International Journal of Engineering Research & Science (IJOER) [Vol-1, Issue-9, December- 2015]

Transcription

1 Internatnal Jurnal f Engneerng Research & Scence (IJOER) [Vl, Issue9, December 05] Unstraned Element LengthBased Methds fr Determnng One Optmzed Intal State f CableStayed Brdges MyungRag Jung, Dngju Mn, Mar M. Attard 3, MnYung Km 4 Graduate Student, Dept. f Cvl and Envrnmental Engneerng, Sungkyunkwan Unversty, Graduate Student, Dept. f Cvl and Envrnmental Engneerng, Sungkyunkwan Unversty, 3 Asscate Prfessr, Schl f Cvl and Envrnmental Engneerng, he Unversty f New Suth Wales, Sydney, NSW 05, Australa, 4 Prfessr, Dept. f Cvl and Envrnmental Engneerng, Sungkyunkwan Unversty, 066, SebuR, Jangangu, Suwns, , S. Krea(crrespndng authr) Abstract w rbust prcedures evaluatng all unstraned element lengths are presented t fnd ne practcally ptmzed ntal shape f cablestayed brdges under dead lads. An analytcal methd based n the cntnuus grder mdel accuntng fr PΔeffects due t staycable tensns s frst prpsed t calculate ptmzed cable tensns and unstraned element lengths wthut recurse t refned nnlnear FE analyss methd. And then t s addressed hw the G.CUD methd [0] develped fr suspensn brdges shuld be appled t determne an ptmzed ntal state f cablestayed brdges. Fr ths, the extended nnlnear frmulatns f the crtatnal frame element as well as the elastc catenary cable element are brefly summarzed by addng unstraned lengths f all fnte elements t the unknwn. Fnally, based n the unstraned lengths determned frm tw methds, the unstraned length methds are presented t effectvely perfrm nnlnear FE analyss f cable stayed brdges subjected t varus lad cmbnatns. Cnsequently accuracy and effectveness f the prpsed schemes are demnstrated by shwng that nt nly the unstraned lengths f a lngspan cablestayed brdge mdel by the analytcal methd are nearly same as thse by the G.CUD methd but als these tw methds lead t essentally ne ptmzed ntal cnfguratn whch s n sut wth the target gemetry. Keywrds Intal shapng, G.CUD, elastc catenary cable element, crtatnal frame element, unstraned length, cablestayed brdge I. INRODUCION Generally ne ntal cnfguratn satsfyng the equlbrum cndtn between external dead lads and nternal member frces ncludng cable tensns shuld be predetermned n the prelmnary desgn stage f cablesupprted brdges because cable members cannt be defned n the stressfree state. Mrever t s f extreme mprtance t btan the mnmzed bendng mment dstrbutns by determnng ptmzed cable tensnng frces because the nternal frces due t dead lads can be sgnfcantly large as the span length f cable brdges s ncreased. hs analyss prcess fndng ne ntal equlbrum state clse t the target cnfguratn f cable structures under full dead lads s referred t as shape fndng, frm fndng r ntal shape analyss. Wth relatn t shape fndng prblems f cablestayed brdges, a set f ptmzed tensnng frces fr staycables shuld be fund such that the vertcal dsplacements f the man grder vansh except fr the fabrcatn camber and the hrzntal dsplacements f the pyln are mnmzed wthn the allwable lmt. Otherwse, huge bendng mments n the deck and pylns f cablestayed brdges under dead lads can be nduced due t the P effect by hrzntal r vertcal cmpnents f the cable tensn. Furthermre, n case f fan and harptyped cablestayed brdges, ne practcally ptmzed ntal state shuld be searched because there can exst several ntal cnfguratns. Partcularly as the span length f cablesupprted brdges s greatly ncreased, the maxmum bendng mment ccurrng n the man grder and the pyln can becme rapdly utszed dependng n the fabrcatn camber and the balanced cndtn wth respect t selfweghts. Untl nw, t fnd the ntal state slutn f cablestayed brdges, varus analyss methds have been develped such as the zer dsplacement methd [], the frce equlbrum methd [], the ptmzatn methd [3, 4, 5], the ntal frce methd [6], the CUD (arget Cnfguratn Under Dead lads) methd [7], and the cmbnatn methd f ntal frce methd and CUD methd [8, 9]. Hwever, t s judged that the ptmzed cable tensnng prblem f cablestayed brdges s stll Page 73

2 Internatnal Jurnal f Engneerng Research & Scence (IJOER) [Vl, Issue9, December 05] challengng because the slght varatn f cable tensn frces can result n massve bendng mments n the man grder r pylns. Partcularly t s wrth mentnng the unstraned element lengthbased methd [0] generalzng CUD methd recently prpsed fr fndng an ptmzed ntal shape f suspensn brdges under dead lad. In that study, the extended tangental stffness matrces f the frame element as well as the cable element were derved by addng unstraned lengths f all fnte elements t the unknwn. And the unstraned element lengthbased methds ncludng the G.CUD methd were then prpsed based n Newtn teratn methd. Eventually t was demnstrated thrugh numercal applcatn that ne deally ptmzed ntal cnfguratn fr typcal suspensn brdges subjected t full dead lads can be successfully fund such that nt nly the cnverged state well cnfrmed t the desgned cnfguratn but als bendng mments n the man grder were mnmzed and mments n the twer were neglgbly small. Fr cablestayed brdges cntrary t suspensn brdges, t s questnable whether the nnlnear analyss methds prpsed fr suspensn brdges can be straghtfrwardly appled t lngspan cablestayed brdges r nt. On the ther hand, sme analytcal methd has been prpsed t get the tral ntal state slutn f cable brdges. he study by Chen et. al [], whch s based n the cntnuus beam mdel vrtually supprted at the pnts anchred by stay cables and a cnstrant cndtn f hrzntal dsplacements at the tp f the pyln, s wrth referrng n case f cablestayed brdges. Hwever, mst f ntal state slutns btaned analytcally by these methds mght nt prvde ne ptmzed ntal cnfguratn due t the cmbned actn f fabrcatn cambers and hrzntal tensn cmpnents f cable members n case f selfanchred cablestayed brdges. In ther wrds, fr cablestayed brdges havng fabrcatn cambers, bendng mments and reactn frces f the cntnuus stffenng grder supprted vrtually at the pnts anchrng by cable elements can be naccurately evaluated due t hrzntal tensns because t s subjected t hrzntal tensn cmpnents f cable members as well as selfweghts. Furthermre, staycables n case f lngspan cablestayed brdges are s lng that t can be smetmes requred t mprve the accuracy n calculatng ther unstraned lengths. hs paper ntends t prpse tw rbust prcedures evaluatng all unstraned element lengths t fnd ne practcally ptmzed ntal shape f cablestayed brdges under dead lads:. An mprved analytcal methd based n the cntnuus grder mdel accuntng fr P effects f the man grder due t cable tensns s frst prpsed t calculate ptmzed cable tensns and unstraned element lengths wthut recurse t refned nnlnear FE methd.. And then t s addressed hw the G.CUD methd [0] develped fr suspensn brdges shuld be appled t determne an ptmzed ntal state f cablestayed brdges. 3. Fr ths, the extended nnlnear frmulatns f the frame element as well as the cable element are brefly summarzed by addng unstraned lengths f all fnte elements t the unknwn. 4. Fnally, based n the unstraned lengths determned frm tw methds, the unstraned length methds are presented t effectvely perfrm nnlnear FE analyss f cablestayed brdges subjected t varus lad cmbnatns. 5. Fr a lngspan cablestayed brdge example havng tw ntermedate pers, accuracy and effectveness f the prpsed tw schemes are demnstrated by shwng that nt nly the unstraned lengths by the analytcal methd are nearly same as thse by the G.CUD methd but als these tw methds lead t essentally ne ptmzed ntal cnfguratn whch s n sut wth the target gemetry. 6. In partcular, amplfed effects f the fabrcatn camber and the weght balancng between center and sde spans n the ntal state slutn are carefully nvestgated thrugh the brdge example. II. NONLINEAR ELASIC CAENARY CABLE AND FRAME ELEMENS Jayaraman and Knudsn [] have frstly prpsed an elastc catenary cable element frm the exact slutn (Irvne []) f the elastc catenary cable equatn under ts selfweght. And a frame element fr gemetrcally nnlnear analyss f plane frames has been develped by several researchers (Pacsteand Erkssn [3], Crsfeld [4], Leand Battn [5]). In ths sectn, the unstraned lengthbased frame and cable element presented n Jung el al. [0] are brefly frmulated t develp the G.CUD methd and the crrespndng unstraned length methd n the subsequent sectns. Page 74

