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Analytic Model of Long Span Self Shored Arch Bridge Zhong Liu1 Fang Li2 and W M Kim Roddis3 Abstract An innovative self shoring staged construction method was developed to build the world s longest reinforced composite concrete arch bridge across the Yangtze River at Wanxian in Chongqing China The method uses a steel tube truss frame constructed by the conventional cantilever launching technique This steel frame with concrete fi lled tubes performs the dual role of arch falsework and arch main reinforcement for the fi nal reinforced concrete arch bridge An optimized schedule for concrete placement was proposed to control the stresses defl ections and stability of the arch rib during construction The time dependent effects of concrete the nonlinear stress strain relationship of steel and concrete as well as the geometric nonlinearility were considered Control information at various stages of construction can be provided using the model developed A program was developed to conduct parametric studies for selection of the fi nal construction scheme and to direct the construction progress by monitoring and comparing actual and predicted stress and defl ection DOI 10 1061 ASCE 1084 0702 2002 7 1 14 CE Database keywords Bridges arch Bridges spans Analytical techniques China Steel frames Concrete reinforced Falsework Introduction The primary advantage of an arch bridge is that compression is the dominant stress induced in the arch under uniform loading Materials such as stone and concrete with low cost and high compressive strength are well suited to the arch form These heavy material arch bridges have historically been used in small and medium spans and were constructed using a full shoring sys tem Advances in the use of high strength concrete steel and concrete steel composites in recent years have signifi cantly re duced the weight of the structure and have extended the limits of arch bridges to longer spans The concrete fi lled tubular steel composite section combines the primary advantages of steel and concrete for arch structures The thin walled steel tube provides the required structural stiffness while minimizing steel weight The concrete core provides the required compressive strength as well as support to prevent local buckling of the steel tube In return the confi nement provided by the steel tube to the concrete core improves its capacity and performance Much recent re search has been conducted in the use of concrete fi lled tubular steel composites to establish acceptable design and construction procedures for both buildings and bridges Zhou and Zhu 1997 For long span arch bridge construction it is often diffi cult and costly to erect a temporary shoring system especially for bridges across deep water channels Construction methods developed to reduce and even eliminate the requirement for shoring include the cantilever launching method as well as the horizontal and vertical swing methods The cantilever launching method uses main and auxiliary cables to maintain stability and balance during construc tion The horizontal swing method begins with prefabricating a complete half leaf of an arch rib parallel to the river on both banks with their ends supported on spherical hinges at the abut ments With the help of balance weights and hydraulic jacking equipment the two prefabricated ribs can be rotated horizontally to the closure position Using the vertical swing method two half leaves of arch ribs are fabricated at ground level to save shoring cost and then rotated to the closure position The com bined use of horizontal and vertical swing techniques is also fea sible for lightweight arch bridges Construction methods to main tain balance and stability become more diffi cult as the span increases and as the arch weight increases The Wanxian Yangtze River Bridge Yan and Yang 1997 is a record breaking design for a reinforced concrete arch bridge with a main arch span of 420 M Fig 1 It was completed in July 1997 Fig 2 The total length of the bridge is 856 meters The north approach consists of eight simple spans of 30 7 m and the south approach consists of fi ve simple spans of 30 7 m The bridge carries four lane traffi c and two pedestrian sidewalks The arch is a catenary with a rise to span ratio of 1 5 A 16 m wide and 7 m deep three cell reinforced concrete box section was selected for the arch rib Fig 3 If the conventional cantilever launching method were used crane lifting capacity as well as number of precasted units would be very high In addition a very large temporary balance tower system would be required to maintain the balance and stability of the massive cantilever arch rib For these reasons construction cost of the arch would be much higher than that for other design alternatives To reduce the cost and complexity of construction a new self shoring construction method was developed The method uses a truss frame fabricated with steel tubes by the conventional canti lever launching technique This steel tube frame performs the dual role of arch falsework and arch main reinforcement After the steel tube truss frame is completed concrete fi ll is pumped into 1Professor Dept of Bridge Engineering Chongqing Jiaotong Univ Chongqing 400074 People s Republic of China and Graduate Research Assistant Dept of Civil and Environmental Engineering Univ of Kan sas Lawrence KS 66045 E mail zliu falcon cc ukans edu 2Senior Bridge Engineer Imbsen and Associates Inc Sacramento CA 95827 3Professor Dept of Civil and Environmental Engineering Univ of Kansas 2008 Learned Hall Lawrence KS 66045 Note Discussion open until June 1 2002 Separate discussions must be submitted for individual papers To extend the closing date by one month a written request must be fi led with the ASCE Managing Editor The manuscript for this paper was submitted for review and possible publication on August 4 1998 approved on March 12 2001 This paper is part of the Journal of Bridge Engineering Vol 