ab

Tus/n

Department of Bridge Engineering, Tongji University, 1239, Siping Road, 200092 Shanghai, ChinadDepartment of Construction Engineering, Universitat P

a r t i c l e i n f o

Article history:Received 26 April 2011Revised 2 December 2011Accepted 5 February 2012Available online 28 March 2012

tion task and leads to lower costs [6]. This technique has beenemployed in many cable-stayed bridges such as the Val-BenoitBridge [7] or the Sanhao Bridge [8]. When environmental factorsor the requirements of the foundations prevent the placement of

many authors [1116]. Nevertheless, the erection procedures ofthe cable-stayed bridges are not so studied. Many works [1724],have been presented in order to either optimize or simulate theconstruction process of cable-stayed bridges using the cantilevermethod. Nevertheless, no specic works based on the temporarysupports erection method have been found. This paper aims to llthis gap by providing a new procedure that models the construc-tion process of cable-stayed bridges built on temporary supports.

Fig. 1 presents the temporary support erection method for anN = 6 six stay cable-stayed bridge built on K construction stages.

Corresponding author. Tel.: +34 902 204 100x3277.E-mail addresses: joseantonio.lozano@uclm.es (J.A. Lozano-Galant), igpaza@

upvnet.upv.es (I. Pay-Zaforteza), xu_dong@tongji.edu.cn (D. Xu), Jose.turmo@uclm.edu (J. Turmo).

Engineering Structures 40 (2012) 95106

Contents lists available at

Engineering

lse1 Tel.: +34 963 877 000x75623.trol the correct and safe prestressing of the stay on site. In addition, it also help the designer to control ifthe anchor wedge bites the strand in the same position several times during the prestressing process.

2012 Elsevier Ltd. All rights reserved.

1. Introduction

One of the most important causes of the rapid progress ofcable-stayed bridges in recent decades is the development of theconstruction techniques that made its erection possible [15].Wherever it is possible, the temporary support method is the fast-est way of building cable-stayed bridges because conventionalconstruction techniques may be used. This fact simplies the erec-

the temporary supports during construction, the cantilever erec-tion method [9] is used. This technique has been employed inthe construction of the longest cable-stayed bridges such as SutongBridge [10] with a main span of 1088 m. The cantilever erectionmethod consists on the placement of deck segments in cantilevereither in both sides of the pylon or in one side only and balancedby backstays located on the opposite side of the deck.

The structural behavior of these structures has been studied byKeywords:Cable-stayed bridgesConstruction processTemporary supports erection methodBackward modeling0141-0296/$ - see front matter 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.engstruct.2012.02.005olitcnica de Catalunya, BarcelonaTech, c/Jordi Girona 1-3, C1, 08034 Barcelona, Spain

a b s t r a c t

The temporary supports erection method is a fast and economical way of building cable-stayed bridges.In this method the bridge deck is rst erected on a set of temporary and permanent supports and then,the stays are successively placed and tensioned according to a predened tensioning sequence. A properdenition and analysis of this sequence is very complex as the structure is highly statically indetermi-nate, exhibits a non linear behavior and has a changing static scheme.Despite its importance, no specic research referring to the modeling of the temporary support erec-

tion method has been found as most of the modeling procedures are proposed for the alternative erectiontechnique, the cantilever erection method. The modeling carried out by most of these methods is basedon the opposite construction sequence followed on site, this is to say, the structure is disassembled fromthe desired nal stage (Objective Completion Stage, OCS).A procedure, the Backward Algorithm (BA), is formally presented in this paper for calculation of the

erection of cable-stayed bridges built on temporary supports. Because of its simplicity the BA can bereproduced by any structural code that enables the modeling of the prestresses of the stays by meansof imposed strains or imposed temperature increments. Another advantage is that no separate modelsare needed to calculate the evolution of stresses in the strands when the strand by strand tensioningtechnique is used. Furthermore, the stay elongations when prestressed can be easily obtained whenthe stays are prestressed in a single operation or strand by strand. This information is important to con-b Instituto de Ciencia y Tecnologa del Hormign (ICITECH), Departamento de Ingeniera de la Construccin y Proyectos de Ingeniera Civil, Universidad Politcnica de Valencia, Caminode Vera s/n, 46023 Valencia, SpaincAnalysis of the construction process of con temporary supports

J.A. Lozano-Galant a,, I. Pay-Zaforteza b,1, D. Xu c, J.aDepartment of Civil Engineering, Castilla-La Mancha University, Avda. Camilo Jos Cela

journal homepage: www.ell rights reserved.le-stayed bridges built

rmo d

, 13071 Ciudad Real, Spain

SciVerse ScienceDirect

Structures

vier .com/ locate /engstruct

erinNomenclature

BA Backward AlgorithmAuxiliary Modelk,i ith local iteration of the kth auxiliary modelCn nth stayCn,1 Connection between the pylon and the nth stayCn,2 Connection between the deck and the nth stayCPm mth Comparison ParameterFEM Finite Element Method[FM] Force Matrix[IM] Inuence MatrixK number of construction stagesL length of the stayN number of staysNBAkCn axial force in the nth stay in the kth construction stage

obtained by the BANSoftwarekCn axial force in the nth stay in the kth construction stage

obtained by the commercial softwareNOCSCn axial force in the nth stay in the OCS

Nk;iCn axial force in the nth stay in the kth construction stagein the ith local iteration

NkTM axial force dened for the kth row of the [TM]

NeUCmCn

axial force in the nth stay produced by an imposed uni-tary strain in the mth stay

{NTL} vector of axial forces in the stays produced by the TL

96 J.A. Lozano-Galant et al. / EngineIn the initial stage (Stage k = 0), the self weight of the structure, g1,is counterbalanced by a set of T = 2 temporary supports and P = 3permanent supports (Fig. 1a). This way, vertical reactions in boththe temporary supports, R0Tt , and in the permanent supports, R

