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Free vibration analysis of functionally graded conical shell panelsby a meshless methodX. Zhao, K.M. LiewDepartment of Building and Construction, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kongarti cle i nfoArticle history:Available online 15 September 2010Keywords:Functionally graded materialsFree vibrationConical shellShear deformationElement-freeMeshless methodabstractA free vibration analysis of metal and ceramic functionally graded conical shell panels is presented usingthe element-free kp-Ritz method. The rst-order shear deformation shell theory is used to account for thetransverseshearstrainsandrotaryinertia, andmesh-freekernel particlefunctionsareemployedtoapproximate the two-dimensional displacement elds. The material properties of the conical shell panelsare assumed to vary continuously through their thickness in accordance with a power-law distribution ofthe volume fractions of their constituents. Convergence studies are performed in terms of the number ofnodes, and comparisons of the current solutions and those reported in literature are provided to verifythe accuracy of the proposed method. Two types of functionally graded conical shell panels, includingAl/ZrO2 and Ti6Al4V/aluminum oxide, are chosen in the study, and the effects of the volume fraction,boundary condition, semi-vertex angle, and length-to-thickness ratio on their frequency characteristicsare discussed in detail. 2010 Elsevier Ltd. All rights reserved.1. IntroductionConical shells and panels have been widely used in a variety ofengineering elds as important structural components due to theirspecial geometric shapes, especially in aerospace and marineindustries. Extensive investigations have been conducted to exam-ine the dynamic responses of such structures. An earlier survey onthe free vibration of conical shells was provided by Leissa [1], theeffectsof different boundaryconditionsandsemi-vertexangleson the frequency characteristics of conical shells were investigated,but most result are for thin conical shells. Liew and Lim [2,3] car-ried out the vibration analysis of shallow conical shells by a globalRitzformulationbasedontheenergyprinciple. Later, theyalsopresented a formulation for the free vibration of moderately thickconical shell panels based on shear deformable theory [4,5]. Finiteelementanalysisofshellandshellpanelssubjectedtovibrationwere also examined by Heet al. [6], Ng etal. [7] and Liew et al.[8]. Shu [9] employed the generalized differential quadraturemethod to study the free vibration of composite laminated conicalshells, and Bardell et al. [10] investigated the vibration characteris-tics of open conically curved, isotropic shell panels using a hp ver-sion of nite element method.Functionallygradedmaterials (FGM) are special compositesthat are formed by mixing two or more different materials accord-ing to a pre-determined formula that depends on the volume frac-tions of constituents. Such materials possess continuous andsmooth material properties, which make them preferable in engi-neeringapplications. Mucheffort has beendevotedtovariousstructural analyses of functionally graded structures, such as ther-mal stresses, static, buckling, andvibrationanalyses. Noda[11]presentedareviewonthermal stresses infunctionallygradedmaterials, thethermalstressesonthefunctionallygradedplatesand the thermal stress intensity factor in the functionally gradedplateswithcrackwerediscussed. Fukuietal. [12]examinedthestresses and strains in a functionally thick-walled tube under uni-form thermal loading, and Ng et al. [13] studied the effects of FGMmaterialsontheparametricresonanceof platestructures. Liewet al. [14] investigated the thermal stress behavior of functionallygradedhollowcircular cylinders, andPraveenandReddy[15]examined the static and dynamic response of functionally gradedplates in terms of the combination of the rst-order shear defor-mation plate theory and the von Krmn strains. He et al. [16] pro-vided a nite element formulation for the active control offunctionallygradedplateswithintegratedpiezoelectricsensorsandactuators,andEfraimandEisenberger [17]conducted vibra-tionanalysis of variable thickness annular functionallygradedplates.A few but not many publications on the analysis of functionallygraded conical shells and panels have been reported in literature.Soyev [18,19] investigated the stability of truncated conical shellsof functionally gradedmaterial subjectedto external pressurevaryingasapowerfunctionof time, aswell asuniformlateraland hydrostatic pressures. Tornabene [20] carried out free0263-8223/$ - see front matter 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.compstruct.2010.08.014Corresponding author.E-mail address: [email protected] (K.M. Liew).Composite Structures 93 (2011) 649664ContentslistsavailableatScienceDirectComposite Structuresj our nal homepage: www. el sevi er . com/ l ocat e/ compst r uctvibration analysis of moderately thick functionally graded conical,cylindrical shell and annular plate structures with a four-parame-ter power-law distribution. The formulation was based on the rst-order shear deformation theory and generalized differentialquadrature method. Tornbene et al. [21] also examined the samestructures with two different power-law distributions. The appli-cations of functionallygradedconical shell panels canbeverybroad. Duetotheirhighstrengthandresistancetotemperaturechange, the functionally graded conical shell and panels can be ap-plied to military aircraft