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Direct simulation of the tensioning process of cable stayed bridges Jose Antonio Lozano Galant a XU Dong b Ignacio Pay Zaforteza c Jose Turmo d aDepartment of Civil Engineering University of Castilla La Mancha 13071 Ciudad Real Spain bDepartment of Bridge Engineering Tongji University 1239 Siping Road 200092 Shanghai China cInstituto de Ciencia y Tecnolog a del Hormig n ICITECH Departamento de Ingenier a de la Construcci n y Proyectos de Ingenier a Civil Universitat Polit cnica de Val ncia 46023 Valencia Spain dDepartment of Construction Engineering Universitat Polit cnica de Catalunya BarcelonaTech 08034 Barcelona Spain a r t i c l ei n f o Article history Received 31 October 2012 Accepted 16 March 2013 Available online 13 April 2013 Keywords Construction simulation Cable stayed bridges Temporary supports Tensioning process Analysis Unstressed length a b s t r a c t This paper proposes a new and innovative algorithm the Direct Algorithm DA which introduces for the very fi rst time the unstressed length of the stays concept into the modeling of the construction process of cable stayed bridges This assumption enables a fast and direct simulation of construction stages by ana lyzing independent Finite Element Models when time dependent phenomena are neglected The compu tation al speed and the limited computer storing requirements of the DA make it especially indicated for optimi zation problems Furthermore it can be implemented in any structural analysis software 2013 Elsevier Ltd All rights reserved 1 Introduction One of the fi rst stages in the design of a cable stayed bridge is the defi nition of a target stress state or geometry to be achieved in service in a stage known as the Objective Service Stage OSS 1 This in practice leads to defi ne an appropriate set of stay forces 2 3 The defi nition of an adequate construction process that en ables the achievemen t of the OSS on site is a complex nonlinear problem as during construction the structural system does not re main constant and partial structures arise These partial structures are more fl exible than the complete d bridge and are subjected to construction loads Thus the effects of dynamic loads such as wind 4 5 or earthquake 6 during construction can be even more sig nifi cant than in the completed bridge Furthermor e as cable stayed bridges are highly statically redundant structures tension ing one single cable or removing a temporary support affect the stresses in cables supports pylon and deck For these reasons many researchers have recommended the complete simulation of the construction process of cable stayed bridges see 7 9 The objective of this simulation is double On the one hand guarantee ing that the limit states are not exceeded during erection and on the other hand defi ning a tensioning process that guarante es the achievemen t of the OSS at the end of the construction process In the last decades the development the Health Monitoring fi eld 10 enables the structural control 6 and damage detection 11 12 of complex structure s Cable stayed bridges used to be monitore d both during constructi on and in service The informa tion obtained by monitoring systems can be used to calibrate sim ulation models that must be linked with maintenanc e and structura l managemen t to assure structural safety and functiona lly of structures 13 This can also be used to monitor the construc tion process ensuring that the tensioning process is followed accordin g to the designer specifi cations The traditional method to simulate the construction process of cable sta yed bridges is to start at the OSS and gradually reduce the structure backward simulation stage by stage until the fi rst constructi on stage is reached Several authors have proposed meth ods based on the backward approach for the cantilever see 14 16 and the temporary support erection method see 1 The methods based on the backward approach are the most likely to achieve the fastest result as they start from a structurally correct solution the fi nal stage Nevertheless these methods are not ade quate when deviations during the tensioning process of the struc ture appear Furthermore they can only approximat e the effects of the time depend ent phenomena such as creep and shrinkage in concrete elements see 17 18 when a global iterative process or a backward fo rward analysis is performed To overcome all these problems a forward simulation which follows the erection se quence on site has been proposed for the cantilever see 14 15 and the temporary supports erection method see 19 20 The main trade off of the forward simulatio n is that it is time consuming 0045 7949 see front matter 2013 Elsevier Ltd All rights reserved http dx doi org 10 1016 pstruc 2013 03 010 Corresponding author E mail addresses joseantonio lozano uclm es J A Lozano Galant xu dong XU Dong igpaza upvnet upv es I Pay Zaforteza jose turmo upc edu J Turmo Computers and Structures 121 2013 64 75 Contents lists available at SciVerse ScienceDi rect Computers and Structu res journal homepage as an Overall Iterative Process is required to assure the achievemen t of the OSS at completion Most of the simulation methods presented in the literature as sume that any construction stage can be obtained by deactivati ng or activating group of elements loads or boundary condition s from the following or the preceding construction stages This hypothes is assumes that the construction process can be simulated by linear superposition of stages see 1 19 The drawback of this simula tion is that informat ion from the following backward approach or the preceding forward approach construction stages is re quired Therefore geometry and stress state of intermedi ate con struction stages cannot be directly analyzed that is without simulating all the following or preceding constructi on stages and the simulation is more time consumin g Furthermore the whole geometry and stress state history of the preceding