Research ArticlePC Cable-Stayed Bridge Main Girder Shear Lag EffectsAssessment of Single Cable Plane in Construction Stage

Yan-feng Li 12 Xing-long Sun 1 and Long-sheng Bao1

1School of Transportation Engineering Shenyang Jianzhu University Shenyang 110168 China2Key Laboratory of Ministry of Education on Safe Mining of Deep Metal Mines Northeastern UniversityShenyang 110819 China

Correspondence should be addressed to Yan-feng Li lyfneu126com

Received 9 January 2020 Revised 3 March 2020 Accepted 4 March 2020 Published 31 March 2020

Academic Editor Dimitrios E Manolakos

Copyright copy 2020 Yan-feng Li et al is 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

A model test and finite element analysis were conducted in this study to determine the distribution law of shear lag effect in themain beam section a box girder during the cable-stayed bridge construction process e experimental and theoretical resultswere compared in an example of loading the control sectione stress value of the cable tension area of themain beam upper edgewas found to markedly change when tensiling the cable force and was accompanied by prominent shear lag effect After a hangingbasket load was applied the main beam of certain sections showed alternating positive and negative shear lag characteristics eshear lag distribution law in the box girder of the single-cable-plane prestressed concrete cable-stayed bridge along the lon-gitudinal direction was determined in order to observe the stress distribution of the girder e results show that finite elementanalysis of the plane bar system should be conducted at different positions in the bridge under construction the calculated shearlag coefficient of the cable force acting at the cable end of the cantilever reflects the actual force In the beam segments between thecable forces the shear lag coefficient determined by the ratio of the bending moment to the axial force reflects the actual stress atthe cable force action point In the midspan beam section between the action points of cable forces the shear lag coefficient of thebending moment reflects the actual stress e section shear lag coefficient can be obtained by linear interpolation of the beamsection between the cable action point and the middle of the span

1 Introduction

e main girder a single-cable plane cable- (PC-) stayedbridge is generally in box section form e shear lag effectof the box girder flange is serious under compression andbending loads and the normal stress is highly complex[1 2] Shear lag causes uneven distribution of normal stressin the flange plates which is one of the main causes ofcracks in concrete box section girders [3 4] e effectivewidth is generally used to replace the actual width of theflange to calculate the effect of shear lag on box girders ebridge codes of various countries prescribe effective widthcalculation methods for special structures such as simplysupported beams continuous beams and cantilever beamsbut there is no clear stipulation for cable-stayed bridgedesigns

e variational principle of potential energy has beenused by previous researchers to analyze the theoretical shearlag effects in concrete box beams [5 6] Chang et al forexample used the principle of minimum potential energy toanalyze the shear lag effects of thin-walled trapezoidal boxsections with inclined stiffeners in the cell they found thatthe bridge segment presents bending stress and axial stresssimultaneously and assumed the parabolic transverse changeof the axial force on the section [7] Wu and Hong estab-lished a 3D finite element model of a bridge to study theshear lag effects of cable-stayed bridges where shear lagcoefficients varied in different locations along the longitu-dinal axis of the bridge [8] Pengzhen et al used the finiteelement method to observe the influence of a concrete boxbeam with various flange depths along the orientation ofcross section [9] Yamaguchi et al established a plate and

HindawiAdvances in Materials Science and EngineeringVolume 2020 Article ID 2646513 16 pageshttpsdoiorg10115520202646513

shell finite element model for simply supported and con-tinuous beams to assess the influence of finite element meshsize load mode width to height ratio and width to spanratio on the stress of the box girderrsquos wing slab [10] Shenanalyzed the shear lag effect of the main beam of a wide andshort tower cable-stayed bridge to explore the warpingdisplacement function caused by shear lag [11]

ere have been many other valuable contributions tothe literature as well Xingmin used a finite elementanalysis method considering shear lag degrees of freedom(DOFs) to observe the influence of the constructionprocess on box girder shear lag effects [12] Muyu et alused spatial finite element analysis to find that the shearlag effect of the main beam section exerts a greater impacton the structure during the construction process thanafter the completion of the bridge In the cantileverconstruction stage of the ldquoTrdquo structure the main beam ofthe low-tower cable-stayed bridge is subjected to bendingas well as massive axial force which creates an unevendistribution of the section stress during the diffusionprocess For the large-flange wide box girder this unevendistribution of cross section stress is even more signifi-cant [13] Qianshu et al established an ANSYS solid finiteelement model to simulate the cantilever constructionprocess they found that the shear lag effect of the roof inthe maximum cantilever stage changes from positive tonegative in the cantilever direction e shear lag coef-ficient trends toward stability however as cantileverconstruction progresses [14]

e extant research shows that the shear lag effect ofcable-stayed bridge under cantilever construction is indeedsignificant but no specific theoretical numerical results havebeen given for the beam sections at different positions In thecable-stayed bridge discussed here the beam segment is in acantilever state during the construction process e boxgirder shear lag coefficient fluctuates as the stay cable isplaced continuously under tension the shear lag effect be-comes increasingly prominent as construction progresses sothat the distribution law of shear lag effect on the main beamof a single-cable-plane cable-stayed bridge during con-struction was systematically analyzed in this study emodel test and numerical analysis results were used toanalyze the distribution of shear lag in the transverse di-rection of the main beam section Characteristic points withprominent shear lag along the longitudinal direction in thetransverse section of the main beam were analyzed specif-ically in different stages of construction and the distributionlaw was obtained

Due to space limitations this paper only presentsWorking Conditions 1 and 4 of the construction process foranalysis e shear lag coefficients λΜ caused by the bendingmoment and λN caused by the axial force were respectivelyanalyzed to solve the shear lag coefficient λ after superpo-sition e relationship between the axial force ratio of thebending moment and the shear lag coefficient λ is discussedin detail below We find that different shear lag coefficientsare appropriate for different beam sections of cable-stayedbridges during construction e results presented here mayhave a certain reference value for similar projects

2 Research Background

e Fumin Bridge (Shenyang China) served as the researchobject for this study Fumin is a single-cable PC cable-stayedbridge with a length of 42000m and a span of8900m+ 24200m+ 8900m as shown in Figure 1(a) Itsmain beam is an approximate triangular section with strongwind resistance a single box and three chambers Its beamheight is 341m e dimensions of each part of the middlespan are shown in Figure 1(b) e thickness of each partnear the root of the tower and the closing section of themiddle span increase appropriately e dimensions of eachpart of the side span are shown in Figure 1(c) A diaphragmcan be found along every 315m of the side span ethickness of the middle compartment diaphragm at thesection anchored by cable is 40 cm e thickness of themiddle compartment diaphragm at the section without acable anchorage is 28 cm e main tower also has a boxsection with a height of 6750m

e stay cable of this bridge was fabricated from Φ7galvanized high-strength steel wire e wire arrangement is301-Φ7 241-Φ7 211-Φ7 and 151-Φ7 across four specifi-cations in a fan layout Each main tower has a layout of 15pairs making 120 stay cables in total e cable spacing onthe main beam is arranged as the middle span at 14times 740mthe side span cable distance is 10times 630m and the tail cablearea is 150m+ 150m+ 575m+ 700m e distance be-tween cable number 1 and the intersection point of the toweraxis and main beam center line is 1590m e cablespacing on the tower from top to bottom is4times150m+ 300m+ 400m+ 8times 450m e tower beamand pier are consolidated at bridge pier number 4e towerand beam are consolidated while the beam and pier areseparate at bridge pier number 5e cantilever constructionmethod was adopted In the figure the letter ldquoCrdquo representsthe number of the stay cables and ldquoGrdquo represents the numberof test cross sections in the main beam

3 Experimental Study on Shear Lag Effect ofMain Beam in Construction Stage of PCCable-Stayed Bridge with Single Cable Plane

31 Design Model ree tenets of similarity theory wereused as the basis for the structural model in this study toensure results reflective of objective reality [15ndash18] esection stiffness of the Fumin Bridge test model corre-sponding to the bridge prototype is similare test model isgeometrically similar to the bridge prototype and was builtwith as large a scale as possible e load of the test modelwas reduced according to the actual load ratio e bridgetesting model and prototype material compositions aresimilar as well e test model was given the same boundaryconditions as the real bridge

e model test of cable-stayed bridge requires certainsite conditions loading conditions and test costs Selectingthe optimal geometric scale is the primary problem to besolved in the model design process Large-scale models areeasy to make and experimental data are easy to collect fromthem but costly labor-intensive large test sites and loading

2 Advances in Materials Science and Engineering

equipment are required A small-scale model requires only alightweight load but it is relatively difficult to build or equipwith test instruments further the measurement error isrelatively large It is necessary to weigh various test con-ditions and requirements before selecting the most appro-priate proportion for model making roughcomprehensive consideration the geometric scaling ratio ofthe wholemodel in this study was determined to be 1 40 andthe length of the cable-stayed bridge model was 105m

311 Box Girder Design e main girder of this model ismade of aluminum alloy Each section of the box girder wasmade from plates pulled through rivets and glued togethere elastic modulus ratio between C50 concrete and alu-minum alloy is 050 e section area and the moment ofinertia for the model box girder can be calculated under theprinciple of similarity

Am Ap middot12

1113874 11138751402

1113874 1113875 (1)

Im Ip

12

1113874 11138751404

1113874 1113875 (2)

According to geometric similarity the size of the boxgirder section model is basically unchanged e wallthickness of the box girder was halved to ensure that the

section area and the moment of inertia were similar in thisanalysis Some section similarity deviation was controlledwithin an allowable range e error of modulus only affectsthe section stress so the model and prototype were calcu-lated under corresponding working conditions eir ac-curate similarity ratios were determined according to thestrain and stress calculation results

e box girder of the test model consists of 19 sectionswhich may be assembled via a typical construction processDead load compensation was conducted by hanging heavyobjects directly onto a predefined hanging point the con-struction load and live load were imposed by adding heavyloads on the bridge deck e full-bridge model is shown inFigure 2

312 Stay Cable Design e stay cable of model is made ofhigh-strength steel wires similar to the prototype cable eelastic modulus ratio in this case is equal to 1 e sectionalarea of the model stay cable is

Am Ap middot1402

1113874 1113875 (3)

e girder and stay cable were connected by screw threadto allow us to adjust the stay cables easily A vibrating wire-type load cell was installed at the joint tomeasure the force ofthe stay cable e self-weight of the stay cable markedly

G6G9G4

Stay cablesnumbers C15 to C1

420

G5G3

5

Stay cablesnumbers C15prime to C1prime

89 242G7G1

4

Stay cablesnumbers C1 to C15

89

6

G8G2

Stay cablesnumbers C1prime to C15prime

3

(a)

150

320

304

16 50

500

15 15

9253250

2525

40

400 925

15040

40

341

4

30

500

(b)

4030

151630

432

0

500 925

250

50 15

50

3250400

4050

925

50

341

4

500

250

(c)

Figure 1 Structural drawing of bridge (a) main bridge arrangement (unit m) (b) cross section of midspan girder (unit cm) and (c) crosssection of side-span girder (unit cm)

Advances in Materials Science and Engineering 3

influences its deformation in order to accurately reflect thestress characteristics of the structural nonlinearity we im-posed load compensation for the dead weight of the cable viaadditional mass method

313 Main Tower Design e main tower of the cable-stayed bridge was welded with Q235B carbon structural steele ratio of elastic modulus to C50 concrete is 6 According tosimilarity theory the section area and moment of inertia ofthe main tower of the model can be calculated as follows

Am Ap middot16

1113874 11138751402

1113874 1113875 (4)

Im Ip

16

1113874 11138751404

1113874 1113875 (5)

e outer contour of the main tower section remainedbasically unchanged and the wall thickness was reduced by16 throughout this analysis to satisfy the similarity betweenthe moment of inertia and the area of the cross section emain tower was made in one piece and bolted into con-nection with the box girder Because the main tower of thebridge was a broken line the dead load of the main towerproduced not only the stress of the tower body but also thebending moment of the tower body which affect the entirestructural system of the cable-stayed bridge internal forcestate We carried out dead load compensation for the maintower to account for this

314 Design of Bridge Pier and Bearing emain piers andside pier were fabricated by steel weldinge box girder andPier 4 were connected by bolts with set bearing for Pier 3

80

100220 220

8125

1362513625

(a)

2225 2225605010500

Stay cablesC15prime~C1prime

Stay cablesC1~C15 Stay cables

C1prime~C15prime

Stay cablesC15~C1

Pier 4 Pier 5

1688

3543210123 211234

53752 times 630 = 12601575 2 times 740 = 1480 555740 2 times 630 = 1260 15752 times 740 = 1480555 537574050

(b)

Stay cablesC15prime~C1prime

Stay cablesC1~C15

Stay cablesC1prime~C15prime

Stay cablesC15~C1 16

8814

00

2225 2225605010500

3553554 5

(c)

Figure 2 General arrangement drawing of model (unit mm) (a) midspan section diagram (b) block diagram and (c) temporary supportand loading diagram

4 Advances in Materials Science and Engineering

Pier 5 and Pier 6 We set a load sensor in each bearing Atension-applying device was used to simulate the dead load

32 Model Fabrication and Installation We used a specialvibrating wire sensor to simulate the cable sensor which wascalibrated before installation to ensure long-term stabilityaccuracy and reliability In order to accurately simulate theconstruction process we installed the box girder in sub-section form

According to the real-world construction requirementsPier 5 was set as a temporary pier and a load sensor wasinstalled to measure the counter force of the temporarysupport under the action of the construction load emodel installation process is shown in Figure 3

33 Loading System and Dead Load Compensation

331 Dead Load Compensation of theMain Tower edeadload compensation of the main tower can be divided intotwo partsere is a concentrated load which was applied vialever system e upper segment load was applied to thecenter of gravity on the upper segment of the main towere lower segment load compensation accords with thesimilarity of the tower rootrsquos bending moment whichguaranteed that the tower section and tower angle sectionload effects were similare quality of the compensation forthe upper segment in this test was 83710 kg and that of thelower segment was 62000 kg

332 Load Compensation of Main Girder e load com-pensation of the main beam was applied to the diaphragm inconcentrated form e compensation quality of the inter-mediate span standard segment in this case was 26030 kgthe compensation quality of the intermediate span towerroot segment was 38530 kg the compensation quality of theside span standard segment was 27720 kg and the inter-mediate span tower root segment compensation quality was33580 kg e second-stage load compensation for eachlifting point included a side span of 6500 kg and a middlespan of 7630 kg

34 Test Load Working Conditions and Test Contents Weinstalled a stay cable and box girder in accordance with theactual construction process of the original bridge e staycable box girder and main tower were gradually com-pensated for dead loads and the stay cable was tensioned andadjusted as necessary e model test was carried out tosimulate the construction load and stress of the girdersection under the working conditions of a midspan closurein the construction stage

35 Test System and Measuring Point Arrangement Ourmodel testing system includes a support reaction forcetesting component dead load testing component stresstesting component displacement testing component andcable force testing component Each fulcrum was equippedwith a calibrated strain tester and a load cell to form a

reaction force test system e dead load test was performeddirectly on the calibrated scale e stress testing systemconsists of strain gauge and resistance strain gauge eelectrical test method was used to test the live load and deadload structural stress distributions e displacement testwas performed using a level gauge a dial gauge and atheodolite Our cable force test system consists of a cali-brated special frequency tester and a steel string load celle section and measuring point arrangement we used areshown in Figures 4 and 5

36 Construction Stage Working Conditions We exampledsix typical construction working conditions correspondingto real-world construction phases In Working Condition 1(double cantilever stage) we installed 1 (1prime) beam sectiontensioned C1ndashC4 (C1prime-C4prime) cables and applied a hangingbasket load In Working Condition 2 (maximum doublecantilever stage) we installed 2 (2prime) and 3 (3prime) beam sectionstensioned C5ndashC10 (C5prime-C10prime) cables and applied a hangingbasket load InWorking Condition 3 we closed the side spanof number 5 pier line and applied a hanging basket load tothe C11 cable In Working Condition 4 (maximum singlecantilever stage) we installed Section 4 tensioned C12ndashC14(C12prime-C14prime) cables and applied a hanging basket load at theC14 cable In Working Condition 5 we tensioned the C15(C15prime) cable and applied a closing weight to complete themidspan closure Working Condition 6 included cabletension adjustment and application of a second-stage deadload

4 Test Results versus Finite ElementAnalysis Results

41 Establishment of Finite Element Model We used MidasFEA finite element software to conduct our numerical an-alyses e construction scheme investigated in this studywas considered a sliding formwork with a combination ofthe stress state of the main girder and control the maintower e bridge was constructed via forward installationmethod [19ndash22] In the finite element analysis the cable wassimulated by the truss unit only the tower was simulated bythe beam element e main beam in this analysis is a solidelement and a six-node tetrahedron was used for divisione full-bridge truss unit is 120 the beam unit is 54 the solidunit is 29787 and there are 52631 nodes

After tensioning the C4 (C4prime) cable force the hangingbasket load was 5100 kN the boundary condition was a fixedbottom section of Pier 4 restraining the line displacementand angular displacement in the X Y and Z directions ebeam and Pier 5 were separated the beam bottom con-strained the linear displacement in X Y and Z directions inthis case as well Temporary support was simulated at1485m on the right side of the beam section of Pier 5which constrained the linear displacement in the Z direction

Next we tensioned C10 (C10prime) after the cable force ehanging basket load was 5100 kN and the boundary con-dition was the same as the first working condition e sidespan was closed the cable force of Pier 5 was adjusted and

Advances in Materials Science and Engineering 5

the hanging basket load was 5100 kN after the C11 (C11prime)cable tension was applied e boundary condition in thiscase involved removed the temporary support placing thebottom of Piers 4 and 5 under the same working con-dition e line displacement in the Z direction was re-strained at the end of the main beam

After the tension of the C14 (C14prime) was applied thehanging basket load was applied At 5100 kN the boundaryconditions were the same as the previous working conditionAfter applying the tension of C15 (C15prime) cable force the

closing weight of 1950 kN was applied e middle span wasclosed and the closing weight was unloaded e boundarycondition in this case was an unchanged constraint on thebottom of Pier 4 while the line displacement of the bottomof Pier 5 was constrained in the Z direction

e finite element model as established under thesespecific working conditions is shown in Figure 6 Due tolimited space only the results of Working Condition 1and Working Condition 4 are presented here forcomparison

(a) (b)

(c) (d)

Figure 3 Installing the model (a) installing four pairs of stay cables at Pier 4 (b) closure of side span of Pier 5 (c) second stage dead loadapplied by model and (d) model exerts live load

2225 2225605010500

Stay cablesC15prime~C1prime

Stay cablesC1~C15

Stay cablesC1prime~C15prime

Stay cablesC15~C1

Pier 4 Pier 5

1688

G4 G2 G1 G3 G9 G8G7 G5 G6

Figure 4 Diagram of testing section arrangement about girder (unit mm)

1 2

18

3

13

8

17

16

4 5

15 14

6 7 10

12

9 11

Figure 5 Arrangement of stiff girder stress measuring point

6 Advances in Materials Science and Engineering

We conducted forward installation calculations reflect-ing the construction process of the entire bridge e leastsquare method was applied to calculate of the cable force ofthe bridge during forward installation and the results werecompared with the reasonable cable force of the bridge untilall design requirements were satisfied At this point thecorresponding construction cable force could be consideredthe initial tension of the stay cable To accelerate conver-gence the initial tension of the cable was selected per thestatic equilibrium determined in the first forward installa-tion iteration Table 1 lists the initial tension of each staycable after the iterative calculation the number of stay cablesis shown in Figure 1

e stay cable at Pier 5 was adjusted once the side spanwas closed to bring the internal force at the root of the maintower closer to the deformation at the cantilever end of themain beam e C6 cable force was adjusted to 8000 kN C7cable force to 8000 kN C8 cable force to 8000 kN and C9cable force to 7500 kN at this time Before the constructionof bridge deck pavement to ensure uniform internal forceand deformation of the main girder across the whole bridgethe cable at Pier 4 was adjusted by bringing C1prime cable forceto 9000 kN C2prime cable force to 9000 kN C1 cable force to11000 kN C2 cable force to 11500 kN C3 cable force to9000 kN and C4 cable force to 9000 kN e cable force ofpart of the stay cable at Pier 5 was also adjusted by bringingC1 cable force to 10000 kN C2 cable force to 10000 kN C1primecable force to 10000 kN and C2prime cable force to 10500 kN

42 Main Beam Cross Section Shear Lag Coefficient underWorking Condition 1 e model test results and the shearlag coefficient of the upper edge of the main beam section ofWorking Condition 1 in the spatial finite element weresubjected to comparative analysis After tensioning the C4cable the full beam of the main beams G1 and G2 waspressed (tensile stress is positive compressive stress isnegative) the average measured stress of the upper edge ofthe main beams G1 and G2 was minus207MPa and minus077MPa atthis point After applying a hanging basket load on thecantilever end of the box girder the upper edge of the maingirders G1 and G2 showed a large amount of stress changese midabdominal segment bore compressive stress whilethe upper edge of section G1 of main beam bore tensilestress e upper edge of section G2 of main beam boretensile stress e average measured stress of the upper edgeof the main beams G1 and G2 was 024MPa and 049MPaAfter the tensioning force of the main girders G1 and G2 andapplication of the basket load the shear lag coefficient ateach point of the upper edge was as shown in Figure 7

After tensioning the C4 cable the main beams G5 andG6 were pressed in the full section and the average measuredstress of the upper edge of the main beams G5 and G6 wasminus189MPa and minus053MPa After applying the hangingbasket load on the cantilever end of the box girder the upperedge of the main beam G5 and G6 showed a large amount ofstress change e midabdominal segment bore compressivestress and the upper edge of section G5 and G6 of main beam

(a) (b)

(c) (d)

(e) (f )

Figure 6 Special finite element model for main construction operating conditions (a) Tension C4 (C4prime) applying a basket load (b) TensionC10 (C10prime) applying a basket load (c) Cable tension adjustment tensioning C11(C11prime) applying a hanging basket load (d) TensionC14(C14prime) applying a hanging basket load (e) Tension C15 (C15prime) applying counterweight midspan closure (f ) Grid division of mainbeam section

Advances in Materials Science and Engineering 7

bore tensile stress e average measured stress of the upperedge of the main beams G5 and G6 was 053MPa and055MPae shear lag coefficient at each point of the upperedge was as shown in Figure 8 after the tensioning force ofthe main girders G5 and G6 and the application of the basketload

As tension was applied the concentrated load was placedin the middle of the box girder to intuitively observe theshear lag effect in the middle of the box girder section ehanging basket load was applied in the form of a concen-trated load so that the upper edge of sections G1 G5 and G6of the main beam presented tensile stress though the middleweb section did not After the tensile force and the hangingbasket load were applied positive and negative shear lageffects appeared alternately on the upper edge of the mainbeam

43 Main Beam Cross Section Shear Lag Coefficient underWorking Condition 4 e model test results and the shearlag coefficient of the upper edge of the main beams G1ndashG8section under Working Condition 4 for the spatial finiteelement was also extracted for comparative analysis Asshown in Figures 9 and 10 the shear lag coefficient of theupper edge of the section G7 of themain beam changed from009ndash229 to minus293ndash210 after the tension of the cable and theapplication of the hanging basket load e shear lag

coefficient of the upper edge of the main beam G3 changedfrom 055ndash135 to 039ndash232 the shear lag coefficient of theupper edge of the main beam G4 changed from 063ndash159 to066ndash166 and the shear lag coefficient of the upper edge ofthe main beam G8 changed from 034ndash152 to 021ndash192 Inthis case after the tension was applied to the C14 cable andthe hanging basket load was applied the shear lag coefficientof the upper edge decelerated and the changes were slighte shear lag coefficient of the upper edge of the sections G2and G1 G5 and G6 of the main girder near the tower rootwas weak In each section the shear lag coefficient of theupper edge decelerated as tension cable force increased etest results are consistent with the finite element calculationresults

5 Longitudinal Shear Lag Effect in MainBeam of Single-Cable Plane P C Cable-Stayed Bridge

e stress on the main girder section of cable-stayed bridgeis highly complex e construction of the cable-stayedbridge is a dynamic process e stress in the main beamsection under different working conditions and differentbeam sections may widely differ e shear lag coefficientcalculation results along the longitudinal length of severalkey nodes on the upper edge of the cross section of the main

Table 1 Initial cable tension of each stay cable

Pier 4 Pier 5Cablenumber

Tension cable force(left) (kN)

Cablenumber

Tension cable force(right) (kN)

Cablenumber

Tension cable force(left) (kN)

Cablenumber

Tension cable force(right) (kN)

C1prime 7500 C1 9000 C1 9000 C1prime 4500C2prime 6500 C2 9500 C2 9500 C2prime 6500C3prime 5500 C3 8000 C3 8000 C3prime 5500C4prime 5500 C4 8000 C4 8000 C4prime 5000C5prime 5500 C5 8500 C5 8000 C5prime 6000C6prime 5500 C6 8000 C6 7400 C6prime 6500C7prime 5500 C7 8000 C7 7000 C7prime 5500C8prime 6000 C8 8000 C8 7000 C8prime 6000C9prime 6000 C9 7500 C9 7000 C9prime 6000C10prime 6500 C10 7500 C10 7500 C10prime 6500C11prime 6500 C11 7500 C11 7500 C11prime 6500C12prime 9500 C12 7500 C12 7500 C12prime 9000C13prime 9500 C13 7500 C13 7500 C13prime 9000C14prime 10000 C14 7500 C14 7500 C14prime 9500C15prime 9500 C15 6500 C15 6500 C15prime 9500

ndash100ndash050

000050100150

Shea

r lag

coeffi

cien

t

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(a)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

ndash200ndash100

000100200300

Shea

r lag

coeffi

cien

t

Node lateral coordinate (m)

(b)

Figure 7 Shear lag coefficient variations in G1 and G2 (Condition 1) (a) G1 section (b) G2 section

8 Advances in Materials Science and Engineering

beam can be used to understand the shear lag coefficientdistribution along the longitudinal length of the upper edgeof the main beam section during the construction stage Forconvenience the key points on the top edge of the crosssection of the main beam are marked here with capitalletters e distances from A-D points to the transversesymmetry lines are 0m minus2m minus11m and minus16m respec-tively e lateral position of each point is shown inFigure 11

Due to space limitations only the distribution of theshear lag coefficient along the longitudinal length of theupper edge of the main beam section under the followingtwo typical working conditions is described here WorkingCondition 1 is defined by tensiling the C4 (C4prime) cable forceWorking Condition 4 is defined by tensiling the C14 (C14prime)cable force e boundary conditions of each workingcondition were kept consistent

51Working Condition 1 Shear Lag Effect along LongitudinalBridge Under the action of self-weight and cable force the

shear lag coefficient of the flange of the main beam sectionalong the longitudinal bridge was observed as shown inFigure 12 In the longitudinal distribution of the shear lagcoefficient at the key point of the upper flange the midpointof the cross section of the main beam (Point A) and theboundary point between the upper edge and the middle web(Point B) change most sharply

At Pier 4 a sudden change in shear lag coefficient at theend of the cantilever occurred due to the tension of the C4(C4prime) cable A large stress concentration occurred at theseconcentrated load points which altered the upper flangestress distribution law Under the action of cable force theupper edge of the main beam section shows a more sig-nificant shear lag effect of the cable tension zone at locationscloser to the cable action point At Point A the shear lagcoefficient reaches 418 at Point B it is 307 e stressdistribution of the main beam appears to grow more uni-form as the distance between the main beam and the cableforce increases

Pier 4 C3ndashC1 (C3prime-C1prime) shows a similar distributionpattern in each beam section e shear lag coefficient of

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

ndash100ndash050

000050100150200

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

(a)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

ndash200ndash100

000100200300

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

(b)

Figure 8 Shear lag coefficient variations in G5 and G6 (Condition 1) (a) G5 section (b) G6 section

000

050

100

150

200

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16Node lateral coordinate (m)

Shea

r lag

coef

ficie

nt

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(a)

000050100150200250

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16Node lateral coordinate (m)

Shea

r lag

coef

ficie

ntTensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(b)

000050100150200250

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(c)

Shea

r lag

coef

ficie

nt

000050100150200250

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(d)

Figure 9 Contrast diagram of shear lag coefficient in G1-G4 (Condition 4) (a) G1 section (b) G2 section (c) G3 section and (d) G4 section

Advances in Materials Science and Engineering 9

Point A and Point B is prominent at the point of action of thecable forcee change range of shear lag coefficient of PointA at the point of cable force action is 222ndash271 and at Point Bis 122ndash212 e stress distribution of the beam sectionbetween the points of cable force is relatively uniform eA-point shear lag coefficient λ ranges from 115 to 116 theB-point shear lag coefficient λ from 112 to 140 and theC-point shear lag coefficient λ from 109 to 126

e stress distribution of the upper flange of the Pier 4tower root section changed due to the existence of the crossbeame change appears to be most dramatic at Point D atthe end of the upper extension arm of the main beam crosssection e shear lag coefficient λ at Point D near the towerranges from 114 to 154 marking a very significant variationcompared with other beam sections

e variation of the cantilever end the cable force pointand the C3ndashC1 (C3prime-C1prime) beam section of Pier 5 is similarto that of Pier 4 Under the action of the tension C4 (C4prime)Point A here shows a shear lag coefficient λ of 300 while thatat Point B reaches 250 In our analysis the stress distri-bution of the main beam grew more uniform as the distancebetween the main beam and the cable force increased

Pier 5 C3ndashC1(C3prime-C1prime) shows a similar distributionpattern in each beam section e shear force lag coeffi-cient of Point A and Point B is prominent at the point ofaction of the cable force e variation range of lag

coefficient λ is 157ndash244 and the shear lag coefficient λ ofPoint B is 138ndash184 at the point of action of cable forcee stress distribution of the beam section between thecable force points is relatively uniform e changes inshear lag coefficient λ of Point A range from 141 to 166that at Point B varies from 107 to 157 and that at Point Cranges from 075 to 108 Again the most significantchange was observed at Point D the shear lag coefficient λof Pier 5 at point D near the tower root varies from 120to 133

At the C1prime cable force on the right side of Pier 5 theshear lag coefficient of Points A and B is weaker than at otherpoints of cable force mainly because of the temporarysupport e shear lag effects on both sides were similar dueto the small reaction force of the temporary pier

Under the action of self-weight and cable force with thisworking condition the main characteristics of the shear lageffect of the upper edge of the main beam section of thesingle-cable PC cable-stayed bridge in the longitudinal di-rection can be summarized as follows

(1) In the construction stage under the action of cableforce and near the point of cable tension the upperedge of the girder section shows more significantshear lag effect in the cable force tension area egirder stress distribution grows more uniform as thedistance between the girder section and the cableforce action point increases

(2) In the C3ndashC1(C3prime-C1prime) beam section the upper edgeof the main beam section at the cable force actionpoint presents shear lag effects in the tension zonee stress distribution in the beam section betweenthe points of the cable force is relatively uniformwhich characterizes the shear lag distribution in theordinary curved box girder

000020040060080100120140160

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(a)

000

050

100

150

200

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(b)

ndash400ndash300ndash200ndash100

000100200300

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(c)

000

050

100

150

200

250

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(d)

Figure 10 Contrast diagram of shear lag coefficient in G1-G4 (Condition 4) (a) G5 section (b) G6 section (c) G7 section and (d) G8 section

B ACD

Figure 11 Main girder cross-section diagram at every point

10 Advances in Materials Science and Engineering

(3) e shear lag coefficient at the end of the upperextension arm of the cross section of the main beamat the root of the tower changes most dramaticallyamong the areas we observed We also observedconversion from a positive shear lag effect to anegative shear lag effect near the tower

(4) Under the influence of temporary support the shearforce lag coefficient of the action point of the C1primecable force on the right side of Pier 5 is weaker thanother cable forces e temporary pier reaction forceis small whichmakes the shear lag on both sideseeffects were still relatively similar in this case

52Working Condition 4 Shear Lag Effect along LongitudinalBridge is working condition represents the largestsingle cantilever stage e side spans at this point areclosed and the shear lag effects near their supports havemarkedly increased due to tensioning of C12primendashC14prime tailse shear lag effect of the inner beam section of the sidespan C11primendashC1prime is lower than under the previous workingcondition

As shown in Figure 13 at Pier 4 the cantilever end fallsunder the action of tensioning C14 cable force e shear lagcoefficient λ at Point A reaches 340 and at Point B is 240 Inthe beam section of Pier 4 C11ndashC1 (C11primendashC1prime) the shearlag coefficient of Points A and B is prominent at the point ofaction of the cable force Point A shows a shear lag coefficientλ of 151ndash281 at the cable force action point where Point Bshows a shear lag coefficient λ of 435ndash212 the stress dis-tribution of the beam section between the points of the cableforce is relatively uniform e shear lag coefficient λ ofPoint A varies from 105 to 144 and that of Point B from 119to 110 e shear lag coefficient λ of Point C ranges from065 to 095

At Pier 5 the end of the cantilever is under the action ofthe tensioning force of C14 e shear lag coefficient λ ofPoint A reaches 357 and that of Point B is 222 In the beamsection of C11ndashC1 (C11primendashC1prime) in Pier 5 the shear lagcoefficients of Points A and B at the point of action of thecable force are prominent Point A has a shear lag coefficientλ of 148ndash281 at the action point of the cable force wherePoint B has a shear lag coefficient λ of 133ndash212 e stressdistribution of beam section between the points of cable

4rsquo 3rsquo 2rsquo 1rsquo 4 1 2 3 4Longitudinal bridge position

Point APoint B

Point CPoint D

Pier 4

000050100150200250300350400450

Shea

r lag

coef

ficie

nt

(a)

Point APoint B

Point CPoint D

4 3 2 1 5 1rsquo 2rsquo 3rsquo 4rsquoLongitudinal bridge position

Pier 5

000

050

100

150

200

250

300

350

Shea

r lag

coef

ficie

nt

(b)

Figure 12 Distribution of shear lag coefficient along longitudinal direction (Condition 1) (a) Pier 4 (b) Pier 5

Advances in Materials Science and Engineering 11

Point APoint B

Point CPoint D

3 14rsquo13rsquo12rsquo 11rsquo 10rsquo 9rsquo 8rsquo 7rsquo 6rsquo 5rsquo 4rsquo 3rsquo 2rsquo 1rsquo 4Longitudinal bridge position

Pier 4

000

050

100

150

200

250

300

Shea

r lag

coef

ficie

nt

(a)

Point APoint B

Point CPoint D

4 1 2 3 4 5 6 7 8 9 10 11 12 13 14Longitudinal bridge position

000

050

100

150

200

250

300

350

400

Shea

r lag

coef

ficie

nt

(b)

Point APoint B

Point CPoint D

14 13 12 11 10 9 8 7 6 5 4 3 2 1 5Longitudinal bridge position

Pier 5

000

050

100

150

200

250

300

350

400

Shea

r lag

coef

ficie

nt

(c)

Figure 13 Continued

12 Advances in Materials Science and Engineering

force is relatively uniform e shear lag coefficient λ ofPoint A ranges from 120 to 155 that of Point B from 096 to135 and that of Point C from 076 to 101

Under this working condition the action of self-weightand cable force make the main characteristics of the shear lagat the upper edge of the main beam section of the single-cable PC cable-stayed bridge along the vertical direction verysimilar to those of underWorking Condition 1e shear lageffect at the main beam section is reduced as cable forceincreases

6 Distribution of Shear Lag Effect duringConstruction Stage

61 Shear Lag Coefficient of Main Beam of Single-Cable PlanePC Cable-Stayed Bridge e solution to the shear lag coef-ficient under the compressive bending load involves decom-posing the compressive bending load into a superposition ofthe bending moment action and the axial force action thencalculating the normal stress of themain beamunder the actionof the bending moment and the axial force alone [22] It isassumed that the axial force is borne by the whole section thatis the normal stress generated by the axial force is evenlydistributed along the cross section e results can then belinearly superposed to obtain the comprehensive shear lageffect coefficient Our previous analysis indicated that whenthere is cable tension of the main beam of the cable-stayedbridge in the single-cable PC there is a large stress concen-tration at the point of action of the concentrated loads estress distribution law of the roof plate changes accordingly

Assume that the axial force which is borne by the wholesection does not reflect the actual internal force of the maingirder of the single-cable PC cable-stayed bridge e dis-tribution of the longitudinal normal stress generated byinternal force is not uniform e size and distribution laware related to the internal force state namely the bendingmoment and axial force In this study the shear lag coef-ficient λΜ generated by bending moment and the shear lag

coefficient λΜ generated by axial force were respectivelycalculated to solve the shear lag coefficient λ aftersuperposition

e shear lag coefficient is calculated as follows

λ σσ

(6)

where σ is actual stress and σ( ) is nominal stresse shear lag coefficient is related to the wide span ratio

support conditions load form cross-sectional shape andsize material properties and location of the section [23ndash25]In the final analysis the shear lag is caused by unevendistribution of the longitudinal normal stress caused byinternal force its size and distribution are related to theinternal force state that is the bending moment is related tothe axial force

e actual stress σ and nominal stress σ( ) are thesuperposition of the bending moment and the normalstress generated by the axial force If the shear lag coeffi-cient λΜ produced by the bending moment is definedproperly the shear lag coefficient generated by the axialforce is λN and the shear lag coefficient after the super-position is λ us

λ σN + σM

σN + σM

λNσN + λMσM

σN + σM

(7)

where σ( )N and σ( )M are the nominal normal stressgenerated by bending moment and axial force respectivelyσ( )N and σ( )M are the actual normal stress generated bybending moment and axial force respectively e nu-merator and denominator at the right end of the aboveformula are divided by σ( )N to rewrite (7) as

λ λN + λM σMσN( 1113857

1 + σMσN( 1113857 (8)

ere are σM (MI)y and σN (NA) so(σMσN) (yAI) middot (MN) and order yAI C can beplugged into (8) to obtain

Point APoint B

Point CPoint D

000

050

100

150

200

250

300

350

5 1prime 2prime 3prime 4prime 5prime 6prime 7prime 8prime 9prime 10prime 11prime 12prime13prime14prime6

Shea

r lag

coef

ficie

nt

Longitudinal bridge position

(d)

Figure 13 Distribution of shear lag coefficient along longitudinal direction (Condition 4) (a) Left side of Pier 4 (b) Right side of Pier 4(c) Left side of Pier 5 (d) Right side of Pier 5

Advances in Materials Science and Engineering 13

λ λN + CλM(MN)

1 + C(MN) (9)

where y is the distance from the centroid of the section to theupper and lower edges A is the sectional area I is the sectionmoment of inertia and M and N represent the bendingmoment and axial force of the section

Equation (9) indicates that y A and I are the parametersinherent to the section under analysis λN and λΜ are theshear lag coefficients caused by the axial force and thebending moment respectively which are independent of themagnitude of the internal force Once λN and λΜ are de-termined the total shear lag coefficient of λ is only related tothe ratio of bending moment to axial force

Figure 14 shows the curve of the shear lag coefficientwith the bending moment axial force ratio when λΜ 200and λN 100 Taking the upper flange of the main beam ofthe Fumin cable-stayed bridge as an example C yAI 101

As shown in Figure 14 the total shear lag coefficient λgradually approaches the bending moment shear lag coef-ficient λΜ as the bending moment force ratio increases Asthe bending moment axial force ratio gradually decreases to0 the total shear lag coefficient λ is also gradually reduceduntil it is equal to the axial force shear lag coefficient λNWhen the bending moment force ratio is between 0 and 2the total shear lag coefficient λ changes substantially Alarger λΜ results in a greater change in the total shear lagcoefficient λ When the bending moment force ratio isgreater than 2 the total shear lag coefficient λ changesslowly at is to say in the internal force of this sectionwhether the bending moment or axial force accounts for alarger proportion affects how close the total shear lag co-efficient λ of the section is to the shear lag coefficient (λΜ orλN) caused by this item of internal force

For a general beam bridge the bending moment is themain internal force and so the section shear lag coefficient λis close to λΜ However for concrete cable-stayed bridgesthe main beam is in a compression state and is mainlysubjected to axial force its shear lag coefficient λ is closer toλN

62 eoretical versus Finite Element Main Beam Shear LagResults e shear lag variation law of the main beam of thesingle-cable PC cable-stayed bridge along the longitudinaldirection was discussed above as per our theoretical analysise shear lag coefficient and normal stress value of the mainbeam cross section are mainly concentrated in the range ofthe force point of the cable e effective width of the topplate of the upper flange in our analysis was set to 400m andthe full section of the bottom plate to σN 032N andσ( )N 009N (where N is the axial component of the cableforce) so that λN σNσ( )N 356 Shear lag coefficient λΜwas only considered a momentary action based on thedefinition of the effective width of the bridge e effectivewidth of the top plate in this case was 2400m so λΜ 135

We established the full-bridge plane bar modelaccording to finite element analysis e diagram of the

bending moment and axial force of the main beam underWorking Condition 1 suggests relatively small internalforces in Pier 4 and the main girder of Pier 5 e internalforce diagram of the main beam at Pier 4 was isolated forfurther analysis as shown in Figure 15

As shown in Figure 15 the bending moment force ratioMN is between λΜ and λN that is between 135 and 356e bending moment axial force ratio MN at the actionpoint of the cantilever end cable force C4 (C4prime) is 120 and118 According to formula (6) the shear lag coefficient λ ofthe upper edge of the main beam section is 235 and 236respectively

In the construction section the axial moment ratioMNof the main beam section of the C3primendashC1prime cable force actionpoint is 149 159 and 164 According to formula (6) theshear lag coefficients of the upper edge of the main beamsection of C3primendashC1prime cable force point are 223 220 and 218respectively e axial force ratio MN of the main beamsection of the C3ndashC1 cable force action point is 147 181and 203 Again according to formula (6) the shear lagcoefficients λ of the upper edge of the main beam section ofthe C3ndashC1 cable force action point are 224 213 and 207Our finite element results of the shear lag effect of the mainbeam along the longitudinal direction suggest that the shearlag coefficient at Point A of the action point of cantilever endcable force C4 (C4prime) is 418 and 361 the shear lag coefficientλ at Point B reaches 307 and 306

In the Pier 4 C3primendashC1prime beam section the shear lagcoefficients λ of Point A at the cable force action point are232 222 and 249 respectively those of Point B at the cableforce action point are 126 123 and 129 respectivelyWithin the C3ndashC1 beam section the shear lag coefficients λof Point A at the cable force action point are 257 271 and271 and those corresponding to Point B are 211 122 and212 respectively

In the beam section construction stage the analysisresults of the C3ndashC1 (C3primendashC1prime) cable force action pointtreatment theory agree with the finite element analysisresults In this position the theoretically derived shearforce lag coefficient λ can be used to reflect the actual forceat that point e finite element analysis results at the C4(C4prime) action point of the cantilever end cable force exceedthe theoretical analysis results e cantilever end cableforce action point is mainly subjected to axial force so itsshear lag coefficient λ should be closer to λN In this po-sition the shear lag factor λN can be used to reflect theactual force

005

115

225

0 1 2 3 4 5 6 7 8 9 10

Shea

r lag

coef

ficie

nt

MN

λM = 2 λN = 1

Figure 14 Relationship between shear lag coefficient and bendingmoment axial force ratio

14 Advances in Materials Science and Engineering

63 Variation Law of Shear Lag Effect of PC Cable-StayedBridge Main Beam along Longitudinal Direction e shearlag variations in the main beam of the single-cable PC cable-stayed bridge along the longitudinal direction can besummarized as follows

(1) In the construction stage at the cantilever end thecable force action point is the main recipient of axialforce e shear lag coefficient λN of the axial forceacting at this position reflects the actual force at thatpoint

(2) In the beam segments between the cable forces at thecable force action point the theoretical shear lagcoefficient λ determined by the ratio MN of thebending moment to axial force reflects the actualstress

(3) In the midspan beam section between the actionpoints of cable forces the shear lag coefficient ofbending moment λM is used to reflect the actualstress For the beam section between the action pointof cable force and the middle of the span the sectionshear lag coefficient is obtained by linearinterpolation

7 Conclusion

e structural control section stress at work in a typicalconstruction process was measured on a full cable-stayedbridge model in this study Sets of finite element and modeltest results were respectively analyzed and compared Ourconclusions can be summarized as follows

(1) In the construction stage the stress value of the cabletension area of the main beam upper edge markedlychanges after tensiling the cable force the shear lageffect is very prominent After a hanging basket loadis applied the main beams of certain sections presentalternating positive and negative shear lag

(2) e shear lag coefficient of the upper edge of themain beam in G1ndashG8 sections under WorkingConditions 1 and 4 was extracted from the spatialfinite element analysis for comparison to find thatthe experimental and theoretical results are in ac-cordance After tension of the cable force and theapplication of a hanging basket load positive andnegative shear lag effects alternate on the upper edgeof the box girder in certain areas As tension cableforce increases in each section the shear lag coef-ficient of the section upper edge decelerates

(3) In the construction stage at the cantilever end cableforce action point is mainly subjected to axial forcee shear lag coefficient λN of the axial force on thisposition reflects the actual force situation at thatpoint In the beam segments between the cableforces the shear lag coefficient λ determined by theratio of the bending moment to the axial force re-flects the actual stress at the cable force action pointIn the midspan beam section between the actionpoints of cable forces the shear lag coefficient λM ofthe bendingmoment reflects the actual stress For thebeam section between the cable force action pointand the middle of the span the section shear lagcoefficient can be obtained by linear interpolation

C2prime C3

4250kNmiddotm

17300kNmiddotm 19100kNmiddotm

4

C1prime C4

24400kNmiddotm

36800kNmiddotm

C4prime C3prime C2

10500kNmiddotm

18600kNmiddotm

C1

17700kNmiddotm

55000kNmiddotm

6820kNmiddotm

(a)

C1prime C4

13000kN7030kN

C4prime C1

5680kN

20300kN

3600kN

27000kN

C2prime C3

10900kN

C3prime C2

14900kN

4

(b)

Figure 15 Internal force of main girder on Pier 4 (Condition 1) (a) Moment diagram (b) Axial force diagram

Advances in Materials Science and Engineering 15

(4) e shear lag effect of the main beam is prominent inthe construction process e shear lag effect is mostobvious at the cable force operation point of thecantilever end As construction progresses and thecable force increases the shear lag effect of otherbeam sections decreases at each point except for thecantilever end

Data Availability

Part of the shear lag coefficient data used to support thefindings of this study are included within the article All ofthe shear lag coefficient data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors would like to thank the financial supports fromthe National Key RampD Program of China (2018YFC0809600and 2018YFC0809606) and Natural Science Foundation ofLiaoning Province (2019-MS-265)

References

[1] Z Li J G Nie and W Y Ji ldquoPositive and negative shear lagbehaviors of composite twin-girder decks with varying cross-sectionrdquo Science China Technological Sciences vol 60 no 1pp 116ndash132 2017

[2] S-J Zhou ldquoFinite beam element considering shear-lag effectin box girderrdquo Journal of Engineering Mechanics vol 136no 9 pp 1115ndash1122 2010

[3] Z Li and R K L Su ldquoAnalytical solutions for compositebeams with slip shear-lag and time-dependent effectsrdquo En-gineering Structures vol 152 pp 559ndash578 2017

[4] Q Z Luo Y M Wu Q S Li J Tang and G D Liu ldquoA finitesegment model for shear lag analysisrdquo Engineering Structuresvol 26 no 14 pp 2113ndash2124 2004

[5] P Z Lin F K Liu W Ji et al ldquoAnalysis on shear lag effect ofconcrete box beam by variational principlerdquo Journal of theChina Railway Society vol 35 no 2 pp 93ndash98 2013

[6] Y Y Zhang H Zhang W Li et al ldquoAnalysis on shear-lageffect of box girders based on different shear-lag generalizeddisplacementrdquo Journal of Railway Science and Engineeringvol 13 no 6 pp 1083ndash1090 2016

[7] S T Chang Q Zhang and S Zhang ldquoShear lag effect in singleplane cable-stayed bridgerdquo Advances in Structural Engi-neering vol 1 no 4 pp 301ndash306 1998

[8] G-F Wu and X Hong ldquoeoretical and experimental studyon shear lag effect of partially cable-stayed bridgerdquo Journal ofZhejiang University-Science A vol 6 no 8 pp 875ndash877 2005

[9] L Pengzhen Y Zijiang L Fengkui et al ldquoShear lag effect ofconcrete box girders considered varying flange-depthsrdquoJournal of Civil Architecturalamp Environmental Engineeringvol 35 no 1 pp 76ndash79 2013

[10] E Yamaguchi T Chaisomphob J Sa-Nguanmanasak andC Lertsima ldquoStress concentration and deflection of simplysupported box girder including shear lag effectrdquo StructuralEngineering and Mechanics vol 28 no 2 pp 207ndash220 2008

[11] X B Shen ldquoAnalysis of shear lag effect in the main girder ofwide low-pylon cable stayed bridgerdquo Construction Technologyvol 44 no 24 pp 76ndash79 2015

[12] L Xingmin ldquoAnalysis of influences of construction processeson shear lag effect of concrete box girdersrdquo Bridge Con-struction vol 43 no 1 pp 30ndash34 2013

[13] L Muyu W H Sun X D Sun et al ldquoStress analysis of widergirder of an extradosed cable-stayed bridge based on thelargest cantilever construction stagerdquo Journal of HuazhongUniversity of Science and Technology (Urban Science Edition)vol 27 no 2 pp 11ndash14 2010

[14] C Qianshu H Wenlong and Y Menggang ldquoAnalysis ofshear lag effect in construction stage of wide box girderextradosed cable-stayed bridge with large flangesrdquo Journal ofRailway Science and Engineering vol 15 no 12 pp 3158ndash3164 2018

[15] D R Li B M Wang and Y C Lin Structural Model Ex-periment Science Press Beijing China 1996

[16] J W Dali andW F Laili Experiment and Analysis of ResidualStress Ocean Press Beijing China 1987

[17] H Hossdort Structural Model Analysis China Architecture ampBuilding Press Beijing China 1986

[18] S K Mei Structural Experiments and Structural DesignChina Communications Press Beijing China 1980

[19] B Atmaca and S Ates ldquoConstruction stage analysis of three-dimensional cable-stayed bridgesrdquo Steel amp Composite Struc-tures vol 12 no 5 pp 413ndash426 2012

[20] S W Park M R Jung D J Min and M Y Kim ldquoCon-struction stage analysis of cable-stayed bridges using theunstrained element length methodrdquo Journal of e KoreanSociety of Civil Engineers vol 36 no 6 pp 991ndash998 2016

