STATIČKA ANALIZA KABLOVA STATIC CABLE ANALYSIS

Transcription

1 STATIČKA ANALIZA KABLOVA STATIC CABLE ANALYSIS Špro GOPČEVIĆ Stanko BRČIĆ Ljljana ŽUGIĆ ORIGINALNI NAUČNI RАD UDK: = 86 UVOD Kablov, kao konstruktvn element, upotrebljavaju se u mnogm oblastma nženjerstva predstavljaju vtaln noseć deo ranh konstrukcja kao što su: mostov velkh raspona sa kablovma - vseć mostov mostov sa kosm kablovma, krovnh konstrukcja sa kablovskm mrežama, komunkacjskh tornjeva sa kosm ategama, konstrukcja a eksploatacju nafte u morma sa kablovma a sdrenje, vodov a prenos elektrčne energje td. Postoje generalno dva prlaa u ravoju konačnh elemenata a numerčko modelranje kablova. Prv prla je upotreba polnoma u opsu oblka kablova polja pomeranja. Drug prla je upotreba analtčkh raa a lančancu, koj u matematčkom smslu tačno opsuju kabl pod ralčtm uslovma opterećenja, a u skladu sa odgovarajućom teorjom lančance. U prvom prlau, usvajaju se nterpolacone funkcje koje treba da opšu nelnearno ponašanje kabla ravjen su konačn element: prost štap sa dva čvora [,,9,], prost štap sa unutrašnjm čvorovma [,,], gredn element sa dva čvora sa unutrašnjm čvorovma [,8,]. Prost štap sa dva čvora je element koj se najčešće upotrebljava kod modelovanja kablova. Ovaj elemenat je pogodan a modelovanje kablova koj su ategnut sa vsokom slom ateanja. Da b se u obr ueo ugb kabla, moduo elastčnost kabla se amenjuje ekvvalentnm (Ernstovm) modulom elastčnost, [6]. Na INTRODUCTION Cables, as the structural elements, are beng used n many felds of engneerng and present the vtal structural elements of varous cable-supported structures, such as: the long-span brdges, lke suspenson and cable-stayed brdges, roof structures wth cable nets, communcaton towers wth cable stays (masts), floatng off-shore ol platforms wth anchorng cables, transmsson power lnes etc. Generally, there are two approaches n development of the fnte elements related to numercal modelng of cables. The frst approach s the use of polynomals to descrbe the shape of cable and the dsplacement feld. The second approach s the use of analytcal catenary relatons, whch n the mathematcal sense eactly descrbe the cable under varous loadng condtons, n accordance wth the correspondng catenary theory. In the frst approach, the nterpolaton functons whch should descrbe the non-lnear cable behavor are assumed and the fnte elements are developed: truss beam wth two nodes [,,9,], truss beam wth nternal nodes [,,], beam wth two nodes and wth nternal nodes [,8,]. The truss element wth two nodes s the element that s mostly used n modelng of cables. Ths element s convenent for cables that are eposed to hgh tenson forces. In order to take nto account the sag of the cable, the real modulus of elastcty of the cable s Dr Špro Gopčevć, dpl.nž.građ. AD Želence Srbje, Nemanjna 6, Beograd, Srbja; e-mal: spro.gopcevc@srbral.rs Prof. dr Stanko Brčć, dpl.nž.građ.unvertet u Beogradu, Građevnsk fakultet, Bulevar kralja Aleksandra 7, Beograd, Srbja; e-mal: stanko@grf.bg.ac.rs Doc. dr. Ljljana Žugć, dpl.nž.građ. Unvertet Crne Gore, Građevnsk fakultet, Cetnjsk put bb, 8 Podgorca, Crna Gora; e-mal: ljlja@ac.me Dr Špro Gopčevć, Cv.Eng., Želence Srbje AD, Nemanjna 6, Belgrade, Serba; e-mal: spro.gopcevc@srbral.rs Prof. dr Stanko Brčć, Cv.Eng. Unversty of Belgrade, Faculty of Cvl Eng., Bulevar kralja Aleksandra 7, Belgrade, Serba; e-mal: stanko@grf.bg.ac.rs Ass. Prof. dr Ljljana Žugć, Cv.Eng., Unversty of Montenegro, Faculty of Cvl Eng., cetnjsk put bb, 8 Podgorca, Montenegro; e-mal: ljlja@ac.me GRAĐEVINSKI MATERIJALI I KONSTRUKCIJE 54 () (9-4) BUILDING MATERIALS AND STRUCTURES 54 () (9-4) 9

