Three dimensional Free Vibration and Transient Analysis of Two Directional Functionally Graded Thick Cylindrical Panels Under Impact Loading

Transcription

1 5 hee dimensional Fee Vibation and ansient Analysis of wo Diectional Functionally Gaded hick Cylindical Panels Unde Impact Loading Abstact In this pape thee dimensional fee vibation and tansient esponse of a cylindical panel made of two diectional functionally gaded mateials (D-FGMs) based on thee dimensional equations of elasticity and subjected to intenal impact loading is consideed. Mateial popeties vay though both adial and axial diections continuously. he 3D gaded finite element method (GFEM) based on Rayleigh-Rit enegy fomulation and Newmak diect integation method has been applied to solve the equations in space and time domains. he fundamental nomalied natual fequency, time histoy of displacements and stesses in thee diections and velocity of adial stess wave popagation fo vaious values of span angel of cylindical panel and diffeent powe law exponents have been investigated. he pesent esults show that using D-FGMs leads to a moe flexible design than conventional D-FGMs. he GFEM solution have been compaed with the esults of an FG thick hollow cylinde and an FG cuved panel, whee a good ageement between them is obseved. Keywods Dynamic Analysis, Impact Loading, Cylindical Panel, heoy of Elasticity, D Functionally Gaded Mateials, Gaded Finite Element Method Hassan Zafamand a Manoucheh Salehi b Kaman Asemi c a Mechanical Engineeing Depatment, Amikabi Univesity of echnology, ehan, Ian. (Pemanent Adess: Depatment of Mechanical Engineeing, Fedowsi Univesity of Mashhad, Mashhad, Ian.) Ha.afamand@stu.um.ac.i b Mechanical Engineeing Depatment, Amikabi Univesity of echnology, ehan, Ian. Msalehi@aut.ac.i c Mechanical Engineeing Depatment, Amikabi Univesity of echnology, ehan, Ian. Kamian64@yahoo.com Received 6..3 In evised fom Accepted..4 Available online 3..4

2 6 H.Zafamand et al./ hee dimensional fee vibation and tansient analysis of two diectional panels unde impact loading INRODUCION In ecent yeas, the composition of seveal diffeent mateials is often used in stuctual components in ode to optimie the esponses of stuctues subjected to themal and mechanical loads. Functionally gaded mateials (FGMs) ae suitable to achieve this pupose. FGMs ae composite mateials, micoscopically inhomogeneous, in which the mechanical popeties vay smoothly and continuously fom one suface to the othe. his idea was used fo the fist time by Japanese eseaches (Koiumi, 993), which led to the concept of FGMs. Cylindical panels ae of pactical inteest in many fields of engineeing, including civil, mechanical, and aeospace, as they give ise to optimum conditions fo dynamic behavio. he dynamic loadings on these stuctues can have seious consequences fo thei stength and safety. Many papes studying FGMs have been published ecently. A citical litatue eview of these woks is pesented by Biman and Byd (7) and Jha et al. (3). Also, a wide ange of eseaches has been caied out on dynamic analysis of FGM stuctues (Qian et al., 4; Asemi et al., ; Akbaadeh et al., ; Sun and Luo, ; Malekadeh and Monajjemadeh, 3; Uyma and Aydogdu, 7) which mostly use the plate and shell theoies. Among the FGM stuctues, little attention has been given to analysis of FGM cylindical stuctues than those of ectangula and cicula ones. Fo example, Shakei et al. (6) pesented the vibation and adial wave popagation velocity in an FG thick hollow cylinde unde dynamic loading by dividing the cylinde into some homogeneous sub-cylindes. Hosseini et al. (7) studied the dynamic esponses and natual fequencies of an infinite FG cylinde unde intenal pessue. Asgai et al. (9) studied the dynamic behavio of a D-FG thick hollow cylinde with finite length unde impact loading. Asemi et al. () investigated the dynamic behavio and adial wave popagation of thick shot length FG cylindes. Shao et al. (4; 5) analytically obtained the elastic behavio of finite length FG hollow cylindes with simply suppoted end conditions unde the hydostatic pessues. In these publications, in ode to satisfy the bounday conditions, the sinusoidal load along the axial diection can only be used. Bodaghi and Shakei () by using Hamilton s pinciple and fist ode shea defomation theoy (FSD) studied the fee vibation and dynamic esponse of simply suppoted FG pieoelectic cylindical panel impacted by timedependent blast pulses. Yas and Sobhani Aagh () investigated the thee dimensional steady state esponse of an FG fibe einfoced othotopic cylindical panel simply suppoted at the adial edges. Wang and Sudak (8) pesented a geneal method fo the analysis of a themoelastic multi-layeed cylindical panel made of an oblique pile of FG layes having othotopic mateial popeties. he two edges of the panel ae simply-suppoted. Bahtui and Eslami (7) studied the coupled themoelastic esponse of an FG axisymmetic cylindical shell. A second ode shea defomation shell theoy and Galekin finite element fomulation ae used to fomulate the poblem. Dava et al. (3) investigated the esponse of simply suppoted cicula cylindical shell made of FGM subjected to lateal impulse load. Fist ode shea defomation theoy and Love s fist appoximation theoy ae utilied in the equilibium equations. Qu et al. (3) studied vibational esponse of FG shells of evolution with abitay bounday conditions. he fomulation is deived by means of a modified vaiational pinciple in conjunction with a multisegment patitioning pocedue on the basis of the fist-ode shea defomation shell theoy. Ebahimi and Najafiadeh (4) analyed the fee vibation of a F-FG cicula cylindical shell. Latin Ameican Jounal of Solids and Stuctues (5) 5-5