0Pp ,

are found. Then, during the tensioning process, the stays are suc-cessively placed and tensioned by the jacks and the deck is raisedfrom the temporary supports (Fig. 1b). In these stages, the load g1 iscounterbalanced by the non-raised supports, RkTt , and by the tensileforces introduced into the placed stays, NkCn . When the tensioningprocess is completed after K tensioning stages (Fig. 1c), the naldesired stage, known as the Objective Completion Stage (OCS), isachieved. This stage can be easily calculated from the ObjectiveService Stage (OSS), which satises the stress distribution pursuedby the designer in such a way that under a certain load hypothesis,target load, TL, the stays present a given vector of forces {NOSS}(Fig. 1d). This stage will only be achieved when TL is applied intothe structure.

Calculation of the construction process of cable-stayed bridges,this is to say, the tensioning process that has to be followed duringconstruction, is very complicated as the structure is staticallyhighly redundant, non linear and is continuously changing its staticscheme during its construction. The forward simulation of the ac-tual construction sequence on site is associated with a number ofcomputational difculties. For example, each time that any stayis being prestressed the axial forces of the rest of stays are changed(see Fig. 8 in Section 5.2). Because of these difculties the back-ward simulation is commonly used, as it is much simpler. In fact,this technique has been employed by many authors for the

{NOCS} vector of axial forces in the stays in the OCSOCS Objective Completion StageOSS Objective Service StageP number of permanent supportsRTLTt vertical reaction in the tth temporary support when TL

is appliedRTLCn;2 vertical reaction in a ctitious temporary support lo-

cated at the connection between the nth stay and thebridge girder when TL is appliedRk;iTt vertical reaction in the tth temporary support in the kthconstruction stage and ith local iteration

RkPp vertical reaction in the pth permanent support in thekth construction stage

Stagek kth construction stageStagek1 k 1th construction stageT number of temporary supportsTL Target Load[TM] Tensioning Matrixi indicator of the local iterative processk indicator of the construction stagep indicator of the permanent supportt indicator of the temporary supportuTLCn;1 horizontal deection at the connection between the nth

stay and the pylon when TL is appliedwk;iTt vertical deection at the tth temporary support pro-

duced in the kth construction stage in the ith local iter-ation

acn inclination of the nth stayDNk;iCn increment of axial force at the nth stay produced in the

ith local iteration of the kth auxiliary modelDL elongation of the stayDWk;iTt increment of vertical deection at the tth temporary

produced in the ith local iteration of the kth auxiliary

g Structures 40 (2012) 95106cantilever method: Behin in [17,18] proposed a substructure-fron-tal technique that started the calculation in the reference congu-ration of the completed bridge. In this technique, nonlinearities forP-Delta effects were included by a continuous updating of the geo-metric conguration and nonlinearities from cables were includedby using catenary equations. Fan et al. in [25] proposed a methodto dene the optimum stay forces from a backward analysis thatincluded the creep effect. Mao et al. in [26] proposed a back-ward-analysis based on the creep aging theory for erection of con-crete cable-stayed bridges. Reddy et al. in [20] proposed anonlinear nite-element methodology for the stage by stage con-struction. The results of this method were compared with eldmeasurements of a long-span cable-stayed bridge. Wang et al. in[23] proposed a method for nding the initial shape of bridgestructures during the cantilever erection procedure. In the back-ward approach the structure is disassembled from the OCS. The se-quence of events in the disassembly analysis is the opposite of thatwhich occurs during erection. The tensioning process that has to befollowed during construction can be dened by a Tensioning Ma-trix [TM], like the one shown in Fig. 2. As the backward approachis used, this matrix is calculated from the bottom up. Nevertheless,the erection direction is the opposite. With K being the number ofconstruction stages and N the number of stays, this matrix isformed of K rows and two columns; the rst column describesthe stay that is prestressed at each stage and the second describesthe axial force to be introduced by the jack. Usually each stay istensioned several times through the tensioning process. For a Kstage construction process, the last K N + 1 axial forces of [TM]

modeleUCm imposed unitary strain in the mth stay{eCP} vector of imposed strains in the stays during construc-

tion processek;iCP imposed strain in the kth construction stage of the con-

struction process and ith iteration of the local iterativeprocess

erinJ.A. Lozano-Galant et al. / Enginecan be directly dened by the designer. Nevertheless, the remain-ing N 1 axial forces, highlighted in bold in Fig. 2, are unknowndue to innate evolutionary nature of the construction process ofthe cable-stayed bridges. For this reason, it can be said that the[TM] is incomplete or not fully known. The calculation of these un-known forces is indirect and must take into account all the preced-ing and subsequent tensioning operations. In fact, the axial forcethat has to be introduced into each stay when placed must be cal-culated in such a way that the achievement of the OCS is assuredafter completion. [TM] can be enlarged into the Force Matrix,[FM], which describes the axial forces of all placed stays

Fig. 1. Temporary supports erection method: (a) Stagek=0, (b) Stagek=k, (c) Stagek=K orObjective Completion Stage (OCS), (d) Objective Service Stage (OSS).throughout the construction process. This matrix, K N size, canbe dened from the [TM] as presented in Fig. 2. In [FM] the tensileforces introduced by the jack in each stage, this is to say, the valuesof the [TM], are framed. The last row of the [FM] is known and rep-resents the tensile forces in the stays at the OCS. Nevertheless, un-known values appear in the rest of stages. In fact, in addition to theinitial unknown forces dened in [TM], the evolution of axial forcesin placed stays, highlighted in bold in Fig. 2, is also unknown andmust be calculated throughout the analysis of the constructionprocess.

