geometric nonlinearity of cable stayed bridges

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Egyptian Society of Engineers


Bridge Engineering Conference, March 2000

Sharm El Sheikh, Egypt

GEOMETRIC NONLINEARITY OF SINGLE-PLANE CABLE-STAYED BRIDGES DURING CANTILEVERING

Hassan HEGAB1, Osama TAWFIK2, Mohammed T. NEMIR3,and Hesham NOUR El-DEEN4

Summary

The geometric nonlinearity of single-plane cable-stayed bridges during cantilevering isinvestigated. A three-dimensional geometric nonlinear finite element implementation for atwo-node beam-column element based on the theory of torsional-flexural behaviour isapplied. A real bridge (Aswan cable-stayed bridge) is taken for the sake of numerical study.The results show that cable-stayed bridges, during cantilevering, are highly geometricallynonlinear. The results also clarify that while the vertical deflections of the deck duringconstruction are much more than those developed due to live loads, the forces in the staycables are slightly less than those caused by live loads on the completed bridge.

Keywords: Cable-stayed bridges, Single-plane, Construction, Cantilevering, Finite element.

1. Introduction

During the construction of cable-stayed bridges by the cantilevering method, the erectedpart of the bridge carries its own weight and the weight of the construction equipment. Atthis phase, the statical system of the bridge is double-stayed cantilever girder, which has aremarkably less stiffness than the final statical system of the entire bridge. The objective ofthis investigation is to study the behaviour of single-plane cable-stayed bridges duringcantilevering.A three-dimensional geometric nonlinear finite element formulation is used. The threesources of geometric nonlinearity in cable-stayed bridges namely; axial force-bendingmoment interaction, sag of stay cables, and change of bridge geometry due to largedisplacements, are taken into account in the analysis.

2. Construction by the Cantilevering Method

1 Professor of theory of structures, Faculty of Engineering, Ain Shams University, Egypt2 Professor of theory of structures, Faculty of Engineering, Menoufia University, Egypt3 Associate Professor of structural engineering, Faculty of Engineering, Menoufia University, Egypt4 Structural engineer, Arab Consulting Engineers (ACE), Egypt, E-mail [email protected]

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Bridge Engineering Conference 2000

This method is very often employed in the construction of cable-stayed bridges wheretemporary supports are not recommended. It may increase the required cross section of thedeck compared with that required for the final stage to accommodate the increasedmoments and shear forces during construction (Podolny 1976). Cantilever construction of abridge deck is carried out by providing a succession of segments, where each segmentplaced carries the weight of the next segment and, on occasion, the weight of the formwork.Each segment is integrated with the previous one as soon as it becomes strong enough. Itthen becomes self-supported and, in its turn, is the starting base for a new segment. Thestability of the resulting cantilever is secured, at each step of construction, either by hightensile bolts and welding in case of a steel deck, or by longitudinal prestressing cables incase of a concrete deck. These prestressing cables are set on the upper fibres of the deck(Mathivat 1983). For concrete bridges, segments may be casted in-situ in mobile forms, orprefabricated, transported, and set in place with an appropriate lifting device.

3. Stiffness Matrices for Beam-Column Element

For the bridge deck and pylon elements, it is efficient to use a geometric stiffness matrix tomodify the elastic stiffness matrix of each element. The element stiffness equation isexpressed by:

( 1 )

in which { }P is the nodal forces column vector (Barsoum 1970), [ ]EK is the elasticstiffness matrix, [ ]GK is the geometric stiffness matrix, and { }i∆ is the column vector ofnodal displacements. [ ]EK and [ ]GK are given by (Nour El-deen 1997) as:

( 2 )

( 3 )

in which { }id is a shape functions column vector, { }D is the matrix of properties of theelement, and V is the element volume (Barsoum1970). Substituting for the shape functionsand their derivatives in Eqs. 3 and 2, respectively, and integrating with respect to thevolume of the element, the elastic stiffness matrix, and the geometric stiffness matrix can bederived. The details of the derivation, and the full matrices are explained in (Nemir 1985,and Nour El-deen 1997).

