Composite Structures

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1 Composite Structures 0 (03) 8 Contents lists ville t SciVerse ScienceDirect Composite Structures journl homepge: Advnces in the Ritz formultion for free virtion response of douly-curved nisotropic lminted composite shllow nd deep shells Fiorenzo A. Fzzolri,,, Ersmo Crrer, City University London, Northmpton Squre, London ECV 0HB, United Kingdom Politecnico di Torino, Corso Duc degli Aruzzi 4, 09 Torino, Itly rticle info strct Article history: Aville online 8 Ferury 03 Keywords: Advnced hierrchicl shell theories Douly-curved nisotropic lminted shells Ritz method Free virtion The hierrchicl trigonometric Ritz formultion (HTRF) developed in the frmewor of the Crrer unified formultion (CUF), for the first time, is extended to shell structures in order to cope with the free virtion response of douly-curved nisotropic lminted composite shells. The HTRF is the outcome of the comintion of dvnced shell theories hierrchiclly generted vi the CUF with the trigonometric Ritz method. It is sed on so-clled Ritz fundmentl primry nuclei otined y virtue of the principle of virtul displcements (PVD). The PVD is further used to derive the governing differentil equtions nd nturl oundry conditions. Donnell Mushtri s shllow shell-type equtions re given s prticulr condition. Severl shell geometries ccounting for thin nd thic shllow cylindricl nd sphericl shells, deep cylindricl shells nd hollow circulr cylindricl shells, with cross-ply nd ngle-ply sting sequences re investigted. CUF-sed refined shell models re ssessed y comprison with the 3D elsticity solution. Convergence nd ccurcy of the presented formultion re exmined. The effects of significnt prmeters such s stcing sequence, length-to-thicness rtio nd rdius-to-length rtio on the circulr frequency prmeters re discussed. Ó 03 Elsevier Ltd. All rights reserved.. Introduction Shell Structures re widely used in common erospce pplictions nd n ccurte evlution of their sttic nd dynmic ehvior is required in order to provide relile dt to design engineers. They hve een reserched for mny yers due to his high stiffness-to-weight nd strength-to-weight rtios. During the yers mny theories hve een developed to ccurtely nlyze the sttic nd dynmic ehvior of douly-curved shell structures such s those provided y Donnell [,], Mushtri [3,4], Love [5,6], Timosheno [7], Gol denveizer [8], Novozhilov [9], Flügge [0,], Lur ye [], Byrne [3], Reissner [4], Nghdy-Berry [5], Snders [6] nd Vlsov [7,8] others were otined expressly for circulr cylindricl shells such s Arnold nd Wrurton [9,0] nd Houghton nd hons [] (see Leiss [] for further detils). Alwys within the frmewor of the shell theories it ws pointed out y Koiter [3] (Koiter s recommendtion (KR)), tht for trditionl isotropic one-lyer, refinements of Love s first pproximtion theory re meningless unless the effects of trnsverse sher nd norml stresses re oth ten into ccount in refined theory. Crrer Corresponding uthor. Tel.: +44 (0) ; fx: +44 (0) E-mil ddress: Fiorenzo.Fzzolri.city.c.u (F.A. Fzzolri). Ph.D. Cndidte, School of Engineering nd Mthemticl Sciences. Professor, Deprtment of Mechnicl nd Aerospce Engineering. [4] mended the KR extending it to composite structures y proposing the following sttement: ny refinements of clssicl models re meningless, in generl, unless the effects of interlminr continuous trnsverse sher nd norml stresses (strins) re oth ten into ccount in multilyered shell theory. The sme uthor [5 9] proposed unified formultion le to generte wide clss of D nd qusi-3d equivlent single lyer, zig-zg nd lyer-wise shell models which ccurtely descrie the free virtion ehvior of douly-curved lminted composite shells. Both the xiomtic nd the symptotic-lie [30,3] pproches hve een emedded in the CUF. The cpility to study the effectiveness of ech single term in the displcement field independently from its loction hs een investigted oth to virtion [30] nd ending [3] nlyses. Comprehensive documenttions on shell structures cn e found in mny pulished ppers. Reviews on finite element shell formultions hve een given y Denis nd Plzzotto [3] nd Di nd Rmm [33]. Exhustive reviews on clssicl theories cn e found in Bert [34] nd Lirescu [35]. Theories regrding to the fulfillment of the C z 0 -requirements were reviewed y Grigolyu nd Kuliov [36]. As mny rticles on the ppliction of symptotic methods hve een collected y Fetthlioglu nd Steel [37], Wider nd Logn [38], Wider nd Fn [39], Spencer et l. [40] nd Cicl [4]. A complete overview of different prolems relted to multilyered shells modeling hs een provided y Kpni [4] nd Noor nd Burton [43]. From the huge literture presents on this /$ - see front mtter Ó 03 Elsevier Ltd. All rights reserved.

2 F.A. Fzzolri, E. Crrer / Composite Structures 0 (03) 8 suject only few ttempts hve een mde in order to solve the governing differentil equtions in n exct wy, Forserg [44] y using the Donnell nd Flügge ssumptions nd Soedel [] for orthotropic circulr cylindricl shells nd simply supported (sher diphrgms) oundry condition, mongst others [46,47]. A systemtic procedure for otining the closed-form eigensolution ws given y Cllhn nd Bruh [48]. Recently, n nlyticl procedure to otin the exct chrcteristic eqution for free virtion nlysis of circulr cylindricl shells hs een give y Liu et l. [49]. Although most of them re sed on strong simplifictions, these efforts re however necessry due to the high vriility with which the pproximte solutions mtch the exct one. Approximte solutions re strongly used in the nlysis of shell structures. ho et l. [50] used mesh-free p-ritz method for the nlysis of lminted cylindricl pnels with two side simply-supported. The sme uthor [5] employed meshfree pproch to underte free virtion nlysis of composite cylindricl pnels. Liew nd Lim [5] delt with the virtion nlysis of douly-curved rectngulr shllow shells vi the Ritz method nd refined first order theory. Lim nd Liew [53] used higher order theory for the free virtion nlysis of cylindricl shllow shells. Soldtos nd Messin [54] studied the influence of oundry conditions nd trnsverse sher on the free virtion ehvior of symmetric nd ntisymmetric ngle-ply lminted circulr cylinders nd cylindricl pnels y Ritz formultion. Messin nd Soldtos [55] investigted the effect of different oundry conditions on the dynmic chrcteristics of cross-ply lminted circulr cylinders. Accurte equtions sed on the FSDT were provided y Qtu [56] for lminted composite deep thic shells. Qtu nd Asdi [57] ddressed the virtion nlysis of douly-curved shllow shells with ritrry oundry conditions y using the Ritz method with lgeric polynomil displcement functions. Asdi et l. [58] employed 3D nd severl sher deformtion theories in order to crry out sttic nd virtion nlysis of thic deep lminted cylindricl shells. Ferreir et l. [59] used wvelet colloction method for the nlysis of lminted shells. The sme uthor [60] comined sinusoidl sher deformtion theory with the rdil sis functions colloction method to del with sttic nd virtion nlyses of lminted composite shells. Tornene [6] studied the free virtion ehvior of douly-curved nisotropic lminted composite shells nd revolution pnels y mens of the generlized differentil qudrture (GDQ). Recently some reviews on the recent reserch on the nlysis of composite shells hve een presented. Qtu [6] reviews most of the reserch done on the dynmic ehvior of composite shells including free virtion, impct, trnsient, shoc, etc. Liew et l. [63] proposed review of meshless methods for lminted nd functionlly grded pltes nd shells. In the present rticle the hierrchicl trigonometric Ritz formultion (HTRF), which ws introduced for the first time y Fzzolri nd Crrer in [30,64 67] in order to investigte the sttic nd dynmic ehvior of nisotropic lminted composite nd sndwich pltes, hs een extended to shell structures. The trigonometric Ritz method is used in conjunction with the dvnced hierrchicl shell models provided y the CUF to cope with free virtion nlysis of douly-curved nisotropic lminted composite shells. In the prticulr cse of lyer-wise shell models, eing the governing differentil equtions written t lyer level further refinement hs een introduced in order to ccurtely descrie the reference surfces t ech lyer. However, on the other hnd, refinement in results cnnot e chieved only y virtue of n enhncement in the inemtics description of the displcement model, ut comining it with n dequte description of the curvture terms h=r i with i ¼ ;. The ccurcy of the presented formultion hs een widely demonstrted in the results Section 7 y mens of severl numericl enchmrs. Different shell configurtions hve een ccounted for. An ccurte convergence nlysis hs een crried out evluting the effect of the inemtics descriptions nd higher order terms on the rte of convergence. In prticulr, thin nd thic shllow cylindricl nd sphericl shells, deep cylindricl shells nd hollow circulr cylindricl shells, with cross-ply nd ngle-ply sting sequences hve een investigted. Some relile enchmrs hve een provided nd the effects of ey prmeters such s stcing sequence, length-to-thicness rtio nd rdius-tolength rtio on the circulr frequency prmeters hve een commented.. Lminted composite shell geometries The slient fetures of lminted composite shell geometries re shown in Fig.. A lminted shell composed of N l lyers is considered. The integer, used s superscript or suscript, denotes the lyer numer which strts from the ottom of the shell. The lyer geometry is denoted y the sme symols s those used for the whole multilyered shell nd vice vers. With nd the curviliner orthogonl coordintes (coinciding with the lines of principl curvture) on the lyer reference surfce (middle surfce of the lyer). The z denotes the rectiliner coordinte in the norml direction with respect to the lyer middle surfce. The ngle / is commonly referred to s shllowness ngle. The C is the oundry: C g nd Cm re those prts of C on which the geometricl nd mechnicl oundry conditions re imposed, respectively. These oundries re herein considered prllel to or. For convenience the further dimensionless thicness coordinte is introduced f ¼ z, where h h denotes the thicness in A domin. The following reltionships hold in the given orthogonl system of curviliner coordintes [68,69]: Squre of line elements ds ¼ d d H þ H þ dz H z Are of n infinitesiml rectngle on d ¼ H H d d Infinitesiml volume where H ¼ A dv ¼ H H H z d d dz þ z R H ¼ B þ z R H z ¼ The R nd R re the rdii of curvture in the nd directions, respectively. The coefficient of the first fundmentl form of d re A nd B. Attention is herein focused on shells with constnt curvture, i.e., douly-curved shells (cylindricl, sphericl, toroidl geometries) for which A ¼ B ¼. 3. Constitutive equtions nd geometricl reltionships The nottion for the displcement vector is: u ¼ ½u u u z Š T ð5þ Superscript T represents the trnsposition opertor. The stresses, r, nd the strins, e, re expressed s follows: h i T; h i T r ph ¼ r r s e pg ¼ e e c h i T; h i T ð6þ r nh ¼ s z s z r zz e ng ¼ c z c z e zz ðþ ðþ ð3þ ð4þ

