Dorian Janjic Managing Director TDV GmbH, Graz, Austria

Transcription

1 Optimisatio of the Tesioig Schedule for Cable-Stayed Bridges usig Dyamic Software Heiz Boka Maager Project Cetre TDV GmbH, Graz, Austria Heiz Boka, bor 947 civil egieerig degree from the Techical Uiversity of Civil Egieerig, Graz. Over 30 years of experiece i structural aalysis i a wide rage of applicatios. Doria Jajic Maagig Director TDV GmbH, Graz, Austria office@tdv.at Doria Jajic, bor 960, civil egieerig degree from the Faculty of Civil Egieerig, Sarajevo. 5 years of experiece i techical research, software developmet. Marko HEIDEN Project Egieer TDV GesmbH. Graz, Austria office@tdv.at Marko Heide, bor 973, civil egieerig degree from the Techical Uiversity of Graz i 000. Curretly workig as a project egieer with TDV- Austria. Ivolved i may iteratioal high-speed railway projects durig the last few years Summary The desig of log spa cable-stayed bridges ca prove tedious whe it comes to fidig a appropriate strategy for the stay cable tesioig procedure. The desig cocept for achievig the appropriate tesioig procedure i cable stay bridges is ofte based o fidig the forces i the idividual cables that give rise to certai allowable structural displacemets, momets or stress distributios i the girder ad the pylos at the ed of costructio. The stressig forces ad the sequece of stressig for all the cables eeds to be optimised to meet these pre-defied requiremets as closely as possible. The calculatio procedure described i this paper models every costructio stage i detail with the tesioig of each idividual cable beig firstly cosidered as uit-loadig cases actig o the curretly active structural system ad ifluecig all previously applied uit-loadig cases. The effects of the other loadig cases appropriate to the costructio procedure affectig all previously costructed parts (such as self weight of the ew segmet, traveller relocatio etc.) are also calculated. The displacemets ad the iteral forces from each costructio stage are accumulated ad the values are sub-divided ito costat (i.e. self weight.) ad several variable compoets - each variable compoet beig coected to oe of the uit loadig cases. A system of equatios is built up by comparig these accumulated values with the iitial desig requiremets. The result from the equatio reductio is the itesity factors for all the uit-loadig cases to achieve the predefied costraits (displacemets, momets, stresses). The dyamic software procedure works equally well for both liear ad o-liear structures with the effects of Creep & Shrikage beig fully cosidered. The beefit from this method is achievig a optimal tesioig strategy which results i reduced stressig actios with cosequet huge time savig ad cost savig durig costructio. The cocept is illustrated by the aalysis of the Verige Bridge that crosses the Bay of Boka Kotorska i Moteegro. Keywords: Cable Stayed Bridge, tesioig strategy, costructio sequece, o-liear, software. Itroductio The AddCo Method (The Additioal Costrait Method) is a ovel solutio for optimisatio problems i structural egieerig. This is a extesio of the Uit Load Method for o-liear problems. Bridge desig ad aalysis is a iterative process. Durig this process the egieer is lookig for the best solutio for give criteria by chagig specific system parameters. Egieer experiece helps to reduce the time required, but there will still be a eed for may iteratio steps util the de-

2 sig criteria are met. Computer programs owadays should provide the best possible support of this desig process. It should be metioed here that it is ot possible to have a computer program complete all egieerig tasks. The best computer support that ca be expected (eve with a supercomputer) is to fid the best solutio for some give costraits, but it is still the egieer s duty to fid correct ad logical costraits ad to prepare the iput properly for the computer. There are various costraits used i structural egieerig. Most obviously, costraits ca be applied to calculatio results (deformatios, stresses, forces, etc.). But also all other desig parameters ca be used as costraits, e.g. geometric parameters, material properties, etc. It is importat that a computer supported desig methodology supports various types of costraits.. Liear Optimisatio of the Cable Tesioig Fig. Bed. Momet Mz at fial stage without optimisatio Fig. Bed. Momet Mz at fial stage after optimisatio. Geeral Stadard bridge desig processes begis with preparig a structural model, defiig the loads ad the costructio schedule (costructio stages). The egieer will the ru a computer aalysis. Two result types of the computer aalysis are of mai iterest: The loadig case result, ad the evelope result. The loadig case result represets the structural state at a defiite time. The evelope result provides iformatio about maximal/miimal peaks of a give result together with other correspodig results. Based o the results, the desig criteria ca be checked ad optimisatio ca be started. A simple example is give to demostrate the priciple of optimised cable tesioig: I Fig. the bedig momet diagram is show for the fial stage without optimisatio of the cable tesioig. I Fig. the same results are show after optimisatio.