3 H.Zafamand et al./ hee dimensional fee vibation and tansient analysis of two diectional panels unde impact loading 7 he equations of motion ae based on the Love s fist appoximation classical shell theoy and the equations of motion and bounday conditions ae discetied by the methods of genealied diffeential quadatue (GDQ) and genealied integal quadatue (GIQ). he liteatue eview denotes that most of the studies deal with axisymmetic FG cylindical shells and they ae based on the shea defomation theoies. he application of these theoies to thick stuctues can cause consideable eos. heefoe, fo eliminating the lack of these studies, the 3D elasticity solutions not only povides ealistic and accuate esults but also allows futhe physical insights, which cannot othewise be estimated by the othe two dimensional o plate and shell theoies. Also, in these studies, the mateial popeties ae assumed to have a smooth vaiation usually in one diection and majoity of eseaches woks into static analyses. Investigations into dynamic analysis of FG cylindical stuctues ae limited to one and two dimensional analyses. Also, analytical o semi-analytical solutions ae available only though a numbe of poblems with simple bounday conditions o fo cylindical panels with simply suppoted adial edges. Moeove, conventional functionally gaded mateials may also not be so effective in such design poblems since all oute suface of the body will have the same composition distibution. heefoe, vaiation of volume faction in two diections has a highe capability to educe the mechanical, themal and esidual stesses and leads to a moe flexible design than D-FGMs. Some studies have been caied out about static and dynamic analyses of stuctues made of D-FGMs (Asgai and Akhlaghi, ; Nemat-Allah, 9; Zafamand and Hassani, 4). Due to difficulty in obtaining analytical solutions fo dynamic analysis of D-FG cylindical panel based on 3D elasticity and fo abitay bounday conditions, poweful numeical methods such as gaded finite element method (GFEM) ae needed. he gaded finite elements, which incopoate the mateial popety gadient at the sie scale of the element, have been employed a genealied isopaametic fomulation. Some woks can be found in the liteatue on modeling nonhomogeneous stuctues by using GFEM (Kim and Paulino, 9; Ashafi et al., ; Zafamand and Kadkhodayan, 4). In these studies, it is shown that the conventional FEM fomulation cause a discontinuous stess field in the diection pependicula to the mateial popety gadation, while the gaded elements give a continuous and smooth vaiation. Howeve, when the loading is paallel to the mateial gadation diection, the GFEM fomulation estimates shap ises in stess at the element boundaies while the conventional FEM fomulation does not. o the best knowledge of the authos, thee ae no studies available in the liteatue on dynamic analysis of D-FG cylindical panel with clamped bounday conditions based on 3D elasticity and gaded finite elements. he govening equations ae deived based on pinciple of minimum potential enegy and Rayleigh-Rit method. o solve the time dependent equations, Newmak diect integation method with suitable time steps has been used. he effects of volume faction exponents and diffeent span angles on the time histoies of displacements, stesses, velocities of adial stess wave popagation and fundamental nomalied natual fequency have been investigated. Latin Ameican Jounal of Solids and Stuctues (5) 5-5