3 F.A. Fzzolri, E. Crrer / Composite Structures 0 (03) 8 3 Fig.. Lminted composite singly nd douly curved shells. The suscripts n nd p denote trnsverse (out-of-plne, norml), respectively, whilst the suscript H nd G stte tht Hooe s lw nd geometric reltions re used. The strin displcement reltions re: e pg ¼ D pu þ A p u e ng ¼ D nu þ d D A n u ¼ D np u þ d D A n u þ D nz u where d D is trcer used to introduce Donnell Mushtri s shllow shell-type pproximtion, D p ; D np nd D nz re differentil mtrix opertors: D p ¼ ; D n ¼ 4 0 z 0 z 7 5; 0 0 z z D np ¼ ; D nz ¼ z z A p nd A n re mtrices contining geometricl prmeters opertors: H R H R A p ¼ H 5 ; A R n ¼ H 5 ð9þ R In the cse of orthotropic mterils, Hooe s lw holds: r ¼ C e According to Eq. (6), the previous eqution ecomes: r ph ¼ C ~ pp e pg þ C ~ pn e ng r nh ¼ C ~ np e pg þ C ~ nn e ng where mtrices ~ C pp ; ~ C nn, ~ C pn nd ~ C np re: ð7þ ð8þ ð0þ ðþ 3 3 C e C e C e 6 C e 55 ec 0 ~C pp ¼ 6 4 ec C e C e ; C ~ nn ¼ 6 ec e 7 4 C ; ec 6 C e 6 C e C e e 3 3 C ~C pn ¼ C e ; C ~ np ¼ C e 36 ec 3 C e 3 C e 36 ðþ nd re expressed in the lminte coordinte system following the stress tensor trnsformtion rule. For the se of conciseness, the dependence of the coefficients e C ij versus Young s moduli, Poisson s rtio, the sher moduli nd the fier ngle is not reported. It cn e found in Tsi [70], Reddy [7] or ones [7]. 4. Advnced nd refined hierrchicl shell models The theories of thin shells which re commonly used re sed on the ssumption tht the trnsverse norml stress my e neglected in the mteril constitutive equtions. This ssumption is sustntited y the fct tht in usul pplictions in-plne stresses hve generlly igger of mgnitude. Furthermore from Love s hypothesis the trnsverse norml strin is zero. However these ssumptions re no longer vlid when 3D locl effects pper. Consequently, s highlighted in [73], theory which includes r zz ¼ 0 nd e zz ¼ 0 leds to wrong results. To remove the inconsistency completely, it is compulsory to expnd the displcement field t higher order with respect to the z coordinte. According to the ove considertions the CUF, well nown in the sttic nd dynmics nlysis of lyered ems, pltes nd shells, removes the inconsistency generting lrge vriety of D nd qusi-3d hierrchicl models using unified pproch. According to the CUF, we nd strong formultions of the governing differentil equtions re written in terms of primry fundmentl nuclei, which re mthemticlly nd formlly independent oth from the expnsion orders used in the displcement field for ech unnown vrile nd from the used inemtics description such s equivlent single

4 4 F.A. Fzzolri, E. Crrer / Composite Structures 0 (03) 8 lyer, zig-zg or lyer-wise. Moreover, in the cse of we formultion the Ritz primry fundmentl nuclei re invrint with respect to the chosen tril functions. By using n xiomtic pproch, the prolem relted to the shell structures cn e reduced from 3D to D. Mthemticlly, this mens tht, the unnown vriles cn e expressed s set of thicness functions tht only depend on the thicness coordinte z (or f in the cse of lyer-wise shell theories) nd the ssocited vrile depending on the in-plne orthogonl curviliner coordintes nd. The cpility to use different expnsion orders N u ; N u nd N uz in the thicness-shell-direction for ech unnown vrile in the displcement field [30,74 80] is included in the presented shell models. Such rtifice permits to tret ech vrile independently form the others nd this ecomes extremely useful when multifield nlyses re crried out such s in thermoelstic [66,74] nd piezoelectric [8] pplictions. Therefore, following this pproch the displcement vriles cn e written in compct form s: u ¼ F s u s ; s ¼ s u ; s u ; ð3þ nd their virtul vrition: du ¼ F s du s ; s ¼ s u ; s u ; ð4þ where F su 0 0 u su du su 6 F s ¼ 4 0 F su 0 7 5; u s ¼ 6 4 u 7 su 5 ; du s ¼ 6 du 7 4 su F suz u zsuz du zsuz ð5þ F su ; F su ; F suz re functions of z. The xiomtic hierrchicl shell models here employed re sed on the chosen of polynomils which re introduced y the forementioned thicness functions nd descrie the trend of the displcement components trough the thicness-shell-direction. The functions u su ; u su ; u zsuz re displcement vectors. According to Einstein s nottion, the repeted suscripts s u ; s u ; indicte summtion. Since superscript ppers in the primry unnown vriles lyer-wise inemtics description is implied. The thicness functions F su ðf Þ; F su ðf Þ; F suz ðf Þ hve now een defined t the -lyer level s follows: F tu ¼ P 0 þ P ; F u ¼ P 0 P ; F ru ¼ P ru P ru ; r u ¼ ; ; 3;...; N u F tu ¼ P 0 þ P ; F u ¼ P 0 P ; F ru ¼ P ru P ru ; r u ¼ ; ; 3;...; N u F tuz ¼ P 0 þ P ; F uz ¼ P 0 P ; F ruz ¼ P ruz P ruz ; r uz ¼ ; ; 3;...; N uz ð6þ where suscript t nd refer to the lyer top nd ottom respectively, furthermore the P j ¼ P j ðf is the Legendre polynomil of the j-order defined in the f -domin: 6 f 6. A prolic displcement field is shown in Fig.. The relted polynomils re: P 0 ¼ ; P ¼ f ; P ¼ 3f ; P 3 ¼ 5f3 3f ; P 4 ¼ 35f4 8 5f 4 þ 3 8 ; P 5 ¼ 63f5 8 35f3 4 þ 5f 8 ð7þ The chosen functions hve the following properties: ( f ¼ : F u ; F u ; F uz ¼ 0; F ru ; F ru ; F ruz ¼ 0; F tu ; F tu ; F tuz ¼ ; : F u ; F u ; F uz ¼ ; F ru ; F ru ; F ruz ¼ 0; F tu ; F tu ; F tuz ¼ 0; ð8þ The top nd ottom vlues hve een used s unnown vriles. The interlminr comptiility of displcements t ech interfce is esily lined: u t u ¼ u þ u ; u t u ¼ u þ u ; u t uz ¼ u þ uz ; ¼ ;...; N l ð9þ However, on the other hnd, it is possile to employ n equivlent single lyer pproch for the primry displcement vriles, in tht cse the thicness functions re expressed in Tylor series nd using the sme nottion dopted in the lyer-wise cse, the thicness functions re written s follows: F u ¼ ; F ru ¼ z ru ; F tu ¼ z Nu ; r u ¼ ; ; 3;...; N u F u ¼ ; F ru ¼ z ru ; Ftu ¼ z Nu ; ru ¼ ; ; 3;...; N u F uz ¼ ; F ruz ¼ z ruz ; F tuz ¼ z Nuz ; r uz ¼ ; ; 3;...; N uz ð0þ Possile inemtics descriptions in the HTRF sed on the CUF Introduction of the Murmi zig-zg function Fig.. Advnced shell models.

