash4 0 I 6416 mdash 6354 mdash 63650 4 I 6416 mdash 6354 mdash 63654 1 I 6816 6801 mdash 6815 mdash1 4 I 6816 6801 mdash 6815 mdash119886ℎ = 201 0 I 2002 mdash 1951 mdash 19510 1 I 2002 mdash 1951 mdash 19511 1 I 3989 3995 mdash 3988 mdash2 0 I 7916 mdash 7864 mdash 77970 2 I 7916 mdash 7864 mdash 77972 1 I 9858 9954 mdash 9857 mdash1 2 I 9858 9954 mdash 9857 mdash2 2 I 1560 1588 mdash 1560 mdash3 0 I 1749 mdash 1767 mdash 17290 3 I 1749 mdash 1767 mdash 17293 1 I 1936 1980 mdash 1936 mdash1 3 I 1936 1980 mdash 1936 mdash3 2 I 2490 2566 mdash 2490 mdash2 3 I 2490 2566 mdash 2490 mdash1 0 II (119908 = 0) 2775 2773 mdash 2775 mdash0 1 II (119908 = 0) 2775 2773 mdash 2775 mdash4 0 I 3034 mdash 3124 mdash 30070 4 I 3034 mdash 3124 mdash 3007

where 119901 is the power exponent which varies the volumefraction of the ceramic component through the thicknessTwo limit cases occur for (24) 119881119888 = 1 when 119901 = 0 and119881119888 rarr 0when119901 = infin In the following computations it is set as119901 = 05 Note that the volume fraction function in (24) givesa material which is ceramic (119888) rich at the top and metal (119898)rich at the bottomof the structureThemechanical properties(Youngmodulus andmass density) are given by the theory ofmixtures as

119864 (119911) = 119864119898 + (119864119888 minus 119864119898) 119881119888120588 (119911) = 120588119898 + (120588119888 minus 120588119898) 119881119888 (25)

where 119864119898 and 120588119898 are the elastic modulus and mass densityof the metallic phase respectively 119864119888 and 120588119888 are the corre-sponding properties of the ceramic phase Note that in allthe following computations the Poisson ratio is consideredconstant through the thickness thus ]119888 = ]119898 = ] = 03 Thetwo constituents have the following mechanical properties

119864119898 = 73GPa120588119898 = 2800 kgm3119864119888 = 380GPa120588119888 = 3800 kgm3

(26)

Shock and Vibration 9

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(a) Mode I (0 1) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

000501

01502

(b) Mode I (0 1) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(c) Mode II (119908 = 0) (0 1) 3D

002minus002

090908

08070706

06050504

04030302

020101

00minus01

1

(d) Mode II (119908 = 0) (0 1) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(e) Mode I (1 0) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

0minus005minus01

minus015minus02

(f) Mode I (1 0) 2D

Figure 2 Continued

10 Shock and Vibration

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(g) Mode II (119908 = 0) (1 0) 3D

002minus002

0908 08

0706 06

0504 04

0302 02

01 0 0

11

(h) Mode II (119908 = 0) (1 0) 2D

Figure 2 Differences between in-plane and cylindrical bending vibration modes for the 119886ℎ = 20 one-layered FGM plate with 119901 = 05 3Dmodes versus 2D modes

Since the FE commercial software does not have anembedded tool for the computation of functionally gradedproperties through the thickness the present FE applicationsare carried out building a fictitious laminated structure withisotropic plies In detail the thickness of the FGM lamina hasbeen divided into 100 fictitious layers and for each layer thecorresponding mechanical properties have been computedthrough (25) The numerical model has been made of auniform element division of 40 times 40 Quad8 elements for atotal number of 29280 degrees of freedom (dofs) The GDQsolution is computed with a not-uniform Chebyshev-Gauss-Lobatto grid [41] with 31 times 31 points for both thicknessratios It is underlined that since a single ply is set no zigzagfunction is needed in the computation It is recalled that thepresent higher order GDQ model considers a third orderexpansion with 120591 = 0 1 2 3 whereas the FE one has afirst order model (also known as Reissner-Mindlin model)Table 1 reports the results in terms of natural frequency(Hz) wherein the first column reports the number of half-waves (119898 119899) of the 3D exact solution Note that for a fixedset of half-wave numbers different mode shapes can occur(indicated by the second column ldquoModerdquo) In particular Iindicates the first mode and II indicates the second mode Inthe case of in-plane modes the caption (119908 = 0) is added tounderline that the vibration occurs in the plane and not outof plane The third column shows all the results for the 3Dexact solution both classical and cylindrical bending (CB)results It is noted that these frequencies are obtained witha single run of the 3D exact code keeping simply supportedboundary conditions for all the four edges of the plate Theother four columns of Table 1 refer to the numerical solutionsfor simply supported structures and for cylindrical bendingconditions (setting two free opposite edges) The cylindricalbending results are indicated by the acronym CB in the sametable

There is a very good agreement among the three solutionswhen simply supported conditions are considered On thecontrary a small error is observed in cylindrical bendingbecause the actual CB condition is not completely satisfieddue to the fact that the plate has a limited dimension in thesimply supported direction This feature can be detected inFigures 2 and 3 because the deformed shapes of the plate inthe CB configuration are not properly correct In fact thefibers parallel to the supported edges are not parallel amongthem but there is a curvature effect due to the free boundariesand to the limited length of the plate On the contrary thein-plane mode shapes (119908 = 0) when the plate is simplysupported on the four edges have all the fibers parallel amongthem also after the deformation These modes are the resultsobtained in the 3D solution with 119898 = 0 or 119899 = 0 combinedwith 119908 = 0 They are not cylindrical bending modes infact they can be found with 2D numerical models withoutmodifying the boundary conditions 3D frequencieswith119898 =0 or 119899 = 0 (with w different from 0) are CB modes which canbe obtained considering two simply supported edges and twofree edges in 2D numerical models

As it is well-known the ldquoperfectrdquo CB would occur whenthe simply supported direction has an ldquoinfiniterdquo lengthThis cannot be proven numerically but a trend can beobserved modeling different plates in cylindrical bendingconfiguration with different lengthsThese analyses are madekeeping one side of the plate constant and enlarging thelength of the other edge In the present computations 119886 iskept constant and 119887 = 119871 varies from 1 to 20 The results ofthe aforementioned analysis are listed in Table 2 and depictedin Figures 4 and 5 Table 2 includes the numerical valuesof the first mode shape in cylindrical bending for the 3Dexact solution the 2D FE and the 2D GDQmodels Figure 4shows that both GDQ and FE solutions tend to the 3D exactone increasing the length of the simply supported edge In

Shock and Vibration 11

ulowast

lowast

wlowast

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

(a) Mode I (1 0) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

000501

01502

(b) Mode I (1 0) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(c) Mode I (2 0) 3D

109

0807

060504

0302

01 0

109

0807

0605

0403

02010

0201501

0050

minus005minus01

minus015minus02

(d) Mode I (2 0) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(e) Mode I (3 0) 3D

10908

0706

050403

0201 0

109

0807

0605

0403

02010

0201501

0050

minus005minus01

minus015minus02

(f) Mode I (3 0) 2D

Figure 3 Continued

12 Shock and Vibration

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(g) Mode I (4 0) 3D

10908

0706050403

0201 0

10908

0706

0504

0302

010

0201501

0050

minus005minus01

minus015minus02

(h) Mode I (4 0) 2D

Figure 3 First four cylindrical bending vibration modes for the 119886ℎ = 20 one-layered FGM plate with 119901 = 05 3D modes versus 2D modes

3D exact2D GDQ2D FEM

f(H

z)

39

392

394

396

398

40

402

404

5 10 15 200La

Figure 4 First frequency 119891 in Hz for the 119886ℎ = 100 one-layeredFGM plate with 119901 = 05 convergence of 2D GDQ and 2D FEsolutions increasing the length of the simply supported sides

particular the GDQ solution is slightly more accurate thanthe FE one The 2D GDQ solution is carried out by enlargingthe length of one of the sides of the plate and the samenumberof dofs is kept On the contrary the FE model is generatedby 40 times 40 blocks of mesh In other words the plate with119871 = 2119886 has a mesh of 40 times 80 the one with 119871 = 3119886 has amesh 40 times 120 and so on When 119871 = 20119886 the error amongthe 3D exact and the 2D numerical solutions is very small andthis is proven also by the mode shapes of Figure 5 whereinthe boundary effect is negligible with respect the length of

the simply supported edges It must be underlined that it isimpossible to completely avoid the boundary effects due tothe free edges

The second example is related to a simply supportedcylinder with 119877120572 = 10m 119877120573 = infin 119886 = 2120587119877120572 and 119887 =20m and thickness ratios 119877120572ℎ = 100 and 119877120572ℎ = 10 Aone-layered FGM scheme is considered with ℎ1 = ℎ andthe same volume fraction function of (24) for the previousexample but setting 119901 = 20 The present material propertiesare the same as the flat plate The FE model has a 20 times80 Quad8 mesh with 27966 dofs and the GDQ solutionhas a 15 times 51 Chebyshev-Gauss-Lobatto grid Numericallyspeaking compatibility conditions must be enforced on theclosing meridian in order to obtain a closed revolution shell[42] To obtain the CB conditions numerically the cylinderis modeled completely free and the first 6 rigid mode shapesare neglected and removed from the numerical solution Itis noted that the GDQ model uses a third-order kinematicexpansion with no zigzag effect as the previous case sincea one-layer structure has been taken into account Table 3reports the results with the same meaning of the symbolsseen in Table 1 A very good agreement is observed when thecylinder is simply supported whereas when the cylinder iscompletely free some small errors can be observed but thenumerical solutions follow the 3D exact one

Results with 119899 = 0 and 119908 = 0 indicate the CB conditionsSuch results are obtained by means of 2D numerical modelsmodifying the boundary conditions (see 2D FE(CB) and 2DGDQ(CB)models)The cases 119899 = 0 combined with119908 = 0 arenot CB conditions and the frequencies can be found withoutmodifying the boundary conditions in the 2D numericalmodels

The final test is related to a cylindrical panel with 119877120572 =10m 119877120573 = infin 119886 = 1205873119877120572 and 119887 = 20m and thickness ratios119877120572ℎ = 100 and 119877120572ℎ = 10 The present shell is a sandwichstructure embedding a FGM core with ℎ2 = 07ℎ and two

Shock and Vibration 13

218

1614

121 1

08 0806 06

04 0402 02

0 0

0minus01minus02minus03minus04

(a) 119871119886 = 2

3

25

2

151

0806

040502

0 0

1

0minus01minus02minus03minus04minus05

(b) 119871119886 = 3

435

325

10505

0 0

15

0

minus02minus04minus06

1

2

(c) 119871119886 = 4

545

35

10505

0 0

0

minus04minus02

minus06minus08

1

25

4

3

215

(d) 119871119886 = 5

5

1

0 005

0

minus04minus02

minus06minus08

1

4

3

2

6

minus1

(e) 119871119886 = 6

5

10 0 051

43

2

67

89

100

minus05minus1

minus15

(f) 119871119886 = 10

Figure 5 First vibration mode for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 2D GDQ solution and edge effects versus the lengthof the simply supported sides

external skins with ℎ1 = ℎ3 = 015ℎ (ℎ1 + ℎ2 + ℎ3 = ℎ) Thebottom layer ismade of isotropicmetallicmaterial and the toplayer is made of ceramic material the mechanical propertiesare

119864119898 = 73GPa120588119898 = 2800 kgm3119864119888 = 200GPa120588119888 = 5700 kgm3

(27)

The FGM core has a volume fraction function defined as

119881119888 (119911) = (05 + 11991107ℎ)119901 with 119911 isin [minus035ℎ 035ℎ] (28)

where 119901 = 10 is employed in the present computations ThePoisson ratio is constant (]119888 = ]119898 = ] = 03) Equations (25)are still valid with 119881119888 defined as in (28)

The FEmodel is made of 40times40Quad8 elements for total29280 dofs and the GDQ model has a 31 times 31 Chebyshev-Gauss-Lobatto grid Since a sandwich structure is analyzedin the following a third-order theory with zigzag functionis considered in the computations for the GDQ method Thepresent natural frequencies are reported in Table 4 with the

14 Shock and Vibration

Table 2 First frequency 119891 in Hz for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 convergence of 2D GDQ and 2D FE solutionsincreasing the length of the simply supported sides

119871119886 1 2 3 4 5 6 8 10 15 20 3D exact2D GDQ 3921 3964 3980 3988 3993 3997 4001 4004 4007 4008 40192D FEM 3907 3949 3966 3974 3979 3983 3987 3989 3993 3994 4019

Table 3 One-layered FGM cylinder with 119901 = 20 Analytical 3D versus 2D numerical models Frequency 119891 in Hz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1004 0 I 0896 mdash 0897 mdash 08836 0 I 2532 mdash 2538 mdash 25118 0 I 4854 mdash 4867 mdash 481910 0 I 7848 mdash 7871 mdash 780112 0 I 1151 mdash 1155 mdash 114610 1 I 1356 1360 mdash 1357 mdash12 1 I 1441 1449 mdash 1445 mdash14 0 I 1584 mdash 1589 mdash 15778 1 I 1638 1640 mdash 1636 mdash14 1 I 1756 1767 mdash 1763 mdash16 0 I 2083 mdash 2090 mdash 207016 1 I 2207 2222 mdash 2216 mdash6 1 I 2441 2442 mdash 2438 mdash18 0 I 2649 mdash 2658 mdash 264014 2 I 2742 2752 mdash 2746 mdash18 1 I 2751 2772 mdash 2763 mdash16 2 I 2875 2890 mdash 2883 mdash12 2 I 2921 2926 mdash 2922 mdash119877120572ℎ = 104 0 I 8774 mdash 9085 mdash 86796 0 I 2456 mdash 2571 mdash 24306 1 I 3978 4089 mdash 3979 mdash4 1 I 4388 4357 mdash 4388 mdash8 0 I 4654 mdash 4920 mdash 46028 1 I 5575 5905 mdash 5576 mdash2 1 I 7202 7101 mdash 7202 mdash2 0 I (119908 = 0) 7322 7390 mdash 7319 mdash10 0 I 7419 mdash 7933 mdash 73476 2 I 8110 8261 mdash 8111 mdash10 1 I 8189 8811 mdash 8191 mdash8 2 I 8783 9173 mdash 8785 mdash4 2 I 9016 9084 mdash 9017 mdash12 0 I 1071 mdash 1159 mdash 106110 2 I 1081 1155 mdash 1081 mdash2 2 I 1089 1105 mdash 1089 mdash12 1 I 1142 1245 mdash 1143 mdash4 3 I 1263 1286 mdash 1263 mdash

same meaning of the columns seen in the previous casesIn order to achieve the cylindrical bending conditions thenumerical models have the straight edges simply supportedand the curved edges as free Due to the CB definitionit is not possible to achieve CB conditions when 119898 = 0

and 119899 = 1 2 3 because the cylindrical bending shouldoccur on the curved edge and this is not physically possiblebecause of the variation of displacement components due tothe curvature presence This aspect is indicated as NA (NotApplicable) solution in Table 4 Therefore this kind of 3D

Shock and Vibration 15

Table 4 Sandwich cylindrical shell panel embedding FGM core with 119901 = 10 Analytical 3D versus 2D numerical models Frequency 119891 inHz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1001 0 I 1948 mdash 1951 mdash 19412 0 I 8856 mdash 8872 mdash 88232 1 I 1104 1110 mdash 1104 mdash1 1 I 1850 1873 mdash 1850 mdash3 0 I 2038 mdash 2042 mdash 20303 1 I 2116 2128 mdash 2116 mdash2 2 I 2221 2242 mdash 2221 mdash3 2 I 2484 2496 mdash 2484 mdash3 3 I 3235 3254 mdash 3235 mdash4 0 I 3649 mdash 3660 mdash 36342 3 I 3692 3739 mdash 3692 mdash4 1 I 3714 3745 mdash 3714 mdash4 2 I 3940 3965 mdash 3941 mdash3 4 I 4226 4261 mdash 4226 mdash4 3 I 4378 4403 mdash 4378 mdash1 2 I 4568 4664 mdash 4568 mdash4 4 I 5035 5064 mdash 5035 mdash2 4 I 5040 5128 mdash 5040 mdash119877120572ℎ = 101 0 I 1905 mdash 1963 mdash 18841 1 I 3054 3145 mdash 3055 mdash1 2 I 6209 6409 mdash 6209 mdash2 0 I 8326 mdash 8857 mdash 82540 1 I 8710 mdash NA mdash NA0 1 II (119908 = 0) 8820 8956 mdash 8820 mdash2 1 I 8880 9531 mdash 8882 mdash0 2 I 9176 mdash NA mdash NA1 3 I 9576 9983 mdash 9567 mdash0 3 I 1036 mdash NA mdash NA2 2 I 1066 1147 mdash 1067 mdash0 4 I 1291 mdash NA mdash NA1 4 I 1335 1413 mdash 1335 mdash2 3 I 1364 1475 mdash 1364 mdash1 0 II (119908 = 0) 1665 1738 mdash 1665 mdash0 5 I 1683 mdash NA mdash NA2 4 I 1755 1918 mdash 1755 mdash0 2 II (119908 = 0) 1764 1791 mdash 1764 mdash

exact solutions cannot be achieved by any numerical modelOnce again solution with 119899 = 0 or 119898 = 0 combined with119908 = 0 (in-plane modes) can be found with 2D numericalmodels without modifying the simply supported boundaryconditions

5 Conclusions

Thepresent paper analyzes the cylindrical bending in the freefrequency analysis of functionally graded material (FGM)plates and cylindrical shells This condition is investigatedby means of a 3D exact shell model and two different 2D

numerical models a weak formulation based on classicalFinite Elements (FEs) and a strong formulation based onthe GDQ method The cylindrical bending (CB) behavior inthe 3D exact shell model can be analyzed using completelysimply supported structures and supposing one of the twohalf-wave numbers equals zeroThe numerical models obtainthe CB behavior using mixed simply supported and freeboundary conditions It has been demonstrated that theactual cylindrical bending (CB) is possible only with respectto straight edgesTheCBnatural frequencies of the numericalsolutions converge to the 3D exact CB ones if the length ofstraight simply supported edges is increased up to 20 times

16 Shock and Vibration

the value of the original edges The CB conditions are notpossible in 2D numericalmodels when the half-wave numberis zero in the curvilinear edge in this case the modificationof boundary conditions does not properly work In-planevibrationmodeswith zero transverse displacement combinedwith 119898 = 0 or 119899 = 0 cannot be defined as CB frequencies andthey can be calculated via 2D numerical models without themodification of the simply supported conditions for the fouredges All the considerations about the cylindrical bendingare valid for each geometry (plate cylinder and cylindricalshell) lamination scheme FGM law and thickness ratio

Competing Interests

The authors declare that they have no competing interests

References

[1] W Q Chen J B Cai and G R Ye ldquoResponses of cross-ply laminates with viscous interfaces in cylindrical bendingrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 823ndash835 2005

[2] W Q Chen and K Y Lee ldquoThree-dimensional exact analysisof angle-ply laminates in cylindrical bending with interfacialdamage via state-space methodrdquo Composite Structures vol 64no 3-4 pp 275ndash283 2004

[3] W Q Chen and K Y Lee ldquoTime-dependent behaviors of angle-ply laminates with viscous interfaces in cylindrical bendingrdquoEuropean Journal ofMechanicsmdashASolids vol 23 no 2 pp 235ndash245 2004

[4] A H Gandhi and H K Raval ldquoAnalytical and empiricalmodeling of top roller position for three-roller cylindricalbending of plates and its experimental verificationrdquo Journal ofMaterials Processing Technology vol 197 no 1ndash3 pp 268ndash2782008

[5] S Oral and H Darendeliler ldquoThe optimum die profile for thecylindrical bending of platesrdquo Journal of Materials ProcessingTechnology vol 70 no 1ndash3 pp 151ndash155 1997

[6] V A Jairazbhoy P Petukhov and J Qu ldquoLarge deflectionof thin plates in cylindrical bendingmdashnon-unique solutionsrdquoInternational Journal of Solids and Structures vol 45 no 11-12pp 3203ndash3218 2008

[7] A Lebee and K Sab ldquoA bending-gradient model for thickplates part II closed-form solutions for cylindrical bending oflaminatesrdquo International Journal of Solids and Structures vol 48no 20 pp 2889ndash2901 2011

[8] P V Nimbolkar and I M Jain ldquoCylindrical bending of elasticplatesrdquo Procedia Materials Science vol 10 pp 793ndash802 2015

[9] A S Sayyad and Y M Ghugal ldquoA nth-order shear deformationtheory for composite laminates in cylindrical bendingrdquo Curvedand Layered Structures vol 2 no 1 pp 290ndash300 2015

[10] N J Pagano ldquoExact solutions for composite laminates incylindrical bendingrdquo Journal of Composite Materials vol 3 no3 pp 398ndash411 1969

[11] N J Pagano and A S D Wang ldquoFurther study of compositelaminates under cylindrical bendingrdquo Journal of CompositeMaterials vol 5 pp 421ndash428 1971

[12] N Saeedi K Sab and J-F Caron ldquoCylindrical bending ofmultilayered plates with multi-delamination via a layerwisestress approachrdquo Composite Structures vol 95 pp 728ndash7392013

[13] X-P Shu and K P Soldatos ldquoCylindrical bending of angle-ply laminates subjected to different sets of edge boundaryconditionsrdquo International Journal of Solids and Structures vol37 no 31 pp 4289ndash4307 2000

[14] Z G Bian W Q Chen C W Lim and N Zhang ldquoAnalyticalsolutions for single- and multi-span functionally graded platesin cylindrical bendingrdquo International Journal of Solids andStructures vol 42 no 24-25 pp 6433ndash6456 2005

[15] H M Navazi and H Haddadpour ldquoNonlinear cylindricalbending analysis of shear deformable functionally graded platesunder different loadings using analytical methodsrdquo Interna-tional Journal of Mechanical Sciences vol 50 no 12 pp 1650ndash1657 2008

[16] W Q Chen J Ying J B Cai and G R Ye ldquoBenchmarksolution of imperfect angle-ply laminated rectangular plates incylindrical bending with surface piezoelectric layers as actuatorand sensorrdquo Computers and Structures vol 82 no 22 pp 1773ndash1784 2004

[17] W Q Chen and K Y Lee ldquoExact solution of angle-plypiezoelectric laminates in cylindrical bending with interfacialimperfectionsrdquo Composite Structures vol 65 no 3-4 pp 329ndash337 2004

[18] T Kant and SM Shiyekar ldquoCylindrical bending of piezoelectriclaminates with a higher order shear and normal deformationtheoryrdquo Computers and Structures vol 86 no 15-16 pp 1594ndash1603 2008

[19] P Lu H P Lee and C Lu ldquoAn exact solution for functionallygraded piezoelectric laminates in cylindrical bendingrdquo Interna-tional Journal of Mechanical Sciences vol 47 no 3 pp 437ndash4382005

[20] W Yan J Wang and W Q Chen ldquoCylindrical bendingresponses of angle-ply piezoelectric laminates with viscoelasticinterfacesrdquo Applied Mathematical Modelling vol 38 no 24 pp6018ndash6030 2014

[21] Y Y Zhou W Q Chen and C F Lu ldquoSemi-analytical solutionfor orthotropic piezoelectric laminates in cylindrical bendingwith interfacial imperfectionsrdquo Composite Structures vol 92no 4 pp 1009ndash1018 2010

[22] G P Dube S Kapuria and P C Dumir ldquoExact piezother-moelastic solution of a simply-supported orthotropic flat panelin cylindrical bendingrdquo International Journal of MechanicalSciences vol 38 pp 374ndash387 1996

[23] V N Pilipchuk V L Berdichevsky and R A IbrahimldquoThermo-mechanical coupling in cylindrical bending of sand-wich platesrdquoComposite Structures vol 92 no 11 pp 2632ndash26402010

[24] C Zhang S Di and N Zhang ldquoA new procedure for staticanalysis of thermo-electric laminated composite plates undercylindrical bendingrdquo Composite Structures vol 56 no 2 pp131ndash140 2002

[25] W Q Chen and K Y Lee ldquoState-space approach for statics anddynamics of angle-ply laminated cylindrical panels in cylindri-cal bendingrdquo International Journal of Mechanical Sciences vol47 no 3 pp 374ndash387 2005

[26] C-P Wu and Y-S Syu ldquoExact solutions of functionally gradedpiezoelectric shells under cylindrical bendingrdquo InternationalJournal of Solids and Structures vol 44 no 20 pp 6450ndash64722007

[27] W Yan J Ying and W Q Chen ldquoThe behavior of angle-ply laminated cylindrical shells with viscoelastic interfaces incylindrical bendingrdquo Composite Structures vol 78 no 4 pp551ndash559 2007

Shock and Vibration 17

[28] S Brischetto F Tornabene N Fantuzzi and E Viola ldquo3Dexact and 2D generalized differential quadraturemodels for freevibration analysis of functionally graded plates and cylindersrdquoMeccanica vol 51 no 9 pp 2059ndash2098 2016

[29] N Fantuzzi S Brischetto F Tornabene and E Viola ldquo2D and3D shell models for the free vibration investigation of func-tionally graded cylindrical and spherical panelsrdquo CompositeStructures vol 154 pp 573ndash590 2016

[30] S Brischetto and R Torre ldquoExact 3D solutions and finiteelement 2D models for free vibration analysis of plates andcylindersrdquo Curved and Layered Structures vol 1 no 1 pp 59ndash92 2014

[31] F Tornabene S Brischetto N Fantuzzi and E Viola ldquoNumer-ical and exact models for free vibration analysis of cylindricaland spherical shell panelsrdquo Composites Part B Engineering vol81 pp 231ndash250 2015

[32] S Brischetto ldquoExact elasticity solution for natural frequenciesof functionally graded simply-supported structuresrdquo ComputerModeling in Engineering amp Sciences vol 95 no 5 pp 391ndash4302013

[33] S Brischetto ldquoThree-dimensional exact free vibration analysisof spherical cylindrical and flat one-layered panelsrdquo Shock andVibration vol 2014 Article ID 479738 29 pages 2014

[34] S Brischetto ldquoA continuum elastic three-dimensional modelfor natural frequencies of single-walled carbon nanotubesrdquoComposites Part B Engineering vol 61 pp 222ndash228 2014

[35] S Brischetto ldquoAn exact 3d solution for free vibrations ofmultilayered cross-ply composite and sandwich plates andshellsrdquo International Journal of Applied Mechanics vol 6 no 6Article ID 1450076 42 pages 2014

[36] S Brischetto ldquoA continuumshellmodel including vanderWaalsinteraction for free vibrations of double-walled carbon nan-otubesrdquoCMES-ComputerModeling in Engineering and Sciencesvol 104 no 4 pp 305ndash327 2015

[37] S Brischetto F Tornabene N Fantuzzi and M BacciocchildquoRefined 2D and exact 3D shell models for the free vibrationanalysis of single- and double-walled carbon nanotubesrdquo Tech-nologies vol 3 no 4 pp 259ndash284 2015

[38] S Brischetto ldquoConvergence analysis of the exponential matrixmethod for the solution of 3D equilibrium equations for freevibration analysis of plates and shellsrdquo Composites Part BEngineering vol 98 pp 453ndash471 2016

[39] S Brischetto ldquoExact and approximate shell geometry in the freevibration analysis of one-layered and multilayered structuresrdquoInternational Journal of Mechanical Sciences vol 113 pp 81ndash932016

[40] S Brischetto ldquoCurvature approximation effects in the freevibration analysis of functionally graded shellsrdquo InternationalJournal of Applied Mechanics 2016

[41] F Tornabene N Fantuzzi F Ubertini and E Viola ldquoStrong for-mulation finite element method based on differential quadra-ture a surveyrdquo Applied Mechanics Reviews vol 67 no 2 ArticleID 020801 pp 1ndash55 2015

[42] F Tornabene E Viola and N Fantuzzi ldquoGeneral higher-orderequivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panelsrdquo CompositeStructures vol 104 pp 94ndash117 2013

[43] N Fantuzzi F Tornabene E Viola and A J Ferreira ldquoAstrong formulation finite element method (SFEM) based onRBF and GDQ techniques for the static and dynamic analysesof laminated plates of arbitrary shaperdquo Meccanica vol 49 no10 pp 2503ndash2542 2014

[44] F Tornabene N Fantuzzi and M Bacciocchi ldquoFree vibrationsof free-form doubly-curved shells made of functionally gradedmaterials using higher-order equivalent single layer theoriesrdquoComposites Part B Engineering vol 67 pp 490ndash509 2014

[45] F Tornabene and E Viola ldquoStatic analysis of functionallygraded doubly-curved shells and panels of revolutionrdquo Mecca-nica vol 48 no 4 pp 901ndash930 2013

[46] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoDynamic analysis of thick and thin elliptic shell structuresmade of laminated composite materialsrdquo Composite Structuresvol 133 pp 278ndash299 2015

[47] F TornabeneN FantuzziM Bacciocchi and EViola ldquoHigher-order theories for the free vibrations of doubly-curved lam-inated panels with curvilinear reinforcing fibers by meansof a local version of the GDQ methodrdquo Composites Part BEngineering vol 81 pp 196ndash230 2015

[48] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoFree vibrations of composite oval and elliptic cylinders bythe generalized differential quadrature methodrdquo Thin-WalledStructures vol 97 pp 114ndash129 2015

[49] F Tornabene and J N Reddy ldquoFGM and laminated doubly-curved and degenerate shells resting on nonlinear elastic foun-dations a GDQ solution for static analysis with a posterioristress and strain recoveryrdquo Journal of the Indian Institute ofScience vol 93 no 4 pp 635ndash688 2013

[50] N Fantuzzi F Tornabene and E Viola ldquoFour-parameter func-tionally graded cracked plates of arbitrary shape a GDQFEMsolution for free vibrationsrdquo Mechanics of Advanced Materialsand Structures vol 23 no 1 pp 89ndash107 2016

International Journal of

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Submit your manuscripts athttpwwwhindawicom

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Shock and Vibration

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International Journal of

Shock and Vibration 7

119860( )120585120578(984858120577)

= 119873119871sum119896=1

int119911119896+1119911119896

119862(119896)120585120578 119904119911119904minus1120591119911120591minus1119867120572119867120573119867984858120572119867120577120573 119889119911for 120591 119904 = 0 1 2 119873119888 119873119888 + 1 120585 120578 = 1 2 3 4 5 6 984858 120577 = 0 1 2

(19)

It is recalled that the exponent 119873119888 + 1 is reserved for theMurakami function otherwise this order is neglected Thesubscripts and indicate the derivatives of the thicknessfunctions 119865120591 and 119865119904 with respect to 119911 The subscripts 984858 and 120577are the exponents of the quantities119867120572 and 119867120573 whereas 120585 and120578 are the indices of the material constants 119862(119896)120585120578 for the 119896th ply

[42ndash49]Note that119862(119896)120585120578 119867120572 and119867120573 are quantities that dependon the thickness 119911 Therefore the integrals of (19) cannotbe computed analytically The present solution for solvingsuch integrals is to employ a numerical integration algorithmIn particular the so-called Generalized Integral Quadrature(GIQ) method is utilized For an accurate evaluation of theintegrals 51 points per layer are considered in the integration

The Hamilton principle supplies for the dynamic equilib-rium equations and the correspondent boundary conditionsIn detail the fundamental nucleus of three motion equationstakes the form

D⋆ΩS(120591) = 119873119888+1sum

119904=0

M(120591119904)u(119904)

for 120591 = 0 1 2 119873119888 119873119888 + 1(20)

whereD⋆Ω is the equilibrium operator andM(120591119904) is the inertialmatrix Their explicit expressions have been presented in aprevious authorsrsquo paper [42]

Combining the kinematic equation (14) constitutiveequation (17) and themotion equation (20) the fundamentalgoverning system in terms of displacement parameters iscarried out

119873119888+1sum119904=0

L(120591119904)u(119904) = 119873119888+1sum119904=0

M(120591119904)u(119904)

for 120591 = 0 1 2 119873119888 119873119888 + 1(21)

where L(120591119904) = D⋆ΩA(120591119904)DΩ is the fundamental operator [42]

Boundary conditions must be included for the solutionof the differential equations (21) The present paper dealsonly with free and simply supported boundary conditionsThe former is used to reproduce the cylindrical bendingconditions considering alternatively one free edge and onesimply supported edge When the four edges are all simplysupported the problem is the same as the ones investigated

in the previous papers [29ndash31] Simply supported conditionsare

119873(120591)120572 = 0119906(120591)120573 = 119906(120591)119911 = 0

120591 = 0 1 2 119873119888 119873119888 + 1 at 120572 = 1205720 or 120572 = 1205721 1205730 le 120573 le 1205731119906(120591)120572 = 0

119873(120591)120573 = 0119906(120591)119911 = 0

120591 = 0 1 2 119873119888 119873119888 + 1 at 120573 = 1205730 or 120573 = 1205731 1205720 le 120572 le 1205721

(22)

Free edge conditions are

119873(120591)120572 = 0119873(120591)120573 = 0119879(120591)120572 = 0120591 = 0 1 2 119873119888 119873119888 + 1 at 120572 = 1205720 or 120572 = 1205721 1205730 le 120573 le 1205731119873(120591)120572 = 0119873(120591)120573 = 0119879(120591)120573 = 0120591 = 0 1 2 119873119888 119873119888 + 1 at 120573 = 1205730 or 120573 = 1205731 1205720 le 120572 le 1205721