[21] M Gunaydin S Adanur A C Altunisik B Sevim andE Turker ldquoDetermination of structural behavior of Bosporussuspension bridge considering construction stages and dif-ferent soil conditionsrdquo Steel and Composite Structures vol 17no 4 pp 405ndash429 2014

[22] S Adanur M Gunaydin A C Altunisik and B SevimldquoConstruction stage analysis of humber suspension bridgerdquoApplied Mathematical Modelling vol 36 no 11 pp 5492ndash5505 2012

[23] J B Mander M J N Priestley and R Park ldquoeoreticalstress-strain model for confined concreterdquo Journal of Struc-tural Engineering vol 114 no 8 pp 1804ndash1826 1988

[24] Y A Khalid C L Chan B B Sahari and A M S HamoudaldquoBending behaviour of corrugated web beamsrdquo Journal ofMaterials Processing Technology vol 150 no 3 pp 242ndash2542004

[25] A Demir N Caglar H Ozturk and Y Sumer ldquoNonlinearfinite element study on the improvement of shear capacity inreinforced concrete T-Section beams by an alternative diag-onal shear reinforcementrdquo Engineering Structures vol 120pp 158ndash165 2016

16 Advances in Materials Science and Engineering

equipment are required A small-scale model requires only alightweight load but it is relatively difficult to build or equipwith test instruments further the measurement error isrelatively large It is necessary to weigh various test con-ditions and requirements before selecting the most appro-priate proportion for model making roughcomprehensive consideration the geometric scaling ratio ofthe wholemodel in this study was determined to be 1 40 andthe length of the cable-stayed bridge model was 105m

311 Box Girder Design e main girder of this model ismade of aluminum alloy Each section of the box girder wasmade from plates pulled through rivets and glued togethere elastic modulus ratio between C50 concrete and alu-minum alloy is 050 e section area and the moment ofinertia for the model box girder can be calculated under theprinciple of similarity

Am Ap middot12

1113874 11138751402

1113874 1113875 (1)

Im Ip

12

1113874 11138751404

1113874 1113875 (2)

According to geometric similarity the size of the boxgirder section model is basically unchanged e wallthickness of the box girder was halved to ensure that the

section area and the moment of inertia were similar in thisanalysis Some section similarity deviation was controlledwithin an allowable range e error of modulus only affectsthe section stress so the model and prototype were calcu-lated under corresponding working conditions eir ac-curate similarity ratios were determined according to thestrain and stress calculation results

e box girder of the test model consists of 19 sectionswhich may be assembled via a typical construction processDead load compensation was conducted by hanging heavyobjects directly onto a predefined hanging point the con-struction load and live load were imposed by adding heavyloads on the bridge deck e full-bridge model is shown inFigure 2

312 Stay Cable Design e stay cable of model is made ofhigh-strength steel wires similar to the prototype cable eelastic modulus ratio in this case is equal to 1 e sectionalarea of the model stay cable is

Am Ap middot1402

1113874 1113875 (3)

e girder and stay cable were connected by screw threadto allow us to adjust the stay cables easily A vibrating wire-type load cell was installed at the joint tomeasure the force ofthe stay cable e self-weight of the stay cable markedly

G6G9G4

Stay cablesnumbers C15 to C1

420

G5G3

5

Stay cablesnumbers C15prime to C1prime

89 242G7G1

4

Stay cablesnumbers C1 to C15

89

6

G8G2

Stay cablesnumbers C1prime to C15prime

3

(a)

150

320

304

16 50

500

15 15

9253250

2525

40

400 925

15040

40

341

4

30

500

(b)

4030

151630

432

0

500 925

250

50 15

50

3250400

4050

925

50

341

4

500

250

(c)

Figure 1 Structural drawing of bridge (a) main bridge arrangement (unit m) (b) cross section of midspan girder (unit cm) and (c) crosssection of side-span girder (unit cm)

Advances in Materials Science and Engineering 3

influences its deformation in order to accurately reflect thestress characteristics of the structural nonlinearity we im-posed load compensation for the dead weight of the cable viaadditional mass method

313 Main Tower Design e main tower of the cable-stayed bridge was welded with Q235B carbon structural steele ratio of elastic modulus to C50 concrete is 6 According tosimilarity theory the section area and moment of inertia ofthe main tower of the model can be calculated as follows

Am Ap middot16

1113874 11138751402

1113874 1113875 (4)

Im Ip

16

1113874 11138751404

1113874 1113875 (5)

e outer contour of the main tower section remainedbasically unchanged and the wall thickness was reduced by16 throughout this analysis to satisfy the similarity betweenthe moment of inertia and the area of the cross section emain tower was made in one piece and bolted into con-nection with the box girder Because the main tower of thebridge was a broken line the dead load of the main towerproduced not only the stress of the tower body but also thebending moment of the tower body which affect the entirestructural system of the cable-stayed bridge internal forcestate We carried out dead load compensation for the maintower to account for this

314 Design of Bridge Pier and Bearing emain piers andside pier were fabricated by steel weldinge box girder andPier 4 were connected by bolts with set bearing for Pier 3

80

100220 220

8125

1362513625

(a)

2225 2225605010500

Stay cablesC15prime~C1prime

Stay cablesC1~C15 Stay cables

C1prime~C15prime

Stay cablesC15~C1

Pier 4 Pier 5

1688

3543210123 211234

53752 times 630 = 12601575 2 times 740 = 1480 555740 2 times 630 = 1260 15752 times 740 = 1480555 537574050

(b)

Stay cablesC15prime~C1prime

Stay cablesC1~C15

Stay cablesC1prime~C15prime

Stay cablesC15~C1 16

8814

00

2225 2225605010500

3553554 5

(c)

Figure 2 General arrangement drawing of model (unit mm) (a) midspan section diagram (b) block diagram and (c) temporary supportand loading diagram

4 Advances in Materials Science and Engineering

Pier 5 and Pier 6 We set a load sensor in each bearing Atension-applying device was used to simulate the dead load

32 Model Fabrication and Installation We used a specialvibrating wire sensor to simulate the cable sensor which wascalibrated before installation to ensure long-term stabilityaccuracy and reliability In order to accurately simulate theconstruction process we installed the box girder in sub-section form

According to the real-world construction requirementsPier 5 was set as a temporary pier and a load sensor wasinstalled to measure the counter force of the temporarysupport under the action of the construction load emodel installation process is shown in Figure 3

33 Loading System and Dead Load Compensation

331 Dead Load Compensation of theMain Tower edeadload compensation of the main tower can be divided intotwo partsere is a concentrated load which was applied vialever system e upper segment load was applied to thecenter of gravity on the upper segment of the main towere lower segment load compensation accords with thesimilarity of the tower rootrsquos bending moment whichguaranteed that the tower section and tower angle sectionload effects were similare quality of the compensation forthe upper segment in this test was 83710 kg and that of thelower segment was 62000 kg

332 Load Compensation of Main Girder e load com-pensation of the main beam was applied to the diaphragm inconcentrated form e compensation quality of the inter-mediate span standard segment in this case was 26030 kgthe compensation quality of the intermediate span towerroot segment was 38530 kg the compensation quality of theside span standard segment was 27720 kg and the inter-mediate span tower root segment compensation quality was33580 kg e second-stage load compensation for eachlifting point included a side span of 6500 kg and a middlespan of 7630 kg

34 Test Load Working Conditions and Test Contents Weinstalled a stay cable and box girder in accordance with theactual construction process of the original bridge e staycable box girder and main tower were gradually com-pensated for dead loads and the stay cable was tensioned andadjusted as necessary e model test was carried out tosimulate the construction load and stress of the girdersection under the working conditions of a midspan closurein the construction stage

35 Test System and Measuring Point Arrangement Ourmodel testing system includes a support reaction forcetesting component dead load testing component stresstesting component displacement testing component andcable force testing component Each fulcrum was equippedwith a calibrated strain tester and a load cell to form a

reaction force test system e dead load test was performeddirectly on the calibrated scale e stress testing systemconsists of strain gauge and resistance strain gauge eelectrical test method was used to test the live load and deadload structural stress distributions e displacement testwas performed using a level gauge a dial gauge and atheodolite Our cable force test system consists of a cali-brated special frequency tester and a steel string load celle section and measuring point arrangement we used areshown in Figures 4 and 5

36 Construction Stage Working Conditions We exampledsix typical construction working conditions correspondingto real-world construction phases In Working Condition 1(double cantilever stage) we installed 1 (1prime) beam sectiontensioned C1ndashC4 (C1prime-C4prime) cables and applied a hangingbasket load In Working Condition 2 (maximum doublecantilever stage) we installed 2 (2prime) and 3 (3prime) beam sectionstensioned C5ndashC10 (C5prime-C10prime) cables and applied a hangingbasket load InWorking Condition 3 we closed the side spanof number 5 pier line and applied a hanging basket load tothe C11 cable In Working Condition 4 (maximum singlecantilever stage) we installed Section 4 tensioned C12ndashC14(C12prime-C14prime) cables and applied a hanging basket load at theC14 cable In Working Condition 5 we tensioned the C15(C15prime) cable and applied a closing weight to complete themidspan closure Working Condition 6 included cabletension adjustment and application of a second-stage deadload

4 Test Results versus Finite ElementAnalysis Results

41 Establishment of Finite Element Model We used MidasFEA finite element software to conduct our numerical an-alyses e construction scheme investigated in this studywas considered a sliding formwork with a combination ofthe stress state of the main girder and control the maintower e bridge was constructed via forward installationmethod [19ndash22] In the finite element analysis the cable wassimulated by the truss unit only the tower was simulated bythe beam element e main beam in this analysis is a solidelement and a six-node tetrahedron was used for divisione full-bridge truss unit is 120 the beam unit is 54 the solidunit is 29787 and there are 52631 nodes

After tensioning the C4 (C4prime) cable force the hangingbasket load was 5100 kN the boundary condition was a fixedbottom section of Pier 4 restraining the line displacementand angular displacement in the X Y and Z directions ebeam and Pier 5 were separated the beam bottom con-strained the linear displacement in X Y and Z directions inthis case as well Temporary support was simulated at1485m on the right side of the beam section of Pier 5which constrained the linear displacement in the Z direction

Next we tensioned C10 (C10prime) after the cable force ehanging basket load was 5100 kN and the boundary con-dition was the same as the first working condition e sidespan was closed the cable force of Pier 5 was adjusted and

Advances in Materials Science and Engineering 5

the hanging basket load was 5100 kN after the C11 (C11prime)cable tension was applied e boundary condition in thiscase involved removed the temporary support placing thebottom of Piers 4 and 5 under the same working con-dition e line displacement in the Z direction was re-strained at the end of the main beam

After the tension of the C14 (C14prime) was applied thehanging basket load was applied At 5100 kN the boundaryconditions were the same as the previous working conditionAfter applying the tension of C15 (C15prime) cable force the

closing weight of 1950 kN was applied e middle span wasclosed and the closing weight was unloaded e boundarycondition in this case was an unchanged constraint on thebottom of Pier 4 while the line displacement of the bottomof Pier 5 was constrained in the Z direction

e finite element model as established under thesespecific working conditions is shown in Figure 6 Due tolimited space only the results of Working Condition 1and Working Condition 4 are presented here forcomparison

(a) (b)

(c) (d)

Figure 3 Installing the model (a) installing four pairs of stay cables at Pier 4 (b) closure of side span of Pier 5 (c) second stage dead loadapplied by model and (d) model exerts live load

2225 2225605010500

Stay cablesC15prime~C1prime

Stay cablesC1~C15

Stay cablesC1prime~C15prime

Stay cablesC15~C1

Pier 4 Pier 5

1688

G4 G2 G1 G3 G9 G8G7 G5 G6

Figure 4 Diagram of testing section arrangement about girder (unit mm)

1 2

18

3

13

8

17

16

4 5

15 14

6 7 10

12

9 11

Figure 5 Arrangement of stiff girder stress measuring point

6 Advances in Materials Science and Engineering

We conducted forward installation calculations reflect-ing the construction process of the entire bridge e leastsquare method was applied to calculate of the cable force ofthe bridge during forward installation and the results werecompared with the reasonable cable force of the bridge untilall design requirements were satisfied At this point thecorresponding construction cable force could be consideredthe initial tension of the stay cable To accelerate conver-gence the initial tension of the cable was selected per thestatic equilibrium determined in the first forward installa-tion iteration Table 1 lists the initial tension of each staycable after the iterative calculation the number of stay cablesis shown in Figure 1

e stay cable at Pier 5 was adjusted once the side spanwas closed to bring the internal force at the root of the maintower closer to the deformation at the cantilever end of themain beam e C6 cable force was adjusted to 8000 kN C7cable force to 8000 kN C8 cable force to 8000 kN and C9cable force to 7500 kN at this time Before the constructionof bridge deck pavement to ensure uniform internal forceand deformation of the main girder across the whole bridgethe cable at Pier 4 was adjusted by bringing C1prime cable forceto 9000 kN C2prime cable force to 9000 kN C1 cable force to11000 kN C2 cable force to 11500 kN C3 cable force to9000 kN and C4 cable force to 9000 kN e cable force ofpart of the stay cable at Pier 5 was also adjusted by bringingC1 cable force to 10000 kN C2 cable force to 10000 kN C1primecable force to 10000 kN and C2prime cable force to 10500 kN

42 Main Beam Cross Section Shear Lag Coefficient underWorking Condition 1 e model test results and the shearlag coefficient of the upper edge of the main beam section ofWorking Condition 1 in the spatial finite element weresubjected to comparative analysis After tensioning the C4cable the full beam of the main beams G1 and G2 waspressed (tensile stress is positive compressive stress isnegative) the average measured stress of the upper edge ofthe main beams G1 and G2 was minus207MPa and minus077MPa atthis point After applying a hanging basket load on thecantilever end of the box girder the upper edge of the maingirders G1 and G2 showed a large amount of stress changese midabdominal segment bore compressive stress whilethe upper edge of section G1 of main beam bore tensilestress e upper edge of section G2 of main beam boretensile stress e average measured stress of the upper edgeof the main beams G1 and G2 was 024MPa and 049MPaAfter the tensioning force of the main girders G1 and G2 andapplication of the basket load the shear lag coefficient ateach point of the upper edge was as shown in Figure 7

After tensioning the C4 cable the main beams G5 andG6 were pressed in the full section and the average measuredstress of the upper edge of the main beams G5 and G6 wasminus189MPa and minus053MPa After applying the hangingbasket load on the cantilever end of the box girder the upperedge of the main beam G5 and G6 showed a large amount ofstress change e midabdominal segment bore compressivestress and the upper edge of section G5 and G6 of main beam

(a) (b)

(c) (d)

(e) (f )

Figure 6 Special finite element model for main construction operating conditions (a) Tension C4 (C4prime) applying a basket load (b) TensionC10 (C10prime) applying a basket load (c) Cable tension adjustment tensioning C11(C11prime) applying a hanging basket load (d) TensionC14(C14prime) applying a hanging basket load (e) Tension C15 (C15prime) applying counterweight midspan closure (f ) Grid division of mainbeam section

Advances in Materials Science and Engineering 7

bore tensile stress e average measured stress of the upperedge of the main beams G5 and G6 was 053MPa and055MPae shear lag coefficient at each point of the upperedge was as shown in Figure 8 after the tensioning force ofthe main girders G5 and G6 and the application of the basketload

As tension was applied the concentrated load was placedin the middle of the box girder to intuitively observe theshear lag effect in the middle of the box girder section ehanging basket load was applied in the form of a concen-trated load so that the upper edge of sections G1 G5 and G6of the main beam presented tensile stress though the middleweb section did not After the tensile force and the hangingbasket load were applied positive and negative shear lageffects appeared alternately on the upper edge of the mainbeam

43 Main Beam Cross Section Shear Lag Coefficient underWorking Condition 4 e model test results and the shearlag coefficient of the upper edge of the main beams G1ndashG8section under Working Condition 4 for the spatial finiteelement was also extracted for comparative analysis Asshown in Figures 9 and 10 the shear lag coefficient of theupper edge of the section G7 of themain beam changed from009ndash229 to minus293ndash210 after the tension of the cable and theapplication of the hanging basket load e shear lag

coefficient of the upper edge of the main beam G3 changedfrom 055ndash135 to 039ndash232 the shear lag coefficient of theupper edge of the main beam G4 changed from 063ndash159 to066ndash166 and the shear lag coefficient of the upper edge ofthe main beam G8 changed from 034ndash152 to 021ndash192 Inthis case after the tension was applied to the C14 cable andthe hanging basket load was applied the shear lag coefficientof the upper edge decelerated and the changes were slighte shear lag coefficient of the upper edge of the sections G2and G1 G5 and G6 of the main girder near the tower rootwas weak In each section the shear lag coefficient of theupper edge decelerated as tension cable force increased etest results are consistent with the finite element calculationresults

5 Longitudinal Shear Lag Effect in MainBeam of Single-Cable Plane P C Cable-Stayed Bridge

e stress on the main girder section of cable-stayed bridgeis highly complex e construction of the cable-stayedbridge is a dynamic process e stress in the main beamsection under different working conditions and differentbeam sections may widely differ e shear lag coefficientcalculation results along the longitudinal length of severalkey nodes on the upper edge of the cross section of the main

Table 1 Initial cable tension of each stay cable

Pier 4 Pier 5Cablenumber

Tension cable force(left) (kN)

Cablenumber

Tension cable force(right) (kN)

Cablenumber

Tension cable force(left) (kN)

Cablenumber

Tension cable force(right) (kN)

C1prime 7500 C1 9000 C1 9000 C1prime 4500C2prime 6500 C2 9500 C2 9500 C2prime 6500C3prime 5500 C3 8000 C3 8000 C3prime 5500C4prime 5500 C4 8000 C4 8000 C4prime 5000C5prime 5500 C5 8500 C5 8000 C5prime 6000C6prime 5500 C6 8000 C6 7400 C6prime 6500C7prime 5500 C7 8000 C7 7000 C7prime 5500C8prime 6000 C8 8000 C8 7000 C8prime 6000C9prime 6000 C9 7500 C9 7000 C9prime 6000C10prime 6500 C10 7500 C10 7500 C10prime 6500C11prime 6500 C11 7500 C11 7500 C11prime 6500C12prime 9500 C12 7500 C12 7500 C12prime 9000C13prime 9500 C13 7500 C13 7500 C13prime 9000C14prime 10000 C14 7500 C14 7500 C14prime 9500C15prime 9500 C15 6500 C15 6500 C15prime 9500

ndash100ndash050

000050100150

Shea

r lag

coeffi

cien

t

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(a)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

ndash200ndash100

000100200300

Shea

r lag

coeffi

cien

t

Node lateral coordinate (m)

(b)

Figure 7 Shear lag coefficient variations in G1 and G2 (Condition 1) (a) G1 section (b) G2 section

8 Advances in Materials Science and Engineering

beam can be used to understand the shear lag coefficientdistribution along the longitudinal length of the upper edgeof the main beam section during the construction stage Forconvenience the key points on the top edge of the crosssection of the main beam are marked here with capitalletters e distances from A-D points to the transversesymmetry lines are 0m minus2m minus11m and minus16m respec-tively e lateral position of each point is shown inFigure 11

Due to space limitations only the distribution of theshear lag coefficient along the longitudinal length of theupper edge of the main beam section under the followingtwo typical working conditions is described here WorkingCondition 1 is defined by tensiling the C4 (C4prime) cable forceWorking Condition 4 is defined by tensiling the C14 (C14prime)cable force e boundary conditions of each workingcondition were kept consistent

51Working Condition 1 Shear Lag Effect along LongitudinalBridge Under the action of self-weight and cable force the

shear lag coefficient of the flange of the main beam sectionalong the longitudinal bridge was observed as shown inFigure 12 In the longitudinal distribution of the shear lagcoefficient at the key point of the upper flange the midpointof the cross section of the main beam (Point A) and theboundary point between the upper edge and the middle web(Point B) change most sharply

At Pier 4 a sudden change in shear lag coefficient at theend of the cantilever occurred due to the tension of the C4(C4prime) cable A large stress concentration occurred at theseconcentrated load points which altered the upper flangestress distribution law Under the action of cable force theupper edge of the main beam section shows a more sig-nificant shear lag effect of the cable tension zone at locationscloser to the cable action point At Point A the shear lagcoefficient reaches 418 at Point B it is 307 e stressdistribution of the main beam appears to grow more uni-form as the distance between the main beam and the cableforce increases

Pier 4 C3ndashC1 (C3prime-C1prime) shows a similar distributionpattern in each beam section e shear lag coefficient of

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

ndash100ndash050

000050100150200

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

(a)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

ndash200ndash100

000100200300

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

(b)

Figure 8 Shear lag coefficient variations in G5 and G6 (Condition 1) (a) G5 section (b) G6 section

000

050

100

150

200

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16Node lateral coordinate (m)

Shea

r lag

coef

ficie

nt

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(a)

000050100150200250

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16Node lateral coordinate (m)

Shea

r lag

coef

ficie

ntTensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(b)

000050100150200250

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(c)

Shea

r lag

coef

ficie

nt

000050100150200250

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(d)

Figure 9 Contrast diagram of shear lag coefficient in G1-G4 (Condition 4) (a) G1 section (b) G2 section (c) G3 section and (d) G4 section

Advances in Materials Science and Engineering 9

Point A and Point B is prominent at the point of action of thecable forcee change range of shear lag coefficient of PointA at the point of cable force action is 222ndash271 and at Point Bis 122ndash212 e stress distribution of the beam sectionbetween the points of cable force is relatively uniform eA-point shear lag coefficient λ ranges from 115 to 116 theB-point shear lag coefficient λ from 112 to 140 and theC-point shear lag coefficient λ from 109 to 126

e stress distribution of the upper flange of the Pier 4tower root section changed due to the existence of the crossbeame change appears to be most dramatic at Point D atthe end of the upper extension arm of the main beam crosssection e shear lag coefficient λ at Point D near the towerranges from 114 to 154 marking a very significant variationcompared with other beam sections

e variation of the cantilever end the cable force pointand the C3ndashC1 (C3prime-C1prime) beam section of Pier 5 is similarto that of Pier 4 Under the action of the tension C4 (C4prime)Point A here shows a shear lag coefficient λ of 300 while thatat Point B reaches 250 In our analysis the stress distri-bution of the main beam grew more uniform as the distancebetween the main beam and the cable force increased

Pier 5 C3ndashC1(C3prime-C1prime) shows a similar distributionpattern in each beam section e shear force lag coeffi-cient of Point A and Point B is prominent at the point ofaction of the cable force e variation range of lag

coefficient λ is 157ndash244 and the shear lag coefficient λ ofPoint B is 138ndash184 at the point of action of cable forcee stress distribution of the beam section between thecable force points is relatively uniform e changes inshear lag coefficient λ of Point A range from 141 to 166that at Point B varies from 107 to 157 and that at Point Cranges from 075 to 108 Again the most significantchange was observed at Point D the shear lag coefficient λof Pier 5 at point D near the tower root varies from 120to 133

At the C1prime cable force on the right side of Pier 5 theshear lag coefficient of Points A and B is weaker than at otherpoints of cable force mainly because of the temporarysupport e shear lag effects on both sides were similar dueto the small reaction force of the temporary pier

Under the action of self-weight and cable force with thisworking condition the main characteristics of the shear lageffect of the upper edge of the main beam section of thesingle-cable PC cable-stayed bridge in the longitudinal di-rection can be summarized as follows

(1) In the construction stage under the action of cableforce and near the point of cable tension the upperedge of the girder section shows more significantshear lag effect in the cable force tension area egirder stress distribution grows more uniform as thedistance between the girder section and the cableforce action point increases

(2) In the C3ndashC1(C3prime-C1prime) beam section the upper edgeof the main beam section at the cable force actionpoint presents shear lag effects in the tension zonee stress distribution in the beam section betweenthe points of the cable force is relatively uniformwhich characterizes the shear lag distribution in theordinary curved box girder

000020040060080100120140160

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(a)

000

050

100

150

200

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(b)

ndash400ndash300ndash200ndash100

000100200300

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(c)

000

050

100

150

200

250

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(d)

Figure 10 Contrast diagram of shear lag coefficient in G1-G4 (Condition 4) (a) G5 section (b) G6 section (c) G7 section and (d) G8 section

B ACD

Figure 11 Main girder cross-section diagram at every point

10 Advances in Materials Science and Engineering

(3) e shear lag coefficient at the end of the upperextension arm of the cross section of the main beamat the root of the tower changes most dramaticallyamong the areas we observed We also observedconversion from a positive shear lag effect to anegative shear lag effect near the tower

(4) Under the influence of temporary support the shearforce lag coefficient of the action point of the C1primecable force on the right side of Pier 5 is weaker thanother cable forces e temporary pier reaction forceis small whichmakes the shear lag on both sideseeffects were still relatively similar in this case

52Working Condition 4 Shear Lag Effect along LongitudinalBridge is working condition represents the largestsingle cantilever stage e side spans at this point areclosed and the shear lag effects near their supports havemarkedly increased due to tensioning of C12primendashC14prime tailse shear lag effect of the inner beam section of the sidespan C11primendashC1prime is lower than under the previous workingcondition

As shown in Figure 13 at Pier 4 the cantilever end fallsunder the action of tensioning C14 cable force e shear lagcoefficient λ at Point A reaches 340 and at Point B is 240 Inthe beam section of Pier 4 C11ndashC1 (C11primendashC1prime) the shearlag coefficient of Points A and B is prominent at the point ofaction of the cable force Point A shows a shear lag coefficientλ of 151ndash281 at the cable force action point where Point Bshows a shear lag coefficient λ of 435ndash212 the stress dis-tribution of the beam section between the points of the cableforce is relatively uniform e shear lag coefficient λ ofPoint A varies from 105 to 144 and that of Point B from 119to 110 e shear lag coefficient λ of Point C ranges from065 to 095

At Pier 5 the end of the cantilever is under the action ofthe tensioning force of C14 e shear lag coefficient λ ofPoint A reaches 357 and that of Point B is 222 In the beamsection of C11ndashC1 (C11primendashC1prime) in Pier 5 the shear lagcoefficients of Points A and B at the point of action of thecable force are prominent Point A has a shear lag coefficientλ of 148ndash281 at the action point of the cable force wherePoint B has a shear lag coefficient λ of 133ndash212 e stressdistribution of beam section between the points of cable

4rsquo 3rsquo 2rsquo 1rsquo 4 1 2 3 4Longitudinal bridge position

Point APoint B

Point CPoint D

Pier 4

000050100150200250300350400450

Shea

r lag

coef

ficie

nt

(a)

Point APoint B

Point CPoint D

4 3 2 1 5 1rsquo 2rsquo 3rsquo 4rsquoLongitudinal bridge position

Pier 5

000

050

100

150

200

250

300

350

Shea

r lag

coef

ficie

nt

(b)

Figure 12 Distribution of shear lag coefficient along longitudinal direction (Condition 1) (a) Pier 4 (b) Pier 5

Advances in Materials Science and Engineering 11

Point APoint B

Point CPoint D

3 14rsquo13rsquo12rsquo 11rsquo 10rsquo 9rsquo 8rsquo 7rsquo 6rsquo 5rsquo 4rsquo 3rsquo 2rsquo 1rsquo 4Longitudinal bridge position

Pier 4

000

050

100

150

200

250

300

Shea

r lag

coef

ficie

nt

(a)

Point APoint B

Point CPoint D

4 1 2 3 4 5 6 7 8 9 10 11 12 13 14Longitudinal bridge position

000

050

100

150

200

250

300

350

400

Shea

r lag

coef

ficie

nt

(b)

Point APoint B

Point CPoint D

14 13 12 11 10 9 8 7 6 5 4 3 2 1 5Longitudinal bridge position

Pier 5

000

050

100

150

200

250

300

350

400

Shea

r lag

coef

ficie

nt

(c)

Figure 13 Continued

12 Advances in Materials Science and Engineering

force is relatively uniform e shear lag coefficient λ ofPoint A ranges from 120 to 155 that of Point B from 096 to135 and that of Point C from 076 to 101

Under this working condition the action of self-weightand cable force make the main characteristics of the shear lagat the upper edge of the main beam section of the single-cable PC cable-stayed bridge along the vertical direction verysimilar to those of underWorking Condition 1e shear lageffect at the main beam section is reduced as cable forceincreases

6 Distribution of Shear Lag Effect duringConstruction Stage

61 Shear Lag Coefficient of Main Beam of Single-Cable PlanePC Cable-Stayed Bridge e solution to the shear lag coef-ficient under the compressive bending load involves decom-posing the compressive bending load into a superposition ofthe bending moment action and the axial force action thencalculating the normal stress of themain beamunder the actionof the bending moment and the axial force alone [22] It isassumed that the axial force is borne by the whole section thatis the normal stress generated by the axial force is evenlydistributed along the cross section e results can then belinearly superposed to obtain the comprehensive shear lageffect coefficient Our previous analysis indicated that whenthere is cable tension of the main beam of the cable-stayedbridge in the single-cable PC there is a large stress concen-tration at the point of action of the concentrated loads estress distribution law of the roof plate changes accordingly

Assume that the axial force which is borne by the wholesection does not reflect the actual internal force of the maingirder of the single-cable PC cable-stayed bridge e dis-tribution of the longitudinal normal stress generated byinternal force is not uniform e size and distribution laware related to the internal force state namely the bendingmoment and axial force In this study the shear lag coef-ficient λΜ generated by bending moment and the shear lag

coefficient λΜ generated by axial force were respectivelycalculated to solve the shear lag coefficient λ aftersuperposition

e shear lag coefficient is calculated as follows

λ σσ

(6)

where σ is actual stress and σ( ) is nominal stresse shear lag coefficient is related to the wide span ratio

support conditions load form cross-sectional shape andsize material properties and location of the section [23ndash25]In the final analysis the shear lag is caused by unevendistribution of the longitudinal normal stress caused byinternal force its size and distribution are related to theinternal force state that is the bending moment is related tothe axial force

e actual stress σ and nominal stress σ( ) are thesuperposition of the bending moment and the normalstress generated by the axial force If the shear lag coeffi-cient λΜ produced by the bending moment is definedproperly the shear lag coefficient generated by the axialforce is λN and the shear lag coefficient after the super-position is λ us

λ σN + σM

σN + σM

λNσN + λMσM

σN + σM

(7)

where σ( )N and σ( )M are the nominal normal stressgenerated by bending moment and axial force respectivelyσ( )N and σ( )M are the actual normal stress generated bybending moment and axial force respectively e nu-merator and denominator at the right end of the aboveformula are divided by σ( )N to rewrite (7) as

λ λN + λM σMσN( 1113857

1 + σMσN( 1113857 (8)

ere are σM (MI)y and σN (NA) so(σMσN) (yAI) middot (MN) and order yAI C can beplugged into (8) to obtain

Point APoint B

Point CPoint D

000

050

100

150

200

250

300

350

5 1prime 2prime 3prime 4prime 5prime 6prime 7prime 8prime 9prime 10prime 11prime 12prime13prime14prime6

Shea

r lag

coef

ficie

nt

Longitudinal bridge position

(d)

Figure 13 Distribution of shear lag coefficient along longitudinal direction (Condition 4) (a) Left side of Pier 4 (b) Right side of Pier 4(c) Left side of Pier 5 (d) Right side of Pier 5

Advances in Materials Science and Engineering 13

λ λN + CλM(MN)

1 + C(MN) (9)

where y is the distance from the centroid of the section to theupper and lower edges A is the sectional area I is the sectionmoment of inertia and M and N represent the bendingmoment and axial force of the section

Equation (9) indicates that y A and I are the parametersinherent to the section under analysis λN and λΜ are theshear lag coefficients caused by the axial force and thebending moment respectively which are independent of themagnitude of the internal force Once λN and λΜ are de-termined the total shear lag coefficient of λ is only related tothe ratio of bending moment to axial force

Figure 14 shows the curve of the shear lag coefficientwith the bending moment axial force ratio when λΜ 200and λN 100 Taking the upper flange of the main beam ofthe Fumin cable-stayed bridge as an example C yAI 101

As shown in Figure 14 the total shear lag coefficient λgradually approaches the bending moment shear lag coef-ficient λΜ as the bending moment force ratio increases Asthe bending moment axial force ratio gradually decreases to0 the total shear lag coefficient λ is also gradually reduceduntil it is equal to the axial force shear lag coefficient λNWhen the bending moment force ratio is between 0 and 2the total shear lag coefficient λ changes substantially Alarger λΜ results in a greater change in the total shear lagcoefficient λ When the bending moment force ratio isgreater than 2 the total shear lag coefficient λ changesslowly at is to say in the internal force of this sectionwhether the bending moment or axial force accounts for alarger proportion affects how close the total shear lag co-efficient λ of the section is to the shear lag coefficient (λΜ orλN) caused by this item of internal force

For a general beam bridge the bending moment is themain internal force and so the section shear lag coefficient λis close to λΜ However for concrete cable-stayed bridgesthe main beam is in a compression state and is mainlysubjected to axial force its shear lag coefficient λ is closer toλN

62 eoretical versus Finite Element Main Beam Shear LagResults e shear lag variation law of the main beam of thesingle-cable PC cable-stayed bridge along the longitudinaldirection was discussed above as per our theoretical analysise shear lag coefficient and normal stress value of the mainbeam cross section are mainly concentrated in the range ofthe force point of the cable e effective width of the topplate of the upper flange in our analysis was set to 400m andthe full section of the bottom plate to σN 032N andσ( )N 009N (where N is the axial component of the cableforce) so that λN σNσ( )N 356 Shear lag coefficient λΜwas only considered a momentary action based on thedefinition of the effective width of the bridge e effectivewidth of the top plate in this case was 2400m so λΜ 135

We established the full-bridge plane bar modelaccording to finite element analysis e diagram of the

bending moment and axial force of the main beam underWorking Condition 1 suggests relatively small internalforces in Pier 4 and the main girder of Pier 5 e internalforce diagram of the main beam at Pier 4 was isolated forfurther analysis as shown in Figure 15

As shown in Figure 15 the bending moment force ratioMN is between λΜ and λN that is between 135 and 356e bending moment axial force ratio MN at the actionpoint of the cantilever end cable force C4 (C4prime) is 120 and118 According to formula (6) the shear lag coefficient λ ofthe upper edge of the main beam section is 235 and 236respectively

In the construction section the axial moment ratioMNof the main beam section of the C3primendashC1prime cable force actionpoint is 149 159 and 164 According to formula (6) theshear lag coefficients of the upper edge of the main beamsection of C3primendashC1prime cable force point are 223 220 and 218respectively e axial force ratio MN of the main beamsection of the C3ndashC1 cable force action point is 147 181and 203 Again according to formula (6) the shear lagcoefficients λ of the upper edge of the main beam section ofthe C3ndashC1 cable force action point are 224 213 and 207Our finite element results of the shear lag effect of the mainbeam along the longitudinal direction suggest that the shearlag coefficient at Point A of the action point of cantilever endcable force C4 (C4prime) is 418 and 361 the shear lag coefficientλ at Point B reaches 307 and 306

In the Pier 4 C3primendashC1prime beam section the shear lagcoefficients λ of Point A at the cable force action point are232 222 and 249 respectively those of Point B at the cableforce action point are 126 123 and 129 respectivelyWithin the C3ndashC1 beam section the shear lag coefficients λof Point A at the cable force action point are 257 271 and271 and those corresponding to Point B are 211 122 and212 respectively

In the beam section construction stage the analysisresults of the C3ndashC1 (C3primendashC1prime) cable force action pointtreatment theory agree with the finite element analysisresults In this position the theoretically derived shearforce lag coefficient λ can be used to reflect the actual forceat that point e finite element analysis results at the C4(C4prime) action point of the cantilever end cable force exceedthe theoretical analysis results e cantilever end cableforce action point is mainly subjected to axial force so itsshear lag coefficient λ should be closer to λN In this po-sition the shear lag factor λN can be used to reflect theactual force

005

115

225

0 1 2 3 4 5 6 7 8 9 10

Shea

r lag

coef

ficie

nt

MN

λM = 2 λN = 1

Figure 14 Relationship between shear lag coefficient and bendingmoment axial force ratio

14 Advances in Materials Science and Engineering

63 Variation Law of Shear Lag Effect of PC Cable-StayedBridge Main Beam along Longitudinal Direction e shearlag variations in the main beam of the single-cable PC cable-stayed bridge along the longitudinal direction can besummarized as follows

(1) In the construction stage at the cantilever end thecable force action point is the main recipient of axialforce e shear lag coefficient λN of the axial forceacting at this position reflects the actual force at thatpoint

(2) In the beam segments between the cable forces at thecable force action point the theoretical shear lagcoefficient λ determined by the ratio MN of thebending moment to axial force reflects the actualstress

(3) In the midspan beam section between the actionpoints of cable forces the shear lag coefficient ofbending moment λM is used to reflect the actualstress For the beam section between the action pointof cable force and the middle of the span the sectionshear lag coefficient is obtained by linearinterpolation

7 Conclusion

e structural control section stress at work in a typicalconstruction process was measured on a full cable-stayedbridge model in this study Sets of finite element and modeltest results were respectively analyzed and compared Ourconclusions can be summarized as follows

(1) In the construction stage the stress value of the cabletension area of the main beam upper edge markedlychanges after tensiling the cable force the shear lageffect is very prominent After a hanging basket loadis applied the main beams of certain sections presentalternating positive and negative shear lag

(2) e shear lag coefficient of the upper edge of themain beam in G1ndashG8 sections under WorkingConditions 1 and 4 was extracted from the spatialfinite element analysis for comparison to find thatthe experimental and theoretical results are in ac-cordance After tension of the cable force and theapplication of a hanging basket load positive andnegative shear lag effects alternate on the upper edgeof the box girder in certain areas As tension cableforce increases in each section the shear lag coef-ficient of the section upper edge decelerates

(3) In the construction stage at the cantilever end cableforce action point is mainly subjected to axial forcee shear lag coefficient λN of the axial force on thisposition reflects the actual force situation at thatpoint In the beam segments between the cableforces the shear lag coefficient λ determined by theratio of the bending moment to the axial force re-flects the actual stress at the cable force action pointIn the midspan beam section between the actionpoints of cable forces the shear lag coefficient λM ofthe bendingmoment reflects the actual stress For thebeam section between the cable force action pointand the middle of the span the section shear lagcoefficient can be obtained by linear interpolation

C2prime C3

4250kNmiddotm

17300kNmiddotm 19100kNmiddotm

4

C1prime C4

24400kNmiddotm

36800kNmiddotm

C4prime C3prime C2

10500kNmiddotm

18600kNmiddotm

C1

17700kNmiddotm

55000kNmiddotm

6820kNmiddotm

(a)

C1prime C4

13000kN7030kN

C4prime C1

5680kN

20300kN

3600kN

27000kN

C2prime C3

10900kN

C3prime C2

14900kN

4

(b)

Figure 15 Internal force of main girder on Pier 4 (Condition 1) (a) Moment diagram (b) Axial force diagram

Advances in Materials Science and Engineering 15

(4) e shear lag effect of the main beam is prominent inthe construction process e shear lag effect is mostobvious at the cable force operation point of thecantilever end As construction progresses and thecable force increases the shear lag effect of otherbeam sections decreases at each point except for thecantilever end

Data Availability

Part of the shear lag coefficient data used to support thefindings of this study are included within the article All ofthe shear lag coefficient data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors would like to thank the financial supports fromthe National Key RampD Program of China (2018YFC0809600and 2018YFC0809606) and Natural Science Foundation ofLiaoning Province (2019-MS-265)

References

[1] Z Li J G Nie and W Y Ji ldquoPositive and negative shear lagbehaviors of composite twin-girder decks with varying cross-sectionrdquo Science China Technological Sciences vol 60 no 1pp 116ndash132 2017

[2] S-J Zhou ldquoFinite beam element considering shear-lag effectin box girderrdquo Journal of Engineering Mechanics vol 136no 9 pp 1115ndash1122 2010

[3] Z Li and R K L Su ldquoAnalytical solutions for compositebeams with slip shear-lag and time-dependent effectsrdquo En-gineering Structures vol 152 pp 559ndash578 2017

[4] Q Z Luo Y M Wu Q S Li J Tang and G D Liu ldquoA finitesegment model for shear lag analysisrdquo Engineering Structuresvol 26 no 14 pp 2113ndash2124 2004

[5] P Z Lin F K Liu W Ji et al ldquoAnalysis on shear lag effect ofconcrete box beam by variational principlerdquo Journal of theChina Railway Society vol 35 no 2 pp 93ndash98 2013

[6] Y Y Zhang H Zhang W Li et al ldquoAnalysis on shear-lageffect of box girders based on different shear-lag generalizeddisplacementrdquo Journal of Railway Science and Engineeringvol 13 no 6 pp 1083ndash1090 2016

[7] S T Chang Q Zhang and S Zhang ldquoShear lag effect in singleplane cable-stayed bridgerdquo Advances in Structural Engi-neering vol 1 no 4 pp 301ndash306 1998

[8] G-F Wu and X Hong ldquoeoretical and experimental studyon shear lag effect of partially cable-stayed bridgerdquo Journal ofZhejiang University-Science A vol 6 no 8 pp 875ndash877 2005

[9] L Pengzhen Y Zijiang L Fengkui et al ldquoShear lag effect ofconcrete box girders considered varying flange-depthsrdquoJournal of Civil Architecturalamp Environmental Engineeringvol 35 no 1 pp 76ndash79 2013

[10] E Yamaguchi T Chaisomphob J Sa-Nguanmanasak andC Lertsima ldquoStress concentration and deflection of simplysupported box girder including shear lag effectrdquo StructuralEngineering and Mechanics vol 28 no 2 pp 207ndash220 2008

[11] X B Shen ldquoAnalysis of shear lag effect in the main girder ofwide low-pylon cable stayed bridgerdquo Construction Technologyvol 44 no 24 pp 76ndash79 2015

[12] L Xingmin ldquoAnalysis of influences of construction processeson shear lag effect of concrete box girdersrdquo Bridge Con-struction vol 43 no 1 pp 30ndash34 2013

[13] L Muyu W H Sun X D Sun et al ldquoStress analysis of widergirder of an extradosed cable-stayed bridge based on thelargest cantilever construction stagerdquo Journal of HuazhongUniversity of Science and Technology (Urban Science Edition)vol 27 no 2 pp 11ndash14 2010

[14] C Qianshu H Wenlong and Y Menggang ldquoAnalysis ofshear lag effect in construction stage of wide box girderextradosed cable-stayed bridge with large flangesrdquo Journal ofRailway Science and Engineering vol 15 no 12 pp 3158ndash3164 2018

[15] D R Li B M Wang and Y C Lin Structural Model Ex-periment Science Press Beijing China 1996

[16] J W Dali andW F Laili Experiment and Analysis of ResidualStress Ocean Press Beijing China 1987

[17] H Hossdort Structural Model Analysis China Architecture ampBuilding Press Beijing China 1986

[18] S K Mei Structural Experiments and Structural DesignChina Communications Press Beijing China 1980

[19] B Atmaca and S Ates ldquoConstruction stage analysis of three-dimensional cable-stayed bridgesrdquo Steel amp Composite Struc-tures vol 12 no 5 pp 413ndash426 2012

[20] S W Park M R Jung D J Min and M Y Kim ldquoCon-struction stage analysis of cable-stayed bridges using theunstrained element length methodrdquo Journal of e KoreanSociety of Civil Engineers vol 36 no 6 pp 991ndash998 2016

[21] M Gunaydin S Adanur A C Altunisik B Sevim andE Turker ldquoDetermination of structural behavior of Bosporussuspension bridge considering construction stages and dif-ferent soil conditionsrdquo Steel and Composite Structures vol 17no 4 pp 405ndash429 2014

[22] S Adanur M Gunaydin A C Altunisik and B SevimldquoConstruction stage analysis of humber suspension bridgerdquoApplied Mathematical Modelling vol 36 no 11 pp 5492ndash5505 2012

[23] J B Mander M J N Priestley and R Park ldquoeoreticalstress-strain model for confined concreterdquo Journal of Struc-tural Engineering vol 114 no 8 pp 1804ndash1826 1988

[24] Y A Khalid C L Chan B B Sahari and A M S HamoudaldquoBending behaviour of corrugated web beamsrdquo Journal ofMaterials Processing Technology vol 150 no 3 pp 242ndash2542004

[25] A Demir N Caglar H Ozturk and Y Sumer ldquoNonlinearfinite element study on the improvement of shear capacity inreinforced concrete T-Section beams by an alternative diag-onal shear reinforcementrdquo Engineering Structures vol 120pp 158ndash165 2016

16 Advances in Materials Science and Engineering

influences its deformation in order to accurately reflect thestress characteristics of the structural nonlinearity we im-posed load compensation for the dead weight of the cable viaadditional mass method

313 Main Tower Design e main tower of the cable-stayed bridge was welded with Q235B carbon structural steele ratio of elastic modulus to C50 concrete is 6 According tosimilarity theory the section area and moment of inertia ofthe main tower of the model can be calculated as follows

Am Ap middot16

1113874 11138751402

1113874 1113875 (4)

Im Ip

16

1113874 11138751404

1113874 1113875 (5)

e outer contour of the main tower section remainedbasically unchanged and the wall thickness was reduced by16 throughout this analysis to satisfy the similarity betweenthe moment of inertia and the area of the cross section emain tower was made in one piece and bolted into con-nection with the box girder Because the main tower of thebridge was a broken line the dead load of the main towerproduced not only the stress of the tower body but also thebending moment of the tower body which affect the entirestructural system of the cable-stayed bridge internal forcestate We carried out dead load compensation for the maintower to account for this

314 Design of Bridge Pier and Bearing emain piers andside pier were fabricated by steel weldinge box girder andPier 4 were connected by bolts with set bearing for Pier 3

80

100220 220

8125

1362513625

(a)

2225 2225605010500

Stay cablesC15prime~C1prime

Stay cablesC1~C15 Stay cables

C1prime~C15prime

Stay cablesC15~C1

Pier 4 Pier 5

1688

3543210123 211234

53752 times 630 = 12601575 2 times 740 = 1480 555740 2 times 630 = 1260 15752 times 740 = 1480555 537574050

(b)

Stay cablesC15prime~C1prime

Stay cablesC1~C15

Stay cablesC1prime~C15prime

Stay cablesC15~C1 16

8814

00

2225 2225605010500

3553554 5

(c)

Figure 2 General arrangement drawing of model (unit mm) (a) midspan section diagram (b) block diagram and (c) temporary supportand loading diagram

4 Advances in Materials Science and Engineering

Pier 5 and Pier 6 We set a load sensor in each bearing Atension-applying device was used to simulate the dead load

32 Model Fabrication and Installation We used a specialvibrating wire sensor to simulate the cable sensor which wascalibrated before installation to ensure long-term stabilityaccuracy and reliability In order to accurately simulate theconstruction process we installed the box girder in sub-section form

According to the real-world construction requirementsPier 5 was set as a temporary pier and a load sensor wasinstalled to measure the counter force of the temporarysupport under the action of the construction load emodel installation process is shown in Figure 3

33 Loading System and Dead Load Compensation

331 Dead Load Compensation of theMain Tower edeadload compensation of the main tower can be divided intotwo partsere is a concentrated load which was applied vialever system e upper segment load was applied to thecenter of gravity on the upper segment of the main towere lower segment load compensation accords with thesimilarity of the tower rootrsquos bending moment whichguaranteed that the tower section and tower angle sectionload effects were similare quality of the compensation forthe upper segment in this test was 83710 kg and that of thelower segment was 62000 kg

332 Load Compensation of Main Girder e load com-pensation of the main beam was applied to the diaphragm inconcentrated form e compensation quality of the inter-mediate span standard segment in this case was 26030 kgthe compensation quality of the intermediate span towerroot segment was 38530 kg the compensation quality of theside span standard segment was 27720 kg and the inter-mediate span tower root segment compensation quality was33580 kg e second-stage load compensation for eachlifting point included a side span of 6500 kg and a middlespan of 7630 kg

34 Test Load Working Conditions and Test Contents Weinstalled a stay cable and box girder in accordance with theactual construction process of the original bridge e staycable box girder and main tower were gradually com-pensated for dead loads and the stay cable was tensioned andadjusted as necessary e model test was carried out tosimulate the construction load and stress of the girdersection under the working conditions of a midspan closurein the construction stage

35 Test System and Measuring Point Arrangement Ourmodel testing system includes a support reaction forcetesting component dead load testing component stresstesting component displacement testing component andcable force testing component Each fulcrum was equippedwith a calibrated strain tester and a load cell to form a

reaction force test system e dead load test was performeddirectly on the calibrated scale e stress testing systemconsists of strain gauge and resistance strain gauge eelectrical test method was used to test the live load and deadload structural stress distributions e displacement testwas performed using a level gauge a dial gauge and atheodolite Our cable force test system consists of a cali-brated special frequency tester and a steel string load celle section and measuring point arrangement we used areshown in Figures 4 and 5

36 Construction Stage Working Conditions We exampledsix typical construction working conditions correspondingto real-world construction phases In Working Condition 1(double cantilever stage) we installed 1 (1prime) beam sectiontensioned C1ndashC4 (C1prime-C4prime) cables and applied a hangingbasket load In Working Condition 2 (maximum doublecantilever stage) we installed 2 (2prime) and 3 (3prime) beam sectionstensioned C5ndashC10 (C5prime-C10prime) cables and applied a hangingbasket load InWorking Condition 3 we closed the side spanof number 5 pier line and applied a hanging basket load tothe C11 cable In Working Condition 4 (maximum singlecantilever stage) we installed Section 4 tensioned C12ndashC14(C12prime-C14prime) cables and applied a hanging basket load at theC14 cable In Working Condition 5 we tensioned the C15(C15prime) cable and applied a closing weight to complete themidspan closure Working Condition 6 included cabletension adjustment and application of a second-stage deadload

4 Test Results versus Finite ElementAnalysis Results

41 Establishment of Finite Element Model We used MidasFEA finite element software to conduct our numerical an-alyses e construction scheme investigated in this studywas considered a sliding formwork with a combination ofthe stress state of the main girder and control the maintower e bridge was constructed via forward installationmethod [19ndash22] In the finite element analysis the cable wassimulated by the truss unit only the tower was simulated bythe beam element e main beam in this analysis is a solidelement and a six-node tetrahedron was used for divisione full-bridge truss unit is 120 the beam unit is 54 the solidunit is 29787 and there are 52631 nodes

