A Four-Noded Quadrilateral Element for Composite Laminated Plates/Shells using the Refined Zigzag Theory. A. Eijo E. Oñate S.

Transcription

1 A Four-Noded Quadrlateral Element for Compote Lamnated Plate/Shell ung the Refned Zgag Theor A. Ejo E. Oñate S. Oller Publcaton CIMNE Nº-378, June 2012

2 A Four-Noded Quadrlateral Element for Compote Lamnated Plate/Shell ung the Refned Zgag Theor A. Ejo E. Oñate S. Oller Publcaton CIMNE Nº-378, June 2012 Internatonal Center for Numercal Method n Engneerng Gran Captán /n, Barcelona, Span

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4 A FOUR-NODED QUADRILATERAL ELEMENT FOR COMPOSITE LAMINATED PLATES/SHELLS USING THE REFINED ZIGZAG THEORY A. Ejo, E. Oñate and S. Oller Internatonal Center for Numercal Method n Engneerng (CIMNE) Unvertat Poltècnca de Cataluna (UPC) Campu Norte UPC, Barcelona, Span Abtract. A new blnear 4-noded quadrlateral element (called QLRZ) for the anal of compote lamnated and andwch plate/hell baed on the refned gag theor (RZT) propoed b Teler et al. [1] preented. The element ha even nematc varable per node. Shear locng avoded b ntroducng an aumed lnear hear tran feld. The performance of the element tuded n everal eample where the reference oluton the 3D fnte element anal ung 20-noded heahedral element. Fnall, the capablt for capturng delamnaton effect analed. 1 INTRODUCTION Clacal plate thn theor, nown a Krchhoff theor [2], and the more advanced Rener-Mndln theor (RMT) [3, 4], alo called Frt Order Shear Deformaton Theor (FSDT), were the frt mplfed theore capable to precel model a plate tructure of homogeneou materal. However, when appled to hghl heterogeneou lamnated compote plate t nown that both theore gve poor predcton. The caue of th drawbac due to that thee theore propoe a lnear thcne dtrbuton of the aal dplacement, whch unable to repreent the comple real nematc of a compote lamnate. 3D fnte element anal the more approprate tool to accuratel modelng plate and hell of lamnated compote materal. However, for compote wth hundred of ple, 3D anal become prohbtvel epenve. Improved FSDT model have been obtaned b the o-called Hgher Order Shear Deformaton Theor (HSDT) [5, 6]. In thee model hgher-order nematc term wth repect to the plate thcne are added to the epreon for the aal dplacement. However, thee model are not effectve for comple cae wth localed load or hgh tranvere anotrop. More accurate model are gven b Laer-We Theore (LWT) [5, 7] n whch the thcne coordnate dvded nto a number of anal laer (that ma be not concdent wth the number of lamnate laer) aumng eparate dplacement feld epanon wthn each pl. LWT eld hgh qualt predcton. However, the number of unnown proportonal to the number of anal laer, whch largel ncreae the computatonal cot of the method. An attractve alternatve between the accurac of LWT and the computatonal effcenc of FSDT and ome HSDT are the ZgZag (ZZ) theore [5, 6, 8]. In ZZ theore the n-plane dplacement a uperpoton of a pecewe lnear dplacement functon (the gag functon) over a lnear, quadratc or cubc dplacement feld along the thcne drecton. It mportant to note that the number of nematc varable n ZZ theore ndependent of the number of laer. Man of the ZZ formulaton uffer from ther nablt to model correctl a clamped boundar condton, whch mae t dffcult to atf equlbrum of force at a upport. In addton man ZZ theore requre C 1 contnut for the deflecton feld, whch a dadvantage veru mpler C 0 contnut plate theore, uch a RMT. 1

5 Teler et al. [1, 9] have recentl developed a Refned ZgZag Theor (RZT) for beam and plate that adopt Tmoheno and RMT dplacement feld a the baelne for beam and plate anal, repectvel. The e attrbute of the RZT are, frt, a lnear pecewe gag functon that vanhe at top and bottom urface of the beam and plate ecton. Second, t doe not requre full tranvere hear tre contnut acro the lamnated plate depth. Thrd, C 0 contnut onl requred for the fnte element method (FEM) appromaton of the nematc varable and fnall, all boundar condton can be effectvel mulated [1, 9-11]. Oñate et al. [10] have taen the RZT a the ba for developng a mple two-noded C 0 beam element named LRZ. The accurac of the LRZ beam element for analng compote lamnated beam ha been demontrated for mple upport and clamped beam under dfferent load. The poblt of the LRZ beam element for modelng delamnaton effect ha alo been teted [10, 12]. More recentl, anoparametrc two and three-noded C 0 beam element baed n the RZT have been preented b Gherlone et al. [11]. In th wor we preent the formulaton of an oparametrc four-noded C 0 quadrlateral plate element named QLRZ wth even nematc varable per node baed on the RZT [1]. Shear locng avoded b ung an aumed lnear hear tran feld. A deep tud of th phenomenon preented n anne I. The good performance of QLRZ hown n three dfferent tude: verfcaton, convergence, and comparon. The verfcaton ecton am at evaluatng the performance of th element when the materal homogenou,.e., when the gag functon vanhe. The nfluence of compote materal on the convergence and accurac of the QLRZ element analed n the convergence ecton. Net, we preent everal eample of the good performance of the QLRZ element hghl heterogeneou materal n the comparon ecton. Fnall, the poblt of the QLRZ element for predctng delamnaton effect n compote lamnated plate demontrated n a mple eample. 2 GENERAL CONCEPTS OF ZIGZAG PLATE THEORY 2.1 ZIGZAG KINEMATICS The nematc feld n gag plate theor generall wrtten a u (,, ) = u (, ) θ (, ) + u (,, ) 0 v (,, ) = v (, ) θ (, ) + v (,, ) w (, ) = w0 (, ) where the aal dplacement functon are u = φ ( ) ψ (, ) ; = 1, N 0 v = φ ( ) ψ (, ) and upercrpt ndcate quantte wthn the th laer wth + 1, the vertcal coordnate of the th nterface and N the number of laer. The unform aal dplacement along the coordnate drecton and are u 0 and v 0, repectvel; θ and θ repreent the average bendng rotaton of the tranvere normal about the potve and drecton; and w 0 the tranvere deflecton. φ (, ) (1a) (1b) = denote a nown pecewe lnear gag functon, and ψ a prmar nematc varable defnng the ampltude of the gag functon on the plate. Summarng, the nematc varable are 2