During the initial N stages of the construction process on site,the structural schemes change since new members (stays) areadded to the structure. Also, the load-carrying system is usuallychanged from the temporary supports to the stays [27]. This way,the temporary supports are successively raised. Once any tempo-rary support has been raised, according to our direct interviewswith different contractors and designers, two different tendenciesare observed: some usually keep it on site in order to control thedeections in the following construction stages. On the other hand,other designers prefer to remove it, to better control the stiffness ofthe bridge. In this paper, the former approach has been followed.

Despite the fact that the OCS can be modeled easily with anystructural program, the modeling of the sequence of events thathas to be followed on site in order to assure its achievement isquite complicated as the structure is highly evolutionary and stat-ically redundant. The modeling of the construction process carriedout by the commercial programs consists of applying the superpo-sition of stages in reverse. Some advanced commercial programse.g. Midas or Wiseplus [28], include special features such as thenonlinear analysis for the raising of the supports, or the fact thatthe prestresses of the stays are introduced as imposed forces. Thislast characteristic has two disadvantages. First of all, separate mod-els are needed to calculate the stresses in the rst placed strandwhen the strand by strand tensioning technique is used. Secondly,the stay elongations when these elements are prestressed in a sin-gle operation or strand by strand cannot be easily calculated. Fur-thermore, most of the ordinary structural programs do not includesuch rened features and, therefore, the analysis of the construc-tion process is even more complex.

In this paper, an algorithm based on the backward approach forstudying the construction of cable-stayed bridges built on tempo-rary supports, the Backward Algorithm, (BA), is formally presented.The three objectives of the algorithm are: (1) To introduce a proce-dure for designing the construction of cable-stayed bridges built ontemporary supports that can be applied using any structural soft-ware, (2) To dene a method to calculate the prestressing stressthat is to be applied to the rst strand when using a strand bystrand tensioning technique, (3) To dene a procedure to calculatethe stay elongations when these elements are prestressed.

This paper starts with the analysis of the modeling of the OSS byimposed strains according to the Rigidly Continuous Beam Crite-rion [29,30]. After that, the main hypotheses of the modeling ofthe construction process of cable-stayed bridges carried out bytwo commercial software and the BA are described. Next, the re-sults of a simplied cable-stayed bridge obtained by the analysiscarried out by commercial programs are compared with those ob-tained by the BA. It is to point out that the study presented hereconsists of an initial analysis that can be used for designing theconstruction process of cable-stayed bridges built on temporarysupports and for this reason the effects of time-dependent phe-nomena, such as creep, shrinkage or stay relaxation, or geometricalnonlinearities, such as beam-column effect or large displacementsare not taken into account. Due to the limited lengths of the stays

g Structures 40 (2012) 95106 97in the bridges built on temporary supports, usually cable sag is notan issue that needs to be modeled. Only when very low stresses areused to prestress the cables in the initial stages, Ernst Modulus can

brided b

erinFig. 2. Example of Tensioning Matrix, [TM], and Force Matrix, [FM], for a N = 6 stayand the prestressed stays in each stage are framed. The modeling direction is show98 J.A. Lozano-Galant et al. / Enginebe taken into account. If this is the case, a proposed approximationthat will keep the presented algorithm still valid is to modify theErnst Modulus in a stay only when it is prestressed. This will bean acceptable approximation providing that the prestress of a gi-ven stays does not signicantly affect the stresses of the others.If it is not the case, an iterative process will be needed and the algo-rithm should be modied. Only the changes of static scheme andthe raising of the temporary supports nonlinearities have beenmodeled. Finally, the main conclusions of the work are drawn.

2. Modeling of the OSS using strains

The Objective Service Stage (OSS) is a target scheme of forcesand/or deections dened according to the designer criterion,which has to be achieved when a given load hypothesis, the targetload, TL, is applied in the completed structure without any tempo-rary support. Given the TL, this stage can be dened by a vector ofaxial forces in the stays, {NOSS}. Once the OSS has been dened, thelast construction stage that counterbalances the self weight, g1, canbe easily obtained. This stage is known as the Objective Comple-tion Stage (OCS). The tensioning strategy that is described intothe Tensioning Matrix, [TM], should assure the achievement ofthe OCS at the end of the construction process.

In the literature there are ve main structural criteria to denethe OSS of cable-stayed bridges: (1) Rigidly Continuous Beam[29,30], (2) Zero Displacement Criterion [29,31], (3) Minimal Bend-ing Energy Criterion [32], and also some (4) Optimization Criterion[19,21]. This last criterion includes some restriction to achievepractical feasible solutions. Another innovative criterion was pro-posed by Janjic et al. in [22] who developed a method to take intoaccount the construction process and the effect of the time-depen-dent phenomena into the denition of the OSS. In this paper theRigidly Continuous Supported Beam criterion has been used.ge with K = 12 tensioning stages modeled backwards. The unknown forces are boldy the continuous arrow and the erection direction by the dotted one.

g Structures 40 (2012) 95106Nevertheless, without any lack of generality any of the other de-scribed criteria could have been used. The Rigidly Continuous Sup-ported Beam criterion is based on the assumption that the cable-stayed bridge deck behaves, in a long-term, like a rigidly continu-ous beam borne on ctitious rigid supports at cable anchor points.In order to dene the axial forces in the stays in the OSS, NOSSCn , thefollowing procedure is followed: rst of all, the vertical reaction,RTLCn;2 , obtained in the equivalent beam model in each temporarysupport, Cn,2, under the TL is obtained, as presented in Fig. 3a. Then,as is shown in Fig. 3b, the axial forces of the stays, NOSSCn , are ob-tained considering that RTLCn;2 is its vertical projection. Mathemati-cally, the value of NOSSCn can be deduced from Eq. (1), wheretensile forces and upward reactions are considered positive.