(23)

4 Results

The present section includes some numerical applicationsfor the comparison of the 3D exact shell vibrations withtwo numerical 2D models In particular classical simplysupported solutions and cylindrical bending (CB) behaviorof plates and shells are presentedThe following computationsfocus on plates circular cylinders and cylindrical panels dueto the fact that the CB effect occurs only when at least twoparallel straight edges are present in the structure

The first case is a square plate (119886 = 119887 = 1m) withsimply supported edges Two thickness ratios are analyzedthin 119886ℎ = 100 and moderately thick 119886ℎ = 20 structure Theplate is made of a single functionally graded layer (ℎ1 = ℎ)with volume fraction of ceramic (119888) phase

119881119888 (119911) = (05 + 119911ℎ)119901 with 119911 isin [minusℎ2 ℎ2 ] (24)

8 Shock and Vibration

Table 1 One-layered FGM plate with 119901 = 05 Analytical 3D versus 2D numerical models Frequency 119891 in Hz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119886ℎ = 1001 0 I 4019 mdash 3907 mdash 39210 1 I 4019 mdash 3907 mdash 39211 1 I 8037 8006 mdash 8036 mdash2 0 I 1607 mdash 1580 mdash 15840 2 I 1607 mdash 1580 mdash 15842 1 I 2008 2001 mdash 2008 mdash1 2 I 2008 2001 mdash 2008 mdash2 2 I 3212 3202 mdash 3211 mdash3 0 I 3613 mdash 3568 mdash 35770 3 I 3613 mdash 3568 mdash 35773 1 I 4014 4002 mdash 4013 mdash1 3 I 4014 4002 mdash 4013 mdash3 2 I 5215 5202 mdash 5215 mdash2 3 I 5215 5202 mdash 5215 mdash4 0 I 6416 mdash 6354 mdash 63650 4 I 6416 mdash 6354 mdash 63654 1 I 6816 6801 mdash 6815 mdash1 4 I 6816 6801 mdash 6815 mdash119886ℎ = 201 0 I 2002 mdash 1951 mdash 19510 1 I 2002 mdash 1951 mdash 19511 1 I 3989 3995 mdash 3988 mdash2 0 I 7916 mdash 7864 mdash 77970 2 I 7916 mdash 7864 mdash 77972 1 I 9858 9954 mdash 9857 mdash1 2 I 9858 9954 mdash 9857 mdash2 2 I 1560 1588 mdash 1560 mdash3 0 I 1749 mdash 1767 mdash 17290 3 I 1749 mdash 1767 mdash 17293 1 I 1936 1980 mdash 1936 mdash1 3 I 1936 1980 mdash 1936 mdash3 2 I 2490 2566 mdash 2490 mdash2 3 I 2490 2566 mdash 2490 mdash1 0 II (119908 = 0) 2775 2773 mdash 2775 mdash0 1 II (119908 = 0) 2775 2773 mdash 2775 mdash4 0 I 3034 mdash 3124 mdash 30070 4 I 3034 mdash 3124 mdash 3007

where 119901 is the power exponent which varies the volumefraction of the ceramic component through the thicknessTwo limit cases occur for (24) 119881119888 = 1 when 119901 = 0 and119881119888 rarr 0when119901 = infin In the following computations it is set as119901 = 05 Note that the volume fraction function in (24) givesa material which is ceramic (119888) rich at the top and metal (119898)rich at the bottomof the structureThemechanical properties(Youngmodulus andmass density) are given by the theory ofmixtures as

119864 (119911) = 119864119898 + (119864119888 minus 119864119898) 119881119888120588 (119911) = 120588119898 + (120588119888 minus 120588119898) 119881119888 (25)

where 119864119898 and 120588119898 are the elastic modulus and mass densityof the metallic phase respectively 119864119888 and 120588119888 are the corre-sponding properties of the ceramic phase Note that in allthe following computations the Poisson ratio is consideredconstant through the thickness thus ]119888 = ]119898 = ] = 03 Thetwo constituents have the following mechanical properties

119864119898 = 73GPa120588119898 = 2800 kgm3119864119888 = 380GPa120588119888 = 3800 kgm3

(26)

Shock and Vibration 9

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(a) Mode I (0 1) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

000501

01502

(b) Mode I (0 1) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(c) Mode II (119908 = 0) (0 1) 3D

002minus002

090908

08070706

06050504

04030302

020101

00minus01

1

(d) Mode II (119908 = 0) (0 1) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(e) Mode I (1 0) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

0minus005minus01

minus015minus02

(f) Mode I (1 0) 2D

Figure 2 Continued

10 Shock and Vibration

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(g) Mode II (119908 = 0) (1 0) 3D

002minus002

0908 08

0706 06

0504 04

0302 02

01 0 0

11

(h) Mode II (119908 = 0) (1 0) 2D

Figure 2 Differences between in-plane and cylindrical bending vibration modes for the 119886ℎ = 20 one-layered FGM plate with 119901 = 05 3Dmodes versus 2D modes

Since the FE commercial software does not have anembedded tool for the computation of functionally gradedproperties through the thickness the present FE applicationsare carried out building a fictitious laminated structure withisotropic plies In detail the thickness of the FGM lamina hasbeen divided into 100 fictitious layers and for each layer thecorresponding mechanical properties have been computedthrough (25) The numerical model has been made of auniform element division of 40 times 40 Quad8 elements for atotal number of 29280 degrees of freedom (dofs) The GDQsolution is computed with a not-uniform Chebyshev-Gauss-Lobatto grid [41] with 31 times 31 points for both thicknessratios It is underlined that since a single ply is set no zigzagfunction is needed in the computation It is recalled that thepresent higher order GDQ model considers a third orderexpansion with 120591 = 0 1 2 3 whereas the FE one has afirst order model (also known as Reissner-Mindlin model)Table 1 reports the results in terms of natural frequency(Hz) wherein the first column reports the number of half-waves (119898 119899) of the 3D exact solution Note that for a fixedset of half-wave numbers different mode shapes can occur(indicated by the second column ldquoModerdquo) In particular Iindicates the first mode and II indicates the second mode Inthe case of in-plane modes the caption (119908 = 0) is added tounderline that the vibration occurs in the plane and not outof plane The third column shows all the results for the 3Dexact solution both classical and cylindrical bending (CB)results It is noted that these frequencies are obtained witha single run of the 3D exact code keeping simply supportedboundary conditions for all the four edges of the plate Theother four columns of Table 1 refer to the numerical solutionsfor simply supported structures and for cylindrical bendingconditions (setting two free opposite edges) The cylindricalbending results are indicated by the acronym CB in the sametable

There is a very good agreement among the three solutionswhen simply supported conditions are considered On thecontrary a small error is observed in cylindrical bendingbecause the actual CB condition is not completely satisfieddue to the fact that the plate has a limited dimension in thesimply supported direction This feature can be detected inFigures 2 and 3 because the deformed shapes of the plate inthe CB configuration are not properly correct In fact thefibers parallel to the supported edges are not parallel amongthem but there is a curvature effect due to the free boundariesand to the limited length of the plate On the contrary thein-plane mode shapes (119908 = 0) when the plate is simplysupported on the four edges have all the fibers parallel amongthem also after the deformation These modes are the resultsobtained in the 3D solution with 119898 = 0 or 119899 = 0 combinedwith 119908 = 0 They are not cylindrical bending modes infact they can be found with 2D numerical models withoutmodifying the boundary conditions 3D frequencieswith119898 =0 or 119899 = 0 (with w different from 0) are CB modes which canbe obtained considering two simply supported edges and twofree edges in 2D numerical models

As it is well-known the ldquoperfectrdquo CB would occur whenthe simply supported direction has an ldquoinfiniterdquo lengthThis cannot be proven numerically but a trend can beobserved modeling different plates in cylindrical bendingconfiguration with different lengthsThese analyses are madekeeping one side of the plate constant and enlarging thelength of the other edge In the present computations 119886 iskept constant and 119887 = 119871 varies from 1 to 20 The results ofthe aforementioned analysis are listed in Table 2 and depictedin Figures 4 and 5 Table 2 includes the numerical valuesof the first mode shape in cylindrical bending for the 3Dexact solution the 2D FE and the 2D GDQmodels Figure 4shows that both GDQ and FE solutions tend to the 3D exactone increasing the length of the simply supported edge In

Shock and Vibration 11

ulowast

lowast

wlowast

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

(a) Mode I (1 0) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

000501

01502

(b) Mode I (1 0) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(c) Mode I (2 0) 3D

109

0807

060504

0302

01 0

109

0807

0605

0403

02010

0201501

0050

minus005minus01

minus015minus02

(d) Mode I (2 0) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(e) Mode I (3 0) 3D

10908

0706

050403

0201 0

109

0807

0605

0403

02010

0201501

0050

minus005minus01

minus015minus02

(f) Mode I (3 0) 2D

Figure 3 Continued

12 Shock and Vibration

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(g) Mode I (4 0) 3D

10908

0706050403

0201 0

10908

0706

0504

0302

010

0201501

0050

minus005minus01

minus015minus02

(h) Mode I (4 0) 2D

Figure 3 First four cylindrical bending vibration modes for the 119886ℎ = 20 one-layered FGM plate with 119901 = 05 3D modes versus 2D modes

3D exact2D GDQ2D FEM

f(H

z)

39

392

394

396

398

40

402

404

5 10 15 200La

Figure 4 First frequency 119891 in Hz for the 119886ℎ = 100 one-layeredFGM plate with 119901 = 05 convergence of 2D GDQ and 2D FEsolutions increasing the length of the simply supported sides

particular the GDQ solution is slightly more accurate thanthe FE one The 2D GDQ solution is carried out by enlargingthe length of one of the sides of the plate and the samenumberof dofs is kept On the contrary the FE model is generatedby 40 times 40 blocks of mesh In other words the plate with119871 = 2119886 has a mesh of 40 times 80 the one with 119871 = 3119886 has amesh 40 times 120 and so on When 119871 = 20119886 the error amongthe 3D exact and the 2D numerical solutions is very small andthis is proven also by the mode shapes of Figure 5 whereinthe boundary effect is negligible with respect the length of

the simply supported edges It must be underlined that it isimpossible to completely avoid the boundary effects due tothe free edges

The second example is related to a simply supportedcylinder with 119877120572 = 10m 119877120573 = infin 119886 = 2120587119877120572 and 119887 =20m and thickness ratios 119877120572ℎ = 100 and 119877120572ℎ = 10 Aone-layered FGM scheme is considered with ℎ1 = ℎ andthe same volume fraction function of (24) for the previousexample but setting 119901 = 20 The present material propertiesare the same as the flat plate The FE model has a 20 times80 Quad8 mesh with 27966 dofs and the GDQ solutionhas a 15 times 51 Chebyshev-Gauss-Lobatto grid Numericallyspeaking compatibility conditions must be enforced on theclosing meridian in order to obtain a closed revolution shell[42] To obtain the CB conditions numerically the cylinderis modeled completely free and the first 6 rigid mode shapesare neglected and removed from the numerical solution Itis noted that the GDQ model uses a third-order kinematicexpansion with no zigzag effect as the previous case sincea one-layer structure has been taken into account Table 3reports the results with the same meaning of the symbolsseen in Table 1 A very good agreement is observed when thecylinder is simply supported whereas when the cylinder iscompletely free some small errors can be observed but thenumerical solutions follow the 3D exact one

Results with 119899 = 0 and 119908 = 0 indicate the CB conditionsSuch results are obtained by means of 2D numerical modelsmodifying the boundary conditions (see 2D FE(CB) and 2DGDQ(CB)models)The cases 119899 = 0 combined with119908 = 0 arenot CB conditions and the frequencies can be found withoutmodifying the boundary conditions in the 2D numericalmodels

The final test is related to a cylindrical panel with 119877120572 =10m 119877120573 = infin 119886 = 1205873119877120572 and 119887 = 20m and thickness ratios119877120572ℎ = 100 and 119877120572ℎ = 10 The present shell is a sandwichstructure embedding a FGM core with ℎ2 = 07ℎ and two

Shock and Vibration 13

218

1614

121 1

08 0806 06

04 0402 02

0 0

0minus01minus02minus03minus04

(a) 119871119886 = 2

3

25

2

151

0806

040502

0 0

1

0minus01minus02minus03minus04minus05

(b) 119871119886 = 3

435

325

10505

0 0

15

0

minus02minus04minus06

1

2

(c) 119871119886 = 4

545

35

10505

0 0

0

minus04minus02

minus06minus08

1

25

4

3

215

(d) 119871119886 = 5

5

1

0 005

0

minus04minus02

minus06minus08

1

4

3

2

6

minus1

(e) 119871119886 = 6

5

10 0 051

43

2

67

89

100

minus05minus1

minus15

(f) 119871119886 = 10

Figure 5 First vibration mode for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 2D GDQ solution and edge effects versus the lengthof the simply supported sides

external skins with ℎ1 = ℎ3 = 015ℎ (ℎ1 + ℎ2 + ℎ3 = ℎ) Thebottom layer ismade of isotropicmetallicmaterial and the toplayer is made of ceramic material the mechanical propertiesare

119864119898 = 73GPa120588119898 = 2800 kgm3119864119888 = 200GPa120588119888 = 5700 kgm3

(27)

The FGM core has a volume fraction function defined as

119881119888 (119911) = (05 + 11991107ℎ)119901 with 119911 isin [minus035ℎ 035ℎ] (28)

where 119901 = 10 is employed in the present computations ThePoisson ratio is constant (]119888 = ]119898 = ] = 03) Equations (25)are still valid with 119881119888 defined as in (28)

The FEmodel is made of 40times40Quad8 elements for total29280 dofs and the GDQ model has a 31 times 31 Chebyshev-Gauss-Lobatto grid Since a sandwich structure is analyzedin the following a third-order theory with zigzag functionis considered in the computations for the GDQ method Thepresent natural frequencies are reported in Table 4 with the

14 Shock and Vibration

Table 2 First frequency 119891 in Hz for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 convergence of 2D GDQ and 2D FE solutionsincreasing the length of the simply supported sides

119871119886 1 2 3 4 5 6 8 10 15 20 3D exact2D GDQ 3921 3964 3980 3988 3993 3997 4001 4004 4007 4008 40192D FEM 3907 3949 3966 3974 3979 3983 3987 3989 3993 3994 4019

Table 3 One-layered FGM cylinder with 119901 = 20 Analytical 3D versus 2D numerical models Frequency 119891 in Hz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1004 0 I 0896 mdash 0897 mdash 08836 0 I 2532 mdash 2538 mdash 25118 0 I 4854 mdash 4867 mdash 481910 0 I 7848 mdash 7871 mdash 780112 0 I 1151 mdash 1155 mdash 114610 1 I 1356 1360 mdash 1357 mdash12 1 I 1441 1449 mdash 1445 mdash14 0 I 1584 mdash 1589 mdash 15778 1 I 1638 1640 mdash 1636 mdash14 1 I 1756 1767 mdash 1763 mdash16 0 I 2083 mdash 2090 mdash 207016 1 I 2207 2222 mdash 2216 mdash6 1 I 2441 2442 mdash 2438 mdash18 0 I 2649 mdash 2658 mdash 264014 2 I 2742 2752 mdash 2746 mdash18 1 I 2751 2772 mdash 2763 mdash16 2 I 2875 2890 mdash 2883 mdash12 2 I 2921 2926 mdash 2922 mdash119877120572ℎ = 104 0 I 8774 mdash 9085 mdash 86796 0 I 2456 mdash 2571 mdash 24306 1 I 3978 4089 mdash 3979 mdash4 1 I 4388 4357 mdash 4388 mdash8 0 I 4654 mdash 4920 mdash 46028 1 I 5575 5905 mdash 5576 mdash2 1 I 7202 7101 mdash 7202 mdash2 0 I (119908 = 0) 7322 7390 mdash 7319 mdash10 0 I 7419 mdash 7933 mdash 73476 2 I 8110 8261 mdash 8111 mdash10 1 I 8189 8811 mdash 8191 mdash8 2 I 8783 9173 mdash 8785 mdash4 2 I 9016 9084 mdash 9017 mdash12 0 I 1071 mdash 1159 mdash 106110 2 I 1081 1155 mdash 1081 mdash2 2 I 1089 1105 mdash 1089 mdash12 1 I 1142 1245 mdash 1143 mdash4 3 I 1263 1286 mdash 1263 mdash

same meaning of the columns seen in the previous casesIn order to achieve the cylindrical bending conditions thenumerical models have the straight edges simply supportedand the curved edges as free Due to the CB definitionit is not possible to achieve CB conditions when 119898 = 0

and 119899 = 1 2 3 because the cylindrical bending shouldoccur on the curved edge and this is not physically possiblebecause of the variation of displacement components due tothe curvature presence This aspect is indicated as NA (NotApplicable) solution in Table 4 Therefore this kind of 3D

Shock and Vibration 15

Table 4 Sandwich cylindrical shell panel embedding FGM core with 119901 = 10 Analytical 3D versus 2D numerical models Frequency 119891 inHz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1001 0 I 1948 mdash 1951 mdash 19412 0 I 8856 mdash 8872 mdash 88232 1 I 1104 1110 mdash 1104 mdash1 1 I 1850 1873 mdash 1850 mdash3 0 I 2038 mdash 2042 mdash 20303 1 I 2116 2128 mdash 2116 mdash2 2 I 2221 2242 mdash 2221 mdash3 2 I 2484 2496 mdash 2484 mdash3 3 I 3235 3254 mdash 3235 mdash4 0 I 3649 mdash 3660 mdash 36342 3 I 3692 3739 mdash 3692 mdash4 1 I 3714 3745 mdash 3714 mdash4 2 I 3940 3965 mdash 3941 mdash3 4 I 4226 4261 mdash 4226 mdash4 3 I 4378 4403 mdash 4378 mdash1 2 I 4568 4664 mdash 4568 mdash4 4 I 5035 5064 mdash 5035 mdash2 4 I 5040 5128 mdash 5040 mdash119877120572ℎ = 101 0 I 1905 mdash 1963 mdash 18841 1 I 3054 3145 mdash 3055 mdash1 2 I 6209 6409 mdash 6209 mdash2 0 I 8326 mdash 8857 mdash 82540 1 I 8710 mdash NA mdash NA0 1 II (119908 = 0) 8820 8956 mdash 8820 mdash2 1 I 8880 9531 mdash 8882 mdash0 2 I 9176 mdash NA mdash NA1 3 I 9576 9983 mdash 9567 mdash0 3 I 1036 mdash NA mdash NA2 2 I 1066 1147 mdash 1067 mdash0 4 I 1291 mdash NA mdash NA1 4 I 1335 1413 mdash 1335 mdash2 3 I 1364 1475 mdash 1364 mdash1 0 II (119908 = 0) 1665 1738 mdash 1665 mdash0 5 I 1683 mdash NA mdash NA2 4 I 1755 1918 mdash 1755 mdash0 2 II (119908 = 0) 1764 1791 mdash 1764 mdash

exact solutions cannot be achieved by any numerical modelOnce again solution with 119899 = 0 or 119898 = 0 combined with119908 = 0 (in-plane modes) can be found with 2D numericalmodels without modifying the simply supported boundaryconditions

5 Conclusions

Thepresent paper analyzes the cylindrical bending in the freefrequency analysis of functionally graded material (FGM)plates and cylindrical shells This condition is investigatedby means of a 3D exact shell model and two different 2D

numerical models a weak formulation based on classicalFinite Elements (FEs) and a strong formulation based onthe GDQ method The cylindrical bending (CB) behavior inthe 3D exact shell model can be analyzed using completelysimply supported structures and supposing one of the twohalf-wave numbers equals zeroThe numerical models obtainthe CB behavior using mixed simply supported and freeboundary conditions It has been demonstrated that theactual cylindrical bending (CB) is possible only with respectto straight edgesTheCBnatural frequencies of the numericalsolutions converge to the 3D exact CB ones if the length ofstraight simply supported edges is increased up to 20 times

16 Shock and Vibration

the value of the original edges The CB conditions are notpossible in 2D numericalmodels when the half-wave numberis zero in the curvilinear edge in this case the modificationof boundary conditions does not properly work In-planevibrationmodeswith zero transverse displacement combinedwith 119898 = 0 or 119899 = 0 cannot be defined as CB frequencies andthey can be calculated via 2D numerical models without themodification of the simply supported conditions for the fouredges All the considerations about the cylindrical bendingare valid for each geometry (plate cylinder and cylindricalshell) lamination scheme FGM law and thickness ratio

Competing Interests

The authors declare that they have no competing interests

References

[1] W Q Chen J B Cai and G R Ye ldquoResponses of cross-ply laminates with viscous interfaces in cylindrical bendingrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 823ndash835 2005

[2] W Q Chen and K Y Lee ldquoThree-dimensional exact analysisof angle-ply laminates in cylindrical bending with interfacialdamage via state-space methodrdquo Composite Structures vol 64no 3-4 pp 275ndash283 2004

[3] W Q Chen and K Y Lee ldquoTime-dependent behaviors of angle-ply laminates with viscous interfaces in cylindrical bendingrdquoEuropean Journal ofMechanicsmdashASolids vol 23 no 2 pp 235ndash245 2004

[4] A H Gandhi and H K Raval ldquoAnalytical and empiricalmodeling of top roller position for three-roller cylindricalbending of plates and its experimental verificationrdquo Journal ofMaterials Processing Technology vol 197 no 1ndash3 pp 268ndash2782008

[5] S Oral and H Darendeliler ldquoThe optimum die profile for thecylindrical bending of platesrdquo Journal of Materials ProcessingTechnology vol 70 no 1ndash3 pp 151ndash155 1997

[6] V A Jairazbhoy P Petukhov and J Qu ldquoLarge deflectionof thin plates in cylindrical bendingmdashnon-unique solutionsrdquoInternational Journal of Solids and Structures vol 45 no 11-12pp 3203ndash3218 2008

[7] A Lebee and K Sab ldquoA bending-gradient model for thickplates part II closed-form solutions for cylindrical bending oflaminatesrdquo International Journal of Solids and Structures vol 48no 20 pp 2889ndash2901 2011

[8] P V Nimbolkar and I M Jain ldquoCylindrical bending of elasticplatesrdquo Procedia Materials Science vol 10 pp 793ndash802 2015

[9] A S Sayyad and Y M Ghugal ldquoA nth-order shear deformationtheory for composite laminates in cylindrical bendingrdquo Curvedand Layered Structures vol 2 no 1 pp 290ndash300 2015

[10] N J Pagano ldquoExact solutions for composite laminates incylindrical bendingrdquo Journal of Composite Materials vol 3 no3 pp 398ndash411 1969

[11] N J Pagano and A S D Wang ldquoFurther study of compositelaminates under cylindrical bendingrdquo Journal of CompositeMaterials vol 5 pp 421ndash428 1971

[12] N Saeedi K Sab and J-F Caron ldquoCylindrical bending ofmultilayered plates with multi-delamination via a layerwisestress approachrdquo Composite Structures vol 95 pp 728ndash7392013

[13] X-P Shu and K P Soldatos ldquoCylindrical bending of angle-ply laminates subjected to different sets of edge boundaryconditionsrdquo International Journal of Solids and Structures vol37 no 31 pp 4289ndash4307 2000

[14] Z G Bian W Q Chen C W Lim and N Zhang ldquoAnalyticalsolutions for single- and multi-span functionally graded platesin cylindrical bendingrdquo International Journal of Solids andStructures vol 42 no 24-25 pp 6433ndash6456 2005

[15] H M Navazi and H Haddadpour ldquoNonlinear cylindricalbending analysis of shear deformable functionally graded platesunder different loadings using analytical methodsrdquo Interna-tional Journal of Mechanical Sciences vol 50 no 12 pp 1650ndash1657 2008

[16] W Q Chen J Ying J B Cai and G R Ye ldquoBenchmarksolution of imperfect angle-ply laminated rectangular plates incylindrical bending with surface piezoelectric layers as actuatorand sensorrdquo Computers and Structures vol 82 no 22 pp 1773ndash1784 2004

[17] W Q Chen and K Y Lee ldquoExact solution of angle-plypiezoelectric laminates in cylindrical bending with interfacialimperfectionsrdquo Composite Structures vol 65 no 3-4 pp 329ndash337 2004

[18] T Kant and SM Shiyekar ldquoCylindrical bending of piezoelectriclaminates with a higher order shear and normal deformationtheoryrdquo Computers and Structures vol 86 no 15-16 pp 1594ndash1603 2008

[19] P Lu H P Lee and C Lu ldquoAn exact solution for functionallygraded piezoelectric laminates in cylindrical bendingrdquo Interna-tional Journal of Mechanical Sciences vol 47 no 3 pp 437ndash4382005

[20] W Yan J Wang and W Q Chen ldquoCylindrical bendingresponses of angle-ply piezoelectric laminates with viscoelasticinterfacesrdquo Applied Mathematical Modelling vol 38 no 24 pp6018ndash6030 2014

[21] Y Y Zhou W Q Chen and C F Lu ldquoSemi-analytical solutionfor orthotropic piezoelectric laminates in cylindrical bendingwith interfacial imperfectionsrdquo Composite Structures vol 92no 4 pp 1009ndash1018 2010

[22] G P Dube S Kapuria and P C Dumir ldquoExact piezother-moelastic solution of a simply-supported orthotropic flat panelin cylindrical bendingrdquo International Journal of MechanicalSciences vol 38 pp 374ndash387 1996

[23] V N Pilipchuk V L Berdichevsky and R A IbrahimldquoThermo-mechanical coupling in cylindrical bending of sand-wich platesrdquoComposite Structures vol 92 no 11 pp 2632ndash26402010

[24] C Zhang S Di and N Zhang ldquoA new procedure for staticanalysis of thermo-electric laminated composite plates undercylindrical bendingrdquo Composite Structures vol 56 no 2 pp131ndash140 2002

[25] W Q Chen and K Y Lee ldquoState-space approach for statics anddynamics of angle-ply laminated cylindrical panels in cylindri-cal bendingrdquo International Journal of Mechanical Sciences vol47 no 3 pp 374ndash387 2005

[26] C-P Wu and Y-S Syu ldquoExact solutions of functionally gradedpiezoelectric shells under cylindrical bendingrdquo InternationalJournal of Solids and Structures vol 44 no 20 pp 6450ndash64722007

[27] W Yan J Ying and W Q Chen ldquoThe behavior of angle-ply laminated cylindrical shells with viscoelastic interfaces incylindrical bendingrdquo Composite Structures vol 78 no 4 pp551ndash559 2007

Shock and Vibration 17

[28] S Brischetto F Tornabene N Fantuzzi and E Viola ldquo3Dexact and 2D generalized differential quadraturemodels for freevibration analysis of functionally graded plates and cylindersrdquoMeccanica vol 51 no 9 pp 2059ndash2098 2016

[29] N Fantuzzi S Brischetto F Tornabene and E Viola ldquo2D and3D shell models for the free vibration investigation of func-tionally graded cylindrical and spherical panelsrdquo CompositeStructures vol 154 pp 573ndash590 2016

[30] S Brischetto and R Torre ldquoExact 3D solutions and finiteelement 2D models for free vibration analysis of plates andcylindersrdquo Curved and Layered Structures vol 1 no 1 pp 59ndash92 2014

[31] F Tornabene S Brischetto N Fantuzzi and E Viola ldquoNumer-ical and exact models for free vibration analysis of cylindricaland spherical shell panelsrdquo Composites Part B Engineering vol81 pp 231ndash250 2015

[32] S Brischetto ldquoExact elasticity solution for natural frequenciesof functionally graded simply-supported structuresrdquo ComputerModeling in Engineering amp Sciences vol 95 no 5 pp 391ndash4302013

[33] S Brischetto ldquoThree-dimensional exact free vibration analysisof spherical cylindrical and flat one-layered panelsrdquo Shock andVibration vol 2014 Article ID 479738 29 pages 2014

[34] S Brischetto ldquoA continuum elastic three-dimensional modelfor natural frequencies of single-walled carbon nanotubesrdquoComposites Part B Engineering vol 61 pp 222ndash228 2014

[35] S Brischetto ldquoAn exact 3d solution for free vibrations ofmultilayered cross-ply composite and sandwich plates andshellsrdquo International Journal of Applied Mechanics vol 6 no 6Article ID 1450076 42 pages 2014

[36] S Brischetto ldquoA continuumshellmodel including vanderWaalsinteraction for free vibrations of double-walled carbon nan-otubesrdquoCMES-ComputerModeling in Engineering and Sciencesvol 104 no 4 pp 305ndash327 2015

[37] S Brischetto F Tornabene N Fantuzzi and M BacciocchildquoRefined 2D and exact 3D shell models for the free vibrationanalysis of single- and double-walled carbon nanotubesrdquo Tech-nologies vol 3 no 4 pp 259ndash284 2015

[38] S Brischetto ldquoConvergence analysis of the exponential matrixmethod for the solution of 3D equilibrium equations for freevibration analysis of plates and shellsrdquo Composites Part BEngineering vol 98 pp 453ndash471 2016

[39] S Brischetto ldquoExact and approximate shell geometry in the freevibration analysis of one-layered and multilayered structuresrdquoInternational Journal of Mechanical Sciences vol 113 pp 81ndash932016

[40] S Brischetto ldquoCurvature approximation effects in the freevibration analysis of functionally graded shellsrdquo InternationalJournal of Applied Mechanics 2016

[41] F Tornabene N Fantuzzi F Ubertini and E Viola ldquoStrong for-mulation finite element method based on differential quadra-ture a surveyrdquo Applied Mechanics Reviews vol 67 no 2 ArticleID 020801 pp 1ndash55 2015

[42] F Tornabene E Viola and N Fantuzzi ldquoGeneral higher-orderequivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panelsrdquo CompositeStructures vol 104 pp 94ndash117 2013

[43] N Fantuzzi F Tornabene E Viola and A J Ferreira ldquoAstrong formulation finite element method (SFEM) based onRBF and GDQ techniques for the static and dynamic analysesof laminated plates of arbitrary shaperdquo Meccanica vol 49 no10 pp 2503ndash2542 2014

[44] F Tornabene N Fantuzzi and M Bacciocchi ldquoFree vibrationsof free-form doubly-curved shells made of functionally gradedmaterials using higher-order equivalent single layer theoriesrdquoComposites Part B Engineering vol 67 pp 490ndash509 2014

[45] F Tornabene and E Viola ldquoStatic analysis of functionallygraded doubly-curved shells and panels of revolutionrdquo Mecca-nica vol 48 no 4 pp 901ndash930 2013

[46] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoDynamic analysis of thick and thin elliptic shell structuresmade of laminated composite materialsrdquo Composite Structuresvol 133 pp 278ndash299 2015

[47] F TornabeneN FantuzziM Bacciocchi and EViola ldquoHigher-order theories for the free vibrations of doubly-curved lam-inated panels with curvilinear reinforcing fibers by meansof a local version of the GDQ methodrdquo Composites Part BEngineering vol 81 pp 196ndash230 2015

[48] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoFree vibrations of composite oval and elliptic cylinders bythe generalized differential quadrature methodrdquo Thin-WalledStructures vol 97 pp 114ndash129 2015

[49] F Tornabene and J N Reddy ldquoFGM and laminated doubly-curved and degenerate shells resting on nonlinear elastic foun-dations a GDQ solution for static analysis with a posterioristress and strain recoveryrdquo Journal of the Indian Institute ofScience vol 93 no 4 pp 635ndash688 2013

[50] N Fantuzzi F Tornabene and E Viola ldquoFour-parameter func-tionally graded cracked plates of arbitrary shape a GDQFEMsolution for free vibrationsrdquo Mechanics of Advanced Materialsand Structures vol 23 no 1 pp 89ndash107 2016

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Shock and Vibration

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8 Shock and Vibration

Table 1 One-layered FGM plate with 119901 = 05 Analytical 3D versus 2D numerical models Frequency 119891 in Hz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119886ℎ = 1001 0 I 4019 mdash 3907 mdash 39210 1 I 4019 mdash 3907 mdash 39211 1 I 8037 8006 mdash 8036 mdash2 0 I 1607 mdash 1580 mdash 15840 2 I 1607 mdash 1580 mdash 15842 1 I 2008 2001 mdash 2008 mdash1 2 I 2008 2001 mdash 2008 mdash2 2 I 3212 3202 mdash 3211 mdash3 0 I 3613 mdash 3568 mdash 35770 3 I 3613 mdash 3568 mdash 35773 1 I 4014 4002 mdash 4013 mdash1 3 I 4014 4002 mdash 4013 mdash3 2 I 5215 5202 mdash 5215 mdash2 3 I 5215 5202 mdash 5215 mdash4 0 I 6416 mdash 6354 mdash 63650 4 I 6416 mdash 6354 mdash 63654 1 I 6816 6801 mdash 6815 mdash1 4 I 6816 6801 mdash 6815 mdash119886ℎ = 201 0 I 2002 mdash 1951 mdash 19510 1 I 2002 mdash 1951 mdash 19511 1 I 3989 3995 mdash 3988 mdash2 0 I 7916 mdash 7864 mdash 77970 2 I 7916 mdash 7864 mdash 77972 1 I 9858 9954 mdash 9857 mdash1 2 I 9858 9954 mdash 9857 mdash2 2 I 1560 1588 mdash 1560 mdash3 0 I 1749 mdash 1767 mdash 17290 3 I 1749 mdash 1767 mdash 17293 1 I 1936 1980 mdash 1936 mdash1 3 I 1936 1980 mdash 1936 mdash3 2 I 2490 2566 mdash 2490 mdash2 3 I 2490 2566 mdash 2490 mdash1 0 II (119908 = 0) 2775 2773 mdash 2775 mdash0 1 II (119908 = 0) 2775 2773 mdash 2775 mdash4 0 I 3034 mdash 3124 mdash 30070 4 I 3034 mdash 3124 mdash 3007

where 119901 is the power exponent which varies the volumefraction of the ceramic component through the thicknessTwo limit cases occur for (24) 119881119888 = 1 when 119901 = 0 and119881119888 rarr 0when119901 = infin In the following computations it is set as119901 = 05 Note that the volume fraction function in (24) givesa material which is ceramic (119888) rich at the top and metal (119898)rich at the bottomof the structureThemechanical properties(Youngmodulus andmass density) are given by the theory ofmixtures as

