Analytical modeling of synthetic fiber ropes subjected to axial loads. Part I: A new continuum model for multilayered fibrous structures

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1 Please nte that this is an auth-pdued PDF f an atile aepted f publiatin fllwing pee eview. The definitive publishe-authentiated vesin is available n the publishe Web site Intenatinal Junal f Slids and Stutues Vlume 44, Issue 9, 1 May 007, Pages hp://dx.di.g/ /j.ijslst Elsevie Ltd All ights eseved Ahime, ahive institutinnelle de l Ifeme hp:// Analytial mdeling f syntheti fibe pes subjeted t axial lads. Pat I: A new ntinuum mdel f multilayeed fibus stutues Seyed Rea Gheishi 1,, Patie Cataud 1,, Pete Davies,, Tanguy Message, 1. Institut de ehehe en Génie ivil et Méanique (GéM), Ele Centale de Nantes, BP9101, 441 Nantes, Fane. IFREMER, Mateials and Stutues gup BP70, 980 Pluané, Fane. Univesité de Nantes, Nantes Atlantique Univesités, Institut de ehehe en Génie ivil et Méanique (GéM), Ele Centale de Nantes, BP9101, 441 Nantes, Fane Cespnding auth: Patie CARTRAUD ; Ele Centale de Nantes ; BP 9101 ; 441 Nantes édex Fane ; Tel: ; Fax: ; patie.ataud@e-nantes.f Abstat: Syntheti fibe pes ae haateied by a vey mplex ahitetue and a hieahial stutue. Cnsideing the fibe pe ahitetue, t pass fm fibe t pe stutue behavi, tw sale tansitin mdels ae neessay, used in sequene: ne is devted t an assembly f a lage numbe f twisted mpnents (multilayeed), wheeas the send is suitable f a stutue with a ental staight e and six helial wies (1 + 6). The pat I f this pape fist desibes the develpment f a mdel f the stati behavi f a fibus stutue with a lage numbe f twisted mpnents. Tests wee then pefmed n tw diffeent stutues subjeted t axial lads. Using the mdel pesented hee the axial stiffness f the stutues has been pedited and gd ageement with measued values is btained. A mpanin pape (Gheishi, S.R. et al., in pess. Analytial mdeling f syntheti fibe pes, pat II: A linea elasti mdel f fibus stutues, Intenatinal Junal f Slids and Stutues, di: /j.ijslst ) pesents the send mdel t pedit the mehanial behavi f a fibus stutue. Keywds: Fibe pe; Yan; Aamid; Multilayeed stutues; Analytial mdel; Testing 1

2 1. Intdutin Syntheti fibe pe ming systems, whih ae ften mpsed f steel hain at the ends and a ental syntheti fibe pe, ae ineasingly finding appliatins as ffshe il explatin ges t deepe sites. Pevius eseahes have shwn that suh ming lines pvide numeus advantages ve steel ming lines (steel wie pes and hains), patiulaly in deep wate appliatins f whih the lage self-weight f steel lines is phibitive (Beltan et al., 004; Fste, 00). It is theefe essential t be able t mdel the mehanial behavi f vey lng syntheti ming lines in de t edue the need f expensive tests unde vaying paametes and peating nditins. Lage syntheti fibe pes ae assemblies f millins f fibes and haateied by a vey mplex ahitetue and a hieahial stutue in whih the base mpnents (fibe yan) ae mdified by twisting peatins. This stutue is then a base mpnent f the next highe stutue. Its hieahial stutue leads t the hieahial appah whee the tp is the fibe pe and the bm is the base mpnents, with seveal diffeent types f elements between the base mpnent and the fibe pe, i.e. yan, assembled yan and stand. Figue 1 illustates this hieahial stutue. Cnsideing the fibe pe ahitetue, it nsists f tw diffeent types f stutue: ne is a stutue with a ental staight e and 6 helial mpnents (1+6), wheeas the send is an assembly f a lage numbe f twisted mpnents (multilayeed), see Figue. S t pass fm fibe t pe stutue, tw sale tansitin mdels ae neessay, used in sequene. The esults f the mdel at eah level an be used as input data f the mdel at the next highe level. Use f this appah fm the lwest level, at whih mehanial ppeties ae given as input, t the highest level f the pe detemines the pe ppeties. Based n this stategy, the tansitin mdels an be used t analye syntheti fibe pes f mplex ss setin. Figue shws the typial hieahy aning fm the smallest level t the

3 highest level f a 05 tn bea lad fibe pe. The fus f this pape is the mdeling f the stati behavi f a fibus stutue with a lage numbe f twisted mpnents subjeted t axial lads, stating fm the mehanial behavi f the base mpnent, and the gmeti desiptin f the pe stutue. In setin, a desiptin f the stutue (gmety and behavi) is given, then, in setin, we pesent an veview f the existing mehanial mdels f suh stutues. In setin 4, a new ntinuum mdel is develped. The analytial mdels ae mpaed in setin 5. Tensile tests have been pefmed, t pvide the expeimental data that ae desibed in setin 6. In setin 7, we demnstate the auay f the mdels by mpaing thei peditins t expeimental esults.. Stutue desiptin Let us nside a multilayeed stutue in whih eah mpnent fllws a egula helial path und a ental axis f the stutue. The gmety f eah mpnent is haateied by the pith length, P (length f ne tun f the twist, eipal f twists pe unit length) and the lay angle, α, measued with espet t the stutue Z axis. The mpnent s enteline is then an helial uve f adius. The pith length P, is the same at all adial psitins, but the lay angle will inease fm e at the ente t a maximum at the extenal sufae f the stutue, Figue 4. α, as shwn in e It may be nted that the mpnent ss-setins ae elliptial in the plane pependiula t the Z axis. Theefe, the lay angle f a mpnent at a adial psitin, dented by α an be alulated using the fllwing expessin: i i

