Keywords: Form-finding; Exact coordinates ; Iteration; Suspension cable.

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1 Bluchr Mchanical nginring Procdings May 4, vol., num. APPLICATION O XACT LMNT-MTHOD ON CALCULATION O ORM-INDING AND UNSTRSSD LNGTH O CABL Li Zhou,Wang Ronghui,,Wi Yuanchng 3,Huang Xiaofng School of Civil nginring and Transportation, South China univrsity of tchnology(97654@qq.com) Stat Ky Laboratory of Subtropical Building Scinc 3 JiangXi lctric Powr Dsign Institut Abstract. Th problm of form-finding for th suspndd cabl is actually th problm of dtrmining all ky points coordinats on main cabl, which ar by quilibrium rlation on th horiontal forc, main cabl sagitta and lifting point forc undr th prcondition of dtrmining th ndpoint s boundary conditions of cabl sgmnt. According from th static quilibrium rlationship of cabl lmnt, basd on th analysis of its analytical solution procss, in this papr, th cabl lmnts ar dividd into two typs in accordanc withth vrtical distribution load along th arc lngth and along th string lngth, th corrsponding shap curv of cabl lmnt is th parabola and th catnary, and with parabolic rsults as its initial valu for th itration of nonlinar solution, thn cabl lmnt vntually convrg for th catnary. And basd on th act coordinats rsults,th calculation mthod of th lngth without strss is prsntd,and compild corrsponding computational procdurs. By comparing th rsults of form-finding and th cabl-lngth in non-strss according to program compild and th rsults from th finit lmnt softwar and th masurd valu of Aihai suspnsion bridg, compard with th nonlinar finit lmnt mthod,it confirmd th mthod rquiring smallr dividing lmnt dnsity, th convrgnc spd is quickr and th rsults can nsur th prcision. Kywords: orm-finding; act coordinats ; Itration; Suspnsion cabl.. INTRODUCTION inding Cabl form is to dtrmin th initial quilibrium stat of cabl, th cor issu in static analysis on suspnsion bridg. Basd on th diffrnt tims, th static analysis mthods can b classifid into thr catgoris: lastic Thory Mthod, Dflction Mthod and Nonlinar init lmnt Mthod [-3] which is widly applid intriorly, whil mor calculation mthods hav prsntd abroad such as orc Dnsity Mthod [4] and Dynamic Rlaation Mthod [5]. In thos mthods cabl sgmnts ar usually takn considr into approimat shap, for ampl, to rgard

2 th main cabl form undr horiontal of loads as scond-dgr parabola in lastic thory mthod, parabola in dflction mthod and bar lmnts [6] in unit dnsity nough cass in nonlinar finit lmnt mthod. Howvr, th cabl actually has a sag. Th papr mainly introducs a mthod on th basis of th nonlinar finit lmnt mthod, in which th accuratly ndpoint coordinats of cabl unit could b dtrmind by itration with th initial parabolic valu into th accurat curv of th catnary, in othr words, to obtain ultimatly prcis cabl form undr dad loads. Th mthod sts out from ordinary diffrntial quations of quilibrium for cabl unit, stablishs th quations of th rlationship btwn cabl unit trminal points prpndicular coordinats and loads on it, and gts th prcis coordinats through solving th quations. vntually, th form of th whol cabl sgmnt undr constant loads can b obtaind by mans of intgrating cabl units into whol cabl sgmnt. Compard to th Nonlinar init lmnt Mthod, this mthod can rduc dividd unit dnsity so as to improv th opration spd and will not affct th accuracy of th rsults, for it Just nd divid unit on th lifting point.. GNRAL SITUATION O NGINRING XAMPL This articl taks Aihai suspnsion bridg as a nginring cas for Analysing. Th main cabls span arrangmnt of Aihai suspnsion bridg is 4m+76m+6m, th full lngth of main bam is.5m, th cross dirction of main bridg has st % cross slop, th nt width of bridg floor systm is 4m, th full width stl truss stiffning girdr is 7m. This bridg us two main cabls with plan arrangmnt, th ris-span ratio of main cabl is /L=/9.6, th cntr distanc is 7m.Th bridg us 68 pairs of slings, and th standard spacing of slings is 4.5m. urthrmor, a pair of sling has bn anchord in rock away from th first pair of sling 9m in th JISHOU sid, and two pairs of slings with 9m spacing hav bn anchord in rock away from th first pair of sling 9m in CHADONG sid. Th hight of girdr in main span (main truss cntr lin) is 7.5m. Vrtical baring and rsistd wind baring hav bn st on th abutmnt, and lastic inclind cabl-stayd buckls hav bn st on th mid-span. Th cabl towr of JISHOU sid usd doubl column typ gat-shapd fram, which including sprad foundation, towr bas, towr column (uppr towr column, middl towr column, lowr towr column) and bam (uppr bam, middl cross bam). Th hight of th towr is 34.6 m(including 4m as to th hight of th hood) from pil cap abov. Th cross dirction of th towr is outward inclin from up and down, and two bams (uppr bam, middl cross bam) hav bn st on th towr. Th cabl towr of CHADONG sid also usd doubl column typ gat-shapd fram, which including sprad foundation, towr bas, towr column (uppr towr column, lowr towr column) and bam. Th hight of th towr is 6.94m (including 3.6m as to th hight of th hood) from pil cap abov. Th cross dirction of th towr is vrtical, it has st uppr bam and towr bas which locatd on th sparatd sprad foundation. Th main girdr is using stl truss

