Non-Linear Dynamics of Reinforced Laminated Plates and Shells considering a Consistent Mass Matrix for Fibers

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1 Paper Non-Lnear Dynamcs of Renforced Lamnated Plates and hells consderng a Consstent Mass Matrx for Fbers M..M. ampao, R.R. Paccola and H.B. Coda ão Carlos chool of Engneerng Unversty of ão Paulo, Brazl Cvl-Comp Press, 5 Proceedngs of the Ffteenth Internatonal Conference on Cvl, tructural and Envronmental Engneerng Computng, J. Krus,. Tsompanas and B.H.V. Toppng, (Edtors), Cvl-Comp Press, trlngshre, cotland Abstract In ths paper the non-lnear dynamc formulaton proposed by [] for the analyss of fber-renforced lamnated plates and shells s mproved substtutng the adopted homogenzed mass matrx for embedded fbers by a consstent mass matrx developed for each fber fnte element. Ths complete technque separates the contnuum and fbers mechancal propertes and naturally assembles the consstent renforced meda for dynamc analyss. To valdate the proposed formulaton the present results are compared to reference values n whch homogenzed data were adopted for dynamc propertes. Keywords: dynamcs analyss, fber-renforced lamnates plates and shells, physcal and geometrcal non-lnear analyss, fnte element. Introducton The ncreased applcaton of renforced lamnated composte plates and shells n many structures and devces developng large dsplacements has generated much research to predct ther pre and post-crtcal behavour appled to varous branches of engneerng as, for example, cvl, mechancal, medcal, aeronautc, aerospace and mechatroncs. Despte ts mportance some dffcultes are stll present regardng ts numercal smulaton. ome formulatons for non-lnear dynamc analyss on large dsplacements, as the co-rotatonal and the updated Lagrangan, present varable mass matrx. Ths feature requres the avalablty of specfc tme ntegrators to solve the problem assocated wth energy conservaton [,,3]. Moreover, regardng the renforcement ncluson to consttute composte materals, t s usual to follow the homogenzaton technque that maes dffcult the mechancal characterzaton of the homogeneous equvalent materal, reducng the confdence n general results and the correct modellng of heterogeneous meda [4,5]

2 and mae t dffcult to dentfy the fber-matrx contact stresses necessary for some falure analyss, see [6,7] for example. Furthermore, n some cases, these formulatons are restrcted to geometrc lnear analyss. An extensve lterature revew on the dynamc analyss of composte shells can be seen n [8,9]. In ths wor the non-lnear dynamc formulaton proposed by [] for the analyss of fbers renforced lamnated plates and shells s mproved substtutng the adopted homogenzed mass matrx for embedded fbers by a consstent mass matrx developed for each fber fnte element. Ths complete technque separates the contnuum and fbers mechancal propertes and naturally assembles the consstent renforced meda for dynamc analyss. The trangular lamnated shell fnte element used to dscretze the matrx has ten nodes and seven degrees of freedom per node,.e., three translatons, three components of a generalzed vector and the lnear rate of stran varaton along the thcness. The curved fbers, long or short, are ntroduced n any layer of the lamnated plate or shell by means of nematc relatons to ensure ts adherence to the lamnated matrx wthout ntroducng new degrees of freedom n the global system of equatons. Although the system of equatons to be solved has the sze correspondng to the number of degrees of freedom of the plate or shell, the fbers contrbuton for the total energy of the system nvolves the manpulaton of a large number of terms requrng an effcent strategy to avod an unfeasble computatonal effort. The any-order one-dmensonal fnte elements wth three degrees of freedom per node used to dscretze fbers are developed to consder geometrcal nonlnearty by means of a deformaton functon based on postonal mappng. The Total Lagrangan descrpton of moton and the ant-venant Krchhoff Consttutve Law are adopted. The equlbrum s acheved from the Prncple of tatonary Energy and the Newton-Raphson teratve procedure s used to solve the nonlnear system of equatons. A non-assocatve orthotropc elastoplastc consttutve relaton for the matrx presented by [,] s used together elastc fbers to consttute the lamnated structure. The adopted general falure surface degenerates nto the Von-Mses or Drucer-Prager surfaces for sotropc materals. Ths elastoplastc formulaton s based on the control of the volume change at plastc phase reducng possble locng problems and the tangent stffness tensor s used to mprove the convergence rate of the Newton-Raphson procedure adopted to solve the nonlnear equlbrum problem. As the proposed fnte element formulaton s total Lagrangan and uses generalzed vectors [3,4], the acheved mass matrx s constant; therefore, the Newmar tme ntegrator conserves energy n elastc problems developng large dsplacements [5,6]. To present the proposed formulaton the paper s organzed as follows: ecton descrbes the dynamc non-lnear equlbrum problem for fber-renforced lamnated bodes; ecton 3 presents the nematcs, nternal force vector, Hessan matrx and a consstent mass matrx developed for each fber fnte element; n ecton 4 the lamnated shell fnte element were fbers are embedded s descrbed; ecton 5 descrbes the tme marchng process and the Newton-Raphson teratve procedure; n ecton 6 the dynamc couplng that does not ncrease the number of degrees of