or follow ing construction stages have to be stored It is to highlight that both the computation time and the computer storing capacity are of critical importance when the construction process is intro duced into any optimization process 21 24 An alternative to the superpos ition principle in which each construction stage is analyzed by an iterative process named shape fi nding analysis is proposed in 14 The fact that an iterative process is required to defi ne each construction stage is time consumin g To overcome all these problems a new and innovative algo rithm the Direct Algorithm DA is proposed in this paper for the direct simulation of the construction process of cable stayed bridges This algorithm is applied to the temporary support erec tion method In this construction process the bridge superstruc ture is fi rst erected on a set of temporary and permanent supports and then during the tensionin g process the load counter balanced by the temporary supports is successively transmitted to the stay system see 25 The DA introduces for the very fi rst time the unstressed length of the stays concept into the simulation of the construction process of cable stayed bridges Using this concept enables a simulatio n of the geometry and stress state of intermediate constructi on stages without applying the superposition of stages principle A simula tion without the superposition principle has many advantages For example any construction stage can be analyzed directly by an independen t Finite Element Model FEM as no information from the following or the preceding construction stages is re quired Furthermore forward simulation without any overall iter ative process can be carried out The main trade off of not using the superposition principle is that time dependent phenomena cannot be easily simulated Hence DA is proposed here for steel bridges However as in many bridges designers neglect the effects of these phenomena the method still might be applied by many practitio ners in the initial design and construction of concrete structure s The DA is characterized by its simplicity and it can be implemented in any structural analysis software that computes temperature increments or imposed strains All these characteristics make the DA especially suited for optimizati on problems The paper is organized as follows In Section 2 the main hypotheses of the DA are explained In Section 3 the DA is com pared with two alternative methods proposed in the literature the Backward Algorithm BA and the Forward Algorithm FA In Section 4 the application of the DA to simulate the construction process of a cable stayed bridge is presented Furthermor e the re sults of the DA are compared with those obtained by the FA Finally in Section 5 some conclusions are drawn 2 Direct algorithm In this section one of the main innovations of the Direct Algorithm DA the applicati on of the unstressed length of the stays concept into the simulation of the construction process of cable sta yed bridges is fi rst described Then the simulatio n of the unstressed lengths by mean of imposed strains in the stays in the Objective Service Stage OSS eOSS is presente d Next the simulatio n of the tensioning process of the DA is described Then the Local Iterative Process Local IP required to simulate the rais ing of the temporary supports during the tensining process is pre sented Next the computation time of the DA is analyzed Finally the fl owchart of the DA is presented the algorithm is applied to a specifi c case study and its results are compare d to those given by other alternativ e methods proposed in the literature 2 1 Unstressed length of the stays concept A length based adjustment is sometimes used to introduce the tensionin g of the stays on site This is the case of cable stayed bridges with prefabricated cables In this procedure the stay n is made to accurately measure a so called unstressed or neutral length L0n This is the length of a given cable n when it does not have any axial stress or strain This length is measure d when the cable rests horizontally on a support that counterbala nces the effects of its own self weight see Fig 1 A The unstressed length is an intrinsic parameter that is independent of the condi tions to which the stay is subjected on site In Fig 1 A L0nis com pared with the length of the same stay in the un deforme d geometry on the FEM of the bridge Ln this is to say the length gi ven to the stay element in the stiffness matrix when neither loads nor imposed strains are applied in any element of the model In the constructi on process it is necessary to stress the stay on site from L0nuntil its ends occupy the position of the anchorages in the de formed geometry This stress changes the geometry of the bridge and the stay length is changed from L0nto the stressed length LSn as presented in Fig 1 B This elongation of the stay is equivalent of introducing an imposed strain en With DLnbeing the increment of length in the stay the value of this strain can be calculated as presente d in Eq 1 Fig 1 Defi nition of the different lengths of a stay A Installation of the fi rst stay with a length L0n on site in the un deformed geometry Ln B Stressed length Lsn when the stay is elongated and length Ln Unfi lled pylon shows the un deformed geometry and fi lled pylon the deformed one when the stay is installed and stressed J A Lozano Galant et al Computers and Structures 121 2013 64 7565 en DLn L0n LSn L0n L0n 1 According to 26 in ordinary cable stayed bridges enpresents a value close to 0 003 The main inconven ient of the length based adjustment to stay tensionin g on site is its high sensitivity to geo metrical tolerances 26 Therefore this technique is rarely used on practice and only few applicati ons such as the Oresund Bridge 27 or the Hwamyung Bridge 28 can be found in the literature With Nnbeing the axial force of this stay E being the Elastic modulus of the cable material and Anits cross sectional area L0n can be determined from LSnas