4 8 H.Zafamand et al./ hee dimensional fee vibation and tansient analysis of two diectional panels unde impact loading PROBLEM FORMULAION. Volume faction and mateial distibution in D-FG cylindical panel A functionally gaded cylindical panel with inne adius a, oute adius b, length L and span angle subjected to an intenal impact loading is consideed. he geomety and coodinate system of the cylindical panel is shown in Figue. Figue : Geomety of the cylindical panel D-FGMs ae usually made by continuous gadation of thee o fou diffeent mateial phases whee one o two of them ae ceamics and the othes ae metal alloy phases. Significant developments in fabication and pocessing methods have made it possible to poduce FGMs with two and thee dimensional gadients using compute aided manufactuing pocesses. heefoe, the inne suface of the cylindical panel is made of two distinct ceamics and the oute suface of two metals. he volume factions of the constituent mateials vay continuously though adial and axial diections in a pedetemined composition pofile. he volume faction distibution function of each mateial can be expessed as: a Vc (, ) b a L n n () Vc (, n n ) a b a L () a Vm (, ) b a L n n (3) Latin Ameican Jounal of Solids and Stuctues (5) 5-5

5 H.Zafamand et al./ hee dimensional fee vibation and tansient analysis of two diectional panels unde impact loading 9 n n m (, ) a V (4) b a L whee c, c, m and m denote fist ceamic, second ceamic, fist metal and second metal, espectively. n and n ae non-negative volume faction exponents though the and diections. Fo instant, the volume faction distibutions of the fist metal ( V m ) fo the typical values of n n 3 is shown in Figue. In this case a. m, b.4 m and L m. Accoding to this figue, V m is equal to one at b, and equal to eo at othe cones and its vaiation in plane depends on n and n. Similaly V m, V c and V c ae equal to one at b, L, a, and a, L, espectively. Figue : Volume faction distibution of m with n n 3 Mateial popeties at each point can be obtained by using the linea ule of mixtues, theefoe the mateial popety ( R ) such as modulus of elasticity and mass density in the D-FG cylindical panel is detemined by linea combination of volume factions and mateial popeties of the basic mateials as: R R V R V R V R V (5) c c c c m m m m Accodingly, the cylindical panel is pue c, c, m and m at a,, a, L, b, and b, L, espectively and the vaiation of desied mateial popety depends on the volume faction distibutions of the constituents in which the volume factions depends on n and n. Fo example, the vaiation of a mateial popety such as modulus of elasticity based on the mentioned appoach fo the typical values of n n 3 is shown in Figue 3. As it can Latin Ameican Jounal of Solids and Stuctues (5) 5-5

6 H.Zafamand et al./ hee dimensional fee vibation and tansient analysis of two diectional panels unde impact loading be seen fom this figue, the elastic modulus is equal to E c, E c, E m and E m at a,, a, L, b, and b, L, espectively and vaies in plane based on Eq. (5). he volume factions in Eqs. (-4) educe to the conventional D-FGMs fo n and in this case, the mateial popeties vay only though the thickness diection. he basic constituents of the D-FG cylindical panel ae pesented in able. It should be noted that Poisson s atio is assumed to be a constant value of.3. his assumption is easonable because of the small diffeences between the Poisson s atios of basic mateials. Figue 3: Distibution of elasticity modulus with n n 3 Constituents M ateial E ( GPa ) 3 ( kg m ) m i6a4v m Al, 7 77 c MnO c ZO 5 3 able : Basic constituents of the D-FG cylindical panel. Govening Equations Neglecting body foces, the equations of motion in cylindical coodinates ae obtained as: u (, ) (, ) t v t (6-) (6-) Latin Ameican Jounal of Solids and Stuctues (5) 5-5

7 H.Zafamand et al./ hee dimensional fee vibation and tansient analysis of two diectional panels unde impact loading w (, ) t (6-3) whee u, v and w ae adial, cicumfeential and axial components of displacement, espectively and is the mass density that depends on and coodinates. he stess-stain elations fom the Hook s law in the matix fom ae as: D (7) whee the stess and stain components and the coefficients of elasticity D, ae as the following elations: { } (8) { } (9) E(, ) D () ( )( ) whee denotes the Poisson s atio and E is the Young s modulus of elasticity that depends on and coodinates. he stain-displacement equations based on the theoy of linea elasticity in cylindical coodinate ae: u, v w v u w u, () w, u v v and in the matix fom ae as: U () Latin Ameican Jounal of Solids and Stuctues (5) 5-5