3 freedom of the problem and does not requre matchng nodes n the dscretzaton, for fber-renforced lamnated plates and shells s presented; n ecton 7, the results obtaned wth the present formulaton are compared to reference values n whch homogenzed data were adopted for dynamc propertes [] demonstratng the good behavour and the potental of the proposed formulaton and, fnally, conclusons are presented n ecton 8. Upper symbols ( ) and ( ) are used to dstngush, respectvely, shell and fber varables. Dynamc nonlnear equlbrum problem To solve the present problem, the deal potental energy functon ( ) can be wrtten as a composton of the stran energy ( W ), the potental energy of appled conservatve forces ( P ), the netc energy ( K ) and dsspaton ( Q ), as follows: W PK Q () The stran energy ( W ) stored n the body ncludng matrx stran energy W and fber stran energy W contrbutons,.e., W W W, s gven by: W w( )dv w ( ) dv () l p l j V V n whch w and w are, respectvely, shell and fber stran energy per unt of ntal volume; are shell nodal postons; are fber nodal postons; V s the ntal l p j volume of shell fnte elements and V s the ntal volume of fber fnte elements. The potental energy of appled conservatve forces ( P ) s gven by: P F (3) where F represents forces appled n drecton ; s the th current poston of the pont where the load s appled. mlarly, the netc energy ( K ) stored n the body ncludng matrx netc energy K and fber netc energy K,.e., K K K, s gven by: K y ydv y y dv V j j (4) V where y and y are, respectvely, matrx and fber veloctes and and are matrx and fber mass denstes, respectvely. 3

4 The dsspatve term ( Q ), ncludng matrx dsspaton Q and fber dsspaton Q,.e., QQQ, s wrtten n ts dfferental form as: Q( t,x ) y dv y dv (5) V m m V where m and m are, respectvely, matrx and fber proportonal dampng constants. The prncple of statonary energy s appled by dfferentatng Eq. () regardng a generc nodal poston,.e.: W P K Q g (6) Equaton (6) at an nstant t s nonlnear regardng. In order to solve t, a Taylor expanson regardng s used as follows: g g() g( ) O j j l j l j ( l ) (7) Neglectng hgher order terms O j n Eq. (7) results, g j g j( l ) ( l ) (8) where, for conservatve loads: g W K Q s s s s s (9) To solve Eq. (6) for a specfc tme step one calculates g for an arbtrary g (usually the last nown one) usng Eq. (6), calculates usng Eq. (9) and s usng Eq. (8), updates, returns to Eq. (6) and repeats all the procedure untl could be neglected. 4

5 3 Fber consstent mass matrx The fbers for whch a consstent mass matrx s developed n ths secton are dscretzed by any-order curved fnte elements wth three degrees of freedom per node and consder geometrcal nonlnearty usng the ant-venant-krchhoff consttutve Law gven by: w E () where s the one-dmensonal elastc modulus and E s the one-dmensonal Green stran. Fgure : Fber fnte element mappng for ntal and current confguratons. Fgure shows the fber mappng from ntal to current confguraton. The ponts wth coordnates x n ntal confguraton and the ponts wth coordnates y n the current confguraton are mapped from the auxlary confguraton, respectvely, by the expressons: x X and y () p p p p p where X p are coordnates of fber nodes p n the ntal confguraton, are coordnates of fber nodes n the current confguraton and p are any-order shape functons evaluated n dmensonless coordnates. Index p,...,n and,,3 n Eq. () represent, respectvely, the fber fnte element nodes and fber coordnate drectons. 5