4 π tan α = i ( 1 ) i P F maine appliatins, the fibe pes ae subjeted t axial lads, and the axial behavi f suh stutues exhibits upling between tensin and tsin due t the helial design f the mpnents. Thus, the veall behavi an be expessed in the fm: F M = u θ, εε εθ, ( ) θε θθ whee u, dentes the veall axial stain, θ the twist angle pe unit length, F the, axial fe and M the tque. The fu stiffness matix mpnents,, and ae pue tensile, pue tsin and upling tems espetively. Mve, the stiffness matix shuld be symmeti, as an be shwn fm Bei s eipal them. εε θθ θε εθ. Ealie mdels This w is nentated n stutues with a lage numbe f mpnents (nstitutive elements). As nted by Raf and Hbbs (1988), sine the stutue nsists f a lage numbe f mpnents, the bending mments and tque in individual mpnents an be negleted. Seveal auths have develped analytial mdels t pedit the glbal elasti nstants pviding the elatinship between lads and stains f suh multi-layeed stutues, based n a nwledge f the mpnent mateial and gmety f the stutue. Tw ategies f these mdels ae pesented: semi-ntinuus mdels develped f metalli ables and mdels speifially pesented f syntheti ables. semi-ntinuus mdels Hmgeniatin is a well nwn methd in slid mehanis, and an be used f the ntinuum mdelling f a disete system mpsed f a lage numbe f idential epetitive 4

5 elements. With an apppiate hie f the mateial paametes, ne an auately epesent the glbal behavi f the eal system. This methd was fist applied t able mdelling by Hbbs and Raf (198). It is the thtpi sheet mdel that has been desibed in detail by Raf (198) and then extended by Raf and his assiates ve tw deades. In this mdel the lassial twisted d thies f the behavi f helial laid wies has been extended t inlude a set f inemati mpatibility nditins. The individual laye f wies is eplaed by an equivalent ylindial thtpi sheet, whih is assumed t be thin and t be in a plane stess state. As in the ase f mpsite laminates, fu elasti nstants ae neessay. Tw f them ae btained dietly fm the mehanial ppeties f the wies. The the tw ae elated t the ntat stiffness between adjaent wies in the laye. The mplete stutue is then teated as a disete set f nenti thtpi ylindes. The thtpy axes espnd t thse f a fibe mpsite mateial in whih the fibes have the same lay angle as the wies in the espnding laye. Anthe semi-ntinuus mdel was develped by Bluin and Cadu (1989), and late extended by Jlieu and Cadu (1994; 1996). This als nsists f eplaing eah laye with a ylinde f thtpi, tansvesely istpi mateial. In this mdel the elasti nstants an be used as fee, adjustable, paametes, else estimated atinally fm ntat mehanis equatins as in the ase f the thtpi sheet mdel. One the able is mdelled using suh ntinuum appah, analytial slutins f elementay ladings an be deived (Cssley et al., 00a, 00b). These semi ntinuus mdels tae int aunt fitin between nstituents. Hweve, sme elasti nstants ae btained fm ntat mehanis, nsideing laye mpnents have iula ss-setin. It an be seen fm Figue that it s nt the ase f fibe pes. Mve, due t the hmgeniatin pess, the auay f this mdel 5

6 ineases when the numbe f wies in a given laye ineases. Lastly, these mdels ae tedius t use, and sine they ae nn-linea, they equie a numeial slving. Despite these limitatins, the mdel f Raf and Hbbes (1988), biefly pesented in setin.1, will be applied hee in setin 5. syntheti fibe pes mdels In this ategy the simplest mdel is that f Hppe (1991) in whih the stutue and the mpnents ae assumed t be subjeted t pue tensile fes, the bending and tsinal stiffness f bth f them being negleted. Cntat and fitin between the mpnents ae als negleted., but suh an appximatin is justified f mntni axial lading. It shuld be nted that this analytial mdel pvides nly the veall tensile behavi. Leeh et al. (199), pesented a me mplex quasi-stati analysis f fibe pes and inluded it in a mmeial sftwae: Fibe Rpe Mdelle FRM. Thei analysis is based n the piniple f vitual w and an tae fitinal effets int aunt. The pgam mputes tensin and tque fm thei dependene n elngatin and twist. Anthe mdel was develped by Rungamnat et al. (00), and late extended by Beltan et al. (00) and Beltan and Williamsn (004). These mdels ae vey simila with that f Leeh but they have nentated n a damage mdel t tae int aunt the degadatin f pe ppeties as a funtin f lading histy. The Leeh s mdel appeas t be vey sphistiated, with an auate mehanial mdelling f the mpnents f the fibe pes behavi and thei inteatins. Mve, the ss-setin gmety an be desibed using diffeent fms f aangement f mpnents (see setin.). Theefe, the Leeh s mdel an be nsideed as a efeene mdel, but it s a mpute-based mdel. Heeafte, the syntheti fibe pes mdels f Hppe (1991) and Leeh et al. (199) 6

