Research Article Nonlinear Stability Analysis of a

Research ArticleNonlinear Stability Analysis of a Composite Girder Cable-StayedBridge with Three Pylons during Construction

Xiaoguang Deng and Muyu Liu

Hubei Key Laboratory of Roadway Bridge amp Structure Engineering Wuhan University of Technology Wuhan 430070 China

Correspondence should be addressed to Muyu Liu liumuyuwhuteducn

Received 1 August 2014 Accepted 8 August 2014

Academic Editor Zhiqiang Hu

Copyright copy 2015 X Deng and M LiuThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Based on the nonlinear stability analysis method the 3D nonlinear finite element model of a composite girder cable-stayed bridgewith three pylons is established to research the effect of factors including geometric nonlinearity material nonlinearity static windload and unbalanced construction load on the structural stability during construction Besides the structural nonlinear stabilityin different construction schemes and the determination of temporary pier position are also studied The nonlinear stability safetyfactors are calculated to demonstrate the rationality and safety of construction schemesThe results show that the nonlinear stabilitysafety factors of this bridge during construction meet the design requirement and the minimum value occurs in the maximumdouble cantilever stage Besides the nonlinear stability of the structure in the side of edge-pylon meets the design requirementin the two construction schemes Furthermore the temporary pier can improve the structure stability effectively and the actualposition is reasonable In addition the local buckling of steel girder occurs earlier than overall instability under load in some cabletension stages Finally static wind load and the unbalanced construction load should be considered in the stability analysis for theadverse impact

1 Introduction

Due to advanced manufacturing technology and efficientutilization of structural materials cable-stayed bridge hasbeen demonstrated as an economical solution for long-spanbridges and widely applied in recent decades [1] Since cablesinstead of interval piers support cable-stayed bridges they aremuchmore flexible than conventional continuous bridges [2]The axial forces arising from cable tension make the pylonsand girders to be compression-flexure members [3] Whenthe span length increases the structural stability problem ofcable-stayed bridge is more prominent

Studies on the stability of cable-stayed bridges have beencarried out by many researchers Shu and Wang investigatedthe stability characteristics of box-girder cable-stayed bridgesby three-dimensional finite element methods taking intoaccount geometric nonlinearity andmany design parameterssuch as the main span length the cable arrangement andthe type of pylons [4] Yoo et al proposed a modifiedeigenvalue analysis by introducing the concept of a fictitious

axial force to obtain buckling lengths of steel girder membersin cable-stayed bridges which could avoid the problemof generating excessively large buckling lengths in somegirder members having small axial forces [5] Thai and Kimproposed a fiber hinge beam-column model to predict ulti-mate load-carrying capacity and ultimate behavior of cable-stayed bridges which included both geometric and materialnonlinearities [6] Xi et al adopted an energy method ofanalysis for the in-plane ultimate load capacity of cable-stayedbridgeswith different deck andpylon connection patterns [7]Although a number of researchers have investigated this issuefrom different perspectives and some important findingsare generalized almost all of them focus on the stabilityproblem of cable-stayed bridge in completion stage Actuallythe structure of cable-stayed bridge should undergo severalsystem conversions and sustain changing construction loadsduring the cantilever construction stage With the increaseof the cantilever length geometric nonlinearity and materialnonlinearity will cause secondary internal force and reducestructure stiffness which make the structural deformation

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 978514 9 pageshttpdxdoiorg1011552015978514

2 Mathematical Problems in Engineering

and internal force increase As a result the structural stabilityproblem of cable-stayed bridge during construction is moreimportant and more complex

This paper takes Wuhan Erqi Yangtze River Bridge asbackground which adopts the structural style of doublecable plane composite girder cable-stayed bridge with threepylons Cantilever splicing construction method is used toerect composite girder and the maximum cantilever lengthreaches 308m the largest span in the same type of bridgesin the world By using corotational formulation incrementalmethod to analyze beam-column effect and large displace-ment effect ideal elastic-plastic model and broken-linemodel to analyze the material nonlinearity of steel andconcrete respectively the structural nonlinear stability of thisbridge during construction has been researched based on thenonlinear stability analysis method

2 Nonlinear Stability Analysis Method

21 Geometric Nonlinearity Analysis Method Geometricnonlinearity of cable-stayed bridge includes sag effect of staycable beam-column effect and large displacement effect [8]

The cables are assumed to be perfectly flexible and to resistthe tensile force only Under the action of its own dead loadand axial tensile force the deformation of a stay cable consistsof two parts namely the sag and elastic deformation Thetension stiffness of the table which varies depending on thesag is modeled by using an equivalent straight truss elementwith an equivalent modulus of elasticity This concept wasfirst proposed by Ernst [9] The equivalent cable modulus ofelasticity can be given as

119864ep =119864119890

1 + (119860(119902119897 cos120572)2121198793) 119864119890

(1)

where119864ep is the equivalentmodulus of cable119864119890is the Youngrsquos

modulus of cable 119897 cos120572 is the horizontal projected length ofthe cable 119902 is the cable weight per unite length of cable 119860 isthe cross-sectional area of cable and 119879 is the cable tension