119864 (119911) = 119864119898 + (119864119888 minus 119864119898) 119881119888120588 (119911) = 120588119898 + (120588119888 minus 120588119898) 119881119888 (25)

where 119864119898 and 120588119898 are the elastic modulus and mass densityof the metallic phase respectively 119864119888 and 120588119888 are the corre-sponding properties of the ceramic phase Note that in allthe following computations the Poisson ratio is consideredconstant through the thickness thus ]119888 = ]119898 = ] = 03 Thetwo constituents have the following mechanical properties

119864119898 = 73GPa120588119898 = 2800 kgm3119864119888 = 380GPa120588119888 = 3800 kgm3

(26)

Shock and Vibration 9

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(a) Mode I (0 1) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

000501

01502

(b) Mode I (0 1) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(c) Mode II (119908 = 0) (0 1) 3D

002minus002

090908

08070706

06050504

04030302

020101

00minus01

1

(d) Mode II (119908 = 0) (0 1) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(e) Mode I (1 0) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

0minus005minus01

minus015minus02

(f) Mode I (1 0) 2D

Figure 2 Continued

10 Shock and Vibration

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(g) Mode II (119908 = 0) (1 0) 3D

002minus002

0908 08

0706 06

0504 04

0302 02

01 0 0

11

(h) Mode II (119908 = 0) (1 0) 2D

Figure 2 Differences between in-plane and cylindrical bending vibration modes for the 119886ℎ = 20 one-layered FGM plate with 119901 = 05 3Dmodes versus 2D modes

Since the FE commercial software does not have anembedded tool for the computation of functionally gradedproperties through the thickness the present FE applicationsare carried out building a fictitious laminated structure withisotropic plies In detail the thickness of the FGM lamina hasbeen divided into 100 fictitious layers and for each layer thecorresponding mechanical properties have been computedthrough (25) The numerical model has been made of auniform element division of 40 times 40 Quad8 elements for atotal number of 29280 degrees of freedom (dofs) The GDQsolution is computed with a not-uniform Chebyshev-Gauss-Lobatto grid [41] with 31 times 31 points for both thicknessratios It is underlined that since a single ply is set no zigzagfunction is needed in the computation It is recalled that thepresent higher order GDQ model considers a third orderexpansion with 120591 = 0 1 2 3 whereas the FE one has afirst order model (also known as Reissner-Mindlin model)Table 1 reports the results in terms of natural frequency(Hz) wherein the first column reports the number of half-waves (119898 119899) of the 3D exact solution Note that for a fixedset of half-wave numbers different mode shapes can occur(indicated by the second column ldquoModerdquo) In particular Iindicates the first mode and II indicates the second mode Inthe case of in-plane modes the caption (119908 = 0) is added tounderline that the vibration occurs in the plane and not outof plane The third column shows all the results for the 3Dexact solution both classical and cylindrical bending (CB)results It is noted that these frequencies are obtained witha single run of the 3D exact code keeping simply supportedboundary conditions for all the four edges of the plate Theother four columns of Table 1 refer to the numerical solutionsfor simply supported structures and for cylindrical bendingconditions (setting two free opposite edges) The cylindricalbending results are indicated by the acronym CB in the sametable

There is a very good agreement among the three solutionswhen simply supported conditions are considered On thecontrary a small error is observed in cylindrical bendingbecause the actual CB condition is not completely satisfieddue to the fact that the plate has a limited dimension in thesimply supported direction This feature can be detected inFigures 2 and 3 because the deformed shapes of the plate inthe CB configuration are not properly correct In fact thefibers parallel to the supported edges are not parallel amongthem but there is a curvature effect due to the free boundariesand to the limited length of the plate On the contrary thein-plane mode shapes (119908 = 0) when the plate is simplysupported on the four edges have all the fibers parallel amongthem also after the deformation These modes are the resultsobtained in the 3D solution with 119898 = 0 or 119899 = 0 combinedwith 119908 = 0 They are not cylindrical bending modes infact they can be found with 2D numerical models withoutmodifying the boundary conditions 3D frequencieswith119898 =0 or 119899 = 0 (with w different from 0) are CB modes which canbe obtained considering two simply supported edges and twofree edges in 2D numerical models

As it is well-known the ldquoperfectrdquo CB would occur whenthe simply supported direction has an ldquoinfiniterdquo lengthThis cannot be proven numerically but a trend can beobserved modeling different plates in cylindrical bendingconfiguration with different lengthsThese analyses are madekeeping one side of the plate constant and enlarging thelength of the other edge In the present computations 119886 iskept constant and 119887 = 119871 varies from 1 to 20 The results ofthe aforementioned analysis are listed in Table 2 and depictedin Figures 4 and 5 Table 2 includes the numerical valuesof the first mode shape in cylindrical bending for the 3Dexact solution the 2D FE and the 2D GDQmodels Figure 4shows that both GDQ and FE solutions tend to the 3D exactone increasing the length of the simply supported edge In

Shock and Vibration 11

ulowast

lowast

wlowast

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

(a) Mode I (1 0) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

000501

01502

(b) Mode I (1 0) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(c) Mode I (2 0) 3D

109

0807

060504

0302

01 0

109

0807

0605

0403

02010

0201501

0050

minus005minus01

minus015minus02

(d) Mode I (2 0) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(e) Mode I (3 0) 3D

10908

0706

050403

0201 0

109

0807

0605

0403

02010

0201501

0050

minus005minus01

minus015minus02

(f) Mode I (3 0) 2D

Figure 3 Continued

12 Shock and Vibration

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(g) Mode I (4 0) 3D

10908

0706050403

0201 0

10908

0706

0504

0302

010

0201501

0050

minus005minus01

minus015minus02

(h) Mode I (4 0) 2D

Figure 3 First four cylindrical bending vibration modes for the 119886ℎ = 20 one-layered FGM plate with 119901 = 05 3D modes versus 2D modes

3D exact2D GDQ2D FEM

f(H

z)

39

392

394

396

398

40

402

404

5 10 15 200La

Figure 4 First frequency 119891 in Hz for the 119886ℎ = 100 one-layeredFGM plate with 119901 = 05 convergence of 2D GDQ and 2D FEsolutions increasing the length of the simply supported sides

particular the GDQ solution is slightly more accurate thanthe FE one The 2D GDQ solution is carried out by enlargingthe length of one of the sides of the plate and the samenumberof dofs is kept On the contrary the FE model is generatedby 40 times 40 blocks of mesh In other words the plate with119871 = 2119886 has a mesh of 40 times 80 the one with 119871 = 3119886 has amesh 40 times 120 and so on When 119871 = 20119886 the error amongthe 3D exact and the 2D numerical solutions is very small andthis is proven also by the mode shapes of Figure 5 whereinthe boundary effect is negligible with respect the length of

the simply supported edges It must be underlined that it isimpossible to completely avoid the boundary effects due tothe free edges

The second example is related to a simply supportedcylinder with 119877120572 = 10m 119877120573 = infin 119886 = 2120587119877120572 and 119887 =20m and thickness ratios 119877120572ℎ = 100 and 119877120572ℎ = 10 Aone-layered FGM scheme is considered with ℎ1 = ℎ andthe same volume fraction function of (24) for the previousexample but setting 119901 = 20 The present material propertiesare the same as the flat plate The FE model has a 20 times80 Quad8 mesh with 27966 dofs and the GDQ solutionhas a 15 times 51 Chebyshev-Gauss-Lobatto grid Numericallyspeaking compatibility conditions must be enforced on theclosing meridian in order to obtain a closed revolution shell[42] To obtain the CB conditions numerically the cylinderis modeled completely free and the first 6 rigid mode shapesare neglected and removed from the numerical solution Itis noted that the GDQ model uses a third-order kinematicexpansion with no zigzag effect as the previous case sincea one-layer structure has been taken into account Table 3reports the results with the same meaning of the symbolsseen in Table 1 A very good agreement is observed when thecylinder is simply supported whereas when the cylinder iscompletely free some small errors can be observed but thenumerical solutions follow the 3D exact one

Results with 119899 = 0 and 119908 = 0 indicate the CB conditionsSuch results are obtained by means of 2D numerical modelsmodifying the boundary conditions (see 2D FE(CB) and 2DGDQ(CB)models)The cases 119899 = 0 combined with119908 = 0 arenot CB conditions and the frequencies can be found withoutmodifying the boundary conditions in the 2D numericalmodels

The final test is related to a cylindrical panel with 119877120572 =10m 119877120573 = infin 119886 = 1205873119877120572 and 119887 = 20m and thickness ratios119877120572ℎ = 100 and 119877120572ℎ = 10 The present shell is a sandwichstructure embedding a FGM core with ℎ2 = 07ℎ and two

Shock and Vibration 13

218

1614

121 1

08 0806 06

04 0402 02

0 0

0minus01minus02minus03minus04

(a) 119871119886 = 2

3

25

2

151

0806

040502

0 0

1

0minus01minus02minus03minus04minus05

(b) 119871119886 = 3

435

325

10505

0 0

15

0

minus02minus04minus06

1

2

(c) 119871119886 = 4

545

35

10505

0 0

0

minus04minus02

minus06minus08

1

25

4

3

215

(d) 119871119886 = 5

5

1

0 005

0

minus04minus02

minus06minus08

1

4

3

2

6

minus1

(e) 119871119886 = 6

5

10 0 051

43

2

67

89

100

minus05minus1

minus15

(f) 119871119886 = 10

Figure 5 First vibration mode for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 2D GDQ solution and edge effects versus the lengthof the simply supported sides

external skins with ℎ1 = ℎ3 = 015ℎ (ℎ1 + ℎ2 + ℎ3 = ℎ) Thebottom layer ismade of isotropicmetallicmaterial and the toplayer is made of ceramic material the mechanical propertiesare

119864119898 = 73GPa120588119898 = 2800 kgm3119864119888 = 200GPa120588119888 = 5700 kgm3

(27)

The FGM core has a volume fraction function defined as

119881119888 (119911) = (05 + 11991107ℎ)119901 with 119911 isin [minus035ℎ 035ℎ] (28)

where 119901 = 10 is employed in the present computations ThePoisson ratio is constant (]119888 = ]119898 = ] = 03) Equations (25)are still valid with 119881119888 defined as in (28)

The FEmodel is made of 40times40Quad8 elements for total29280 dofs and the GDQ model has a 31 times 31 Chebyshev-Gauss-Lobatto grid Since a sandwich structure is analyzedin the following a third-order theory with zigzag functionis considered in the computations for the GDQ method Thepresent natural frequencies are reported in Table 4 with the

14 Shock and Vibration

Table 2 First frequency 119891 in Hz for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 convergence of 2D GDQ and 2D FE solutionsincreasing the length of the simply supported sides

119871119886 1 2 3 4 5 6 8 10 15 20 3D exact2D GDQ 3921 3964 3980 3988 3993 3997 4001 4004 4007 4008 40192D FEM 3907 3949 3966 3974 3979 3983 3987 3989 3993 3994 4019

Table 3 One-layered FGM cylinder with 119901 = 20 Analytical 3D versus 2D numerical models Frequency 119891 in Hz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1004 0 I 0896 mdash 0897 mdash 08836 0 I 2532 mdash 2538 mdash 25118 0 I 4854 mdash 4867 mdash 481910 0 I 7848 mdash 7871 mdash 780112 0 I 1151 mdash 1155 mdash 114610 1 I 1356 1360 mdash 1357 mdash12 1 I 1441 1449 mdash 1445 mdash14 0 I 1584 mdash 1589 mdash 15778 1 I 1638 1640 mdash 1636 mdash14 1 I 1756 1767 mdash 1763 mdash16 0 I 2083 mdash 2090 mdash 207016 1 I 2207 2222 mdash 2216 mdash6 1 I 2441 2442 mdash 2438 mdash18 0 I 2649 mdash 2658 mdash 264014 2 I 2742 2752 mdash 2746 mdash18 1 I 2751 2772 mdash 2763 mdash16 2 I 2875 2890 mdash 2883 mdash12 2 I 2921 2926 mdash 2922 mdash119877120572ℎ = 104 0 I 8774 mdash 9085 mdash 86796 0 I 2456 mdash 2571 mdash 24306 1 I 3978 4089 mdash 3979 mdash4 1 I 4388 4357 mdash 4388 mdash8 0 I 4654 mdash 4920 mdash 46028 1 I 5575 5905 mdash 5576 mdash2 1 I 7202 7101 mdash 7202 mdash2 0 I (119908 = 0) 7322 7390 mdash 7319 mdash10 0 I 7419 mdash 7933 mdash 73476 2 I 8110 8261 mdash 8111 mdash10 1 I 8189 8811 mdash 8191 mdash8 2 I 8783 9173 mdash 8785 mdash4 2 I 9016 9084 mdash 9017 mdash12 0 I 1071 mdash 1159 mdash 106110 2 I 1081 1155 mdash 1081 mdash2 2 I 1089 1105 mdash 1089 mdash12 1 I 1142 1245 mdash 1143 mdash4 3 I 1263 1286 mdash 1263 mdash

same meaning of the columns seen in the previous casesIn order to achieve the cylindrical bending conditions thenumerical models have the straight edges simply supportedand the curved edges as free Due to the CB definitionit is not possible to achieve CB conditions when 119898 = 0

and 119899 = 1 2 3 because the cylindrical bending shouldoccur on the curved edge and this is not physically possiblebecause of the variation of displacement components due tothe curvature presence This aspect is indicated as NA (NotApplicable) solution in Table 4 Therefore this kind of 3D

Shock and Vibration 15

Table 4 Sandwich cylindrical shell panel embedding FGM core with 119901 = 10 Analytical 3D versus 2D numerical models Frequency 119891 inHz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1001 0 I 1948 mdash 1951 mdash 19412 0 I 8856 mdash 8872 mdash 88232 1 I 1104 1110 mdash 1104 mdash1 1 I 1850 1873 mdash 1850 mdash3 0 I 2038 mdash 2042 mdash 20303 1 I 2116 2128 mdash 2116 mdash2 2 I 2221 2242 mdash 2221 mdash3 2 I 2484 2496 mdash 2484 mdash3 3 I 3235 3254 mdash 3235 mdash4 0 I 3649 mdash 3660 mdash 36342 3 I 3692 3739 mdash 3692 mdash4 1 I 3714 3745 mdash 3714 mdash4 2 I 3940 3965 mdash 3941 mdash3 4 I 4226 4261 mdash 4226 mdash4 3 I 4378 4403 mdash 4378 mdash1 2 I 4568 4664 mdash 4568 mdash4 4 I 5035 5064 mdash 5035 mdash2 4 I 5040 5128 mdash 5040 mdash119877120572ℎ = 101 0 I 1905 mdash 1963 mdash 18841 1 I 3054 3145 mdash 3055 mdash1 2 I 6209 6409 mdash 6209 mdash2 0 I 8326 mdash 8857 mdash 82540 1 I 8710 mdash NA mdash NA0 1 II (119908 = 0) 8820 8956 mdash 8820 mdash2 1 I 8880 9531 mdash 8882 mdash0 2 I 9176 mdash NA mdash NA1 3 I 9576 9983 mdash 9567 mdash0 3 I 1036 mdash NA mdash NA2 2 I 1066 1147 mdash 1067 mdash0 4 I 1291 mdash NA mdash NA1 4 I 1335 1413 mdash 1335 mdash2 3 I 1364 1475 mdash 1364 mdash1 0 II (119908 = 0) 1665 1738 mdash 1665 mdash0 5 I 1683 mdash NA mdash NA2 4 I 1755 1918 mdash 1755 mdash0 2 II (119908 = 0) 1764 1791 mdash 1764 mdash

exact solutions cannot be achieved by any numerical modelOnce again solution with 119899 = 0 or 119898 = 0 combined with119908 = 0 (in-plane modes) can be found with 2D numericalmodels without modifying the simply supported boundaryconditions

5 Conclusions

Thepresent paper analyzes the cylindrical bending in the freefrequency analysis of functionally graded material (FGM)plates and cylindrical shells This condition is investigatedby means of a 3D exact shell model and two different 2D

numerical models a weak formulation based on classicalFinite Elements (FEs) and a strong formulation based onthe GDQ method The cylindrical bending (CB) behavior inthe 3D exact shell model can be analyzed using completelysimply supported structures and supposing one of the twohalf-wave numbers equals zeroThe numerical models obtainthe CB behavior using mixed simply supported and freeboundary conditions It has been demonstrated that theactual cylindrical bending (CB) is possible only with respectto straight edgesTheCBnatural frequencies of the numericalsolutions converge to the 3D exact CB ones if the length ofstraight simply supported edges is increased up to 20 times

16 Shock and Vibration

the value of the original edges The CB conditions are notpossible in 2D numericalmodels when the half-wave numberis zero in the curvilinear edge in this case the modificationof boundary conditions does not properly work In-planevibrationmodeswith zero transverse displacement combinedwith 119898 = 0 or 119899 = 0 cannot be defined as CB frequencies andthey can be calculated via 2D numerical models without themodification of the simply supported conditions for the fouredges All the considerations about the cylindrical bendingare valid for each geometry (plate cylinder and cylindricalshell) lamination scheme FGM law and thickness ratio

Competing Interests

The authors declare that they have no competing interests

References

[1] W Q Chen J B Cai and G R Ye ldquoResponses of cross-ply laminates with viscous interfaces in cylindrical bendingrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 823ndash835 2005

[2] W Q Chen and K Y Lee ldquoThree-dimensional exact analysisof angle-ply laminates in cylindrical bending with interfacialdamage via state-space methodrdquo Composite Structures vol 64no 3-4 pp 275ndash283 2004

[3] W Q Chen and K Y Lee ldquoTime-dependent behaviors of angle-ply laminates with viscous interfaces in cylindrical bendingrdquoEuropean Journal ofMechanicsmdashASolids vol 23 no 2 pp 235ndash245 2004

[4] A H Gandhi and H K Raval ldquoAnalytical and empiricalmodeling of top roller position for three-roller cylindricalbending of plates and its experimental verificationrdquo Journal ofMaterials Processing Technology vol 197 no 1ndash3 pp 268ndash2782008

[5] S Oral and H Darendeliler ldquoThe optimum die profile for thecylindrical bending of platesrdquo Journal of Materials ProcessingTechnology vol 70 no 1ndash3 pp 151ndash155 1997

[6] V A Jairazbhoy P Petukhov and J Qu ldquoLarge deflectionof thin plates in cylindrical bendingmdashnon-unique solutionsrdquoInternational Journal of Solids and Structures vol 45 no 11-12pp 3203ndash3218 2008

[7] A Lebee and K Sab ldquoA bending-gradient model for thickplates part II closed-form solutions for cylindrical bending oflaminatesrdquo International Journal of Solids and Structures vol 48no 20 pp 2889ndash2901 2011

[8] P V Nimbolkar and I M Jain ldquoCylindrical bending of elasticplatesrdquo Procedia Materials Science vol 10 pp 793ndash802 2015

[9] A S Sayyad and Y M Ghugal ldquoA nth-order shear deformationtheory for composite laminates in cylindrical bendingrdquo Curvedand Layered Structures vol 2 no 1 pp 290ndash300 2015

[10] N J Pagano ldquoExact solutions for composite laminates incylindrical bendingrdquo Journal of Composite Materials vol 3 no3 pp 398ndash411 1969

[11] N J Pagano and A S D Wang ldquoFurther study of compositelaminates under cylindrical bendingrdquo Journal of CompositeMaterials vol 5 pp 421ndash428 1971

[12] N Saeedi K Sab and J-F Caron ldquoCylindrical bending ofmultilayered plates with multi-delamination via a layerwisestress approachrdquo Composite Structures vol 95 pp 728ndash7392013

[13] X-P Shu and K P Soldatos ldquoCylindrical bending of angle-ply laminates subjected to different sets of edge boundaryconditionsrdquo International Journal of Solids and Structures vol37 no 31 pp 4289ndash4307 2000

[14] Z G Bian W Q Chen C W Lim and N Zhang ldquoAnalyticalsolutions for single- and multi-span functionally graded platesin cylindrical bendingrdquo International Journal of Solids andStructures vol 42 no 24-25 pp 6433ndash6456 2005

[15] H M Navazi and H Haddadpour ldquoNonlinear cylindricalbending analysis of shear deformable functionally graded platesunder different loadings using analytical methodsrdquo Interna-tional Journal of Mechanical Sciences vol 50 no 12 pp 1650ndash1657 2008

[16] W Q Chen J Ying J B Cai and G R Ye ldquoBenchmarksolution of imperfect angle-ply laminated rectangular plates incylindrical bending with surface piezoelectric layers as actuatorand sensorrdquo Computers and Structures vol 82 no 22 pp 1773ndash1784 2004

[17] W Q Chen and K Y Lee ldquoExact solution of angle-plypiezoelectric laminates in cylindrical bending with interfacialimperfectionsrdquo Composite Structures vol 65 no 3-4 pp 329ndash337 2004

[18] T Kant and SM Shiyekar ldquoCylindrical bending of piezoelectriclaminates with a higher order shear and normal deformationtheoryrdquo Computers and Structures vol 86 no 15-16 pp 1594ndash1603 2008

[19] P Lu H P Lee and C Lu ldquoAn exact solution for functionallygraded piezoelectric laminates in cylindrical bendingrdquo Interna-tional Journal of Mechanical Sciences vol 47 no 3 pp 437ndash4382005

[20] W Yan J Wang and W Q Chen ldquoCylindrical bendingresponses of angle-ply piezoelectric laminates with viscoelasticinterfacesrdquo Applied Mathematical Modelling vol 38 no 24 pp6018ndash6030 2014

[21] Y Y Zhou W Q Chen and C F Lu ldquoSemi-analytical solutionfor orthotropic piezoelectric laminates in cylindrical bendingwith interfacial imperfectionsrdquo Composite Structures vol 92no 4 pp 1009ndash1018 2010

[22] G P Dube S Kapuria and P C Dumir ldquoExact piezother-moelastic solution of a simply-supported orthotropic flat panelin cylindrical bendingrdquo International Journal of MechanicalSciences vol 38 pp 374ndash387 1996

[23] V N Pilipchuk V L Berdichevsky and R A IbrahimldquoThermo-mechanical coupling in cylindrical bending of sand-wich platesrdquoComposite Structures vol 92 no 11 pp 2632ndash26402010

[24] C Zhang S Di and N Zhang ldquoA new procedure for staticanalysis of thermo-electric laminated composite plates undercylindrical bendingrdquo Composite Structures vol 56 no 2 pp131ndash140 2002

[25] W Q Chen and K Y Lee ldquoState-space approach for statics anddynamics of angle-ply laminated cylindrical panels in cylindri-cal bendingrdquo International Journal of Mechanical Sciences vol47 no 3 pp 374ndash387 2005

[26] C-P Wu and Y-S Syu ldquoExact solutions of functionally gradedpiezoelectric shells under cylindrical bendingrdquo InternationalJournal of Solids and Structures vol 44 no 20 pp 6450ndash64722007

[27] W Yan J Ying and W Q Chen ldquoThe behavior of angle-ply laminated cylindrical shells with viscoelastic interfaces incylindrical bendingrdquo Composite Structures vol 78 no 4 pp551ndash559 2007

Shock and Vibration 17

[28] S Brischetto F Tornabene N Fantuzzi and E Viola ldquo3Dexact and 2D generalized differential quadraturemodels for freevibration analysis of functionally graded plates and cylindersrdquoMeccanica vol 51 no 9 pp 2059ndash2098 2016

[29] N Fantuzzi S Brischetto F Tornabene and E Viola ldquo2D and3D shell models for the free vibration investigation of func-tionally graded cylindrical and spherical panelsrdquo CompositeStructures vol 154 pp 573ndash590 2016

[30] S Brischetto and R Torre ldquoExact 3D solutions and finiteelement 2D models for free vibration analysis of plates andcylindersrdquo Curved and Layered Structures vol 1 no 1 pp 59ndash92 2014

[31] F Tornabene S Brischetto N Fantuzzi and E Viola ldquoNumer-ical and exact models for free vibration analysis of cylindricaland spherical shell panelsrdquo Composites Part B Engineering vol81 pp 231ndash250 2015

[32] S Brischetto ldquoExact elasticity solution for natural frequenciesof functionally graded simply-supported structuresrdquo ComputerModeling in Engineering amp Sciences vol 95 no 5 pp 391ndash4302013

[33] S Brischetto ldquoThree-dimensional exact free vibration analysisof spherical cylindrical and flat one-layered panelsrdquo Shock andVibration vol 2014 Article ID 479738 29 pages 2014

[34] S Brischetto ldquoA continuum elastic three-dimensional modelfor natural frequencies of single-walled carbon nanotubesrdquoComposites Part B Engineering vol 61 pp 222ndash228 2014

[35] S Brischetto ldquoAn exact 3d solution for free vibrations ofmultilayered cross-ply composite and sandwich plates andshellsrdquo International Journal of Applied Mechanics vol 6 no 6Article ID 1450076 42 pages 2014

[36] S Brischetto ldquoA continuumshellmodel including vanderWaalsinteraction for free vibrations of double-walled carbon nan-otubesrdquoCMES-ComputerModeling in Engineering and Sciencesvol 104 no 4 pp 305ndash327 2015

[37] S Brischetto F Tornabene N Fantuzzi and M BacciocchildquoRefined 2D and exact 3D shell models for the free vibrationanalysis of single- and double-walled carbon nanotubesrdquo Tech-nologies vol 3 no 4 pp 259ndash284 2015

[38] S Brischetto ldquoConvergence analysis of the exponential matrixmethod for the solution of 3D equilibrium equations for freevibration analysis of plates and shellsrdquo Composites Part BEngineering vol 98 pp 453ndash471 2016

[39] S Brischetto ldquoExact and approximate shell geometry in the freevibration analysis of one-layered and multilayered structuresrdquoInternational Journal of Mechanical Sciences vol 113 pp 81ndash932016

[40] S Brischetto ldquoCurvature approximation effects in the freevibration analysis of functionally graded shellsrdquo InternationalJournal of Applied Mechanics 2016

[41] F Tornabene N Fantuzzi F Ubertini and E Viola ldquoStrong for-mulation finite element method based on differential quadra-ture a surveyrdquo Applied Mechanics Reviews vol 67 no 2 ArticleID 020801 pp 1ndash55 2015

[42] F Tornabene E Viola and N Fantuzzi ldquoGeneral higher-orderequivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panelsrdquo CompositeStructures vol 104 pp 94ndash117 2013

[43] N Fantuzzi F Tornabene E Viola and A J Ferreira ldquoAstrong formulation finite element method (SFEM) based onRBF and GDQ techniques for the static and dynamic analysesof laminated plates of arbitrary shaperdquo Meccanica vol 49 no10 pp 2503ndash2542 2014

[44] F Tornabene N Fantuzzi and M Bacciocchi ldquoFree vibrationsof free-form doubly-curved shells made of functionally gradedmaterials using higher-order equivalent single layer theoriesrdquoComposites Part B Engineering vol 67 pp 490ndash509 2014

[45] F Tornabene and E Viola ldquoStatic analysis of functionallygraded doubly-curved shells and panels of revolutionrdquo Mecca-nica vol 48 no 4 pp 901ndash930 2013

[46] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoDynamic analysis of thick and thin elliptic shell structuresmade of laminated composite materialsrdquo Composite Structuresvol 133 pp 278ndash299 2015

[47] F TornabeneN FantuzziM Bacciocchi and EViola ldquoHigher-order theories for the free vibrations of doubly-curved lam-inated panels with curvilinear reinforcing fibers by meansof a local version of the GDQ methodrdquo Composites Part BEngineering vol 81 pp 196ndash230 2015

[48] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoFree vibrations of composite oval and elliptic cylinders bythe generalized differential quadrature methodrdquo Thin-WalledStructures vol 97 pp 114ndash129 2015

[49] F Tornabene and J N Reddy ldquoFGM and laminated doubly-curved and degenerate shells resting on nonlinear elastic foun-dations a GDQ solution for static analysis with a posterioristress and strain recoveryrdquo Journal of the Indian Institute ofScience vol 93 no 4 pp 635ndash688 2013

[50] N Fantuzzi F Tornabene and E Viola ldquoFour-parameter func-tionally graded cracked plates of arbitrary shape a GDQFEMsolution for free vibrationsrdquo Mechanics of Advanced Materialsand Structures vol 23 no 1 pp 89ndash107 2016

International Journal of

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Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Shock and Vibration

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International Journal of

Shock and Vibration 9

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(a) Mode I (0 1) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

000501

01502

(b) Mode I (0 1) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(c) Mode II (119908 = 0) (0 1) 3D

002minus002

090908

08070706

06050504

04030302

020101

00minus01

1

(d) Mode II (119908 = 0) (0 1) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(e) Mode I (1 0) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

0minus005minus01

minus015minus02

(f) Mode I (1 0) 2D

Figure 2 Continued

10 Shock and Vibration

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(g) Mode II (119908 = 0) (1 0) 3D

002minus002

0908 08

0706 06

0504 04

0302 02

01 0 0

11

(h) Mode II (119908 = 0) (1 0) 2D

Figure 2 Differences between in-plane and cylindrical bending vibration modes for the 119886ℎ = 20 one-layered FGM plate with 119901 = 05 3Dmodes versus 2D modes

Since the FE commercial software does not have anembedded tool for the computation of functionally gradedproperties through the thickness the present FE applicationsare carried out building a fictitious laminated structure withisotropic plies In detail the thickness of the FGM lamina hasbeen divided into 100 fictitious layers and for each layer thecorresponding mechanical properties have been computedthrough (25) The numerical model has been made of auniform element division of 40 times 40 Quad8 elements for atotal number of 29280 degrees of freedom (dofs) The GDQsolution is computed with a not-uniform Chebyshev-Gauss-Lobatto grid [41] with 31 times 31 points for both thicknessratios It is underlined that since a single ply is set no zigzagfunction is needed in the computation It is recalled that thepresent higher order GDQ model considers a third orderexpansion with 120591 = 0 1 2 3 whereas the FE one has afirst order model (also known as Reissner-Mindlin model)Table 1 reports the results in terms of natural frequency(Hz) wherein the first column reports the number of half-waves (119898 119899) of the 3D exact solution Note that for a fixedset of half-wave numbers different mode shapes can occur(indicated by the second column ldquoModerdquo) In particular Iindicates the first mode and II indicates the second mode Inthe case of in-plane modes the caption (119908 = 0) is added tounderline that the vibration occurs in the plane and not outof plane The third column shows all the results for the 3Dexact solution both classical and cylindrical bending (CB)results It is noted that these frequencies are obtained witha single run of the 3D exact code keeping simply supportedboundary conditions for all the four edges of the plate Theother four columns of Table 1 refer to the numerical solutionsfor simply supported structures and for cylindrical bendingconditions (setting two free opposite edges) The cylindricalbending results are indicated by the acronym CB in the sametable

There is a very good agreement among the three solutionswhen simply supported conditions are considered On thecontrary a small error is observed in cylindrical bendingbecause the actual CB condition is not completely satisfieddue to the fact that the plate has a limited dimension in thesimply supported direction This feature can be detected inFigures 2 and 3 because the deformed shapes of the plate inthe CB configuration are not properly correct In fact thefibers parallel to the supported edges are not parallel amongthem but there is a curvature effect due to the free boundariesand to the limited length of the plate On the contrary thein-plane mode shapes (119908 = 0) when the plate is simplysupported on the four edges have all the fibers parallel amongthem also after the deformation These modes are the resultsobtained in the 3D solution with 119898 = 0 or 119899 = 0 combinedwith 119908 = 0 They are not cylindrical bending modes infact they can be found with 2D numerical models withoutmodifying the boundary conditions 3D frequencieswith119898 =0 or 119899 = 0 (with w different from 0) are CB modes which canbe obtained considering two simply supported edges and twofree edges in 2D numerical models

As it is well-known the ldquoperfectrdquo CB would occur whenthe simply supported direction has an ldquoinfiniterdquo lengthThis cannot be proven numerically but a trend can beobserved modeling different plates in cylindrical bendingconfiguration with different lengthsThese analyses are madekeeping one side of the plate constant and enlarging thelength of the other edge In the present computations 119886 iskept constant and 119887 = 119871 varies from 1 to 20 The results ofthe aforementioned analysis are listed in Table 2 and depictedin Figures 4 and 5 Table 2 includes the numerical valuesof the first mode shape in cylindrical bending for the 3Dexact solution the 2D FE and the 2D GDQmodels Figure 4shows that both GDQ and FE solutions tend to the 3D exactone increasing the length of the simply supported edge In

Shock and Vibration 11

ulowast

lowast

wlowast

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

(a) Mode I (1 0) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

000501

01502

(b) Mode I (1 0) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(c) Mode I (2 0) 3D

