Research ArticleDynamic Response and Robustness Evaluation ofCable-Supported Arch Bridges Subjected to Cable Breaking

Guotao Shao 12 Hui Jin 2 Ruinian Jiang 3 and Yue Xu 1

1School of Highway Changrsquoan University Xirsquoan Shanxi China2Collage of Civil Engineering and Architecture Taizhou University Taizhou Zhejiang China3Department of Engineering and Technology and Surveying Engineering New Mexico State University Las Cruces NM USA

Correspondence should be addressed to Hui Jin jinhuitzceducn

Received 29 October 2020 Revised 10 January 2021 Accepted 29 January 2021 Published 9 February 2021

Academic Editor Salvatore Caddemi

Copyright copy 2021 Guotao Shao 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

Cable-supported arch bridges have had many cable break accidents which led to dramatic deck damage and even progressivecollapse To investigate the dynamic response and robustness of cable-supported arch bridges subjected to cable breakingnumerical simulation methods were developed compared and analyzed and an effective and accurate simulation method waspresented (en the cable fracture of a prototype bridge was simulated and the dynamic response of the cable system deck andarch rib was illustrated Finally the robustness evaluation indexes of the cable system deck and arch rib were constructed andtheir robustness was evaluated (e results show that the dynamic response of the adjacent cables is proportional to the length ofthe broken cable while the dynamic response of the deck is inversely proportional to the length of the broken cable (e dynamicamplification factor of the cable tension and deck displacement is within 20 while that of the arch rib bending moment exceeds20(e break of a single cable will not trigger progressive collapseWhen subjected to cable breaking the deck system has the leastrobustness (e proposed cable break simulation procedure and the robustness evaluation method are applicable to both existingand new cable-supported bridges

1 Introduction

Cables are the key components of cable-supported archbridges Due to corrosion abrasion and fatigue many cablefracture accidents of cable-supported arch bridges happenedin the past few years For example the deck of Sichuan YibinSouth-Gate Bridge collapsed into river after the fracture ofits eight short cables in 2001 and the Taiwan NanfangrsquoaoBridge encountered a progressive collapse triggered by cablefracture in 2019 To prevent the accidental event of cablebreaking many codes have introduced an accidental loadfor example the Post Tension Institution (PTI) [1] rec-ommends using a dynamic amplification factor (DAF) forcable-stayed bridges However the values of DAF varyamong the codes some of which have been found to beunreasonable in some cases

For determining rational values of DAF many re-searches have been conducted Zoli and Woodward [2]pointed out that the result of an equivalent static analysis

with a DAF of 20 has a larger deviation than that of adynamic analysis Ruiz-Teran and Aparicio [3] indicatedthat the DAFs of some components were larger than 20 Inthe research of cable breaking of cable-stayed bridges bothMozos [4 5] and Zhou and Chen [6] concluded that anequivalent static analysis with a DAF of 20 cannot capturethe extreme values of both moment and stress and shouldnot be used in the response analysis of cable breakage events(ere are also some studies that support the equivalent staticmethod using DAF for example Park et al [7] found thatthe DAF values were below 15 for the critical sections in thecable rupture of Seohae Bridge and Cai et al [8] pointed outthat the DAF of 20 is a good estimate for an equivalent staticanalysis procedure for cable breakage of the studied cable-stayed bridges Due to the varying DAF results further studyon DAF values for cable-supported arch bridges is needed

While dynamic analysis methods are recommended forsimulation of cable breaking some critical issues in thedynamic analysis procedure need further study including

HindawiShock and VibrationVolume 2021 Article ID 6689630 14 pageshttpsdoiorg10115520216689630

the cable loss mode [9] cable loss duration [3 10] and theeffect of the material and geometry nonlinearity[6 8 10ndash12] Some experimental studies have been con-ducted such as the seven-wire steel strand specimenbreakage experiment by Mozos and Aparicio [13] and theimpact fracture test of a single steel wire by Hoang et al [14](ese studies were based on specimens which did not reflectthe response of cable-supported bridge structures

(e structural performance under component lossscenarios is often evaluated by robustness and redundancyRobustness is defined as the ability to withstand a given levelof damage without suffering degradation or loss of functionRedundancy is defined as the quality of having alternativepaths by which forces can be transferred [15] (e study ofrobustness evaluation of cable-supported bridges subjectedto cable breaking has been very limited Wolff and Starossek[16] and Cai et al [8] concluded respectively that the failureof one cable would cause significant bendingmoments in thestiffening girder but would not lead to a zipper-type pro-gressive collapse Hoang et al [14] pointed out that aprogressive collapse would happen when four cables break inthe studied cable-stayed bridge (e influences of cablecorrosion and fatigue in the progressive collapse were alsotaken into consideration byWu et al [17] and Morgese et al[18] Shoghijavan et al [19 20] derived an approximationfunction for the stress increase ratio of the adjacent cablesand found that the robustness can be improved by adjustingthe rigidity of the main girder at different positionsDomaneschi et al [15 21] used the applied element methodin the study of disproportionate collapse of cable-stayedbridges and found that the structural redundancy was astrategic measure for avoiding disproportionate collapse andimproving robustness No guidance has been developed forcarrying out detailed robustness evaluation or optimizationof cable-supported arch bridges subjected to cable breaking

Most researchers have focused on studying the ruptureof cables in cable-stayed bridges However different fromcable-stayed bridges that have strong longitudinal stiffeninggirders cable-supported arch bridges always use a grid deckwith transverse girders in which cable rupture may lead toserious damage to the bridges

(e main aim of this paper is to investigate the dynamicresponse and robustness of cable-supported arch bridgessubjected to cable breaking (e geometry and numericalmodel of a prototype cable-supported arch bridge are de-scribed in Section 2(ree key issues of cable loss simulationare studied and discussed in Section 3 (e dynamic re-sponses of the cable system deck and arch rib are dem-onstrated and discussed in Section 4 (e robustnessevaluation indexes are calculated and applied in the bridgerobustness evaluation in Section 5

2 System and Modeling

Cable-supported arch bridges have excellent spanningability which always use half-through trusses and float decksystems(ere are hundreds of cable-supported arch bridgeswith a span of more than 200m worldwide Two examplesare the Bayonne Bridge with a span of 5036m between New

Jersey and New York and the Wushan Yangtze River Bridgewith a span of 492m in Chongqing China

(e deck system of long-span arch bridges usuallyadopts longitudinal beams simply supported on strangertransverse beams or transverse beams fixed on strangerlongitudinal beams If a simply supported deck is used thebeam will directly collapse after the cable breaks And if afixed deck is used its response under the cable fracture issimilar to that of a cable-stayed bridge (erefore theJiantiao Bridge was selected in this study which is a large-span arch bridge with decks composed of box-shaped beamsand flange plates

21 Bridge System (e Jiantiao Bridge is a half-throughconcrete-filled steel tubular (CFST) arch bridge with a spanof 245m as shown in Figure 1 (e deck width is 228m andthe distance between the transverse girders is 59m (centerto center) Double cables are used on each end of thetransverse girders (e arch axis is a quadratic parabola witha rise-to-span ratio of 15

(e arch rib is a CFST truss (e cross section of theupper and lower chords has a flat dumbbell shape and theouter diameter of the circular tube is 800mm (e arch footis protected by outer concrete as shown in Figure 1 sectionsIII-III (e prestressed concrete transverse girders have abox section with wide flanges that are used as part of thedeck

(e cable system consists of 55 galvanized high-strengthsteel wires with a diameter of 7mm Cables were installed onthe upstream and downstream sides of the transversegirders(e downstream cables are numbered S1 to S60 andthe upstream cables are numbered X1 to X60 (e distancebetween the upstream and downstream cables is 174m asshown in sections I-I in Figure 1 (ere are two cables (apair) anchored on each side of every transverse girder with adistance of 15m (e transverse girders are connected witha 29m wide cast-in-place concrete slab as shown in Fig-ure 1 (ere are 120 cables in total (e end cables (S1 S2X1 X2 S59 S60 X59 and X60) are anchored to the upperchord of the arch rib while the remaining cables are an-chored to the lower chord of the arch rib Cables S3 X3 S58and X58 are the shortest cables Due to the symmetry of thestructure the breakage of downstream S1 to S30 cables wasstudied in this paper

22 FE Model A three-dimension (3D) finite element (FE)model of Jiantiao Bridge was built using ANSYS in whichthe arch rib columns and cross braces were modeled withbeam elements and the cables were modeled with trusselements (e transverse girders and the deck were modeledwith shell elements

(e boundary conditions of the FE model used a fixedarch foot and the cables were hinged to the anchorage pointson the transverse girders and the arch rib To simulate theboundary conditions of the deck the contact pairs were usedbetween the expansion joints (e material properties of thearch rib transverse girders and steel wires are listed inTable 1

2 Shock and Vibration

To consider the material nonlinearity a multilinearisotropic hardening criterion in ANSYS was chosen for theconcrete material and a bilinear isotropic reinforcementcriterion was chosen for the steel material (e steel wires ofthe cables were considered to be linearly elastic (e geo-metric nonlinearity of the structure was also considered byapplying a large deformation effect

(e analysis process of the cable fracture started with astatic analysis to obtain the initial state and a transient analysisfor cable breaking was then conducted After cable breakage alonger transient analysis with 50 seconds was used to trace thepostbreakage dynamic response(e cable breakage simulationmethod is presented in Section 3 Considering the fact thatmost of the cable break accidents occurred because of cor-rosion the dynamic analysis of cable fracture was performedunder the dead load Because cable breakage is a local effect andthe previous cable breakage accidents did not cause arch bridgeinstability this study does not address the stability issues

23 Validation of the FEModel To verify the accuracy of theFE model the simulated cable tensions under the dead loadwere compared with those measured in the field (e on-site

cable forces were measured by a cable force meter in whichan acceleration sensor was used to measure the naturalfrequency of cable and then converted the natural frequencyto cable force (e relationship between natural frequencyand cable force was expressed as

T 4mLfn

n1113888 1113889

2

minusn2π2

EI

L2 (1)

where T is the cable force m is the linear density L is theeffective length fn is the nth natural frequency and EI is thebending stiffness of the cable

Figure 2 shows the comparison of the cable forces betweenthe FE model and the field measurement (e average value ofthe cable forces calculated by the FE model is 3677 kN whilethat of the field measurement is 3736 kN(e results of the FEmodel are close to the actual state of the bridge and themaximum difference between the two is within 5

3 Cable Break Simulation Method

31 Cable Break Mode (ere are two main methods forsimulating cable breaks One method is to remove the

Cable Cable

Precast

Cable Cable

(Unit m)

(Unit m)

Precast

Cast-in-place Cast-in-place15

30

I-III-II

11

174

1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

I

I

29 times 59 895 times 59 5 times 5989

III

II

II

III23

36

48

III-III

29 30

15 Cast-in-place

Figure 1 Configuration of the prototype cable-supported arch bridge

Table 1 Properties of the materials in the prototype arch bridge

Properties Unit Steel Concrete Cable steel wireWeight density kNm3 78 25 75Elasticity modulus GPa 210 345 195Poissonrsquos ratio μ 03 02 03Compressive strength MPa 210 324 mdashTensile strength MPa 210 264 1670

Shock and Vibration 3

broken cable element and use the equivalent functionsHoang et al [14] and Olamigoke [22] simulated the cablebreakage by using an impact waveform after removing thecable elements Wolff and Starossek [16] and Park et al [7]simulated the cable breakage by using a transient load to theanchorage nodes Zoli and Woodward [2] simulated a cableloss event by using an abrupt drop in a forcing functionZhou and Chen [6] demonstrated that it is more accurate toremove the broken cable element in the static analysis stagebefore the dynamic analysis (e other method is to directlyremove the cable element or reduce the cable element areaMozos [4 13] and Wu et al [17] simulated a cable loss bydirectly reducing the area of the damaged cable in the firststep of a dynamic analysis (e main difference between thetwo methods lies in whether or not using a force function torepresent the cable force Zhou and Chen [10] used a re-duced cable area and a force function to simulate the cablefracture process but ignored the fact that the loss of cablearea is not proportional to the loss of force which wasdemonstrated by Xu et al [23]

When a force function is used the form of the functiondetermines the accuracy of the calculation Ruiz-Teran andAparicio [3] compared seven types of force functions andpointed out that when the breakage time is sufficiently shortthe maximum response is irrespective of the way in whichthe cable breaks To avoid an inappropriate function directlyremoving the damaged cables in a suitable cable loss du-ration is less error-prone and consistent with the actualsituation

It should be noted that removing an element directly isappropriate for simulating fractures caused by rust andfatigue however it is not applicable to the situation wherethe cable fractured by hitting Because most of the long-spanarch bridge cable fractures are caused by corrosion andfatigue the mode of directly removing the failed cable ischosen in this paper to simulate the cable loss

32 Cable Break Duration (ere is currently limited re-search on the duration of cable breakage Fracture experi-ments were conducted by Mozos and Aparicio [13] on aseven-wire strand of a specific steel type the result showsthat the average rupture time is 00055 s(e impact fracturetest by Hoang et al [14] on a single wire indicated that theduration of the cable breaking process is 0017 s Because thecable types differ for different bridges the fracture durationvaries

A parametric analysis was carried out for determining areasonable cable break duration for the arch bridge in thisstudy Six break durations of 001 s 01 s 10 s 20 s 40 s and100 s were selected and the tension of S30 caused by thecable break of S29 was used for comparison

Figure 3 depicts the dynamic response for each cablebreak duration As observed the maximum dynamic re-sponse was reached in the durations of 001 s 01 s and 10 sWhen the fracture duration increases from 20 s to 100 s thepeak response gradually decreases (e dynamic curves of20 s 40 s and 100 s show that the cable force increaseslinearly in the initial stage which reflected the shortage ofthe direct-removing-cable method It also shows that if thefracture duration is less than 10 s the dynamic responsecurves almost coincide and the maximum response isconsistent which was also demonstrated by Ruiz-Teran andAparicio [3] (erefore the abrupt breakage duration wasassumed to be 01 s in this study

33 Damping Coefficient In long-span cable-supportedbridges the effect of damping on the dynamic response issignificant (e Rayleigh damping function is defined via thefollowing equations as

[C] α[M] + β[K] (2)

α ζ2ωiωj

ωi + ωj

(3)

β ζ2

ωi + ωj

(4)

where the damping matrix C is proportional to the massmatrixM and the stiffness matrix K α is the mass factor andβ is the stiffness factor α and β can be computed based onthe undamped modal frequencies ωi and ωj and the pre-scribed damping ratio ζ For cable-supported bridges thestructural vibration excited by cable rupture is a local vi-bration around the broken cable Wolff and Starossek [16]pointed out that the high-order modes characterized by alocal vibration should be considered in Rayleigh dampingZoli and Woodward [2] recommend taking at least 20thfrequency (preferably 50th frequency) to calculate α and β(rough the modal analysis of the Jiantiao Bridge it wasfound that the local vibration corresponds to modes greaterthan the 30th order To obtain the appropriate dampingcoefficient the damping ratios of the 1st 2nd 20th 50thand 100th natural frequencies were selected and the value ofζ was set as 5

FE model

Field test

0

100

200

300

400

500

Tens

ion

(kN

)

5 10 15 20 25 300Cable number

Figure 2 Comparison between the calculated and measured cableforces

4 Shock and Vibration

Figure 4 displays the damping ratio curves obtainedunder each frequency combination (e damping ratio ζ12which was calculated using ω1 and ω2 showed significantsuppression of high-order vibrations (e value of ζ12 in the50th order mode is 22 and the value of the 100th order is33 which greatly exceeded the design value of 5 (edamping ratios ζ120 and ζ150 show a better fit within the first50th order (e damping ratio ζ1100 keeps at 3ndash5 on ahigh-order frequency which matches the damping char-acteristics of reinforced concrete structures therefore ζ1100was chosen as the damping ratio in this study

4 Dynamic Response Results

41 Tension in Adjacent Cables Figure 5 presents the ad-jacent cable tension time-history curves subjected to thebreakage of short cables S1(5a) and S2 (5b) respectively Itcan be observed in Figures 5(a) and 5(b) that the breakage ofcable S1 has the largest impact on cable S2 while thebreakage of cable S2 has the largest impact on cable S1 (isindicates that after a single cable breaks the closest cablereceives the maximum dynamic impact (e maximumdynamic response is 22 times as big as the cable tensionunder the dead load (e breaking of the downstream cableshas little effect on the cables on the upstream side (X1 and X2in the figure) and the maximum dynamic response of thecables on the same side is within 12 times of the tensionunder the dead load

Figure 6 illustrates the adjacent cable tension time-history curves subjected to the break of long cables S29 (6a)and S30 (6b) respectively It can be observed that thebreakage of cable S29 has the largest impact on cable S30and that of cable S30 has the largest impact on cable S29which is similar to the short-cable cases However themaximum dynamic response of the cable tension was 17times the tension of the dead load for the long-cablebreakage and the peak value of the dynamic response is

significantly smaller than that of the short cables (ebreakage of long cables results in a greater dynamic responseof the farther cables for example the maximum response ofS28 which was anchored on the adjacent transverse grid is14 times of the dead-load tension when cable S29 breaks(e effect on the cables on the upstream side is slight whichis the same as that in the short-cable cases

Figure 7 depicts the maximum tension in the adjacentcables caused by a single cable break It can be observed thatthe maximum cable force of 821 kN was caused by the breakof S3 which is the shortest cable As the length of the cableincreases the impact effect due to the break of the cablegradually decreases For the tension of all cables caused by asingle cable break the maximum value is 16 to 22 times asbig as the tension under the dead load

42 Displacement of theDeck (e displacement of the lowercable anchorage points was selected to investigate the vi-bration of the deck (e displacement time-history curves ofthe deck due to the break of a short cable (S1 and S2) and along cable (S29 and S30) are presented in Figures 8 and 9respectively

Figure 8 shows the displacement time-history curves dueto the break of cables S1 (8a) and S2 (8b) respectively Asshown in Figure 8 the breakage of a cable causes themaximum dynamic displacement at its own anchor point(e maximum displacement of S1 is minus113mm and that ofS2 is minus119mm (e displacement at the anchorage point ofthe adjacent cables located at the same transverse girder isslightly smaller than that of the broken cable (e dis-placements of the anchorage points that are not on the sametransverse girder of the broken cable are significantlysmaller

Figure 9 shows the displacement time-history curves dueto the break of long cables S29 (9a) and S30 (9b) respec-tively Compared with the short-cable break the long-cablebreak also caused the largest displacement at its own an-chorage point but the deck displacement was larger (emaximum displacement of S29 is minus277mm and S30 isminus279mm Moreover the maximum displacement of theanchorage points of the adjacent cables anchored on dif-ferent transverse girders also exceeds minus10mm

(e maximum deck displacement caused by thebreakage of each cable is shown in Figure 10 It is noticedthat the minimum deck displacement is 84mm which iscaused by the break of the shortest cable (S3) and themaximum deck displacement is 279mm which is caused bythe break of the longest cable (S30) (e displacement of thebridge deck almost increases proportionally with the lengthof the cable

43Moments of theArch Rib To determine the impact effectof cable fracture on the arch rib the bending moment time-history curves at the six nodes around the upper anchoragepoint of the broken cable were selected (e six rigid framenode bending moments are the upper chord node bendingmomentMT lower chord node bending momentMB upperleft node bending moment MLT lower left node bending

001s01s10s

20s40s100s

350

400

450

500

550

600

Tens

ion

of th

e adj

acen

t cab

le (k

N)

5 10 15 20 25 300Time (s)

Figure 3 Dynamic response to six cable break durations

Shock and Vibration 5

moment MLB upper right node bending moment MRT andlower right node bending moment MRB

Figure 11 depicts the moment time-history curves due tothe breakage of short cables S1 (11a) and S2 (11b) re-spectively Because cables S1 and S2 are anchored on theupper chord of the arch rib the dynamic response variationof MT is the most significant (e maximum changes of MTare 135 kNmiddotm due to the break of S1 and 150 kNmiddotmdue to thebreak of S2 (e bending moment changes at other positionsare small all are within 100 kNmiddotm

Figure 12 indicates the bending moment time-historycurves due to the breakage of long cables S29 (12a) and S30

(12b) respectively Because cables S29 and S30 are anchoredon the lower chord of the arch rib the variation of thedynamic response of MB is the most significant (e max-imum changes of MB are 168 kNmiddotm due to the break of S29and 170 kNmiddotmdue to the break of S30(emaximum changeof MLB and MRB is 145 kNmiddotm and the changes of MLT MTand MRT are within 100 kNmiddotm

(e bending moment in the upper and lower chords ofthe arch rib is investigated Figure 13 displays the envelope ofthe bending moments of the arch rib sections caused by thebreakage of each cable (e bending moment of the upperchord sections is between minus100 kNmiddotm and 210 kNmiddotm and

Dam

ping

ratio

ζ (

)

ζ12

ζ120

ζ150

ζ1100

ω1ω2 ω20 ω50 ω100

2 4 6 8 100Natural frequencies ω (Hz)

0

10

20

30

40

50

Figure 4 Damping ratio curves obtained under four frequency combinations

S2S3

X1X2

300

400

500

600

700

Tens

ion

of ca

bles

(kN

)

10 20 30 40 500Time (s)

(a)

S2S3

X1X2

300

400

500

600

700

Tens

ion

of ca

bles

(kN

)

10 20 30 40 500Time (s)

(b)

Figure 5 Tension time-history curves due to short-cable fracture (a) break of cable S1 (b) break of cable S2

6 Shock and Vibration

that of the lower chord is between minus100 kNmiddotm and 450 kNmiddotm(is indicates that the bending moment of the lower chordof the arch rib is greater than that of the upper chord

In order to reflect the variation amplitude of the arch ribbending moment during the breakage of each cable Fig-ure 14 shows the bending moment variation of the upperand lower chords of the arch rib caused by a single cablebreak (e variation of the bending moment of the upperchord is between 113 kNmiddotm and 188 kNmiddotm with an averagevalue of 137 kNmiddotm and the bending moment variation of thelower arch rib is between 100 kN and 280 kNmiddotm with anaverage value of 229 kNmiddotm Because most cables are an-chored to the lower chord except S1 and S2 the bendingmoments of the lower chord vary more than that of theupper chord Moreover as the length of the cable increasesthe fracture impact effect increases

44 DAF of the Cable Deck and Rib No specifications haveprovided details for calculating the DAF of cable-supportedarch bridges For the design of cable-stayed bridges the DAFis set between 15 and 20 in some codes and recommen-dations In this section the cable failure of the JiantiaoBridge is investigated to discuss the DAF for the cabletension deck displacement and arch rib bendingmoment ofcable-supported arch bridges Equation (1) is used to cal-culate DAF

DAF RD minus R0

RP D minus R0 (5)

where RD represents themaximum dynamic response R0 theinitial state under the dead load before cable break and RPDthe stable value after cable break

S28S30

X29X30

300

350

400

450

500

550

600

650Te

nsio

n of

cabl

es (k

N)

10 20 30 40 500Time (s)

(a)

S29S31

X29X30

10 20 30 40 500Time (s)

300

350

400

450

500

550

600

650

Tens

ion

of ca

bles

(kN

)(b)

Figure 6 Time-history curves due to long-cable fracture (a) break of cable S29 (b) break of cable S30

0

200

400

600

800

Tens

ion

of ca

bles

(kN

)

5 10 15 20 25 300Cable number

Figure 7 Maximum tension of the adjacent cables caused by the break of a single cable

Shock and Vibration 7

Figure 15 shows the calculated DAF for the cable tensiondue to the break of the adjacent cable It can be observed thatthe DAF of the adjacent cable tension is distributed between14 and 16 with an average value of 154

Figure 16 presents the DAF for the deck displacementwhich is between 130 and 165 with an average value of 140(ere is a tendency for the deck displacement DAF to de-crease as the cable length increases

Figure 17 displays the DAF for the arch rib bendingmoment which is between 15 and 24 with an average value

of 193 (e DAF exceeds 20 in some cases which is therecommended value of PTI [1]

Compared with the recommended common DAF valueof 20 for cable-stayed bridges by PTI [1] the DAF of thecable-supported arch bridge in this study varies for eachcomponent (e maximum DAFs of the cable tension anddeck displacement are less than 20 while the maximumDAF of the bending moment of the arch rib exceeds 20(isshows that adopting a common DAF for all the componentsof cable-supported arch bridge structures is not advisable

S1S2S3

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4D

eck

disp

lace

men

t (m

m)

10 20 30 40 500Time (s)

(a)

S1S2S3

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

Dec

k di

spla

cem

ent (

mm

)

10 20 30 40 500Time (s)

(b)

Figure 8 Displacement time-history curves of the deck (a) cable S1 break (b) cable S2 break

S28S29S30

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

Dec

k di

spla

cem

ent (

mm

)

10 20 30 40 500Time (s)

(a)

S29S30S31

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

Dec

k di

spla

cem

ent (

mm

)

10 20 30 40 500Time (s)

