6 H INERNAIONAL CONFERENCE ON COMPOSIE MAERIALS OPIMIZAION OF OW-PLACED AILORED COMPOSIE LAMINAES Adiana W. Blom* Mosafa M. Abdalla* Zafe Güdal* *Delf Univesi of echnolog he Nehelands Kewods: vaiable siffness ow placemen cuved fibes seamline hickness ovelap Absac Fibe-einfoced composies ae usuall designed ug consan fibe oienaion in each pl. In ceain cases howeve a vaing fibe angle migh be favoable fo sucual pefomance. his possibili can be full uilized ug ow placemen echnolog. Because of he fibe angle vaiaion ow-placed couses ma ovelap and pl hickness will build up on he suface. his hickness build up affecs manufacuing ime sucual esponse and suface quali of he finished poduc. his pape will pesen a mehod fo designing composie plaes and shells ug vaing fibe angles. he hickness build-up is pediced as funcion of pl angle vaiaion ug a smeaed appoach. I is found ha he hickness build-up is no unique and depends on he chosen sa locaions of fibe couses. Opimal fibe couses ae fomulaed in ems of minimizing he maimum pl hickness maimizing suface smoohness o combining hese objecives wih and wihou peiodic bounda condiions. Inoducion In indus fibe-einfoced composies ae usuall designed ug a consan fibe oienaion in each pl. he fibe angles in hese laminaes ae picall 0 90 and ±45 degees. adiionall he choice of hese la-ups was moivaed b manufacuabili while nowadas la-ups wih changing o even nonconvenional fibe angles ae avoided because of he lack of allowables. Howeve eseach on composies wih a vaing in-plane fibe oienaion has shown ha vaiable siffness can be beneficial fo sucual pefomance [-] because vaiable-siffness laminaes ae able o edisibue he loading as opposed o consansiffness laminaes. In mos cases cuvilinea fibe pahs manufacued b ow placemen ae used o consuc he vaiable-siffness laminaes [458-02]. Jegle aing and Güdal [8-0] designed vaiable-siffness fla plaes wih holes and demonsaed hei effeciveness b building and eg seveal specimens. Due o fibe angle vaiaion a ow-placed shell picall ehibis gaps and/o ovelaps beween adjacen couses and pl hickness will change along he suface [8-02]. he amoun of gap/ovelap affecs sucual esponse manufacuing ime and suface quali of he finished poduc. his pape pesens a mehod fo designing composie plaes and shells ug vaing fibe angles. he hickness build-up is pediced as funcion of pl angle vaiaion ug a seamline analog. I is found ha he hickness build-up is no unique and depends on he chosen sa locaions of fibe couses. Opimal disibuions of fibe couses ae fomulaed in ems of minimizing he maimum pl hickness o maimizing suface smoohness eihe wih o wihou peiodic bounda condiions. Subsequenl he discee hickness build-up esuling fom he ow placemen pocess will be shown as compaison o he smeaed hickness appoimaion. Finall a numbe of applicaions fo he developed mehods and suggesions fo fuue eseach will be given. 2 Seamline Analog Fo he consucion of discee fibe pahs a seamline analog is being used. In his case
ADRIANA W. BLOM Mosafa M. ABDALLA and Z. Güdal each seamline epesens he ceneline of a couse o if he couse widh is made infiniel small each seamline will epesen a gle fibe. Mahemaicall a seamline is epesened b a seam funcion ( ) C () which connecs all he poins wih a consan value C. Fo a given fibe angle vaiaion () he seamlines can be found b solving he following paial diffeenial equaion: d ds d d + ds ds + 0 (2) A unique soluion fo he seam funcion (and hus he locaion of he seam lines) depends on he bounda condiions. Befoe seeking a soluion o he seam funcion addiional consideaions elevan o he phsical epesenaion of he fibe pahs ae in ode. ae discee a fis appoimaion