109

0807

060504

0302

01 0

109

0807

0605

0403

02010

0201501

0050

minus005minus01

minus015minus02

(d) Mode I (2 0) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(e) Mode I (3 0) 3D

10908

0706

050403

0201 0

109

0807

0605

0403

02010

0201501

0050

minus005minus01

minus015minus02

(f) Mode I (3 0) 2D

Figure 3 Continued

12 Shock and Vibration

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(g) Mode I (4 0) 3D

10908

0706050403

0201 0

10908

0706

0504

0302

010

0201501

0050

minus005minus01

minus015minus02

(h) Mode I (4 0) 2D

Figure 3 First four cylindrical bending vibration modes for the 119886ℎ = 20 one-layered FGM plate with 119901 = 05 3D modes versus 2D modes

3D exact2D GDQ2D FEM

f(H

z)

39

392

394

396

398

40

402

404

5 10 15 200La

Figure 4 First frequency 119891 in Hz for the 119886ℎ = 100 one-layeredFGM plate with 119901 = 05 convergence of 2D GDQ and 2D FEsolutions increasing the length of the simply supported sides

particular the GDQ solution is slightly more accurate thanthe FE one The 2D GDQ solution is carried out by enlargingthe length of one of the sides of the plate and the samenumberof dofs is kept On the contrary the FE model is generatedby 40 times 40 blocks of mesh In other words the plate with119871 = 2119886 has a mesh of 40 times 80 the one with 119871 = 3119886 has amesh 40 times 120 and so on When 119871 = 20119886 the error amongthe 3D exact and the 2D numerical solutions is very small andthis is proven also by the mode shapes of Figure 5 whereinthe boundary effect is negligible with respect the length of

the simply supported edges It must be underlined that it isimpossible to completely avoid the boundary effects due tothe free edges

The second example is related to a simply supportedcylinder with 119877120572 = 10m 119877120573 = infin 119886 = 2120587119877120572 and 119887 =20m and thickness ratios 119877120572ℎ = 100 and 119877120572ℎ = 10 Aone-layered FGM scheme is considered with ℎ1 = ℎ andthe same volume fraction function of (24) for the previousexample but setting 119901 = 20 The present material propertiesare the same as the flat plate The FE model has a 20 times80 Quad8 mesh with 27966 dofs and the GDQ solutionhas a 15 times 51 Chebyshev-Gauss-Lobatto grid Numericallyspeaking compatibility conditions must be enforced on theclosing meridian in order to obtain a closed revolution shell[42] To obtain the CB conditions numerically the cylinderis modeled completely free and the first 6 rigid mode shapesare neglected and removed from the numerical solution Itis noted that the GDQ model uses a third-order kinematicexpansion with no zigzag effect as the previous case sincea one-layer structure has been taken into account Table 3reports the results with the same meaning of the symbolsseen in Table 1 A very good agreement is observed when thecylinder is simply supported whereas when the cylinder iscompletely free some small errors can be observed but thenumerical solutions follow the 3D exact one

Results with 119899 = 0 and 119908 = 0 indicate the CB conditionsSuch results are obtained by means of 2D numerical modelsmodifying the boundary conditions (see 2D FE(CB) and 2DGDQ(CB)models)The cases 119899 = 0 combined with119908 = 0 arenot CB conditions and the frequencies can be found withoutmodifying the boundary conditions in the 2D numericalmodels

The final test is related to a cylindrical panel with 119877120572 =10m 119877120573 = infin 119886 = 1205873119877120572 and 119887 = 20m and thickness ratios119877120572ℎ = 100 and 119877120572ℎ = 10 The present shell is a sandwichstructure embedding a FGM core with ℎ2 = 07ℎ and two

Shock and Vibration 13

218

1614

121 1

08 0806 06

04 0402 02

0 0

0minus01minus02minus03minus04

(a) 119871119886 = 2

3

25

2

151

0806

040502

0 0

1

0minus01minus02minus03minus04minus05

(b) 119871119886 = 3

435

325

10505

0 0

15

0

minus02minus04minus06

1

2

(c) 119871119886 = 4

545

35

10505

0 0

0

minus04minus02

minus06minus08

1

25

4

3

215

(d) 119871119886 = 5

5

1

0 005

0

minus04minus02

minus06minus08

1

4

3

2

6

minus1

(e) 119871119886 = 6

5

10 0 051

43

2

67

89

100

minus05minus1

minus15

(f) 119871119886 = 10

Figure 5 First vibration mode for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 2D GDQ solution and edge effects versus the lengthof the simply supported sides

external skins with ℎ1 = ℎ3 = 015ℎ (ℎ1 + ℎ2 + ℎ3 = ℎ) Thebottom layer ismade of isotropicmetallicmaterial and the toplayer is made of ceramic material the mechanical propertiesare

119864119898 = 73GPa120588119898 = 2800 kgm3119864119888 = 200GPa120588119888 = 5700 kgm3

(27)

The FGM core has a volume fraction function defined as

119881119888 (119911) = (05 + 11991107ℎ)119901 with 119911 isin [minus035ℎ 035ℎ] (28)

where 119901 = 10 is employed in the present computations ThePoisson ratio is constant (]119888 = ]119898 = ] = 03) Equations (25)are still valid with 119881119888 defined as in (28)

The FEmodel is made of 40times40Quad8 elements for total29280 dofs and the GDQ model has a 31 times 31 Chebyshev-Gauss-Lobatto grid Since a sandwich structure is analyzedin the following a third-order theory with zigzag functionis considered in the computations for the GDQ method Thepresent natural frequencies are reported in Table 4 with the

14 Shock and Vibration

Table 2 First frequency 119891 in Hz for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 convergence of 2D GDQ and 2D FE solutionsincreasing the length of the simply supported sides

119871119886 1 2 3 4 5 6 8 10 15 20 3D exact2D GDQ 3921 3964 3980 3988 3993 3997 4001 4004 4007 4008 40192D FEM 3907 3949 3966 3974 3979 3983 3987 3989 3993 3994 4019

Table 3 One-layered FGM cylinder with 119901 = 20 Analytical 3D versus 2D numerical models Frequency 119891 in Hz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1004 0 I 0896 mdash 0897 mdash 08836 0 I 2532 mdash 2538 mdash 25118 0 I 4854 mdash 4867 mdash 481910 0 I 7848 mdash 7871 mdash 780112 0 I 1151 mdash 1155 mdash 114610 1 I 1356 1360 mdash 1357 mdash12 1 I 1441 1449 mdash 1445 mdash14 0 I 1584 mdash 1589 mdash 15778 1 I 1638 1640 mdash 1636 mdash14 1 I 1756 1767 mdash 1763 mdash16 0 I 2083 mdash 2090 mdash 207016 1 I 2207 2222 mdash 2216 mdash6 1 I 2441 2442 mdash 2438 mdash18 0 I 2649 mdash 2658 mdash 264014 2 I 2742 2752 mdash 2746 mdash18 1 I 2751 2772 mdash 2763 mdash16 2 I 2875 2890 mdash 2883 mdash12 2 I 2921 2926 mdash 2922 mdash119877120572ℎ = 104 0 I 8774 mdash 9085 mdash 86796 0 I 2456 mdash 2571 mdash 24306 1 I 3978 4089 mdash 3979 mdash4 1 I 4388 4357 mdash 4388 mdash8 0 I 4654 mdash 4920 mdash 46028 1 I 5575 5905 mdash 5576 mdash2 1 I 7202 7101 mdash 7202 mdash2 0 I (119908 = 0) 7322 7390 mdash 7319 mdash10 0 I 7419 mdash 7933 mdash 73476 2 I 8110 8261 mdash 8111 mdash10 1 I 8189 8811 mdash 8191 mdash8 2 I 8783 9173 mdash 8785 mdash4 2 I 9016 9084 mdash 9017 mdash12 0 I 1071 mdash 1159 mdash 106110 2 I 1081 1155 mdash 1081 mdash2 2 I 1089 1105 mdash 1089 mdash12 1 I 1142 1245 mdash 1143 mdash4 3 I 1263 1286 mdash 1263 mdash

same meaning of the columns seen in the previous casesIn order to achieve the cylindrical bending conditions thenumerical models have the straight edges simply supportedand the curved edges as free Due to the CB definitionit is not possible to achieve CB conditions when 119898 = 0

and 119899 = 1 2 3 because the cylindrical bending shouldoccur on the curved edge and this is not physically possiblebecause of the variation of displacement components due tothe curvature presence This aspect is indicated as NA (NotApplicable) solution in Table 4 Therefore this kind of 3D

Shock and Vibration 15

Table 4 Sandwich cylindrical shell panel embedding FGM core with 119901 = 10 Analytical 3D versus 2D numerical models Frequency 119891 inHz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1001 0 I 1948 mdash 1951 mdash 19412 0 I 8856 mdash 8872 mdash 88232 1 I 1104 1110 mdash 1104 mdash1 1 I 1850 1873 mdash 1850 mdash3 0 I 2038 mdash 2042 mdash 20303 1 I 2116 2128 mdash 2116 mdash2 2 I 2221 2242 mdash 2221 mdash3 2 I 2484 2496 mdash 2484 mdash3 3 I 3235 3254 mdash 3235 mdash4 0 I 3649 mdash 3660 mdash 36342 3 I 3692 3739 mdash 3692 mdash4 1 I 3714 3745 mdash 3714 mdash4 2 I 3940 3965 mdash 3941 mdash3 4 I 4226 4261 mdash 4226 mdash4 3 I 4378 4403 mdash 4378 mdash1 2 I 4568 4664 mdash 4568 mdash4 4 I 5035 5064 mdash 5035 mdash2 4 I 5040 5128 mdash 5040 mdash119877120572ℎ = 101 0 I 1905 mdash 1963 mdash 18841 1 I 3054 3145 mdash 3055 mdash1 2 I 6209 6409 mdash 6209 mdash2 0 I 8326 mdash 8857 mdash 82540 1 I 8710 mdash NA mdash NA0 1 II (119908 = 0) 8820 8956 mdash 8820 mdash2 1 I 8880 9531 mdash 8882 mdash0 2 I 9176 mdash NA mdash NA1 3 I 9576 9983 mdash 9567 mdash0 3 I 1036 mdash NA mdash NA2 2 I 1066 1147 mdash 1067 mdash0 4 I 1291 mdash NA mdash NA1 4 I 1335 1413 mdash 1335 mdash2 3 I 1364 1475 mdash 1364 mdash1 0 II (119908 = 0) 1665 1738 mdash 1665 mdash0 5 I 1683 mdash NA mdash NA2 4 I 1755 1918 mdash 1755 mdash0 2 II (119908 = 0) 1764 1791 mdash 1764 mdash

exact solutions cannot be achieved by any numerical modelOnce again solution with 119899 = 0 or 119898 = 0 combined with119908 = 0 (in-plane modes) can be found with 2D numericalmodels without modifying the simply supported boundaryconditions

5 Conclusions

Thepresent paper analyzes the cylindrical bending in the freefrequency analysis of functionally graded material (FGM)plates and cylindrical shells This condition is investigatedby means of a 3D exact shell model and two different 2D

numerical models a weak formulation based on classicalFinite Elements (FEs) and a strong formulation based onthe GDQ method The cylindrical bending (CB) behavior inthe 3D exact shell model can be analyzed using completelysimply supported structures and supposing one of the twohalf-wave numbers equals zeroThe numerical models obtainthe CB behavior using mixed simply supported and freeboundary conditions It has been demonstrated that theactual cylindrical bending (CB) is possible only with respectto straight edgesTheCBnatural frequencies of the numericalsolutions converge to the 3D exact CB ones if the length ofstraight simply supported edges is increased up to 20 times

16 Shock and Vibration

the value of the original edges The CB conditions are notpossible in 2D numericalmodels when the half-wave numberis zero in the curvilinear edge in this case the modificationof boundary conditions does not properly work In-planevibrationmodeswith zero transverse displacement combinedwith 119898 = 0 or 119899 = 0 cannot be defined as CB frequencies andthey can be calculated via 2D numerical models without themodification of the simply supported conditions for the fouredges All the considerations about the cylindrical bendingare valid for each geometry (plate cylinder and cylindricalshell) lamination scheme FGM law and thickness ratio

Competing Interests

The authors declare that they have no competing interests

References

[1] W Q Chen J B Cai and G R Ye ldquoResponses of cross-ply laminates with viscous interfaces in cylindrical bendingrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 823ndash835 2005

[2] W Q Chen and K Y Lee ldquoThree-dimensional exact analysisof angle-ply laminates in cylindrical bending with interfacialdamage via state-space methodrdquo Composite Structures vol 64no 3-4 pp 275ndash283 2004

[3] W Q Chen and K Y Lee ldquoTime-dependent behaviors of angle-ply laminates with viscous interfaces in cylindrical bendingrdquoEuropean Journal ofMechanicsmdashASolids vol 23 no 2 pp 235ndash245 2004

[4] A H Gandhi and H K Raval ldquoAnalytical and empiricalmodeling of top roller position for three-roller cylindricalbending of plates and its experimental verificationrdquo Journal ofMaterials Processing Technology vol 197 no 1ndash3 pp 268ndash2782008

[5] S Oral and H Darendeliler ldquoThe optimum die profile for thecylindrical bending of platesrdquo Journal of Materials ProcessingTechnology vol 70 no 1ndash3 pp 151ndash155 1997

[6] V A Jairazbhoy P Petukhov and J Qu ldquoLarge deflectionof thin plates in cylindrical bendingmdashnon-unique solutionsrdquoInternational Journal of Solids and Structures vol 45 no 11-12pp 3203ndash3218 2008

[7] A Lebee and K Sab ldquoA bending-gradient model for thickplates part II closed-form solutions for cylindrical bending oflaminatesrdquo International Journal of Solids and Structures vol 48no 20 pp 2889ndash2901 2011

[8] P V Nimbolkar and I M Jain ldquoCylindrical bending of elasticplatesrdquo Procedia Materials Science vol 10 pp 793ndash802 2015

[9] A S Sayyad and Y M Ghugal ldquoA nth-order shear deformationtheory for composite laminates in cylindrical bendingrdquo Curvedand Layered Structures vol 2 no 1 pp 290ndash300 2015

[10] N J Pagano ldquoExact solutions for composite laminates incylindrical bendingrdquo Journal of Composite Materials vol 3 no3 pp 398ndash411 1969

[11] N J Pagano and A S D Wang ldquoFurther study of compositelaminates under cylindrical bendingrdquo Journal of CompositeMaterials vol 5 pp 421ndash428 1971

[12] N Saeedi K Sab and J-F Caron ldquoCylindrical bending ofmultilayered plates with multi-delamination via a layerwisestress approachrdquo Composite Structures vol 95 pp 728ndash7392013

[13] X-P Shu and K P Soldatos ldquoCylindrical bending of angle-ply laminates subjected to different sets of edge boundaryconditionsrdquo International Journal of Solids and Structures vol37 no 31 pp 4289ndash4307 2000

[14] Z G Bian W Q Chen C W Lim and N Zhang ldquoAnalyticalsolutions for single- and multi-span functionally graded platesin cylindrical bendingrdquo International Journal of Solids andStructures vol 42 no 24-25 pp 6433ndash6456 2005

[15] H M Navazi and H Haddadpour ldquoNonlinear cylindricalbending analysis of shear deformable functionally graded platesunder different loadings using analytical methodsrdquo Interna-tional Journal of Mechanical Sciences vol 50 no 12 pp 1650ndash1657 2008

[16] W Q Chen J Ying J B Cai and G R Ye ldquoBenchmarksolution of imperfect angle-ply laminated rectangular plates incylindrical bending with surface piezoelectric layers as actuatorand sensorrdquo Computers and Structures vol 82 no 22 pp 1773ndash1784 2004

[17] W Q Chen and K Y Lee ldquoExact solution of angle-plypiezoelectric laminates in cylindrical bending with interfacialimperfectionsrdquo Composite Structures vol 65 no 3-4 pp 329ndash337 2004

[18] T Kant and SM Shiyekar ldquoCylindrical bending of piezoelectriclaminates with a higher order shear and normal deformationtheoryrdquo Computers and Structures vol 86 no 15-16 pp 1594ndash1603 2008

[19] P Lu H P Lee and C Lu ldquoAn exact solution for functionallygraded piezoelectric laminates in cylindrical bendingrdquo Interna-tional Journal of Mechanical Sciences vol 47 no 3 pp 437ndash4382005

[20] W Yan J Wang and W Q Chen ldquoCylindrical bendingresponses of angle-ply piezoelectric laminates with viscoelasticinterfacesrdquo Applied Mathematical Modelling vol 38 no 24 pp6018ndash6030 2014

[21] Y Y Zhou W Q Chen and C F Lu ldquoSemi-analytical solutionfor orthotropic piezoelectric laminates in cylindrical bendingwith interfacial imperfectionsrdquo Composite Structures vol 92no 4 pp 1009ndash1018 2010

[22] G P Dube S Kapuria and P C Dumir ldquoExact piezother-moelastic solution of a simply-supported orthotropic flat panelin cylindrical bendingrdquo International Journal of MechanicalSciences vol 38 pp 374ndash387 1996

[23] V N Pilipchuk V L Berdichevsky and R A IbrahimldquoThermo-mechanical coupling in cylindrical bending of sand-wich platesrdquoComposite Structures vol 92 no 11 pp 2632ndash26402010

[24] C Zhang S Di and N Zhang ldquoA new procedure for staticanalysis of thermo-electric laminated composite plates undercylindrical bendingrdquo Composite Structures vol 56 no 2 pp131ndash140 2002

[25] W Q Chen and K Y Lee ldquoState-space approach for statics anddynamics of angle-ply laminated cylindrical panels in cylindri-cal bendingrdquo International Journal of Mechanical Sciences vol47 no 3 pp 374ndash387 2005

[26] C-P Wu and Y-S Syu ldquoExact solutions of functionally gradedpiezoelectric shells under cylindrical bendingrdquo InternationalJournal of Solids and Structures vol 44 no 20 pp 6450ndash64722007

[27] W Yan J Ying and W Q Chen ldquoThe behavior of angle-ply laminated cylindrical shells with viscoelastic interfaces incylindrical bendingrdquo Composite Structures vol 78 no 4 pp551ndash559 2007

Shock and Vibration 17

[28] S Brischetto F Tornabene N Fantuzzi and E Viola ldquo3Dexact and 2D generalized differential quadraturemodels for freevibration analysis of functionally graded plates and cylindersrdquoMeccanica vol 51 no 9 pp 2059ndash2098 2016

[29] N Fantuzzi S Brischetto F Tornabene and E Viola ldquo2D and3D shell models for the free vibration investigation of func-tionally graded cylindrical and spherical panelsrdquo CompositeStructures vol 154 pp 573ndash590 2016

[30] S Brischetto and R Torre ldquoExact 3D solutions and finiteelement 2D models for free vibration analysis of plates andcylindersrdquo Curved and Layered Structures vol 1 no 1 pp 59ndash92 2014

[31] F Tornabene S Brischetto N Fantuzzi and E Viola ldquoNumer-ical and exact models for free vibration analysis of cylindricaland spherical shell panelsrdquo Composites Part B Engineering vol81 pp 231ndash250 2015

[32] S Brischetto ldquoExact elasticity solution for natural frequenciesof functionally graded simply-supported structuresrdquo ComputerModeling in Engineering amp Sciences vol 95 no 5 pp 391ndash4302013

[33] S Brischetto ldquoThree-dimensional exact free vibration analysisof spherical cylindrical and flat one-layered panelsrdquo Shock andVibration vol 2014 Article ID 479738 29 pages 2014

[34] S Brischetto ldquoA continuum elastic three-dimensional modelfor natural frequencies of single-walled carbon nanotubesrdquoComposites Part B Engineering vol 61 pp 222ndash228 2014

[35] S Brischetto ldquoAn exact 3d solution for free vibrations ofmultilayered cross-ply composite and sandwich plates andshellsrdquo International Journal of Applied Mechanics vol 6 no 6Article ID 1450076 42 pages 2014

[36] S Brischetto ldquoA continuumshellmodel including vanderWaalsinteraction for free vibrations of double-walled carbon nan-otubesrdquoCMES-ComputerModeling in Engineering and Sciencesvol 104 no 4 pp 305ndash327 2015

[37] S Brischetto F Tornabene N Fantuzzi and M BacciocchildquoRefined 2D and exact 3D shell models for the free vibrationanalysis of single- and double-walled carbon nanotubesrdquo Tech-nologies vol 3 no 4 pp 259ndash284 2015

[38] S Brischetto ldquoConvergence analysis of the exponential matrixmethod for the solution of 3D equilibrium equations for freevibration analysis of plates and shellsrdquo Composites Part BEngineering vol 98 pp 453ndash471 2016

[39] S Brischetto ldquoExact and approximate shell geometry in the freevibration analysis of one-layered and multilayered structuresrdquoInternational Journal of Mechanical Sciences vol 113 pp 81ndash932016

[40] S Brischetto ldquoCurvature approximation effects in the freevibration analysis of functionally graded shellsrdquo InternationalJournal of Applied Mechanics 2016

[41] F Tornabene N Fantuzzi F Ubertini and E Viola ldquoStrong for-mulation finite element method based on differential quadra-ture a surveyrdquo Applied Mechanics Reviews vol 67 no 2 ArticleID 020801 pp 1ndash55 2015

[42] F Tornabene E Viola and N Fantuzzi ldquoGeneral higher-orderequivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panelsrdquo CompositeStructures vol 104 pp 94ndash117 2013

[43] N Fantuzzi F Tornabene E Viola and A J Ferreira ldquoAstrong formulation finite element method (SFEM) based onRBF and GDQ techniques for the static and dynamic analysesof laminated plates of arbitrary shaperdquo Meccanica vol 49 no10 pp 2503ndash2542 2014

[44] F Tornabene N Fantuzzi and M Bacciocchi ldquoFree vibrationsof free-form doubly-curved shells made of functionally gradedmaterials using higher-order equivalent single layer theoriesrdquoComposites Part B Engineering vol 67 pp 490ndash509 2014

[45] F Tornabene and E Viola ldquoStatic analysis of functionallygraded doubly-curved shells and panels of revolutionrdquo Mecca-nica vol 48 no 4 pp 901ndash930 2013

[46] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoDynamic analysis of thick and thin elliptic shell structuresmade of laminated composite materialsrdquo Composite Structuresvol 133 pp 278ndash299 2015

[47] F TornabeneN FantuzziM Bacciocchi and EViola ldquoHigher-order theories for the free vibrations of doubly-curved lam-inated panels with curvilinear reinforcing fibers by meansof a local version of the GDQ methodrdquo Composites Part BEngineering vol 81 pp 196ndash230 2015

[48] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoFree vibrations of composite oval and elliptic cylinders bythe generalized differential quadrature methodrdquo Thin-WalledStructures vol 97 pp 114ndash129 2015

[49] F Tornabene and J N Reddy ldquoFGM and laminated doubly-curved and degenerate shells resting on nonlinear elastic foun-dations a GDQ solution for static analysis with a posterioristress and strain recoveryrdquo Journal of the Indian Institute ofScience vol 93 no 4 pp 635ndash688 2013

[50] N Fantuzzi F Tornabene and E Viola ldquoFour-parameter func-tionally graded cracked plates of arbitrary shape a GDQFEMsolution for free vibrationsrdquo Mechanics of Advanced Materialsand Structures vol 23 no 1 pp 89ndash107 2016

International Journal of

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Shock and Vibration

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Electrical and Computer Engineering

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DistributedSensor Networks

International Journal of

10 Shock and Vibration

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(g) Mode II (119908 = 0) (1 0) 3D

002minus002

0908 08

0706 06

0504 04

0302 02

01 0 0

11

(h) Mode II (119908 = 0) (1 0) 2D

Figure 2 Differences between in-plane and cylindrical bending vibration modes for the 119886ℎ = 20 one-layered FGM plate with 119901 = 05 3Dmodes versus 2D modes

Since the FE commercial software does not have anembedded tool for the computation of functionally gradedproperties through the thickness the present FE applicationsare carried out building a fictitious laminated structure withisotropic plies In detail the thickness of the FGM lamina hasbeen divided into 100 fictitious layers and for each layer thecorresponding mechanical properties have been computedthrough (25) The numerical model has been made of auniform element division of 40 times 40 Quad8 elements for atotal number of 29280 degrees of freedom (dofs) The GDQsolution is computed with a not-uniform Chebyshev-Gauss-Lobatto grid [41] with 31 times 31 points for both thicknessratios It is underlined that since a single ply is set no zigzagfunction is needed in the computation It is recalled that thepresent higher order GDQ model considers a third orderexpansion with 120591 = 0 1 2 3 whereas the FE one has afirst order model (also known as Reissner-Mindlin model)Table 1 reports the results in terms of natural frequency(Hz) wherein the first column reports the number of half-waves (119898 119899) of the 3D exact solution Note that for a fixedset of half-wave numbers different mode shapes can occur(indicated by the second column ldquoModerdquo) In particular Iindicates the first mode and II indicates the second mode Inthe case of in-plane modes the caption (119908 = 0) is added tounderline that the vibration occurs in the plane and not outof plane The third column shows all the results for the 3Dexact solution both classical and cylindrical bending (CB)results It is noted that these frequencies are obtained witha single run of the 3D exact code keeping simply supportedboundary conditions for all the four edges of the plate Theother four columns of Table 1 refer to the numerical solutionsfor simply supported structures and for cylindrical bendingconditions (setting two free opposite edges) The cylindricalbending results are indicated by the acronym CB in the sametable

There is a very good agreement among the three solutionswhen simply supported conditions are considered On thecontrary a small error is observed in cylindrical bendingbecause the actual CB condition is not completely satisfieddue to the fact that the plate has a limited dimension in thesimply supported direction This feature can be detected inFigures 2 and 3 because the deformed shapes of the plate inthe CB configuration are not properly correct In fact thefibers parallel to the supported edges are not parallel amongthem but there is a curvature effect due to the free boundariesand to the limited length of the plate On the contrary thein-plane mode shapes (119908 = 0) when the plate is simplysupported on the four edges have all the fibers parallel amongthem also after the deformation These modes are the resultsobtained in the 3D solution with 119898 = 0 or 119899 = 0 combinedwith 119908 = 0 They are not cylindrical bending modes infact they can be found with 2D numerical models withoutmodifying the boundary conditions 3D frequencieswith119898 =0 or 119899 = 0 (with w different from 0) are CB modes which canbe obtained considering two simply supported edges and twofree edges in 2D numerical models

As it is well-known the ldquoperfectrdquo CB would occur whenthe simply supported direction has an ldquoinfiniterdquo lengthThis cannot be proven numerically but a trend can beobserved modeling different plates in cylindrical bendingconfiguration with different lengthsThese analyses are madekeeping one side of the plate constant and enlarging thelength of the other edge In the present computations 119886 iskept constant and 119887 = 119871 varies from 1 to 20 The results ofthe aforementioned analysis are listed in Table 2 and depictedin Figures 4 and 5 Table 2 includes the numerical valuesof the first mode shape in cylindrical bending for the 3Dexact solution the 2D FE and the 2D GDQmodels Figure 4shows that both GDQ and FE solutions tend to the 3D exactone increasing the length of the simply supported edge In

Shock and Vibration 11

ulowast

lowast

wlowast

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

(a) Mode I (1 0) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

000501

01502

(b) Mode I (1 0) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(c) Mode I (2 0) 3D

109

0807

060504

0302

01 0

109

0807

0605

0403

02010

0201501

0050

minus005minus01

minus015minus02

(d) Mode I (2 0) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(e) Mode I (3 0) 3D

10908

0706

050403

0201 0

109

0807

0605

0403

02010

0201501

0050

minus005minus01

minus015minus02

(f) Mode I (3 0) 2D

Figure 3 Continued

12 Shock and Vibration

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(g) Mode I (4 0) 3D

10908

0706050403

0201 0

10908

0706

0504

0302

010

0201501

0050

minus005minus01

minus015minus02

(h) Mode I (4 0) 2D

Figure 3 First four cylindrical bending vibration modes for the 119886ℎ = 20 one-layered FGM plate with 119901 = 05 3D modes versus 2D modes

3D exact2D GDQ2D FEM

f(H

z)

39

392

394

396

398

40

402

404

5 10 15 200La

Figure 4 First frequency 119891 in Hz for the 119886ℎ = 100 one-layeredFGM plate with 119901 = 05 convergence of 2D GDQ and 2D FEsolutions increasing the length of the simply supported sides

particular the GDQ solution is slightly more accurate thanthe FE one The 2D GDQ solution is carried out by enlargingthe length of one of the sides of the plate and the samenumberof dofs is kept On the contrary the FE model is generatedby 40 times 40 blocks of mesh In other words the plate with119871 = 2119886 has a mesh of 40 times 80 the one with 119871 = 3119886 has amesh 40 times 120 and so on When 119871 = 20119886 the error amongthe 3D exact and the 2D numerical solutions is very small andthis is proven also by the mode shapes of Figure 5 whereinthe boundary effect is negligible with respect the length of

the simply supported edges It must be underlined that it isimpossible to completely avoid the boundary effects due tothe free edges

The second example is related to a simply supportedcylinder with 119877120572 = 10m 119877120573 = infin 119886 = 2120587119877120572 and 119887 =20m and thickness ratios 119877120572ℎ = 100 and 119877120572ℎ = 10 Aone-layered FGM scheme is considered with ℎ1 = ℎ andthe same volume fraction function of (24) for the previousexample but setting 119901 = 20 The present material propertiesare the same as the flat plate The FE model has a 20 times80 Quad8 mesh with 27966 dofs and the GDQ solutionhas a 15 times 51 Chebyshev-Gauss-Lobatto grid Numericallyspeaking compatibility conditions must be enforced on theclosing meridian in order to obtain a closed revolution shell[42] To obtain the CB conditions numerically the cylinderis modeled completely free and the first 6 rigid mode shapesare neglected and removed from the numerical solution Itis noted that the GDQ model uses a third-order kinematicexpansion with no zigzag effect as the previous case sincea one-layer structure has been taken into account Table 3reports the results with the same meaning of the symbolsseen in Table 1 A very good agreement is observed when thecylinder is simply supported whereas when the cylinder iscompletely free some small errors can be observed but thenumerical solutions follow the 3D exact one

Results with 119899 = 0 and 119908 = 0 indicate the CB conditionsSuch results are obtained by means of 2D numerical modelsmodifying the boundary conditions (see 2D FE(CB) and 2DGDQ(CB)models)The cases 119899 = 0 combined with119908 = 0 arenot CB conditions and the frequencies can be found withoutmodifying the boundary conditions in the 2D numericalmodels

The final test is related to a cylindrical panel with 119877120572 =10m 119877120573 = infin 119886 = 1205873119877120572 and 119887 = 20m and thickness ratios119877120572ℎ = 100 and 119877120572ℎ = 10 The present shell is a sandwichstructure embedding a FGM core with ℎ2 = 07ℎ and two

Shock and Vibration 13

218

1614

121 1

08 0806 06

04 0402 02

0 0

0minus01minus02minus03minus04

(a) 119871119886 = 2

3

25

2

151

0806

040502

0 0

1

0minus01minus02minus03minus04minus05

(b) 119871119886 = 3

435

325

10505

0 0

15

0

minus02minus04minus06

1

2

(c) 119871119886 = 4

545

35

10505

0 0

0

minus04minus02

minus06minus08

1

25

4

3

215

(d) 119871119886 = 5

5

1

0 005

0

minus04minus02

minus06minus08

1

4

3

2

6

minus1

(e) 119871119886 = 6

5

10 0 051

43

2

67

89

100

minus05minus1

minus15

(f) 119871119886 = 10

Figure 5 First vibration mode for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 2D GDQ solution and edge effects versus the lengthof the simply supported sides

external skins with ℎ1 = ℎ3 = 015ℎ (ℎ1 + ℎ2 + ℎ3 = ℎ) Thebottom layer ismade of isotropicmetallicmaterial and the toplayer is made of ceramic material the mechanical propertiesare

119864119898 = 73GPa120588119898 = 2800 kgm3119864119888 = 200GPa120588119888 = 5700 kgm3

(27)

The FGM core has a volume fraction function defined as

119881119888 (119911) = (05 + 11991107ℎ)119901 with 119911 isin [minus035ℎ 035ℎ] (28)

where 119901 = 10 is employed in the present computations ThePoisson ratio is constant (]119888 = ]119898 = ] = 03) Equations (25)are still valid with 119881119888 defined as in (28)

The FEmodel is made of 40times40Quad8 elements for total29280 dofs and the GDQ model has a 31 times 31 Chebyshev-Gauss-Lobatto grid Since a sandwich structure is analyzedin the following a third-order theory with zigzag functionis considered in the computations for the GDQ method Thepresent natural frequencies are reported in Table 4 with the

14 Shock and Vibration

Table 2 First frequency 119891 in Hz for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 convergence of 2D GDQ and 2D FE solutionsincreasing the length of the simply supported sides

119871119886 1 2 3 4 5 6 8 10 15 20 3D exact2D GDQ 3921 3964 3980 3988 3993 3997 4001 4004 4007 4008 40192D FEM 3907 3949 3966 3974 3979 3983 3987 3989 3993 3994 4019