3 Internatnal Jurnal f Engneerng Research & Scence (IJOER) [Vl, Issue9, December 05] Cnsder an elastc catenary cable element suspended between tw pnts (0,0) and j( L, L ) as shwn n Fg.. By ntegratng equlbrum equatns exactly, the fllwng cmpatblty cndtn can be derved as a functn f the ndal frces F and F at the nde and the unstraned length L as x y L x F L F F wl = + snh snh EA w F F F (a) L y F L = EA wl ( ) + + (b) q p EA w FIGURE AN ELASIC CAENARY CABLE ELEMEN SUBJECED O IS SELFWEIGH AND NODAL FORCES where p F + F = ; ( ) = F + wl F ; EA = the axal rgdty; w = selfweght per unt length. q Nw partal dfferentatn f bth sdes f Eq. () yelds the fllwng ncremental relatnshps: Lx Lx / F Lx / F F L / x L = + L L L / F L / F F L / L y y y y 0 (a) L x = U 3 U and L y = U 4 U (b, c) Cnsequently the nverse f the flexblty matrx n Eq. (a) leads t ncremental equlbrum equatns f an elastc catenary cable element as fllws; Fc = K c U c + K cu L (3) Where F c = the ncremental ndal frce vectr; K = the tangental stffness matrx; c U c = the ncremental dsplacement vectr; K = the stffness matrx related t the unstraned length. It shuld be emphaszed that all cu the stffness terms n Eq. (3) are fully used fr calculatng the extended tangental stffness matrx n case f the CUD methds but the last term n Eq. (3) vanshes n the unstraned length methd because the unstraned cable length L s kept cnstant. In addtn t ths, ne f ndal frces F and F f lng staycable members s assumed t reman unchanged n develpng an analytcal methd n sectn 3.. Cnsequently t s wrth pntng ut that Eq. () shuld be teratvely slved wth keepng ne f F, F and L a fxed value n rder t reslve the state determnatn prblem f elastc catenary cable elements. Page 75

4 Internatnal Jurnal f Engneerng Research & Scence (IJOER) [Vl, Issue9, December 05] On the ther hand, Fg. shws ndal dsplacements and defrmatn cmpnents f a frame element wth respect t the crtatnal crdnate system at the ntal and the defrmed state where the ndal dsplacement and frce vectr may be defned as fllws; ( U U U U U U ) U =,,,,, (4a) f ( F F F F F F ) F =,,,,, (4b) f FIGURE NODAL DISPLACEMENS AND PURE DEFORMAIONS OF A NONLINEAR FRAME ELEMEN Large rgdbdy mtns but the small defrmatns are assumed n ths frmulatn. Remvng rgd bdy mdes frm ndal dsplacement, the three pure defrmatns cnsstng f the axal defrmatn and relatve rtatns can be determned as fllws; D L L D = D = U 3 α D 3 U 6 α where L = the chrd length between tw element ndes. Nte that L s the element length cmputed as the dstance between the ndal pnts n the CUD methd but the unstraned element length whch shuld be updated teratvely n the G.CUD methd [0]. hen the chrd length L and rgd bdy rtatn α are calculated as ( j 4 ) ( j 5 ) L = x x + U U + y y + U U y j y + U U tanα = x x + U U j 5 4 where ( x, y ), ( x j, y j ) = the ndal crdnates n the glbal crdnate system. Nw frcedefrmatn relatnshps f the beamclumn element cnsderng the bwng effect and P δ effect can be expressed as fllws; D P EA L = + D D D + D (5) (6a) P 4EI P L = + D EI P L + D 3 L 5 L 30 (6b) Page 76

5 Internatnal Jurnal f Engneerng Research & Scence (IJOER) [Vl, Issue9, December 05] EI P L P = D 4EI P L + + D 3 3 L 30 L 5 (6c) Referrng t [0], the ncremental equlbrum equatn f aframe element cans bebtanedn the glbal crdnate system as fllws; Where K f, K fu can be expressed as ΔFf = K f ΔUf + K fu L (7) = the tangental stffness and the unstraned lengthrelated stffness matrx, respectvely, whch * * * K = R (k + k + k )R f e d g K fu = R k * fu (8a, b) where R = the crdnate transfrmatn matrx; * * * k e, k d, k and g * k fu = the elastc stffness, stffness due t member defrmatns, gemetrc stffness due t member frces and the unstraned lengthrelated stffness matrces, respectvely, n the crtatnal crdnate system. her detaled frms are presented n Appendx. III. AN ANALYICAL MEHOD FOR HE INIIAL SHAPING ANALYSIS OF CABLESAYED BRIDGES fnd ne ptmzed ntal cnfguratn f cablestayed brdges under dead lads analytcally wthut nnlnear FE analyss, basc assumptns are gven n sectn 3. and then an analytcal prcedure determnng all the unstraned element lengths s prpsed n sectn Basc assumptns fr develpng the analytcal methd Frst f all, t s assumed that n rder t lcalze bendng mments n the man grder due t dead lads, the stffenng grder s vrtually supprted at the ndal pnts anchred by stay cables s that vertcal dsplacements shuld nt ccur at thse pnts except fr the fabrcatn camber. hs assumptn usually leads t the mnmzed bendng mment dstrbutn f the stffenng grder. Secnd, the selfweghts f a center span and tw sde spans n case f selfanchred cablestayed brdges shuld be well balanced whch can result n mnmzatn f bendng mments n pylns. hrd, t s assumed that the staycable element havng relatvely small cable lengths s parablc under selfweghts and ndal frces can be decmpsed nt the pretensn θ and the vertcal reactn cmpnent wl / as shwn n Fg. 3. Als, the unstraned length L f the nclned stay cables can be evaluated by slvng the cubc equatn f Eq. (9) when ther nmnal tensn θ and the chrd length l are gven: 3 EA EA ( wl cs α) θ + ( L l ) θ = 0 L 4 w L 0 (9) θ w L 0 w L 0 θ FIGURE 3 FREE BODY DIAGRAM OF A PARABOLIC CABLE ELEMEN UNDER IS SELFWEIGH Page 77