7 ae biefly pesented and then a new ntinuum mdel will be develped fm the Hppe s ne t analysis the stutue with a lage numbe f twisted mpnents..1 Raf s mdel Raf and his assiates have wed extensively n the behavi f metalli stutues with a lage numbe f wies s that the bending mments and tque in individual wies beme muh less signifiant than they ae in six and seven wie ables (Hbbs and Raf, 198; Raf, 198; Raf and Hbbs, 1988; Raf, 1991; Raf and Kainani, 1995a; Raf and Kainani, 1995b). In these studies a geat deal f aentin has been paid t the inte-wie ntat phenmena and fitin has been taen fully int aunt. By teating eah laye f wies as an thtpi sheet with nn-linea ppeties detemined using the mehanial ntat thies and assuming Culmb fitin, it has been pssible t establish the stiffness matix in the pesene f an axial lad. The main featues f this mdel ae pesented heeafte, in the ase f metalli multilayeed stutue with an istpi mateial. These auths have established a set f nnlinea simultanus equatins t analyse the inematis f eah laye f wies, pviding a set f mpatible stains in the anistpi ylinde with a e (f me details see Raf and Hbbs (1988)). The elasti behavi f eah thtpi sheet in the lal dinate system (t,b,n), see Figue 5, an be expessed in the fllwing matix fm: ε S ε = S bb ε tb S S S 66 σ σ bb σ tb ( ) whee S, ij ε and σ ae the mpliane, the stains and stesses efeed t the axes f thtpy paallel and nmal t the wie axes, espetively. 7

8 The mpliane paallel t the wie axis S 11 is staightfwad, efleting the ati between the sheet aea and the wie aea ( 4 /π ): S 4 π E 11 = ( 4 ) whee E is the Yung s mdulus f the wie mateial, and the upling tem S 1 is given by: S 1 = ν S ( 5 ) 11 whee ν is Pissn s ati. The mpessin mpliane, S, has been expessed as: S = D 4 (1 ν ) + ln (1 ν ) π E 1 P D(1 ν ) ( ) E ( 6 ) whee D is the wie diamete and, P is the ntat lad pe unit length n the ntat aea whih is btained fm Hetian ntat thy f the ntat f tw paallel ylindes. The ntat lad, P The shea mpliane, 1949):, is detemined numeially by using an iteative methd. S 66, is detemined fm the esults f the ntat thy, (Mindlin, S 66 S δ l = 1 1 ν δ l max 1/ ( 7 ) whee δ is the line ntat displaement f a given al petubatin in stutue axial l stain, and δ l max is the espnding displaement at the nset f full-sliding nditin. The stiffness ( mpliane) has been shwn t be a funtin f the amplitude f the lad vaiatin abut the mean. F small hanges f axial fe the stiffness is lage than it is f bigge vaiatins. Small hanges d nt veme the inte-wie fitin, while lage hanges d, ausing sliding and a lwe effetive mdulus. 8

9 In this study, the stiffness matix esults f this mdel f tw exteme ases ae pesented: the lwe bund full-slip, espnd t δ l f δ = 0. l = δ and the uppe bund n-slip One the stiffness matix f all the layes (f a given axial pelad) has been fund, in de t btain the behavi f the stutue, the stiffness matix f eah laye is tansfmed int the glbal dinate system f the stutue (t', b', n'), see Figue 5, and the summatin f the stiffness f all the layes enables the glbal behavi f the stutue t be established. It shuld be nted that t apply this mdel t a multilayeed fibus stutue, Yung s mdulus f the mpnent mateial is btained fm axial stiffness f mpnents in the dietin f thei axis, see setin 4. In additin, Pissn s ati, ν, ading t the vlume nstant defmatin assumptin, has been set t 0.5. l max. Hppe s mdel The w f Hppe (1991) based n puely gmetial nsideatins, allws a mdel f behavi f this type f stutue unde a simple tensile fe t be detemined. This mdel equies the nwledge f the tensile ppeties f the mpnents and the nstutin paametes f the stutue, i.e. the numbe f layes, the numbe f mpnents in eah laye and the lay angle f eah laye. This mdel is based n the fllwing hyptheses: the gmety f the stutue is multilayeed with the helial mpnent having iula setin; at the lal and glbal levels, the base mpnents and the stutue w nly in tatin in the dietin f thei axis (bending and tsin ae negleted); the setin f the stutue emains plane, and pependiula t its axis afte defmatin; defmatin f the stutue is at nstant vlume; stains and fitin effets due t ntat between mpnents ae negleted. Using these hyptheses, the elngatin f eah mpnent is detemined as a funtin 9

10 f thse f the stutue, and then the axial fe in eah mpnent is detemined. The pjetin f the fe n the stutue axis and summing f all the mpnents enables a lsed-fm expessin f the glbal behavi (nly axial stiffness) f the stutue t be established. In setin 4, a lsed-fm analytial slutin, f stiffness matix mpnents, will be develped whih is based n Hppe s mdel.. Leeh s mdel Leeh and al. (199) pesented a mdel whse fmulatin is based n the piniple f vitual w t analye fibe pes. This mdel is integated in a mmeial sftwae (FRM, 00) t pedit the behavi f the syntheti ables subjeted t an axial lad. This mdel diffes fm Hppe s mdel by the fllwing aspets: at the glbal level, the behavi f the stutue is haateied by upling between tensin and tsin phenmena using a stiffness matix; fitin effets due t ntat and the elative mtins between mpnents ae nsideed; the gmety f the stutue is multilayeed, and tw exteme pe gmety desiptins in tansvese defmatin have been nsideed: Layeed paing gmety and Wedge gmety, see Figue 6. F layeed paing gmety, it is assumed that a bundle f paallel idential mpnents with iula ss-setin is twisted in the assembly t fm a stutue with a e, suunded by a laye f equally wund mpnents, this laye enlsed by anthe laye and s n until all the mpnents ae used. Eah laye is a helial stutue f many mpnents and eah helix has the same pith length but a diffeent lay angle. F wedge gmety, the mpnents in the same level ae allwed t defm tansvesely and hange thei shape t a wedge tunated wedge. The equivalent helix adius is the adius f the ente f aea f the wedge. Within eah laye the paing fat (PF) is intdued t tae int aunt the pesene f the vids in the laye that an be 10