After tensioning the C4 (C4prime) cable force the hangingbasket load was 5100 kN the boundary condition was a fixedbottom section of Pier 4 restraining the line displacementand angular displacement in the X Y and Z directions ebeam and Pier 5 were separated the beam bottom con-strained the linear displacement in X Y and Z directions inthis case as well Temporary support was simulated at1485m on the right side of the beam section of Pier 5which constrained the linear displacement in the Z direction

Next we tensioned C10 (C10prime) after the cable force ehanging basket load was 5100 kN and the boundary con-dition was the same as the first working condition e sidespan was closed the cable force of Pier 5 was adjusted and

Advances in Materials Science and Engineering 5

the hanging basket load was 5100 kN after the C11 (C11prime)cable tension was applied e boundary condition in thiscase involved removed the temporary support placing thebottom of Piers 4 and 5 under the same working con-dition e line displacement in the Z direction was re-strained at the end of the main beam

After the tension of the C14 (C14prime) was applied thehanging basket load was applied At 5100 kN the boundaryconditions were the same as the previous working conditionAfter applying the tension of C15 (C15prime) cable force the

closing weight of 1950 kN was applied e middle span wasclosed and the closing weight was unloaded e boundarycondition in this case was an unchanged constraint on thebottom of Pier 4 while the line displacement of the bottomof Pier 5 was constrained in the Z direction

e finite element model as established under thesespecific working conditions is shown in Figure 6 Due tolimited space only the results of Working Condition 1and Working Condition 4 are presented here forcomparison

(a) (b)

(c) (d)

Figure 3 Installing the model (a) installing four pairs of stay cables at Pier 4 (b) closure of side span of Pier 5 (c) second stage dead loadapplied by model and (d) model exerts live load

2225 2225605010500

Stay cablesC15prime~C1prime

Stay cablesC1~C15

Stay cablesC1prime~C15prime

Stay cablesC15~C1

Pier 4 Pier 5

1688

G4 G2 G1 G3 G9 G8G7 G5 G6

Figure 4 Diagram of testing section arrangement about girder (unit mm)

1 2

18

3

13

8

17

16

4 5

15 14

6 7 10

12

9 11

Figure 5 Arrangement of stiff girder stress measuring point

6 Advances in Materials Science and Engineering

We conducted forward installation calculations reflect-ing the construction process of the entire bridge e leastsquare method was applied to calculate of the cable force ofthe bridge during forward installation and the results werecompared with the reasonable cable force of the bridge untilall design requirements were satisfied At this point thecorresponding construction cable force could be consideredthe initial tension of the stay cable To accelerate conver-gence the initial tension of the cable was selected per thestatic equilibrium determined in the first forward installa-tion iteration Table 1 lists the initial tension of each staycable after the iterative calculation the number of stay cablesis shown in Figure 1

e stay cable at Pier 5 was adjusted once the side spanwas closed to bring the internal force at the root of the maintower closer to the deformation at the cantilever end of themain beam e C6 cable force was adjusted to 8000 kN C7cable force to 8000 kN C8 cable force to 8000 kN and C9cable force to 7500 kN at this time Before the constructionof bridge deck pavement to ensure uniform internal forceand deformation of the main girder across the whole bridgethe cable at Pier 4 was adjusted by bringing C1prime cable forceto 9000 kN C2prime cable force to 9000 kN C1 cable force to11000 kN C2 cable force to 11500 kN C3 cable force to9000 kN and C4 cable force to 9000 kN e cable force ofpart of the stay cable at Pier 5 was also adjusted by bringingC1 cable force to 10000 kN C2 cable force to 10000 kN C1primecable force to 10000 kN and C2prime cable force to 10500 kN

42 Main Beam Cross Section Shear Lag Coefficient underWorking Condition 1 e model test results and the shearlag coefficient of the upper edge of the main beam section ofWorking Condition 1 in the spatial finite element weresubjected to comparative analysis After tensioning the C4cable the full beam of the main beams G1 and G2 waspressed (tensile stress is positive compressive stress isnegative) the average measured stress of the upper edge ofthe main beams G1 and G2 was minus207MPa and minus077MPa atthis point After applying a hanging basket load on thecantilever end of the box girder the upper edge of the maingirders G1 and G2 showed a large amount of stress changese midabdominal segment bore compressive stress whilethe upper edge of section G1 of main beam bore tensilestress e upper edge of section G2 of main beam boretensile stress e average measured stress of the upper edgeof the main beams G1 and G2 was 024MPa and 049MPaAfter the tensioning force of the main girders G1 and G2 andapplication of the basket load the shear lag coefficient ateach point of the upper edge was as shown in Figure 7

After tensioning the C4 cable the main beams G5 andG6 were pressed in the full section and the average measuredstress of the upper edge of the main beams G5 and G6 wasminus189MPa and minus053MPa After applying the hangingbasket load on the cantilever end of the box girder the upperedge of the main beam G5 and G6 showed a large amount ofstress change e midabdominal segment bore compressivestress and the upper edge of section G5 and G6 of main beam

(a) (b)

(c) (d)

(e) (f )

Figure 6 Special finite element model for main construction operating conditions (a) Tension C4 (C4prime) applying a basket load (b) TensionC10 (C10prime) applying a basket load (c) Cable tension adjustment tensioning C11(C11prime) applying a hanging basket load (d) TensionC14(C14prime) applying a hanging basket load (e) Tension C15 (C15prime) applying counterweight midspan closure (f ) Grid division of mainbeam section

Advances in Materials Science and Engineering 7

bore tensile stress e average measured stress of the upperedge of the main beams G5 and G6 was 053MPa and055MPae shear lag coefficient at each point of the upperedge was as shown in Figure 8 after the tensioning force ofthe main girders G5 and G6 and the application of the basketload

As tension was applied the concentrated load was placedin the middle of the box girder to intuitively observe theshear lag effect in the middle of the box girder section ehanging basket load was applied in the form of a concen-trated load so that the upper edge of sections G1 G5 and G6of the main beam presented tensile stress though the middleweb section did not After the tensile force and the hangingbasket load were applied positive and negative shear lageffects appeared alternately on the upper edge of the mainbeam

43 Main Beam Cross Section Shear Lag Coefficient underWorking Condition 4 e model test results and the shearlag coefficient of the upper edge of the main beams G1ndashG8section under Working Condition 4 for the spatial finiteelement was also extracted for comparative analysis Asshown in Figures 9 and 10 the shear lag coefficient of theupper edge of the section G7 of themain beam changed from009ndash229 to minus293ndash210 after the tension of the cable and theapplication of the hanging basket load e shear lag

coefficient of the upper edge of the main beam G3 changedfrom 055ndash135 to 039ndash232 the shear lag coefficient of theupper edge of the main beam G4 changed from 063ndash159 to066ndash166 and the shear lag coefficient of the upper edge ofthe main beam G8 changed from 034ndash152 to 021ndash192 Inthis case after the tension was applied to the C14 cable andthe hanging basket load was applied the shear lag coefficientof the upper edge decelerated and the changes were slighte shear lag coefficient of the upper edge of the sections G2and G1 G5 and G6 of the main girder near the tower rootwas weak In each section the shear lag coefficient of theupper edge decelerated as tension cable force increased etest results are consistent with the finite element calculationresults

5 Longitudinal Shear Lag Effect in MainBeam of Single-Cable Plane P C Cable-Stayed Bridge

e stress on the main girder section of cable-stayed bridgeis highly complex e construction of the cable-stayedbridge is a dynamic process e stress in the main beamsection under different working conditions and differentbeam sections may widely differ e shear lag coefficientcalculation results along the longitudinal length of severalkey nodes on the upper edge of the cross section of the main

Table 1 Initial cable tension of each stay cable

Pier 4 Pier 5Cablenumber

Tension cable force(left) (kN)

Cablenumber

Tension cable force(right) (kN)

Cablenumber

Tension cable force(left) (kN)

Cablenumber

Tension cable force(right) (kN)

C1prime 7500 C1 9000 C1 9000 C1prime 4500C2prime 6500 C2 9500 C2 9500 C2prime 6500C3prime 5500 C3 8000 C3 8000 C3prime 5500C4prime 5500 C4 8000 C4 8000 C4prime 5000C5prime 5500 C5 8500 C5 8000 C5prime 6000C6prime 5500 C6 8000 C6 7400 C6prime 6500C7prime 5500 C7 8000 C7 7000 C7prime 5500C8prime 6000 C8 8000 C8 7000 C8prime 6000C9prime 6000 C9 7500 C9 7000 C9prime 6000C10prime 6500 C10 7500 C10 7500 C10prime 6500C11prime 6500 C11 7500 C11 7500 C11prime 6500C12prime 9500 C12 7500 C12 7500 C12prime 9000C13prime 9500 C13 7500 C13 7500 C13prime 9000C14prime 10000 C14 7500 C14 7500 C14prime 9500C15prime 9500 C15 6500 C15 6500 C15prime 9500

ndash100ndash050

000050100150

Shea

r lag

coeffi

cien

t

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(a)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

ndash200ndash100

000100200300

Shea

r lag

coeffi

cien

t

Node lateral coordinate (m)

(b)

Figure 7 Shear lag coefficient variations in G1 and G2 (Condition 1) (a) G1 section (b) G2 section

8 Advances in Materials Science and Engineering

beam can be used to understand the shear lag coefficientdistribution along the longitudinal length of the upper edgeof the main beam section during the construction stage Forconvenience the key points on the top edge of the crosssection of the main beam are marked here with capitalletters e distances from A-D points to the transversesymmetry lines are 0m minus2m minus11m and minus16m respec-tively e lateral position of each point is shown inFigure 11

Due to space limitations only the distribution of theshear lag coefficient along the longitudinal length of theupper edge of the main beam section under the followingtwo typical working conditions is described here WorkingCondition 1 is defined by tensiling the C4 (C4prime) cable forceWorking Condition 4 is defined by tensiling the C14 (C14prime)cable force e boundary conditions of each workingcondition were kept consistent

51Working Condition 1 Shear Lag Effect along LongitudinalBridge Under the action of self-weight and cable force the

shear lag coefficient of the flange of the main beam sectionalong the longitudinal bridge was observed as shown inFigure 12 In the longitudinal distribution of the shear lagcoefficient at the key point of the upper flange the midpointof the cross section of the main beam (Point A) and theboundary point between the upper edge and the middle web(Point B) change most sharply

At Pier 4 a sudden change in shear lag coefficient at theend of the cantilever occurred due to the tension of the C4(C4prime) cable A large stress concentration occurred at theseconcentrated load points which altered the upper flangestress distribution law Under the action of cable force theupper edge of the main beam section shows a more sig-nificant shear lag effect of the cable tension zone at locationscloser to the cable action point At Point A the shear lagcoefficient reaches 418 at Point B it is 307 e stressdistribution of the main beam appears to grow more uni-form as the distance between the main beam and the cableforce increases

Pier 4 C3ndashC1 (C3prime-C1prime) shows a similar distributionpattern in each beam section e shear lag coefficient of

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

ndash100ndash050

000050100150200

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

(a)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

ndash200ndash100

000100200300

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

(b)

Figure 8 Shear lag coefficient variations in G5 and G6 (Condition 1) (a) G5 section (b) G6 section

000

050

100

150

200

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16Node lateral coordinate (m)

Shea

r lag

coef

ficie

nt

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(a)

000050100150200250

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16Node lateral coordinate (m)

Shea

r lag

coef

ficie

ntTensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(b)

000050100150200250

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(c)

Shea

r lag

coef

ficie

nt

000050100150200250

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(d)

Figure 9 Contrast diagram of shear lag coefficient in G1-G4 (Condition 4) (a) G1 section (b) G2 section (c) G3 section and (d) G4 section

Advances in Materials Science and Engineering 9

Point A and Point B is prominent at the point of action of thecable forcee change range of shear lag coefficient of PointA at the point of cable force action is 222ndash271 and at Point Bis 122ndash212 e stress distribution of the beam sectionbetween the points of cable force is relatively uniform eA-point shear lag coefficient λ ranges from 115 to 116 theB-point shear lag coefficient λ from 112 to 140 and theC-point shear lag coefficient λ from 109 to 126

e stress distribution of the upper flange of the Pier 4tower root section changed due to the existence of the crossbeame change appears to be most dramatic at Point D atthe end of the upper extension arm of the main beam crosssection e shear lag coefficient λ at Point D near the towerranges from 114 to 154 marking a very significant variationcompared with other beam sections

e variation of the cantilever end the cable force pointand the C3ndashC1 (C3prime-C1prime) beam section of Pier 5 is similarto that of Pier 4 Under the action of the tension C4 (C4prime)Point A here shows a shear lag coefficient λ of 300 while thatat Point B reaches 250 In our analysis the stress distri-bution of the main beam grew more uniform as the distancebetween the main beam and the cable force increased

Pier 5 C3ndashC1(C3prime-C1prime) shows a similar distributionpattern in each beam section e shear force lag coeffi-cient of Point A and Point B is prominent at the point ofaction of the cable force e variation range of lag

coefficient λ is 157ndash244 and the shear lag coefficient λ ofPoint B is 138ndash184 at the point of action of cable forcee stress distribution of the beam section between thecable force points is relatively uniform e changes inshear lag coefficient λ of Point A range from 141 to 166that at Point B varies from 107 to 157 and that at Point Cranges from 075 to 108 Again the most significantchange was observed at Point D the shear lag coefficient λof Pier 5 at point D near the tower root varies from 120to 133

At the C1prime cable force on the right side of Pier 5 theshear lag coefficient of Points A and B is weaker than at otherpoints of cable force mainly because of the temporarysupport e shear lag effects on both sides were similar dueto the small reaction force of the temporary pier

Under the action of self-weight and cable force with thisworking condition the main characteristics of the shear lageffect of the upper edge of the main beam section of thesingle-cable PC cable-stayed bridge in the longitudinal di-rection can be summarized as follows

(1) In the construction stage under the action of cableforce and near the point of cable tension the upperedge of the girder section shows more significantshear lag effect in the cable force tension area egirder stress distribution grows more uniform as thedistance between the girder section and the cableforce action point increases

(2) In the C3ndashC1(C3prime-C1prime) beam section the upper edgeof the main beam section at the cable force actionpoint presents shear lag effects in the tension zonee stress distribution in the beam section betweenthe points of the cable force is relatively uniformwhich characterizes the shear lag distribution in theordinary curved box girder

000020040060080100120140160

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(a)

000

050

100

150

200

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(b)

ndash400ndash300ndash200ndash100

000100200300

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(c)

000

050

100

150

200

250

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(d)

Figure 10 Contrast diagram of shear lag coefficient in G1-G4 (Condition 4) (a) G5 section (b) G6 section (c) G7 section and (d) G8 section

B ACD

Figure 11 Main girder cross-section diagram at every point

10 Advances in Materials Science and Engineering

(3) e shear lag coefficient at the end of the upperextension arm of the cross section of the main beamat the root of the tower changes most dramaticallyamong the areas we observed We also observedconversion from a positive shear lag effect to anegative shear lag effect near the tower

(4) Under the influence of temporary support the shearforce lag coefficient of the action point of the C1primecable force on the right side of Pier 5 is weaker thanother cable forces e temporary pier reaction forceis small whichmakes the shear lag on both sideseeffects were still relatively similar in this case

52Working Condition 4 Shear Lag Effect along LongitudinalBridge is working condition represents the largestsingle cantilever stage e side spans at this point areclosed and the shear lag effects near their supports havemarkedly increased due to tensioning of C12primendashC14prime tailse shear lag effect of the inner beam section of the sidespan C11primendashC1prime is lower than under the previous workingcondition

As shown in Figure 13 at Pier 4 the cantilever end fallsunder the action of tensioning C14 cable force e shear lagcoefficient λ at Point A reaches 340 and at Point B is 240 Inthe beam section of Pier 4 C11ndashC1 (C11primendashC1prime) the shearlag coefficient of Points A and B is prominent at the point ofaction of the cable force Point A shows a shear lag coefficientλ of 151ndash281 at the cable force action point where Point Bshows a shear lag coefficient λ of 435ndash212 the stress dis-tribution of the beam section between the points of the cableforce is relatively uniform e shear lag coefficient λ ofPoint A varies from 105 to 144 and that of Point B from 119to 110 e shear lag coefficient λ of Point C ranges from065 to 095

At Pier 5 the end of the cantilever is under the action ofthe tensioning force of C14 e shear lag coefficient λ ofPoint A reaches 357 and that of Point B is 222 In the beamsection of C11ndashC1 (C11primendashC1prime) in Pier 5 the shear lagcoefficients of Points A and B at the point of action of thecable force are prominent Point A has a shear lag coefficientλ of 148ndash281 at the action point of the cable force wherePoint B has a shear lag coefficient λ of 133ndash212 e stressdistribution of beam section between the points of cable

4rsquo 3rsquo 2rsquo 1rsquo 4 1 2 3 4Longitudinal bridge position

Point APoint B

Point CPoint D

Pier 4

000050100150200250300350400450

Shea

r lag

coef

ficie

nt

(a)

Point APoint B

Point CPoint D

4 3 2 1 5 1rsquo 2rsquo 3rsquo 4rsquoLongitudinal bridge position

Pier 5

000

050

100

150

200

250

300

350

Shea

r lag

coef

ficie

nt

(b)

Figure 12 Distribution of shear lag coefficient along longitudinal direction (Condition 1) (a) Pier 4 (b) Pier 5

Advances in Materials Science and Engineering 11

Point APoint B

Point CPoint D

3 14rsquo13rsquo12rsquo 11rsquo 10rsquo 9rsquo 8rsquo 7rsquo 6rsquo 5rsquo 4rsquo 3rsquo 2rsquo 1rsquo 4Longitudinal bridge position

Pier 4

000

050

100

150

200

250

300

Shea

r lag

coef

ficie

nt

(a)

Point APoint B

Point CPoint D

4 1 2 3 4 5 6 7 8 9 10 11 12 13 14Longitudinal bridge position

000

050

100

150

200

250

300

350

400

Shea

r lag

coef

ficie

nt

(b)

Point APoint B

Point CPoint D

14 13 12 11 10 9 8 7 6 5 4 3 2 1 5Longitudinal bridge position

Pier 5

000

050

100

150

200

250

300

350

400

Shea

r lag

coef

ficie

nt

(c)

Figure 13 Continued

12 Advances in Materials Science and Engineering

force is relatively uniform e shear lag coefficient λ ofPoint A ranges from 120 to 155 that of Point B from 096 to135 and that of Point C from 076 to 101

Under this working condition the action of self-weightand cable force make the main characteristics of the shear lagat the upper edge of the main beam section of the single-cable PC cable-stayed bridge along the vertical direction verysimilar to those of underWorking Condition 1e shear lageffect at the main beam section is reduced as cable forceincreases

6 Distribution of Shear Lag Effect duringConstruction Stage

61 Shear Lag Coefficient of Main Beam of Single-Cable PlanePC Cable-Stayed Bridge e solution to the shear lag coef-ficient under the compressive bending load involves decom-posing the compressive bending load into a superposition ofthe bending moment action and the axial force action thencalculating the normal stress of themain beamunder the actionof the bending moment and the axial force alone [22] It isassumed that the axial force is borne by the whole section thatis the normal stress generated by the axial force is evenlydistributed along the cross section e results can then belinearly superposed to obtain the comprehensive shear lageffect coefficient Our previous analysis indicated that whenthere is cable tension of the main beam of the cable-stayedbridge in the single-cable PC there is a large stress concen-tration at the point of action of the concentrated loads estress distribution law of the roof plate changes accordingly

Assume that the axial force which is borne by the wholesection does not reflect the actual internal force of the maingirder of the single-cable PC cable-stayed bridge e dis-tribution of the longitudinal normal stress generated byinternal force is not uniform e size and distribution laware related to the internal force state namely the bendingmoment and axial force In this study the shear lag coef-ficient λΜ generated by bending moment and the shear lag

coefficient λΜ generated by axial force were respectivelycalculated to solve the shear lag coefficient λ aftersuperposition

e shear lag coefficient is calculated as follows

λ σσ

(6)

where σ is actual stress and σ( ) is nominal stresse shear lag coefficient is related to the wide span ratio

support conditions load form cross-sectional shape andsize material properties and location of the section [23ndash25]In the final analysis the shear lag is caused by unevendistribution of the longitudinal normal stress caused byinternal force its size and distribution are related to theinternal force state that is the bending moment is related tothe axial force

e actual stress σ and nominal stress σ( ) are thesuperposition of the bending moment and the normalstress generated by the axial force If the shear lag coeffi-cient λΜ produced by the bending moment is definedproperly the shear lag coefficient generated by the axialforce is λN and the shear lag coefficient after the super-position is λ us

λ σN + σM

σN + σM

λNσN + λMσM

σN + σM

(7)

where σ( )N and σ( )M are the nominal normal stressgenerated by bending moment and axial force respectivelyσ( )N and σ( )M are the actual normal stress generated bybending moment and axial force respectively e nu-merator and denominator at the right end of the aboveformula are divided by σ( )N to rewrite (7) as

λ λN + λM σMσN( 1113857

1 + σMσN( 1113857 (8)

ere are σM (MI)y and σN (NA) so(σMσN) (yAI) middot (MN) and order yAI C can beplugged into (8) to obtain

Point APoint B

Point CPoint D

000

050

100

150

200

250

300

350

5 1prime 2prime 3prime 4prime 5prime 6prime 7prime 8prime 9prime 10prime 11prime 12prime13prime14prime6

Shea

r lag

coef

ficie

nt

Longitudinal bridge position

(d)

Figure 13 Distribution of shear lag coefficient along longitudinal direction (Condition 4) (a) Left side of Pier 4 (b) Right side of Pier 4(c) Left side of Pier 5 (d) Right side of Pier 5

Advances in Materials Science and Engineering 13

λ λN + CλM(MN)

1 + C(MN) (9)

where y is the distance from the centroid of the section to theupper and lower edges A is the sectional area I is the sectionmoment of inertia and M and N represent the bendingmoment and axial force of the section

Equation (9) indicates that y A and I are the parametersinherent to the section under analysis λN and λΜ are theshear lag coefficients caused by the axial force and thebending moment respectively which are independent of themagnitude of the internal force Once λN and λΜ are de-termined the total shear lag coefficient of λ is only related tothe ratio of bending moment to axial force

Figure 14 shows the curve of the shear lag coefficientwith the bending moment axial force ratio when λΜ 200and λN 100 Taking the upper flange of the main beam ofthe Fumin cable-stayed bridge as an example C yAI 101

As shown in Figure 14 the total shear lag coefficient λgradually approaches the bending moment shear lag coef-ficient λΜ as the bending moment force ratio increases Asthe bending moment axial force ratio gradually decreases to0 the total shear lag coefficient λ is also gradually reduceduntil it is equal to the axial force shear lag coefficient λNWhen the bending moment force ratio is between 0 and 2the total shear lag coefficient λ changes substantially Alarger λΜ results in a greater change in the total shear lagcoefficient λ When the bending moment force ratio isgreater than 2 the total shear lag coefficient λ changesslowly at is to say in the internal force of this sectionwhether the bending moment or axial force accounts for alarger proportion affects how close the total shear lag co-efficient λ of the section is to the shear lag coefficient (λΜ orλN) caused by this item of internal force

For a general beam bridge the bending moment is themain internal force and so the section shear lag coefficient λis close to λΜ However for concrete cable-stayed bridgesthe main beam is in a compression state and is mainlysubjected to axial force its shear lag coefficient λ is closer toλN

62 eoretical versus Finite Element Main Beam Shear LagResults e shear lag variation law of the main beam of thesingle-cable PC cable-stayed bridge along the longitudinaldirection was discussed above as per our theoretical analysise shear lag coefficient and normal stress value of the mainbeam cross section are mainly concentrated in the range ofthe force point of the cable e effective width of the topplate of the upper flange in our analysis was set to 400m andthe full section of the bottom plate to σN 032N andσ( )N 009N (where N is the axial component of the cableforce) so that λN σNσ( )N 356 Shear lag coefficient λΜwas only considered a momentary action based on thedefinition of the effective width of the bridge e effectivewidth of the top plate in this case was 2400m so λΜ 135

We established the full-bridge plane bar modelaccording to finite element analysis e diagram of the

bending moment and axial force of the main beam underWorking Condition 1 suggests relatively small internalforces in Pier 4 and the main girder of Pier 5 e internalforce diagram of the main beam at Pier 4 was isolated forfurther analysis as shown in Figure 15

As shown in Figure 15 the bending moment force ratioMN is between λΜ and λN that is between 135 and 356e bending moment axial force ratio MN at the actionpoint of the cantilever end cable force C4 (C4prime) is 120 and118 According to formula (6) the shear lag coefficient λ ofthe upper edge of the main beam section is 235 and 236respectively

In the construction section the axial moment ratioMNof the main beam section of the C3primendashC1prime cable force actionpoint is 149 159 and 164 According to formula (6) theshear lag coefficients of the upper edge of the main beamsection of C3primendashC1prime cable force point are 223 220 and 218respectively e axial force ratio MN of the main beamsection of the C3ndashC1 cable force action point is 147 181and 203 Again according to formula (6) the shear lagcoefficients λ of the upper edge of the main beam section ofthe C3ndashC1 cable force action point are 224 213 and 207Our finite element results of the shear lag effect of the mainbeam along the longitudinal direction suggest that the shearlag coefficient at Point A of the action point of cantilever endcable force C4 (C4prime) is 418 and 361 the shear lag coefficientλ at Point B reaches 307 and 306

In the Pier 4 C3primendashC1prime beam section the shear lagcoefficients λ of Point A at the cable force action point are232 222 and 249 respectively those of Point B at the cableforce action point are 126 123 and 129 respectivelyWithin the C3ndashC1 beam section the shear lag coefficients λof Point A at the cable force action point are 257 271 and271 and those corresponding to Point B are 211 122 and212 respectively

In the beam section construction stage the analysisresults of the C3ndashC1 (C3primendashC1prime) cable force action pointtreatment theory agree with the finite element analysisresults In this position the theoretically derived shearforce lag coefficient λ can be used to reflect the actual forceat that point e finite element analysis results at the C4(C4prime) action point of the cantilever end cable force exceedthe theoretical analysis results e cantilever end cableforce action point is mainly subjected to axial force so itsshear lag coefficient λ should be closer to λN In this po-sition the shear lag factor λN can be used to reflect theactual force

005

115

225

0 1 2 3 4 5 6 7 8 9 10

Shea

r lag

coef

ficie

nt

MN

λM = 2 λN = 1

Figure 14 Relationship between shear lag coefficient and bendingmoment axial force ratio

14 Advances in Materials Science and Engineering

63 Variation Law of Shear Lag Effect of PC Cable-StayedBridge Main Beam along Longitudinal Direction e shearlag variations in the main beam of the single-cable PC cable-stayed bridge along the longitudinal direction can besummarized as follows

(1) In the construction stage at the cantilever end thecable force action point is the main recipient of axialforce e shear lag coefficient λN of the axial forceacting at this position reflects the actual force at thatpoint

(2) In the beam segments between the cable forces at thecable force action point the theoretical shear lagcoefficient λ determined by the ratio MN of thebending moment to axial force reflects the actualstress

(3) In the midspan beam section between the actionpoints of cable forces the shear lag coefficient ofbending moment λM is used to reflect the actualstress For the beam section between the action pointof cable force and the middle of the span the sectionshear lag coefficient is obtained by linearinterpolation

7 Conclusion

e structural control section stress at work in a typicalconstruction process was measured on a full cable-stayedbridge model in this study Sets of finite element and modeltest results were respectively analyzed and compared Ourconclusions can be summarized as follows

(1) In the construction stage the stress value of the cabletension area of the main beam upper edge markedlychanges after tensiling the cable force the shear lageffect is very prominent After a hanging basket loadis applied the main beams of certain sections presentalternating positive and negative shear lag

(2) e shear lag coefficient of the upper edge of themain beam in G1ndashG8 sections under WorkingConditions 1 and 4 was extracted from the spatialfinite element analysis for comparison to find thatthe experimental and theoretical results are in ac-cordance After tension of the cable force and theapplication of a hanging basket load positive andnegative shear lag effects alternate on the upper edgeof the box girder in certain areas As tension cableforce increases in each section the shear lag coef-ficient of the section upper edge decelerates

(3) In the construction stage at the cantilever end cableforce action point is mainly subjected to axial forcee shear lag coefficient λN of the axial force on thisposition reflects the actual force situation at thatpoint In the beam segments between the cableforces the shear lag coefficient λ determined by theratio of the bending moment to the axial force re-flects the actual stress at the cable force action pointIn the midspan beam section between the actionpoints of cable forces the shear lag coefficient λM ofthe bendingmoment reflects the actual stress For thebeam section between the cable force action pointand the middle of the span the section shear lagcoefficient can be obtained by linear interpolation

C2prime C3

4250kNmiddotm

17300kNmiddotm 19100kNmiddotm

4

C1prime C4

24400kNmiddotm

36800kNmiddotm

C4prime C3prime C2

10500kNmiddotm

18600kNmiddotm

C1

17700kNmiddotm

55000kNmiddotm

6820kNmiddotm

(a)

C1prime C4

13000kN7030kN

C4prime C1

5680kN

20300kN

3600kN

27000kN

C2prime C3

10900kN

C3prime C2

14900kN

4

(b)

Figure 15 Internal force of main girder on Pier 4 (Condition 1) (a) Moment diagram (b) Axial force diagram

Advances in Materials Science and Engineering 15

(4) e shear lag effect of the main beam is prominent inthe construction process e shear lag effect is mostobvious at the cable force operation point of thecantilever end As construction progresses and thecable force increases the shear lag effect of otherbeam sections decreases at each point except for thecantilever end

Data Availability

Part of the shear lag coefficient data used to support thefindings of this study are included within the article All ofthe shear lag coefficient data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors would like to thank the financial supports fromthe National Key RampD Program of China (2018YFC0809600and 2018YFC0809606) and Natural Science Foundation ofLiaoning Province (2019-MS-265)

References

[1] Z Li J G Nie and W Y Ji ldquoPositive and negative shear lagbehaviors of composite twin-girder decks with varying cross-sectionrdquo Science China Technological Sciences vol 60 no 1pp 116ndash132 2017

[2] S-J Zhou ldquoFinite beam element considering shear-lag effectin box girderrdquo Journal of Engineering Mechanics vol 136no 9 pp 1115ndash1122 2010

[3] Z Li and R K L Su ldquoAnalytical solutions for compositebeams with slip shear-lag and time-dependent effectsrdquo En-gineering Structures vol 152 pp 559ndash578 2017

[4] Q Z Luo Y M Wu Q S Li J Tang and G D Liu ldquoA finitesegment model for shear lag analysisrdquo Engineering Structuresvol 26 no 14 pp 2113ndash2124 2004

[5] P Z Lin F K Liu W Ji et al ldquoAnalysis on shear lag effect ofconcrete box beam by variational principlerdquo Journal of theChina Railway Society vol 35 no 2 pp 93ndash98 2013

[6] Y Y Zhang H Zhang W Li et al ldquoAnalysis on shear-lageffect of box girders based on different shear-lag generalizeddisplacementrdquo Journal of Railway Science and Engineeringvol 13 no 6 pp 1083ndash1090 2016

[7] S T Chang Q Zhang and S Zhang ldquoShear lag effect in singleplane cable-stayed bridgerdquo Advances in Structural Engi-neering vol 1 no 4 pp 301ndash306 1998

[8] G-F Wu and X Hong ldquoeoretical and experimental studyon shear lag effect of partially cable-stayed bridgerdquo Journal ofZhejiang University-Science A vol 6 no 8 pp 875ndash877 2005

[9] L Pengzhen Y Zijiang L Fengkui et al ldquoShear lag effect ofconcrete box girders considered varying flange-depthsrdquoJournal of Civil Architecturalamp Environmental Engineeringvol 35 no 1 pp 76ndash79 2013

[10] E Yamaguchi T Chaisomphob J Sa-Nguanmanasak andC Lertsima ldquoStress concentration and deflection of simplysupported box girder including shear lag effectrdquo StructuralEngineering and Mechanics vol 28 no 2 pp 207ndash220 2008

[11] X B Shen ldquoAnalysis of shear lag effect in the main girder ofwide low-pylon cable stayed bridgerdquo Construction Technologyvol 44 no 24 pp 76ndash79 2015

[12] L Xingmin ldquoAnalysis of influences of construction processeson shear lag effect of concrete box girdersrdquo Bridge Con-struction vol 43 no 1 pp 30ndash34 2013

[13] L Muyu W H Sun X D Sun et al ldquoStress analysis of widergirder of an extradosed cable-stayed bridge based on thelargest cantilever construction stagerdquo Journal of HuazhongUniversity of Science and Technology (Urban Science Edition)vol 27 no 2 pp 11ndash14 2010

[14] C Qianshu H Wenlong and Y Menggang ldquoAnalysis ofshear lag effect in construction stage of wide box girderextradosed cable-stayed bridge with large flangesrdquo Journal ofRailway Science and Engineering vol 15 no 12 pp 3158ndash3164 2018

[15] D R Li B M Wang and Y C Lin Structural Model Ex-periment Science Press Beijing China 1996

[16] J W Dali andW F Laili Experiment and Analysis of ResidualStress Ocean Press Beijing China 1987

[17] H Hossdort Structural Model Analysis China Architecture ampBuilding Press Beijing China 1986

[18] S K Mei Structural Experiments and Structural DesignChina Communications Press Beijing China 1980

[19] B Atmaca and S Ates ldquoConstruction stage analysis of three-dimensional cable-stayed bridgesrdquo Steel amp Composite Struc-tures vol 12 no 5 pp 413ndash426 2012

[20] S W Park M R Jung D J Min and M Y Kim ldquoCon-struction stage analysis of cable-stayed bridges using theunstrained element length methodrdquo Journal of e KoreanSociety of Civil Engineers vol 36 no 6 pp 991ndash998 2016

[21] M Gunaydin S Adanur A C Altunisik B Sevim andE Turker ldquoDetermination of structural behavior of Bosporussuspension bridge considering construction stages and dif-ferent soil conditionsrdquo Steel and Composite Structures vol 17no 4 pp 405ndash429 2014

[22] S Adanur M Gunaydin A C Altunisik and B SevimldquoConstruction stage analysis of humber suspension bridgerdquoApplied Mathematical Modelling vol 36 no 11 pp 5492ndash5505 2012

[23] J B Mander M J N Priestley and R Park ldquoeoreticalstress-strain model for confined concreterdquo Journal of Struc-tural Engineering vol 114 no 8 pp 1804ndash1826 1988

[24] Y A Khalid C L Chan B B Sahari and A M S HamoudaldquoBending behaviour of corrugated web beamsrdquo Journal ofMaterials Processing Technology vol 150 no 3 pp 242ndash2542004

[25] A Demir N Caglar H Ozturk and Y Sumer ldquoNonlinearfinite element study on the improvement of shear capacity inreinforced concrete T-Section beams by an alternative diag-onal shear reinforcementrdquo Engineering Structures vol 120pp 158ndash165 2016

16 Advances in Materials Science and Engineering

Pier 5 and Pier 6 We set a load sensor in each bearing Atension-applying device was used to simulate the dead load

32 Model Fabrication and Installation We used a specialvibrating wire sensor to simulate the cable sensor which wascalibrated before installation to ensure long-term stabilityaccuracy and reliability In order to accurately simulate theconstruction process we installed the box girder in sub-section form

According to the real-world construction requirementsPier 5 was set as a temporary pier and a load sensor wasinstalled to measure the counter force of the temporarysupport under the action of the construction load emodel installation process is shown in Figure 3

33 Loading System and Dead Load Compensation

331 Dead Load Compensation of theMain Tower edeadload compensation of the main tower can be divided intotwo partsere is a concentrated load which was applied vialever system e upper segment load was applied to thecenter of gravity on the upper segment of the main towere lower segment load compensation accords with thesimilarity of the tower rootrsquos bending moment whichguaranteed that the tower section and tower angle sectionload effects were similare quality of the compensation forthe upper segment in this test was 83710 kg and that of thelower segment was 62000 kg

332 Load Compensation of Main Girder e load com-pensation of the main beam was applied to the diaphragm inconcentrated form e compensation quality of the inter-mediate span standard segment in this case was 26030 kgthe compensation quality of the intermediate span towerroot segment was 38530 kg the compensation quality of theside span standard segment was 27720 kg and the inter-mediate span tower root segment compensation quality was33580 kg e second-stage load compensation for eachlifting point included a side span of 6500 kg and a middlespan of 7630 kg

34 Test Load Working Conditions and Test Contents Weinstalled a stay cable and box girder in accordance with theactual construction process of the original bridge e staycable box girder and main tower were gradually com-pensated for dead loads and the stay cable was tensioned andadjusted as necessary e model test was carried out tosimulate the construction load and stress of the girdersection under the working conditions of a midspan closurein the construction stage

35 Test System and Measuring Point Arrangement Ourmodel testing system includes a support reaction forcetesting component dead load testing component stresstesting component displacement testing component andcable force testing component Each fulcrum was equippedwith a calibrated strain tester and a load cell to form a

reaction force test system e dead load test was performeddirectly on the calibrated scale e stress testing systemconsists of strain gauge and resistance strain gauge eelectrical test method was used to test the live load and deadload structural stress distributions e displacement testwas performed using a level gauge a dial gauge and atheodolite Our cable force test system consists of a cali-brated special frequency tester and a steel string load celle section and measuring point arrangement we used areshown in Figures 4 and 5

36 Construction Stage Working Conditions We exampledsix typical construction working conditions correspondingto real-world construction phases In Working Condition 1(double cantilever stage) we installed 1 (1prime) beam sectiontensioned C1ndashC4 (C1prime-C4prime) cables and applied a hangingbasket load In Working Condition 2 (maximum doublecantilever stage) we installed 2 (2prime) and 3 (3prime) beam sectionstensioned C5ndashC10 (C5prime-C10prime) cables and applied a hangingbasket load InWorking Condition 3 we closed the side spanof number 5 pier line and applied a hanging basket load tothe C11 cable In Working Condition 4 (maximum singlecantilever stage) we installed Section 4 tensioned C12ndashC14(C12prime-C14prime) cables and applied a hanging basket load at theC14 cable In Working Condition 5 we tensioned the C15(C15prime) cable and applied a closing weight to complete themidspan closure Working Condition 6 included cabletension adjustment and application of a second-stage deadload

4 Test Results versus Finite ElementAnalysis Results

41 Establishment of Finite Element Model We used MidasFEA finite element software to conduct our numerical an-alyses e construction scheme investigated in this studywas considered a sliding formwork with a combination ofthe stress state of the main girder and control the maintower e bridge was constructed via forward installationmethod [19ndash22] In the finite element analysis the cable wassimulated by the truss unit only the tower was simulated bythe beam element e main beam in this analysis is a solidelement and a six-node tetrahedron was used for divisione full-bridge truss unit is 120 the beam unit is 54 the solidunit is 29787 and there are 52631 nodes

After tensioning the C4 (C4prime) cable force the hangingbasket load was 5100 kN the boundary condition was a fixedbottom section of Pier 4 restraining the line displacementand angular displacement in the X Y and Z directions ebeam and Pier 5 were separated the beam bottom con-strained the linear displacement in X Y and Z directions inthis case as well Temporary support was simulated at1485m on the right side of the beam section of Pier 5which constrained the linear displacement in the Z direction

Next we tensioned C10 (C10prime) after the cable force ehanging basket load was 5100 kN and the boundary con-dition was the same as the first working condition e sidespan was closed the cable force of Pier 5 was adjusted and

Advances in Materials Science and Engineering 5

the hanging basket load was 5100 kN after the C11 (C11prime)cable tension was applied e boundary condition in thiscase involved removed the temporary support placing thebottom of Piers 4 and 5 under the same working con-dition e line displacement in the Z direction was re-strained at the end of the main beam

After the tension of the C14 (C14prime) was applied thehanging basket load was applied At 5100 kN the boundaryconditions were the same as the previous working conditionAfter applying the tension of C15 (C15prime) cable force the

closing weight of 1950 kN was applied e middle span wasclosed and the closing weight was unloaded e boundarycondition in this case was an unchanged constraint on thebottom of Pier 4 while the line displacement of the bottomof Pier 5 was constrained in the Z direction

e finite element model as established under thesespecific working conditions is shown in Figure 6 Due tolimited space only the results of Working Condition 1and Working Condition 4 are presented here forcomparison

(a) (b)

(c) (d)

Figure 3 Installing the model (a) installing four pairs of stay cables at Pier 4 (b) closure of side span of Pier 5 (c) second stage dead loadapplied by model and (d) model exerts live load

2225 2225605010500

Stay cablesC15prime~C1prime

Stay cablesC1~C15

Stay cablesC1prime~C15prime

Stay cablesC15~C1

Pier 4 Pier 5

1688

G4 G2 G1 G3 G9 G8G7 G5 G6

Figure 4 Diagram of testing section arrangement about girder (unit mm)

1 2

18

3

13

8

17

16

4 5

15 14

6 7 10

12

9 11

Figure 5 Arrangement of stiff girder stress measuring point

6 Advances in Materials Science and Engineering

We conducted forward installation calculations reflect-ing the construction process of the entire bridge e leastsquare method was applied to calculate of the cable force ofthe bridge during forward installation and the results werecompared with the reasonable cable force of the bridge untilall design requirements were satisfied At this point thecorresponding construction cable force could be consideredthe initial tension of the stay cable To accelerate conver-gence the initial tension of the cable was selected per thestatic equilibrium determined in the first forward installa-tion iteration Table 1 lists the initial tension of each staycable after the iterative calculation the number of stay cablesis shown in Figure 1

e stay cable at Pier 5 was adjusted once the side spanwas closed to bring the internal force at the root of the maintower closer to the deformation at the cantilever end of themain beam e C6 cable force was adjusted to 8000 kN C7cable force to 8000 kN C8 cable force to 8000 kN and C9cable force to 7500 kN at this time Before the constructionof bridge deck pavement to ensure uniform internal forceand deformation of the main girder across the whole bridgethe cable at Pier 4 was adjusted by bringing C1prime cable forceto 9000 kN C2prime cable force to 9000 kN C1 cable force to11000 kN C2 cable force to 11500 kN C3 cable force to9000 kN and C4 cable force to 9000 kN e cable force ofpart of the stay cable at Pier 5 was also adjusted by bringingC1 cable force to 10000 kN C2 cable force to 10000 kN C1primecable force to 10000 kN and C2prime cable force to 10500 kN

42 Main Beam Cross Section Shear Lag Coefficient underWorking Condition 1 e model test results and the shearlag coefficient of the upper edge of the main beam section ofWorking Condition 1 in the spatial finite element weresubjected to comparative analysis After tensioning the C4cable the full beam of the main beams G1 and G2 waspressed (tensile stress is positive compressive stress isnegative) the average measured stress of the upper edge ofthe main beams G1 and G2 was minus207MPa and minus077MPa atthis point After applying a hanging basket load on thecantilever end of the box girder the upper edge of the maingirders G1 and G2 showed a large amount of stress changese midabdominal segment bore compressive stress whilethe upper edge of section G1 of main beam bore tensilestress e upper edge of section G2 of main beam boretensile stress e average measured stress of the upper edgeof the main beams G1 and G2 was 024MPa and 049MPaAfter the tensioning force of the main girders G1 and G2 andapplication of the basket load the shear lag coefficient ateach point of the upper edge was as shown in Figure 7

After tensioning the C4 cable the main beams G5 andG6 were pressed in the full section and the average measuredstress of the upper edge of the main beams G5 and G6 wasminus189MPa and minus053MPa After applying the hangingbasket load on the cantilever end of the box girder the upperedge of the main beam G5 and G6 showed a large amount ofstress change e midabdominal segment bore compressivestress and the upper edge of section G5 and G6 of main beam

(a) (b)

(c) (d)

(e) (f )

Figure 6 Special finite element model for main construction operating conditions (a) Tension C4 (C4prime) applying a basket load (b) TensionC10 (C10prime) applying a basket load (c) Cable tension adjustment tensioning C11(C11prime) applying a hanging basket load (d) TensionC14(C14prime) applying a hanging basket load (e) Tension C15 (C15prime) applying counterweight midspan closure (f ) Grid division of mainbeam section

Advances in Materials Science and Engineering 7

bore tensile stress e average measured stress of the upperedge of the main beams G5 and G6 was 053MPa and055MPae shear lag coefficient at each point of the upperedge was as shown in Figure 8 after the tensioning force ofthe main girders G5 and G6 and the application of the basketload

As tension was applied the concentrated load was placedin the middle of the box girder to intuitively observe theshear lag effect in the middle of the box girder section ehanging basket load was applied in the form of a concen-trated load so that the upper edge of sections G1 G5 and G6of the main beam presented tensile stress though the middleweb section did not After the tensile force and the hangingbasket load were applied positive and negative shear lageffects appeared alternately on the upper edge of the mainbeam

43 Main Beam Cross Section Shear Lag Coefficient underWorking Condition 4 e model test results and the shearlag coefficient of the upper edge of the main beams G1ndashG8section under Working Condition 4 for the spatial finiteelement was also extracted for comparative analysis Asshown in Figures 9 and 10 the shear lag coefficient of theupper edge of the section G7 of themain beam changed from009ndash229 to minus293ndash210 after the tension of the cable and theapplication of the hanging basket load e shear lag

coefficient of the upper edge of the main beam G3 changedfrom 055ndash135 to 039ndash232 the shear lag coefficient of theupper edge of the main beam G4 changed from 063ndash159 to066ndash166 and the shear lag coefficient of the upper edge ofthe main beam G8 changed from 034ndash152 to 021ndash192 Inthis case after the tension was applied to the C14 cable andthe hanging basket load was applied the shear lag coefficientof the upper edge decelerated and the changes were slighte shear lag coefficient of the upper edge of the sections G2and G1 G5 and G6 of the main girder near the tower rootwas weak In each section the shear lag coefficient of theupper edge decelerated as tension cable force increased etest results are consistent with the finite element calculationresults

5 Longitudinal Shear Lag Effect in MainBeam of Single-Cable Plane P C Cable-Stayed Bridge

e stress on the main girder section of cable-stayed bridgeis highly complex e construction of the cable-stayedbridge is a dynamic process e stress in the main beamsection under different working conditions and differentbeam sections may widely differ e shear lag coefficientcalculation results along the longitudinal length of severalkey nodes on the upper edge of the cross section of the main

Table 1 Initial cable tension of each stay cable

Pier 4 Pier 5Cablenumber

Tension cable force(left) (kN)

Cablenumber

Tension cable force(right) (kN)

Cablenumber

Tension cable force(left) (kN)

Cablenumber

Tension cable force(right) (kN)

C1prime 7500 C1 9000 C1 9000 C1prime 4500C2prime 6500 C2 9500 C2 9500 C2prime 6500C3prime 5500 C3 8000 C3 8000 C3prime 5500C4prime 5500 C4 8000 C4 8000 C4prime 5000C5prime 5500 C5 8500 C5 8000 C5prime 6000C6prime 5500 C6 8000 C6 7400 C6prime 6500C7prime 5500 C7 8000 C7 7000 C7prime 5500C8prime 6000 C8 8000 C8 7000 C8prime 6000C9prime 6000 C9 7500 C9 7000 C9prime 6000C10prime 6500 C10 7500 C10 7500 C10prime 6500C11prime 6500 C11 7500 C11 7500 C11prime 6500C12prime 9500 C12 7500 C12 7500 C12prime 9000C13prime 9500 C13 7500 C13 7500 C13prime 9000C14prime 10000 C14 7500 C14 7500 C14prime 9500C15prime 9500 C15 6500 C15 6500 C15prime 9500

ndash100ndash050

000050100150

Shea

r lag

coeffi

cien

t

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(a)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

ndash200ndash100

000100200300

Shea

r lag

coeffi

cien

t

Node lateral coordinate (m)

(b)

Figure 7 Shear lag coefficient variations in G1 and G2 (Condition 1) (a) G1 section (b) G2 section

8 Advances in Materials Science and Engineering

beam can be used to understand the shear lag coefficientdistribution along the longitudinal length of the upper edgeof the main beam section during the construction stage Forconvenience the key points on the top edge of the crosssection of the main beam are marked here with capitalletters e distances from A-D points to the transversesymmetry lines are 0m minus2m minus11m and minus16m respec-tively e lateral position of each point is shown inFigure 11

Due to space limitations only the distribution of theshear lag coefficient along the longitudinal length of theupper edge of the main beam section under the followingtwo typical working conditions is described here WorkingCondition 1 is defined by tensiling the C4 (C4prime) cable forceWorking Condition 4 is defined by tensiling the C14 (C14prime)cable force e boundary conditions of each workingcondition were kept consistent

51Working Condition 1 Shear Lag Effect along LongitudinalBridge Under the action of self-weight and cable force the

shear lag coefficient of the flange of the main beam sectionalong the longitudinal bridge was observed as shown inFigure 12 In the longitudinal distribution of the shear lagcoefficient at the key point of the upper flange the midpointof the cross section of the main beam (Point A) and theboundary point between the upper edge and the middle web(Point B) change most sharply

At Pier 4 a sudden change in shear lag coefficient at theend of the cantilever occurred due to the tension of the C4(C4prime) cable A large stress concentration occurred at theseconcentrated load points which altered the upper flangestress distribution law Under the action of cable force theupper edge of the main beam section shows a more sig-nificant shear lag effect of the cable tension zone at locationscloser to the cable action point At Point A the shear lagcoefficient reaches 418 at Point B it is 307 e stressdistribution of the main beam appears to grow more uni-form as the distance between the main beam and the cableforce increases

Pier 4 C3ndashC1 (C3prime-C1prime) shows a similar distributionpattern in each beam section e shear lag coefficient of

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

ndash100ndash050

000050100150200

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

(a)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

ndash200ndash100

000100200300

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

(b)

Figure 8 Shear lag coefficient variations in G5 and G6 (Condition 1) (a) G5 section (b) G6 section

000

050

100

150

200

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16Node lateral coordinate (m)

Shea

r lag

coef

ficie

nt

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(a)

000050100150200250

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16Node lateral coordinate (m)

Shea

r lag

coef

ficie

ntTensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(b)

000050100150200250

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(c)

Shea

r lag

coef

ficie

nt

000050100150200250

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(d)

Figure 9 Contrast diagram of shear lag coefficient in G1-G4 (Condition 4) (a) G1 section (b) G2 section (c) G3 section and (d) G4 section