5 F.A. Fzzolri, E. Crrer / Composite Structures 0 (03) 8 5 A prolic displcement field is shown in Fig.. Nevertheless, it should er in mind tht equivlent single lyer models violte interlminr equilirium of trnsverse stresses nd they do not descrie the zig-zg trend of the displcement components in the thicness direction. Nonetheless, the zig-zg trends which hinges on the trnsverse nisotropy of composite structures, cn e ccounted for in the equivlent single lyer shell models y simply dding the Murmi zig-zg function (MF), MðzÞ ¼ð Þ f, which reproduces the discontinuity of the first derivte of the displcement vriles in the z-direction, nd the displcement field in Eq. (0) ecomes, F u ¼ ; F ru ¼ z ru ; F tu ¼ð Þ f ; r u ¼ ; ; 3;...; N u F u ¼ ; F ru ¼ z ru ; Ftu ¼ð Þ f ; r u ¼ ; ; 3;...; N u F uz ¼ ; F ruz ¼ z ruz ; F tuz ¼ð Þ f ; r uz ¼ ; ; 3;...; N uz ðþ It should e noticed tht F tu ; F tu ; F tuz ssume the vlues in correspondence to the ottom nd the top interfces of the -lyer (see Fig. ). An exhustive nd comprehensive documenttion on the MF cn e found in [8 85]. 5. Theoreticl formultion The vritionl sttement used in the derivtion of wht follows is the principle of virtul displcements (PVD). The PVD is powerful tool employed oth to develop pproximte solution methods nd to derive the governing differentil equtions with nturl oundry conditions when pplied in conjunction with the Guss theorem. The explicit form of the PVD t multilyer level, cn e written s: N l ¼ A de T pg r pc þ T de ng r nc d dz ¼ N l dl F in ¼ 5.. The hierrchicl trigonometric Ritz formultion ðþ The HTRF, herein proposed for douly-curved shells, is sed on the Ritz fundmentl primry nuclei, which cn e developed following four steps [67]:. The choice of the vritionl sttement.. The introduction of the stress strin constitutive reltionships. 3. The choice of the Ritz functions. 4. The use of the geometric reltions. The PVD hs lredy een chosen s vritionl sttement (see Eq. ()), the stress strin constitutive reltionships hve een given in Eq. (). The third step is the definition of the displcement field in terms of Ritz functions. In prticulr, in the Ritz method the displcement mplitude vector u s is tht which individutes the mximum mplitude in the oscilltion, tht mximizes the relted wor nd is expressed in series expnsion s: u s ¼ U si W i where i ¼ ;...; N s ¼ s u ; s u ; ð3þ p N indictes the order of expnsion in the pproximtion, ı ¼ ffiffiffiffiffiffiffi, t is the time nd x ij the circulr frequency. Consequently the hrmonic displcement field, in compct wy, ssumes the following form: u ¼ F s U si W i ð4þ where Usu i eı x 3 ij t 3 3 w i 0 0 F su 0 0 U si ¼ U su i eı x ij t ; W 6 i ¼ 0 w i ; F s ¼ 0 F su U zsuz i eı x ij t 0 0 w zi 0 0 F suz ð5þ U su i; U su i ; U zsuz i re the unnown coefficients, w i ; w i ; w zi re the Ritz functions ppropritely chosen on the type of prolem. Convergence to the exct solution is gurnteed if the sis functions re dmissile functions in the used vritionl principle [64,86,87]. In the fourth step, y coupling the geometricl reltions in Eq. (7) with Eq. (4) the strin vectors ecome: e pg e ng ¼ D p ðf s W i ¼ D np ðf s W i ÞU si þ A pðf s W i ÞU si ÞU si þ d D A n ðf s W i ÞU si þ D nzðf s W i ÞU si ð6þ By sustituting the previous expression in Eq. () the explicit expressions of the internl wor nd the wor done y the inertil forces in terms of Ritz functions nd unnown coefficients re otined: dl int ¼ du T T A si D p ðf s W i Þ C ~ pp D p F s W j U d sj dzþ du T T A si D p ðf s W i Þ C ~ pp A p F s W j U d sj dzþ du T T A si D p ðf s W i Þ C ~ pn D np F s W j U d sj dzþ du T T A si D p ðf s W i Þ C ~ pn d D A n F s W j U d sj dzþ du T T A si D p ðf s W i Þ C ~ pn D nz F s W j U d sj dzþ du T T A si A p ðf s W i Þ C ~ pp D p F s W j U d sj dzþ du T T A si A p ðf s W i Þ C ~ pp A p F s W j U d sj dzþ du T T A si A p ðf s W i Þ C ~ pn D np F s W j U d sj dzþ du T T A si A p ðf s W i Þ C ~ pn d D A n F s W j U d sj dzþ du T T A si A p ðf s W i Þ C ~ pn D nz F s W j U d sj dzþ du T T A si D np ðf s W i Þ C ~ np D p F s W j U d sj dzþ ð7þ du T T A si D np ðf s W i Þ C ~ np A p F s W j U d sj dzþ du T T A si D np ðf s W i Þ C ~ nn D np F s W j U d sj dzþ du T T A si D np ðf s W i Þ C ~ nn d D A n F s W j U d sj dzþ du T T A si D np ðf s W i Þ C ~ nn D nz F s W j U d sj dzþ du T A si ½ du T A si ½ d D A n ðf s W i d D A n ðf s W i ÞŠ T C ~ ÞŠ T C ~ np D p np A p F s W j U d sj dzþ F s W j U d sj dzþ