(b)

Figure 9 Displacement time-history curves of the deck (a) cable S29 break (b) cable S30 break

8 Shock and Vibration

5 Robustness Evaluation

For long-span cable-supported arch bridges the failure of asingle cable may trigger progressive fracture of adjacentcables and damage to the deck and arch ribs (erefore it isnecessary to evaluate the robustness of cable-supported archbridges subjected to cable breaking (is section discussesthe robustness evaluation of the cable system deck and archrib subjected to a single cable break

51 Robustness of the Cable System To study the possibilityof a progressive fracture of adjacent cables subjected to asingle cable break the impact effect of the cable breakage is

investigated and compared with the carrying capacity of thecable system (e cable system of the Jiantiao Bridge iscomposed of 55 parallel steel wires that have a diameter of7mm and a tensile strength of 1670MPa (e cablersquos ulti-mate bearing capability fu is 3530 kN which is much biggerthan the maximum dynamic load of 821 kN caused by asingle cable break (is indicates that a single cable breakageunder the dead load will not trigger a progressive fracture ofthe adjacent cables

Furthermore to consider the load combination with ref-erence to the PTI [1] recommendation and the study by Zoliand Woodward [2] the limit state design combination Td andthe accidental combination Tf of the cable failure are defined as

0

5

10

15

20

25

30

35

Max

imum

disp

lace

men

t (m

m)

5 10 15 20 25 300Cable number

Figure 10 Maximum displacement of the deck caused by a single cable break

0

MLTMTMRT

MLBMBMRB

ndash200

ndash100

100

200

300

Mom

ent o

f arc

h rib

(kN

middotm)

10 20 30 40 500Time (s)

(a)

MLTMTMRT

MLBMBMRB

ndash200

ndash100

0

100

200

300M

omen

t of a

rch

rib (k

Nmiddotm

)

10 20 30 40 500Time (s)

(b)

Figure 11 Bending moment time-history curves of the arch rib (a) cable S1 break (b) cable S2 break

Shock and Vibration 9

Td 12Ts + 14Tv + 075 Tp + Tt1113872 1113873

Tf 12Ts + 075 Tv + Tp + Tt1113872 1113873 + Tc(6)

where Ts Tv Tp and Tt are the tension of the dead loadvehicle load pedestrian load and temperature load re-spectively Tc is the net dynamic effect of the cable breakwhich is equal to the maximum tension of the dynamicresponse minus the cable tension under the dead load

Figure 18 shows the comparison of the design limitcombination Td and the accidental combination Tf of each

cable It is observed that the values of cable fracture ac-cidental combination are all higher than the carryingcapability limit state combination (is means that thecable breakage accidental combination is the most un-favorable situation and should not be ignored in thedesign stage

(e maximum value of the accidental combination Tf is1130 kN and the minimum safety factor of the cablecompared with the ultimate bearing capability fu is 31 Itshows that the cable system has good robustness under acable breakage accidental combination and a single cable

MLTMTMRT

MLBMBMRB

0

100

200

300

400

500M

omen

t of a

rch

rib (k

Nmiddotm

)

10 20 30 40 500Time (s)

(a)

MLTMTMRT

MLBMBMRB

10 20 30 40 500Time(s)

0

100

200

300

400

500

Mom

ent o

f arc

h rib

(kN

middotm)

(b)

Figure 12 Bending moment time-history curves of the arch rib (a) cable S29 break (b) cable S30 break

Midspan

MaxMin

ndash100

0

100

200

300

Mom

ent o

f upp

er ch

ords

(kN

middotm)

5 10 15 20 25 300Cable number

(a)

MaxMin

Midspan

ndash100

0

100

200

300

400

500

Mom

ent o

f low

er ch

ords

(kN

middotm)

5 10 15 20 25 300Cable number

(b)

Figure 13 Envelope of the arch rib bending moment subjected to a single cable break (a) upper chords (b) lower chords

10 Shock and Vibration

break will not trigger a progressive breakage of adjacentcables

52 Robustness of the Deck Different from cable-stayedbridges that have a strong longitudinal stiffening girdercable-supported arch bridge deck systems always usetransverse girders as the main load-bearing members Toevaluate the damage to the deck system the capability of thedeck was obtained by a pushover analysis

(e pushover model was composed of three transversegirders with its flange plate An elastic support was used tosimulate the rigidity of the cables at the anchoring positions(e horizontal side displacement of the deck was notallowed by fixing the left and right ends of the deck system toavoid any rigid movement (e vertical pushover load wasapplied to the anchored position of the broken cable on the

right side of the middle girder as indicated by the arrow inFigure 19 (e relative displacements between the transversegirders were chosen as the capability index to represent thebearing capacity of the deck system

Figure 20 depicts the load-displacement curve of thedeck It can be noticed that the failure process of the deck canbe divided into three stages elastic stage cracked stage andfailure stage When the relative displacement is less than235mm the deck is in the elastic stage When the relativedisplacement increases from 235mm to 315mm cracksgradually increase and the deck works with the cracks Whenthe relative displacement exceeds 315mm the longitudinalreinforcement yields and the displacement increases rapidly

(e robustness of the deck was evaluated by comparingthe relative displacement of the cable breakage with that ofthe load-displacement curve As displayed in Figure 21 themaximum relative displacement of the deck is between 8mmand 15mm subjected to a single cable break which increases

Midspan

Upper chordsLower chords

0

100

200

300

Mom

ent v

aria

tion

(kN

middotm)

5 10 15 20 25 300Cable number

Figure 14 Moment variation of arch rib chords caused by a singlecable break

0 5 10 15 20 25 3010

12

14

16

18

DA

F of

adja

cent

cabl

e ten

sion

Cable number

Figure 15 DAF of the cable tension caused by a single cable break

10

12

14

16

18

DA

F of

dec

k di

spla

cem

ent

5 10 15 20 25 300Cable number

Figure 16 DAF of the deck displacement caused by a single cablebreak

10

12

14

16

18

20

22

24D

AF

of ar

ch ri

b be

ndin

g m

omen

t

5 10 15 20 25 300Cable number

Figure 17 DAF of the arch rib moment caused by a single cablebreak

Shock and Vibration 11

with the length of the broken cable Compared with that ofthe load-displacement curve in Figure 20 the relative dis-placement of the deck subjected to a single cable breakage iswithin the cracked stage and will not lead to a deck collapseUnder the worst conditions the deck system has a safety

factor of approximately 20 which indicates that the decksystem has relatively lower robustness than the cable system

53 Robustness of theArch Rib To evaluate the robustness ofthe arch rib which is an eccentric compression member anM-N curve was drawn to evaluate its bearing capacityFigures 22 and 23 show the axial force and bending momentcurves of the arch rib subjected to S1 and S30 cable breakrespectively (e curves in the circle are the internal forcecurves of the arch rib which are enlarged at the upper rightof the figure It is observed that the arch ribs are mainlyunder compression (e axial force is between 14000 kNand 17500 kN and the bending moment is between minus150and 450 kNmiddotm (e impact effect caused by the cable break

TdTf

400

600

800

1000

1200

1400Te

nsio

n of

cabl

es (k

N)

5 10 15 20 25 300Cable number

Figure 18 Comparison of the limit state combination Td andaccidental combination Tf of each cable

Figure 19 Pushover model of the deck system

0

500

1000

1500

2000

Push

forc

e (kN

)

10 20 30 400Relative displacement of anchorage end (mm)

Figure 20 Load-displacement curve of the deck

5 10 15 20 25 300Cable number

8

10

12

14

16

Max

imum

relat

ive d

ispla

cem

ent (

mm

)

Figure 21 Maximum relative deck displacements caused by asingle cable break

ndash200 ndash100 0 100 200 30014000

15000

16000

17000

MLTMLBMT

MBMRTMRB

Axi

al fo

rce (

kN)

Moments (kNmiddotm)

ndash50000

0

50000

100000

150000A

xial

forc

e (kN

)

0 10000 20000 30000ndash10000Moments (kNmiddotm)

Figure 22 Internal force change curves of the arch rib due to cableS1 break within its M-N curves

12 Shock and Vibration

was within 20 of the arch rib bearing capability indicatingthat the failure of a single cable does not cause failure of thearch rib (e safety factor of the arch rib is greater than 50which reveals that the arch rib has sufficient robustnesssubjected to single cable breakage

6 Conclusions

Cable breaking is common damage in cable-supported archbridges as a result of aging and erosion caused by the harshenvironment To evaluate the dynamic response and robust-ness of cable-supported arch bridges subjected to cablebreaking a nonlinear dynamic simulation procedure and ro-bustness evaluation method are developed in this study Aprototype arch bridge with a single cable break was analyzed asan example Several critical factors for cable breakage simu-lation such as the cable break mode cable break duration anddamping are studied (e dynamic responses of the cablesystem deck and rib are evaluated and the DAF of differentcomponents of the cable-supported arch bridge is also dis-cussed Robustness evaluation indexes of the cable systemdeck and arch rib are calculated and their robustness levels areevaluated (e main conclusions are as follows

(1) Using an appropriate simulation method is essentialto the dynamic analysis of a cable breaking processDirectly removing the failed cable in an appropriatetime step of a transient analysis can effectivelysimulate the cable breaking effect When the cablebreak duration is sufficiently short the maximumimpact effect can be fully attained regardless of thetype of the break simulation mode used (e Ray-leigh damping coefficient calculated by a higher-order frequency can avoid the suppression effect of alocal vibration and better reflect the local vibrationcaused by a cable break

(2) (e closest cable to the broken cable receives thelargest impact effect (e impact effect of the cable

tension caused by the break of a short cable is greaterthan that of a long cable (e deck displacement andbending moment increase with the length of thefailure cable(e breaking of a downstream cable haslittle effect on the upstream cables

(3) (e DAF of each component is different(e DAF ofcable tension is between 14 and 16 while that of thedeck displacement is between 13 and 165 (e archrib bending moment has the largest DAF that isbetween 15 and 24 which is greater than the PTIrecommended value of 20 in some cases Using aunified DAF of 20 cannot satisfy all the componentsof cable-supported arch bridges

(4) (e deck system is the least robust component ofcable-supported arch bridges (e cable system has asafety factor of 31 under the cable breakage acci-dental combination a single cable break will nottrigger progressive damage (e arch rib has suffi-cient robustness subjected to a single cable breakagethough it has the largest DAF

(e results obtained by this analysis are conducive to thedesign and robustness evaluation of cable-supported archbridges Although only the breakage of a single cable wasdemonstrated the proposed methodology can be applied tomultiple cable breakage scenario of cable-supported bridgesFuture research can be carried out in different types of cable-supported arch bridges subjected to multiple cablebreakages

Data Availability

(e data that support the findings of this study are availablefrom the corresponding author upon reasonable request

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(e authors would like to acknowledge Professor RuinianJiang of NMSU for providing help during the preparation ofthe manuscript and the authors also gratefully acknowledgethe support of Taizhou Highway Administration (is re-search was funded by the Zhejiang Provincial Transportationand Transportation Office Technology Funding Project(2012H47)

References

[1] PTI Recommendations for Stay Cable Design Testing andInstallation Post-Tensioning Institute Farmington Hills MIUSA 2012

[2] T Zoli and R Woodward ldquoDesign of long span bridges forcable lossrdquo IABSE Symposium Report vol 90 no 9 pp 17ndash252005

[3] A M Ruiz-Teran and A C Aparicio ldquoResponse of under-deck cable-stayed bridges to the accidental breakage of stay

MLTMLBMT

MBMRTMRB

0 100 200 300 400 50013000

14000

15000

16000

17000

18000

Axi

al fo

rce (

kN)

Moments (kNmiddotm)

ndash50000

0

50000

100000

150000A

xial

forc

e (kN

)

0 10000 20000 30000ndash10000Moments (kNmiddotm)

Figure 23 Internal force change curves of the arch rib due to cableS30 break within its M-N curves

Shock and Vibration 13

cablesrdquo Engineering Structures vol 31 no 7 pp 1425ndash14342009

[4] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part I bending moment acting on the deckrdquo En-gineering Structures vol 32 no 10 pp 3288ndash3300 2010

[5] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part II bending moment acting on the pylons andstress on the staysrdquo Engineering Structures vol 32 no 10pp 3301ndash3312 2010

[6] Y Zhou and S Chen ldquoTime-progressive dynamic assessmentof abrupt cable-breakage events on cable-stayed bridgesrdquoJournal of Bridge Engineering vol 19 no 2 pp 159ndash171 2014

[7] Y Park U Starossek H-M Koh J F Choo H Kim andS W Lee ldquoEffect of cable loss in cable stayed bridgesmdashfocuson dynamic amplificationrdquo IABSE Symposium Report vol 93no 5 pp 9ndash16 2007

[8] J-g Cai Y-x Xu L-p Zhuang J Feng and J ZhangldquoComparison of various procedures for progressive collapseanalysis of cable-stayed bridgesrdquo Journal of Zhejiang Uni-versity Science A vol 13 no 5 pp 323ndash334 2012

[9] Y Xiang Z Chen Y Yang H Lin and S Zhu ldquoDynamicresponse analysis for submerged floating tunnel with anchor-cables subjected to sudden cable breakagerdquoMarine Structuresvol 59 pp 179ndash191 2018

[10] Y F Zhou and S R Chen ldquoFramework of nonlinear dynamicsimulation of long-span cable-stayed bridge and traffic systemsubjected to cable-loss incidentsrdquo Journal of Structural En-gineering vol 142 no 3 2016

[11] W L Qiu M Jiang and C L Huang ldquoParametric study onresponses of a self-anchored suspension bridge to suddenbreakage of a hangerrdquo Scientific World Journal vol 2014Article ID 512120 10 pages 2014

[12] W Qiu M Jiang and Z Zhang ldquoResponses of self-anchoredsuspension bridge to sudden breakage of hangersrdquo StructuralEngineering and Mechanics vol 50 no 2 pp 241ndash255 2014

[13] C M Mozos and A C Aparicio ldquoNumerical and experi-mental study on the interaction cable structure during thefailure of a stay in a cable stayed bridgerdquo EngineeringStructures vol 33 no 8 pp 2330ndash2341 2011

[14] V Hoang O Kiyomiya and T An ldquoExperimental and nu-merical study of lateral cable rupture in cable-stayed bridgescase studyrdquo Journal of Bridge Engineering vol 23 no 6 2018

[15] M Domaneschi G P Cimellaro and G Scutiero ldquoDispro-portionate collapse of a cable-stayed bridgerdquo Proceedings ofthe Institution of Civil EngineersmdashBridge Engineering vol 172no 1 pp 13ndash26 2019

[16] M Wolff and U Starossek ldquoCable loss and progressivecollapse in cable-stayed bridgesrdquo Bridge Structures vol 5no 1 pp 17ndash28 2009

[17] G Wu W Qiu and T Wu ldquoNonlinear dynamic analysis ofthe self-anchored suspension bridge subjected to suddenbreakage of a hangerrdquo Engineering Failure Analysis vol 97pp 701ndash717 2019

[18] M Morgese F Ansari M Domaneschi and G P CimellaroldquoPost-collapse analysis of morandirsquos polcevera viaduct ingenoa Italyrdquo Journal of Civil Structural Health Monitoringvol 10 no 1 pp 69ndash85 2020

[19] M Shoghijavan and U Starossek ldquoStructural robustness oflong-span cable-supported bridges in a cable-loss scenariordquoJournal of Bridge Engineering vol 23 no 2 2018

[20] M Shoghijavan and U Starossek ldquoAn analytical study on thebending moment acting on the girder of a long-span cable-

supported bridge suffering from cable failurerdquo EngineeringStructures vol 167 pp 166ndash174 2018

[21] M Domaneschi C Pellecchia E De Iuliis et al ldquoCollapseanalysis of the Polcevera viaduct by the applied elementmethodrdquo Engineering Structures vol 214 p 110659 2020

[22] O Olamigoke Structural Response of Cable-Stayed Bridges toCable Loss University of Surrey Guildford UK 2018

[23] J Xu H H Sun and S Y Cai ldquoEffect of symmetrical brokenwires damage on mechanical characteristics of stay cablerdquoJournal of Sound and Vibration vol 461 2019

14 Shock and Vibration

To consider the material nonlinearity a multilinearisotropic hardening criterion in ANSYS was chosen for theconcrete material and a bilinear isotropic reinforcementcriterion was chosen for the steel material (e steel wires ofthe cables were considered to be linearly elastic (e geo-metric nonlinearity of the structure was also considered byapplying a large deformation effect

(e analysis process of the cable fracture started with astatic analysis to obtain the initial state and a transient analysisfor cable breaking was then conducted After cable breakage alonger transient analysis with 50 seconds was used to trace thepostbreakage dynamic response(e cable breakage simulationmethod is presented in Section 3 Considering the fact thatmost of the cable break accidents occurred because of cor-rosion the dynamic analysis of cable fracture was performedunder the dead load Because cable breakage is a local effect andthe previous cable breakage accidents did not cause arch bridgeinstability this study does not address the stability issues

23 Validation of the FEModel To verify the accuracy of theFE model the simulated cable tensions under the dead loadwere compared with those measured in the field (e on-site

cable forces were measured by a cable force meter in whichan acceleration sensor was used to measure the naturalfrequency of cable and then converted the natural frequencyto cable force (e relationship between natural frequencyand cable force was expressed as

T 4mLfn

n1113888 1113889

2

minusn2π2

EI

L2 (1)

where T is the cable force m is the linear density L is theeffective length fn is the nth natural frequency and EI is thebending stiffness of the cable

Figure 2 shows the comparison of the cable forces betweenthe FE model and the field measurement (e average value ofthe cable forces calculated by the FE model is 3677 kN whilethat of the field measurement is 3736 kN(e results of the FEmodel are close to the actual state of the bridge and themaximum difference between the two is within 5

3 Cable Break Simulation Method

31 Cable Break Mode (ere are two main methods forsimulating cable breaks One method is to remove the

Cable Cable

Precast

Cable Cable

(Unit m)

(Unit m)

Precast

Cast-in-place Cast-in-place15

30

I-III-II

11

174

1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

I

I

29 times 59 895 times 59 5 times 5989

III

II

II

III23

36

48

III-III

29 30

15 Cast-in-place

Figure 1 Configuration of the prototype cable-supported arch bridge

Table 1 Properties of the materials in the prototype arch bridge

Properties Unit Steel Concrete Cable steel wireWeight density kNm3 78 25 75Elasticity modulus GPa 210 345 195Poissonrsquos ratio μ 03 02 03Compressive strength MPa 210 324 mdashTensile strength MPa 210 264 1670

Shock and Vibration 3

broken cable element and use the equivalent functionsHoang et al [14] and Olamigoke [22] simulated the cablebreakage by using an impact waveform after removing thecable elements Wolff and Starossek [16] and Park et al [7]simulated the cable breakage by using a transient load to theanchorage nodes Zoli and Woodward [2] simulated a cableloss event by using an abrupt drop in a forcing functionZhou and Chen [6] demonstrated that it is more accurate toremove the broken cable element in the static analysis stagebefore the dynamic analysis (e other method is to directlyremove the cable element or reduce the cable element areaMozos [4 13] and Wu et al [17] simulated a cable loss bydirectly reducing the area of the damaged cable in the firststep of a dynamic analysis (e main difference between thetwo methods lies in whether or not using a force function torepresent the cable force Zhou and Chen [10] used a re-duced cable area and a force function to simulate the cablefracture process but ignored the fact that the loss of cablearea is not proportional to the loss of force which wasdemonstrated by Xu et al [23]

When a force function is used the form of the functiondetermines the accuracy of the calculation Ruiz-Teran andAparicio [3] compared seven types of force functions andpointed out that when the breakage time is sufficiently shortthe maximum response is irrespective of the way in whichthe cable breaks To avoid an inappropriate function directlyremoving the damaged cables in a suitable cable loss du-ration is less error-prone and consistent with the actualsituation

It should be noted that removing an element directly isappropriate for simulating fractures caused by rust andfatigue however it is not applicable to the situation wherethe cable fractured by hitting Because most of the long-spanarch bridge cable fractures are caused by corrosion andfatigue the mode of directly removing the failed cable ischosen in this paper to simulate the cable loss

32 Cable Break Duration (ere is currently limited re-search on the duration of cable breakage Fracture experi-ments were conducted by Mozos and Aparicio [13] on aseven-wire strand of a specific steel type the result showsthat the average rupture time is 00055 s(e impact fracturetest by Hoang et al [14] on a single wire indicated that theduration of the cable breaking process is 0017 s Because thecable types differ for different bridges the fracture durationvaries

A parametric analysis was carried out for determining areasonable cable break duration for the arch bridge in thisstudy Six break durations of 001 s 01 s 10 s 20 s 40 s and100 s were selected and the tension of S30 caused by thecable break of S29 was used for comparison

Figure 3 depicts the dynamic response for each cablebreak duration As observed the maximum dynamic re-sponse was reached in the durations of 001 s 01 s and 10 sWhen the fracture duration increases from 20 s to 100 s thepeak response gradually decreases (e dynamic curves of20 s 40 s and 100 s show that the cable force increaseslinearly in the initial stage which reflected the shortage ofthe direct-removing-cable method It also shows that if thefracture duration is less than 10 s the dynamic responsecurves almost coincide and the maximum response isconsistent which was also demonstrated by Ruiz-Teran andAparicio [3] (erefore the abrupt breakage duration wasassumed to be 01 s in this study

33 Damping Coefficient In long-span cable-supportedbridges the effect of damping on the dynamic response issignificant (e Rayleigh damping function is defined via thefollowing equations as

[C] α[M] + β[K] (2)

α ζ2ωiωj

ωi + ωj

(3)

β ζ2

ωi + ωj

(4)

where the damping matrix C is proportional to the massmatrixM and the stiffness matrix K α is the mass factor andβ is the stiffness factor α and β can be computed based onthe undamped modal frequencies ωi and ωj and the pre-scribed damping ratio ζ For cable-supported bridges thestructural vibration excited by cable rupture is a local vi-bration around the broken cable Wolff and Starossek [16]pointed out that the high-order modes characterized by alocal vibration should be considered in Rayleigh dampingZoli and Woodward [2] recommend taking at least 20thfrequency (preferably 50th frequency) to calculate α and β(rough the modal analysis of the Jiantiao Bridge it wasfound that the local vibration corresponds to modes greaterthan the 30th order To obtain the appropriate dampingcoefficient the damping ratios of the 1st 2nd 20th 50thand 100th natural frequencies were selected and the value ofζ was set as 5

FE model

Field test

0

100

200

300

400

500

Tens

ion

(kN

)

5 10 15 20 25 300Cable number

Figure 2 Comparison between the calculated and measured cableforces

4 Shock and Vibration

Figure 4 displays the damping ratio curves obtainedunder each frequency combination (e damping ratio ζ12which was calculated using ω1 and ω2 showed significantsuppression of high-order vibrations (e value of ζ12 in the50th order mode is 22 and the value of the 100th order is33 which greatly exceeded the design value of 5 (edamping ratios ζ120 and ζ150 show a better fit within the first50th order (e damping ratio ζ1100 keeps at 3ndash5 on ahigh-order frequency which matches the damping char-acteristics of reinforced concrete structures therefore ζ1100was chosen as the damping ratio in this study

4 Dynamic Response Results

41 Tension in Adjacent Cables Figure 5 presents the ad-jacent cable tension time-history curves subjected to thebreakage of short cables S1(5a) and S2 (5b) respectively Itcan be observed in Figures 5(a) and 5(b) that the breakage ofcable S1 has the largest impact on cable S2 while thebreakage of cable S2 has the largest impact on cable S1 (isindicates that after a single cable breaks the closest cablereceives the maximum dynamic impact (e maximumdynamic response is 22 times as big as the cable tensionunder the dead load (e breaking of the downstream cableshas little effect on the cables on the upstream side (X1 and X2in the figure) and the maximum dynamic response of thecables on the same side is within 12 times of the tensionunder the dead load

Figure 6 illustrates the adjacent cable tension time-history curves subjected to the break of long cables S29 (6a)and S30 (6b) respectively It can be observed that thebreakage of cable S29 has the largest impact on cable S30and that of cable S30 has the largest impact on cable S29which is similar to the short-cable cases However themaximum dynamic response of the cable tension was 17times the tension of the dead load for the long-cablebreakage and the peak value of the dynamic response is

significantly smaller than that of the short cables (ebreakage of long cables results in a greater dynamic responseof the farther cables for example the maximum response ofS28 which was anchored on the adjacent transverse grid is14 times of the dead-load tension when cable S29 breaks(e effect on the cables on the upstream side is slight whichis the same as that in the short-cable cases

Figure 7 depicts the maximum tension in the adjacentcables caused by a single cable break It can be observed thatthe maximum cable force of 821 kN was caused by the breakof S3 which is the shortest cable As the length of the cableincreases the impact effect due to the break of the cablegradually decreases For the tension of all cables caused by asingle cable break the maximum value is 16 to 22 times asbig as the tension under the dead load