Advances in Materials Science and Engineering 9

Point A and Point B is prominent at the point of action of thecable forcee change range of shear lag coefficient of PointA at the point of cable force action is 222ndash271 and at Point Bis 122ndash212 e stress distribution of the beam sectionbetween the points of cable force is relatively uniform eA-point shear lag coefficient λ ranges from 115 to 116 theB-point shear lag coefficient λ from 112 to 140 and theC-point shear lag coefficient λ from 109 to 126

e stress distribution of the upper flange of the Pier 4tower root section changed due to the existence of the crossbeame change appears to be most dramatic at Point D atthe end of the upper extension arm of the main beam crosssection e shear lag coefficient λ at Point D near the towerranges from 114 to 154 marking a very significant variationcompared with other beam sections

e variation of the cantilever end the cable force pointand the C3ndashC1 (C3prime-C1prime) beam section of Pier 5 is similarto that of Pier 4 Under the action of the tension C4 (C4prime)Point A here shows a shear lag coefficient λ of 300 while thatat Point B reaches 250 In our analysis the stress distri-bution of the main beam grew more uniform as the distancebetween the main beam and the cable force increased

Pier 5 C3ndashC1(C3prime-C1prime) shows a similar distributionpattern in each beam section e shear force lag coeffi-cient of Point A and Point B is prominent at the point ofaction of the cable force e variation range of lag

coefficient λ is 157ndash244 and the shear lag coefficient λ ofPoint B is 138ndash184 at the point of action of cable forcee stress distribution of the beam section between thecable force points is relatively uniform e changes inshear lag coefficient λ of Point A range from 141 to 166that at Point B varies from 107 to 157 and that at Point Cranges from 075 to 108 Again the most significantchange was observed at Point D the shear lag coefficient λof Pier 5 at point D near the tower root varies from 120to 133

At the C1prime cable force on the right side of Pier 5 theshear lag coefficient of Points A and B is weaker than at otherpoints of cable force mainly because of the temporarysupport e shear lag effects on both sides were similar dueto the small reaction force of the temporary pier

Under the action of self-weight and cable force with thisworking condition the main characteristics of the shear lageffect of the upper edge of the main beam section of thesingle-cable PC cable-stayed bridge in the longitudinal di-rection can be summarized as follows

(1) In the construction stage under the action of cableforce and near the point of cable tension the upperedge of the girder section shows more significantshear lag effect in the cable force tension area egirder stress distribution grows more uniform as thedistance between the girder section and the cableforce action point increases

(2) In the C3ndashC1(C3prime-C1prime) beam section the upper edgeof the main beam section at the cable force actionpoint presents shear lag effects in the tension zonee stress distribution in the beam section betweenthe points of the cable force is relatively uniformwhich characterizes the shear lag distribution in theordinary curved box girder

000020040060080100120140160

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(a)

000

050

100

150

200

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(b)

ndash400ndash300ndash200ndash100

000100200300

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(c)

000

050

100

150

200

250

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(d)

Figure 10 Contrast diagram of shear lag coefficient in G1-G4 (Condition 4) (a) G5 section (b) G6 section (c) G7 section and (d) G8 section

B ACD

Figure 11 Main girder cross-section diagram at every point

10 Advances in Materials Science and Engineering

(3) e shear lag coefficient at the end of the upperextension arm of the cross section of the main beamat the root of the tower changes most dramaticallyamong the areas we observed We also observedconversion from a positive shear lag effect to anegative shear lag effect near the tower

(4) Under the influence of temporary support the shearforce lag coefficient of the action point of the C1primecable force on the right side of Pier 5 is weaker thanother cable forces e temporary pier reaction forceis small whichmakes the shear lag on both sideseeffects were still relatively similar in this case

52Working Condition 4 Shear Lag Effect along LongitudinalBridge is working condition represents the largestsingle cantilever stage e side spans at this point areclosed and the shear lag effects near their supports havemarkedly increased due to tensioning of C12primendashC14prime tailse shear lag effect of the inner beam section of the sidespan C11primendashC1prime is lower than under the previous workingcondition

As shown in Figure 13 at Pier 4 the cantilever end fallsunder the action of tensioning C14 cable force e shear lagcoefficient λ at Point A reaches 340 and at Point B is 240 Inthe beam section of Pier 4 C11ndashC1 (C11primendashC1prime) the shearlag coefficient of Points A and B is prominent at the point ofaction of the cable force Point A shows a shear lag coefficientλ of 151ndash281 at the cable force action point where Point Bshows a shear lag coefficient λ of 435ndash212 the stress dis-tribution of the beam section between the points of the cableforce is relatively uniform e shear lag coefficient λ ofPoint A varies from 105 to 144 and that of Point B from 119to 110 e shear lag coefficient λ of Point C ranges from065 to 095

At Pier 5 the end of the cantilever is under the action ofthe tensioning force of C14 e shear lag coefficient λ ofPoint A reaches 357 and that of Point B is 222 In the beamsection of C11ndashC1 (C11primendashC1prime) in Pier 5 the shear lagcoefficients of Points A and B at the point of action of thecable force are prominent Point A has a shear lag coefficientλ of 148ndash281 at the action point of the cable force wherePoint B has a shear lag coefficient λ of 133ndash212 e stressdistribution of beam section between the points of cable

4rsquo 3rsquo 2rsquo 1rsquo 4 1 2 3 4Longitudinal bridge position

Point APoint B

Point CPoint D

Pier 4

000050100150200250300350400450

Shea

r lag

coef

ficie

nt

(a)

Point APoint B

Point CPoint D

4 3 2 1 5 1rsquo 2rsquo 3rsquo 4rsquoLongitudinal bridge position

Pier 5

000

050

100

150

200

250

300

350

Shea

r lag

coef

ficie

nt

(b)

Figure 12 Distribution of shear lag coefficient along longitudinal direction (Condition 1) (a) Pier 4 (b) Pier 5

Advances in Materials Science and Engineering 11

Point APoint B

Point CPoint D

3 14rsquo13rsquo12rsquo 11rsquo 10rsquo 9rsquo 8rsquo 7rsquo 6rsquo 5rsquo 4rsquo 3rsquo 2rsquo 1rsquo 4Longitudinal bridge position

Pier 4

000

050

100

150

200

250

300

Shea

r lag

coef

ficie

nt

(a)

Point APoint B

Point CPoint D

4 1 2 3 4 5 6 7 8 9 10 11 12 13 14Longitudinal bridge position

000

050

100

150

200

250

300

350

400

Shea

r lag

coef

ficie

nt

(b)

Point APoint B

Point CPoint D

14 13 12 11 10 9 8 7 6 5 4 3 2 1 5Longitudinal bridge position

Pier 5

000

050

100

150

200

250

300

350

400

Shea

r lag

coef

ficie

nt

(c)

Figure 13 Continued

12 Advances in Materials Science and Engineering

force is relatively uniform e shear lag coefficient λ ofPoint A ranges from 120 to 155 that of Point B from 096 to135 and that of Point C from 076 to 101

Under this working condition the action of self-weightand cable force make the main characteristics of the shear lagat the upper edge of the main beam section of the single-cable PC cable-stayed bridge along the vertical direction verysimilar to those of underWorking Condition 1e shear lageffect at the main beam section is reduced as cable forceincreases

6 Distribution of Shear Lag Effect duringConstruction Stage

61 Shear Lag Coefficient of Main Beam of Single-Cable PlanePC Cable-Stayed Bridge e solution to the shear lag coef-ficient under the compressive bending load involves decom-posing the compressive bending load into a superposition ofthe bending moment action and the axial force action thencalculating the normal stress of themain beamunder the actionof the bending moment and the axial force alone [22] It isassumed that the axial force is borne by the whole section thatis the normal stress generated by the axial force is evenlydistributed along the cross section e results can then belinearly superposed to obtain the comprehensive shear lageffect coefficient Our previous analysis indicated that whenthere is cable tension of the main beam of the cable-stayedbridge in the single-cable PC there is a large stress concen-tration at the point of action of the concentrated loads estress distribution law of the roof plate changes accordingly

Assume that the axial force which is borne by the wholesection does not reflect the actual internal force of the maingirder of the single-cable PC cable-stayed bridge e dis-tribution of the longitudinal normal stress generated byinternal force is not uniform e size and distribution laware related to the internal force state namely the bendingmoment and axial force In this study the shear lag coef-ficient λΜ generated by bending moment and the shear lag

coefficient λΜ generated by axial force were respectivelycalculated to solve the shear lag coefficient λ aftersuperposition

e shear lag coefficient is calculated as follows

λ σσ

(6)

where σ is actual stress and σ( ) is nominal stresse shear lag coefficient is related to the wide span ratio

support conditions load form cross-sectional shape andsize material properties and location of the section [23ndash25]In the final analysis the shear lag is caused by unevendistribution of the longitudinal normal stress caused byinternal force its size and distribution are related to theinternal force state that is the bending moment is related tothe axial force

e actual stress σ and nominal stress σ( ) are thesuperposition of the bending moment and the normalstress generated by the axial force If the shear lag coeffi-cient λΜ produced by the bending moment is definedproperly the shear lag coefficient generated by the axialforce is λN and the shear lag coefficient after the super-position is λ us

λ σN + σM

σN + σM

λNσN + λMσM

σN + σM

(7)

where σ( )N and σ( )M are the nominal normal stressgenerated by bending moment and axial force respectivelyσ( )N and σ( )M are the actual normal stress generated bybending moment and axial force respectively e nu-merator and denominator at the right end of the aboveformula are divided by σ( )N to rewrite (7) as

λ λN + λM σMσN( 1113857

1 + σMσN( 1113857 (8)

ere are σM (MI)y and σN (NA) so(σMσN) (yAI) middot (MN) and order yAI C can beplugged into (8) to obtain

Point APoint B

Point CPoint D

000

050

100

150

200

250

300

350

5 1prime 2prime 3prime 4prime 5prime 6prime 7prime 8prime 9prime 10prime 11prime 12prime13prime14prime6

Shea

r lag

coef

ficie

nt

Longitudinal bridge position

(d)

Figure 13 Distribution of shear lag coefficient along longitudinal direction (Condition 4) (a) Left side of Pier 4 (b) Right side of Pier 4(c) Left side of Pier 5 (d) Right side of Pier 5

Advances in Materials Science and Engineering 13

λ λN + CλM(MN)

1 + C(MN) (9)

where y is the distance from the centroid of the section to theupper and lower edges A is the sectional area I is the sectionmoment of inertia and M and N represent the bendingmoment and axial force of the section

Equation (9) indicates that y A and I are the parametersinherent to the section under analysis λN and λΜ are theshear lag coefficients caused by the axial force and thebending moment respectively which are independent of themagnitude of the internal force Once λN and λΜ are de-termined the total shear lag coefficient of λ is only related tothe ratio of bending moment to axial force

Figure 14 shows the curve of the shear lag coefficientwith the bending moment axial force ratio when λΜ 200and λN 100 Taking the upper flange of the main beam ofthe Fumin cable-stayed bridge as an example C yAI 101

As shown in Figure 14 the total shear lag coefficient λgradually approaches the bending moment shear lag coef-ficient λΜ as the bending moment force ratio increases Asthe bending moment axial force ratio gradually decreases to0 the total shear lag coefficient λ is also gradually reduceduntil it is equal to the axial force shear lag coefficient λNWhen the bending moment force ratio is between 0 and 2the total shear lag coefficient λ changes substantially Alarger λΜ results in a greater change in the total shear lagcoefficient λ When the bending moment force ratio isgreater than 2 the total shear lag coefficient λ changesslowly at is to say in the internal force of this sectionwhether the bending moment or axial force accounts for alarger proportion affects how close the total shear lag co-efficient λ of the section is to the shear lag coefficient (λΜ orλN) caused by this item of internal force

For a general beam bridge the bending moment is themain internal force and so the section shear lag coefficient λis close to λΜ However for concrete cable-stayed bridgesthe main beam is in a compression state and is mainlysubjected to axial force its shear lag coefficient λ is closer toλN

62 eoretical versus Finite Element Main Beam Shear LagResults e shear lag variation law of the main beam of thesingle-cable PC cable-stayed bridge along the longitudinaldirection was discussed above as per our theoretical analysise shear lag coefficient and normal stress value of the mainbeam cross section are mainly concentrated in the range ofthe force point of the cable e effective width of the topplate of the upper flange in our analysis was set to 400m andthe full section of the bottom plate to σN 032N andσ( )N 009N (where N is the axial component of the cableforce) so that λN σNσ( )N 356 Shear lag coefficient λΜwas only considered a momentary action based on thedefinition of the effective width of the bridge e effectivewidth of the top plate in this case was 2400m so λΜ 135

We established the full-bridge plane bar modelaccording to finite element analysis e diagram of the

bending moment and axial force of the main beam underWorking Condition 1 suggests relatively small internalforces in Pier 4 and the main girder of Pier 5 e internalforce diagram of the main beam at Pier 4 was isolated forfurther analysis as shown in Figure 15

As shown in Figure 15 the bending moment force ratioMN is between λΜ and λN that is between 135 and 356e bending moment axial force ratio MN at the actionpoint of the cantilever end cable force C4 (C4prime) is 120 and118 According to formula (6) the shear lag coefficient λ ofthe upper edge of the main beam section is 235 and 236respectively

In the construction section the axial moment ratioMNof the main beam section of the C3primendashC1prime cable force actionpoint is 149 159 and 164 According to formula (6) theshear lag coefficients of the upper edge of the main beamsection of C3primendashC1prime cable force point are 223 220 and 218respectively e axial force ratio MN of the main beamsection of the C3ndashC1 cable force action point is 147 181and 203 Again according to formula (6) the shear lagcoefficients λ of the upper edge of the main beam section ofthe C3ndashC1 cable force action point are 224 213 and 207Our finite element results of the shear lag effect of the mainbeam along the longitudinal direction suggest that the shearlag coefficient at Point A of the action point of cantilever endcable force C4 (C4prime) is 418 and 361 the shear lag coefficientλ at Point B reaches 307 and 306

In the Pier 4 C3primendashC1prime beam section the shear lagcoefficients λ of Point A at the cable force action point are232 222 and 249 respectively those of Point B at the cableforce action point are 126 123 and 129 respectivelyWithin the C3ndashC1 beam section the shear lag coefficients λof Point A at the cable force action point are 257 271 and271 and those corresponding to Point B are 211 122 and212 respectively

In the beam section construction stage the analysisresults of the C3ndashC1 (C3primendashC1prime) cable force action pointtreatment theory agree with the finite element analysisresults In this position the theoretically derived shearforce lag coefficient λ can be used to reflect the actual forceat that point e finite element analysis results at the C4(C4prime) action point of the cantilever end cable force exceedthe theoretical analysis results e cantilever end cableforce action point is mainly subjected to axial force so itsshear lag coefficient λ should be closer to λN In this po-sition the shear lag factor λN can be used to reflect theactual force

005

115

225

0 1 2 3 4 5 6 7 8 9 10

Shea

r lag

coef

ficie

nt

MN

λM = 2 λN = 1

Figure 14 Relationship between shear lag coefficient and bendingmoment axial force ratio

14 Advances in Materials Science and Engineering

63 Variation Law of Shear Lag Effect of PC Cable-StayedBridge Main Beam along Longitudinal Direction e shearlag variations in the main beam of the single-cable PC cable-stayed bridge along the longitudinal direction can besummarized as follows

(1) In the construction stage at the cantilever end thecable force action point is the main recipient of axialforce e shear lag coefficient λN of the axial forceacting at this position reflects the actual force at thatpoint

(2) In the beam segments between the cable forces at thecable force action point the theoretical shear lagcoefficient λ determined by the ratio MN of thebending moment to axial force reflects the actualstress

(3) In the midspan beam section between the actionpoints of cable forces the shear lag coefficient ofbending moment λM is used to reflect the actualstress For the beam section between the action pointof cable force and the middle of the span the sectionshear lag coefficient is obtained by linearinterpolation

7 Conclusion

e structural control section stress at work in a typicalconstruction process was measured on a full cable-stayedbridge model in this study Sets of finite element and modeltest results were respectively analyzed and compared Ourconclusions can be summarized as follows

(1) In the construction stage the stress value of the cabletension area of the main beam upper edge markedlychanges after tensiling the cable force the shear lageffect is very prominent After a hanging basket loadis applied the main beams of certain sections presentalternating positive and negative shear lag

(2) e shear lag coefficient of the upper edge of themain beam in G1ndashG8 sections under WorkingConditions 1 and 4 was extracted from the spatialfinite element analysis for comparison to find thatthe experimental and theoretical results are in ac-cordance After tension of the cable force and theapplication of a hanging basket load positive andnegative shear lag effects alternate on the upper edgeof the box girder in certain areas As tension cableforce increases in each section the shear lag coef-ficient of the section upper edge decelerates

(3) In the construction stage at the cantilever end cableforce action point is mainly subjected to axial forcee shear lag coefficient λN of the axial force on thisposition reflects the actual force situation at thatpoint In the beam segments between the cableforces the shear lag coefficient λ determined by theratio of the bending moment to the axial force re-flects the actual stress at the cable force action pointIn the midspan beam section between the actionpoints of cable forces the shear lag coefficient λM ofthe bendingmoment reflects the actual stress For thebeam section between the cable force action pointand the middle of the span the section shear lagcoefficient can be obtained by linear interpolation

C2prime C3

4250kNmiddotm

17300kNmiddotm 19100kNmiddotm

4

C1prime C4

24400kNmiddotm

36800kNmiddotm

C4prime C3prime C2

10500kNmiddotm

18600kNmiddotm

C1

17700kNmiddotm

55000kNmiddotm

6820kNmiddotm

(a)

C1prime C4

13000kN7030kN

C4prime C1

5680kN

20300kN

3600kN

27000kN

C2prime C3

10900kN

C3prime C2

14900kN

4

(b)

Figure 15 Internal force of main girder on Pier 4 (Condition 1) (a) Moment diagram (b) Axial force diagram

Advances in Materials Science and Engineering 15

(4) e shear lag effect of the main beam is prominent inthe construction process e shear lag effect is mostobvious at the cable force operation point of thecantilever end As construction progresses and thecable force increases the shear lag effect of otherbeam sections decreases at each point except for thecantilever end

Data Availability

Part of the shear lag coefficient data used to support thefindings of this study are included within the article All ofthe shear lag coefficient data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors would like to thank the financial supports fromthe National Key RampD Program of China (2018YFC0809600and 2018YFC0809606) and Natural Science Foundation ofLiaoning Province (2019-MS-265)

References

[1] Z Li J G Nie and W Y Ji ldquoPositive and negative shear lagbehaviors of composite twin-girder decks with varying cross-sectionrdquo Science China Technological Sciences vol 60 no 1pp 116ndash132 2017

[2] S-J Zhou ldquoFinite beam element considering shear-lag effectin box girderrdquo Journal of Engineering Mechanics vol 136no 9 pp 1115ndash1122 2010

[3] Z Li and R K L Su ldquoAnalytical solutions for compositebeams with slip shear-lag and time-dependent effectsrdquo En-gineering Structures vol 152 pp 559ndash578 2017

[4] Q Z Luo Y M Wu Q S Li J Tang and G D Liu ldquoA finitesegment model for shear lag analysisrdquo Engineering Structuresvol 26 no 14 pp 2113ndash2124 2004

[5] P Z Lin F K Liu W Ji et al ldquoAnalysis on shear lag effect ofconcrete box beam by variational principlerdquo Journal of theChina Railway Society vol 35 no 2 pp 93ndash98 2013

[6] Y Y Zhang H Zhang W Li et al ldquoAnalysis on shear-lageffect of box girders based on different shear-lag generalizeddisplacementrdquo Journal of Railway Science and Engineeringvol 13 no 6 pp 1083ndash1090 2016

[7] S T Chang Q Zhang and S Zhang ldquoShear lag effect in singleplane cable-stayed bridgerdquo Advances in Structural Engi-neering vol 1 no 4 pp 301ndash306 1998

[8] G-F Wu and X Hong ldquoeoretical and experimental studyon shear lag effect of partially cable-stayed bridgerdquo Journal ofZhejiang University-Science A vol 6 no 8 pp 875ndash877 2005

[9] L Pengzhen Y Zijiang L Fengkui et al ldquoShear lag effect ofconcrete box girders considered varying flange-depthsrdquoJournal of Civil Architecturalamp Environmental Engineeringvol 35 no 1 pp 76ndash79 2013

[10] E Yamaguchi T Chaisomphob J Sa-Nguanmanasak andC Lertsima ldquoStress concentration and deflection of simplysupported box girder including shear lag effectrdquo StructuralEngineering and Mechanics vol 28 no 2 pp 207ndash220 2008

[11] X B Shen ldquoAnalysis of shear lag effect in the main girder ofwide low-pylon cable stayed bridgerdquo Construction Technologyvol 44 no 24 pp 76ndash79 2015

[12] L Xingmin ldquoAnalysis of influences of construction processeson shear lag effect of concrete box girdersrdquo Bridge Con-struction vol 43 no 1 pp 30ndash34 2013

[13] L Muyu W H Sun X D Sun et al ldquoStress analysis of widergirder of an extradosed cable-stayed bridge based on thelargest cantilever construction stagerdquo Journal of HuazhongUniversity of Science and Technology (Urban Science Edition)vol 27 no 2 pp 11ndash14 2010

[14] C Qianshu H Wenlong and Y Menggang ldquoAnalysis ofshear lag effect in construction stage of wide box girderextradosed cable-stayed bridge with large flangesrdquo Journal ofRailway Science and Engineering vol 15 no 12 pp 3158ndash3164 2018

[15] D R Li B M Wang and Y C Lin Structural Model Ex-periment Science Press Beijing China 1996

[16] J W Dali andW F Laili Experiment and Analysis of ResidualStress Ocean Press Beijing China 1987

[17] H Hossdort Structural Model Analysis China Architecture ampBuilding Press Beijing China 1986

[18] S K Mei Structural Experiments and Structural DesignChina Communications Press Beijing China 1980

[19] B Atmaca and S Ates ldquoConstruction stage analysis of three-dimensional cable-stayed bridgesrdquo Steel amp Composite Struc-tures vol 12 no 5 pp 413ndash426 2012

[20] S W Park M R Jung D J Min and M Y Kim ldquoCon-struction stage analysis of cable-stayed bridges using theunstrained element length methodrdquo Journal of e KoreanSociety of Civil Engineers vol 36 no 6 pp 991ndash998 2016

[21] M Gunaydin S Adanur A C Altunisik B Sevim andE Turker ldquoDetermination of structural behavior of Bosporussuspension bridge considering construction stages and dif-ferent soil conditionsrdquo Steel and Composite Structures vol 17no 4 pp 405ndash429 2014

[22] S Adanur M Gunaydin A C Altunisik and B SevimldquoConstruction stage analysis of humber suspension bridgerdquoApplied Mathematical Modelling vol 36 no 11 pp 5492ndash5505 2012

[23] J B Mander M J N Priestley and R Park ldquoeoreticalstress-strain model for confined concreterdquo Journal of Struc-tural Engineering vol 114 no 8 pp 1804ndash1826 1988

[24] Y A Khalid C L Chan B B Sahari and A M S HamoudaldquoBending behaviour of corrugated web beamsrdquo Journal ofMaterials Processing Technology vol 150 no 3 pp 242ndash2542004

[25] A Demir N Caglar H Ozturk and Y Sumer ldquoNonlinearfinite element study on the improvement of shear capacity inreinforced concrete T-Section beams by an alternative diag-onal shear reinforcementrdquo Engineering Structures vol 120pp 158ndash165 2016

16 Advances in Materials Science and Engineering

the hanging basket load was 5100 kN after the C11 (C11prime)cable tension was applied e boundary condition in thiscase involved removed the temporary support placing thebottom of Piers 4 and 5 under the same working con-dition e line displacement in the Z direction was re-strained at the end of the main beam

After the tension of the C14 (C14prime) was applied thehanging basket load was applied At 5100 kN the boundaryconditions were the same as the previous working conditionAfter applying the tension of C15 (C15prime) cable force the

closing weight of 1950 kN was applied e middle span wasclosed and the closing weight was unloaded e boundarycondition in this case was an unchanged constraint on thebottom of Pier 4 while the line displacement of the bottomof Pier 5 was constrained in the Z direction

e finite element model as established under thesespecific working conditions is shown in Figure 6 Due tolimited space only the results of Working Condition 1and Working Condition 4 are presented here forcomparison

(a) (b)

(c) (d)

Figure 3 Installing the model (a) installing four pairs of stay cables at Pier 4 (b) closure of side span of Pier 5 (c) second stage dead loadapplied by model and (d) model exerts live load

2225 2225605010500

Stay cablesC15prime~C1prime

Stay cablesC1~C15

Stay cablesC1prime~C15prime

Stay cablesC15~C1

Pier 4 Pier 5

1688

G4 G2 G1 G3 G9 G8G7 G5 G6

Figure 4 Diagram of testing section arrangement about girder (unit mm)

1 2

18

3

13

8

17

16

4 5

15 14

6 7 10

12

9 11

Figure 5 Arrangement of stiff girder stress measuring point

6 Advances in Materials Science and Engineering

We conducted forward installation calculations reflect-ing the construction process of the entire bridge e leastsquare method was applied to calculate of the cable force ofthe bridge during forward installation and the results werecompared with the reasonable cable force of the bridge untilall design requirements were satisfied At this point thecorresponding construction cable force could be consideredthe initial tension of the stay cable To accelerate conver-gence the initial tension of the cable was selected per thestatic equilibrium determined in the first forward installa-tion iteration Table 1 lists the initial tension of each staycable after the iterative calculation the number of stay cablesis shown in Figure 1

e stay cable at Pier 5 was adjusted once the side spanwas closed to bring the internal force at the root of the maintower closer to the deformation at the cantilever end of themain beam e C6 cable force was adjusted to 8000 kN C7cable force to 8000 kN C8 cable force to 8000 kN and C9cable force to 7500 kN at this time Before the constructionof bridge deck pavement to ensure uniform internal forceand deformation of the main girder across the whole bridgethe cable at Pier 4 was adjusted by bringing C1prime cable forceto 9000 kN C2prime cable force to 9000 kN C1 cable force to11000 kN C2 cable force to 11500 kN C3 cable force to9000 kN and C4 cable force to 9000 kN e cable force ofpart of the stay cable at Pier 5 was also adjusted by bringingC1 cable force to 10000 kN C2 cable force to 10000 kN C1primecable force to 10000 kN and C2prime cable force to 10500 kN

42 Main Beam Cross Section Shear Lag Coefficient underWorking Condition 1 e model test results and the shearlag coefficient of the upper edge of the main beam section ofWorking Condition 1 in the spatial finite element weresubjected to comparative analysis After tensioning the C4cable the full beam of the main beams G1 and G2 waspressed (tensile stress is positive compressive stress isnegative) the average measured stress of the upper edge ofthe main beams G1 and G2 was minus207MPa and minus077MPa atthis point After applying a hanging basket load on thecantilever end of the box girder the upper edge of the maingirders G1 and G2 showed a large amount of stress changese midabdominal segment bore compressive stress whilethe upper edge of section G1 of main beam bore tensilestress e upper edge of section G2 of main beam boretensile stress e average measured stress of the upper edgeof the main beams G1 and G2 was 024MPa and 049MPaAfter the tensioning force of the main girders G1 and G2 andapplication of the basket load the shear lag coefficient ateach point of the upper edge was as shown in Figure 7

After tensioning the C4 cable the main beams G5 andG6 were pressed in the full section and the average measuredstress of the upper edge of the main beams G5 and G6 wasminus189MPa and minus053MPa After applying the hangingbasket load on the cantilever end of the box girder the upperedge of the main beam G5 and G6 showed a large amount ofstress change e midabdominal segment bore compressivestress and the upper edge of section G5 and G6 of main beam

(a) (b)

(c) (d)

(e) (f )

Figure 6 Special finite element model for main construction operating conditions (a) Tension C4 (C4prime) applying a basket load (b) TensionC10 (C10prime) applying a basket load (c) Cable tension adjustment tensioning C11(C11prime) applying a hanging basket load (d) TensionC14(C14prime) applying a hanging basket load (e) Tension C15 (C15prime) applying counterweight midspan closure (f ) Grid division of mainbeam section

Advances in Materials Science and Engineering 7

bore tensile stress e average measured stress of the upperedge of the main beams G5 and G6 was 053MPa and055MPae shear lag coefficient at each point of the upperedge was as shown in Figure 8 after the tensioning force ofthe main girders G5 and G6 and the application of the basketload

As tension was applied the concentrated load was placedin the middle of the box girder to intuitively observe theshear lag effect in the middle of the box girder section ehanging basket load was applied in the form of a concen-trated load so that the upper edge of sections G1 G5 and G6of the main beam presented tensile stress though the middleweb section did not After the tensile force and the hangingbasket load were applied positive and negative shear lageffects appeared alternately on the upper edge of the mainbeam

43 Main Beam Cross Section Shear Lag Coefficient underWorking Condition 4 e model test results and the shearlag coefficient of the upper edge of the main beams G1ndashG8section under Working Condition 4 for the spatial finiteelement was also extracted for comparative analysis Asshown in Figures 9 and 10 the shear lag coefficient of theupper edge of the section G7 of themain beam changed from009ndash229 to minus293ndash210 after the tension of the cable and theapplication of the hanging basket load e shear lag

coefficient of the upper edge of the main beam G3 changedfrom 055ndash135 to 039ndash232 the shear lag coefficient of theupper edge of the main beam G4 changed from 063ndash159 to066ndash166 and the shear lag coefficient of the upper edge ofthe main beam G8 changed from 034ndash152 to 021ndash192 Inthis case after the tension was applied to the C14 cable andthe hanging basket load was applied the shear lag coefficientof the upper edge decelerated and the changes were slighte shear lag coefficient of the upper edge of the sections G2and G1 G5 and G6 of the main girder near the tower rootwas weak In each section the shear lag coefficient of theupper edge decelerated as tension cable force increased etest results are consistent with the finite element calculationresults

5 Longitudinal Shear Lag Effect in MainBeam of Single-Cable Plane P C Cable-Stayed Bridge

e stress on the main girder section of cable-stayed bridgeis highly complex e construction of the cable-stayedbridge is a dynamic process e stress in the main beamsection under different working conditions and differentbeam sections may widely differ e shear lag coefficientcalculation results along the longitudinal length of severalkey nodes on the upper edge of the cross section of the main

Table 1 Initial cable tension of each stay cable

Pier 4 Pier 5Cablenumber

Tension cable force(left) (kN)

Cablenumber

Tension cable force(right) (kN)

Cablenumber

Tension cable force(left) (kN)

Cablenumber

Tension cable force(right) (kN)

C1prime 7500 C1 9000 C1 9000 C1prime 4500C2prime 6500 C2 9500 C2 9500 C2prime 6500C3prime 5500 C3 8000 C3 8000 C3prime 5500C4prime 5500 C4 8000 C4 8000 C4prime 5000C5prime 5500 C5 8500 C5 8000 C5prime 6000C6prime 5500 C6 8000 C6 7400 C6prime 6500C7prime 5500 C7 8000 C7 7000 C7prime 5500C8prime 6000 C8 8000 C8 7000 C8prime 6000C9prime 6000 C9 7500 C9 7000 C9prime 6000C10prime 6500 C10 7500 C10 7500 C10prime 6500C11prime 6500 C11 7500 C11 7500 C11prime 6500C12prime 9500 C12 7500 C12 7500 C12prime 9000C13prime 9500 C13 7500 C13 7500 C13prime 9000C14prime 10000 C14 7500 C14 7500 C14prime 9500C15prime 9500 C15 6500 C15 6500 C15prime 9500

ndash100ndash050

000050100150

Shea

r lag

coeffi

cien

t

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(a)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

ndash200ndash100

000100200300

Shea

r lag

coeffi

cien

t

Node lateral coordinate (m)

(b)

Figure 7 Shear lag coefficient variations in G1 and G2 (Condition 1) (a) G1 section (b) G2 section

8 Advances in Materials Science and Engineering

beam can be used to understand the shear lag coefficientdistribution along the longitudinal length of the upper edgeof the main beam section during the construction stage Forconvenience the key points on the top edge of the crosssection of the main beam are marked here with capitalletters e distances from A-D points to the transversesymmetry lines are 0m minus2m minus11m and minus16m respec-tively e lateral position of each point is shown inFigure 11

Due to space limitations only the distribution of theshear lag coefficient along the longitudinal length of theupper edge of the main beam section under the followingtwo typical working conditions is described here WorkingCondition 1 is defined by tensiling the C4 (C4prime) cable forceWorking Condition 4 is defined by tensiling the C14 (C14prime)cable force e boundary conditions of each workingcondition were kept consistent

51Working Condition 1 Shear Lag Effect along LongitudinalBridge Under the action of self-weight and cable force the

shear lag coefficient of the flange of the main beam sectionalong the longitudinal bridge was observed as shown inFigure 12 In the longitudinal distribution of the shear lagcoefficient at the key point of the upper flange the midpointof the cross section of the main beam (Point A) and theboundary point between the upper edge and the middle web(Point B) change most sharply

At Pier 4 a sudden change in shear lag coefficient at theend of the cantilever occurred due to the tension of the C4(C4prime) cable A large stress concentration occurred at theseconcentrated load points which altered the upper flangestress distribution law Under the action of cable force theupper edge of the main beam section shows a more sig-nificant shear lag effect of the cable tension zone at locationscloser to the cable action point At Point A the shear lagcoefficient reaches 418 at Point B it is 307 e stressdistribution of the main beam appears to grow more uni-form as the distance between the main beam and the cableforce increases

Pier 4 C3ndashC1 (C3prime-C1prime) shows a similar distributionpattern in each beam section e shear lag coefficient of

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

ndash100ndash050

000050100150200

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

(a)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

ndash200ndash100

000100200300

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

(b)

Figure 8 Shear lag coefficient variations in G5 and G6 (Condition 1) (a) G5 section (b) G6 section

000

050

100

150

200

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16Node lateral coordinate (m)

Shea

r lag

coef

ficie

nt

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(a)

000050100150200250

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16Node lateral coordinate (m)

Shea

r lag

coef

ficie

ntTensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(b)

000050100150200250

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(c)

Shea

r lag

coef

ficie

nt

000050100150200250

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(d)

Figure 9 Contrast diagram of shear lag coefficient in G1-G4 (Condition 4) (a) G1 section (b) G2 section (c) G3 section and (d) G4 section

Advances in Materials Science and Engineering 9

Point A and Point B is prominent at the point of action of thecable forcee change range of shear lag coefficient of PointA at the point of cable force action is 222ndash271 and at Point Bis 122ndash212 e stress distribution of the beam sectionbetween the points of cable force is relatively uniform eA-point shear lag coefficient λ ranges from 115 to 116 theB-point shear lag coefficient λ from 112 to 140 and theC-point shear lag coefficient λ from 109 to 126

e stress distribution of the upper flange of the Pier 4tower root section changed due to the existence of the crossbeame change appears to be most dramatic at Point D atthe end of the upper extension arm of the main beam crosssection e shear lag coefficient λ at Point D near the towerranges from 114 to 154 marking a very significant variationcompared with other beam sections

e variation of the cantilever end the cable force pointand the C3ndashC1 (C3prime-C1prime) beam section of Pier 5 is similarto that of Pier 4 Under the action of the tension C4 (C4prime)Point A here shows a shear lag coefficient λ of 300 while thatat Point B reaches 250 In our analysis the stress distri-bution of the main beam grew more uniform as the distancebetween the main beam and the cable force increased

Pier 5 C3ndashC1(C3prime-C1prime) shows a similar distributionpattern in each beam section e shear force lag coeffi-cient of Point A and Point B is prominent at the point ofaction of the cable force e variation range of lag

coefficient λ is 157ndash244 and the shear lag coefficient λ ofPoint B is 138ndash184 at the point of action of cable forcee stress distribution of the beam section between thecable force points is relatively uniform e changes inshear lag coefficient λ of Point A range from 141 to 166that at Point B varies from 107 to 157 and that at Point Cranges from 075 to 108 Again the most significantchange was observed at Point D the shear lag coefficient λof Pier 5 at point D near the tower root varies from 120to 133

At the C1prime cable force on the right side of Pier 5 theshear lag coefficient of Points A and B is weaker than at otherpoints of cable force mainly because of the temporarysupport e shear lag effects on both sides were similar dueto the small reaction force of the temporary pier

Under the action of self-weight and cable force with thisworking condition the main characteristics of the shear lageffect of the upper edge of the main beam section of thesingle-cable PC cable-stayed bridge in the longitudinal di-rection can be summarized as follows

(1) In the construction stage under the action of cableforce and near the point of cable tension the upperedge of the girder section shows more significantshear lag effect in the cable force tension area egirder stress distribution grows more uniform as thedistance between the girder section and the cableforce action point increases

(2) In the C3ndashC1(C3prime-C1prime) beam section the upper edgeof the main beam section at the cable force actionpoint presents shear lag effects in the tension zonee stress distribution in the beam section betweenthe points of the cable force is relatively uniformwhich characterizes the shear lag distribution in theordinary curved box girder

000020040060080100120140160

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(a)

000

050

100

150

200

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(b)

ndash400ndash300ndash200ndash100

000100200300

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(c)

000

050

100

150

200

250

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(d)

Figure 10 Contrast diagram of shear lag coefficient in G1-G4 (Condition 4) (a) G5 section (b) G6 section (c) G7 section and (d) G8 section

B ACD

Figure 11 Main girder cross-section diagram at every point

10 Advances in Materials Science and Engineering

(3) e shear lag coefficient at the end of the upperextension arm of the cross section of the main beamat the root of the tower changes most dramaticallyamong the areas we observed We also observedconversion from a positive shear lag effect to anegative shear lag effect near the tower

(4) Under the influence of temporary support the shearforce lag coefficient of the action point of the C1primecable force on the right side of Pier 5 is weaker thanother cable forces e temporary pier reaction forceis small whichmakes the shear lag on both sideseeffects were still relatively similar in this case

52Working Condition 4 Shear Lag Effect along LongitudinalBridge is working condition represents the largestsingle cantilever stage e side spans at this point areclosed and the shear lag effects near their supports havemarkedly increased due to tensioning of C12primendashC14prime tailse shear lag effect of the inner beam section of the sidespan C11primendashC1prime is lower than under the previous workingcondition

As shown in Figure 13 at Pier 4 the cantilever end fallsunder the action of tensioning C14 cable force e shear lagcoefficient λ at Point A reaches 340 and at Point B is 240 Inthe beam section of Pier 4 C11ndashC1 (C11primendashC1prime) the shearlag coefficient of Points A and B is prominent at the point ofaction of the cable force Point A shows a shear lag coefficientλ of 151ndash281 at the cable force action point where Point Bshows a shear lag coefficient λ of 435ndash212 the stress dis-tribution of the beam section between the points of the cableforce is relatively uniform e shear lag coefficient λ ofPoint A varies from 105 to 144 and that of Point B from 119to 110 e shear lag coefficient λ of Point C ranges from065 to 095

At Pier 5 the end of the cantilever is under the action ofthe tensioning force of C14 e shear lag coefficient λ ofPoint A reaches 357 and that of Point B is 222 In the beamsection of C11ndashC1 (C11primendashC1prime) in Pier 5 the shear lagcoefficients of Points A and B at the point of action of thecable force are prominent Point A has a shear lag coefficientλ of 148ndash281 at the action point of the cable force wherePoint B has a shear lag coefficient λ of 133ndash212 e stressdistribution of beam section between the points of cable

4rsquo 3rsquo 2rsquo 1rsquo 4 1 2 3 4Longitudinal bridge position

Point APoint B

Point CPoint D

Pier 4

000050100150200250300350400450

Shea

r lag

coef

ficie

nt

(a)

Point APoint B

Point CPoint D

4 3 2 1 5 1rsquo 2rsquo 3rsquo 4rsquoLongitudinal bridge position

Pier 5

000

050

100

150

200

250

300

350

Shea

r lag

coef

ficie

nt

(b)

Figure 12 Distribution of shear lag coefficient along longitudinal direction (Condition 1) (a) Pier 4 (b) Pier 5

Advances in Materials Science and Engineering 11

Point APoint B

Point CPoint D

3 14rsquo13rsquo12rsquo 11rsquo 10rsquo 9rsquo 8rsquo 7rsquo 6rsquo 5rsquo 4rsquo 3rsquo 2rsquo 1rsquo 4Longitudinal bridge position

Pier 4

000

050

100

150

200

250

300

Shea

r lag

coef

ficie

nt

(a)

Point APoint B

Point CPoint D

4 1 2 3 4 5 6 7 8 9 10 11 12 13 14Longitudinal bridge position

000

050

100

150

200

250

300

350

400

Shea

r lag

coef

ficie

nt

(b)

Point APoint B

Point CPoint D

14 13 12 11 10 9 8 7 6 5 4 3 2 1 5Longitudinal bridge position

Pier 5

000

050

100

150

200

250

300

350

400

Shea

r lag

coef

ficie

nt

(c)

Figure 13 Continued

12 Advances in Materials Science and Engineering

force is relatively uniform e shear lag coefficient λ ofPoint A ranges from 120 to 155 that of Point B from 096 to135 and that of Point C from 076 to 101

Under this working condition the action of self-weightand cable force make the main characteristics of the shear lagat the upper edge of the main beam section of the single-cable PC cable-stayed bridge along the vertical direction verysimilar to those of underWorking Condition 1e shear lageffect at the main beam section is reduced as cable forceincreases

6 Distribution of Shear Lag Effect duringConstruction Stage

61 Shear Lag Coefficient of Main Beam of Single-Cable PlanePC Cable-Stayed Bridge e solution to the shear lag coef-ficient under the compressive bending load involves decom-posing the compressive bending load into a superposition ofthe bending moment action and the axial force action thencalculating the normal stress of themain beamunder the actionof the bending moment and the axial force alone [22] It isassumed that the axial force is borne by the whole section thatis the normal stress generated by the axial force is evenlydistributed along the cross section e results can then belinearly superposed to obtain the comprehensive shear lageffect coefficient Our previous analysis indicated that whenthere is cable tension of the main beam of the cable-stayedbridge in the single-cable PC there is a large stress concen-tration at the point of action of the concentrated loads estress distribution law of the roof plate changes accordingly

Assume that the axial force which is borne by the wholesection does not reflect the actual internal force of the maingirder of the single-cable PC cable-stayed bridge e dis-tribution of the longitudinal normal stress generated byinternal force is not uniform e size and distribution laware related to the internal force state namely the bendingmoment and axial force In this study the shear lag coef-ficient λΜ generated by bending moment and the shear lag

coefficient λΜ generated by axial force were respectivelycalculated to solve the shear lag coefficient λ aftersuperposition

e shear lag coefficient is calculated as follows

λ σσ

(6)

where σ is actual stress and σ( ) is nominal stresse shear lag coefficient is related to the wide span ratio

support conditions load form cross-sectional shape andsize material properties and location of the section [23ndash25]In the final analysis the shear lag is caused by unevendistribution of the longitudinal normal stress caused byinternal force its size and distribution are related to theinternal force state that is the bending moment is related tothe axial force

e actual stress σ and nominal stress σ( ) are thesuperposition of the bending moment and the normalstress generated by the axial force If the shear lag coeffi-cient λΜ produced by the bending moment is definedproperly the shear lag coefficient generated by the axialforce is λN and the shear lag coefficient after the super-position is λ us

λ σN + σM

σN + σM

λNσN + λMσM

σN + σM

(7)

where σ( )N and σ( )M are the nominal normal stressgenerated by bending moment and axial force respectivelyσ( )N and σ( )M are the actual normal stress generated bybending moment and axial force respectively e nu-merator and denominator at the right end of the aboveformula are divided by σ( )N to rewrite (7) as

λ λN + λM σMσN( 1113857

1 + σMσN( 1113857 (8)

ere are σM (MI)y and σN (NA) so(σMσN) (yAI) middot (MN) and order yAI C can beplugged into (8) to obtain

Point APoint B

Point CPoint D

000

050

100

150

200

250

300

350

5 1prime 2prime 3prime 4prime 5prime 6prime 7prime 8prime 9prime 10prime 11prime 12prime13prime14prime6

Shea

r lag

coef

ficie

nt

Longitudinal bridge position

(d)

Figure 13 Distribution of shear lag coefficient along longitudinal direction (Condition 4) (a) Left side of Pier 4 (b) Right side of Pier 4(c) Left side of Pier 5 (d) Right side of Pier 5

Advances in Materials Science and Engineering 13

λ λN + CλM(MN)

1 + C(MN) (9)

where y is the distance from the centroid of the section to theupper and lower edges A is the sectional area I is the sectionmoment of inertia and M and N represent the bendingmoment and axial force of the section

Equation (9) indicates that y A and I are the parametersinherent to the section under analysis λN and λΜ are theshear lag coefficients caused by the axial force and thebending moment respectively which are independent of themagnitude of the internal force Once λN and λΜ are de-termined the total shear lag coefficient of λ is only related tothe ratio of bending moment to axial force

Figure 14 shows the curve of the shear lag coefficientwith the bending moment axial force ratio when λΜ 200and λN 100 Taking the upper flange of the main beam ofthe Fumin cable-stayed bridge as an example C yAI 101

As shown in Figure 14 the total shear lag coefficient λgradually approaches the bending moment shear lag coef-ficient λΜ as the bending moment force ratio increases Asthe bending moment axial force ratio gradually decreases to0 the total shear lag coefficient λ is also gradually reduceduntil it is equal to the axial force shear lag coefficient λNWhen the bending moment force ratio is between 0 and 2the total shear lag coefficient λ changes substantially Alarger λΜ results in a greater change in the total shear lagcoefficient λ When the bending moment force ratio isgreater than 2 the total shear lag coefficient λ changesslowly at is to say in the internal force of this sectionwhether the bending moment or axial force accounts for alarger proportion affects how close the total shear lag co-efficient λ of the section is to the shear lag coefficient (λΜ orλN) caused by this item of internal force

For a general beam bridge the bending moment is themain internal force and so the section shear lag coefficient λis close to λΜ However for concrete cable-stayed bridgesthe main beam is in a compression state and is mainlysubjected to axial force its shear lag coefficient λ is closer toλN

62 eoretical versus Finite Element Main Beam Shear LagResults e shear lag variation law of the main beam of thesingle-cable PC cable-stayed bridge along the longitudinaldirection was discussed above as per our theoretical analysise shear lag coefficient and normal stress value of the mainbeam cross section are mainly concentrated in the range ofthe force point of the cable e effective width of the topplate of the upper flange in our analysis was set to 400m andthe full section of the bottom plate to σN 032N andσ( )N 009N (where N is the axial component of the cableforce) so that λN σNσ( )N 356 Shear lag coefficient λΜwas only considered a momentary action based on thedefinition of the effective width of the bridge e effectivewidth of the top plate in this case was 2400m so λΜ 135

We established the full-bridge plane bar modelaccording to finite element analysis e diagram of the

bending moment and axial force of the main beam underWorking Condition 1 suggests relatively small internalforces in Pier 4 and the main girder of Pier 5 e internalforce diagram of the main beam at Pier 4 was isolated forfurther analysis as shown in Figure 15

As shown in Figure 15 the bending moment force ratioMN is between λΜ and λN that is between 135 and 356e bending moment axial force ratio MN at the actionpoint of the cantilever end cable force C4 (C4prime) is 120 and118 According to formula (6) the shear lag coefficient λ ofthe upper edge of the main beam section is 235 and 236respectively

In the construction section the axial moment ratioMNof the main beam section of the C3primendashC1prime cable force actionpoint is 149 159 and 164 According to formula (6) theshear lag coefficients of the upper edge of the main beamsection of C3primendashC1prime cable force point are 223 220 and 218respectively e axial force ratio MN of the main beamsection of the C3ndashC1 cable force action point is 147 181and 203 Again according to formula (6) the shear lagcoefficients λ of the upper edge of the main beam section ofthe C3ndashC1 cable force action point are 224 213 and 207Our finite element results of the shear lag effect of the mainbeam along the longitudinal direction suggest that the shearlag coefficient at Point A of the action point of cantilever endcable force C4 (C4prime) is 418 and 361 the shear lag coefficientλ at Point B reaches 307 and 306

In the Pier 4 C3primendashC1prime beam section the shear lagcoefficients λ of Point A at the cable force action point are232 222 and 249 respectively those of Point B at the cableforce action point are 126 123 and 129 respectivelyWithin the C3ndashC1 beam section the shear lag coefficients λof Point A at the cable force action point are 257 271 and271 and those corresponding to Point B are 211 122 and212 respectively

In the beam section construction stage the analysisresults of the C3ndashC1 (C3primendashC1prime) cable force action pointtreatment theory agree with the finite element analysisresults In this position the theoretically derived shearforce lag coefficient λ can be used to reflect the actual forceat that point e finite element analysis results at the C4(C4prime) action point of the cantilever end cable force exceedthe theoretical analysis results e cantilever end cableforce action point is mainly subjected to axial force so itsshear lag coefficient λ should be closer to λN In this po-sition the shear lag factor λN can be used to reflect theactual force

005

115

225

0 1 2 3 4 5 6 7 8 9 10

Shea

r lag

coef

ficie

nt

MN

λM = 2 λN = 1

Figure 14 Relationship between shear lag coefficient and bendingmoment axial force ratio

14 Advances in Materials Science and Engineering

63 Variation Law of Shear Lag Effect of PC Cable-StayedBridge Main Beam along Longitudinal Direction e shearlag variations in the main beam of the single-cable PC cable-stayed bridge along the longitudinal direction can besummarized as follows

(1) In the construction stage at the cantilever end thecable force action point is the main recipient of axialforce e shear lag coefficient λN of the axial forceacting at this position reflects the actual force at thatpoint

(2) In the beam segments between the cable forces at thecable force action point the theoretical shear lagcoefficient λ determined by the ratio MN of thebending moment to axial force reflects the actualstress

(3) In the midspan beam section between the actionpoints of cable forces the shear lag coefficient ofbending moment λM is used to reflect the actualstress For the beam section between the action pointof cable force and the middle of the span the sectionshear lag coefficient is obtained by linearinterpolation

7 Conclusion

e structural control section stress at work in a typicalconstruction process was measured on a full cable-stayedbridge model in this study Sets of finite element and modeltest results were respectively analyzed and compared Ourconclusions can be summarized as follows

(1) In the construction stage the stress value of the cabletension area of the main beam upper edge markedlychanges after tensiling the cable force the shear lageffect is very prominent After a hanging basket loadis applied the main beams of certain sections presentalternating positive and negative shear lag