propulsion system, fuselage structures ofcivil airliners, and other machine parts.Thedistinctivenode-basedfeaturemakesmesh-freemethodspromising numerical methods in engineering analysis. As alterna-tives to nite element methods, their applications have expandedto various engineering problems, such as the deformation of thinshells[2225], foldedplatestructures[26,27], staticanalysisoffunctionally gradedplates[28], transient thermoelasticdeforma-tions of thick functionally graded plates [29], thermal analysis ofReissnerMindlinshallowshellswithFGMproperties[30], staticand dynamic analysis of shells with orthotropic material properties[31,32], free vibration analysis of rotating shells [33], elasto-plas-ticity problems [34], and biomechanics problems [35].In thispaper,the free vibration of functionally graded conicalshell panels is investigated using the element-free kp-Ritz method[3639]. Therst-order shear deformationshell theoryis em-ployedtoaccountforthetransverseshearstrains. Thematerialproperties are assumedtovarycontinuouslyacross the depthdirection according to a power-law distribution, and are expressedin terms of the volume fractions of the constituents. The evaluationof the system stiffness is treated separately: the bending stiffnessis estimated using a full Gaussian integration, whereas the mem-brane and shear terms are evaluated using a single-point Gaussianintegration. Therefore, themembranelockingandshearlockingcan be eliminated for a very thin conical shell panel. The inuencesof the semi-vertex angle, boundarycondition, volume fractionexponent, and length-to-thickness ratio on the natural frequencyof thefunctionallygradedconical shell panelsarediscussedindetail.2. Functionally graded conical shell panelsAfunctionallygradedcircularconicalshellpanelisshowninFig. 1, where a coordinate system (x, h, z) is established on the mid-dlesurfaceoftheconicalpanel. Thegeometricpropertiesoftheconical panel are represented by length L, semi-vertex angle a, sub-tended angle h0, thickness h, and the radii at the two ends R1 andR2. The cone radius at any point along its length is given byRx R1 x sina: 1The material propertiesareassumed to vary according tothe fol-lowing power-law distribution:Fig. 1. Geometry of the functionally graded conical shell panel.-0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.50.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0n = 0.5n = 0.3n = 0.1n = 10n = 5n = 3n = 1Vcz / hFig. 2. Variation in the volume fraction through the panel thickness.650 X. Zhao, K.M. Liew/ Composite Structures 93 (2011) 649664Pz Pc PmVc Pm; 2aVc 12 zh_ _nn P0; 2bwhere P represents the effective material properties includingYoungs modulus E, density q, Poissons ratio m, thermal conductiv-ity k, and thermal expansion a. Pc and Pm denote the properties oftheceramicandmetal, respectively, Vcisthevolumefractionofthe ceramic, and n is the volume fraction exponent. The variationin the volume fraction through the thickness with different volumefraction exponents is illustrated in Fig. 2. The properties of temper-ature-dependent materials are determined by [40]P P0P1T11 P1T P2T2 P3T3; 3where P0, P1, P1, P2, and P3 represent the temperature coefcients.In this study, the material properties are calculated at roomtemper-ature (T = 27 C) unless otherwise specied. Table 1 gives the mate-rial properties of aluminum and zirconia (Al/ZrO2)and Ti6Al4Vand aluminum oxide (Ti6Al4V/Al2O3).3. Theoretical formulation3.1. Energy functionalAccording to the rst-order shear deformation shell theory, thedisplacement eld is expressed asux; h; z u0x; h z/xx; h;vx; h; z v0x; h z/hx; h;wx; h; z w0x; h;4where u0, v0, and w0 are the displacements at the middle surface ofthe conical panel in the x, h, and z directions, and /x and /h repre-sent the transverse normal rotations about the h and x axes, respec-tively. The linear straindisplacement relationship is written asexxehhcxh______ e0 zj;chzcxz_ _ c0; 5wheree0 @u0@x1Rx@v0@h u0 sinaRx w0 cos aRx1Rx@u0@h @v0@x v0 sinaRx______; 6aj @/x@x1Rx@/h@h /x sinaRx1Rx@/x@h @/h@x /h sinaRx______; c0 /h 1Rx@w0@h v0 cos aRx/x @w0@x_ _:6b; cThus, the constitutive equations are expressed asrxxrhhrxhrhzrxz______Q11Q120 0 0Q12Q110 0 00 0 Q660 00 0 0 Q4400 0 0 0 Q55____exxehhcxhchzcxz______11000______^aDT________________; 7whereTable 1Properties of the FGM components.Material PropertiesE (N/m2) m q (kg/m3)Aluminum (Al) 70.0 1090.30 2707Zirconia (ZrO2) 151 1090.30 3000Ti6Al4V 105.7 1090.298 4429Aluminum oxide (Al2O3) 320.2 1090.26 3750Table 2Comparison of the rst four non-dimensional frequency parameters ~ x xa2qh=D_for a clampedconical shell panel (L/s = 0.6, L/h = 100, L/R0 = 0.3, a = 30, h0 = 60,m = 0.3).Mode Present Cheunget al. [43]Bardellet al. [10]8 8 10 10 14 14 16 161 206.75 206.30 206.46 206.69 213.4 209.842 251.51 250.28 250.52 250.85 262.5 257.113 300.70 301.98 302.86 303.21 314.7 307.904 344.84 344.08 344.37 344.68 358.6 351.90Table 3Comparison of the rst four non-dimensional frequency parameters~ x xLb0qh=D_for a clamped conical shell panel (L/s = 0.2, s/h = 1000, a = 7.5, h0 = 30, m = 0.3).Mode Present Lim and Liew [4]10 10 12 12 14 14 16 161 211.32 211.26 211.46 211.77 221.682 242.69 242.92 243.32 243.76 254.403 274.44 275.75 276.12 276.77 287.874 288.07 288.35 288.64 288.92 297.895 308.27 308.54 308.80 309.04 317.716 310.29 311.69 312.80 313.73 325.597 325.09 325.52 325.82 326.06 334.648 340.28 341.34 341.82 342.13 350.81Table 4Comparison of the rst four non-dimensional frequency parameters ~ x xLb0qh=D_for a clamped conical shell