6 Internatnal Jurnal f Engneerng Research & Scence (IJOER) [Vl, Issue9, December 05] Furth, n case f extremely lngspan cablestayed brdges, the length f stay cables near the backstay cable s s lng that sme devatn frm the accurate slutn can ccur f nly ne parablc cable element s used fr each stay cable member. In ths case, ne stay cable member needs t be regarded as an elastc catenary cable element nstead f a parablc cable element. Ffth, the chrd length l n Eq. (9) n applyng the analytcal methd s evaluated usng the ntal dstance between tw anchr pnts f each stay cable because man grder and the pyln subjected t dead lads and ptmzed cable tensns are well balanced enugh t experence neglgbly small dsplacements. 3. An analytcal methd fr determnng an ptmzed ntal state f cablestayed brdges In case f cablestayed brdges, nmnal tensns and unstraned lengths f stay cables are usually determned frm reactn frces R btaned thrugh lnear elastc analyss f the cntnuus grder mdel vrtually supprted at anchred pnts under nly dead lads (see Fg. 4(a)). Hwever, f gemetrcally nnlnear analyss fr the full brdge mdel s perfrmed based n them, glbally huge bendng mments due t P effect by hrzntal tensn cmpnents f stay cables can be nevtably nduced n the man grder havng the fabrcatn camber.,back,back w g R R 3 R R 4 R5 R6 R7 8 R FIG 4 (A) A MAIN GIRDER SUBJECED O IS SELFWEIGH ONLY get the ptmzed ntal state slutn, tensn cmpnents f stay cables are sutably mdfed s that thse bendng mment n the grder shuld be cmpletely excluded except fr lcal mments. hs prblem can be vercme by analyzng the cntnuus beam subjected t ts selfweght and updated tensn cmpnents f stay cables smultaneusly thrugh sme teratn prcess. In ther wrds, the cntnuus grder mdel subjected t nt nly dead lads but als hrzntal cable tensns as shwn n Fg. 4(b) s newly cnsdered t get rd f thse glbal mments and t generate nly lcal mments n the ntal cnfguratn f the man grder. In that case, mdfed reactn frces R and crrespndng cable tensns can be evaluated frm lnear elastc analyss f the mprved grder mdel. And t s necessary t update hrzntal cable tensns thrugh sme teratn lp because thse nmnal tensns f stay cables are nt knwn n advance.,back,back w g H,back H H H H3 H H 4 5 H,back 6 H H 7 8 R ' R ' R ' 3 R 4 ' R ' 5 R ' 6 R ' 7 R ' 8 FIG 4 (B) A MAIN GIRDER SUBJECED O BOH IS SELFWEIGH AND HE HORIZONAL ENSIONS OF SAY CABLES FIGURE 4 HALF MODELS OF HE CONINUOUS MAIN GIRDER WIH FABRICAION CAMBERS IN A CABLE SAYED BRIDGE Page 78

7 Internatnal Jurnal f Engneerng Research & Scence (IJOER) [Vl, Issue9, December 05] Nw fr cablestayed brdges wth fabrcatn cambers subjected t dead lads, an analytcal calculatn prcedure t make nternal mment dstrbutns mnmzed and t prvde the crrespndng unstraned lengths f all elements s gven ncludng the teratn lp f cable tensns as fllws; Step ) Buld a structural mdel fr the cntnuus stffenng grdersupprted vertcally at the pnts anchrng by staycables as shwn n Fg. 4. Step ) after the teratn ndex k s set t be zer, calculate the ntal reactn frces R ( ) at the anchr pnts f the cntnuus man grder subjected t nt nly ts selfweght but als the ntal hrzntal tensn cmpnent ( ) H subsequent teratns. f stay cables whch are zer n the frst teratn prcess but are newly updated n the Step 3) Enter the teratn prcess: k = k + Step 4) Calculate the updated reactn frce R at the anchr pntsf the cntnuus man grder subjected t ( k ) nt nly ts selfweght but als the hrzntal tensn H. Step 5) Evaluate the updated hrzntal tensn H f the th stay cable usng ne f the fllwng tw cases: I) In case f relatvely shrt stay cables (see assumptn 3), determne the nmnal tensn ( k ) f the stay cable and the axal frces P, P f the man grder and the pyln, respectvely, by nvkngthe fllwng, g, p equlbrum cndtnat tw anchrage pntsf the stay cable (see Fg. 5): s, s, θ = + s, sn R W / s, cs θ = H = g, + g, H P P s, θ = p, + p, s, sn P P W / (0) W s, s, θ W s, P P, P g, P g, + θ P P, + R s, FIGURE 5 FREE BODY DIAGRAMS A WO ANCHORAGE POINS OF ONE SAY CABLE where θ = the nclnatn angle; W s, = the selfweght f the thstay cable. II) In case f relatvely lng stay cables (see assumptn 4), slve the cmpatblty equatn () f an elastc catenary cable ( k) element derved as a functn f the ndal frces and the unstraned length where nte that F ( = R ) s a knwn value calculated n Step 4) (refer t Fg. 6). Accrdngly Newtn teratn prcess s executed usng the ncremental equatn () ( k) t fnd tw unknwns L, F ( = H ) where ther ntal values are chsen frm Eq. (9) and (0). L L L L L = F + L L = F + L x x y y x, y F L F L () Page 79