(2) e shear lag coefficient of the upper edge of themain beam in G1ndashG8 sections under WorkingConditions 1 and 4 was extracted from the spatialfinite element analysis for comparison to find thatthe experimental and theoretical results are in ac-cordance After tension of the cable force and theapplication of a hanging basket load positive andnegative shear lag effects alternate on the upper edgeof the box girder in certain areas As tension cableforce increases in each section the shear lag coef-ficient of the section upper edge decelerates

(3) In the construction stage at the cantilever end cableforce action point is mainly subjected to axial forcee shear lag coefficient λN of the axial force on thisposition reflects the actual force situation at thatpoint In the beam segments between the cableforces the shear lag coefficient λ determined by theratio of the bending moment to the axial force re-flects the actual stress at the cable force action pointIn the midspan beam section between the actionpoints of cable forces the shear lag coefficient λM ofthe bendingmoment reflects the actual stress For thebeam section between the cable force action pointand the middle of the span the section shear lagcoefficient can be obtained by linear interpolation

C2prime C3

4250kNmiddotm

17300kNmiddotm 19100kNmiddotm

4

C1prime C4

24400kNmiddotm

36800kNmiddotm

C4prime C3prime C2

10500kNmiddotm

18600kNmiddotm

C1

17700kNmiddotm

55000kNmiddotm

6820kNmiddotm

(a)

C1prime C4

13000kN7030kN

C4prime C1

5680kN

20300kN

3600kN

27000kN

C2prime C3

10900kN

C3prime C2

14900kN

4

(b)

Figure 15 Internal force of main girder on Pier 4 (Condition 1) (a) Moment diagram (b) Axial force diagram

Advances in Materials Science and Engineering 15

(4) e shear lag effect of the main beam is prominent inthe construction process e shear lag effect is mostobvious at the cable force operation point of thecantilever end As construction progresses and thecable force increases the shear lag effect of otherbeam sections decreases at each point except for thecantilever end

Data Availability

Part of the shear lag coefficient data used to support thefindings of this study are included within the article All ofthe shear lag coefficient data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors would like to thank the financial supports fromthe National Key RampD Program of China (2018YFC0809600and 2018YFC0809606) and Natural Science Foundation ofLiaoning Province (2019-MS-265)

References

[1] Z Li J G Nie and W Y Ji ldquoPositive and negative shear lagbehaviors of composite twin-girder decks with varying cross-sectionrdquo Science China Technological Sciences vol 60 no 1pp 116ndash132 2017

[2] S-J Zhou ldquoFinite beam element considering shear-lag effectin box girderrdquo Journal of Engineering Mechanics vol 136no 9 pp 1115ndash1122 2010

[3] Z Li and R K L Su ldquoAnalytical solutions for compositebeams with slip shear-lag and time-dependent effectsrdquo En-gineering Structures vol 152 pp 559ndash578 2017

[4] Q Z Luo Y M Wu Q S Li J Tang and G D Liu ldquoA finitesegment model for shear lag analysisrdquo Engineering Structuresvol 26 no 14 pp 2113ndash2124 2004

[5] P Z Lin F K Liu W Ji et al ldquoAnalysis on shear lag effect ofconcrete box beam by variational principlerdquo Journal of theChina Railway Society vol 35 no 2 pp 93ndash98 2013

[6] Y Y Zhang H Zhang W Li et al ldquoAnalysis on shear-lageffect of box girders based on different shear-lag generalizeddisplacementrdquo Journal of Railway Science and Engineeringvol 13 no 6 pp 1083ndash1090 2016

[7] S T Chang Q Zhang and S Zhang ldquoShear lag effect in singleplane cable-stayed bridgerdquo Advances in Structural Engi-neering vol 1 no 4 pp 301ndash306 1998

[8] G-F Wu and X Hong ldquoeoretical and experimental studyon shear lag effect of partially cable-stayed bridgerdquo Journal ofZhejiang University-Science A vol 6 no 8 pp 875ndash877 2005

[9] L Pengzhen Y Zijiang L Fengkui et al ldquoShear lag effect ofconcrete box girders considered varying flange-depthsrdquoJournal of Civil Architecturalamp Environmental Engineeringvol 35 no 1 pp 76ndash79 2013

[10] E Yamaguchi T Chaisomphob J Sa-Nguanmanasak andC Lertsima ldquoStress concentration and deflection of simplysupported box girder including shear lag effectrdquo StructuralEngineering and Mechanics vol 28 no 2 pp 207ndash220 2008

[11] X B Shen ldquoAnalysis of shear lag effect in the main girder ofwide low-pylon cable stayed bridgerdquo Construction Technologyvol 44 no 24 pp 76ndash79 2015

[12] L Xingmin ldquoAnalysis of influences of construction processeson shear lag effect of concrete box girdersrdquo Bridge Con-struction vol 43 no 1 pp 30ndash34 2013

[13] L Muyu W H Sun X D Sun et al ldquoStress analysis of widergirder of an extradosed cable-stayed bridge based on thelargest cantilever construction stagerdquo Journal of HuazhongUniversity of Science and Technology (Urban Science Edition)vol 27 no 2 pp 11ndash14 2010

[14] C Qianshu H Wenlong and Y Menggang ldquoAnalysis ofshear lag effect in construction stage of wide box girderextradosed cable-stayed bridge with large flangesrdquo Journal ofRailway Science and Engineering vol 15 no 12 pp 3158ndash3164 2018

[15] D R Li B M Wang and Y C Lin Structural Model Ex-periment Science Press Beijing China 1996

[16] J W Dali andW F Laili Experiment and Analysis of ResidualStress Ocean Press Beijing China 1987

[17] H Hossdort Structural Model Analysis China Architecture ampBuilding Press Beijing China 1986

[18] S K Mei Structural Experiments and Structural DesignChina Communications Press Beijing China 1980

[19] B Atmaca and S Ates ldquoConstruction stage analysis of three-dimensional cable-stayed bridgesrdquo Steel amp Composite Struc-tures vol 12 no 5 pp 413ndash426 2012

[20] S W Park M R Jung D J Min and M Y Kim ldquoCon-struction stage analysis of cable-stayed bridges using theunstrained element length methodrdquo Journal of e KoreanSociety of Civil Engineers vol 36 no 6 pp 991ndash998 2016

[21] M Gunaydin S Adanur A C Altunisik B Sevim andE Turker ldquoDetermination of structural behavior of Bosporussuspension bridge considering construction stages and dif-ferent soil conditionsrdquo Steel and Composite Structures vol 17no 4 pp 405ndash429 2014

[22] S Adanur M Gunaydin A C Altunisik and B SevimldquoConstruction stage analysis of humber suspension bridgerdquoApplied Mathematical Modelling vol 36 no 11 pp 5492ndash5505 2012

[23] J B Mander M J N Priestley and R Park ldquoeoreticalstress-strain model for confined concreterdquo Journal of Struc-tural Engineering vol 114 no 8 pp 1804ndash1826 1988

[24] Y A Khalid C L Chan B B Sahari and A M S HamoudaldquoBending behaviour of corrugated web beamsrdquo Journal ofMaterials Processing Technology vol 150 no 3 pp 242ndash2542004

[25] A Demir N Caglar H Ozturk and Y Sumer ldquoNonlinearfinite element study on the improvement of shear capacity inreinforced concrete T-Section beams by an alternative diag-onal shear reinforcementrdquo Engineering Structures vol 120pp 158ndash165 2016

16 Advances in Materials Science and Engineering

We conducted forward installation calculations reflect-ing the construction process of the entire bridge e leastsquare method was applied to calculate of the cable force ofthe bridge during forward installation and the results werecompared with the reasonable cable force of the bridge untilall design requirements were satisfied At this point thecorresponding construction cable force could be consideredthe initial tension of the stay cable To accelerate conver-gence the initial tension of the cable was selected per thestatic equilibrium determined in the first forward installa-tion iteration Table 1 lists the initial tension of each staycable after the iterative calculation the number of stay cablesis shown in Figure 1

e stay cable at Pier 5 was adjusted once the side spanwas closed to bring the internal force at the root of the maintower closer to the deformation at the cantilever end of themain beam e C6 cable force was adjusted to 8000 kN C7cable force to 8000 kN C8 cable force to 8000 kN and C9cable force to 7500 kN at this time Before the constructionof bridge deck pavement to ensure uniform internal forceand deformation of the main girder across the whole bridgethe cable at Pier 4 was adjusted by bringing C1prime cable forceto 9000 kN C2prime cable force to 9000 kN C1 cable force to11000 kN C2 cable force to 11500 kN C3 cable force to9000 kN and C4 cable force to 9000 kN e cable force ofpart of the stay cable at Pier 5 was also adjusted by bringingC1 cable force to 10000 kN C2 cable force to 10000 kN C1primecable force to 10000 kN and C2prime cable force to 10500 kN

42 Main Beam Cross Section Shear Lag Coefficient underWorking Condition 1 e model test results and the shearlag coefficient of the upper edge of the main beam section ofWorking Condition 1 in the spatial finite element weresubjected to comparative analysis After tensioning the C4cable the full beam of the main beams G1 and G2 waspressed (tensile stress is positive compressive stress isnegative) the average measured stress of the upper edge ofthe main beams G1 and G2 was minus207MPa and minus077MPa atthis point After applying a hanging basket load on thecantilever end of the box girder the upper edge of the maingirders G1 and G2 showed a large amount of stress changese midabdominal segment bore compressive stress whilethe upper edge of section G1 of main beam bore tensilestress e upper edge of section G2 of main beam boretensile stress e average measured stress of the upper edgeof the main beams G1 and G2 was 024MPa and 049MPaAfter the tensioning force of the main girders G1 and G2 andapplication of the basket load the shear lag coefficient ateach point of the upper edge was as shown in Figure 7

After tensioning the C4 cable the main beams G5 andG6 were pressed in the full section and the average measuredstress of the upper edge of the main beams G5 and G6 wasminus189MPa and minus053MPa After applying the hangingbasket load on the cantilever end of the box girder the upperedge of the main beam G5 and G6 showed a large amount ofstress change e midabdominal segment bore compressivestress and the upper edge of section G5 and G6 of main beam

(a) (b)

(c) (d)

(e) (f )

Figure 6 Special finite element model for main construction operating conditions (a) Tension C4 (C4prime) applying a basket load (b) TensionC10 (C10prime) applying a basket load (c) Cable tension adjustment tensioning C11(C11prime) applying a hanging basket load (d) TensionC14(C14prime) applying a hanging basket load (e) Tension C15 (C15prime) applying counterweight midspan closure (f ) Grid division of mainbeam section

Advances in Materials Science and Engineering 7

bore tensile stress e average measured stress of the upperedge of the main beams G5 and G6 was 053MPa and055MPae shear lag coefficient at each point of the upperedge was as shown in Figure 8 after the tensioning force ofthe main girders G5 and G6 and the application of the basketload

As tension was applied the concentrated load was placedin the middle of the box girder to intuitively observe theshear lag effect in the middle of the box girder section ehanging basket load was applied in the form of a concen-trated load so that the upper edge of sections G1 G5 and G6of the main beam presented tensile stress though the middleweb section did not After the tensile force and the hangingbasket load were applied positive and negative shear lageffects appeared alternately on the upper edge of the mainbeam

43 Main Beam Cross Section Shear Lag Coefficient underWorking Condition 4 e model test results and the shearlag coefficient of the upper edge of the main beams G1ndashG8section under Working Condition 4 for the spatial finiteelement was also extracted for comparative analysis Asshown in Figures 9 and 10 the shear lag coefficient of theupper edge of the section G7 of themain beam changed from009ndash229 to minus293ndash210 after the tension of the cable and theapplication of the hanging basket load e shear lag

coefficient of the upper edge of the main beam G3 changedfrom 055ndash135 to 039ndash232 the shear lag coefficient of theupper edge of the main beam G4 changed from 063ndash159 to066ndash166 and the shear lag coefficient of the upper edge ofthe main beam G8 changed from 034ndash152 to 021ndash192 Inthis case after the tension was applied to the C14 cable andthe hanging basket load was applied the shear lag coefficientof the upper edge decelerated and the changes were slighte shear lag coefficient of the upper edge of the sections G2and G1 G5 and G6 of the main girder near the tower rootwas weak In each section the shear lag coefficient of theupper edge decelerated as tension cable force increased etest results are consistent with the finite element calculationresults

5 Longitudinal Shear Lag Effect in MainBeam of Single-Cable Plane P C Cable-Stayed Bridge

e stress on the main girder section of cable-stayed bridgeis highly complex e construction of the cable-stayedbridge is a dynamic process e stress in the main beamsection under different working conditions and differentbeam sections may widely differ e shear lag coefficientcalculation results along the longitudinal length of severalkey nodes on the upper edge of the cross section of the main

Table 1 Initial cable tension of each stay cable

Pier 4 Pier 5Cablenumber

Tension cable force(left) (kN)

Cablenumber

Tension cable force(right) (kN)

Cablenumber

Tension cable force(left) (kN)

Cablenumber

Tension cable force(right) (kN)

C1prime 7500 C1 9000 C1 9000 C1prime 4500C2prime 6500 C2 9500 C2 9500 C2prime 6500C3prime 5500 C3 8000 C3 8000 C3prime 5500C4prime 5500 C4 8000 C4 8000 C4prime 5000C5prime 5500 C5 8500 C5 8000 C5prime 6000C6prime 5500 C6 8000 C6 7400 C6prime 6500C7prime 5500 C7 8000 C7 7000 C7prime 5500C8prime 6000 C8 8000 C8 7000 C8prime 6000C9prime 6000 C9 7500 C9 7000 C9prime 6000C10prime 6500 C10 7500 C10 7500 C10prime 6500C11prime 6500 C11 7500 C11 7500 C11prime 6500C12prime 9500 C12 7500 C12 7500 C12prime 9000C13prime 9500 C13 7500 C13 7500 C13prime 9000C14prime 10000 C14 7500 C14 7500 C14prime 9500C15prime 9500 C15 6500 C15 6500 C15prime 9500

ndash100ndash050

000050100150

Shea

r lag

coeffi

cien

t

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(a)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

ndash200ndash100

000100200300

Shea

r lag

coeffi

cien

t

Node lateral coordinate (m)

(b)

Figure 7 Shear lag coefficient variations in G1 and G2 (Condition 1) (a) G1 section (b) G2 section

8 Advances in Materials Science and Engineering

beam can be used to understand the shear lag coefficientdistribution along the longitudinal length of the upper edgeof the main beam section during the construction stage Forconvenience the key points on the top edge of the crosssection of the main beam are marked here with capitalletters e distances from A-D points to the transversesymmetry lines are 0m minus2m minus11m and minus16m respec-tively e lateral position of each point is shown inFigure 11

Due to space limitations only the distribution of theshear lag coefficient along the longitudinal length of theupper edge of the main beam section under the followingtwo typical working conditions is described here WorkingCondition 1 is defined by tensiling the C4 (C4prime) cable forceWorking Condition 4 is defined by tensiling the C14 (C14prime)cable force e boundary conditions of each workingcondition were kept consistent

51Working Condition 1 Shear Lag Effect along LongitudinalBridge Under the action of self-weight and cable force the

shear lag coefficient of the flange of the main beam sectionalong the longitudinal bridge was observed as shown inFigure 12 In the longitudinal distribution of the shear lagcoefficient at the key point of the upper flange the midpointof the cross section of the main beam (Point A) and theboundary point between the upper edge and the middle web(Point B) change most sharply

At Pier 4 a sudden change in shear lag coefficient at theend of the cantilever occurred due to the tension of the C4(C4prime) cable A large stress concentration occurred at theseconcentrated load points which altered the upper flangestress distribution law Under the action of cable force theupper edge of the main beam section shows a more sig-nificant shear lag effect of the cable tension zone at locationscloser to the cable action point At Point A the shear lagcoefficient reaches 418 at Point B it is 307 e stressdistribution of the main beam appears to grow more uni-form as the distance between the main beam and the cableforce increases

Pier 4 C3ndashC1 (C3prime-C1prime) shows a similar distributionpattern in each beam section e shear lag coefficient of

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

ndash100ndash050

000050100150200

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

(a)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

ndash200ndash100

000100200300

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

(b)

Figure 8 Shear lag coefficient variations in G5 and G6 (Condition 1) (a) G5 section (b) G6 section

000

050

100

150

200

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16Node lateral coordinate (m)

Shea

r lag

coef

ficie

nt

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(a)

000050100150200250

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16Node lateral coordinate (m)

Shea

r lag

coef

ficie

ntTensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(b)

000050100150200250

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(c)

Shea

r lag

coef

ficie

nt

000050100150200250

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(d)

Figure 9 Contrast diagram of shear lag coefficient in G1-G4 (Condition 4) (a) G1 section (b) G2 section (c) G3 section and (d) G4 section

Advances in Materials Science and Engineering 9

Point A and Point B is prominent at the point of action of thecable forcee change range of shear lag coefficient of PointA at the point of cable force action is 222ndash271 and at Point Bis 122ndash212 e stress distribution of the beam sectionbetween the points of cable force is relatively uniform eA-point shear lag coefficient λ ranges from 115 to 116 theB-point shear lag coefficient λ from 112 to 140 and theC-point shear lag coefficient λ from 109 to 126

e stress distribution of the upper flange of the Pier 4tower root section changed due to the existence of the crossbeame change appears to be most dramatic at Point D atthe end of the upper extension arm of the main beam crosssection e shear lag coefficient λ at Point D near the towerranges from 114 to 154 marking a very significant variationcompared with other beam sections

e variation of the cantilever end the cable force pointand the C3ndashC1 (C3prime-C1prime) beam section of Pier 5 is similarto that of Pier 4 Under the action of the tension C4 (C4prime)Point A here shows a shear lag coefficient λ of 300 while thatat Point B reaches 250 In our analysis the stress distri-bution of the main beam grew more uniform as the distancebetween the main beam and the cable force increased

Pier 5 C3ndashC1(C3prime-C1prime) shows a similar distributionpattern in each beam section e shear force lag coeffi-cient of Point A and Point B is prominent at the point ofaction of the cable force e variation range of lag

coefficient λ is 157ndash244 and the shear lag coefficient λ ofPoint B is 138ndash184 at the point of action of cable forcee stress distribution of the beam section between thecable force points is relatively uniform e changes inshear lag coefficient λ of Point A range from 141 to 166that at Point B varies from 107 to 157 and that at Point Cranges from 075 to 108 Again the most significantchange was observed at Point D the shear lag coefficient λof Pier 5 at point D near the tower root varies from 120to 133

At the C1prime cable force on the right side of Pier 5 theshear lag coefficient of Points A and B is weaker than at otherpoints of cable force mainly because of the temporarysupport e shear lag effects on both sides were similar dueto the small reaction force of the temporary pier

Under the action of self-weight and cable force with thisworking condition the main characteristics of the shear lageffect of the upper edge of the main beam section of thesingle-cable PC cable-stayed bridge in the longitudinal di-rection can be summarized as follows

(1) In the construction stage under the action of cableforce and near the point of cable tension the upperedge of the girder section shows more significantshear lag effect in the cable force tension area egirder stress distribution grows more uniform as thedistance between the girder section and the cableforce action point increases

(2) In the C3ndashC1(C3prime-C1prime) beam section the upper edgeof the main beam section at the cable force actionpoint presents shear lag effects in the tension zonee stress distribution in the beam section betweenthe points of the cable force is relatively uniformwhich characterizes the shear lag distribution in theordinary curved box girder

000020040060080100120140160

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(a)

000

050

100

150

200

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(b)

ndash400ndash300ndash200ndash100

000100200300

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(c)

000

050

100

150

200

250

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(d)

Figure 10 Contrast diagram of shear lag coefficient in G1-G4 (Condition 4) (a) G5 section (b) G6 section (c) G7 section and (d) G8 section

B ACD

Figure 11 Main girder cross-section diagram at every point

10 Advances in Materials Science and Engineering

(3) e shear lag coefficient at the end of the upperextension arm of the cross section of the main beamat the root of the tower changes most dramaticallyamong the areas we observed We also observedconversion from a positive shear lag effect to anegative shear lag effect near the tower

(4) Under the influence of temporary support the shearforce lag coefficient of the action point of the C1primecable force on the right side of Pier 5 is weaker thanother cable forces e temporary pier reaction forceis small whichmakes the shear lag on both sideseeffects were still relatively similar in this case

52Working Condition 4 Shear Lag Effect along LongitudinalBridge is working condition represents the largestsingle cantilever stage e side spans at this point areclosed and the shear lag effects near their supports havemarkedly increased due to tensioning of C12primendashC14prime tailse shear lag effect of the inner beam section of the sidespan C11primendashC1prime is lower than under the previous workingcondition

As shown in Figure 13 at Pier 4 the cantilever end fallsunder the action of tensioning C14 cable force e shear lagcoefficient λ at Point A reaches 340 and at Point B is 240 Inthe beam section of Pier 4 C11ndashC1 (C11primendashC1prime) the shearlag coefficient of Points A and B is prominent at the point ofaction of the cable force Point A shows a shear lag coefficientλ of 151ndash281 at the cable force action point where Point Bshows a shear lag coefficient λ of 435ndash212 the stress dis-tribution of the beam section between the points of the cableforce is relatively uniform e shear lag coefficient λ ofPoint A varies from 105 to 144 and that of Point B from 119to 110 e shear lag coefficient λ of Point C ranges from065 to 095

At Pier 5 the end of the cantilever is under the action ofthe tensioning force of C14 e shear lag coefficient λ ofPoint A reaches 357 and that of Point B is 222 In the beamsection of C11ndashC1 (C11primendashC1prime) in Pier 5 the shear lagcoefficients of Points A and B at the point of action of thecable force are prominent Point A has a shear lag coefficientλ of 148ndash281 at the action point of the cable force wherePoint B has a shear lag coefficient λ of 133ndash212 e stressdistribution of beam section between the points of cable

4rsquo 3rsquo 2rsquo 1rsquo 4 1 2 3 4Longitudinal bridge position

Point APoint B

Point CPoint D

Pier 4

000050100150200250300350400450

Shea

r lag

coef

ficie

nt

(a)

Point APoint B

Point CPoint D

4 3 2 1 5 1rsquo 2rsquo 3rsquo 4rsquoLongitudinal bridge position

Pier 5

000

050

100

150

200

250

300

350

Shea

r lag

coef

ficie

nt

(b)

Figure 12 Distribution of shear lag coefficient along longitudinal direction (Condition 1) (a) Pier 4 (b) Pier 5

Advances in Materials Science and Engineering 11

Point APoint B

Point CPoint D

3 14rsquo13rsquo12rsquo 11rsquo 10rsquo 9rsquo 8rsquo 7rsquo 6rsquo 5rsquo 4rsquo 3rsquo 2rsquo 1rsquo 4Longitudinal bridge position

Pier 4

000

050

100

150

200

250

300

Shea

r lag

coef

ficie

nt

(a)

Point APoint B

Point CPoint D

4 1 2 3 4 5 6 7 8 9 10 11 12 13 14Longitudinal bridge position

000

050

100

150

200

250

300

350

400

Shea

r lag

coef

ficie

nt

(b)

Point APoint B

Point CPoint D

14 13 12 11 10 9 8 7 6 5 4 3 2 1 5Longitudinal bridge position

Pier 5

000

050

100

150

200

250

300

350

400

Shea

r lag

coef

ficie

nt

(c)

Figure 13 Continued

12 Advances in Materials Science and Engineering

force is relatively uniform e shear lag coefficient λ ofPoint A ranges from 120 to 155 that of Point B from 096 to135 and that of Point C from 076 to 101

Under this working condition the action of self-weightand cable force make the main characteristics of the shear lagat the upper edge of the main beam section of the single-cable PC cable-stayed bridge along the vertical direction verysimilar to those of underWorking Condition 1e shear lageffect at the main beam section is reduced as cable forceincreases

6 Distribution of Shear Lag Effect duringConstruction Stage

61 Shear Lag Coefficient of Main Beam of Single-Cable PlanePC Cable-Stayed Bridge e solution to the shear lag coef-ficient under the compressive bending load involves decom-posing the compressive bending load into a superposition ofthe bending moment action and the axial force action thencalculating the normal stress of themain beamunder the actionof the bending moment and the axial force alone [22] It isassumed that the axial force is borne by the whole section thatis the normal stress generated by the axial force is evenlydistributed along the cross section e results can then belinearly superposed to obtain the comprehensive shear lageffect coefficient Our previous analysis indicated that whenthere is cable tension of the main beam of the cable-stayedbridge in the single-cable PC there is a large stress concen-tration at the point of action of the concentrated loads estress distribution law of the roof plate changes accordingly

Assume that the axial force which is borne by the wholesection does not reflect the actual internal force of the maingirder of the single-cable PC cable-stayed bridge e dis-tribution of the longitudinal normal stress generated byinternal force is not uniform e size and distribution laware related to the internal force state namely the bendingmoment and axial force In this study the shear lag coef-ficient λΜ generated by bending moment and the shear lag

coefficient λΜ generated by axial force were respectivelycalculated to solve the shear lag coefficient λ aftersuperposition

e shear lag coefficient is calculated as follows

λ σσ

(6)

where σ is actual stress and σ( ) is nominal stresse shear lag coefficient is related to the wide span ratio

support conditions load form cross-sectional shape andsize material properties and location of the section [23ndash25]In the final analysis the shear lag is caused by unevendistribution of the longitudinal normal stress caused byinternal force its size and distribution are related to theinternal force state that is the bending moment is related tothe axial force

e actual stress σ and nominal stress σ( ) are thesuperposition of the bending moment and the normalstress generated by the axial force If the shear lag coeffi-cient λΜ produced by the bending moment is definedproperly the shear lag coefficient generated by the axialforce is λN and the shear lag coefficient after the super-position is λ us

λ σN + σM

σN + σM

λNσN + λMσM

σN + σM

(7)

where σ( )N and σ( )M are the nominal normal stressgenerated by bending moment and axial force respectivelyσ( )N and σ( )M are the actual normal stress generated bybending moment and axial force respectively e nu-merator and denominator at the right end of the aboveformula are divided by σ( )N to rewrite (7) as

λ λN + λM σMσN( 1113857

1 + σMσN( 1113857 (8)

ere are σM (MI)y and σN (NA) so(σMσN) (yAI) middot (MN) and order yAI C can beplugged into (8) to obtain

Point APoint B

Point CPoint D

000

050

100

150

200

250

300

350

5 1prime 2prime 3prime 4prime 5prime 6prime 7prime 8prime 9prime 10prime 11prime 12prime13prime14prime6

Shea

r lag

coef

ficie

nt

Longitudinal bridge position

(d)

Figure 13 Distribution of shear lag coefficient along longitudinal direction (Condition 4) (a) Left side of Pier 4 (b) Right side of Pier 4(c) Left side of Pier 5 (d) Right side of Pier 5

Advances in Materials Science and Engineering 13

λ λN + CλM(MN)

1 + C(MN) (9)

where y is the distance from the centroid of the section to theupper and lower edges A is the sectional area I is the sectionmoment of inertia and M and N represent the bendingmoment and axial force of the section

Equation (9) indicates that y A and I are the parametersinherent to the section under analysis λN and λΜ are theshear lag coefficients caused by the axial force and thebending moment respectively which are independent of themagnitude of the internal force Once λN and λΜ are de-termined the total shear lag coefficient of λ is only related tothe ratio of bending moment to axial force

Figure 14 shows the curve of the shear lag coefficientwith the bending moment axial force ratio when λΜ 200and λN 100 Taking the upper flange of the main beam ofthe Fumin cable-stayed bridge as an example C yAI 101

As shown in Figure 14 the total shear lag coefficient λgradually approaches the bending moment shear lag coef-ficient λΜ as the bending moment force ratio increases Asthe bending moment axial force ratio gradually decreases to0 the total shear lag coefficient λ is also gradually reduceduntil it is equal to the axial force shear lag coefficient λNWhen the bending moment force ratio is between 0 and 2the total shear lag coefficient λ changes substantially Alarger λΜ results in a greater change in the total shear lagcoefficient λ When the bending moment force ratio isgreater than 2 the total shear lag coefficient λ changesslowly at is to say in the internal force of this sectionwhether the bending moment or axial force accounts for alarger proportion affects how close the total shear lag co-efficient λ of the section is to the shear lag coefficient (λΜ orλN) caused by this item of internal force

For a general beam bridge the bending moment is themain internal force and so the section shear lag coefficient λis close to λΜ However for concrete cable-stayed bridgesthe main beam is in a compression state and is mainlysubjected to axial force its shear lag coefficient λ is closer toλN

62 eoretical versus Finite Element Main Beam Shear LagResults e shear lag variation law of the main beam of thesingle-cable PC cable-stayed bridge along the longitudinaldirection was discussed above as per our theoretical analysise shear lag coefficient and normal stress value of the mainbeam cross section are mainly concentrated in the range ofthe force point of the cable e effective width of the topplate of the upper flange in our analysis was set to 400m andthe full section of the bottom plate to σN 032N andσ( )N 009N (where N is the axial component of the cableforce) so that λN σNσ( )N 356 Shear lag coefficient λΜwas only considered a momentary action based on thedefinition of the effective width of the bridge e effectivewidth of the top plate in this case was 2400m so λΜ 135

We established the full-bridge plane bar modelaccording to finite element analysis e diagram of the

bending moment and axial force of the main beam underWorking Condition 1 suggests relatively small internalforces in Pier 4 and the main girder of Pier 5 e internalforce diagram of the main beam at Pier 4 was isolated forfurther analysis as shown in Figure 15

As shown in Figure 15 the bending moment force ratioMN is between λΜ and λN that is between 135 and 356e bending moment axial force ratio MN at the actionpoint of the cantilever end cable force C4 (C4prime) is 120 and118 According to formula (6) the shear lag coefficient λ ofthe upper edge of the main beam section is 235 and 236respectively

In the construction section the axial moment ratioMNof the main beam section of the C3primendashC1prime cable force actionpoint is 149 159 and 164 According to formula (6) theshear lag coefficients of the upper edge of the main beamsection of C3primendashC1prime cable force point are 223 220 and 218respectively e axial force ratio MN of the main beamsection of the C3ndashC1 cable force action point is 147 181and 203 Again according to formula (6) the shear lagcoefficients λ of the upper edge of the main beam section ofthe C3ndashC1 cable force action point are 224 213 and 207Our finite element results of the shear lag effect of the mainbeam along the longitudinal direction suggest that the shearlag coefficient at Point A of the action point of cantilever endcable force C4 (C4prime) is 418 and 361 the shear lag coefficientλ at Point B reaches 307 and 306

In the Pier 4 C3primendashC1prime beam section the shear lagcoefficients λ of Point A at the cable force action point are232 222 and 249 respectively those of Point B at the cableforce action point are 126 123 and 129 respectivelyWithin the C3ndashC1 beam section the shear lag coefficients λof Point A at the cable force action point are 257 271 and271 and those corresponding to Point B are 211 122 and212 respectively

In the beam section construction stage the analysisresults of the C3ndashC1 (C3primendashC1prime) cable force action pointtreatment theory agree with the finite element analysisresults In this position the theoretically derived shearforce lag coefficient λ can be used to reflect the actual forceat that point e finite element analysis results at the C4(C4prime) action point of the cantilever end cable force exceedthe theoretical analysis results e cantilever end cableforce action point is mainly subjected to axial force so itsshear lag coefficient λ should be closer to λN In this po-sition the shear lag factor λN can be used to reflect theactual force

005

115

225

0 1 2 3 4 5 6 7 8 9 10

Shea

r lag

coef

ficie

nt

MN

λM = 2 λN = 1

Figure 14 Relationship between shear lag coefficient and bendingmoment axial force ratio

14 Advances in Materials Science and Engineering

63 Variation Law of Shear Lag Effect of PC Cable-StayedBridge Main Beam along Longitudinal Direction e shearlag variations in the main beam of the single-cable PC cable-stayed bridge along the longitudinal direction can besummarized as follows

(1) In the construction stage at the cantilever end thecable force action point is the main recipient of axialforce e shear lag coefficient λN of the axial forceacting at this position reflects the actual force at thatpoint

(2) In the beam segments between the cable forces at thecable force action point the theoretical shear lagcoefficient λ determined by the ratio MN of thebending moment to axial force reflects the actualstress

(3) In the midspan beam section between the actionpoints of cable forces the shear lag coefficient ofbending moment λM is used to reflect the actualstress For the beam section between the action pointof cable force and the middle of the span the sectionshear lag coefficient is obtained by linearinterpolation

7 Conclusion

e structural control section stress at work in a typicalconstruction process was measured on a full cable-stayedbridge model in this study Sets of finite element and modeltest results were respectively analyzed and compared Ourconclusions can be summarized as follows

(1) In the construction stage the stress value of the cabletension area of the main beam upper edge markedlychanges after tensiling the cable force the shear lageffect is very prominent After a hanging basket loadis applied the main beams of certain sections presentalternating positive and negative shear lag

(2) e shear lag coefficient of the upper edge of themain beam in G1ndashG8 sections under WorkingConditions 1 and 4 was extracted from the spatialfinite element analysis for comparison to find thatthe experimental and theoretical results are in ac-cordance After tension of the cable force and theapplication of a hanging basket load positive andnegative shear lag effects alternate on the upper edgeof the box girder in certain areas As tension cableforce increases in each section the shear lag coef-ficient of the section upper edge decelerates

(3) In the construction stage at the cantilever end cableforce action point is mainly subjected to axial forcee shear lag coefficient λN of the axial force on thisposition reflects the actual force situation at thatpoint In the beam segments between the cableforces the shear lag coefficient λ determined by theratio of the bending moment to the axial force re-flects the actual stress at the cable force action pointIn the midspan beam section between the actionpoints of cable forces the shear lag coefficient λM ofthe bendingmoment reflects the actual stress For thebeam section between the cable force action pointand the middle of the span the section shear lagcoefficient can be obtained by linear interpolation

C2prime C3

4250kNmiddotm

17300kNmiddotm 19100kNmiddotm

4

C1prime C4

24400kNmiddotm

36800kNmiddotm

C4prime C3prime C2

10500kNmiddotm

18600kNmiddotm

C1

17700kNmiddotm

55000kNmiddotm

6820kNmiddotm

(a)

C1prime C4

13000kN7030kN

C4prime C1

5680kN

20300kN

3600kN

27000kN

C2prime C3

10900kN

C3prime C2

14900kN

4

(b)

Figure 15 Internal force of main girder on Pier 4 (Condition 1) (a) Moment diagram (b) Axial force diagram

Advances in Materials Science and Engineering 15

(4) e shear lag effect of the main beam is prominent inthe construction process e shear lag effect is mostobvious at the cable force operation point of thecantilever end As construction progresses and thecable force increases the shear lag effect of otherbeam sections decreases at each point except for thecantilever end

Data Availability

Part of the shear lag coefficient data used to support thefindings of this study are included within the article All ofthe shear lag coefficient data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors would like to thank the financial supports fromthe National Key RampD Program of China (2018YFC0809600and 2018YFC0809606) and Natural Science Foundation ofLiaoning Province (2019-MS-265)

References

[1] Z Li J G Nie and W Y Ji ldquoPositive and negative shear lagbehaviors of composite twin-girder decks with varying cross-sectionrdquo Science China Technological Sciences vol 60 no 1pp 116ndash132 2017

[2] S-J Zhou ldquoFinite beam element considering shear-lag effectin box girderrdquo Journal of Engineering Mechanics vol 136no 9 pp 1115ndash1122 2010

[3] Z Li and R K L Su ldquoAnalytical solutions for compositebeams with slip shear-lag and time-dependent effectsrdquo En-gineering Structures vol 152 pp 559ndash578 2017

[4] Q Z Luo Y M Wu Q S Li J Tang and G D Liu ldquoA finitesegment model for shear lag analysisrdquo Engineering Structuresvol 26 no 14 pp 2113ndash2124 2004

[5] P Z Lin F K Liu W Ji et al ldquoAnalysis on shear lag effect ofconcrete box beam by variational principlerdquo Journal of theChina Railway Society vol 35 no 2 pp 93ndash98 2013

[6] Y Y Zhang H Zhang W Li et al ldquoAnalysis on shear-lageffect of box girders based on different shear-lag generalizeddisplacementrdquo Journal of Railway Science and Engineeringvol 13 no 6 pp 1083ndash1090 2016

[7] S T Chang Q Zhang and S Zhang ldquoShear lag effect in singleplane cable-stayed bridgerdquo Advances in Structural Engi-neering vol 1 no 4 pp 301ndash306 1998

[8] G-F Wu and X Hong ldquoeoretical and experimental studyon shear lag effect of partially cable-stayed bridgerdquo Journal ofZhejiang University-Science A vol 6 no 8 pp 875ndash877 2005

[9] L Pengzhen Y Zijiang L Fengkui et al ldquoShear lag effect ofconcrete box girders considered varying flange-depthsrdquoJournal of Civil Architecturalamp Environmental Engineeringvol 35 no 1 pp 76ndash79 2013

[10] E Yamaguchi T Chaisomphob J Sa-Nguanmanasak andC Lertsima ldquoStress concentration and deflection of simplysupported box girder including shear lag effectrdquo StructuralEngineering and Mechanics vol 28 no 2 pp 207ndash220 2008

[11] X B Shen ldquoAnalysis of shear lag effect in the main girder ofwide low-pylon cable stayed bridgerdquo Construction Technologyvol 44 no 24 pp 76ndash79 2015

[12] L Xingmin ldquoAnalysis of influences of construction processeson shear lag effect of concrete box girdersrdquo Bridge Con-struction vol 43 no 1 pp 30ndash34 2013

[13] L Muyu W H Sun X D Sun et al ldquoStress analysis of widergirder of an extradosed cable-stayed bridge based on thelargest cantilever construction stagerdquo Journal of HuazhongUniversity of Science and Technology (Urban Science Edition)vol 27 no 2 pp 11ndash14 2010

[14] C Qianshu H Wenlong and Y Menggang ldquoAnalysis ofshear lag effect in construction stage of wide box girderextradosed cable-stayed bridge with large flangesrdquo Journal ofRailway Science and Engineering vol 15 no 12 pp 3158ndash3164 2018

[15] D R Li B M Wang and Y C Lin Structural Model Ex-periment Science Press Beijing China 1996

[16] J W Dali andW F Laili Experiment and Analysis of ResidualStress Ocean Press Beijing China 1987

[17] H Hossdort Structural Model Analysis China Architecture ampBuilding Press Beijing China 1986

[18] S K Mei Structural Experiments and Structural DesignChina Communications Press Beijing China 1980

[19] B Atmaca and S Ates ldquoConstruction stage analysis of three-dimensional cable-stayed bridgesrdquo Steel amp Composite Struc-tures vol 12 no 5 pp 413ndash426 2012

[20] S W Park M R Jung D J Min and M Y Kim ldquoCon-struction stage analysis of cable-stayed bridges using theunstrained element length methodrdquo Journal of e KoreanSociety of Civil Engineers vol 36 no 6 pp 991ndash998 2016

[21] M Gunaydin S Adanur A C Altunisik B Sevim andE Turker ldquoDetermination of structural behavior of Bosporussuspension bridge considering construction stages and dif-ferent soil conditionsrdquo Steel and Composite Structures vol 17no 4 pp 405ndash429 2014

[22] S Adanur M Gunaydin A C Altunisik and B SevimldquoConstruction stage analysis of humber suspension bridgerdquoApplied Mathematical Modelling vol 36 no 11 pp 5492ndash5505 2012

[23] J B Mander M J N Priestley and R Park ldquoeoreticalstress-strain model for confined concreterdquo Journal of Struc-tural Engineering vol 114 no 8 pp 1804ndash1826 1988

[24] Y A Khalid C L Chan B B Sahari and A M S HamoudaldquoBending behaviour of corrugated web beamsrdquo Journal ofMaterials Processing Technology vol 150 no 3 pp 242ndash2542004

[25] A Demir N Caglar H Ozturk and Y Sumer ldquoNonlinearfinite element study on the improvement of shear capacity inreinforced concrete T-Section beams by an alternative diag-onal shear reinforcementrdquo Engineering Structures vol 120pp 158ndash165 2016

16 Advances in Materials Science and Engineering

bore tensile stress e average measured stress of the upperedge of the main beams G5 and G6 was 053MPa and055MPae shear lag coefficient at each point of the upperedge was as shown in Figure 8 after the tensioning force ofthe main girders G5 and G6 and the application of the basketload

As tension was applied the concentrated load was placedin the middle of the box girder to intuitively observe theshear lag effect in the middle of the box girder section ehanging basket load was applied in the form of a concen-trated load so that the upper edge of sections G1 G5 and G6of the main beam presented tensile stress though the middleweb section did not After the tensile force and the hangingbasket load were applied positive and negative shear lageffects appeared alternately on the upper edge of the mainbeam

43 Main Beam Cross Section Shear Lag Coefficient underWorking Condition 4 e model test results and the shearlag coefficient of the upper edge of the main beams G1ndashG8section under Working Condition 4 for the spatial finiteelement was also extracted for comparative analysis Asshown in Figures 9 and 10 the shear lag coefficient of theupper edge of the section G7 of themain beam changed from009ndash229 to minus293ndash210 after the tension of the cable and theapplication of the hanging basket load e shear lag

coefficient of the upper edge of the main beam G3 changedfrom 055ndash135 to 039ndash232 the shear lag coefficient of theupper edge of the main beam G4 changed from 063ndash159 to066ndash166 and the shear lag coefficient of the upper edge ofthe main beam G8 changed from 034ndash152 to 021ndash192 Inthis case after the tension was applied to the C14 cable andthe hanging basket load was applied the shear lag coefficientof the upper edge decelerated and the changes were slighte shear lag coefficient of the upper edge of the sections G2and G1 G5 and G6 of the main girder near the tower rootwas weak In each section the shear lag coefficient of theupper edge decelerated as tension cable force increased etest results are consistent with the finite element calculationresults

5 Longitudinal Shear Lag Effect in MainBeam of Single-Cable Plane P C Cable-Stayed Bridge

e stress on the main girder section of cable-stayed bridgeis highly complex e construction of the cable-stayedbridge is a dynamic process e stress in the main beamsection under different working conditions and differentbeam sections may widely differ e shear lag coefficientcalculation results along the longitudinal length of severalkey nodes on the upper edge of the cross section of the main

Table 1 Initial cable tension of each stay cable

Pier 4 Pier 5Cablenumber

Tension cable force(left) (kN)

Cablenumber

Tension cable force(right) (kN)

Cablenumber

Tension cable force(left) (kN)

Cablenumber

Tension cable force(right) (kN)

C1prime 7500 C1 9000 C1 9000 C1prime 4500C2prime 6500 C2 9500 C2 9500 C2prime 6500C3prime 5500 C3 8000 C3 8000 C3prime 5500C4prime 5500 C4 8000 C4 8000 C4prime 5000C5prime 5500 C5 8500 C5 8000 C5prime 6000C6prime 5500 C6 8000 C6 7400 C6prime 6500C7prime 5500 C7 8000 C7 7000 C7prime 5500C8prime 6000 C8 8000 C8 7000 C8prime 6000C9prime 6000 C9 7500 C9 7000 C9prime 6000C10prime 6500 C10 7500 C10 7500 C10prime 6500C11prime 6500 C11 7500 C11 7500 C11prime 6500C12prime 9500 C12 7500 C12 7500 C12prime 9000C13prime 9500 C13 7500 C13 7500 C13prime 9000C14prime 10000 C14 7500 C14 7500 C14prime 9500C15prime 9500 C15 6500 C15 6500 C15prime 9500

ndash100ndash050

000050100150

Shea

r lag

coeffi

cien

t

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(a)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

ndash200ndash100

000100200300

Shea

r lag

coeffi

cien

t

Node lateral coordinate (m)

(b)

Figure 7 Shear lag coefficient variations in G1 and G2 (Condition 1) (a) G1 section (b) G2 section

8 Advances in Materials Science and Engineering

beam can be used to understand the shear lag coefficientdistribution along the longitudinal length of the upper edgeof the main beam section during the construction stage Forconvenience the key points on the top edge of the crosssection of the main beam are marked here with capitalletters e distances from A-D points to the transversesymmetry lines are 0m minus2m minus11m and minus16m respec-tively e lateral position of each point is shown inFigure 11

Due to space limitations only the distribution of theshear lag coefficient along the longitudinal length of theupper edge of the main beam section under the followingtwo typical working conditions is described here WorkingCondition 1 is defined by tensiling the C4 (C4prime) cable forceWorking Condition 4 is defined by tensiling the C14 (C14prime)cable force e boundary conditions of each workingcondition were kept consistent

51Working Condition 1 Shear Lag Effect along LongitudinalBridge Under the action of self-weight and cable force the

shear lag coefficient of the flange of the main beam sectionalong the longitudinal bridge was observed as shown inFigure 12 In the longitudinal distribution of the shear lagcoefficient at the key point of the upper flange the midpointof the cross section of the main beam (Point A) and theboundary point between the upper edge and the middle web(Point B) change most sharply

At Pier 4 a sudden change in shear lag coefficient at theend of the cantilever occurred due to the tension of the C4(C4prime) cable A large stress concentration occurred at theseconcentrated load points which altered the upper flangestress distribution law Under the action of cable force theupper edge of the main beam section shows a more sig-nificant shear lag effect of the cable tension zone at locationscloser to the cable action point At Point A the shear lagcoefficient reaches 418 at Point B it is 307 e stressdistribution of the main beam appears to grow more uni-form as the distance between the main beam and the cableforce increases

Pier 4 C3ndashC1 (C3prime-C1prime) shows a similar distributionpattern in each beam section e shear lag coefficient of

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

ndash100ndash050

000050100150200

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

(a)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

ndash200ndash100

000100200300

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

(b)

Figure 8 Shear lag coefficient variations in G5 and G6 (Condition 1) (a) G5 section (b) G6 section

000

050

100

150

200

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16Node lateral coordinate (m)

Shea

r lag

coef

ficie

nt

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(a)

000050100150200250

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16Node lateral coordinate (m)

Shea

r lag

coef

ficie

ntTensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(b)

000050100150200250

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(c)

Shea

r lag

coef

ficie

nt

000050100150200250

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(d)

Figure 9 Contrast diagram of shear lag coefficient in G1-G4 (Condition 4) (a) G1 section (b) G2 section (c) G3 section and (d) G4 section

Advances in Materials Science and Engineering 9

Point A and Point B is prominent at the point of action of thecable forcee change range of shear lag coefficient of PointA at the point of cable force action is 222ndash271 and at Point Bis 122ndash212 e stress distribution of the beam sectionbetween the points of cable force is relatively uniform eA-point shear lag coefficient λ ranges from 115 to 116 theB-point shear lag coefficient λ from 112 to 140 and theC-point shear lag coefficient λ from 109 to 126

e stress distribution of the upper flange of the Pier 4tower root section changed due to the existence of the crossbeame change appears to be most dramatic at Point D atthe end of the upper extension arm of the main beam crosssection e shear lag coefficient λ at Point D near the towerranges from 114 to 154 marking a very significant variationcompared with other beam sections

e variation of the cantilever end the cable force pointand the C3ndashC1 (C3prime-C1prime) beam section of Pier 5 is similarto that of Pier 4 Under the action of the tension C4 (C4prime)Point A here shows a shear lag coefficient λ of 300 while thatat Point B reaches 250 In our analysis the stress distri-bution of the main beam grew more uniform as the distancebetween the main beam and the cable force increased

Pier 5 C3ndashC1(C3prime-C1prime) shows a similar distributionpattern in each beam section e shear force lag coeffi-cient of Point A and Point B is prominent at the point ofaction of the cable force e variation range of lag

coefficient λ is 157ndash244 and the shear lag coefficient λ ofPoint B is 138ndash184 at the point of action of cable forcee stress distribution of the beam section between thecable force points is relatively uniform e changes inshear lag coefficient λ of Point A range from 141 to 166that at Point B varies from 107 to 157 and that at Point Cranges from 075 to 108 Again the most significantchange was observed at Point D the shear lag coefficient λof Pier 5 at point D near the tower root varies from 120to 133

At the C1prime cable force on the right side of Pier 5 theshear lag coefficient of Points A and B is weaker than at otherpoints of cable force mainly because of the temporarysupport e shear lag effects on both sides were similar dueto the small reaction force of the temporary pier

Under the action of self-weight and cable force with thisworking condition the main characteristics of the shear lageffect of the upper edge of the main beam section of thesingle-cable PC cable-stayed bridge in the longitudinal di-rection can be summarized as follows

(1) In the construction stage under the action of cableforce and near the point of cable tension the upperedge of the girder section shows more significantshear lag effect in the cable force tension area egirder stress distribution grows more uniform as thedistance between the girder section and the cableforce action point increases

(2) In the C3ndashC1(C3prime-C1prime) beam section the upper edgeof the main beam section at the cable force actionpoint presents shear lag effects in the tension zonee stress distribution in the beam section betweenthe points of the cable force is relatively uniformwhich characterizes the shear lag distribution in theordinary curved box girder

000020040060080100120140160

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(a)

000

050

100

150

200

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(b)

ndash400ndash300ndash200ndash100

000100200300

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(c)

000

050

100

150

200

250

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(d)

Figure 10 Contrast diagram of shear lag coefficient in G1-G4 (Condition 4) (a) G5 section (b) G6 section (c) G7 section and (d) G8 section

B ACD

Figure 11 Main girder cross-section diagram at every point

10 Advances in Materials Science and Engineering

(3) e shear lag coefficient at the end of the upperextension arm of the cross section of the main beamat the root of the tower changes most dramaticallyamong the areas we observed We also observedconversion from a positive shear lag effect to anegative shear lag effect near the tower

(4) Under the influence of temporary support the shearforce lag coefficient of the action point of the C1primecable force on the right side of Pier 5 is weaker thanother cable forces e temporary pier reaction forceis small whichmakes the shear lag on both sideseeffects were still relatively similar in this case

52Working Condition 4 Shear Lag Effect along LongitudinalBridge is working condition represents the largestsingle cantilever stage e side spans at this point areclosed and the shear lag effects near their supports havemarkedly increased due to tensioning of C12primendashC14prime tailse shear lag effect of the inner beam section of the sidespan C11primendashC1prime is lower than under the previous workingcondition

