Computer Aded Optmum Desgn n Engneerng XI 107 Cable optmzaton of a long span cable stayed brdge n La Coruña (Span) A. Baldomr & S. Hernández School of Cvl Engneerng, Unversty of Coruña, La Coruña, Span Abstract Ths document descrbes an optmzaton problem of cable cross secton of a cable stayed brdge consderng constrants of cable stress and deck dsplacement. Snce the brdge s stll n study phase, the geometry and the mechancal characterstcs are subjected to changes. In order to avod creatng dfferent structural models, a program was produced to construct a model from geometrcal and mechancal data and resolve optmzaton problem. At the end of the document, two examples are presented to show how ths program works. Keywords: optmzaton, cable stayed brdges, cvl engneerng. 1 Introducton The optmzaton technques are not commonly employed n professonal lfe n cvl engneerng, specfcally n the area of structural calculaton, and the majorty of consultng frms do not apply these technques to ther desgns. Nevertheless, n the desgn of great structures, these technques are ganng more mportance due to ther mpact on reducton n materal cost. The presented work comes from the general study of traffc systems around A Coruña, n partcular, the connecton of the lttoral area of Oleros and Sada to the cty. After analyzng traffc n dfferent roads, ther ntensty, and future projects of hub connectons, t was concluded that t would be necessary to construct a new road to be able to effcently releve heavy traffc. The new connecton requres a constructon of a brdge over the Coruña estuary, whch s the most mportant part of the road. Then a study was conducted on the brdge typology that fts the best to the techncal, envronmental, and aesthetc condtons of the area. After consderng dfferent proposals, the typology of cable stayed brdge was found to be the most adequate. do:10.2495/op090101
108 Computer Aded Optmum Desgn n Engneerng XI Cable stayed brdges are used more and more to overcome long spans and there already exsts one wth one klometer of span length [1]. The reasons to use such typology are for ts good structural functonalty, vsual lghtness, great aesthetc component, and ts low mpact on the envronment, whch was the key ssue at the locaton of the brdge over the Coruña estuary. The work presented n ths document takes part n the project from ths pont, and optmzaton of the cable cross secton of the cable stayed brdge s developed. 2 Brdge descrpton and structural model After the study the road soluton consders the constructon of a cable stayed brdge of 1198 m of total length whch s dvded nto two lateral spans of 270 m and the man span of 658 m. Fgure 1: Sde vew of the brdge. The man elements that compose the brdge are: Deck: made of steel wth an aerodynamc secton of 3 m of heght and 34 meter of wdth, whch permts a confguraton of 3 lanes n each drecton. The dfference n heght between two end ponts s 15 m. Towers: the brdge has two towers: each fxed n the foundaton wth two twn concrete pers separated by the deck wdth as shown n Fgure 3. From the deck two steel masts come out n the form of λ unted by transversal bracng and ther upper part contnue as a sngle vertcal mast. The cables are anchored to ths part of the tower. Moreover, there are other transversal bracng at the lowest part of the vertcal mast to avod bucklng problem. Cables: the brdge deck s sustaned by 80 par of cables that connect to the towers. The cable confguraton s hybrd of harp and fan. Fgure 2: Cross secton of the deck. The optmzaton process of the cables requres successve structural analyss of the brdge. A fnte element model wth beam elements was generated n ABAQUS code [2]. As wll be seen later, ths model s not fxed, but parameterzed for avodng to remodel due to eventual changes n the brdge desgn.
Computer Aded Optmum Desgn n Engneerng XI 109 Bastaguero Tower Oza Tower Fgure 3: Tower detals. Fgure 4: Vsualzaton of the new brdge over the Coruña estuary. Fgure 5: Structural model of the brdge.
