CABLE STRUCTURE WITH LOAD-ADAPTING GEOMETRY

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1 Compostes n Constructon Thrd Internatona Conerence Lon, France, Ju 3, CABLE STRUCTURE WITH LOAD-ADAPTING GEOMETRY A.S. Jüch, J.F. Caron and O. Bavere Insttut Naver - Lam, ENPC-LCPC 6 et, avenue Base Pasca Cté Descartes Champs-sur-Marne Marne-a-Vaée Cedex, France juch@enpc.am.r, caron@enpc.am.r, bavere@enpc.am.r ABSTRACT: For an appcaton n a composte cabe-staed ootbrdge a sstem s deveoped, whch s to ensure an eua dstrbuton o statc or uas-statc e oads n the compostes retaners. Thus the aowed e oad can be mzed or ths knd o structures whe mantanng the necessar wde saet margn. The structure s optma geometr or the contro procedure s determned b means o an agorthmc ormndng process, based on the method o orce denst. The resuts o the shape optmzaton seem to match wth a mechanca devce to be deveoped.. INTRODUCTION The rage behavour o composte materas (carbon and gass bres and the ack o experence reure hgh saet actors, whch are prejudca to the ntroducton o compostes n cv engneerng. The stud s context s an a-composte cabe-staed ootbrdge or whch a shape contro sstem s deveoped. The shape o the ootbrdge w adapt tse to externa e-oads to euaze as much as possbe the stresses wthn the derent tpes o eements. As a resut the aowabe e oads can be mzed even hgh saet actors are to be kept. Ths paper presents bre the method o orce denst. Then ths method s apped n an agorthmc orm to a smped D structure nspred b the na cabe-staed ootbrdge. Fna the optmsaton resuts are nterpreted or an appcaton to a mechanca contro devce.. FORCE DENSITY METHOD Wth the orce denst method, ntroduced b Lnkwtz n 9 [,], eubrum shapes o prestressed cabe-structures can eas be ound. As a hgh non-near sstem o euatons s nearsed, ths method s proved to be hgh ecent when determnng consstent eubrum orms or tenson nets. The method o orce denstes s bre ntroduced means a smped structure whose geometr s chosen to be cose to the ootbrdge s geometr we pan to stud. Fg. shows the Fnte Eement mode created wth ANSYS or the cabe-staed ootbrdge. Fg. shows the smped mode chosen or the shape optmsaton wth Scab and the notatons used.

2 Compostes n Constructon Thrd Internatona Conerence Lon, France, Ju 3, 3 S S3 S S C C C3 F 6 x Fgure FE-Mode n 3D. Fgure Smped mode wth notatons The smped structure conssts o our stas (S, S, S3 and S and a cabe made o three parts (C, C and C3. Both eement tpes work n tenson on. Ponts,, 3,, and 6 are constraned n both x- and -drecton, thus ther coordnates are known. We w determne then the coordnates o ponts and accordng to the eubrum shape or a oad set. The structure oads we consder are prestress and eventua a orce on nodes and/or. In the oowng euatons we chose to app a vertca oad F on node.. The euatons o the method o orce denst Consderng the structures eubrum on nodes and, we can wrte down the oowng euatons: Node Node (x ( (x ( x ( ( ( ( x + ( (' + ( x ( ( ( ( x3 3 + ( (3 (3 (3 (3 + ( x ( ( ( ( x + ( ( ( ( ( + ( x x6 6 = = F (6 (6 (6 (6 = = ( ( wth: xk x -coordnate o node k ( jk k -coordnate o node k ( jk orce wthn the eement between node j and k ength o the stress-extended eement between node j and k The dstance between node j and k,.e. the ength o the stress-extended eement, s descrbed usng Pthagoras aw ( jk = (x j xk + ( j k. The sstem o euatons gven b ( and ( s then obvous hgh non-near regardng the geometrca varabes (x,, x and. Introducng orce denstes dened as (jk = (jk / (jk the euatons ( and ( can be nearzed as shown n euatons (3 and (. Node Node x ( ( x ( ( + + ( ( ( ( ( ( (3 (3 x + + ( ( + + = x = (6 (6 ( ( = x = (3 (3 ( ( + + ( ( + ( ( F (6 (6 (3 (