8 Internatnal Jurnal f Engneerng Research & Scence (IJOER) [Vl, Issue9, December 05] R wl O (k ) H R (k ) H FIGURE6A SAY CABLE REAED AS AN ELASIC CAENARY CABLE IN HE KH IERAION PROCESS ( k) Step 6) Evaluate the hrzntal tensn H back f the backstay cable: ts nmnal tensns cannt be evaluated frm Eq. (0) due t exstence f real vertcal supprts. Mrever t s well knwn that backstay cable tensns cannt be unquely determned n case f fan r harptyped cablestayed brdges. Practcally ths can be calculated by analyzng the pyln mdel wth the rller supprt at the nde anchred by backstay cables and subjected t hrzntal tensn cmpnents f staycables evaluated n Step 5) as shwn n Fg. 7., Back, Back H 8 ( k) H7 H H ( k) H H 5 H H ( k) 4 ( k) 6 ( k) ( k) 3 H H, Back, Back 3 4 (A) A PYLON SUBJECED O NOMINAL CABLE ENSIONS (B) FREE BODY DIAGRAM OF A PYLON UNDER HORIZONAL ENSIONS FIGURE 7A PYLON SUBJECED O NOMINAL ENSIONS BYHE SAY CABLE Step 7) Check whether the nmnal tensns f stay cables cnverge r nt: If t s nt cnverged, g t Step 3) and repeat the teratn prcess t Step 7). If t s cnverged, ext t Step 8). Step 8) Determne the cnverged tensns and the unstraned lengths f all stay cables usng ne f the fllwng tw cases: I) In case f relatvely shrt stay cables, the unknwn values are easly calculated usng Eq. (9) and (0). II) If the backstay cable s relatvely lng, the cmpatblty cndtn f the elastc catenary cable s appled smlarly t II) f Step 4) namely, Eq. () s teratvely slved wth respect t F and determned n Step 6) (see Fg. 8). L as shwn n Eq. () because L L L L L = F + L L = F + L x x y y x, y F L F L H ( k) back ( = F ) s Step 9) Fnally evaluate the axal frce and the unstraned length f frame members: the axal frce f all frame members can () Page 80

9 Internatnal Jurnal f Engneerng Research & Scence (IJOER) [Vl, Issue9, December 05] be determned frm statcs and the crrespndng unstraned length usng Hke s law. F wl O H back F H back FIGURE: 8 A BACKSAYCABLE REAED AS AN ELASIC CAENARY CABLE In ths study, three analytcal prcedures are taken nt accunt fr cmparsn. Fg. 9 t shws flwcharts f three algrthms t determne unstraned lengths f all cable and frame elements analytcally. Here the analytcal methd (AM) wthut any teratn prcess neglects P effect by hrzntal tensn cmpnents f stay cables whch can cause huge bendng mments n the man grder and the pylns due t the hrzntal r the vertcal cmpnents f stay cable tensns. On the ther hand, bth the analytcal methd (AM) and the analytcal methd 3 (AM3) can greatly reduce P effects f cable tensns wng t the updated teratn prcess f stay cable tensns. Here the man dfference between the AM and AM3 s that each stay cable s mdeled as an apprxmate parablc cable element n the frmer but as an accurate elastc catenary cable n the latter. herefre, the AM3 s expected t prvde the mst ptmzed ntal state slutn by treatng lng staycables as a catenary cable member. Input the gemetrc and dead lad data f cable stayed brdges Calculate reactn frces R f the cntnuus man grder under ts selfweght nly Determne the nmnal tensn s, and the hrzntal tensn usng Eq. (0) H Determne the hrzntal tensn H, back f backstay cables referrng t Fg. 7(b) Calculate unstraned lengths f all cable elements usng Eq. (9) Calculate axal frces f all frame elements and the crrespndng unstraned lengths usng Hke s law Cmplete AM FIGURE 9 A FLOWCHAR O DEERMINE UNSRAINED ELEMEN LENGHS BY HE ANALYICAL MEHOD IV. HE G.CUD MEHOD FOR HE INIIAL SHAPING ANALYSIS OF CABLESAYED BRIDGES In ths sectn, the generalzed CUD methd s presented fr ntal shapng analyss f cablestayed brdges. he CUD and G.CUD methds fr ntal shapng analyss f suspensn brdges have well develped n the prevus papers [7, 0]. Bascally thse prcedures can be appled t determne the ntal state f lngspan cablestayed brdges under dead lads wthut majr mdfcatn. Accrdngly n ths sectn, the nnlnear frmulatn f the G.CUD methd s cmpactly summarzed and sme dfferences between suspensn brdges and cablestayed brdges n applyng these methds are mentned. Page 8

10 Internatnal Jurnal f Engneerng Research & Scence (IJOER) [Vl, Issue9, December 05] ( ) H R H (k) H s, H, back FIGURE 0 A FLOW CHAR O DEERMINE UNSRAINED ELEMEN LENGHS BY HE ANALYICAL MEHOD he extended ncremental equlbrum equatn fr the whle structural system accuntng fr unstraned element lengths as the unknwn can be wrtten as ΔF = K tδu + K ulδl (3) where ΔF ( n ) = the unbalanced lad vectr; K ( n n) and K ( n m) unstraned lengthrelated stffness matrx, respectvely; ΔU ( n ) t ul = the tangental stffness matrx and the = the ncremental ndal dsplacement vectr; n = the number f ttal degree f freedm; ΔL ( m ) = the ncremental unstraned length vectr where nte that m s equal t the ttal number f all the cable elements n CUD methd and the number f bth cable and frame elements n G.CUD methd, respectvely. Clearly addtnal cnstrant cndtns shuld be ntrduced t slve the ncremental equatn (3) snce the ttal number f unknwn varables n Eq. (3) exceeds the ttal number f equatns. Fg. llustrates ne example f the cnstrants appled t a cablestayed brdge havng tw ntermedate pers n whch the arrwed degrees f freedm are addtnal restrants due t unstraned lengths f cable and frame elements ntrduced n G.CUD methd. In ther wrds, addtnal gemetrc restrants due t cable members are the vertcal dsplacements f the man grder at the pnts anchred by stay cables and the hrzntal dsplacements f the pyln at ndal pnts cnnectng t backstay cables. And cnstrants due t frame members nclude the axal dsplacements f all the ndal pnts n bth the man grder and the pyln. In case f suspensn brdges, the hrzntal dsplacements at the tp f the twer are deally zer and the bendng mments are lcalzed n the man grder under dead lads rrespectvely f balanced r unbalanced cndtns. Hwever, t shuld be pnted ut that all hrzntal mvements f the pyln cannt be perfectly Page 8

11 Internatnal Jurnal f Engneerng Research & Scence (IJOER) [Vl, Issue9, December 05] suppressed n fan r harptyped cablestayed brdges. mnmze these bendng mments ccurrng n pylns, t s f central mprtance fr ttal selfweghts between the center span and the sde spans t be well balanced. Input the gemetrc and dead lad data f cable stayed brdges Enter the teratn lp: k=0 H (O ) Set = 0 Yes k = 0? Calculate reactn frces R f the cntnuus man grder (k ) under bth ts selfweght and the hrzntal tensn N H Calculate ntal values f tw unknwns usng Eq. (9) and (0) L,, F ( = H ) Slve Eq.() and () teratvely by keepng (k ) R cnstant If cnverged, n the element level update L, H Determne the hrzntal tensn H, back f backstay cables referrng t Fg 7(b) k = k+ N Are the hrzntal tensns cnverged n the structure level? Yes Intalzed unstraned lengths and vertcal reactns f backstay cables usng Eq.(9) and (0) Iteratvely slve Eq.() and () wth keepng If cnverged, n the element level update H, back Calculate axal frces f all frame elements and the crrespndng unstraned lengths usng Hke s law L, back cnstant. Cmplete AM3 FIGURE A FLOW CHAR O DEERMINE UNSRAINED ELEMEN LENGHS BY HE ANALYICAL MEHOD 3 FIGURE ADDIIONAL GEOMERIC CONSRAINS IN HE G.CUD MEHOD FOR A HALF MODEL OF A CABLESAYED BRIDGE HAVING ONE INERMEDIAE PIER Page 83