7 No 1 January 1 2002 ASCE ISSN 1084 0702 2002 1 14 21 8 001 50 per page 14 JOURNAL OF BRIDGE ENGINEERING JANUARY FEBRUARY 2002 the steel tubes to increase the capacity of the truss frame system The stiffened truss frame is then encased by subsequent concrete placements to become the main reinforcement of the completed arch section Once the reinforced concrete arch is in place the columns spandrel beams and deck system are constructed The self shoring staged construction method leads to reduced weight incorporation of shoring into the fi nal load carrying structure and savings in construction equipment and labor The procedure and primary advantages of this technique are summarized in Table 1 and Fig 4 Each time a new concrete lift gains strength the stiffness and capacity of the cross section increases so that the weight of each subsequently placed lift is carried by the steel tube frame and the preceding concrete lifts The stress distribution in the steel and concrete depends on the selected construction sequence The con crete material properties at each stage depend on the current strain as well as time dependent effects such as creep shrinkage aging and temperature Geometric nonlinearity due to large dis placements may also affect the stress distributions A study was conducted to select a concrete placement se quence that would lead to a better load distribution minimized defl ection and minimized steel requirements without allowing premature yielding of steel truss frame As a result of this study a nine stage erection scheme was selected Fig 5 The subdivision of the arch for concrete placement is shown in Fig 6 Constitutive Modeling Since the steel tubes frame members comprise the initial self shoring structure of bridge they have large stresses and elasto plastic behavior must be taken into account Since the concrete layers are placed at different times they have not only large dif ferences in stress levels but also large differences in tangent modulus The constitutive laws adopted for the analysis corre spond to those specifi ed in Sargin s equations Sargin et al 1971 for concrete and the bilinear equation for steel Fig 7 Since the concrete layers were placed in stages time dependent effects such as temperature loading age creep and shrinkage need to be taken into account to predict the stress dis tribution and defl ection at various construction stages High strength concrete 60 MPa with relatively low ductility was used for the construction of the Wanxian Yangtze River Bridge ACI 2091992 The stress strain relationship for the concrete was lim ited to the elastic range However for the well confi ned concrete core in the steel tubes the ideal elastic plastic relationship was used Neville et al 1983 Bazant and Wittmann 1982 The labo ratory tests for loading were conducted to obtain discrete values of creep and shrinkage at 7 14 28 and 90 days Using the ex perimental data the creep and shrinkage functions may be ap proximated by a Dirichlet series as given below Ketchum 1984 The creep compliance function J t t may be expressed as J t t 5 1 Ec i51 4 ai t 12e2li t2t 1 Fig 3 Cross section of arch rib Table 1 Construction Schedule Age days Construction steps 12Completion of steel truss frame by cable crane 12 29 Pumping concrete fi ll into steel tubes 25 109Placing bottom slab of middle cell in twelve steps 109 154Placing lower portion of interior webs in six steps 154 205Placing upper portion of interior webs in six steps 209 253Placing top slab of middle cell in eight steps 263 275Placing bottom slab of exterior cells in four steps 278 322Placing exterior web in four steps 327 339Placing top slab of exterior cells in four steps 344 398Erecting columns of bridge 400 426Erecting top spandrel beams of bridge 429 500Erecting bridge deck 865One year late after completion 1 230Two years late after completion 1 595Three years late after completion Fig 1 Wanxian Yangtze River Bridge layout Fig 2 Bridge open to traffi c JOURNAL OF BRIDGE ENGINEERING JANUARY FEBRUARY 2002 15 The shrinkage function sh t t may be expressed as sh t t 5 i51 4 shi t 12e2li t2t 2 where retardation coeffi cients are l151 l250 1 l350 01 and l450 001 respectively Ec5elastic modulus of concrete ai t 5creep compliance coeffi cients shi t 5shrinkage coeffi cients Table 2 t5observation time in days and t5loading age in days Finite Element Formulations Staged construction presents engineers with a diffi cult analysis problem At each stage of construction the cross section changes its weight strength and stiffness as the new layers of concrete are placed An explicit model requires the use of different types of elements and assumptions for compatibility between layer inter faces Such an explicit approach often becomes impractical when a large number of parameter studies are required for the selection of a better solution among possible alternatives A simplifi ed ap proach is to model the composite section as an equivalent thin walled beam member for each stage from bare steel to partial or complete composite cross section This equivalent thin walled beam approach has been proposed elsewhere and the analytical results based on this approach has been experientially verifi ed to be quite effective The 3D beam element was used to model the composite box section of the arch ring The element geometry is defi ned with respect to a local coordinate system Each element is divided into a discrete number of layers The geometry of each layer is defi ned by its area and position with respect to a fi xed local coordinate system and each layer is in the state of uniaxial stress defi ned by a given nonlinear stress strain relationship and time dependent Fig 4 Construction scheme a Construction of truss frame b Pumping concrete fi ll into steel tubes c Construction of concrete layers Fig 5 Construction scheme of cross section Fig 6 Working subdivision for concrete placements Fig 7 Constitutive laws for a concrete b steel 16 JOURNAL OF BRIDGE ENGINEERING JANUARY