3 wth: ( jk the orce denst o the eement between nodes j and k,.e. ( jk ( jk. We obtan a near sstem o euatons wth euatons and unknowns (x,, x and. A partcuar set o orce denstes (jk reates to a unue eubrum shape.. Appcaton to the prestressed structure As Lnkwtz notced [3], nvestgatons and practca experments just the choce o ver smpe tpes o orce denstes to create eubrum shapes as an nta approach to ormndng: a orce denst = c s assgned to eements o eua ength whe a orce denst o = c / s used or structures wth eements o rreguar ength. In both cases c s a constant vaue, but whe the constant c means a constant orce denst when usng the rst tpe o orce denstes, the constant c stands or a constant orce when appng orce denstes o the second tpe. The structure we stud has eements o rreguar ength. Hence or the shape optmzaton the orce denstes = c / o the second tpe are chosen. Two derent sets o orce denstes s or the stas and c or the cabe eements are chosen,.e. a constant orce c s = s or a the stas and a constant orce c c = c or the cabe. These orce denstes state a prestress dstrbuton wthn the structure. Rememberng the orce denst denton = /, t becomes obvous that mpementng a non-zero orce denst to an eement resuts n an eubrum orm wth a non-zero orce or ths eement. Thus a eements are subjected to a orce. The two derent sets o orce denstes s / c orm two derent states o prestress dstrbuton. No orce F s apped. The eubrum orms or the nta shape are gven n Fg 3 and Fg. The orce denstes chosen or Fg. 3 are s = / and c = /, wth reatng to the ength o the stress-extended eement n the nta geometr. The eubrum shape o Fg. resuts rom choce o the orce denstes s = / and c = /. In Fg. the rato o s / c = / s preserved but the eubrum s cacuated or an nta shape. The nta geometr s drawn n a dotted ne and the eubrum orm s represented as an unbroken ne. Fgure 3 Inta shape and eubrum or s = and c =. Fgure Inta shape and eubrum or s = and c =. Fgure Inta shape and eubrum or s = and c =. It can be seen that: the orce denst rato contros the radus o curvature o the cabe (eements C, C and C3. the nta shape has an nuence on the eubrum shape. The normaton o the nta shape s not ncuded through the coordnates o the unknown (x,, x and n euatons (3 and ( but through the engths o each eement n the orce denstes. Actua, to determne the eubrum shape, the orce denstes are cacuated dependng on the nta extended ength o the eements and evdent derent nta geometres match wth derent nta eement engths. 3

4 3. ITERATIVE STRESS-EQUALIZING METHOD The chosen agorthm s ver smpe. Other tpes have been proposed b [,] but as the am s not to deveop a hgh ecent and ast agorthm but a stress contro devce, ths approach seems justabe. The am o the presented agorthm s to determne geometres provdng a (uas-eua stress dstrbuton wthn each tpe o eement. As each tpe o eement has a determned cross-secton, to euaze stresses means to euaze orces wthn the eements no matter what ther ength. Fg. 6 shows the teraton process chosen; troughout a the teratons each c (c s and c c s kept constant and thus corresponds to the euazed orces s and c to be reached ( possbe wthn the eements. INITIAL SHAPE, x (k,nta (k,nta Cacuaton o the stress extended engths Cacuaton o the orce denstes Sove orce denst euatons (jk (jk =c/ (jk new x (k and (k no convergence? or mum teraton number? x (k,opt es OPTIMIZED SHAPE,, (k,opt (jk,opt Fgure 6 Iteraton process to generate eubrum orms euazng stresses wthn eements. However, t shoud be notced that t s not awas easbe to determne a shape that eads to the mposed orces n each eement. Nevertheess the teratve process can determne a shape as cose as possbe to the reurements; ths w be the optmzed geometr. Ths s wh two cassca crtera are tested to stop the teraton procedure. On the one hand we have a convergence test and on the other hand a mum teraton step. 3. Appcaton : prestressed structure For ths evauaton the nta shape s chosen because o the greater varaton o the eement s ength. Ths aows an easer evauaton o the nuence o the nta geometr. Fg. shows the mprovement o the eubrum shape or s = / and c = / throughout the teratons,.e. teratons =,,. Tabe gves the mum eement ength derence o a eements or these teratons. Ths mum derence d s the mum derence between the dstance o nodes j and k (or the stress extended ength o the eement between nodes j and k o teraton and the dstance o nodes j - and k - o the prevous teraton step -,.e. d = ( jk, ( jk,. Studng the mum ength derence, the mprovement o the eubrum geometr can be observed. When the new geometr does not partcuar der rom the prevous one,.e. the mum ength derence d gets cose to zero, the orce denstes, dependng on the engths, do not var much ether. The agorthm converges. Ths can eas be notced n Tabe on the act that the average orces o the stas c and o the cabe = ~ c s, s =