6 T = u0 v0 w0 θ θ ψ ψ a (1c) The gag dplacement feld of Eq.(1a) a uperpoton between the tandard nematc of the frt order Rener-Mndln theor (RMT) and the lnear pecewe gag functon (Eq.(1b)). Note that the gag dplacement vanhe for homogeneou materal leadng to the dplacement feld of the RMT. The n-plane ( ε ) and tranvere hear ( ε ) tran are defned a p u v ε ε ε p u v ε = = γ = + ε t γ u w γ + v w + θ ψ φ ( ) u 0 θ ψ φ ( ) v 0 θ θ ψ ψ = φ ( ) + φ ( ) u0 v 0 = + w φ 0 0 θ ψ 0 w 0 φ θ ψ ε mb m εb ε φ S p 0 ε p = + + = (2a) 0 ε ε φ 0 St εt where ε m, ε b and ε are the tran vector duo to membrane, bendng and tranvere hear effect of the RMT, repectvel. The n-plane and tranvere hear tran vector emanatng from the RZT are denoted b ε mbφ and ε φ, and ε p and ε t are the generaled n-plane and tranvere hear tran vector defned a ε m ε ε ; p = εb εt = φ ε (2b) εmbφ S p = Sm Sb S mbφ ; S t = S S φ t where (.) denote the generaled tran vector gven b 3

7 ε ε 0 T u0 v0 u0 v0 γ ; m = + ε = w = 0 γ θ b θ θ θ θ ψ = + ; ε φ = ψ T T w θ ψ ψ ψ ψ ε mbφ = Sm = = I3 ; Sb = I3 ; S = = I φ ( ) φ (2d) 0 Smb φ = 0 φ ( ) 0 0 ; φ = S φ 0 0 φ ( ) φ ( ) 0 where γ ( =, ) the average tranvere hear tran of RMT. Note that φ φ pecewe lnear, hence, t dervatve = β contant wthn each laer. 2.2 CONSTITUTIVE RELATIONSHIPS The relatonhp between the n-plane and the tranvere hear tree and the tran for the th laer are epreed n matr form a wth σ σ σ p p p τ D 0 ε = = = = Dε σt 0 Dt εt τ σ D D τ E υe 0 1 = υ E E 0 1 υυ 0 0 ( 1 υυ ) G G 0 = 0 G p t The reultant tre vector are defned a Membrane force (2c) (3a) (3b) 4

8 N T σ m = N = Sm σ pd (3c) N T T T ( d) ( d) ( φd) σ = S D S ε + S D S ε + S D S ε m m p m m m p b b m p mb mbφ Bendng moment σ = D ε + D ε + D ε m m m mb b mmbφ mbφ T D = S D S d m m p m T D = S D S d mb m p b D = S D S T d mmbφ m p mbφ M T σ b = M = Sb σ pd (3d) M T T T ( d) ( d) ( φd) σ = S D S ε + S D S ε + S D S ε b b p m m b p b b b p mb mbφ Tranvere hear force σ = D ε + Dε + D ε b bm m b b bmbφ mbφ T D = S D S d bm b p m T D = S D S d b b p b D = S D S T d bmbφ b p mbφ Q T σ = = t d Q S σ (3e) T T ( d) ( φd) σ = S D S ε + S D S ε t t φ σ = Dε + D ε φ φ T D = S D S d D t T φ = S D d t S φ Net, we defne the addtonal peudo-bendng moment and the peudo-hear force emanatng from the RZT, whch are conjugate to the new generaled tran ψ (, j =, ) and the varable ψ, repectvel. j The peudo-bendng moment are defned b M φ M φ T σ mbφ = mbφ pd M = S σ (3f) φ M φ 5

9 T T T ( d) ( d) ( d) σ = S D S ε + S D S ε + S D S ε mbφ mbφ p m m mbφ p b b mbφ p mbφ mbφ σ = D ε + D ε + D ε mbφ mbφm m mbφb b mbφ mbφ T D = S D S d mbφ m mbφ p m T D = S D S d mbφb mbφ p b D = S D S T mbφ mbφ p mbφ and the peudo-hear force b Qψ T σ = = φ t d Q S σ (3g) ψ d ( ) ( ) T T d d σ = S D S ε + S D S ε φ φ t φ t φ φ σ = D ε + D ε φ φ φ φ D D T φ = S φ Dt S T = S Dt S φ φ φ The overall conttutve epreon for the reultant tree can be wrtten n matr form a σ p D p 0 ε p σ = = (3h) σt 0 Dt εt where σ p and σ t contan the n-plane and tranvere hear reultant tree, repectvel. σ m σ σ ; p = σb σt = φ σ σmbφ The n-plane and tranvere hear generaled conttutve matrce, are gven b D m Dmb D mmbφ D D φ D p = Dbm Db Dbmb φ ; D t = D φ D φ Dmb φm Dmb φb Dmb φ d d D p and 2.3 PRINCIPLE OF VIRTUAL WORK (PVW) The vrtual wor prncple for a dtrbuted load q and pont load f can be tated a V npl T T T dv = da + A = 1 (3) δε σ δa q δa f (4a) where the l.h.. of Eq.(4a) epree the nternal vrtual wor performed b the tree and the r.h.. the eternal vrtual wor of the dtrbuted and pont load. V the volume of the plate, A the area of applcaton of the dtrbuted load, and npl the number of pont load. Subttutng Eq.(2a) nto Eq.(4a) gve D t 6

10 V T ( ) V φ φ T T T T + ( δε S + δε φ S φ ) σt dv δε σ dv = δε S + δε S + δε S σ dv + T T T T T T m m b b mb mb p V T T T T T T ( δε m Sm σ p δεb Sb σ p δεmb φsmbφ σ p ) V T T T T ( δε S σt δε φ S φ σt ) dv = + + dv V Ung Eq. (3c), (3d), (3e), (3f), (3g) eld T T T T T T δε σ = ( ) ( m m b b mb mb ) V δε σ + δε σ + δε A φ σ φ + δε σ + δε A φ σ φ dv da da The vrtual wor can be therefore wrtten a ( ) A npl T + T = T T p p t t da + A = 1 δε σ δε σ δa q δa f (4b) The ntegrand n Eq.(4b) contan nematc varable dervatve up to frt order onl, whch allow u to ue C 0 contnuou element. 3 DERIVATION OF THE ZIGZAG FUNCTION The gag functon defned wthn each laer b φ + φ φ φ φ = ( 1 ξ ) φ + ( 1+ ξ ) φ = + ξ =, where φ and 1 φ the gag functon valued at and 1 nterface, 1 ( ) 0 N repectvel wth φ = φ = 0 and ξ = 2 1. h Fgure (1) how the gag functon φ, the gag dplacement dplacement functon φ. (5) u, and the aal u, for the drecton. A mlar dtrbuton found for the gag a) b) c) Fgure 1 Thcne dtrbuton of the gag functon dplacement φ a), gag dplacement u c) n the RZT. u b), and aal 7