o he amoun of ovelap could be made b smeaing ou his discee ovelap o fom a coninuous hickness disibuion. In his case he smeaed hickness will be invesel popoional o he disance beween adjacen couses which can be eplained as follows. If a numbe of N couses wih a given widh w c and hickness has a fied volume V and if hese successive couses ae placed close han he widh of he couses hen he oal widh coveed is less hen N w c and he hickness has o be inceased in ode o mainain he same maeial volume V. When he disance beween wo seamlines is dn hen / dn (as eplained above). Since d/dn and d beween wo seamlines is consan accoding o Eq.. he hickness will be popoional o n as follows: (3) dn n ( d ) d If d is assumed o be hen. which can be used o deive a diec coelaion beween he hickness disibuion and he fibe angle vaiaion: ( ) n s ln (4) Fig.. Definiions As saed ealie he seamlines epesen he cenal pah of a finie widh couse. Unless he seamlines ae paallel he successive couses will alwas ovelap each ohe when no gaps ae allowed beween hem (o alenaivel he gaps will fom beween he passes if wo successive finie widh passes ae no allowed o ovelap). he amoun of ovelap depends on he disance beween he couse cenelines: when he disance is deceased he ovelap aea is inceased. Alhough in eali hese ovelaps in which s and n epesen he angen and nomal ve o a seamline especivel as shown in Fig.. he phsical eplanaion of Eq. 4 is ha he change in hickness along a seamline depends on he change of he fibe oienaion pependicula o ha seamline. Since boh ve s and n depend on he given fibe angle disibuion () he onl unknown in Eq. 4 is he hickness. he hickness can now be deemined b solving his equaion bu ce i is a diffeenial equaion bounda condiions ae needed in ode o obain a unique soluion. In accodance wih seamline heo bounda condiions ae onl needed a he inflow bounda whee he inflow bounda is defined b: 2
OPIMIZAION OF OW-PLACED AILORED COMPOSIE LAMINAES s N 0 (5) whee s is he veco angen o he seamline and N is he ouwad nomal veco o he bounda as shown in Fig.. B changing he hickness a he inflow bounda he hickness disibuion inside he domain and a he ouflow boundaies will change. 3. Deemining Bounda Condiions hee eis an infinie numbe of possible bounda condiions fo which he hickness disibuion associaed wih he seamlines can be found bu he mos difficul pa is o find he ones ha ae phsicall sensible fo he poblem in hand. In his pape he bounda condiions ae esablished such ha he fulfill a ceain opimali cieion. he opimali cieia demonsaed in his pape ae minimum maimum hickness maimum smoohness and combinaions of hese wo. In addiion o he opimali cieion consains such as a minimum of one o peiodic bounda condiions can be enfoced as well. 3. Geneal Soluion B ug he following change of vaiables: τ ln Eq. 4 becomes: s τ n (6) Above equaion is solved numeicall b disceizing he deivaives so ha i is wien as: [ M ] τ B (7) whee [M] is he mai ha epesens he lef hand side of Eq. 6 τ is he veco ha epesens τ a eve gid poin and B is he veco ha epesens he igh hand side of Eq. 6 as well as he bounda condiions. If he hickness a he inflow boundaies is assumed o be one evewhee (τ 0) a nominal soluion can be found fo τ which will be efeed o as τ 0. A geneal soluion of Eq. 7 can be epessed as: [ ] τ in τ τ 0 + (8) whee each column j in mai [] epesens he influence of bounda gid poin j on he hickness disibuion in he complee domain while saisfing Eq. 7. Since hese columns ae independen of each ohe and ce Eq. 7 is a linea equaion an linea combinaion of hese columns also epesens a soluion as given b Eq. 8. he enies in τ in all ende he hickness a a gle poin on he inflow bounda. B subsiuing Eq. 8 in Eq. 7 he hickness can be opimized fo one of he cieia menioned ealie b ug τ in as design vaiables. Ofen i is desied o have a leas one lae of maeial evewhee so ha no gaps eis. heefoe i is equied ha he hickness ove he enie domain is a leas one ( τ 0) in all opimizaions descibed below. 