NOSSCn RTLCn;2

SinaCn1

The axial force in the backstays is calculated in order to avoidthe horizontal deections of the top of the pylon, uOSSC1;1 0 or tominimize shear or bending moment at the base of the pylon. Posi-tive horizontal (u) and vertical (w) deections followed the posi-tive direction of axes X and Z, as presented in Fig. 3a.

Once the NOSSCn have been obtained, modeling the OSS in some ad-vanced commercial programs, e.g. Midas or Wiseplus, is direct be-cause advanced features that allow the modeling of the tensileforces in the stays by means of imposed forces are available. Nev-ertheless, in not so rened programs or in the algorithm that hasbeen developed in this paper, modeling of the OSS should be donethrough imposed strains in the cables (or temperature decre-ments). The imposed strains, eCm , which have to be introduced intoeach stay to achieve the OSS are unknown. To dene these param-eters the effect of the TL on the stays must be separated from theprestress introduced by the jack. Therefore, the target axial forcefor the nth stay, NOSSCn , can be calculated from the axial force in

erinthe passive nth stay produced by the TL in the model without tem-porary supports, NTLCn , and the effect on the nth stay of the prestressof all N stays, as presented in Eq. (2).

NOSSCn NTLCn PNm1

NeUCmCn

eCm 2

eU

Fig. 3. Rigidly continuous beam criterion: (a) Continuous beam reactions. (b)Projection of the reaction in the stays direction. The second subindex shows the stayextremity, 1 is for the pylon and 2 for the deck.

J.A. Lozano-Galant et al. / EngineThe term N CmCn represents the axial force produced in the nth stay bya unitary strain introduced in mth stay of the structure, eUCm . Thisrelation can be expressed more compactly in a matrix form.

fNOSSgN1 fNTLgN1 IMNN fegN1 3

The vector of axial forces, fNOSSg; N 1 size, is obtained by addingthe vector of axial forces produced by the TL in an unprestressedstructure, {NTL}, N 1 size, and the effect of the prestress. This pre-stress can be dened by the product of an Inuence Matrix, [IM],N N size, and a vector of imposed strains in the stays, {e}, N 1size. [IM] shows how the axial forces in all the stays vary when aunitary strain is introduced into each stay. The vector of imposedstrains is the only unknown in Eq. (3) and, therefore, can be easilydetermined from a direct [29] or an iterative [33] process. Althoughany of these methods could have been used, in this work the vector{e} has been dened using the inverse of the Inuence Matrix,[IM]1, as presented in Eq. (4).

feg IM1 fNOSSg fNTLg

4

Once {e} has been determined, the OSS can be modeled directly bystrains. After calculating the OSS, the last construction stage knownas the OCS can be obtained easily. The importance of calculating theOCS is that it is the starting point from which the modeling of theconstruction process in the backward approach should start.

It is worthy to notice that in current practice {e} is usually ob-tained through a trial and error process what can be time consum-ing and not always accurate. In any case, to the best of the authorsknowledge, no such detailed explanation can be found anywhere[22,33].3. Commercial software

Nowadays, the modeling of the construction process of cable-stayed bridges built on temporary supports can be carried out byadvanced commercial programs which are usually based on thebackward approach. In this approach, the structure is successivelydisassembled from the OCS (according to the opposite constructionsequence) until the initial stage, where the bridge deck is sup-ported by the set of temporary and permanent supports, isachieved. An intuitive interface is available to dene the plannedconstruction schedule exactly, including all changes in the struc-tural behavior of the bridge. This tool allows the users to activateor deactivate loads, elements and boundary groups throughoutthe modeling of the construction process. The superposition ofstages principle is used. The temporary supports are usually mod-eled by means of special elements that are only able to counterbal-ance compressive axial forces. Hence, if tensile stresses areobtained in any temporary support during the modeling of the con-struction process, the element is deactivated from the structure bymeans of a local iterative process, affecting only this precise con-struction stage, where its force is redistributed to the rest of thestructure. These programs also include several advanced featuressuch as powerful solver modules to analyze the optimum forcesof the stays during construction. Another sophisticated feature isthat the stay prestress is introduced by means of imposed forces.This way of modeling the tensile forces in the stays, although cor-rect, does not provide too much information to the designer. Infact, it is necessary to develop separate models to dene the stres-ses in the strands when the strand by strand tensioning procedureis used. Furthermore, the vast majority of the ordinary calculationprograms do not include such rened features and therefore, themodeling of the construction process of cable-stayed bridges ismore complex. In the next section a procedure developed by theauthors that solves all these problems, the Backward Algorithm(BA), is described.

4. Backward Algorithm

A nite element computation procedure that models the con-struction process of cable-stayed bridges built on temporary sup-ports, the Backward Algorithm (BA), in formally presented in thissection. Its application is limited for an initial design of the con-struction process. If differences between the predicted and the ac-tual behavior on site are observed, a forward approach [34] issuggested. The BA, as most of the advanced commercial programs,is based on the backward approach from which the algorithm re-ceives its name. In the backward approach, the conguration ofany partial structure is determined by disassembling the bridgefrom the OCS. The sequence of events in a disassembly analysis isthe opposite of that which occurs during erection. Thus, everykth stage, Stagek, can be calculated by subtracting an auxiliarymodel to the following construction stage as shown in Fig. 4 orby subtracting Kk auxiliary models from the OCS as presented inEq. (5)

Stagek Stagek1 Auxiliary Modelk

OCSPKkk

Auxiliary Modelk 5

Each of these K Auxiliary Modelsk represents the effect of the ten-sile force introduced by the jack in each of the rows of the Tension-ing Matrix [TM].