As shown in Figure 13 at Pier 4 the cantilever end fallsunder the action of tensioning C14 cable force e shear lagcoefficient λ at Point A reaches 340 and at Point B is 240 Inthe beam section of Pier 4 C11ndashC1 (C11primendashC1prime) the shearlag coefficient of Points A and B is prominent at the point ofaction of the cable force Point A shows a shear lag coefficientλ of 151ndash281 at the cable force action point where Point Bshows a shear lag coefficient λ of 435ndash212 the stress dis-tribution of the beam section between the points of the cableforce is relatively uniform e shear lag coefficient λ ofPoint A varies from 105 to 144 and that of Point B from 119to 110 e shear lag coefficient λ of Point C ranges from065 to 095

At Pier 5 the end of the cantilever is under the action ofthe tensioning force of C14 e shear lag coefficient λ ofPoint A reaches 357 and that of Point B is 222 In the beamsection of C11ndashC1 (C11primendashC1prime) in Pier 5 the shear lagcoefficients of Points A and B at the point of action of thecable force are prominent Point A has a shear lag coefficientλ of 148ndash281 at the action point of the cable force wherePoint B has a shear lag coefficient λ of 133ndash212 e stressdistribution of beam section between the points of cable

4rsquo 3rsquo 2rsquo 1rsquo 4 1 2 3 4Longitudinal bridge position

Point APoint B

Point CPoint D

Pier 4

000050100150200250300350400450

Shea

r lag

coef

ficie

nt

(a)

Point APoint B

Point CPoint D

4 3 2 1 5 1rsquo 2rsquo 3rsquo 4rsquoLongitudinal bridge position

Pier 5

000

050

100

150

200

250

300

350

Shea

r lag

coef

ficie

nt

(b)

Figure 12 Distribution of shear lag coefficient along longitudinal direction (Condition 1) (a) Pier 4 (b) Pier 5

Advances in Materials Science and Engineering 11

Point APoint B

Point CPoint D

3 14rsquo13rsquo12rsquo 11rsquo 10rsquo 9rsquo 8rsquo 7rsquo 6rsquo 5rsquo 4rsquo 3rsquo 2rsquo 1rsquo 4Longitudinal bridge position

Pier 4

000

050

100

150

200

250

300

Shea

r lag

coef

ficie

nt

(a)

Point APoint B

Point CPoint D

4 1 2 3 4 5 6 7 8 9 10 11 12 13 14Longitudinal bridge position

000

050

100

150

200

250

300

350

400

Shea

r lag

coef

ficie

nt

(b)

Point APoint B

Point CPoint D

14 13 12 11 10 9 8 7 6 5 4 3 2 1 5Longitudinal bridge position

Pier 5

000

050

100

150

200

250

300

350

400

Shea

r lag

coef

ficie

nt

(c)

Figure 13 Continued

12 Advances in Materials Science and Engineering

force is relatively uniform e shear lag coefficient λ ofPoint A ranges from 120 to 155 that of Point B from 096 to135 and that of Point C from 076 to 101

Under this working condition the action of self-weightand cable force make the main characteristics of the shear lagat the upper edge of the main beam section of the single-cable PC cable-stayed bridge along the vertical direction verysimilar to those of underWorking Condition 1e shear lageffect at the main beam section is reduced as cable forceincreases

6 Distribution of Shear Lag Effect duringConstruction Stage

61 Shear Lag Coefficient of Main Beam of Single-Cable PlanePC Cable-Stayed Bridge e solution to the shear lag coef-ficient under the compressive bending load involves decom-posing the compressive bending load into a superposition ofthe bending moment action and the axial force action thencalculating the normal stress of themain beamunder the actionof the bending moment and the axial force alone [22] It isassumed that the axial force is borne by the whole section thatis the normal stress generated by the axial force is evenlydistributed along the cross section e results can then belinearly superposed to obtain the comprehensive shear lageffect coefficient Our previous analysis indicated that whenthere is cable tension of the main beam of the cable-stayedbridge in the single-cable PC there is a large stress concen-tration at the point of action of the concentrated loads estress distribution law of the roof plate changes accordingly

Assume that the axial force which is borne by the wholesection does not reflect the actual internal force of the maingirder of the single-cable PC cable-stayed bridge e dis-tribution of the longitudinal normal stress generated byinternal force is not uniform e size and distribution laware related to the internal force state namely the bendingmoment and axial force In this study the shear lag coef-ficient λΜ generated by bending moment and the shear lag

coefficient λΜ generated by axial force were respectivelycalculated to solve the shear lag coefficient λ aftersuperposition

e shear lag coefficient is calculated as follows

λ σσ

(6)

where σ is actual stress and σ( ) is nominal stresse shear lag coefficient is related to the wide span ratio

support conditions load form cross-sectional shape andsize material properties and location of the section [23ndash25]In the final analysis the shear lag is caused by unevendistribution of the longitudinal normal stress caused byinternal force its size and distribution are related to theinternal force state that is the bending moment is related tothe axial force

e actual stress σ and nominal stress σ( ) are thesuperposition of the bending moment and the normalstress generated by the axial force If the shear lag coeffi-cient λΜ produced by the bending moment is definedproperly the shear lag coefficient generated by the axialforce is λN and the shear lag coefficient after the super-position is λ us

λ σN + σM

σN + σM

λNσN + λMσM

σN + σM

(7)

where σ( )N and σ( )M are the nominal normal stressgenerated by bending moment and axial force respectivelyσ( )N and σ( )M are the actual normal stress generated bybending moment and axial force respectively e nu-merator and denominator at the right end of the aboveformula are divided by σ( )N to rewrite (7) as

λ λN + λM σMσN( 1113857

1 + σMσN( 1113857 (8)

ere are σM (MI)y and σN (NA) so(σMσN) (yAI) middot (MN) and order yAI C can beplugged into (8) to obtain

Point APoint B

Point CPoint D

000

050

100

150

200

250

300

350

5 1prime 2prime 3prime 4prime 5prime 6prime 7prime 8prime 9prime 10prime 11prime 12prime13prime14prime6

Shea

r lag

coef

ficie

nt

Longitudinal bridge position

(d)

Figure 13 Distribution of shear lag coefficient along longitudinal direction (Condition 4) (a) Left side of Pier 4 (b) Right side of Pier 4(c) Left side of Pier 5 (d) Right side of Pier 5

Advances in Materials Science and Engineering 13

λ λN + CλM(MN)

1 + C(MN) (9)

where y is the distance from the centroid of the section to theupper and lower edges A is the sectional area I is the sectionmoment of inertia and M and N represent the bendingmoment and axial force of the section

Equation (9) indicates that y A and I are the parametersinherent to the section under analysis λN and λΜ are theshear lag coefficients caused by the axial force and thebending moment respectively which are independent of themagnitude of the internal force Once λN and λΜ are de-termined the total shear lag coefficient of λ is only related tothe ratio of bending moment to axial force

Figure 14 shows the curve of the shear lag coefficientwith the bending moment axial force ratio when λΜ 200and λN 100 Taking the upper flange of the main beam ofthe Fumin cable-stayed bridge as an example C yAI 101

As shown in Figure 14 the total shear lag coefficient λgradually approaches the bending moment shear lag coef-ficient λΜ as the bending moment force ratio increases Asthe bending moment axial force ratio gradually decreases to0 the total shear lag coefficient λ is also gradually reduceduntil it is equal to the axial force shear lag coefficient λNWhen the bending moment force ratio is between 0 and 2the total shear lag coefficient λ changes substantially Alarger λΜ results in a greater change in the total shear lagcoefficient λ When the bending moment force ratio isgreater than 2 the total shear lag coefficient λ changesslowly at is to say in the internal force of this sectionwhether the bending moment or axial force accounts for alarger proportion affects how close the total shear lag co-efficient λ of the section is to the shear lag coefficient (λΜ orλN) caused by this item of internal force

For a general beam bridge the bending moment is themain internal force and so the section shear lag coefficient λis close to λΜ However for concrete cable-stayed bridgesthe main beam is in a compression state and is mainlysubjected to axial force its shear lag coefficient λ is closer toλN

62 eoretical versus Finite Element Main Beam Shear LagResults e shear lag variation law of the main beam of thesingle-cable PC cable-stayed bridge along the longitudinaldirection was discussed above as per our theoretical analysise shear lag coefficient and normal stress value of the mainbeam cross section are mainly concentrated in the range ofthe force point of the cable e effective width of the topplate of the upper flange in our analysis was set to 400m andthe full section of the bottom plate to σN 032N andσ( )N 009N (where N is the axial component of the cableforce) so that λN σNσ( )N 356 Shear lag coefficient λΜwas only considered a momentary action based on thedefinition of the effective width of the bridge e effectivewidth of the top plate in this case was 2400m so λΜ 135

We established the full-bridge plane bar modelaccording to finite element analysis e diagram of the

bending moment and axial force of the main beam underWorking Condition 1 suggests relatively small internalforces in Pier 4 and the main girder of Pier 5 e internalforce diagram of the main beam at Pier 4 was isolated forfurther analysis as shown in Figure 15

As shown in Figure 15 the bending moment force ratioMN is between λΜ and λN that is between 135 and 356e bending moment axial force ratio MN at the actionpoint of the cantilever end cable force C4 (C4prime) is 120 and118 According to formula (6) the shear lag coefficient λ ofthe upper edge of the main beam section is 235 and 236respectively

In the construction section the axial moment ratioMNof the main beam section of the C3primendashC1prime cable force actionpoint is 149 159 and 164 According to formula (6) theshear lag coefficients of the upper edge of the main beamsection of C3primendashC1prime cable force point are 223 220 and 218respectively e axial force ratio MN of the main beamsection of the C3ndashC1 cable force action point is 147 181and 203 Again according to formula (6) the shear lagcoefficients λ of the upper edge of the main beam section ofthe C3ndashC1 cable force action point are 224 213 and 207Our finite element results of the shear lag effect of the mainbeam along the longitudinal direction suggest that the shearlag coefficient at Point A of the action point of cantilever endcable force C4 (C4prime) is 418 and 361 the shear lag coefficientλ at Point B reaches 307 and 306

In the Pier 4 C3primendashC1prime beam section the shear lagcoefficients λ of Point A at the cable force action point are232 222 and 249 respectively those of Point B at the cableforce action point are 126 123 and 129 respectivelyWithin the C3ndashC1 beam section the shear lag coefficients λof Point A at the cable force action point are 257 271 and271 and those corresponding to Point B are 211 122 and212 respectively

In the beam section construction stage the analysisresults of the C3ndashC1 (C3primendashC1prime) cable force action pointtreatment theory agree with the finite element analysisresults In this position the theoretically derived shearforce lag coefficient λ can be used to reflect the actual forceat that point e finite element analysis results at the C4(C4prime) action point of the cantilever end cable force exceedthe theoretical analysis results e cantilever end cableforce action point is mainly subjected to axial force so itsshear lag coefficient λ should be closer to λN In this po-sition the shear lag factor λN can be used to reflect theactual force

005

115

225

0 1 2 3 4 5 6 7 8 9 10

Shea

r lag

coef

ficie

nt

MN

λM = 2 λN = 1

Figure 14 Relationship between shear lag coefficient and bendingmoment axial force ratio

14 Advances in Materials Science and Engineering

63 Variation Law of Shear Lag Effect of PC Cable-StayedBridge Main Beam along Longitudinal Direction e shearlag variations in the main beam of the single-cable PC cable-stayed bridge along the longitudinal direction can besummarized as follows

(1) In the construction stage at the cantilever end thecable force action point is the main recipient of axialforce e shear lag coefficient λN of the axial forceacting at this position reflects the actual force at thatpoint

(2) In the beam segments between the cable forces at thecable force action point the theoretical shear lagcoefficient λ determined by the ratio MN of thebending moment to axial force reflects the actualstress

(3) In the midspan beam section between the actionpoints of cable forces the shear lag coefficient ofbending moment λM is used to reflect the actualstress For the beam section between the action pointof cable force and the middle of the span the sectionshear lag coefficient is obtained by linearinterpolation

7 Conclusion

e structural control section stress at work in a typicalconstruction process was measured on a full cable-stayedbridge model in this study Sets of finite element and modeltest results were respectively analyzed and compared Ourconclusions can be summarized as follows

(1) In the construction stage the stress value of the cabletension area of the main beam upper edge markedlychanges after tensiling the cable force the shear lageffect is very prominent After a hanging basket loadis applied the main beams of certain sections presentalternating positive and negative shear lag

(2) e shear lag coefficient of the upper edge of themain beam in G1ndashG8 sections under WorkingConditions 1 and 4 was extracted from the spatialfinite element analysis for comparison to find thatthe experimental and theoretical results are in ac-cordance After tension of the cable force and theapplication of a hanging basket load positive andnegative shear lag effects alternate on the upper edgeof the box girder in certain areas As tension cableforce increases in each section the shear lag coef-ficient of the section upper edge decelerates

(3) In the construction stage at the cantilever end cableforce action point is mainly subjected to axial forcee shear lag coefficient λN of the axial force on thisposition reflects the actual force situation at thatpoint In the beam segments between the cableforces the shear lag coefficient λ determined by theratio of the bending moment to the axial force re-flects the actual stress at the cable force action pointIn the midspan beam section between the actionpoints of cable forces the shear lag coefficient λM ofthe bendingmoment reflects the actual stress For thebeam section between the cable force action pointand the middle of the span the section shear lagcoefficient can be obtained by linear interpolation

C2prime C3

4250kNmiddotm

17300kNmiddotm 19100kNmiddotm

4

C1prime C4

24400kNmiddotm

36800kNmiddotm

C4prime C3prime C2

10500kNmiddotm

18600kNmiddotm

C1

17700kNmiddotm

55000kNmiddotm

6820kNmiddotm

(a)

C1prime C4

13000kN7030kN

C4prime C1

5680kN

20300kN

3600kN

27000kN

C2prime C3

10900kN

C3prime C2

14900kN

4

(b)

Figure 15 Internal force of main girder on Pier 4 (Condition 1) (a) Moment diagram (b) Axial force diagram

Advances in Materials Science and Engineering 15

(4) e shear lag effect of the main beam is prominent inthe construction process e shear lag effect is mostobvious at the cable force operation point of thecantilever end As construction progresses and thecable force increases the shear lag effect of otherbeam sections decreases at each point except for thecantilever end

Data Availability

Part of the shear lag coefficient data used to support thefindings of this study are included within the article All ofthe shear lag coefficient data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors would like to thank the financial supports fromthe National Key RampD Program of China (2018YFC0809600and 2018YFC0809606) and Natural Science Foundation ofLiaoning Province (2019-MS-265)

References

[1] Z Li J G Nie and W Y Ji ldquoPositive and negative shear lagbehaviors of composite twin-girder decks with varying cross-sectionrdquo Science China Technological Sciences vol 60 no 1pp 116ndash132 2017

[2] S-J Zhou ldquoFinite beam element considering shear-lag effectin box girderrdquo Journal of Engineering Mechanics vol 136no 9 pp 1115ndash1122 2010

[3] Z Li and R K L Su ldquoAnalytical solutions for compositebeams with slip shear-lag and time-dependent effectsrdquo En-gineering Structures vol 152 pp 559ndash578 2017

[4] Q Z Luo Y M Wu Q S Li J Tang and G D Liu ldquoA finitesegment model for shear lag analysisrdquo Engineering Structuresvol 26 no 14 pp 2113ndash2124 2004

[5] P Z Lin F K Liu W Ji et al ldquoAnalysis on shear lag effect ofconcrete box beam by variational principlerdquo Journal of theChina Railway Society vol 35 no 2 pp 93ndash98 2013

[6] Y Y Zhang H Zhang W Li et al ldquoAnalysis on shear-lageffect of box girders based on different shear-lag generalizeddisplacementrdquo Journal of Railway Science and Engineeringvol 13 no 6 pp 1083ndash1090 2016

[7] S T Chang Q Zhang and S Zhang ldquoShear lag effect in singleplane cable-stayed bridgerdquo Advances in Structural Engi-neering vol 1 no 4 pp 301ndash306 1998

[8] G-F Wu and X Hong ldquoeoretical and experimental studyon shear lag effect of partially cable-stayed bridgerdquo Journal ofZhejiang University-Science A vol 6 no 8 pp 875ndash877 2005

[9] L Pengzhen Y Zijiang L Fengkui et al ldquoShear lag effect ofconcrete box girders considered varying flange-depthsrdquoJournal of Civil Architecturalamp Environmental Engineeringvol 35 no 1 pp 76ndash79 2013

[10] E Yamaguchi T Chaisomphob J Sa-Nguanmanasak andC Lertsima ldquoStress concentration and deflection of simplysupported box girder including shear lag effectrdquo StructuralEngineering and Mechanics vol 28 no 2 pp 207ndash220 2008

[11] X B Shen ldquoAnalysis of shear lag effect in the main girder ofwide low-pylon cable stayed bridgerdquo Construction Technologyvol 44 no 24 pp 76ndash79 2015

[12] L Xingmin ldquoAnalysis of influences of construction processeson shear lag effect of concrete box girdersrdquo Bridge Con-struction vol 43 no 1 pp 30ndash34 2013

[13] L Muyu W H Sun X D Sun et al ldquoStress analysis of widergirder of an extradosed cable-stayed bridge based on thelargest cantilever construction stagerdquo Journal of HuazhongUniversity of Science and Technology (Urban Science Edition)vol 27 no 2 pp 11ndash14 2010

[14] C Qianshu H Wenlong and Y Menggang ldquoAnalysis ofshear lag effect in construction stage of wide box girderextradosed cable-stayed bridge with large flangesrdquo Journal ofRailway Science and Engineering vol 15 no 12 pp 3158ndash3164 2018

[15] D R Li B M Wang and Y C Lin Structural Model Ex-periment Science Press Beijing China 1996

[16] J W Dali andW F Laili Experiment and Analysis of ResidualStress Ocean Press Beijing China 1987

[17] H Hossdort Structural Model Analysis China Architecture ampBuilding Press Beijing China 1986

[18] S K Mei Structural Experiments and Structural DesignChina Communications Press Beijing China 1980

[19] B Atmaca and S Ates ldquoConstruction stage analysis of three-dimensional cable-stayed bridgesrdquo Steel amp Composite Struc-tures vol 12 no 5 pp 413ndash426 2012

[20] S W Park M R Jung D J Min and M Y Kim ldquoCon-struction stage analysis of cable-stayed bridges using theunstrained element length methodrdquo Journal of e KoreanSociety of Civil Engineers vol 36 no 6 pp 991ndash998 2016

[21] M Gunaydin S Adanur A C Altunisik B Sevim andE Turker ldquoDetermination of structural behavior of Bosporussuspension bridge considering construction stages and dif-ferent soil conditionsrdquo Steel and Composite Structures vol 17no 4 pp 405ndash429 2014

[22] S Adanur M Gunaydin A C Altunisik and B SevimldquoConstruction stage analysis of humber suspension bridgerdquoApplied Mathematical Modelling vol 36 no 11 pp 5492ndash5505 2012

[23] J B Mander M J N Priestley and R Park ldquoeoreticalstress-strain model for confined concreterdquo Journal of Struc-tural Engineering vol 114 no 8 pp 1804ndash1826 1988

[24] Y A Khalid C L Chan B B Sahari and A M S HamoudaldquoBending behaviour of corrugated web beamsrdquo Journal ofMaterials Processing Technology vol 150 no 3 pp 242ndash2542004

[25] A Demir N Caglar H Ozturk and Y Sumer ldquoNonlinearfinite element study on the improvement of shear capacity inreinforced concrete T-Section beams by an alternative diag-onal shear reinforcementrdquo Engineering Structures vol 120pp 158ndash165 2016

16 Advances in Materials Science and Engineering

beam can be used to understand the shear lag coefficientdistribution along the longitudinal length of the upper edgeof the main beam section during the construction stage Forconvenience the key points on the top edge of the crosssection of the main beam are marked here with capitalletters e distances from A-D points to the transversesymmetry lines are 0m minus2m minus11m and minus16m respec-tively e lateral position of each point is shown inFigure 11

Due to space limitations only the distribution of theshear lag coefficient along the longitudinal length of theupper edge of the main beam section under the followingtwo typical working conditions is described here WorkingCondition 1 is defined by tensiling the C4 (C4prime) cable forceWorking Condition 4 is defined by tensiling the C14 (C14prime)cable force e boundary conditions of each workingcondition were kept consistent

51Working Condition 1 Shear Lag Effect along LongitudinalBridge Under the action of self-weight and cable force the

shear lag coefficient of the flange of the main beam sectionalong the longitudinal bridge was observed as shown inFigure 12 In the longitudinal distribution of the shear lagcoefficient at the key point of the upper flange the midpointof the cross section of the main beam (Point A) and theboundary point between the upper edge and the middle web(Point B) change most sharply

At Pier 4 a sudden change in shear lag coefficient at theend of the cantilever occurred due to the tension of the C4(C4prime) cable A large stress concentration occurred at theseconcentrated load points which altered the upper flangestress distribution law Under the action of cable force theupper edge of the main beam section shows a more sig-nificant shear lag effect of the cable tension zone at locationscloser to the cable action point At Point A the shear lagcoefficient reaches 418 at Point B it is 307 e stressdistribution of the main beam appears to grow more uni-form as the distance between the main beam and the cableforce increases

Pier 4 C3ndashC1 (C3prime-C1prime) shows a similar distributionpattern in each beam section e shear lag coefficient of

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

ndash100ndash050

000050100150200

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

(a)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

ndash200ndash100

000100200300

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

(b)

Figure 8 Shear lag coefficient variations in G5 and G6 (Condition 1) (a) G5 section (b) G6 section

000

050

100

150

200

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16Node lateral coordinate (m)

Shea

r lag

coef

ficie

nt

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(a)

000050100150200250

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16Node lateral coordinate (m)

Shea

r lag

coef

ficie

ntTensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(b)

000050100150200250

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(c)

Shea

r lag

coef

ficie

nt

000050100150200250

ndash16 ndash14 ndash11 ndash6 ndash2 0 2 6 11 14 16Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(d)

Figure 9 Contrast diagram of shear lag coefficient in G1-G4 (Condition 4) (a) G1 section (b) G2 section (c) G3 section and (d) G4 section

Advances in Materials Science and Engineering 9

Point A and Point B is prominent at the point of action of thecable forcee change range of shear lag coefficient of PointA at the point of cable force action is 222ndash271 and at Point Bis 122ndash212 e stress distribution of the beam sectionbetween the points of cable force is relatively uniform eA-point shear lag coefficient λ ranges from 115 to 116 theB-point shear lag coefficient λ from 112 to 140 and theC-point shear lag coefficient λ from 109 to 126

e stress distribution of the upper flange of the Pier 4tower root section changed due to the existence of the crossbeame change appears to be most dramatic at Point D atthe end of the upper extension arm of the main beam crosssection e shear lag coefficient λ at Point D near the towerranges from 114 to 154 marking a very significant variationcompared with other beam sections

e variation of the cantilever end the cable force pointand the C3ndashC1 (C3prime-C1prime) beam section of Pier 5 is similarto that of Pier 4 Under the action of the tension C4 (C4prime)Point A here shows a shear lag coefficient λ of 300 while thatat Point B reaches 250 In our analysis the stress distri-bution of the main beam grew more uniform as the distancebetween the main beam and the cable force increased

Pier 5 C3ndashC1(C3prime-C1prime) shows a similar distributionpattern in each beam section e shear force lag coeffi-cient of Point A and Point B is prominent at the point ofaction of the cable force e variation range of lag

coefficient λ is 157ndash244 and the shear lag coefficient λ ofPoint B is 138ndash184 at the point of action of cable forcee stress distribution of the beam section between thecable force points is relatively uniform e changes inshear lag coefficient λ of Point A range from 141 to 166that at Point B varies from 107 to 157 and that at Point Cranges from 075 to 108 Again the most significantchange was observed at Point D the shear lag coefficient λof Pier 5 at point D near the tower root varies from 120to 133

At the C1prime cable force on the right side of Pier 5 theshear lag coefficient of Points A and B is weaker than at otherpoints of cable force mainly because of the temporarysupport e shear lag effects on both sides were similar dueto the small reaction force of the temporary pier

Under the action of self-weight and cable force with thisworking condition the main characteristics of the shear lageffect of the upper edge of the main beam section of thesingle-cable PC cable-stayed bridge in the longitudinal di-rection can be summarized as follows

(1) In the construction stage under the action of cableforce and near the point of cable tension the upperedge of the girder section shows more significantshear lag effect in the cable force tension area egirder stress distribution grows more uniform as thedistance between the girder section and the cableforce action point increases

(2) In the C3ndashC1(C3prime-C1prime) beam section the upper edgeof the main beam section at the cable force actionpoint presents shear lag effects in the tension zonee stress distribution in the beam section betweenthe points of the cable force is relatively uniformwhich characterizes the shear lag distribution in theordinary curved box girder

000020040060080100120140160

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(a)

000

050

100

150

200

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(b)

ndash400ndash300ndash200ndash100

000100200300

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(c)

000

050

100

150

200

250

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(d)

Figure 10 Contrast diagram of shear lag coefficient in G1-G4 (Condition 4) (a) G5 section (b) G6 section (c) G7 section and (d) G8 section

B ACD

Figure 11 Main girder cross-section diagram at every point

10 Advances in Materials Science and Engineering

(3) e shear lag coefficient at the end of the upperextension arm of the cross section of the main beamat the root of the tower changes most dramaticallyamong the areas we observed We also observedconversion from a positive shear lag effect to anegative shear lag effect near the tower

(4) Under the influence of temporary support the shearforce lag coefficient of the action point of the C1primecable force on the right side of Pier 5 is weaker thanother cable forces e temporary pier reaction forceis small whichmakes the shear lag on both sideseeffects were still relatively similar in this case

52Working Condition 4 Shear Lag Effect along LongitudinalBridge is working condition represents the largestsingle cantilever stage e side spans at this point areclosed and the shear lag effects near their supports havemarkedly increased due to tensioning of C12primendashC14prime tailse shear lag effect of the inner beam section of the sidespan C11primendashC1prime is lower than under the previous workingcondition

As shown in Figure 13 at Pier 4 the cantilever end fallsunder the action of tensioning C14 cable force e shear lagcoefficient λ at Point A reaches 340 and at Point B is 240 Inthe beam section of Pier 4 C11ndashC1 (C11primendashC1prime) the shearlag coefficient of Points A and B is prominent at the point ofaction of the cable force Point A shows a shear lag coefficientλ of 151ndash281 at the cable force action point where Point Bshows a shear lag coefficient λ of 435ndash212 the stress dis-tribution of the beam section between the points of the cableforce is relatively uniform e shear lag coefficient λ ofPoint A varies from 105 to 144 and that of Point B from 119to 110 e shear lag coefficient λ of Point C ranges from065 to 095

At Pier 5 the end of the cantilever is under the action ofthe tensioning force of C14 e shear lag coefficient λ ofPoint A reaches 357 and that of Point B is 222 In the beamsection of C11ndashC1 (C11primendashC1prime) in Pier 5 the shear lagcoefficients of Points A and B at the point of action of thecable force are prominent Point A has a shear lag coefficientλ of 148ndash281 at the action point of the cable force wherePoint B has a shear lag coefficient λ of 133ndash212 e stressdistribution of beam section between the points of cable

4rsquo 3rsquo 2rsquo 1rsquo 4 1 2 3 4Longitudinal bridge position

Point APoint B

Point CPoint D

Pier 4

000050100150200250300350400450

Shea

r lag

coef

ficie

nt

(a)

Point APoint B

Point CPoint D

4 3 2 1 5 1rsquo 2rsquo 3rsquo 4rsquoLongitudinal bridge position

Pier 5

000

050

100

150

200

250

300

350

Shea

r lag

coef

ficie

nt

(b)

Figure 12 Distribution of shear lag coefficient along longitudinal direction (Condition 1) (a) Pier 4 (b) Pier 5

Advances in Materials Science and Engineering 11

Point APoint B

Point CPoint D

3 14rsquo13rsquo12rsquo 11rsquo 10rsquo 9rsquo 8rsquo 7rsquo 6rsquo 5rsquo 4rsquo 3rsquo 2rsquo 1rsquo 4Longitudinal bridge position

Pier 4

000

050

100

150

200

250

300

Shea

r lag

coef

ficie

nt

(a)

Point APoint B

Point CPoint D

4 1 2 3 4 5 6 7 8 9 10 11 12 13 14Longitudinal bridge position

000

050

100

150

200

250

300

350

400

Shea

r lag

coef

ficie

nt

(b)

Point APoint B

Point CPoint D

14 13 12 11 10 9 8 7 6 5 4 3 2 1 5Longitudinal bridge position

Pier 5

000

050

100

150

200

250

300

350

400

Shea

r lag

coef

ficie

nt

(c)

Figure 13 Continued

12 Advances in Materials Science and Engineering

force is relatively uniform e shear lag coefficient λ ofPoint A ranges from 120 to 155 that of Point B from 096 to135 and that of Point C from 076 to 101

Under this working condition the action of self-weightand cable force make the main characteristics of the shear lagat the upper edge of the main beam section of the single-cable PC cable-stayed bridge along the vertical direction verysimilar to those of underWorking Condition 1e shear lageffect at the main beam section is reduced as cable forceincreases

6 Distribution of Shear Lag Effect duringConstruction Stage

61 Shear Lag Coefficient of Main Beam of Single-Cable PlanePC Cable-Stayed Bridge e solution to the shear lag coef-ficient under the compressive bending load involves decom-posing the compressive bending load into a superposition ofthe bending moment action and the axial force action thencalculating the normal stress of themain beamunder the actionof the bending moment and the axial force alone [22] It isassumed that the axial force is borne by the whole section thatis the normal stress generated by the axial force is evenlydistributed along the cross section e results can then belinearly superposed to obtain the comprehensive shear lageffect coefficient Our previous analysis indicated that whenthere is cable tension of the main beam of the cable-stayedbridge in the single-cable PC there is a large stress concen-tration at the point of action of the concentrated loads estress distribution law of the roof plate changes accordingly

Assume that the axial force which is borne by the wholesection does not reflect the actual internal force of the maingirder of the single-cable PC cable-stayed bridge e dis-tribution of the longitudinal normal stress generated byinternal force is not uniform e size and distribution laware related to the internal force state namely the bendingmoment and axial force In this study the shear lag coef-ficient λΜ generated by bending moment and the shear lag

coefficient λΜ generated by axial force were respectivelycalculated to solve the shear lag coefficient λ aftersuperposition

e shear lag coefficient is calculated as follows

λ σσ

(6)

where σ is actual stress and σ( ) is nominal stresse shear lag coefficient is related to the wide span ratio

support conditions load form cross-sectional shape andsize material properties and location of the section [23ndash25]In the final analysis the shear lag is caused by unevendistribution of the longitudinal normal stress caused byinternal force its size and distribution are related to theinternal force state that is the bending moment is related tothe axial force

e actual stress σ and nominal stress σ( ) are thesuperposition of the bending moment and the normalstress generated by the axial force If the shear lag coeffi-cient λΜ produced by the bending moment is definedproperly the shear lag coefficient generated by the axialforce is λN and the shear lag coefficient after the super-position is λ us

λ σN + σM

σN + σM

λNσN + λMσM

σN + σM

(7)

where σ( )N and σ( )M are the nominal normal stressgenerated by bending moment and axial force respectivelyσ( )N and σ( )M are the actual normal stress generated bybending moment and axial force respectively e nu-merator and denominator at the right end of the aboveformula are divided by σ( )N to rewrite (7) as

λ λN + λM σMσN( 1113857

1 + σMσN( 1113857 (8)

ere are σM (MI)y and σN (NA) so(σMσN) (yAI) middot (MN) and order yAI C can beplugged into (8) to obtain

Point APoint B

Point CPoint D

000

050

100

150

200

250

300

350

5 1prime 2prime 3prime 4prime 5prime 6prime 7prime 8prime 9prime 10prime 11prime 12prime13prime14prime6

Shea

r lag

coef

ficie

nt

Longitudinal bridge position

(d)

Figure 13 Distribution of shear lag coefficient along longitudinal direction (Condition 4) (a) Left side of Pier 4 (b) Right side of Pier 4(c) Left side of Pier 5 (d) Right side of Pier 5

Advances in Materials Science and Engineering 13

λ λN + CλM(MN)

1 + C(MN) (9)

where y is the distance from the centroid of the section to theupper and lower edges A is the sectional area I is the sectionmoment of inertia and M and N represent the bendingmoment and axial force of the section

Equation (9) indicates that y A and I are the parametersinherent to the section under analysis λN and λΜ are theshear lag coefficients caused by the axial force and thebending moment respectively which are independent of themagnitude of the internal force Once λN and λΜ are de-termined the total shear lag coefficient of λ is only related tothe ratio of bending moment to axial force

Figure 14 shows the curve of the shear lag coefficientwith the bending moment axial force ratio when λΜ 200and λN 100 Taking the upper flange of the main beam ofthe Fumin cable-stayed bridge as an example C yAI 101

As shown in Figure 14 the total shear lag coefficient λgradually approaches the bending moment shear lag coef-ficient λΜ as the bending moment force ratio increases Asthe bending moment axial force ratio gradually decreases to0 the total shear lag coefficient λ is also gradually reduceduntil it is equal to the axial force shear lag coefficient λNWhen the bending moment force ratio is between 0 and 2the total shear lag coefficient λ changes substantially Alarger λΜ results in a greater change in the total shear lagcoefficient λ When the bending moment force ratio isgreater than 2 the total shear lag coefficient λ changesslowly at is to say in the internal force of this sectionwhether the bending moment or axial force accounts for alarger proportion affects how close the total shear lag co-efficient λ of the section is to the shear lag coefficient (λΜ orλN) caused by this item of internal force

For a general beam bridge the bending moment is themain internal force and so the section shear lag coefficient λis close to λΜ However for concrete cable-stayed bridgesthe main beam is in a compression state and is mainlysubjected to axial force its shear lag coefficient λ is closer toλN

62 eoretical versus Finite Element Main Beam Shear LagResults e shear lag variation law of the main beam of thesingle-cable PC cable-stayed bridge along the longitudinaldirection was discussed above as per our theoretical analysise shear lag coefficient and normal stress value of the mainbeam cross section are mainly concentrated in the range ofthe force point of the cable e effective width of the topplate of the upper flange in our analysis was set to 400m andthe full section of the bottom plate to σN 032N andσ( )N 009N (where N is the axial component of the cableforce) so that λN σNσ( )N 356 Shear lag coefficient λΜwas only considered a momentary action based on thedefinition of the effective width of the bridge e effectivewidth of the top plate in this case was 2400m so λΜ 135

We established the full-bridge plane bar modelaccording to finite element analysis e diagram of the

bending moment and axial force of the main beam underWorking Condition 1 suggests relatively small internalforces in Pier 4 and the main girder of Pier 5 e internalforce diagram of the main beam at Pier 4 was isolated forfurther analysis as shown in Figure 15

As shown in Figure 15 the bending moment force ratioMN is between λΜ and λN that is between 135 and 356e bending moment axial force ratio MN at the actionpoint of the cantilever end cable force C4 (C4prime) is 120 and118 According to formula (6) the shear lag coefficient λ ofthe upper edge of the main beam section is 235 and 236respectively

In the construction section the axial moment ratioMNof the main beam section of the C3primendashC1prime cable force actionpoint is 149 159 and 164 According to formula (6) theshear lag coefficients of the upper edge of the main beamsection of C3primendashC1prime cable force point are 223 220 and 218respectively e axial force ratio MN of the main beamsection of the C3ndashC1 cable force action point is 147 181and 203 Again according to formula (6) the shear lagcoefficients λ of the upper edge of the main beam section ofthe C3ndashC1 cable force action point are 224 213 and 207Our finite element results of the shear lag effect of the mainbeam along the longitudinal direction suggest that the shearlag coefficient at Point A of the action point of cantilever endcable force C4 (C4prime) is 418 and 361 the shear lag coefficientλ at Point B reaches 307 and 306

In the Pier 4 C3primendashC1prime beam section the shear lagcoefficients λ of Point A at the cable force action point are232 222 and 249 respectively those of Point B at the cableforce action point are 126 123 and 129 respectivelyWithin the C3ndashC1 beam section the shear lag coefficients λof Point A at the cable force action point are 257 271 and271 and those corresponding to Point B are 211 122 and212 respectively

In the beam section construction stage the analysisresults of the C3ndashC1 (C3primendashC1prime) cable force action pointtreatment theory agree with the finite element analysisresults In this position the theoretically derived shearforce lag coefficient λ can be used to reflect the actual forceat that point e finite element analysis results at the C4(C4prime) action point of the cantilever end cable force exceedthe theoretical analysis results e cantilever end cableforce action point is mainly subjected to axial force so itsshear lag coefficient λ should be closer to λN In this po-sition the shear lag factor λN can be used to reflect theactual force

005

115

225

0 1 2 3 4 5 6 7 8 9 10

Shea

r lag

coef

ficie

nt

MN

λM = 2 λN = 1

Figure 14 Relationship between shear lag coefficient and bendingmoment axial force ratio

14 Advances in Materials Science and Engineering

63 Variation Law of Shear Lag Effect of PC Cable-StayedBridge Main Beam along Longitudinal Direction e shearlag variations in the main beam of the single-cable PC cable-stayed bridge along the longitudinal direction can besummarized as follows

(1) In the construction stage at the cantilever end thecable force action point is the main recipient of axialforce e shear lag coefficient λN of the axial forceacting at this position reflects the actual force at thatpoint

(2) In the beam segments between the cable forces at thecable force action point the theoretical shear lagcoefficient λ determined by the ratio MN of thebending moment to axial force reflects the actualstress

(3) In the midspan beam section between the actionpoints of cable forces the shear lag coefficient ofbending moment λM is used to reflect the actualstress For the beam section between the action pointof cable force and the middle of the span the sectionshear lag coefficient is obtained by linearinterpolation

7 Conclusion

e structural control section stress at work in a typicalconstruction process was measured on a full cable-stayedbridge model in this study Sets of finite element and modeltest results were respectively analyzed and compared Ourconclusions can be summarized as follows

(1) In the construction stage the stress value of the cabletension area of the main beam upper edge markedlychanges after tensiling the cable force the shear lageffect is very prominent After a hanging basket loadis applied the main beams of certain sections presentalternating positive and negative shear lag

(2) e shear lag coefficient of the upper edge of themain beam in G1ndashG8 sections under WorkingConditions 1 and 4 was extracted from the spatialfinite element analysis for comparison to find thatthe experimental and theoretical results are in ac-cordance After tension of the cable force and theapplication of a hanging basket load positive andnegative shear lag effects alternate on the upper edgeof the box girder in certain areas As tension cableforce increases in each section the shear lag coef-ficient of the section upper edge decelerates

(3) In the construction stage at the cantilever end cableforce action point is mainly subjected to axial forcee shear lag coefficient λN of the axial force on thisposition reflects the actual force situation at thatpoint In the beam segments between the cableforces the shear lag coefficient λ determined by theratio of the bending moment to the axial force re-flects the actual stress at the cable force action pointIn the midspan beam section between the actionpoints of cable forces the shear lag coefficient λM ofthe bendingmoment reflects the actual stress For thebeam section between the cable force action pointand the middle of the span the section shear lagcoefficient can be obtained by linear interpolation

C2prime C3

4250kNmiddotm

17300kNmiddotm 19100kNmiddotm

4

C1prime C4

24400kNmiddotm

36800kNmiddotm

C4prime C3prime C2

10500kNmiddotm

18600kNmiddotm

C1

17700kNmiddotm

55000kNmiddotm

6820kNmiddotm

(a)

C1prime C4

13000kN7030kN

C4prime C1

5680kN

20300kN

3600kN

27000kN

C2prime C3

10900kN

C3prime C2

14900kN

4

(b)

Figure 15 Internal force of main girder on Pier 4 (Condition 1) (a) Moment diagram (b) Axial force diagram

Advances in Materials Science and Engineering 15

(4) e shear lag effect of the main beam is prominent inthe construction process e shear lag effect is mostobvious at the cable force operation point of thecantilever end As construction progresses and thecable force increases the shear lag effect of otherbeam sections decreases at each point except for thecantilever end

Data Availability

Part of the shear lag coefficient data used to support thefindings of this study are included within the article All ofthe shear lag coefficient data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors would like to thank the financial supports fromthe National Key RampD Program of China (2018YFC0809600and 2018YFC0809606) and Natural Science Foundation ofLiaoning Province (2019-MS-265)

References

[1] Z Li J G Nie and W Y Ji ldquoPositive and negative shear lagbehaviors of composite twin-girder decks with varying cross-sectionrdquo Science China Technological Sciences vol 60 no 1pp 116ndash132 2017

[2] S-J Zhou ldquoFinite beam element considering shear-lag effectin box girderrdquo Journal of Engineering Mechanics vol 136no 9 pp 1115ndash1122 2010

[3] Z Li and R K L Su ldquoAnalytical solutions for compositebeams with slip shear-lag and time-dependent effectsrdquo En-gineering Structures vol 152 pp 559ndash578 2017

[4] Q Z Luo Y M Wu Q S Li J Tang and G D Liu ldquoA finitesegment model for shear lag analysisrdquo Engineering Structuresvol 26 no 14 pp 2113ndash2124 2004

[5] P Z Lin F K Liu W Ji et al ldquoAnalysis on shear lag effect ofconcrete box beam by variational principlerdquo Journal of theChina Railway Society vol 35 no 2 pp 93ndash98 2013

[6] Y Y Zhang H Zhang W Li et al ldquoAnalysis on shear-lageffect of box girders based on different shear-lag generalizeddisplacementrdquo Journal of Railway Science and Engineeringvol 13 no 6 pp 1083ndash1090 2016

[7] S T Chang Q Zhang and S Zhang ldquoShear lag effect in singleplane cable-stayed bridgerdquo Advances in Structural Engi-neering vol 1 no 4 pp 301ndash306 1998

[8] G-F Wu and X Hong ldquoeoretical and experimental studyon shear lag effect of partially cable-stayed bridgerdquo Journal ofZhejiang University-Science A vol 6 no 8 pp 875ndash877 2005

[9] L Pengzhen Y Zijiang L Fengkui et al ldquoShear lag effect ofconcrete box girders considered varying flange-depthsrdquoJournal of Civil Architecturalamp Environmental Engineeringvol 35 no 1 pp 76ndash79 2013

[10] E Yamaguchi T Chaisomphob J Sa-Nguanmanasak andC Lertsima ldquoStress concentration and deflection of simplysupported box girder including shear lag effectrdquo StructuralEngineering and Mechanics vol 28 no 2 pp 207ndash220 2008

[11] X B Shen ldquoAnalysis of shear lag effect in the main girder ofwide low-pylon cable stayed bridgerdquo Construction Technologyvol 44 no 24 pp 76ndash79 2015

[12] L Xingmin ldquoAnalysis of influences of construction processeson shear lag effect of concrete box girdersrdquo Bridge Con-struction vol 43 no 1 pp 30ndash34 2013

[13] L Muyu W H Sun X D Sun et al ldquoStress analysis of widergirder of an extradosed cable-stayed bridge based on thelargest cantilever construction stagerdquo Journal of HuazhongUniversity of Science and Technology (Urban Science Edition)vol 27 no 2 pp 11ndash14 2010

[14] C Qianshu H Wenlong and Y Menggang ldquoAnalysis ofshear lag effect in construction stage of wide box girderextradosed cable-stayed bridge with large flangesrdquo Journal ofRailway Science and Engineering vol 15 no 12 pp 3158ndash3164 2018

[15] D R Li B M Wang and Y C Lin Structural Model Ex-periment Science Press Beijing China 1996

[16] J W Dali andW F Laili Experiment and Analysis of ResidualStress Ocean Press Beijing China 1987

[17] H Hossdort Structural Model Analysis China Architecture ampBuilding Press Beijing China 1986

[18] S K Mei Structural Experiments and Structural DesignChina Communications Press Beijing China 1980

[19] B Atmaca and S Ates ldquoConstruction stage analysis of three-dimensional cable-stayed bridgesrdquo Steel amp Composite Struc-tures vol 12 no 5 pp 413ndash426 2012

[20] S W Park M R Jung D J Min and M Y Kim ldquoCon-struction stage analysis of cable-stayed bridges using theunstrained element length methodrdquo Journal of e KoreanSociety of Civil Engineers vol 36 no 6 pp 991ndash998 2016

[21] M Gunaydin S Adanur A C Altunisik B Sevim andE Turker ldquoDetermination of structural behavior of Bosporussuspension bridge considering construction stages and dif-ferent soil conditionsrdquo Steel and Composite Structures vol 17no 4 pp 405ndash429 2014

[22] S Adanur M Gunaydin A C Altunisik and B SevimldquoConstruction stage analysis of humber suspension bridgerdquoApplied Mathematical Modelling vol 36 no 11 pp 5492ndash5505 2012

[23] J B Mander M J N Priestley and R Park ldquoeoreticalstress-strain model for confined concreterdquo Journal of Struc-tural Engineering vol 114 no 8 pp 1804ndash1826 1988

[24] Y A Khalid C L Chan B B Sahari and A M S HamoudaldquoBending behaviour of corrugated web beamsrdquo Journal ofMaterials Processing Technology vol 150 no 3 pp 242ndash2542004

[25] A Demir N Caglar H Ozturk and Y Sumer ldquoNonlinearfinite element study on the improvement of shear capacity inreinforced concrete T-Section beams by an alternative diag-onal shear reinforcementrdquo Engineering Structures vol 120pp 158ndash165 2016

16 Advances in Materials Science and Engineering

Point A and Point B is prominent at the point of action of thecable forcee change range of shear lag coefficient of PointA at the point of cable force action is 222ndash271 and at Point Bis 122ndash212 e stress distribution of the beam sectionbetween the points of cable force is relatively uniform eA-point shear lag coefficient λ ranges from 115 to 116 theB-point shear lag coefficient λ from 112 to 140 and theC-point shear lag coefficient λ from 109 to 126

e stress distribution of the upper flange of the Pier 4tower root section changed due to the existence of the crossbeame change appears to be most dramatic at Point D atthe end of the upper extension arm of the main beam crosssection e shear lag coefficient λ at Point D near the towerranges from 114 to 154 marking a very significant variationcompared with other beam sections

e variation of the cantilever end the cable force pointand the C3ndashC1 (C3prime-C1prime) beam section of Pier 5 is similarto that of Pier 4 Under the action of the tension C4 (C4prime)Point A here shows a shear lag coefficient λ of 300 while thatat Point B reaches 250 In our analysis the stress distri-bution of the main beam grew more uniform as the distancebetween the main beam and the cable force increased

Pier 5 C3ndashC1(C3prime-C1prime) shows a similar distributionpattern in each beam section e shear force lag coeffi-cient of Point A and Point B is prominent at the point ofaction of the cable force e variation range of lag

coefficient λ is 157ndash244 and the shear lag coefficient λ ofPoint B is 138ndash184 at the point of action of cable forcee stress distribution of the beam section between thecable force points is relatively uniform e changes inshear lag coefficient λ of Point A range from 141 to 166that at Point B varies from 107 to 157 and that at Point Cranges from 075 to 108 Again the most significantchange was observed at Point D the shear lag coefficient λof Pier 5 at point D near the tower root varies from 120to 133

At the C1prime cable force on the right side of Pier 5 theshear lag coefficient of Points A and B is weaker than at otherpoints of cable force mainly because of the temporarysupport e shear lag effects on both sides were similar dueto the small reaction force of the temporary pier

Under the action of self-weight and cable force with thisworking condition the main characteristics of the shear lageffect of the upper edge of the main beam section of thesingle-cable PC cable-stayed bridge in the longitudinal di-rection can be summarized as follows

(1) In the construction stage under the action of cableforce and near the point of cable tension the upperedge of the girder section shows more significantshear lag effect in the cable force tension area egirder stress distribution grows more uniform as thedistance between the girder section and the cableforce action point increases

(2) In the C3ndashC1(C3prime-C1prime) beam section the upper edgeof the main beam section at the cable force actionpoint presents shear lag effects in the tension zonee stress distribution in the beam section betweenthe points of the cable force is relatively uniformwhich characterizes the shear lag distribution in theordinary curved box girder

000020040060080100120140160

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(a)

000

050

100

150

200

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(b)

ndash400ndash300ndash200ndash100

000100200300

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(c)

000

050

100

150

200

250

Shea

r lag

coef

ficie

nt

Node lateral coordinate (m)

Tensile forceTensile force (test)

Hanging basket loadHanging basket load (test)

(d)

Figure 10 Contrast diagram of shear lag coefficient in G1-G4 (Condition 4) (a) G5 section (b) G6 section (c) G7 section and (d) G8 section

B ACD

Figure 11 Main girder cross-section diagram at every point

10 Advances in Materials Science and Engineering

(3) e shear lag coefficient at the end of the upperextension arm of the cross section of the main beamat the root of the tower changes most dramaticallyamong the areas we observed We also observedconversion from a positive shear lag effect to anegative shear lag effect near the tower

(4) Under the influence of temporary support the shearforce lag coefficient of the action point of the C1primecable force on the right side of Pier 5 is weaker thanother cable forces e temporary pier reaction forceis small whichmakes the shear lag on both sideseeffects were still relatively similar in this case

52Working Condition 4 Shear Lag Effect along LongitudinalBridge is working condition represents the largestsingle cantilever stage e side spans at this point areclosed and the shear lag effects near their supports havemarkedly increased due to tensioning of C12primendashC14prime tailse shear lag effect of the inner beam section of the sidespan C11primendashC1prime is lower than under the previous workingcondition

As shown in Figure 13 at Pier 4 the cantilever end fallsunder the action of tensioning C14 cable force e shear lagcoefficient λ at Point A reaches 340 and at Point B is 240 Inthe beam section of Pier 4 C11ndashC1 (C11primendashC1prime) the shearlag coefficient of Points A and B is prominent at the point ofaction of the cable force Point A shows a shear lag coefficientλ of 151ndash281 at the cable force action point where Point Bshows a shear lag coefficient λ of 435ndash212 the stress dis-tribution of the beam section between the points of the cableforce is relatively uniform e shear lag coefficient λ ofPoint A varies from 105 to 144 and that of Point B from 119to 110 e shear lag coefficient λ of Point C ranges from065 to 095

At Pier 5 the end of the cantilever is under the action ofthe tensioning force of C14 e shear lag coefficient λ ofPoint A reaches 357 and that of Point B is 222 In the beamsection of C11ndashC1 (C11primendashC1prime) in Pier 5 the shear lagcoefficients of Points A and B at the point of action of thecable force are prominent Point A has a shear lag coefficientλ of 148ndash281 at the action point of the cable force wherePoint B has a shear lag coefficient λ of 133ndash212 e stressdistribution of beam section between the points of cable