11 defined by the ati f the aea f mateial t the laye ss setinal aea. It an be expessed by: PF n A / sα i i = ( 8 ) i π W i i whee n, A and W ae the numbe f mpnents in laye i, mpnent ss i i setin aea and the width f the laye i espetively. It shuld be nted that f a given PF, the width f the laye will be defined and vie vesa. The estimatin f the fitinal fes that develp between the mpnents in a stutue is based n the lassial slip-sti mdel whee the fitin fe is assumed t develp between tw ntat sufaes in the dietin ppsite t the elative slip f these tw sufaes. Six sliding mdes have been pesented and it was nted that, f the twisted stutue unde axial lading, the nly signifiant fitinal ntibutin, (and even that is small), mes fm the axial sliding mde (Leeh et al., 199; Leeh, 00). In the pesent study, FRM sftwae was used t btain the esults f Leeh s mdel, with the wedge gmety ptin. Fist, the stutue is defined. Essentially, this nsists f speifying the numbe f mpnents in eah laye with the apppiate twist and the natue f the paing at that laye. Send, the dimensinal and tensile ppeties f the mpnents must be pvided. Mst ae single paametes, but the nn-linea fe-stain elatins an be defined in the sftwae by the effiients f futh de plynmials. In this study the festain elatins wee nsideed linea and deived fm test data. The stiffness matix is btained in tw steps. Fist, we let θ = 0 and vay the axial stain,, abut a given value (0.01), t alulate F and M thugh the FRM sftwae, u, whih leads t and fm Eq. (1). In the same way, and will be btained by εε θε εθ, θθ seing the axial stain, u,, t a given value (0.01) and vaying θ., 11

12 4. Cntinuum mdel All the mdels pesented abve equie the nstutin paametes f the stutue suh as numbe f layes, numbe f mpnents in eah laye (see Figue 6) and lay angle f eah laye (see Figue 5). These ae nt always easy t define peisely f fibe pe stutues, see Figue, whee it appeas diffiult t mdel the stand ss-setin as a multilayeed stutue. In additin, these mdels ae integated in pgams and numeial analysis is neessay, (exept f the Hppe mdel whih pesented a lsed-fm expessin but nly f the pue tensile behavi f the stutue with n tsin and upling tems). Hee, an analytial mdel with a lsed-fm expessin and mdel gmety me in ageement with the eal gmety f the stutue will be established. This invlves an extensin f the Hppe s mdel (Hppe, 1991) whih is based n the same hyptheses, as in the initial mdel, with the exeptin f: In the liteatue, the stutues studied ae multilayeed, but in the pesent mdel we dn't nside them lie an assembly f layes, but athe as a set f axial helixes. These helixes have the same numbe f tuns pe unit length, and thei setin amunts t a mateial pint, and that desibe the gmety f a nstituent element. It is in this sense that this mdel is temed a ntinuum mdel. Mve, within the stutue the paing is assumed t be unifm. Theefe, the gmeti input data f this mdel ae estited t the extenal stutue adius, the pith length and a paing fat value. In additin, the pesent mdel an desibe upling behavi between tatin and tsin. The stess-stain (fe-stain) ppeties f the mateial whih ae intdued int the mdel ae, in geneal, taen t espnd t the atual fe-stain ppeties f the mpnent as btained fm expeiments. The elatin between fe-stain is assumed linea and Yung s mdulus f the mpnent mateial is given by: E = A (9 ) 1

13 whee is the mpnent axial stiffness (slpe f the fe-stain uve) and A is the ss setin aea f the mpnent. 4.1 Axial stain f mpnents In the pesent mdel, the mpnents ae assumed t be subjeted t pue tensile fes, the bending and tsin stiffness ae negleted. In axial lading, with tatin and tsin, the axial stain f eah mpnent is mpsed f tw diffeent pats: the fist esults fm the elngatin f the stutue, wheeas the send is due t its tatin. F small stains, it is pssible t sepaate these phenmena, the axial defmatin f the mpnent is expessed theefe by: A R ε = ε + ε ( 10 ) A R whee t is the tangent t the mpnent ente line, ε and ε ae the axial stains f the mpnent due t the elngatin and t the tatin f the stutue espetively. Elngatin Let λ be the extensin ati (ati f defmed length t initial length f the stutue) measued alng the stutue axis, and λ the espnding extensin ati f a mpnent whse initial and final adial psitins ae The extensin atis, λ and λ, ae defined as fllws: and, espetively, see Figue 7 a). λ = λ = L L l l 0 = 1 + u = 1 + ε, A ( 11 ) As the vlume is suppsed t emain nstant, the initial and final adial psitins f eah 1

14 mpnent an be elated by the fllwing expessin: λ ( 1 ) = ( ) If α is the lay angle f this mpnent in the initial state, ne has: π tan α = ( 1 ) P sine afte defmatin the pith length, P, detemine by P = P λ, the espnding lay angle α in the defmed state is given by: tanα tanα = ( 14 ) λ / Let us nside a stutue having the initial length L, and bunded by planes pependiula t the stutue axis. The initial length f a mpnent f lay angle l = L / sα ( 15 ) the axial length in the defmed state being λ L, the espnding mpnent length in the defmed state is l = ( λ ) / sα ( 16 ) L using equatins (1-16), the mpnent extensin ati λ an be expessed as fllws: α is λ sα sin α = ( λ ) = λ s α + ( 17 ) sα λ A whih yields ε fm (11). Rtatin length When the stutue undeges a elative tatin, θ, between the tw end setins f L, the axial stain f the mpnent due t this tatin is expessed by: 14