panel (s/h = 1000, a = 7.5, h0 = 20, m = 0.3).L/s Mode1 2 3 4 5 6 7 80.2 Present 233.82 245.71 256.64 267.02 277.60 290.58 293.66 308.50Bardell et al. [10] 235.35 247.45 258.64 269.32 280.39 294.21 302.82 312.74Lim and Liew [5] 239.10 251.32 262.61 273.37 284.64 298.19 306.17 316.560.8 Present 931.05 978.38 1026.4 1081.1 1139.6 1162.2 1203.8 1275.1Bardell et al. [10] 941.79 990.29 1035.1 1077.8 1119.3 1160.3 1200.9 1212.9Lim and Liew [5] 956.57 1006.0 1055.8 1120.3 1206.1 1225.0 1284.8 1337.6X. Zhao, K.M. Liew/ Composite Structures 93 (2011) 649664 651Q11 Ez1 mz2 ; Q12 mQ11; Q22 Q11;Q44 Q55 Q66 Ez21 mz; 8DT is the temperature change fromastress-free state(T = 0 C), and ^ aisthethermal coefcient of expansion. Thetotal in-planeforceresultants,moment resultants, and transverse force resultants are dened asN NxxNhhNxh_______h=2h=2rxxrhhrxh______dz; M MxxMhhMxh_______h=2h=2rxxrhhrxh______zdz; 9a; bQs QhQx_ __h=2h=2rhzrxz_ _dz: 9cThey can be written in matrix form asNM______A BB D____e0j_ _NTMT______; Qs Asc0; 10a; bwhereNT_h=2h=21 1 0 TQ11 Q12^ aDT dz; 11aMT_h=2h=21 1 0 TQ11 Q12^ aDTzdz; 11bTable 5Comparison of the rst four non-dimensional frequency parameters ~ x xab0qh=D_for a conical shell panel (CFFF, L/s = 0.2, s/h = 1000, L/R1 = 0.3, a = 7.5, h0 = 30, m = 0.3).Mode1 2 3 4 5 6 7 8Present 5.5187 8.8141 26.532 28.892 49.251 64.197 75.031 78.158Bardell et al. [10] 5.5130 8.9563 26.989 28.852 50.174 64.497 75.479 79.578Lim and Liew [5] 6.1727 9.0708 27.299 29.758 50.669 65.171 74.499 80.201Table 6Comparison of the rst 10 frequencies for the functionally graded conical panel with different volume fraction exponents (CFFF, R1 = 0.5 m, h = 0.1 m, a = 2 m, a = 40, h0 = 120).n Mode1 2 3 4 5 6 7 80 Present 79.60 111.04 157.51 192.68 258.41 271.04 319.27 355.14Tornabene et al. [21] 80.36 111.46 159.21 195.79 260.08 274.26 322.79 361.920.6 Present 76.96 108.33 152.24 185.42 249.81 259.72 309.64 341.60Tornabene et al. [21] 78.18 109.02 154.95 189.60 253.29 265.15 314.65 350.181 Present 76.56 107.75 151.43 184.73 248.41 258.31 308.09 340.49Tornabene et al. [21] 77.81 108.45 154.18 188.71 252.08 263.94 313.01 348.485 Present 77.64 107.65 153.03 188.79 251.11 265.06 309.70 348.05Tornabene et al. [21] 78.96 108.27 156.11 193.10 254.47 271.26 315.45 356.75Table 7The rst eight frequency parameters ~ x xa2qc=Ec h2_=2p for the FGM Al/ZrO2 conical shell panels (SSSS, R1 = 0.2 m, h = 0.01 m, L = 0.8 m, h0 = 120).a () n Mode1 2 3 4 5 6 7 815 0 5.1897 6.9241 10.4463 10.5224 11.4045 12.7374 14.8945 14.93760.5 4.7323 6.2408 9.3967 9.6898 10.3344 11.6854 13.4229 13.42361 4.5051 5.9435 8.9503 9.2239 9.8399 11.1227 12.7841 12.78545 4.0739 5.5182 8.1626 8.3507 9.0249 9.9201 11.8726 11.959810 3.9505 5.3691 7.8902 8.1297 8.7679 9.6013 11.5524 11.646930 0 4.9602 4.9847 6.5849 8.7806 9.6585 10.6078 10.9323 11.48790.5 4.4895 4.5660 5.9411 7.9064 8.7956 9.7692 9.8912 10.33561 4.2761 4.3468 5.6599 7.5332 8.3753 9.2997 9.4214 9.84835 3.8927 3.9372 5.2521 7.0243 7.6042 8.2291 8.6848 8.961210 3.7694 3.8259 5.1083 6.8358 7.3763 7.9543 8.4404 8.669645 0 4.0262 4.5282 4.9630 6.3582 8.0063 8.1230 8.7646 8.86700.5 3.6531 4.1528 4.4852 5.7319 7.3003 7.3123 7.9403 8.16381 3.4797 3.9536 4.2734 5.4620 6.9523 6.9686 7.5643 7.77145 3.1886 3.5325 3.9540 5.0845 6.2994 6.5094 6.8811 6.958710 3.0961 3.4192 3.8437 4.9462 6.1082 6.3349 6.6519 6.759660 0 3.2521 3.5779 3.9732 5.0811 5.7963 6.2338 6.3819 6.47610.5 2.9521 3.2801 3.5905 4.5798 5.3319 5.6706 5.8280 5.84691 2.8123 3.1231 3.4215 4.3649 5.0762 5.4018 5.5546 5.56825 2.5764 2.7945 3.1688 4.0679 4.5052 4.9259 4.9933 5.194610 2.5012 2.7051 3.0804 3.9573 4.3563 4.7797 4.8349 5.0551652 X. Zhao, K.M. Liew/ Composite Structures 93 (2011) 649664A A11A12A16A12A22A26A16A26A66____; B B11B12B16B12B22B26B16B26B66____; 12a; b D D11D12D16D12D22D26D16D26D66____; AsA44A45A45A55_ _; 13a; b0 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.4Fig. 3. The rst eight modes for a simply-supported Al/ZrO2 conical shell panel (SSSS, R0 = 0.2 m, h = 0.01 m, a = 0.8 m, a = 15, h0 = 120).0 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.5Fig. 4. The rst eight modes for a simply-supported Al/ZrO2 conical shell panel (SSSS, R0 = 0.2 m, h = 0.01 m, a = 0.8 m, a = 30, h0 = 120).0 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.5Fig. 5. The rst eight modes for a simply-supported Al/ZrO2 conical shell panel (SSSS, R0 = 0.2 m, h = 0.01 m, a = 0.8 m, a = 45, h0 = 120).0 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.5Fig. 6. The rst eight modes for a simply-supported Al/ZrO2 conical shell panel (SSSS, R0 = 0.2 m, h = 0.01 m, a = 0.8 m, a = 60, h0 = 120).X. Zhao, K.M. Liew/ Composite Structures 93 (2011) 649664 653and NTandMTrepresent the thermal force and moment resultants,respectively. Aij, Bij, Dij, and Asijare dened asAij; Bij; Dij _h=2h=2Qij1; z; z2dz; Asij K_h=2h=2Qijdz; 14a; bwhere Aij, Bij, and Dij are dened for i, j = 1, 2, 6, and Asij is dened fori, j = 4, 5. K denotes the transverse shear correction coefcient and isgiven by [17]K 56 v1V1 v2V2; 15where V1 and V2 represent the volume fraction of each material inthe entire cross-section, and m1 and m2 are the original Poisson ratiosof ceramic and metal. Note that the volume fractions V1 and V2, andthe Poisson ratios m1 and m2 are different with that expressed in Eq.(2), which a function of thickness coordinate z.The strain energy of the conical panel is expressed asUe 12_XNTedX; 16where N and e are expressed asN NMQs______; e e0jc0______: 17a; bThe kinetic energy of the conical panel is given byH 12_X_h=2h=2qz _ u2_ v2_ w2dzdX: 18The total energy functional is thus given byP Ue H: 193.2. Two-dimensional kernel particle shape functionsThetwo-dimensionalshapefunctionsareconstructedaccord-ingtoakernelparticleestimationconcept, andtheconstructionprocedureisbrieyreviewedinthissection. ConsideradomainXthatisdiscretizedbyasetofscatterednodesxI, I = 1, . . . , NP.Theapproximationof afunctionu(x) isdenotedbyuhandex-pressed asuh