11 The lope of the gag functon (Eq.(5)) gve a contant value for each laer defned a 1 φ ( φ φ ) β = = (6a) h Becaue the gag functon vanhe on the top and bottom urface, the throughthe-thcne ntegral of the lope functon β equal to ero,.e. β d = 0 (6b) It convenent to defne a new dfferent functon η a η = γ ψ (7) whch lead to the followng epreon for the th laer tranvere hear tran and tree a γ = 1+ β γ β η (8a) ( ) ( 1 ) τ = G + β γ G β η (8b) The average hear tran over the plate thcne are obtaned b ntegratng the tranvere hear tran γ (Eq.(8a)) over the thcne and ung Eq.(6b). Th gve 1 γ γ = d h (9) The nterfacal contnut of the frt term, aocated wth the average hear tran γ, enforced n the tranvere hear tre dtrbuton (Eq.(8b)),.e. G ( 1 β ) G ( 1 β ) + = + (10a) whch lead to a contant hear modulu acro the plate thcne defned b G = G 1+ β (10b) Then, from Eq. (10b) ( ) G β = 1 (11) G the eplct form of G obtaned b ubttutng β n the ntegral of Eq.(6b),.e. N h G = h (12) = 1 G Fnall, the gag functon obtaned b replacng Eq.(6a) nto Eq.(5), th gve 1 h β φ = φ + ( ζ + 1) (13) 2 wth β defned b Eq.(11). 1 8

12 4 QUADRILATERAL LINEAR REFINED ZIGZAG PLATE ELEMENT (QLRZ) The QLRZ element (fgure AI.2) a Lagrangan oparametrc 4-noded fnte element derved from the refned gag theor decrbed above. 4.1 DISCRETIZATION OF THE DISPLACEMENT FIELD The mddle urface of a plate dcreted nto 4-node 2D oparametrc fnte element of quadrlateral hape. The nematc varable (Eq.(1c)) can be nterpolated wthn each element a u0 v 0 (e) a 1 w 0 4 (e) (e) 2 (e) θ [ ] a a = = N a = N N N N = Na (14) (e) = 1 3 θ a (e) 4 ψ a ψ where beng (, ) 77 unt matr. ( e) N = NI7 ; a = u0 v0 w0 θ θ ψ ψ N ξ η (Eq.(21)) the C 0 contnuou hape functon of node th and 7 T I the 4.2 GENERALIZED STRAIN FIELD The generaled n-plane tran are obtaned n term of the nodal nematc varable b ubttutng Eq.(14) nto the generaled n-plane hear tran p ε (Eq.(2b)), 9

13 m p b mb u N u v N v u v N N u v N N N φ θ θ θ θ θ θ θ ψ ψ ψ ψ + + = = = + + ε ε ε ε 4 4 ( ) ( ) 1 1 e e p p N N N N N θ ψ ψ ψ ψ = = = = B a B a (15a) where p B and p B are the n-plane generaled tran matrce for the element and the th node, repectvel. The matr p B can be pltted nto membrane ( m ), bendng (b ) and gag ( mbφ ) contrbuton. Th lead to m p b mb φ = B B B B (15b) wth N N N N m b N N N N N N mb N N φ = = = B B B (15c) Replacng Eq.(14) nto (Eq.(2b)) the generaled tranvere tran are obtaned a

14 w0 N θ w 0 N θ ε w N ( e) ( e) θ w 0 N θ ε t = t t = = = B a = B a (16a) εφ = 1 = 1 ψ Nψ ψ Nψ where B t and B t are the tranvere generaled tran matrce for the e element and the th node, repectvel. Matr B t can be plt nto hear ( ) and gag ( φ ) contrbuton a B Bt = (16b) Bφ where N 0 0 N B = N N 0 0 (16c) N 0 Bφ = N 4.3 ELEMENT STIFFNESS MATRIX The equlbrum equaton relatng nodal force and dplacement are obtaned b ubttutng the dcreted equaton (15a) and (16a) nto the vrtual wor prncple ( ) A npl T + T = T T p p t t da + A = 1 δε σ δε σ δa q δa f (17a) Subttutng Eq.(3h) nto the l.h. of Eq.(17a) gve δε T σ + δε T σ = δε T D ε + δε T Dε da (17b) Conderng that ( p p t t ) ( p p p t t t ) A A 4 T ( e) T T (e) T T p = p = p = 1 δε δa B δa B 4 T ( e) T T (e) T T t = t = t = 1 δε δa B δa B and ubttutng Eq.(15a), (16a), (17c) nto Eq.(17b) eld T T (e) T T ( e) ( δε pd pε p + δεt Dε t t ) da = ( p p p ) A δa B D B a A (e) T T ( e) ( δa B t DtBta ) da A = ( p p p + t t t ) A (e) T T T ( e) (e) T ( e) ( e) da + = δa B D B B D B da a = δa K a (17c) 11

15 Fnall Eq.(17a) reduced to where beng - npl ( e) ( e) K a qda f A = 1 = 0 ( e) K the ought element tffne matr. Th matr can be epreed a ( e) ( e) ( e) = p + t K and ( e ) p Thee are gven b ( e ) t (18) K K K (19a) K the n-plane and tranvere tffne matrce, repectvel. K = B D B K ( e) T p p p p A = ( e) T t A t t t da B D B da To facltate ubequent hear locng tude, matr wth 4.4 BOUNDARY CONDITIONS The boundar condton are: A. Clamped de: ( e) ( e) ( e) ( e) ( e) T t φ φ φ K plt a follow ( e ) t (19b) K = K + K + K + K (20a) K = B D B da ( e) T A K = B D B ( e) T φ A φ φ φ K = B D B ( e) T φ φ φ A w = 0 u = θ = ψ = 0 v = θ = ψ = 0 B. Smpl upported de: Hard upport: w = u = θ = ψ = 0 Soft upport: w = 0 where the drecton of the de. C. Smmetr a: un = θn = ψ n = 0 where n the orthogonal drecton to the mmetr a. da da (20b) 4.5 SHEAR LOCKING The orgnal form of the QLRZ element uffer hear locng for lender compote lamnated plate. In order to remove th defect two dfferent alternatve are analed ( e) n Anne I: 1) b ung a reduced ntegraton of the tranvere tffne matr K t (Eq.(20a), and 2) b ung an aumed tranvere hear tran feld [13]. The tud howed that the aumed tranvere hear tran technque a more content alternatve for avodng the hear locng problem. Matrce B m, B b, B mbφ from Eq.(15c) and B φ (Eq.(16c)) are computed ung blnear hape functon (Eq.(21)) whle matr B (Eq. (16b) and (AI.16.b)) replaced b the ubttute tranvere hear tran matr B of Eq.(AI.18). 12