4rsquo 3rsquo 2rsquo 1rsquo 4 1 2 3 4Longitudinal bridge position

Point APoint B

Point CPoint D

Pier 4

000050100150200250300350400450

Shea

r lag

coef

ficie

nt

(a)

Point APoint B

Point CPoint D

4 3 2 1 5 1rsquo 2rsquo 3rsquo 4rsquoLongitudinal bridge position

Pier 5

000

050

100

150

200

250

300

350

Shea

r lag

coef

ficie

nt

(b)

Figure 12 Distribution of shear lag coefficient along longitudinal direction (Condition 1) (a) Pier 4 (b) Pier 5

Advances in Materials Science and Engineering 11

Point APoint B

Point CPoint D

3 14rsquo13rsquo12rsquo 11rsquo 10rsquo 9rsquo 8rsquo 7rsquo 6rsquo 5rsquo 4rsquo 3rsquo 2rsquo 1rsquo 4Longitudinal bridge position

Pier 4

000

050

100

150

200

250

300

Shea

r lag

coef

ficie

nt

(a)

Point APoint B

Point CPoint D

4 1 2 3 4 5 6 7 8 9 10 11 12 13 14Longitudinal bridge position

000

050

100

150

200

250

300

350

400

Shea

r lag

coef

ficie

nt

(b)

Point APoint B

Point CPoint D

14 13 12 11 10 9 8 7 6 5 4 3 2 1 5Longitudinal bridge position

Pier 5

000

050

100

150

200

250

300

350

400

Shea

r lag

coef

ficie

nt

(c)

Figure 13 Continued

12 Advances in Materials Science and Engineering

force is relatively uniform e shear lag coefficient λ ofPoint A ranges from 120 to 155 that of Point B from 096 to135 and that of Point C from 076 to 101

Under this working condition the action of self-weightand cable force make the main characteristics of the shear lagat the upper edge of the main beam section of the single-cable PC cable-stayed bridge along the vertical direction verysimilar to those of underWorking Condition 1e shear lageffect at the main beam section is reduced as cable forceincreases

6 Distribution of Shear Lag Effect duringConstruction Stage

61 Shear Lag Coefficient of Main Beam of Single-Cable PlanePC Cable-Stayed Bridge e solution to the shear lag coef-ficient under the compressive bending load involves decom-posing the compressive bending load into a superposition ofthe bending moment action and the axial force action thencalculating the normal stress of themain beamunder the actionof the bending moment and the axial force alone [22] It isassumed that the axial force is borne by the whole section thatis the normal stress generated by the axial force is evenlydistributed along the cross section e results can then belinearly superposed to obtain the comprehensive shear lageffect coefficient Our previous analysis indicated that whenthere is cable tension of the main beam of the cable-stayedbridge in the single-cable PC there is a large stress concen-tration at the point of action of the concentrated loads estress distribution law of the roof plate changes accordingly

Assume that the axial force which is borne by the wholesection does not reflect the actual internal force of the maingirder of the single-cable PC cable-stayed bridge e dis-tribution of the longitudinal normal stress generated byinternal force is not uniform e size and distribution laware related to the internal force state namely the bendingmoment and axial force In this study the shear lag coef-ficient λΜ generated by bending moment and the shear lag

coefficient λΜ generated by axial force were respectivelycalculated to solve the shear lag coefficient λ aftersuperposition

e shear lag coefficient is calculated as follows

λ σσ

(6)

where σ is actual stress and σ( ) is nominal stresse shear lag coefficient is related to the wide span ratio

support conditions load form cross-sectional shape andsize material properties and location of the section [23ndash25]In the final analysis the shear lag is caused by unevendistribution of the longitudinal normal stress caused byinternal force its size and distribution are related to theinternal force state that is the bending moment is related tothe axial force

e actual stress σ and nominal stress σ( ) are thesuperposition of the bending moment and the normalstress generated by the axial force If the shear lag coeffi-cient λΜ produced by the bending moment is definedproperly the shear lag coefficient generated by the axialforce is λN and the shear lag coefficient after the super-position is λ us

λ σN + σM

σN + σM

λNσN + λMσM

σN + σM

(7)

where σ( )N and σ( )M are the nominal normal stressgenerated by bending moment and axial force respectivelyσ( )N and σ( )M are the actual normal stress generated bybending moment and axial force respectively e nu-merator and denominator at the right end of the aboveformula are divided by σ( )N to rewrite (7) as

λ λN + λM σMσN( 1113857

1 + σMσN( 1113857 (8)

ere are σM (MI)y and σN (NA) so(σMσN) (yAI) middot (MN) and order yAI C can beplugged into (8) to obtain

Point APoint B

Point CPoint D

000

050

100

150

200

250

300

350

5 1prime 2prime 3prime 4prime 5prime 6prime 7prime 8prime 9prime 10prime 11prime 12prime13prime14prime6

Shea

r lag

coef

ficie

nt

Longitudinal bridge position

(d)

Figure 13 Distribution of shear lag coefficient along longitudinal direction (Condition 4) (a) Left side of Pier 4 (b) Right side of Pier 4(c) Left side of Pier 5 (d) Right side of Pier 5

Advances in Materials Science and Engineering 13

λ λN + CλM(MN)

1 + C(MN) (9)

where y is the distance from the centroid of the section to theupper and lower edges A is the sectional area I is the sectionmoment of inertia and M and N represent the bendingmoment and axial force of the section

Equation (9) indicates that y A and I are the parametersinherent to the section under analysis λN and λΜ are theshear lag coefficients caused by the axial force and thebending moment respectively which are independent of themagnitude of the internal force Once λN and λΜ are de-termined the total shear lag coefficient of λ is only related tothe ratio of bending moment to axial force

Figure 14 shows the curve of the shear lag coefficientwith the bending moment axial force ratio when λΜ 200and λN 100 Taking the upper flange of the main beam ofthe Fumin cable-stayed bridge as an example C yAI 101

As shown in Figure 14 the total shear lag coefficient λgradually approaches the bending moment shear lag coef-ficient λΜ as the bending moment force ratio increases Asthe bending moment axial force ratio gradually decreases to0 the total shear lag coefficient λ is also gradually reduceduntil it is equal to the axial force shear lag coefficient λNWhen the bending moment force ratio is between 0 and 2the total shear lag coefficient λ changes substantially Alarger λΜ results in a greater change in the total shear lagcoefficient λ When the bending moment force ratio isgreater than 2 the total shear lag coefficient λ changesslowly at is to say in the internal force of this sectionwhether the bending moment or axial force accounts for alarger proportion affects how close the total shear lag co-efficient λ of the section is to the shear lag coefficient (λΜ orλN) caused by this item of internal force

For a general beam bridge the bending moment is themain internal force and so the section shear lag coefficient λis close to λΜ However for concrete cable-stayed bridgesthe main beam is in a compression state and is mainlysubjected to axial force its shear lag coefficient λ is closer toλN

62 eoretical versus Finite Element Main Beam Shear LagResults e shear lag variation law of the main beam of thesingle-cable PC cable-stayed bridge along the longitudinaldirection was discussed above as per our theoretical analysise shear lag coefficient and normal stress value of the mainbeam cross section are mainly concentrated in the range ofthe force point of the cable e effective width of the topplate of the upper flange in our analysis was set to 400m andthe full section of the bottom plate to σN 032N andσ( )N 009N (where N is the axial component of the cableforce) so that λN σNσ( )N 356 Shear lag coefficient λΜwas only considered a momentary action based on thedefinition of the effective width of the bridge e effectivewidth of the top plate in this case was 2400m so λΜ 135

We established the full-bridge plane bar modelaccording to finite element analysis e diagram of the

bending moment and axial force of the main beam underWorking Condition 1 suggests relatively small internalforces in Pier 4 and the main girder of Pier 5 e internalforce diagram of the main beam at Pier 4 was isolated forfurther analysis as shown in Figure 15

As shown in Figure 15 the bending moment force ratioMN is between λΜ and λN that is between 135 and 356e bending moment axial force ratio MN at the actionpoint of the cantilever end cable force C4 (C4prime) is 120 and118 According to formula (6) the shear lag coefficient λ ofthe upper edge of the main beam section is 235 and 236respectively

In the construction section the axial moment ratioMNof the main beam section of the C3primendashC1prime cable force actionpoint is 149 159 and 164 According to formula (6) theshear lag coefficients of the upper edge of the main beamsection of C3primendashC1prime cable force point are 223 220 and 218respectively e axial force ratio MN of the main beamsection of the C3ndashC1 cable force action point is 147 181and 203 Again according to formula (6) the shear lagcoefficients λ of the upper edge of the main beam section ofthe C3ndashC1 cable force action point are 224 213 and 207Our finite element results of the shear lag effect of the mainbeam along the longitudinal direction suggest that the shearlag coefficient at Point A of the action point of cantilever endcable force C4 (C4prime) is 418 and 361 the shear lag coefficientλ at Point B reaches 307 and 306

In the Pier 4 C3primendashC1prime beam section the shear lagcoefficients λ of Point A at the cable force action point are232 222 and 249 respectively those of Point B at the cableforce action point are 126 123 and 129 respectivelyWithin the C3ndashC1 beam section the shear lag coefficients λof Point A at the cable force action point are 257 271 and271 and those corresponding to Point B are 211 122 and212 respectively

In the beam section construction stage the analysisresults of the C3ndashC1 (C3primendashC1prime) cable force action pointtreatment theory agree with the finite element analysisresults In this position the theoretically derived shearforce lag coefficient λ can be used to reflect the actual forceat that point e finite element analysis results at the C4(C4prime) action point of the cantilever end cable force exceedthe theoretical analysis results e cantilever end cableforce action point is mainly subjected to axial force so itsshear lag coefficient λ should be closer to λN In this po-sition the shear lag factor λN can be used to reflect theactual force

005

115

225

0 1 2 3 4 5 6 7 8 9 10

Shea

r lag

coef

ficie

nt

MN

λM = 2 λN = 1

Figure 14 Relationship between shear lag coefficient and bendingmoment axial force ratio

14 Advances in Materials Science and Engineering

63 Variation Law of Shear Lag Effect of PC Cable-StayedBridge Main Beam along Longitudinal Direction e shearlag variations in the main beam of the single-cable PC cable-stayed bridge along the longitudinal direction can besummarized as follows

(1) In the construction stage at the cantilever end thecable force action point is the main recipient of axialforce e shear lag coefficient λN of the axial forceacting at this position reflects the actual force at thatpoint

(2) In the beam segments between the cable forces at thecable force action point the theoretical shear lagcoefficient λ determined by the ratio MN of thebending moment to axial force reflects the actualstress

(3) In the midspan beam section between the actionpoints of cable forces the shear lag coefficient ofbending moment λM is used to reflect the actualstress For the beam section between the action pointof cable force and the middle of the span the sectionshear lag coefficient is obtained by linearinterpolation

7 Conclusion

e structural control section stress at work in a typicalconstruction process was measured on a full cable-stayedbridge model in this study Sets of finite element and modeltest results were respectively analyzed and compared Ourconclusions can be summarized as follows

(1) In the construction stage the stress value of the cabletension area of the main beam upper edge markedlychanges after tensiling the cable force the shear lageffect is very prominent After a hanging basket loadis applied the main beams of certain sections presentalternating positive and negative shear lag

(2) e shear lag coefficient of the upper edge of themain beam in G1ndashG8 sections under WorkingConditions 1 and 4 was extracted from the spatialfinite element analysis for comparison to find thatthe experimental and theoretical results are in ac-cordance After tension of the cable force and theapplication of a hanging basket load positive andnegative shear lag effects alternate on the upper edgeof the box girder in certain areas As tension cableforce increases in each section the shear lag coef-ficient of the section upper edge decelerates

(3) In the construction stage at the cantilever end cableforce action point is mainly subjected to axial forcee shear lag coefficient λN of the axial force on thisposition reflects the actual force situation at thatpoint In the beam segments between the cableforces the shear lag coefficient λ determined by theratio of the bending moment to the axial force re-flects the actual stress at the cable force action pointIn the midspan beam section between the actionpoints of cable forces the shear lag coefficient λM ofthe bendingmoment reflects the actual stress For thebeam section between the cable force action pointand the middle of the span the section shear lagcoefficient can be obtained by linear interpolation

C2prime C3

4250kNmiddotm

17300kNmiddotm 19100kNmiddotm

4

C1prime C4

24400kNmiddotm

36800kNmiddotm

C4prime C3prime C2

10500kNmiddotm

18600kNmiddotm

C1

17700kNmiddotm

55000kNmiddotm

6820kNmiddotm

(a)

C1prime C4

13000kN7030kN

C4prime C1

5680kN

20300kN

3600kN

27000kN

C2prime C3

10900kN

C3prime C2

14900kN

4

(b)

Figure 15 Internal force of main girder on Pier 4 (Condition 1) (a) Moment diagram (b) Axial force diagram

Advances in Materials Science and Engineering 15

(4) e shear lag effect of the main beam is prominent inthe construction process e shear lag effect is mostobvious at the cable force operation point of thecantilever end As construction progresses and thecable force increases the shear lag effect of otherbeam sections decreases at each point except for thecantilever end

Data Availability

Part of the shear lag coefficient data used to support thefindings of this study are included within the article All ofthe shear lag coefficient data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors would like to thank the financial supports fromthe National Key RampD Program of China (2018YFC0809600and 2018YFC0809606) and Natural Science Foundation ofLiaoning Province (2019-MS-265)

References

[1] Z Li J G Nie and W Y Ji ldquoPositive and negative shear lagbehaviors of composite twin-girder decks with varying cross-sectionrdquo Science China Technological Sciences vol 60 no 1pp 116ndash132 2017

[2] S-J Zhou ldquoFinite beam element considering shear-lag effectin box girderrdquo Journal of Engineering Mechanics vol 136no 9 pp 1115ndash1122 2010

[3] Z Li and R K L Su ldquoAnalytical solutions for compositebeams with slip shear-lag and time-dependent effectsrdquo En-gineering Structures vol 152 pp 559ndash578 2017

[4] Q Z Luo Y M Wu Q S Li J Tang and G D Liu ldquoA finitesegment model for shear lag analysisrdquo Engineering Structuresvol 26 no 14 pp 2113ndash2124 2004

[5] P Z Lin F K Liu W Ji et al ldquoAnalysis on shear lag effect ofconcrete box beam by variational principlerdquo Journal of theChina Railway Society vol 35 no 2 pp 93ndash98 2013

[6] Y Y Zhang H Zhang W Li et al ldquoAnalysis on shear-lageffect of box girders based on different shear-lag generalizeddisplacementrdquo Journal of Railway Science and Engineeringvol 13 no 6 pp 1083ndash1090 2016

[7] S T Chang Q Zhang and S Zhang ldquoShear lag effect in singleplane cable-stayed bridgerdquo Advances in Structural Engi-neering vol 1 no 4 pp 301ndash306 1998

[8] G-F Wu and X Hong ldquoeoretical and experimental studyon shear lag effect of partially cable-stayed bridgerdquo Journal ofZhejiang University-Science A vol 6 no 8 pp 875ndash877 2005

[9] L Pengzhen Y Zijiang L Fengkui et al ldquoShear lag effect ofconcrete box girders considered varying flange-depthsrdquoJournal of Civil Architecturalamp Environmental Engineeringvol 35 no 1 pp 76ndash79 2013

[10] E Yamaguchi T Chaisomphob J Sa-Nguanmanasak andC Lertsima ldquoStress concentration and deflection of simplysupported box girder including shear lag effectrdquo StructuralEngineering and Mechanics vol 28 no 2 pp 207ndash220 2008

[11] X B Shen ldquoAnalysis of shear lag effect in the main girder ofwide low-pylon cable stayed bridgerdquo Construction Technologyvol 44 no 24 pp 76ndash79 2015

[12] L Xingmin ldquoAnalysis of influences of construction processeson shear lag effect of concrete box girdersrdquo Bridge Con-struction vol 43 no 1 pp 30ndash34 2013

[13] L Muyu W H Sun X D Sun et al ldquoStress analysis of widergirder of an extradosed cable-stayed bridge based on thelargest cantilever construction stagerdquo Journal of HuazhongUniversity of Science and Technology (Urban Science Edition)vol 27 no 2 pp 11ndash14 2010

[14] C Qianshu H Wenlong and Y Menggang ldquoAnalysis ofshear lag effect in construction stage of wide box girderextradosed cable-stayed bridge with large flangesrdquo Journal ofRailway Science and Engineering vol 15 no 12 pp 3158ndash3164 2018

[15] D R Li B M Wang and Y C Lin Structural Model Ex-periment Science Press Beijing China 1996

[16] J W Dali andW F Laili Experiment and Analysis of ResidualStress Ocean Press Beijing China 1987

[17] H Hossdort Structural Model Analysis China Architecture ampBuilding Press Beijing China 1986

[18] S K Mei Structural Experiments and Structural DesignChina Communications Press Beijing China 1980

[19] B Atmaca and S Ates ldquoConstruction stage analysis of three-dimensional cable-stayed bridgesrdquo Steel amp Composite Struc-tures vol 12 no 5 pp 413ndash426 2012

[20] S W Park M R Jung D J Min and M Y Kim ldquoCon-struction stage analysis of cable-stayed bridges using theunstrained element length methodrdquo Journal of e KoreanSociety of Civil Engineers vol 36 no 6 pp 991ndash998 2016

[21] M Gunaydin S Adanur A C Altunisik B Sevim andE Turker ldquoDetermination of structural behavior of Bosporussuspension bridge considering construction stages and dif-ferent soil conditionsrdquo Steel and Composite Structures vol 17no 4 pp 405ndash429 2014

[22] S Adanur M Gunaydin A C Altunisik and B SevimldquoConstruction stage analysis of humber suspension bridgerdquoApplied Mathematical Modelling vol 36 no 11 pp 5492ndash5505 2012

[23] J B Mander M J N Priestley and R Park ldquoeoreticalstress-strain model for confined concreterdquo Journal of Struc-tural Engineering vol 114 no 8 pp 1804ndash1826 1988

[24] Y A Khalid C L Chan B B Sahari and A M S HamoudaldquoBending behaviour of corrugated web beamsrdquo Journal ofMaterials Processing Technology vol 150 no 3 pp 242ndash2542004

[25] A Demir N Caglar H Ozturk and Y Sumer ldquoNonlinearfinite element study on the improvement of shear capacity inreinforced concrete T-Section beams by an alternative diag-onal shear reinforcementrdquo Engineering Structures vol 120pp 158ndash165 2016

16 Advances in Materials Science and Engineering

(3) e shear lag coefficient at the end of the upperextension arm of the cross section of the main beamat the root of the tower changes most dramaticallyamong the areas we observed We also observedconversion from a positive shear lag effect to anegative shear lag effect near the tower

(4) Under the influence of temporary support the shearforce lag coefficient of the action point of the C1primecable force on the right side of Pier 5 is weaker thanother cable forces e temporary pier reaction forceis small whichmakes the shear lag on both sideseeffects were still relatively similar in this case

52Working Condition 4 Shear Lag Effect along LongitudinalBridge is working condition represents the largestsingle cantilever stage e side spans at this point areclosed and the shear lag effects near their supports havemarkedly increased due to tensioning of C12primendashC14prime tailse shear lag effect of the inner beam section of the sidespan C11primendashC1prime is lower than under the previous workingcondition

As shown in Figure 13 at Pier 4 the cantilever end fallsunder the action of tensioning C14 cable force e shear lagcoefficient λ at Point A reaches 340 and at Point B is 240 Inthe beam section of Pier 4 C11ndashC1 (C11primendashC1prime) the shearlag coefficient of Points A and B is prominent at the point ofaction of the cable force Point A shows a shear lag coefficientλ of 151ndash281 at the cable force action point where Point Bshows a shear lag coefficient λ of 435ndash212 the stress dis-tribution of the beam section between the points of the cableforce is relatively uniform e shear lag coefficient λ ofPoint A varies from 105 to 144 and that of Point B from 119to 110 e shear lag coefficient λ of Point C ranges from065 to 095

At Pier 5 the end of the cantilever is under the action ofthe tensioning force of C14 e shear lag coefficient λ ofPoint A reaches 357 and that of Point B is 222 In the beamsection of C11ndashC1 (C11primendashC1prime) in Pier 5 the shear lagcoefficients of Points A and B at the point of action of thecable force are prominent Point A has a shear lag coefficientλ of 148ndash281 at the action point of the cable force wherePoint B has a shear lag coefficient λ of 133ndash212 e stressdistribution of beam section between the points of cable

4rsquo 3rsquo 2rsquo 1rsquo 4 1 2 3 4Longitudinal bridge position

Point APoint B

Point CPoint D

Pier 4

000050100150200250300350400450

Shea

r lag

coef

ficie

nt

(a)

Point APoint B

Point CPoint D

4 3 2 1 5 1rsquo 2rsquo 3rsquo 4rsquoLongitudinal bridge position

Pier 5

000

050

100

150

200

250

300

350

Shea

r lag

coef

ficie

nt

(b)

Figure 12 Distribution of shear lag coefficient along longitudinal direction (Condition 1) (a) Pier 4 (b) Pier 5

Advances in Materials Science and Engineering 11

Point APoint B

Point CPoint D

3 14rsquo13rsquo12rsquo 11rsquo 10rsquo 9rsquo 8rsquo 7rsquo 6rsquo 5rsquo 4rsquo 3rsquo 2rsquo 1rsquo 4Longitudinal bridge position

Pier 4

000

050

100

150

200

250

300

Shea

r lag

coef

ficie

nt

(a)

Point APoint B

Point CPoint D

4 1 2 3 4 5 6 7 8 9 10 11 12 13 14Longitudinal bridge position

000

050

100

150

200

250

300

350

400

Shea

r lag

coef

ficie

nt

(b)

Point APoint B

Point CPoint D

14 13 12 11 10 9 8 7 6 5 4 3 2 1 5Longitudinal bridge position

Pier 5

000

050

100

150

200

250

300

350

400

Shea

r lag

coef

ficie

nt

(c)

Figure 13 Continued

12 Advances in Materials Science and Engineering

force is relatively uniform e shear lag coefficient λ ofPoint A ranges from 120 to 155 that of Point B from 096 to135 and that of Point C from 076 to 101

Under this working condition the action of self-weightand cable force make the main characteristics of the shear lagat the upper edge of the main beam section of the single-cable PC cable-stayed bridge along the vertical direction verysimilar to those of underWorking Condition 1e shear lageffect at the main beam section is reduced as cable forceincreases

6 Distribution of Shear Lag Effect duringConstruction Stage

61 Shear Lag Coefficient of Main Beam of Single-Cable PlanePC Cable-Stayed Bridge e solution to the shear lag coef-ficient under the compressive bending load involves decom-posing the compressive bending load into a superposition ofthe bending moment action and the axial force action thencalculating the normal stress of themain beamunder the actionof the bending moment and the axial force alone [22] It isassumed that the axial force is borne by the whole section thatis the normal stress generated by the axial force is evenlydistributed along the cross section e results can then belinearly superposed to obtain the comprehensive shear lageffect coefficient Our previous analysis indicated that whenthere is cable tension of the main beam of the cable-stayedbridge in the single-cable PC there is a large stress concen-tration at the point of action of the concentrated loads estress distribution law of the roof plate changes accordingly

Assume that the axial force which is borne by the wholesection does not reflect the actual internal force of the maingirder of the single-cable PC cable-stayed bridge e dis-tribution of the longitudinal normal stress generated byinternal force is not uniform e size and distribution laware related to the internal force state namely the bendingmoment and axial force In this study the shear lag coef-ficient λΜ generated by bending moment and the shear lag

coefficient λΜ generated by axial force were respectivelycalculated to solve the shear lag coefficient λ aftersuperposition

e shear lag coefficient is calculated as follows

λ σσ

(6)

where σ is actual stress and σ( ) is nominal stresse shear lag coefficient is related to the wide span ratio

support conditions load form cross-sectional shape andsize material properties and location of the section [23ndash25]In the final analysis the shear lag is caused by unevendistribution of the longitudinal normal stress caused byinternal force its size and distribution are related to theinternal force state that is the bending moment is related tothe axial force

e actual stress σ and nominal stress σ( ) are thesuperposition of the bending moment and the normalstress generated by the axial force If the shear lag coeffi-cient λΜ produced by the bending moment is definedproperly the shear lag coefficient generated by the axialforce is λN and the shear lag coefficient after the super-position is λ us

λ σN + σM

σN + σM

λNσN + λMσM

σN + σM

(7)

where σ( )N and σ( )M are the nominal normal stressgenerated by bending moment and axial force respectivelyσ( )N and σ( )M are the actual normal stress generated bybending moment and axial force respectively e nu-merator and denominator at the right end of the aboveformula are divided by σ( )N to rewrite (7) as

λ λN + λM σMσN( 1113857

1 + σMσN( 1113857 (8)

ere are σM (MI)y and σN (NA) so(σMσN) (yAI) middot (MN) and order yAI C can beplugged into (8) to obtain

Point APoint B

Point CPoint D

000

050

100

150

200

250

300

350

5 1prime 2prime 3prime 4prime 5prime 6prime 7prime 8prime 9prime 10prime 11prime 12prime13prime14prime6

Shea

r lag

coef

ficie

nt

Longitudinal bridge position

(d)

Figure 13 Distribution of shear lag coefficient along longitudinal direction (Condition 4) (a) Left side of Pier 4 (b) Right side of Pier 4(c) Left side of Pier 5 (d) Right side of Pier 5

Advances in Materials Science and Engineering 13

λ λN + CλM(MN)

1 + C(MN) (9)

where y is the distance from the centroid of the section to theupper and lower edges A is the sectional area I is the sectionmoment of inertia and M and N represent the bendingmoment and axial force of the section

Equation (9) indicates that y A and I are the parametersinherent to the section under analysis λN and λΜ are theshear lag coefficients caused by the axial force and thebending moment respectively which are independent of themagnitude of the internal force Once λN and λΜ are de-termined the total shear lag coefficient of λ is only related tothe ratio of bending moment to axial force

Figure 14 shows the curve of the shear lag coefficientwith the bending moment axial force ratio when λΜ 200and λN 100 Taking the upper flange of the main beam ofthe Fumin cable-stayed bridge as an example C yAI 101

As shown in Figure 14 the total shear lag coefficient λgradually approaches the bending moment shear lag coef-ficient λΜ as the bending moment force ratio increases Asthe bending moment axial force ratio gradually decreases to0 the total shear lag coefficient λ is also gradually reduceduntil it is equal to the axial force shear lag coefficient λNWhen the bending moment force ratio is between 0 and 2the total shear lag coefficient λ changes substantially Alarger λΜ results in a greater change in the total shear lagcoefficient λ When the bending moment force ratio isgreater than 2 the total shear lag coefficient λ changesslowly at is to say in the internal force of this sectionwhether the bending moment or axial force accounts for alarger proportion affects how close the total shear lag co-efficient λ of the section is to the shear lag coefficient (λΜ orλN) caused by this item of internal force

For a general beam bridge the bending moment is themain internal force and so the section shear lag coefficient λis close to λΜ However for concrete cable-stayed bridgesthe main beam is in a compression state and is mainlysubjected to axial force its shear lag coefficient λ is closer toλN

62 eoretical versus Finite Element Main Beam Shear LagResults e shear lag variation law of the main beam of thesingle-cable PC cable-stayed bridge along the longitudinaldirection was discussed above as per our theoretical analysise shear lag coefficient and normal stress value of the mainbeam cross section are mainly concentrated in the range ofthe force point of the cable e effective width of the topplate of the upper flange in our analysis was set to 400m andthe full section of the bottom plate to σN 032N andσ( )N 009N (where N is the axial component of the cableforce) so that λN σNσ( )N 356 Shear lag coefficient λΜwas only considered a momentary action based on thedefinition of the effective width of the bridge e effectivewidth of the top plate in this case was 2400m so λΜ 135

We established the full-bridge plane bar modelaccording to finite element analysis e diagram of the

bending moment and axial force of the main beam underWorking Condition 1 suggests relatively small internalforces in Pier 4 and the main girder of Pier 5 e internalforce diagram of the main beam at Pier 4 was isolated forfurther analysis as shown in Figure 15

As shown in Figure 15 the bending moment force ratioMN is between λΜ and λN that is between 135 and 356e bending moment axial force ratio MN at the actionpoint of the cantilever end cable force C4 (C4prime) is 120 and118 According to formula (6) the shear lag coefficient λ ofthe upper edge of the main beam section is 235 and 236respectively

In the construction section the axial moment ratioMNof the main beam section of the C3primendashC1prime cable force actionpoint is 149 159 and 164 According to formula (6) theshear lag coefficients of the upper edge of the main beamsection of C3primendashC1prime cable force point are 223 220 and 218respectively e axial force ratio MN of the main beamsection of the C3ndashC1 cable force action point is 147 181and 203 Again according to formula (6) the shear lagcoefficients λ of the upper edge of the main beam section ofthe C3ndashC1 cable force action point are 224 213 and 207Our finite element results of the shear lag effect of the mainbeam along the longitudinal direction suggest that the shearlag coefficient at Point A of the action point of cantilever endcable force C4 (C4prime) is 418 and 361 the shear lag coefficientλ at Point B reaches 307 and 306

In the Pier 4 C3primendashC1prime beam section the shear lagcoefficients λ of Point A at the cable force action point are232 222 and 249 respectively those of Point B at the cableforce action point are 126 123 and 129 respectivelyWithin the C3ndashC1 beam section the shear lag coefficients λof Point A at the cable force action point are 257 271 and271 and those corresponding to Point B are 211 122 and212 respectively

In the beam section construction stage the analysisresults of the C3ndashC1 (C3primendashC1prime) cable force action pointtreatment theory agree with the finite element analysisresults In this position the theoretically derived shearforce lag coefficient λ can be used to reflect the actual forceat that point e finite element analysis results at the C4(C4prime) action point of the cantilever end cable force exceedthe theoretical analysis results e cantilever end cableforce action point is mainly subjected to axial force so itsshear lag coefficient λ should be closer to λN In this po-sition the shear lag factor λN can be used to reflect theactual force

005

115

225

0 1 2 3 4 5 6 7 8 9 10

Shea

r lag

coef

ficie

nt

MN

λM = 2 λN = 1

Figure 14 Relationship between shear lag coefficient and bendingmoment axial force ratio

14 Advances in Materials Science and Engineering

63 Variation Law of Shear Lag Effect of PC Cable-StayedBridge Main Beam along Longitudinal Direction e shearlag variations in the main beam of the single-cable PC cable-stayed bridge along the longitudinal direction can besummarized as follows

(1) In the construction stage at the cantilever end thecable force action point is the main recipient of axialforce e shear lag coefficient λN of the axial forceacting at this position reflects the actual force at thatpoint

(2) In the beam segments between the cable forces at thecable force action point the theoretical shear lagcoefficient λ determined by the ratio MN of thebending moment to axial force reflects the actualstress

(3) In the midspan beam section between the actionpoints of cable forces the shear lag coefficient ofbending moment λM is used to reflect the actualstress For the beam section between the action pointof cable force and the middle of the span the sectionshear lag coefficient is obtained by linearinterpolation

7 Conclusion

e structural control section stress at work in a typicalconstruction process was measured on a full cable-stayedbridge model in this study Sets of finite element and modeltest results were respectively analyzed and compared Ourconclusions can be summarized as follows

(1) In the construction stage the stress value of the cabletension area of the main beam upper edge markedlychanges after tensiling the cable force the shear lageffect is very prominent After a hanging basket loadis applied the main beams of certain sections presentalternating positive and negative shear lag

(2) e shear lag coefficient of the upper edge of themain beam in G1ndashG8 sections under WorkingConditions 1 and 4 was extracted from the spatialfinite element analysis for comparison to find thatthe experimental and theoretical results are in ac-cordance After tension of the cable force and theapplication of a hanging basket load positive andnegative shear lag effects alternate on the upper edgeof the box girder in certain areas As tension cableforce increases in each section the shear lag coef-ficient of the section upper edge decelerates

(3) In the construction stage at the cantilever end cableforce action point is mainly subjected to axial forcee shear lag coefficient λN of the axial force on thisposition reflects the actual force situation at thatpoint In the beam segments between the cableforces the shear lag coefficient λ determined by theratio of the bending moment to the axial force re-flects the actual stress at the cable force action pointIn the midspan beam section between the actionpoints of cable forces the shear lag coefficient λM ofthe bendingmoment reflects the actual stress For thebeam section between the cable force action pointand the middle of the span the section shear lagcoefficient can be obtained by linear interpolation

C2prime C3

4250kNmiddotm

17300kNmiddotm 19100kNmiddotm

4

C1prime C4

24400kNmiddotm

36800kNmiddotm

C4prime C3prime C2

10500kNmiddotm

18600kNmiddotm

C1

17700kNmiddotm

55000kNmiddotm

6820kNmiddotm

(a)

C1prime C4

13000kN7030kN

C4prime C1

5680kN

20300kN

3600kN

27000kN

C2prime C3

10900kN

C3prime C2

14900kN

4

(b)

Figure 15 Internal force of main girder on Pier 4 (Condition 1) (a) Moment diagram (b) Axial force diagram

Advances in Materials Science and Engineering 15

(4) e shear lag effect of the main beam is prominent inthe construction process e shear lag effect is mostobvious at the cable force operation point of thecantilever end As construction progresses and thecable force increases the shear lag effect of otherbeam sections decreases at each point except for thecantilever end

Data Availability

Part of the shear lag coefficient data used to support thefindings of this study are included within the article All ofthe shear lag coefficient data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors would like to thank the financial supports fromthe National Key RampD Program of China (2018YFC0809600and 2018YFC0809606) and Natural Science Foundation ofLiaoning Province (2019-MS-265)

References

[1] Z Li J G Nie and W Y Ji ldquoPositive and negative shear lagbehaviors of composite twin-girder decks with varying cross-sectionrdquo Science China Technological Sciences vol 60 no 1pp 116ndash132 2017

[2] S-J Zhou ldquoFinite beam element considering shear-lag effectin box girderrdquo Journal of Engineering Mechanics vol 136no 9 pp 1115ndash1122 2010

[3] Z Li and R K L Su ldquoAnalytical solutions for compositebeams with slip shear-lag and time-dependent effectsrdquo En-gineering Structures vol 152 pp 559ndash578 2017

[4] Q Z Luo Y M Wu Q S Li J Tang and G D Liu ldquoA finitesegment model for shear lag analysisrdquo Engineering Structuresvol 26 no 14 pp 2113ndash2124 2004

[5] P Z Lin F K Liu W Ji et al ldquoAnalysis on shear lag effect ofconcrete box beam by variational principlerdquo Journal of theChina Railway Society vol 35 no 2 pp 93ndash98 2013

[6] Y Y Zhang H Zhang W Li et al ldquoAnalysis on shear-lageffect of box girders based on different shear-lag generalizeddisplacementrdquo Journal of Railway Science and Engineeringvol 13 no 6 pp 1083ndash1090 2016

[7] S T Chang Q Zhang and S Zhang ldquoShear lag effect in singleplane cable-stayed bridgerdquo Advances in Structural Engi-neering vol 1 no 4 pp 301ndash306 1998

[8] G-F Wu and X Hong ldquoeoretical and experimental studyon shear lag effect of partially cable-stayed bridgerdquo Journal ofZhejiang University-Science A vol 6 no 8 pp 875ndash877 2005

[9] L Pengzhen Y Zijiang L Fengkui et al ldquoShear lag effect ofconcrete box girders considered varying flange-depthsrdquoJournal of Civil Architecturalamp Environmental Engineeringvol 35 no 1 pp 76ndash79 2013

[10] E Yamaguchi T Chaisomphob J Sa-Nguanmanasak andC Lertsima ldquoStress concentration and deflection of simplysupported box girder including shear lag effectrdquo StructuralEngineering and Mechanics vol 28 no 2 pp 207ndash220 2008

[11] X B Shen ldquoAnalysis of shear lag effect in the main girder ofwide low-pylon cable stayed bridgerdquo Construction Technologyvol 44 no 24 pp 76ndash79 2015

[12] L Xingmin ldquoAnalysis of influences of construction processeson shear lag effect of concrete box girdersrdquo Bridge Con-struction vol 43 no 1 pp 30ndash34 2013

[13] L Muyu W H Sun X D Sun et al ldquoStress analysis of widergirder of an extradosed cable-stayed bridge based on thelargest cantilever construction stagerdquo Journal of HuazhongUniversity of Science and Technology (Urban Science Edition)vol 27 no 2 pp 11ndash14 2010

[14] C Qianshu H Wenlong and Y Menggang ldquoAnalysis ofshear lag effect in construction stage of wide box girderextradosed cable-stayed bridge with large flangesrdquo Journal ofRailway Science and Engineering vol 15 no 12 pp 3158ndash3164 2018

[15] D R Li B M Wang and Y C Lin Structural Model Ex-periment Science Press Beijing China 1996

[16] J W Dali andW F Laili Experiment and Analysis of ResidualStress Ocean Press Beijing China 1987

[17] H Hossdort Structural Model Analysis China Architecture ampBuilding Press Beijing China 1986

[18] S K Mei Structural Experiments and Structural DesignChina Communications Press Beijing China 1980

[19] B Atmaca and S Ates ldquoConstruction stage analysis of three-dimensional cable-stayed bridgesrdquo Steel amp Composite Struc-tures vol 12 no 5 pp 413ndash426 2012

[20] S W Park M R Jung D J Min and M Y Kim ldquoCon-struction stage analysis of cable-stayed bridges using theunstrained element length methodrdquo Journal of e KoreanSociety of Civil Engineers vol 36 no 6 pp 991ndash998 2016

[21] M Gunaydin S Adanur A C Altunisik B Sevim andE Turker ldquoDetermination of structural behavior of Bosporussuspension bridge considering construction stages and dif-ferent soil conditionsrdquo Steel and Composite Structures vol 17no 4 pp 405ndash429 2014

[22] S Adanur M Gunaydin A C Altunisik and B SevimldquoConstruction stage analysis of humber suspension bridgerdquoApplied Mathematical Modelling vol 36 no 11 pp 5492ndash5505 2012

[23] J B Mander M J N Priestley and R Park ldquoeoreticalstress-strain model for confined concreterdquo Journal of Struc-tural Engineering vol 114 no 8 pp 1804ndash1826 1988

[24] Y A Khalid C L Chan B B Sahari and A M S HamoudaldquoBending behaviour of corrugated web beamsrdquo Journal ofMaterials Processing Technology vol 150 no 3 pp 242ndash2542004

[25] A Demir N Caglar H Ozturk and Y Sumer ldquoNonlinearfinite element study on the improvement of shear capacity inreinforced concrete T-Section beams by an alternative diag-onal shear reinforcementrdquo Engineering Structures vol 120pp 158ndash165 2016

16 Advances in Materials Science and Engineering

Point APoint B

Point CPoint D

3 14rsquo13rsquo12rsquo 11rsquo 10rsquo 9rsquo 8rsquo 7rsquo 6rsquo 5rsquo 4rsquo 3rsquo 2rsquo 1rsquo 4Longitudinal bridge position

Pier 4

000

050

100

150

200

250

300

Shea

r lag

coef

ficie

nt

(a)

Point APoint B

Point CPoint D

4 1 2 3 4 5 6 7 8 9 10 11 12 13 14Longitudinal bridge position

000

050

100

150

200

250

300

350

400

Shea

r lag

coef

ficie

nt

(b)

Point APoint B

Point CPoint D

14 13 12 11 10 9 8 7 6 5 4 3 2 1 5Longitudinal bridge position

Pier 5

000

050

100

150

200

250

300

350

400

Shea

r lag

coef

ficie

nt

(c)

Figure 13 Continued

12 Advances in Materials Science and Engineering

force is relatively uniform e shear lag coefficient λ ofPoint A ranges from 120 to 155 that of Point B from 096 to135 and that of Point C from 076 to 101

Under this working condition the action of self-weightand cable force make the main characteristics of the shear lagat the upper edge of the main beam section of the single-cable PC cable-stayed bridge along the vertical direction verysimilar to those of underWorking Condition 1e shear lageffect at the main beam section is reduced as cable forceincreases

6 Distribution of Shear Lag Effect duringConstruction Stage

61 Shear Lag Coefficient of Main Beam of Single-Cable PlanePC Cable-Stayed Bridge e solution to the shear lag coef-ficient under the compressive bending load involves decom-posing the compressive bending load into a superposition ofthe bending moment action and the axial force action thencalculating the normal stress of themain beamunder the actionof the bending moment and the axial force alone [22] It isassumed that the axial force is borne by the whole section thatis the normal stress generated by the axial force is evenlydistributed along the cross section e results can then belinearly superposed to obtain the comprehensive shear lageffect coefficient Our previous analysis indicated that whenthere is cable tension of the main beam of the cable-stayedbridge in the single-cable PC there is a large stress concen-tration at the point of action of the concentrated loads estress distribution law of the roof plate changes accordingly

Assume that the axial force which is borne by the wholesection does not reflect the actual internal force of the maingirder of the single-cable PC cable-stayed bridge e dis-tribution of the longitudinal normal stress generated byinternal force is not uniform e size and distribution laware related to the internal force state namely the bendingmoment and axial force In this study the shear lag coef-ficient λΜ generated by bending moment and the shear lag

coefficient λΜ generated by axial force were respectivelycalculated to solve the shear lag coefficient λ aftersuperposition

e shear lag coefficient is calculated as follows

λ σσ

(6)

where σ is actual stress and σ( ) is nominal stresse shear lag coefficient is related to the wide span ratio

support conditions load form cross-sectional shape andsize material properties and location of the section [23ndash25]In the final analysis the shear lag is caused by unevendistribution of the longitudinal normal stress caused byinternal force its size and distribution are related to theinternal force state that is the bending moment is related tothe axial force

e actual stress σ and nominal stress σ( ) are thesuperposition of the bending moment and the normalstress generated by the axial force If the shear lag coeffi-cient λΜ produced by the bending moment is definedproperly the shear lag coefficient generated by the axialforce is λN and the shear lag coefficient after the super-position is λ us

λ σN + σM

σN + σM

λNσN + λMσM

σN + σM

(7)

where σ( )N and σ( )M are the nominal normal stressgenerated by bending moment and axial force respectivelyσ( )N and σ( )M are the actual normal stress generated bybending moment and axial force respectively e nu-merator and denominator at the right end of the aboveformula are divided by σ( )N to rewrite (7) as

λ λN + λM σMσN( 1113857

1 + σMσN( 1113857 (8)

ere are σM (MI)y and σN (NA) so(σMσN) (yAI) middot (MN) and order yAI C can beplugged into (8) to obtain

Point APoint B

Point CPoint D

000

050

100

150

200

250

300

350

5 1prime 2prime 3prime 4prime 5prime 6prime 7prime 8prime 9prime 10prime 11prime 12prime13prime14prime6

Shea

r lag

coef

ficie

nt

Longitudinal bridge position

(d)

Figure 13 Distribution of shear lag coefficient along longitudinal direction (Condition 4) (a) Left side of Pier 4 (b) Right side of Pier 4(c) Left side of Pier 5 (d) Right side of Pier 5

Advances in Materials Science and Engineering 13

λ λN + CλM(MN)

1 + C(MN) (9)

where y is the distance from the centroid of the section to theupper and lower edges A is the sectional area I is the sectionmoment of inertia and M and N represent the bendingmoment and axial force of the section

Equation (9) indicates that y A and I are the parametersinherent to the section under analysis λN and λΜ are theshear lag coefficients caused by the axial force and thebending moment respectively which are independent of themagnitude of the internal force Once λN and λΜ are de-termined the total shear lag coefficient of λ is only related tothe ratio of bending moment to axial force

Figure 14 shows the curve of the shear lag coefficientwith the bending moment axial force ratio when λΜ 200and λN 100 Taking the upper flange of the main beam ofthe Fumin cable-stayed bridge as an example C yAI 101

As shown in Figure 14 the total shear lag coefficient λgradually approaches the bending moment shear lag coef-ficient λΜ as the bending moment force ratio increases Asthe bending moment axial force ratio gradually decreases to0 the total shear lag coefficient λ is also gradually reduceduntil it is equal to the axial force shear lag coefficient λNWhen the bending moment force ratio is between 0 and 2the total shear lag coefficient λ changes substantially Alarger λΜ results in a greater change in the total shear lagcoefficient λ When the bending moment force ratio isgreater than 2 the total shear lag coefficient λ changesslowly at is to say in the internal force of this sectionwhether the bending moment or axial force accounts for alarger proportion affects how close the total shear lag co-efficient λ of the section is to the shear lag coefficient (λΜ orλN) caused by this item of internal force

For a general beam bridge the bending moment is themain internal force and so the section shear lag coefficient λis close to λΜ However for concrete cable-stayed bridgesthe main beam is in a compression state and is mainlysubjected to axial force its shear lag coefficient λ is closer toλN

62 eoretical versus Finite Element Main Beam Shear LagResults e shear lag variation law of the main beam of thesingle-cable PC cable-stayed bridge along the longitudinaldirection was discussed above as per our theoretical analysise shear lag coefficient and normal stress value of the mainbeam cross section are mainly concentrated in the range ofthe force point of the cable e effective width of the topplate of the upper flange in our analysis was set to 400m andthe full section of the bottom plate to σN 032N andσ( )N 009N (where N is the axial component of the cableforce) so that λN σNσ( )N 356 Shear lag coefficient λΜwas only considered a momentary action based on thedefinition of the effective width of the bridge e effectivewidth of the top plate in this case was 2400m so λΜ 135

We established the full-bridge plane bar modelaccording to finite element analysis e diagram of the

bending moment and axial force of the main beam underWorking Condition 1 suggests relatively small internalforces in Pier 4 and the main girder of Pier 5 e internalforce diagram of the main beam at Pier 4 was isolated forfurther analysis as shown in Figure 15

As shown in Figure 15 the bending moment force ratioMN is between λΜ and λN that is between 135 and 356e bending moment axial force ratio MN at the actionpoint of the cantilever end cable force C4 (C4prime) is 120 and118 According to formula (6) the shear lag coefficient λ ofthe upper edge of the main beam section is 235 and 236respectively

In the construction section the axial moment ratioMNof the main beam section of the C3primendashC1prime cable force actionpoint is 149 159 and 164 According to formula (6) theshear lag coefficients of the upper edge of the main beamsection of C3primendashC1prime cable force point are 223 220 and 218respectively e axial force ratio MN of the main beamsection of the C3ndashC1 cable force action point is 147 181and 203 Again according to formula (6) the shear lagcoefficients λ of the upper edge of the main beam section ofthe C3ndashC1 cable force action point are 224 213 and 207Our finite element results of the shear lag effect of the mainbeam along the longitudinal direction suggest that the shearlag coefficient at Point A of the action point of cantilever endcable force C4 (C4prime) is 418 and 361 the shear lag coefficientλ at Point B reaches 307 and 306

In the Pier 4 C3primendashC1prime beam section the shear lagcoefficients λ of Point A at the cable force action point are232 222 and 249 respectively those of Point B at the cableforce action point are 126 123 and 129 respectivelyWithin the C3ndashC1 beam section the shear lag coefficients λof Point A at the cable force action point are 257 271 and271 and those corresponding to Point B are 211 122 and212 respectively

In the beam section construction stage the analysisresults of the C3ndashC1 (C3primendashC1prime) cable force action pointtreatment theory agree with the finite element analysisresults In this position the theoretically derived shearforce lag coefficient λ can be used to reflect the actual forceat that point e finite element analysis results at the C4(C4prime) action point of the cantilever end cable force exceedthe theoretical analysis results e cantilever end cableforce action point is mainly subjected to axial force so itsshear lag coefficient λ should be closer to λN In this po-sition the shear lag factor λN can be used to reflect theactual force

005

115

225

0 1 2 3 4 5 6 7 8 9 10

Shea

r lag

coef

ficie

nt

MN

λM = 2 λN = 1

Figure 14 Relationship between shear lag coefficient and bendingmoment axial force ratio

14 Advances in Materials Science and Engineering

63 Variation Law of Shear Lag Effect of PC Cable-StayedBridge Main Beam along Longitudinal Direction e shearlag variations in the main beam of the single-cable PC cable-stayed bridge along the longitudinal direction can besummarized as follows

(1) In the construction stage at the cantilever end thecable force action point is the main recipient of axialforce e shear lag coefficient λN of the axial forceacting at this position reflects the actual force at thatpoint

(2) In the beam segments between the cable forces at thecable force action point the theoretical shear lagcoefficient λ determined by the ratio MN of thebending moment to axial force reflects the actualstress

(3) In the midspan beam section between the actionpoints of cable forces the shear lag coefficient ofbending moment λM is used to reflect the actualstress For the beam section between the action pointof cable force and the middle of the span the sectionshear lag coefficient is obtained by linearinterpolation

7 Conclusion

e structural control section stress at work in a typicalconstruction process was measured on a full cable-stayedbridge model in this study Sets of finite element and modeltest results were respectively analyzed and compared Ourconclusions can be summarized as follows

(1) In the construction stage the stress value of the cabletension area of the main beam upper edge markedlychanges after tensiling the cable force the shear lageffect is very prominent After a hanging basket loadis applied the main beams of certain sections presentalternating positive and negative shear lag

(2) e shear lag coefficient of the upper edge of themain beam in G1ndashG8 sections under WorkingConditions 1 and 4 was extracted from the spatialfinite element analysis for comparison to find thatthe experimental and theoretical results are in ac-cordance After tension of the cable force and theapplication of a hanging basket load positive andnegative shear lag effects alternate on the upper edgeof the box girder in certain areas As tension cableforce increases in each section the shear lag coef-ficient of the section upper edge decelerates

(3) In the construction stage at the cantilever end cableforce action point is mainly subjected to axial forcee shear lag coefficient λN of the axial force on thisposition reflects the actual force situation at thatpoint In the beam segments between the cableforces the shear lag coefficient λ determined by theratio of the bending moment to the axial force re-flects the actual stress at the cable force action pointIn the midspan beam section between the actionpoints of cable forces the shear lag coefficient λM ofthe bendingmoment reflects the actual stress For thebeam section between the cable force action pointand the middle of the span the section shear lagcoefficient can be obtained by linear interpolation

C2prime C3

4250kNmiddotm

17300kNmiddotm 19100kNmiddotm

4

C1prime C4

24400kNmiddotm

36800kNmiddotm

C4prime C3prime C2

10500kNmiddotm

18600kNmiddotm

C1

17700kNmiddotm

55000kNmiddotm

6820kNmiddotm

(a)