42 Displacement of theDeck (e displacement of the lowercable anchorage points was selected to investigate the vi-bration of the deck (e displacement time-history curves ofthe deck due to the break of a short cable (S1 and S2) and along cable (S29 and S30) are presented in Figures 8 and 9respectively

Figure 8 shows the displacement time-history curves dueto the break of cables S1 (8a) and S2 (8b) respectively Asshown in Figure 8 the breakage of a cable causes themaximum dynamic displacement at its own anchor point(e maximum displacement of S1 is minus113mm and that ofS2 is minus119mm (e displacement at the anchorage point ofthe adjacent cables located at the same transverse girder isslightly smaller than that of the broken cable (e dis-placements of the anchorage points that are not on the sametransverse girder of the broken cable are significantlysmaller

Figure 9 shows the displacement time-history curves dueto the break of long cables S29 (9a) and S30 (9b) respec-tively Compared with the short-cable break the long-cablebreak also caused the largest displacement at its own an-chorage point but the deck displacement was larger (emaximum displacement of S29 is minus277mm and S30 isminus279mm Moreover the maximum displacement of theanchorage points of the adjacent cables anchored on dif-ferent transverse girders also exceeds minus10mm

(e maximum deck displacement caused by thebreakage of each cable is shown in Figure 10 It is noticedthat the minimum deck displacement is 84mm which iscaused by the break of the shortest cable (S3) and themaximum deck displacement is 279mm which is caused bythe break of the longest cable (S30) (e displacement of thebridge deck almost increases proportionally with the lengthof the cable

43Moments of theArch Rib To determine the impact effectof cable fracture on the arch rib the bending moment time-history curves at the six nodes around the upper anchoragepoint of the broken cable were selected (e six rigid framenode bending moments are the upper chord node bendingmomentMT lower chord node bending momentMB upperleft node bending moment MLT lower left node bending

001s01s10s

20s40s100s

350

400

450

500

550

600

Tens

ion

of th

e adj

acen

t cab

le (k

N)

5 10 15 20 25 300Time (s)

Figure 3 Dynamic response to six cable break durations

Shock and Vibration 5

moment MLB upper right node bending moment MRT andlower right node bending moment MRB

Figure 11 depicts the moment time-history curves due tothe breakage of short cables S1 (11a) and S2 (11b) re-spectively Because cables S1 and S2 are anchored on theupper chord of the arch rib the dynamic response variationof MT is the most significant (e maximum changes of MTare 135 kNmiddotm due to the break of S1 and 150 kNmiddotmdue to thebreak of S2 (e bending moment changes at other positionsare small all are within 100 kNmiddotm

Figure 12 indicates the bending moment time-historycurves due to the breakage of long cables S29 (12a) and S30

(12b) respectively Because cables S29 and S30 are anchoredon the lower chord of the arch rib the variation of thedynamic response of MB is the most significant (e max-imum changes of MB are 168 kNmiddotm due to the break of S29and 170 kNmiddotmdue to the break of S30(emaximum changeof MLB and MRB is 145 kNmiddotm and the changes of MLT MTand MRT are within 100 kNmiddotm

(e bending moment in the upper and lower chords ofthe arch rib is investigated Figure 13 displays the envelope ofthe bending moments of the arch rib sections caused by thebreakage of each cable (e bending moment of the upperchord sections is between minus100 kNmiddotm and 210 kNmiddotm and

Dam

ping

ratio

ζ (

)

ζ12

ζ120

ζ150

ζ1100

ω1ω2 ω20 ω50 ω100

2 4 6 8 100Natural frequencies ω (Hz)

0

10

20

30

40

50

Figure 4 Damping ratio curves obtained under four frequency combinations

S2S3

X1X2

300

400

500

600

700

Tens

ion

of ca

bles

(kN

)

10 20 30 40 500Time (s)

(a)

S2S3

X1X2

300

400

500

600

700

Tens

ion

of ca

bles

(kN

)

10 20 30 40 500Time (s)

(b)

Figure 5 Tension time-history curves due to short-cable fracture (a) break of cable S1 (b) break of cable S2

6 Shock and Vibration

that of the lower chord is between minus100 kNmiddotm and 450 kNmiddotm(is indicates that the bending moment of the lower chordof the arch rib is greater than that of the upper chord

In order to reflect the variation amplitude of the arch ribbending moment during the breakage of each cable Fig-ure 14 shows the bending moment variation of the upperand lower chords of the arch rib caused by a single cablebreak (e variation of the bending moment of the upperchord is between 113 kNmiddotm and 188 kNmiddotm with an averagevalue of 137 kNmiddotm and the bending moment variation of thelower arch rib is between 100 kN and 280 kNmiddotm with anaverage value of 229 kNmiddotm Because most cables are an-chored to the lower chord except S1 and S2 the bendingmoments of the lower chord vary more than that of theupper chord Moreover as the length of the cable increasesthe fracture impact effect increases

44 DAF of the Cable Deck and Rib No specifications haveprovided details for calculating the DAF of cable-supportedarch bridges For the design of cable-stayed bridges the DAFis set between 15 and 20 in some codes and recommen-dations In this section the cable failure of the JiantiaoBridge is investigated to discuss the DAF for the cabletension deck displacement and arch rib bendingmoment ofcable-supported arch bridges Equation (1) is used to cal-culate DAF

DAF RD minus R0

RP D minus R0 (5)

where RD represents themaximum dynamic response R0 theinitial state under the dead load before cable break and RPDthe stable value after cable break

S28S30

X29X30

300

350

400

450

500

550

600

650Te

nsio

n of

cabl

es (k

N)

10 20 30 40 500Time (s)

(a)

S29S31

X29X30

10 20 30 40 500Time (s)

300

350

400

450

500

550

600

650

Tens

ion

of ca

bles

(kN

)(b)

Figure 6 Time-history curves due to long-cable fracture (a) break of cable S29 (b) break of cable S30

0

200

400

600

800

Tens

ion

of ca

bles

(kN

)

5 10 15 20 25 300Cable number

Figure 7 Maximum tension of the adjacent cables caused by the break of a single cable

Shock and Vibration 7

Figure 15 shows the calculated DAF for the cable tensiondue to the break of the adjacent cable It can be observed thatthe DAF of the adjacent cable tension is distributed between14 and 16 with an average value of 154

Figure 16 presents the DAF for the deck displacementwhich is between 130 and 165 with an average value of 140(ere is a tendency for the deck displacement DAF to de-crease as the cable length increases

Figure 17 displays the DAF for the arch rib bendingmoment which is between 15 and 24 with an average value

of 193 (e DAF exceeds 20 in some cases which is therecommended value of PTI [1]

Compared with the recommended common DAF valueof 20 for cable-stayed bridges by PTI [1] the DAF of thecable-supported arch bridge in this study varies for eachcomponent (e maximum DAFs of the cable tension anddeck displacement are less than 20 while the maximumDAF of the bending moment of the arch rib exceeds 20(isshows that adopting a common DAF for all the componentsof cable-supported arch bridge structures is not advisable

S1S2S3

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4D

eck

disp

lace

men

t (m

m)

10 20 30 40 500Time (s)

(a)

S1S2S3

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

Dec

k di

spla

cem

ent (

mm

)

10 20 30 40 500Time (s)

(b)

Figure 8 Displacement time-history curves of the deck (a) cable S1 break (b) cable S2 break

S28S29S30

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

Dec

k di

spla

cem

ent (

mm

)

10 20 30 40 500Time (s)

(a)

S29S30S31

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

Dec

k di

spla

cem

ent (

mm

)

10 20 30 40 500Time (s)

(b)

Figure 9 Displacement time-history curves of the deck (a) cable S29 break (b) cable S30 break

8 Shock and Vibration

5 Robustness Evaluation

For long-span cable-supported arch bridges the failure of asingle cable may trigger progressive fracture of adjacentcables and damage to the deck and arch ribs (erefore it isnecessary to evaluate the robustness of cable-supported archbridges subjected to cable breaking (is section discussesthe robustness evaluation of the cable system deck and archrib subjected to a single cable break

51 Robustness of the Cable System To study the possibilityof a progressive fracture of adjacent cables subjected to asingle cable break the impact effect of the cable breakage is

investigated and compared with the carrying capacity of thecable system (e cable system of the Jiantiao Bridge iscomposed of 55 parallel steel wires that have a diameter of7mm and a tensile strength of 1670MPa (e cablersquos ulti-mate bearing capability fu is 3530 kN which is much biggerthan the maximum dynamic load of 821 kN caused by asingle cable break (is indicates that a single cable breakageunder the dead load will not trigger a progressive fracture ofthe adjacent cables

Furthermore to consider the load combination with ref-erence to the PTI [1] recommendation and the study by Zoliand Woodward [2] the limit state design combination Td andthe accidental combination Tf of the cable failure are defined as

0

5

10

15

20

25

30

35

Max

imum

disp

lace

men

t (m

m)

5 10 15 20 25 300Cable number

Figure 10 Maximum displacement of the deck caused by a single cable break

0

MLTMTMRT

MLBMBMRB

ndash200

ndash100

100

200

300

Mom

ent o

f arc

h rib

(kN

middotm)

10 20 30 40 500Time (s)

(a)

MLTMTMRT

MLBMBMRB

ndash200

ndash100

0

100

200

300M

omen

t of a

rch

rib (k

Nmiddotm

)

10 20 30 40 500Time (s)

(b)

Figure 11 Bending moment time-history curves of the arch rib (a) cable S1 break (b) cable S2 break

Shock and Vibration 9

Td 12Ts + 14Tv + 075 Tp + Tt1113872 1113873

Tf 12Ts + 075 Tv + Tp + Tt1113872 1113873 + Tc(6)

where Ts Tv Tp and Tt are the tension of the dead loadvehicle load pedestrian load and temperature load re-spectively Tc is the net dynamic effect of the cable breakwhich is equal to the maximum tension of the dynamicresponse minus the cable tension under the dead load

Figure 18 shows the comparison of the design limitcombination Td and the accidental combination Tf of each

cable It is observed that the values of cable fracture ac-cidental combination are all higher than the carryingcapability limit state combination (is means that thecable breakage accidental combination is the most un-favorable situation and should not be ignored in thedesign stage

(e maximum value of the accidental combination Tf is1130 kN and the minimum safety factor of the cablecompared with the ultimate bearing capability fu is 31 Itshows that the cable system has good robustness under acable breakage accidental combination and a single cable

MLTMTMRT

MLBMBMRB

0

100

200

300

400

500M

omen

t of a

rch

rib (k

Nmiddotm

)

10 20 30 40 500Time (s)

(a)

MLTMTMRT

MLBMBMRB

10 20 30 40 500Time(s)

0

100

200

300

400

500

Mom

ent o

f arc

h rib

(kN

middotm)

(b)

Figure 12 Bending moment time-history curves of the arch rib (a) cable S29 break (b) cable S30 break

Midspan

MaxMin

ndash100

0

100

200

300

Mom

ent o

f upp

er ch

ords

(kN

middotm)

5 10 15 20 25 300Cable number

(a)

MaxMin

Midspan

ndash100

0

100

200

300

400

500

Mom

ent o

f low

er ch

ords

(kN

middotm)

5 10 15 20 25 300Cable number

(b)

Figure 13 Envelope of the arch rib bending moment subjected to a single cable break (a) upper chords (b) lower chords

10 Shock and Vibration

break will not trigger a progressive breakage of adjacentcables

52 Robustness of the Deck Different from cable-stayedbridges that have a strong longitudinal stiffening girdercable-supported arch bridge deck systems always usetransverse girders as the main load-bearing members Toevaluate the damage to the deck system the capability of thedeck was obtained by a pushover analysis

(e pushover model was composed of three transversegirders with its flange plate An elastic support was used tosimulate the rigidity of the cables at the anchoring positions(e horizontal side displacement of the deck was notallowed by fixing the left and right ends of the deck system toavoid any rigid movement (e vertical pushover load wasapplied to the anchored position of the broken cable on the

right side of the middle girder as indicated by the arrow inFigure 19 (e relative displacements between the transversegirders were chosen as the capability index to represent thebearing capacity of the deck system

Figure 20 depicts the load-displacement curve of thedeck It can be noticed that the failure process of the deck canbe divided into three stages elastic stage cracked stage andfailure stage When the relative displacement is less than235mm the deck is in the elastic stage When the relativedisplacement increases from 235mm to 315mm cracksgradually increase and the deck works with the cracks Whenthe relative displacement exceeds 315mm the longitudinalreinforcement yields and the displacement increases rapidly

(e robustness of the deck was evaluated by comparingthe relative displacement of the cable breakage with that ofthe load-displacement curve As displayed in Figure 21 themaximum relative displacement of the deck is between 8mmand 15mm subjected to a single cable break which increases

Midspan

Upper chordsLower chords

0

100

200

300

Mom

ent v

aria

tion

(kN

middotm)

5 10 15 20 25 300Cable number

Figure 14 Moment variation of arch rib chords caused by a singlecable break

0 5 10 15 20 25 3010

12

14

16

18

DA

F of

adja

cent

cabl

e ten

sion

Cable number

Figure 15 DAF of the cable tension caused by a single cable break

10

12

14

16

18

DA

F of

dec

k di

spla

cem

ent

5 10 15 20 25 300Cable number

Figure 16 DAF of the deck displacement caused by a single cablebreak

10

12

14

16

18

20

22

24D

AF

of ar

ch ri

b be

ndin

g m

omen

t

5 10 15 20 25 300Cable number

Figure 17 DAF of the arch rib moment caused by a single cablebreak

Shock and Vibration 11

with the length of the broken cable Compared with that ofthe load-displacement curve in Figure 20 the relative dis-placement of the deck subjected to a single cable breakage iswithin the cracked stage and will not lead to a deck collapseUnder the worst conditions the deck system has a safety

factor of approximately 20 which indicates that the decksystem has relatively lower robustness than the cable system

53 Robustness of theArch Rib To evaluate the robustness ofthe arch rib which is an eccentric compression member anM-N curve was drawn to evaluate its bearing capacityFigures 22 and 23 show the axial force and bending momentcurves of the arch rib subjected to S1 and S30 cable breakrespectively (e curves in the circle are the internal forcecurves of the arch rib which are enlarged at the upper rightof the figure It is observed that the arch ribs are mainlyunder compression (e axial force is between 14000 kNand 17500 kN and the bending moment is between minus150and 450 kNmiddotm (e impact effect caused by the cable break

TdTf

400

600

800

1000

1200

1400Te

nsio

n of

cabl

es (k

N)

5 10 15 20 25 300Cable number

Figure 18 Comparison of the limit state combination Td andaccidental combination Tf of each cable

Figure 19 Pushover model of the deck system

0

500

1000

1500

2000

Push

forc

e (kN

)

10 20 30 400Relative displacement of anchorage end (mm)

Figure 20 Load-displacement curve of the deck

5 10 15 20 25 300Cable number

8

10

12

14

16

Max

imum

relat

ive d

ispla

cem

ent (

mm

)

Figure 21 Maximum relative deck displacements caused by asingle cable break

ndash200 ndash100 0 100 200 30014000

15000

16000

17000

MLTMLBMT

MBMRTMRB

Axi

al fo

rce (

kN)

Moments (kNmiddotm)

ndash50000

0

50000

100000

150000A

xial

forc

e (kN

)

0 10000 20000 30000ndash10000Moments (kNmiddotm)

Figure 22 Internal force change curves of the arch rib due to cableS1 break within its M-N curves

12 Shock and Vibration

was within 20 of the arch rib bearing capability indicatingthat the failure of a single cable does not cause failure of thearch rib (e safety factor of the arch rib is greater than 50which reveals that the arch rib has sufficient robustnesssubjected to single cable breakage

6 Conclusions

Cable breaking is common damage in cable-supported archbridges as a result of aging and erosion caused by the harshenvironment To evaluate the dynamic response and robust-ness of cable-supported arch bridges subjected to cablebreaking a nonlinear dynamic simulation procedure and ro-bustness evaluation method are developed in this study Aprototype arch bridge with a single cable break was analyzed asan example Several critical factors for cable breakage simu-lation such as the cable break mode cable break duration anddamping are studied (e dynamic responses of the cablesystem deck and rib are evaluated and the DAF of differentcomponents of the cable-supported arch bridge is also dis-cussed Robustness evaluation indexes of the cable systemdeck and arch rib are calculated and their robustness levels areevaluated (e main conclusions are as follows

(1) Using an appropriate simulation method is essentialto the dynamic analysis of a cable breaking processDirectly removing the failed cable in an appropriatetime step of a transient analysis can effectivelysimulate the cable breaking effect When the cablebreak duration is sufficiently short the maximumimpact effect can be fully attained regardless of thetype of the break simulation mode used (e Ray-leigh damping coefficient calculated by a higher-order frequency can avoid the suppression effect of alocal vibration and better reflect the local vibrationcaused by a cable break

(2) (e closest cable to the broken cable receives thelargest impact effect (e impact effect of the cable

tension caused by the break of a short cable is greaterthan that of a long cable (e deck displacement andbending moment increase with the length of thefailure cable(e breaking of a downstream cable haslittle effect on the upstream cables

(3) (e DAF of each component is different(e DAF ofcable tension is between 14 and 16 while that of thedeck displacement is between 13 and 165 (e archrib bending moment has the largest DAF that isbetween 15 and 24 which is greater than the PTIrecommended value of 20 in some cases Using aunified DAF of 20 cannot satisfy all the componentsof cable-supported arch bridges

(4) (e deck system is the least robust component ofcable-supported arch bridges (e cable system has asafety factor of 31 under the cable breakage acci-dental combination a single cable break will nottrigger progressive damage (e arch rib has suffi-cient robustness subjected to a single cable breakagethough it has the largest DAF

(e results obtained by this analysis are conducive to thedesign and robustness evaluation of cable-supported archbridges Although only the breakage of a single cable wasdemonstrated the proposed methodology can be applied tomultiple cable breakage scenario of cable-supported bridgesFuture research can be carried out in different types of cable-supported arch bridges subjected to multiple cablebreakages

Data Availability

(e data that support the findings of this study are availablefrom the corresponding author upon reasonable request

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(e authors would like to acknowledge Professor RuinianJiang of NMSU for providing help during the preparation ofthe manuscript and the authors also gratefully acknowledgethe support of Taizhou Highway Administration (is re-search was funded by the Zhejiang Provincial Transportationand Transportation Office Technology Funding Project(2012H47)

References

[1] PTI Recommendations for Stay Cable Design Testing andInstallation Post-Tensioning Institute Farmington Hills MIUSA 2012

[2] T Zoli and R Woodward ldquoDesign of long span bridges forcable lossrdquo IABSE Symposium Report vol 90 no 9 pp 17ndash252005

[3] A M Ruiz-Teran and A C Aparicio ldquoResponse of under-deck cable-stayed bridges to the accidental breakage of stay

MLTMLBMT

MBMRTMRB

0 100 200 300 400 50013000

14000

15000

16000

17000

18000

Axi

al fo

rce (

kN)

Moments (kNmiddotm)

ndash50000

0

50000

100000

150000A

xial

forc

e (kN

)

0 10000 20000 30000ndash10000Moments (kNmiddotm)

Figure 23 Internal force change curves of the arch rib due to cableS30 break within its M-N curves

Shock and Vibration 13

cablesrdquo Engineering Structures vol 31 no 7 pp 1425ndash14342009

[4] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part I bending moment acting on the deckrdquo En-gineering Structures vol 32 no 10 pp 3288ndash3300 2010

[5] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part II bending moment acting on the pylons andstress on the staysrdquo Engineering Structures vol 32 no 10pp 3301ndash3312 2010

[6] Y Zhou and S Chen ldquoTime-progressive dynamic assessmentof abrupt cable-breakage events on cable-stayed bridgesrdquoJournal of Bridge Engineering vol 19 no 2 pp 159ndash171 2014

[7] Y Park U Starossek H-M Koh J F Choo H Kim andS W Lee ldquoEffect of cable loss in cable stayed bridgesmdashfocuson dynamic amplificationrdquo IABSE Symposium Report vol 93no 5 pp 9ndash16 2007

[8] J-g Cai Y-x Xu L-p Zhuang J Feng and J ZhangldquoComparison of various procedures for progressive collapseanalysis of cable-stayed bridgesrdquo Journal of Zhejiang Uni-versity Science A vol 13 no 5 pp 323ndash334 2012

[9] Y Xiang Z Chen Y Yang H Lin and S Zhu ldquoDynamicresponse analysis for submerged floating tunnel with anchor-cables subjected to sudden cable breakagerdquoMarine Structuresvol 59 pp 179ndash191 2018

[10] Y F Zhou and S R Chen ldquoFramework of nonlinear dynamicsimulation of long-span cable-stayed bridge and traffic systemsubjected to cable-loss incidentsrdquo Journal of Structural En-gineering vol 142 no 3 2016

[11] W L Qiu M Jiang and C L Huang ldquoParametric study onresponses of a self-anchored suspension bridge to suddenbreakage of a hangerrdquo Scientific World Journal vol 2014Article ID 512120 10 pages 2014

[12] W Qiu M Jiang and Z Zhang ldquoResponses of self-anchoredsuspension bridge to sudden breakage of hangersrdquo StructuralEngineering and Mechanics vol 50 no 2 pp 241ndash255 2014

[13] C M Mozos and A C Aparicio ldquoNumerical and experi-mental study on the interaction cable structure during thefailure of a stay in a cable stayed bridgerdquo EngineeringStructures vol 33 no 8 pp 2330ndash2341 2011

[14] V Hoang O Kiyomiya and T An ldquoExperimental and nu-merical study of lateral cable rupture in cable-stayed bridgescase studyrdquo Journal of Bridge Engineering vol 23 no 6 2018

[15] M Domaneschi G P Cimellaro and G Scutiero ldquoDispro-portionate collapse of a cable-stayed bridgerdquo Proceedings ofthe Institution of Civil EngineersmdashBridge Engineering vol 172no 1 pp 13ndash26 2019

[16] M Wolff and U Starossek ldquoCable loss and progressivecollapse in cable-stayed bridgesrdquo Bridge Structures vol 5no 1 pp 17ndash28 2009

[17] G Wu W Qiu and T Wu ldquoNonlinear dynamic analysis ofthe self-anchored suspension bridge subjected to suddenbreakage of a hangerrdquo Engineering Failure Analysis vol 97pp 701ndash717 2019

[18] M Morgese F Ansari M Domaneschi and G P CimellaroldquoPost-collapse analysis of morandirsquos polcevera viaduct ingenoa Italyrdquo Journal of Civil Structural Health Monitoringvol 10 no 1 pp 69ndash85 2020

[19] M Shoghijavan and U Starossek ldquoStructural robustness oflong-span cable-supported bridges in a cable-loss scenariordquoJournal of Bridge Engineering vol 23 no 2 2018

[20] M Shoghijavan and U Starossek ldquoAn analytical study on thebending moment acting on the girder of a long-span cable-

supported bridge suffering from cable failurerdquo EngineeringStructures vol 167 pp 166ndash174 2018

[21] M Domaneschi C Pellecchia E De Iuliis et al ldquoCollapseanalysis of the Polcevera viaduct by the applied elementmethodrdquo Engineering Structures vol 214 p 110659 2020

[22] O Olamigoke Structural Response of Cable-Stayed Bridges toCable Loss University of Surrey Guildford UK 2018

[23] J Xu H H Sun and S Y Cai ldquoEffect of symmetrical brokenwires damage on mechanical characteristics of stay cablerdquoJournal of Sound and Vibration vol 461 2019

14 Shock and Vibration

broken cable element and use the equivalent functionsHoang et al [14] and Olamigoke [22] simulated the cablebreakage by using an impact waveform after removing thecable elements Wolff and Starossek [16] and Park et al [7]simulated the cable breakage by using a transient load to theanchorage nodes Zoli and Woodward [2] simulated a cableloss event by using an abrupt drop in a forcing functionZhou and Chen [6] demonstrated that it is more accurate toremove the broken cable element in the static analysis stagebefore the dynamic analysis (e other method is to directlyremove the cable element or reduce the cable element areaMozos [4 13] and Wu et al [17] simulated a cable loss bydirectly reducing the area of the damaged cable in the firststep of a dynamic analysis (e main difference between thetwo methods lies in whether or not using a force function torepresent the cable force Zhou and Chen [10] used a re-duced cable area and a force function to simulate the cablefracture process but ignored the fact that the loss of cablearea is not proportional to the loss of force which wasdemonstrated by Xu et al [23]

When a force function is used the form of the functiondetermines the accuracy of the calculation Ruiz-Teran andAparicio [3] compared seven types of force functions andpointed out that when the breakage time is sufficiently shortthe maximum response is irrespective of the way in whichthe cable breaks To avoid an inappropriate function directlyremoving the damaged cables in a suitable cable loss du-ration is less error-prone and consistent with the actualsituation