3 stiffning girdr of.5m long, which is comprisd of main truss, top bracing, lowr bracing and transvrs truss. Th bridg floor systm adoptd th stl-concrt composit structur which is th combination of vrtical flangd bam and concrt slab. Th ris-span ratio of main cabl is /9.6, th main cabls computational span of main span is 76m, th main cabls computational span of sid span in JISHOU sid is 4m, th main cabls computational span of sid span in CHADONG sid is 6m. Th gnral arrangmnt of this bridg has shown in igur. igur. Th ovrall layout of AiZhai suspnsion bridg 3. THORTICAL LABORATION O SHAP INDING As figur shows, it s a chart of quilibrium rlation of cabl lmnt. or plan problm, ar th latral and vrtical distributd load along th dirction of cabl lmnt, T, dt ar aial tnsion of cabl lmnt, H, dh, Q, dq ar th and dirction componnt of aial tnsion. T Q q q H QdQ HdH TdT d ig. Cabl lmnt quilibrium diagram According to th quilibrium rlation of horiontal and vrtical [7], w hav: Hnc: H q d H dh. () Q q d Q dq () dh / d q (a)

4 dq / d q (b) Whr Q H d / d,thrfor, quation (b) can b chang as follows: ' ' ( ) H q (b) In this papr only th condition of in-plan and q is considrd. According to quation (a), w can know that H is a constant which has nothing to do with indpndnt variabl. Th quation (b) is a diffrntial quation about, whn q is assumd to b th uniform load q which distribut along th chord lngth of cabl, and q dg / dl, thr dg is th dadwight of cabl lmnt, thn th shap of cabl lmnt will b parabola in this cas. And whn q is assumd to b th uniform load which distribut along th cabl, thr q q ( '), in this cas th shap of cabl lmnt will b catnary which approach th ral shap. inding th solution of ( ) in quation (b) is so calld shap finding. 3.. To find th horiontal forc H As shown in igur, th cabl unit is undr th action of th uniformly distributd Load along th cabl arc lngth q, th curv quation(3) for th initial stat of th cabl lmnt can b solvd from formula(b). H ( ) ch ch( q ) (3) q l Hypothsiing that th lft nd point is th origin point in th local coordinat systm,thn,formula(3)can b changd into formula(3a): H ch ch( ) q l (3a) In th formula(3a), h/ l Arcsh( ), sh ql ; (3b) H Thn th rlationship btwn th cabl sag and th horiontal cabl forc can b dscribd as formula(3c),which is drivd by th quation(3a). H h f ch ch( ) (3c) q Th cabl sag for initial stat of cabl can b dtrmind by dsign, thn th horiontal cabl forc can b dtrmind by quation (3c).Thrfor, th forc H is supposd as a known condition in th following.