C1prime C4

13000kN7030kN

C4prime C1

5680kN

20300kN

3600kN

27000kN

C2prime C3

10900kN

C3prime C2

14900kN

4

(b)

Figure 15 Internal force of main girder on Pier 4 (Condition 1) (a) Moment diagram (b) Axial force diagram

Advances in Materials Science and Engineering 15

(4) e shear lag effect of the main beam is prominent inthe construction process e shear lag effect is mostobvious at the cable force operation point of thecantilever end As construction progresses and thecable force increases the shear lag effect of otherbeam sections decreases at each point except for thecantilever end

Data Availability

Part of the shear lag coefficient data used to support thefindings of this study are included within the article All ofthe shear lag coefficient data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors would like to thank the financial supports fromthe National Key RampD Program of China (2018YFC0809600and 2018YFC0809606) and Natural Science Foundation ofLiaoning Province (2019-MS-265)

References

[1] Z Li J G Nie and W Y Ji ldquoPositive and negative shear lagbehaviors of composite twin-girder decks with varying cross-sectionrdquo Science China Technological Sciences vol 60 no 1pp 116ndash132 2017

[2] S-J Zhou ldquoFinite beam element considering shear-lag effectin box girderrdquo Journal of Engineering Mechanics vol 136no 9 pp 1115ndash1122 2010

[3] Z Li and R K L Su ldquoAnalytical solutions for compositebeams with slip shear-lag and time-dependent effectsrdquo En-gineering Structures vol 152 pp 559ndash578 2017

[4] Q Z Luo Y M Wu Q S Li J Tang and G D Liu ldquoA finitesegment model for shear lag analysisrdquo Engineering Structuresvol 26 no 14 pp 2113ndash2124 2004

[5] P Z Lin F K Liu W Ji et al ldquoAnalysis on shear lag effect ofconcrete box beam by variational principlerdquo Journal of theChina Railway Society vol 35 no 2 pp 93ndash98 2013

[6] Y Y Zhang H Zhang W Li et al ldquoAnalysis on shear-lageffect of box girders based on different shear-lag generalizeddisplacementrdquo Journal of Railway Science and Engineeringvol 13 no 6 pp 1083ndash1090 2016

[7] S T Chang Q Zhang and S Zhang ldquoShear lag effect in singleplane cable-stayed bridgerdquo Advances in Structural Engi-neering vol 1 no 4 pp 301ndash306 1998

[8] G-F Wu and X Hong ldquoeoretical and experimental studyon shear lag effect of partially cable-stayed bridgerdquo Journal ofZhejiang University-Science A vol 6 no 8 pp 875ndash877 2005

[9] L Pengzhen Y Zijiang L Fengkui et al ldquoShear lag effect ofconcrete box girders considered varying flange-depthsrdquoJournal of Civil Architecturalamp Environmental Engineeringvol 35 no 1 pp 76ndash79 2013

[10] E Yamaguchi T Chaisomphob J Sa-Nguanmanasak andC Lertsima ldquoStress concentration and deflection of simplysupported box girder including shear lag effectrdquo StructuralEngineering and Mechanics vol 28 no 2 pp 207ndash220 2008

[11] X B Shen ldquoAnalysis of shear lag effect in the main girder ofwide low-pylon cable stayed bridgerdquo Construction Technologyvol 44 no 24 pp 76ndash79 2015

[12] L Xingmin ldquoAnalysis of influences of construction processeson shear lag effect of concrete box girdersrdquo Bridge Con-struction vol 43 no 1 pp 30ndash34 2013

[13] L Muyu W H Sun X D Sun et al ldquoStress analysis of widergirder of an extradosed cable-stayed bridge based on thelargest cantilever construction stagerdquo Journal of HuazhongUniversity of Science and Technology (Urban Science Edition)vol 27 no 2 pp 11ndash14 2010

[14] C Qianshu H Wenlong and Y Menggang ldquoAnalysis ofshear lag effect in construction stage of wide box girderextradosed cable-stayed bridge with large flangesrdquo Journal ofRailway Science and Engineering vol 15 no 12 pp 3158ndash3164 2018

[15] D R Li B M Wang and Y C Lin Structural Model Ex-periment Science Press Beijing China 1996

[16] J W Dali andW F Laili Experiment and Analysis of ResidualStress Ocean Press Beijing China 1987

[17] H Hossdort Structural Model Analysis China Architecture ampBuilding Press Beijing China 1986

[18] S K Mei Structural Experiments and Structural DesignChina Communications Press Beijing China 1980

[19] B Atmaca and S Ates ldquoConstruction stage analysis of three-dimensional cable-stayed bridgesrdquo Steel amp Composite Struc-tures vol 12 no 5 pp 413ndash426 2012

[20] S W Park M R Jung D J Min and M Y Kim ldquoCon-struction stage analysis of cable-stayed bridges using theunstrained element length methodrdquo Journal of e KoreanSociety of Civil Engineers vol 36 no 6 pp 991ndash998 2016

[21] M Gunaydin S Adanur A C Altunisik B Sevim andE Turker ldquoDetermination of structural behavior of Bosporussuspension bridge considering construction stages and dif-ferent soil conditionsrdquo Steel and Composite Structures vol 17no 4 pp 405ndash429 2014

[22] S Adanur M Gunaydin A C Altunisik and B SevimldquoConstruction stage analysis of humber suspension bridgerdquoApplied Mathematical Modelling vol 36 no 11 pp 5492ndash5505 2012

[23] J B Mander M J N Priestley and R Park ldquoeoreticalstress-strain model for confined concreterdquo Journal of Struc-tural Engineering vol 114 no 8 pp 1804ndash1826 1988

[24] Y A Khalid C L Chan B B Sahari and A M S HamoudaldquoBending behaviour of corrugated web beamsrdquo Journal ofMaterials Processing Technology vol 150 no 3 pp 242ndash2542004

[25] A Demir N Caglar H Ozturk and Y Sumer ldquoNonlinearfinite element study on the improvement of shear capacity inreinforced concrete T-Section beams by an alternative diag-onal shear reinforcementrdquo Engineering Structures vol 120pp 158ndash165 2016

16 Advances in Materials Science and Engineering

force is relatively uniform e shear lag coefficient λ ofPoint A ranges from 120 to 155 that of Point B from 096 to135 and that of Point C from 076 to 101

Under this working condition the action of self-weightand cable force make the main characteristics of the shear lagat the upper edge of the main beam section of the single-cable PC cable-stayed bridge along the vertical direction verysimilar to those of underWorking Condition 1e shear lageffect at the main beam section is reduced as cable forceincreases

6 Distribution of Shear Lag Effect duringConstruction Stage

61 Shear Lag Coefficient of Main Beam of Single-Cable PlanePC Cable-Stayed Bridge e solution to the shear lag coef-ficient under the compressive bending load involves decom-posing the compressive bending load into a superposition ofthe bending moment action and the axial force action thencalculating the normal stress of themain beamunder the actionof the bending moment and the axial force alone [22] It isassumed that the axial force is borne by the whole section thatis the normal stress generated by the axial force is evenlydistributed along the cross section e results can then belinearly superposed to obtain the comprehensive shear lageffect coefficient Our previous analysis indicated that whenthere is cable tension of the main beam of the cable-stayedbridge in the single-cable PC there is a large stress concen-tration at the point of action of the concentrated loads estress distribution law of the roof plate changes accordingly

Assume that the axial force which is borne by the wholesection does not reflect the actual internal force of the maingirder of the single-cable PC cable-stayed bridge e dis-tribution of the longitudinal normal stress generated byinternal force is not uniform e size and distribution laware related to the internal force state namely the bendingmoment and axial force In this study the shear lag coef-ficient λΜ generated by bending moment and the shear lag

coefficient λΜ generated by axial force were respectivelycalculated to solve the shear lag coefficient λ aftersuperposition

e shear lag coefficient is calculated as follows

λ σσ

(6)

where σ is actual stress and σ( ) is nominal stresse shear lag coefficient is related to the wide span ratio

support conditions load form cross-sectional shape andsize material properties and location of the section [23ndash25]In the final analysis the shear lag is caused by unevendistribution of the longitudinal normal stress caused byinternal force its size and distribution are related to theinternal force state that is the bending moment is related tothe axial force

e actual stress σ and nominal stress σ( ) are thesuperposition of the bending moment and the normalstress generated by the axial force If the shear lag coeffi-cient λΜ produced by the bending moment is definedproperly the shear lag coefficient generated by the axialforce is λN and the shear lag coefficient after the super-position is λ us

λ σN + σM

σN + σM

λNσN + λMσM

σN + σM

(7)

where σ( )N and σ( )M are the nominal normal stressgenerated by bending moment and axial force respectivelyσ( )N and σ( )M are the actual normal stress generated bybending moment and axial force respectively e nu-merator and denominator at the right end of the aboveformula are divided by σ( )N to rewrite (7) as

λ λN + λM σMσN( 1113857

1 + σMσN( 1113857 (8)

ere are σM (MI)y and σN (NA) so(σMσN) (yAI) middot (MN) and order yAI C can beplugged into (8) to obtain

Point APoint B

Point CPoint D

000

050

100

150

200

250

300

350

5 1prime 2prime 3prime 4prime 5prime 6prime 7prime 8prime 9prime 10prime 11prime 12prime13prime14prime6

Shea

r lag

coef

ficie

nt

Longitudinal bridge position

(d)

Figure 13 Distribution of shear lag coefficient along longitudinal direction (Condition 4) (a) Left side of Pier 4 (b) Right side of Pier 4(c) Left side of Pier 5 (d) Right side of Pier 5

Advances in Materials Science and Engineering 13

λ λN + CλM(MN)

1 + C(MN) (9)

where y is the distance from the centroid of the section to theupper and lower edges A is the sectional area I is the sectionmoment of inertia and M and N represent the bendingmoment and axial force of the section

Equation (9) indicates that y A and I are the parametersinherent to the section under analysis λN and λΜ are theshear lag coefficients caused by the axial force and thebending moment respectively which are independent of themagnitude of the internal force Once λN and λΜ are de-termined the total shear lag coefficient of λ is only related tothe ratio of bending moment to axial force

Figure 14 shows the curve of the shear lag coefficientwith the bending moment axial force ratio when λΜ 200and λN 100 Taking the upper flange of the main beam ofthe Fumin cable-stayed bridge as an example C yAI 101

As shown in Figure 14 the total shear lag coefficient λgradually approaches the bending moment shear lag coef-ficient λΜ as the bending moment force ratio increases Asthe bending moment axial force ratio gradually decreases to0 the total shear lag coefficient λ is also gradually reduceduntil it is equal to the axial force shear lag coefficient λNWhen the bending moment force ratio is between 0 and 2the total shear lag coefficient λ changes substantially Alarger λΜ results in a greater change in the total shear lagcoefficient λ When the bending moment force ratio isgreater than 2 the total shear lag coefficient λ changesslowly at is to say in the internal force of this sectionwhether the bending moment or axial force accounts for alarger proportion affects how close the total shear lag co-efficient λ of the section is to the shear lag coefficient (λΜ orλN) caused by this item of internal force

For a general beam bridge the bending moment is themain internal force and so the section shear lag coefficient λis close to λΜ However for concrete cable-stayed bridgesthe main beam is in a compression state and is mainlysubjected to axial force its shear lag coefficient λ is closer toλN

62 eoretical versus Finite Element Main Beam Shear LagResults e shear lag variation law of the main beam of thesingle-cable PC cable-stayed bridge along the longitudinaldirection was discussed above as per our theoretical analysise shear lag coefficient and normal stress value of the mainbeam cross section are mainly concentrated in the range ofthe force point of the cable e effective width of the topplate of the upper flange in our analysis was set to 400m andthe full section of the bottom plate to σN 032N andσ( )N 009N (where N is the axial component of the cableforce) so that λN σNσ( )N 356 Shear lag coefficient λΜwas only considered a momentary action based on thedefinition of the effective width of the bridge e effectivewidth of the top plate in this case was 2400m so λΜ 135

We established the full-bridge plane bar modelaccording to finite element analysis e diagram of the

bending moment and axial force of the main beam underWorking Condition 1 suggests relatively small internalforces in Pier 4 and the main girder of Pier 5 e internalforce diagram of the main beam at Pier 4 was isolated forfurther analysis as shown in Figure 15

As shown in Figure 15 the bending moment force ratioMN is between λΜ and λN that is between 135 and 356e bending moment axial force ratio MN at the actionpoint of the cantilever end cable force C4 (C4prime) is 120 and118 According to formula (6) the shear lag coefficient λ ofthe upper edge of the main beam section is 235 and 236respectively

In the construction section the axial moment ratioMNof the main beam section of the C3primendashC1prime cable force actionpoint is 149 159 and 164 According to formula (6) theshear lag coefficients of the upper edge of the main beamsection of C3primendashC1prime cable force point are 223 220 and 218respectively e axial force ratio MN of the main beamsection of the C3ndashC1 cable force action point is 147 181and 203 Again according to formula (6) the shear lagcoefficients λ of the upper edge of the main beam section ofthe C3ndashC1 cable force action point are 224 213 and 207Our finite element results of the shear lag effect of the mainbeam along the longitudinal direction suggest that the shearlag coefficient at Point A of the action point of cantilever endcable force C4 (C4prime) is 418 and 361 the shear lag coefficientλ at Point B reaches 307 and 306

In the Pier 4 C3primendashC1prime beam section the shear lagcoefficients λ of Point A at the cable force action point are232 222 and 249 respectively those of Point B at the cableforce action point are 126 123 and 129 respectivelyWithin the C3ndashC1 beam section the shear lag coefficients λof Point A at the cable force action point are 257 271 and271 and those corresponding to Point B are 211 122 and212 respectively

In the beam section construction stage the analysisresults of the C3ndashC1 (C3primendashC1prime) cable force action pointtreatment theory agree with the finite element analysisresults In this position the theoretically derived shearforce lag coefficient λ can be used to reflect the actual forceat that point e finite element analysis results at the C4(C4prime) action point of the cantilever end cable force exceedthe theoretical analysis results e cantilever end cableforce action point is mainly subjected to axial force so itsshear lag coefficient λ should be closer to λN In this po-sition the shear lag factor λN can be used to reflect theactual force

005

115

225

0 1 2 3 4 5 6 7 8 9 10

Shea

r lag

coef

ficie

nt

MN

λM = 2 λN = 1

Figure 14 Relationship between shear lag coefficient and bendingmoment axial force ratio

14 Advances in Materials Science and Engineering

63 Variation Law of Shear Lag Effect of PC Cable-StayedBridge Main Beam along Longitudinal Direction e shearlag variations in the main beam of the single-cable PC cable-stayed bridge along the longitudinal direction can besummarized as follows

(1) In the construction stage at the cantilever end thecable force action point is the main recipient of axialforce e shear lag coefficient λN of the axial forceacting at this position reflects the actual force at thatpoint

(2) In the beam segments between the cable forces at thecable force action point the theoretical shear lagcoefficient λ determined by the ratio MN of thebending moment to axial force reflects the actualstress

(3) In the midspan beam section between the actionpoints of cable forces the shear lag coefficient ofbending moment λM is used to reflect the actualstress For the beam section between the action pointof cable force and the middle of the span the sectionshear lag coefficient is obtained by linearinterpolation

7 Conclusion

e structural control section stress at work in a typicalconstruction process was measured on a full cable-stayedbridge model in this study Sets of finite element and modeltest results were respectively analyzed and compared Ourconclusions can be summarized as follows

(1) In the construction stage the stress value of the cabletension area of the main beam upper edge markedlychanges after tensiling the cable force the shear lageffect is very prominent After a hanging basket loadis applied the main beams of certain sections presentalternating positive and negative shear lag

(2) e shear lag coefficient of the upper edge of themain beam in G1ndashG8 sections under WorkingConditions 1 and 4 was extracted from the spatialfinite element analysis for comparison to find thatthe experimental and theoretical results are in ac-cordance After tension of the cable force and theapplication of a hanging basket load positive andnegative shear lag effects alternate on the upper edgeof the box girder in certain areas As tension cableforce increases in each section the shear lag coef-ficient of the section upper edge decelerates

(3) In the construction stage at the cantilever end cableforce action point is mainly subjected to axial forcee shear lag coefficient λN of the axial force on thisposition reflects the actual force situation at thatpoint In the beam segments between the cableforces the shear lag coefficient λ determined by theratio of the bending moment to the axial force re-flects the actual stress at the cable force action pointIn the midspan beam section between the actionpoints of cable forces the shear lag coefficient λM ofthe bendingmoment reflects the actual stress For thebeam section between the cable force action pointand the middle of the span the section shear lagcoefficient can be obtained by linear interpolation

C2prime C3

4250kNmiddotm

17300kNmiddotm 19100kNmiddotm

4

C1prime C4

24400kNmiddotm

36800kNmiddotm

C4prime C3prime C2

10500kNmiddotm

18600kNmiddotm

C1

17700kNmiddotm

55000kNmiddotm

6820kNmiddotm

(a)

C1prime C4

13000kN7030kN

C4prime C1

5680kN

20300kN

3600kN

27000kN

C2prime C3

10900kN

C3prime C2

14900kN

4

(b)

Figure 15 Internal force of main girder on Pier 4 (Condition 1) (a) Moment diagram (b) Axial force diagram

Advances in Materials Science and Engineering 15

(4) e shear lag effect of the main beam is prominent inthe construction process e shear lag effect is mostobvious at the cable force operation point of thecantilever end As construction progresses and thecable force increases the shear lag effect of otherbeam sections decreases at each point except for thecantilever end

Data Availability

Part of the shear lag coefficient data used to support thefindings of this study are included within the article All ofthe shear lag coefficient data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors would like to thank the financial supports fromthe National Key RampD Program of China (2018YFC0809600and 2018YFC0809606) and Natural Science Foundation ofLiaoning Province (2019-MS-265)

References

[1] Z Li J G Nie and W Y Ji ldquoPositive and negative shear lagbehaviors of composite twin-girder decks with varying cross-sectionrdquo Science China Technological Sciences vol 60 no 1pp 116ndash132 2017

[2] S-J Zhou ldquoFinite beam element considering shear-lag effectin box girderrdquo Journal of Engineering Mechanics vol 136no 9 pp 1115ndash1122 2010

[3] Z Li and R K L Su ldquoAnalytical solutions for compositebeams with slip shear-lag and time-dependent effectsrdquo En-gineering Structures vol 152 pp 559ndash578 2017

[4] Q Z Luo Y M Wu Q S Li J Tang and G D Liu ldquoA finitesegment model for shear lag analysisrdquo Engineering Structuresvol 26 no 14 pp 2113ndash2124 2004

[5] P Z Lin F K Liu W Ji et al ldquoAnalysis on shear lag effect ofconcrete box beam by variational principlerdquo Journal of theChina Railway Society vol 35 no 2 pp 93ndash98 2013

[6] Y Y Zhang H Zhang W Li et al ldquoAnalysis on shear-lageffect of box girders based on different shear-lag generalizeddisplacementrdquo Journal of Railway Science and Engineeringvol 13 no 6 pp 1083ndash1090 2016

[7] S T Chang Q Zhang and S Zhang ldquoShear lag effect in singleplane cable-stayed bridgerdquo Advances in Structural Engi-neering vol 1 no 4 pp 301ndash306 1998

[8] G-F Wu and X Hong ldquoeoretical and experimental studyon shear lag effect of partially cable-stayed bridgerdquo Journal ofZhejiang University-Science A vol 6 no 8 pp 875ndash877 2005

[9] L Pengzhen Y Zijiang L Fengkui et al ldquoShear lag effect ofconcrete box girders considered varying flange-depthsrdquoJournal of Civil Architecturalamp Environmental Engineeringvol 35 no 1 pp 76ndash79 2013

[10] E Yamaguchi T Chaisomphob J Sa-Nguanmanasak andC Lertsima ldquoStress concentration and deflection of simplysupported box girder including shear lag effectrdquo StructuralEngineering and Mechanics vol 28 no 2 pp 207ndash220 2008

[11] X B Shen ldquoAnalysis of shear lag effect in the main girder ofwide low-pylon cable stayed bridgerdquo Construction Technologyvol 44 no 24 pp 76ndash79 2015

[12] L Xingmin ldquoAnalysis of influences of construction processeson shear lag effect of concrete box girdersrdquo Bridge Con-struction vol 43 no 1 pp 30ndash34 2013

[13] L Muyu W H Sun X D Sun et al ldquoStress analysis of widergirder of an extradosed cable-stayed bridge based on thelargest cantilever construction stagerdquo Journal of HuazhongUniversity of Science and Technology (Urban Science Edition)vol 27 no 2 pp 11ndash14 2010

[14] C Qianshu H Wenlong and Y Menggang ldquoAnalysis ofshear lag effect in construction stage of wide box girderextradosed cable-stayed bridge with large flangesrdquo Journal ofRailway Science and Engineering vol 15 no 12 pp 3158ndash3164 2018

[15] D R Li B M Wang and Y C Lin Structural Model Ex-periment Science Press Beijing China 1996

[16] J W Dali andW F Laili Experiment and Analysis of ResidualStress Ocean Press Beijing China 1987

[17] H Hossdort Structural Model Analysis China Architecture ampBuilding Press Beijing China 1986

[18] S K Mei Structural Experiments and Structural DesignChina Communications Press Beijing China 1980

[19] B Atmaca and S Ates ldquoConstruction stage analysis of three-dimensional cable-stayed bridgesrdquo Steel amp Composite Struc-tures vol 12 no 5 pp 413ndash426 2012

[20] S W Park M R Jung D J Min and M Y Kim ldquoCon-struction stage analysis of cable-stayed bridges using theunstrained element length methodrdquo Journal of e KoreanSociety of Civil Engineers vol 36 no 6 pp 991ndash998 2016

[21] M Gunaydin S Adanur A C Altunisik B Sevim andE Turker ldquoDetermination of structural behavior of Bosporussuspension bridge considering construction stages and dif-ferent soil conditionsrdquo Steel and Composite Structures vol 17no 4 pp 405ndash429 2014

[22] S Adanur M Gunaydin A C Altunisik and B SevimldquoConstruction stage analysis of humber suspension bridgerdquoApplied Mathematical Modelling vol 36 no 11 pp 5492ndash5505 2012

[23] J B Mander M J N Priestley and R Park ldquoeoreticalstress-strain model for confined concreterdquo Journal of Struc-tural Engineering vol 114 no 8 pp 1804ndash1826 1988

[24] Y A Khalid C L Chan B B Sahari and A M S HamoudaldquoBending behaviour of corrugated web beamsrdquo Journal ofMaterials Processing Technology vol 150 no 3 pp 242ndash2542004

[25] A Demir N Caglar H Ozturk and Y Sumer ldquoNonlinearfinite element study on the improvement of shear capacity inreinforced concrete T-Section beams by an alternative diag-onal shear reinforcementrdquo Engineering Structures vol 120pp 158ndash165 2016

16 Advances in Materials Science and Engineering

λ λN + CλM(MN)

1 + C(MN) (9)

where y is the distance from the centroid of the section to theupper and lower edges A is the sectional area I is the sectionmoment of inertia and M and N represent the bendingmoment and axial force of the section

Equation (9) indicates that y A and I are the parametersinherent to the section under analysis λN and λΜ are theshear lag coefficients caused by the axial force and thebending moment respectively which are independent of themagnitude of the internal force Once λN and λΜ are de-termined the total shear lag coefficient of λ is only related tothe ratio of bending moment to axial force

Figure 14 shows the curve of the shear lag coefficientwith the bending moment axial force ratio when λΜ 200and λN 100 Taking the upper flange of the main beam ofthe Fumin cable-stayed bridge as an example C yAI 101

As shown in Figure 14 the total shear lag coefficient λgradually approaches the bending moment shear lag coef-ficient λΜ as the bending moment force ratio increases Asthe bending moment axial force ratio gradually decreases to0 the total shear lag coefficient λ is also gradually reduceduntil it is equal to the axial force shear lag coefficient λNWhen the bending moment force ratio is between 0 and 2the total shear lag coefficient λ changes substantially Alarger λΜ results in a greater change in the total shear lagcoefficient λ When the bending moment force ratio isgreater than 2 the total shear lag coefficient λ changesslowly at is to say in the internal force of this sectionwhether the bending moment or axial force accounts for alarger proportion affects how close the total shear lag co-efficient λ of the section is to the shear lag coefficient (λΜ orλN) caused by this item of internal force

For a general beam bridge the bending moment is themain internal force and so the section shear lag coefficient λis close to λΜ However for concrete cable-stayed bridgesthe main beam is in a compression state and is mainlysubjected to axial force its shear lag coefficient λ is closer toλN

62 eoretical versus Finite Element Main Beam Shear LagResults e shear lag variation law of the main beam of thesingle-cable PC cable-stayed bridge along the longitudinaldirection was discussed above as per our theoretical analysise shear lag coefficient and normal stress value of the mainbeam cross section are mainly concentrated in the range ofthe force point of the cable e effective width of the topplate of the upper flange in our analysis was set to 400m andthe full section of the bottom plate to σN 032N andσ( )N 009N (where N is the axial component of the cableforce) so that λN σNσ( )N 356 Shear lag coefficient λΜwas only considered a momentary action based on thedefinition of the effective width of the bridge e effectivewidth of the top plate in this case was 2400m so λΜ 135

We established the full-bridge plane bar modelaccording to finite element analysis e diagram of the

bending moment and axial force of the main beam underWorking Condition 1 suggests relatively small internalforces in Pier 4 and the main girder of Pier 5 e internalforce diagram of the main beam at Pier 4 was isolated forfurther analysis as shown in Figure 15

As shown in Figure 15 the bending moment force ratioMN is between λΜ and λN that is between 135 and 356e bending moment axial force ratio MN at the actionpoint of the cantilever end cable force C4 (C4prime) is 120 and118 According to formula (6) the shear lag coefficient λ ofthe upper edge of the main beam section is 235 and 236respectively

In the construction section the axial moment ratioMNof the main beam section of the C3primendashC1prime cable force actionpoint is 149 159 and 164 According to formula (6) theshear lag coefficients of the upper edge of the main beamsection of C3primendashC1prime cable force point are 223 220 and 218respectively e axial force ratio MN of the main beamsection of the C3ndashC1 cable force action point is 147 181and 203 Again according to formula (6) the shear lagcoefficients λ of the upper edge of the main beam section ofthe C3ndashC1 cable force action point are 224 213 and 207Our finite element results of the shear lag effect of the mainbeam along the longitudinal direction suggest that the shearlag coefficient at Point A of the action point of cantilever endcable force C4 (C4prime) is 418 and 361 the shear lag coefficientλ at Point B reaches 307 and 306

In the Pier 4 C3primendashC1prime beam section the shear lagcoefficients λ of Point A at the cable force action point are232 222 and 249 respectively those of Point B at the cableforce action point are 126 123 and 129 respectivelyWithin the C3ndashC1 beam section the shear lag coefficients λof Point A at the cable force action point are 257 271 and271 and those corresponding to Point B are 211 122 and212 respectively

In the beam section construction stage the analysisresults of the C3ndashC1 (C3primendashC1prime) cable force action pointtreatment theory agree with the finite element analysisresults In this position the theoretically derived shearforce lag coefficient λ can be used to reflect the actual forceat that point e finite element analysis results at the C4(C4prime) action point of the cantilever end cable force exceedthe theoretical analysis results e cantilever end cableforce action point is mainly subjected to axial force so itsshear lag coefficient λ should be closer to λN In this po-sition the shear lag factor λN can be used to reflect theactual force

005

115

225

0 1 2 3 4 5 6 7 8 9 10

Shea

r lag

coef

ficie

nt

MN

λM = 2 λN = 1

Figure 14 Relationship between shear lag coefficient and bendingmoment axial force ratio

14 Advances in Materials Science and Engineering

63 Variation Law of Shear Lag Effect of PC Cable-StayedBridge Main Beam along Longitudinal Direction e shearlag variations in the main beam of the single-cable PC cable-stayed bridge along the longitudinal direction can besummarized as follows

(1) In the construction stage at the cantilever end thecable force action point is the main recipient of axialforce e shear lag coefficient λN of the axial forceacting at this position reflects the actual force at thatpoint

(2) In the beam segments between the cable forces at thecable force action point the theoretical shear lagcoefficient λ determined by the ratio MN of thebending moment to axial force reflects the actualstress

(3) In the midspan beam section between the actionpoints of cable forces the shear lag coefficient ofbending moment λM is used to reflect the actualstress For the beam section between the action pointof cable force and the middle of the span the sectionshear lag coefficient is obtained by linearinterpolation

7 Conclusion

e structural control section stress at work in a typicalconstruction process was measured on a full cable-stayedbridge model in this study Sets of finite element and modeltest results were respectively analyzed and compared Ourconclusions can be summarized as follows

(1) In the construction stage the stress value of the cabletension area of the main beam upper edge markedlychanges after tensiling the cable force the shear lageffect is very prominent After a hanging basket loadis applied the main beams of certain sections presentalternating positive and negative shear lag

(2) e shear lag coefficient of the upper edge of themain beam in G1ndashG8 sections under WorkingConditions 1 and 4 was extracted from the spatialfinite element analysis for comparison to find thatthe experimental and theoretical results are in ac-cordance After tension of the cable force and theapplication of a hanging basket load positive andnegative shear lag effects alternate on the upper edgeof the box girder in certain areas As tension cableforce increases in each section the shear lag coef-ficient of the section upper edge decelerates

(3) In the construction stage at the cantilever end cableforce action point is mainly subjected to axial forcee shear lag coefficient λN of the axial force on thisposition reflects the actual force situation at thatpoint In the beam segments between the cableforces the shear lag coefficient λ determined by theratio of the bending moment to the axial force re-flects the actual stress at the cable force action pointIn the midspan beam section between the actionpoints of cable forces the shear lag coefficient λM ofthe bendingmoment reflects the actual stress For thebeam section between the cable force action pointand the middle of the span the section shear lagcoefficient can be obtained by linear interpolation

C2prime C3

4250kNmiddotm

17300kNmiddotm 19100kNmiddotm

4

C1prime C4

24400kNmiddotm

36800kNmiddotm

C4prime C3prime C2

10500kNmiddotm

18600kNmiddotm

C1

17700kNmiddotm

55000kNmiddotm

6820kNmiddotm

(a)

C1prime C4

13000kN7030kN

C4prime C1

5680kN

20300kN

3600kN

27000kN

C2prime C3

10900kN

C3prime C2

14900kN

4

(b)

Figure 15 Internal force of main girder on Pier 4 (Condition 1) (a) Moment diagram (b) Axial force diagram

Advances in Materials Science and Engineering 15

(4) e shear lag effect of the main beam is prominent inthe construction process e shear lag effect is mostobvious at the cable force operation point of thecantilever end As construction progresses and thecable force increases the shear lag effect of otherbeam sections decreases at each point except for thecantilever end

Data Availability

Part of the shear lag coefficient data used to support thefindings of this study are included within the article All ofthe shear lag coefficient data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors would like to thank the financial supports fromthe National Key RampD Program of China (2018YFC0809600and 2018YFC0809606) and Natural Science Foundation ofLiaoning Province (2019-MS-265)

References

[1] Z Li J G Nie and W Y Ji ldquoPositive and negative shear lagbehaviors of composite twin-girder decks with varying cross-sectionrdquo Science China Technological Sciences vol 60 no 1pp 116ndash132 2017

[2] S-J Zhou ldquoFinite beam element considering shear-lag effectin box girderrdquo Journal of Engineering Mechanics vol 136no 9 pp 1115ndash1122 2010

[3] Z Li and R K L Su ldquoAnalytical solutions for compositebeams with slip shear-lag and time-dependent effectsrdquo En-gineering Structures vol 152 pp 559ndash578 2017

[4] Q Z Luo Y M Wu Q S Li J Tang and G D Liu ldquoA finitesegment model for shear lag analysisrdquo Engineering Structuresvol 26 no 14 pp 2113ndash2124 2004

[5] P Z Lin F K Liu W Ji et al ldquoAnalysis on shear lag effect ofconcrete box beam by variational principlerdquo Journal of theChina Railway Society vol 35 no 2 pp 93ndash98 2013

[6] Y Y Zhang H Zhang W Li et al ldquoAnalysis on shear-lageffect of box girders based on different shear-lag generalizeddisplacementrdquo Journal of Railway Science and Engineeringvol 13 no 6 pp 1083ndash1090 2016

[7] S T Chang Q Zhang and S Zhang ldquoShear lag effect in singleplane cable-stayed bridgerdquo Advances in Structural Engi-neering vol 1 no 4 pp 301ndash306 1998

[8] G-F Wu and X Hong ldquoeoretical and experimental studyon shear lag effect of partially cable-stayed bridgerdquo Journal ofZhejiang University-Science A vol 6 no 8 pp 875ndash877 2005

[9] L Pengzhen Y Zijiang L Fengkui et al ldquoShear lag effect ofconcrete box girders considered varying flange-depthsrdquoJournal of Civil Architecturalamp Environmental Engineeringvol 35 no 1 pp 76ndash79 2013

[10] E Yamaguchi T Chaisomphob J Sa-Nguanmanasak andC Lertsima ldquoStress concentration and deflection of simplysupported box girder including shear lag effectrdquo StructuralEngineering and Mechanics vol 28 no 2 pp 207ndash220 2008

[11] X B Shen ldquoAnalysis of shear lag effect in the main girder ofwide low-pylon cable stayed bridgerdquo Construction Technologyvol 44 no 24 pp 76ndash79 2015

[12] L Xingmin ldquoAnalysis of influences of construction processeson shear lag effect of concrete box girdersrdquo Bridge Con-struction vol 43 no 1 pp 30ndash34 2013

[13] L Muyu W H Sun X D Sun et al ldquoStress analysis of widergirder of an extradosed cable-stayed bridge based on thelargest cantilever construction stagerdquo Journal of HuazhongUniversity of Science and Technology (Urban Science Edition)vol 27 no 2 pp 11ndash14 2010

[14] C Qianshu H Wenlong and Y Menggang ldquoAnalysis ofshear lag effect in construction stage of wide box girderextradosed cable-stayed bridge with large flangesrdquo Journal ofRailway Science and Engineering vol 15 no 12 pp 3158ndash3164 2018

[15] D R Li B M Wang and Y C Lin Structural Model Ex-periment Science Press Beijing China 1996

[16] J W Dali andW F Laili Experiment and Analysis of ResidualStress Ocean Press Beijing China 1987

[17] H Hossdort Structural Model Analysis China Architecture ampBuilding Press Beijing China 1986

[18] S K Mei Structural Experiments and Structural DesignChina Communications Press Beijing China 1980

[19] B Atmaca and S Ates ldquoConstruction stage analysis of three-dimensional cable-stayed bridgesrdquo Steel amp Composite Struc-tures vol 12 no 5 pp 413ndash426 2012

[20] S W Park M R Jung D J Min and M Y Kim ldquoCon-struction stage analysis of cable-stayed bridges using theunstrained element length methodrdquo Journal of e KoreanSociety of Civil Engineers vol 36 no 6 pp 991ndash998 2016

[21] M Gunaydin S Adanur A C Altunisik B Sevim andE Turker ldquoDetermination of structural behavior of Bosporussuspension bridge considering construction stages and dif-ferent soil conditionsrdquo Steel and Composite Structures vol 17no 4 pp 405ndash429 2014

[22] S Adanur M Gunaydin A C Altunisik and B SevimldquoConstruction stage analysis of humber suspension bridgerdquoApplied Mathematical Modelling vol 36 no 11 pp 5492ndash5505 2012

[23] J B Mander M J N Priestley and R Park ldquoeoreticalstress-strain model for confined concreterdquo Journal of Struc-tural Engineering vol 114 no 8 pp 1804ndash1826 1988

[24] Y A Khalid C L Chan B B Sahari and A M S HamoudaldquoBending behaviour of corrugated web beamsrdquo Journal ofMaterials Processing Technology vol 150 no 3 pp 242ndash2542004

[25] A Demir N Caglar H Ozturk and Y Sumer ldquoNonlinearfinite element study on the improvement of shear capacity inreinforced concrete T-Section beams by an alternative diag-onal shear reinforcementrdquo Engineering Structures vol 120pp 158ndash165 2016

16 Advances in Materials Science and Engineering

63 Variation Law of Shear Lag Effect of PC Cable-StayedBridge Main Beam along Longitudinal Direction e shearlag variations in the main beam of the single-cable PC cable-stayed bridge along the longitudinal direction can besummarized as follows

(1) In the construction stage at the cantilever end thecable force action point is the main recipient of axialforce e shear lag coefficient λN of the axial forceacting at this position reflects the actual force at thatpoint

(2) In the beam segments between the cable forces at thecable force action point the theoretical shear lagcoefficient λ determined by the ratio MN of thebending moment to axial force reflects the actualstress

(3) In the midspan beam section between the actionpoints of cable forces the shear lag coefficient ofbending moment λM is used to reflect the actualstress For the beam section between the action pointof cable force and the middle of the span the sectionshear lag coefficient is obtained by linearinterpolation

7 Conclusion

e structural control section stress at work in a typicalconstruction process was measured on a full cable-stayedbridge model in this study Sets of finite element and modeltest results were respectively analyzed and compared Ourconclusions can be summarized as follows

(1) In the construction stage the stress value of the cabletension area of the main beam upper edge markedlychanges after tensiling the cable force the shear lageffect is very prominent After a hanging basket loadis applied the main beams of certain sections presentalternating positive and negative shear lag

(2) e shear lag coefficient of the upper edge of themain beam in G1ndashG8 sections under WorkingConditions 1 and 4 was extracted from the spatialfinite element analysis for comparison to find thatthe experimental and theoretical results are in ac-cordance After tension of the cable force and theapplication of a hanging basket load positive andnegative shear lag effects alternate on the upper edgeof the box girder in certain areas As tension cableforce increases in each section the shear lag coef-ficient of the section upper edge decelerates

(3) In the construction stage at the cantilever end cableforce action point is mainly subjected to axial forcee shear lag coefficient λN of the axial force on thisposition reflects the actual force situation at thatpoint In the beam segments between the cableforces the shear lag coefficient λ determined by theratio of the bending moment to the axial force re-flects the actual stress at the cable force action pointIn the midspan beam section between the actionpoints of cable forces the shear lag coefficient λM ofthe bendingmoment reflects the actual stress For thebeam section between the cable force action pointand the middle of the span the section shear lagcoefficient can be obtained by linear interpolation

C2prime C3

4250kNmiddotm

17300kNmiddotm 19100kNmiddotm

4

C1prime C4

24400kNmiddotm

36800kNmiddotm

C4prime C3prime C2

10500kNmiddotm

18600kNmiddotm

C1

17700kNmiddotm

55000kNmiddotm

6820kNmiddotm

(a)

C1prime C4

13000kN7030kN

C4prime C1

5680kN

20300kN

3600kN

27000kN

C2prime C3

10900kN

C3prime C2

14900kN

4

(b)

Figure 15 Internal force of main girder on Pier 4 (Condition 1) (a) Moment diagram (b) Axial force diagram

Advances in Materials Science and Engineering 15

(4) e shear lag effect of the main beam is prominent inthe construction process e shear lag effect is mostobvious at the cable force operation point of thecantilever end As construction progresses and thecable force increases the shear lag effect of otherbeam sections decreases at each point except for thecantilever end

Data Availability

Part of the shear lag coefficient data used to support thefindings of this study are included within the article All ofthe shear lag coefficient data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors would like to thank the financial supports fromthe National Key RampD Program of China (2018YFC0809600and 2018YFC0809606) and Natural Science Foundation ofLiaoning Province (2019-MS-265)

References

[1] Z Li J G Nie and W Y Ji ldquoPositive and negative shear lagbehaviors of composite twin-girder decks with varying cross-sectionrdquo Science China Technological Sciences vol 60 no 1pp 116ndash132 2017

[2] S-J Zhou ldquoFinite beam element considering shear-lag effectin box girderrdquo Journal of Engineering Mechanics vol 136no 9 pp 1115ndash1122 2010

[3] Z Li and R K L Su ldquoAnalytical solutions for compositebeams with slip shear-lag and time-dependent effectsrdquo En-gineering Structures vol 152 pp 559ndash578 2017

[4] Q Z Luo Y M Wu Q S Li J Tang and G D Liu ldquoA finitesegment model for shear lag analysisrdquo Engineering Structuresvol 26 no 14 pp 2113ndash2124 2004

[5] P Z Lin F K Liu W Ji et al ldquoAnalysis on shear lag effect ofconcrete box beam by variational principlerdquo Journal of theChina Railway Society vol 35 no 2 pp 93ndash98 2013

[6] Y Y Zhang H Zhang W Li et al ldquoAnalysis on shear-lageffect of box girders based on different shear-lag generalizeddisplacementrdquo Journal of Railway Science and Engineeringvol 13 no 6 pp 1083ndash1090 2016

[7] S T Chang Q Zhang and S Zhang ldquoShear lag effect in singleplane cable-stayed bridgerdquo Advances in Structural Engi-neering vol 1 no 4 pp 301ndash306 1998

[8] G-F Wu and X Hong ldquoeoretical and experimental studyon shear lag effect of partially cable-stayed bridgerdquo Journal ofZhejiang University-Science A vol 6 no 8 pp 875ndash877 2005

[9] L Pengzhen Y Zijiang L Fengkui et al ldquoShear lag effect ofconcrete box girders considered varying flange-depthsrdquoJournal of Civil Architecturalamp Environmental Engineeringvol 35 no 1 pp 76ndash79 2013

[10] E Yamaguchi T Chaisomphob J Sa-Nguanmanasak andC Lertsima ldquoStress concentration and deflection of simplysupported box girder including shear lag effectrdquo StructuralEngineering and Mechanics vol 28 no 2 pp 207ndash220 2008

[11] X B Shen ldquoAnalysis of shear lag effect in the main girder ofwide low-pylon cable stayed bridgerdquo Construction Technologyvol 44 no 24 pp 76ndash79 2015

[12] L Xingmin ldquoAnalysis of influences of construction processeson shear lag effect of concrete box girdersrdquo Bridge Con-struction vol 43 no 1 pp 30ndash34 2013

[13] L Muyu W H Sun X D Sun et al ldquoStress analysis of widergirder of an extradosed cable-stayed bridge based on thelargest cantilever construction stagerdquo Journal of HuazhongUniversity of Science and Technology (Urban Science Edition)vol 27 no 2 pp 11ndash14 2010

[14] C Qianshu H Wenlong and Y Menggang ldquoAnalysis ofshear lag effect in construction stage of wide box girderextradosed cable-stayed bridge with large flangesrdquo Journal ofRailway Science and Engineering vol 15 no 12 pp 3158ndash3164 2018

[15] D R Li B M Wang and Y C Lin Structural Model Ex-periment Science Press Beijing China 1996

[16] J W Dali andW F Laili Experiment and Analysis of ResidualStress Ocean Press Beijing China 1987

[17] H Hossdort Structural Model Analysis China Architecture ampBuilding Press Beijing China 1986

[18] S K Mei Structural Experiments and Structural DesignChina Communications Press Beijing China 1980

[19] B Atmaca and S Ates ldquoConstruction stage analysis of three-dimensional cable-stayed bridgesrdquo Steel amp Composite Struc-tures vol 12 no 5 pp 413ndash426 2012

[20] S W Park M R Jung D J Min and M Y Kim ldquoCon-struction stage analysis of cable-stayed bridges using theunstrained element length methodrdquo Journal of e KoreanSociety of Civil Engineers vol 36 no 6 pp 991ndash998 2016

[21] M Gunaydin S Adanur A C Altunisik B Sevim andE Turker ldquoDetermination of structural behavior of Bosporussuspension bridge considering construction stages and dif-ferent soil conditionsrdquo Steel and Composite Structures vol 17no 4 pp 405ndash429 2014

[22] S Adanur M Gunaydin A C Altunisik and B SevimldquoConstruction stage analysis of humber suspension bridgerdquoApplied Mathematical Modelling vol 36 no 11 pp 5492ndash5505 2012

[23] J B Mander M J N Priestley and R Park ldquoeoreticalstress-strain model for confined concreterdquo Journal of Struc-tural Engineering vol 114 no 8 pp 1804ndash1826 1988

[24] Y A Khalid C L Chan B B Sahari and A M S HamoudaldquoBending behaviour of corrugated web beamsrdquo Journal ofMaterials Processing Technology vol 150 no 3 pp 242ndash2542004

[25] A Demir N Caglar H Ozturk and Y Sumer ldquoNonlinearfinite element study on the improvement of shear capacity inreinforced concrete T-Section beams by an alternative diag-onal shear reinforcementrdquo Engineering Structures vol 120pp 158ndash165 2016

16 Advances in Materials Science and Engineering

(4) e shear lag effect of the main beam is prominent inthe construction process e shear lag effect is mostobvious at the cable force operation point of thecantilever end As construction progresses and thecable force increases the shear lag effect of otherbeam sections decreases at each point except for thecantilever end

Data Availability

Part of the shear lag coefficient data used to support thefindings of this study are included within the article All ofthe shear lag coefficient data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors would like to thank the financial supports fromthe National Key RampD Program of China (2018YFC0809600and 2018YFC0809606) and Natural Science Foundation ofLiaoning Province (2019-MS-265)

References

[1] Z Li J G Nie and W Y Ji ldquoPositive and negative shear lagbehaviors of composite twin-girder decks with varying cross-sectionrdquo Science China Technological Sciences vol 60 no 1pp 116ndash132 2017

[2] S-J Zhou ldquoFinite beam element considering shear-lag effectin box girderrdquo Journal of Engineering Mechanics vol 136no 9 pp 1115ndash1122 2010

[3] Z Li and R K L Su ldquoAnalytical solutions for compositebeams with slip shear-lag and time-dependent effectsrdquo En-gineering Structures vol 152 pp 559ndash578 2017

[4] Q Z Luo Y M Wu Q S Li J Tang and G D Liu ldquoA finitesegment model for shear lag analysisrdquo Engineering Structuresvol 26 no 14 pp 2113ndash2124 2004

[5] P Z Lin F K Liu W Ji et al ldquoAnalysis on shear lag effect ofconcrete box beam by variational principlerdquo Journal of theChina Railway Society vol 35 no 2 pp 93ndash98 2013

[6] Y Y Zhang H Zhang W Li et al ldquoAnalysis on shear-lageffect of box girders based on different shear-lag generalizeddisplacementrdquo Journal of Railway Science and Engineeringvol 13 no 6 pp 1083ndash1090 2016

[7] S T Chang Q Zhang and S Zhang ldquoShear lag effect in singleplane cable-stayed bridgerdquo Advances in Structural Engi-neering vol 1 no 4 pp 301ndash306 1998

[8] G-F Wu and X Hong ldquoeoretical and experimental studyon shear lag effect of partially cable-stayed bridgerdquo Journal ofZhejiang University-Science A vol 6 no 8 pp 875ndash877 2005

[9] L Pengzhen Y Zijiang L Fengkui et al ldquoShear lag effect ofconcrete box girders considered varying flange-depthsrdquoJournal of Civil Architecturalamp Environmental Engineeringvol 35 no 1 pp 76ndash79 2013

[10] E Yamaguchi T Chaisomphob J Sa-Nguanmanasak andC Lertsima ldquoStress concentration and deflection of simplysupported box girder including shear lag effectrdquo StructuralEngineering and Mechanics vol 28 no 2 pp 207ndash220 2008

[11] X B Shen ldquoAnalysis of shear lag effect in the main girder ofwide low-pylon cable stayed bridgerdquo Construction Technologyvol 44 no 24 pp 76ndash79 2015

[12] L Xingmin ldquoAnalysis of influences of construction processeson shear lag effect of concrete box girdersrdquo Bridge Con-struction vol 43 no 1 pp 30ndash34 2013

[13] L Muyu W H Sun X D Sun et al ldquoStress analysis of widergirder of an extradosed cable-stayed bridge based on thelargest cantilever construction stagerdquo Journal of HuazhongUniversity of Science and Technology (Urban Science Edition)vol 27 no 2 pp 11ndash14 2010

[14] C Qianshu H Wenlong and Y Menggang ldquoAnalysis ofshear lag effect in construction stage of wide box girderextradosed cable-stayed bridge with large flangesrdquo Journal ofRailway Science and Engineering vol 15 no 12 pp 3158ndash3164 2018

[15] D R Li B M Wang and Y C Lin Structural Model Ex-periment Science Press Beijing China 1996

[16] J W Dali andW F Laili Experiment and Analysis of ResidualStress Ocean Press Beijing China 1987

[17] H Hossdort Structural Model Analysis China Architecture ampBuilding Press Beijing China 1986

[18] S K Mei Structural Experiments and Structural DesignChina Communications Press Beijing China 1980

[19] B Atmaca and S Ates ldquoConstruction stage analysis of three-dimensional cable-stayed bridgesrdquo Steel amp Composite Struc-tures vol 12 no 5 pp 413ndash426 2012

[20] S W Park M R Jung D J Min and M Y Kim ldquoCon-struction stage analysis of cable-stayed bridges using theunstrained element length methodrdquo Journal of e KoreanSociety of Civil Engineers vol 36 no 6 pp 991ndash998 2016

[21] M Gunaydin S Adanur A C Altunisik B Sevim andE Turker ldquoDetermination of structural behavior of Bosporussuspension bridge considering construction stages and dif-ferent soil conditionsrdquo Steel and Composite Structures vol 17no 4 pp 405ndash429 2014

[22] S Adanur M Gunaydin A C Altunisik and B SevimldquoConstruction stage analysis of humber suspension bridgerdquoApplied Mathematical Modelling vol 36 no 11 pp 5492ndash5505 2012

[23] J B Mander M J N Priestley and R Park ldquoeoreticalstress-strain model for confined concreterdquo Journal of Struc-tural Engineering vol 114 no 8 pp 1804ndash1826 1988

[24] Y A Khalid C L Chan B B Sahari and A M S HamoudaldquoBending behaviour of corrugated web beamsrdquo Journal ofMaterials Processing Technology vol 150 no 3 pp 242ndash2542004

[25] A Demir N Caglar H Ozturk and Y Sumer ldquoNonlinearfinite element study on the improvement of shear capacity inreinforced concrete T-Section beams by an alternative diag-onal shear reinforcementrdquo Engineering Structures vol 120pp 158ndash165 2016

16 Advances in Materials Science and Engineering