Corotational formulation which developed in recentyears can separate the rigid motion of the unit from thedeformation under load by building corotational coordinatesystemwhich canmove with the deformation of the structureand then the rigid motion can be deducted exactly in theincremental formulation which makes the internal forcecalculation more accurate [10] Thus it can be used toanalyze the beam-column effect and large displacement effectaccurately

In Figure 1 119883-119884 is the main coordinate system wherenode coordinate node displacement and corner incre-ment are built 1198831015840-1198841015840 is a corotational coordinate systemwhere the element displacement 120575 can be denoted as[0 0 120579

1015840

1198941199061015840

1198940 1205791015840

119895]119879

In 1199052moment the included angle between corotational

coordinate system and 119883-axis in main coordinate system isas follows

120572 = arctan(V119894minus V119895+ 1198710sin1205720

119906119894minus 119906119895+ 1198710cos1205720

) (2)

0

Y

X

ui

i

j

uj

Configuration in t1 moment

Configuration in t2 moment120579i

120579j

1205720

1205720

1205720

120572

120572120579998400i

120579998400j

Figure 1 Corotational coordinate system

O

E0

EC

E = 01205900

120590c

120590

120576120576u1205760120576c

Figure 2 Stress-strain curve of concrete

The rigid motion of the unit 120573 is as follows

120573 = 120572 minus 1205720 (3)

So the degrees of freedomof plane beam element decreaseto 3 and the relationship with main coordinate system is asfollows

1205791015840

119894= 120579119894minus 120573

1205791015840

119895= 120579119895minus 120573

1199061015840

119895= 119878 minus 119878

0

119878 asymp radic(V119894minus V119895+ 1198710sin1205720)2

+ (119906119894minus 119906119895+ 1198710cos1205720)2

(4)

where 1198780and 119878 are arc length of element before and after

deformation respectively

22 Material Nonlinearity Analysis Method The relationshipbetween stress and strain of concrete and steel is nonlinearAs shown in Figures 2 and 3 thematerial nonlinearity of steeland concrete are analyzed by ideal elastic-plastic model andbroken-line model respectively

ForWuhan Erqi Yangtze River Bridge in the stress-straincurve of deck lab concrete 120590

0is 385MPa and 120590

119888= 144MPa

119864119888= 36GPa 120576

119906= 00033 120576

0= 0002 120576

119888= 00004 and 119864

0

= 150625GPa In the stress-strain curve of pylons 1205900=

324MPa 120590119888= 138MPa 119864

119888= 345GPa 120576

119906= 00033 120576

0=

0002 120576119888= 00004 and 119864

0= 11625GPa In the stress-strain

curve of steel girder 120590119904= 370MPa and 119864

119892= 210GPa

Mathematical Problems in Engineering 3

O

Eg

E = 0120590s

120590

120576120576s

Figure 3 Stress-strain curve of steel

23 Stability Analysis Method The essence of Second-orderstability of point type is to obtain the structural ultimate loadby solving the load-displacement curve based on nonlineartheory

Considering geometric nonlinearity andmaterial nonlin-earity the basic equation of nonlinear stability is given as

(1198700+ 119870120590+ 119870119897) 120575 = 119875 (5)

where 1198700is the small displacement elastoplastic stiffness

matrix 119870120590is geometric stiffness matrix and 119870

119897is large

displacement elastoplastic stiffness matrixWhen the structure integral rigidity matrix 119870

119905(119870119905=

1198700+119870120590+119870119897) is not positive definite the cable-stayed bridge

reaches the instability ultimate stateThe stability safety factor 120582 during construction process

is defined as

120582 =119875119888119903

119875119905

(6)

where 119875119905is load base during construction process which

includes dead load construction load and so forth 119875119888119903

isthe ultimate bearing capacity which can be obtained byincreasing the load until the structure integral rigidity matrix|119870119905| = 0

3 Calculation FEM Model

31 Engineering Situation Wuhan Erqi Yangtze River Bridgewhose layout is shown in Figure 4 adopts the structural styleof double cable plane composite girder cable-stayed bridgewith three pylons The span consists of 90 + 160 + 616 +616 + 160 + 90m and the main span is 616m which is thelargest span bridge of the same type in the worldThe I cross-section of steel girder has two longitudinal stiffeners arrangedat the lateral part of web and the thickness of the concretebridge deck is 26 cm Shear studs are used to connect themand the standard cross-section of composite girder is shownin Figure 5

There are two construction schemes applied on thestructure in the side of edge-pylon Scheme A steel-pipesupport method is adopted on the construction of 160mcomposite girder then the crane is used to erect the 308mcomposite girder in single cantilever untilmidspan closure asshown in Figure 6(a) SchemeB the cranes are used to erect

the composite girder in double cantilever at the beginningwhen the 160m composite girder is erected and the side-spanclosure is done and the crane is used only in one directionuntil midspan closure as shown in Figure 6(b)