The BA models the tensile forces in the stays by means of im-

g Structures 40 (2012) 95106 99posed strains instead of imposed forces, as in the case of the mostadvanced commercial software. This way of modeling the tensileforces in the stays has two advantages. First of all, it is not

necessary to develop separate models to analyze stresses in thestrands when the strand by strand tensioning technique is used.The second advantage is that the vast majority of the structuralprograms include this feature and therefore, the procedure pro-posed by the BA can be easily reproduced with almost anysoftware.

The BA includes two hypotheses also considered in the ad-vanced commercial software. First of all, the superposition be-tween stages applies. Secondly, the nonlinear effect of the raisingof the temporary supports is based on a local iterative process. In-stead of using special elements that are only able to counterbal-ance compressive stresses, the temporary supports can bemodeled as vertical xities that are activated if the vertical deec-tion at its connection with the bridge deck in the kth constructionstage, wkTt , is positive. This can be done manually or with a verysimple programming. Fig. 4a, shows the evolution from Stagek+1,which includes axial forces in the stays, Nk1Cn , and the self weight,g1, to Stagek. The subtraction from Stagek+1 of the results of the Aux-iliary Modelk,i, can be dened as Stagek,i as presented in Eq. (5). Thenal structural scheme of Stagek is dened by the local iterativeprocess. At the initial iteration, (i = 1), the Auxiliary Modelk,i hasthe same structural scheme (same number of stays and temporarysupports) as Stagek+1. The stay that is being prestressed betweenstages k and k + 1 is represented by a lled arrow. The rest ofplaced stays, represented by a unlled arrow, are passive. The ac-tive forces are modeled by imposed strains, eCP, in the stays thatare being prestressed in each stage. This strain is calculated in such

a way that the axial force dened in the kth row of the [TM] isachieved. The analysis of each auxiliary model produces someincrements of vertical deections in the bridge deck at the locationof the temporary support T;Dwk;iTt , and axial forces in the stays,DNk;iCn . Once the effect of the tensile force has been subtracted fromStagek+1, the ith iteration of the local iterative process is nishedand the Stagek,i is obtained. If, in this stage, positive deections intemporary supports, wk;iTt , are obtained, the structural system ofthe auxiliary model is changed adding additional temporary sup-ports in the next iteration of the local iterative process as shownin Fig. 4b. The requirement to evaluate if any temporary supporthas been borne is presented in Eq. (6).

wk;iTt wk1Tt Dwk;iTt P 0 6The local iterative process stops when non-positive deections

are found in any temporary support of Stagek.The input data of the BA consists of the geometry and mechan-

ical properties of the cable-stayed bridge, the location of the tem-porary supports in the bridge deck, the target loads, TL, the vectorof axial forces in the stays in the OSS, {NOSS}, as well as the incom-plete [TM]. This matrix is incomplete because the axial forces of therst N 1 stages are unknown and must be calculated taking intoaccount the entire construction process of the bridge. At the end ofthe computation, the lled-in Force Matrix, [FM], is provided. Thismatrix is related with [TM] and describes the axial forces in theplaced stays in each of the construction stages. Other output dataconsists of the deection, reactions and effort matrices for the dif-

100 J.A. Lozano-Galant et al. / Engineering Structures 40 (2012) 95106Fig. 4. Superposition of stages in the BA and modeling of temporary supports bearing: (a)arrow and the passive stays by unlled arrows.ith local iteration, (b) i + 1th local iteration. The active stays are presented by a lled

The ow chart presented in Fig. 5 summarizes the procedure

the following auxiliary model (Auxiliary Modelk,i+1). If no positive

are listed in Table 2.

pressive stresses in these programs. The prestressing of the stayshas been modeled by means of imposed axial forces acting in theanchorages following the direction of the stays and acting inwards,with no actual cable element present in the structure. This is to say,the stay effect on the structure is modeled by the forces introducedby the cable in the anchorages. Once the forces are introduced inthe structure and it is deformed, by means of advanced calculationtools, the cable element is introduced into the deformed or unde-formed structure with an axial force.

The modeling of the construction process starts by dening thevector of axial forces in the stays in the OSS. From this stage, theaxial forces in the stays in the OCS can be deduced easily. Then,

erindeections, wk;iTt , nor tensile reactions, Rk;iTt are found, the next con-

struction stage, k 1, is calculated. The process stops when the ini-tial stage (k = 0), in which the deck is supported by the set of thetemporary and permanent supports, is achieved. The main advan-tage of this procedure is that it can be easily reproduced by anystructural software that is able to include either imposed temper-ature increments or strains in the stays.

In the following section the results of the modeling of the con-struction process of a simplied cable-stayed bridge obtained byseveral advanced commercial programs and those obtained bythe BA are presented.

5. Application of the algorithm

The construction process of a cable-stayed bridge analyzed bythe Backward Algorithm is described in this section. The maincharacteristics of this structure and its modeled tensioning processare rst described. It is to point out that this structure is erected ona set of temporary supports placed below every stay. Nevertheless,and without any lack of generality, the algorithm could have beenapplied for any other distribution of the temporary supports. Then,the results obtained by several advanced commercial programs arepresented. Next, these results are compared with those obtainedby the BA. Finally, the BA is applied to obtain the stresses in the rststrand when using the strand by strand tensioning technique aswell as the stay elongations when prestressed.