3 . Result storage The results of oe structural state i (e.g result of loadig case i) ca be writte as a vector of dimesio e: 0i T { E } { E E E3. E } = e () Each item E j, j = e i vector {E} represets oe result of ay type, e.g. displacemet, itegral force/momet, stress, etc. Usually ot oly the basic results but also liear combiatios of result vectors E j are of iterest. The sigificat results for the user ca be writte i vector form as well, where these results are calculated as liear combiatios of the basic results: { i 0 E } [ L]{ E i } = () Vector {E i } has the dimesio where << e. Matrix [L] has dimesio x e ad coverts result from vector {E 0i } to vector {E i }. The result value for which a costrait ca be defied is calculated as the liear combiatio of all system state results, e.g. as liear sum. { } { } = m i E E i = (3) Costat part: Mz due to permaet loads Fig.3 Sigle results Mz for additioal dead loads at fial stage (Liear calculatio) Variable part: Cable Stressig Fig.4 Forces, displacemets, stresses etc. ca be cotrolled with cable stressig Stressig of Cables ca be cotrolled Mz limits Fig.5 Bedig momet Mz at fial stage without optimisatio I each desig step, a ew, differet result vector for the same chose result will be produced. Parts of the results {E i } may be chaged; other parts may be costat, depedig o the differet system parameters. For the further aalysis it is ecessary to split the m result vector ito mv variable ad mc costat results (m = mc + mv). Stress limits Fig.6 Stress results at fial stage at the bottom edge of the mai girder cross-sectio without optimisatio The mc costat results ca be summed up directly: co, co, co mc, co { E } { E } + { E } + { E } = (4) All mv variable results ca be writte i matrix form. Matrix [M] is of dimesio x mv: var, var [ ] [{ }{ }{ } { }], var, var 3, mv M = E E E E (5) iput The sum of the costat results ad the variable results should match the iput costraits E. The goal of the liear optimisatio is to fid which system parameter must be chaged i order to meet the costraits.

4 I the liear system model, each system parameter results i a variable result. A vector with liear weight for the variable results describes the system parameters. The costrait ca ow be writte as show i the followig equatio: iput co { E } = [ M ]*{ f } + { E } Where the weightig factors {f} are basic ukows. M 55 z M z M z M = z Solutio of the costraits problem E v E v E 3v E 4v f f f f The costrait equatio is a system of liear equatios. The solutio is trivial for the special case where =mv: user cost { f } = [ M ] * ({ E } { E }) (7) It is ecessary that matrix [M] is o-sigular. If matrix [M] is sigular, o physical solutio exists. This may happe if the chose costraits are coupled. Due to umerical effects, it is also possible to get solutios that have o practical meaig because matrix [M] is early sigular. 3. No-liear effects: Provided that the results are calculated i a liear aalysis, a liear optimisatio solutio ca be expected, ad for o-liear aalysis a o-liear optimisatio solutio must be applied. Ufortuately, this is ot a case. Eve with simple liear structural aalysis, o-liear optimisatio methods must be applied. There are two major reasos for the o-liear compoets i the results: Cosiderig time effects i calculatio ad cotiuous chagig of the structural system. 3. Cosiderig time effects I practical egieerig, time effects (creep, shrikage ad relaxatio) must be cosidered i the aalysis. I most codes, the time effects are described to have quasi superpositio behaviour. But these effects produce o-liear results i the fial state. The reaso for this is the forth dimesio (time): A chage of weightig factor for oe loadig will creep over time. These creep results will be icreased with the same weightig factor if creep has quasi superpositio behaviour. Lookig at the structural state after some time, chages occur due to the iitial chage of loadig ad chages from all followig creep steps up to this state. Chages due to creep are ot kow iitially. These o-liear effects chage the results oly slightly, but liear optimisatio ca ot be applied ay more. I order to cover these effects i the described system, the matrix [M] is split ito a liear ad a o-liear (time-effect) part: [ M] [ MLIN + M] = (8) Equatio 6 chages to: (6)