It should be noted that removing an element directly isappropriate for simulating fractures caused by rust andfatigue however it is not applicable to the situation wherethe cable fractured by hitting Because most of the long-spanarch bridge cable fractures are caused by corrosion andfatigue the mode of directly removing the failed cable ischosen in this paper to simulate the cable loss

32 Cable Break Duration (ere is currently limited re-search on the duration of cable breakage Fracture experi-ments were conducted by Mozos and Aparicio [13] on aseven-wire strand of a specific steel type the result showsthat the average rupture time is 00055 s(e impact fracturetest by Hoang et al [14] on a single wire indicated that theduration of the cable breaking process is 0017 s Because thecable types differ for different bridges the fracture durationvaries

A parametric analysis was carried out for determining areasonable cable break duration for the arch bridge in thisstudy Six break durations of 001 s 01 s 10 s 20 s 40 s and100 s were selected and the tension of S30 caused by thecable break of S29 was used for comparison

Figure 3 depicts the dynamic response for each cablebreak duration As observed the maximum dynamic re-sponse was reached in the durations of 001 s 01 s and 10 sWhen the fracture duration increases from 20 s to 100 s thepeak response gradually decreases (e dynamic curves of20 s 40 s and 100 s show that the cable force increaseslinearly in the initial stage which reflected the shortage ofthe direct-removing-cable method It also shows that if thefracture duration is less than 10 s the dynamic responsecurves almost coincide and the maximum response isconsistent which was also demonstrated by Ruiz-Teran andAparicio [3] (erefore the abrupt breakage duration wasassumed to be 01 s in this study

33 Damping Coefficient In long-span cable-supportedbridges the effect of damping on the dynamic response issignificant (e Rayleigh damping function is defined via thefollowing equations as

[C] α[M] + β[K] (2)

α ζ2ωiωj

ωi + ωj

(3)

β ζ2

ωi + ωj

(4)

where the damping matrix C is proportional to the massmatrixM and the stiffness matrix K α is the mass factor andβ is the stiffness factor α and β can be computed based onthe undamped modal frequencies ωi and ωj and the pre-scribed damping ratio ζ For cable-supported bridges thestructural vibration excited by cable rupture is a local vi-bration around the broken cable Wolff and Starossek [16]pointed out that the high-order modes characterized by alocal vibration should be considered in Rayleigh dampingZoli and Woodward [2] recommend taking at least 20thfrequency (preferably 50th frequency) to calculate α and β(rough the modal analysis of the Jiantiao Bridge it wasfound that the local vibration corresponds to modes greaterthan the 30th order To obtain the appropriate dampingcoefficient the damping ratios of the 1st 2nd 20th 50thand 100th natural frequencies were selected and the value ofζ was set as 5

FE model

Field test

0

100

200

300

400

500

Tens

ion

(kN

)

5 10 15 20 25 300Cable number

Figure 2 Comparison between the calculated and measured cableforces

4 Shock and Vibration

Figure 4 displays the damping ratio curves obtainedunder each frequency combination (e damping ratio ζ12which was calculated using ω1 and ω2 showed significantsuppression of high-order vibrations (e value of ζ12 in the50th order mode is 22 and the value of the 100th order is33 which greatly exceeded the design value of 5 (edamping ratios ζ120 and ζ150 show a better fit within the first50th order (e damping ratio ζ1100 keeps at 3ndash5 on ahigh-order frequency which matches the damping char-acteristics of reinforced concrete structures therefore ζ1100was chosen as the damping ratio in this study

4 Dynamic Response Results

41 Tension in Adjacent Cables Figure 5 presents the ad-jacent cable tension time-history curves subjected to thebreakage of short cables S1(5a) and S2 (5b) respectively Itcan be observed in Figures 5(a) and 5(b) that the breakage ofcable S1 has the largest impact on cable S2 while thebreakage of cable S2 has the largest impact on cable S1 (isindicates that after a single cable breaks the closest cablereceives the maximum dynamic impact (e maximumdynamic response is 22 times as big as the cable tensionunder the dead load (e breaking of the downstream cableshas little effect on the cables on the upstream side (X1 and X2in the figure) and the maximum dynamic response of thecables on the same side is within 12 times of the tensionunder the dead load

Figure 6 illustrates the adjacent cable tension time-history curves subjected to the break of long cables S29 (6a)and S30 (6b) respectively It can be observed that thebreakage of cable S29 has the largest impact on cable S30and that of cable S30 has the largest impact on cable S29which is similar to the short-cable cases However themaximum dynamic response of the cable tension was 17times the tension of the dead load for the long-cablebreakage and the peak value of the dynamic response is

significantly smaller than that of the short cables (ebreakage of long cables results in a greater dynamic responseof the farther cables for example the maximum response ofS28 which was anchored on the adjacent transverse grid is14 times of the dead-load tension when cable S29 breaks(e effect on the cables on the upstream side is slight whichis the same as that in the short-cable cases

Figure 7 depicts the maximum tension in the adjacentcables caused by a single cable break It can be observed thatthe maximum cable force of 821 kN was caused by the breakof S3 which is the shortest cable As the length of the cableincreases the impact effect due to the break of the cablegradually decreases For the tension of all cables caused by asingle cable break the maximum value is 16 to 22 times asbig as the tension under the dead load

42 Displacement of theDeck (e displacement of the lowercable anchorage points was selected to investigate the vi-bration of the deck (e displacement time-history curves ofthe deck due to the break of a short cable (S1 and S2) and along cable (S29 and S30) are presented in Figures 8 and 9respectively

Figure 8 shows the displacement time-history curves dueto the break of cables S1 (8a) and S2 (8b) respectively Asshown in Figure 8 the breakage of a cable causes themaximum dynamic displacement at its own anchor point(e maximum displacement of S1 is minus113mm and that ofS2 is minus119mm (e displacement at the anchorage point ofthe adjacent cables located at the same transverse girder isslightly smaller than that of the broken cable (e dis-placements of the anchorage points that are not on the sametransverse girder of the broken cable are significantlysmaller

Figure 9 shows the displacement time-history curves dueto the break of long cables S29 (9a) and S30 (9b) respec-tively Compared with the short-cable break the long-cablebreak also caused the largest displacement at its own an-chorage point but the deck displacement was larger (emaximum displacement of S29 is minus277mm and S30 isminus279mm Moreover the maximum displacement of theanchorage points of the adjacent cables anchored on dif-ferent transverse girders also exceeds minus10mm

(e maximum deck displacement caused by thebreakage of each cable is shown in Figure 10 It is noticedthat the minimum deck displacement is 84mm which iscaused by the break of the shortest cable (S3) and themaximum deck displacement is 279mm which is caused bythe break of the longest cable (S30) (e displacement of thebridge deck almost increases proportionally with the lengthof the cable

43Moments of theArch Rib To determine the impact effectof cable fracture on the arch rib the bending moment time-history curves at the six nodes around the upper anchoragepoint of the broken cable were selected (e six rigid framenode bending moments are the upper chord node bendingmomentMT lower chord node bending momentMB upperleft node bending moment MLT lower left node bending

001s01s10s

20s40s100s

350

400

450

500

550

600

Tens

ion

of th

e adj

acen

t cab

le (k

N)

5 10 15 20 25 300Time (s)

Figure 3 Dynamic response to six cable break durations

Shock and Vibration 5

moment MLB upper right node bending moment MRT andlower right node bending moment MRB

Figure 11 depicts the moment time-history curves due tothe breakage of short cables S1 (11a) and S2 (11b) re-spectively Because cables S1 and S2 are anchored on theupper chord of the arch rib the dynamic response variationof MT is the most significant (e maximum changes of MTare 135 kNmiddotm due to the break of S1 and 150 kNmiddotmdue to thebreak of S2 (e bending moment changes at other positionsare small all are within 100 kNmiddotm

Figure 12 indicates the bending moment time-historycurves due to the breakage of long cables S29 (12a) and S30

(12b) respectively Because cables S29 and S30 are anchoredon the lower chord of the arch rib the variation of thedynamic response of MB is the most significant (e max-imum changes of MB are 168 kNmiddotm due to the break of S29and 170 kNmiddotmdue to the break of S30(emaximum changeof MLB and MRB is 145 kNmiddotm and the changes of MLT MTand MRT are within 100 kNmiddotm

(e bending moment in the upper and lower chords ofthe arch rib is investigated Figure 13 displays the envelope ofthe bending moments of the arch rib sections caused by thebreakage of each cable (e bending moment of the upperchord sections is between minus100 kNmiddotm and 210 kNmiddotm and

Dam

ping

ratio

ζ (

)

ζ12

ζ120

ζ150

ζ1100

ω1ω2 ω20 ω50 ω100

2 4 6 8 100Natural frequencies ω (Hz)

0

10

20

30

40

50

Figure 4 Damping ratio curves obtained under four frequency combinations

S2S3

X1X2

300

400

500

600

700

Tens

ion

of ca

bles

(kN

)

10 20 30 40 500Time (s)

(a)

S2S3

X1X2

300

400

500

600

700

Tens

ion

of ca

bles

(kN

)

10 20 30 40 500Time (s)

(b)

Figure 5 Tension time-history curves due to short-cable fracture (a) break of cable S1 (b) break of cable S2

6 Shock and Vibration

that of the lower chord is between minus100 kNmiddotm and 450 kNmiddotm(is indicates that the bending moment of the lower chordof the arch rib is greater than that of the upper chord

In order to reflect the variation amplitude of the arch ribbending moment during the breakage of each cable Fig-ure 14 shows the bending moment variation of the upperand lower chords of the arch rib caused by a single cablebreak (e variation of the bending moment of the upperchord is between 113 kNmiddotm and 188 kNmiddotm with an averagevalue of 137 kNmiddotm and the bending moment variation of thelower arch rib is between 100 kN and 280 kNmiddotm with anaverage value of 229 kNmiddotm Because most cables are an-chored to the lower chord except S1 and S2 the bendingmoments of the lower chord vary more than that of theupper chord Moreover as the length of the cable increasesthe fracture impact effect increases

44 DAF of the Cable Deck and Rib No specifications haveprovided details for calculating the DAF of cable-supportedarch bridges For the design of cable-stayed bridges the DAFis set between 15 and 20 in some codes and recommen-dations In this section the cable failure of the JiantiaoBridge is investigated to discuss the DAF for the cabletension deck displacement and arch rib bendingmoment ofcable-supported arch bridges Equation (1) is used to cal-culate DAF

DAF RD minus R0

RP D minus R0 (5)

where RD represents themaximum dynamic response R0 theinitial state under the dead load before cable break and RPDthe stable value after cable break

S28S30

X29X30

300

350

400

450

500

550

600

650Te

nsio

n of

cabl

es (k

N)

10 20 30 40 500Time (s)

(a)

S29S31

X29X30

10 20 30 40 500Time (s)

300

350

400

450

500

550

600

650

Tens

ion

of ca

bles

(kN

)(b)

Figure 6 Time-history curves due to long-cable fracture (a) break of cable S29 (b) break of cable S30

0

200

400

600

800

Tens

ion

of ca

bles

(kN

)

5 10 15 20 25 300Cable number

Figure 7 Maximum tension of the adjacent cables caused by the break of a single cable

Shock and Vibration 7

Figure 15 shows the calculated DAF for the cable tensiondue to the break of the adjacent cable It can be observed thatthe DAF of the adjacent cable tension is distributed between14 and 16 with an average value of 154

Figure 16 presents the DAF for the deck displacementwhich is between 130 and 165 with an average value of 140(ere is a tendency for the deck displacement DAF to de-crease as the cable length increases

Figure 17 displays the DAF for the arch rib bendingmoment which is between 15 and 24 with an average value

of 193 (e DAF exceeds 20 in some cases which is therecommended value of PTI [1]

Compared with the recommended common DAF valueof 20 for cable-stayed bridges by PTI [1] the DAF of thecable-supported arch bridge in this study varies for eachcomponent (e maximum DAFs of the cable tension anddeck displacement are less than 20 while the maximumDAF of the bending moment of the arch rib exceeds 20(isshows that adopting a common DAF for all the componentsof cable-supported arch bridge structures is not advisable

S1S2S3

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4D

eck

disp

lace

men

t (m

m)

10 20 30 40 500Time (s)

(a)

S1S2S3

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

Dec

k di

spla

cem

ent (

mm

)

10 20 30 40 500Time (s)

(b)

Figure 8 Displacement time-history curves of the deck (a) cable S1 break (b) cable S2 break

S28S29S30

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

Dec

k di

spla

cem

ent (

mm

)

10 20 30 40 500Time (s)

(a)

S29S30S31

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

Dec

k di

spla

cem

ent (

mm

)

10 20 30 40 500Time (s)

(b)

Figure 9 Displacement time-history curves of the deck (a) cable S29 break (b) cable S30 break

8 Shock and Vibration

5 Robustness Evaluation

For long-span cable-supported arch bridges the failure of asingle cable may trigger progressive fracture of adjacentcables and damage to the deck and arch ribs (erefore it isnecessary to evaluate the robustness of cable-supported archbridges subjected to cable breaking (is section discussesthe robustness evaluation of the cable system deck and archrib subjected to a single cable break

51 Robustness of the Cable System To study the possibilityof a progressive fracture of adjacent cables subjected to asingle cable break the impact effect of the cable breakage is

investigated and compared with the carrying capacity of thecable system (e cable system of the Jiantiao Bridge iscomposed of 55 parallel steel wires that have a diameter of7mm and a tensile strength of 1670MPa (e cablersquos ulti-mate bearing capability fu is 3530 kN which is much biggerthan the maximum dynamic load of 821 kN caused by asingle cable break (is indicates that a single cable breakageunder the dead load will not trigger a progressive fracture ofthe adjacent cables

Furthermore to consider the load combination with ref-erence to the PTI [1] recommendation and the study by Zoliand Woodward [2] the limit state design combination Td andthe accidental combination Tf of the cable failure are defined as

0

5

10

15

20

25

30

35

Max

imum

disp

lace

men

t (m

m)

5 10 15 20 25 300Cable number

Figure 10 Maximum displacement of the deck caused by a single cable break

0

MLTMTMRT

MLBMBMRB

ndash200

ndash100

100

200

300

Mom

ent o

f arc

h rib

(kN

middotm)

10 20 30 40 500Time (s)

(a)

MLTMTMRT

MLBMBMRB

ndash200

ndash100

0

100

200

300M

omen

t of a

rch

rib (k

Nmiddotm

)

10 20 30 40 500Time (s)

(b)

Figure 11 Bending moment time-history curves of the arch rib (a) cable S1 break (b) cable S2 break

Shock and Vibration 9

Td 12Ts + 14Tv + 075 Tp + Tt1113872 1113873

Tf 12Ts + 075 Tv + Tp + Tt1113872 1113873 + Tc(6)

where Ts Tv Tp and Tt are the tension of the dead loadvehicle load pedestrian load and temperature load re-spectively Tc is the net dynamic effect of the cable breakwhich is equal to the maximum tension of the dynamicresponse minus the cable tension under the dead load

Figure 18 shows the comparison of the design limitcombination Td and the accidental combination Tf of each

cable It is observed that the values of cable fracture ac-cidental combination are all higher than the carryingcapability limit state combination (is means that thecable breakage accidental combination is the most un-favorable situation and should not be ignored in thedesign stage

(e maximum value of the accidental combination Tf is1130 kN and the minimum safety factor of the cablecompared with the ultimate bearing capability fu is 31 Itshows that the cable system has good robustness under acable breakage accidental combination and a single cable

MLTMTMRT

MLBMBMRB

0

100

200

300

400

500M

omen

t of a

rch

rib (k

Nmiddotm

)

10 20 30 40 500Time (s)

(a)

MLTMTMRT

MLBMBMRB

10 20 30 40 500Time(s)

0

100

200

300

400

500

Mom

ent o

f arc

h rib

(kN

middotm)

(b)

Figure 12 Bending moment time-history curves of the arch rib (a) cable S29 break (b) cable S30 break

Midspan

MaxMin

ndash100

0

100

200

300

Mom

ent o

f upp

er ch

ords

(kN

middotm)

5 10 15 20 25 300Cable number

(a)

MaxMin

Midspan

ndash100

0

100

200

300

400

500

Mom

ent o

f low

er ch

ords

(kN

middotm)

5 10 15 20 25 300Cable number

(b)

Figure 13 Envelope of the arch rib bending moment subjected to a single cable break (a) upper chords (b) lower chords

10 Shock and Vibration

break will not trigger a progressive breakage of adjacentcables

52 Robustness of the Deck Different from cable-stayedbridges that have a strong longitudinal stiffening girdercable-supported arch bridge deck systems always usetransverse girders as the main load-bearing members Toevaluate the damage to the deck system the capability of thedeck was obtained by a pushover analysis

(e pushover model was composed of three transversegirders with its flange plate An elastic support was used tosimulate the rigidity of the cables at the anchoring positions(e horizontal side displacement of the deck was notallowed by fixing the left and right ends of the deck system toavoid any rigid movement (e vertical pushover load wasapplied to the anchored position of the broken cable on the

right side of the middle girder as indicated by the arrow inFigure 19 (e relative displacements between the transversegirders were chosen as the capability index to represent thebearing capacity of the deck system

Figure 20 depicts the load-displacement curve of thedeck It can be noticed that the failure process of the deck canbe divided into three stages elastic stage cracked stage andfailure stage When the relative displacement is less than235mm the deck is in the elastic stage When the relativedisplacement increases from 235mm to 315mm cracksgradually increase and the deck works with the cracks Whenthe relative displacement exceeds 315mm the longitudinalreinforcement yields and the displacement increases rapidly

(e robustness of the deck was evaluated by comparingthe relative displacement of the cable breakage with that ofthe load-displacement curve As displayed in Figure 21 themaximum relative displacement of the deck is between 8mmand 15mm subjected to a single cable break which increases

Midspan

Upper chordsLower chords

0

100

200

300

Mom

ent v

aria

tion

(kN

middotm)

5 10 15 20 25 300Cable number

Figure 14 Moment variation of arch rib chords caused by a singlecable break

0 5 10 15 20 25 3010

12

14

16

18

DA

F of

adja

cent

cabl

e ten

sion

Cable number

Figure 15 DAF of the cable tension caused by a single cable break

10

12

14

16

18

DA

F of

dec

k di

spla

cem

ent

5 10 15 20 25 300Cable number

Figure 16 DAF of the deck displacement caused by a single cablebreak

10

12

14

16

18

20

22

24D

AF

of ar

ch ri

b be

ndin

g m

omen

t

5 10 15 20 25 300Cable number

Figure 17 DAF of the arch rib moment caused by a single cablebreak

Shock and Vibration 11

with the length of the broken cable Compared with that ofthe load-displacement curve in Figure 20 the relative dis-placement of the deck subjected to a single cable breakage iswithin the cracked stage and will not lead to a deck collapseUnder the worst conditions the deck system has a safety

factor of approximately 20 which indicates that the decksystem has relatively lower robustness than the cable system

53 Robustness of theArch Rib To evaluate the robustness ofthe arch rib which is an eccentric compression member anM-N curve was drawn to evaluate its bearing capacityFigures 22 and 23 show the axial force and bending momentcurves of the arch rib subjected to S1 and S30 cable breakrespectively (e curves in the circle are the internal forcecurves of the arch rib which are enlarged at the upper rightof the figure It is observed that the arch ribs are mainlyunder compression (e axial force is between 14000 kNand 17500 kN and the bending moment is between minus150and 450 kNmiddotm (e impact effect caused by the cable break

TdTf

400

600

800

1000

1200

1400Te

nsio

n of

cabl

es (k

N)

5 10 15 20 25 300Cable number

Figure 18 Comparison of the limit state combination Td andaccidental combination Tf of each cable

Figure 19 Pushover model of the deck system

0

500

1000

1500

2000

Push

forc

e (kN

)

10 20 30 400Relative displacement of anchorage end (mm)

Figure 20 Load-displacement curve of the deck

5 10 15 20 25 300Cable number

8

10

12

14

16

Max

imum

relat

ive d

ispla

cem

ent (

mm

)

Figure 21 Maximum relative deck displacements caused by asingle cable break

ndash200 ndash100 0 100 200 30014000

15000

16000

17000

MLTMLBMT

MBMRTMRB

Axi

al fo

rce (

kN)

Moments (kNmiddotm)

ndash50000

0

50000

100000

150000A

xial

forc

e (kN

)

0 10000 20000 30000ndash10000Moments (kNmiddotm)

Figure 22 Internal force change curves of the arch rib due to cableS1 break within its M-N curves

12 Shock and Vibration

was within 20 of the arch rib bearing capability indicatingthat the failure of a single cable does not cause failure of thearch rib (e safety factor of the arch rib is greater than 50which reveals that the arch rib has sufficient robustnesssubjected to single cable breakage

6 Conclusions

Cable breaking is common damage in cable-supported archbridges as a result of aging and erosion caused by the harshenvironment To evaluate the dynamic response and robust-ness of cable-supported arch bridges subjected to cablebreaking a nonlinear dynamic simulation procedure and ro-bustness evaluation method are developed in this study Aprototype arch bridge with a single cable break was analyzed asan example Several critical factors for cable breakage simu-lation such as the cable break mode cable break duration anddamping are studied (e dynamic responses of the cablesystem deck and rib are evaluated and the DAF of differentcomponents of the cable-supported arch bridge is also dis-cussed Robustness evaluation indexes of the cable systemdeck and arch rib are calculated and their robustness levels areevaluated (e main conclusions are as follows

(1) Using an appropriate simulation method is essentialto the dynamic analysis of a cable breaking processDirectly removing the failed cable in an appropriatetime step of a transient analysis can effectivelysimulate the cable breaking effect When the cablebreak duration is sufficiently short the maximumimpact effect can be fully attained regardless of thetype of the break simulation mode used (e Ray-leigh damping coefficient calculated by a higher-order frequency can avoid the suppression effect of alocal vibration and better reflect the local vibrationcaused by a cable break

(2) (e closest cable to the broken cable receives thelargest impact effect (e impact effect of the cable

tension caused by the break of a short cable is greaterthan that of a long cable (e deck displacement andbending moment increase with the length of thefailure cable(e breaking of a downstream cable haslittle effect on the upstream cables

(3) (e DAF of each component is different(e DAF ofcable tension is between 14 and 16 while that of thedeck displacement is between 13 and 165 (e archrib bending moment has the largest DAF that isbetween 15 and 24 which is greater than the PTIrecommended value of 20 in some cases Using aunified DAF of 20 cannot satisfy all the componentsof cable-supported arch bridges

(4) (e deck system is the least robust component ofcable-supported arch bridges (e cable system has asafety factor of 31 under the cable breakage acci-dental combination a single cable break will nottrigger progressive damage (e arch rib has suffi-cient robustness subjected to a single cable breakagethough it has the largest DAF

(e results obtained by this analysis are conducive to thedesign and robustness evaluation of cable-supported archbridges Although only the breakage of a single cable wasdemonstrated the proposed methodology can be applied tomultiple cable breakage scenario of cable-supported bridgesFuture research can be carried out in different types of cable-supported arch bridges subjected to multiple cablebreakages

Data Availability

(e data that support the findings of this study are availablefrom the corresponding author upon reasonable request

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(e authors would like to acknowledge Professor RuinianJiang of NMSU for providing help during the preparation ofthe manuscript and the authors also gratefully acknowledgethe support of Taizhou Highway Administration (is re-search was funded by the Zhejiang Provincial Transportationand Transportation Office Technology Funding Project(2012H47)

References

[1] PTI Recommendations for Stay Cable Design Testing andInstallation Post-Tensioning Institute Farmington Hills MIUSA 2012

[2] T Zoli and R Woodward ldquoDesign of long span bridges forcable lossrdquo IABSE Symposium Report vol 90 no 9 pp 17ndash252005

[3] A M Ruiz-Teran and A C Aparicio ldquoResponse of under-deck cable-stayed bridges to the accidental breakage of stay

MLTMLBMT

MBMRTMRB

0 100 200 300 400 50013000

14000

15000

16000

17000

18000

Axi

al fo

rce (

kN)

Moments (kNmiddotm)

ndash50000

0

50000

100000

150000A

xial

forc

e (kN

)

0 10000 20000 30000ndash10000Moments (kNmiddotm)

Figure 23 Internal force change curves of the arch rib due to cableS30 break within its M-N curves

Shock and Vibration 13

cablesrdquo Engineering Structures vol 31 no 7 pp 1425ndash14342009

[4] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part I bending moment acting on the deckrdquo En-gineering Structures vol 32 no 10 pp 3288ndash3300 2010

[5] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part II bending moment acting on the pylons andstress on the staysrdquo Engineering Structures vol 32 no 10pp 3301ndash3312 2010

[6] Y Zhou and S Chen ldquoTime-progressive dynamic assessmentof abrupt cable-breakage events on cable-stayed bridgesrdquoJournal of Bridge Engineering vol 19 no 2 pp 159ndash171 2014

[7] Y Park U Starossek H-M Koh J F Choo H Kim andS W Lee ldquoEffect of cable loss in cable stayed bridgesmdashfocuson dynamic amplificationrdquo IABSE Symposium Report vol 93no 5 pp 9ndash16 2007