110 Computer Aded Optmum Desgn n Engneerng XI To model the structure, 3D beam element, B31 s used wth ts rotatons released at the beam ends for the cables. The materal utlzed s steel for the deck and the masts of the towers, and concrete for the lower part of the towers. A lnear elastc model s consdered wth sotropc character for Young s module and Posson s rato. 3 Formulaton of optmzaton problem The objectve of ths work s: to create a program that s able to generate a generc fnte element model of a cable stayed brdge (accordng to the structural scheme proposed n the prevous secton), to be calculated n ABAQUS, and from the data obtaned n ths calculaton, to perform optmzaton process on the cable cross sectonal area. The optmzaton problem s defned by the followng elements [3 4]: Desgn varables The desgn varables are each of the cable cross sectonal areas. The number of desgn varable s reduced to half of the number of cables due to symmetry about the longtudnal plane; however, t s not symmetrcal about the perpendcular plane. For better performance of the optmzaton process, the nverse of the areas are used to lnearze the stress constrants of the cables. x = 1/ A = 1,,n From now on, we call n total number of desgn varables. Desgn constrants Three types of desgn constrants were taken nto account: 1. Stress constrants of the cables k σ ( ) σ = 1,,n k=1,,lc x M where: σ k ( x ) : normal tensle stress n the cable, for the load case, k σ : maxmum allowable normal tensle stress n the cables M LC : total number of load cases 2. Dsplacement constrants of the deck w k ( x ) w = 2,,n-1 k=1,,lc max where: w k ( x ): deck dsplacement at the cable poston for the load case k. w : maxmum allowable dsplacement (postve value) max 3. Dsplacement constrants of towers n the longtudnal drecton
Computer Aded Optmum Desgn n Engneerng XI 111 u k Towerj u j = 1,2 k=1,,lc max u : dsplacement n x drecton at the top of the tower j for k Towerj the load case k. u : maxmum dsplacement n x drecton at the top of the tower max (postve value). Objectve functon F(X) The objectve of optmzaton n ths case s to reduce the quantty of steel for the cables. Thus the objectve functon s expressed as: n 1 mn F = 2 L x where: L : cable length As we can see, ths s an optmzaton problem wth n desgn varables, 2n x LC nequalty constrants, and an objectve functon. 4 Program descrpton The resoluton of the problem formulated prevously s carred out by a man program created n MATLAB [5] and the use of software ABAQUS for the structural calculaton. Ths program bascally conssts of three parts: 1. Generaton of fnte element models. 2. Calculaton of ntal stresses n the cables for the self weght. 3. Optmzaton of the cable sectons accordng to the establshed load hypotheses. 4.1 Generaton of structural model In ths part of the program, we assgn dmensons, mechancal characterstcs, and dscretzaton for each of the elements that compose the structural model. Ths part of program conssts of three subroutnes, nputtablero, nputtorres and nputcables whch are n charge of wrtng Abaqus code accordng to the data provded ntally. Intal values of desgn varables are assgned to each cable secton n the program. 4.2 Intal tenson force n the cable The constructon of the brdge s executed by balanced cantlever from each of the towers. The constrant to be satsfed n each phase of the process s the dsplacement of the deck where cables are attached should be null. Therefore, = 1
112 Computer Aded Optmum Desgn n Engneerng XI just before placng the last center deck secton, the brdge should have zero dsplacement, and after the last deck secton s placed, statc analyss calculaton should show only slght axal forces n such deck secton. If the calculaton of ntal tenson wth the complete model was carred out, ths last deck secton would have axal forces above the actual ones. Therefore the calculaton of ntal tenson s carred out on the brdge wth ts sectons wthout jonng and enterng the weght of ths last deck secton. Fgure 6: Structural model to obtan the ntal tenson forces. These stresses are calculated as follows: 1. Deck dsplacement s calculated where cables are attached to the deck, whch s produced by self-weght wthout ntal stresses n the cables. 