2 taj načn aksjalna krutost pravolnjskog elementa postaje ekvvalentna aksjalnoj krutost krvolnjskog elementa. Prost štap sa unutrašnjm čvorovma uključuje efekte geometrje kabla preko uvođenja nternh čvorova. Kod ovh elemenata se kao nterpolacone funkcje upotrebljavaju polnom všeg reda. Da b se begao dskontnutet u nagbu među dva prosta štapa u čvoru gde ne deluje koncentrsnao opterećenje, jer se sstem tada ponaša kao mehanam može da prourokuje numerčke probleme veane a konvergencju, upotrebljavaju se gredn element koj uspostavljaju kontnutet u nagbu na spoju dva elementa jer sadrže rotacone stepene slobode pomeranja u krajnjm čvorovma. Element asnovan na polnomma pogodn su a modelovanje kablova sa malom strelom, odn. a tv. pltke lančance. Za kablove sa velkom strelom, odn. a duboke lančance, kabl je potrebno delt na velk broj elemenata. U drugom prlau, analtčk ra a lančancu se upotrebljavaju da realno opšu ponašanje kabla [5,7,8,9,,,4,5]. Glavna prednost ovh elemenata je što kod statčke anale jedan kabl može da se predstav samo jednm elementom ovoga tpa da se pr tome dobju reultat vsoke tačnost. U slučaju dnamčke anale, svak kabl treba da se modeluje sa vše ovh elemenata. Konačn element, prkaan u ovome radu, je veden na osnovu drugog prlaa to na osnovu analtčkh raa a elastčnu hperbolčku lančancu. Tangentna matrca krutost vektor nternh čvornh sla elementa su veden tačnh analtčkh raa a elastčnu lančancu. Sopstvena težna kabla se ramatra drektno be aproksmacja. Efekt predhodnog napreanja kabla takođe su uključen u formulacju elementa. Sa ovm konačnm elementom može da se opše ponašanje, kako pltkog, tako dubokog kabla. Opsan konačn element metod rešavanja nkrementalnh jednačna ravnoteže su ugrađen u ravjen računarsk program ELAN [4], koj omogućava lnearnu nelnearnu analu konstrukcja sa kablovma, usled dejstva statčkog dnamčkog opterećenja. Valdnost formulacje konačnog elementa a kablove je proverena kro test prmere u kojma se ramatra statčk odgovor kabla: pomeranja horontalna sla u kablu usled statčkog opterećenja: jednakopodeljenm opterećenjem koncentrsanom slom. Vrednost dobjene programom su upoređvane sa teorjskm vrednostma vrednostma lterature. HIPERBOLIČKA TEORIJA DUBOKE LANČANICE - ANALITIČKA REŠENJA. Neelastčna lančanca Lančanca se defnše kao dealno savtljva nerastegljva materjalna lnja koja ne pruža nkakav otpor promen svog oblka. Ovo je osnovna defncja lančance, koja je u skladu sa pretpostavkom o krutom telu. Međutm, kao što su sva tela vše l manje deformablna, tako je lančanca do određene mere restegljva, al se pak prvo posmatra nerastegljva, odn. neelastčna lančanca. Za dalja ramatranja usvaja se dspocja lančance kao na slc. Neka je vertkalna ravan u kojoj se nala lančanca onačena sa, pr čemu je osa vertkalna, sa smerom na gore. Usvaja se da je lančanca na svojm krajevma veana a replaced by the equvalent (Ernst's) modulus of elastcty, [6]. In such a way the aal stffness of straght element becomes equvalent to the aal stffness of curved (saged) element. The truss element wth nternal nodes ncludes the effects of cable geometry through nternal nodes. As the nterpolaton functons polynomals of the hgher order are beng used. In order to avod the slope dscontnuty between two truss elements n the node wthout the concentrc loadng, snce n that case the system s behavng as the mechansm and could produce numercal problems related to convergence, the beam elements are used, snce they also have rotatonal degrees of freedom and enable the slope contnuty between elements. Fnte elements based upon polynomals are sutable for modelng of cables wth small sag, the so-called shallow catenares. For cables wth large sag,.e. for deep catenares, the cable should be dvded nto the large number of elements. In the second approach, analytcal catenary epressons are beng used n order to descrbe the realstc cable behavor [5,7,8,9,,,4,5]. The man advantage of such elements s that n the statc cable analyss a sngle fnte element could be used for the cable and also to obtan the results of hgh accuracy. In dynamc cable analyss, each cable should be modelled wth more such fnte elements. The cable fnte element descrbed n ths paper s based upon the second approach, usng analytcal epressons related to elastc hyperbolc catenary. The tangent stffness matr and the vector of nternal forces of the element are derved from the eact analytcal epressons for elastc catenary. The self weght of the cable s drectly consdered, wthout appromatons. The effects of pretenson of the cable are also ncluded nto formulaton of the element. Wth ths fnte element t s possble to descrbe behavor of both the shallow and the deep cable. Presented fnte element and the method of soluton of the ncremental equatons of equlbrum are mplemented nto the computer code ELAN [4], whch enables the lnear and non-lnear analyss of cable supported structures, due to statc and also dynamc loadng. Verfcaton of the fnte element formulaton s checked through test eamples of statc cable analyss, determnaton of dsplacements and horontal cable force due to unformly dstrbuted and concentrated loadng. The values obtaned by the program are compared wth the theoretcal results and the results presented n the lterature. HYPERBOLIC THEORY OF DEEP CATENARY ANALYTICAL SOLUTIONS. Nonelastc catenary The catenary s defned as an deally fleble netensble materal lne whch does not resst to any change of ts shape. Ths s the basc defnton of the catenary, whch corresponds to the noton of the rgd body. However, snce all the bodes are more or less deformable, so the catenary s also, up to the pont, etensble, but stll, an unetensble,.e. nonelastc catenary s consdered frst. In further dscusson the catenary layout as assumed as shown n Fg.. The vertcal plane n whch the catenary belongs to s GRAĐEVINSKI MATERIJALI I KONSTRUKCIJE 54 () (9-4) BUILDING MATERIALS AND STRUCTURES 54 () (9-4)

3 nepokretne oslonce. Početak O koordnatnog sstema O usvojen je u osloncu na levom kraju lančance. Pr tome je horontalan ramak među oslonaca, dakle raspon lančance, onačen sa l, dok je vertkalna denvelacja oslonaca onačena sa l. denoted as -, wth as beng vertcal and the postve sense upwards. The ends of the catenary are connected to the fed supports and. The orgn O of the system O s assumed at the left end of the catenary. The horontal dstance betwen the supports,.e. the span of the catenary, s denoted as l, whle the vertcal denvelaton of supports s denoted by l. Slka Lančanca opterećena sopstvenom težnom u koordnatnom sstemu O Fgure Gravtatonaly loaded catenary n the coordnate system O Posmatrana lančanca opterećena je raspodeljenm opterećenjem sopstvenom težnom q(s ) duž luka s nerastegljve lančance, koj se mer od levog kraja lančance. Predpostavlja se da je operećenje konstantno: q(s ) = q = const. Takođe se predpostavlja se da je lančanca dealno eksblna (EI ), nerastegljva (AE ), da nema toronu krutost da može da prm samo slu ateanja T. Sla ateanja ma pravac tangente u svakoj tačk luka lančance. Dferencjalna jednačna ravnoteže dvojenog elementa luka lančance glas Consdered catenary s loaded by the dstrbuted loadng due to the self weght q(s ) along the arc s of netensble catenary, whch s measured from the left end of the catenary. It s assumed that the loadng s constant: q(s ) = q = const. It s also assumed that the catenary s deally fleble (EI ), netensble (AE ), that lacks the torsonal stffness and can be eposed only to the tenson force T. The tenson force has the drecton of the tangent n any pont along the arc of the catenary. Dfferental equaton of equlbrum of solated arc element of the catenary s gven by gde je H horontalna komponenta sle ateanja u lančanc koja ma konstantnu vrednost duž cele lančance, jer je lančanca opterećena raspodeljenm opterećenjem stalnog (u ovom slučaju vertkalnog) pravca. Sa (..) je onačeno dferencranje po koordnat. Ako se leva desna strana jednačne () podele sa d, ds = ( + ) preured unese da je d, jednačna () se transformše u oblk Ako se uvedu onake λ = ql H d H = q( s ) ds where H s the horontal component of the tenson force n the catenary, whch has the constant value along entre catenary, snce the catenary s loaded by dstrbuted loadng wth constant (n ths case vertcal) drecton. By (..) the dfferentaton wth respect to coordnate s denoted. If both sdes of Eq. () are dvded by d, after some transformaton and consderng ds = ( + ) that d, Eq. () may be transformed nto ( ) H = q + () Introducng notatons λl Φ= arcsn h λ =Θ λ l snh λ () () GRAĐEVINSKI MATERIJALI I KONSTRUKCIJE 54 () (9-4) BUILDING MATERIALS AND STRUCTURES 54 () (9-4)