8 H.Zafamand et al./ hee dimensional fee vibation and tansient analysis of two diectional panels unde impact loading is a matix containing patial diffeentiating equa- whee U is the displacements vecto and tions as: U { u v w} (3) (4) he cylindical panel is clamped on its two end edges and it is subjected to intenal pessue that is a function of time. heefoe, the bounday conditions ae as: U (5), L he intenal pessue function p () t is an abitay function that will be pesented late..3 Gaded finite element modeling In ode to solve the govening equations the gaded finite element method is used. In this method, in addition to displacement field, the heteogeneity of the mateial popeties of the FGM may also be detemined based on thei nodal values. In this egad, shape functions simila to those of the displacement field may be used. his pocedue can be used even fo multidiectional functionally gaded mateials. heefoe, a gaded finite element method (GFEM) is used to effectively tace smooth vaiations of the mateial popeties at the element level. Using the gaded elements fo modeling of gadation of the mateial leads to moe accuate esults than dividing the solution domain into homogenous elements. he cylindical panel is divided into a numbe of 8 node linea elements. Fo element () e, the displacements in thee diections ae appoximated as follow: ( e ) ( e) U (6) whee is the matix of linea shape functions in cylindical coodinate and displacement vecto of the element that ae as: () e is the nodal 8 8 (7) () 8 8 e { U V W U V W } (8) 8 8 the components of Φ ae given in the appendix. Latin Ameican Jounal of Solids and Stuctues (5) 5-5

9 H.Zafamand et al./ hee dimensional fee vibation and tansient analysis of two diectional panels unde impact loading 3 o teat the mateial inhomogeneity by using the GFEM, it may be witten: ( e) 8 (9) i i i () e whee is the mateial popety of the element. Substituting Eq. (6) in Eq. () gives the stain matix of element ( e ) as: ( e) ( e) B () whee: () e B () Now by using Hamilton s pincipal and Rayleigh-Rit enegy fomulation, the GFEM is imposed and finally the mass, stiffness and foce matices ae obtained as following: t t W dt () whee, and W ae potential enegy, kinetic enegy and vitual wok done by suface tactions, espectively. hese functions and thei vaiations ae: V dv (3) V dv (4) V U U dv (5) U U dv V (6) W A P UdA (7) W A P UdA (8) whee V and A ae the volume and aea of the domain unde consideation and P is the vecto of suface taction and in the case of intenal pessue is as: P { p } (9) Latin Ameican Jounal of Solids and Stuctues (5) 5-5

10 4 H.Zafamand et al./ hee dimensional fee vibation and tansient analysis of two diectional panels unde impact loading Substituting Eqs. (3-8) in Hamilton s Pincipal, applying side conditions U and using pat integation: t, t dv U UdV P UdA V V A (3) Fo each element, by imposing Eqs. (7, 6 and ) into Eq. (3), it can be achieved: ( e) ( e) ( e) ( e) V dv () e A V PdA B DBdV (3) () e Since is the vaiation of the nodal displacements and is abitay, it can be omitted fom Eq. (3), thus this equation can be witten as: whee the chaacteistic matices ae defined as: ( e) ( e) ( e) ( e) ( e) M K F (3) M () e V dv (33) () e K B DB dv V (34) As D and () e F P da A ae not constant, fo obtaining the chaacteistic matices fo each element, numeical integation should be applied. Now by assembling the element matices and imposing the bounday conditions, the global equations of motion fo the D-FG cylindical panel can be witten as: (35) M K F (36) Once the finite element equations ae detemined, fo fee vibation analysis whee the foce matix ( F ) is eo, by substituting i t e ( is the natual fequency) into Eq. (36), an eigen value poblem would be obtained that can be solved using standad eigen value extaction pocedues. Also, in the case of tansient analysis, the Newmak diect integation method with suitable time steps is utilied to solve the equations. Newmak integation paametes ae taken as (Zienkiewic and aylo, 5):, 4 Newmak Newmak (37) Latin Ameican Jounal of Solids and Stuctues (5) 5-5