12 Internatnal Jurnal f Engneerng Research & Scence (IJOER) [Vl, Issue9, December 05] Nw takng nt accunt thse addtnal restrants, Eq. (3) may be rewrtten as where (( n m) ) u ΔF = K tuδu u + K tsδus + K ulδl (4) ΔU = the unknwn dsplacement vectr t be determned; ΔU ( m ) = the cnstraned dsplacement vectr mpsed by the desgner t fulfll the target shape f the brdge; K ( n ( n m)) and K ( n m) = parttned stffness matrces crrespndng tδuuandδu s, respectvely. tu ts Accrdngly, the secnd term n the rghthand sde f Eq. (4) vanshes and the ther tw terms result n a nnsymmetrc stffness frmulatn as s ΔU = tu ul ΔL ( ) [ ] u ΔF K K (5) Cnsequently the teratve G.CUD algrthm can be represented as fllws; () ΔU () () u () K tu K ul = ( W F ) fr =,, () ΔL () () () U u U u ΔU u = + () () () L L ΔL (6) () where W = the dead lad vectr; F = the equvalent nternal frce. After the smultaneus equatn (6a) havng the nnsymmetrc stffness matrx s slved, the ttal ndal dsplacement and the unstraned length vectr are updated as seen n Eq. (6b) and the nternal frce vectr () F (0) based n the ttal dsplacements and the unstraned lengths. Partcularly F s evaluated by the state determnatn prcedure dentes the nternal frce due t ntal cable tensns and axal frces determned by the analytcal prcedure n sectn 3..Generally ths vectr vanshes n case f grder brdges but shuld be cnsstently calculated fr cable brdges because t may be slwly cnverged r dverged f t s neglected. Remark : Nt nly the G.CUD methd prvdes an ptmzed ntal state f balanced cablestayed brdges but als lnear analyses based n t can be cnducted under varus lad cmbnatns. Hwever, smlarly t analytcal methds ntrduced n sectn 3, sme dffcultes can be caused n perfrmng nnlnear FE analyses under extreme lads. Remark : Nnlnear analyses under lmt lad cmbnatns ncludng the gemetry cntrl can be easly and accurately executed thrugh the unstraned length element methd presented n the next sectn. V. UNSRAINED LENGH MEHOD FOR NONLINEAR ANALYSIS OF CABLESAYED BRIDGES Basc cncept f the unstraned length methd (ULM) fr the ntal state analyss f cablestayed brdges s smlar t that fr suspensn brdges [0]. he unstraned length methd cnssts f tw stages. In the frst stage, unstraned lengths f bth cable and frame elements are predetermned n the reasnable way and n the secnd stage, nnlnear FE analyss based n Newtn teratn methd s perfrmed under dead lads by keepng unstraned element lengths cnstant. Cnsequently NewtnRaphsn teratn algrthm fr the secnd stage can be represented as fllws; K () ΔU () = W F () fr =,, L t Page 84

13 Internatnal Jurnal f Engneerng Research & Scence (IJOER) [Vl, Issue9, December 05] () () () U = U + ΔU (7) () L = L () where W and F = the dead lad and the nternal frce vectr dentcal t thse n Eq. (6). It shuld be agan emphaszed that the tangental stffness matrx s symmetrc and the unstraned lengths f all fnte elements reman cnstant n the teratn prcess. In the last stage, ncremental nnlnear analyses under addtnal lve lad cmbnatns are perfrmed by smply addng the lve lad λw t the dead lad n Eq. (7a) as fllws; L K () ΔU () = W + λw F () fr =,, L (8) t L Partcularly f sme temperature ncrease f specfc elements ccurs, the unstraned length f the crrespndng elements shuld be adjusted dependng n the thermal change n whch the ncremental unstraned length s calculated as where L = α L (9) α = the ceffcent f lnear thermal expansn. Wth relatn t the frst stage f ULM, three analytcal schemes determnng all the unstraned lengths have been prpsed n sectn 3. and G.CUD presented n sectn 4. In ths study, fur ULMs are taken nt accunt dependng n the scheme evaluatng the unstraned length as fllws;. ULM: the unstraned lengths f cable and frame elements btaned frm the AM n sectn 3. are drectly used n gemetrcally nnlnear FE analyss.. ULM: the unstraned element lengths btaned frm the AM are used. 3. ULM3: the unstraned lengths frm the AM3 are used. 4. ULM4: the slutn determned by G.CUD s fully used. Fg. 3 represents a flw chart f fur ULMs. In cnnectn wth the cncept f unstraned cable lengths, t s wrth referrng t the study by LzanGalant et al. [6] because t well explans what the unstraned length f cable elements means n cnstructn stage analyss even thugh t s based n lnear analyss. VI. NUMERICAL EXAMPLES In sectn 3 and 4, the analytcal methds and G.CUD methd determnng all the unstraned element lengths have been presented t fnd ne ptmzed ntal state f cablestayed brdges and n sectn 5, fur ULMs fr nnlnear analyss f cable brdges subjected t addtnal lad cmbnatns have been prpsed based n the tw ntal shapng analyss methds. In ths sectn, ne example determnng the ntal shape f selfanchred cablestayed brdge wth an ntermedate per s prvded t demnstrate effcency and effectveness f thse methds. Inchen Brdge whch cnnects Inchen Internatnal Arprt and Sngd Internatnal cty n Inchens s the bggest cablestayed brdge n Krea. Fgure 4 shws a structural mdel f Inchen brdge, and t s a lngspan cablestayed brdge wth fve spans f m, and the cncrete twer s an nverse Y shape wth 38.5m heght. Streamlned steel bx grder s suspended by 08 cables wth duble cable planes. Page 85

14 Internatnal Jurnal f Engneerng Research & Scence (IJOER) [Vl, Issue9, December 05] Als Fgure 5 dentes the vertcal camber f the man grder whch s lnear alng the sde spans and parablc thrughut the center span. F = W + λwl Slve KΔU t =ΔF FIGURE 3 FLOW CHAR OF HE UNSRAINED LENGH MEHOD FOR CABLESAYED BRIDGES Page 86