15 R Δl ε = ( 18 ) l 0 whee Δl is defined by (see Figue 7 b)): Δ l = θ sinα ( 19 ) substituting ( 1 ), ( 15 ) and (19 ) int the expessin ( 18 ), we btain : ε R = θ sinα sα ( 0 ), λ whee θ is the twist angle pe unit length defined by:, θ θ = ( 1 ), L 0 Hweve, in geneal, f a given stutue, its ute diamete, the value f the lay angle n the ute laye,, is nwn, as well as α. Sine f all the mpnents the pith length, P, is the same, the lay angle f an abitay mpnent with a adial psitin f be wien as a funtin f the paametes f the ute laye:, an tan α = tanα ( ) using equatins ( 14 ) and ( ), ne btains: s α = sin α = λ + 0 λ + λ tan tan α tan α α ( ) While taing int aunt the expessins (11) and substituting the elatins (17) and (0) int equatin ( 10 ), the al axial stain f the mpnent is given by: ε sin α A R = ε + ε = [ λ s α + 1] + θ sinα sα ( 4 ) λ, λ whee sin α is given ading t the equatin (), whih ae funtins f extensin ati, 15

16 λ, and the ute laye paametes assiated t the initial gmety ( and α ). Othewise, s α and α sin ae given by substituting λ = 1 int the elatin ( ). Theefe, f an abitay pint at a adial psitin funtin f tw independent vaiables, λ and., the axial stain in the lal dinate system, ε, is a 4. Stiffness matix deivatin In this mdel the mpnents ae assumed t be puely tensile elements with a uniaxial behavi that an be epesented by: σ = E ε ( 5 ) whee t is the tangent t the mpnent enteline (see Figue 5). In de t btain the stiffness matix the stess in the lal dinate system, σ, is tansfmed t the glbal ylindial dinate system (, θ, ) : σ σ = σ = σ θ s α sα sinα ( 6 ) theefe, the al axial fe and tque ae btained by integatin f the stesses n the ss setin aea f the stutue in the initial state: F M = PF g = PF g π 0 π 0 0 σ 0 s α d dθ σ sα sinα d dθ ( 7 ) whee σ is btained fm ( 4 ) and ( 5 ) and s α and sin α fm ( ). The glbal paing fat ( PF g ) is intdued t tae int aunt the pesene f the vids in the whle f the ss setinal aea f the stutue. It an be expessed by: PF g = [ N A π 0 π 0 0 d dθ 0 ] / ( π ) s α d dθ ( 8 ) 16

17 whee N is the al numbe f mpnents in the stutue. Afte integatin f the elatins (7) using the Maple sftwae, and ewiting the esults in the matix fm, equatin (), the stiffness matix mpnents, f the linea mateial, ae expessed as fllws: εε εθ θε θθ = π E = π E = π E 1 λ 1 λ (( 1 λ ) ln( + + λ 4 = π E 4 PF PF PF g g g PF g λ λ ln( [ + tan λ ln( λ + tan [ ln(( λ + tan 1 λ α ) + ln( + + λ tan α ( λ 1) 1 λ 1 [ λ (( 1 λ ) ln( + tan α + + (1 + tan α ) ( λ + tan α ) (1 + tan α ) ( λ + tan α ) ) λ (tan α λ ln( λ + tan α )) λ λ α ) + ( λ + tan 1 ) λ 6 λ tan α λ + tan α λ λ α + + (1 + tan α ) ( λ + tan α tan α ( λ 1) 1.5 tan α 4.5 ln( λ )] ) ln( λ ) 1 α ) ] + λ λ (ln( λ + tan tan 4 α 1.5 α λ ln( λ ) ] / [tan ( λ 1)] ) ) α ) ln( λ )) the stiffness matix is a funtin f nly the extensin ati f the stutue, (9) λ, glbal paing fat, PF g, and the ute laye gmetial paametes f the stutue in the initial state ( and α ). Sine the stiffness matix mpnents depend n the stain, this mdel is essentially nnlinea, but f the inteval [ ] f extensin ati (patial stain ange f aamid), λ, the esults an be nsideed as nstant. In the fllwing, the esults f the same axial stain ( λ = 1.01) ae pesented. 17

18 5. Mdels mpaisn The pevius mdels have been applied t a stand f a 05 tn aamid able f nwn nstutin paametes (given by the able supplie) shwn in Table 1. Table shws the input data neessay f all mdels. Cmpaing tables 1 and shws that input data ae missing f all the mdels. A sensitivity analysis has been pefmed elsewhee by Gheishi (005), and the esults have shwn that the veall behavi is nt sensitive t these missing values f the patial stutues f inteest hee ( α 15 ). Sme illustative pats f this sensitivity analysis ae epted heeafte. As it has been peviusly mentined, it s patially diffiult t epesent the stand ss-setin with a multilayeed stutue. Theefe, seveal multilayeed disetiatins an be used. Fm the value f the stand adius and assembled yan sufae, it has been nsideed that the stand was made with 4 layes. The esults btained with the Leeh s mdel espnding t diffeent multilayeed disetiatins ae given in Table, with vey small diffeenes. The influene f the paing fat has als been studied, sine this paamete is nt defined at the lal sale (i.e. in eah laye) when the Leeh s mdel is used. A fu layes disetiatin with espetively 1, 6, 14 and 1 assembled yans in eah laye, has been nsideed, with diffeent values f the adius f the assembled yan. F a given value f this adius, the paing fat f the layes t 4 was nstant and alibated in de t btain a ss-setin adius nsistent with the stand adius value. The esults ae listed in Table 4, whee it an be heed that they ae slightly sensitive t the paing fat values. 18