6 6 F.A. Fzzolri, E. Crrer / Composite Structures 0 (03) 8 dl F in ¼ du T A si ½d D A n ðf s W i ÞŠ T C ~ nn D np F s W j U d sj dzþ du T A si ½d D A n ðf s W i ÞŠ T C ~ nn d D A n F s W j U d sj dzþ du T A si ½d D A n ðf s W i ÞŠ T C ~ nn D nz F s W j U d sj dzþ du T A si ½D nz ðf s W i ÞŠ T C ~ np D p F s W j U d sj dzþ du T A si ½D nz ðf s W i ÞŠ T C ~ np A p F s W j U d sj dzþ du T A si ½D nz ðf s W i ÞŠ T C ~ nn D np F s W j U d sj dzþ du T A si ½D nz ðf s W i ÞŠ T C ~ nn d D A n F s W j U d sj dzþ du T A si ½D nz ðf s W i ÞŠ T C ~ nn D nz F s W j U d sj dz h du T i A si q ðf s W i Þ T F s W Usj j d dz The qudrtic forms of the internl wor nd the wor done y the inertil forces cn e written s: dl int ¼ dut si Kssij U sj ; dl F in ¼ du T si Mssij U sj ð8þ the Ritz fundmentl primry nuclei re esily otined compring Eq. (7) with Eq. (8): h K ssij T i h T ¼ D p ðf s W i Þ þ A p ðf s W i Þ ~C A pp D p F s W j þ C ~ pp A p F s W j þ C ~ pn D np F s W j þ C ~ pn d D A n F s W j þ C ~ pn D i h T nz F s W j þ D np ðf s W i Þ þþd ½ D A n ðf s W i ÞŠ T i h þþ½d nz ðf s W i ÞŠ T C ~ np D p F s W j þ C ~ np A p F s W j þ C ~ nn D np F s W j þ C ~ nn d D A n F s W j þ C ~ nn D i nz F s W j d dz M ssij h i ¼ q ðf s W i Þ T F s W j d dz A ð9þ At this point it is useful to introduce the following integrls in order to write in concise mnner the explicit form of the Ritz fundmentl secondry nuclei: ss ; ss szs ; szs ssz ; ssz ; ss szsz ; szsz ; szs ; ssz ; ss ; szs ; szsz ; ssz F s F s ¼ A z z ; ss ; szs ; ssz ; szsz ; ss ; szs ; ssz ; szsz ¼ F s F s A ; H ; H ; H F s ¼ A z F s F s ¼ F s A z ; szsz ; H ; H ; H ; H ; H H H H H ; H ; H H H dz ; H ; H ; H ; H ; H H H H dz ;H ;H ;H ; H ;H H H H dz dz ð30þ where s ¼ s u ; s u ; nd s ¼ s u ; s u ;. The explicit forms of the Ritz fundmentl secondry stiffness nd mss nuclei re following reported: su su K ¼C e su su w i ; w j ; d d 6 su su w i ; w j ; d d 6 su su w i ; w j ; d d su su 66 w i ; w j ; d d su;z su;z 55 w i w j d d þ d D C e su;z su 55 w i w j d d K su su ec su su;z 55 R R w i w j d d þc e su su 55 R w i w j d d ¼C e su su 6 w i ; w j ; d d su su w i ; w j ; d d þ ec 66 su su w i ; w j ; d d su su 6 w i ; w j ; d d su su w i w j d d þ d D C e ec R su su ;z þc e R R R su;z su ;z su;z su w i w j w i w j d d d d w i w j d d su suz Ku uz ¼C e suz 3 su w i ; w zj d d suz su;z 55 w i w zj ; d d ec su suz R w i ; w zj d d su suz w i ; w zj d d R 6 su suz w R i ; w zj d d su suz 6 w i ; w zj d d R d D C e su suz 55 R w i w zj ; d d þc e su suz w R i w zj ; d d

7 K su su u u K su su u u K su suz u u z ¼C e su 6 su w i ; w j ; d d su su w i ; w j ; d d 66 su su w i ; w j ; d d su 6 su w i ; w j ; d d ec su su w i w j d d ec R þ d D C e R su su ;z R R su;z su ;z su;z su w i w j þ w i w j d d d d w i w j d d ¼C e su 66 su w i ; w j ; d d 6 su su w i ; w j ; d d 6 su su w i ; w j ; d d su su w i ; w j ; d d su;z su;z 44 w i w j d d þ d D C e 44 su;z su w i w j d d ec 44 R su su;z R w i w j d d þc e 44 su su w i w j d d R ¼C e su ;z suz w i w zj ; d d su 44 ;z suz w i w zj ; d d su 3 suz;z w i ; w zj d d su R suz w i ; w zj d d su su w i ; w zj d d R su 36 suz;z w i ; w zj d d 6 su suz R w i ; w zj d d 6 su su w i ; w zj d d R d D C e su suz w i w zj ; d d þc e 44 R R su su;z w i w zj ; d d F.A. Fzzolri, E. Crrer / Composite Structures 0 (03) 8 7 suz su Ku zu ¼C e su 3 ;z su w zi w j ; d d su 36 ;z su w zi w j ; d d suz suz;z w zi ; w j d d 6 suz su w R zi ; w j d d su su 6 w zi ; w j d d K suz su u zu R suz suz;z 55 w zi ; w j d d suz su R w zi ; w j d d su su w zi ; w j d d R d D C e su su 55 R w zi w j ; d d þc e suz su;z w R zi w j ; d d ¼ e C 36 su ;z su þ e C 3 su ;z su suz suz;z 44 R R suz suz;z 6 R 6 R w zi w j ; d d w zi w j ; d d w zi ; w j d d suz su w zi ; w j d d su su w zi ; w j d d w zi ; w j d d suz su w zi ; w j d d su su w zi ; w j d d d D C e su su R w zi w j ; d d þc e 44 suz su;z w R zi w j ; d d suz suz Ku zu z ¼C e suz suz 55 w zi ; w j ; d d suz suz w zi ; w j ; d d suz suz w zi ; w j ; d d suz suz 44 w zi ; w j ; d d su;z su;z 33 w zi w j d d suz;z suz 3 R w zi w j d d suz;z suz 3 w zi w j d d R