4 konačno rešenje jednačne (), vodeć računa o grančnm uslovma () = (l ) = l, dato je sa q H the fnal soluton of Eq. (), havng n mnd the boundary condtons () = and (l ) = l, s gven as λ (4) l = snh +Φ = snh +Φ H q H = cosh cosh cosh cosh q +Φ Φ = +Φ Φ H q l Rešenje =() predstavlja jednačnu hperbole, pa se ato ovakav prstup nava hperbolčka teorja lančance. Ukupna dužna luka nerastegljve hperbolčke lančance L, vodeć računa o dobjenom rešenju (4) odgovarajućm rama a transformacju hperbolčkh funkcja, dobja se da je jednaka λ (5) The soluton =() represents the equaton of the hyperbola, so ths approach s called the hyperbolc theory of the catenary. The total arc length of netensble catenary L, keepng n mnd obtaned soluton (4) and the correspondng relatons for transformatons of hyperbolc functons, s obtaned as ds q H L = ds s = d= + d = +Φ = +Φ d H q l l ' l cosh snh λ cosh( λ ) (6) Sle duž luka lančance, a hperbolčko rešenje, mogu se rat na sledeć načn The forces along the arc of catenary, for hyperbolc soluton, may be epressed as H( ) H V( ) H ( ) H ( ) H snh = = = λ +Φ l T( ) H + V ( ) + ( ) cosh λ +Φ l gde je V vertkalna komponenta sle ateanja u lančanc. Vertkalne komponente reakcja u osloncma, onačene sa F F 6 ražene preko ukupne dužne luka lančance L, a dobjene transformacjom raa (7-), posle sređvanja se prkauju u oblku q V() = F = HsnhΦ= l coth L where V s the vertcal component of the catenary tenson force. The vertcal components of reacton forces n supports, denoted as F and F 6 and epressed by the total arc length of the catenary L, obtaned by transformaton of epresson (7-), after some transformaton are presented as ( λ ) q Vl ( ) = F = Hsnh +Φ = l coth + L Sle ateanja T T, na krajevma lančance, ražene preko dužne luka lančance L, a dobjene transformacjom raa (7-), posle sređvanja glase ( λ ) ( λ ) 6 q T() = T = Hcosh( Φ ) = ( Lcoth λ l ) q Tl ( ) = T = Hcosh λ+φ = Lcoth λ + l Korsteć rae (8)-(), mogu da se vedu sledeće jednakost The tenson forces T and T, at catenary ends, epressed by the arc length of the catenary L, obtaned by transformaton of epresson (7-), mght be gven as ( ) ( ) F+ F6 = ql F6 F = ql coth λ T+ T = qlcoth λ T T = ql (7) (8) (9) () () Usng epressons (8)-(), the followng relatons may be derved () () I jednačne (-) dobja se From Eq. (-) one obtans GRAĐEVINSKI MATERIJALI I KONSTRUKCIJE 54 () (9-4) BUILDING MATERIALS AND STRUCTURES 54 () (9-4)

5 gde je ln prrodn logartam sa osnovom e =.7... Ijednačavajuć (-) (4) dobja se Kombnujuć rae (-) (-), elmnacjom q, dobja se se T + T T + T + ql ql T + T ql λ = coth = ln (4) l H T + T + ql = ln q T + T ql l T = F where ln s the natural logarthm wth bass e=.7... Equatng epressons (-) and (4) one obtans T L + F 6 (5) Usng epressons (-) and (-), after elmnatng q, one obtans I raa (5), ako se unese da je = l, dobja se From epresson (5), when ntroducng that = l, one obtans H q ( ) = l = ( ( +Φ ) Φ ) = ( +Φ ) l Kada se ra (6) (7) kvadrraju odumu, dobja H cosh λ cosh snh λ snh λ (7) q 4H snh L l l q λ a posle preuređenja, dobja se da je kvadrat dužne luka nerastegljve lančance jednak = snh λ = = + L l l Kao što se vd, oblk luka lančance u ravnotežnoj konfguracj avs od horontalne komponente sle u lančanc H, koja je, u prncpu, neponata velčna, tako da je anala lančance složen statčk neodređen problem. Ist aključak sled same defncje lančance kao dealno savtljve materjalne lnje (vše l manje nerastegljve), kod koje je oblk u ravnotežnoj konfguracj unapred neponat avs od aplcranog opterećenja. Jedan od načna rešavanja problema određvanja neponate horontalne komponente sle u lančanc je da se unapred ponate početne dužne luka lančance, date sa (9), teratvno odred sla H, majuć u vdu ra (-) a parametar λ u kome fgurše sla H.. Elastčna lančanca snh λ (6) If the epressons (6) and (7) are squared and added, then λ (8) and, after some transformaton, one obtans that the square of the arc length of netensble catenary s obtaned as λ (9) As may be seen, the shape of the arc of the catenary n ts equlbrum poston depends on the horontal component H of the catenary tenson, whch s, n prncple, the unknown quantty, so the catenary analyss s the comple statcally ndetermned problem. The same concluson follows also from the very defnton of the catenary as deally fleble materal lne (more or less netensble), whose shape n the equlbrum poston s unknown n advance and depends on the appled loadng. One of the ways to deal wth the problem to determne the unkonwn horontal component of the catenary tenson force s that from the ntally known total arc length of the catenary, gven by (9), the force H s teratvely determned, havng n mnd the epresson (-) for the parameter λ, whch also contans the unknown force H. Stvarna lančanca ma konačnu aksjalnu krutost. Usvaja se da se materjal od kojeg je napravljena lančanca ponaša lnearno elastčno, odn. da adovoljava Hooke-ov akon, što je sasvm prhvatljvo a većnu realnh kablova u praks. Jednačne ravnoteže dela kabla (od početka kabla, do neke provoljne dužne luka s ), u horontalnom vertkalnom pravcu su, vdet slku,. Elastc catenary The actual catenay has the fnte aal stffness. It s assumed that the materal propertes of the catenary are lneraly elastc,.e. to obey the Hooke s law, whch s qute acceptable for most real cables n engneerng practce. Equlbrum equatons of the part of the catenary (from the begnng of the cable up to some arbtrary arc length s ), n horontal and vertcal drectons, are, see Fg., GRAĐEVINSKI MATERIJALI I KONSTRUKCIJE 54 () (9-4) BUILDING MATERIALS AND STRUCTURES 54 () (9-4)

6 T d H ds = d T qs F ds = () Može da se vd da se bog konervacje mase ukupna težna dela lančance ne menja usled duženja kabla [4]. Takođe, geometrjsko ogrančenje It could be seen that, due to conservaton of mass, the total weght of the part of the catenary unchanged due to lengthenng of the cable [4]. Also, the geometrc restrant d d + = ds ds () mora da bude adovoljeno. Kako je dlatacja elementa luka lančance (prblžno) jednaka kao pošto materjal lančance adovoljava Hook-ov akon, dobja se U rau () sa A je onačena površna poprečnog preseka neopterećene lančance, σ je napon u poprečnom preseku E je Young-ov moduo elastčnost. Ako se jednačne () kvadrraju amene u jednačn (), dobja se sla ateanja u blo kojoj tačk s lančance, u avsnost od njene horontalne H vertkalne komponente F, kao Slka Segment elastčne lančance Fgure Segment of elastc catenary ε ds ds ds ds ds must be satsfed. Snce the dlataton of the arc element of the catenary s (appromately) equal to = = () ds T = Aσ = AEε = AE ds ( ) / and also snce the catenary materal s obeys the Hooke's law, one obtans () In epresson () A s the cross-sectonal area of unloaded catenary, σ stress n the cross-sectonal area and E s the Young's modulus of elastcty. If Equatons () are squared and substtuted nto Eq. (), one obtans the tenson force n any pont s of the catenary, n dependance of ts horontal H and vertcal component F, as T( s ) = H + ( qs F ) (4) Vodeć računa da je d ds d ds = ds ds d ds Havng n mnd that d ds ds ds = (5) korsteć jednačne () (), dobjaju se parametarske jednačne lančance u oblku and usng Eqs. () and (), one obtans the parametrc catenary equatons n the form 4 GRAĐEVINSKI MATERIJALI I KONSTRUKCIJE 54 () (9-4) BUILDING MATERIALS AND STRUCTURES 54 () (9-4)