2. A unt force s appled to the frst par of cables and dsplacement s obtaned at each pont of the deck. Ths operaton s repeated for the rest of the cable pars. 3. Intal tenson forces are obtaned by resolvng the followng system of lnear equatons: w pp j n + P w = 0 j = 1,,n j = 1 pp w : deck dsplacement j due to the self-weght load j P : ntal tenson force n the cable par, w : deck dsplacement at poston, j due to the tenson force of the j cable par, 4.3 Optmzaton The optmzaton process s carred out usng an optmzaton module mplemented n MATLAB. The functon to perform ths process s called fmncon that mnmzes functons wth varous varables subject to nequalty constrants. Ths optmzaton subroutne requres certan nput values such as desgn varables, constrants, and an objectve functon. For obtanng better results, t
Computer Aded Optmum Desgn n Engneerng XI 113 also provdes gradents of the objectve functon and the desgn constrants. Those gradents are calculated by fnte dfference, the most costly part of the program. Fgure 7: Flow chart of optmzaton process. 5 Applcaton examples The frst example we present s a fcttous case of a brdge wth the same longtudnal profle as the ntal brdge descrbed n secton 2, but wth notceably reduced number of desgn varables (n=8). The second example shows the soluton obtaned for the brdge confguraton descrbed at the begnnng of the document. The number of varables n ths case s much greater than the prevous example (n=80). In both examples the optmzaton of the cable s performed for three smple load cases as: Overload of use, 4 KN/m 2 on the left lateral span Overload of use, 4 KN/m 2 on the man span Overload of use, 4 KN/m 2 on the rght lateral span
114 Computer Aded Optmum Desgn n Engneerng XI Obvously those load cases are not determnant for the lmt state defned n the regulaton, however, they serve to verfy the program performance. The ntroducton of new load cases n the program does not present any dffculty snce these cases smply need to be entered n the subroutne, casos_carga. 5.1 Brdge wth 16 cables Fgure 8: Structural model (16 cables). Desgn varables The desgn varables are the nverse of the areas of eght cables n the model. We use 0.1m 2 for the ntal value of the area. 0 x 1 1 = = = 10 = 1,,8 () 0.1 0 A cable Desgn constrants The desgn constrants consdered for ths example are summarzed n the followng equatons: k KN σ ( x ) 800000 2 m = 1,,8 k=1,2,3 k L1 k L w ( x ) = 2 k=1,2,3; 2 w ( x ) = 3,,6 550 550 k=1,2,3 k L H 3 k j w ( x ) = 7 k=1,2,3; u Tower j =1, 2 j 550 550 k=1,2,3 where L1, L2 and L3 are the length of the left span, man span, and rght span respectvely and H1 and H2, the tower heght. The dmensons are all n meter. Objectve functon The objectve functon s the total volume of steel cables that we ntend to mnmze. 8 1 F = 2 L = 1 x The optmzaton process converges after 28 teratons gvng results shown n fgure 10 to 12.
Computer Aded Optmum Desgn n Engneerng XI 115 Area (m 2 ) 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 1 6 11 16 21 26 Iteraton Cable 1 Cable2 Cable3 Cable4 Cable5 Cable6 Cable7 Cable8 Fgure 9: Evoluton of the cable areas. Cable Intal Area Optmal area Area (m 2 ) Cable Fgure 10: Dstrbuton of cable area. 2900 2400 Volume (m 3 ) 1900 1400 900 400 1 6 11 16 21 26 Iteraton Fgure 11: Evoluton of the objectve functon. The optmzaton results lead to larger area for cables attached to both ends of the deck just as occurs n the desgn of cable stayed brdges. Those cables are the ones that brace the towers. Lkewse, the cables close to the center of the man span have a cross secton larger than the rest, whch can be observed n dmensonng of those brdges due to lmtng the deck deflecton. In regard to the objectve functon, we can observe that the ntal model has nsuffcent volume movng from 400m 3 to a value close to 900 m 3. Obvously these results do not correspond to any real cases snce brdges are never desgned
116 Computer Aded Optmum Desgn n Engneerng XI wth such small number of cables. We smply want to see the program performance. After the optmzaton, we checked to see f all the desgn constrants are satsfed. The dsplacement constrant n the man span s actve for the overload case on ths secton as well as the stress n the cables adjacent to those wth the largest secton. None of the constrants are volated. 5.2 Brdge wth 160 cables Ths example corresponds to the structural model descrbed at the begnnng of the document. Fgure 12: Structural model (160 cables). Area (m 2 ) Iteraton Fgure 13: Evoluton of the cross sectons.