5 { E } [ M + M ] { f} + { E cost } = * (9) LIN The o-liear part of Matrix [M] ca be added to the costat results. The costat results get quai-costat results (marked with a asterisk * ): * cost cost { E } = [ M ]*{ f } + { E } (0) Fig.7 Optimised Aalysis without creep ad shrikage effects for t=0 Fig.8 Results icl. Creep & shrikage effects after a optimised liear calculatio (t=, 85 years) ad Equatio 9 chages to: * cost { E} [ M ]*{ f } + { E } = () LIN The differece of the costat parts of the results (which actually should be zero) is ow a measure for the o-liear part of [M]. err cost cost { E } { E } { E } = * () Fig.9 Optimised aalysis icludig creep ad shrikage effects for the optimisatio 3. Cosiderig cotiuous chage of structural systems Cotiuous chage of structural systems is aother major reaso for gettig o-liear optimisatio problems. To uderstad the physical reaso, a very simple example of usig a temporary cable i the costructio schedule is show i Fig. 0 &. This o-liearity has a differet ature to the time-effects o-liearity but it ca be treated with the same optimisatio method. 3.3 No-liear structural behaviour If structural respose is ot liear, the optimisatio problem is o-liear from the very begiig. I practical cases this o-liearity is ot too far away from a liear solutio. Desig experiece shows that o-liear effects are usually withi 0% of liear solutio. This is the same order of magitude as the oliearity due to time effects. Agai, these effects ca be treated with the same method described above. With a mild o-liearity grade we cover almost all problems. Temporary support F=000kN Fig.0 Bedig momets o the mai girder before closure (with temporary primary support active) F=000kN Fig. Bedig momets after closure o the mai girder due to temporary support removig 4. No-Liear optimisatio for mild o-liear problems 4. Liearizatio of the o-liear part I geeral case the matrix [M] depeds o weightig factors {f}. This depedecy is ot give directly. Oe way of describig it mathematically is to produce chages i {f} ad to watch correspodig chages i [E]. The ukow o-liear part of [M] ca be computed by solvig the liear system:

6 δe δe δe δe δe δe = δe δe * δe (3) This equatio is the the basis for a o-liear optimisatio solutio. 4. Iterative search for solutio Each full calculatio loop provides oe additioal piece of umerical iformatio about the o-liear behaviour. As the umber of iteratios grows, more ad more data is available to fid the proper o-liear part of [M]. The algorithm must fid the best result to calculate from the solutios available (pivotig) ad approximate all other elemets of [M]. The more results that become available, the better the solutios. With a good selectio algorithm, it is possible to fid a solutio for the oliear problem for ukow parameters withi a fractio of iteratio steps. Durig iteratio step i, k = i- differeces for results ad differeces for factors are already available. The pivotig algorithm chooses the k biggest lies from {E error } ad extracts a k x k matrix [de*]. All other errors are approximated by a diagoal matrix [de**] cosistig of the diagoal elemet of [de]. The Matrix [dm] is the recostructed out of the two results. i = 0 {f } = iput i = i + {E} = RECALC (fi) {E} - {E u } < {Tol} NO [M li ] = [M li ] ({f i }) [ M] = [ E cost ] * [ f] - {E cost } = { * E cost } - [ M] *{ f i } {f i+ } = [M+ M] - *{E user -E cost } START NEXT ITERATION FINISH