[8] J-g Cai Y-x Xu L-p Zhuang J Feng and J ZhangldquoComparison of various procedures for progressive collapseanalysis of cable-stayed bridgesrdquo Journal of Zhejiang Uni-versity Science A vol 13 no 5 pp 323ndash334 2012

[9] Y Xiang Z Chen Y Yang H Lin and S Zhu ldquoDynamicresponse analysis for submerged floating tunnel with anchor-cables subjected to sudden cable breakagerdquoMarine Structuresvol 59 pp 179ndash191 2018

[10] Y F Zhou and S R Chen ldquoFramework of nonlinear dynamicsimulation of long-span cable-stayed bridge and traffic systemsubjected to cable-loss incidentsrdquo Journal of Structural En-gineering vol 142 no 3 2016

[11] W L Qiu M Jiang and C L Huang ldquoParametric study onresponses of a self-anchored suspension bridge to suddenbreakage of a hangerrdquo Scientific World Journal vol 2014Article ID 512120 10 pages 2014

[12] W Qiu M Jiang and Z Zhang ldquoResponses of self-anchoredsuspension bridge to sudden breakage of hangersrdquo StructuralEngineering and Mechanics vol 50 no 2 pp 241ndash255 2014

[13] C M Mozos and A C Aparicio ldquoNumerical and experi-mental study on the interaction cable structure during thefailure of a stay in a cable stayed bridgerdquo EngineeringStructures vol 33 no 8 pp 2330ndash2341 2011

[14] V Hoang O Kiyomiya and T An ldquoExperimental and nu-merical study of lateral cable rupture in cable-stayed bridgescase studyrdquo Journal of Bridge Engineering vol 23 no 6 2018

[15] M Domaneschi G P Cimellaro and G Scutiero ldquoDispro-portionate collapse of a cable-stayed bridgerdquo Proceedings ofthe Institution of Civil EngineersmdashBridge Engineering vol 172no 1 pp 13ndash26 2019

[16] M Wolff and U Starossek ldquoCable loss and progressivecollapse in cable-stayed bridgesrdquo Bridge Structures vol 5no 1 pp 17ndash28 2009

[17] G Wu W Qiu and T Wu ldquoNonlinear dynamic analysis ofthe self-anchored suspension bridge subjected to suddenbreakage of a hangerrdquo Engineering Failure Analysis vol 97pp 701ndash717 2019

[18] M Morgese F Ansari M Domaneschi and G P CimellaroldquoPost-collapse analysis of morandirsquos polcevera viaduct ingenoa Italyrdquo Journal of Civil Structural Health Monitoringvol 10 no 1 pp 69ndash85 2020

[19] M Shoghijavan and U Starossek ldquoStructural robustness oflong-span cable-supported bridges in a cable-loss scenariordquoJournal of Bridge Engineering vol 23 no 2 2018

[20] M Shoghijavan and U Starossek ldquoAn analytical study on thebending moment acting on the girder of a long-span cable-

supported bridge suffering from cable failurerdquo EngineeringStructures vol 167 pp 166ndash174 2018

[21] M Domaneschi C Pellecchia E De Iuliis et al ldquoCollapseanalysis of the Polcevera viaduct by the applied elementmethodrdquo Engineering Structures vol 214 p 110659 2020

[22] O Olamigoke Structural Response of Cable-Stayed Bridges toCable Loss University of Surrey Guildford UK 2018

[23] J Xu H H Sun and S Y Cai ldquoEffect of symmetrical brokenwires damage on mechanical characteristics of stay cablerdquoJournal of Sound and Vibration vol 461 2019

14 Shock and Vibration

Figure 4 displays the damping ratio curves obtainedunder each frequency combination (e damping ratio ζ12which was calculated using ω1 and ω2 showed significantsuppression of high-order vibrations (e value of ζ12 in the50th order mode is 22 and the value of the 100th order is33 which greatly exceeded the design value of 5 (edamping ratios ζ120 and ζ150 show a better fit within the first50th order (e damping ratio ζ1100 keeps at 3ndash5 on ahigh-order frequency which matches the damping char-acteristics of reinforced concrete structures therefore ζ1100was chosen as the damping ratio in this study

4 Dynamic Response Results

41 Tension in Adjacent Cables Figure 5 presents the ad-jacent cable tension time-history curves subjected to thebreakage of short cables S1(5a) and S2 (5b) respectively Itcan be observed in Figures 5(a) and 5(b) that the breakage ofcable S1 has the largest impact on cable S2 while thebreakage of cable S2 has the largest impact on cable S1 (isindicates that after a single cable breaks the closest cablereceives the maximum dynamic impact (e maximumdynamic response is 22 times as big as the cable tensionunder the dead load (e breaking of the downstream cableshas little effect on the cables on the upstream side (X1 and X2in the figure) and the maximum dynamic response of thecables on the same side is within 12 times of the tensionunder the dead load

Figure 6 illustrates the adjacent cable tension time-history curves subjected to the break of long cables S29 (6a)and S30 (6b) respectively It can be observed that thebreakage of cable S29 has the largest impact on cable S30and that of cable S30 has the largest impact on cable S29which is similar to the short-cable cases However themaximum dynamic response of the cable tension was 17times the tension of the dead load for the long-cablebreakage and the peak value of the dynamic response is

significantly smaller than that of the short cables (ebreakage of long cables results in a greater dynamic responseof the farther cables for example the maximum response ofS28 which was anchored on the adjacent transverse grid is14 times of the dead-load tension when cable S29 breaks(e effect on the cables on the upstream side is slight whichis the same as that in the short-cable cases

Figure 7 depicts the maximum tension in the adjacentcables caused by a single cable break It can be observed thatthe maximum cable force of 821 kN was caused by the breakof S3 which is the shortest cable As the length of the cableincreases the impact effect due to the break of the cablegradually decreases For the tension of all cables caused by asingle cable break the maximum value is 16 to 22 times asbig as the tension under the dead load

42 Displacement of theDeck (e displacement of the lowercable anchorage points was selected to investigate the vi-bration of the deck (e displacement time-history curves ofthe deck due to the break of a short cable (S1 and S2) and along cable (S29 and S30) are presented in Figures 8 and 9respectively

Figure 8 shows the displacement time-history curves dueto the break of cables S1 (8a) and S2 (8b) respectively Asshown in Figure 8 the breakage of a cable causes themaximum dynamic displacement at its own anchor point(e maximum displacement of S1 is minus113mm and that ofS2 is minus119mm (e displacement at the anchorage point ofthe adjacent cables located at the same transverse girder isslightly smaller than that of the broken cable (e dis-placements of the anchorage points that are not on the sametransverse girder of the broken cable are significantlysmaller

Figure 9 shows the displacement time-history curves dueto the break of long cables S29 (9a) and S30 (9b) respec-tively Compared with the short-cable break the long-cablebreak also caused the largest displacement at its own an-chorage point but the deck displacement was larger (emaximum displacement of S29 is minus277mm and S30 isminus279mm Moreover the maximum displacement of theanchorage points of the adjacent cables anchored on dif-ferent transverse girders also exceeds minus10mm

(e maximum deck displacement caused by thebreakage of each cable is shown in Figure 10 It is noticedthat the minimum deck displacement is 84mm which iscaused by the break of the shortest cable (S3) and themaximum deck displacement is 279mm which is caused bythe break of the longest cable (S30) (e displacement of thebridge deck almost increases proportionally with the lengthof the cable

43Moments of theArch Rib To determine the impact effectof cable fracture on the arch rib the bending moment time-history curves at the six nodes around the upper anchoragepoint of the broken cable were selected (e six rigid framenode bending moments are the upper chord node bendingmomentMT lower chord node bending momentMB upperleft node bending moment MLT lower left node bending

001s01s10s

20s40s100s

350

400

450

500

550

600

Tens

ion

of th

e adj

acen

t cab

le (k

N)

5 10 15 20 25 300Time (s)

Figure 3 Dynamic response to six cable break durations

Shock and Vibration 5

moment MLB upper right node bending moment MRT andlower right node bending moment MRB

Figure 11 depicts the moment time-history curves due tothe breakage of short cables S1 (11a) and S2 (11b) re-spectively Because cables S1 and S2 are anchored on theupper chord of the arch rib the dynamic response variationof MT is the most significant (e maximum changes of MTare 135 kNmiddotm due to the break of S1 and 150 kNmiddotmdue to thebreak of S2 (e bending moment changes at other positionsare small all are within 100 kNmiddotm

Figure 12 indicates the bending moment time-historycurves due to the breakage of long cables S29 (12a) and S30

(12b) respectively Because cables S29 and S30 are anchoredon the lower chord of the arch rib the variation of thedynamic response of MB is the most significant (e max-imum changes of MB are 168 kNmiddotm due to the break of S29and 170 kNmiddotmdue to the break of S30(emaximum changeof MLB and MRB is 145 kNmiddotm and the changes of MLT MTand MRT are within 100 kNmiddotm

(e bending moment in the upper and lower chords ofthe arch rib is investigated Figure 13 displays the envelope ofthe bending moments of the arch rib sections caused by thebreakage of each cable (e bending moment of the upperchord sections is between minus100 kNmiddotm and 210 kNmiddotm and

Dam

ping

ratio

ζ (

)

ζ12

ζ120

ζ150

ζ1100

ω1ω2 ω20 ω50 ω100

2 4 6 8 100Natural frequencies ω (Hz)

0

10

20

30

40

50

Figure 4 Damping ratio curves obtained under four frequency combinations

S2S3

X1X2

300

400

500

600

700

Tens

ion

of ca

bles

(kN

)

10 20 30 40 500Time (s)

(a)

S2S3

X1X2

300

400

500

600

700

Tens

ion

of ca

bles

(kN

)

10 20 30 40 500Time (s)

(b)

Figure 5 Tension time-history curves due to short-cable fracture (a) break of cable S1 (b) break of cable S2

6 Shock and Vibration

that of the lower chord is between minus100 kNmiddotm and 450 kNmiddotm(is indicates that the bending moment of the lower chordof the arch rib is greater than that of the upper chord

In order to reflect the variation amplitude of the arch ribbending moment during the breakage of each cable Fig-ure 14 shows the bending moment variation of the upperand lower chords of the arch rib caused by a single cablebreak (e variation of the bending moment of the upperchord is between 113 kNmiddotm and 188 kNmiddotm with an averagevalue of 137 kNmiddotm and the bending moment variation of thelower arch rib is between 100 kN and 280 kNmiddotm with anaverage value of 229 kNmiddotm Because most cables are an-chored to the lower chord except S1 and S2 the bendingmoments of the lower chord vary more than that of theupper chord Moreover as the length of the cable increasesthe fracture impact effect increases

44 DAF of the Cable Deck and Rib No specifications haveprovided details for calculating the DAF of cable-supportedarch bridges For the design of cable-stayed bridges the DAFis set between 15 and 20 in some codes and recommen-dations In this section the cable failure of the JiantiaoBridge is investigated to discuss the DAF for the cabletension deck displacement and arch rib bendingmoment ofcable-supported arch bridges Equation (1) is used to cal-culate DAF

DAF RD minus R0

RP D minus R0 (5)

where RD represents themaximum dynamic response R0 theinitial state under the dead load before cable break and RPDthe stable value after cable break

S28S30

X29X30

300

350

400

450

500

550

600

650Te

nsio

n of

cabl

es (k

N)

10 20 30 40 500Time (s)

(a)

S29S31

X29X30

10 20 30 40 500Time (s)

300

350

400

450

500

550

600

650

Tens

ion

of ca

bles

(kN

)(b)

Figure 6 Time-history curves due to long-cable fracture (a) break of cable S29 (b) break of cable S30

0

200

400

600

800

Tens

ion

of ca

bles

(kN

)

5 10 15 20 25 300Cable number

Figure 7 Maximum tension of the adjacent cables caused by the break of a single cable

Shock and Vibration 7

Figure 15 shows the calculated DAF for the cable tensiondue to the break of the adjacent cable It can be observed thatthe DAF of the adjacent cable tension is distributed between14 and 16 with an average value of 154

Figure 16 presents the DAF for the deck displacementwhich is between 130 and 165 with an average value of 140(ere is a tendency for the deck displacement DAF to de-crease as the cable length increases

Figure 17 displays the DAF for the arch rib bendingmoment which is between 15 and 24 with an average value

of 193 (e DAF exceeds 20 in some cases which is therecommended value of PTI [1]

Compared with the recommended common DAF valueof 20 for cable-stayed bridges by PTI [1] the DAF of thecable-supported arch bridge in this study varies for eachcomponent (e maximum DAFs of the cable tension anddeck displacement are less than 20 while the maximumDAF of the bending moment of the arch rib exceeds 20(isshows that adopting a common DAF for all the componentsof cable-supported arch bridge structures is not advisable

S1S2S3

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4D

eck

disp

lace

men

t (m

m)

10 20 30 40 500Time (s)

(a)

S1S2S3

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

Dec

k di

spla

cem

ent (

mm

)

10 20 30 40 500Time (s)

(b)

Figure 8 Displacement time-history curves of the deck (a) cable S1 break (b) cable S2 break

S28S29S30

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

Dec

k di

spla

cem

ent (

mm

)

10 20 30 40 500Time (s)

(a)

S29S30S31

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

Dec

k di

spla

cem

ent (

mm

)

10 20 30 40 500Time (s)

(b)

Figure 9 Displacement time-history curves of the deck (a) cable S29 break (b) cable S30 break

8 Shock and Vibration

5 Robustness Evaluation

For long-span cable-supported arch bridges the failure of asingle cable may trigger progressive fracture of adjacentcables and damage to the deck and arch ribs (erefore it isnecessary to evaluate the robustness of cable-supported archbridges subjected to cable breaking (is section discussesthe robustness evaluation of the cable system deck and archrib subjected to a single cable break

51 Robustness of the Cable System To study the possibilityof a progressive fracture of adjacent cables subjected to asingle cable break the impact effect of the cable breakage is

investigated and compared with the carrying capacity of thecable system (e cable system of the Jiantiao Bridge iscomposed of 55 parallel steel wires that have a diameter of7mm and a tensile strength of 1670MPa (e cablersquos ulti-mate bearing capability fu is 3530 kN which is much biggerthan the maximum dynamic load of 821 kN caused by asingle cable break (is indicates that a single cable breakageunder the dead load will not trigger a progressive fracture ofthe adjacent cables

Furthermore to consider the load combination with ref-erence to the PTI [1] recommendation and the study by Zoliand Woodward [2] the limit state design combination Td andthe accidental combination Tf of the cable failure are defined as

0

5

10

15

20

25

30

35

Max

imum

disp

lace

men

t (m

m)

5 10 15 20 25 300Cable number

Figure 10 Maximum displacement of the deck caused by a single cable break

0

MLTMTMRT

MLBMBMRB

ndash200

ndash100

100

200

300

Mom

ent o

f arc

h rib

(kN

middotm)

10 20 30 40 500Time (s)

(a)

MLTMTMRT

MLBMBMRB

ndash200

ndash100

0

100

200

300M

omen

t of a

rch

rib (k

Nmiddotm

)

10 20 30 40 500Time (s)

(b)

Figure 11 Bending moment time-history curves of the arch rib (a) cable S1 break (b) cable S2 break

Shock and Vibration 9

Td 12Ts + 14Tv + 075 Tp + Tt1113872 1113873

Tf 12Ts + 075 Tv + Tp + Tt1113872 1113873 + Tc(6)

where Ts Tv Tp and Tt are the tension of the dead loadvehicle load pedestrian load and temperature load re-spectively Tc is the net dynamic effect of the cable breakwhich is equal to the maximum tension of the dynamicresponse minus the cable tension under the dead load

Figure 18 shows the comparison of the design limitcombination Td and the accidental combination Tf of each

cable It is observed that the values of cable fracture ac-cidental combination are all higher than the carryingcapability limit state combination (is means that thecable breakage accidental combination is the most un-favorable situation and should not be ignored in thedesign stage

(e maximum value of the accidental combination Tf is1130 kN and the minimum safety factor of the cablecompared with the ultimate bearing capability fu is 31 Itshows that the cable system has good robustness under acable breakage accidental combination and a single cable

MLTMTMRT

MLBMBMRB

0

100

200

300

400

500M

omen

t of a

rch

rib (k

Nmiddotm

)

10 20 30 40 500Time (s)

(a)

MLTMTMRT

MLBMBMRB

10 20 30 40 500Time(s)

0

100

200

300

400

500

Mom

ent o

f arc

h rib

(kN

middotm)

(b)

Figure 12 Bending moment time-history curves of the arch rib (a) cable S29 break (b) cable S30 break

Midspan

MaxMin

ndash100

0

100

200

300

Mom

ent o

f upp

er ch

ords

(kN

middotm)

5 10 15 20 25 300Cable number

(a)

MaxMin

Midspan

ndash100

0

100

200

300

400

500

Mom

ent o

f low

er ch

ords

(kN

middotm)

5 10 15 20 25 300Cable number

(b)

Figure 13 Envelope of the arch rib bending moment subjected to a single cable break (a) upper chords (b) lower chords

10 Shock and Vibration

break will not trigger a progressive breakage of adjacentcables

52 Robustness of the Deck Different from cable-stayedbridges that have a strong longitudinal stiffening girdercable-supported arch bridge deck systems always usetransverse girders as the main load-bearing members Toevaluate the damage to the deck system the capability of thedeck was obtained by a pushover analysis

(e pushover model was composed of three transversegirders with its flange plate An elastic support was used tosimulate the rigidity of the cables at the anchoring positions(e horizontal side displacement of the deck was notallowed by fixing the left and right ends of the deck system toavoid any rigid movement (e vertical pushover load wasapplied to the anchored position of the broken cable on the

right side of the middle girder as indicated by the arrow inFigure 19 (e relative displacements between the transversegirders were chosen as the capability index to represent thebearing capacity of the deck system

Figure 20 depicts the load-displacement curve of thedeck It can be noticed that the failure process of the deck canbe divided into three stages elastic stage cracked stage andfailure stage When the relative displacement is less than235mm the deck is in the elastic stage When the relativedisplacement increases from 235mm to 315mm cracksgradually increase and the deck works with the cracks Whenthe relative displacement exceeds 315mm the longitudinalreinforcement yields and the displacement increases rapidly

(e robustness of the deck was evaluated by comparingthe relative displacement of the cable breakage with that ofthe load-displacement curve As displayed in Figure 21 themaximum relative displacement of the deck is between 8mmand 15mm subjected to a single cable break which increases

Midspan

Upper chordsLower chords

0

100

200

300

Mom

ent v

aria

tion

(kN

middotm)

5 10 15 20 25 300Cable number

Figure 14 Moment variation of arch rib chords caused by a singlecable break

0 5 10 15 20 25 3010

12

14

16

18

DA

F of

adja

cent

cabl

e ten

sion

Cable number

Figure 15 DAF of the cable tension caused by a single cable break

10

12

14

16

18

DA

F of

dec

k di

spla

cem

ent

5 10 15 20 25 300Cable number

Figure 16 DAF of the deck displacement caused by a single cablebreak

10

12

14

16

18

20

22

24D

AF

of ar

ch ri

b be

ndin

g m

omen

t

5 10 15 20 25 300Cable number

Figure 17 DAF of the arch rib moment caused by a single cablebreak

Shock and Vibration 11

with the length of the broken cable Compared with that ofthe load-displacement curve in Figure 20 the relative dis-placement of the deck subjected to a single cable breakage iswithin the cracked stage and will not lead to a deck collapseUnder the worst conditions the deck system has a safety

factor of approximately 20 which indicates that the decksystem has relatively lower robustness than the cable system

53 Robustness of theArch Rib To evaluate the robustness ofthe arch rib which is an eccentric compression member anM-N curve was drawn to evaluate its bearing capacityFigures 22 and 23 show the axial force and bending momentcurves of the arch rib subjected to S1 and S30 cable breakrespectively (e curves in the circle are the internal forcecurves of the arch rib which are enlarged at the upper rightof the figure It is observed that the arch ribs are mainlyunder compression (e axial force is between 14000 kNand 17500 kN and the bending moment is between minus150and 450 kNmiddotm (e impact effect caused by the cable break

TdTf

400

600

800

1000

1200

1400Te

nsio

n of

cabl

es (k

N)

5 10 15 20 25 300Cable number

Figure 18 Comparison of the limit state combination Td andaccidental combination Tf of each cable

Figure 19 Pushover model of the deck system

0

500

1000

1500

2000

Push

forc

e (kN

)

10 20 30 400Relative displacement of anchorage end (mm)

Figure 20 Load-displacement curve of the deck

5 10 15 20 25 300Cable number

8

10

12

14

16

Max

imum

relat

ive d

ispla

cem

ent (

mm

)

Figure 21 Maximum relative deck displacements caused by asingle cable break

ndash200 ndash100 0 100 200 30014000

15000

16000

17000

MLTMLBMT

MBMRTMRB

Axi

al fo

rce (

kN)

Moments (kNmiddotm)

ndash50000

0

50000

100000

150000A

xial

forc

e (kN

)

0 10000 20000 30000ndash10000Moments (kNmiddotm)

Figure 22 Internal force change curves of the arch rib due to cableS1 break within its M-N curves

12 Shock and Vibration

was within 20 of the arch rib bearing capability indicatingthat the failure of a single cable does not cause failure of thearch rib (e safety factor of the arch rib is greater than 50which reveals that the arch rib has sufficient robustnesssubjected to single cable breakage

6 Conclusions

Cable breaking is common damage in cable-supported archbridges as a result of aging and erosion caused by the harshenvironment To evaluate the dynamic response and robust-ness of cable-supported arch bridges subjected to cablebreaking a nonlinear dynamic simulation procedure and ro-bustness evaluation method are developed in this study Aprototype arch bridge with a single cable break was analyzed asan example Several critical factors for cable breakage simu-lation such as the cable break mode cable break duration anddamping are studied (e dynamic responses of the cablesystem deck and rib are evaluated and the DAF of differentcomponents of the cable-supported arch bridge is also dis-cussed Robustness evaluation indexes of the cable systemdeck and arch rib are calculated and their robustness levels areevaluated (e main conclusions are as follows

(1) Using an appropriate simulation method is essentialto the dynamic analysis of a cable breaking processDirectly removing the failed cable in an appropriatetime step of a transient analysis can effectivelysimulate the cable breaking effect When the cablebreak duration is sufficiently short the maximumimpact effect can be fully attained regardless of thetype of the break simulation mode used (e Ray-leigh damping coefficient calculated by a higher-order frequency can avoid the suppression effect of alocal vibration and better reflect the local vibrationcaused by a cable break

(2) (e closest cable to the broken cable receives thelargest impact effect (e impact effect of the cable

tension caused by the break of a short cable is greaterthan that of a long cable (e deck displacement andbending moment increase with the length of thefailure cable(e breaking of a downstream cable haslittle effect on the upstream cables

(3) (e DAF of each component is different(e DAF ofcable tension is between 14 and 16 while that of thedeck displacement is between 13 and 165 (e archrib bending moment has the largest DAF that isbetween 15 and 24 which is greater than the PTIrecommended value of 20 in some cases Using aunified DAF of 20 cannot satisfy all the componentsof cable-supported arch bridges

(4) (e deck system is the least robust component ofcable-supported arch bridges (e cable system has asafety factor of 31 under the cable breakage acci-dental combination a single cable break will nottrigger progressive damage (e arch rib has suffi-cient robustness subjected to a single cable breakagethough it has the largest DAF

(e results obtained by this analysis are conducive to thedesign and robustness evaluation of cable-supported archbridges Although only the breakage of a single cable wasdemonstrated the proposed methodology can be applied tomultiple cable breakage scenario of cable-supported bridgesFuture research can be carried out in different types of cable-supported arch bridges subjected to multiple cablebreakages

Data Availability

(e data that support the findings of this study are availablefrom the corresponding author upon reasonable request

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(e authors would like to acknowledge Professor RuinianJiang of NMSU for providing help during the preparation ofthe manuscript and the authors also gratefully acknowledgethe support of Taizhou Highway Administration (is re-search was funded by the Zhejiang Provincial Transportationand Transportation Office Technology Funding Project(2012H47)

References

[1] PTI Recommendations for Stay Cable Design Testing andInstallation Post-Tensioning Institute Farmington Hills MIUSA 2012

[2] T Zoli and R Woodward ldquoDesign of long span bridges forcable lossrdquo IABSE Symposium Report vol 90 no 9 pp 17ndash252005

[3] A M Ruiz-Teran and A C Aparicio ldquoResponse of under-deck cable-stayed bridges to the accidental breakage of stay

MLTMLBMT

MBMRTMRB

0 100 200 300 400 50013000

14000

15000

16000

17000

18000

Axi

al fo

rce (

kN)

Moments (kNmiddotm)

ndash50000

0

50000

100000

150000A

xial

forc

e (kN

)

0 10000 20000 30000ndash10000Moments (kNmiddotm)