8 8 F.A. Fzzolri, E. Crrer / Composite Structures 0 (03) 8 3 R 3 R suz suz;z suz suz;z w i w j d d w i w j d d suz suz R w i w j d d suz suz w R i w j d d R suz suz w R i w j d d R suz suz w i w j d d R 6. Governing differentil equtions of douly-curved shells In order to derive the governing differentil equtions nd nturl oundry conditions the Guss theorem is pplied: D p d T d ¼ d T D p d d þ d T I p d dc C ðd Þd T d ¼ d ðd Þ T d d þ d ði Þ T d dc C ð37þ where cn e displcement or stress vriles nd the introduced I p nd I rrys re: n H 0 0 n H n I p ¼ H ; I ¼ 0 0 n H 5 ð38þ n H n H su su M ¼ q su su w i w j d d M su su u u ¼ q su su w i w j d d suz suz Mu zu z ¼ q suz suz w zi w zj d d ð3þ By exploiting the use of the reltions dt ¼ dl in nd du ¼ dl int (see [30]), the PVD in Eq. () for furthe detils cn e rewritten s: d T U ¼ 0 ð3þ which represents minimiztion of the totl energy of the system with respect to the introduced degrees of freedom [30,64], therefore it cn e equivlently written s: T U ¼ 0 with i ¼ ;...;N ; s U u ¼ u ;r u ;t u r u ¼ ;3;...;N u su i T U ¼ 0 with i ¼ ;...;N ; U su i s u ¼ u ;r u ;t u r u ¼ ;3;...;N u T U U zsuz i ¼ 0 with i ¼ ;...;N ; ¼ uz ; uz ;t uz r uz ¼ ;3;...;N uz ð33þ The Ritz method leds to the discrete form of the governing differentil equtions in terms of the Ritz primry fundmentl nuclei: du T si h : K ssij x ij Mssij i U sj ¼ 0 ð34þ The free-virtion response of the multilyered shells leds to the following eigenvlues prolem: K ssij x ij Mssij ¼0 ð35þ The doule rs denote the determinnt. Ech Ritz fundmentl secondry nucleus of the CUF hs to e expnded individully ccording to the expnsion order chosen for the displcement components [30]. When the expnsions hve een performed then they cn e rrnged s in Eq. (36) nd generte the Ritz fundmentl primry nuclei relted t the prticulr theory used. 3 3 K u u K u u K u u z M 0 0 K ssij 6 ¼ 4 K u u K u u K 7 u u z 5 M ssij 6 7 ¼ 4 0 M u u 0 5 The norml to the oundry of domin is: ^n ¼ n " # cos ðu Þ ¼ cos n u ð39þ where u nd u re the direction cosines, nmely, the ngles etween the norml ^n nd the directions nd, respectively. The governing differentil equtions nd nturl oundry conditions (Neumnn-type) on C m, for douly-curved nisotropic composite shell t multilyer level cn e written s: h N l du s ðf s Þ T T i D p þ ðf s Þ T T A p ½ C ~ A pp D pðf s Þþ C ~ pp A pðf s Þ ¼ þ C ~ pn D npðf s Þþ C ~ pn d ~ D A n D npðf s Þþ C ~ pn D nzðf s ÞŠ ½ F s ð Þ T D np þðf s Þ T ðd D A n Þ T þ ðf s Þ T ðd nz Þ T Šþ½ C ~ np D pðf s Þþ C ~ np A pðf s Þ þ C ~ nn D npðf s Þþ C ~ nn d ~ D A n D npðf s Þþ C ~ nn D nzðf s ÞŠ u s d dzþ N l du s ðf s Þ T T I p ½ C ~ ¼ A pp D pðf s Þþ C ~ pp A pðf s Þþ C ~ pn D npðf s Þ þ C ~ pn d D A n ðf s Þþ C ~ pn D nzðf s ÞŠþðF s Þ T T I np ½ C ~ np D pðf s Þþ C ~ np A pðf s Þ þ C ~ nn D npðf s Þþ C ~ nn d D A n ðf s Þþ C ~ nn D nzðf s ÞŠ u s d dz ¼ N l du s q ðf s Þ T ðf s Þ u s d dz ð40þ A ¼ nd in compct form re: du s : K ss u s þ M ss u s ¼ 0 C m : Pss u s ¼ P ss u s C g : u ð4þ s ¼ u s where h K ss A ¼ ðf s Þ T T i h D p þ ðf s Þ T T A p ~C pp D pðf s Þþ C ~ pp A pðf s Þ þ C ~ pn D npðf s Þþ C ~ pn d ~ D A n D npðf s Þþ C ~ i h pn D nzðf s Þ ðf s Þ T D np þðf s Þ T ðd D A n Þ T þ ðf s Þ T ðd nz Þ Ti h þ C ~ np D pðf s Þþ C ~ np A pðf s Þ þ C ~ nn D npðf s Þþ C ~ nn d ~ D A n D npðf s Þþ C ~ i nn D nzðf s Þ H H dz P ss A ¼ h ðf s Þ T T I p C ~ pp D pðf s Þþ C ~ pp A pðf s Þþ C ~ pn D npðf s Þ þ C ~ pn d D A n ðf s Þþ C ~ i pn D h nzðf s Þ þ ðf s Þ T T I np C ~ np D pðf s Þ þ C ~ np A pðf s Þþ C ~ nn D npðf s Þþ C ~ nn d D A n ðf s Þþ C ~ i nn D nzðf s Þ H H dz T T K uzu K uzu K uzu z 0 0 M uzuz ð36þ M ss A ¼ q ðf s Þ T ðf s ÞH H dz ð4þ

9 For the se of completeness the nine components of the fundmentl primry differentil nucleus K re following reported: su su K ¼ C e su su C s u e 6 su su s u s u s u C e su 6 su C s u e 66 su su s u s u s u K su su su suz K þ e C 55 suz suz d D R ¼ C e su su C e su 6 su þ e C ¼ e C suz su z R su suz C e suzz 3 su e C 6 þ e C þ e C 55 R su suz suz suz suz suz s u s u d D R suz s u d D susu z R K su su ¼ C e su su s u K su su K su suz e C 6 su su þ e C s u þ d D R C e u 6 s su s u e C 66 su su s u s u su su su z s u d D susuz þ d D su su R R R C e su suz s u R C e 6 su suz s u R s u C e 36 su suz s u s u su suz R su suz R su s uz z s u d D R ¼ C e su su C e 6 su su þ e C 44 su z su z s u s u d D R s u s u su d D R s u s u e C 6 s u s u C e s u 66 su s u s u s u s u s u s u s u suz su þ d D su su R R C e s u 6 su e C 66 s u s u suz s u d D susuz R s u s u þ d D su su ¼ C e su suz C R e su s u R C e 3 su suzz C e 6 su s u R s u C e 6 su suz C R e 36 su suz s u s u su z suz su suz R 44 su z suz su R F.A. Fzzolri, E. Crrer / Composite Structures 0 (03) 8 9 R s u s u s u suz su K ¼ e C K suz su R suz su su suzz 3 R e C e C 55 ¼þ e C suz suz K suz s suz sz þ C e suz su s u R s u þ C e 6 suz sz þ C s u R e 6 s u þ C e 36 z s z s u s u suz su R suz sz R suz suz suz su R þ e C 3 suzz su suz su R e C e C 44 ¼þC e suz su z suz su z R þ C e s u R þ C e 6 s u R þ C e 36 z s u s u R suz suz suz su suz su suz su R R þ C e suz suz þ C R R e 3 R 3 suzz suz suzz þ R C e suz suz 44 C e suz suz 55 C e suz suz C e suz suz suz su s u þ C e 6 s u s u suz suz suzz suzz 33 suzz þ suz suzz ð43þ The nine components of the fundmentl primry oundry nucleus P cn e written s follows: su su P ¼ n C e P su su su su þ n C e 6 su su ¼ n e C su 6 su þ n e C 66 su su su su P ¼ n C e R s u su suz þ n C e 6 su su s u þ n C e su 66 su s u þ n e C 6 su su s u þ n e C su su s u s u þ n C e R su suz þ n C e suzz 3 su þ n C e suz 6 su þ n C e suz R 6 su þ n C e suzz R 36 su s u s u

10 0 F.A. Fzzolri, E. Crrer / Composite Structures 0 (03) 8 P su su ¼ n C e P su su 6 su su þ n C e su su ¼ n e C 66 su su þ n e C 6 su su P su su ¼ n C e R s u 6 su suz s u þ n C e 66 su su s u þ n C e 6 su su s u þ n e C su su s u þ n e C 6 su su s u þ n C e 6 R su suz þ n C e 36 su suzz þ n C e R su suz þ n C e R su suz þ n C e 3 su suzz suz su P ¼ n C e 55 R suz su þ n C e suz suz s u s u þ n C e suz 55 su n C e su suz R P suz su ¼ n C e R suz su þ n C e su su z n C e s 44 R suz þ n C e su 44 suz suz suz P ¼ n C e suz suz 55 þ n C e suz suz þ n C e suz suz 44 þ n C e suz suz ð44þ Finlly the fundmentl primry mss nucleus M components cn e written s: su su M ¼ q su su M su su su suz ¼ 0 M ¼ 0 M su su ¼ 0 M su su ¼ q su su M su suz ¼ 0 suz su M ¼ 0 M suz su ¼ 0 M suz suz 7. Numericl results nd discussion ¼ q suz suz ðþ Numericl results hve een computed ting into ccount cross-ply nd ngle-ply simply supported squre shells, mterils in Tle nd set of trigonometric tril functions following defined: w mn ð; Þ ¼ cos m p sin n p m;n w mn ð; Þ ¼ sin m p cos n p ð46þ m;n w zmn ð; Þ ¼ sin m p sin n p m;n Tle List of cronyms used in tles to denote the shell theories. Acronym Description Cross-ply hollow circulr cylindricl shells 3D exct 3D exct elsticity solution y Ye nd Soldtos, [88] PAR ds Discontinuous interlminr stresses distriuted proliclly y Timrci nd Soldtos, [89] HYT ds Discontinuous interlminr stresses distriuted hyperoliclly y Timrci nd Soldtos, [89] UNI cs Continuous interlminr stresses distriuted uniformly y Timrci nd Soldtos, [55] PAR cs Continuous interlminr stresses distriuted proliclly y Timrci nd Soldtos, [89] HYT cs Continuous interlminr stresses distriuted hyperoliclly y Timrci nd Soldtos, [89] D Refined zig-zg theory y Di Sciuv nd Crrer, [90] HSDTM Refined higher Order theory y Mtsung, [9] Cross-ply shllow cylindricl nd sphericl shells WLC Wvelet colloction y Ferreir, [59] RBF Rdil sis functions y Ferreir, [9] SRBC Sinusoidl rdil sis colloction y Ferreir, [93] LWRBC Lyer-wise rdil sis colloction y Ferreir, [60] Cross-ply deep cylindricl shells FSDTQ First order sher deformtion theory y Qtu, [58] 3D FEM 3D FEM solution y Qtu, [58] Angle-ply shllow cylindricl nd sphericl shells R-FSDT Ritz method using lgeric polynomils y Qtu, [94] Angle-ply deep cylindricl shells R-FSDT Ritz method using lgeric polynomils y Qtu, [94] Tle Mterils. Mt- E =E E 3 =E G =E ; G 3 =E G 3 =E m ; m 3 m 3, Mt- E-glss/epoxy (E/E) E ½GPŠ E ; E 3 ½GPŠ G ; G 3 ½GPŠ G 3 ½GPŠ m ; m 3, m 3,