4. Stiffness Matrices for Stay Cable Element

The inclined single cable forms the basic element of cable-stayed bridges. A rightunderstanding of the deformational characteristics of such cable is therefore essential for thesynthesis as well as the analysis of this type of structures. The use of equivalent modulus ofelasticity to idealize the cable element to a straight linear elastic element is one of the mostpractical methods to analyse stay cables. This concept was firstly introduced by Ernst andhas been verified by some investigators (Podolny 1976, and Gimsing 83). The equivalentmodulus of elasticity over a certain load increment is given by:

{ } [ ] [ ][ ]{ }iGE KKP ∆+=

[ ] { } { }{ }dVdDdK iT

iE ′′= ∫2

1

[ ] { } { }{ }dVdPdK i

T

iG ∫=

3


( 5 )

in which E is the modulus of elasticity of the cable material, A is the cross-sectional area, Ti

and Tf are the initial and final tensile forces in the cable during the load increment, wc is theunit weight of the cable, and Lh is the horizontal projection of the cable. Based on theconcept of equivalent modulus of elasticity, the cable element is idealized to a truss element,consequently, the stiffness matrix in local coordinates of the cable element is given by:

( 6 )

5. Numerical Study

The following example illustrates the behaviour of cable-stayed bridges during cantilevering(Nour El-deen 1997). Aswan cable-stayed bridge over the Nile∗ is chosen for the sake ofnumerical study. The cable-stayed part of Aswan bridge over the Nile is 500 m long. Itconsists of five spans with lengths of 48.92, 76.08, 250, 76.08, and 48.92 m, respectively.The width of the deck is 24.3 m with four lanes of 3.75 m each, median strip of 3 m, and twosidewalks; each of 2.5 m. The bridge has two pylons of height 53 ms over the deck. Thebridge is supported by a single plane of stay cables. Fourteen pairs of stay cables are attachedto each pylon. Figure 1 shows the longitudinal profile of the bridge.

5.1 Construction Stages

Aswan cable-stayed bridge is built by the cantilevering method, with segments cast in-situ inmobile carriage. The segment length is 3.906 m. The construction is divided into 16 stages.Except stages 0, and 15, other stages are standard. Each standard stage is devoted to theconstruction of two stayed segments, two unstayed segments, and the installation of two staycables. Figures 2, 3, and 4 show the bridge at the end of stages 0, 1, and 8, respectively.Stage 0 involves the construction of the pier segment and the first three segments in each ofthe main and side spans. Construction of the key segment is the construction stagenumber 15. At the end of this stage, the bridge is completely built and takes the longitudinalprofile shown in Fig. 1.

5.2 Vertical Deflection during Cantilevering

The bridge is solved during cantilevering at the end of each construction stage four times.Firstly, it is solved linearly using the designed prestressing forces in the stay cables. Secondly,it is solved nonlinearly using the same prestressing forces in the stay cables. Thirdly, thebridge is solved linearly without any prestressing forces in the stay cables, and finally, it issolved nonlinearly without any prestressing forces in the stay cables. At each time, the verticaldeflections of the points of attachment of the stay cables with the deck are determined.

∗ Designed by EEG (France), under construction by Nile company (Egypt) and Freyssinet international(France)

( ) ( )

++

=

2f

2i

2.

T T 421

AETTLw

EE

fihc

eq

[ ]

−

−=

11

11

L

AEK eq

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Although, each stage i of construction, is built on the displaced shape resulting from theprevious stage i -1, it is preferable, especially for comparison purposes, that each stage issolved using the reference shape of the bridge. In order to investigate the vertical deflections, the bridge is resolved under uniform distributedlive load of 700 kg/m2 for traffic lanes and uniform distributed live load of 400 kg/m2 for sidewalks. The vertical deflections of the points of attachment of the stay cables are plotted in theform of curves. Each curve represents a historical study of one point, along the period ofconstruction, and in the case where the bridge is totally loaded by the live loads mentionedabove. Figures 5 to 8 show this historical study for points S1, M1, S9, and M9. The completehistorical studies for all points of stay cable attachments are given in (Nour El-deen 1997).Observation of these Figs. yields to the following comments:1. In case of non-prestressed stay cables, the structure tends to behave linearly during

construction or in service alike.2. In case of applying prestressing forces to the stay cables, the structure tends to behave

nonlinearly under construction or in service alike.3. The main span tends to nonlinearity much more than the side spans.4. Up to construction stage No. 9, the linear and nonlinear results are approximately the

same. From the 10th stage, the nonlinear deflections are much more than the lineardeflections.