6 From Fgure the one-dmensonal Green stran n Eq. () can be stated as: where T and T T T E () T are the modulus of the fber tangent vector n ntal and current confguratons, calculated respectvely, as: d p( ) d p p( ) d p p( ) p T X X X 3 d d d (3) d p( ) d p p( ) d p p( ) p T 3 d d d (4) The frst and the second dervatves of stran energy n Eq. () regardng fber nodal postons,.e., the natural nternal fber force vector and the Hessan matrx, result, after developng the necessary calculatons, n the followng smple expressons to be solved usng the Gauss Legendre quadrature: d ( ) d j( ) d E J ( )Ad (5) p p W d j T W d p( ) d p ( ) d p( ) d p j( ) j J ( )Ad 4 T d d d d (6) E d ( ) d j( ) J ( )Ad T d d where dx dx dx3 J( ) T (7) d d d s the dfferental Jacoban of, see Fgure. 6

7 The fber nertal contrbuton for the dynamc equlbrum equaton gven n Eq. (6) s obtaned by the frst dervatve of the fber netc energy and the fber mass matrx s obtaned by the second dervatve of the fber netc energy,.e.: K M and K s M (8) where are nodal fber acceleratons and M s a fber consstent mass matrx. In another way, the frst and the second dervatves of fber netc energy gven n Eq. (4) can be wrtten respectvely, as: K y j V y dv j K y y y y j V j j where y s approxmated by the expresson: dv p p (9) () y () As fber veloctes are functon of tme t and fber postons p, after the necessary calculatons n Eq. (9), the frst dervatve of fber netc energy to be assembled n the nertal term of the dynamc equlbrum equaton can be acheve as: K j V dv p j p () From Eq. (), a fber consstent mass matrx results n: K M dv A ds L j j j j V A J ( )d j (3) to be solved usng the Gauss Legendre quadrature. Fnally, the frst and the second dervatves of fber dsspaton energy are gven, respectvely, as: Q j V y dv m and Q y j V m dv (4) Rememberng that dampng s consdered proportonal to mass. 7

8 4 Lamnated hell Fnte Element Fgure shows the nematc descrpton used to map the lamnate plate or shell lam fnte element where fbers are embedded. The deformaton functon from the auxlary confguraton to the ntal confguraton and the deformaton functon lam from auxlary confguraton to current confguraton can be wrtten, respectvely, as: lam lam lam h x l(, )Xl d 3l(, )Nl (5) lam y l(, )l lam lam lam h lam h (6) d 3l(, )ald 3 (, )G m where y l(, ) X m l and y l(, ) l are, respectvely, ponts n the reference surface n ntal and current confguratons, and lam lam h g d 3l(, ) Nl (7) lam lam lam h lam h g d 3l(, ) ald 3 (, ) G (8) are generc vectors n ntal and current confguratons, respectvely, d lam s the dstance from the reference surface to the center of a consdered lamna followng drecton g lam n the ntal confguraton and h s the ntal thcness of ths lamna (see Fgure b). h lam. 3 d lam Reference surface (a) (b) Fgure : Knematc descrpton for a lamnate shell fnte element: (a) gradent mappng; (b) cross-secton. 8

9 In Eqs. (5)-(8), l(, ) are general shape functons evaluated n dmensonless coordnates ; X l l and are the coordnates of the nodes l n the ntal and current confguratons, respectvely; N l and G are, respectvely, nodal values of ntal and current generalzed vectors and a l are nodal values of the lnear rato of thcness varaton that avod volumetrc and shear locng from usual homogeneous or lamnate shell formulatons [4]. The stran energy per unt of ntal volume used for the lamnated shell element w descrbes a ant-venant- Krchhoff materal,.e., ts consttutve Law s gven by the lnear relatonshp among the second Pola-Krchhoff stress tensor j and the Green-Lagrange stran tensor E j gven, respectvely, by: w Ej jlel (9) w j jlel (3) Ej j j j E C A Aj j (3) where jl, C j, A and are, respectvely, the lnear elastc consttutve fourthorder tensor, the rght Cauchy-Green stretch tensor, the deformaton gradent and the j Identty tensor. lam Functons lam and lam are used to fnd the gradent of mappngs, A j and lam A j shown n Fg. a. The composton of these two values for each ntegraton lam pont gves the numercal value of the deformaton gradent A,.e.: lam lam A j, A j, A A ( A ) lam lam lam lam lam j j (3) The frst and the second dervatves of the stran energy n Eq. (9) regardng shell nodal postons, results, respectvely, n the natural nternal force vector and the Hessan matrx for plate or shell elements and can be seen n [7]. 5 Tme marchng process & Newton-Raphson procedure From the prevous developments, Eq. (6) can be wrtten as: W g F M C (33) 9