5 eements c c, c = ute reach the reured orces c n teraton step =. Addtona the standard devatons s s, and s c,. ndcate that a good euazaton o the orces s archved or =. Tabe Trends o d, s, and c,, ss, and s c, s, s s, c, s c, d nta shape shape = shape = optmzed shape Fgure Improved eubrum shape or teraton steps =,,. Consderng the evouton o the eubrum geometr throughout teraton steps, t can be xed that: the greatest varaton o shape occurs between nta geometr and eubrum geometr o the rst teraton step =. the optmzed geometr s determned b the nta shape. Ths can be notced consderng that even though a smmetrca prestress oad s apped the engths o eements C and C3 are not eua. 3. Appcaton : oaded and prestressed structure The nuence o externa oad on the optmzed shape o the prestressed structure s now studed. As notced n the prevous chapter, the nta geometr s sgncant when determnng new optmzed shapes. Ths s wh the nta prestress shape o the structure has to be chosen careu. The seecton procedure or an optmzed nta shape w not urther be dscussed n ths paper, but t shoud be kept n mnd that ths deta w ead to addtona nvestgaton. In ths paper, the seected optmzed prestressed shape s the eubrum orm shown n Fg.. The orce denst rato that eads to ths optmzed shape s cose to the rato resutng rom the prevous Fnte Eement stud o the ootbrdge. The structure s oaded n two derent manners. Frst oad case (LC: The structure s oaded smmetrca wth a oad F = apped on nodes and. These orces mode the pedestran oad. To take nto account the new nterna stress dstrbuton resutng rom externa oadng the adapted orce denstes aotted are s = / and c = /. These orce denstes are aso determned based on the Fnte Eement computaton resuts. Second oad case (LC: The structure s oaded asmmetrca wth F = apped on on node. The matchng orce denstes are s = / and c = /. The Fg. and 9 ustrate the eubrum shapes or both oad cases at teraton step =. As or the prevous gures, the dotted ne shows the nta geometr and the unbroken ne the eubrum orm. Fgure Eubrum or LC wth and c =. s = Fgure 9 Eubrum or LC wth and c = s =