7 Hs H qs F F s ( ) = + snh snh AE q H H Fs qs H qs F F s ( o) = AE F q H H / / (6) (7) Ako se u jednačnama (6) (7) stav da je s = L, vodeć računa o ve hperbolčkh logartamskh funkcja, kao o rama () (), dobjaju se pogodnj ra a raspon vertkalnu denvelacju oslonaca lančance u oblku L F + T 6 L ( ) = l = H + ln EA q T F U relacjama (8) su raspon vertkalna denvelacja oslonaca lančance ražen preko sla na krajevma, kao drugh podataka o lančanc (modul elastčnost E, površna poprečnog preseka A, gravtacono opterećenje q ukupna početna dužna luka lančance L ). Ove relacje su načajne u vođenju elemenata matrce fleksblnost, odn. matrce krutost, u numerčkom prstupu na ba metode konačnh elemenata. If one ntroduces s = L nto Eqs. (6) and (7), keepng n mnd relatons between hyperbolc and logarthmc functons, as well as epressons () and (), one obtans the more convenent epressons for the span and the vertcal denvelaton of supports n the form T T L ( ) = l = + ( T T ) (8) q AEq In Eqs. (8) the span and vertcal denvelaton of supports of the catenary are epressed through the forces at catenary ends, as well as through some other catenary data (modulus of elastcty E, cross-sectonal area A, gravtatonal loadng q and the total ntal arc length of the catenary L ). Ths relatons are mportant n dervng the elements of the fleblty matr n the numercal approach usng the fnte element method. PARABOLIČKA TEORIJA LANČANICE - ANALITIČKA REŠENJA. Neelastčna lančanca U osnov prkaane hperbolčke teorje lančance se usvaja da je gravtacono opterećenje q ravnomerno raspoređeno po luku lančance, q=const, slka, što predstavlja realnu stuacju, jer je kabl celom svojom dužnom stog preseka od stog materjala. U slučaju tv. pltkh lančanca, gde je odnos strele lančance prema rasponu relatvno mal, občno f/l <.5, (pod strelom f se podraumeva vertkalan ugb na sredn lančance, mereno od tetve luka), može da se usvoj da je gravtacono opterećenje konstantno, al raspoređeno po horontalnoj projekcj luka lančance: q =const. Dferencjalna jednačna ravnoteže dvojenog elementa luka lančance (), je jednostavnja, vdet, npr.[4], data je sa PARABOLIC CATENARY THEORY ANALYTICAL SOLUTIONS. Nonelastc catenary Wthn presented hyperbolc catenary theory t s assumed that the gravtatonal loadng q s unformly dstrbuted along the arc length of the catenary, q=const, Fg.. In the case of the so-called shallow catenary, where the rato of the sag and the span s relatvely small, usually f/l <.5 (the sag f of catenary s the vertcal deflecton measured n the mddle of the span from the chord to the arc), t s assumed that the gravtatonal loadng s constant, but when epressed as dstrbuted along the horontal projecton of the arc length (.e. span): q =const. In that case the dfferental equaton of equlbrum of solated arc element of the catenary () s more smple [4] Imajuć u vdu grančne uslove ()= (l )=l, dobjaju se rešenja dferencjalne jednačne (9) u oblku H = q (9) q l = ( l ) + H l q l = ( l ) + H l Havng n mnd the boundary condtons ()= and (l )=l, the soluton of dfferental equaton (9) s obtaned n the form () () Kao što se vd, rešenje =() je dobjeno u oblku kvadratne parabole, a ralku od hperbolčkh relacja dath sa (4) (5). Rešenje () predstavlja oblk luka As may be notced, the soluton =() s obtaned n the form of quadratc parabola, as opposed to hyperbolc relatons gven by (4) and (5). The soluton () GRAĐEVINSKI MATERIJALI I KONSTRUKCIJE 54 () (9-4) BUILDING MATERIALS AND STRUCTURES 54 () (9-4) 5

8 nerastegljve parabolčke, odn. pltke, lančance koj odgovara sopstvenoj težn lančance q koja je konstantno raspodeljena po horontalnoj projekcj luka lančance. Naravno, kao u slučaju duboke lančance hperbolčkh relacja, u parabolčkom rešenju fgurše neponata horontalna komponenta sle H u lančanc, koja se, kao kod duboke lančance, može da odred teratvnm putem ponate dužne luka lančance. Dužna luka lančance je data sa ntegralom (6), koj se, unoseć parabolčke relacje () (), nešto složenje rešava, a ralku od hperbolčkh relacja. Kao reultat, dobja se prhvatljvo prblžno rešenje a dužnu luka parabolčke (odn. pltke) lančance u oblku: represents the arc of the parabolc or shallow catenary, whch corresponds to the self-weght q whch s unformly dstrbuted along the horontal projecton of the arc of the catenary. Of course, the same as n the case of deep catenary and hyperbolc relatons, the parabolc soluton s epressed through unknown horontal component of the catenary tenson force. As n the case of deep catenary, the unknown force H may be determned n teratve fashon from the known arc length of the catenary. The total arc length of the cable s gven by the ntegral (6), whch, when ntroducng the parabolc relatons () and (), s somewhat more complcated than n the case of hyperbolc relatons. As the result, an acceptably good appromate soluton for the total arc length of the parabolc (or shallow) catenary s gven n the form ' l ' ds l l ql l L = ds = d d d l s = + + = + + () d 6 H l. Elastčna lančanca Za opterećenje usled sopstvene težne lančanca auma položaj opsan sa (). Ako se na lančancu aplcra neko dodatno gravtacono opterećenje, onda se lančanca pomera prvobtne ravnotežne konfguracje date sa () u neku novu ravnotežnu konfguracju (smatra se da se dodatno opterećenje aplcra dovoljno polako, tako da nercjalne sle mogu da se anemare). Sve tačke luka lančance usled dodatnog opterećenja dobjaju neka dodatna pomeranja u w u pravcma osa dlatacja ε elementa luka lančance može da se prkaže u oblku gde su uključene male velčne drugog reda. U slučaju anemarvanja malh velčna drugog reda, poslednj član u jedn. () može da se anemar. Ako se sa τ obelež nastal prraštaj sle ateanja kabla T, koj se javlja usled dodatnog opterećenja, onda je horontalna komponenta prraštaja sle ateanja lančance data sa jer je, a pltku lančancu, nagb tangente na luk prblžno jednak nagbu tetve luka. Imajuć u vdu pretpostavku da se materjal lačance ponaša u skladu sa Hooke-ovm akonom, onda je ε. Elastc catenary ds ds d du d dw dw ( ) ds ds ds ds ds ds For the loadng due to the self-weght the catenary takes the poston descrbed by Eq. (). If an addtonal gravtatonal loadng s appled, then the catenary s dsplaced from the ntal equlbroum confguraton gven by () nto another new equlbrum confguraton (t s consdered that the addtonal loadng s appled suffcently slowly, so the nertal forces may be neglected). All ponts of the arc of the catenary due to addtoal loadng obtan some addtonal dspalcements u and w n drecton of aes and and the dlataton ε of the arc element may be presented n the form = = + + () where the small quanttes of the second order are ncluded too. In the case when small quanttes of the second order are beng neglected, the last term n Eq. () may be neglected. If by τ one denotes the addtonal ncrement of the catenary tenson force T, whch s developng due to addtonal loadng, then the horontal component of ncrement of the catenary tenson force s gven by d h = τ = τ cosθ τ cos β (4) ds τ EA snce, for the shallow catenary, the slope of the tangent to the arc s appromately equal to the slope of the arc chord. Havng n mnd that t s assumed that the catenary materal s behavng accordng to the Hooke s law, and then t s = ε (5) pr čemu se ne posmatraju dlatacje usled moguće promene temperature. Unoseć relacje () (4) u (5), under the assumpton that dlatatons due to the temperature change are dsregarded. Introducng the 6 GRAĐEVINSKI MATERIJALI I KONSTRUKCIJE 54 () (9-4) BUILDING MATERIALS AND STRUCTURES 54 () (9-4)