5.1. Description of the model

In order to evaluate the efciency of the developed algorithm,the cable-stayed bridge shown in Fig. 6, is analyzed. This structureis a simplied model of a project for the city of Wuxi in China. Thebridge has one 54 m high concrete pylon, a 180 m length steel boxgirder deck and 18 stayed cables arranged in a semi-harp symmet-rical form. The self weight of the bridge deck, g1, and the targetload, TL, are 135 kN/m and 202.5 kN/m respectively. The anchoragefollowed by the BA. Once the input data has been introduced intothe program, the OSS is stored. Then, auxiliary models that includethe effect of each tensioning stage by means of imposed strains, ek;iCP ,are successively subtracted in order to obtain the preceding con-struction stage. This strain ek;iCP is calculated in such a way thatthe axial force dened in the kth row of the [TM] is achieved. A lo-cal iterative process is used to model the nonlinear behavior of thetemporary supports bearing. At the beginning of this iterative pro-cess, the Auxiliary Modelk,i has the same number of temporary sup-ports and stays that the Stagek+1. After subtracting Auxiliary Modelk,i

from Stagek+1, the Stagek,i is obtained. In this stage, if positivedeections,wk;iTt in any temporary support T are measured, AuxiliaryModelk,i is changed into Auxiliary Modelk,i+1 activating the bornetemporary supports. In addition to this, if tensile reactions, Rk;iTt ,are found in any active temporary support, this is deactivated inferent elements of the structure. Among all these matrices, they areremarkable the two ones that show the evolution of the raising ofthe temporary supports. One of these matrices presents the com-pressive force of each temporary support when borne and theother one their vertical deection when raised. Finally, comparedwith advanced commercial programs, an additional output is pro-vided. This is a vector of strains in the stay that has to be intro-duced by the jack along the Construction Process, {eCP}. Asummary of the input and output data can be found in Table 1.

J.A. Lozano-Galant et al. / Engineof the two central stays in the bridge deck is separated 15 m fromthe pylon. The anchorage of these elements in the pylon is sepa-rated 28.8 m from bridge deck. The rest of the stays are uniformlyThe cable-stayed bridge is built by means of the temporary sup-ports erection method. The tensioning process has K = 35 stagesand its OSS has been dened by means of the Rigidly ContinuousBeam Criterion [29,30]. It has been assumed that non evolutionaryconstruction process has been needed to reach this initial stage.The initial stage on site consists of the bridge deck supported bya set of T = 18 temporary supports. Then, in the next N 1 stagesthe rst 17 stays are placed and prestressed by the jack. The axialforces to be introduced by the jack in these stages are unknown aspresented in Fig. 2. Those forces correspond with the rst N 1stages of the Tensioning Matrix, [TM]. During the modeling of theevolutionary process, once a temporary support has been raisedit is removed from the structure. In the Nth stage the 18th stay isplaced. Afterwards, in the nal N 1 stages the axial forces of allthe rest of stays are successively modied. It is worth noting thatthere are innity tensioning strategies that can be applied to thestructure in order to assure the achievement of a certain stage aftercompletion. This strategy can be characterized by the denition ofthe N known axial forces of [TM]. In this example, these forces werechosen to assure that any bending moments, shear forces, axialforces and nor deection of the structure exceeded safety rangesduring construction.

5.2. Commercial software

The results of the modeling of the construction process of thecable-stayed bridge presented in Fig. 6, obtained by two advancedcommercial programs, Midas V7.01 and Wiseplus V1.2 [28], arepresented in this section. The temporary supports have been mod-eled as special elements that are only able to counterbalance com-anchored every 9 m along the bridge deck and every 1.8 m alongthe pylon.

The structural Finite Element Model (FEM) of the whole bridgeconsists of 20 beam elements for the girder and 12 beam elementsfor the pylon and 18 special elements for the stays. These last ele-ments have no bending stiffness. The values of the Elasticity Mod-ulus, Inertia and Area of the different elements used in the model

Table 1Input and Output data of the BA.

Input data Output data

Geometry and mechanical properties Force Matrix [FM]Location of the temporary supports

in the deckDeections, reactions and effortsmatrices

Target Load, TL Raising of temporary support matrixAxial forces in the stays in the OSS {NOSS} Strains vector during Construction

Process {eCP}Uncompleted Tensioning Matrix [TM]

g Structures 40 (2012) 95106 101the incomplete Tensioning Matrix, [TM], is dened by the designer.Next, the construction process is modeled. The results of thismodeling can be dened by the axial forces in the stays in each

erin102 J.A. Lozano-Galant et al. / Engineconstruction stage, which are summarized by the Force Matrix,[FM]. No signicant differences in bending moments, deectionsand stay forces between results of both commercial programsand BA were found. For this reason, only [TM], [FM] and {ecp} cal-culated by the BA are presented in Fig. 7.

The variation of the axial forces in a certain stay throughout theconstruction process can be obtained from [FM] according to theerection direction. This variation is studied in the rst placed stay,the 9th one, and the results are presented in Fig. 8. This gure canbe divided into two distinct regions which are separated by thedotted line. The rst region, located on the left hand side of the g-ure, starts with the placing and tensioning of the studied stay, inthe construction stage 1. As the stay is being prestressed this stageis presented by a dotted line. In the following stages of this region,the structural system is successively changed as new stays areintroduced and the raised temporary supports are removed fromthe structure. The location of the prestressed stay has great inu-ence in the variation of the axial force of all stays. In fact, in thecase of the 9th stay, its axial force is highly increased when the pre-stressed stay is not located on the same side as the pylon and de-creased when located on the same side. The second region, locatedon the right hand side of the gure, corresponds with the second

Fig. 5. Flow chag Structures 40 (2012) 95106tensioning operation. As all the stays have been already placedand the set of temporary supports has been removed in the preced-ing stages, the structural system in these stages remains constant.As the stay is being prestressed in the 35th construction stage, thisvalue is represented by a dotted line.

5.3. Backward Algorithm

In this section the cable-stayed bridge presented in Fig. 6 is ana-lyzed according to the Backward Algorithm, (BA). The procedureused to calculate the construction process by the BA is similar tothat described in the previous section. Nevertheless, the prestressof the stays has been modeled by means of imposed strains. Thisway of modeling has, among some others, the advantage that thestresses in the strands of the stays can be easily calculated withoutthe need of separate models even when the strand by strand ten-sioning technique is used. The BA was implemented in a FortranFEM code developed at Technical University of Catalonia and de-scribed by Cho in [35].