Figure 23 Internal force change curves of the arch rib due to cableS30 break within its M-N curves

Shock and Vibration 13

cablesrdquo Engineering Structures vol 31 no 7 pp 1425ndash14342009

[4] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part I bending moment acting on the deckrdquo En-gineering Structures vol 32 no 10 pp 3288ndash3300 2010

[5] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part II bending moment acting on the pylons andstress on the staysrdquo Engineering Structures vol 32 no 10pp 3301ndash3312 2010

[6] Y Zhou and S Chen ldquoTime-progressive dynamic assessmentof abrupt cable-breakage events on cable-stayed bridgesrdquoJournal of Bridge Engineering vol 19 no 2 pp 159ndash171 2014

[7] Y Park U Starossek H-M Koh J F Choo H Kim andS W Lee ldquoEffect of cable loss in cable stayed bridgesmdashfocuson dynamic amplificationrdquo IABSE Symposium Report vol 93no 5 pp 9ndash16 2007

[8] J-g Cai Y-x Xu L-p Zhuang J Feng and J ZhangldquoComparison of various procedures for progressive collapseanalysis of cable-stayed bridgesrdquo Journal of Zhejiang Uni-versity Science A vol 13 no 5 pp 323ndash334 2012

[9] Y Xiang Z Chen Y Yang H Lin and S Zhu ldquoDynamicresponse analysis for submerged floating tunnel with anchor-cables subjected to sudden cable breakagerdquoMarine Structuresvol 59 pp 179ndash191 2018

[10] Y F Zhou and S R Chen ldquoFramework of nonlinear dynamicsimulation of long-span cable-stayed bridge and traffic systemsubjected to cable-loss incidentsrdquo Journal of Structural En-gineering vol 142 no 3 2016

[11] W L Qiu M Jiang and C L Huang ldquoParametric study onresponses of a self-anchored suspension bridge to suddenbreakage of a hangerrdquo Scientific World Journal vol 2014Article ID 512120 10 pages 2014

[12] W Qiu M Jiang and Z Zhang ldquoResponses of self-anchoredsuspension bridge to sudden breakage of hangersrdquo StructuralEngineering and Mechanics vol 50 no 2 pp 241ndash255 2014

[13] C M Mozos and A C Aparicio ldquoNumerical and experi-mental study on the interaction cable structure during thefailure of a stay in a cable stayed bridgerdquo EngineeringStructures vol 33 no 8 pp 2330ndash2341 2011

[14] V Hoang O Kiyomiya and T An ldquoExperimental and nu-merical study of lateral cable rupture in cable-stayed bridgescase studyrdquo Journal of Bridge Engineering vol 23 no 6 2018

[15] M Domaneschi G P Cimellaro and G Scutiero ldquoDispro-portionate collapse of a cable-stayed bridgerdquo Proceedings ofthe Institution of Civil EngineersmdashBridge Engineering vol 172no 1 pp 13ndash26 2019

[16] M Wolff and U Starossek ldquoCable loss and progressivecollapse in cable-stayed bridgesrdquo Bridge Structures vol 5no 1 pp 17ndash28 2009

[17] G Wu W Qiu and T Wu ldquoNonlinear dynamic analysis ofthe self-anchored suspension bridge subjected to suddenbreakage of a hangerrdquo Engineering Failure Analysis vol 97pp 701ndash717 2019

[18] M Morgese F Ansari M Domaneschi and G P CimellaroldquoPost-collapse analysis of morandirsquos polcevera viaduct ingenoa Italyrdquo Journal of Civil Structural Health Monitoringvol 10 no 1 pp 69ndash85 2020

[19] M Shoghijavan and U Starossek ldquoStructural robustness oflong-span cable-supported bridges in a cable-loss scenariordquoJournal of Bridge Engineering vol 23 no 2 2018

[20] M Shoghijavan and U Starossek ldquoAn analytical study on thebending moment acting on the girder of a long-span cable-

supported bridge suffering from cable failurerdquo EngineeringStructures vol 167 pp 166ndash174 2018

[21] M Domaneschi C Pellecchia E De Iuliis et al ldquoCollapseanalysis of the Polcevera viaduct by the applied elementmethodrdquo Engineering Structures vol 214 p 110659 2020

[22] O Olamigoke Structural Response of Cable-Stayed Bridges toCable Loss University of Surrey Guildford UK 2018

[23] J Xu H H Sun and S Y Cai ldquoEffect of symmetrical brokenwires damage on mechanical characteristics of stay cablerdquoJournal of Sound and Vibration vol 461 2019

14 Shock and Vibration

moment MLB upper right node bending moment MRT andlower right node bending moment MRB

Figure 11 depicts the moment time-history curves due tothe breakage of short cables S1 (11a) and S2 (11b) re-spectively Because cables S1 and S2 are anchored on theupper chord of the arch rib the dynamic response variationof MT is the most significant (e maximum changes of MTare 135 kNmiddotm due to the break of S1 and 150 kNmiddotmdue to thebreak of S2 (e bending moment changes at other positionsare small all are within 100 kNmiddotm

Figure 12 indicates the bending moment time-historycurves due to the breakage of long cables S29 (12a) and S30

(12b) respectively Because cables S29 and S30 are anchoredon the lower chord of the arch rib the variation of thedynamic response of MB is the most significant (e max-imum changes of MB are 168 kNmiddotm due to the break of S29and 170 kNmiddotmdue to the break of S30(emaximum changeof MLB and MRB is 145 kNmiddotm and the changes of MLT MTand MRT are within 100 kNmiddotm

(e bending moment in the upper and lower chords ofthe arch rib is investigated Figure 13 displays the envelope ofthe bending moments of the arch rib sections caused by thebreakage of each cable (e bending moment of the upperchord sections is between minus100 kNmiddotm and 210 kNmiddotm and

Dam

ping

ratio

ζ (

)

ζ12

ζ120

ζ150

ζ1100

ω1ω2 ω20 ω50 ω100

2 4 6 8 100Natural frequencies ω (Hz)

0

10

20

30

40

50

Figure 4 Damping ratio curves obtained under four frequency combinations

S2S3

X1X2

300

400

500

600

700

Tens

ion

of ca

bles

(kN

)

10 20 30 40 500Time (s)

(a)

S2S3

X1X2

300

400

500

600

700

Tens

ion

of ca

bles

(kN

)

10 20 30 40 500Time (s)

(b)

Figure 5 Tension time-history curves due to short-cable fracture (a) break of cable S1 (b) break of cable S2

6 Shock and Vibration

that of the lower chord is between minus100 kNmiddotm and 450 kNmiddotm(is indicates that the bending moment of the lower chordof the arch rib is greater than that of the upper chord

In order to reflect the variation amplitude of the arch ribbending moment during the breakage of each cable Fig-ure 14 shows the bending moment variation of the upperand lower chords of the arch rib caused by a single cablebreak (e variation of the bending moment of the upperchord is between 113 kNmiddotm and 188 kNmiddotm with an averagevalue of 137 kNmiddotm and the bending moment variation of thelower arch rib is between 100 kN and 280 kNmiddotm with anaverage value of 229 kNmiddotm Because most cables are an-chored to the lower chord except S1 and S2 the bendingmoments of the lower chord vary more than that of theupper chord Moreover as the length of the cable increasesthe fracture impact effect increases

44 DAF of the Cable Deck and Rib No specifications haveprovided details for calculating the DAF of cable-supportedarch bridges For the design of cable-stayed bridges the DAFis set between 15 and 20 in some codes and recommen-dations In this section the cable failure of the JiantiaoBridge is investigated to discuss the DAF for the cabletension deck displacement and arch rib bendingmoment ofcable-supported arch bridges Equation (1) is used to cal-culate DAF

DAF RD minus R0

RP D minus R0 (5)

where RD represents themaximum dynamic response R0 theinitial state under the dead load before cable break and RPDthe stable value after cable break

S28S30

X29X30

300

350

400

450

500

550

600

650Te

nsio

n of

cabl

es (k

N)

10 20 30 40 500Time (s)

(a)

S29S31

X29X30

10 20 30 40 500Time (s)

300

350

400

450

500

550

600

650

Tens

ion

of ca

bles

(kN

)(b)

Figure 6 Time-history curves due to long-cable fracture (a) break of cable S29 (b) break of cable S30

0

200

400

600

800

Tens

ion

of ca

bles

(kN

)

5 10 15 20 25 300Cable number

Figure 7 Maximum tension of the adjacent cables caused by the break of a single cable

Shock and Vibration 7

Figure 15 shows the calculated DAF for the cable tensiondue to the break of the adjacent cable It can be observed thatthe DAF of the adjacent cable tension is distributed between14 and 16 with an average value of 154

Figure 16 presents the DAF for the deck displacementwhich is between 130 and 165 with an average value of 140(ere is a tendency for the deck displacement DAF to de-crease as the cable length increases

Figure 17 displays the DAF for the arch rib bendingmoment which is between 15 and 24 with an average value

of 193 (e DAF exceeds 20 in some cases which is therecommended value of PTI [1]

Compared with the recommended common DAF valueof 20 for cable-stayed bridges by PTI [1] the DAF of thecable-supported arch bridge in this study varies for eachcomponent (e maximum DAFs of the cable tension anddeck displacement are less than 20 while the maximumDAF of the bending moment of the arch rib exceeds 20(isshows that adopting a common DAF for all the componentsof cable-supported arch bridge structures is not advisable

S1S2S3

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4D

eck

disp

lace

men

t (m

m)

10 20 30 40 500Time (s)

(a)

S1S2S3

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

Dec

k di

spla

cem

ent (

mm

)

10 20 30 40 500Time (s)

(b)

Figure 8 Displacement time-history curves of the deck (a) cable S1 break (b) cable S2 break

S28S29S30

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

Dec

k di

spla

cem

ent (

mm

)

10 20 30 40 500Time (s)

(a)

S29S30S31

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

Dec

k di

spla

cem

ent (

mm

)

10 20 30 40 500Time (s)

(b)

Figure 9 Displacement time-history curves of the deck (a) cable S29 break (b) cable S30 break

8 Shock and Vibration

5 Robustness Evaluation

For long-span cable-supported arch bridges the failure of asingle cable may trigger progressive fracture of adjacentcables and damage to the deck and arch ribs (erefore it isnecessary to evaluate the robustness of cable-supported archbridges subjected to cable breaking (is section discussesthe robustness evaluation of the cable system deck and archrib subjected to a single cable break

51 Robustness of the Cable System To study the possibilityof a progressive fracture of adjacent cables subjected to asingle cable break the impact effect of the cable breakage is

investigated and compared with the carrying capacity of thecable system (e cable system of the Jiantiao Bridge iscomposed of 55 parallel steel wires that have a diameter of7mm and a tensile strength of 1670MPa (e cablersquos ulti-mate bearing capability fu is 3530 kN which is much biggerthan the maximum dynamic load of 821 kN caused by asingle cable break (is indicates that a single cable breakageunder the dead load will not trigger a progressive fracture ofthe adjacent cables

Furthermore to consider the load combination with ref-erence to the PTI [1] recommendation and the study by Zoliand Woodward [2] the limit state design combination Td andthe accidental combination Tf of the cable failure are defined as

0

5

10

15

20

25

30

35

Max

imum

disp

lace

men

t (m

m)

5 10 15 20 25 300Cable number

Figure 10 Maximum displacement of the deck caused by a single cable break

0

MLTMTMRT

MLBMBMRB

ndash200

ndash100

100

200

300

Mom

ent o

f arc

h rib

(kN

middotm)

10 20 30 40 500Time (s)

(a)

MLTMTMRT

MLBMBMRB

ndash200

ndash100

0

100

200

300M

omen

t of a

rch

rib (k

Nmiddotm

)

10 20 30 40 500Time (s)

(b)

Figure 11 Bending moment time-history curves of the arch rib (a) cable S1 break (b) cable S2 break

Shock and Vibration 9

Td 12Ts + 14Tv + 075 Tp + Tt1113872 1113873

Tf 12Ts + 075 Tv + Tp + Tt1113872 1113873 + Tc(6)

where Ts Tv Tp and Tt are the tension of the dead loadvehicle load pedestrian load and temperature load re-spectively Tc is the net dynamic effect of the cable breakwhich is equal to the maximum tension of the dynamicresponse minus the cable tension under the dead load

Figure 18 shows the comparison of the design limitcombination Td and the accidental combination Tf of each

cable It is observed that the values of cable fracture ac-cidental combination are all higher than the carryingcapability limit state combination (is means that thecable breakage accidental combination is the most un-favorable situation and should not be ignored in thedesign stage

(e maximum value of the accidental combination Tf is1130 kN and the minimum safety factor of the cablecompared with the ultimate bearing capability fu is 31 Itshows that the cable system has good robustness under acable breakage accidental combination and a single cable

MLTMTMRT

MLBMBMRB

0

100

200

300

400

500M

omen

t of a

rch

rib (k

Nmiddotm

)

10 20 30 40 500Time (s)

(a)

MLTMTMRT

MLBMBMRB

10 20 30 40 500Time(s)

0

100

200

300

400

500

Mom

ent o

f arc

h rib

(kN

middotm)

(b)

Figure 12 Bending moment time-history curves of the arch rib (a) cable S29 break (b) cable S30 break

Midspan

MaxMin

ndash100

0

100

200

300

Mom

ent o

f upp

er ch

ords

(kN

middotm)

5 10 15 20 25 300Cable number

(a)

MaxMin

Midspan

ndash100

0

100

200

300

400

500

Mom

ent o

f low

er ch

ords

(kN

middotm)

5 10 15 20 25 300Cable number

(b)

Figure 13 Envelope of the arch rib bending moment subjected to a single cable break (a) upper chords (b) lower chords

10 Shock and Vibration

break will not trigger a progressive breakage of adjacentcables

52 Robustness of the Deck Different from cable-stayedbridges that have a strong longitudinal stiffening girdercable-supported arch bridge deck systems always usetransverse girders as the main load-bearing members Toevaluate the damage to the deck system the capability of thedeck was obtained by a pushover analysis

(e pushover model was composed of three transversegirders with its flange plate An elastic support was used tosimulate the rigidity of the cables at the anchoring positions(e horizontal side displacement of the deck was notallowed by fixing the left and right ends of the deck system toavoid any rigid movement (e vertical pushover load wasapplied to the anchored position of the broken cable on the

right side of the middle girder as indicated by the arrow inFigure 19 (e relative displacements between the transversegirders were chosen as the capability index to represent thebearing capacity of the deck system

Figure 20 depicts the load-displacement curve of thedeck It can be noticed that the failure process of the deck canbe divided into three stages elastic stage cracked stage andfailure stage When the relative displacement is less than235mm the deck is in the elastic stage When the relativedisplacement increases from 235mm to 315mm cracksgradually increase and the deck works with the cracks Whenthe relative displacement exceeds 315mm the longitudinalreinforcement yields and the displacement increases rapidly

(e robustness of the deck was evaluated by comparingthe relative displacement of the cable breakage with that ofthe load-displacement curve As displayed in Figure 21 themaximum relative displacement of the deck is between 8mmand 15mm subjected to a single cable break which increases

Midspan

Upper chordsLower chords

0

100

200

300

Mom

ent v

aria

tion

(kN

middotm)

5 10 15 20 25 300Cable number

Figure 14 Moment variation of arch rib chords caused by a singlecable break

0 5 10 15 20 25 3010

12

14

16

18

DA

F of

adja

cent

cabl

e ten

sion

Cable number

Figure 15 DAF of the cable tension caused by a single cable break

10

12

14

16

18

DA

F of

dec

k di

spla

cem

ent

5 10 15 20 25 300Cable number

Figure 16 DAF of the deck displacement caused by a single cablebreak

10

12

14

16

18

20

22

24D

AF

of ar

ch ri

b be

ndin

g m

omen

t

5 10 15 20 25 300Cable number

Figure 17 DAF of the arch rib moment caused by a single cablebreak

Shock and Vibration 11

with the length of the broken cable Compared with that ofthe load-displacement curve in Figure 20 the relative dis-placement of the deck subjected to a single cable breakage iswithin the cracked stage and will not lead to a deck collapseUnder the worst conditions the deck system has a safety

factor of approximately 20 which indicates that the decksystem has relatively lower robustness than the cable system

53 Robustness of theArch Rib To evaluate the robustness ofthe arch rib which is an eccentric compression member anM-N curve was drawn to evaluate its bearing capacityFigures 22 and 23 show the axial force and bending momentcurves of the arch rib subjected to S1 and S30 cable breakrespectively (e curves in the circle are the internal forcecurves of the arch rib which are enlarged at the upper rightof the figure It is observed that the arch ribs are mainlyunder compression (e axial force is between 14000 kNand 17500 kN and the bending moment is between minus150and 450 kNmiddotm (e impact effect caused by the cable break

TdTf

400

600

800

1000

1200

1400Te

nsio

n of

cabl

es (k

N)

5 10 15 20 25 300Cable number

Figure 18 Comparison of the limit state combination Td andaccidental combination Tf of each cable

Figure 19 Pushover model of the deck system

0

500

1000

1500

2000

Push

forc

e (kN

)

10 20 30 400Relative displacement of anchorage end (mm)

Figure 20 Load-displacement curve of the deck

5 10 15 20 25 300Cable number

8

10

12

14

16

Max

imum

relat

ive d

ispla

cem

ent (

mm

)

Figure 21 Maximum relative deck displacements caused by asingle cable break

ndash200 ndash100 0 100 200 30014000

15000

16000

17000

MLTMLBMT

MBMRTMRB

Axi

al fo

rce (

kN)

Moments (kNmiddotm)

ndash50000

0

50000

100000

150000A

xial

forc

e (kN

)

0 10000 20000 30000ndash10000Moments (kNmiddotm)

Figure 22 Internal force change curves of the arch rib due to cableS1 break within its M-N curves

12 Shock and Vibration

was within 20 of the arch rib bearing capability indicatingthat the failure of a single cable does not cause failure of thearch rib (e safety factor of the arch rib is greater than 50which reveals that the arch rib has sufficient robustnesssubjected to single cable breakage

6 Conclusions

Cable breaking is common damage in cable-supported archbridges as a result of aging and erosion caused by the harshenvironment To evaluate the dynamic response and robust-ness of cable-supported arch bridges subjected to cablebreaking a nonlinear dynamic simulation procedure and ro-bustness evaluation method are developed in this study Aprototype arch bridge with a single cable break was analyzed asan example Several critical factors for cable breakage simu-lation such as the cable break mode cable break duration anddamping are studied (e dynamic responses of the cablesystem deck and rib are evaluated and the DAF of differentcomponents of the cable-supported arch bridge is also dis-cussed Robustness evaluation indexes of the cable systemdeck and arch rib are calculated and their robustness levels areevaluated (e main conclusions are as follows

(1) Using an appropriate simulation method is essentialto the dynamic analysis of a cable breaking processDirectly removing the failed cable in an appropriatetime step of a transient analysis can effectivelysimulate the cable breaking effect When the cablebreak duration is sufficiently short the maximumimpact effect can be fully attained regardless of thetype of the break simulation mode used (e Ray-leigh damping coefficient calculated by a higher-order frequency can avoid the suppression effect of alocal vibration and better reflect the local vibrationcaused by a cable break

(2) (e closest cable to the broken cable receives thelargest impact effect (e impact effect of the cable

tension caused by the break of a short cable is greaterthan that of a long cable (e deck displacement andbending moment increase with the length of thefailure cable(e breaking of a downstream cable haslittle effect on the upstream cables

(3) (e DAF of each component is different(e DAF ofcable tension is between 14 and 16 while that of thedeck displacement is between 13 and 165 (e archrib bending moment has the largest DAF that isbetween 15 and 24 which is greater than the PTIrecommended value of 20 in some cases Using aunified DAF of 20 cannot satisfy all the componentsof cable-supported arch bridges

(4) (e deck system is the least robust component ofcable-supported arch bridges (e cable system has asafety factor of 31 under the cable breakage acci-dental combination a single cable break will nottrigger progressive damage (e arch rib has suffi-cient robustness subjected to a single cable breakagethough it has the largest DAF

(e results obtained by this analysis are conducive to thedesign and robustness evaluation of cable-supported archbridges Although only the breakage of a single cable wasdemonstrated the proposed methodology can be applied tomultiple cable breakage scenario of cable-supported bridgesFuture research can be carried out in different types of cable-supported arch bridges subjected to multiple cablebreakages

Data Availability

(e data that support the findings of this study are availablefrom the corresponding author upon reasonable request

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(e authors would like to acknowledge Professor RuinianJiang of NMSU for providing help during the preparation ofthe manuscript and the authors also gratefully acknowledgethe support of Taizhou Highway Administration (is re-search was funded by the Zhejiang Provincial Transportationand Transportation Office Technology Funding Project(2012H47)

References

[1] PTI Recommendations for Stay Cable Design Testing andInstallation Post-Tensioning Institute Farmington Hills MIUSA 2012

[2] T Zoli and R Woodward ldquoDesign of long span bridges forcable lossrdquo IABSE Symposium Report vol 90 no 9 pp 17ndash252005

[3] A M Ruiz-Teran and A C Aparicio ldquoResponse of under-deck cable-stayed bridges to the accidental breakage of stay

MLTMLBMT

MBMRTMRB

0 100 200 300 400 50013000

14000

15000

16000

17000

18000

Axi

al fo

rce (

kN)

Moments (kNmiddotm)

ndash50000

0

50000

100000

150000A

xial

forc

e (kN

)

0 10000 20000 30000ndash10000Moments (kNmiddotm)

Figure 23 Internal force change curves of the arch rib due to cableS30 break within its M-N curves

Shock and Vibration 13

cablesrdquo Engineering Structures vol 31 no 7 pp 1425ndash14342009

[4] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part I bending moment acting on the deckrdquo En-gineering Structures vol 32 no 10 pp 3288ndash3300 2010

[5] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part II bending moment acting on the pylons andstress on the staysrdquo Engineering Structures vol 32 no 10pp 3301ndash3312 2010

[6] Y Zhou and S Chen ldquoTime-progressive dynamic assessmentof abrupt cable-breakage events on cable-stayed bridgesrdquoJournal of Bridge Engineering vol 19 no 2 pp 159ndash171 2014

[7] Y Park U Starossek H-M Koh J F Choo H Kim andS W Lee ldquoEffect of cable loss in cable stayed bridgesmdashfocuson dynamic amplificationrdquo IABSE Symposium Report vol 93no 5 pp 9ndash16 2007

[8] J-g Cai Y-x Xu L-p Zhuang J Feng and J ZhangldquoComparison of various procedures for progressive collapseanalysis of cable-stayed bridgesrdquo Journal of Zhejiang Uni-versity Science A vol 13 no 5 pp 323ndash334 2012

[9] Y Xiang Z Chen Y Yang H Lin and S Zhu ldquoDynamicresponse analysis for submerged floating tunnel with anchor-cables subjected to sudden cable breakagerdquoMarine Structuresvol 59 pp 179ndash191 2018

[10] Y F Zhou and S R Chen ldquoFramework of nonlinear dynamicsimulation of long-span cable-stayed bridge and traffic systemsubjected to cable-loss incidentsrdquo Journal of Structural En-gineering vol 142 no 3 2016

[11] W L Qiu M Jiang and C L Huang ldquoParametric study onresponses of a self-anchored suspension bridge to suddenbreakage of a hangerrdquo Scientific World Journal vol 2014Article ID 512120 10 pages 2014

[12] W Qiu M Jiang and Z Zhang ldquoResponses of self-anchoredsuspension bridge to sudden breakage of hangersrdquo StructuralEngineering and Mechanics vol 50 no 2 pp 241ndash255 2014

[13] C M Mozos and A C Aparicio ldquoNumerical and experi-mental study on the interaction cable structure during thefailure of a stay in a cable stayed bridgerdquo EngineeringStructures vol 33 no 8 pp 2330ndash2341 2011

[14] V Hoang O Kiyomiya and T An ldquoExperimental and nu-merical study of lateral cable rupture in cable-stayed bridgescase studyrdquo Journal of Bridge Engineering vol 23 no 6 2018

[15] M Domaneschi G P Cimellaro and G Scutiero ldquoDispro-portionate collapse of a cable-stayed bridgerdquo Proceedings ofthe Institution of Civil EngineersmdashBridge Engineering vol 172no 1 pp 13ndash26 2019

[16] M Wolff and U Starossek ldquoCable loss and progressivecollapse in cable-stayed bridgesrdquo Bridge Structures vol 5no 1 pp 17ndash28 2009

[17] G Wu W Qiu and T Wu ldquoNonlinear dynamic analysis ofthe self-anchored suspension bridge subjected to suddenbreakage of a hangerrdquo Engineering Failure Analysis vol 97pp 701ndash717 2019

[18] M Morgese F Ansari M Domaneschi and G P CimellaroldquoPost-collapse analysis of morandirsquos polcevera viaduct ingenoa Italyrdquo Journal of Civil Structural Health Monitoringvol 10 no 1 pp 69ndash85 2020