NPI1wIxuI; 20where wI(x) and uI are the shape function and coefcient associatedwith node I, and the shape function wI(x) is dened as [41,42]wIx CxUax xI; 21where Ua(x xI) is the kernel function, C(x) is the correction func-tion and is used to satisfy the reproducing condition

NPI1wIxxrIhsI xrhsfor r s 0; 1; 2: 22ThecorrectionfunctionC(x)isexpressedasacombinationofthecomplete second-order monomial functionsCx HTx xIbx; 23bx b0x; h; b1x; h; b2x; h; b3x; h; b4x; h; b5x; hT; 24a10 20 30 40 50 6012345678n = 0.5Frequency parameter Semi-vertex angle mode (1,2)mode (1,3)mode (1,4)mode (1,1)Fig. 7. Variation in the frequencies of the modes with semi-vertex angle for the Al/ZrO2 conical panel (SSSS).Table 8The rst eight frequency parameters ~ x xa2qc=Ec h2_=2p for the FGM Al/ZrO2 conical shell panels (CCCC, R1 = 0.2 m, h = 0.01 m, L = 0.8 m, h0 = 120).a () n Mode1 2 3 4 5 6 7 815 0 9.7829 10.3196 13.943 14.621 19.939 18.039 19.568 19.7890.5 8.8563 9.4030 12.6377 13.2701 15.3608 16.2689 17.7874 17.97971 8.4331 8.9521 12.035 12.635 14.626 15.495 16.937 17.1195 7.7516 8.1041 11.041 11.547 13.387 14.372 15.433 15.61210 7.5327 7.8581 10.726 11.212 12.999 13.981 14.978 15.15430 0 7.3268 7.8105 10.4890 10.7231 12.1514 12.5098 13.9850 15.36030.5 6.6612 7.0890 9.5649 9.6868 11.0581 11.3591 12.6374 13.89491 6.3434 6.7515 9.1082 9.2280 10.5306 10.8183 12.0391 13.23605 5.7807 6.1747 8.2453 8.5379 9.5792 9.8865 11.1342 12.204310 5.6099 5.9941 7.9943 8.3005 9.2939 9.5964 10.8238 11.858845 0 5.8266 6.0593 8.0163 8.1664 9.6259 9.8110 10.435 11.4680.5 5.3040 5.5010 7.2469 7.4459 8.7642 8.9099 9.4443 10.4231 5.0515 5.2399 6.9046 7.0913 8.3473 8.4873 8.9979 9.92955 4.5935 4.7945 6.3851 6.4233 7.5892 7.7593 8.3063 9.041710 4.4558 4.6540 6.2063 6.2268 7.3612 7.5305 8.0732 8.756260 0 4.5034 4.7074 6.0648 6.3268 7.5224 7.6575 7.8665 8.19730.5 4.0971 4.2776 5.5275 5.7213 6.8377 6.9785 7.2002 7.39871 3.9028 4.0750 5.2649 5.4519 6.5139 6.6382 6.8571 7.05115 3.5567 3.7222 4.7798 5.0394 5.9500 6.0429 6.1631 6.552710 3.4507 3.6112 4.6352 4.8969 5.7740 5.8608 5.9688 6.3727654 X. Zhao, K.M. Liew/ Composite Structures 93 (2011) 649664HTx xI 1; x xI; h hI; x xIh hI; x xI2; h hI2;24bwhere H is a vector with a quadratic basis and b(x) is a coefcientvector that has yet to be determined. The shape function can nowbe written as0 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.4Fig. 8. The rst eight modes for a clamped Al/ZrO2 conical shell panel (CCCC, R0 = 0.2 m, h = 0.01 m, a = 0.8 m, a = 15, h0 = 120).0 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.5Fig. 9. The rst eight modes for a clamped Al/ZrO2 conical shell panel (CCCC, R0 = 0.2 m, h = 0.01 m, a = 0.8 m, a = 30, h0 = 120).0 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.5Fig. 10. The rst eight modes for a clamped Al/ZrO2 conical shell panel (CCCC, R0 = 0.2 m, h = 0.01 m, a = 0.8 m, a = 45, h0 = 120).0 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.5Fig. 11. The rst eight modes for a clamped Al/ZrO2 conical shell panel (CCCC, R0 = 0.2 m, h = 0.01 m, a = 0.8 m, a = 60, h0 = 120).X. Zhao, K.M. Liew/ Composite Structures 93 (2011) 649664 655wIx bTxHx xIUax xI: 25The coefcient b(x) is obtained by substituting Eq. (25) into Eq. (22),bx M1xH0; 26whereMisamomentmatrixandH(0)isaconstantvector. Theexpressions for M and H(0) are given byMx

NPI1Hx xIHTx xIUax xI; 27aH0 1; 0; 0; 0; 0; 0T: 27bThe kernel function Ua(x xI) is expressed asUax xI Uax Uah; 28in whichUax ux xIa_ _; 29where u(x) is a weight function. The cubic spline function is chosenas that weight function, and is given byuzzI 23 4z2I 4z3Ifor 0 6 jzIj 61243 4zI 4z2I 43z3Ifor12 < jzIj 6 10 otherwise______; 30where zI xxIdI, dI is the size of the support and is given bydI dmaxcI; 31where dmax is a scaling factor and distance cI is chosen so that thereare sufcient nodes to avoid the singularity of matrix M.Table 9The rst eight frequency parameters ~ x xa2qc=Ec h2_=2p for the FGM Al/ZrO2 conical shell panels (CFFF, R1 = 0.2 m, h = 0.01 m, L = 0.8 m, h0 = 120).a () n Mode1 2 3 4 5 6 7 815 0 1.3666 2.2649 3.5513 3.8597 4.3713 6.3943 6.7488 6.78040.5 1.2486 2.0499 3.2488 3.5035 3.9679 5.8128 6.1144 6.11881 1.1893 1.9534 3.0937 3.3382 3.7785 5.5355 5.8266 5.82875 1.0737 1.8024 2.7790 3.0595 3.4571 5.0466 5.3662 5.403210 1.0404 1.7509 2.6914 2.9695 3.3573 4.8979 5.2120 5.254530 0 1.6631 2.0007 3.2983 3.6771 5.1960 5.6438 5.9398 6.10550.5 1.5208 1.8121 2.9883 3.3348 4.7683 5.1053 5.3751 5.56321 1.4486 1.7267 2.8474 3.1777 4.5386 4.8630 5.1210 5.29915 1.3052 1.5905 2.6213 2.9188 4.0448 4.4808 4.7256 4.814310 1.2642 1.5449 2.5461 2.8337 3.9146 4.3535 4.5923 4.668945 0 1.6557 1.7819 2.9411 3.3529 4.9051 5.0306 5.5111 5.55250.5 1.5119 1.6157 2.6675 3.0344 4.4463 4.5668 5.0509 5.05301 1.4402 1.5395 2.5416 2.8918 4.2356 4.3504 4.8082 4.81365 1.3018 1.4148 2.3346 2.6700 3.8865 3.9837 4.2949 4.387210 1.2614 1.3738 2.2671 2.5939 3.7735 3.8671 4.1573 4.256260 0 1.4235 1.5319 2.5956 2.9899 4.0192 4.1012 4.6332 4.81770.5 1.2983 1.3889 2.3517 2.7046 3.6630 3.7193 4.2528 4.36031 1.2368 1.3235 2.2409 2.5776 3.4889 3.5435 4.0487 4.25525 1.1213 1.2171 2.0641 2.3828 3.1637 