11 F.A. Fzzolri, E. Crrer / Composite Structures 0 (03) 8 Tle 3 qffiffiffiffi Dimensionless fundmentl circulr frequency prmeter ^x ¼ x q E, of squre circulr cylindricl shells with sting sequence ½0 =90 =90 =0 Š, m =,n in rcets, length-to thicness rtio /h = 0, ED shell models nd vrying the rdius-to-length rtios. Theory Ave. err. (%) Mx err. (%) R = 3D exct (4) 0.07 () 9.90 (30) (4) 9.85 () PAR ds (.97) (.0) HYT ds (.00) (.07) UNI cs (.6) (.65) PAR cs (0.5) (0.5) HYT cs (0.3) (0.4) D (.60) (.65) HSDTM (0.63) (0.65) Present ED models C-E ED.383 (4) c.06 () (30) (4) () (0.40) (0.67) ED (0.8) (0.47) ED (.53) (.57) ED (.5) (.57) ED (0.8) (0.84) ED (0.8) (0.84) ED (0.8) (0.84) ED (0.8) (0.84) Present ED models MD-E ED (0.4) (0.67) ED (0.9) (0.47) ED (.54) (.57) ED (.53) (.57) ED (0.83) (0.84) ED (0.83) (0.84) ED (0.8) (0.84) ED (0.8) (0.84) c C-E (Complete equtions). DM-E (Donnell Mushtri s shllow shell-type equtions). The hlf-wve numer n is the sme for ll the higher order models nd for the C-E nd DM-E. Tle 4 qffiffiffiffi Dimensionless fundmentl circulr frequency prmeter ^x ¼ x q E, of squre circulr cylindricl shells with sting sequence ½0 =90 =90 =0 Š, m =,n in rcets, length-to thicness rtio /h = 0, ED shell models nd vrying the rdius-to-length rtios. Theory Ave. err. (%) Mx err. (%) R = 3D exct (4) 0.07 () 9.90 (30) (4) 9.85 () PAR ds (.97) (.0) HYT ds (.00) (.07) UNI cs (.6) (.65) PAR cs (0.5) (0.5) HYT cs (0.3) (0.4) D (.60) (.65) HSDTM (0.63) (0.65) Present ED models C-E ED.333 (4) c.0599 () (30) (4) () (0.36) (0.63) ED (0.8) (0.47) ED (.53) (.57) ED (.5) (.57) ED (0.8) (0.84) ED (0.8) (0.84) ED (0.8) (0.84) ED (0.8) (0.84) Present ED models MD-E ED (0.37) (0.63) ED (0.9) (0.47) ED (.54) (.57) ED (.53) (.57) ED (0.83) (0.84) ED (0.83) (0.84) ED (0.8) (0.84) ED (0.8) (0.84) c C-E (Complete equtions). DM-E (Donnell Mushtri s shllow shell-type equtions). The hlf-wve numer n is the sme for ll the higher order models nd for the C-E nd DM-E.

12 F.A. Fzzolri, E. Crrer / Composite Structures 0 (03) 8 Tle 5 qffiffiffiffi Dimensionless fundmentl circulr frequency prmeter ^x ¼ x q E, of squre circulr cylindricl shells with sting sequence ½0 =90 =90 =0 Š, m =,n in rcets, length-to thicness rtio /h = 0, LD shell models nd vrying the rdius-to-length rtios. Theory Ave. err. (%) Mx err. (%) R = 3D exct (4) 0.07 () 9.90 (30) (4) 9.85 () PAR ds (.97) (.0) HYT ds (.00) (.07) UNI cs (.6) (.65) PAR cs (0.5) (0.5) HYT cs (0.3) (0.4) D (.60) (.65) HSDTM (0.63) (0.65) Present LD models C-E LD (4) c () (30) (4) () (0.65) (0.66) LD (0.0) (0.0) LD (0.00) (0.00) LD (0.00) (0.00) Present LD models MD-E LD (0.65) (0.66) LD (0.03) (0.06) LD (0.0) (0.05) LD (0.0) (0.05) c C-E (Complete equtions). DM-E (Donnell Mushtri s shllow shell-type equtions). The hlf-wve numer n is the sme for ll the higher order models nd for the C-E nd DM-E. Tle 6 qffiffiffiffi Dimensionless fundmentl circulr frequency prmeter ^x ¼ x q, of squre cross-ply shllow cylindricl shells with stcing sequence ½0 =90 =0 Š. h E /h 0 00 Theory R = WLC RBF SRBC LWRBC Present models C-E LD LD ED ED ED ED ED ED Present models DM-E LD LD ED ED ED ED ED ED C-E (Complete equtions). DM-E (Donnell Mushtri s shllow shell-type equtions). When the tril functions in Eq. (46) re used s solution functions in Eq. (4) long with the condition C e 6 ¼ C e 6 ¼ C e 36 ¼ ec ¼ 0 (condition fulfilled for cross-ply lmintion schemes) the Nvier-type closed-form solution is otined. Then when crossply stcing sequences re investigted, the present HTRF leds to the Nvier-type closed-form solution. This phenomenon is usully referred to s one-term Ritz solution. The results will e given using the usul cronyms system used in the CUF [7,30]. Therefore, the equivlent single lyer theories re indicted s ED Nu Nu, where E mens the equivlent single lyer pproch, Nuz D mens tht the principle of virtul displcements hs een employed nd N u ; N u ; N uz re the three different expnsion orders used in the displcement field. Similrly the cronym used to descrie the zig-zg theories is ED Nu Nu, where sttes tht Nuz Murmi s zig-zg function hs een introduced. Lyer-wise theories re defined s LD Nu Nu where L indictes tht lyer-wise Nuz pproch hs een used. As highlighted in Section 6, the governing differentil equtions for lyer-wise shell models re written for ech lyer. It ecomes necessry for the shell structures to hve the sme product d d in ech governing eqution t -lyer