16 The b-lnear hape functon N are 1 N = 41+ ξξ 1+ ηη (21) ( )( ) Node ξ η Table 1 Value of ξ and η for each node. Fgure 2 Blnear hape functon. The tffne matrce K and ( e ) K K of = ( e ) φ K (Eq.(20a)) are computed a ( e ) t B D B da ( e) T A K = B D B ( e) T da φ A φ φ (22) 4.6 A POSTERIORI COMPUTATION OF TRANSVERSE SHEAR STRESSES Whle n-plane tree (, and ) tranvere hear tree ( and ) σ σ τ are well predcted b equaton (3a), the τ τ are not. The reaon that the conttutve Eq. (3a) eld a contant value nto each laer, leadng to a dcontnuou thcne dtrbuton of τ and τ. An ueful alternatve to compute τ and τ a poteror from the n-plane tree ung the equlbrum equaton, σ τ τ + + = 0 (23a) τ σ τ + + = 0 from whch, the tranvere hear tree at a pont P acro the thcne coordnate are computed b σ τ τ ( ) = P d d h h 2 P 2 P (23b) σ τ τ ( ) = d d P h h 2 2 P P 13

17 The n-plane tree at pont P n the QLRZ element are appromated b 4 = P σ P = 1 σ ( ) N ( ) 4 = P σ P = 1 σ ( ) N ( ) 4 = P τ P = 1 τ ( ) N ( ) (23c) where N the hape functon (Eq.(21)) and denote the th node. Fnall, the tranvere hear tree are obtaned b replacng Eq.(23c) nto Eq(23b), 4 4 N N τ ( ) = ( ) ( ) P σ τ h 2 h 2 = 1 P 1 = P (24) 4 4 N N τ ( ) = σ ( ) τ ( ) P h 2 h 2 = 1 P = 1 P 5 VERIFICATION STUDIES The accurac of the QLRZ element for otropc homogeneou materal tuded n th ecton. The am to evaluate the behavor of the QLRZ element when φ (, ) = vanhe whch lead to ψ = 0 and the nematc of Eq.(1a) concde wth that of RMT. Th tud cont n analng a SS and a clamped quare plate of de length L = 2 and thcne h = 0.05 ( λ = L = 40) under a unforml dtrbuted load q = 1 h and a pont load P = 4 actng at the center (Fgure 3). Iotropc homogeneou materal properte are aumed wth: E = 0.219, 0.25 G = E 2 1+ µ. µ =, and ( ) a) b) c) d) Fgure 3 Square plate ( λ = 40 ) for verfcaton and convergence anal. SS plate under unforml dtrbuted load a) and pont load b). Clamped plate under unforml dtrbuted load c) and pont load d). 14

18 Aumng mmetr along both ae, onl one quarter of the plate analed. Fve dfferent mehe of QLRZ element (fgure 4) whoe properte are lted n table 2 are emploed. a) b) c) d) e) Fgure 4 Mehe of NN QLRZ element emploed for verfcaton and convergence anal. a) N = 2; b) N = 4; c) N = 8; d) N = 16; e) N = 32. Mehe Properte Meh N Element Node DOF Table 2 QLRZ mehe properte. In order to ae the element accurac, the followng relatve error defned a w wk er = (25) w where K w the vertcal deflecton at the center pont computed wth the th meh ( = 1, 2,...,5) and w K the analtcal Krchhoff oluton defned a n F L E h wk = α wth D = D ( µ ) where F denote the load andα a coeffcent dependent on the boundar condton and the load. The analtcal oluton of the problem are hown n Table 3. (26) Analtcal Krchhoff oluton Boundar Load Α F n w (Krchhoff) SS Clamped Dtrbuted 0,00406 q Pont 0,01160 P Dtrbuted 0,00122 q Pont 0,00560 P Table 3 Analtcal Krchhoff oluton. The QLRZ oluton of the problem and the relatve error are preented n Table 4. Fgure 5 how the behavor of the error. Label SS-P, SS-q, C-P, and C-q n Fgure 5 15

19 refer to mpl-upported-pont-load, mpl-upported-dtrbuted-load, clamped-pontload, and clamped-dtrbuted-load, repectvel. Relatve error (e r %) of w at center pont Load Dtrbuted Pont Meh SS Clamped w er (%) w er (%) Table 4 Relatve error e r of w at center pont. Fgure 5 Relatve error e r of central deflecton w value. Table 5 how that the converged oluton obtaned for the 88 meh (Nº 3). Good accurac obtaned alread for the 44 meh ( e r le than 2.5%). Reult for the SS cae (error 1.0%) are better than for the clamped one. The wort reult obtaned for the clamped plate under central pont load for the 22 meh ( e r = 12.42% ). 6 CONVERGENCE STUDIES We tud net the nfluence of the heterogenet of the compote materal on the convergence and accurac of the QLRZ element, a SS and clamped quare plate of length de L = 2m and thcne h 0.1m λ = 20 under unforml dtrbuted load q = ( ) 2 = 1N m (Fgure 3a and 3c). Three dfferent compote lamnated materal, whoe 16

20 properte are hown n Table 5, are condered for each eample. The degree of heterogenet ncreae from compote C1 to C3. Properte Laer 1 (Top) Laer 2 Laer 3 (Bottom) Compote C1 Compote C2 Compote C3 h [m] h/3 h/3 h/3 E [MPa] h [m] h/3 h/3 h/3 E [MPa] h [m] h/10 h/1.25 h/10 E [MPa] E Table 5 Compote materal properte. Tang advantage of mmetr onl one quarter of plate analed ung the QLRZ mehe decrbed n ecton 5 (Fgure 4). The reference oluton wa obtaned b a 3D fnte element anal ung a meh of (3 element per pl) 20-noded heahedral element nvolvng 4499 node and DOF (Fgure 6). a) b) Fgure HEXA20 mehe emploed to compute the reference oluton for compote C1 and C2 a), and compote C3 b). Convergence quantfed b the relatve error defned a m m3 D er = (27) m 3D 17