11 H.Zafamand et al./ hee dimensional fee vibation and tansient analysis of two diectional panels unde impact loading 5 which lead to a constant aveage acceleation. his choice of paametes coesponds to a tapeoidal ule which is unconditionally stable in linea analyes. 3 NUMERICAL RESULS AND DISCUSSION 3. Validation o validate the tansient analysis of the cuent wok, the data of an FG thick hollow cylinde unde intenal impact loading can be used (Asemi et al., ). Mateial distibution is assumed to be as D-FGM and it vaies in adial diection with a powe law function. In ode to validate the esults of this pape with the esults of the pesent study, the span angel of the panel should be taken to become a hollow cylinde, n is taken eo, a = m, b =.5 m, L = m, 3 3 Em 7 GPa, m 77 kg m, Ec 38 GPa, c 38 kg m and the intenal loading function is assumed to be as: p () t Pt t.5 s t.5 s (38) whee P is 4 GPa s. he compaison of the adial displacement at.5 m and.5 fo n.5 with the published data is shown in Figue 4 and a good ageement between these two esults is obseved. m x -4 Radial displacement (m) - Pesent Asemi et al ime (s) Figue 4: ime histoy of the adial displacement at.5 m and.5 m fo n.5 compaed with (Asemi et al., ) Also, in ode to validate the fee vibation analysis of this study, the data of an FG cuved panel can be used (Zahedinejad, ). he mateial popeties ae assumed to be vaied fom ceamic ( 3 3 Ec 38 GPa, c 38 kg m ) at inne adius to metal ( Em 7 GPa, m 7 kg m ) at Latin Ameican Jounal of Solids and Stuctues (5) 5-5

12 6 H.Zafamand et al./ hee dimensional fee vibation and tansient analysis of two diectional panels unde impact loading oute adius with a powe law function. he bounday conditions ae simply suppoted in fou edges and the geomety is defined as: a.75 m, L m b.5 m, ad (39) he compaison of fundamental fequency paamete ( ( b a) c E c ) with the published esults fo diffeent powe law exponents is made in able. Accoding to this table, a closed ageement between the pesent esults and those obtained fom Zahedinejad () is obseved. n.5 4 Pesent Zahedinejad () Diffeence (% ) able : he fundamental fequency paamete fo FG cylindical panel with fou edges simply suppoted compaed with (Zahedinejad, ) 3.3 Numeical Results Conside a D-FG cylindical panel with inne adius a =. m, oute adius b =.4 m and length of L =.5 m. he panel is clamped at, L and subjected to an impact loading at its inne suface. Constituent mateials ae two distinct ceamics and two distinct metals descibed in able. he Poisson s atio is taken as a constant value of.3 and the loading function is assumed to be as: p () t P sin( t t ) t t t t (4) 7 4 whee P and t ae loading constants that ae assumed to be as Pa and 5 s, espectively. Vaiation of the intenal pessue with time is displayed in Figue 5. Accoding to this 4 figue, the cylindical panel is excited by unloading in t 5 s. It is obvious that afte the unloading, a tansient vibation which is affected by the wave popagation, eflection and intefeence would be occued. he thee dimensional tansient and fee vibation analysis fo diffeent values of the powe law exponents and span angles ae pesented and discussed as following. Latin Ameican Jounal of Solids and Stuctues (5) 5-5

13 H.Zafamand et al./ hee dimensional fee vibation and tansient analysis of two diectional panels unde impact loading 7 x 6 Pessue (Pa) ime (s) x -3 Figue 5: Vaiation of intenal pessue ( p ) with time. Figues 6 and 7 show the time histoy of adial displacement at.3 m, 3,.5 m and.35 m, 5,.5 m, espectively afte the unloading fo diffeent values of the powe law exponents. he span angle of the cylindical panel is assumed to be 3. hese figues denote that the maximum amplitude of vibation belongs to the panel with powe law exponents of n, n 3. In this case the cylinde is made of fist and second metal and the mateial distibution vay though the axial diection only. Also it can be seen fom this figues that the minimum amplitude of vibation is fo n 3, n 3. 6 x -5 n =n =.5 n =n =3 n =3, n = n =, n =3 Radial displacement (m) ime (s) x -3 Figue 6: ime histoy of the adial displacement at.3 m, 3,.5 m with 3 fo diffeent values of n and n Latin Ameican Jounal of Solids and Stuctues (5) 5-5

14 8 H.Zafamand et al./ hee dimensional fee vibation and tansient analysis of two diectional panels unde impact loading Radial displacement (m) 4 x n =n =.5 n =n =3 n =3, n = n =, n = ime (s) x -3 Figue 7: ime histoy of the adial displacement at.35 m, 5,.5 m with 3 fo diffeent values of n and n he time histoy of cicumfeential displacement at.5 m, 45,.375m fo a cylindical panel with 3 afte the unloading fo diffeent values of powe law exponent is demonstated in Figue 8. Accoding to this figue, due to asymmety of geomety, the cicumfeential displacement cannot be neglected. hese esults show how the time histoies of displacements field ae affected by the vaiation of volume faction of constituent mateials in two diections. Cicumfeential displacement (m) 5-5 x -6 n =n =.5 n =n =3 n =3, n = n =, n =3.5.5 ime (s) x -3 Figue 8: ime histoy of the cicumfeential displacement at.5 m, 45,.375 m with 3 fo diffeent values of n and n Latin Ameican Jounal of Solids and Stuctues (5) 5-5