13 F.A. Fzzolri, E. Crrer / Composite Structures 0 (03) 8 3 Tle 7 qffiffiffiffi Dimensionless fundmentl circulr frequency prmeter ^x ¼ x q, of squre cross-ply shllow sphericl shells ðr h ¼ R ¼ RÞ with stcing sequence ½0 =90 =0 Š. E /h 0 00 Theory R/ WLC RBF SRBC LWRBC Present models C-E LD LD ED ED ED ED ED ED Present models DM-E LD LD ED ED ED ED ED ED C-E (Complete equtions). DM-E (Donnell-Mushtri s shllow shell-type equtions). Tle 8 qffiffiffiffiffiffi First five dimensionless circulr frequency prmeters ^x ¼ x qe, of squre cross-ply deep cylindricl shells with sting sequence ½0 =90 =90 =0 Š nd R = ¼ 0:5. h /h Theory Circulr frequency prmeters Ave. err. (%) Mx err. (%) ^x ^x ^x 3 ^x 4 ^x 5 0 3D FEM FSDT (4.05) (7.40) FSDTQ (3.99) (7.8) Present models C-E LD ( 0.5) ( 0.) LD ( 0.5) ( 0.) ED (0.07) (0.9) ED (0.9) (0.73) ED (0.5) (0.79) ED (.79) (.86) ED (7.4) (.47) ED (7.80) (.60) 0 3D FEM FSDT (.7) (4.78) FSDTQ (.64) (4.74) Present models C-E LD ( 0.05) ( 0.09) LD ( 0.05) ( 0.09) ED (0.04) (0.) ED (0.) (0.30) ED (0.9) (0.5) ED (0.94) (.99) ED (.88) (7.4) ED (3.74) (7.55) C-E (Complete equtions). level due to the curvture vritions. This opertion cn e underten choosing the vlue on the reference shell surfce, dd s d d. Then, ech d d defined on cn therefore e written s: d d ¼ R R dd R R ð47þ The proposed dvnced shell models hve een ssessed y comprison with the 3D elsticity solution nd other results present in literture nd listed in Tle. 7.. Cross-ply lminted hollow circulr cylindricl shells A preliminry ssessment of the CUF-sed equivlent single lyer, zig-zg nd lyer-wise theories is underten in Tles 3 5, respectively. Simply-supported cross-ply circulr cylindricl shells with mteril Mt- given in Tle re studied. Results re compred towrds the exct 3D elsticity solution provided y Ye nd Soldtos [88] nd other theories from the literture. The theories PAR ds, HYT ds, UNI cs, PAR cs nd HYT cs re given y Timrci nd Soldtos [89], in prticulr PAR ds nd HYT ds ccount

14 4 F.A. Fzzolri, E. Crrer / Composite Structures 0 (03) 8 for prolic nd hyperolic discontinuous distriution of the interlminr stresses, respectively. In shrp contrst, the theories UNI cs, PAR cs nd HYT cs hve uniform, prolic nd hyperolic continuous distriution of the interlminr stresses, respectively. Further theories considered for comprison purpose re the refined zig-zg theory D given y Di Sciuv nd Crrer [90] nd the refined higher order model provided y Mtsung [9]. As cn een seen from Tles 3 nd 4 equivlent single lyer nd zig-zg shell modes provide ccurte results in terms of verge nd mximum error percentges of the first six dimensionless circulr frequency prmeters if t lest cuic expnsions is used. Indeed, in this lst cse for equivlent single lyer shell models verge nd mximum error percentges re.565% nd.5705%, respectively, when using the complete shell equtions (C-E), whilst.5374% nd.5705% when pplying the Donnell-Mushtri s shllow shell-type equtions (DM-E). The introduction of the Murmin zig-zg function (MF) in the equivlent single lyer displcement field does not ffect considerly the results, oth in the cse of C-E, where the error percentges re.554% nd.5693%, nd in the cse of DM-E where their vlues re.536% nd.5693%, respetively. In the sme Tles, it is interesting to note tht if higher ccurcy is required, in oth cses the fifth order is needed, indeed in this cse the verge nd mximum error percentges re lower thn the %. Lyer-wise shell models hve een ssessed in Tle 5. As cn e seen in the sme Tle, y using cuic trend of the displcement components t ech lyer through the thicness nd the C-E, the solution perfectly mtches the 3D exct one. A similr oservtion cn e drwn when pplying the DM- E, lthough in this cse, s expected, for lower vlues of the rdius-to-length rtio smll error percentges cn e oserved. 7.. Cross-ply lminted shllow cylindricl nd sphericl shells Simply-supported three-lyered cross-ply shllow cylindricl nd sphericl shells mde of Mt- (see Tle ) re nlyzed in Tles 6 nd 7. The results, otined y using the dvnced shell models, re presented in terms of dimensionless fundmentl circulr frequency prmeter nd re in good greement with those proposed y Ferreir et l. [59,60,9,93] y using different solution methods such s wvelet colloction, rdil sis, sinusoidl rdil sis colloction nd lyer-wise rdil sis colloction, respectively. With respect to the circulr cylindricl shell cse, the use of the MF introduces conspicuous refinement in results ove ll when thic shllow shells re nlyzed. DM-E in this cse led to similr ccurcy of the C-E Cross-ply lminted deep cylindricl shells In Tle 8 the developed shell models re used to compute the first five dimensionless circulr frequency prmeters of four-lyers symmetric cross-ply deep thic nd modertely thic cylindricl shells mde of Mt- (see Tle ). The ccurcy of the present models, in terms of verge nd mximum error percentges, is proved y comprison with 3D FEM solution [58] nd other theories present in literture [58]. As cn e seen from the Tle 8 the present higher order equivlent single lyer, zig-zg nd lyerwise shell models provide etter ccurcy compred to the FSDT, sed on plte-lie stiffness coefficients, nd the FSDTQ y Qtu [58], where the stiffness coefficients hve een integrted exctly. In prticulr, for /h = 0 the shell models ED 999, ED 333, ED 888, LD 333 nd LD 555 led to n verge error percentge which is equl or lower thn the 0.5% ginst the 4% of the forementioned FSDT. A similr ehvior cn e oserved for /h = 0 in this cse the verge error percentge for the models ED 333, ED 888, LD 333 nd LD 555 is equl or lower thn the 0.%. In the uthors opinion, the results otined represent further confirmtion of the Tle 9 Convergence nlysis of the first six dimensionless circulr frequency prmeters q ^x ¼ x ffiffiffiffiffiffiffiffi 4 q, of squre ngle-ply shllow cylindricl shells with lmintion scheme E h ½30 = 30 =30 Š; =h ¼ 0; R = ¼ nd vrying ED shell models. Theory M, N Circulr frequency prmeters ^x ^x ^x 3 ^x 4 ^x 5 ^x 6 ED ED ED ED importnce of develop shell theories, which ccount for not only of correct description of the curvture terms nd higher order expnsion in the displcement model ut lso of the zig-zg trend of the displcement components through the thicness due to the trnsverse nisotropy of lminted composite structures. Tle 0 Convergence nlysis of the first six dimensionless circulr frequency prmeters q ^x ¼ x ffiffiffiffiffiffiffiffi 4 q, of squre ngle-ply shllow cylindricl shells with lmintion scheme E h ½30 = 30 =30 Š; =h ¼ 0; R = ¼ nd vrying LD shell models. Theory M,N Circulr frequency prmeters ^x ^x ^x 3 ^x 4 ^x 5 ^x 6 LD LD LD LD

15 F.A. Fzzolri, E. Crrer / Composite Structures 0 (03) 8 5 Tle q The first six dimensionless circulr frequency prmeters ^x ¼ x ffiffiffiffiffiffiffi 4 q, of squre ngle-ply shllow cylindricl shells with lmintion scheme ½h = h =h Š; =h ¼ 0; R E h = ¼ nd vrying lmintion ngle. h Theory Circulr frequency prmeters ^x ^x ^x 3 ^x 4 ^x 5 ^x 6 0 R-FSDT Present models ED ED LD R-FSDT Present models ED ED LD R-FSDT Present models ED ED LD R-FSDT Present models ED ED LD R-FSDT Present models ED ED LD R-FSDT Present models ED ED LD R-FSDT Present models ED ED LD Tle Convergence nlysis of the first six dimensionless circulr frequency prmeters q ^x ¼ x ffiffiffiffiffiffiffiffi 4 q, of squre ngle-ply shllow sphericl shells with lmintion scheme E h ½30 = 30 =30 Š; =h ¼ 0; R = ¼ nd vrying ED shell models. Tle 3 Convergence nlysis of the first six dimensionless circulr frequency prmeters q ^x ¼ x ffiffiffiffiffiffiffiffi 4 q, of squre ngle-ply shllow sphericl shells with lmintion scheme E h ½30 = 30 =30 Š; =h ¼ 0; R = ¼ nd vrying LD shell models. Theory M, N Circulr frequency prmeters Theory M, N Circulr frequency prmeters ^x ^x ^x 3 ^x 4 ^x 5 ^x 6 ^x ^x ^x 3 ^x 4 ^x 5 ^x 6 ED ED ED ED LD LD LD LD