21 where m and m 3D are the magntude of nteret obtaned wth the th QLRZ meh ( = 1, 2,...,5) and the 3D reference oluton, repectvel. The magntude tuded m are: the vertcal deflecton w at the center pont C (Fgure 3), the aal tre σ on the top urface of pl 1 at pont E, and ψ at pont E. Snce ψ doe not appear n 3D fnte element anal, m and 3D m are the value of th magntude obtaned ung the th QLRZ meh ( = 1,..., 4) and the fnet meh (3232), repectvel. The reult obtaned are hown n Table 6-7, and Fgure 7-8. It clearl een that convergence alwa lower for the more heterogeneou materal and for the clamped plate. For the clamped plate and the three materal (Table 6) error are le than 10% for the 1616 meh for all varable. For the SS plate (Table 7) error are le than 2.3% for the 88 meh n all cae. For compote C1 (the more homogeneou one) error are le than 2.9% for the 88 meh n all cae and le than 6.3% for the 44 meh n all cae ecept for σ n the clamped plate. For the more heterogeneou materal (compote C3), the dfference n the reult between the SS and the clamped plate larger. For the SS plate (Table 7) error are le than 2.3% for the 88 meh n all varable. For the clamped plate (Table 6) error are le than 23% for the 88 meh and le than 10% for the 1616 meh n all cae. The qualt of the reult obtaned for the compote C2 are between that of compote C1 and C3. Relatve error e r (%) n clamped plate Meh w at pont C σ at pont E ψ at pont E C1 C2 C3 C1 C2 C3 C1 C2 C ,71 50,28 60,99 99, ,13 80,09 86, ,65 30,16 43,47 20,86 44,14 45,53-6,28 43,34 54, ,60 12,32 22,44 2,90 14,35 17,24-1,47 13,68 18, ,29 3,67 9,25-1,21-0,40-1,15-0,30 2,58 2, ,14 0,69 2,85-2,22-4,70-4,62 0,00 0,00 0,00 Table 6 Clamped quare plate ( λ = 20 ) under unforml dtrbuted load. Relatve error e (%) for w, r Meh σ, and ψ. Relatve error e r (%) n SS plate w at pont C σ at pont E ψ at pont E C1 C2 C3 C1 C2 C3 C1 C2 C3 22 2,69 19,36 25,83 26,98 32,89 33,24-9,11 41,06 51, ,68 6,50 10,14 4,86 7,70 9,05-3,99 8,95 13, ,25 1,54 2,22-0,30-0,79 0,44-0,71-0,40-1, ,15 0,38 0,35-1,55-3,04-1,92 0,07-0,45-1, ,12 0,12-0,02-1,86-3,49-2,07 0,00 0,00 0,00 Table 7 SS quare plate ( λ = 20 ) under unforml dtrbuted load. Relatve error e (%) for w, r and ψ. σ, 18

22 (a) (b) (c) Fgure 7 Clamped quare plate ( λ = 20 ) under unforml dtrbuted load. Relatve error e (%) for w r a), σ b), and ψ c). (a) (b) (c) Fgure 8 SS quare plate ( λ = 20 ) under unforml dtrbuted load. Relatve error e (%) for w a), r σ b), and ψ c). 7 COMPARISON STUDIED FOR SS SQUARE AND CIRCULAR COMPOSITE LAMINATED PLATES In order to how the performance of the QLRZ element for hghl heterogeneou compote materal, a quare SS plate of length L = 2m and thcne h = 0.1m, and a crcular SS plate of dameter d = 2m and thcne h = 0.1m are tuded. The tructure 2 are loaded under a unforml dtrbuted load, q = N m (Fgure 9). Each plate tuded for dfferent compote lamnated materal wth properte hown n Table 8 and 9. The quare plate analed for compote C4-7 and the crcular plate for compote C6-7. Do to mmetr onl one quarter of plate analed ung the QLRZ mehe hown n Fgure 10 whoe properte are lted n Table 10. The reference oluton the 3D fnte element anal ung HEXA20 element. The dfferent 3D mehe for each cae are hown n Fgure 11. Detal of each meh are gven n Table 11. The RMT reult for the quare plate of compote C4 are alo hown n Fgure 12. The RMT oluton wa obtaned b ung a meh of 1616 four-noded QLLL plate element [13, 14] 19

23 a) b) Fgure 9 Square SS plate a) and crcular SS plate b) under unforml dtrbuted load. Laer materal properte A B C D E E E G G G Table 8 Laer materal properte. E and G are gven n MPa. Compote lamnated materal Compote Laer dtrbuton h C4 (A/C/A) (0.1/0.8/0.1) C5 (A/B) (0.5/0.5) C6 (A/B/C/D) (0.1/0.3/0.5/0.1) C7 (A/C/A/C/B/C/A/C/A) (0.1/0.1/0.1/0.1/0.2/0.1/0.1/0.1/0.1) Table 9 Laer dtrbuton of the compote materal. h 20

24 a) b) c) d) Fgure 10 QLRZ mehe. Square plate: 88 a) and 1616 element b). Crcular plate: 40 c) and 168 d) element. 21

25 a) b) c) d) e) f) Fgure 11 HEXA20 reference mehe. Square mehe for compote C4 a), C5 b), C6 c), C7 d), and crcular mehe for compote C6 e) and C7 f). 22

26 QLRZ mehe properte Mehe (Fgure 10) NN Number of element Node DOF a b c d Table 10 QLRZ meh properte. Meh (Fgure 11) HEXA20 meh properte Compote Number of element Node DOF a C b C c C d C e C f C Table 11 HEXA20 meh properte. Fgure how the computed vertcal deflecton w (a), the thcne dtrbuton of the aal dplacement u (b), the aal tre σ (c), the tranvere hear tre τ (d) for each plate under tud. The vertcal deflecton accuratel captured. At the center of plate, the mamum error (14%) gven b the crcular plate of compote C6 ung the 40-element meh (Fgure 16a). For the fnet meh (168 element) the computed error are le than 10%. The thcne dtrbuton of the aal dplacement accuratel predcted n all cae. The ablt to capture the comple nematc of lamnated compote materal a e feature of the QLRZ plate element. The ucceful aal dplacement predcton lead to accurate aal tre value a hown n Fgure c). Fgure d) dpla the good reult for the thcne dtrbuton of tranvere hear tree computed a poteror ung Eq.(24). Reult demontrate the good performance of the QLRZ element. Fgure 12 how the naccurate reult when modelng a compote lamnated plate ung QLLL element baed on RMT. The deflecton at the plate center three tme tffer than the reference oluton (Fgure 12a). The RMT oluton alo eld an erroneou lnear thcne dtrbuton of the aal dplacement (Fgure 12b), whch lead to a dtorted dtrbuton of the aal tre (Fgure 12c). Fnall, the RMT unable to capture the correct tranvere hear tre dtrbuton (Fgure 12d). 23

27 a) b) c) d) Fgure 12 SS quare plate under unforml dtrbuted load. Compote C4. a) Vertcal deflecton along central lne BC. Thcne dtrbuton of: b) aal dplacement u at pont B, c) aal tre the center pont C, and d) tranvere hear tre τ at pont E. σ at 24

28 a) b) c) d) Fgure 13 SS quare plate under unforml dtrbuted load. Compote C5. a) Vertcal deflecton along central lne BC. Thcne dtrbuton of: b) aal dplacement u at pont B, c) aal tre the center pont C, and d) tranvere hear tre τ at pont E. σ at 25