7 4.3 Covergece criteria The iteratio algorithm i the AddCo Method stops whe the results are withi a give tolerace {T} of the desig criteria: { E user } { E} <= { Tol}. Chagig of factors f over the iteratio process ca be see i Fig Number of Iterat ios [] Fig. chagig of the factors f over the iteratio process (till user defied costraits are fulfilled) f f f3 5. Practical Applicatio of the Uit Load Method: The Verige Bridge 5. Geeral All the project egieerig work was performed by the Gradis, Sloweia. This compay has used the TDV-software [] icludig features described i [,3] for may years i their costructio offices for the desig of differet type of bridges. For the curret project TDV ad GRADIS had a close cooperatio for all the electroic aalysis work. Static ad dyamic stability verificatio had bee carried out by meas of the software system RM004 for all costructio stages as well as for the fial stage. With this program system a stage by stage aalysis ca be performed. I each calculatio stage the results of all stages should be summed up. Thus a cotrol of all iteral forces ad momets arisig i a certai rime period is give. For cocrete cross-sectios pre-stressig with actual tedo characteristics ad ay geometry layout ca be take ito accout. For the desig calculatio of the pylo the d order theory had bee adopted with the software system RM004. The costructio stage aalysis had bee performed for overall 8 costructio stages. 5. Project Descriptio The Verige Bridge is a cable-stayed bridge located i the southwest of Moteegro ad crosses the Bay of Boka Kotorska. The bridge cosists of 7 spas icludig the mai cable-stayed part ad the adjacet spas with a total legth of 98m. The mai spa betwee the two pylos is 450m. The height of the bridge above sea level is about 50m. Fig.3 Verige Bridge, spas of m, Moteegro [4]

8 Fig.4 Verige Bridge, cross sectio [4] 5.3 Costructio Stage Aalysis The cable-stayed bridge was to be erected by the catilever method startig from both pylos i both directios util the approach-bridges were reached ad a moolithic coectio could be established after the closure of the mai. The structural system for the fial stage aalysis as well as the structural system for the idividual costructio stages was modelled with RM004 []. The mai girder of the Cable-Stayed-Bridge was costructed symmetrically by free catileverig. First, base parts above the piers of the approach spas were executed followed by a symmetrical costructio of idividual segmets. At the same time, the first elemets (hammer head) were costructed at both pylos as well from where free catileverig was carried out. The legth of a idividual segmet amouted to 5.0 metres. Every secod segmet was coected with a stay cable. The remaiig portio of the deck was also costructed accordig to the cast-i-situ free catilever method while the deck portios at the abutmet the piers of the approach spas were executed by meas of a formwork. Fig.5 to Fig.7 show the bedig momet diagrams for differet costructio stages, icludig all the o-liear time-depedet effects, i the last iteratio cycle of the cable stressig optimizatio procedure. Fig.5 Costructio stage 4 after stressig, bedig momets i the superstructure Fig.6 Costructio stage 7 after closig of the approach spas ad 3, bedig momets i the superstructure

9 Fig.7 Costructio stage 8 after applyig permaet loads, bedig momets i the superstructure & pylo: momets are as defied i the ADDCON Method 6. Coclusios A method to fid the optimal tesioig strategy for the costructio of cable-stayed bridges has bee derived. This paper explais this method called the Addco Method ad explais how oliear ad time-depedet effects which are relevat for the desig of bridges ca be icluded. The Addco Method computes the correct tesioig forces for the stay cables which lead exactly to a pre-determied momet distributio withi the deck ad the pylo ad also to the iteded geometry of the bridge rederig the traditioal trial-ad-error approach to this problem obsolete. The method has bee implemeted ito a bridge-desig software package ad has bee used i practice o several occasios. Oe of these practical applicatios, the aalysis of the Verige Bridge i Moteegro, serves as a example i this paper. The method is ot restricted to bridge desig, may other applicatios exist as the method has bee formulated ad implemeted i a very geeral way. 7. Refereces [] TDV GmbH. RM004 & RM006 - Techical Descriptio, Graz, Austria, 006 [] JANJIC D, PIRCHER H., FEMBRIDGE - Techical Project Descriptio, 004, TDV- Austria [3] JANJIC D., PIRCHER M., PIRCHER H., BRIDGE R.Q. Towards a Holistic Approach to Bridge Desig, Proceedigs: IABSE-Symposium 00, Melboure, pp [4] Klobucar A., Verige Bridge across the Bay of Boka Kotorska i Moeegro, TDV- Cablestayed Semiar 003, Graz