Table 3 One-layered FGM cylinder with 119901 = 20 Analytical 3D versus 2D numerical models Frequency 119891 in Hz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1004 0 I 0896 mdash 0897 mdash 08836 0 I 2532 mdash 2538 mdash 25118 0 I 4854 mdash 4867 mdash 481910 0 I 7848 mdash 7871 mdash 780112 0 I 1151 mdash 1155 mdash 114610 1 I 1356 1360 mdash 1357 mdash12 1 I 1441 1449 mdash 1445 mdash14 0 I 1584 mdash 1589 mdash 15778 1 I 1638 1640 mdash 1636 mdash14 1 I 1756 1767 mdash 1763 mdash16 0 I 2083 mdash 2090 mdash 207016 1 I 2207 2222 mdash 2216 mdash6 1 I 2441 2442 mdash 2438 mdash18 0 I 2649 mdash 2658 mdash 264014 2 I 2742 2752 mdash 2746 mdash18 1 I 2751 2772 mdash 2763 mdash16 2 I 2875 2890 mdash 2883 mdash12 2 I 2921 2926 mdash 2922 mdash119877120572ℎ = 104 0 I 8774 mdash 9085 mdash 86796 0 I 2456 mdash 2571 mdash 24306 1 I 3978 4089 mdash 3979 mdash4 1 I 4388 4357 mdash 4388 mdash8 0 I 4654 mdash 4920 mdash 46028 1 I 5575 5905 mdash 5576 mdash2 1 I 7202 7101 mdash 7202 mdash2 0 I (119908 = 0) 7322 7390 mdash 7319 mdash10 0 I 7419 mdash 7933 mdash 73476 2 I 8110 8261 mdash 8111 mdash10 1 I 8189 8811 mdash 8191 mdash8 2 I 8783 9173 mdash 8785 mdash4 2 I 9016 9084 mdash 9017 mdash12 0 I 1071 mdash 1159 mdash 106110 2 I 1081 1155 mdash 1081 mdash2 2 I 1089 1105 mdash 1089 mdash12 1 I 1142 1245 mdash 1143 mdash4 3 I 1263 1286 mdash 1263 mdash

same meaning of the columns seen in the previous casesIn order to achieve the cylindrical bending conditions thenumerical models have the straight edges simply supportedand the curved edges as free Due to the CB definitionit is not possible to achieve CB conditions when 119898 = 0

and 119899 = 1 2 3 because the cylindrical bending shouldoccur on the curved edge and this is not physically possiblebecause of the variation of displacement components due tothe curvature presence This aspect is indicated as NA (NotApplicable) solution in Table 4 Therefore this kind of 3D

Shock and Vibration 15

Table 4 Sandwich cylindrical shell panel embedding FGM core with 119901 = 10 Analytical 3D versus 2D numerical models Frequency 119891 inHz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1001 0 I 1948 mdash 1951 mdash 19412 0 I 8856 mdash 8872 mdash 88232 1 I 1104 1110 mdash 1104 mdash1 1 I 1850 1873 mdash 1850 mdash3 0 I 2038 mdash 2042 mdash 20303 1 I 2116 2128 mdash 2116 mdash2 2 I 2221 2242 mdash 2221 mdash3 2 I 2484 2496 mdash 2484 mdash3 3 I 3235 3254 mdash 3235 mdash4 0 I 3649 mdash 3660 mdash 36342 3 I 3692 3739 mdash 3692 mdash4 1 I 3714 3745 mdash 3714 mdash4 2 I 3940 3965 mdash 3941 mdash3 4 I 4226 4261 mdash 4226 mdash4 3 I 4378 4403 mdash 4378 mdash1 2 I 4568 4664 mdash 4568 mdash4 4 I 5035 5064 mdash 5035 mdash2 4 I 5040 5128 mdash 5040 mdash119877120572ℎ = 101 0 I 1905 mdash 1963 mdash 18841 1 I 3054 3145 mdash 3055 mdash1 2 I 6209 6409 mdash 6209 mdash2 0 I 8326 mdash 8857 mdash 82540 1 I 8710 mdash NA mdash NA0 1 II (119908 = 0) 8820 8956 mdash 8820 mdash2 1 I 8880 9531 mdash 8882 mdash0 2 I 9176 mdash NA mdash NA1 3 I 9576 9983 mdash 9567 mdash0 3 I 1036 mdash NA mdash NA2 2 I 1066 1147 mdash 1067 mdash0 4 I 1291 mdash NA mdash NA1 4 I 1335 1413 mdash 1335 mdash2 3 I 1364 1475 mdash 1364 mdash1 0 II (119908 = 0) 1665 1738 mdash 1665 mdash0 5 I 1683 mdash NA mdash NA2 4 I 1755 1918 mdash 1755 mdash0 2 II (119908 = 0) 1764 1791 mdash 1764 mdash

exact solutions cannot be achieved by any numerical modelOnce again solution with 119899 = 0 or 119898 = 0 combined with119908 = 0 (in-plane modes) can be found with 2D numericalmodels without modifying the simply supported boundaryconditions

5 Conclusions

Thepresent paper analyzes the cylindrical bending in the freefrequency analysis of functionally graded material (FGM)plates and cylindrical shells This condition is investigatedby means of a 3D exact shell model and two different 2D

numerical models a weak formulation based on classicalFinite Elements (FEs) and a strong formulation based onthe GDQ method The cylindrical bending (CB) behavior inthe 3D exact shell model can be analyzed using completelysimply supported structures and supposing one of the twohalf-wave numbers equals zeroThe numerical models obtainthe CB behavior using mixed simply supported and freeboundary conditions It has been demonstrated that theactual cylindrical bending (CB) is possible only with respectto straight edgesTheCBnatural frequencies of the numericalsolutions converge to the 3D exact CB ones if the length ofstraight simply supported edges is increased up to 20 times

16 Shock and Vibration

the value of the original edges The CB conditions are notpossible in 2D numericalmodels when the half-wave numberis zero in the curvilinear edge in this case the modificationof boundary conditions does not properly work In-planevibrationmodeswith zero transverse displacement combinedwith 119898 = 0 or 119899 = 0 cannot be defined as CB frequencies andthey can be calculated via 2D numerical models without themodification of the simply supported conditions for the fouredges All the considerations about the cylindrical bendingare valid for each geometry (plate cylinder and cylindricalshell) lamination scheme FGM law and thickness ratio

Competing Interests

The authors declare that they have no competing interests

References

[1] W Q Chen J B Cai and G R Ye ldquoResponses of cross-ply laminates with viscous interfaces in cylindrical bendingrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 823ndash835 2005

[2] W Q Chen and K Y Lee ldquoThree-dimensional exact analysisof angle-ply laminates in cylindrical bending with interfacialdamage via state-space methodrdquo Composite Structures vol 64no 3-4 pp 275ndash283 2004

[3] W Q Chen and K Y Lee ldquoTime-dependent behaviors of angle-ply laminates with viscous interfaces in cylindrical bendingrdquoEuropean Journal ofMechanicsmdashASolids vol 23 no 2 pp 235ndash245 2004

[4] A H Gandhi and H K Raval ldquoAnalytical and empiricalmodeling of top roller position for three-roller cylindricalbending of plates and its experimental verificationrdquo Journal ofMaterials Processing Technology vol 197 no 1ndash3 pp 268ndash2782008

[5] S Oral and H Darendeliler ldquoThe optimum die profile for thecylindrical bending of platesrdquo Journal of Materials ProcessingTechnology vol 70 no 1ndash3 pp 151ndash155 1997

[6] V A Jairazbhoy P Petukhov and J Qu ldquoLarge deflectionof thin plates in cylindrical bendingmdashnon-unique solutionsrdquoInternational Journal of Solids and Structures vol 45 no 11-12pp 3203ndash3218 2008

[7] A Lebee and K Sab ldquoA bending-gradient model for thickplates part II closed-form solutions for cylindrical bending oflaminatesrdquo International Journal of Solids and Structures vol 48no 20 pp 2889ndash2901 2011

[8] P V Nimbolkar and I M Jain ldquoCylindrical bending of elasticplatesrdquo Procedia Materials Science vol 10 pp 793ndash802 2015

[9] A S Sayyad and Y M Ghugal ldquoA nth-order shear deformationtheory for composite laminates in cylindrical bendingrdquo Curvedand Layered Structures vol 2 no 1 pp 290ndash300 2015

[10] N J Pagano ldquoExact solutions for composite laminates incylindrical bendingrdquo Journal of Composite Materials vol 3 no3 pp 398ndash411 1969

[11] N J Pagano and A S D Wang ldquoFurther study of compositelaminates under cylindrical bendingrdquo Journal of CompositeMaterials vol 5 pp 421ndash428 1971

[12] N Saeedi K Sab and J-F Caron ldquoCylindrical bending ofmultilayered plates with multi-delamination via a layerwisestress approachrdquo Composite Structures vol 95 pp 728ndash7392013

[13] X-P Shu and K P Soldatos ldquoCylindrical bending of angle-ply laminates subjected to different sets of edge boundaryconditionsrdquo International Journal of Solids and Structures vol37 no 31 pp 4289ndash4307 2000

[14] Z G Bian W Q Chen C W Lim and N Zhang ldquoAnalyticalsolutions for single- and multi-span functionally graded platesin cylindrical bendingrdquo International Journal of Solids andStructures vol 42 no 24-25 pp 6433ndash6456 2005

[15] H M Navazi and H Haddadpour ldquoNonlinear cylindricalbending analysis of shear deformable functionally graded platesunder different loadings using analytical methodsrdquo Interna-tional Journal of Mechanical Sciences vol 50 no 12 pp 1650ndash1657 2008

[16] W Q Chen J Ying J B Cai and G R Ye ldquoBenchmarksolution of imperfect angle-ply laminated rectangular plates incylindrical bending with surface piezoelectric layers as actuatorand sensorrdquo Computers and Structures vol 82 no 22 pp 1773ndash1784 2004

[17] W Q Chen and K Y Lee ldquoExact solution of angle-plypiezoelectric laminates in cylindrical bending with interfacialimperfectionsrdquo Composite Structures vol 65 no 3-4 pp 329ndash337 2004

[18] T Kant and SM Shiyekar ldquoCylindrical bending of piezoelectriclaminates with a higher order shear and normal deformationtheoryrdquo Computers and Structures vol 86 no 15-16 pp 1594ndash1603 2008

[19] P Lu H P Lee and C Lu ldquoAn exact solution for functionallygraded piezoelectric laminates in cylindrical bendingrdquo Interna-tional Journal of Mechanical Sciences vol 47 no 3 pp 437ndash4382005

[20] W Yan J Wang and W Q Chen ldquoCylindrical bendingresponses of angle-ply piezoelectric laminates with viscoelasticinterfacesrdquo Applied Mathematical Modelling vol 38 no 24 pp6018ndash6030 2014

[21] Y Y Zhou W Q Chen and C F Lu ldquoSemi-analytical solutionfor orthotropic piezoelectric laminates in cylindrical bendingwith interfacial imperfectionsrdquo Composite Structures vol 92no 4 pp 1009ndash1018 2010

[22] G P Dube S Kapuria and P C Dumir ldquoExact piezother-moelastic solution of a simply-supported orthotropic flat panelin cylindrical bendingrdquo International Journal of MechanicalSciences vol 38 pp 374ndash387 1996

[23] V N Pilipchuk V L Berdichevsky and R A IbrahimldquoThermo-mechanical coupling in cylindrical bending of sand-wich platesrdquoComposite Structures vol 92 no 11 pp 2632ndash26402010

[24] C Zhang S Di and N Zhang ldquoA new procedure for staticanalysis of thermo-electric laminated composite plates undercylindrical bendingrdquo Composite Structures vol 56 no 2 pp131ndash140 2002

[25] W Q Chen and K Y Lee ldquoState-space approach for statics anddynamics of angle-ply laminated cylindrical panels in cylindri-cal bendingrdquo International Journal of Mechanical Sciences vol47 no 3 pp 374ndash387 2005

[26] C-P Wu and Y-S Syu ldquoExact solutions of functionally gradedpiezoelectric shells under cylindrical bendingrdquo InternationalJournal of Solids and Structures vol 44 no 20 pp 6450ndash64722007

[27] W Yan J Ying and W Q Chen ldquoThe behavior of angle-ply laminated cylindrical shells with viscoelastic interfaces incylindrical bendingrdquo Composite Structures vol 78 no 4 pp551ndash559 2007

Shock and Vibration 17

[28] S Brischetto F Tornabene N Fantuzzi and E Viola ldquo3Dexact and 2D generalized differential quadraturemodels for freevibration analysis of functionally graded plates and cylindersrdquoMeccanica vol 51 no 9 pp 2059ndash2098 2016

[29] N Fantuzzi S Brischetto F Tornabene and E Viola ldquo2D and3D shell models for the free vibration investigation of func-tionally graded cylindrical and spherical panelsrdquo CompositeStructures vol 154 pp 573ndash590 2016

[30] S Brischetto and R Torre ldquoExact 3D solutions and finiteelement 2D models for free vibration analysis of plates andcylindersrdquo Curved and Layered Structures vol 1 no 1 pp 59ndash92 2014

[31] F Tornabene S Brischetto N Fantuzzi and E Viola ldquoNumer-ical and exact models for free vibration analysis of cylindricaland spherical shell panelsrdquo Composites Part B Engineering vol81 pp 231ndash250 2015

[32] S Brischetto ldquoExact elasticity solution for natural frequenciesof functionally graded simply-supported structuresrdquo ComputerModeling in Engineering amp Sciences vol 95 no 5 pp 391ndash4302013

[33] S Brischetto ldquoThree-dimensional exact free vibration analysisof spherical cylindrical and flat one-layered panelsrdquo Shock andVibration vol 2014 Article ID 479738 29 pages 2014

[34] S Brischetto ldquoA continuum elastic three-dimensional modelfor natural frequencies of single-walled carbon nanotubesrdquoComposites Part B Engineering vol 61 pp 222ndash228 2014

[35] S Brischetto ldquoAn exact 3d solution for free vibrations ofmultilayered cross-ply composite and sandwich plates andshellsrdquo International Journal of Applied Mechanics vol 6 no 6Article ID 1450076 42 pages 2014

[36] S Brischetto ldquoA continuumshellmodel including vanderWaalsinteraction for free vibrations of double-walled carbon nan-otubesrdquoCMES-ComputerModeling in Engineering and Sciencesvol 104 no 4 pp 305ndash327 2015

[37] S Brischetto F Tornabene N Fantuzzi and M BacciocchildquoRefined 2D and exact 3D shell models for the free vibrationanalysis of single- and double-walled carbon nanotubesrdquo Tech-nologies vol 3 no 4 pp 259ndash284 2015

[38] S Brischetto ldquoConvergence analysis of the exponential matrixmethod for the solution of 3D equilibrium equations for freevibration analysis of plates and shellsrdquo Composites Part BEngineering vol 98 pp 453ndash471 2016

[39] S Brischetto ldquoExact and approximate shell geometry in the freevibration analysis of one-layered and multilayered structuresrdquoInternational Journal of Mechanical Sciences vol 113 pp 81ndash932016

[40] S Brischetto ldquoCurvature approximation effects in the freevibration analysis of functionally graded shellsrdquo InternationalJournal of Applied Mechanics 2016

[41] F Tornabene N Fantuzzi F Ubertini and E Viola ldquoStrong for-mulation finite element method based on differential quadra-ture a surveyrdquo Applied Mechanics Reviews vol 67 no 2 ArticleID 020801 pp 1ndash55 2015

[42] F Tornabene E Viola and N Fantuzzi ldquoGeneral higher-orderequivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panelsrdquo CompositeStructures vol 104 pp 94ndash117 2013

[43] N Fantuzzi F Tornabene E Viola and A J Ferreira ldquoAstrong formulation finite element method (SFEM) based onRBF and GDQ techniques for the static and dynamic analysesof laminated plates of arbitrary shaperdquo Meccanica vol 49 no10 pp 2503ndash2542 2014

[44] F Tornabene N Fantuzzi and M Bacciocchi ldquoFree vibrationsof free-form doubly-curved shells made of functionally gradedmaterials using higher-order equivalent single layer theoriesrdquoComposites Part B Engineering vol 67 pp 490ndash509 2014

[45] F Tornabene and E Viola ldquoStatic analysis of functionallygraded doubly-curved shells and panels of revolutionrdquo Mecca-nica vol 48 no 4 pp 901ndash930 2013

[46] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoDynamic analysis of thick and thin elliptic shell structuresmade of laminated composite materialsrdquo Composite Structuresvol 133 pp 278ndash299 2015

[47] F TornabeneN FantuzziM Bacciocchi and EViola ldquoHigher-order theories for the free vibrations of doubly-curved lam-inated panels with curvilinear reinforcing fibers by meansof a local version of the GDQ methodrdquo Composites Part BEngineering vol 81 pp 196ndash230 2015

[48] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoFree vibrations of composite oval and elliptic cylinders bythe generalized differential quadrature methodrdquo Thin-WalledStructures vol 97 pp 114ndash129 2015

[49] F Tornabene and J N Reddy ldquoFGM and laminated doubly-curved and degenerate shells resting on nonlinear elastic foun-dations a GDQ solution for static analysis with a posterioristress and strain recoveryrdquo Journal of the Indian Institute ofScience vol 93 no 4 pp 635ndash688 2013

[50] N Fantuzzi F Tornabene and E Viola ldquoFour-parameter func-tionally graded cracked plates of arbitrary shape a GDQFEMsolution for free vibrationsrdquo Mechanics of Advanced Materialsand Structures vol 23 no 1 pp 89ndash107 2016

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

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Advances inOptoElectronics

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

Shock and Vibration 11

ulowast

lowast

wlowast

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

(a) Mode I (1 0) 3D

109

0807

0605

0403

0201 0

109

0807

0605

0403

02010

000501

01502

(b) Mode I (1 0) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(c) Mode I (2 0) 3D

109

0807

060504

0302

01 0

109

0807

0605

0403

02010

0201501

0050

minus005minus01

minus015minus02

(d) Mode I (2 0) 2D

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(e) Mode I (3 0) 3D

10908

0706

050403

0201 0

109

0807

0605

0403

02010

0201501

0050

minus005minus01

minus015minus02

(f) Mode I (3 0) 2D

Figure 3 Continued

12 Shock and Vibration

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(g) Mode I (4 0) 3D

10908

0706050403

0201 0

10908

0706

0504

0302

010

0201501

0050

minus005minus01

minus015minus02

(h) Mode I (4 0) 2D

Figure 3 First four cylindrical bending vibration modes for the 119886ℎ = 20 one-layered FGM plate with 119901 = 05 3D modes versus 2D modes

3D exact2D GDQ2D FEM

f(H

z)

39

392

394

396

398

40

402

404

5 10 15 200La

Figure 4 First frequency 119891 in Hz for the 119886ℎ = 100 one-layeredFGM plate with 119901 = 05 convergence of 2D GDQ and 2D FEsolutions increasing the length of the simply supported sides

particular the GDQ solution is slightly more accurate thanthe FE one The 2D GDQ solution is carried out by enlargingthe length of one of the sides of the plate and the samenumberof dofs is kept On the contrary the FE model is generatedby 40 times 40 blocks of mesh In other words the plate with119871 = 2119886 has a mesh of 40 times 80 the one with 119871 = 3119886 has amesh 40 times 120 and so on When 119871 = 20119886 the error amongthe 3D exact and the 2D numerical solutions is very small andthis is proven also by the mode shapes of Figure 5 whereinthe boundary effect is negligible with respect the length of

the simply supported edges It must be underlined that it isimpossible to completely avoid the boundary effects due tothe free edges

The second example is related to a simply supportedcylinder with 119877120572 = 10m 119877120573 = infin 119886 = 2120587119877120572 and 119887 =20m and thickness ratios 119877120572ℎ = 100 and 119877120572ℎ = 10 Aone-layered FGM scheme is considered with ℎ1 = ℎ andthe same volume fraction function of (24) for the previousexample but setting 119901 = 20 The present material propertiesare the same as the flat plate The FE model has a 20 times80 Quad8 mesh with 27966 dofs and the GDQ solutionhas a 15 times 51 Chebyshev-Gauss-Lobatto grid Numericallyspeaking compatibility conditions must be enforced on theclosing meridian in order to obtain a closed revolution shell[42] To obtain the CB conditions numerically the cylinderis modeled completely free and the first 6 rigid mode shapesare neglected and removed from the numerical solution Itis noted that the GDQ model uses a third-order kinematicexpansion with no zigzag effect as the previous case sincea one-layer structure has been taken into account Table 3reports the results with the same meaning of the symbolsseen in Table 1 A very good agreement is observed when thecylinder is simply supported whereas when the cylinder iscompletely free some small errors can be observed but thenumerical solutions follow the 3D exact one

Results with 119899 = 0 and 119908 = 0 indicate the CB conditionsSuch results are obtained by means of 2D numerical modelsmodifying the boundary conditions (see 2D FE(CB) and 2DGDQ(CB)models)The cases 119899 = 0 combined with119908 = 0 arenot CB conditions and the frequencies can be found withoutmodifying the boundary conditions in the 2D numericalmodels

The final test is related to a cylindrical panel with 119877120572 =10m 119877120573 = infin 119886 = 1205873119877120572 and 119887 = 20m and thickness ratios119877120572ℎ = 100 and 119877120572ℎ = 10 The present shell is a sandwichstructure embedding a FGM core with ℎ2 = 07ℎ and two

Shock and Vibration 13

218

1614

121 1

08 0806 06

04 0402 02

0 0

0minus01minus02minus03minus04

(a) 119871119886 = 2

3

25

2

151

0806

040502

0 0

1

0minus01minus02minus03minus04minus05

(b) 119871119886 = 3

435

325

10505

0 0

15

0

minus02minus04minus06

1

2

(c) 119871119886 = 4

545

35

10505

0 0

0

minus04minus02

minus06minus08

1

25

4

3

215

(d) 119871119886 = 5

5

1

0 005

0

minus04minus02

minus06minus08

1

4

3

2

6

minus1

(e) 119871119886 = 6

5

10 0 051

43

2

67

89

100

minus05minus1

minus15

(f) 119871119886 = 10

Figure 5 First vibration mode for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 2D GDQ solution and edge effects versus the lengthof the simply supported sides

external skins with ℎ1 = ℎ3 = 015ℎ (ℎ1 + ℎ2 + ℎ3 = ℎ) Thebottom layer ismade of isotropicmetallicmaterial and the toplayer is made of ceramic material the mechanical propertiesare

119864119898 = 73GPa120588119898 = 2800 kgm3119864119888 = 200GPa120588119888 = 5700 kgm3

(27)

The FGM core has a volume fraction function defined as

119881119888 (119911) = (05 + 11991107ℎ)119901 with 119911 isin [minus035ℎ 035ℎ] (28)

where 119901 = 10 is employed in the present computations ThePoisson ratio is constant (]119888 = ]119898 = ] = 03) Equations (25)are still valid with 119881119888 defined as in (28)

The FEmodel is made of 40times40Quad8 elements for total29280 dofs and the GDQ model has a 31 times 31 Chebyshev-Gauss-Lobatto grid Since a sandwich structure is analyzedin the following a third-order theory with zigzag functionis considered in the computations for the GDQ method Thepresent natural frequencies are reported in Table 4 with the

14 Shock and Vibration

Table 2 First frequency 119891 in Hz for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 convergence of 2D GDQ and 2D FE solutionsincreasing the length of the simply supported sides

119871119886 1 2 3 4 5 6 8 10 15 20 3D exact2D GDQ 3921 3964 3980 3988 3993 3997 4001 4004 4007 4008 40192D FEM 3907 3949 3966 3974 3979 3983 3987 3989 3993 3994 4019

Table 3 One-layered FGM cylinder with 119901 = 20 Analytical 3D versus 2D numerical models Frequency 119891 in Hz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1004 0 I 0896 mdash 0897 mdash 08836 0 I 2532 mdash 2538 mdash 25118 0 I 4854 mdash 4867 mdash 481910 0 I 7848 mdash 7871 mdash 780112 0 I 1151 mdash 1155 mdash 114610 1 I 1356 1360 mdash 1357 mdash12 1 I 1441 1449 mdash 1445 mdash14 0 I 1584 mdash 1589 mdash 15778 1 I 1638 1640 mdash 1636 mdash14 1 I 1756 1767 mdash 1763 mdash16 0 I 2083 mdash 2090 mdash 207016 1 I 2207 2222 mdash 2216 mdash6 1 I 2441 2442 mdash 2438 mdash18 0 I 2649 mdash 2658 mdash 264014 2 I 2742 2752 mdash 2746 mdash18 1 I 2751 2772 mdash 2763 mdash16 2 I 2875 2890 mdash 2883 mdash12 2 I 2921 2926 mdash 2922 mdash119877120572ℎ = 104 0 I 8774 mdash 9085 mdash 86796 0 I 2456 mdash 2571 mdash 24306 1 I 3978 4089 mdash 3979 mdash4 1 I 4388 4357 mdash 4388 mdash8 0 I 4654 mdash 4920 mdash 46028 1 I 5575 5905 mdash 5576 mdash2 1 I 7202 7101 mdash 7202 mdash2 0 I (119908 = 0) 7322 7390 mdash 7319 mdash10 0 I 7419 mdash 7933 mdash 73476 2 I 8110 8261 mdash 8111 mdash10 1 I 8189 8811 mdash 8191 mdash8 2 I 8783 9173 mdash 8785 mdash4 2 I 9016 9084 mdash 9017 mdash12 0 I 1071 mdash 1159 mdash 106110 2 I 1081 1155 mdash 1081 mdash2 2 I 1089 1105 mdash 1089 mdash12 1 I 1142 1245 mdash 1143 mdash4 3 I 1263 1286 mdash 1263 mdash

same meaning of the columns seen in the previous casesIn order to achieve the cylindrical bending conditions thenumerical models have the straight edges simply supportedand the curved edges as free Due to the CB definitionit is not possible to achieve CB conditions when 119898 = 0

and 119899 = 1 2 3 because the cylindrical bending shouldoccur on the curved edge and this is not physically possiblebecause of the variation of displacement components due tothe curvature presence This aspect is indicated as NA (NotApplicable) solution in Table 4 Therefore this kind of 3D

Shock and Vibration 15

Table 4 Sandwich cylindrical shell panel embedding FGM core with 119901 = 10 Analytical 3D versus 2D numerical models Frequency 119891 inHz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1001 0 I 1948 mdash 1951 mdash 19412 0 I 8856 mdash 8872 mdash 88232 1 I 1104 1110 mdash 1104 mdash1 1 I 1850 1873 mdash 1850 mdash3 0 I 2038 mdash 2042 mdash 20303 1 I 2116 2128 mdash 2116 mdash2 2 I 2221 2242 mdash 2221 mdash3 2 I 2484 2496 mdash 2484 mdash3 3 I 3235 3254 mdash 3235 mdash4 0 I 3649 mdash 3660 mdash 36342 3 I 3692 3739 mdash 3692 mdash4 1 I 3714 3745 mdash 3714 mdash4 2 I 3940 3965 mdash 3941 mdash3 4 I 4226 4261 mdash 4226 mdash4 3 I 4378 4403 mdash 4378 mdash1 2 I 4568 4664 mdash 4568 mdash4 4 I 5035 5064 mdash 5035 mdash2 4 I 5040 5128 mdash 5040 mdash119877120572ℎ = 101 0 I 1905 mdash 1963 mdash 18841 1 I 3054 3145 mdash 3055 mdash1 2 I 6209 6409 mdash 6209 mdash2 0 I 8326 mdash 8857 mdash 82540 1 I 8710 mdash NA mdash NA0 1 II (119908 = 0) 8820 8956 mdash 8820 mdash2 1 I 8880 9531 mdash 8882 mdash0 2 I 9176 mdash NA mdash NA1 3 I 9576 9983 mdash 9567 mdash0 3 I 1036 mdash NA mdash NA2 2 I 1066 1147 mdash 1067 mdash0 4 I 1291 mdash NA mdash NA1 4 I 1335 1413 mdash 1335 mdash2 3 I 1364 1475 mdash 1364 mdash1 0 II (119908 = 0) 1665 1738 mdash 1665 mdash0 5 I 1683 mdash NA mdash NA2 4 I 1755 1918 mdash 1755 mdash0 2 II (119908 = 0) 1764 1791 mdash 1764 mdash

exact solutions cannot be achieved by any numerical modelOnce again solution with 119899 = 0 or 119898 = 0 combined with119908 = 0 (in-plane modes) can be found with 2D numericalmodels without modifying the simply supported boundaryconditions

5 Conclusions

Thepresent paper analyzes the cylindrical bending in the freefrequency analysis of functionally graded material (FGM)plates and cylindrical shells This condition is investigatedby means of a 3D exact shell model and two different 2D

numerical models a weak formulation based on classicalFinite Elements (FEs) and a strong formulation based onthe GDQ method The cylindrical bending (CB) behavior inthe 3D exact shell model can be analyzed using completelysimply supported structures and supposing one of the twohalf-wave numbers equals zeroThe numerical models obtainthe CB behavior using mixed simply supported and freeboundary conditions It has been demonstrated that theactual cylindrical bending (CB) is possible only with respectto straight edgesTheCBnatural frequencies of the numericalsolutions converge to the 3D exact CB ones if the length ofstraight simply supported edges is increased up to 20 times

16 Shock and Vibration

the value of the original edges The CB conditions are notpossible in 2D numericalmodels when the half-wave numberis zero in the curvilinear edge in this case the modificationof boundary conditions does not properly work In-planevibrationmodeswith zero transverse displacement combinedwith 119898 = 0 or 119899 = 0 cannot be defined as CB frequencies andthey can be calculated via 2D numerical models without themodification of the simply supported conditions for the fouredges All the considerations about the cylindrical bendingare valid for each geometry (plate cylinder and cylindricalshell) lamination scheme FGM law and thickness ratio

Competing Interests

The authors declare that they have no competing interests

References

[1] W Q Chen J B Cai and G R Ye ldquoResponses of cross-ply laminates with viscous interfaces in cylindrical bendingrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 823ndash835 2005

[2] W Q Chen and K Y Lee ldquoThree-dimensional exact analysisof angle-ply laminates in cylindrical bending with interfacialdamage via state-space methodrdquo Composite Structures vol 64no 3-4 pp 275ndash283 2004

[3] W Q Chen and K Y Lee ldquoTime-dependent behaviors of angle-ply laminates with viscous interfaces in cylindrical bendingrdquoEuropean Journal ofMechanicsmdashASolids vol 23 no 2 pp 235ndash245 2004

[4] A H Gandhi and H K Raval ldquoAnalytical and empiricalmodeling of top roller position for three-roller cylindricalbending of plates and its experimental verificationrdquo Journal ofMaterials Processing Technology vol 197 no 1ndash3 pp 268ndash2782008

[5] S Oral and H Darendeliler ldquoThe optimum die profile for thecylindrical bending of platesrdquo Journal of Materials ProcessingTechnology vol 70 no 1ndash3 pp 151ndash155 1997

[6] V A Jairazbhoy P Petukhov and J Qu ldquoLarge deflectionof thin plates in cylindrical bendingmdashnon-unique solutionsrdquoInternational Journal of Solids and Structures vol 45 no 11-12pp 3203ndash3218 2008

[7] A Lebee and K Sab ldquoA bending-gradient model for thickplates part II closed-form solutions for cylindrical bending oflaminatesrdquo International Journal of Solids and Structures vol 48no 20 pp 2889ndash2901 2011

[8] P V Nimbolkar and I M Jain ldquoCylindrical bending of elasticplatesrdquo Procedia Materials Science vol 10 pp 793ndash802 2015

[9] A S Sayyad and Y M Ghugal ldquoA nth-order shear deformationtheory for composite laminates in cylindrical bendingrdquo Curvedand Layered Structures vol 2 no 1 pp 290ndash300 2015

[10] N J Pagano ldquoExact solutions for composite laminates incylindrical bendingrdquo Journal of Composite Materials vol 3 no3 pp 398ndash411 1969

[11] N J Pagano and A S D Wang ldquoFurther study of compositelaminates under cylindrical bendingrdquo Journal of CompositeMaterials vol 5 pp 421ndash428 1971

[12] N Saeedi K Sab and J-F Caron ldquoCylindrical bending ofmultilayered plates with multi-delamination via a layerwisestress approachrdquo Composite Structures vol 95 pp 728ndash7392013

[13] X-P Shu and K P Soldatos ldquoCylindrical bending of angle-ply laminates subjected to different sets of edge boundaryconditionsrdquo International Journal of Solids and Structures vol37 no 31 pp 4289ndash4307 2000

[14] Z G Bian W Q Chen C W Lim and N Zhang ldquoAnalyticalsolutions for single- and multi-span functionally graded platesin cylindrical bendingrdquo International Journal of Solids andStructures vol 42 no 24-25 pp 6433ndash6456 2005

[15] H M Navazi and H Haddadpour ldquoNonlinear cylindricalbending analysis of shear deformable functionally graded platesunder different loadings using analytical methodsrdquo Interna-tional Journal of Mechanical Sciences vol 50 no 12 pp 1650ndash1657 2008

[16] W Q Chen J Ying J B Cai and G R Ye ldquoBenchmarksolution of imperfect angle-ply laminated rectangular plates incylindrical bending with surface piezoelectric layers as actuatorand sensorrdquo Computers and Structures vol 82 no 22 pp 1773ndash1784 2004

[17] W Q Chen and K Y Lee ldquoExact solution of angle-plypiezoelectric laminates in cylindrical bending with interfacialimperfectionsrdquo Composite Structures vol 65 no 3-4 pp 329ndash337 2004

[18] T Kant and SM Shiyekar ldquoCylindrical bending of piezoelectriclaminates with a higher order shear and normal deformationtheoryrdquo Computers and Structures vol 86 no 15-16 pp 1594ndash1603 2008