29 a) b) c) d) Fgure 14 SS quare plate under unforml dtrbuted load. Compote C6. a) Vertcal deflecton along central lne BC. Thcne dtrbuton of: b) aal dplacement u at pont B, c) aal tre the center pont C, and d) tranvere hear tre τ at pont E. σ at 26

30 a) b) c) d) Fgure 15 SS quare plate under unforml dtrbuted load. Compote C7. a) Vertcal deflecton along central lne BC. Thcne dtrbuton of: b) aal dplacement u at pont B, c) aal tre the center pont C, and d) tranvere hear tre τ at pont E. σ at 27

31 a) b) c) d) Fgure 16 SS crcular plate under unforml dtrbuted load. Compote C6. a) Vertcal deflecton along lne BC. Thcne dtrbuton of: b) aal dplacement u at pont D, c) aal tre center pont C, and d) tranvere hear tre τ at pont D. σ at the 28

32 a) b) c) d) Fgure 17 SS crcular plate under unforml dtrbuted load. Compote C7. a) Vertcal deflecton along lne BC. Thcne dtrbuton of: b) aal dplacement u at pont D, c) aal tre center pont C, and d) tranvere hear tre τ at pont D. σ at the 8 MODELING OF DELAMINATION WITH THE QLRZ ELEMENT Delamnaton,.e. nterlamnar crac, a common and dangerou ource of damage n lamnated compote materal [15], hence the hgher nteret of centfc communt n t modelng. The mulaton of delamnaton n plate and hell tructure tll a challenger n computatonal old mechanc. Among the mot popular technque to model the delamnaton phenomenon are the Vrtual Crac Cloure Technque (VCCT) [16], the Coheve Fnte Element [17-19] and the Contnuou Mechanc ung damage model [20]. All of them need a "3D dcretaton" and, a mentoned above, th ma lead to epenve computatonal cot for lamnated compote of hundred of ple. Therefore, the ue of plate model able to mulate delamnaton a motvatng alternatve. 29

33 In th ecton, the capablt of the QLRZ element to model effectvel delamnaton effect n lamnated compote plate hown. The delamnaton model choen here mpl mple ntroducng a ver thn pl between adjacent materal laer n the lamnated compote ecton. Delamnaton occur when the materal properte of the added pl are reduced due to nterlamnar falure b ung a contnuou damage model. The QLRZ element wth th mple delamnaton model can tae nto account the reducton of the plate tffne do to the nterlamnar falure. Moreover, the QLRZ element can alo accuratel repreent the jump n the aal dplacement feld acro the laer. Net we preent an eample where the SS quare plate of Fgure 9a ( L = 2m and 2 h = 0.1m ) analed under unforml dtrbuted load q = 10000N m. The C7 compote lamnated materal emploed (Table 9), and t materal properte are hown n Table 8. The objectve to mulate delamnaton between laer 3 and 4 b ntroducng a ver thn laer ( h = 0.001m ) between thee two laer (Fgure 18), whoe ntal properte concde wth thoe of the laer 4. The delamnaton progrevel nduced b reducng the hear modulu of the nterface laer up to 5 order of 3 2 magntude from GIL1 = MPa (model 1) to GIL6 = MPa (model 6) (Table 12). Th reducton appled over the whole plate urface. The QLRZ oluton obtaned b ung the 1616 meh of Fgure 10. The reference oluton emploed obtaned b a 3D fnte element anal ung a meh of 8828 HEXA20 element (3 element per laer L1-9, and 1 element for the nterface laer). Fgure 18 Lamnated compote C7 to whch a thn nterface laer between laer 3 and 4 added. Shear modulu value for nterface laer Model G IL [MPa] Model G IL [MPa] 1 0, , , , , , Table 12 Induced hear modulu value of the nterface laer for delamnaton tud. 30

34 Fgure 19 how the evoluton of the vertcal deflecton at the plate center n term of the hear modulu = 1,...,6 of the nterface laer (Table 12). Note that the G ( ) IL deflecton ncreae a the hear modulu decreae. Alo, t nteretng that the 1 deflecton doe not change after model 5 ( GIL5 = MPa ). Th ma mean that after th value of G IL5 the compote C7 eparated nto two completel dconnected part (laer 1-3 and laer 4-9), and, therefore, unable to tranmt the hear tre acro the ecton. The relatve error for the central deflecton, between the QLRZ and the HEXA20 oluton, near to 25% for model 6. Fgure 20 how the thcne dtrbuton of the aal dplacement along drecton meaured at pont E (Fgure 9a). A can be oberved, the jump of aal dplacement between laer 3 and 4 durng delamnaton well captured. Note that the jump reman tatonar after model 5 accordng to the no change n the deflecton value. Fgure 19 Vertcal deflecton at center of plate for the dfferent hear modulu value of the nterface laer. 31

35 a) b) c) d) e) f) Fgure 20 Thcne dtrbuton of aal dplacement along drecton at pont E for each model. 32

36 9 CONCLUSIONS A mple, robut, hear locng free and accurate 4-noded plate element (called QLRZ) baed on the refned gag theor ha been preented. The hear locng defect wa overcome b ntroducng an aumed lnear hear tran feld. The element ha onl even unnown per node whch are nterpolated b tandard C 0 lnear hape functon. The thcne dtrbuton of the tranvere hear tree accuratel reproduced b a poteror computatonal proce. The verfcaton anal ha hown that the element able to accuratel model plate of homogeneou materal for dfferent load and boundar condton. The nfluence of the heterogenet of compote lamnated materal on the convergence and accurac of QLRZ oluton ha been tuded. An mportant feature of the QLRZ element t ablt to capture the gag dtrbuton of aal dplacement and the ubequent comple tran and tre dtrbuton acro the thcne wth the mple appromaton choen. Th propert mae poble to predct delamnaton effect a t ha been hown n a mple demontratve eample. 10 ACKNOWLEDGEMENTS Th reearch wa partall fnancal upported b the SAFECON project of the European Reearch Councl (ERC) of the European Common. ANNEX I SHEAR LOCKING SOLUTION It nown that the tandard four node plate element baed on the Rener-Mndln theor (RMT) ehbt hear locng for thn plate [13]. Tang nto account that the nematc of RZT a um between the RMT and the gag dplacement, we aume that the QLRZ element alo uffer th defect. Th aumpton wll be later evdenced n th ecton. In the net we preent two alternatve to overcome th drawbac. The frt baed on a reduced ntegraton of the tranvere tffne matr K t (Eq.(20a)). The econd baed on an aumed tranvere hear tran feld [13]. AI.1 REDUCED INTEGRATION It well nown that hear locng can be elmnated b ung reduced ntegraton of the tranvere hear tffne matr. Let u remember that K t ha econd-order polnomal for rectangular element, th mean the eact ntegraton obtaned ung a 22 Gau quadrature. The reduced ntegraton mple ung a 11 Gau quadrature. Let u retae the prevou matr K t from Eq.(20a) K = K + K + K + K T (AI.1) t φ φ φ In order to ae the nfluence of the reduced ntegraton of the matr K t, the followng ntegraton combnaton are elected. 33