3.2 Minimized Maimum hickness Minimizing he maimum hickness of he plae is he fis opimali cieion ha will be elaboaed on in his pape. his cieion is elevan fo judging how pacical he esuling hickness disibuion would be in a eal life sucue as well as fo deemining if i is possible o manufacue a plae wih consan hickness fo he given fibe angle vaiaion. If he hickness build-up is oo sevee (i.e. if one poin is 00 o 000 imes hicke han anohe poin) i will no be applicable o ealisic sucues. In ode o solve he min-ma poblem he bound fomulaion as inoduced b Olhoff [3] is used. his fomulaion inoduces a new vaiable α which epesens he maimum hickness and which also seves as he new objecive funcion fo he minimizaion. Addiionall a consain on he hickness a each gid poin is being inoduced so ha he hickness neve eceeds he minimized maimum hickness α: 3
ADRIANA W. BLOM Mosafa M. ABDALLA and Z. Güdal s.. F min τ α α i 2... i N g (9) whee N g is he numbe of gid poins. he design vaiables ha esul fom he opimizaion ae subsiued in Eq. 8 and hen he hickness disibuion is found b changing τ i vaiables again: e. i 3.3 Maimized Smoohness Anohe possible opimizaion objecive is o maimize he smoohness of he hickness disibuion of he composie panel. Alhough in eali he change in hickness will alwas be discee due o he discee naue of ow couses i would sill be desiable fo pl dops/ovelaps o be disibued houghou he panel ahe han o be concenaed a paicula egions. In ode o achieve his smoohness is defined as he nom of he ae of change of hickness. Smoohness is maimized b minimizing he H -nom of he hickness: min τ [ K]τ (0) 2 whee [K] is he mai ha disceizes he Laplacian. Subsiuing he epession fo τ (Eq. 8) in Eq. 0 gives: 2 τ [ K] τ 2τ 0 [ K ] τ 0 + τ 0 [ K ] + τ in [ ] [ K ][ ] τ in 2 τ in () he fis em in his equaion is consan so ha he objecive funcion o be minimized is: wih F [ K ] τ in f τ in min 2 τ in (2) [ K ] [ ] [ K ][ ] f τ 0 [ K ][ ] (3) he minimum of Eq. 2 can be found b diffeeniaing i and equaing i o zeo so ha: [ K ] in f τ (4) his is a linea ssem ha can be solved fo τ in. Howeve he [K ]-mai is one ime gula and heefoe one en of τ in is given an assumed value so ha he ssem can be solved. Afe he soluion is subsiued in Eq. 8 a consan can be added o τ such ha he condiion of τ 0 is being me (his will no change he H -nom bu will change he absolue value of hickness). 3.4 Combined Objecive Funcion Since boh minimizing he maimum hickness and maimizing he smoohness ae valid opimizaion cieia designes migh conside combining he wo in ode o obain a bee design. Depending on he designe diffeen weighs can be assigned o he individual cieia. he objecive funcions of Eq. 9 and Eq. 2 can hen be combined o fom a new objecive funcion: F min ( w) α + w α 2τ in 2τ in [ K ] τ in f τ in [ K ] τ in f τ in (5) In his equaion w is he weighing funcion ha indicaes he impoance of he smoohness in he opimizaion. Fuhemoe he wo objecive funcions ae nomalized b α * and τ 2 in [ K ] τ in f τ in especivel whee α * is he minimum maimum hickness obained fom Eq. 9 and τ in ae he design vaiables fo maimum smoohness as obained b Eq. 2. 3.5 Peiodic Bounda condiions he pesen fomulaion would also be valid fo clindical shells ce poins on a clindical suface ae in one o one coespondence o poins on a ecangula panel. Neveheless an impoan diffeence eiss; in 4