In this section the axial forces in the stays obtained by the com-mercial software are compared with those obtained by the BA. Thiscomparison has been based on the analysis of two Comparison

rt of the BA.

ridg

erinParameters, (CP). The rst one, CP1, is based on the differences inabsolute value along the K stages and the N stays. As these valueswill be different for each stay and each construction stage, themaximum value has been used. The denition of this parameterfor the comparison between the results obtained by the BA andthose obtained by the commercial software is presented in Eq. (7).

CP1 MaximumjNBAkCn N

SoftwarekCn

j !

n : 1N; k : 1K 7

c1 c2 c3 c4 c5 c6 c7 c8 c9

Fig. 6. Cable-stayed b

Table 2Properties of the elements of the FEM.

Elasticity modulus (MPa) Inertia (m4) Area (m2)

Girder 206,000 4.2 1.7200Pylon 33,500 14.4 8.5400Cable 195,000 0.0 0.0072

J.A. Lozano-Galant et al. / EngineNBAkCn

The term NBAkCn represents the axial force in the nth stay calculated bythe BA in the kth construction stage while NSoftwarekCn represents thesame force calculated by the commercial software. The second com-parison parameter, CP2, is based on the differences of axial forces inall placed stays for a certain construction stage k in absolute value.As this parameter is different for each construction stage, the max-imum value is used. Eq. (8) shows the denition of this parameterfor the quadratic comparison between the BA and the commercialsoftware.

CP2 Maximum 1PN

n1NBAkCn NSoftwarekCn

PNn1NBAkCn N

BAkCn

!k : 1K 8

The comparison of axial forces in the stays can be summarized aspresented in Table 3. This table shows the value of the comparisonparameters obtained after comparing the results obtained by thecommercial programs and those obtained by the BA. Both parame-ters are presented in terms of a percentage of the value obtained bythe BA.

The analysis of these results showed negligible differences be-tween the results obtained by the two commercial programs andthose obtained by the BA.

5.3.1. Analysis of the strand by strand tensioning techniqueThe BA has, among some others, the advantage that the stresses

in the strands of the stays can be easily calculated without theneed of separate models, even when the strand by strand tension-ing technique is used. This prestressing method is more and moreoften used as it eliminates the need for heavy erection equipment.The method has several variants according to the patent used byeach company, but basically consists on prestressing a rst singlestrand up to a dened stress. Once the strand is anchored, the restof the strands are successively prestressed one by one until theirstresses match the stress of the rst strand. It is to highlight thatthe rst strand is losing tensile stresses when the others strandsare prestressed, due to the elastic shortening of the stays.

The strand by strand tensioning technique has been modeled inthis section. The fty strands forming each stay are successivelyintroduced into the structure and prestressed in the rst N stages.At the end of every stage, the complete stay will have been intro-duced and prestressed. The strain that is needed to achieve inthe full stay the axial force dened in the corresponding kth rowof [TM], that is, ekCP is successively applied into each of the strandswhen placed to simulate its tensioning. In the modeling, when anew strand is introduced and prestressed the axial forces of allplaced strands are modied. At the end of the stage, when all thestrands are placed and prestressed with the same ekCP , all the forcesin the strands are equal and the force of the stay matches the valuedened in the kth row of [TM].

Fig. 9a shows the stress in the rst placed strand of the 9th stayin the rst construction stage throughout the strand by strand ten-sioning stages when the imposed strain is achieved in each of the

c18c17c16c15c14c13c12c11c10

e. Dimensions in m.

g Structures 40 (2012) 95106 103strands. In this construction stage, the structural system consists ofthe set of the temporary and permanent supports. When the rststrand is placed, a certain stress is obtained. Then, as new strandsare successively introduced into the structure, the stress of the rstplaced strand is successively reduced. When all the strands havebeen placed and prestressed, the stress of the rst strand is re-duced 84.3 MPa respect to its value when rst prestressed. This im-plies a prestressing loss of the rst strand of 20.2%. When the 28thstrand is introduced and prestressed, highlighted in Fig. 9a, the ax-ial force introduced into the stay is such that is able to unload therst temporary support. Therefore, the structural system of thebridge changes as the unloaded temporary support is removedfrom the model. In the following KN stages, the stays will be re-stressed, using the strand by strand tensioning technique. In orderto do so, the imposed strain in the strands when they were rstprestressed is successively changed to the one that correspondswith the stage where the strands are being re-stressed. Fig. 9bshows the stress in the rst re-stressed strand of the 9th stay inthe 35th construction stage, after its imposed strain is changedfrom e1CP to e35CP . As all the stays were placed and the temporary sup-ports were unloaded and removed from the model in the precedingconstruction stages, the structural system is that presented inFig. 9b.

erin104 J.A. Lozano-Galant et al. / EngineThis structural system remains constant throughout the re-stressing stage. When all the strands have been re-stressed, thestress of the rst strand is reduced 1.70 MPa respect to its valuewhen re-stressed. The increment of axial force along the re-stressed stage is 12.5 kN, which is the 5.2% of that axial force intro-duced by the jack, that is, N35C9 N

34C9. The comparison of the reduc-

tion percentages of the stress in the rst strand obtained in the 1stand the 35th construction stages showed that lower reductions ofstress were calculated when the stay is re-stressed. This can be ex-plained by the fact that in the re-stressing stage the axial forceintroduced by the jack is lower as well the structural system ofthe bridge is stiffer as all the stays were placed in the precedingstages.

Fig. 7. Tensioning Matrix, [TM], and Force Matrix, [FM], and strain vector feCPg obtainedin kN and strains dimensionless.