3.2549 3.6088 3.837610 1.0869 1.1819 2.0049 2.3152 3.0670 3.1610 3.4917 3.7288Table 10The rst eight frequency parameters ~ x xa2qc=Ec h2_=2p for the FGM Al/ZrO2 conical shell panels (SSSS, R1 = 0.2 m, h = 0.04 m, L = 0.8 m, h0 = 120).a () n Mode1 2 3 4 5 6 7 815 0 2.7379 3.0581 5.3306 5.6405 5.6978 6.2502 7.1774 7.95520.5 2.5141 2.7439 4.8084 5.0975 5.1710 5.7903 6.6065 7.18741 2.3924 2.6077 4.5713 4.8440 4.9200 5.5249 6.2880 6.83565 2.1265 2.4187 4.2016 4.3899 4.5172 4.8854 5.5670 6.239410 2.0576 2.3579 4.0895 4.2508 4.4072 4.7108 5.3823 6.050030 0 2.2884 2.7364 3.6376 4.2568 4.4615 5.4357 5.5066 6.06600.5 2.0654 2.5145 3.2655 3.9019 4.0371 4.9395 4.9950 5.45161 1.9647 2.3929 3.1064 3.7135 3.8411 4.6989 4.7542 5.18565 1.8084 2.1249 2.8934 3.3200 3.5217 4.2242 4.3844 4.817410 1.7594 2.0556 2.8196 3.2142 3.4238 4.0865 4.2735 4.693145 0 1.9071 2.3457 2.7787 3.3826 3.7895 4.0626 4.3761 5.03580.5 1.7261 2.1535 2.4988 3.0951 3.4274 3.6489 4.0198 4.53071 1.6429 2.0494 2.3787 2.9471 3.2627 3.4736 3.8267 4.31255 1.5076 1.8224 2.2148 2.6477 3.0027 3.2434 3.4070 4.010810 1.4654 1.7645 2.1568 2.5643 2.9192 3.1596 3.2963 3.905460 0 1.5922 1.6666 2.3216 2.8818 3.2971 3.3584 3.7565 4.44470.5 1.4404 1.6269 2.0887 2.6222 2.9752 3.0187 3.4579 3.99911 1.3716 1.4535 1.9894 2.4973 2.8337 2.8752 3.2924 3.80865 1.2634 1.3007 1.8555 2.2733 2.6265 2.6874 2.9202 3.550910 1.2281 1.2594 1.8064 2.2056 2.5548 2.6169 2.8232 3.4569656 X. Zhao, K.M. Liew/ Composite Structures 93 (2011) 649664The nal form of the shape function is expressed aswIx HT0M1xHx xIUax xI: 323.3. Discrete eigen equationFor a conical shell panel discretized by a set of discrete nodes xI,I = 1, . . ., NP, the discrete displacement approximations of its mid-dle surface are expressed as0 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.4Fig. 12. The rst eight modes for a simply-supported Al/ZrO2 conical shell panel (SSSS, R0 = 0.2 m, h = 0.04 m, a = 0.8 m, a = 15, h0 = 120).0 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.5Fig. 13. The rst eight modes for a simply-supported Al/ZrO2 conical shell panel (SSSS, R0 = 0.2 m, h = 0.04 m, a = 0.8 m, a = 30, h0 = 120).0 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.5Fig. 14. The rst eight modes for a simply-supported Al/ZrO2 conical shell panel (SSSS, R0 = 0.2 m, h = 0.04 m, a = 0.8 m, a = 45, h0 = 120).0 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.5Fig. 15. The rst eight modes for a simply-supported Al/ZrO2 conical shell panel (SSSS, R0 = 0.2 m, h = 0.04 m, a = 0.8 m, a = 60, h0 = 120).X. Zhao, K.M. Liew/ Composite Structures 93 (2011) 649664 657uh0 uh0vh0wh0/hx/hh________________

NPI1wIuIvIwI/xI/hI________________eixt

NPI1wIxuIeixt: 33SubstitutingEq. (33)intoEq. (19)andtakingthevariationintheenergy functional yields the eigen equationK x2M~u 0; 34whereK K1KKT;

M K1M_;~u Ku 35a; b; cKIJ wIxJI; I is the identity matrix 36K KbKmKs; 37Table 11The rst eight frequency parameters ~ x xa2qc=Ec h2_=2p for the FGM Al/ZrO2 conical shell panels (CCCC, R1 = 0.2 m, h = 0.04 m, L = 0.8 m, h0 = 120).a () n Mode1 2 3 4 5 6 7 815 0 4.8641 6.0555 7.3079 7.7461 8.5324 9.6142 9.9226 10.60310.5 4.3892 5.4945 6.6095 7.0526 7.7355 8.7429 8.9759 9.58391 4.1726 5.2239 6.2851 6.7072 7.3571 8.3169 8.5372 9.11145 3.8307 4.7411 5.7517 6.0540 6.7055 7.5398 7.8204 8.338710 3.7278 4.6039 5.5929 5.8738 6.5167 7.3184 7.6037 8.110730 0 3.6473 4.1975 5.5426 5.8765 5.9949 6.9373 7.6488 8.11230.5 3.3029 3.8083 5.0365 5.3197 5.4588 6.2395 6.9102 7.36691 3.1422 3.6232 4.7934 5.0621 5.1944 5.9356 6.5747 7.01155 2.8737 3.2990 4.3631 4.6399 4.7014 5.5063 6.0487 6.392110 2.7930 3.2032 4.2366 4.5098 4.5609 5.3626 5.8827 6.207145 0 2.9321 3.1713 4.0803 4.9819 5.2217 6.3029 6.8697 6.95610.5 2.6569 2.8858 3.6975 4.3503 4.5064 4.6991 5.6841 6.17281 2.5289 2.7469 3.5202 4.1415 4.2903 4.4732 5.4108 5.87585 2.3158 2.4918 3.2286 3.7622 3.9484 4.1584 5.0089 5.461610 2.2501 2.4170 3.1374 3.6503 3.8380 4.0486 4.8736 5.316260 0 2.3403 2.3951 3.2697 3.8743 4.3247 4.3816 5.5253 5.68780.5 2.1347 2.1667 2.9497 3.5144 3.8981 3.9505 4.9779 5.11981 2.0327 2.0633 2.8094 3.3472 3.7127 3.7626 4.7409 4.87585 1.8389 1.9013 2.6055 3.0686 3.4481 3.4946 4.4078 4.541310 1.7827 1.8481 2.5345 2.9800 3.3543 3.4005 4.2893 4.42030 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.40 0.5-0.4-0.200.20.4Fig. 16. The rst eight modes for a clamped Al/ZrO2 conical shell panel (CCCC, R0 = 0.2 m, h = 0.04 m, a = 0.8 m, a = 15, h0 = 120).0 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.5Fig. 17. The rst eight modes for a clamped Al/ZrO2 conical shell panel (CCCC, R0 = 0.2 m, h = 0.04 m, a = 0.8 m, a = 30, h0 = 120).658 X. Zhao, K.M. Liew/ Composite Structures 93 (2011) 649664KbIJ _XBbTIDBbJ dX; 38KmIJ _XBmTIABmJdX_XBmTIBBbJ dX_XBbTIBBmJdX; 39KsIJ _XBsTIAsBsJ dX; 40M_IJ _XGTI

mGJdX; 41BbI 0 0 0@wI@x00 0 0sinaRx wI1Rx@wI@h0 0 01Rx@wI@h@wI@x sinaRx wI____; 42BmI@wI@x0 0 0 0sinaRx wI1Rx@wI@hcos aRx wI0 01Rx@wI@h@wI@x sinaRx wI0 0 0____; 43BsI 0 0@wI@xwI00cosaRxwI1Rx@wI@h0 wI_ _; GI wI0 0 0 00 wI0 0 00 0 wI0 00 0 0 wI00 0 0 0 wI____; 44a; b