19 Theefe, f the pesent study, the values f the missing data wee taen as fllws: Numbe f layes is hsen t be 4, Cmpnent numbes f eah laye ae 1, 6, 14 and 1 and the PF in eah laye ae 1, 0.75, 0.88 and 0.89 espetively. On the the hand, equatin ( 8 ) gives a glbal paing fat, PF = This value is in ageement with the pevius values used in the Leeh s mdel, whih shws that bth mdels have the same quantity f mateial in the ss-setin f the stutue. g Cmpnent adius: 1.1 mm whih yields a value f mpnent N / mm f Yung s mdulus f Table 5 pesents the esults btained f the diffeent mdels. Besides the alulated stiffness matix mpnents, the peentage f asymmety between upling tems, εθ and θε, is shwn f eah mdel. The influene f fitin is pesented f the Raf and Leeh mdels. It shuld be nted that in syntheti fibe pes, the fitin effiient between the diffeent mpnents is nt a well nwn paamete. F the yan n yan, and the aamid mateial, fitin effiient values ae given between (FRM, 00). These values have been btained fm tests n the diffeent yans. It shuld be als mentined that, in Raf s mdel, the paing fat in eah laye is assumed t be 4 π (f metalli mpnents), but hee this value is mdified by using the value espnding t that hsen in the FRM sftwae, as well as the glbal paing fat f the ntinuum mdel. Indeed, the same stutue is defined f all the mdels (same mateial quantity in the stutue). The main nlusin fm table 5 is that all the mdels yield vey simila esults f the axial stiffness, εε. The diffeene f the upling tems is visible. Only the tsin tem esults, θθ, ae signifiantly diffeent f the diffeent mdels. 19

20 T shw whih mdel gives me eliable esults (patiulaly f the tsin tem, ), it wuld be neessay t be able t mpae them t expeimental esults. θθ In Raf s mdel, the stutue in the n-slip ase is muh stiffe than in full-slip, hweve the upling tems ae smalle in the n-slip ase. On the the hand, exept f the axial stiffness whee the tw limit ase esults ae simila, the diffeenes between the tw ases ae signifiant, patiulaly f the tsin tem. It is inteesting t nte that the thtpi sheet thy pesented f the multilayeed metalli ables by Raf, and based n the ntat thy between the metalli mpnents with iula ss setin, yields esults mpletely mpaable with thse btained fm the speifi mdels f syntheti ables. The mdel f Hppe pvides a simila value f the axial stiffness but des nt allw the the stiffness tems t be btained. The esults fm Leeh s mdel shw that the fitin effet an be negleted f axial lading. Hweve, it shuld be mentined that while the fitin effet plays a small le in glbal stiffness behavi f suh stutues, the effet f fitin n the lng-tem pefmane and duability f a stutue unde yli lading an be signifiant. Then, the thetial peditins will be mpaed t expeimental esults that btained fm tatin test n tw diffeent stutues. 6. Expeimental esults Expeimental studies have been pefmed at tw sale levels, fist n yans t detemine the base mpnent ppeties and then n tw diffeent assembled yans whih epesent the multi-laye stutue. Tensile tests at the yan level give an indiatin f the mateial behavi withut the effets f twist and nstutin. They wee pefmed n a 10 N test mahine at an applied sshead displaement ate f 50 mm/minute. Elngatin was measued using tw digital 0

21 ameas, whih ed the mvements f mas n the yans, as shwn in Figue 8. The test pedue f these and all subsequent tests was t apply five bedding-in lad-unlad yles up t 50% f the nminal bea lad, befe the lad yle whih was used f the mdeling. This is standad patie in pe testing and stabilies the mateial and nstutin. An example f the yan test esults inluding the five bedding-in yles and the test t failue is shwn in Figue 9. Chailleux and Davies (00) have als used yan tests t identify the intinsi viselasti and visplasti behaviu f the aamid fibes used in the pesent study (Twan 1000). In de t pvide data f elatin with the mdels, tests wee then pefmed n tw diffeent assembled yans taen fm a 5 tn bea lad pe, Figue 10 (at least 5 speimens wee haateied f eah), f whih the nstutin paametes ae given in Table 6. All the samples wee made with the same aamid fibe gade. The lad was intdued thugh ne and spie end nnetins. Tests invlved applying five initial bedding-in yles, as f the yan tests, by lading the samples t 50% f thei nminal bea lad at a lading and ate f 50 mm/minute then unlading at the same ate. The same image analysis system was used, measuing the displaements f tw mas bnded t the assembled yan (Figue 10). Fm the tests n the mpnent (yan) and the stutues (assembled yans 1 and ) the axial stiffness values wee measued as shwn in table 7. The stiffness values pesented ae thse fm the 6 th lading. 7. Cmpaisn between peditin and tests In this setin the pevius expeimental esults will be mpaed t mdels peditins. F mdeling the assembled yan 1, the numbe f layes is assumed t be, f whih the mpnent numbes f eah laye ae and 9. The PF s f eah laye ae bth 1