OPIMIZAION OF OW-PLACED AILORED COMPOSIE LAMINAES he case of a clindical shell he soluion mus be peiodic. When he pl angle vaiaion is peiodic coninui in hickness is obained b including he hickness peiodici consains in he ealie descibed opimizaion ouines. Fo peiodici in -diecion his akes he fom: ( 0) τ ( b) 0 l τ (6) i i whee l is he lengh and b is he widh of he panel. 4. Discee Fibe Couses Once he smeaed hickness disibuion is obained hough one of he opimizaions descibed above he coesponding seam funcion can be obained b inegaing n ove dn: ( ) ndn d d dn + d dn d + i d d d d dn dn (7) he deivaives of wih espec o and can be epessed as funcions of s and n as follows: s s + (8) Since s 0 and n he combinaion of Eqs. 8 and 9 will give: ( ) + 0 0 ( ) ( ) d ( ) ( ) d (9) Boh () and () ae known funcions so ha () can be solved. B ploing he conou lines of his funcion a fied values d fom each ohe he seamlines ae found ha could epesen he cenelines of he acual fibe couses. he consan of inegaion will deemine he eac locaion of he fibe couses which can be used fo saggeing in case of muliple plies wih he same fibe angle disibuion. Once he couse cenelines ae known discee couses can be placed on op of hem and he discee hickness disibuions can be found. 5. Resuls o illusae he diffeences beween he vaious opimali cieia descibed in secion 3 an eample panel is analzed which has he following linea angle vaiaion in -diecion: ( ) 30 30 l (20) such ha he fibe oienaion is a -30 degees a he lef side of he panel ( 0) and -60 degees a he igh side of he panel ( l ). he lengh o widh aio of he panels is 3. Fig. 2a. shows he hickness disibuion of a panel fo which he maimum hickness is minimized. he hickness along he lef and op edge is one which means ha hee ae no ovelaps on hese sides. he maimum hickness occus in he lowe igh cone of he panel. he hickness disibuion of a panel wih maimized smoohness is pesened in Fig. 2b. Compaed o he fis panel he maimum hickness is inceased b appoimael 20 pecen while smoohness is impoved b 40 pecen. In Fig. 2c. he smeaed hickness fo he combined objecive wih w 0.5 is ploed. Since boh maimum hickness and smoohness ae included in he objecive funcion he incease in maimum hickness is onl 7 pecen while he impovemen in smoohness is 30 pecen when compaed o he fis panel. Finall a panel wih peiodic bounda condiions is shown in Fig. 2d. he maimum hickness of his panel is moe han 40 pecen lage han he minimum maimum hickness and also smoohness is deceased. he discee hickness disibuions coesponding o he fou smeaed hickness disibuions of Fig. 2. ae shown in Fig. 3. he widh of hese couses was assumed o be /6 of 5
ADRIANA W. BLOM Mosafa M. ABDALLA and Z. Güdal Fig. 2. hickness disibuion fo vaious opimizaion cieia Fig. 3. Discee hickness build-up (black lae whie 2 laes) he panel widh. Fig. 3a. cleal shows he leas amoun of ovelap. If a laminae wih consan hickness was desied he fibe pahs obained b his opimizaion can be used as basic pahs and he ovelaps could be eliminaed b cuing individual ows on he sides of he couses. he smoohness of he laminae in Fig. 3b. is no ve appaen unil muliple plies ae sacked on op of each ohe and saggeed wih espec o each ohe. he combined objecive laminae of Fig. 3c. is indeed in beween he laminaes of Fig. 3a. and Fig. 3b. Finall he elaivel lage hickness build-up of he laminae wih peiodic bounda condiions is anslaed in lage ovelap aeas as shown in Fig. 3d. 6. Fuue Reseach In his pape i was successfull demonsaed ha a seam line analog can be used o pedic and influence he hickness disibuion in a laminae wih vaiable fibe angles. Diffeen objecives fo opimizaion 6