[19] M Shoghijavan and U Starossek ldquoStructural robustness oflong-span cable-supported bridges in a cable-loss scenariordquoJournal of Bridge Engineering vol 23 no 2 2018

[20] M Shoghijavan and U Starossek ldquoAn analytical study on thebending moment acting on the girder of a long-span cable-

supported bridge suffering from cable failurerdquo EngineeringStructures vol 167 pp 166ndash174 2018

[21] M Domaneschi C Pellecchia E De Iuliis et al ldquoCollapseanalysis of the Polcevera viaduct by the applied elementmethodrdquo Engineering Structures vol 214 p 110659 2020

[22] O Olamigoke Structural Response of Cable-Stayed Bridges toCable Loss University of Surrey Guildford UK 2018

[23] J Xu H H Sun and S Y Cai ldquoEffect of symmetrical brokenwires damage on mechanical characteristics of stay cablerdquoJournal of Sound and Vibration vol 461 2019

14 Shock and Vibration

that of the lower chord is between minus100 kNmiddotm and 450 kNmiddotm(is indicates that the bending moment of the lower chordof the arch rib is greater than that of the upper chord

In order to reflect the variation amplitude of the arch ribbending moment during the breakage of each cable Fig-ure 14 shows the bending moment variation of the upperand lower chords of the arch rib caused by a single cablebreak (e variation of the bending moment of the upperchord is between 113 kNmiddotm and 188 kNmiddotm with an averagevalue of 137 kNmiddotm and the bending moment variation of thelower arch rib is between 100 kN and 280 kNmiddotm with anaverage value of 229 kNmiddotm Because most cables are an-chored to the lower chord except S1 and S2 the bendingmoments of the lower chord vary more than that of theupper chord Moreover as the length of the cable increasesthe fracture impact effect increases

44 DAF of the Cable Deck and Rib No specifications haveprovided details for calculating the DAF of cable-supportedarch bridges For the design of cable-stayed bridges the DAFis set between 15 and 20 in some codes and recommen-dations In this section the cable failure of the JiantiaoBridge is investigated to discuss the DAF for the cabletension deck displacement and arch rib bendingmoment ofcable-supported arch bridges Equation (1) is used to cal-culate DAF

DAF RD minus R0

RP D minus R0 (5)

where RD represents themaximum dynamic response R0 theinitial state under the dead load before cable break and RPDthe stable value after cable break

S28S30

X29X30

300

350

400

450

500

550

600

650Te

nsio

n of

cabl

es (k

N)

10 20 30 40 500Time (s)

(a)

S29S31

X29X30

10 20 30 40 500Time (s)

300

350

400

450

500

550

600

650

Tens

ion

of ca

bles

(kN

)(b)

Figure 6 Time-history curves due to long-cable fracture (a) break of cable S29 (b) break of cable S30

0

200

400

600

800

Tens

ion

of ca

bles

(kN

)

5 10 15 20 25 300Cable number

Figure 7 Maximum tension of the adjacent cables caused by the break of a single cable

Shock and Vibration 7

Figure 15 shows the calculated DAF for the cable tensiondue to the break of the adjacent cable It can be observed thatthe DAF of the adjacent cable tension is distributed between14 and 16 with an average value of 154

Figure 16 presents the DAF for the deck displacementwhich is between 130 and 165 with an average value of 140(ere is a tendency for the deck displacement DAF to de-crease as the cable length increases

Figure 17 displays the DAF for the arch rib bendingmoment which is between 15 and 24 with an average value

of 193 (e DAF exceeds 20 in some cases which is therecommended value of PTI [1]

Compared with the recommended common DAF valueof 20 for cable-stayed bridges by PTI [1] the DAF of thecable-supported arch bridge in this study varies for eachcomponent (e maximum DAFs of the cable tension anddeck displacement are less than 20 while the maximumDAF of the bending moment of the arch rib exceeds 20(isshows that adopting a common DAF for all the componentsof cable-supported arch bridge structures is not advisable

S1S2S3

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4D

eck

disp

lace

men

t (m

m)

10 20 30 40 500Time (s)

(a)

S1S2S3

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

Dec

k di

spla

cem

ent (

mm

)

10 20 30 40 500Time (s)

(b)

Figure 8 Displacement time-history curves of the deck (a) cable S1 break (b) cable S2 break

S28S29S30

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

Dec

k di

spla

cem

ent (

mm

)

10 20 30 40 500Time (s)

(a)

S29S30S31

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

Dec

k di

spla

cem

ent (

mm

)

10 20 30 40 500Time (s)

(b)

Figure 9 Displacement time-history curves of the deck (a) cable S29 break (b) cable S30 break

8 Shock and Vibration

5 Robustness Evaluation

For long-span cable-supported arch bridges the failure of asingle cable may trigger progressive fracture of adjacentcables and damage to the deck and arch ribs (erefore it isnecessary to evaluate the robustness of cable-supported archbridges subjected to cable breaking (is section discussesthe robustness evaluation of the cable system deck and archrib subjected to a single cable break

51 Robustness of the Cable System To study the possibilityof a progressive fracture of adjacent cables subjected to asingle cable break the impact effect of the cable breakage is

investigated and compared with the carrying capacity of thecable system (e cable system of the Jiantiao Bridge iscomposed of 55 parallel steel wires that have a diameter of7mm and a tensile strength of 1670MPa (e cablersquos ulti-mate bearing capability fu is 3530 kN which is much biggerthan the maximum dynamic load of 821 kN caused by asingle cable break (is indicates that a single cable breakageunder the dead load will not trigger a progressive fracture ofthe adjacent cables

Furthermore to consider the load combination with ref-erence to the PTI [1] recommendation and the study by Zoliand Woodward [2] the limit state design combination Td andthe accidental combination Tf of the cable failure are defined as

0

5

10

15

20

25

30

35

Max

imum

disp

lace

men

t (m

m)

5 10 15 20 25 300Cable number

Figure 10 Maximum displacement of the deck caused by a single cable break

0

MLTMTMRT

MLBMBMRB

ndash200

ndash100

100

200

300

Mom

ent o

f arc

h rib

(kN

middotm)

10 20 30 40 500Time (s)

(a)

MLTMTMRT

MLBMBMRB

ndash200

ndash100

0

100

200

300M

omen

t of a

rch

rib (k

Nmiddotm

)

10 20 30 40 500Time (s)

(b)

Figure 11 Bending moment time-history curves of the arch rib (a) cable S1 break (b) cable S2 break

Shock and Vibration 9

Td 12Ts + 14Tv + 075 Tp + Tt1113872 1113873

Tf 12Ts + 075 Tv + Tp + Tt1113872 1113873 + Tc(6)

where Ts Tv Tp and Tt are the tension of the dead loadvehicle load pedestrian load and temperature load re-spectively Tc is the net dynamic effect of the cable breakwhich is equal to the maximum tension of the dynamicresponse minus the cable tension under the dead load

Figure 18 shows the comparison of the design limitcombination Td and the accidental combination Tf of each

cable It is observed that the values of cable fracture ac-cidental combination are all higher than the carryingcapability limit state combination (is means that thecable breakage accidental combination is the most un-favorable situation and should not be ignored in thedesign stage

(e maximum value of the accidental combination Tf is1130 kN and the minimum safety factor of the cablecompared with the ultimate bearing capability fu is 31 Itshows that the cable system has good robustness under acable breakage accidental combination and a single cable

MLTMTMRT

MLBMBMRB

0

100

200

300

400

500M

omen

t of a

rch

rib (k

Nmiddotm

)

10 20 30 40 500Time (s)

(a)

MLTMTMRT

MLBMBMRB

10 20 30 40 500Time(s)

0

100

200

300

400

500

Mom

ent o

f arc

h rib

(kN

middotm)

(b)

Figure 12 Bending moment time-history curves of the arch rib (a) cable S29 break (b) cable S30 break

Midspan

MaxMin

ndash100

0

100

200

300

Mom

ent o

f upp

er ch

ords

(kN

middotm)

5 10 15 20 25 300Cable number

(a)

MaxMin

Midspan

ndash100

0

100

200

300

400

500

Mom

ent o

f low

er ch

ords

(kN

middotm)

5 10 15 20 25 300Cable number

(b)

Figure 13 Envelope of the arch rib bending moment subjected to a single cable break (a) upper chords (b) lower chords

10 Shock and Vibration

break will not trigger a progressive breakage of adjacentcables

52 Robustness of the Deck Different from cable-stayedbridges that have a strong longitudinal stiffening girdercable-supported arch bridge deck systems always usetransverse girders as the main load-bearing members Toevaluate the damage to the deck system the capability of thedeck was obtained by a pushover analysis

(e pushover model was composed of three transversegirders with its flange plate An elastic support was used tosimulate the rigidity of the cables at the anchoring positions(e horizontal side displacement of the deck was notallowed by fixing the left and right ends of the deck system toavoid any rigid movement (e vertical pushover load wasapplied to the anchored position of the broken cable on the

right side of the middle girder as indicated by the arrow inFigure 19 (e relative displacements between the transversegirders were chosen as the capability index to represent thebearing capacity of the deck system

Figure 20 depicts the load-displacement curve of thedeck It can be noticed that the failure process of the deck canbe divided into three stages elastic stage cracked stage andfailure stage When the relative displacement is less than235mm the deck is in the elastic stage When the relativedisplacement increases from 235mm to 315mm cracksgradually increase and the deck works with the cracks Whenthe relative displacement exceeds 315mm the longitudinalreinforcement yields and the displacement increases rapidly

(e robustness of the deck was evaluated by comparingthe relative displacement of the cable breakage with that ofthe load-displacement curve As displayed in Figure 21 themaximum relative displacement of the deck is between 8mmand 15mm subjected to a single cable break which increases

Midspan

Upper chordsLower chords

0

100

200

300

Mom

ent v

aria

tion

(kN

middotm)

5 10 15 20 25 300Cable number

Figure 14 Moment variation of arch rib chords caused by a singlecable break

0 5 10 15 20 25 3010

12

14

16

18

DA

F of

adja

cent

cabl

e ten

sion

Cable number

Figure 15 DAF of the cable tension caused by a single cable break

10

12

14

16

18

DA

F of

dec

k di

spla

cem

ent

5 10 15 20 25 300Cable number

Figure 16 DAF of the deck displacement caused by a single cablebreak

10

12

14

16

18

20

22

24D

AF

of ar

ch ri

b be

ndin

g m

omen

t

5 10 15 20 25 300Cable number

Figure 17 DAF of the arch rib moment caused by a single cablebreak

Shock and Vibration 11

with the length of the broken cable Compared with that ofthe load-displacement curve in Figure 20 the relative dis-placement of the deck subjected to a single cable breakage iswithin the cracked stage and will not lead to a deck collapseUnder the worst conditions the deck system has a safety

factor of approximately 20 which indicates that the decksystem has relatively lower robustness than the cable system

53 Robustness of theArch Rib To evaluate the robustness ofthe arch rib which is an eccentric compression member anM-N curve was drawn to evaluate its bearing capacityFigures 22 and 23 show the axial force and bending momentcurves of the arch rib subjected to S1 and S30 cable breakrespectively (e curves in the circle are the internal forcecurves of the arch rib which are enlarged at the upper rightof the figure It is observed that the arch ribs are mainlyunder compression (e axial force is between 14000 kNand 17500 kN and the bending moment is between minus150and 450 kNmiddotm (e impact effect caused by the cable break

TdTf

400

600

800

1000

1200

1400Te

nsio

n of

cabl

es (k

N)

5 10 15 20 25 300Cable number

Figure 18 Comparison of the limit state combination Td andaccidental combination Tf of each cable

Figure 19 Pushover model of the deck system

0

500

1000

1500

2000

Push

forc

e (kN

)

10 20 30 400Relative displacement of anchorage end (mm)

Figure 20 Load-displacement curve of the deck

5 10 15 20 25 300Cable number

8

10

12

14

16

Max

imum

relat

ive d

ispla

cem

ent (

mm

)

Figure 21 Maximum relative deck displacements caused by asingle cable break

ndash200 ndash100 0 100 200 30014000

15000

16000

17000

MLTMLBMT

MBMRTMRB

Axi

al fo

rce (

kN)

Moments (kNmiddotm)

ndash50000

0

50000

100000

150000A

xial

forc

e (kN

)

0 10000 20000 30000ndash10000Moments (kNmiddotm)

Figure 22 Internal force change curves of the arch rib due to cableS1 break within its M-N curves

12 Shock and Vibration

was within 20 of the arch rib bearing capability indicatingthat the failure of a single cable does not cause failure of thearch rib (e safety factor of the arch rib is greater than 50which reveals that the arch rib has sufficient robustnesssubjected to single cable breakage

6 Conclusions

Cable breaking is common damage in cable-supported archbridges as a result of aging and erosion caused by the harshenvironment To evaluate the dynamic response and robust-ness of cable-supported arch bridges subjected to cablebreaking a nonlinear dynamic simulation procedure and ro-bustness evaluation method are developed in this study Aprototype arch bridge with a single cable break was analyzed asan example Several critical factors for cable breakage simu-lation such as the cable break mode cable break duration anddamping are studied (e dynamic responses of the cablesystem deck and rib are evaluated and the DAF of differentcomponents of the cable-supported arch bridge is also dis-cussed Robustness evaluation indexes of the cable systemdeck and arch rib are calculated and their robustness levels areevaluated (e main conclusions are as follows

(1) Using an appropriate simulation method is essentialto the dynamic analysis of a cable breaking processDirectly removing the failed cable in an appropriatetime step of a transient analysis can effectivelysimulate the cable breaking effect When the cablebreak duration is sufficiently short the maximumimpact effect can be fully attained regardless of thetype of the break simulation mode used (e Ray-leigh damping coefficient calculated by a higher-order frequency can avoid the suppression effect of alocal vibration and better reflect the local vibrationcaused by a cable break

(2) (e closest cable to the broken cable receives thelargest impact effect (e impact effect of the cable

tension caused by the break of a short cable is greaterthan that of a long cable (e deck displacement andbending moment increase with the length of thefailure cable(e breaking of a downstream cable haslittle effect on the upstream cables

(3) (e DAF of each component is different(e DAF ofcable tension is between 14 and 16 while that of thedeck displacement is between 13 and 165 (e archrib bending moment has the largest DAF that isbetween 15 and 24 which is greater than the PTIrecommended value of 20 in some cases Using aunified DAF of 20 cannot satisfy all the componentsof cable-supported arch bridges

(4) (e deck system is the least robust component ofcable-supported arch bridges (e cable system has asafety factor of 31 under the cable breakage acci-dental combination a single cable break will nottrigger progressive damage (e arch rib has suffi-cient robustness subjected to a single cable breakagethough it has the largest DAF

(e results obtained by this analysis are conducive to thedesign and robustness evaluation of cable-supported archbridges Although only the breakage of a single cable wasdemonstrated the proposed methodology can be applied tomultiple cable breakage scenario of cable-supported bridgesFuture research can be carried out in different types of cable-supported arch bridges subjected to multiple cablebreakages

Data Availability

(e data that support the findings of this study are availablefrom the corresponding author upon reasonable request

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(e authors would like to acknowledge Professor RuinianJiang of NMSU for providing help during the preparation ofthe manuscript and the authors also gratefully acknowledgethe support of Taizhou Highway Administration (is re-search was funded by the Zhejiang Provincial Transportationand Transportation Office Technology Funding Project(2012H47)

References

[1] PTI Recommendations for Stay Cable Design Testing andInstallation Post-Tensioning Institute Farmington Hills MIUSA 2012

[2] T Zoli and R Woodward ldquoDesign of long span bridges forcable lossrdquo IABSE Symposium Report vol 90 no 9 pp 17ndash252005

[3] A M Ruiz-Teran and A C Aparicio ldquoResponse of under-deck cable-stayed bridges to the accidental breakage of stay

MLTMLBMT

MBMRTMRB

0 100 200 300 400 50013000

14000

15000

16000

17000

18000

Axi

al fo

rce (

kN)

Moments (kNmiddotm)

ndash50000

0

50000

100000

150000A

xial

forc

e (kN

)

0 10000 20000 30000ndash10000Moments (kNmiddotm)

Figure 23 Internal force change curves of the arch rib due to cableS30 break within its M-N curves

Shock and Vibration 13

cablesrdquo Engineering Structures vol 31 no 7 pp 1425ndash14342009

[4] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part I bending moment acting on the deckrdquo En-gineering Structures vol 32 no 10 pp 3288ndash3300 2010

[5] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part II bending moment acting on the pylons andstress on the staysrdquo Engineering Structures vol 32 no 10pp 3301ndash3312 2010

[6] Y Zhou and S Chen ldquoTime-progressive dynamic assessmentof abrupt cable-breakage events on cable-stayed bridgesrdquoJournal of Bridge Engineering vol 19 no 2 pp 159ndash171 2014

[7] Y Park U Starossek H-M Koh J F Choo H Kim andS W Lee ldquoEffect of cable loss in cable stayed bridgesmdashfocuson dynamic amplificationrdquo IABSE Symposium Report vol 93no 5 pp 9ndash16 2007

[8] J-g Cai Y-x Xu L-p Zhuang J Feng and J ZhangldquoComparison of various procedures for progressive collapseanalysis of cable-stayed bridgesrdquo Journal of Zhejiang Uni-versity Science A vol 13 no 5 pp 323ndash334 2012

[9] Y Xiang Z Chen Y Yang H Lin and S Zhu ldquoDynamicresponse analysis for submerged floating tunnel with anchor-cables subjected to sudden cable breakagerdquoMarine Structuresvol 59 pp 179ndash191 2018

[10] Y F Zhou and S R Chen ldquoFramework of nonlinear dynamicsimulation of long-span cable-stayed bridge and traffic systemsubjected to cable-loss incidentsrdquo Journal of Structural En-gineering vol 142 no 3 2016

[11] W L Qiu M Jiang and C L Huang ldquoParametric study onresponses of a self-anchored suspension bridge to suddenbreakage of a hangerrdquo Scientific World Journal vol 2014Article ID 512120 10 pages 2014

[12] W Qiu M Jiang and Z Zhang ldquoResponses of self-anchoredsuspension bridge to sudden breakage of hangersrdquo StructuralEngineering and Mechanics vol 50 no 2 pp 241ndash255 2014

[13] C M Mozos and A C Aparicio ldquoNumerical and experi-mental study on the interaction cable structure during thefailure of a stay in a cable stayed bridgerdquo EngineeringStructures vol 33 no 8 pp 2330ndash2341 2011

[14] V Hoang O Kiyomiya and T An ldquoExperimental and nu-merical study of lateral cable rupture in cable-stayed bridgescase studyrdquo Journal of Bridge Engineering vol 23 no 6 2018

[15] M Domaneschi G P Cimellaro and G Scutiero ldquoDispro-portionate collapse of a cable-stayed bridgerdquo Proceedings ofthe Institution of Civil EngineersmdashBridge Engineering vol 172no 1 pp 13ndash26 2019

[16] M Wolff and U Starossek ldquoCable loss and progressivecollapse in cable-stayed bridgesrdquo Bridge Structures vol 5no 1 pp 17ndash28 2009

[17] G Wu W Qiu and T Wu ldquoNonlinear dynamic analysis ofthe self-anchored suspension bridge subjected to suddenbreakage of a hangerrdquo Engineering Failure Analysis vol 97pp 701ndash717 2019

[18] M Morgese F Ansari M Domaneschi and G P CimellaroldquoPost-collapse analysis of morandirsquos polcevera viaduct ingenoa Italyrdquo Journal of Civil Structural Health Monitoringvol 10 no 1 pp 69ndash85 2020

[19] M Shoghijavan and U Starossek ldquoStructural robustness oflong-span cable-supported bridges in a cable-loss scenariordquoJournal of Bridge Engineering vol 23 no 2 2018

[20] M Shoghijavan and U Starossek ldquoAn analytical study on thebending moment acting on the girder of a long-span cable-

supported bridge suffering from cable failurerdquo EngineeringStructures vol 167 pp 166ndash174 2018

[21] M Domaneschi C Pellecchia E De Iuliis et al ldquoCollapseanalysis of the Polcevera viaduct by the applied elementmethodrdquo Engineering Structures vol 214 p 110659 2020

[22] O Olamigoke Structural Response of Cable-Stayed Bridges toCable Loss University of Surrey Guildford UK 2018

[23] J Xu H H Sun and S Y Cai ldquoEffect of symmetrical brokenwires damage on mechanical characteristics of stay cablerdquoJournal of Sound and Vibration vol 461 2019

14 Shock and Vibration

Figure 15 shows the calculated DAF for the cable tensiondue to the break of the adjacent cable It can be observed thatthe DAF of the adjacent cable tension is distributed between14 and 16 with an average value of 154

Figure 16 presents the DAF for the deck displacementwhich is between 130 and 165 with an average value of 140(ere is a tendency for the deck displacement DAF to de-crease as the cable length increases

Figure 17 displays the DAF for the arch rib bendingmoment which is between 15 and 24 with an average value

of 193 (e DAF exceeds 20 in some cases which is therecommended value of PTI [1]

Compared with the recommended common DAF valueof 20 for cable-stayed bridges by PTI [1] the DAF of thecable-supported arch bridge in this study varies for eachcomponent (e maximum DAFs of the cable tension anddeck displacement are less than 20 while the maximumDAF of the bending moment of the arch rib exceeds 20(isshows that adopting a common DAF for all the componentsof cable-supported arch bridge structures is not advisable

S1S2S3

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4D

eck

disp

lace

men

t (m

m)

10 20 30 40 500Time (s)

(a)

S1S2S3

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

Dec

k di

spla

cem

ent (

mm

)

10 20 30 40 500Time (s)

(b)

Figure 8 Displacement time-history curves of the deck (a) cable S1 break (b) cable S2 break

S28S29S30

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

Dec

k di

spla

cem

ent (

mm

)

10 20 30 40 500Time (s)

(a)

S29S30S31

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

Dec

k di

spla

cem

ent (

mm

)

10 20 30 40 500Time (s)

(b)

Figure 9 Displacement time-history curves of the deck (a) cable S29 break (b) cable S30 break

8 Shock and Vibration

5 Robustness Evaluation

For long-span cable-supported arch bridges the failure of asingle cable may trigger progressive fracture of adjacentcables and damage to the deck and arch ribs (erefore it isnecessary to evaluate the robustness of cable-supported archbridges subjected to cable breaking (is section discussesthe robustness evaluation of the cable system deck and archrib subjected to a single cable break

51 Robustness of the Cable System To study the possibilityof a progressive fracture of adjacent cables subjected to asingle cable break the impact effect of the cable breakage is

investigated and compared with the carrying capacity of thecable system (e cable system of the Jiantiao Bridge iscomposed of 55 parallel steel wires that have a diameter of7mm and a tensile strength of 1670MPa (e cablersquos ulti-mate bearing capability fu is 3530 kN which is much biggerthan the maximum dynamic load of 821 kN caused by asingle cable break (is indicates that a single cable breakageunder the dead load will not trigger a progressive fracture ofthe adjacent cables

Furthermore to consider the load combination with ref-erence to the PTI [1] recommendation and the study by Zoliand Woodward [2] the limit state design combination Td andthe accidental combination Tf of the cable failure are defined as

0

5

10

15

20

25

30

35

Max

imum

disp

lace

men

t (m

m)

5 10 15 20 25 300Cable number

Figure 10 Maximum displacement of the deck caused by a single cable break

0

MLTMTMRT

MLBMBMRB

ndash200

ndash100

100

200

300

Mom

ent o

f arc

h rib

(kN

middotm)

10 20 30 40 500Time (s)

(a)

MLTMTMRT

MLBMBMRB

ndash200

ndash100

0

100

200

300M

omen

t of a

rch

rib (k

Nmiddotm

)

10 20 30 40 500Time (s)

(b)

Figure 11 Bending moment time-history curves of the arch rib (a) cable S1 break (b) cable S2 break

Shock and Vibration 9

Td 12Ts + 14Tv + 075 Tp + Tt1113872 1113873

Tf 12Ts + 075 Tv + Tp + Tt1113872 1113873 + Tc(6)

where Ts Tv Tp and Tt are the tension of the dead loadvehicle load pedestrian load and temperature load re-spectively Tc is the net dynamic effect of the cable breakwhich is equal to the maximum tension of the dynamicresponse minus the cable tension under the dead load

Figure 18 shows the comparison of the design limitcombination Td and the accidental combination Tf of each

cable It is observed that the values of cable fracture ac-cidental combination are all higher than the carryingcapability limit state combination (is means that thecable breakage accidental combination is the most un-favorable situation and should not be ignored in thedesign stage

(e maximum value of the accidental combination Tf is1130 kN and the minimum safety factor of the cablecompared with the ultimate bearing capability fu is 31 Itshows that the cable system has good robustness under acable breakage accidental combination and a single cable

MLTMTMRT

MLBMBMRB

0

100

200

300

400

500M

omen

t of a

rch

rib (k

Nmiddotm

)

10 20 30 40 500Time (s)