22 0.95, and the espnding glbal paing fat, fm equatin ( 8 ), is als In the assembled yan, the numbe f layes is assumed t be in whih the mpnent numbes f eah laye ae 1, 5 and 10. The PF s in eah laye ae 1, 0.96 and 0.96 espetively and the espnding glbal paing fat, fm equatin (8 ), is The yan axial stiffness and the gmetial paametes then enable a peditin t be made f the stiffness effiients f the stutues using the ntinuum mdel (Eq. (9) 1 ), and this gives axial stiffness values f 5.7 N and 6.7 N f assembled yans 1 and espetively. The stutues wee als mdeled with the FRM sftwae, and this gives esults vey lse t thse f the ntinuum mdel (5.6 N and 6.9 N espetively). Raf s mdel was nt applied t these stutues beause thee ae nt a lage numbe f wies in eah f the layes hee. The mpaisn is shwn gaphially belw in Figue11. S fa all the tests pefmed have nentated n the axial stiffness εε by testing stutues with fixed end lading nditins. Hweve, a small numbe f tests have shwn that thee is nt measuable tensin-tsin upling tems and tsin stiffness f the small diamete assembled yans at this level. In de t detemine the the effiients (upling tems and tsin tem) and t mpae them with pedited values test esults f the highe level suh as stands f 05T fibe pe wuld be neessay. 8. Cnlusin A nnlinea elasti ntinuum mdel has been develped f analysis the veall axial stiffness f fibus stutue with a lage numbe f twisted mpnents. By ntast with multilayeed appahes, the stutue unde nsideatin is heein depited as a set f axial helixes nly haateied by thei extenal lay angle and espnding adius. The nstitutive mateial is assumed t be linea. Cnsideing stati mntni axial lads, the

23 inte-fibe fitin effets ae nt taen int aunt. Mve, the studied stutues exhibiting small lay angles, the veall diametal ntatins ae negleted, whih may ntibute t the veestimatin f stiffness. The develped analytial mdel leads t useful lsed-fm expessins thus allwing t ptimie pe nstutins. Due t la f published expeimental data, the mdel has fist been mpaed with mdels f the liteatue. The esults btained, have shwn that all the mdels give esults that agee easnably well with eah the, exept with espet t the tsin stiffness, f whih thee is a signifiant diffeene. In additin, stiffness maties f all the mdels deviate slightly fm symmety and this la f symmety is due t a etain la f nsisteny in the vaius simplifying hyptheses. Tensile tests have then been pefmed n aamid fibe assemblies with tw stutues, t btain the axial stiffness. This peliminay test esults indiate a gd elatin with the mdel. Additinal test data, espeially t examine tensin-tsin and pue tsin lading, ae needed t gauge pefmane f the mdels. The integatin f these esults in a mdel f a lage aamid wie pe and mpaisn with tensin and tensin-tsin upling test esults will be desibed in Pat f this pape.

24 REFERENCES Amaniampng, G., and Bugyne, C. J., Analysis f the tensile stength f paallel-lay pes and bundles f paallel elements by pbability thy. Intenatinal Junal f Slids and Stutues (4), Beltan, J. F., Rungamnat, J., and Williamsn, E. B., 00. Cmputatinal mdel f the analysis f damage pes. In: Peedings f The thiteenth Intenatinal Offshe and Pla Engineeing Cnfeene, Hnlulu, Hawai, USA. Beltan, J. F., and Williamsn, E. B., 004. Investigatin f the Damage-Dependent Respnse f Ming Rpes. In: Peedings f The Futeenth Intenatinal Offshe and Pla Engineeing Cnfeene Tuln, Fane. Bluin, F., and Cadu, A., A study f helially einfed ylindes unde axially symmeti lads mathematial mdelling. Intenatinal Junal f Slids and Stutues 5 (), Chailleux E., and Davies P., 00. Mdelling the nn-linea viselasti and visplasti behaviu f aamid fibe yans. Mehanis f Time dependent mateials junal 7 (-4), Chudba, R., Vehvsy, M., and Knad, M., 006. Sthasti mdeling f multi-filament yans: I. Randm ppeties within the ss-setin and sie effet. Intenatinal Junal f Slids and Stutues 4 (), Cssley, J. A., Spene A. J. M., and England A. H., 00a. Analytial slutins f bending and flexue f helially einfed ylindes. Intenatinal Junal f Slids and Stutues 40 (4), Cssley, J. A., England A. H., and Spene A. J. M., 00b. Bending and flexue f ylindially mnlini elasti ylindes. Intenatinal Junal f Slids and Stutues 40 (5), Fste G.P., 00 "Advantages f fibe pe ve wie pe," Junal f industial textiles (1), FRM, Fibe Rpe Mdelle, vesin 1.1.5, 00. Sftwae develpment f Tensin Tehnlgy Intenatinal Ltd.(TTI). Gheishi S. R., 005. Mdélisatin analytique et aatéisatin expéimentale du mptement de âbles synthétiques. Ph.D. thesis, Ele Centale de Nantes, Fane. Gheishi, S. R., Davies, P., Cataud, P., and Message, T., 006. Analytial mdeling f syntheti fibe pes, pat II: A linea elasti mdel f 1+6 fibus stutues. Submied t Intenatinal Junal f Slids and Stutues. Hbbs, R.E., and Raf, M., 198. Intewie slippage and fatigue peditin in standed ables f TLP tethes. Behaviu f Offshe Stutues, Hemisphee publishing/mgaw- Hill, New Y, Vl,