10 Equaton (33) represents the dynamc equlbrum equaton at any tme and has to be solved. In order to do so the frst step s to wrte ths equlbrum for a specfc nstant ( ), as follows: W F M C (34) Replacng Newmar approxmatons for poston descrpton gven by: t t (35) t( ) t (36) nto Eq. (34) results n: W M C g( ) F MQ CR tcq t t (37) n whch vectors Q and R represent the dynamc contrbuton of the past, and are gven, respectvely, by: Q (38) t t R t( ) (39) Equaton (37) understood smply by g ( ) s nonlnear regardng ( ). A Taylor expanson to solve t regardng postons s appled as follows: W M C g ( ) t t (4) where C s assumed proportonal to M. A Taylor seres of frst order s bult as: ( ) ( ) ( ) g g g (4) and the Newton-Raphson procedure to solve Eq. (37) s derved as: g ( ) g ( ) (4)

11 where s a tral poston, usually, for used n Eq. (37) to calculate olvng, a new tral for s calculated as: g ( ). (43) The acceleraton must be corrected for each teraton by an expresson obtaned from Eq. (35),. e., Q t (44) Equaton (44) s used n Eq. (36) to correct velocty. The stop crteron s gven by Eq. (45) when a chosen tolerance (TOL) s satsfed,.e.: or (45) g( ) TOL TOL 6 Fber-lamnated matrx dynamc couplng Fbers are embedded at any poston n the doman (plate or shell) wthout ncreasng the degrees of freedom and wthout matchng nodes n the dscretzaton, wrtng the postons of fbers nodes as a functon of the postons of lamnated shell nodes by the expresson: (, ) p p p l l lam lam lam h p p p l lam h p p p d 3 l(, ) 4 d 3 (, ) (4) (46) n whch the symbols ( ) and () dstngush, respectvely, shell and fber varables and p ndcates a fber node that belongs to a shell element. nce shell equlbrum postons are nown for a specfc tme t, fbers postons are updated usng Eq. (48). From updated fbers postons t s possble calculate acceleratons, veloctes and dynamc contrbuton of the past, Q and R, for fbers, respectvely, by the expressons: Q t (47) t( ) t (48) Q t t (49) R t( ) (5)

12 Equatons (49) and (5) are used to determne the fbers dynamc contrbuton to be coupled to shell dynamc contrbuton,.e., mass and dampng. To couple fbers to matrx, t s necessary to perfume the dervatves of the stran energy, netc energy and dsspaton of fbers regardng shell nodal parameters. These dervatves results n a transformaton matrx analogous to that presented by [7] and are not repeated here. Fnally, the procedure adopted to consder plastcty n ths paper follows [,,8]. Therefore, the elastoplastc evoluton procedure and the tangent stffness tensor requred to mprove the convergence rate of the Newton-Raphson teratve procedure adopted to solve the nonlnear equlbrum problem s not repeated here. 7 Numercal example A rng subjected to tenson represented by letter q n Fg. 3a, orgnally ntroduced by [] for a homogenzed mass matrx, s analyzed consderng the consstent mass matrx ntroduced n ths paper. The rng cross secton s consttuted by 5 lamnas. Three consttuted of alumnum and two of carbon/epoxy composte, Fg. 3b, representng a recent fbre-metal lamnate materal (FML) used n arcraft structures. Alumnum s consdered sotropc and the carbon/epoxy ply arrangement s consdered n two ways: and 9 as shown n Fgs. 3c and 3d, respectvely. Axs z s taen as the orgn for. Two analyses are performed: (a) carbon fbers dsposed n the epoxy materal and (b) carbon/epoxy homogenzed materal. When the fbers are dsposed n the carbon/epoxy materal (present formulaton) the adopted unaxal propertes are gven n Fg. 4a. When the carbon/epoxy materal s consdered homogenzed (reference formulaton) the adopted unaxal propertes are gven n Fg. 5. The numercal experment adopted to acheved the homogenzed carbon/epoxy materal propertes was performed by [] consderng 8 fbers of dameter f.mmdsposed as n Fg. 3c for the ply arrangement and as n Fg. 3d for the ply arrangement 9. From the performed experments the followng propertes for the carbon/epoxy homogenzed materal are acheved: E 64 N mm, E 59.8 N mm,.53 and.43. (a) (b) (c) (d) Fgure 3: Lamnated rng structural scheme and cross secton.