[19] P Lu H P Lee and C Lu ldquoAn exact solution for functionallygraded piezoelectric laminates in cylindrical bendingrdquo Interna-tional Journal of Mechanical Sciences vol 47 no 3 pp 437ndash4382005

[20] W Yan J Wang and W Q Chen ldquoCylindrical bendingresponses of angle-ply piezoelectric laminates with viscoelasticinterfacesrdquo Applied Mathematical Modelling vol 38 no 24 pp6018ndash6030 2014

[21] Y Y Zhou W Q Chen and C F Lu ldquoSemi-analytical solutionfor orthotropic piezoelectric laminates in cylindrical bendingwith interfacial imperfectionsrdquo Composite Structures vol 92no 4 pp 1009ndash1018 2010

[22] G P Dube S Kapuria and P C Dumir ldquoExact piezother-moelastic solution of a simply-supported orthotropic flat panelin cylindrical bendingrdquo International Journal of MechanicalSciences vol 38 pp 374ndash387 1996

[23] V N Pilipchuk V L Berdichevsky and R A IbrahimldquoThermo-mechanical coupling in cylindrical bending of sand-wich platesrdquoComposite Structures vol 92 no 11 pp 2632ndash26402010

[24] C Zhang S Di and N Zhang ldquoA new procedure for staticanalysis of thermo-electric laminated composite plates undercylindrical bendingrdquo Composite Structures vol 56 no 2 pp131ndash140 2002

[25] W Q Chen and K Y Lee ldquoState-space approach for statics anddynamics of angle-ply laminated cylindrical panels in cylindri-cal bendingrdquo International Journal of Mechanical Sciences vol47 no 3 pp 374ndash387 2005

[26] C-P Wu and Y-S Syu ldquoExact solutions of functionally gradedpiezoelectric shells under cylindrical bendingrdquo InternationalJournal of Solids and Structures vol 44 no 20 pp 6450ndash64722007

[27] W Yan J Ying and W Q Chen ldquoThe behavior of angle-ply laminated cylindrical shells with viscoelastic interfaces incylindrical bendingrdquo Composite Structures vol 78 no 4 pp551ndash559 2007

Shock and Vibration 17

[28] S Brischetto F Tornabene N Fantuzzi and E Viola ldquo3Dexact and 2D generalized differential quadraturemodels for freevibration analysis of functionally graded plates and cylindersrdquoMeccanica vol 51 no 9 pp 2059ndash2098 2016

[29] N Fantuzzi S Brischetto F Tornabene and E Viola ldquo2D and3D shell models for the free vibration investigation of func-tionally graded cylindrical and spherical panelsrdquo CompositeStructures vol 154 pp 573ndash590 2016

[30] S Brischetto and R Torre ldquoExact 3D solutions and finiteelement 2D models for free vibration analysis of plates andcylindersrdquo Curved and Layered Structures vol 1 no 1 pp 59ndash92 2014

[31] F Tornabene S Brischetto N Fantuzzi and E Viola ldquoNumer-ical and exact models for free vibration analysis of cylindricaland spherical shell panelsrdquo Composites Part B Engineering vol81 pp 231ndash250 2015

[32] S Brischetto ldquoExact elasticity solution for natural frequenciesof functionally graded simply-supported structuresrdquo ComputerModeling in Engineering amp Sciences vol 95 no 5 pp 391ndash4302013

[33] S Brischetto ldquoThree-dimensional exact free vibration analysisof spherical cylindrical and flat one-layered panelsrdquo Shock andVibration vol 2014 Article ID 479738 29 pages 2014

[34] S Brischetto ldquoA continuum elastic three-dimensional modelfor natural frequencies of single-walled carbon nanotubesrdquoComposites Part B Engineering vol 61 pp 222ndash228 2014

[35] S Brischetto ldquoAn exact 3d solution for free vibrations ofmultilayered cross-ply composite and sandwich plates andshellsrdquo International Journal of Applied Mechanics vol 6 no 6Article ID 1450076 42 pages 2014

[36] S Brischetto ldquoA continuumshellmodel including vanderWaalsinteraction for free vibrations of double-walled carbon nan-otubesrdquoCMES-ComputerModeling in Engineering and Sciencesvol 104 no 4 pp 305ndash327 2015

[37] S Brischetto F Tornabene N Fantuzzi and M BacciocchildquoRefined 2D and exact 3D shell models for the free vibrationanalysis of single- and double-walled carbon nanotubesrdquo Tech-nologies vol 3 no 4 pp 259ndash284 2015

[38] S Brischetto ldquoConvergence analysis of the exponential matrixmethod for the solution of 3D equilibrium equations for freevibration analysis of plates and shellsrdquo Composites Part BEngineering vol 98 pp 453ndash471 2016

[39] S Brischetto ldquoExact and approximate shell geometry in the freevibration analysis of one-layered and multilayered structuresrdquoInternational Journal of Mechanical Sciences vol 113 pp 81ndash932016

[40] S Brischetto ldquoCurvature approximation effects in the freevibration analysis of functionally graded shellsrdquo InternationalJournal of Applied Mechanics 2016

[41] F Tornabene N Fantuzzi F Ubertini and E Viola ldquoStrong for-mulation finite element method based on differential quadra-ture a surveyrdquo Applied Mechanics Reviews vol 67 no 2 ArticleID 020801 pp 1ndash55 2015

[42] F Tornabene E Viola and N Fantuzzi ldquoGeneral higher-orderequivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panelsrdquo CompositeStructures vol 104 pp 94ndash117 2013

[43] N Fantuzzi F Tornabene E Viola and A J Ferreira ldquoAstrong formulation finite element method (SFEM) based onRBF and GDQ techniques for the static and dynamic analysesof laminated plates of arbitrary shaperdquo Meccanica vol 49 no10 pp 2503ndash2542 2014

[44] F Tornabene N Fantuzzi and M Bacciocchi ldquoFree vibrationsof free-form doubly-curved shells made of functionally gradedmaterials using higher-order equivalent single layer theoriesrdquoComposites Part B Engineering vol 67 pp 490ndash509 2014

[45] F Tornabene and E Viola ldquoStatic analysis of functionallygraded doubly-curved shells and panels of revolutionrdquo Mecca-nica vol 48 no 4 pp 901ndash930 2013

[46] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoDynamic analysis of thick and thin elliptic shell structuresmade of laminated composite materialsrdquo Composite Structuresvol 133 pp 278ndash299 2015

[47] F TornabeneN FantuzziM Bacciocchi and EViola ldquoHigher-order theories for the free vibrations of doubly-curved lam-inated panels with curvilinear reinforcing fibers by meansof a local version of the GDQ methodrdquo Composites Part BEngineering vol 81 pp 196ndash230 2015

[48] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoFree vibrations of composite oval and elliptic cylinders bythe generalized differential quadrature methodrdquo Thin-WalledStructures vol 97 pp 114ndash129 2015

[49] F Tornabene and J N Reddy ldquoFGM and laminated doubly-curved and degenerate shells resting on nonlinear elastic foun-dations a GDQ solution for static analysis with a posterioristress and strain recoveryrdquo Journal of the Indian Institute ofScience vol 93 no 4 pp 635ndash688 2013

[50] N Fantuzzi F Tornabene and E Viola ldquoFour-parameter func-tionally graded cracked plates of arbitrary shape a GDQFEMsolution for free vibrationsrdquo Mechanics of Advanced Materialsand Structures vol 23 no 1 pp 89ndash107 2016

International Journal of

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Control Scienceand Engineering

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RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

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Electrical and Computer Engineering

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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DistributedSensor Networks

International Journal of

12 Shock and Vibration

050minus05 1minus1

ulowast lowast wlowast

0

01

02

03

04

05

06

07

08

09

1

zlowast

ulowast

lowast

wlowast

(g) Mode I (4 0) 3D

10908

0706050403

0201 0

10908

0706

0504

0302

010

0201501

0050

minus005minus01

minus015minus02

(h) Mode I (4 0) 2D

Figure 3 First four cylindrical bending vibration modes for the 119886ℎ = 20 one-layered FGM plate with 119901 = 05 3D modes versus 2D modes

3D exact2D GDQ2D FEM

f(H

z)

39

392

394

396

398

40

402

404

5 10 15 200La

Figure 4 First frequency 119891 in Hz for the 119886ℎ = 100 one-layeredFGM plate with 119901 = 05 convergence of 2D GDQ and 2D FEsolutions increasing the length of the simply supported sides

particular the GDQ solution is slightly more accurate thanthe FE one The 2D GDQ solution is carried out by enlargingthe length of one of the sides of the plate and the samenumberof dofs is kept On the contrary the FE model is generatedby 40 times 40 blocks of mesh In other words the plate with119871 = 2119886 has a mesh of 40 times 80 the one with 119871 = 3119886 has amesh 40 times 120 and so on When 119871 = 20119886 the error amongthe 3D exact and the 2D numerical solutions is very small andthis is proven also by the mode shapes of Figure 5 whereinthe boundary effect is negligible with respect the length of

the simply supported edges It must be underlined that it isimpossible to completely avoid the boundary effects due tothe free edges

The second example is related to a simply supportedcylinder with 119877120572 = 10m 119877120573 = infin 119886 = 2120587119877120572 and 119887 =20m and thickness ratios 119877120572ℎ = 100 and 119877120572ℎ = 10 Aone-layered FGM scheme is considered with ℎ1 = ℎ andthe same volume fraction function of (24) for the previousexample but setting 119901 = 20 The present material propertiesare the same as the flat plate The FE model has a 20 times80 Quad8 mesh with 27966 dofs and the GDQ solutionhas a 15 times 51 Chebyshev-Gauss-Lobatto grid Numericallyspeaking compatibility conditions must be enforced on theclosing meridian in order to obtain a closed revolution shell[42] To obtain the CB conditions numerically the cylinderis modeled completely free and the first 6 rigid mode shapesare neglected and removed from the numerical solution Itis noted that the GDQ model uses a third-order kinematicexpansion with no zigzag effect as the previous case sincea one-layer structure has been taken into account Table 3reports the results with the same meaning of the symbolsseen in Table 1 A very good agreement is observed when thecylinder is simply supported whereas when the cylinder iscompletely free some small errors can be observed but thenumerical solutions follow the 3D exact one

Results with 119899 = 0 and 119908 = 0 indicate the CB conditionsSuch results are obtained by means of 2D numerical modelsmodifying the boundary conditions (see 2D FE(CB) and 2DGDQ(CB)models)The cases 119899 = 0 combined with119908 = 0 arenot CB conditions and the frequencies can be found withoutmodifying the boundary conditions in the 2D numericalmodels

The final test is related to a cylindrical panel with 119877120572 =10m 119877120573 = infin 119886 = 1205873119877120572 and 119887 = 20m and thickness ratios119877120572ℎ = 100 and 119877120572ℎ = 10 The present shell is a sandwichstructure embedding a FGM core with ℎ2 = 07ℎ and two

Shock and Vibration 13

218

1614

121 1

08 0806 06

04 0402 02

0 0

0minus01minus02minus03minus04

(a) 119871119886 = 2

3

25

2

151

0806

040502

0 0

1

0minus01minus02minus03minus04minus05

(b) 119871119886 = 3

435

325

10505

0 0

15

0

minus02minus04minus06

1

2

(c) 119871119886 = 4

545

35

10505

0 0

0

minus04minus02

minus06minus08

1

25

4

3

215

(d) 119871119886 = 5

5

1

0 005

0

minus04minus02

minus06minus08

1

4

3

2

6

minus1

(e) 119871119886 = 6

5

10 0 051

43

2

67

89

100

minus05minus1

minus15

(f) 119871119886 = 10

Figure 5 First vibration mode for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 2D GDQ solution and edge effects versus the lengthof the simply supported sides

external skins with ℎ1 = ℎ3 = 015ℎ (ℎ1 + ℎ2 + ℎ3 = ℎ) Thebottom layer ismade of isotropicmetallicmaterial and the toplayer is made of ceramic material the mechanical propertiesare

119864119898 = 73GPa120588119898 = 2800 kgm3119864119888 = 200GPa120588119888 = 5700 kgm3

(27)

The FGM core has a volume fraction function defined as

119881119888 (119911) = (05 + 11991107ℎ)119901 with 119911 isin [minus035ℎ 035ℎ] (28)

where 119901 = 10 is employed in the present computations ThePoisson ratio is constant (]119888 = ]119898 = ] = 03) Equations (25)are still valid with 119881119888 defined as in (28)

The FEmodel is made of 40times40Quad8 elements for total29280 dofs and the GDQ model has a 31 times 31 Chebyshev-Gauss-Lobatto grid Since a sandwich structure is analyzedin the following a third-order theory with zigzag functionis considered in the computations for the GDQ method Thepresent natural frequencies are reported in Table 4 with the

14 Shock and Vibration

Table 2 First frequency 119891 in Hz for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 convergence of 2D GDQ and 2D FE solutionsincreasing the length of the simply supported sides

119871119886 1 2 3 4 5 6 8 10 15 20 3D exact2D GDQ 3921 3964 3980 3988 3993 3997 4001 4004 4007 4008 40192D FEM 3907 3949 3966 3974 3979 3983 3987 3989 3993 3994 4019

Table 3 One-layered FGM cylinder with 119901 = 20 Analytical 3D versus 2D numerical models Frequency 119891 in Hz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1004 0 I 0896 mdash 0897 mdash 08836 0 I 2532 mdash 2538 mdash 25118 0 I 4854 mdash 4867 mdash 481910 0 I 7848 mdash 7871 mdash 780112 0 I 1151 mdash 1155 mdash 114610 1 I 1356 1360 mdash 1357 mdash12 1 I 1441 1449 mdash 1445 mdash14 0 I 1584 mdash 1589 mdash 15778 1 I 1638 1640 mdash 1636 mdash14 1 I 1756 1767 mdash 1763 mdash16 0 I 2083 mdash 2090 mdash 207016 1 I 2207 2222 mdash 2216 mdash6 1 I 2441 2442 mdash 2438 mdash18 0 I 2649 mdash 2658 mdash 264014 2 I 2742 2752 mdash 2746 mdash18 1 I 2751 2772 mdash 2763 mdash16 2 I 2875 2890 mdash 2883 mdash12 2 I 2921 2926 mdash 2922 mdash119877120572ℎ = 104 0 I 8774 mdash 9085 mdash 86796 0 I 2456 mdash 2571 mdash 24306 1 I 3978 4089 mdash 3979 mdash4 1 I 4388 4357 mdash 4388 mdash8 0 I 4654 mdash 4920 mdash 46028 1 I 5575 5905 mdash 5576 mdash2 1 I 7202 7101 mdash 7202 mdash2 0 I (119908 = 0) 7322 7390 mdash 7319 mdash10 0 I 7419 mdash 7933 mdash 73476 2 I 8110 8261 mdash 8111 mdash10 1 I 8189 8811 mdash 8191 mdash8 2 I 8783 9173 mdash 8785 mdash4 2 I 9016 9084 mdash 9017 mdash12 0 I 1071 mdash 1159 mdash 106110 2 I 1081 1155 mdash 1081 mdash2 2 I 1089 1105 mdash 1089 mdash12 1 I 1142 1245 mdash 1143 mdash4 3 I 1263 1286 mdash 1263 mdash

same meaning of the columns seen in the previous casesIn order to achieve the cylindrical bending conditions thenumerical models have the straight edges simply supportedand the curved edges as free Due to the CB definitionit is not possible to achieve CB conditions when 119898 = 0

and 119899 = 1 2 3 because the cylindrical bending shouldoccur on the curved edge and this is not physically possiblebecause of the variation of displacement components due tothe curvature presence This aspect is indicated as NA (NotApplicable) solution in Table 4 Therefore this kind of 3D

Shock and Vibration 15

Table 4 Sandwich cylindrical shell panel embedding FGM core with 119901 = 10 Analytical 3D versus 2D numerical models Frequency 119891 inHz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1001 0 I 1948 mdash 1951 mdash 19412 0 I 8856 mdash 8872 mdash 88232 1 I 1104 1110 mdash 1104 mdash1 1 I 1850 1873 mdash 1850 mdash3 0 I 2038 mdash 2042 mdash 20303 1 I 2116 2128 mdash 2116 mdash2 2 I 2221 2242 mdash 2221 mdash3 2 I 2484 2496 mdash 2484 mdash3 3 I 3235 3254 mdash 3235 mdash4 0 I 3649 mdash 3660 mdash 36342 3 I 3692 3739 mdash 3692 mdash4 1 I 3714 3745 mdash 3714 mdash4 2 I 3940 3965 mdash 3941 mdash3 4 I 4226 4261 mdash 4226 mdash4 3 I 4378 4403 mdash 4378 mdash1 2 I 4568 4664 mdash 4568 mdash4 4 I 5035 5064 mdash 5035 mdash2 4 I 5040 5128 mdash 5040 mdash119877120572ℎ = 101 0 I 1905 mdash 1963 mdash 18841 1 I 3054 3145 mdash 3055 mdash1 2 I 6209 6409 mdash 6209 mdash2 0 I 8326 mdash 8857 mdash 82540 1 I 8710 mdash NA mdash NA0 1 II (119908 = 0) 8820 8956 mdash 8820 mdash2 1 I 8880 9531 mdash 8882 mdash0 2 I 9176 mdash NA mdash NA1 3 I 9576 9983 mdash 9567 mdash0 3 I 1036 mdash NA mdash NA2 2 I 1066 1147 mdash 1067 mdash0 4 I 1291 mdash NA mdash NA1 4 I 1335 1413 mdash 1335 mdash2 3 I 1364 1475 mdash 1364 mdash1 0 II (119908 = 0) 1665 1738 mdash 1665 mdash0 5 I 1683 mdash NA mdash NA2 4 I 1755 1918 mdash 1755 mdash0 2 II (119908 = 0) 1764 1791 mdash 1764 mdash

exact solutions cannot be achieved by any numerical modelOnce again solution with 119899 = 0 or 119898 = 0 combined with119908 = 0 (in-plane modes) can be found with 2D numericalmodels without modifying the simply supported boundaryconditions

5 Conclusions

Thepresent paper analyzes the cylindrical bending in the freefrequency analysis of functionally graded material (FGM)plates and cylindrical shells This condition is investigatedby means of a 3D exact shell model and two different 2D

numerical models a weak formulation based on classicalFinite Elements (FEs) and a strong formulation based onthe GDQ method The cylindrical bending (CB) behavior inthe 3D exact shell model can be analyzed using completelysimply supported structures and supposing one of the twohalf-wave numbers equals zeroThe numerical models obtainthe CB behavior using mixed simply supported and freeboundary conditions It has been demonstrated that theactual cylindrical bending (CB) is possible only with respectto straight edgesTheCBnatural frequencies of the numericalsolutions converge to the 3D exact CB ones if the length ofstraight simply supported edges is increased up to 20 times

16 Shock and Vibration

the value of the original edges The CB conditions are notpossible in 2D numericalmodels when the half-wave numberis zero in the curvilinear edge in this case the modificationof boundary conditions does not properly work In-planevibrationmodeswith zero transverse displacement combinedwith 119898 = 0 or 119899 = 0 cannot be defined as CB frequencies andthey can be calculated via 2D numerical models without themodification of the simply supported conditions for the fouredges All the considerations about the cylindrical bendingare valid for each geometry (plate cylinder and cylindricalshell) lamination scheme FGM law and thickness ratio

Competing Interests

The authors declare that they have no competing interests

References

[1] W Q Chen J B Cai and G R Ye ldquoResponses of cross-ply laminates with viscous interfaces in cylindrical bendingrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 823ndash835 2005

[2] W Q Chen and K Y Lee ldquoThree-dimensional exact analysisof angle-ply laminates in cylindrical bending with interfacialdamage via state-space methodrdquo Composite Structures vol 64no 3-4 pp 275ndash283 2004

[3] W Q Chen and K Y Lee ldquoTime-dependent behaviors of angle-ply laminates with viscous interfaces in cylindrical bendingrdquoEuropean Journal ofMechanicsmdashASolids vol 23 no 2 pp 235ndash245 2004

[4] A H Gandhi and H K Raval ldquoAnalytical and empiricalmodeling of top roller position for three-roller cylindricalbending of plates and its experimental verificationrdquo Journal ofMaterials Processing Technology vol 197 no 1ndash3 pp 268ndash2782008

[5] S Oral and H Darendeliler ldquoThe optimum die profile for thecylindrical bending of platesrdquo Journal of Materials ProcessingTechnology vol 70 no 1ndash3 pp 151ndash155 1997

[6] V A Jairazbhoy P Petukhov and J Qu ldquoLarge deflectionof thin plates in cylindrical bendingmdashnon-unique solutionsrdquoInternational Journal of Solids and Structures vol 45 no 11-12pp 3203ndash3218 2008

[7] A Lebee and K Sab ldquoA bending-gradient model for thickplates part II closed-form solutions for cylindrical bending oflaminatesrdquo International Journal of Solids and Structures vol 48no 20 pp 2889ndash2901 2011

[8] P V Nimbolkar and I M Jain ldquoCylindrical bending of elasticplatesrdquo Procedia Materials Science vol 10 pp 793ndash802 2015

[9] A S Sayyad and Y M Ghugal ldquoA nth-order shear deformationtheory for composite laminates in cylindrical bendingrdquo Curvedand Layered Structures vol 2 no 1 pp 290ndash300 2015

[10] N J Pagano ldquoExact solutions for composite laminates incylindrical bendingrdquo Journal of Composite Materials vol 3 no3 pp 398ndash411 1969

[11] N J Pagano and A S D Wang ldquoFurther study of compositelaminates under cylindrical bendingrdquo Journal of CompositeMaterials vol 5 pp 421ndash428 1971

[12] N Saeedi K Sab and J-F Caron ldquoCylindrical bending ofmultilayered plates with multi-delamination via a layerwisestress approachrdquo Composite Structures vol 95 pp 728ndash7392013

[13] X-P Shu and K P Soldatos ldquoCylindrical bending of angle-ply laminates subjected to different sets of edge boundaryconditionsrdquo International Journal of Solids and Structures vol37 no 31 pp 4289ndash4307 2000

[14] Z G Bian W Q Chen C W Lim and N Zhang ldquoAnalyticalsolutions for single- and multi-span functionally graded platesin cylindrical bendingrdquo International Journal of Solids andStructures vol 42 no 24-25 pp 6433ndash6456 2005

[15] H M Navazi and H Haddadpour ldquoNonlinear cylindricalbending analysis of shear deformable functionally graded platesunder different loadings using analytical methodsrdquo Interna-tional Journal of Mechanical Sciences vol 50 no 12 pp 1650ndash1657 2008

[16] W Q Chen J Ying J B Cai and G R Ye ldquoBenchmarksolution of imperfect angle-ply laminated rectangular plates incylindrical bending with surface piezoelectric layers as actuatorand sensorrdquo Computers and Structures vol 82 no 22 pp 1773ndash1784 2004

[17] W Q Chen and K Y Lee ldquoExact solution of angle-plypiezoelectric laminates in cylindrical bending with interfacialimperfectionsrdquo Composite Structures vol 65 no 3-4 pp 329ndash337 2004

[18] T Kant and SM Shiyekar ldquoCylindrical bending of piezoelectriclaminates with a higher order shear and normal deformationtheoryrdquo Computers and Structures vol 86 no 15-16 pp 1594ndash1603 2008

[19] P Lu H P Lee and C Lu ldquoAn exact solution for functionallygraded piezoelectric laminates in cylindrical bendingrdquo Interna-tional Journal of Mechanical Sciences vol 47 no 3 pp 437ndash4382005

[20] W Yan J Wang and W Q Chen ldquoCylindrical bendingresponses of angle-ply piezoelectric laminates with viscoelasticinterfacesrdquo Applied Mathematical Modelling vol 38 no 24 pp6018ndash6030 2014

[21] Y Y Zhou W Q Chen and C F Lu ldquoSemi-analytical solutionfor orthotropic piezoelectric laminates in cylindrical bendingwith interfacial imperfectionsrdquo Composite Structures vol 92no 4 pp 1009ndash1018 2010

[22] G P Dube S Kapuria and P C Dumir ldquoExact piezother-moelastic solution of a simply-supported orthotropic flat panelin cylindrical bendingrdquo International Journal of MechanicalSciences vol 38 pp 374ndash387 1996

[23] V N Pilipchuk V L Berdichevsky and R A IbrahimldquoThermo-mechanical coupling in cylindrical bending of sand-wich platesrdquoComposite Structures vol 92 no 11 pp 2632ndash26402010

[24] C Zhang S Di and N Zhang ldquoA new procedure for staticanalysis of thermo-electric laminated composite plates undercylindrical bendingrdquo Composite Structures vol 56 no 2 pp131ndash140 2002

[25] W Q Chen and K Y Lee ldquoState-space approach for statics anddynamics of angle-ply laminated cylindrical panels in cylindri-cal bendingrdquo International Journal of Mechanical Sciences vol47 no 3 pp 374ndash387 2005

[26] C-P Wu and Y-S Syu ldquoExact solutions of functionally gradedpiezoelectric shells under cylindrical bendingrdquo InternationalJournal of Solids and Structures vol 44 no 20 pp 6450ndash64722007

[27] W Yan J Ying and W Q Chen ldquoThe behavior of angle-ply laminated cylindrical shells with viscoelastic interfaces incylindrical bendingrdquo Composite Structures vol 78 no 4 pp551ndash559 2007

Shock and Vibration 17

[28] S Brischetto F Tornabene N Fantuzzi and E Viola ldquo3Dexact and 2D generalized differential quadraturemodels for freevibration analysis of functionally graded plates and cylindersrdquoMeccanica vol 51 no 9 pp 2059ndash2098 2016

[29] N Fantuzzi S Brischetto F Tornabene and E Viola ldquo2D and3D shell models for the free vibration investigation of func-tionally graded cylindrical and spherical panelsrdquo CompositeStructures vol 154 pp 573ndash590 2016

[30] S Brischetto and R Torre ldquoExact 3D solutions and finiteelement 2D models for free vibration analysis of plates andcylindersrdquo Curved and Layered Structures vol 1 no 1 pp 59ndash92 2014

[31] F Tornabene S Brischetto N Fantuzzi and E Viola ldquoNumer-ical and exact models for free vibration analysis of cylindricaland spherical shell panelsrdquo Composites Part B Engineering vol81 pp 231ndash250 2015

[32] S Brischetto ldquoExact elasticity solution for natural frequenciesof functionally graded simply-supported structuresrdquo ComputerModeling in Engineering amp Sciences vol 95 no 5 pp 391ndash4302013

[33] S Brischetto ldquoThree-dimensional exact free vibration analysisof spherical cylindrical and flat one-layered panelsrdquo Shock andVibration vol 2014 Article ID 479738 29 pages 2014

[34] S Brischetto ldquoA continuum elastic three-dimensional modelfor natural frequencies of single-walled carbon nanotubesrdquoComposites Part B Engineering vol 61 pp 222ndash228 2014

[35] S Brischetto ldquoAn exact 3d solution for free vibrations ofmultilayered cross-ply composite and sandwich plates andshellsrdquo International Journal of Applied Mechanics vol 6 no 6Article ID 1450076 42 pages 2014

[36] S Brischetto ldquoA continuumshellmodel including vanderWaalsinteraction for free vibrations of double-walled carbon nan-otubesrdquoCMES-ComputerModeling in Engineering and Sciencesvol 104 no 4 pp 305ndash327 2015

[37] S Brischetto F Tornabene N Fantuzzi and M BacciocchildquoRefined 2D and exact 3D shell models for the free vibrationanalysis of single- and double-walled carbon nanotubesrdquo Tech-nologies vol 3 no 4 pp 259ndash284 2015

[38] S Brischetto ldquoConvergence analysis of the exponential matrixmethod for the solution of 3D equilibrium equations for freevibration analysis of plates and shellsrdquo Composites Part BEngineering vol 98 pp 453ndash471 2016

[39] S Brischetto ldquoExact and approximate shell geometry in the freevibration analysis of one-layered and multilayered structuresrdquoInternational Journal of Mechanical Sciences vol 113 pp 81ndash932016

[40] S Brischetto ldquoCurvature approximation effects in the freevibration analysis of functionally graded shellsrdquo InternationalJournal of Applied Mechanics 2016

[41] F Tornabene N Fantuzzi F Ubertini and E Viola ldquoStrong for-mulation finite element method based on differential quadra-ture a surveyrdquo Applied Mechanics Reviews vol 67 no 2 ArticleID 020801 pp 1ndash55 2015

[42] F Tornabene E Viola and N Fantuzzi ldquoGeneral higher-orderequivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panelsrdquo CompositeStructures vol 104 pp 94ndash117 2013

[43] N Fantuzzi F Tornabene E Viola and A J Ferreira ldquoAstrong formulation finite element method (SFEM) based onRBF and GDQ techniques for the static and dynamic analysesof laminated plates of arbitrary shaperdquo Meccanica vol 49 no10 pp 2503ndash2542 2014

[44] F Tornabene N Fantuzzi and M Bacciocchi ldquoFree vibrationsof free-form doubly-curved shells made of functionally gradedmaterials using higher-order equivalent single layer theoriesrdquoComposites Part B Engineering vol 67 pp 490ndash509 2014

[45] F Tornabene and E Viola ldquoStatic analysis of functionallygraded doubly-curved shells and panels of revolutionrdquo Mecca-nica vol 48 no 4 pp 901ndash930 2013

[46] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoDynamic analysis of thick and thin elliptic shell structuresmade of laminated composite materialsrdquo Composite Structuresvol 133 pp 278ndash299 2015

[47] F TornabeneN FantuzziM Bacciocchi and EViola ldquoHigher-order theories for the free vibrations of doubly-curved lam-inated panels with curvilinear reinforcing fibers by meansof a local version of the GDQ methodrdquo Composites Part BEngineering vol 81 pp 196ndash230 2015

[48] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoFree vibrations of composite oval and elliptic cylinders bythe generalized differential quadrature methodrdquo Thin-WalledStructures vol 97 pp 114ndash129 2015

[49] F Tornabene and J N Reddy ldquoFGM and laminated doubly-curved and degenerate shells resting on nonlinear elastic foun-dations a GDQ solution for static analysis with a posterioristress and strain recoveryrdquo Journal of the Indian Institute ofScience vol 93 no 4 pp 635ndash688 2013

[50] N Fantuzzi F Tornabene and E Viola ldquoFour-parameter func-tionally graded cracked plates of arbitrary shape a GDQFEMsolution for free vibrationsrdquo Mechanics of Advanced Materialsand Structures vol 23 no 1 pp 89ndash107 2016

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 13

218

1614

121 1

08 0806 06

04 0402 02

0 0

0minus01minus02minus03minus04

(a) 119871119886 = 2

3

25

2

151

0806

040502

0 0

1

0minus01minus02minus03minus04minus05

(b) 119871119886 = 3

435

325

10505

0 0

15

0

minus02minus04minus06

1

2

(c) 119871119886 = 4

545

35

10505

0 0

0

minus04minus02

minus06minus08

1

25

4

3

215

(d) 119871119886 = 5

5

1

0 005

0

minus04minus02

minus06minus08

1

4

3

2

6

minus1

(e) 119871119886 = 6

5

10 0 051

43

2

67

89

100

minus05minus1

minus15

(f) 119871119886 = 10

Figure 5 First vibration mode for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 2D GDQ solution and edge effects versus the lengthof the simply supported sides

external skins with ℎ1 = ℎ3 = 015ℎ (ℎ1 + ℎ2 + ℎ3 = ℎ) Thebottom layer ismade of isotropicmetallicmaterial and the toplayer is made of ceramic material the mechanical propertiesare

119864119898 = 73GPa120588119898 = 2800 kgm3119864119888 = 200GPa120588119888 = 5700 kgm3

(27)

The FGM core has a volume fraction function defined as

119881119888 (119911) = (05 + 11991107ℎ)119901 with 119911 isin [minus035ℎ 035ℎ] (28)

where 119901 = 10 is employed in the present computations ThePoisson ratio is constant (]119888 = ]119898 = ] = 03) Equations (25)are still valid with 119881119888 defined as in (28)

The FEmodel is made of 40times40Quad8 elements for total29280 dofs and the GDQ model has a 31 times 31 Chebyshev-Gauss-Lobatto grid Since a sandwich structure is analyzedin the following a third-order theory with zigzag functionis considered in the computations for the GDQ method Thepresent natural frequencies are reported in Table 4 with the

14 Shock and Vibration

Table 2 First frequency 119891 in Hz for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 convergence of 2D GDQ and 2D FE solutionsincreasing the length of the simply supported sides

119871119886 1 2 3 4 5 6 8 10 15 20 3D exact2D GDQ 3921 3964 3980 3988 3993 3997 4001 4004 4007 4008 40192D FEM 3907 3949 3966 3974 3979 3983 3987 3989 3993 3994 4019