The cranes are used to erect the composite girder indouble cantilever during the construction of the structurein the side of midpylon and a temporary pier is set 204maway from the 4 bridge piers which will be dismantled afterclosure as shown in Figure 7

32 Finite Element Model The 3D nonlinear finite elementmodel of this bridge during construction is established byMIDAS program with 3366 elements (3102 beam elementsand 264 cable elements) The material parameters in thismodel are presented in Table 1 and the finite element modelis presented in Figure 8

33 Construction Process Stimulation On the basis of theactual construction process 127 construction stages are builtcontaining the construction of the pylons cyclic splicingof the girder side-span closure midspan closure systemtransformation and secondary dead load Forward-analysisis carried out by activation and passivation function Typicalconstruction stage is displayed in Figure 9

4 Results and Analysis

41 Influence of Geometrical Nonlinearity on Structural Sta-bility By adopting equivalent modulus method to analyzestay cable sag effect and corotational formulation incre-mental method to analyze beam-column effect and largedisplacement effect the geometrical nonlinear stability of thisbridge during construction is researched Comparisons arepresented between results from linear and nonlinear analysis

Computational vertical load includes dead load and craneload Moreover it increases by the same proportion at thesame location in calculation

As shown in Figures 10 and 11 the change betweenthe geometrical nonlinear stability safety factor of structureand the linear elastic stability safety factor is similar butthe former observably reduces the stability safety factor ofstructure When considering geometrical nonlinearity thestability safety factor of the structure in the side of midpylonin the max double cantilever stage drops from 1388 to 748by 461 And that of the structure in the side of edge-pylonin the max single cantilever stage drops from 1348 to 816 by395

As can be seen in Figures 12 and 13 for the structurein the side of edge-pylon and middle-pylon along with theconstruction process the buckling mode turns from thelongitudinal instability of pylon to combined instability ofpylon and main girder

42 Influence of the Different Construction Schemes on Struc-tural Stability The geometrical nonlinear stability safety


90009450

750

Han Kou Wu Chang

(PC

90009450

750

16000 1600061600 61600

21 times 1350

54321 6 7

10 times 135021 times 135021 times 135021 times 135010 times 135011 times 800 11 times 8001000 100019501950 19501950 19501950

173200

154300

girder)(PC

girder)(composite girder)

Figure 4 Bridge layout of Wuhan Erqi Yangtze River Bridge

140 1350

2 2

3230

2644

280

Bridge centerline

1350100 14075 759 cm asphalt concrete pavement

Figure 5 Standard cross-section of composite girder

s6s7s8

z6z5z4z3z2z1JH1 z7 z8 z9 z10 z11 z12 z13 z14 z15 z16 z17 z18 z19z20 z21

s6s5 s5 s8s7s4 s4s3 s3s2 s2s1 s1

7 65

(a) Construction schemeA

s6

z6 z7 z8 z9 z10 z11 z12 z13 z14 z15 z16 z17 z18

s6s5 s5s4 s4s3 s3s2 s2s1 s1

1 23

(b) Construction schemeB

Figure 6 Construction schemes applied on the structure in the side of edge-pylon

Table 1 Each material parameter of structure in model

Structure Material Modulus of elasticity (MPa) Density (kNm3)Steel girder Q370qD 206 times 10

5 7698Pylon C50 345 times 10

4 26Bridge deck PC beam C60 36 times 10

4 26Stay cable PE galvanized steels trend 195 times 10

5 785


m9

z49 z50 z51 z50 z49z51z52z53z54z55z56z57z52 z53 z54 z55 z56 z57 z58

4

m9m7m8 m8m7m6 m6m5 m5m4 m4m3 m3m2 m2m1 m1

(a) Double cantilever construction scheme

Steel girder Temporary pierCent line of 4 pier

204 m

(b) The set of location of temporary pier

Figure 7 Construction scheme applied on the structure in the side of midpylon

Figure 8 Global finite element model

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

(a) Hoist steel girder

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

(b) First tension of cable

Figure 9 Typical construction stage


0 10 20 30 40 50 60 70 80 90 100 110 120 130

Linear elastic stability safety factorGeometrical nonlinear stability safety factor

Construction stage

Stab

ility

safe

ty fa

ctor

40

36

32

28

24

20

16

12

8

4

Figure 10 Stability safety factors of the structure in the side ofmidpylon

0 10 20 30 40 50 60 70 80 90 100 110 120 130


Construction stage

Stab

ility

safe

ty fa

ctor

40

36

32

28

24

20

16

12

8

4

Figure 11 Stability safety factors of the structure in the side of edge-pylon in SchemeA

factors of structure in the side of edge-pylon in the twoschemes are given in Figure 14