9 ds d dobja se, posle množenja sa ( ) relatons () and (4) nto (5), after multplyng by ds d ( ), one obtans ds h( ) d du d dw dw = + + ( ) EA d d d d (6) Jedn. (6) se nava jednačna promene stanja pltke parabolčke lančance u dferencjalnom oblku, [4,6]. Ako se jednačna (6) ntegral po duž raspona lančance, dobja se jednačna promene stanja pltke parabolčke lančance u ntegralnom oblku gde je uvedana onaka a tv. vrtuelnu dužnu lančance L e Equaton (6) s the so-called catenary equaton of the shallow parabolc catenary n the dfferental form, [4,6]. If Eq. (6) s ntegrated by along the catenary span, one obtans the catenary equaton of the shallow parabolc catenary n the ntegral form hle = + + (7) EA l l ul ( ) u() wd ( w ) d where the notaton for the so-called vrtual catenary length L e s ntroduced 4 l ds f 96 f f Le = ( ) d l tg βtg β (8) d l 5 l l 4 Vrtuelna dužna L e je defnsana kao određen ntegral, a u slučaju parabolčnh relacja se dobja prkaana prblžna vrednost. Parcjalnom ntegracjom prvog ntegrala u jedn. (7), u umanje u obr grančnh uslova o nepokretnost oslonačkh tačaka lančance, jednačna promene stanja u ntegralnom oblku može da se prkaže kao The vrtual length L e s defned as the defnte ntegral, and n the case of the parabolc relatons, one obtans the appromate value gven by (8). By the partal ntegraton of the frst ntegral gven n Eq. (7), takng nto account the boundary condtons related to prevented dsplacements of the end ponts of the catenary (fed supports), the catenary equaton n the ntegral form may be presented as hle q = + ( ) (9) EA H l l wd w d Pr tome, ako se anemar utcaj malh velčna drugog reda u rau a dlatacju, u jednačn promene stanja (9) fgurše samo prv ntegral, dok se drug anemaruje, odn. u jednačn (6) se poslednj član na desnoj stran anemaruje. Tme se dobja jednačna promene stanja u okvru lnerane teorje pltke parabolčne lančance. Also, f one neglects the effect of the small quanttes of the second order n epresson for dlataton, n the catenary equaton (9) only the frst ntegral remans, whle the second one s neglected, or, n Eq. (6), the last term on the rght-hand sde s neglected. In such a way the obtaned catenary equaton corresponds to the lnear theory of the parabolc catenary... Utcaj dodatnog raspodeljenog gravtaconog opterećenja Na lančancu deluje gravtacono opterećenje (sopstvena težna) q ()=q =const, usled kojeg lančanca auma oblk dat sa relacjom (). Pr tome se podraumeva da je horontalna komponenta sle ateanja lančance H takođe ponata, odn. određena. Posmatra se slučaj kada je lančanca opterećena još sa dodatnm statčkm gravtaconm opterećenjem p()=p=const, koje je takođe ravnomerno raspoređeno po horontalnoj projekcj luka lančance, tako da se prmenjuje parabolčka teorja lančance. Usled ovog dodatnog gravtaconog opterećenja lančanca auma nov ravnotežn položaj koj je defnsan sa dodatnm pomeranjem u vertkalnom pravcu w(), koje se mer od položaja prvobtne ravnotežne konfguracje (). Ovom.. The effect of the addtonal gravtatonal loadng Gravtatonal loadng (self-weght) q ()=q =const s actng upon the catenary, due to whch the catenary takes the poston as defned by Eq. (). It s understood that the horontal component of the catenary tenson force H s also known,.e. prevously determned. The case s consdered when the catenary s loaded by the addtonal statc gravtatonal dstrbuted loadng p()=p=const, whch s also unformly dstrbuted along the horontal projecton of the catenary arc length, so the parabolc catenary theory s appled. Due to the addtonal gravtatonal loadng the catenary takes up the new equlbrum poston whch s defned by the addtonal dsplacement n the vertcal drecton w(), meassured from the ntal equlbrum poston (). As the consequence of the addtonal loadng p() there s GRAĐEVINSKI MATERIJALI I KONSTRUKCIJE 54 () (9-4) BUILDING MATERIALS AND STRUCTURES 54 () (9-4) 7

10 dodatnom opterećenju p() odgovara takođe promena sle ateanja lančance, tako da je horontalna komponenta sle ateanja u novoj ravnotežnoj konfguracj jednaka H =H+h, gde je sa h obeležena promena horontalne komponente ateanja lančance. Pr tome je h konstantno duž lančance jer je reultat delovanja dodatnog opterećennja p koje je konstantnog pravca. Dferencjalna jednačna ravnoteže dvojenog elementa luka lančance, posle aplcranog dodatnog opterećenja, je data, analogno sa jednačnom (9), kao also the correspondng change of the catenary tenson force, so the horontal component of the tenson force n the new equlbrum poston s equal to H =H+h, where h denotes the change of the horontal component of the catenary tenson. Also, h s constant along the catenary snce t s the result of acton of the addtonal loadng p, whch has the constant drecton. Dfferental equaton of equlbrum of the solated catenary arc element, after applcaton of the addtonal loadng, s gven, n analogy wth Eq. (9), as Imajuć u vdu df. jednačnu ravnoteže (9) lančance u prvobtnoj ravnotežnoj konfguracj, jednačna (4) dobja oblk ( H + h)( + w ) = ( q + p) (4) ( H hw ) p pr čemu je moguće da se još anemar provod hw, kao relatvno mala velčna. Kako je horontalna komponenta dodatnog ateanja lančance h neponata velčna, al konstantna, dferencjalna jednačna (4) može da se prkaže u oblku Rešenje dferencjalne jednačne (4), u grančne uslove w()=w(l ) =, dobja se kao parabola Konstanta A * data sa (4), koja ma dmenju dužna -, može da se prkaže u oblku gde je α onaka a relatvnu (bedmenonalnu) promenu horontalne komponente sle ateanja lančance h usled dodatnog opterećenja, normrano u odnosu na slu H A Havng n mnd the dfferental equaton (9) of the catenary n ts ntal confguraton, Eq. (4) obtans the form hq H + = (4) p hq = = = H + h H( H + h) w A* const where t s also possble to neglect the term hw, as the relatvely small quantty. Snce the horontal component of the addtonal catenary tenson h s unknown quantty, but whch s constant, the dfferental equaton (4) may be presented n the form (4) The soluton of the dfferental equaton (4), wth the boundary condtons w()=w(l ) =, s obtaned as the parabola w ( ) = A* ( l ) (4) * = = H h H( H h) H( α) q The constant A *, as gven by (4), wth the dmenson length -, may be presented n the form p hq q p α (44) where α denotes the relatve (non-dmensonal) change of the horontal component of the catenary tenson h due to addtonal loadng, normaled wth respect to the force H h α = (45) H U rešenju (4) konstanta A * je neponata velčna, jer je neponata dodatna sla h, odnosno, neponata je relatvna promena horontalne sle data sa koefcjentom α. Međutm, rešenje (4) može da se unese u jednačnu promene stanja (9), koja predstavlja jednačnu kompatblnost elastčne lančance pr dodatnom opterećenju u kojoj fgurše neponata dodatna sla h. Integral na desnoj stran jednačne promene stanja (9) se, posle unošenja rešenja (4), dobjaju kao In the soluton (4) the constant A * s the unknown quantty, snce the addtonal force h s unknown, or, rather, unknown quantty s the relatve change of the horontal tenson force gven by the coeffcent α. However, the soluton (4) may be nserted nto the catenary equaton (9), whch represents the compatblty equaton of elastc catenary, contanng the unknown addtonal force h. The ntegrals on the rghthand sde of the catenary equaton (9), after nsertng the soluton (4), are obtaned as 8 GRAĐEVINSKI MATERIJALI I KONSTRUKCIJE 54 () (9-4) BUILDING MATERIALS AND STRUCTURES 54 () (9-4)