Computer Aded Optmum Desgn n Engneerng XI 117 Area (m 2 ) Cable Fgure 14: Dstrbuton of areas. Area (m 2 ) Iteraton Fgure 15: Evoluton of the objectve functon. Its geometrcal and mechancal characterstcs are dentcal to that of example 1 and t dffers only for the number of cables and ther locatons. Fgure 13 shows the evoluton of the cable areas durng 69 teratons startng from an ntal area of 0.1 m 2. It s dfferent from the other model n terms that the cables at the extreme ends of the brdge have areas much larger than the rest, and two cables close to the center of the man span have approxmately half of that value. Fgure 14 shows the area of each 80 cables after the optmzaton to gve an dea of area dstrbuton along the brdge. Snce the constrants are not volated n any moment, the desgn requrements are satsfed. The most lmtng constrant s the dsplacement at the center of the man span for the dstrbuted load on that span. Although the stress constrants are not actve, some cables have the value close to the mposed lmt.
118 Computer Aded Optmum Desgn n Engneerng XI Obvously we cannot assure that ths optmum soluton s unque, and there can be other local mnmums. Nevertheless, the optmzaton proves to be useful to gve more effcent desgn than that obtaned by heurstc rules. 6 Conclusons In ths work, we have presented the results obtaned from a study on the optmzaton of cable cross secton of a cable stayed brdge. Some conclusons can be drawn from the results: 1. Due to hgh degrees of ndetermnacy of ths type of structure, the optmzaton process s not monotonous presentng dscontnutes n the desgn varables as well as n the objectve functon. A small modfcaton n the desgn varables can affect the fulfllment of constrants far away from the locaton where the changes are produced. 2. We can check the results obtaned n the optmzaton a posteror by a fnte element model. As shown n the examples, we can make sure that the results obtaned satsfy the mposed requrements. 3. The program provdes an dea to a desgn engneer how much cross sectonal area s necessary for each cable under certan stress and dsplacement constrants. Snce these values are not ntutve a pror at all, they serve as a desgn tool. 4. The dstrbuton of optmzed cable cross secton along the brdge agrees wth that of actual cable stayed brdges [6]. 5. After the optmzaton process, t can be observed that the areas of cables attached to the extreme ends of the deck are much larger than the rest of the cables. Those cables serve to mprove the longtudnal bracng of the towers and at the same tme they have an mportant nfluence for vertcal deck dsplacements. 6. The cross secton of the cables located close to the center of the man span have larger area than the rest other than those at the extreme ends, due to fulfllng the deck dsplacement constrants. 7. The optmzaton process provdes a result where the objectve functon has a mnmum (local mnmum), however, t does not guarantee that t s the best soluton among all the possble ones (global maxmum). Nevertheless as long as we acheve to reduce the steel volume n comparson to other conventonal technques, the optmzaton proves to be very useful to reduce materal cost. 8. The developed program permts changes on the fnte element model (geometry, mechancal propertes, boundary condton, etc) n a smple way. Besdes t permts to nclude load hypotheses wthn the optmzaton process whenever one s wllng to pay the much hgher computatonal cost. Ths cost s manly due to obtanng the gradents of desgn constrants. The conclusons drawn here on dmensonng of the cables comply exclusvely wth the terms n whch the optmzaton process s defned. Ths study can be developed amply from many ponts of vew such as:
Computer Aded Optmum Desgn n Engneerng XI 119 1. Includng geometry optmzaton to obtan optmum cable postonng on the deck 2. Includng other structural elements such as the deck, towers, etc as desgn varables. 3. Consderng other types of analyss n optmzaton such as bucklng, vbraton, dynamc analyss, etc. 4. Possblty of employng other types of optmzaton algorthms to acheve faster convergence References [1] N.J. GIMSING. Cable Supported Brdges. J. Wley and Sons, Second Edton 1997. [2] ABAQUS 6.8 Documentaton. [3] HERNÁNDEZ, S., Métodos de Dseño Óptmo de Estructuras. [4] JASBIR S. ARORA, Introducton to Optmum Desgn, Second Edton, 2004. [5] MATLAB Documentaton. [6] Honshu-Shkoku Brdge Authorty, Japan. The Tatara Brdge. Desgn and Constructon Technology for the World s Longest Cable-Stayed Brdge. 1999.