15 H.Zafamand et al./ hee dimensional fee vibation and tansient analysis of two diectional panels unde impact loading 9 he time histoy of adial, cicumfeential, shea ( ) and axial stesses, espectively at.3 m, 3,.5 m fo a cylindical panel with 3 afte the unloading fo diffeent values of powe law exponent ae shown in Figues 9-. It is obvious that vaiation of stesses is highly dependent of powe law exponents. Also, as the same as cicumfeential displacement, the shea stess ( ) cannot be neglected in asymmetic geometies. 8 x 5 n =n =.5 n =n = n =n =3 6 Radial stess (Pa) ime (s) x -3 Figue 9: ime histoy of the adial stess at.3 m, 3,.5 m with 3 fo diffeent values of n and n Cicumfeential stess (Pa) 8 x n =n =.5 n =n = n =n = ime (s) x -3 Figue : ime histoy of the cicumfeential stess at.3 m, 3,.5 m with 3 fo diffeent values of n and n Latin Ameican Jounal of Solids and Stuctues (5) 5-5

16 H.Zafamand et al./ hee dimensional fee vibation and tansient analysis of two diectional panels unde impact loading.5 x 5 n =n =.5 n =n = n =n =3 Shea stess (Pa) ime (s) x -3 Figue : ime histoy of the shea stess ( ) at.3 m, 3,.5 m with 3 fo diffeent values of n and n x 6 n =n =.5 n =n = n =n =3 Axial stess (Pa) ime (s) x -3 Figue : ime histoy of the axial stess at.3 m, 3,.5 m with 3 fo diffeent values of n and n Figues 3 and 4 illustate the time histoy of the adial displacement and stess at.3 m, 3,.5 m, espectively afte the unloading fo diffeent values of with powe law exponents of n, n. It can be seen that by inceasing the span angle of cylindical panel, the amplitude of adial displacement is deceased while the fequency of vibation is inceased. Latin Ameican Jounal of Solids and Stuctues (5) 5-5

17 H.Zafamand et al./ hee dimensional fee vibation and tansient analysis of two diectional panels unde impact loading Radial displacement (m) x beta=6 beta=9 beta= ime (s) x -3 Figue 3: ime histoy of the adial displacement at.3 m, 3,.5 m with n, n fo diffeent values of 8 x 5 6 beta=6 beta=9 beta= Radial stess (Pa) ime (s) x -4 Figue 4: ime histoy of the adial stess at.3 m, 3,.5 m with n, n fo diffeent values of Fom the time histoy of the adial stess, it is possible to extact the appoximate adial stess wave popagation velocity. hus, by computing the time taken by the wave to cove the whole of the cylinde immediately afte the loading, the appoximate adial stess wave popagation velocities can be estimated. able 3 shows the compaison of adial stess wave popagation velocity with analytical esults (Jones, 997) fo cylindical panels made of diffeent homogenous mateials. As it is obvious fom this table, the obtained esults have a good compatibility with analyti- Latin Ameican Jounal of Solids and Stuctues (5) 5-5

18 H.Zafamand et al./ hee dimensional fee vibation and tansient analysis of two diectional panels unde impact loading cal ones. able 4 illustates the vaiation of adial stess wave popagation velocity at diffeent heights of cylindical panel fo diffeent values of powe law exponents. his table denotes that velocity of adial stess wave is stongly affected by the powe law exponents. Also due to the vaiation of mateial popeties though the axial diection, the velocity of adial stess wave at diffeent heights of the cylindical panel is changing. M ateial Pesent ( m s) Analytical ( m s) Diffeence (% ) n =, n = m n = n = m n = n = c n =, n = c able 3: Compaison of adial stess wave popagation velocity of homogenous cylindical panel with analytical esults (Jones, 997) L 4 L 3L 4 n = n = n = n = n = 3, n = n =, n = able 4: Velocity of adial stess wave popagation at diffeent heights of cylindical panel fo diffeent values of powe law exponents Now by intoducing the nomalied natual fequency as: b m E m (4) the fundamental nomalied natual fequency of D-FG cylindical panel fo diffeent values of the powe law exponents and span angles ae pesented. he effect of powe law exponents on the fundamental nomalied natual fequency ( ) of D-FG cylindical panel with 3 is shown in able 5. As it can be seen, by inceasing n and deceasing n the fundamental nomalied natual fequency inceases. n = n = n = 3 n n n 3 n n n 3 n n n able 5: Fundamental nomalied natual fequency fo diffeent values of powe law exponents with 3 Latin Ameican Jounal of Solids and Stuctues (5) 5-5