m I00 0 I100 I00 0 I10 0 I00 0I10 0 I200 I10 0 I2____; I0; I1; I2 _h=2h=2qz1; z; z2dz;45a; bwhere thedensity q(z)isdeterminedbyEq.(2). ThematricesKb,Km, Ksand M_are computed using Gauss integration.4. Numerical examples and discussionIn this section, numerical examples are provided to investigatethefrequencycharacteristicsof thefunctionallygradedconical0 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.5Fig. 18. The rst eight modes for a clamped Al/ZrO2 conical shell panel (CCCC, R0 = 0.2 m, h = 0.04 m, a = 0.8 m, a = 45, h0 = 120).0 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.50 0.5-0.500.5Fig. 19. The rst eight modes for a clamped Al/ZrO2 conical shell panel (CCCC, R0 = 0.2 m, h = 0.04 m, a = 0.8 m, a = 60, h0 = 120).Table 12The rst eight frequency parameters ~ x xa2qc=Ec h2_=2p for the FGM Al/ZrO2 conical shell panels (n = 0.5, R1 = 0.2 m, h = 0.08 m, L = 0.8 m, h0 = 120).Boundary condition a Mode1 2 3 4 5 6 7 8SSSS 15 1.3752 2.3434 2.9111 2.9691 3.3608 3.7701 4.3926 4.707430 1.3526 1.6846 2.1925 2.7955 2.8828 2.9714 3.1492 4.263745 1.1977 1.3891 1.7775 2.2362 2.4036 2.7678 3.0161 3.273260 0.9306 1.2115 1.6508 1.9115 2.1015 2.5403 2.7602 3.0021CCCC 15 3.3115 3.3786 4.3702 4.7737 5.5915 5.7693 5.8876 6.439630 2.4245 2.4844 3.5384 3.8034 3.9771 4.9547 5.1045 5.248345 1.8510 2.0548 2.9524 3.0609 3.5329 3.9452 4.5869 4.728860 1.4424 1.7865 2.5354 2.7260 3.2612 3.3967 4.1922 4.3193X. Zhao, K.M. Liew/ Composite Structures 93 (2011) 649664 659shell panels. Two types of FGMconical panels that consist of alumi-num and zirconia, Ti6Al4V and aluminum oxide are considered,and the material properties of each constituent, including Youngsmodulus, Poissonsratio, anddensityaregiveninTable1. Theapproximations of thetwo-dimensional displacement eldsareexpressed in terms of kernel particle functions. A scaling factor thatpresents the size of the support of 3.5 is used, and a backgroundcell structurebasedonthenodesisconstructedtoperformthenumerical integration. The bending matrix Kbare evaluated usinga 4 4 Gauss integration, and the membrane and shearstiffnessmatricesKm, Ksarecomputedusingone-pointGaussintegrationto eliminate the membrane and shear locking for the very thin con-icalshellpanels. Thetransformationmethodisemployedtoim-posetheessentialboundaryconditions. Threetypesofboundaryconditions, including simply-supported (SSSS), fully clamped(CCCC), and on edge clamped and the other three edges free (CFFF),are given asSSSS:At x 0; L : v0 w0 /h 0; 46aAth 0; h0: u0 w0 /x 0: 46bCCCC:u0 v0 w0 /x /h 0 on all edges: 47CFFF:At x L : u0 v0 w0 /x /h 0; 48a0 0.2 0.4 0.6 0.8-0.4-0.200.20.40 0.2 0.4 0.6 0.8-0.4-0.200.20.40 0.2 0.4 0.6 0.8-0.4-0.200.20.40 0.2 0.4 0.6 0.8-0.4-0.200.20.4Fig. 20. The rst four modes for a clamped Al/ZrO2 conical shell panel (CCCC, n = 0.5, R0 = 0.2 m, h = 0.08 m, a = 0.8 m, a = 15, h0 = 120).0 0.2 0.4 0.6 0.8-0.4-0.200.20.40.60 0.2 0.4 0.6 0.8-0.4-0.200.20.40.60 0.2 0.4 0.6 0.8-0.4-0.200.20.40.60 0.2 0.4 0.6 0.8-0.4-0.200.20.40.6Fig. 21. The rst four modes for a clamped Al/ZrO2 conical shell panel (CCCC, n = 0.5, R0 = 0.2 m, h = 0.08 m, a = 0.8 m, a = 30, h0 = 120).660 X. Zhao, K.M. Liew/ Composite Structures 93 (2011) 649664At x 0; and h 0; h0free: 48b4.1. Convergence studies4.1.1. Isotropic conical shell panelsTo validate the present formulation, comparisons between cur-rentsolutionsandthoseavailableinliteraturearemade. Afullyclamped isotropic conical panel (CCCC) is considered rst. The geo-metricpropertiesof theconical panel areL/s = 0.6, L/h = 100, L/R1 = 3, a = 30, and h0 = 60, and the material properties are Youngsmodulus E = 70 109N/m2, Poissons ratiov = 0.3. Convergencetests are performedusingthe nodal distributionrangingfrom8 8 to 16 16. Table 2 gives the corresponding results of non-dimensional frequency parameterx xL2qh=D_D Eh3=121 m2. For comparison, the solutions reported by Cheung et al.[43] and Bardell et al. [10] using the thin shell theory are also listedin the table. It can be seen that the present method shows a goodconvergence, and the solutions obtained from the proposed meth-od aresmaller than those producedfrom thenitestripmethod[43] and hp nite element method [10]. The maximum difference,however, islessthan4%. It isbelievedthat thediscrepancyisattributedtothedifferent solutionstrategiesandshell theoriesadopted. Next, a clamped conical shell panel that is dened by L/s = 0.2, s/h = 1000,a = 15, h0 = 30, andv = 0.3isconsidered. Theresults of non-dimensional frequency parameter ^ x xLb0qh=D_for the rst eight modes are given in Table 3, whichalsoincludes thesolutions reported byLim and Liew[4]using aglobalRitz formulation and thin shell theory for comparison0 0.2 0.4 0.6 0.8-0.500.50 0.2 0.4 0.6 0.8-0.500.50 0.2 0.4 0.6 0.8-0.500.50 0.2 0.4 0.6 0.8-0.500.5Fig. 22. The rst four modes for a clamped Al/ZrO2 conical shell panel (CCCC, n = 0.5, R0 = 0.2 m, h = 0.08 m, a = 0.8 m, a = 45, h0 = 120).0 0.2 0.4 0.6 0.8-0.500.50 0.2 0.4 0.6 0.8-0.500.50 0.2 0.4 0.6 0.8-0.500.50 0.2 0.4 0.6 0.8-0.500.5Fig. 23. The rst four modes for a clamped Al/ZrO2 conical shell panel (CCCC, n = 0.5, R0 = 0.2 m, h = 0.08 m, a = 0.8 m, a = 60, h0 = 120).X. Zhao, K.M. Liew/ Composite Structures 93 (2011) 649664 661purpose. Thenumber of nodesvariesfrom10 10to16 16.Again, a good convergence is observed and the difference betweentwo sets solutions is relatively less. Table 4 shows the results of thenon-dimensional frequency parameter ^ x xLb0qh=D_of therst eight modes for clamped conical panels with s/h = 1000,a = 7.5, h0 = 20 andv = 0.3. Thepresent solutionsareattainedusinga16 16nodal distributionthroughconvergenttest, andcompared with the solutions given by Limand Liew [5], and Bardellet al. [10]. It can be seen that, for conical shell panels with L/s = 0.2and L/s = 0.8, the present solutions agrees well with those in refer-encesforlowermodes, themaximumdifferenceislessthan4%.AnothercomparisonisprovidedforaconicalshellpanelwithL/s = 0.2 but having a one edge clamped and the other three edgesfree (CFFF), the results