37 INTEGRATION COMBINATIONS Combnaton Eact Reduced C1 K ; K K φ φ C2 K φ K ; K φ C3 - K t Table AI-1 Integraton combnaton ued to ae the nfluence of the reduced ntegraton of K t. The two e attrbute of the reduced ntegraton technque are, frt, ea mplementaton n a fnte element code, and econd, a reduced computatonal cot. A a drawbac, the reduced ntegraton can orgnate new mechanm (n addton to the rgd bod moton) ncompatble wth the boundar condton, whch can propagate wthn the meh. Th problem hown n the eample of ecton AI.3. AI.2 ASSUMED TRANSVERSE SHEAR STRAIN FIELDS A thn plate element mut atf Krchhoff condton of ero tranvere hear tran, that εt = ε + ε φ = 0 (AI.2) where ε and ε φ contan the average tranvere hear tran of RMT ( γ and γ ) and the prmar nematc varable ψ and ψ (Eq.(1b)), repectvel. Although the condton (AI.2) mple ε + ε = 0, we now that hear locng n a 4-noded Rener- φ Mndln element due to ε [13], therefore, we conder that th effect n QLRZ element alo due to the RMT tranvere hear tran ε. Tang nto account that ε = ε, the condton (AI.2) can be reduced to ε = 0 (AI.3) Wth the am of demontrate the orgn of the locng effect n the QLRZ element, ε eplctl developed b ung Eq.(16a) and (21). For mplct, we conder γ onl. From Eq.(16a), 4 γ w0 N w N θ = = 1 θ (AI.4.a) = Subttutng Eq.(16a) nto Eq.(AI.4.a) 4 ξ 1 w ξη w η ξ ξ η γ = θ + θ η + θ ξ θ ξη = 1 4a 4 4a 4 4 4a (AI.4.b) Factorng Eq.(AI.4.b) γ = α ( w, θ ) + α ( w, θ ) η + α ( θ ) ξ + α ( θ ) ξη (AI.4.c) The Krchhoff condton (AI.3) mple 0 = 1, 2,3, 4. α 1 = 0 and α 2 = 0 are phcall poble and the mpoe a relatonhp between θ and w. However, the element unable to atf naturall the condton α 3 = 0 and α 4 = 0, unle θ = 0, α = ( ) whch lead to w = 0 (hear locng effect). Identcal concluon can be found for γ mpl b replacng ξ b η and θ b θ. 34

38 Hence, t poble to avod the hear locng defect b mpong a tranvere hear tran feld (fgure AI-1) le γ α1( w, θ ) + α2 ( w, θ ) η ( e) ε = B a γ = = α3( w, θ ) + α4( w, θ ) η (AI.5) where B the ubttutve hear tran generaled matr. Th matr ued ntead of the orgnal B for computng the hear tffne matrce K and K φ of Eq.(20b),.e. T K = B D B da A T K = B D B da φ A φ φ (AI.6) Thee matrce can now be ntegrated eactl ung a 22 Gau quadrature. Summarng, the dea of th technque to mpoe a pror" a tranvere hear tran feld, whch allowng the vanhng of ε n the thn lmt. The aumed tranvere hear tran nterpolaton where m ε N γ N γ = γ = γ ( e) (AI.7.a) = 1 γ are the average tranvere hear tran value at m pont wthn the element and N γ are the hear nterpolatng functon. B combnng Eq. (16a)and (AI.7.a) obtaned m (e) = γ = ( e) (AI.7.b) = 1 ε N B a B a and the new nterpolaton are (e) a = N a ; ε = N ( e) γ (AI.8) γ Fgure AI-1 Impoed tranvere hear tran feld. AI.2.1 Computaton of ubttutve hear tran generaled matr The natural tranvere hear tran feld gven b [13, 14] α1 ' γ ξ α1 α2η 1 η 0 0 α + 2 ε = A α γ = η α3 α4ξ = = ξ + α3 α 4 B (AI.9) 35

39 The tranvere hear tran ε n the cartean coordnate tem are epreed a γ -1 ' ε J = = ε γ (AI.10.a) where J the 2D Jacoban matr ξ ξ J = (AI.10.b) η η The coeffcent α are obtaned b amplng the natural hear tran (Eq.(AI.9)) at the four pont hown n Fgure AI-2, wth γ = α + α η coδ + α + α ξ n δ ; = 1, 4 (AI.11) ( ) ( ) ξ where δ the angle between ξ drecton and the natural ξ a. Combnng Eq. (AI.9) and (AI.11) gve γ ξ α1 γ ξ α 2 2 γ = P α ξ = = γ α ξ 3 3 (AI.12) γ α4 ξ4 where the tran γ ξ γ ξ are related to γ ξ γ α = P -1 γ ξ and γ ξ γ η b γ 1 γη γ ξ γ 1 2 ξ2 η2 ' = = = T γ γ γ ξ ξ 3 3 γ γ ξ η 4 3 ξ γ ξ 4 γ η4 (AI.13) Combnng Eq. (AI.9), (AI.12) and (AI.13) gve ' -1 ' ε A P T = γ (AI.14) The cartean tranvere hear tran γ at the amplng pont are related to the natural tranvere hear tran ' γ b 1 J γ1 2 ' 0 J 0 0 γ 2 γ γ = = C γ ; γ 3 = (AI.15) 0 0 J 0 γ 3 γ J γ 4 The relatonhp between the cartean hear tran γ at the four amplng pont (fgure AI-2) and the nodal dplacement ( ) a e 36

40 wth where B ( 1, 2,3, 4) B γ = B a (e) (AI.16.a) [ B B B B ] T = (AI.16.b) = the orgnal tranvere generaled tran matr (Eq.(16c)) at the th amplng pont. Combnng Eq. (AI.10.a), (AI.14), (AI.15) and (AI.16.a) gve -1-1 ε = J A P T C B a = B a (AI.17) where B the ought ubttute tranvere hear tran matr gven b -1-1 B J A P T C B = (AI.18) Fgure AI-2 QLRZ plate element. AI.3 STUDY OF ALTERNATIVES In order to how the effcac of the two elected alternatve and to how the mechanm creaton when the reduced ntegraton of Table AI-1 are ued, are analed tow cae. A mpl upported quare plate of length de L = 2 under a unforml dtrbuted load of unt value ( q = 1) analed n the frt eample (Fgure AI-3a). The anal performed for four pan-to-thcne rato: λ = L h = 5,10,50,100. A clamped quare plate of length de L = 2 and thcne h = 0.2 ( λ = 10 ) under a center pont load of value P = 4 tuded n the econd eample (Fgure AI-3b). Both tructure have ung a compote lamnated materal whoe properte are lted n Table AI-2. Onl one quarter of the plate tuded due to mmetr (Fgure AI-3) ung a meh of 1616 QLRZ element (Fgure AI-4a) wht 289 node and 1445 DOF. The reference oluton obtaned b a 3D fnte element anal ung a meh of (3 element per pl) of 20-noded heahedral element (HEXA20) nvolvng 4499 node and DOF (Fgure AI-4b). (a) (b) 37