FEBRUARY 2002 effects of materials In different construction stages the shifting of the centriod and twist center can be accounted for by a trans formation matrix The bracing members of the truss frame can be idealized as an equivalent thin walled layer with conversion thickness d of d5 2EA sin2w cosw Gs 3 where E and G5elastic and shear modulus respectively A 5area of the inclined truss member of truss s5truss spacing and w5inclined angle The geometric nonlinearity due to the relatively large displace ment of the steel truss frame was also included in the formulation Spillers 1990 Chen and Agar 1993 The formulation is based on the following assumptions 1 Plane sections remain plane i e linear strain distribution 2 No slippage between the interface of steel and concrete or concrete layers of different ages 3 The nonlinear stress strain relationship for concrete or steel layers is defi ned in a state of unaxial stress 4 The creep functions of tension and twist are similar to those of compression 5 No distortion of cross sections and 6 Small strains but displacements and rotations can be mod erately large To include all time dependent effects and nonlinearities at the time stage t for concrete layers the incremental total strain D can be obtained as D 5D e1D p1D c1D sh1D Te 4 The incremental elastic strain D e and incremental plastic strain D p satisfy the equation D e1D p 5Ds Ec t 5 where Ds5incremental stress Et t 5tangent modulus for this time and this stress state D c5incremental creep strain D sh 5incremental shrinkage strain D sh and D Te5incremental temperature strain Therefore the linear relation of initial stress and strain is Ds5Et t D 2D 0 6 Here the incremental quasi initial strain D 0is D 05D c1D sh1D Te 7 From these assumptions the linear relation of initial shear stress Dt and shear strain Dg is Dt5Gt t Dg2Dg0 8 The generalized quasi initial stress strain relationship can be used to establish the global analytical model and the incremental equilibrium equation at a given time step can be expressed as TKe1TKG q5T1DTR2TFint1F0 9 where TKe5elastic matrix at time T TKG5geometric matrix at time T q5nodal diplacement T1DTR5external force at the time T1DT TFint5internal force at time T and F05incremental quasi initial strain matrix from T to T1DT Because the centroid and shear center of cross sections change as construction of the arch box proceeds it is necessary to estab lish a new fi nite element method to analyze this kind of structure Referring to Fig 8 consider the displacements of an arbitrary point on the cross section in the y and z directions for small but fi nite deformations The displacement components ui vi and wi may be expressed in terms of axial displacement of the centroid u the shear center displacements vand w and the angle of rotation ux about the shear center Thus ui5u2 y2y0 n x2 z2z0 w x 10 ni5n1ux z2za 11 wi5w2ux y2ya 12 The coordinates of the centroid relative to the original point of elemental orthogonal coordinate system are y0and z0 The coor dinates of the shear center relative to the original point of elemen tal orthogonal coordinate system are yaand za During the fi nite element analysis the forces and displace ments of differing coordinate systems need to be transformed Because the shear center and centroid can be different at each construction stage the fi xed reference local coordinate system has to be used to describe the total beam element The displacements and the element forces will be translated to fi xed reference local coordinate system before being stored If the cross section of an element is composed of different pieces here called layers to emphasis the staged manner of laying down each arch component every layer has its own modulo con struction time sectional features and location in the principal Table 2 Creep and Shrinkage Coeffi cients of Concrete Tested Age days a1a2a3a4sh1 m sh2 m sh3 m sh4 m 70 4270 3160 3750 7292 83154 0243 02127 0 140 2130 2320 4010 2623 6935 7210 0223 5 280 1770 1250 4670 0721 83227 9225 0297 6 900 0810 1530 1970 9768 0426 458 1127 0 Fig 8 Composite beam element JOURNAL OF BRIDGE ENGINEERING JANUARY FEBRUARY 2002 17 reference coordinate system Supposing the section of a layer is a rectangle and based on assumption 1 sectional features can be described in terms of only a few variables with compact data structures as can the stress and strain of a layer Just like a section of normal beam we can establish relation equations between them By means of these data structures the incremental stress and strain for the composite system can be added and stored easily although the centroid and shear center change in each step of the calculation The nonlinear factors of creep shrinkage and temperature are analyzed by quasi initial strain method Geometrical and material nonlinearities are analyzed by Newton Raperson iteration Based on the resulting according to analysis of the design scheme the latter nonlinearities have little infl uence during construction stages Therefore time consuming iteration can be avoided in construction control analysis Design Phase Analysis The analytical model based on the selected construction scheme and working schedule Table 3 was used to provide an initial set of construction control data In the analysis the cross section is subdivided into 44 component layers Fig 9 Layers 1 10 are assigned to steel tubes layers 11 20 are assigned to core concrete fi lling the steel tubes layers 22 24 32 34 36 42 and 44 are assigned to mild reinforced steel and the rest of the layers are assigned to concrete The stability of the arch structure is sensitive to geometric imperfection and load eccentricity The critical load is infl uenced by the location and distribution of the loads and the variation of the fl exural and torsional stiffness of the cross section Two types of analyses were conducted to evaluate the stability and ultimate strength of the global structure at each construction stage and to avoid a loading scheme that add