SchemeA Because the closure construction of side span hasoccurred cable-stayed tensioning enhances the elastic sup-port of the main pylon However the flexibility of the maingirder structure increases rapidly along with the cantileverconstruction and when it comes to a certain degree verticalinstability of the main girder happens so the geometricalnonlinear stability safety factor of structure in the side ofedge-pylon increases firstly and then decreases The lowestgeometrical nonlinear stability safety factor happens in themaximum double cantilever stage of 816 which meets thedesign requirement for being over 4

Scheme B In the beginning of symmetrical cantileversplicing construction the longitudinal instability of pylonhappens firstly and the geometrical nonlinear stability safetyfactor of structure in the side of edge-pylon changes smoothlyuntil the closure construction of side span which provides anelastic support for the main pylon equally and increases thegeometrical nonlinear stability safety factor of structure from736 to 1446 After the closure construction of side span the

Table 2 Influence of construction load on the structural stability

Structure Dead loadonly

Both dead loadand crane load Diff

Structure in the side ofmidpylon 838 748 107

Structure in the side ofedge-pylon 886 816 79

change is the same as Scheme A So the geometrical non-linear stability safety factor of structure in the side of edge-pylon changes smoothly at first and then increased suddenlyfollowed by a decrease The structural stability in both theconstruction schemes meets the stability requirement

43 Influence of Temporary Pier on Structural Stability Thestructure in the side ofmidpylon adopts symmetric cantileverconstruction technology and the unilateral maximum can-tilever length reaches 3045m Between the 3 bridge pier and4 bridge pier there is a temporary pier 204m away from the4 bridge piersGeometrical nonlinear stability analysis of thestructure with the presence or absence of a temporary pier isresearched respectively to get the influence of a temporarypier on the structural stability

Computational vertical load includes dead load and craneload Moreover it increases by the same proportion at thesame location in calculation The geometrical nonlinearstability safety factors of structure in both situations areshown in Figure 15

It can be indicated from the stability safety factor curvewithout temporary pier that the geometrical nonlinear sta-bility safety factors of structure begin to decrease furtherwhen the construction goes to ZL56 segment (almost 200maway from the 4 bridge piers) However the geometricalnonlinear stability safety factors of structure increase from2004 to 2615 when a temporary bridge pier is set at thisplace to provide the vertical and lateral constraints to themain girder enhance the structure stiffness and reduce thecantilever end deformation of the main girder Thereforeit is reasonable to set temporary pier at the location whichis 204m away from 4 bridge piers when considering thestability

44 Influence of Construction Load on Structural StabilityThe main construction load is 22 ton crane load Thestructure in the side ofmidpylon adopts symmetric cantileverconstruction technology and that of edge-pylon adopts singlecantilever construction technology which makes construc-tion load unbalanced

For researching the influence of construction load onthe structural stability the calculation is divided into twosituations considering dead load only considering both deadload and crane load Load increases by the same proportionat the same location in calculation

As shown in Table 2 crane load makes the geometricalnonlinear stability safety factors of structure in the side ofmidpylon decrease from 838 to 748 by 107 and that of


(a) (b)

Figure 12 Buckling mode of structure in the side of edge-pylon in SchemeA

(a) (b)

Figure 13 Buckling mode of structure in the side of middle-pylon

24

20

16

12

8

40 10 20 30 40 50 60 70 80 90 100 110 120 130

SchemeScheme

Construction stage

Stab

ility

safe

ty fa

ctor

12

Figure 14 Geometrical nonlinear stability safety factors of structurein the two schemes

structure in the side of edge-pylon decrease from 886 to 816by 79

45 Influence of Wind Load on Structural Stability Thisbridge is easily affected by wind load because it stretches

40

36

32

28

24

20

16

12

8

40 10 20 30 40 50 60 70 80 90 100 110 120 130

Decrease further

With temporaryWithout temporary pier

Construction stage

Stab

ility

safe

ty fa

ctor

Figure 15 Geometrical nonlinear stability safety factors of structurein the side of middle-pylon

across Yangtze River So it is important to research theinfluence of wind load on the structural stability duringconstruction The three direction component forces of wind


Table 3 Three direction component forces of wind load in the max double cantilever stage

Wind attack angle Drag factor 119862119867

Lift factor 119862119881

Lifting moment factor 119862119872

minus3∘ 13910 minus02680 001770∘ 12680 minus00640 004303∘ 15420 03160 00512

Table 4 Computational formula of the three direction component forces of wind load

Three direction component forces Lateral wind load Vertical wind load Twisting moment

Computational formula 119865119867=1

21205881198812

119892119862119867119867 119865

119881=1

21205881198812

119892119862V119861 119872 =

1

21205881198812

1198921198621198721198612

Table 5 Influence of wind load on geometrical nonlinear stabilitysafety factor

Structure Dead load only Wind attack angleminus3∘ 0∘ 3∘

Structure in the side ofmidpylon 838 773 816 860

Structure in the side ofedge-pylon 886 823 847 893

load in the max double cantilever stage are obtained by windtunnel testing [11]

The three direction component forces of wind load whichact on unit length of main girder are shown in Table 4where 120588 is sir density with the value of 125 kgm3 in thispaper 119881