15 Internatnal Jurnal f Engneerng Research & Scence (IJOER) [Vl, Issue9, December 05] FIGURE 4 PROFILE OF INCHEON BRIDGE MODEL y v = x 0 v 0 = ( x 400) v = x x x x FIGURE 5 VERICAL CAMBER OF INCHEON BRIDGE MODEL nvestgate the effects f cambers and unbalancng f selfweghts, tw types f the cablestayed brdge mdel are bascally analyzed namely, the balanced brdge havng fabrcatn cambers and the balanced brdge wthut cambers. Fr cmparsn, the unbalanced brdge mdels wth r wthut cambers are addtnally explred usng G.CUD methd. able summarzes the materal and crsssectnal prpertes f Inchen brdge mdel. In ths mdel, a cunter weght f 300kN/m between 6.78~36.78m and 443.8~473.m alng the man grder, whch s neglected n case f the unbalanced brdge mdels, s delberately appled t make selfweghts f the center span and sde spans balanced. And the supply pers are lcated n each sde spans whch means that there are fur backstay cables cnnected wth supply pers and end pers. ABLE MAERIAL AND GEOMERIC PROPERIES OF INCHEON BRIDGE MODEL Structural member E(Gpa) A(m ) I(m 4 ) w(kn/m) Remarks Grder Cunter weghtf 87.5kN/m between supplemental and end pers wer ~8m wer m~34.5m Cable C~4, 8~4, 9~56, 7~77, 8~84, Cable C5~7, 5~8, 57~70, 78~80 Page 87

16 Internatnal Jurnal f Engneerng Research & Scence (IJOER) [Vl, Issue9, December 05] get ne ptmzed ntal state slutn, the fllwng cnstrants are ntrduced n the G.CUD methd. he essental bundary cndtn: Ycrdnates f the rllersupprted pnts (N., 9, 65 and n. 78) n the man grder X and Ycrdnates f the hnged pnt (N. ) n the man Ycrdnates f a pnt (N. 34) n the man grder supprted by the twer Fxed end pnts (N. 0 and 34) at the base f tw pylns he addtnal cnstrants ntrduced n G.CUD (dsplayed as arrw n Fg. and as bldface n able ): Ycrdnates f pnts (N. 8, N. 0, N. 3 64, N and N ) n the man grder anchred by stay cables Xcrdnates f pnts (N. 88, 96, and n. 0) n the pylns anchred by back stay cables Xcrdnates f ndal pnts (N., N. 3 86) n the man grder except fr N. Ycrdnates f all ndal pnts n twers except fr N. 0 and 34 ABLE UNSRAINED CABLE ELEMEN LENGHS IN HE CABLESAYED BRIDGE MODEL Cable N. L O (m) by G.CUD L O (m) by AM L O (m) by AM L O (m) by AM3 L O (mm) L O (mm) L O (mm) Remarks () () (3) (4) ()() (3)() (4)() back stay stay cable stay cable stay cable back stay stay cable stay cable stay cable stay cable stay cable stay cable stay cable stay cable stay cable stay cable stay cable stay cable stay cable stay cable stay cable stay cable stay cable Intal slutns ncludng unstraned lengths f all cable and frame elements are frstly determned usng AM, AM, AM3 and G.CUD methds and then fur ULMs are appled t buld an ntal state f the brdge mdel under dead lads. able shws unstraned cable lengths and ther dfferences evaluated by analytcal methds and G.CUD. Als, able 3 dsplay nt nly ntal target crdnates ncludng the vertcal camber f the man grder but als hrzntal and vertcal dsplacements by ULMs at the ndal pnts f the man grder and twers cnnected t stay cables. Here vertcal crdnates f N. 86 n the secnd clumn f able 3 dentes elevatns f fabrcatn cambers f the man grder. Partcularly ndal degrees f freedm crrespndng t bldface and stared values crrespnd t essental bundary cndtns and addtnal Page 88

17 Internatnal Jurnal f Engneerng Research & Scence (IJOER) [Vl, Issue9, December 05] gemetrc restrants ntrduced n G.CUD methd, respectvely (als refer t Fg. ).In addtn, Fg. 6 and 7 shw bendng mment dagrams n the left half f the man grder and the rght pyln, respectvely and Fg. 8 dsplays fluctuatns f stay cable tensns by three ULMs. Fnally able 4 shws the summary f maxmum nternal frces and dsplacements f the man grder and twers analyzed by ntal shapng analyss methds fr the balanced and unbalanced brdge mdels wth the fabrcatn camber. On the ther hand, able 5 summarzes maxmum bendng mments f the man members fr the balanced and unbalanced brdge wthut the camber. Several bservatns and cnclusns can be drawn frm the presented ables and Fgures. ABLE 3 INIIAL COORDINAES AND NODAL DISPLACEMENS A HE POIN OF HE GIRDER AND OWERS CONNECED O SAY CABLES BY HREE UNSRAINED LENGH MEHODS FOR HE BALANCED BRIDGE WIH HE FABRICAION CAMBER Nde N. arget Crd. (X,Y) (m) ΔX by ULM (mm) ΔY by ULM (mm) ΔX by ULM (mm) ΔY by ULM (mm) ΔX by ULM3 (mm) ΔY by ULM3 (mm) ΔX by ULM4 (mm) ΔY by ULM4 (mm) Remark (0.0 *,0.0) rller supprt 5 (40 *,.95 * ) (80 *,.390) rller supprt 3 (60 *,4.780 * ) (40 *,7.70 * ) (340, 0.58) Hnged supprt 30 (500 *,5.08 * ) (648 *,7.35 * ) (7 *,7.76 * ) (784 *,7.66 * ) (90 *,6.5 * ) (080 *,.65 * ) (40 *,0.58) rller supprt 69 (0 *,7.768 * ) (300 *,5.378 * ) (400 *,.390) rller supprt 8 (440 *,.95 * ) (480 *,0.0) rller supprt 88 (340, 74 * ) backstay cable 9 (340, 66 * ) (340, 58 * ) backstay cable 04 (340, 4 * ) (340, 64) fxed supprt (40, 74 * ) backstay cable 6 (40, 66 * ) (40, 58 * ) backstay cable 3 (40, 4 * ) (40, 64) fxed supprt Page 89