OPIMIZAION OF OW-PLACED AILORED COMPOSIE LAMINAES wee consideed and in he fuue ohes such as minimum volume migh be eploed as well. In addiion o one-pl designs he developed mehods could be used o design complee laminaes wih boh vaing fibe angles and vaing hickness. One appoach would be o design he laminae ug laminaion paamees and hickness [674] as spaiall vaing design vaiables and hen as a pos-pocesg sep muliple plies wih vaing fibe angles and hei coesponding hickness disibuions could be fi in ode o mach boh he desied laminaion paamees and hickness disibuion as close as possible. Finall a simila mehod will be developed fo cuved sufaces in ode o epand he applicabili of he mehod. Refeences [] He M.W. and Chaee R.F. he use of cuvilinea fibe foma in composie sucue design. Poceedings of he 30 h AIAA/ASME/ASCE/AHS/ASC Sucues Sucual Dnamics and Maeials (SDM) Confeence New Yok NY pape no. 404 989. [2] He M.W. and Lee H.H. he use of cuvilinea fibe foma o impove buckling esisance of composie plaes wih cenal holes. Composie Sucues Vol. 8 pp 239-26 99. [3] Nagenda S. Kodialam A. Davis J.E. and Pahasaah V.N. Opimizaion of ow fibe pahs fo composie design. Poceedings of he 36 h AIAA/ASME/ASCE/AHS/ASC Sucues Sucual Dnamics and Maeials (SDM) Confeence New Oleans LA pape no. 275 995. [4] Waldha C.J. and Güdal Z. Analsis of ow placed paallel fibe vaiable siffness laminaes. Poceedings of he 37 h AIAA/ASME/ASCE/AHS/ASC Sucues Sucual Dnamics and Maeials (SDM) Confeence Sal Lake Ci U pape no. 569 996. [5] Panas L. Oal S. and Cehan U. Opimum design of composie sucues wih cuved fibe couses. Composie Science and echnolog Vol. 63 pp 07-082 2003. [6] Seoodeh S. Abdalla M.M. and Güdal Z. Design of vaiable siffness laminaes ug laminaion paamees. Composies Pa B: Engineeing Vol. 37 pp 30-309 2006. [7] Abdalla M.M. Seoodeh S. and Güdal Z. Design of vaiable siffness composie panels fo maimum fundamenal fequenc ug laminaion paamees. Poceedings of he Euopean Confeence on Spacecaf Sucues Maeials and Mechanical eg Noodwijk he Nehelands 2005. [8] aing B.F. and Güdal Z. Design and manufacuing of elasicall ailoed ow-placed plaes. NASA/CR-2002-299. [9] aing B.F. and Güdal Z. Auomaed finie elemen analsis of elasicall-ailoed plaes. NASA/CR-2003-22679. [0] Jegle D.C. aing B.F. and Güdal Z. ow-seeed panels wih holes subjeced o compession o shea loading. Poceedings of he 36 h AIAA/ASME/ASCE/AHS/ASC Sucues Sucual Dnamics and Maeials (SDM) Confeence Au X pape no. 208 2005. [] Blom A.W. Seoodeh S. Hol J.M.A.M. and Güdal Z. Design of vaiable-siffness conical shells fo maimum fundamenal fequenc. Poceedings of he 3 d Euopean Confeence on Compuaional Mechanics: Solids Sucues and Coupled Poblems in Engineeing Lisbon Pougal June 2006. [2] Blom A.W. aing B.F. Hol J.M.A.M. and Güdal Z. Pah definiions fo elasicall ailoed conical shells. Poceedings of he 47 h AIAA/ASME/ASCE/AHS/ASC Sucues Sucual Dnamics and Maeials (SDM) Confeence Newpo RI pape no. 940 2006. [3] Olhoff N. Mulicieion sucual opimizaion via bound fomulaion and mahemaical pogamming. Sucual Opimizaion Vol pp -7 989. [4] Seoodeh S. Blom A.W. Abdalla M.M. and Güdal Z. Geneaing cuvilinea fibe pahs fom laminaion paamees disibuion. Poceedings of he 47 h AIAA/ASME/ASCE/AHS/ASC Sucues Sucual Dnamics and Maeials (SDM) Confeence Newpo RI pape no. 875 2006. Appendi A he deivaives of he seam funcion ae: he second deivaives of he seam funcion ae: + Coninui of he second deivaives of he seam funcion implies ha so ha: 7
ADRIANA W. BLOM Mosafa M. ABDALLA and Z. Güdal 8 ( ) ( ) 0 + Reaangemen gives: + + (A) Ug he following definiions: ( ) s n ln Eq. A can be wien as: ( ) n s ln