25 Hppe, L.F.E., Mdeling the stati behavi f Dyneema in wie-pe nstutin. MTS RTM. Jlieu, C., and Cadu, A., An analytial slutin f bending f axial thtpi ylindes. Junal f Engineeing Mehanis 10 (1), Jlieu, C., and Cadu, A., Semintinuus Mathematial Mdel F Bending f Multylayeed Wie Stands. Junal f engineeing Mehanis 1 (7), Leeh C. M., Heale J. W. S., Oveingtn M. S., and Banfield S. J., 199. Mdelling Tensin and tque Ppeties f Fibe Rpes and Splies. In: Peeding f the Thid Intenatinal Offshe and Pla Engineeing Cnfeene Singape. Leeh C. M., 00. The mdeling f fitin in plyme fibe pe. Intenatinal Junal f Mehanial Sienes 44, Mindlin, R., D., Cmpliane f Elasti Bdies in Cntat. Junal f Applied Mehanis 16, Raf, M., 198. Intewie ntat fes and the stati, hysteeti and fatigue ppeties f multi-laye stutual stands. PhD thesis, Impeial Cllage f Siene and Tehnlgy, Lndn, UK. Raf, M., and Hbbs R. E., Analysis f Multilayeed Stutual Stands. Junal f engineeing Mehanis 114 (7), Raf, M., Methd f analysing lage spial stands. Junal f Stain Analysis 6 (), Raf, M., and Kainani I., Simple Deivatin f the Stiffness Matix f Axial/Tsinal Cupling f Spial Stands. Cmputes and Stutues 55 (4), Raf, M., and Kainani I., Analysis f Lage Diamete Steel Rpes. Junal f Engineeing Mehanis 11 (6), Rungamnat, J., Beltan, J. F., Williamsn, E. B., 00. Cmputatinal Mdel f Syntheti-Fibe Rpe Respnse. In: Peeding f fifteenth Engineeing Mehanis Cnfeene, ASCE, New Y. Vehvsy, M., and Chudba, R., 006. Sthasti mdeling f multi-filament yans: II. Randm ppeties ve the length and sie effet. Intenatinal Junal f Slids and Stutues 4 (),

26 Figue 1 : Syntheti fibe pe stutue. Figue : 05 tn bea lad syntheti fibe pe ss setin; the pe epesents a 1+6 stutue, e and stands ae assemblies f a lage numbe f twisted mpnents. 6

27 Figue : diagam shwing the typial hieahy aning fm the smallest level t the highest level f a 05 tn bea lad fibe pe. 7

28 Figue 4: an abitay mpnent at a adial psitin i and a mpnent at the ute sufae f the stutue with a adial psitin. e Figue 5 : lal and glbal dinate systems f a laye f wies. 8

29 Figue 6 : multi layeed gmety f stutue f vaius mdels: a); Raf, Hppe and Leeh (layeed paing gmety) and b) Leeh (wedge gmety). Figue 7: mpnent befe and afte defmatin; a) elngatin and b) tatin f the stutue. 9

30 Figue 8. Testing f yans n 10 N test mahine, tw digital ameas t measue stain. 0

31 Figue 9. Fe-stain plt f tensile test n 6 tex aamid yan (Twan 1000), 5 yles t 50% f bea lad fllwed by test t failue. 1

32 Figue 10. Test n assembled yan sample n 00 N test mahine, shwing sample and tw digital ameas t measue stain.

33 Figue 11: mpaisn between pesent mdel peditins, FRM sftwae esults and espnding expeimental measuements f stain-fe uve f a) assembled yan 1 and b) assembled yan.

34 Tables Oute diamete 18. (mm) Pith length 75 (mm) Cmpnents numbe 4 mpnent axial stiffness, * 46.1 N * btained fm expeiments Table 1: available nstutin paametes f stand f 05T aamid able. Mdels Raf and Hppe Leeh ntinuum Input data Pith length, Numbe f layes, Cmpnents numbe pe laye, Cmpnent adius, Yung s mdulus f mpnent. Pith length, Numbe f layes, Cmpnent numbe pe laye, Cmpnents adius, mpnent axial stiffness,, PF f eah laye. Stand adius, Pith length, Yung s mdulus f mpnent, PF. g Table : neessay input data f all mdels. Multilayeed disetiatin εε (10 N) εθ (N.m) θε (N.m) θθ (N.m ) Table : esults btained f Leeh s mdel f diffeent multilaye disetiatins f the stand made f 4 assembled yans distibuted in 4 layes. Assembled yan adius (mm) PF f layes -4 εε (10 N) εθ (N.m) θε (N.m) θθ (N.m ) Table 4: esults btained f Leeh s mdel f diffeent values f Paing fat.

35 Mdels Raf εε (10 N) εθ (N.m) θε (N.m) θθ (N.m ) εθ εθ (%) Full slip N slip θε Hppe μ = Leeh μ = μ = Cntinuum mdel Table 5: esults btained f diffeent mdels applied t the stand f 05T aamid able. Stutue Cnstutin paametes Stutue Cnstutin paametes Oute diamete.0 (mm) Oute diamete. (mm) Cmpnent diamete 0.57 (mm) Cmpnent diamete 0.57 (mm) Assembled Pith length 5.6 (mm) Assembled Pith length 58.8 (mm) yan 1 Cmpnent yan Cmpnent 1 numbe numbe 16 Cmpnent axial mpnent axial stiffness, 1.4 N stiffness, 1.4 N Table 6: nstutin paametes f diffeent stutues. Samples Test numbe Sample length (mm) Aveage Axial stiffness (N) Aveage Ruptue fe (N) Yan (Twan 1000) ± 1% ± % Assembled yan ±.6% 5.1 ± 8% Assembled yan ± 0.8% 6.88 ± 14% Table 7: test esults n the yans and assembled yans afte five bedding-in yles.