18 Internatnal Jurnal f Engneerng Research & Scence (IJOER) [Vl, Issue9, December 05] Frst f all, the ntal state slutns by AM lk reasnable at frst glance. Hwever, as shwn n Fg. 6(b), Fg. 7(a) and the secnd clumn f able 4 and 5, ULM based n unstraned lengths by AM gves explsvely large bendng mments n bth the man grder and the pyln fr brdge mdels wth camber. Fundamentally ths s because AM cannt remve magnfed bendng mments n the man grder wth the camber and the pylns due t cmpressve frces transmtted by cable tensns. Secnd, t s ntced n able 3 that nt nly vertcal dsplacements ΔY at ndal pnts n the man grder anchred by stay cables and hrzntal dsplacements ΔX at pnts n the pylns anchred by back stay cables but als axal dsplacements at pnts alng the man grder and the twers by ULM4 vansh exactly whch s due t cnstrants ntrduced by G.CUD methd. Partcularly maxmum bendng mments n bth the grder and the pyln by G.CUD are deally small n case f the balanced brdge wthut the camber (able 5) whle t turns ut that n the balanced brdge wth the camber, maxmum mments n the pyln are small wthn the allwable lmt (able 4). ABLE4 SUMMARY OF MAXIMUM INERNAL FORCES AND DISPLACEMENS FOR HE BALANCED AND HE UNBALANCED BRIDGE MODEL SHAVING HE INIIAL CAMBER AM AM AM3 G.CUD G.CUD * (ULM) (ULM) (ULM3) (ULM4) (ULM4 * ) Max. pstve mment 7,85.8 7, , , , * f the man grder(knm) (30,737.) (7,387.6) (7,56.85) (7,863.79) (7, * ) Max. negatve mment 0,47. 0,44.3 0,4.3 0,40. 0,40. * f the man grder(knm) (7,54.) (6,544.8) (0,473.9) (0,40.7) (0,40.0 * ) Max. cmpressve frce 4,694. 0,34. 06,05. 4,687. 4,857. * f the twer (kn) (5,896) (4,683.) (4,703.) (4,687.). (4,857. * ) Max. pstve mment 5,50.4 4, , , ,884.9 * f the rght pyln (knm) (46,3.9) (4,05.) (4,986.) (4,809.) (9,885. * ) Max. negatve mment 6, ,0. 6,68.4 6, ,9.4 * f the rght pyln (knm) (9,87.) (6,998.7) (6,76.) (6,86.5) (67,9.3 * ) Hrzntal dsplacement..3 * at the tp f the twer (mm) (0.7) (9.6) (4.0) (.) (.3 * ) Vertcal dsplacement * at the tp f the twer (mm) (.3) (.4) (0.0) (0.0) (0.0 * ) Axal shrtenng * f the whle man grder (mm) (7.6) (8.5) (0.6) (0.0) (0.0 * ) Max. vertcal dsplacement * f the man grder (mm) (85.8) (5.) (3.) (0.5) (0.6 * ) Page 90

19 Internatnal Jurnal f Engneerng Research & Scence (IJOER) [Vl, Issue9, December 05] Cnsequently bth G.CUD and ULM4 clearly prvde ne ptmzed ntal cnfguratn cnfrmng well t the target gemetry n case f the balanced brdge mdel under dead lads as bserved n Fg. 6 and 7 and able 3. In addtn, nte that the abslute maxmum mment f the pyln frm able 4 s 6.8 MNm whch crrespnds t the small flexural stress f abut 0.36 MPa (=6.88 3/4.7) and als, the maxmum bendng mment f the cntnuus grder wll decrease rapdly as the dstance between anchr pnts f stay cables becmes small. Partcularly t shuld be realzed that bth G.CUD and ULM4 lead t cmpletely dentcal ntal slutns rrespectve f balanced r unbalanced cndtns as bserved n the furth and the ffth clumn f able 4 and 5. y v = x 0 v 0 = ( x 400) v = x x x x FIGURE 5 VERICAL CAMBER OF INCHEON BRIDGE MODEL U L M U L M 3 U L M 4 Grder mment (knm) D stance (m ) FIG 6 (A) BENDING MOMEN DIAGRAMS BY ULM, ULM3, AND ULM U L M U L M Grder mment (knm) D s ta n c e ( m ) FIG 6 (B) BENDING MOMEN DIAGRAMS BY ULM AND ULM4 FIGURE 6 BENDING MOMEN DIAGRAMS IN HE LEF HALFMODEL OFHE MAIN GIRDER IN INCHEON BRIDGE (KNM) hrd, t s nted frm able t 5 and Fg. 6 and 7 that unstraned cable lengths and maxmum bendng mments by AM3 and ULM3 are n extremely gd agreement wth thse by G.CUD and ULM4 whle the results by AMand ULM dsplays large dfference. Furthermre, t shuld be emphaszed that the ntal slutn by AM3 shws lttle dfference wth Page 9

20 Internatnal Jurnal f Engneerng Research & Scence (IJOER) [Vl, Issue9, December 05] that by ULM3. hs means that ne practcally ptmzed ntal state f lngspan cablestayed brdges ncludng unstraned element lengths can be easly determned by adptng the analytcal methd 3 (AM3) wthut recurse t relatvely cmplcated G.CUD. Furth, able t 4 and Fg. 6 and 7 reveal that verall unstraned lengths and bendng mment dstrbutns by AM (ULM) shw gd agreement wth thse by the AM3 (ULM3) and the G.CUD (ULM4) but that the ntal state slutns (the ffth and sxth clumns f able 3) by ULM dsplay sme devatns wth the target cnfguratn. It shuld be pnted ut that each staycable f Inchen brdge mdel s apprxmately mdeled as a parablc cable n AM whch results n lcally unbalanced mment dstrbutns n pylns. Nnetheless, nte that maxmum bendng mments by ULM are very small smlarly t thse by ULM4. herefre, t s judged that AM and ULM can be satsfactrly appled t ntal shapng analyss f cablestayed brdges havng mderate span lengths. ABLE 5 MAXIMUM BENDING MOMENS FOR HE BALANCED AND HE UNBALANCED BRIDGE MODELS WIHOU HE CAMBER AM AM3 G.CUD G.CUD * (ULM) (ULM3) (ULM4) (ULM4 * ) Max. pstve mment 7,85.8 7, ,853. 7,853.3 * fthe man grder (knm) (8,6.76) (7,07.03) (7,85.66) (7,853. * ) Max. negatve mment 0,47. 0,46.8 0,46.9 0,46.9 * f the man grder (knm) (7,849.7) (0,893.) (0,47.6) (0,46.8 * ) Max. pstve mment,498.5,498., ,57.0 * f the rght pyln (knm) (,47.) (,89.98) (,508.6) (5,57. * ) Max. negatve mment,85.6,84.4, ,.7 * f the rght pyln (knm) (7,434.5) (3,0.3) (,58.7) (53,.6 * ) 0000 ULM ULM ULM 3 ULM Cable ensn (kn) Cable N. FIGURE 8 SAY CABLE ENSIONS IN HE INCHE ON BRIDGE MODEL Page 9