19 H.Zafamand et al./ hee dimensional fee vibation and tansient analysis of two diectional panels unde impact loading 3 able 6 illustates the effect of span angle of the D-FG cylindical panel on the fundumental nomalied natual fequency with n and n. It is obvious that by inceasing the span angle of cylindical panel, the fundamental natual fequency gowths able 6: Fundamental nomalied natual fequency fo diffeent values of span angle with n, n Results denote that the distibution of displacements and stesses and natual fequencies ae stongly affected by the powe law exponents and the span angle of cylindical panel. As it can be seen fom the esults, the distibution of stesses has continuous vaiations due to using gaded elements. he use of GFEM has seveal benefits ove the use of conventional FEM in the dynamic and wave popagation analyses. In conventional FEM, the bounday of homogenous elements of an inhomogeneous mateial acts as discete boundaies fo the stess waves. hese boundaies cause atificial wave eflections and have a cumulative effect on the magnitude and speed of popagating stess waves. In addition, these inaccuacies will be moe significant in D-FGM poblems. heefoe, by using gaded finite element in which the mateial popety is gaded continuously though the elements, accuacy can be impoved without efining the mesh sie. 4 CONCLUSIONS he main pupose of the pesent pape is to study the 3D tansient and fee vibational behaviou of D-FG cylindical panels. he 3D gaded finite element method and Rayleigh-Rit enegy fomulation is applied. he poposed method is validated by the esult of a D-FG thick hollow cylinde unde the impact loading and an FG cuved panel which ae extacted fom published liteatue. he compaisons between the esults show that the pesent method has a good ageement with the existing esults. Diffeent types of displacement and stess in thee diections and fundamental nomalied natual fequency ae pesented fo vaious values of powe law exponents and span angle of cylindical panel. Also, velocity of adial stess wave popagation fo diffeent powe law exponents is obtained. Results denote that the maximum stesses, stess distibution, natual fequency and velocities of adial stess wave can be contolled by the vaiation of mateial popeties in two diections. Based on the achieved esults, D-FGMs have poweful potentials fo designing and optimiing stuctues unde multi-functional equiements. Refeences Akbaadeh, A.H., Abbasi, M., Hosseini ad, S.K., Eslami, M.R. (). Dynamic analysis of functionally gaded plates using the hybid Fouie-Laplace tansfom unde themomechanical loading. Meccanica 46: Asemi, K., Salehi, M., Akhlaghi, M. (). Dynamic analysis of functionally gaded thick tuncated cone. Intenational Jounal of Mechanics and Mateials in design 6: Latin Ameican Jounal of Solids and Stuctues (5) 5-5