of the frequency parameter^ x xLb0qh=D_produced from the current method and reportedinliterature[5,10]areshowninTable5. Anodaldistributionof16 16isusedfollowingconvergencestudy. Itisseenthatthesolutionsobtainedfromtheproposedmethodareveryclosetothose in references. For the lower modes, the maximum differenceis less than 3%.4.1.2. Functionally graded conical shell panelsAfunctionallygradedconicalshellpanelwithCFFFboundaryconditionis considered. Thegeometric properties of thepanelare L = 2 m, R1 = 0.5 m, h = 0.1 m, a = 40, and h0 = 120. The mate-rial properties of the constituents are Ec = 168 GPa, vc = 0.3,qc = 5700 kg/m3, Em = 70 GPa, vm = 0.3, andqm = 2707 kg/m3. Thevolume fraction exponent varies from 0 to 5. This case was studiedby Tornabene et al. [21] using two different power-law distributionVC1 12 zh_ _nandVC2 12 zh_ _n. Theirstudyshowsthatthefre-quencies of theconical panel withthepower-lawdistributionVC1areveryclosetothoseoftheconicalpanelwiththepower-lawdistributionVC2, buttheformerisslightlyhigher. Sincethepower-law distribution VC2 is also adopted throughout this study,Table6shows thecorrespondingsolutions of frequencies thatare obtained using a 22 22 nodal distribution. As no numericalsolutionswereprovidedinRef. [21] fortheconical panel withthe power-law distribution VC2, the solutions produced from usingthe power-law distribution VC1 are listed in the table for compari-son purpose. Itis seen that the present results are quite close tothoseinRef. [21]. Itisconcludedthatthepresentsolutionswillmatch those in reference for the case with the power-law distribu-tion VC1.4.2. Free vibration of functionally graded conical shell panelsIn this section, the free vibration of functionally graded conicalshell panels is investigated. Two types of functionally graded con-ical panelsthatconsistof aluminumandzirconia(Al/ZrO2), Ti6Al4V and aluminum oxide (Al2O3) are selected. The conical pan-els that are made from Al/ZrO2 are considered rst. The geometricproperties of the conical panels are L = 0.8 m, R1 = 0.2 m, andh0 = 120. The values of thickness are taken to be h = 0.01 m,0.04 mand0.08 m. The semi-vertexangle ranges from15 to60, andthevolumefractionexponentvariesfrom0to10. Forall cases in this section, a regular nodal distribution 22 22 is usedthrough convergent test.Table 7 shows the results of the frequency parameter~ x xL2qc=Ech2_=2pfor the rst eight modes of simply-sup-ported(SSSS)Al/ZrO2conicalshellpanelswithh = 0.01 m. Itcanbe seen that, for the case with a xed semi-vertex angle, the valueof the frequency parameter for each mode decreases as the volumefractionexponentnincreases;forthecasewithaxedvolumefraction exponent, the value of frequency parameter declines whenthe semi-vertex angle rises. The corresponding mode shapes asso-ciated with the conical panels with semi-vertex angle a = 15, 30,45, and 60 are demonstrated in Figs. 36. It is discerned that themode shape changes with varying semi-vertex angles. For the con-ical panels with certain semi-vertex angle, the variation in volumefraction exponent has no pronounced effect on the mode shapes,but may affect the sequence of modes with high frequency. Notethat the rst four modes for the case with a = 15 appear in a dif-ferent sequence in the modes for the cases with a = 30, 45, and60. Forinstance, themode4inthecasewitha = 15becomesthe mode 6 in the case with a = 30, then changes to the mode 7inthecasewitha = 45, andturnsintothemode5inthecasea = 60. According to the half wave numbers in the x and h direc-tions, therst four modes aredenotedbymodes (1, 2), (1, 3),(1, 4), and (1, 1). The variation in their frequency magnitude withthe semi-vertex angle is illustrated in Fig. 7.Table 8 depicts the variation in the frequency parameter~ xwiththe semi-vertex angle and volume fraction exponent for theclamped(CCCC) conical Al/ZrO2shell panels, whicharedenedby the same geometric properties as the case with SSSS boundarycondition. Asimilarfrequencytrendisobserved, exceptthatthemagnitudeofthefrequencyparameterisgreaterthan thecorre-sponding value in Table 7. The corresponding mode shapes are de-10 20 30 40 50 600.51.01.52.02.53.03.5Frequency parameter Semi-vertex angle mode 1 n = 0.5 mode 2 mode 3 mode 4Fig. 24. Variationinthefrequencyparameterwiththesemi-vertexangleforAl/ZrO2 conical shell panels (SSSS, n = 0.5, R0 = 0.2 m, h = 0.08 m, a = 0.8 m, h0 = 120).10 20 30 40 50 6012345Frequency parameter Semi-vertex angle mode 1 n = 0.5 mode 2 mode 3 mode 4Fig. 25. Variationinthefrequencyparameterwiththesemi-vertexanglefroAl/ZrO2 conical shell panels (CCCC, n = 0.5, R0 = 0.2 m, h = 0.08 m, a = 0.8 m, h0 = 120).662 X. Zhao, K.M. Liew/ Composite Structures 93 (2011) 649664scribedinFigs. 811. Comparedtothemodeshapesinthecasewith SSSS boundary condition, it is found that the correspondingmode shapes alter drastically due to the change of theboundarycondition. Except the rst two modes, the other higher modes varyasthesemi-vertexangleincreases. Table9showsthefrequencycharacteristics of the Al/ZrO2 conical shell panels with CFFF bound-arycondition. Asimilarfrequencypatternisdiscerned, but themagnitudeofthefrequencyparameterislessthanthoseforthecases with SSSS and CCCC boundary conditions.Next, the effects of the thickness on the frequency characteris-tics of functionally graded conical shell panels are also examined.TheAl/ZrO2conical panels that haveathickness of h = 0.04 mand other geometric properties same as those in last case are con-sidered. Table 10 reveals the variation in the frequency parameter~ xwithdifferent