119892is static gust speed with the value of 387ms119867 is

projection height of main girder with the value of 45m and119861 is projection width of main girder with the value of 315m119862119867119862119881 and119862

119872are drag factor lift factor and liftingmoment

factor whose values can be obtained from Table 3As shown in Table 5 the influence is greatest when the

wind attack angle is minus3∘ The geometrical nonlinear stabilitysafety factor of structure in the side of midpylon drops from838 to 773 by 78 and that of structure in the side of edge-pylon drops from 886 to 823 by 71 Actually when thewind attack angle is minus3∘ the wind load provides lateral windload and vertical wind load which intensifies the verticalinstability of main girder for the same direction with deadload

46 Influence of Local Buckling on Structural Stability Com-putational vertical load includes dead load and crane loadMoreover it increases by the same proportion at the samelocation in calculation

In some stages of cable tension the local buckling ofsteel girder occurs earlier than overall instability as shownin Figure 16

For example in the stage of the 13th cable tension thegeometrical nonlinear stability safety factors of structure inview of local buckling is 1552 while the value is 2056 foroverall instability Actually when the cable is tensioned steelgirder in both sides should undertake a large lateral bendingmoment but the lateral bending rigidity of I-steel is small Inaddition the bridge deck is not concreted which can provide

Table 6 The stability safety factors of structure with or withoutmaterial nonlinearity

Construction stage Without materialnonlinearity

With materialnonlinearity Diff

Max double cantileverstage 682 218 680

Max single cantileverstage 759 221 701

24

22

20

18

16

14

12

1045 50 55 60 65 70 75 80

Construction stage

Stab

ility

safe

ty fa

ctor

Local bucklingOverall instability

Figure 16 Geometrical nonlinear stability safety factors of structurein view of local buckling

lateral restraint in consequence the local buckling of steelgirder occurs earlier than overall instability under load

47 Influence of Material Nonlinearity on Structural StabilityThe mode of this bridge in max cantilever stage is built byANSYS program where the nonlinear parameters of stress-strain curve of bridge deck C60 pylon C50 and steel girderQ370qD are imported into it

For researching the influence of construction load on thestability the calculation is divided into two situations allthe above-mentioned factors are considered with or withoutmaterial nonlinearity The stability safety factors of structureare obtained by analyzing the load-displacement curve ofpylon and main girder

As shown in Table 6 material nonlinearity has a greatinfluence on structural stability When the stress of maingirder researches to a certain degree the elasticity moduluswill be 0 The nonlinear stability safety factor of structure in


the side ofmidpylon in consideration ofmaterial nonlinearityis 218 larger than 175 which meets the design requirement

5 Conclusions

Nonlinear structural stability analysis ofWuhan Erqi YangtzeRiver Bridge during construction has been carried out in thispaper and the conclusions are organized as follows

(1) The lowest nonlinear structural stability safety factorof structure occurs in the max double cantilever stagewith the value of 218 larger than 175 which meetsthe design requirement Along with the constructionthe geometrical nonlinear stability safety factors ofstructure in the side of midpylon drops from 2907to 738 and that of structure in the side of edge-pylon increases from 841 to 2295 and then drops to816 The diminishing rate becomes large when thecantilever length increases

(2) Both the two construction schemes meet the stabilityrequirement and the location of temporary pier is204maway from 4 bridge piers which is reasonablefor increasing the geometrical nonlinear stabilitysafety factors of structure from 2004 to 2615

(3) The local buckling of steel girder occurs earlier thanoverall instability under load in the cable tension stagedue to the smaller lateral bending rigidity of I-steeland the lack of lateral restraint provided by bridgedeck

(4) Static wind load and the unbalanced constructionload should be considered in the stability analysis forthe adverse impact

(5) The main girder construction period of Wuhan ErqiYangtze River Bridge was from January 2011 toSeptember 2011 The actual structure state matchedwell with theory state during the whole constructionThe structural stability can be guaranteed in eachconstruction stage for the construction scheme isreasonable

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Fund ldquoStability ofHigh Web in Long-span Steel-concrete Composite BridgeBased on the Elastic Rotational Restraint Boundaryrdquo (no51378405) and the Natural Science Fund of Hubei province(no 2013CFA049) These supports are gratefully acknowl-edged

References

[1] R Karoumi ldquoSome modeling aspects in the nonlinear finiteelement analysis of cable supported bridgesrdquo Computers ampStructures vol 71 no 4 pp 397ndash412 1999

[2] Y-C Wang ldquoNumber of cable effects on buckling analysis ofcable-stayed bridgesrdquo Journal of Bridge Engineering vol 4 no4 pp 242ndash248 1999

[3] D-H Choi H Yoo J-H Koh and K Nogami ldquoStabilityevaluation of steel cable-stayed bridges by elastic and inelasticbuckling analysesrdquo International Journal of Structural Stabilityand Dynamics vol 7 no 4 pp 669ndash691 2007