11 pa se, posle daljeg transformsanja jednačne promene stanja, dobja kubna jednačna po bedmenonalnoj promen horontalne sle u lančanc (u normalovanom oblku po α) U jednačnu (47) je uneta onaka a bedmenonalan parameter lančance λ * dat sa Prema tome, dodatna horontalna komponenta sle ateanja u lančanc, u bedmenonalnom oblku α, određuje se rešavanjem kubne jednačne promene stanja (47), čj se koefcjent određuju preko bedmenonalnog parametra lančance λ *, datm sa (48), kao u avsnost od relatvnog dodatnog opterećenja p/q. Bedmenonalan parametar lančance λ * karakterše mehančko ponašanje lančance predstavlja osnovn parametar elastčne lančance. On uma u obr efekte početne geometrje lančance elastčnost lančance. Ako je parametar λ * relatvno velk broj, onda to predstavlja slučaj nerastegljve lančance, dok slučaj kada je λ * relatvno mal broj nač da se lančanca ponaša kao ategnuta žca. Ira a parametar λ * može da se prkaže u oblku u kome fgurše strela lančance Rešavanjem kubne jednačne (47) se dobja promena horontalne komponente ateanja lančance, odnosno konačna horontalna sla ateanja lančance usled dodatnog opterećenja a atm, prema rama (44) (4), dodatan ugb lančance usled dodatnog gravtaconog opterećenja p()=const. Vd se da je avsnost dodatne sle ateanja lančance usled dodatnog opterećenja, prkaana kubnom jednačnom (47), nelnearna, kao da ta nelnearnost btno avs od parametra λ *. Može da se utvrd da je nelnearnost vše ražena ukolko je parametar lančance λ * manj, a da je a veće vrednost λ *, npr. λ * > 5, nelnearnost sve manja, pa je grančna vrednost dodatnog ateanja lančance, sa porastom parametra λ *, data sa (46) l l wd = Al * ( w ) d = Al * λ* λ* λ * p p α + + α α = + 4 q q After further transformaton, the catenary equaton (9) s reduced to the cubc equaton n the unknown change of the horontal catenary tenson (n the normaled form α) (47) The non-dmensonal catenary parameter λ *, ntroduced nto Eq. (47), s gven by qlea λ * = (48) HL 8 f λ* = l e Therefore, the addtonal horontal catenary tenson force, n the non-dmensonal form α, s determned as the soluton of the cubc catenary equaton (47), whose coeffcents are determned through the non-dmansonal catenary parameter λ *, gven by (48), and also dependng upon the relatve addtonal loadng p/q. The non-dmensonal catenary parameter λ * characteres the mechancal behavor of the catenary. If the catenary parameter λ * s the relatvely large number, then t represents the case of netensble catenary, whle the case when λ * s relatvely small number means that the catenary behaves as the taut wre. The epresson for the catenary parameter λ * may be presented n the form where the catenary sag s eplctly ntroduced EA ql e (49) Upon soluton of the cubc equaton (47) one obtans the change of the horontal component of the catenary tenson, or the fnal horontal catenary tenson due to the addtonal loadng H = H + h = ( + α ) H (5) lm( α) λ * and then, accordng to the epressons (44) and (4), the addtonal catenary deflecton due to the addtonal gravtatonal loadng p. It mght be seen that the dependence of the addtonal catenary tenson to the addtonal loadng, as gven by Eq. (47), s nonlnear, and that nonlnearty strongly depends upon the catenary parameter λ *. It could be establshed that the nonlnearty s more emphased f the catenary parameter λ * s smaller and that for greater values of λ *, for eample λ * > 5, nonlnearty s smaller. It maght be establshed that the lmt value of the addtonal catenary tenson, wth the ncrease of the catenary parameter λ *, s gven as p q = (5) GRAĐEVINSKI MATERIJALI I KONSTRUKCIJE 54 () (9-4) BUILDING MATERIALS AND STRUCTURES 54 () (9-4) 9

12 (asmptotsko prblžavanje je sa donje strane odnosa p/q ). (asymptotc appromaton s from the lower sde of the rato p/q )... Utcaj dodatne koncentrsane gravtacone sle Za ralku od predhodnog slučaja raspodeljenog dodatnog opterećenja (deo..), posmatra se lančanca koja je, osm sopstvenom težnom, opterećena još sa dodatnom koncentrsanom gravtaconom slom nteteta P na rastojanju od levog kraja lančance, slka... The effect of addtonal concentrated gravtatonal force As opposed to the prevous case of dstrbuted addtonal loadng (secton..), the catenary s consdered whch s, besdes the self-weght, also loaded by the addtonal concentrated gravtatonal force of the ntensty P at the dstance from the left end of the catenary, Fg.. Slka Lančanca opterećena dodatnom koncentrsanom gravtaconom slom Fgure Catenary loaded by the addtonal concentrated gravtatonal force Ako su dodatna pomeranja lančance mala, profl lančance je dalje pltak, a ravnoteža sla u pravcu ose koje deluju na dvojen deo lančance provoljne (konačne) dužne na delu levo od sle P, odn. na delu < glas ql ( H + h)( + w ) = P + l l Desna strana jednačne predstavlja ra a transveralnu slu proste grede u preseku na delu < usled sopstvene težne koncentrsane sle, odn. br gravtaconh sla na tom delu lančance, od oslonca u =, pa do beskonačno blsko sa leve strane preseka. Leva strana jedn. (5) je vertkalna projekcja sle ateanja lančance u posmatranom provoljnom preseku na delu <, koja je prkaana preko konstantne horontalne komponente tangensa ugla nagba luka lančance prema horontal. Ravojem raa (5) elmnsanjem, odn. skraćvanjem članova koj avse od sopstvene težne, dobja se H + hw = P h l ( ) If the addtonal dsplacements of the catenary are small, the shape of the catenary s stll shallow, and the equlbrum of forces n drecton of actng upon the solated part of an arbtrary (fnte) length of the catenary on the left sde of the force P,.e. at <<, reads < (5) The rght-hand sde of Eq. (5) represents the epresson for the shear force of the smply supported beam n the cross-secton at the part < of the beam due to the self-weght and the concentrated force. Alternatvely, t represents the sum of all gravtatonal forces at that part of the catenary, from the support at =, up to the nfntely close to the left sde of the crosssecton. The left-hand sde of Eq. (5) s the vertcal projecton of the catenary tenson force at consdered arbtrary cross-secton at the part <, whch s presented by the constant horontal component and the tangent of the slope angle wth respect to the horontal as. By epandng the epresson (5) and upon elmnatng the members that depend upon the selfweght, one obtans < (5) Slčno se dobja da je ravnoteža sla u vertkalnom In the smlar way one obtans the equlbrum of GRAĐEVINSKI MATERIJALI I KONSTRUKCIJE 54 () (9-4) BUILDING MATERIALS AND STRUCTURES 54 () (9-4)