16 6 F.A. Fzzolri, E. Crrer / Composite Structures 0 (03) 8 Tle 4 q The first six dimensionless circulr frequency prmeters ^x ¼ x ffiffiffiffiffiffiffi 4 q, of squre ngle-ply shllow sphericl shells with lmintion scheme ½h = h =h Š; =h ¼ 0; R E h = ¼ nd vrying lmintion ngle. h Theory Circulr frequency prmeters ^x ^x ^x 3 ^x 4 ^x 5 ^x 6 0 R-FSDT Present models ED ED LD R-FSDT Present models ED ED LD R-FSDT Present models ED ED LD R-FSDT Present models ED ED LD Tle 5 q The first six dimensionless circulr frequency prmeters ^x ¼ x ffiffiffiffiffiffiffi 4 q, of squre ngle-ply deep cylindricl shells with lmintion scheme ½h = h =h Š; =h ¼ 0; R E h = ¼ nd vrying lmintion ngle. h Theory Circulr frequency prmeters ^x ^x ^x 3 ^x 4 ^x 5 ^x 6 0 R-FSDT Present models ED ED LD R-FSDT Present models ED ED LD R-FSDT Present models ED ED LD R-FSDT Present models ED ED LD R-FSDT Present models ED ED LD R-FSDT Present models ED ED LD R-FSDT Present models ED ED LD Angle-ply lminted shllow cylindricl nd sphericl shells In Tles 9 modertely thic nd symmetric ngle-ply lminted shllow cylindricl shells re investigted. A convergence nlysis of the first six dimensionless frequency prmeters ting into ccount stcing sequence ½30 = 30 =30 Š, length-to-thicness rtio /h = 0, rdius-to-length rtio R = ¼ nd Mt- (see Tle ) is crried out y exploiting the use of ED nd LD shell models. Both the inemtics descriptions (equivlent single lyer, zig-zg nd lyer-wise) nd the higher order terms do not ffect the rte of convergence. From n overll point of view, the convergence towrds the exct one is quite quic nd smooth, this is due to the choice of the trigonometric tril functions, which hs stted nd proved in [30], show stle ehvior even t higher hlfwve numers. In Tle, results otined ting into ccount three-lyered regulr symmetric ngle-ply ½h = h =h Š hving the sme geometricl nd mteril properties forementioned, vrying the lmintion ngle in rnge strting from 0 to 90 with 5 increments, using hlf-wve numers M = N = nd employing the shell models ED,ED 888,LD 444 re compred with those provided y Qtu [94] using lgeric polynomil tril functions (08 DOFs). An excellent greement is found for the fundmentl

17 F.A. Fzzolri, E. Crrer / Composite Structures 0 (03) 8 7 dimensionless circulr frequency prmeter the difference increses when incresing the frequency. In Tles 4 the sme investigtion is performed for modertely thic nd symmetric ngle-ply lminted shllow sphericl shells. By virtue of the second curvture R long the direction (see Fig. ) which increses the glol stiffness of the structure the dimensionless circulr frequency prmeters re higher with respect to the previous cse (cylindricl shells). Both for the convergence nlysis underten in Tles, 3 nd for the direct comprison of the results with those proposed y Qtu in Tle 4 cn e formulted similr considertions stted in the cse of shllow cylindricl shells. However for sphericl shells the investigtion of the lmintion ngle rnge cn e restricted from 0 to due to the geometricl symmetry of the structures Angle-ply lminted deep cylindricl shells In Tle 5 the first six dimensionless frequency prmeters of modertely thic nd symmetric ngle-ply lminted deep cylindricl shells mde of Mt-, with stcing sequence ½h = h =h Š, lmintion ngle vrying from 0 to 90 with 5 increments, length-to-thicness rtio /h = 0, rdius-to-length rtio R = ¼, hlf-wve numers M = N = nd y using the shell models ED,ED 888,LD 444 re compred with those provided y Qtu [94]. Once gin n excellent greement is found for the dimensionless fundmentl circulr frequency prmeter, ut ting into ccount higher frequencies the differences increse. With respect to shllow cylindricl shells hving the sme geometricl nd mteril properties the frequency re higher nd comprle to those of shllow sphericl shells. 8. Conclusions The hierrchicl trigonometric Ritz formultion for the first time hs een extended to shell structures in order to perform free virtion nlysis of douly-curved nisotropic lminted composite shllow nd deep shells. Refined higher order equivlent single lyer, zig-zg nd lyer-wise shell models re ssessed y comprison with the 3D elsticity nd 3D FEM solutions. Severl shell geometries ccounting for thin nd thic shllow cylindricl nd sphericl shells, deep cylindricl shells nd hollow circulr cylindricl shells, with cross-ply nd ngle-ply sting sequences hve een investigted. The influence of inemtics descriptions nd higher order terms on the rte of convergence hve een exmined. The effects of significnt prmeters such s stcing sequence, length-to-thicness rtio nd rdius-tolength rtio on the circulr frequency prmeters hve een discussed. From the nlyses crried out the following conclusions cn e drwn: The hierrchicl shell models provide the est results in terms of dimensionless circulr frequency prmeters when compred to those present in literture or otined y virtue of commercil FEM softwres. The full description of curvture terms nd higher order terms re needed in order to improve the ccurcy. Lyer-wise nd zig-zg theories, which ccount for the zig-zg trend of the displcement components through the thicness due to the trnsverse nisotropy, re compulsory when thic deep cylindricl shells re nlyzed. Lyer-wise theories emedded in the HTRF led to the 3D elsticity solution with computtionl cost lower thn tht required y complex 3D FEM models. Kinemtics descriptions nd higher order terms do not ffect the rte of convergence. References [] Donnell LH. Stility of thin wlled tues under torsion. Tech. rep. 479, NACA; 933. [] Donnell LH. A discussion of thin shell theory. In: Proceedings of the fifth interntionl congress for pplied mechnics; 938. [3] Mushtri KM. On the stility of cylindricl shells sujected to torsion. Trudy Kznsego vitsionnugo intitut 938; [in Russin]. [4] Mushtri KM. Certin generliztions of the theory of thin shells. Izv Fiz Mt Odd pri Kzn Univ 938;(8) [in Russin]. [5] Love A E H. A tretise on the mthemticl theory of elsticity. 4th ed. New Yor: Dover Pu.; 944. [6] Love AEH. The smll free virtions nd deformtions of thin elstic shell. Philos Trns Roy Soc (Lond) Ser A 888(79): [7] Timosheno SP. Theory of pltes nd shells. New Yor: McGrw Hill; 959. [8] Gol denveizer AL. Theory of thin shells. New Yor: Pergmon Press; 96. [9] Novozhilov VV. The theory of thin elstic shells. Groningen (The Netherlnds): P. Noordhoff Lts; 964. [0] Flügge W. Stti und Dynmi der Schlen. Berlin: ulius Springer; 934 [Reprinted y Edwrds Brothers Inc., Ann Aror, Mich., 943]. [] Flügge W. Stresses in shells. Berlin: Springer Verlg; 96. [] Lur ye AI. Generl theory of elstic shells. Pril Mt Meh 940;4():7 34. [3] Byrne R. Theory of smll deformtions of thin elstic shell. Sem Rep Mth Univ Clif Pu Mth NS 944;():03 5. [4] Reissner E. A new derivtion of the equtions of the deformtion of elstic shells. Am Mth 94;63(): [5] Nghdi PM, Berry G. On the equtions of motion of cylindricl shells. Appl Mech 964;():60 6. [6] Snders L. An improved first pproximtion theory of thin shells. Tech. rep. R4, NASA; 959. [7] Vlsov V. Osnovnye differentsilnye urvneni oshche teorii uprugih ooloche. Pril Mt Meh 944;8 [English trnsltion: NACA TM 4, Bsic differentil equtions in the generl theory of elstic shells, 95]. [8] Vlsov V. Oshchy teoriy ooloche; yeye prilozheniy v tehie. Gos. Izd. Teh.-Teor.-Lit., Moscow-Leningrd, 949 [English trnsltion: NASA TTF-99, Generl theory of shells nd its pplictions in engineering, 964]. [9] Arnold RN, Wrurton GB. Flexurl virtions of the wlls of thin cylindricl shells hving freely supported ends. 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