5. Although, the intermediate pier affects the deflections of the side span, it has noconsiderable effect on the main span. Also, it affects the case when the stay cables are notprestressed much more than those of the case of highly prestressed.

6. The closing of the bridge affects the vertical deflections. The statical system istransformed from a balanced cantilever to a continuous beam, which is much stiffer. Forthis reason, the difference between the nonlinear behaviour and the linear behaviour isremarkably decreased.

7. Always, the deflections for the case of prestressed stay cables are upward. Deflections aredownward for the case of non-prestressed stay cables. This leads to the fact thatdeflections, and accordingly stresses, can be accurately controlled using appropriateprestressing forces.

5.3 Stay Cables Forces during Cantilevering

The stay cable forces corresponding to the vertical deflections discussed in article (5-2) arealso plotted in the form of curve. Each curve represents a historical study of the stay cablealong the period of construction, and in the case where the bridge is totally loaded by liveloads. Figures 9 to 12 show this historical study for SSC1*, MSC1**, SSC9, and MSC9. Thecomplete historical studies for all stay cables are given in (Nour El-deen 1997). Observationof Figs. 9 to 12 yields to the following comments:1. In case of non-prestressed stay cables, the structure tends to behave linearly during

construction or in service alike.2. In case of applying prestressing forces to the stay cables, the structure tends to behave

nonlinearly under construction or in service alike.3. In case of prestressing the stay cables, the stay cable forces are much more than those in

case of non-prestressed stay cables.

* Side span stay cable No. 1.** Main span stay cable No. 1.

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4. Up to construction stage 9, the linear and nonlinear results are approximately the same.After the 9th stage, the cable forces calculated from the nonlinear analysis are muchmore than those calculated from the linear analysis.

5. The closing of the bridge affects the stay cable forces. The statical system is transformedfrom a balanced cantilever to a continuous beam, which is much stiffer. For this reason,the difference between the nonlinear behaviour and the linear behaviour is remarkablydecreased.

6. The effect of nonlineartity on the cable forces is considerable in both main and sidespans.

6. Conclusions

The following general conclusions are recorded from the present study:1. During construction by the cantilevering method, single-plane cable-stayed bridges

are highly nonlinear structures. The nonlinearity increases with the progress ofconstruction as the cantilever length increases.

2. At the end of construction, the statical system of the bridge is transformed frombalanced cantilevers to a continuous beam. This stiffens the structure.Consequently, the vertical deflections along the stayed girder are remarkablyreduced and the nonlinearity effect is also reduced.

3. The higher stay cable pretensioning forces, the higher nonlinearity behaviour. Forthe case of neglecting the pretensioning forces in the stay cables, the linear solutionis quite reasonable.

4. The pretensioning in the stay cable reduces the vertical deflections, and accordinglythe stresses, developed in the girder. This impels to use temporary stay cables forthe construction of other types of bridges, especially for long span bridges.

7. References

Barsoum, R.S., and Gallahger, R.H. (1970), “Finite Element Analysis of Torsional andTorsional-Flexural Stability Problems”, International Journal for Numerical Methods inEngineering, Vol. 2, pp. 335-352.Gimsing, N.J. (1983), “Cable Supported Bridges-Concept and Design” A Wiley-Interscience, New York.Mathivat, J. (1983), “The Cantilever Construction of Prestressed Concrete Bridges” AWiley Intersience Publication, New York.Nemir, M.T. (1985), “Finite Element Stability Analysis of Thin-Walled Steel Structures”Ph.D. Thesis, University of Salford, U.K.Nour El-deen H. (1997), “Finite Element Analysis of Single Plane Cable-Stayed Bridgesunder Construction by Cantilevering Method” M. Sc. Thesis, University of Menoufia.Podolny, W., and Scalzi, J.B. (1976), “Construction and Design of Cable-Stayed Bridges”Wily Interscience, New York.

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Fig. 2. End of Stage 0

Fig. 3. End of Stage 1

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geometric nonlinearity of cable stayed bridges

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