Table 3Maximum differences in the comparison parameters referring to a percentage of theresults obtained by the BA.

Midas (%) Wiseplus (%)

CP1 0.0014 0.0025CP2 0.0001 0.0002g Structures 40 (2012) 951065.3.2. Stay elongationsAnother advantage of modeling the construction process with

imposed strains instead of imposed forces is the fact that the elon-gation of the stays to be measured in the tensioning operation canbe easily predicted. Calculation of stay elongation, DLCn , is veryimportant for controlling the safe and accurate prestress of thestays on site.

The calculation of the stay elongation can be approximated bythe product of the calculated imposed strain introduced in the stayin the kth stage, ekCP , by the unstressed length of the prestressedstay LCn as presented in Eq. (9).

DLCn ekCP LCn 9To facilitate the comparison with other calculation methods, the

stay elongations obtained in the 9th stay are summarized in Ta-ble 4. The comparison of Stage1, where the stay is placed and pre-stressed, and Stage35, where the stay is re-stressed, showed thatlarger elongations are obtained in Stage1 as larger axial forces areintroduced by the jack.

This information is not only important to control the correctand safe prestressing of the stay on site. It also help the designerto control if the anchor wedge bites the strand in the same positionseveral times during the prestressing process. This fact could

by the BA. The stays that are being prestressed in each stage are framed. Axial forces

1 2 3 4 5 6 7 8 9 10 1112 13 14 15 16 1

STR

ages

erinCON

Fig. 8. Axial forces (kN) of the 9th stay along the construction st2500

2600

2700

2800

2900

3000

3100

3200

3300

3400

3500

AXIA

L FO

RC

E (k

N)

J.A. Lozano-Galant et al. / Enginejeopardize the integrity of the strand along the time due to fatigueproblems.

6. Conclusions and future work

When the environmental factors or the requirement of thefoundations do not prevent the placement of the temporary sup-ports during construction, the temporary supports erection meth-od uses to be the most economic way of building cable-stayedbridges. Unlike its alternative construction method, the cantilevererection method, no specic research based on the modeling ofthe construction process of cable-stayed bridges built on tempo-rary supports has been found by the authors. This paper aims to llthis gap by providing a computation procedure, the BackwardAlgorithm (BA).

The modeling proposed by the BA consists of disassembling thecable-stayed bridge from the Objective Completion Stage (OCS)according to the opposite sequence of events which occurs duringerection on site. The BA presents several advantages compared

Fig. 9. Stresses in the rst strand of the 9th stay and structural system of the bridge tconstruction stage, which is 35th.

Table 4DLC9 (mm) in the Stage

1 and Stage35 obtained by different tensioningprocesses.

Stay 9Stage1 80.48Stage35 5.847 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

UCTION STAGE

(kN). Stages 1 and 35, where the stay is prestressed, are dotted.

g Structures 40 (2012) 95106 105with the rest of modeling proposed procedures: (1) Unlike the ad-vanced commercial programs, the BA models the prestress of thestays by means of imposed strains instead of forces. Therefore, itis not necessary to develop separate models to calculate the stressvariation in the rst strand of a stay when the strand by strand ten-sioning technique is used. Knowing these stresses is highly recom-mended because a more efcient control of the constructionprocess can be carried out on site thus increasing safety duringconstruction. (2) The stay elongation when prestressed can be eas-ily calculated when the stay is prestressed. This information is notonly important to control the correct and safe prestressing of thestay on site. It also help the designer to control if the anchor wedgebites the strand in the same position several times during the pre-stressing process. (3) Finally, the main advantage of this process isits simplicity as its results can be easily reproduced by any struc-tural software. This way, faster calculation can be carried out andthe procedure can be efciently used to initially design the con-struction process of cable-stayed bridges.

The numerical analysis of the cable-stayed bridge studied inthis paper showed that non-representative differences were foundbetween the results obtained by two studied commercial pro-grams, and those obtained by the BA. Nevertheless, the BA can onlyapproximate the effects of the time-dependent phenomena, unlessa global iterative process or a backward-forward analysis is per-formed. To take into account these phenomena, the more complexbut more suitable procedure presented in [34] is proposed. In fu-ture works this algorithm will be adapted to take into accountthe evolutionary construction of the bridge superstructure.

hroughout the local tensioning process (MPa): (a) rst construction stage, (b) last

Acknowledgments

The authors wish to thank the support provided by the Ministe-rio de Ciencia e Innovacin and by Junta de Comunidades de Castil-la-La Mancha (Spain) through the research Projects BIA2009-13056 and PII2I09-0129-4085 (Optimization of the constructionprocess of cable-stayed bridges built on temporary supports), di-rected by Jos Turmo.

Part of this work was done through a collaborative agreementbetween University of Castilla-La Mancha (Spain) and Tongji Uni-versity (China). This included an exchange of faculty and scholars.The nancial support from Kwang-Hua Foundation from College ofCivil Engineering of Tongji University and from the InternationalRelation Ofce of University of Castilla-La Mancha is greatlyappreciated.

Finally, the authors also want to thank the support provided byJ.A. LLombart and J. Fernndez from Eipsa (Spain), R. Snchez-deLen and C. Bernal from AIA (Spain), E.W. Vieira from Universityof Castilla-La Mancha and by Y. Zhao from Tongji University.

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Analysis of the construction process of cable-stayed bridges built on temporary supports1 Introduction2 Modeling of the OSS using strains3 Commercial software4 Backward Algorithm5 Application of the algorithm5.1 Description of the model5.2 Commercial software5.3 Backward Algorithm5.3.1 Analysis of the strand by strand tensioning technique5.3.2 Stay elongations

6 Conclusions and future workAcknowledgmentsReferences