41 Fgure AI-3 Smpl upported quare plate under unforml dtrbuted load a). Clamped quare plate under center pont load b). Compote materal properte Properte Laer 1 (top) Laer 2 Laer 3 (bottom) h 0, 25 L 0,50 λ L 0, 25 λ L λ E 2,19E5 2,19E4 4,4E5 G 0,876E5 0,876E4 1,76E5 Table AI-2 Materal properte for hear locng tud. a) b) Fgure AI-4 Mehe ued for the anal of one quarter of the SS plate QLRZ element a) and HEXA20 element b). Fgure AI-4 how the r rato defned a wqlrz r = (AI.19) w 3D 38

42 where w QLRZ and w 3D are the mddle deflecton at the plate center obtaned wth the QLRZ element and the 3D fnte element anal, repectvel. The QLRZ element reult have been obtaned ung eact ntegraton of matr K t (eact), a reduced ntegraton of matrce K, K φ and K φ for the three combnaton of table AI-1 (C1, C2, and C3), and fnall ung the aumed tranvere hear tran feld technque (QLRZ). (a) (b) Fgure AI-5 r rato v. pan-to-thcne λ. Smpl upported quare plate under unforml dtrbuted load. Fgure a): eact ntegraton (eact) and the three ntegraton combnaton (C1, C2, and C3) of table AI-1. Fgure b): eact ntegraton and aumed tranvere hear tran feld (QLRZ). ( e) Fgure AI-5 clearl how the hear locng defect when eact ntegraton of K t ued. However, th defect dappear b ung both technque. Fgure AI-6 how the dtrbuton of the vertcal deflecton w along the plate central lne BC (Fgure AI-3b) obtaned wth eact ntegraton of matr K t, b ung reduced ntegraton (Table AI-1) and aumed tranvere hear tran technque, and 3D anal (HEXA20). Fgure AI-6a reveal the etence of mechanm n meh when reduced ntegraton ued. Thee mechanm do not appear f the aumed tranvere hear tran technque ued. 39

43 (a) (b) Fgure AI-6 Vertcal deflecton w along BC. Clamped quare plate ( λ = 10 ) under a center pont load. Fgure a): eact ntegraton (eact) and the three ntegraton combnaton (C1, C2, and C3) of table AI-1. Fgure b): eact ntegraton, aumed tranvere hear tran feld (QLRZ), and 3D anal (HEXA20). The reult how that the aumed tranvere hear tran technque adequate to develop a robut hear locng free plate element. REFERENCES [1] Teler, A., Scuva, M. D., and Gherlone, M., A content refnement of frt-order hear deformaton theor for lamnated compote and andwch plate ung mproved gag nematc, Mechanc of Materal and Structure, 2010, Vol. 5(2), pp [2] Krchhoff, G., Über da Glechgewcht und de Bewegung ener elathen Schebe, J Angew Math, 1850, Vol. 40(-), pp [3] Rener, E., The effect of tranvere hear deformaton on the bendng of elatc plate, Appl. Mech., 1945, Vol. 12(-), pp [4] Mndln, R. D., Influence of rotator nerta and hear n fleural moton of otropc elatc plate, Appl. Mech., 1951, Vol. 18(-), pp [5] Lu, D. and L, X., An overall vew of lamnate theore baed on dplacement hpothe, Journal of Compote Materal, 1996, Vol. 30(14), pp [6] Wanj, C. and Zhen, W., A Selectve Revew on Recent Development of Dplacement-Baed Lamnated Plate Theore, Recent Patent on Mechancal Engneerng, 2008, Vol. 1(-), pp [7] J. N. Redd and D. H. Robbn, J., Theore and computatonal model for compote lamnate, Appled Mechanc Revew, 1994, Vol. 47(6), pp [8] Carrera, E., Htorcal revew of Zg-Zag theore for multlaered plate and hell, Appled Mechanc Revew, 2003, Vol. 56(3), pp [9] Teler, A., Scuva, M. D., and Gherlone, M., Refnement of Tmoheno Beam Theor for Compote and Sandwch Beam Ung Zgag Knematc, Techncal Publcaton TP , NASA, [10] Oñate, E., Ejo, A., and Oller, S., Smple and accurate two-noded beam element for compote lamnated beam ung a refned gag theor., Computer Method n Appled Mechanc and Engneerng, 2012, pp [11] Gherlone, M., Teler, A., and D Scuva, M., Cº beam element baed on the refned gag theor for multlaered compote and andwch lamnate, Compote Structure. Accepted for publcaton, 2011 [12] Oñate, E., Ejo, A., and Oller, S., Two-noded beam element for compote and andwch beam ung Tmoheno theor and Refned ZgZag Knematc, Publcaton Nº-346, CIMNE, [13] Oñate, E., Structural anal b the fnte element method, Vol. 2: Beam, plate and hell.cimne-sprnger, Barcelona,

44 [14] Oñate, E., Zenewc, O., Suáre, B., and Talor, R. L., A general methodolog for dervng hear contraned Rener-Mndln plate element, Int. Journal for Numercal Method n Engneerng, 1992, Vol. 33(2), pp [15] Bolotn, V. V., Delamnaton n compote tructure: t orgn, buclng, growth and tablt., Compote: Part B 27B, 1996, pp [16] Krueger, R., The Vrtual Crac Cloure Technque: Htor, Approach and Applcaton, Appled Mechanc Revew, 2002, Vol. 57(2), pp [17] Borg, R., Nlon, L., and Smonon, K., Modelng of delamnaton ung a dcreted coheve one and damage formulaton, Compote Scence and Technolog, 2002, Vol. 62(10-11), pp [18] Turon, A., Camanho, P. P., Cota, J., and Da vla, C. G., A damage model for the mulaton of delamnaton n advanced compote under varable-mode loadng, Mechanc of Materal, 2006, Vol. 38(11), pp [19] Balan, C. and Wagner, W., An nterface element for the mulaton of delamnaton n undrectonal fber-renforced compote lamnate, Engneerng Fracture Mechanc, 2008, Vol. 75(9), pp [20] Martne, X., Ratelln, F., Oller, S., Flore, F., and Oñate, E., Computatonall optmed formulaton for the mulaton of compote materal and delamnaton falure, Compote: Part B, 2011, Vol. 47(