presente d in Eq 2 L0n LSn Nn EAn LSn 2 The DA proposes the application of the unstressed length con cept in the simulation of the construction process of cable stayed bridges To do so this algorithm assumes that extendin g a L0n length stay on site to get a given force Nnproduces the same effect that shortening an Lnstay in the model to get such force This shortening can be simulated by an imposed strain in the OSS eOSS n which is related to Ln LSnand L0n The DA includes the un stressed length into the simulation by introducing in the stay ele ment of the FEM a shortening eOSS n This is valid when the same stressed length LSn and the same axial force are obtained by both procedures as presented in Eq 3 LSn L0n1 Nn EAn Ln1 eOSS n Nn EAn 3 The advantage of using the unstressed length concept in the simulation of the construction process of a cable stayed bridge is that it enables a direct simulation of the tensioning operations that does not require the superposition of stages principle Further more it provides a physical meaning to the tensioning operation s In the next section the calculatio n of the strains in the OSS eOSS n is presented 2 2 Calculation of the strains in the OSS In early stages of design a target geometry and or stress state in service known as the OSS is usually defi ned by the designer This stage can be characteri zed by a set of stay forces Some of the main criteria used to defi ne these forces in the OSS are summarized in 1 Among these methods it is remarkable the Rigidly Continuous Beam Criterion 29 30 In this method the stay forces are defi ned by projecting into the stays direction the vertical reaction of an equivalent continuous beam It can be proved that the axial forces of the stays defi ned by this criterion minimize the bending energy of the structure 20 Independentl y of the criteria used once defi ned the stay forces of every stay in the OSS they can be clustered into a vector NOSS N 1 size These forces can be obtained by the sum of a passive NP and an active state NA as presented in Eq 4 The former vector includes the passive stay forces produced when the Target Load TL is applied into the structure and the latter one includes the effect of the prestressi ng of the stays fNOSSg fNPg fNAg fNPg IM feOSSg 4 The active state is simulated in the computer software by intro ducing imposed strains eOSS into the stays The vector of imposed strains eOSS must be calculated taking into account the stiffness of the structure To do so an Infl uence Matrix IM is required This matrix includes the effect of the prestressing of every single stay in the rest of stays in terms of axial forces The only unknown of Eq 4 is eOSS and it can be directly defi ned by mean of the in verse of IM IM 1 as presente d in Eq 5 feOSSg IM 1 fNOSSg fNPg 5 The highest values of IM are found in terms located in the main diagonal This implies that no row of the matrix can be ex pressed as a linear combination of the other rows of the matrix and therefore IM 1usually exists 2 3 Simulatio n of the tensionin g process The tensioning process describes the tensioning operations to be introduced by the jack on site to transfer the load from the tem porary supports to the stay system At the end of these operation s the Objective Completion Stage OCS is reached This stage can be directly obtained from the OSS by subtracting from the TL the effect of permanent and or live loads introduced after the tensioning process The tensioning operations can be summarized in a Tensioning Matrix TM as the one presented in Fig 2 where known values are highlighted in italic and unknown values are highlighted in bold With K being the number of construction stages this matrix is formed of K rows and three columns the fi rst column describes the stay that is prestressed in each stage the second column de scribes the force to be achieved with the jack and the third one de scribes the imposed strain to be introduced in the DA For a forward simulatio n this matrix is computed from up to the bottom To com ply with limit stresses during construction it is not usually possi ble to apply the fi nal prestressing to the stays when they are fi rst installed The balance between the simplifi cation of site works and the structural necessities during constructi on generally results in stays being adjusted at least in two stages 26 The stay forces in the fi rst tensioning operation do not infl uence the achievemen t of the OSS Therefore these forces highlighted in italic in Fig 2 are traditional ly defi ned directly by most of the designers as a percent age b usually between 70 and 85 of the stay forces in the OSS Stay forces in the fi rst tensioning operation can also be calculated indirectly as those obtained when a percentage of the imposed strain in the OSS is introduced Stay forces in the second tensioning operation highlighted in bolt in Fig 2 are always unknown Tradi tionally in a forward approach these forces are defi ned by an Over all Iterative Process Overall IP Nevertheles s the introduction of eOSSas imposed strain in the second tensionin g operation guaran tees the achievemen t of the OSS at completion without the need of any Overall IP As presente d in Fig 2 the stay forces in TM can be enlarged into a Force Matrix FM that describes the stay forces of all in stalled stays throughout the tensioning process The active forces introduce d by the jack in each stage that is the values of TM are framed in FM As cable stayed bridges are highly statically redundant struc tures the prestressing of a single stay modifi es the prestressing of all the previously installed stays For this reason in addition of the unknown forces of TM a new set of unknowns highlight ed in bold in Fig 2 are found in FM Knowing accurate values of these forces is especially advisable for construction control on site The application of the concept of unstressed length during con struction on site implies the installation and prestressing of the L0 long stays However th