5 l q q ig.3 Th diagram of cabl lmnt undr forc As shown in igur 3,it s an cabl lmnt undr forc in a local coordinat systm, in which i is th horiontal forc and i is th vrtical forc for th cabl lmnt nods, i is th vrtical coordinats for th cabl lmnt nods, thy can also b writtn in vctor form as blow:, It is asy to know th rlationship of th nodal forc and th vrtical coordinats for vry nod: ' ( ) i i i (4) 3.. As shown in figur,it s a cabl lmnt undr vrtical distributd load along th chord lngth of th cabl lmnt,, Namly q, q q lmnt s shap is parabola. whn,cabl By th quation (b),it is asy to gt cabl lmnt analytical solution of th form as blow: ( ) N ( ) N ( ) D( ) (5) In formula(5), ( ) N ( ) is a shap function, Dis ( ) a particular solution of th original diffrntial quation; at th tim,th rlationship of ( ) and th vrtical coordinats, of th ndpoint ar linar.ormula(6) is availabl from quation(5) and (4): ' ' ' N( ) N ( ) D ( ) ' ' ' N( ) N ( ) (6) D ( ) It is asy to obtain th shap function as blow:

6 N ( ), l ( ) q( )( ) N, D ( ) ; l H Thn formula (6) chang into : H q l l (6a) It is assumd k H l, which is calld th lmnt stiffnss matri, P ql is known as th lmnt quivalnt nodal loads matri, thn quation (6a) can chang into th following typ of lmnt stiffnss quation: P k (6b) Z Thn th dirct nodal forc vctor Z,quivalnt nodal forc vctor P, stiffnss matri k of ach cabl lmnt can b suprposd into nodal forc vctor and stiffnss matri K for th whol cabl, as a rsult, w can hav th global stiffnss quation as th following form : nod. K Z (7) In quation (7) Z is th act vrtical coordinats vctor of all 3.3. As shown in figur,it s a cabl lmnt undr vrtical distributd load along th arc lngth of th cabl lmnt,, Namly q, cabl lmnt s shap is catnary. q q ( ').whn, Thn Put q into formula (b),it is asy to obtain th analytical solution quation (3);In th quation(3), q H sh Arcsh( ), ql (3d) H It is shown th rlationship of ( ) and, is non-linar in quation (3). Assumd th parabola as th initial shap for th cabl lmnt, w can obtain th approimat coordinats of th initial vrtical vctor of th cabl, thn put of any lmnt into quation (3),w can obtain th curv quation ( ) of th corrsponding lmnt; thn tak ( ) into quation (4), thn Z will b spradd by th Taylor formula at th approimat solution, and omit th scond ordr trac,

7 w can gt formula (8) as blow: ( ) ( ) ( ) ( ) (8) Assumd k ( ), it is calld th lmnt stiffnss matri, and p k ( ) known as th lmnt quivalnt nodal load vctor, thn quation (8) can chang into th following lmnt stiffnss quation: It is asy to know: ( ) p k (9) csh csh q k, p k ( ) (9a) sh csh( ) csh( ) csh( ) In formula(9a), and can b obtaind by putting into quation (3d). Thn th dirct nodal forc vctor Z,quivalnt nodal forc vctor p, stiffnss matri Z of ach cabl lmnt can b suprposd into nodal forc vctor and stiffnss matri K for th whol cabl, as a rsult, w can hav th global stiffnss quation as th following form : K Z () In quation() is th nod forc vctor for th whol cabl, K is th global stiffnss matri, Z is th vrtical coordinats vctor of ach nod for th whol cabl. 4. CALCULATION MTHOD OR UNSTRSSD LNGTH O TH CABL In abov tt, vry act nodal coordinat can b known,thn w can obtain vry cabl lmnt catnary quation f ( ).Also w can gt aial forc of vry cabl lmnt as blow: Namly: + () i i (a) ' i + Assumd th tntion lngth of cabl lmnt undr th aial forc is basd on th rlationship of strss and strain, w can gt quation () as blow: l,