20 4 H.Zafamand et al./ hee dimensional fee vibation and tansient analysis of two diectional panels unde impact loading Asemi, K., Akhlaghi, M., Salehi, M. (). Dynamic analysis of thick shot length FGM cylindes. Meccanica 47: Asgai, M., Akhlaghi, M., Hosseini, S.M. (9). Dynamic analysis of two-dimensional functionally gaded thick hollow cylinde with finite length unde impact loading. Acta Mechanica 8:63-8. Asgai, M., Akhlaghi, M. () ansient themal stesses in two-dimensional functionally gaded thick hollow cylinde with finite length. Achive of Applied Mechanics 8: Ashafi, H., Asemi, K., Shaiyat, M., Salehi, M. () wo dimensional modeling of heteogeneous stuctues using gaded finite element and bounday element methods. Meccanica 48: Bahtui, A., Eslami, M.R. (7) Coupled themoelasticity of functionally gaded cylindical shells. Mechanics Reseach Communications 34:-8. Biman, V., Byd, L. (7) Modeling and analysis of functionally gaded mateials and stuctues. Applied Mechanics Reviews 6:95-6. Bodaghi, M., Shakei, M. () An analytical appoach fo fee vibation and tansient esponse of functionally gaded pieoelectic cylindical panels subjected to impulsive loads. Composite Stuctues 94: Dava, A., Khalili, S.M.R., Malekadeh Fad, K. (3) Dynamic esponse of functionally gaded cicula cylindical shells subjected to adial impulse load. Intenational Jounal of Mechanics and Mateials in Design 9:65-8. Ebahimi, M.J., Najafiadeh, M.M. (4) Fee vibation analysis of two-dimensional functionally gaded cylindical shells. Applied Mathematical Modelling 38: Hosseini, M., Akhlaghi, M., Shakei, M. (7) Dynamic esponse and adial wave popagations of functionally gaded thick hollow cylinde. Intenational Jounal fo Compute-Aided Engineeing and Softwae 4: Jha, D.K., Kant,., Singh, R.K. (3) A citical eview of ecent eseach on functionally gaded plates. Composite Stuctues 96: Jones, N. (997) Stuctual Impact. Univesity Pess, Cambidge. Kim, J.H., Paulino, G.H. () Isopaametic gaded finite elements fo nonhomogeneous isotopic and othotopic mateials. Jounal of Applied Mechanics 69:5-54. Koiumi, M. (993) he concept of FGM. Ceamic ansactions, Functionally Gadient Mateials 34:3-. Malekadeh, P., Monajjemadeh, S.M. (3) Dynamic esponse of functionally gaded plates in themal envionment unde moving load. Composites Pat B: Engineeing 45: Nemat-Alla, M. (9) Reduction of themal stesses by composition optimiation of two-dimensional functionally gaded mateials. Acta Mechanica 8:47-6. Qian, L.F., Bata, R.C, Chen, L.M. (4) Static and dynamic defomations of thick functionally gaded elastic plates by using highe-ode shea and nomal defomable plate theoy and meshless local Petov Galekin method. Composites Pat B: Engineeing 35: Qu, Y., Long, X., Yuan, G., Meng, G. (3) A unified fomulation fo vibation analysis of functionally gaded shells of evolution with abitay bounday conditions. Composites: Pat B 5: Shakei, M., Akhlaghi, M., Hosseini, S.M. (6) Vibation and adial wave popagation velocity in functionally gaded thick hollow cylinde. Composite Stuctues 76:74-8. Shao, Z.S., Fan, L.F., Wang,.J. (4) Analytical solutions of stesses in functionally gaded cicula hollow cylinde with finite length. Key Engineeing Mateials 6 63: Shao, Z.S. (5) Mechanical and themal stesses of a functionally gaded cicula hollow cylinde with finite length. Intenational Jounal of Pessue Vessels and Piping 8: Sun, D., Luo, S.N. () he wave popagation and dynamic esponse of ectangula functionally gaded mateial plates with completed clamped suppots unde impulse load. Euopean Jounal of Mechanics A/Solids 3: Latin Ameican Jounal of Solids and Stuctues (5) 5-5

21 H.Zafamand et al./ hee dimensional fee vibation and tansient analysis of two diectional panels unde impact loading 5 Uyma, B., Aydogdu, M. hee-dimensional Vibation Analyses of Functionally Gaded Plates unde Vaious Bounday Conditions. Jounal of Reinfoce Plastic Composites 6: Wang, X., Sudak, L.J. (8) hee-dimensional analysis of multi-layeed functionally gaded anisotopic cylindical panel unde themomechanical loading. Mechanics of Mateials 4: Yas, M.H., Sobhani Aagh B. () hee-dimensional analysis fo themoelastic esponse of functionally gaded fibe einfoced cylindical panel. Composite Stuctues 9: Zafamand, H., Kadkhodayan, M. (4) hee-dimensional static analysis of thick functionally gaded plates using gaded finite element method. Poceedings of the Institution of Mechanical Enginees, Pat C: Jounal of Mechanical Engineeing Science 8: Zafamand, H., Hassani, B. (4) Analysis of two-dimensional functionally gaded otating disks with vaiable thickness. Acta Mechanica 5: Zahedinejad, P., Malekadeh, P., Faid, M., Kaami, G. () A semi-analytical thee-dimensional fee vibation analysis of functionally gaded cuved panels. Intenational Jounal of Pessue Vessels and Piping 87: Zienkiewic, O.C., aylo, R.L. (5) he finite element method fo solid and stuctual mechanics, 6th edn. Elsevie Buttewoth-Heinemann, Oxfod. Appendix he components of ae: i V X (A) whee V is the volume of each element that is: V (A) and: in which: ( ) i j Α (A3) ij ij X {,,,,,,, } (A4) cos, sin, (A5) i i cos i, i i sin i, i i (A6) whee i, i and i ae nodal coodinates and Α ij is obtained fom eliminating the i th ow and j th column of V. Latin Ameican Jounal of Solids and Stuctues (5) 5-5