[4] H-S Shu and Y-C Wang ldquoStability analysis of box-girdercable-stayed bridgesrdquo Journal of Bridge Engineering vol 6 no1 pp 63ndash68 2001

[5] H Yoo H-S Na E-S Choi and D-H Choi ldquoStability evalua-tion of steel girder members in long-span cable-stayed bridgesby member-based stability conceptrdquo International Journal ofSteel Structures vol 10 no 4 pp 395ndash410 2010

[6] H-T Thai and S-E Kim ldquoSecond-order inelastic analysis ofcable-stayed bridgesrdquo Finite Elements in Analysis and Designvol 53 pp 48ndash55 2012

[7] Z Xi Y Xi and H Xiong ldquoUltimate load capacity of cable-stayed bridges with different deck and pylon connectionsrdquoJournal of Bridge Engineering vol 19 no 1 pp 15ndash33 2014

[8] P-H Wang H-T Lin and T-Y Tang ldquoStudy on nonlinearanalysis of a highly redundant cable-stayed bridgerdquo Computersand Structures vol 80 no 2 pp 165ndash182 2002

[9] H J Ernst ldquoDer E-modul von seilen unter Beruecksichtigungdes durchhangesrdquo Der Bauingenieur vol 40 no 2 pp 52ndash551965

[10] J M Pajot and K Maute ldquoAnalytical sensitivity analysis ofgeometrically nonlinear structures based on the co-rotationalfinite element methodrdquo Finite Elements in Analysis and Designvol 42 no 10 pp 900ndash913 2006

[11] L Yuan Y Yongxin G Zengwei andM Ting Ting ldquoNonlinearwind-induced static instability analysis for long-span cable-stayed bridgeswith three towers and twomain spansrdquo StructuralEngineers vol 28 no 1 pp 87ndash93 2012

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and internal force increase As a result the structural stabilityproblem of cable-stayed bridge during construction is moreimportant and more complex

This paper takes Wuhan Erqi Yangtze River Bridge asbackground which adopts the structural style of doublecable plane composite girder cable-stayed bridge with threepylons Cantilever splicing construction method is used toerect composite girder and the maximum cantilever lengthreaches 308m the largest span in the same type of bridgesin the world By using corotational formulation incrementalmethod to analyze beam-column effect and large displace-ment effect ideal elastic-plastic model and broken-linemodel to analyze the material nonlinearity of steel andconcrete respectively the structural nonlinear stability of thisbridge during construction has been researched based on thenonlinear stability analysis method

2 Nonlinear Stability Analysis Method

21 Geometric Nonlinearity Analysis Method Geometricnonlinearity of cable-stayed bridge includes sag effect of staycable beam-column effect and large displacement effect [8]

The cables are assumed to be perfectly flexible and to resistthe tensile force only Under the action of its own dead loadand axial tensile force the deformation of a stay cable consistsof two parts namely the sag and elastic deformation Thetension stiffness of the table which varies depending on thesag is modeled by using an equivalent straight truss elementwith an equivalent modulus of elasticity This concept wasfirst proposed by Ernst [9] The equivalent cable modulus ofelasticity can be given as

119864ep =119864119890

1 + (119860(119902119897 cos120572)2121198793) 119864119890

(1)

where119864ep is the equivalentmodulus of cable119864119890is the Youngrsquos

modulus of cable 119897 cos120572 is the horizontal projected length ofthe cable 119902 is the cable weight per unite length of cable 119860 isthe cross-sectional area of cable and 119879 is the cable tension

Corotational formulation which developed in recentyears can separate the rigid motion of the unit from thedeformation under load by building corotational coordinatesystemwhich canmove with the deformation of the structureand then the rigid motion can be deducted exactly in theincremental formulation which makes the internal forcecalculation more accurate [10] Thus it can be used toanalyze the beam-column effect and large displacement effectaccurately

In Figure 1 119883-119884 is the main coordinate system wherenode coordinate node displacement and corner incre-ment are built 1198831015840-1198841015840 is a corotational coordinate systemwhere the element displacement 120575 can be denoted as[0 0 120579

1015840

1198941199061015840

1198940 1205791015840

119895]119879

In 1199052moment the included angle between corotational

coordinate system and 119883-axis in main coordinate system isas follows

120572 = arctan(V119894minus V119895+ 1198710sin1205720

119906119894minus 119906119895+ 1198710cos1205720

) (2)

0

Y

X

ui

i

j

uj

Configuration in t1 moment

Configuration in t2 moment120579i

120579j

1205720

1205720

1205720

120572

120572120579998400i

120579998400j

Figure 1 Corotational coordinate system

O

E0

EC

E = 01205900

120590c

120590

120576120576u1205760120576c

Figure 2 Stress-strain curve of concrete

The rigid motion of the unit 120573 is as follows

120573 = 120572 minus 1205720 (3)

So the degrees of freedomof plane beam element decreaseto 3 and the relationship with main coordinate system is asfollows