semi-vertexanglesandvolumefractionexpo-nents. It can be seen that, except a corresponding less magnitudeof frequency parameter, the frequency pattern is similar to that ob-servedfromthe simply-supportedAl/ZrO2conical panels withthicknessh = 0.01 m. Thecorrespondingmodeshapesareshownin Figs. 1215. It is discerned that, compared to the mode shapesfor the case with h = 0.01 m, the current modes change their shapesand sequence due to the increase of the thickness. A similar obser-vation is also made from Table 11 and Figs. 1619, which describethe frequency characteristics of the clamped Al/ZrO2 conical panelswith thickness h = 0.04 m.Table 12 gives the results of the frequency parameter for rsteight modes of the relatively thick Al/ZrO2 conical shell panels withthicknessh = 0.08 mandvolumefractionexponentn = 0.5. BothSSSS and CCCC boundary conditions are considered, and thesemi-vertex angle varies from 15 to 60. The corresponding modeshapes for the rst four modes of the clamped case are depicted inFigs. 2023. ThevariationinthefrequencyparameterwiththeTable 13The rst eight frequency parameters ~ x xa2qc=Ec h2_=2p for the FGM Ti6Al4V/Al2O3 conical shell panels (SSSS, R1 = 0.4 m, h = 0.01 m, L = 1.0 m, h0 = 120).a () n Mode1 2 3 4 5 6 7 815 0 5.0254 6.5523 6.6639 9.1414 10.5186 11.2358 12.2978 12.62230.5 4.2015 5.4265 5.6729 7.5605 8.7825 9.5082 10.1642 10.46361 3.8167 4.9232 5.1692 6.8587 7.9769 8.6548 9.2205 9.49455 3.1731 4.1253 4.2033 5.8868 6.6593 6.9943 7.9362 8.087810 3.0099 3.8684 4.0109 5.6233 6.3227 6.5840 7.5851 7.713830 0 4.4781 4.9694 6.3866 6.6001 8.1803 9.2038 10.1564 10.24740.5 3.7662 4.1309 5.2949 5.6264 6.7719 7.7189 8.4518 8.52931 3.4248 3.7501 4.8054 5.1284 6.1451 7.0166 7.6741 7.73945 2.8085 3.1710 4.0843 4.0987 5.2676 5.7988 6.3696 6.478410 2.6541 3.0187 3.8268 3.9091 5.0294 5.4905 5.9887 6.164545 0 3.9242 4.0868 5.0145 5.8215 6.2476 7.7333 7.8194 8.45230.5 3.3066 3.4048 4.1633 4.9636 5.1769 6.4007 6.5626 7.07741 3.0079 3.0923 3.7795 4.5246 4.6988 5.8089 5.9665 6.42815 2.4574 2.6014 3.2131 3.6029 4.0198 4.9264 4.9898 5.261910 2.3196 2.4729 3.0618 3.3755 3.8358 4.6628 4.7655 4.949360 0 3.1471 3.3677 4.1239 4.3716 5.1349 6.2515 6.2843 6.28780.5 2.6477 2.8057 3.4232 3.7241 4.2539 5.2533 5.2592 5.29821 2.4082 2.5484 3.1079 3.3942 3.8614 4.7682 4.7796 4.82185 1.9759 2.1459 2.6461 2.7073 3.3089 3.8615 3.9292 3.983710 1.8669 2.0402 2.5221 2.5376 3.1583 3.6149 3.7053 3.7791Table 14The rst eight frequency parameters ~ x xa2qc=Ec h2_=2p for the FGM Ti6Al4V/Al2O3 conical shell panels (CCCC, R1 = 0.4 m, h = 0.01 m, L = 1.0 m, h0 = 120).a () n Mode1 2 3 4 5 6 7 815 0 8.1272 8.2602 11.4868 12.3584 13.2458 13.4382 15.6253 16.08480.5 6.8338 6.8968 9.5625 10.3603 11.1163 11.2523 13.0719 13.46581 6.2134 6.2648 8.6832 9.4160 10.1053 10.2252 11.8796 12.23835 5.0989 5.2425 7.3312 7.7858 8.3437 8.4879 9.9219 10.185510 4.8147 4.9752 6.9742 7.3609 7.8875 8.0355 9.4191 9.657230 0 6.6318 6.8175 8.6610 9.4262 10.9338 11.1305 12.0577 12.15320.5 5.5534 5.7246 7.2531 7.8567 9.1763 9.3213 10.0150 10.23971 5.0684 5.2042 6.5922 7.1362 8.3426 8.4716 9.0927 9.31575 4.1916 4.2865 5.4845 6.0058 6.8918 7.0341 7.6338 7.732210 3.9704 4.0492 5.1993 5.7025 6.5207 6.6604 7.2018 7.364545 0 5.4423 5.5874 6.9214 7.3659 9.0501 9.1158 9.4037 9.50230.5 4.5618 4.6887 5.8063 6.1414 7.5940 7.6136 7.8403 8.01081 4.1465 4.2624 5.2789 5.5789 6.9048 6.9177 7.1231 7.28995 3.4356 3.5181 4.3747 4.6963 5.7113 5.7822 5.9768 6.013810 3.2521 3.3248 4.1425 4.4596 5.4049 5.4779 5.6385 5.723960 0 4.3253 4.3692 5.4485 5.8182 7.0284 7.2452 7.2884 7.54020.5 3.6218 3.6710 4.5551 4.8732 5.9066 6.0715 6.1081 6.30331 3.2919 3.3381 4.1400 4.4302 5.3740 5.5199 5.5535 5.72965 2.7365 2.7480 3.4658 3.6872 4.4573 4.5828 4.6142 4.804910 2.5921 2.5957 3.2897 3.4885 4.2229 4.3374 4.3715 4.5529X. Zhao, K.M. Liew/ Composite Structures 93 (2011) 649664 663semi-vertex anglefor therstfourmodes isdepictedin Figs. 24and25fortheSSSSandCCCCcases, respectively. Comparedtothose in cases with thickness h = 0.01 m and h = 0.04 m, it is foundthat the mode shapes change again.The free vibration of the Ti6Al4V/Al2O3 conical shell panels isalsoinvestigated. ThegeometricpropertiesaregivenbyL = 1 m,R1 = 0.4 m, h1 = 0.01 m, and h0 = 120. The semi-vertex anglechangesfrom15to60, andthevaluesof thevolumefractionexponents aretaken toben = 0, 0.5, 1,5, 10. Thefrequency pat-ternsfortheTi6Al4V/Al2O3conicalshell panelswithclampedandsimplysupportedboundaryconditionsareshowninTables13 and 14, respectively. It is seen that, except with a greater mag-nitudeofthecorrespondingfrequencyparameter, thefrequencyresponseis similar tothat discernedfromtheAl/ZrO2conicalpanels.5. ConclusionsThe free vibration of functionally graded conical shell panels isinvestigatedusingthemesh-freekp-Ritzmethod. Theeffectivematerial propertiesaredeterminedaccordingtotheassumptionof a power-law distribution of the volume fraction of constituents.The formulation is based on the rst-order shear deformation shelltheory, which caters for both thin and relatively thick conical shellpanels. The approximations of displacement elds are expressed interms of kernel particle shape functions, and the bending stiffnessandshearstiffnessareseparatelytreatedtoeliminatetheshearlocking. The present formulation is validated by comparisons be-tween the current solutions and those available in literature, andtheeffectsof thevolumefractionexponent, semi-vertexangle,thickness, and boundary condition on the frequency characteristicsare examined. 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