6 The nta stress-extended ength prestress and the stress-extended addtona eement ength LC each eement reures to create the eubrum geometr are dspaed n Tabe 3 or both oad cases. The addtona stress-extended eement ength s LC = LC prestress. The addtona eement ength LC(,C o the entre cabe s cacuated b LC(,C = LC(,C + LC(,C + LC(,C3. Tabe 3 Inta stress-extended ength and reured addtona stress-extended eement ength to create optmzed shape. Eement S S S3 S C C C3 Cabe prestress LC LC EVALUATION OF A CONTROL DEVICE MOCK-UP A the engths gven n the prevous chapters are engths o stress-extended eements. The optmzaton approach takes nto account nether the geometr o the eements nor the eement s matera propertes. In act wth the orce denst method on the structure s geometr s adapted to create a orce ow matchng wth a determned orce dstrbuton. The next step s the constructon o a mock-up o the smped structure to vadate the numerca resuts and to start deveopng a contro devce. When determnng the cuttng pattern o the mockup, the engths o the eements are needed. It becomes necessar to consder the eement s geometr and matera. In order to make the notatons cear, the Fg. shows graphca a notatons used or the engths. j prestress(,jk prestress(,jk j LC(,jk LC(,jk LC(,jk LC(,jk prestress(,jk prestress(,jk LC(,jk LC(,jk prestress(,jk LC(,jk prestress(,jk LC(,jk k k prestress(,jk LC(,jk Prestress Load Case addtona eement engths Fgure Notatons o engths ( jk and ( jk, eongatons ( jk and addtona engths ( jk ( jk.or prestress and oad cases. and Consderng Hooke s near aw to be appcabe or the eement s matera o ths structure, the engths ( jk o the eements can eas be determned b the oowng euatons: ( jk ( jk + ( jk = ( ( jk ( jk ( jk = ( jk ce EEAE EEAE = (6 6

7 ( jk ce = ( jk + EEA ( E wth: ( jk ( jk ( jk ( jk EE A c E E ength o the stress-extended eement between node j and k ength o the eement between node j and k eastc eongaton o the eement between node j and k orce wthn the eement between node j and k stness o the matera o the eement tpe E area o the cross secton o the eement tpe E constant matchng wth the mposed orce or eement E Tabe shows the ength or each eement n the nta state o prestress caed prestress, and or both oad cases caed LC. The addtona eement ength each oad case (LC s aso cacuated wth = LC o each eement or LC LC prestress. Accordng to the FE-mode o the ootbrdge the stness o the stas and o the cabe s E S = E C = MPa, the cross-secton o the stas s A S =. m and the cross-secton o the cabe s A C m. To match wth the dmensons, a engths n Tabe are gven n meters. Tabe Sack engths and reured addtona ength LC to create optmzed shape. Eement S S S3 S C C C3 Cabe prestress LC LC LC LC Because o the smmetr o LC, the ength and addtona ength o the eement S3 are dentca to those o eement S, and smar eement S corresponds to eement S, and C3 to C. The greatest strcture needed s the one to orm the eubrum shape o LC. The eement S (and the eement S has to be shortend b LC(,S =.m ength. The most mportant addtona ength s reured or the eubrum geometr o LC. Here the eement S needs =.63m extra ength. LC(,S The cabe remans amost the same ength: the mum addtona ength s =.9m. LC(,C Consderng the appcaton on the 3D-ootbrdge, the necessar addtona ength w probab be smaer because o a greater number o retaners and cabe eements. The resuts o ths rst evauaton aow urther reectons; t shoud be possbe to desgn a contro devce abe to adapt the eement s ength wthn these margns.. CONCLUSIONS The na am s to contro the stresses wthn eements o a oaded structure b adaptng the shape. In the rst part o ths paper a short snopss o the orce denst method was gven. Then a basc

8 teraton process was presented. The orces n the derent eement tpes are to be euazed adjustng the engths o the eements to the oad stuaton. In the thrd part a premnar approach aowed an evauaton o the eement eongatons reured or ths adaptve shape structure. The order o magntude o the cacuated eongatons seemed sutabe to the whoe geometr and appng such dspacements on the structure shoud be possbe wth a mechanca devce to be deveoped. [] Schek H.J. The orce denst method or orm-ndng and computaton o genera networks. Computer Methods n Apped Mechancs and Engneerng 3; 9; p.-3. [] Lews W.J. Tenson structures, Form and behavour. st ed. London: Thomas Teord; 3; p. -. [3] Lnkwtz K. About ormndng o doube-curved structures. Engneerng Structures; 999; ; p. 9-. [] Lnkwtz K. Formndng b the drect approach and pertnent strateges or the conceptua desgn o prestressed and hangng structures. Internatona Journa o Space Structures; 999; ; No. ; p. 3-. [] Snger P. Anaoges between mnma suraces and membrane constructons (numerca part. Natürche Konstruktonen, 99; SFB 3; p. -.