Table 3 One-layered FGM cylinder with 119901 = 20 Analytical 3D versus 2D numerical models Frequency 119891 in Hz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1004 0 I 0896 mdash 0897 mdash 08836 0 I 2532 mdash 2538 mdash 25118 0 I 4854 mdash 4867 mdash 481910 0 I 7848 mdash 7871 mdash 780112 0 I 1151 mdash 1155 mdash 114610 1 I 1356 1360 mdash 1357 mdash12 1 I 1441 1449 mdash 1445 mdash14 0 I 1584 mdash 1589 mdash 15778 1 I 1638 1640 mdash 1636 mdash14 1 I 1756 1767 mdash 1763 mdash16 0 I 2083 mdash 2090 mdash 207016 1 I 2207 2222 mdash 2216 mdash6 1 I 2441 2442 mdash 2438 mdash18 0 I 2649 mdash 2658 mdash 264014 2 I 2742 2752 mdash 2746 mdash18 1 I 2751 2772 mdash 2763 mdash16 2 I 2875 2890 mdash 2883 mdash12 2 I 2921 2926 mdash 2922 mdash119877120572ℎ = 104 0 I 8774 mdash 9085 mdash 86796 0 I 2456 mdash 2571 mdash 24306 1 I 3978 4089 mdash 3979 mdash4 1 I 4388 4357 mdash 4388 mdash8 0 I 4654 mdash 4920 mdash 46028 1 I 5575 5905 mdash 5576 mdash2 1 I 7202 7101 mdash 7202 mdash2 0 I (119908 = 0) 7322 7390 mdash 7319 mdash10 0 I 7419 mdash 7933 mdash 73476 2 I 8110 8261 mdash 8111 mdash10 1 I 8189 8811 mdash 8191 mdash8 2 I 8783 9173 mdash 8785 mdash4 2 I 9016 9084 mdash 9017 mdash12 0 I 1071 mdash 1159 mdash 106110 2 I 1081 1155 mdash 1081 mdash2 2 I 1089 1105 mdash 1089 mdash12 1 I 1142 1245 mdash 1143 mdash4 3 I 1263 1286 mdash 1263 mdash

same meaning of the columns seen in the previous casesIn order to achieve the cylindrical bending conditions thenumerical models have the straight edges simply supportedand the curved edges as free Due to the CB definitionit is not possible to achieve CB conditions when 119898 = 0

and 119899 = 1 2 3 because the cylindrical bending shouldoccur on the curved edge and this is not physically possiblebecause of the variation of displacement components due tothe curvature presence This aspect is indicated as NA (NotApplicable) solution in Table 4 Therefore this kind of 3D

Shock and Vibration 15

Table 4 Sandwich cylindrical shell panel embedding FGM core with 119901 = 10 Analytical 3D versus 2D numerical models Frequency 119891 inHz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1001 0 I 1948 mdash 1951 mdash 19412 0 I 8856 mdash 8872 mdash 88232 1 I 1104 1110 mdash 1104 mdash1 1 I 1850 1873 mdash 1850 mdash3 0 I 2038 mdash 2042 mdash 20303 1 I 2116 2128 mdash 2116 mdash2 2 I 2221 2242 mdash 2221 mdash3 2 I 2484 2496 mdash 2484 mdash3 3 I 3235 3254 mdash 3235 mdash4 0 I 3649 mdash 3660 mdash 36342 3 I 3692 3739 mdash 3692 mdash4 1 I 3714 3745 mdash 3714 mdash4 2 I 3940 3965 mdash 3941 mdash3 4 I 4226 4261 mdash 4226 mdash4 3 I 4378 4403 mdash 4378 mdash1 2 I 4568 4664 mdash 4568 mdash4 4 I 5035 5064 mdash 5035 mdash2 4 I 5040 5128 mdash 5040 mdash119877120572ℎ = 101 0 I 1905 mdash 1963 mdash 18841 1 I 3054 3145 mdash 3055 mdash1 2 I 6209 6409 mdash 6209 mdash2 0 I 8326 mdash 8857 mdash 82540 1 I 8710 mdash NA mdash NA0 1 II (119908 = 0) 8820 8956 mdash 8820 mdash2 1 I 8880 9531 mdash 8882 mdash0 2 I 9176 mdash NA mdash NA1 3 I 9576 9983 mdash 9567 mdash0 3 I 1036 mdash NA mdash NA2 2 I 1066 1147 mdash 1067 mdash0 4 I 1291 mdash NA mdash NA1 4 I 1335 1413 mdash 1335 mdash2 3 I 1364 1475 mdash 1364 mdash1 0 II (119908 = 0) 1665 1738 mdash 1665 mdash0 5 I 1683 mdash NA mdash NA2 4 I 1755 1918 mdash 1755 mdash0 2 II (119908 = 0) 1764 1791 mdash 1764 mdash

exact solutions cannot be achieved by any numerical modelOnce again solution with 119899 = 0 or 119898 = 0 combined with119908 = 0 (in-plane modes) can be found with 2D numericalmodels without modifying the simply supported boundaryconditions

5 Conclusions

Thepresent paper analyzes the cylindrical bending in the freefrequency analysis of functionally graded material (FGM)plates and cylindrical shells This condition is investigatedby means of a 3D exact shell model and two different 2D

numerical models a weak formulation based on classicalFinite Elements (FEs) and a strong formulation based onthe GDQ method The cylindrical bending (CB) behavior inthe 3D exact shell model can be analyzed using completelysimply supported structures and supposing one of the twohalf-wave numbers equals zeroThe numerical models obtainthe CB behavior using mixed simply supported and freeboundary conditions It has been demonstrated that theactual cylindrical bending (CB) is possible only with respectto straight edgesTheCBnatural frequencies of the numericalsolutions converge to the 3D exact CB ones if the length ofstraight simply supported edges is increased up to 20 times

16 Shock and Vibration

the value of the original edges The CB conditions are notpossible in 2D numericalmodels when the half-wave numberis zero in the curvilinear edge in this case the modificationof boundary conditions does not properly work In-planevibrationmodeswith zero transverse displacement combinedwith 119898 = 0 or 119899 = 0 cannot be defined as CB frequencies andthey can be calculated via 2D numerical models without themodification of the simply supported conditions for the fouredges All the considerations about the cylindrical bendingare valid for each geometry (plate cylinder and cylindricalshell) lamination scheme FGM law and thickness ratio

Competing Interests

The authors declare that they have no competing interests

References

[1] W Q Chen J B Cai and G R Ye ldquoResponses of cross-ply laminates with viscous interfaces in cylindrical bendingrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 823ndash835 2005

[2] W Q Chen and K Y Lee ldquoThree-dimensional exact analysisof angle-ply laminates in cylindrical bending with interfacialdamage via state-space methodrdquo Composite Structures vol 64no 3-4 pp 275ndash283 2004

[3] W Q Chen and K Y Lee ldquoTime-dependent behaviors of angle-ply laminates with viscous interfaces in cylindrical bendingrdquoEuropean Journal ofMechanicsmdashASolids vol 23 no 2 pp 235ndash245 2004

[4] A H Gandhi and H K Raval ldquoAnalytical and empiricalmodeling of top roller position for three-roller cylindricalbending of plates and its experimental verificationrdquo Journal ofMaterials Processing Technology vol 197 no 1ndash3 pp 268ndash2782008

[5] S Oral and H Darendeliler ldquoThe optimum die profile for thecylindrical bending of platesrdquo Journal of Materials ProcessingTechnology vol 70 no 1ndash3 pp 151ndash155 1997

[6] V A Jairazbhoy P Petukhov and J Qu ldquoLarge deflectionof thin plates in cylindrical bendingmdashnon-unique solutionsrdquoInternational Journal of Solids and Structures vol 45 no 11-12pp 3203ndash3218 2008

[7] A Lebee and K Sab ldquoA bending-gradient model for thickplates part II closed-form solutions for cylindrical bending oflaminatesrdquo International Journal of Solids and Structures vol 48no 20 pp 2889ndash2901 2011

[8] P V Nimbolkar and I M Jain ldquoCylindrical bending of elasticplatesrdquo Procedia Materials Science vol 10 pp 793ndash802 2015

[9] A S Sayyad and Y M Ghugal ldquoA nth-order shear deformationtheory for composite laminates in cylindrical bendingrdquo Curvedand Layered Structures vol 2 no 1 pp 290ndash300 2015

[10] N J Pagano ldquoExact solutions for composite laminates incylindrical bendingrdquo Journal of Composite Materials vol 3 no3 pp 398ndash411 1969

[11] N J Pagano and A S D Wang ldquoFurther study of compositelaminates under cylindrical bendingrdquo Journal of CompositeMaterials vol 5 pp 421ndash428 1971

[12] N Saeedi K Sab and J-F Caron ldquoCylindrical bending ofmultilayered plates with multi-delamination via a layerwisestress approachrdquo Composite Structures vol 95 pp 728ndash7392013

[13] X-P Shu and K P Soldatos ldquoCylindrical bending of angle-ply laminates subjected to different sets of edge boundaryconditionsrdquo International Journal of Solids and Structures vol37 no 31 pp 4289ndash4307 2000

[14] Z G Bian W Q Chen C W Lim and N Zhang ldquoAnalyticalsolutions for single- and multi-span functionally graded platesin cylindrical bendingrdquo International Journal of Solids andStructures vol 42 no 24-25 pp 6433ndash6456 2005

[15] H M Navazi and H Haddadpour ldquoNonlinear cylindricalbending analysis of shear deformable functionally graded platesunder different loadings using analytical methodsrdquo Interna-tional Journal of Mechanical Sciences vol 50 no 12 pp 1650ndash1657 2008

[16] W Q Chen J Ying J B Cai and G R Ye ldquoBenchmarksolution of imperfect angle-ply laminated rectangular plates incylindrical bending with surface piezoelectric layers as actuatorand sensorrdquo Computers and Structures vol 82 no 22 pp 1773ndash1784 2004

[17] W Q Chen and K Y Lee ldquoExact solution of angle-plypiezoelectric laminates in cylindrical bending with interfacialimperfectionsrdquo Composite Structures vol 65 no 3-4 pp 329ndash337 2004

[18] T Kant and SM Shiyekar ldquoCylindrical bending of piezoelectriclaminates with a higher order shear and normal deformationtheoryrdquo Computers and Structures vol 86 no 15-16 pp 1594ndash1603 2008

[19] P Lu H P Lee and C Lu ldquoAn exact solution for functionallygraded piezoelectric laminates in cylindrical bendingrdquo Interna-tional Journal of Mechanical Sciences vol 47 no 3 pp 437ndash4382005

[20] W Yan J Wang and W Q Chen ldquoCylindrical bendingresponses of angle-ply piezoelectric laminates with viscoelasticinterfacesrdquo Applied Mathematical Modelling vol 38 no 24 pp6018ndash6030 2014

[21] Y Y Zhou W Q Chen and C F Lu ldquoSemi-analytical solutionfor orthotropic piezoelectric laminates in cylindrical bendingwith interfacial imperfectionsrdquo Composite Structures vol 92no 4 pp 1009ndash1018 2010

[22] G P Dube S Kapuria and P C Dumir ldquoExact piezother-moelastic solution of a simply-supported orthotropic flat panelin cylindrical bendingrdquo International Journal of MechanicalSciences vol 38 pp 374ndash387 1996

[23] V N Pilipchuk V L Berdichevsky and R A IbrahimldquoThermo-mechanical coupling in cylindrical bending of sand-wich platesrdquoComposite Structures vol 92 no 11 pp 2632ndash26402010

[24] C Zhang S Di and N Zhang ldquoA new procedure for staticanalysis of thermo-electric laminated composite plates undercylindrical bendingrdquo Composite Structures vol 56 no 2 pp131ndash140 2002

[25] W Q Chen and K Y Lee ldquoState-space approach for statics anddynamics of angle-ply laminated cylindrical panels in cylindri-cal bendingrdquo International Journal of Mechanical Sciences vol47 no 3 pp 374ndash387 2005

[26] C-P Wu and Y-S Syu ldquoExact solutions of functionally gradedpiezoelectric shells under cylindrical bendingrdquo InternationalJournal of Solids and Structures vol 44 no 20 pp 6450ndash64722007

[27] W Yan J Ying and W Q Chen ldquoThe behavior of angle-ply laminated cylindrical shells with viscoelastic interfaces incylindrical bendingrdquo Composite Structures vol 78 no 4 pp551ndash559 2007

Shock and Vibration 17

[28] S Brischetto F Tornabene N Fantuzzi and E Viola ldquo3Dexact and 2D generalized differential quadraturemodels for freevibration analysis of functionally graded plates and cylindersrdquoMeccanica vol 51 no 9 pp 2059ndash2098 2016

[29] N Fantuzzi S Brischetto F Tornabene and E Viola ldquo2D and3D shell models for the free vibration investigation of func-tionally graded cylindrical and spherical panelsrdquo CompositeStructures vol 154 pp 573ndash590 2016

[30] S Brischetto and R Torre ldquoExact 3D solutions and finiteelement 2D models for free vibration analysis of plates andcylindersrdquo Curved and Layered Structures vol 1 no 1 pp 59ndash92 2014

[31] F Tornabene S Brischetto N Fantuzzi and E Viola ldquoNumer-ical and exact models for free vibration analysis of cylindricaland spherical shell panelsrdquo Composites Part B Engineering vol81 pp 231ndash250 2015

[32] S Brischetto ldquoExact elasticity solution for natural frequenciesof functionally graded simply-supported structuresrdquo ComputerModeling in Engineering amp Sciences vol 95 no 5 pp 391ndash4302013

[33] S Brischetto ldquoThree-dimensional exact free vibration analysisof spherical cylindrical and flat one-layered panelsrdquo Shock andVibration vol 2014 Article ID 479738 29 pages 2014

[34] S Brischetto ldquoA continuum elastic three-dimensional modelfor natural frequencies of single-walled carbon nanotubesrdquoComposites Part B Engineering vol 61 pp 222ndash228 2014

[35] S Brischetto ldquoAn exact 3d solution for free vibrations ofmultilayered cross-ply composite and sandwich plates andshellsrdquo International Journal of Applied Mechanics vol 6 no 6Article ID 1450076 42 pages 2014

[36] S Brischetto ldquoA continuumshellmodel including vanderWaalsinteraction for free vibrations of double-walled carbon nan-otubesrdquoCMES-ComputerModeling in Engineering and Sciencesvol 104 no 4 pp 305ndash327 2015

[37] S Brischetto F Tornabene N Fantuzzi and M BacciocchildquoRefined 2D and exact 3D shell models for the free vibrationanalysis of single- and double-walled carbon nanotubesrdquo Tech-nologies vol 3 no 4 pp 259ndash284 2015

[38] S Brischetto ldquoConvergence analysis of the exponential matrixmethod for the solution of 3D equilibrium equations for freevibration analysis of plates and shellsrdquo Composites Part BEngineering vol 98 pp 453ndash471 2016

[39] S Brischetto ldquoExact and approximate shell geometry in the freevibration analysis of one-layered and multilayered structuresrdquoInternational Journal of Mechanical Sciences vol 113 pp 81ndash932016

[40] S Brischetto ldquoCurvature approximation effects in the freevibration analysis of functionally graded shellsrdquo InternationalJournal of Applied Mechanics 2016

[41] F Tornabene N Fantuzzi F Ubertini and E Viola ldquoStrong for-mulation finite element method based on differential quadra-ture a surveyrdquo Applied Mechanics Reviews vol 67 no 2 ArticleID 020801 pp 1ndash55 2015

[42] F Tornabene E Viola and N Fantuzzi ldquoGeneral higher-orderequivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panelsrdquo CompositeStructures vol 104 pp 94ndash117 2013

[43] N Fantuzzi F Tornabene E Viola and A J Ferreira ldquoAstrong formulation finite element method (SFEM) based onRBF and GDQ techniques for the static and dynamic analysesof laminated plates of arbitrary shaperdquo Meccanica vol 49 no10 pp 2503ndash2542 2014

[44] F Tornabene N Fantuzzi and M Bacciocchi ldquoFree vibrationsof free-form doubly-curved shells made of functionally gradedmaterials using higher-order equivalent single layer theoriesrdquoComposites Part B Engineering vol 67 pp 490ndash509 2014

[45] F Tornabene and E Viola ldquoStatic analysis of functionallygraded doubly-curved shells and panels of revolutionrdquo Mecca-nica vol 48 no 4 pp 901ndash930 2013

[46] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoDynamic analysis of thick and thin elliptic shell structuresmade of laminated composite materialsrdquo Composite Structuresvol 133 pp 278ndash299 2015

[47] F TornabeneN FantuzziM Bacciocchi and EViola ldquoHigher-order theories for the free vibrations of doubly-curved lam-inated panels with curvilinear reinforcing fibers by meansof a local version of the GDQ methodrdquo Composites Part BEngineering vol 81 pp 196ndash230 2015

[48] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoFree vibrations of composite oval and elliptic cylinders bythe generalized differential quadrature methodrdquo Thin-WalledStructures vol 97 pp 114ndash129 2015

[49] F Tornabene and J N Reddy ldquoFGM and laminated doubly-curved and degenerate shells resting on nonlinear elastic foun-dations a GDQ solution for static analysis with a posterioristress and strain recoveryrdquo Journal of the Indian Institute ofScience vol 93 no 4 pp 635ndash688 2013

[50] N Fantuzzi F Tornabene and E Viola ldquoFour-parameter func-tionally graded cracked plates of arbitrary shape a GDQFEMsolution for free vibrationsrdquo Mechanics of Advanced Materialsand Structures vol 23 no 1 pp 89ndash107 2016

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

14 Shock and Vibration

Table 2 First frequency 119891 in Hz for the 119886ℎ = 100 one-layered FGM plate with 119901 = 05 convergence of 2D GDQ and 2D FE solutionsincreasing the length of the simply supported sides

119871119886 1 2 3 4 5 6 8 10 15 20 3D exact2D GDQ 3921 3964 3980 3988 3993 3997 4001 4004 4007 4008 40192D FEM 3907 3949 3966 3974 3979 3983 3987 3989 3993 3994 4019

Table 3 One-layered FGM cylinder with 119901 = 20 Analytical 3D versus 2D numerical models Frequency 119891 in Hz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1004 0 I 0896 mdash 0897 mdash 08836 0 I 2532 mdash 2538 mdash 25118 0 I 4854 mdash 4867 mdash 481910 0 I 7848 mdash 7871 mdash 780112 0 I 1151 mdash 1155 mdash 114610 1 I 1356 1360 mdash 1357 mdash12 1 I 1441 1449 mdash 1445 mdash14 0 I 1584 mdash 1589 mdash 15778 1 I 1638 1640 mdash 1636 mdash14 1 I 1756 1767 mdash 1763 mdash16 0 I 2083 mdash 2090 mdash 207016 1 I 2207 2222 mdash 2216 mdash6 1 I 2441 2442 mdash 2438 mdash18 0 I 2649 mdash 2658 mdash 264014 2 I 2742 2752 mdash 2746 mdash18 1 I 2751 2772 mdash 2763 mdash16 2 I 2875 2890 mdash 2883 mdash12 2 I 2921 2926 mdash 2922 mdash119877120572ℎ = 104 0 I 8774 mdash 9085 mdash 86796 0 I 2456 mdash 2571 mdash 24306 1 I 3978 4089 mdash 3979 mdash4 1 I 4388 4357 mdash 4388 mdash8 0 I 4654 mdash 4920 mdash 46028 1 I 5575 5905 mdash 5576 mdash2 1 I 7202 7101 mdash 7202 mdash2 0 I (119908 = 0) 7322 7390 mdash 7319 mdash10 0 I 7419 mdash 7933 mdash 73476 2 I 8110 8261 mdash 8111 mdash10 1 I 8189 8811 mdash 8191 mdash8 2 I 8783 9173 mdash 8785 mdash4 2 I 9016 9084 mdash 9017 mdash12 0 I 1071 mdash 1159 mdash 106110 2 I 1081 1155 mdash 1081 mdash2 2 I 1089 1105 mdash 1089 mdash12 1 I 1142 1245 mdash 1143 mdash4 3 I 1263 1286 mdash 1263 mdash

same meaning of the columns seen in the previous casesIn order to achieve the cylindrical bending conditions thenumerical models have the straight edges simply supportedand the curved edges as free Due to the CB definitionit is not possible to achieve CB conditions when 119898 = 0

and 119899 = 1 2 3 because the cylindrical bending shouldoccur on the curved edge and this is not physically possiblebecause of the variation of displacement components due tothe curvature presence This aspect is indicated as NA (NotApplicable) solution in Table 4 Therefore this kind of 3D

Shock and Vibration 15

Table 4 Sandwich cylindrical shell panel embedding FGM core with 119901 = 10 Analytical 3D versus 2D numerical models Frequency 119891 inHz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1001 0 I 1948 mdash 1951 mdash 19412 0 I 8856 mdash 8872 mdash 88232 1 I 1104 1110 mdash 1104 mdash1 1 I 1850 1873 mdash 1850 mdash3 0 I 2038 mdash 2042 mdash 20303 1 I 2116 2128 mdash 2116 mdash2 2 I 2221 2242 mdash 2221 mdash3 2 I 2484 2496 mdash 2484 mdash3 3 I 3235 3254 mdash 3235 mdash4 0 I 3649 mdash 3660 mdash 36342 3 I 3692 3739 mdash 3692 mdash4 1 I 3714 3745 mdash 3714 mdash4 2 I 3940 3965 mdash 3941 mdash3 4 I 4226 4261 mdash 4226 mdash4 3 I 4378 4403 mdash 4378 mdash1 2 I 4568 4664 mdash 4568 mdash4 4 I 5035 5064 mdash 5035 mdash2 4 I 5040 5128 mdash 5040 mdash119877120572ℎ = 101 0 I 1905 mdash 1963 mdash 18841 1 I 3054 3145 mdash 3055 mdash1 2 I 6209 6409 mdash 6209 mdash2 0 I 8326 mdash 8857 mdash 82540 1 I 8710 mdash NA mdash NA0 1 II (119908 = 0) 8820 8956 mdash 8820 mdash2 1 I 8880 9531 mdash 8882 mdash0 2 I 9176 mdash NA mdash NA1 3 I 9576 9983 mdash 9567 mdash0 3 I 1036 mdash NA mdash NA2 2 I 1066 1147 mdash 1067 mdash0 4 I 1291 mdash NA mdash NA1 4 I 1335 1413 mdash 1335 mdash2 3 I 1364 1475 mdash 1364 mdash1 0 II (119908 = 0) 1665 1738 mdash 1665 mdash0 5 I 1683 mdash NA mdash NA2 4 I 1755 1918 mdash 1755 mdash0 2 II (119908 = 0) 1764 1791 mdash 1764 mdash

exact solutions cannot be achieved by any numerical modelOnce again solution with 119899 = 0 or 119898 = 0 combined with119908 = 0 (in-plane modes) can be found with 2D numericalmodels without modifying the simply supported boundaryconditions

5 Conclusions

Thepresent paper analyzes the cylindrical bending in the freefrequency analysis of functionally graded material (FGM)plates and cylindrical shells This condition is investigatedby means of a 3D exact shell model and two different 2D

numerical models a weak formulation based on classicalFinite Elements (FEs) and a strong formulation based onthe GDQ method The cylindrical bending (CB) behavior inthe 3D exact shell model can be analyzed using completelysimply supported structures and supposing one of the twohalf-wave numbers equals zeroThe numerical models obtainthe CB behavior using mixed simply supported and freeboundary conditions It has been demonstrated that theactual cylindrical bending (CB) is possible only with respectto straight edgesTheCBnatural frequencies of the numericalsolutions converge to the 3D exact CB ones if the length ofstraight simply supported edges is increased up to 20 times

16 Shock and Vibration

the value of the original edges The CB conditions are notpossible in 2D numericalmodels when the half-wave numberis zero in the curvilinear edge in this case the modificationof boundary conditions does not properly work In-planevibrationmodeswith zero transverse displacement combinedwith 119898 = 0 or 119899 = 0 cannot be defined as CB frequencies andthey can be calculated via 2D numerical models without themodification of the simply supported conditions for the fouredges All the considerations about the cylindrical bendingare valid for each geometry (plate cylinder and cylindricalshell) lamination scheme FGM law and thickness ratio

Competing Interests

The authors declare that they have no competing interests

References

[1] W Q Chen J B Cai and G R Ye ldquoResponses of cross-ply laminates with viscous interfaces in cylindrical bendingrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 823ndash835 2005

[2] W Q Chen and K Y Lee ldquoThree-dimensional exact analysisof angle-ply laminates in cylindrical bending with interfacialdamage via state-space methodrdquo Composite Structures vol 64no 3-4 pp 275ndash283 2004

[3] W Q Chen and K Y Lee ldquoTime-dependent behaviors of angle-ply laminates with viscous interfaces in cylindrical bendingrdquoEuropean Journal ofMechanicsmdashASolids vol 23 no 2 pp 235ndash245 2004

[4] A H Gandhi and H K Raval ldquoAnalytical and empiricalmodeling of top roller position for three-roller cylindricalbending of plates and its experimental verificationrdquo Journal ofMaterials Processing Technology vol 197 no 1ndash3 pp 268ndash2782008

[5] S Oral and H Darendeliler ldquoThe optimum die profile for thecylindrical bending of platesrdquo Journal of Materials ProcessingTechnology vol 70 no 1ndash3 pp 151ndash155 1997

[6] V A Jairazbhoy P Petukhov and J Qu ldquoLarge deflectionof thin plates in cylindrical bendingmdashnon-unique solutionsrdquoInternational Journal of Solids and Structures vol 45 no 11-12pp 3203ndash3218 2008

[7] A Lebee and K Sab ldquoA bending-gradient model for thickplates part II closed-form solutions for cylindrical bending oflaminatesrdquo International Journal of Solids and Structures vol 48no 20 pp 2889ndash2901 2011

[8] P V Nimbolkar and I M Jain ldquoCylindrical bending of elasticplatesrdquo Procedia Materials Science vol 10 pp 793ndash802 2015

[9] A S Sayyad and Y M Ghugal ldquoA nth-order shear deformationtheory for composite laminates in cylindrical bendingrdquo Curvedand Layered Structures vol 2 no 1 pp 290ndash300 2015

[10] N J Pagano ldquoExact solutions for composite laminates incylindrical bendingrdquo Journal of Composite Materials vol 3 no3 pp 398ndash411 1969

[11] N J Pagano and A S D Wang ldquoFurther study of compositelaminates under cylindrical bendingrdquo Journal of CompositeMaterials vol 5 pp 421ndash428 1971

[12] N Saeedi K Sab and J-F Caron ldquoCylindrical bending ofmultilayered plates with multi-delamination via a layerwisestress approachrdquo Composite Structures vol 95 pp 728ndash7392013

[13] X-P Shu and K P Soldatos ldquoCylindrical bending of angle-ply laminates subjected to different sets of edge boundaryconditionsrdquo International Journal of Solids and Structures vol37 no 31 pp 4289ndash4307 2000

[14] Z G Bian W Q Chen C W Lim and N Zhang ldquoAnalyticalsolutions for single- and multi-span functionally graded platesin cylindrical bendingrdquo International Journal of Solids andStructures vol 42 no 24-25 pp 6433ndash6456 2005

[15] H M Navazi and H Haddadpour ldquoNonlinear cylindricalbending analysis of shear deformable functionally graded platesunder different loadings using analytical methodsrdquo Interna-tional Journal of Mechanical Sciences vol 50 no 12 pp 1650ndash1657 2008

[16] W Q Chen J Ying J B Cai and G R Ye ldquoBenchmarksolution of imperfect angle-ply laminated rectangular plates incylindrical bending with surface piezoelectric layers as actuatorand sensorrdquo Computers and Structures vol 82 no 22 pp 1773ndash1784 2004

[17] W Q Chen and K Y Lee ldquoExact solution of angle-plypiezoelectric laminates in cylindrical bending with interfacialimperfectionsrdquo Composite Structures vol 65 no 3-4 pp 329ndash337 2004

[18] T Kant and SM Shiyekar ldquoCylindrical bending of piezoelectriclaminates with a higher order shear and normal deformationtheoryrdquo Computers and Structures vol 86 no 15-16 pp 1594ndash1603 2008

[19] P Lu H P Lee and C Lu ldquoAn exact solution for functionallygraded piezoelectric laminates in cylindrical bendingrdquo Interna-tional Journal of Mechanical Sciences vol 47 no 3 pp 437ndash4382005

[20] W Yan J Wang and W Q Chen ldquoCylindrical bendingresponses of angle-ply piezoelectric laminates with viscoelasticinterfacesrdquo Applied Mathematical Modelling vol 38 no 24 pp6018ndash6030 2014

[21] Y Y Zhou W Q Chen and C F Lu ldquoSemi-analytical solutionfor orthotropic piezoelectric laminates in cylindrical bendingwith interfacial imperfectionsrdquo Composite Structures vol 92no 4 pp 1009ndash1018 2010

[22] G P Dube S Kapuria and P C Dumir ldquoExact piezother-moelastic solution of a simply-supported orthotropic flat panelin cylindrical bendingrdquo International Journal of MechanicalSciences vol 38 pp 374ndash387 1996

[23] V N Pilipchuk V L Berdichevsky and R A IbrahimldquoThermo-mechanical coupling in cylindrical bending of sand-wich platesrdquoComposite Structures vol 92 no 11 pp 2632ndash26402010

[24] C Zhang S Di and N Zhang ldquoA new procedure for staticanalysis of thermo-electric laminated composite plates undercylindrical bendingrdquo Composite Structures vol 56 no 2 pp131ndash140 2002

[25] W Q Chen and K Y Lee ldquoState-space approach for statics anddynamics of angle-ply laminated cylindrical panels in cylindri-cal bendingrdquo International Journal of Mechanical Sciences vol47 no 3 pp 374ndash387 2005

[26] C-P Wu and Y-S Syu ldquoExact solutions of functionally gradedpiezoelectric shells under cylindrical bendingrdquo InternationalJournal of Solids and Structures vol 44 no 20 pp 6450ndash64722007

[27] W Yan J Ying and W Q Chen ldquoThe behavior of angle-ply laminated cylindrical shells with viscoelastic interfaces incylindrical bendingrdquo Composite Structures vol 78 no 4 pp551ndash559 2007

Shock and Vibration 17

[28] S Brischetto F Tornabene N Fantuzzi and E Viola ldquo3Dexact and 2D generalized differential quadraturemodels for freevibration analysis of functionally graded plates and cylindersrdquoMeccanica vol 51 no 9 pp 2059ndash2098 2016

[29] N Fantuzzi S Brischetto F Tornabene and E Viola ldquo2D and3D shell models for the free vibration investigation of func-tionally graded cylindrical and spherical panelsrdquo CompositeStructures vol 154 pp 573ndash590 2016

[30] S Brischetto and R Torre ldquoExact 3D solutions and finiteelement 2D models for free vibration analysis of plates andcylindersrdquo Curved and Layered Structures vol 1 no 1 pp 59ndash92 2014

[31] F Tornabene S Brischetto N Fantuzzi and E Viola ldquoNumer-ical and exact models for free vibration analysis of cylindricaland spherical shell panelsrdquo Composites Part B Engineering vol81 pp 231ndash250 2015

[32] S Brischetto ldquoExact elasticity solution for natural frequenciesof functionally graded simply-supported structuresrdquo ComputerModeling in Engineering amp Sciences vol 95 no 5 pp 391ndash4302013

[33] S Brischetto ldquoThree-dimensional exact free vibration analysisof spherical cylindrical and flat one-layered panelsrdquo Shock andVibration vol 2014 Article ID 479738 29 pages 2014

[34] S Brischetto ldquoA continuum elastic three-dimensional modelfor natural frequencies of single-walled carbon nanotubesrdquoComposites Part B Engineering vol 61 pp 222ndash228 2014

[35] S Brischetto ldquoAn exact 3d solution for free vibrations ofmultilayered cross-ply composite and sandwich plates andshellsrdquo International Journal of Applied Mechanics vol 6 no 6Article ID 1450076 42 pages 2014

[36] S Brischetto ldquoA continuumshellmodel including vanderWaalsinteraction for free vibrations of double-walled carbon nan-otubesrdquoCMES-ComputerModeling in Engineering and Sciencesvol 104 no 4 pp 305ndash327 2015

[37] S Brischetto F Tornabene N Fantuzzi and M BacciocchildquoRefined 2D and exact 3D shell models for the free vibrationanalysis of single- and double-walled carbon nanotubesrdquo Tech-nologies vol 3 no 4 pp 259ndash284 2015

[38] S Brischetto ldquoConvergence analysis of the exponential matrixmethod for the solution of 3D equilibrium equations for freevibration analysis of plates and shellsrdquo Composites Part BEngineering vol 98 pp 453ndash471 2016

[39] S Brischetto ldquoExact and approximate shell geometry in the freevibration analysis of one-layered and multilayered structuresrdquoInternational Journal of Mechanical Sciences vol 113 pp 81ndash932016

[40] S Brischetto ldquoCurvature approximation effects in the freevibration analysis of functionally graded shellsrdquo InternationalJournal of Applied Mechanics 2016

[41] F Tornabene N Fantuzzi F Ubertini and E Viola ldquoStrong for-mulation finite element method based on differential quadra-ture a surveyrdquo Applied Mechanics Reviews vol 67 no 2 ArticleID 020801 pp 1ndash55 2015

[42] F Tornabene E Viola and N Fantuzzi ldquoGeneral higher-orderequivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panelsrdquo CompositeStructures vol 104 pp 94ndash117 2013

[43] N Fantuzzi F Tornabene E Viola and A J Ferreira ldquoAstrong formulation finite element method (SFEM) based onRBF and GDQ techniques for the static and dynamic analysesof laminated plates of arbitrary shaperdquo Meccanica vol 49 no10 pp 2503ndash2542 2014

[44] F Tornabene N Fantuzzi and M Bacciocchi ldquoFree vibrationsof free-form doubly-curved shells made of functionally gradedmaterials using higher-order equivalent single layer theoriesrdquoComposites Part B Engineering vol 67 pp 490ndash509 2014

