Time-dependent analyses of segmentally constructed balanced cantilever bridges
Introduction
Prestressed segmentally constructed concrete bridges are sensitive to a long-term increase in deflection and are often subjected to an increasing long-term deflection. The total vertical displacement of such bridges is a result of a large downward displacement due to the dead load, live loads and a large upward displacement due to prestress. The long-term increase in displacements is of great importance for the serviceability, durability and reliability. Due to this, it is important to be able to obtain accurate predictions of the deformation of these bridges during construction and their service life. Several bridges have been closed or repaired due to excessive deflection before the end of their initially assumed service life. The cost of a reduced service life is tremendous for society, the owners and users.
Box-girder bridges are traditionally analysed according to theory of bending where the cross-sections are assumed to remain plane. This theory is, however, too simplified to capture the deformation of box–girder bridges accurately. The main deficiency of this theory is that it cannot capture the shear lag effect in the slabs due to the dead weight and the prestress. The shear lag causes a nonlinear distribution of normal stresses over the top and bottom flanges in the cross-section. Neglecting the shear lag effect may lead to a considerable underestimation of the long-term deflections of box–girder bridges. A 3D finite element (FE) model consisting of shell or solid elements can automatically capture the effect of shear lag and can also capture the effects of differential shrinkage and drying creep if suitable material descriptions are used.
Few examples where time-dependent effects have resulted in cracking in cast-in-place balanced cantilever bridges are found in the literature. The literature regarding time-dependent effects in this type of bridge mainly focuses on large long-term deflections [1], [2], [3], [4], [5]. In the study of Kristek and Vrablik [6], a program to optimize the tendon layout to counteract the increasing long-term deflections is presented. Previous stepwise analyses of balanced cantilever bridges include visco-elastic creep, but do not include a non-uniform shrinkage rate, in studies of precast concrete [7] and with the prestressing implemented as equivalent nodal forces [8].
After only a few years of service, two similar bridges in Sweden, both segmentally constructed with the balanced cantilever technique, had to be closed to traffic due to extensive cracking in the webs. The hypothesis is that the cracks in these bridges are due to the stressing of the tendons in the bottom flange in combination with the fact that there are no tendons in the web. The purpose of this paper is to report a study of the influence of time-dependent effects in the construction stage. It is particularly important to study which effects are likely to cause cracking and must therefore be included in order to create an accurate model that can describe the cracking. This study is based on a finite element analysis of a segmentally constructed balanced cantilever bridge that describes the stepwise construction with the nonlinear time-dependent development of the material properties.
The Gröndal bridge and the Alvik bridge had to be closed to traffic due to extensive cracking in the web of their box–girder sections. These cracks were first found only a few years after service. The bridges are parts of the light-rail commuter line in Stockholm, Sweden. The inclined web cracks were first observed in an inspection only 2 years after completion. Subsequent bridge inspections showed that the cracks were increasing both in number and in size. The largest cracks were observed near the quarter-point of the webs at the inside of the box–girder section and they were up to 0.6 mm wide. Since the webs were more cracked on the inside of the box–girder section it was considered probable that thermal effects, under summer conditions, might be one of the factors causing the cracks. The designers feared that a shear failure might be imminent unless the bridges were closed to traffic. The inclined web cracks were initially assumed to jeopardise the ultimate limit safety. During a temporary closure, the bridges were strengthened. Information regarding the strengthening using a combination of carbon-fibre laminates and vertical Dywidag tendons can be found in the literature [9], [10], [11]. A previous investigation [12] suggested that the cracking was due to inadequate shear reinforcement in the webs in the serviceability limit state.
Both the Gröndal bridge and the Alvik bridge are prestressed continuous hollow box–girder bridges. The Gröndal bridge consists of 11 spans with a total length of 430 m. Fig. 1 shows the elevation of this bridge. The main and the two adjacent spans were constructed with the balanced cantilever construction technique while the side spans were erected span by span on a supporting scaffold. Ten of the twelve piers of the Gröndal bridge have a rock foundation while the remaining two piers are built on piles. The highest pier on the Gröndal bridge is 34 m.
The cross-sectional height of the superstructure is approximately 7.50 m above the piers and about 2.75 m in the mid-span. The webs are relatively slender with a thickness of 0.35 m and have a rather low amount of reinforcement: horizontal reinforcement with a diameter of 12 mm and 200 mm spacing and vertical web reinforcement with a diameter of 16 mm and 200 mm spacing. The amount of reinforcement is increased in the mid-span to a diameter of 20 mm in the horizontal bars. Prestressing cables are provided in the upper flange as they are necessary in the construction stage, and the cables in the bottom flange are post-tensioned after the completion of the superstructure when the centre segment is cast. The tendon arrangement for the main-span is shown in Fig. 2 together with a sketch of the extent of cracking in the web.
The principle of the free cantilever construction method is that a previously cast segment serves as the work basis for the execution of the next segment. A form traveller is attached to the previously cast segment and carries the form work for the new segment that is to be cast. An illustration of a form traveller is shown in Fig. 4. According to Hewson [13], the weight of the traveller used for in-situ construction with the balanced cantilever technique is usually 40–120 tonnes for spans between 50 and 200 m. This interval in the weight of traveller is slightly smaller according to Takács [3] where it typically weights 500–900 kN. After a segment is poured the traveller remains as a support for the newly cast segment until it has reached sufficient strength and can be stressed to the existing cantilever arm with post-tensioned tendons anchored in the new segment [14]. To compensate for the long-term deflections, an upward displacement during cantilevering occurs due to tensioning of the tendons. These planned displacements are commonly referred to as camber.
The main-span of the Gröndal bridge was symmetrically cast from piers seven and eight, see Fig. 1. The cantilever arms consist of 13 segments, each 4 m long, from the piers, and where two adjacent cantilevers meet they are joined with one 1.4 m long centre segment to close the structure. The segments were cast with a travelling form at intervals of 1 week.
Section snippets
Finite element analysis
The analyses presented in this paper have been performed with the finite element (FE) software Abaqus/Standard 6.7 [15]. The modelling approach used is a three-dimensional model using shell elements. Thereby, the FE model automatically captures the effect of shear lag. A numerical analysis with the program is divided into steps, each corresponding to a load change from one magnitude to another. In this case, a step represents the casting of one segment. The segmental casting has been modelled
Results
To study the influence of the different time-dependent material properties and effects, several different models have been performed. It is of interest to study their impact on the deflections and the stresses in the cantilevers during the construction process. It is especially of interest to see if the built-in stresses from tensioning, the bottom tendons could be sufficient to initiate cracking in the webs. The development of the elastic modulus, creep, weight of the form traveller,
Discussions and conclusions
The results show that to create a finite element model of the Gröndal bridge that can accurately describe the cracking in the webs, the time-dependent effects have to be taken into account. Both creep and shrinkage have a large impact during the construction process and models that neglect these effects underestimate the cracking. Creep is important already during cantilevering and, in the case of the Gröndal bridge, neglecting this effect resulted in an underestimation of the deflections of
Further research
One large problem with most FE material models for describing creep is that they cannot be combined with models which describe cracking. Some sort of approximation has to be made to combine these two effects in a single analysis. The subject for future research will be different methods to combine the visco-elastic effect of creep and the cracking of the webs. One approach might be to assign different properties to different regions, where the regions subjected to high tensile stresses are
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