13 pravcu na dvojenom delu lančance provoljne (konačne) dužne od desnog kraja lančance pa do provoljnog preseka na delu desno od koncentrsane sle P, dakle na delu < l, data u oblku forces n the vertcal drecton actng upon the solated part of the catenary of the arbtrary (fnte) length from the rght end of the catenary up to an arbtrary secton on the rght sde of the concentrated force P,.e. for the part < l. The equlbrum of vertcal forces may be wrtten n the form ( H + hw ) = P h < l (54) l Integraljenjem jednačna (5) (54), u adovoljenje grančnh uslova w()=w(l ) =, dobjaju se relacje Upon ntegraton of Eqs. (5) and (54) and satsfyng the boundary condtons w()=w(l ) =, one obtans the followng relatons w Pl α = H( + α) P * l l l l Pl α w= H( + α) P * l l l l < < l (55) (56) gde je uveden ra a bedmenonalnu gravtaconu koncentrsanu slu where the epresson for the non-dmensonal gravtatonal concentrated force s ntroduced U slučaju kada se problem anale utcaja dodatne koncentrsane sle posmatra kao nelnearan, da b se odredla vrednost parametra α, korst se nelnearna jednačna promene stanja (9). U preseku, gde deluje koncentrsano opterećenje, pa postoj prekd prve vrste u djagramu transveralnh sla, u drugom ntegralu jednačne (9) postoj dskontnutet nagba, pa se a nalaženje tog ntegrala prmenjuje parcjalna ntegracja dva dela l P * P ql = (57) In case when the problem of analyss of the acton of an addtonal concentrated force s treated as non-lnear, n order to determne the value of the parameter α, the non-lnear catenary equaton (9) s used. At the secton, where the concentrated force s actng, so there s the dscontnuty of the frst knd n the shear force dagram, n the second ntegral of Eq. (9) there s the dscontnuty of slope, so, n order to calculate that ntegral, the partal ntegraton n two steps s preformed + '' '' ( w' ) d= ww ' + ( w ) wd+ ( w ) wd (58) l Posle amene raa (55), (56) (58) u rau (9) ntegraljenja, dobja se bedmenonalna kubna jednačna po relatvnoj promen horontalne sle u lančanc (u normalovanom oblku po α ) After ntroducng epressons (55), (56) and (58) nto Eq. (9) and upon ntegraton, one obtans the nondmensonal cubc equaton n the relatve change of the horontal catenary force (n the normaled form n α) λ λ λ + + P = + l l ( P) * * * α α α * * (59) Ako se žel lnearovana anala utcaja koncentrsane dodatne gravtacone sle, potrebno je da se anemare mal članov všeg reda (hw ) koj se pojavljuju u dferencjalnm jednačnama ravnoteže sla (5) (54) na dvojenm konačnm elementma luka lančance. Integraljenjem jednačna (5) (54), u anemarvanje malh članova všeg reda u If one wants to lneare the analyss of the acton of the addtonal concentrated gravtatonal force, t s necessary to neglect the small terms of the hgher order (hw ) whch appear n the dfferental equatons of equlbrum of forces (5) and (54), actng upon the solated fnte lengths of the catenary arc. Upon ntegratons of Eqs. (5) and (54), neglecton of small GRAĐEVINSKI MATERIJALI I KONSTRUKCIJE 54 () (9-4) BUILDING MATERIALS AND STRUCTURES 54 () (9-4)

14 adovoljenje grančnh uslova w()=w(l ) =, dobja se terms of the hgher order and upon satsfacton of the boundary condtons w()=w(l ) =, one obtans Pl α w = H P * l l l l Pl α w= H P * l l l l < < l (6) (6) Ako se ra (6) (6) unesu u jednačnu promene stanja u okvru lnearovane teorje parabolčne lančance, posle ntegraljenja sređvanja se dobja lnearna relacja po α If epressons (6) and (6) are ntroduced nto the catenary equaton correspondng to the lneared theory of the parabolc catenary, upon ntegraton and some transformaton, one obtans the lnear relaton n α α = 6P* l l + λ * (6) Prema tome, dodatna horontalna komponenta sle ateanja u lančanc, u bedmenonalnom oblku α, određuje se rešavanjem l kubne jednačne promene stanja (59) kod prmene nelnearne teorje parabolčne lančance, l lnearne jednačne (6) kod lnearovane teorje parabolčne lančance. I jedna druga jednačna su funkcje bedmenonalnog parametra lančance λ *, datog sa (48), bedmenonalnog koncentrsanog dodatnog opterećenja P* datog sa (57). Lnearovano rešenje će bt prhvatljvo tačno u odnosu na nelnearno samo a relatvno male vrednost P*. Ako je λ * velko, P* ne b trebalo da bude veće od - da b se doblo rešenje u grancama tačnost od %. Ako je λ * relatvno malo, dovoljene su veće vrednost a P* da b se doblo prhvatljvo rešenje u okvru lnearovanog prstupa, [6]. Therefore, the addtonal horontal component of the catenary tenson, n the non-dmensonal form α, s determned by soluton of the cubc catenary equaton (59), applyng the non-lnear theory of the parabolc catenary, or by soluton of the lnear equaton (6), usng the lneared theory of parabolc catenary. Both equatons are the functons of the non-dmensonal catenary parameter λ *, gven by (48), and the nondmensonal concentrated addtonal loadng P* gven by (57). The lneared soluton wll be acceptably eact wth respect to the non-lnear approach only for the relatvely small values of P*. If λ * s large, P* should not be greater then - n order to obtan the soluton wthn the % accuracy lmt. If λ * s relatvely small, one could have larger values of P* and stll obtan the acceptable soluton wthn the lneared approach, [6]. 4 KONAČNI ELEMENTI ZA ELASTIČNU HIPERBOLIČKU LANČANICU Konačn element, asnovan na analtčkm relacjama a elastčnu hperbolčku lančancu, mogu da se upotrebljavaju a analu kablova sa blo kojm odnosom strela-raspon. Ovakvm elementma, sa veoma vsokom tačnošću, mogu da se analraju labav ategnut kablov. U lteratur se mogu nać ovakva rešenja kako a lančancu u ravn [7,8], tako a prostornu lančancu [,]. Čvorne sle čvorna pomeranja prostorne lančance data su na slc 4. Lančanca je opterećena jednakopodeljenm gravtaconm opterećenjem pravcu vertkalne ose. Prvo se vode koefcjent matrce fleksblnost a lančancu u ravn O. Korste se ra (-), (8), (- ), (9) (8) u donekle mod kovanom oblku 4 THE FINITE ELEMENTS FOR ELASTIC HYPERBOLIC CATENARY The fnte elements, based upon the analytcal epressons for elastc hyperbolc catenary, mght be used for the cable analyss wth any sag-span rato. Usng these elements, wth very hgh accuracy, one could analye both the loose and taut cables. Such solutons mght be found n the lterature both for the planar catenary [7,8] and for the spatal catenary [,]. The nodal forces and the nodal dsplacements of the spatal catenary are gven n the Fg.. The catenary s loaded by the unformly dstrbuted gravtatonal loadng n drecton of the vertcal as. At frst, the coeffcents of the fleblty matr for the planar catenary n O plane are derved. Epressons (-), (8), (-), (9) and (8), n the modfed form, are used: snh λ L = l + l λ q cosh F λ = l + L snh λ ql λ = F F6 = ql F (6) (64) GRAĐEVINSKI MATERIJALI I KONSTRUKCIJE 54 () (9-4) BUILDING MATERIALS AND STRUCTURES 54 () (9-4)