13 In both performed analyses the adopted unaxal propertes for the alumnum s gven n Fg. 4b. The oung modulus and the Posson rato for the alumnum are gven, respectvely, by E 69GPa and a.33. The adopted denstes are a 3 for alumnum, 3 3 a 79. g cm e. g cm for epoxy and c 7. g cm for carbon-fbre. The adopted Newmar parameters are.5 and.5. Matrx and fbers dampng are consdered null, m, m Unaxal tress (MPa) Carbon-Epoxy Unaxal tress (MPa) Alumnum..x - 4.x - 6.x - 8.x -.x -.x - Unaxal tran..x -3 4.x -3 6.x -3 8.x -3.x -.x - Unaxal tran (a) (b) Fgure 4: Unaxal stress-stran dagram; a) Carbon/Epoxy; b) Alumnum. 3.5x x 3 4 Unaxal tress (MPa).5x 3.x 3.5x 3.x 3 5.x Renforced wth fbres Homogenzed Unaxal tress (MPa) 3 Renforced wth fbres Homogenzed Unaxal tran Unaxal tran (a) (b) Fgure 5: Homogenzed Carbon/Epoxy unaxal stress-stran dagrams: a) ; b) 9. The reference surface of the cylnder s dscretzed n 8 trangular cubc fnte elements wth 48 nodes and 336 degrees of freedom. For the ply arrangement, 54 cubc fbers are dsposed n lamnas and 4. For the ply arrangement 9, 8 fbers, 9 fbers n each lamna, are dscretzed n 4 cubc fnte elements each. Two values for the load q are consdered: q N mm and q 5.33N mm appled n 3 tme steps wth t 9. s. Fgures 6 and 7 compares the dynamc equlbrum path n the mdpont where the load q s appled for q N mm and q 5.33N mm, respectvely. Fgure 8 shows the rng confguraton n tme step 6 for all consdered cases. The color dfference n the reference confguraton of the rng observed n Fg. 8 s due the 3

14 dscretzaton of the fbers n the present formulaton. The acheved results show the good behavor of the proposed formulaton,.e., the results obtaned wth the formulaton ntroduced n ths paper to consder a consstent mass matrx for each fber fnte element present good agreement wth those obtaned wth the homogenzed formulaton whose lmtaton s the dffculty n determnng contact stresses f a falure analyss s desred, for example Dsplacements (mm) º homogenzed soluton 9º present soluton º homogenzed soluton º present soluton Tme (s).5 Fgure 6: Dynamc equlbrum path of the mdpont under the appled load q for q N mm Dsplacements (mm) º homogenzed soluton 9º present soluton º homogenzed soluton º present soluton Tme (s) Fgure 7: Dynamc equlbrum path of the mdpont under the appled load q for q 5.33N mm. 4

15 Homogenzed oluton Present oluton º º N/mm N/mm 9º 9º º º 5.33N/mm 5.33N/mm 9º 9º 8 Concluson Fgure 8: Rng confguraton n tme step 6. In ths paper the nonlnear dynamc formulaton proposed by [] for the analyss of fber renforced lamnates plates and shells s mproved substtutng the adopted homogenzed mass matrx for embedded fbers by a consstent mass matrx developed for each fber fnte element. Ths technque separates fbers-contnuum mechancal propertes and naturally assembles the consstent renforced meda for dynamc analyss allowng dentfyng the fber-matrx contact stresses necessary for some falure analyss. The mechancal couplng between fbers and the lamnated plates or shell fnte element does not ncrease the number of degrees of freedom and 5