21 Internatnal Jurnal f Engneerng Research & Scence (IJOER) [Vl, Issue9, December 05] Fnally the ntal state slutn f suspensn brdges s usually nsenstve t weghtbalancng between center span and sde spans because the suspensn system cnsstng f the man cable and hangers can effectvely absrbs large bendng mments generated frm the cmbned actn f the fabrcatn camber and the hrzntal cmpressn cmpnent by the man cable even thugh t s a selfanchred suspensn brdge. Hwever, ntal shapng analyss f unbalanced cablestayed brdges by G.CUD can lead t large bendng mments f pylns because cable tensns are drectly transferred frm the man grder t the pylns. Actually t s bserved that bendng mments n the man grder can be always mnmzed by applyng G.CUD but the maxmum bendng mment f the pyln fr the unbalanced brdge mdel by G.CUD becmes abut 4 tmes larger than that fr the balanced brdge (see the ffth and sxth clumns f able 4). hs means that the weght balancng between the center span and sde spans shuld be defntely well preserved n the prelmnary desgn stage. VII. SUMMARY AND CONCLUSIONS w unstranedlength calculatn prcedures fr determnng ne ptmzed ntal state slutn f cablestayed brdges, the analytcal methd and the G.CUD methd, have been presented, n whch the frmer methd s based n the cntnuus beam analyss and the nnlnear algebrac equatns but the latter methd adpts the FE Newtn teratn methd usng the elastc catenary cable element and the cnsstent frame element based n the crtatnal frmulatn. Mrever, the unstraned length methd strngly dependng n the unstranedlength calculatn schemes are presented t effectvely perfrm nnlnear FE analyss f cablestayed brdges subjected t varus lad cmbnatns. Fnally ntal shapng analyss f a cablestayed brdge havng ne ntermedate per s perfrmed and numercal results are analyzed. he mprtant cncludng remarks can be made as fllws:. he ntal state slutns by AM lk reasnable at frst glance but ULM based n unstraned lengths by AM leads t explsvely large bendng mments n bth the man grder and the pyln fr the brdge mdel havng the ntal camber.. he G.CUD ntrduces the crrespndng addtnal bundary cnstrants nstead f addng all the unstraned element lengths t the ndal unknwn whle the ULM adpts Newtn teratn methd wth keepng the predetermned unstraned lengths cnstant. And G.CUD prvdes the ptmzed ntal slutn cnvergng nearly t the target cnfguratn n case f balanced cablestayed brdges under dead lads. 3. Interestngly, even thugh any addtnal cnstrants n the ULM methd are nt enfrced except fr the essental bundary cndtn, the ntal state slutns by ULM3 and ULM4 are nearly dentcal t thse by G.CUD rrespectve f the weghtbalanced cndtn and the fabrcatn camber. 4. Intal state slutns by AM3 and ULM3 are n excellently gd agreement wth thse by G.CUD and ULM3 whle the results by AMand ULM dsplay large dfference. Furthermre, the ntal slutn by AM3 shws lttle dfference wth that by ULM3, whch means that ne ptmzed ntal state f balanced cablestayed brdges can be easly fund by adptng AM3 wthut recurse t relatvely cmplcated G.CUD. 5. Practcally AM and ULM can be appled t the ntal shapng and the cnstructn stage analyss f cablestayed brdges havng mderate span lengths. 6. Bendng mments n the man grder can be always lcalzed by applyng the G.CUD methd but the maxmum mments n pylns n case f the unbalanced cablestayed brdge can be extremely huge than thse n the balanced brdge whch means that the weght balancng between the center span and sde spans shuld be carefully taken nt accunt n the prelmnary desgn. 7. Fnally, t s judged that ULM3 and ULM4 based n the unstranedlengths by AM3 and G.CUD, respectvely, can be the mst pwerful tl fr nt nly the ntal shapng analyss but als the subsequent cnstructn stage analyss and structural nnlnear analyss under varus lad cmbnatns. Page 93

22 Internatnal Jurnal f Engneerng Research & Scence (IJOER) [Vl, Issue9, December 05] REFERENCES [] Wang PH, seng C, Yang CG. Intal shape f cablestayed brdges, Cmput Struct 993;46(6): pp [] Chen DW, Au FK, ham LG, Lee PKK, Determnatn f ntal cable frces n prestressed cncrete cablestayed brdges fr gven desgn deck prfles usng the frce equlbrum methd, Cmput Struct 000;74:pp. 9. [3] akag R, Nakamura, Nakagawa K. A new desgn technque fr prestressed lads f a cablestayed brdge, Cmput Struct995;58:pp [4] Negra JHP, Smes LMC. Optmzatn f cablestayed brdges wth three dmensnal mdelng, Cmput Struct997;64:pp [5] M. M. Hasan, A. O. Nassef, A. A. El Damatty (0), Determnatn f ptmum psttensnng cable frces f cablestayed brdges, Eng StructVl.44, pp [6] Wang PH, Ln H, ang Y. Study n nnlnear analyss f a hghly redundant cablestayed brdge, Cmput Struct 00;80:pp [7] Km KS, Lee HS. Analyss f target cnfguratns under dead lads fr cablesupprted brdges, Cmput Struct 00;79:pp [8] Km HK, Km MY. Effcent cmbnatn f acudmethd and an ntal frce methd fr determnng ntal shapes f cablesupprted brdges, Int J f Steel Structures 0;():pp [9] Km MY, Km DY, Jung MR, Attard MM. Imprved methds fr determnng the 3 dmensnal ntal shapes f cablesupprted brdges, Int J f Steel Structures 04;4():pp [0] Jung MR, Mn DJ, Km MY. Nnlnear analyss methds based n the unstraned element length fr determnng ntal shapng f suspensn brdges under dead lads, Cmput Struct 03;8:pp [] Jayaraman, HB., Knudsn, WC. A curved element fr the analyss f cable structures, Cmput. Struct. 98;4(34):pp [] Irvne, HM. Cable structures, 98. [3] PacsteC, Erkssn A. Beam elements n nstablty prblems, Cmput Methds Appl Mech Engrg, 997; 44:pp [4] Crsfeld,MA. Nnlnear fnte element analyss f slds and structures, vl. Wley, Chschester, 997; pp [5] LeN, Battn JM. Effcent frmulatn fr dynamcs f crtatnal D beams, Cmput Mech 0;48:pp [6] LzanGalant JA, Dng XU, PayaZafrteza I, urm J. Drect smulatn f the tensnng prcess f cablestayed brdges, Cmput Struct 03;:pp APPENDIX he fllwngs are detaled frms f elastc stffness, stffness due t member defrmatns, gemetrc stffness due t member frces and the unstraned lengthrelated stffness cnsstng f the extended tangental stffness matrx f a frame element: * k fu EA D = L D D D D 3 30 D + D3 30 D 4D 3 30 Page 94

23 Internatnal Jurnal f Engneerng Research & Scence (IJOER) [Vl, Issue9, December 05] * k e AL / I AL / I 6L 6L EI 4L 6L L = 3 L AL / I 6L symm. 4L * k d 3( D + D3 ) 3( D + D3 ) 4D + D3 D 4D3 L L 3( D + D3 ) (4 D D3 )( D + D3 ) 3( D + D3 ) 3( D + D3 ) ( D + 4 D3 )( D + D3 ) 0L 0 L 0 L 0 (4 D D3 ) L (4 D D3 )( D + D3 ) L (4D 7D D3 + 4 D3 ) 4D D3 EA = 30 3( D + D3 ) D + 4D 3 L 3( D + D3 ) ( D + 4 D3 )( D + D3 ) sym. 0L 0 ( D 4 D3 ) L 30 * k g 6 / 5 L /0 6 / 5 L /0 L L L P + P 3 L L sym. 6 / 5 L /0 sym. L /5 P /5 /0 / 30 = + Page 95