15 L F + T ln 6 l = F + AE q T F AEq l = + ( T T ) T T q (65) Korste se sledeć ra F = F 4 T = F + F The followng epressons are also used T = F + F (66) 4 6 Ako se ra (64-) (66) amene u relacjama (65), vd se da ra (65) a horontaln raspon vertkalnu denvelacju mogu da se prkažu kao funkcje sla na levom kraju, tj. If epressons (64-) and (66) are substtuted nto relatons (65), t mght be seen that epressons (65) for the horontal span and the vertcal denvelaton could be epressed as functons of the forces at the left end,.e. l l F F = (, ) l l F F = (, ) (67) Slka 4 Čvorne sle čvorna pomeranja lančance Fgure 4 Nodal forces and nodal dsplacements of the catenary Dferencjal dužna l l dath smbolčno sa rama (67), mogu da se prkažu u oblku Dfferentals of the lengths l and l gven symbolcally wth epressons (67) could be presented n the form l l dl F F df f f df df dl = l l df = f f df = df F F f (68) U rau (68) f je matrca fleksblnost lančance u ravn, reda, čj koefcjent mogu da se vedu u oblku In epresson (68) f denotes the fleblty matr of the planar catenary, of order, whose coeffcents mght be derved as f F6 F F6 + T L = + ln q T T T F AE f F = f = q T T f L F6 F = + + AE q T T (69) Za lančancu u trodmenonalnom prostoru postoje tr komponente sle u čvorovma na krajevma lančance. Matrca fleksblnost trodmenonalne lančance može da se vede drektno matrce fleksblnost dvodmenonalne lančance kada joj se doda koefcjent For the catenary n the three-dmensonal space there are three force components at each end of the catenary. The fleblty matr of the spatal catenary could be derved drectly from the fleblty matr of twodmensonal catenary, when addng the fleblty GRAĐEVINSKI MATERIJALI I KONSTRUKCIJE 54 () (9-4) BUILDING MATERIALS AND STRUCTURES 54 () (9-4)

16 fleksblnost f a pomeranja van ravan lančance, vdet [7], coeffcent f correspondng to dsplacements n out-ofplane drecton of the catenary, see [7], f l L F ln 6+ T = = + F AE q T F (7) Matrca krutost k trodmenonalne lančance jednaka je nvernoj matrc fleksblnost f trodmenonalne lančance jednaka je The stffness matr k of the three-dmensonal catenary s equal to the nverse fleblty matr of the three-dmensonal catenary and s gven as f f k k k = f = f = k D f f k k k k = f k = k = f D = k = f f = D f f f (7) Tangentna matrca krutost K T lančance predstavlja veu među vektora nkrementalnh nternh čvornh sla f nt vektora nkrementalnh čvornh pomeranja q. Ova vea je data sa The tangent stffness matr K T of the catenary represents the connecton between the vector of the ncremental nternal nodal forces f nt and the vector of ncremental nodal dsplacements q. Ths relaton s gven by K T q= f nt [ F F F F F F ] T nt f = T [ u v w u v w ] q = K T k k = k k (7) 4. Procedura određvanja tangentne matrce krutost Da b se odredla tangentna matrca krutost K T moraju prvo da se odrede vrednost čvornh sla F F. Te sle su usvojene kao redundantne sle određuju se, u odnosu na datu pocju krajnjh čvorova lančance, upotrebom teratvne procedure. Osm osnovnh podataka o lančanc: q, E, A, kao položaja drugog čvora, odn. raspona vertkalne denvelacje l l, što je, načelno, uvek adato, može da bude ponata još l dužna nerastegljve lančance L, l horontalna sla F. U avsnost od toga da l je još ponato L l F, bra se odgovarajuća teratvna procedura. (A) U prvom slučaju, kada su a lančancu ponate vrednost q, E, A, položaj drugog čvora l l, kao nerastegljva dužna L, postupak određvanja tangentne matrce krutost vektora nternh čvornh sla je sledeć:. Vrednost promenjve λ određuje se raa (6-), kada se dužna L rastegljvog elementa amen sa dužnom L nerastegljvog elementa adrž prv član u ravoju funkcje snh λ/λ u red. To je poslednj slučaj u rau (7) a λ, dok prv slučaj u (7) predstavlja vertkalanu lančancu l =, a drug slučaj se odnos na veoma pltku lančancu, pa se usvaja da je λ mala vrednost. Ukupan ra a λ glas, vdet, npr. [9], 4. The procedure of determnaton of the tangent stffness matr In order to determne the tangent stffness matr K T the values of nodal forces F and F should be determned frst. These forces are assumed as the redundant forces and are determned, wth respect to the gven poston of the end nodes of the catenary, usng the teratve procedure. Besdes the basc catenary data: q, E, A, as well as the poston of the second node,.e. the span and the vertcal denvelaton l and l, whch s, n prncple, always gven, the known quantty mght be also ether the length of the netensble cable L, or the horontal force F. Dependng on what s addtonaly known, L or F, the correspondng teratve procedure s selected. (A) In the frst case, when the known catenary data are q, E, A, poston of the other node l and l, as well as the netensble length L, the procedure to determne the tangent stffness matr and the vector of the nternal nodal forces, s as follows:. The value of the varable λ s determned from epresson (6-), when the length L of the etensble element s substtuted wth the length L of netensble element and keepng the frst term n development of the functon snh λ/λ nto seres. That s the last case n epresson (7) for λ, whle the frst case n (7) represents the vertcal catenary l =, and the second case corresponds to the very 4 GRAĐEVINSKI MATERIJALI I KONSTRUKCIJE 54 () (9-4) BUILDING MATERIALS AND STRUCTURES 54 () (9-4)