1205791015840

119894= 120579119894minus 120573

1205791015840

119895= 120579119895minus 120573

1199061015840

119895= 119878 minus 119878

0

119878 asymp radic(V119894minus V119895+ 1198710sin1205720)2

+ (119906119894minus 119906119895+ 1198710cos1205720)2

(4)

where 1198780and 119878 are arc length of element before and after

deformation respectively

22 Material Nonlinearity Analysis Method The relationshipbetween stress and strain of concrete and steel is nonlinearAs shown in Figures 2 and 3 thematerial nonlinearity of steeland concrete are analyzed by ideal elastic-plastic model andbroken-line model respectively

ForWuhan Erqi Yangtze River Bridge in the stress-straincurve of deck lab concrete 120590

0is 385MPa and 120590

119888= 144MPa

119864119888= 36GPa 120576

119906= 00033 120576

0= 0002 120576

119888= 00004 and 119864

0

= 150625GPa In the stress-strain curve of pylons 1205900=

324MPa 120590119888= 138MPa 119864

119888= 345GPa 120576

119906= 00033 120576

0=

0002 120576119888= 00004 and 119864

0= 11625GPa In the stress-strain

curve of steel girder 120590119904= 370MPa and 119864

119892= 210GPa


O

Eg

E = 0120590s

120590

120576120576s




(1198700+ 119870120590+ 119870119897) 120575 = 119875 (5)



119897is large


119905(119870119905=



is defined as

120582 =119875119888119903

119875119905

(6)


















90009450

750

Han Kou Wu Chang

(PC

90009450

750

16000 1600061600 61600

21 times 1350

54321 6 7


173200

154300

girder)(PC



140 1350

2 2

3230

2644

280

Bridge centerline



s6s7s8



7 65


s6

z6 z7 z8 z9 z10 z11 z12 z13 z14 z15 z16 z17 z18


1 23





5 7698Pylon C50 345 times 10



5 785


m9


4




204 m




minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50


minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50




0 10 20 30 40 50 60 70 80 90 100 110 120 130


Construction stage

Stab

ility

safe

ty fa

ctor

40

36

32

28

24

20

16

12

8

4


0 10 20 30 40 50 60 70 80 90 100 110 120 130


Construction stage

Stab

ility

safe

ty fa

ctor

40

36

32

28

24

20

16

12

8

4


















(a) (b)


(a) (b)


24

20

16

12

8

40 10 20 30 40 50 60 70 80 90 100 110 120 130

SchemeScheme

Construction stage

Stab

ility

safe

ty fa

ctor

12




40

36

32

28

24

20

16

12

8

40 10 20 30 40 50 60 70 80 90 100 110 120 130

Decrease further


Construction stage

Stab

ility

safe

ty fa

ctor












21205881198812

119892119862119867119867 119865

119881=1

21205881198812

119892119862V119861 119872 =

1

21205881198812

1198921198621198721198612




















24

22

20

18

16

14

12

1045 50 55 60 65 70 75 80

Construction stage

Stab

ility

safe

ty fa

ctor









5 Conclusions









Acknowledgments


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










O

Eg

E = 0120590s

120590

120576120576s




(1198700+ 119870120590+ 119870119897) 120575 = 119875 (5)



119897is large


119905(119870119905=



is defined as

120582 =119875119888119903

119875119905

(6)


















90009450

750

Han Kou Wu Chang

(PC

90009450

750

16000 1600061600 61600

21 times 1350

54321 6 7


173200

154300

girder)(PC



140 1350

2 2

3230

2644

280

Bridge centerline



s6s7s8



7 65


s6

z6 z7 z8 z9 z10 z11 z12 z13 z14 z15 z16 z17 z18


1 23





5 7698Pylon C50 345 times 10



5 785


m9


4




204 m




minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50


minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50




0 10 20 30 40 50 60 70 80 90 100 110 120 130


Construction stage

Stab

ility

safe

ty fa

ctor

40

36

32

28

24

20

16

12

8

4


0 10 20 30 40 50 60 70 80 90 100 110 120 130


Construction stage

Stab

ility

safe

ty fa

ctor

40

36

32

28

24

20

16

12

8

4


















(a) (b)


(a) (b)