16 no nodal matchng s necessary to dscretze the problem. As the proposed fnte element formulaton s total Lagrangan and uses generalzed vectors, the acheved consstent mass matrx s constant, and the nonlnear dynamc equlbrum problem s solved usng the Newmar tme ntegrator conservng energy n elastc problems developng large dsplacements. The acheved results show the good behavor of the proposed formulaton. In future wors we ntend to ntroduced nonlnear physcal behavor n the fbers, damage n the matrx and slp between fbers and matrx. Acnowledgements The authors than the Coordnaton for the Improvement of Hgher Educaton Personnel (CAPE) and the ão Paulo Research Foundaton (FAPEP) for the fnancal support of ths research. References [] J.C. mo, M.. Rfa, D.D. Fox, On a stress resultant geometrcally exact shell-model. Part VI: Conservng Algorthms for nonlnear dynamcs, Internatonal Journal for Numercal Methods n Engneerng, 34, 7-64, 99. [] F. Armero, Energy-dsspatve momentum-conservng tme-steppng algorthms for fnte stran multplcatve plastcty, Computer Methods n Appled Mechancs and Engneerng, 95, , 6. [3] E.M.B. Campelo, P.M. Pmenta, P. Wrggers, An exact conservng algorthm for nonlnear dynamcs wth rotatonal DOFs and general hyperelastcty. Part : shells, Computatonal Mechancs, 48, 95-,. [4] G.M. Kulov,.V. Plotnova, Non-lnear geometrcally exact assumed stress-stran four-node sold-shell element wth hgh coarse-mesh accuracy, Fnte Element n Analyss and Desgn, 43, , 7. [5] M. Braun, M. Bschoff, E. Ramm, Nonlnear shell formulatons for complete three-dmensonal consttutve laws ncludng compostes and lamnates, Comput. Mech., 5, -8, 994. [6] H.T.. ang,. agal, A. Masud, R.K. Kapana, A survey of recent shell fnte elemens, Int. J. Numer. Meth. Engng., 47, -7,. [7] W. Wagner, C. Balzan, mulaton of delamnaton n strnger stffened fber-renforced composte shells, Computers & tructures, 86, , 8. [8] M. Qatu, R. ullvan R, W. Wang, Recent research advances on the dynamc analyss of composte shells: -9, Composte tructures, 9, 4-3,. [9] H. ang,. agal, D. Law, Advances of thn shell fnte elements and some applcatons - verson I, Computers & tructures, 35, 48-54, 99. [] M..M. ampao, H.B. Coda, R.R. Paccola, "Geometrcal and Physcal Nonlnear Dynamc Analyss of Renforced Plates and hells", n Proceedngs of the Twelfth Internatonal Conference on Computatonal tructures 6

17 Technology, B.H.V. Toppng and P. Ivány, (Edtors), Cvl-Comp Press, trlngshre, cotland, paper 43, 4. do:.43/ccp [] R. Rgobello, H.B. Coda, J. Munaar Neto, Inelastc analyss of steel frames wth a sold-le fnte element, Journal of Constructonal teel Research, 86, 4-5, 3. [] A.. Botta, R.R. Paccola, W.. Venturn, H.B. Coda, A dscusson on volume change n the plastc phase, Communcatons n Numercal Methods n Engneerng, 4(), 49-6, 8. [3] H.B. Coda, A sold-le FEM for geometrcal non-lnear 3D frames, Computer Methods n Appled Mechancs and Engneerng, 98, 47-48, 9. [4] H.B. Coda, R.R. Paccola, M..M. ampao, Postonal descrpton appled to the soluton of geometrcal non-lnear plates and shells, Fnte Element n Analyss and Desgn, 67, 66-75, 3. [5] H.B. Coda, R.R. Paccola, Unconstraned Fnte Element for Geometrcal Nonlnear Dynamcs of hells, Mathematcal Problems n Engneerng, 5753, 9. do:.55/9/5753. [6] H.B. Coda, R.R. Paccola, A FEM procedure based on postons and unconstraned vectors appled to non-lnear dynamc of 3D frames, Fnte Element n Analyss and Desgn, 47, ,. [7] M..M. ampao, R.R. Paccola, H.B. Coda, A geometrcally nonlnear FEM formulaton for the analyss of fber renforced lamnated plates and shells, Composte tructures, 9, -5, 5. [8] H.B. Coda, M..M. ampao, R.R. Paccola, A FEM contnuous transverse stress dstrbuton for the analyss of geometrcally nonlnear elastoplastc lamnated plates and shells, Fnte Elements n Analyss and Desgn,, 5-33, 5. 7