In th abov quation: l ds () A Thn w can gt: ' ds= + A As a rsult,th tntion lngth of any cabl lmnt is known from th blow quation: d l ' d (a) i i Li ' d (3) i A Thn th tntion lngth of th whol cabl is:

8 In th abov quation: l ds () A Thn w can gt: ' ds= + A As a rsult,th tntion lngth of any cabl lmnt is known from th blow quation: d l ' d (a) i i Li ' d (3) i A Thn th tntion lngth of th whol cabl is: n L Li (3a) i vry cabl lmnt lngth undr uniform loads is: i i LL ds ' ds i So th lngth of th whol cabl is: i n i i i (4) LL LL (4a) inaly w can gt th unstrssd lngth of th whol cabl as blow: L LL L (5) ig.4 Th dsignsktch of AiZhai suspnsion bridg

9 5. LOW CHART O CALCULATION PROCDUR In th procss of practical calculation, firstly tak vrtical uniform loads on lmnt as q q,on th cas, th shap of cabl lmnt is parabolic, so w can gt a group of nodal vrtical coordinats as initial solution ; thn tak q q ( '),and tak as a known initial valu, so w can obtain a group of nw nodal vrtical coordinats, which will b th nw initial valu for th nt calculation. Thn, th linr of cabl lmnt will convrgnc from parabolic to catnary stp by stp. Whn th convrgnc solution can fufill th prcision askd for, th calculation procdur can brak, and w can gt th act vrtical coordinats of all nods. And thn tak th act nodal vrtical coordinats as known conditions, w can gt th curv quation of all cabl lmnts. inally w can obtain th original lngth and tnsion lngth and unstrssd lngth on basis of quations(3),(4),(3a),(4a).dtaild calculation flow chart as shown in figur 5. Tak q q into quation ' ' ( ) H q,gt parabolic analytical solution ( ) Run quation (6a),gt vry lmnts k, P Intgrat k, P into K, Run quation (7),gt initial nodal vrtical coordinats vctor Tak q q ( ') into quation ' ' ( ) H q,gt catnary solution ( ) Tak vry lmnts of as initial coordinats into Tak = Z Run quation (9a),gt vry lmnt s p Intgrat all lmnts p and k, k into and NO Run quation (),gt Z,chck solution fulfil th prcision conditions

10 YS Z is th act nodal vrtical Us solution Z, gt lmnt s catnary quation ( ) Run quation (3)and(4),gt cabl lmnts tnsion lngth Li and original lngth LLi Run quation (3a)and(4a),gt th whol tnsion lngth L and th whol original lngth LL Unstrssd cabl lngth L LL L ig.5 lowchart of th algorithm for form-finding and unstrssd lngth 6. INIT LMNT ANALYSIS O NGINRING XAMPL In th papr, w us two kinds of larg finit lmnt softwar to calculating. In th procdur, th cntr-span suspnsion cabl is takn as analysis objct, and th calculating structur paramtrs ar masurd valu, and tak th tmpratur influnc into account. At th mpty cabl stat, th masurd coordinats of th top points of JISHOU towr and CHADONG towr ar ( , ), ( ,79.69),which will b th start point and finish point of th analysis objct. Only tak th 8, 4,3 8,,5 8,3 4 plac points as kypoints of th cabl, and tak th horiontal tnsion for H= ton. Th calculatd coordinats of all kypoints and unstrssd cabl lngth of finit lmnt analysis as shown in tabl. Tab..orm-finding rsults of finit lmnt softwar/(m) Bridg-span X Z Midas Ansys rror(%) % / % / % 3/ % / % 5/ % 3/ % 7/ % % As shown in tabl,th calculation rsults can show us that th ma