24

20

16

12

8

40 10 20 30 40 50 60 70 80 90 100 110 120 130

SchemeScheme

Construction stage

Stab

ility

safe

ty fa

ctor

12




40

36

32

28

24

20

16

12

8

40 10 20 30 40 50 60 70 80 90 100 110 120 130

Decrease further


Construction stage

Stab

ility

safe

ty fa

ctor












21205881198812

119892119862119867119867 119865

119881=1

21205881198812

119892119862V119861 119872 =

1

21205881198812

1198921198621198721198612




















24

22

20

18

16

14

12

1045 50 55 60 65 70 75 80

Construction stage

Stab

ility

safe

ty fa

ctor









5 Conclusions









Acknowledgments


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










90009450

750

Han Kou Wu Chang

(PC

90009450

750

16000 1600061600 61600

21 times 1350

54321 6 7


173200

154300

girder)(PC



140 1350

2 2

3230

2644

280

Bridge centerline



s6s7s8



7 65


s6

z6 z7 z8 z9 z10 z11 z12 z13 z14 z15 z16 z17 z18


1 23





5 7698Pylon C50 345 times 10



5 785


m9


4




204 m




minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50


minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50




0 10 20 30 40 50 60 70 80 90 100 110 120 130


Construction stage

Stab

ility

safe

ty fa

ctor

40

36

32

28

24

20

16

12

8

4


0 10 20 30 40 50 60 70 80 90 100 110 120 130


Construction stage

Stab

ility

safe

ty fa

ctor

40

36

32

28

24

20

16

12

8

4


















(a) (b)


(a) (b)


24

20

16

12

8

40 10 20 30 40 50 60 70 80 90 100 110 120 130

SchemeScheme

Construction stage

Stab

ility

safe

ty fa

ctor

12




40

36

32

28

24

20

16

12

8

40 10 20 30 40 50 60 70 80 90 100 110 120 130

Decrease further


Construction stage

Stab

ility

safe

ty fa

ctor












21205881198812

119892119862119867119867 119865

119881=1

21205881198812

119892119862V119861 119872 =

1

21205881198812

1198921198621198721198612




















24

22

20

18

16

14

12

1045 50 55 60 65 70 75 80

Construction stage

Stab

ility

safe

ty fa

ctor









5 Conclusions









Acknowledgments


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










m9


4




204 m




minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50


minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50

minus55

50




0 10 20 30 40 50 60 70 80 90 100 110 120 130


Construction stage

Stab

ility

safe

ty fa

ctor

40

36

32

28

24

20

16

12

8

4


0 10 20 30 40 50 60 70 80 90 100 110 120 130


Construction stage

Stab

ility

safe

ty fa

ctor

40

36

32

28

24

20

16

12

8

4


















(a) (b)


(a) (b)


24

20

16

12

8

40 10 20 30 40 50 60 70 80 90 100 110 120 130

SchemeScheme

Construction stage

Stab

ility

safe

ty fa

ctor

12




40

36

32

28

24

20

16

12

8

40 10 20 30 40 50 60 70 80 90 100 110 120 130

Decrease further


Construction stage

Stab

ility

safe

ty fa

ctor












21205881198812

119892119862119867119867 119865

119881=1

21205881198812

119892119862V119861 119872 =

1

21205881198812

1198921198621198721198612




















24

22

20

18

16

14

12

1045 50 55 60 65 70 75 80

Construction stage

Stab

ility

safe

ty fa

ctor









5 Conclusions









Acknowledgments


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 60 70 80 90 100 110 120 130


Construction stage

Stab

ility

safe

ty fa

ctor

40

36

32

28

24

20

16

12

8

4


0 10 20 30 40 50 60 70 80 90 100 110 120 130


Construction stage

Stab

ility

safe

ty fa

ctor

40

36

32

28

24

20

16

12

8

4


















(a) (b)


(a) (b)


24

20

16

12

8

40 10 20 30 40 50 60 70 80 90 100 110 120 130

SchemeScheme

Construction stage

Stab

ility

safe

ty fa

ctor

12




40

36

32

28

24

20

16

12

8

40 10 20 30 40 50 60 70 80 90 100 110 120 130

Decrease further


Construction stage

Stab

ility

safe

ty fa

ctor












21205881198812

119892119862119867119867 119865

119881=1

21205881198812

119892119862V119861 119872 =

1

21205881198812

1198921198621198721198612




















24

22

20

18

16

14

12

1045 50 55 60 65 70 75 80

Construction stage

Stab

ility

safe

ty fa

ctor









5 Conclusions









Acknowledgments


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)


(a) (b)


24

20

16

12

8

40 10 20 30 40 50 60 70 80 90 100 110 120 130

SchemeScheme

Construction stage

Stab

ility

safe

ty fa

ctor

12




40

36

32

28

24

20

16

12

8

40 10 20 30 40 50 60 70 80 90 100 110 120 130

Decrease further


Construction stage

Stab

ility

safe

ty fa

ctor












21205881198812

119892119862119867119867 119865

119881=1

21205881198812

119892119862V119861 119872 =

1

21205881198812

1198921198621198721198612




















24

22

20

18

16

14

12

1045 50 55 60 65 70 75 80

Construction stage

Stab

ility

safe

ty fa

ctor









5 Conclusions









Acknowledgments


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















21205881198812

119892119862119867119867 119865

119881=1

21205881198812

119892119862V119861 119872 =

1

21205881198812

1198921198621198721198612




















24

22

20

18

16

14

12

1045 50 55 60 65 70 75 80

Construction stage

Stab

ility

safe

ty fa

ctor









5 Conclusions









Acknowledgments


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











5 Conclusions









Acknowledgments


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Nonlinear Stability Analysis of a