[45] F Tornabene and E Viola ldquoStatic analysis of functionallygraded doubly-curved shells and panels of revolutionrdquo Mecca-nica vol 48 no 4 pp 901ndash930 2013

[46] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoDynamic analysis of thick and thin elliptic shell structuresmade of laminated composite materialsrdquo Composite Structuresvol 133 pp 278ndash299 2015

[47] F TornabeneN FantuzziM Bacciocchi and EViola ldquoHigher-order theories for the free vibrations of doubly-curved lam-inated panels with curvilinear reinforcing fibers by meansof a local version of the GDQ methodrdquo Composites Part BEngineering vol 81 pp 196ndash230 2015

[48] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoFree vibrations of composite oval and elliptic cylinders bythe generalized differential quadrature methodrdquo Thin-WalledStructures vol 97 pp 114ndash129 2015

[49] F Tornabene and J N Reddy ldquoFGM and laminated doubly-curved and degenerate shells resting on nonlinear elastic foun-dations a GDQ solution for static analysis with a posterioristress and strain recoveryrdquo Journal of the Indian Institute ofScience vol 93 no 4 pp 635ndash688 2013

[50] N Fantuzzi F Tornabene and E Viola ldquoFour-parameter func-tionally graded cracked plates of arbitrary shape a GDQFEMsolution for free vibrationsrdquo Mechanics of Advanced Materialsand Structures vol 23 no 1 pp 89ndash107 2016

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 15

Table 4 Sandwich cylindrical shell panel embedding FGM core with 119901 = 10 Analytical 3D versus 2D numerical models Frequency 119891 inHz

119898 119899 Mode 3D exact 2D FE 2D FE (CB) 2D GDQ 2D GDQ (CB)119877120572ℎ = 1001 0 I 1948 mdash 1951 mdash 19412 0 I 8856 mdash 8872 mdash 88232 1 I 1104 1110 mdash 1104 mdash1 1 I 1850 1873 mdash 1850 mdash3 0 I 2038 mdash 2042 mdash 20303 1 I 2116 2128 mdash 2116 mdash2 2 I 2221 2242 mdash 2221 mdash3 2 I 2484 2496 mdash 2484 mdash3 3 I 3235 3254 mdash 3235 mdash4 0 I 3649 mdash 3660 mdash 36342 3 I 3692 3739 mdash 3692 mdash4 1 I 3714 3745 mdash 3714 mdash4 2 I 3940 3965 mdash 3941 mdash3 4 I 4226 4261 mdash 4226 mdash4 3 I 4378 4403 mdash 4378 mdash1 2 I 4568 4664 mdash 4568 mdash4 4 I 5035 5064 mdash 5035 mdash2 4 I 5040 5128 mdash 5040 mdash119877120572ℎ = 101 0 I 1905 mdash 1963 mdash 18841 1 I 3054 3145 mdash 3055 mdash1 2 I 6209 6409 mdash 6209 mdash2 0 I 8326 mdash 8857 mdash 82540 1 I 8710 mdash NA mdash NA0 1 II (119908 = 0) 8820 8956 mdash 8820 mdash2 1 I 8880 9531 mdash 8882 mdash0 2 I 9176 mdash NA mdash NA1 3 I 9576 9983 mdash 9567 mdash0 3 I 1036 mdash NA mdash NA2 2 I 1066 1147 mdash 1067 mdash0 4 I 1291 mdash NA mdash NA1 4 I 1335 1413 mdash 1335 mdash2 3 I 1364 1475 mdash 1364 mdash1 0 II (119908 = 0) 1665 1738 mdash 1665 mdash0 5 I 1683 mdash NA mdash NA2 4 I 1755 1918 mdash 1755 mdash0 2 II (119908 = 0) 1764 1791 mdash 1764 mdash

exact solutions cannot be achieved by any numerical modelOnce again solution with 119899 = 0 or 119898 = 0 combined with119908 = 0 (in-plane modes) can be found with 2D numericalmodels without modifying the simply supported boundaryconditions

5 Conclusions

Thepresent paper analyzes the cylindrical bending in the freefrequency analysis of functionally graded material (FGM)plates and cylindrical shells This condition is investigatedby means of a 3D exact shell model and two different 2D

numerical models a weak formulation based on classicalFinite Elements (FEs) and a strong formulation based onthe GDQ method The cylindrical bending (CB) behavior inthe 3D exact shell model can be analyzed using completelysimply supported structures and supposing one of the twohalf-wave numbers equals zeroThe numerical models obtainthe CB behavior using mixed simply supported and freeboundary conditions It has been demonstrated that theactual cylindrical bending (CB) is possible only with respectto straight edgesTheCBnatural frequencies of the numericalsolutions converge to the 3D exact CB ones if the length ofstraight simply supported edges is increased up to 20 times

16 Shock and Vibration

the value of the original edges The CB conditions are notpossible in 2D numericalmodels when the half-wave numberis zero in the curvilinear edge in this case the modificationof boundary conditions does not properly work In-planevibrationmodeswith zero transverse displacement combinedwith 119898 = 0 or 119899 = 0 cannot be defined as CB frequencies andthey can be calculated via 2D numerical models without themodification of the simply supported conditions for the fouredges All the considerations about the cylindrical bendingare valid for each geometry (plate cylinder and cylindricalshell) lamination scheme FGM law and thickness ratio

Competing Interests

The authors declare that they have no competing interests

References

[1] W Q Chen J B Cai and G R Ye ldquoResponses of cross-ply laminates with viscous interfaces in cylindrical bendingrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 823ndash835 2005

[2] W Q Chen and K Y Lee ldquoThree-dimensional exact analysisof angle-ply laminates in cylindrical bending with interfacialdamage via state-space methodrdquo Composite Structures vol 64no 3-4 pp 275ndash283 2004

[3] W Q Chen and K Y Lee ldquoTime-dependent behaviors of angle-ply laminates with viscous interfaces in cylindrical bendingrdquoEuropean Journal ofMechanicsmdashASolids vol 23 no 2 pp 235ndash245 2004

[4] A H Gandhi and H K Raval ldquoAnalytical and empiricalmodeling of top roller position for three-roller cylindricalbending of plates and its experimental verificationrdquo Journal ofMaterials Processing Technology vol 197 no 1ndash3 pp 268ndash2782008

[5] S Oral and H Darendeliler ldquoThe optimum die profile for thecylindrical bending of platesrdquo Journal of Materials ProcessingTechnology vol 70 no 1ndash3 pp 151ndash155 1997

[6] V A Jairazbhoy P Petukhov and J Qu ldquoLarge deflectionof thin plates in cylindrical bendingmdashnon-unique solutionsrdquoInternational Journal of Solids and Structures vol 45 no 11-12pp 3203ndash3218 2008

[7] A Lebee and K Sab ldquoA bending-gradient model for thickplates part II closed-form solutions for cylindrical bending oflaminatesrdquo International Journal of Solids and Structures vol 48no 20 pp 2889ndash2901 2011

[8] P V Nimbolkar and I M Jain ldquoCylindrical bending of elasticplatesrdquo Procedia Materials Science vol 10 pp 793ndash802 2015

[9] A S Sayyad and Y M Ghugal ldquoA nth-order shear deformationtheory for composite laminates in cylindrical bendingrdquo Curvedand Layered Structures vol 2 no 1 pp 290ndash300 2015

[10] N J Pagano ldquoExact solutions for composite laminates incylindrical bendingrdquo Journal of Composite Materials vol 3 no3 pp 398ndash411 1969

[11] N J Pagano and A S D Wang ldquoFurther study of compositelaminates under cylindrical bendingrdquo Journal of CompositeMaterials vol 5 pp 421ndash428 1971

[12] N Saeedi K Sab and J-F Caron ldquoCylindrical bending ofmultilayered plates with multi-delamination via a layerwisestress approachrdquo Composite Structures vol 95 pp 728ndash7392013

[13] X-P Shu and K P Soldatos ldquoCylindrical bending of angle-ply laminates subjected to different sets of edge boundaryconditionsrdquo International Journal of Solids and Structures vol37 no 31 pp 4289ndash4307 2000

[14] Z G Bian W Q Chen C W Lim and N Zhang ldquoAnalyticalsolutions for single- and multi-span functionally graded platesin cylindrical bendingrdquo International Journal of Solids andStructures vol 42 no 24-25 pp 6433ndash6456 2005

[15] H M Navazi and H Haddadpour ldquoNonlinear cylindricalbending analysis of shear deformable functionally graded platesunder different loadings using analytical methodsrdquo Interna-tional Journal of Mechanical Sciences vol 50 no 12 pp 1650ndash1657 2008

[16] W Q Chen J Ying J B Cai and G R Ye ldquoBenchmarksolution of imperfect angle-ply laminated rectangular plates incylindrical bending with surface piezoelectric layers as actuatorand sensorrdquo Computers and Structures vol 82 no 22 pp 1773ndash1784 2004

[17] W Q Chen and K Y Lee ldquoExact solution of angle-plypiezoelectric laminates in cylindrical bending with interfacialimperfectionsrdquo Composite Structures vol 65 no 3-4 pp 329ndash337 2004

[18] T Kant and SM Shiyekar ldquoCylindrical bending of piezoelectriclaminates with a higher order shear and normal deformationtheoryrdquo Computers and Structures vol 86 no 15-16 pp 1594ndash1603 2008

[19] P Lu H P Lee and C Lu ldquoAn exact solution for functionallygraded piezoelectric laminates in cylindrical bendingrdquo Interna-tional Journal of Mechanical Sciences vol 47 no 3 pp 437ndash4382005

[20] W Yan J Wang and W Q Chen ldquoCylindrical bendingresponses of angle-ply piezoelectric laminates with viscoelasticinterfacesrdquo Applied Mathematical Modelling vol 38 no 24 pp6018ndash6030 2014

[21] Y Y Zhou W Q Chen and C F Lu ldquoSemi-analytical solutionfor orthotropic piezoelectric laminates in cylindrical bendingwith interfacial imperfectionsrdquo Composite Structures vol 92no 4 pp 1009ndash1018 2010

[22] G P Dube S Kapuria and P C Dumir ldquoExact piezother-moelastic solution of a simply-supported orthotropic flat panelin cylindrical bendingrdquo International Journal of MechanicalSciences vol 38 pp 374ndash387 1996

[23] V N Pilipchuk V L Berdichevsky and R A IbrahimldquoThermo-mechanical coupling in cylindrical bending of sand-wich platesrdquoComposite Structures vol 92 no 11 pp 2632ndash26402010

[24] C Zhang S Di and N Zhang ldquoA new procedure for staticanalysis of thermo-electric laminated composite plates undercylindrical bendingrdquo Composite Structures vol 56 no 2 pp131ndash140 2002

[25] W Q Chen and K Y Lee ldquoState-space approach for statics anddynamics of angle-ply laminated cylindrical panels in cylindri-cal bendingrdquo International Journal of Mechanical Sciences vol47 no 3 pp 374ndash387 2005

[26] C-P Wu and Y-S Syu ldquoExact solutions of functionally gradedpiezoelectric shells under cylindrical bendingrdquo InternationalJournal of Solids and Structures vol 44 no 20 pp 6450ndash64722007

[27] W Yan J Ying and W Q Chen ldquoThe behavior of angle-ply laminated cylindrical shells with viscoelastic interfaces incylindrical bendingrdquo Composite Structures vol 78 no 4 pp551ndash559 2007

Shock and Vibration 17

[28] S Brischetto F Tornabene N Fantuzzi and E Viola ldquo3Dexact and 2D generalized differential quadraturemodels for freevibration analysis of functionally graded plates and cylindersrdquoMeccanica vol 51 no 9 pp 2059ndash2098 2016

[29] N Fantuzzi S Brischetto F Tornabene and E Viola ldquo2D and3D shell models for the free vibration investigation of func-tionally graded cylindrical and spherical panelsrdquo CompositeStructures vol 154 pp 573ndash590 2016

[30] S Brischetto and R Torre ldquoExact 3D solutions and finiteelement 2D models for free vibration analysis of plates andcylindersrdquo Curved and Layered Structures vol 1 no 1 pp 59ndash92 2014

[31] F Tornabene S Brischetto N Fantuzzi and E Viola ldquoNumer-ical and exact models for free vibration analysis of cylindricaland spherical shell panelsrdquo Composites Part B Engineering vol81 pp 231ndash250 2015

[32] S Brischetto ldquoExact elasticity solution for natural frequenciesof functionally graded simply-supported structuresrdquo ComputerModeling in Engineering amp Sciences vol 95 no 5 pp 391ndash4302013

[33] S Brischetto ldquoThree-dimensional exact free vibration analysisof spherical cylindrical and flat one-layered panelsrdquo Shock andVibration vol 2014 Article ID 479738 29 pages 2014

[34] S Brischetto ldquoA continuum elastic three-dimensional modelfor natural frequencies of single-walled carbon nanotubesrdquoComposites Part B Engineering vol 61 pp 222ndash228 2014

[35] S Brischetto ldquoAn exact 3d solution for free vibrations ofmultilayered cross-ply composite and sandwich plates andshellsrdquo International Journal of Applied Mechanics vol 6 no 6Article ID 1450076 42 pages 2014

[36] S Brischetto ldquoA continuumshellmodel including vanderWaalsinteraction for free vibrations of double-walled carbon nan-otubesrdquoCMES-ComputerModeling in Engineering and Sciencesvol 104 no 4 pp 305ndash327 2015

[37] S Brischetto F Tornabene N Fantuzzi and M BacciocchildquoRefined 2D and exact 3D shell models for the free vibrationanalysis of single- and double-walled carbon nanotubesrdquo Tech-nologies vol 3 no 4 pp 259ndash284 2015

[38] S Brischetto ldquoConvergence analysis of the exponential matrixmethod for the solution of 3D equilibrium equations for freevibration analysis of plates and shellsrdquo Composites Part BEngineering vol 98 pp 453ndash471 2016

[39] S Brischetto ldquoExact and approximate shell geometry in the freevibration analysis of one-layered and multilayered structuresrdquoInternational Journal of Mechanical Sciences vol 113 pp 81ndash932016

[40] S Brischetto ldquoCurvature approximation effects in the freevibration analysis of functionally graded shellsrdquo InternationalJournal of Applied Mechanics 2016

[41] F Tornabene N Fantuzzi F Ubertini and E Viola ldquoStrong for-mulation finite element method based on differential quadra-ture a surveyrdquo Applied Mechanics Reviews vol 67 no 2 ArticleID 020801 pp 1ndash55 2015

[42] F Tornabene E Viola and N Fantuzzi ldquoGeneral higher-orderequivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panelsrdquo CompositeStructures vol 104 pp 94ndash117 2013

[43] N Fantuzzi F Tornabene E Viola and A J Ferreira ldquoAstrong formulation finite element method (SFEM) based onRBF and GDQ techniques for the static and dynamic analysesof laminated plates of arbitrary shaperdquo Meccanica vol 49 no10 pp 2503ndash2542 2014

[44] F Tornabene N Fantuzzi and M Bacciocchi ldquoFree vibrationsof free-form doubly-curved shells made of functionally gradedmaterials using higher-order equivalent single layer theoriesrdquoComposites Part B Engineering vol 67 pp 490ndash509 2014

[45] F Tornabene and E Viola ldquoStatic analysis of functionallygraded doubly-curved shells and panels of revolutionrdquo Mecca-nica vol 48 no 4 pp 901ndash930 2013

[46] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoDynamic analysis of thick and thin elliptic shell structuresmade of laminated composite materialsrdquo Composite Structuresvol 133 pp 278ndash299 2015

[47] F TornabeneN FantuzziM Bacciocchi and EViola ldquoHigher-order theories for the free vibrations of doubly-curved lam-inated panels with curvilinear reinforcing fibers by meansof a local version of the GDQ methodrdquo Composites Part BEngineering vol 81 pp 196ndash230 2015

[48] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoFree vibrations of composite oval and elliptic cylinders bythe generalized differential quadrature methodrdquo Thin-WalledStructures vol 97 pp 114ndash129 2015

[49] F Tornabene and J N Reddy ldquoFGM and laminated doubly-curved and degenerate shells resting on nonlinear elastic foun-dations a GDQ solution for static analysis with a posterioristress and strain recoveryrdquo Journal of the Indian Institute ofScience vol 93 no 4 pp 635ndash688 2013

[50] N Fantuzzi F Tornabene and E Viola ldquoFour-parameter func-tionally graded cracked plates of arbitrary shape a GDQFEMsolution for free vibrationsrdquo Mechanics of Advanced Materialsand Structures vol 23 no 1 pp 89ndash107 2016

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

16 Shock and Vibration

the value of the original edges The CB conditions are notpossible in 2D numericalmodels when the half-wave numberis zero in the curvilinear edge in this case the modificationof boundary conditions does not properly work In-planevibrationmodeswith zero transverse displacement combinedwith 119898 = 0 or 119899 = 0 cannot be defined as CB frequencies andthey can be calculated via 2D numerical models without themodification of the simply supported conditions for the fouredges All the considerations about the cylindrical bendingare valid for each geometry (plate cylinder and cylindricalshell) lamination scheme FGM law and thickness ratio

Competing Interests

The authors declare that they have no competing interests

References

[1] W Q Chen J B Cai and G R Ye ldquoResponses of cross-ply laminates with viscous interfaces in cylindrical bendingrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 823ndash835 2005

[2] W Q Chen and K Y Lee ldquoThree-dimensional exact analysisof angle-ply laminates in cylindrical bending with interfacialdamage via state-space methodrdquo Composite Structures vol 64no 3-4 pp 275ndash283 2004

[3] W Q Chen and K Y Lee ldquoTime-dependent behaviors of angle-ply laminates with viscous interfaces in cylindrical bendingrdquoEuropean Journal ofMechanicsmdashASolids vol 23 no 2 pp 235ndash245 2004

[4] A H Gandhi and H K Raval ldquoAnalytical and empiricalmodeling of top roller position for three-roller cylindricalbending of plates and its experimental verificationrdquo Journal ofMaterials Processing Technology vol 197 no 1ndash3 pp 268ndash2782008

[5] S Oral and H Darendeliler ldquoThe optimum die profile for thecylindrical bending of platesrdquo Journal of Materials ProcessingTechnology vol 70 no 1ndash3 pp 151ndash155 1997

[6] V A Jairazbhoy P Petukhov and J Qu ldquoLarge deflectionof thin plates in cylindrical bendingmdashnon-unique solutionsrdquoInternational Journal of Solids and Structures vol 45 no 11-12pp 3203ndash3218 2008

[7] A Lebee and K Sab ldquoA bending-gradient model for thickplates part II closed-form solutions for cylindrical bending oflaminatesrdquo International Journal of Solids and Structures vol 48no 20 pp 2889ndash2901 2011

[8] P V Nimbolkar and I M Jain ldquoCylindrical bending of elasticplatesrdquo Procedia Materials Science vol 10 pp 793ndash802 2015

[9] A S Sayyad and Y M Ghugal ldquoA nth-order shear deformationtheory for composite laminates in cylindrical bendingrdquo Curvedand Layered Structures vol 2 no 1 pp 290ndash300 2015

[10] N J Pagano ldquoExact solutions for composite laminates incylindrical bendingrdquo Journal of Composite Materials vol 3 no3 pp 398ndash411 1969

[11] N J Pagano and A S D Wang ldquoFurther study of compositelaminates under cylindrical bendingrdquo Journal of CompositeMaterials vol 5 pp 421ndash428 1971

[12] N Saeedi K Sab and J-F Caron ldquoCylindrical bending ofmultilayered plates with multi-delamination via a layerwisestress approachrdquo Composite Structures vol 95 pp 728ndash7392013

[13] X-P Shu and K P Soldatos ldquoCylindrical bending of angle-ply laminates subjected to different sets of edge boundaryconditionsrdquo International Journal of Solids and Structures vol37 no 31 pp 4289ndash4307 2000

[14] Z G Bian W Q Chen C W Lim and N Zhang ldquoAnalyticalsolutions for single- and multi-span functionally graded platesin cylindrical bendingrdquo International Journal of Solids andStructures vol 42 no 24-25 pp 6433ndash6456 2005

[15] H M Navazi and H Haddadpour ldquoNonlinear cylindricalbending analysis of shear deformable functionally graded platesunder different loadings using analytical methodsrdquo Interna-tional Journal of Mechanical Sciences vol 50 no 12 pp 1650ndash1657 2008

[16] W Q Chen J Ying J B Cai and G R Ye ldquoBenchmarksolution of imperfect angle-ply laminated rectangular plates incylindrical bending with surface piezoelectric layers as actuatorand sensorrdquo Computers and Structures vol 82 no 22 pp 1773ndash1784 2004

[17] W Q Chen and K Y Lee ldquoExact solution of angle-plypiezoelectric laminates in cylindrical bending with interfacialimperfectionsrdquo Composite Structures vol 65 no 3-4 pp 329ndash337 2004

[18] T Kant and SM Shiyekar ldquoCylindrical bending of piezoelectriclaminates with a higher order shear and normal deformationtheoryrdquo Computers and Structures vol 86 no 15-16 pp 1594ndash1603 2008

[19] P Lu H P Lee and C Lu ldquoAn exact solution for functionallygraded piezoelectric laminates in cylindrical bendingrdquo Interna-tional Journal of Mechanical Sciences vol 47 no 3 pp 437ndash4382005

[20] W Yan J Wang and W Q Chen ldquoCylindrical bendingresponses of angle-ply piezoelectric laminates with viscoelasticinterfacesrdquo Applied Mathematical Modelling vol 38 no 24 pp6018ndash6030 2014

[21] Y Y Zhou W Q Chen and C F Lu ldquoSemi-analytical solutionfor orthotropic piezoelectric laminates in cylindrical bendingwith interfacial imperfectionsrdquo Composite Structures vol 92no 4 pp 1009ndash1018 2010

[22] G P Dube S Kapuria and P C Dumir ldquoExact piezother-moelastic solution of a simply-supported orthotropic flat panelin cylindrical bendingrdquo International Journal of MechanicalSciences vol 38 pp 374ndash387 1996

[23] V N Pilipchuk V L Berdichevsky and R A IbrahimldquoThermo-mechanical coupling in cylindrical bending of sand-wich platesrdquoComposite Structures vol 92 no 11 pp 2632ndash26402010

[24] C Zhang S Di and N Zhang ldquoA new procedure for staticanalysis of thermo-electric laminated composite plates undercylindrical bendingrdquo Composite Structures vol 56 no 2 pp131ndash140 2002

[25] W Q Chen and K Y Lee ldquoState-space approach for statics anddynamics of angle-ply laminated cylindrical panels in cylindri-cal bendingrdquo International Journal of Mechanical Sciences vol47 no 3 pp 374ndash387 2005

[26] C-P Wu and Y-S Syu ldquoExact solutions of functionally gradedpiezoelectric shells under cylindrical bendingrdquo InternationalJournal of Solids and Structures vol 44 no 20 pp 6450ndash64722007

[27] W Yan J Ying and W Q Chen ldquoThe behavior of angle-ply laminated cylindrical shells with viscoelastic interfaces incylindrical bendingrdquo Composite Structures vol 78 no 4 pp551ndash559 2007

Shock and Vibration 17

[28] S Brischetto F Tornabene N Fantuzzi and E Viola ldquo3Dexact and 2D generalized differential quadraturemodels for freevibration analysis of functionally graded plates and cylindersrdquoMeccanica vol 51 no 9 pp 2059ndash2098 2016

[29] N Fantuzzi S Brischetto F Tornabene and E Viola ldquo2D and3D shell models for the free vibration investigation of func-tionally graded cylindrical and spherical panelsrdquo CompositeStructures vol 154 pp 573ndash590 2016

[30] S Brischetto and R Torre ldquoExact 3D solutions and finiteelement 2D models for free vibration analysis of plates andcylindersrdquo Curved and Layered Structures vol 1 no 1 pp 59ndash92 2014

[31] F Tornabene S Brischetto N Fantuzzi and E Viola ldquoNumer-ical and exact models for free vibration analysis of cylindricaland spherical shell panelsrdquo Composites Part B Engineering vol81 pp 231ndash250 2015

[32] S Brischetto ldquoExact elasticity solution for natural frequenciesof functionally graded simply-supported structuresrdquo ComputerModeling in Engineering amp Sciences vol 95 no 5 pp 391ndash4302013

[33] S Brischetto ldquoThree-dimensional exact free vibration analysisof spherical cylindrical and flat one-layered panelsrdquo Shock andVibration vol 2014 Article ID 479738 29 pages 2014

[34] S Brischetto ldquoA continuum elastic three-dimensional modelfor natural frequencies of single-walled carbon nanotubesrdquoComposites Part B Engineering vol 61 pp 222ndash228 2014

[35] S Brischetto ldquoAn exact 3d solution for free vibrations ofmultilayered cross-ply composite and sandwich plates andshellsrdquo International Journal of Applied Mechanics vol 6 no 6Article ID 1450076 42 pages 2014

[36] S Brischetto ldquoA continuumshellmodel including vanderWaalsinteraction for free vibrations of double-walled carbon nan-otubesrdquoCMES-ComputerModeling in Engineering and Sciencesvol 104 no 4 pp 305ndash327 2015

[37] S Brischetto F Tornabene N Fantuzzi and M BacciocchildquoRefined 2D and exact 3D shell models for the free vibrationanalysis of single- and double-walled carbon nanotubesrdquo Tech-nologies vol 3 no 4 pp 259ndash284 2015

[38] S Brischetto ldquoConvergence analysis of the exponential matrixmethod for the solution of 3D equilibrium equations for freevibration analysis of plates and shellsrdquo Composites Part BEngineering vol 98 pp 453ndash471 2016

[39] S Brischetto ldquoExact and approximate shell geometry in the freevibration analysis of one-layered and multilayered structuresrdquoInternational Journal of Mechanical Sciences vol 113 pp 81ndash932016

[40] S Brischetto ldquoCurvature approximation effects in the freevibration analysis of functionally graded shellsrdquo InternationalJournal of Applied Mechanics 2016

[41] F Tornabene N Fantuzzi F Ubertini and E Viola ldquoStrong for-mulation finite element method based on differential quadra-ture a surveyrdquo Applied Mechanics Reviews vol 67 no 2 ArticleID 020801 pp 1ndash55 2015

[42] F Tornabene E Viola and N Fantuzzi ldquoGeneral higher-orderequivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panelsrdquo CompositeStructures vol 104 pp 94ndash117 2013

[43] N Fantuzzi F Tornabene E Viola and A J Ferreira ldquoAstrong formulation finite element method (SFEM) based onRBF and GDQ techniques for the static and dynamic analysesof laminated plates of arbitrary shaperdquo Meccanica vol 49 no10 pp 2503ndash2542 2014

[44] F Tornabene N Fantuzzi and M Bacciocchi ldquoFree vibrationsof free-form doubly-curved shells made of functionally gradedmaterials using higher-order equivalent single layer theoriesrdquoComposites Part B Engineering vol 67 pp 490ndash509 2014

[45] F Tornabene and E Viola ldquoStatic analysis of functionallygraded doubly-curved shells and panels of revolutionrdquo Mecca-nica vol 48 no 4 pp 901ndash930 2013

[46] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoDynamic analysis of thick and thin elliptic shell structuresmade of laminated composite materialsrdquo Composite Structuresvol 133 pp 278ndash299 2015

[47] F TornabeneN FantuzziM Bacciocchi and EViola ldquoHigher-order theories for the free vibrations of doubly-curved lam-inated panels with curvilinear reinforcing fibers by meansof a local version of the GDQ methodrdquo Composites Part BEngineering vol 81 pp 196ndash230 2015

[48] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoFree vibrations of composite oval and elliptic cylinders bythe generalized differential quadrature methodrdquo Thin-WalledStructures vol 97 pp 114ndash129 2015

[49] F Tornabene and J N Reddy ldquoFGM and laminated doubly-curved and degenerate shells resting on nonlinear elastic foun-dations a GDQ solution for static analysis with a posterioristress and strain recoveryrdquo Journal of the Indian Institute ofScience vol 93 no 4 pp 635ndash688 2013

[50] N Fantuzzi F Tornabene and E Viola ldquoFour-parameter func-tionally graded cracked plates of arbitrary shape a GDQFEMsolution for free vibrationsrdquo Mechanics of Advanced Materialsand Structures vol 23 no 1 pp 89ndash107 2016

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 17

[28] S Brischetto F Tornabene N Fantuzzi and E Viola ldquo3Dexact and 2D generalized differential quadraturemodels for freevibration analysis of functionally graded plates and cylindersrdquoMeccanica vol 51 no 9 pp 2059ndash2098 2016

[29] N Fantuzzi S Brischetto F Tornabene and E Viola ldquo2D and3D shell models for the free vibration investigation of func-tionally graded cylindrical and spherical panelsrdquo CompositeStructures vol 154 pp 573ndash590 2016

[30] S Brischetto and R Torre ldquoExact 3D solutions and finiteelement 2D models for free vibration analysis of plates andcylindersrdquo Curved and Layered Structures vol 1 no 1 pp 59ndash92 2014

[31] F Tornabene S Brischetto N Fantuzzi and E Viola ldquoNumer-ical and exact models for free vibration analysis of cylindricaland spherical shell panelsrdquo Composites Part B Engineering vol81 pp 231ndash250 2015

[32] S Brischetto ldquoExact elasticity solution for natural frequenciesof functionally graded simply-supported structuresrdquo ComputerModeling in Engineering amp Sciences vol 95 no 5 pp 391ndash4302013

[33] S Brischetto ldquoThree-dimensional exact free vibration analysisof spherical cylindrical and flat one-layered panelsrdquo Shock andVibration vol 2014 Article ID 479738 29 pages 2014

[34] S Brischetto ldquoA continuum elastic three-dimensional modelfor natural frequencies of single-walled carbon nanotubesrdquoComposites Part B Engineering vol 61 pp 222ndash228 2014

[35] S Brischetto ldquoAn exact 3d solution for free vibrations ofmultilayered cross-ply composite and sandwich plates andshellsrdquo International Journal of Applied Mechanics vol 6 no 6Article ID 1450076 42 pages 2014

[36] S Brischetto ldquoA continuumshellmodel including vanderWaalsinteraction for free vibrations of double-walled carbon nan-otubesrdquoCMES-ComputerModeling in Engineering and Sciencesvol 104 no 4 pp 305ndash327 2015

[37] S Brischetto F Tornabene N Fantuzzi and M BacciocchildquoRefined 2D and exact 3D shell models for the free vibrationanalysis of single- and double-walled carbon nanotubesrdquo Tech-nologies vol 3 no 4 pp 259ndash284 2015

[38] S Brischetto ldquoConvergence analysis of the exponential matrixmethod for the solution of 3D equilibrium equations for freevibration analysis of plates and shellsrdquo Composites Part BEngineering vol 98 pp 453ndash471 2016

[39] S Brischetto ldquoExact and approximate shell geometry in the freevibration analysis of one-layered and multilayered structuresrdquoInternational Journal of Mechanical Sciences vol 113 pp 81ndash932016

[40] S Brischetto ldquoCurvature approximation effects in the freevibration analysis of functionally graded shellsrdquo InternationalJournal of Applied Mechanics 2016

[41] F Tornabene N Fantuzzi F Ubertini and E Viola ldquoStrong for-mulation finite element method based on differential quadra-ture a surveyrdquo Applied Mechanics Reviews vol 67 no 2 ArticleID 020801 pp 1ndash55 2015

[42] F Tornabene E Viola and N Fantuzzi ldquoGeneral higher-orderequivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panelsrdquo CompositeStructures vol 104 pp 94ndash117 2013

[43] N Fantuzzi F Tornabene E Viola and A J Ferreira ldquoAstrong formulation finite element method (SFEM) based onRBF and GDQ techniques for the static and dynamic analysesof laminated plates of arbitrary shaperdquo Meccanica vol 49 no10 pp 2503ndash2542 2014

[44] F Tornabene N Fantuzzi and M Bacciocchi ldquoFree vibrationsof free-form doubly-curved shells made of functionally gradedmaterials using higher-order equivalent single layer theoriesrdquoComposites Part B Engineering vol 67 pp 490ndash509 2014

[45] F Tornabene and E Viola ldquoStatic analysis of functionallygraded doubly-curved shells and panels of revolutionrdquo Mecca-nica vol 48 no 4 pp 901ndash930 2013

[46] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoDynamic analysis of thick and thin elliptic shell structuresmade of laminated composite materialsrdquo Composite Structuresvol 133 pp 278ndash299 2015

[47] F TornabeneN FantuzziM Bacciocchi and EViola ldquoHigher-order theories for the free vibrations of doubly-curved lam-inated panels with curvilinear reinforcing fibers by meansof a local version of the GDQ methodrdquo Composites Part BEngineering vol 81 pp 196ndash230 2015

[48] F Tornabene N Fantuzzi M Bacciocchi and R DimitrildquoFree vibrations of composite oval and elliptic cylinders bythe generalized differential quadrature methodrdquo Thin-WalledStructures vol 97 pp 114ndash129 2015

[49] F Tornabene and J N Reddy ldquoFGM and laminated doubly-curved and degenerate shells resting on nonlinear elastic foun-dations a GDQ solution for static analysis with a posterioristress and strain recoveryrdquo Journal of the Indian Institute ofScience vol 93 no 4 pp 635ndash688 2013

[50] N Fantuzzi F Tornabene and E Viola ldquoFour-parameter func-tionally graded cracked plates of arbitrary shape a GDQFEMsolution for free vibrationsrdquo Mechanics of Advanced Materialsand Structures vol 23 no 1 pp 89ndash107 2016

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of