11 vrtical coordinats rror at th sam horiontal coordinat is.4%, it can prov rsults of th two kinds of finit lmnt softwar ar similar; and th ma unstrssd cabl lngth rror btwn slf-mad program and finit lmnt softwar shown in tabl is.6%,it prov th rsults ar act. Calculation softwar Tab..Rsults of unstrssd cabl lngth/(m) Oringinal tnsion Unstrssd lngth lngth lngth rror with program Midas civil % Ansys % Procdur % 7. COMPARISON BTWN THORTICAL CALCULATION AND MASURD VALU According to th calculation thory for form-finding, mak us of slf-mad program to calculat th coordinats of kypoints, thn compar th rsults with th rsults of finit lmnt softwar and masurd valu in tabl 3 and figur 6, th valu in tabl shows that th ma rror of slf-mad program and masurd valu is.%, it provs th calculation thory act, th rsults can fulfill th nginring accuracy. Tab.3 orm-finding rsults of Slf-mad program and masurd valu/(m) Bridg-span X Z Thortical calculation Masurd valu rror % / % / % 3/ % / % 5/ % 3/ % 7/ % %

12 Rsults of Midas civil Rsults of Ansys Rsults of slf-dit program Masurd rsults ig.6 Th rsults curv of finit softwars and slf-dit program and masurd rsults 8. CONCLUDING RMARKS. Compard to th Nonlinar init lmnt mthod, th mthod of prcis form finding for cabl unit can rduc dividd unit dnsity and numbr of lmnts in matri calculation so as to improv th opration spd.. With th vrtical load distribution changing from along th string lngth to th arc lngth, cabl units convrg into accurat catnary by th nonlinar itration with th initial parabolic valu which has bn nar act solution. Thrfor, th mthod gratly acclrats th convrgnc spd. 3. Th calculation mthod is put forward for cabl-lngth in non-strss on th prmis that th prcis coordinats of suspnsion cabl hav bn achivd according to thory of prcis form-finding for cabl lmnt. 4. By taking Aihai Suspnsion Bridg as an nginring ampl, th papr has analyd th masurmnts and th rsults of finit lmnt softwar, and vrifis th accuracy of calculation thory and th slf-mad calculation program. Acknowldgmnts This papr cost many popl s tim and nrgy. My dpst gratitud gos first and formost to my tutor, Profssor Wang Ronghui, for his constant ncouragmnt and guidanc. H has hlpd m a lot through all th stags of th writing of this papr. Without his constant instruction, this papr can t b managd to rach this prsnt form. My sincr thanks gos to Wi Yuanchng and Huang Xiaofng,thy also hlps m a lot by providing m many usful rfrnc books. I also ow my thanks to my dar frinds and fllow classmats, who hav don m a favor in hlping m work out my problms during th difficult cours of

13 th papr. I would lik to dply thank th all various popl who, during th svral months in which this ndavor lastd, providd m with usful and hlpful assistanc. Without thir car and considration, this ssay would likly not hav finishd. 9. RRNCS [] SHN Shi-hao.XU Chong-bao. Suspnsion Cabl Structur Dsign[M],BIJING:China Architctur and Building Prss,997 [] XIANG Yang,SHN Shi-hao.Initial orm-finding Analysis of Cabl Structurs[J]. Journal of Harbin Univrsity of Civil nginring and Architctur,997,3(3):9-33. [3] YAN Jing-tong,GAO ri.comparison And Analysis Btwn Two orm-finding Mthods or Cabl Nt Structurs[J].Stl Structur,7():4-6. [4] SCHK H K.Th forc dnsitis mthod for form-finding and computation of gnral nts[j].computr Mthods in Applid Mchanics nginring,974.3():5-34. [5] BARNS M R.Dynamic rlaation analysis of tnsion ntworks[a].procdings of th Intrnational Confrnc on Tnsion Structur[C].London,April,974. [6] Zhou mng-bo. Suspnsion bridg manual [M], Bijing:China Communications Prss, 3.8. [7] YUAN Si, CHNG Da-y, Y Kang-shng.. act lmnt Mthod for orm-finding Analysis of Cabl Structurs [J].Journal of Building Structurs 5,6():46-5. [8] Wang Xin-ming. ANSYS nginring structur numrical analysis[m], Bijing:China Communications Prss, 7..

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