(a)

MLTMTMRT

MLBMBMRB

10 20 30 40 500Time(s)

0

100

200

300

400

500

Mom

ent o

f arc

h rib

(kN

middotm)

(b)

Figure 12 Bending moment time-history curves of the arch rib (a) cable S29 break (b) cable S30 break

Midspan

MaxMin

ndash100

0

100

200

300

Mom

ent o

f upp

er ch

ords

(kN

middotm)

5 10 15 20 25 300Cable number

(a)

MaxMin

Midspan

ndash100

0

100

200

300

400

500

Mom

ent o

f low

er ch

ords

(kN

middotm)

5 10 15 20 25 300Cable number

(b)

Figure 13 Envelope of the arch rib bending moment subjected to a single cable break (a) upper chords (b) lower chords

10 Shock and Vibration

break will not trigger a progressive breakage of adjacentcables

52 Robustness of the Deck Different from cable-stayedbridges that have a strong longitudinal stiffening girdercable-supported arch bridge deck systems always usetransverse girders as the main load-bearing members Toevaluate the damage to the deck system the capability of thedeck was obtained by a pushover analysis

(e pushover model was composed of three transversegirders with its flange plate An elastic support was used tosimulate the rigidity of the cables at the anchoring positions(e horizontal side displacement of the deck was notallowed by fixing the left and right ends of the deck system toavoid any rigid movement (e vertical pushover load wasapplied to the anchored position of the broken cable on the

right side of the middle girder as indicated by the arrow inFigure 19 (e relative displacements between the transversegirders were chosen as the capability index to represent thebearing capacity of the deck system

Figure 20 depicts the load-displacement curve of thedeck It can be noticed that the failure process of the deck canbe divided into three stages elastic stage cracked stage andfailure stage When the relative displacement is less than235mm the deck is in the elastic stage When the relativedisplacement increases from 235mm to 315mm cracksgradually increase and the deck works with the cracks Whenthe relative displacement exceeds 315mm the longitudinalreinforcement yields and the displacement increases rapidly

(e robustness of the deck was evaluated by comparingthe relative displacement of the cable breakage with that ofthe load-displacement curve As displayed in Figure 21 themaximum relative displacement of the deck is between 8mmand 15mm subjected to a single cable break which increases

Midspan

Upper chordsLower chords

0

100

200

300

Mom

ent v

aria

tion

(kN

middotm)

5 10 15 20 25 300Cable number

Figure 14 Moment variation of arch rib chords caused by a singlecable break

0 5 10 15 20 25 3010

12

14

16

18

DA

F of

adja

cent

cabl

e ten

sion

Cable number

Figure 15 DAF of the cable tension caused by a single cable break

10

12

14

16

18

DA

F of

dec

k di

spla

cem

ent

5 10 15 20 25 300Cable number

Figure 16 DAF of the deck displacement caused by a single cablebreak

10

12

14

16

18

20

22

24D

AF

of ar

ch ri

b be

ndin

g m

omen

t

5 10 15 20 25 300Cable number

Figure 17 DAF of the arch rib moment caused by a single cablebreak

Shock and Vibration 11

with the length of the broken cable Compared with that ofthe load-displacement curve in Figure 20 the relative dis-placement of the deck subjected to a single cable breakage iswithin the cracked stage and will not lead to a deck collapseUnder the worst conditions the deck system has a safety

factor of approximately 20 which indicates that the decksystem has relatively lower robustness than the cable system

53 Robustness of theArch Rib To evaluate the robustness ofthe arch rib which is an eccentric compression member anM-N curve was drawn to evaluate its bearing capacityFigures 22 and 23 show the axial force and bending momentcurves of the arch rib subjected to S1 and S30 cable breakrespectively (e curves in the circle are the internal forcecurves of the arch rib which are enlarged at the upper rightof the figure It is observed that the arch ribs are mainlyunder compression (e axial force is between 14000 kNand 17500 kN and the bending moment is between minus150and 450 kNmiddotm (e impact effect caused by the cable break

TdTf

400

600

800

1000

1200

1400Te

nsio

n of

cabl

es (k

N)

5 10 15 20 25 300Cable number

Figure 18 Comparison of the limit state combination Td andaccidental combination Tf of each cable

Figure 19 Pushover model of the deck system

0

500

1000

1500

2000

Push

forc

e (kN

)

10 20 30 400Relative displacement of anchorage end (mm)

Figure 20 Load-displacement curve of the deck

5 10 15 20 25 300Cable number

8

10

12

14

16

Max

imum

relat

ive d

ispla

cem

ent (

mm

)

Figure 21 Maximum relative deck displacements caused by asingle cable break

ndash200 ndash100 0 100 200 30014000

15000

16000

17000

MLTMLBMT

MBMRTMRB

Axi

al fo

rce (

kN)

Moments (kNmiddotm)

ndash50000

0

50000

100000

150000A

xial

forc

e (kN

)

0 10000 20000 30000ndash10000Moments (kNmiddotm)

Figure 22 Internal force change curves of the arch rib due to cableS1 break within its M-N curves

12 Shock and Vibration

was within 20 of the arch rib bearing capability indicatingthat the failure of a single cable does not cause failure of thearch rib (e safety factor of the arch rib is greater than 50which reveals that the arch rib has sufficient robustnesssubjected to single cable breakage

6 Conclusions

Cable breaking is common damage in cable-supported archbridges as a result of aging and erosion caused by the harshenvironment To evaluate the dynamic response and robust-ness of cable-supported arch bridges subjected to cablebreaking a nonlinear dynamic simulation procedure and ro-bustness evaluation method are developed in this study Aprototype arch bridge with a single cable break was analyzed asan example Several critical factors for cable breakage simu-lation such as the cable break mode cable break duration anddamping are studied (e dynamic responses of the cablesystem deck and rib are evaluated and the DAF of differentcomponents of the cable-supported arch bridge is also dis-cussed Robustness evaluation indexes of the cable systemdeck and arch rib are calculated and their robustness levels areevaluated (e main conclusions are as follows

(1) Using an appropriate simulation method is essentialto the dynamic analysis of a cable breaking processDirectly removing the failed cable in an appropriatetime step of a transient analysis can effectivelysimulate the cable breaking effect When the cablebreak duration is sufficiently short the maximumimpact effect can be fully attained regardless of thetype of the break simulation mode used (e Ray-leigh damping coefficient calculated by a higher-order frequency can avoid the suppression effect of alocal vibration and better reflect the local vibrationcaused by a cable break

(2) (e closest cable to the broken cable receives thelargest impact effect (e impact effect of the cable

tension caused by the break of a short cable is greaterthan that of a long cable (e deck displacement andbending moment increase with the length of thefailure cable(e breaking of a downstream cable haslittle effect on the upstream cables

(3) (e DAF of each component is different(e DAF ofcable tension is between 14 and 16 while that of thedeck displacement is between 13 and 165 (e archrib bending moment has the largest DAF that isbetween 15 and 24 which is greater than the PTIrecommended value of 20 in some cases Using aunified DAF of 20 cannot satisfy all the componentsof cable-supported arch bridges

(4) (e deck system is the least robust component ofcable-supported arch bridges (e cable system has asafety factor of 31 under the cable breakage acci-dental combination a single cable break will nottrigger progressive damage (e arch rib has suffi-cient robustness subjected to a single cable breakagethough it has the largest DAF

(e results obtained by this analysis are conducive to thedesign and robustness evaluation of cable-supported archbridges Although only the breakage of a single cable wasdemonstrated the proposed methodology can be applied tomultiple cable breakage scenario of cable-supported bridgesFuture research can be carried out in different types of cable-supported arch bridges subjected to multiple cablebreakages

Data Availability

(e data that support the findings of this study are availablefrom the corresponding author upon reasonable request

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(e authors would like to acknowledge Professor RuinianJiang of NMSU for providing help during the preparation ofthe manuscript and the authors also gratefully acknowledgethe support of Taizhou Highway Administration (is re-search was funded by the Zhejiang Provincial Transportationand Transportation Office Technology Funding Project(2012H47)

References

[1] PTI Recommendations for Stay Cable Design Testing andInstallation Post-Tensioning Institute Farmington Hills MIUSA 2012

[2] T Zoli and R Woodward ldquoDesign of long span bridges forcable lossrdquo IABSE Symposium Report vol 90 no 9 pp 17ndash252005

[3] A M Ruiz-Teran and A C Aparicio ldquoResponse of under-deck cable-stayed bridges to the accidental breakage of stay

MLTMLBMT

MBMRTMRB

0 100 200 300 400 50013000

14000

15000

16000

17000

18000

Axi

al fo

rce (

kN)

Moments (kNmiddotm)

ndash50000

0

50000

100000

150000A

xial

forc

e (kN

)

0 10000 20000 30000ndash10000Moments (kNmiddotm)

Figure 23 Internal force change curves of the arch rib due to cableS30 break within its M-N curves

Shock and Vibration 13

cablesrdquo Engineering Structures vol 31 no 7 pp 1425ndash14342009

[4] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part I bending moment acting on the deckrdquo En-gineering Structures vol 32 no 10 pp 3288ndash3300 2010

[5] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part II bending moment acting on the pylons andstress on the staysrdquo Engineering Structures vol 32 no 10pp 3301ndash3312 2010

[6] Y Zhou and S Chen ldquoTime-progressive dynamic assessmentof abrupt cable-breakage events on cable-stayed bridgesrdquoJournal of Bridge Engineering vol 19 no 2 pp 159ndash171 2014

[7] Y Park U Starossek H-M Koh J F Choo H Kim andS W Lee ldquoEffect of cable loss in cable stayed bridgesmdashfocuson dynamic amplificationrdquo IABSE Symposium Report vol 93no 5 pp 9ndash16 2007

[8] J-g Cai Y-x Xu L-p Zhuang J Feng and J ZhangldquoComparison of various procedures for progressive collapseanalysis of cable-stayed bridgesrdquo Journal of Zhejiang Uni-versity Science A vol 13 no 5 pp 323ndash334 2012

[9] Y Xiang Z Chen Y Yang H Lin and S Zhu ldquoDynamicresponse analysis for submerged floating tunnel with anchor-cables subjected to sudden cable breakagerdquoMarine Structuresvol 59 pp 179ndash191 2018

[10] Y F Zhou and S R Chen ldquoFramework of nonlinear dynamicsimulation of long-span cable-stayed bridge and traffic systemsubjected to cable-loss incidentsrdquo Journal of Structural En-gineering vol 142 no 3 2016

[11] W L Qiu M Jiang and C L Huang ldquoParametric study onresponses of a self-anchored suspension bridge to suddenbreakage of a hangerrdquo Scientific World Journal vol 2014Article ID 512120 10 pages 2014

[12] W Qiu M Jiang and Z Zhang ldquoResponses of self-anchoredsuspension bridge to sudden breakage of hangersrdquo StructuralEngineering and Mechanics vol 50 no 2 pp 241ndash255 2014

[13] C M Mozos and A C Aparicio ldquoNumerical and experi-mental study on the interaction cable structure during thefailure of a stay in a cable stayed bridgerdquo EngineeringStructures vol 33 no 8 pp 2330ndash2341 2011

[14] V Hoang O Kiyomiya and T An ldquoExperimental and nu-merical study of lateral cable rupture in cable-stayed bridgescase studyrdquo Journal of Bridge Engineering vol 23 no 6 2018

[15] M Domaneschi G P Cimellaro and G Scutiero ldquoDispro-portionate collapse of a cable-stayed bridgerdquo Proceedings ofthe Institution of Civil EngineersmdashBridge Engineering vol 172no 1 pp 13ndash26 2019

[16] M Wolff and U Starossek ldquoCable loss and progressivecollapse in cable-stayed bridgesrdquo Bridge Structures vol 5no 1 pp 17ndash28 2009

[17] G Wu W Qiu and T Wu ldquoNonlinear dynamic analysis ofthe self-anchored suspension bridge subjected to suddenbreakage of a hangerrdquo Engineering Failure Analysis vol 97pp 701ndash717 2019

[18] M Morgese F Ansari M Domaneschi and G P CimellaroldquoPost-collapse analysis of morandirsquos polcevera viaduct ingenoa Italyrdquo Journal of Civil Structural Health Monitoringvol 10 no 1 pp 69ndash85 2020

[19] M Shoghijavan and U Starossek ldquoStructural robustness oflong-span cable-supported bridges in a cable-loss scenariordquoJournal of Bridge Engineering vol 23 no 2 2018

[20] M Shoghijavan and U Starossek ldquoAn analytical study on thebending moment acting on the girder of a long-span cable-

supported bridge suffering from cable failurerdquo EngineeringStructures vol 167 pp 166ndash174 2018

[21] M Domaneschi C Pellecchia E De Iuliis et al ldquoCollapseanalysis of the Polcevera viaduct by the applied elementmethodrdquo Engineering Structures vol 214 p 110659 2020

[22] O Olamigoke Structural Response of Cable-Stayed Bridges toCable Loss University of Surrey Guildford UK 2018

[23] J Xu H H Sun and S Y Cai ldquoEffect of symmetrical brokenwires damage on mechanical characteristics of stay cablerdquoJournal of Sound and Vibration vol 461 2019

14 Shock and Vibration

5 Robustness Evaluation

For long-span cable-supported arch bridges the failure of asingle cable may trigger progressive fracture of adjacentcables and damage to the deck and arch ribs (erefore it isnecessary to evaluate the robustness of cable-supported archbridges subjected to cable breaking (is section discussesthe robustness evaluation of the cable system deck and archrib subjected to a single cable break

51 Robustness of the Cable System To study the possibilityof a progressive fracture of adjacent cables subjected to asingle cable break the impact effect of the cable breakage is

investigated and compared with the carrying capacity of thecable system (e cable system of the Jiantiao Bridge iscomposed of 55 parallel steel wires that have a diameter of7mm and a tensile strength of 1670MPa (e cablersquos ulti-mate bearing capability fu is 3530 kN which is much biggerthan the maximum dynamic load of 821 kN caused by asingle cable break (is indicates that a single cable breakageunder the dead load will not trigger a progressive fracture ofthe adjacent cables

Furthermore to consider the load combination with ref-erence to the PTI [1] recommendation and the study by Zoliand Woodward [2] the limit state design combination Td andthe accidental combination Tf of the cable failure are defined as

0

5

10

15

20

25

30

35

Max

imum

disp

lace

men

t (m

m)

5 10 15 20 25 300Cable number

Figure 10 Maximum displacement of the deck caused by a single cable break

0

MLTMTMRT

MLBMBMRB

ndash200

ndash100

100

200

300

Mom

ent o

f arc

h rib

(kN

middotm)

10 20 30 40 500Time (s)

(a)

MLTMTMRT

MLBMBMRB

ndash200

ndash100

0

100

200

300M

omen

t of a

rch

rib (k

Nmiddotm

)

10 20 30 40 500Time (s)

(b)

Figure 11 Bending moment time-history curves of the arch rib (a) cable S1 break (b) cable S2 break

Shock and Vibration 9

Td 12Ts + 14Tv + 075 Tp + Tt1113872 1113873

Tf 12Ts + 075 Tv + Tp + Tt1113872 1113873 + Tc(6)

where Ts Tv Tp and Tt are the tension of the dead loadvehicle load pedestrian load and temperature load re-spectively Tc is the net dynamic effect of the cable breakwhich is equal to the maximum tension of the dynamicresponse minus the cable tension under the dead load

Figure 18 shows the comparison of the design limitcombination Td and the accidental combination Tf of each

cable It is observed that the values of cable fracture ac-cidental combination are all higher than the carryingcapability limit state combination (is means that thecable breakage accidental combination is the most un-favorable situation and should not be ignored in thedesign stage

(e maximum value of the accidental combination Tf is1130 kN and the minimum safety factor of the cablecompared with the ultimate bearing capability fu is 31 Itshows that the cable system has good robustness under acable breakage accidental combination and a single cable

MLTMTMRT

MLBMBMRB

0

100

200

300

400

500M

omen

t of a

rch

rib (k

Nmiddotm

)

10 20 30 40 500Time (s)

(a)

MLTMTMRT

MLBMBMRB

10 20 30 40 500Time(s)

0

100

200

300

400

500

Mom

ent o

f arc

h rib

(kN

middotm)

(b)

Figure 12 Bending moment time-history curves of the arch rib (a) cable S29 break (b) cable S30 break

Midspan

MaxMin

ndash100

0

100

200

300

Mom

ent o

f upp

er ch

ords

(kN

middotm)

5 10 15 20 25 300Cable number

(a)

MaxMin

Midspan

ndash100

0

100

200

300

400

500

Mom

ent o

f low

er ch

ords

(kN

middotm)

5 10 15 20 25 300Cable number

(b)

Figure 13 Envelope of the arch rib bending moment subjected to a single cable break (a) upper chords (b) lower chords

10 Shock and Vibration

break will not trigger a progressive breakage of adjacentcables

52 Robustness of the Deck Different from cable-stayedbridges that have a strong longitudinal stiffening girdercable-supported arch bridge deck systems always usetransverse girders as the main load-bearing members Toevaluate the damage to the deck system the capability of thedeck was obtained by a pushover analysis

(e pushover model was composed of three transversegirders with its flange plate An elastic support was used tosimulate the rigidity of the cables at the anchoring positions(e horizontal side displacement of the deck was notallowed by fixing the left and right ends of the deck system toavoid any rigid movement (e vertical pushover load wasapplied to the anchored position of the broken cable on the

right side of the middle girder as indicated by the arrow inFigure 19 (e relative displacements between the transversegirders were chosen as the capability index to represent thebearing capacity of the deck system

Figure 20 depicts the load-displacement curve of thedeck It can be noticed that the failure process of the deck canbe divided into three stages elastic stage cracked stage andfailure stage When the relative displacement is less than235mm the deck is in the elastic stage When the relativedisplacement increases from 235mm to 315mm cracksgradually increase and the deck works with the cracks Whenthe relative displacement exceeds 315mm the longitudinalreinforcement yields and the displacement increases rapidly

(e robustness of the deck was evaluated by comparingthe relative displacement of the cable breakage with that ofthe load-displacement curve As displayed in Figure 21 themaximum relative displacement of the deck is between 8mmand 15mm subjected to a single cable break which increases

Midspan

Upper chordsLower chords

0

100

200

300

Mom

ent v

aria

tion

(kN

middotm)

5 10 15 20 25 300Cable number

Figure 14 Moment variation of arch rib chords caused by a singlecable break

0 5 10 15 20 25 3010

12

14

16

18

DA

F of

adja

cent

cabl

e ten

sion

Cable number

Figure 15 DAF of the cable tension caused by a single cable break

10

12

14

16

18

DA

F of

dec

k di

spla

cem

ent

5 10 15 20 25 300Cable number

Figure 16 DAF of the deck displacement caused by a single cablebreak

10

12

14

16

18

20

22

24D

AF

of ar

ch ri

b be

ndin

g m

omen

t

5 10 15 20 25 300Cable number

Figure 17 DAF of the arch rib moment caused by a single cablebreak

Shock and Vibration 11

with the length of the broken cable Compared with that ofthe load-displacement curve in Figure 20 the relative dis-placement of the deck subjected to a single cable breakage iswithin the cracked stage and will not lead to a deck collapseUnder the worst conditions the deck system has a safety

factor of approximately 20 which indicates that the decksystem has relatively lower robustness than the cable system

53 Robustness of theArch Rib To evaluate the robustness ofthe arch rib which is an eccentric compression member anM-N curve was drawn to evaluate its bearing capacityFigures 22 and 23 show the axial force and bending momentcurves of the arch rib subjected to S1 and S30 cable breakrespectively (e curves in the circle are the internal forcecurves of the arch rib which are enlarged at the upper rightof the figure It is observed that the arch ribs are mainlyunder compression (e axial force is between 14000 kNand 17500 kN and the bending moment is between minus150and 450 kNmiddotm (e impact effect caused by the cable break

TdTf

400

600

800

1000

1200

1400Te

nsio

n of

cabl

es (k

N)

5 10 15 20 25 300Cable number

Figure 18 Comparison of the limit state combination Td andaccidental combination Tf of each cable

Figure 19 Pushover model of the deck system

0

500

1000

1500

2000

Push

forc

e (kN

)

10 20 30 400Relative displacement of anchorage end (mm)

Figure 20 Load-displacement curve of the deck

5 10 15 20 25 300Cable number

8

10

12

14

16

Max

imum

relat

ive d

ispla

cem

ent (

mm

)

Figure 21 Maximum relative deck displacements caused by asingle cable break

ndash200 ndash100 0 100 200 30014000

15000

16000

17000

MLTMLBMT

MBMRTMRB

Axi

al fo

rce (

kN)

Moments (kNmiddotm)

ndash50000

0

50000

100000

150000A

xial

forc

e (kN

)

0 10000 20000 30000ndash10000Moments (kNmiddotm)

Figure 22 Internal force change curves of the arch rib due to cableS1 break within its M-N curves

12 Shock and Vibration

was within 20 of the arch rib bearing capability indicatingthat the failure of a single cable does not cause failure of thearch rib (e safety factor of the arch rib is greater than 50which reveals that the arch rib has sufficient robustnesssubjected to single cable breakage

6 Conclusions

Cable breaking is common damage in cable-supported archbridges as a result of aging and erosion caused by the harshenvironment To evaluate the dynamic response and robust-ness of cable-supported arch bridges subjected to cablebreaking a nonlinear dynamic simulation procedure and ro-bustness evaluation method are developed in this study Aprototype arch bridge with a single cable break was analyzed asan example Several critical factors for cable breakage simu-lation such as the cable break mode cable break duration anddamping are studied (e dynamic responses of the cablesystem deck and rib are evaluated and the DAF of differentcomponents of the cable-supported arch bridge is also dis-cussed Robustness evaluation indexes of the cable systemdeck and arch rib are calculated and their robustness levels areevaluated (e main conclusions are as follows

(1) Using an appropriate simulation method is essentialto the dynamic analysis of a cable breaking processDirectly removing the failed cable in an appropriatetime step of a transient analysis can effectivelysimulate the cable breaking effect When the cablebreak duration is sufficiently short the maximumimpact effect can be fully attained regardless of thetype of the break simulation mode used (e Ray-leigh damping coefficient calculated by a higher-order frequency can avoid the suppression effect of alocal vibration and better reflect the local vibrationcaused by a cable break

(2) (e closest cable to the broken cable receives thelargest impact effect (e impact effect of the cable

tension caused by the break of a short cable is greaterthan that of a long cable (e deck displacement andbending moment increase with the length of thefailure cable(e breaking of a downstream cable haslittle effect on the upstream cables

(3) (e DAF of each component is different(e DAF ofcable tension is between 14 and 16 while that of thedeck displacement is between 13 and 165 (e archrib bending moment has the largest DAF that isbetween 15 and 24 which is greater than the PTIrecommended value of 20 in some cases Using aunified DAF of 20 cannot satisfy all the componentsof cable-supported arch bridges

(4) (e deck system is the least robust component ofcable-supported arch bridges (e cable system has asafety factor of 31 under the cable breakage acci-dental combination a single cable break will nottrigger progressive damage (e arch rib has suffi-cient robustness subjected to a single cable breakagethough it has the largest DAF

(e results obtained by this analysis are conducive to thedesign and robustness evaluation of cable-supported archbridges Although only the breakage of a single cable wasdemonstrated the proposed methodology can be applied tomultiple cable breakage scenario of cable-supported bridgesFuture research can be carried out in different types of cable-supported arch bridges subjected to multiple cablebreakages

Data Availability

(e data that support the findings of this study are availablefrom the corresponding author upon reasonable request

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(e authors would like to acknowledge Professor RuinianJiang of NMSU for providing help during the preparation ofthe manuscript and the authors also gratefully acknowledgethe support of Taizhou Highway Administration (is re-search was funded by the Zhejiang Provincial Transportationand Transportation Office Technology Funding Project(2012H47)

References

[1] PTI Recommendations for Stay Cable Design Testing andInstallation Post-Tensioning Institute Farmington Hills MIUSA 2012

[2] T Zoli and R Woodward ldquoDesign of long span bridges forcable lossrdquo IABSE Symposium Report vol 90 no 9 pp 17ndash252005

[3] A M Ruiz-Teran and A C Aparicio ldquoResponse of under-deck cable-stayed bridges to the accidental breakage of stay

MLTMLBMT

MBMRTMRB

0 100 200 300 400 50013000

14000

15000

16000

17000

18000

Axi

al fo

rce (

kN)

Moments (kNmiddotm)

ndash50000

0

50000

100000

150000A

xial

forc

e (kN

)

0 10000 20000 30000ndash10000Moments (kNmiddotm)

Figure 23 Internal force change curves of the arch rib due to cableS30 break within its M-N curves

Shock and Vibration 13

cablesrdquo Engineering Structures vol 31 no 7 pp 1425ndash14342009

[4] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part I bending moment acting on the deckrdquo En-gineering Structures vol 32 no 10 pp 3288ndash3300 2010

[5] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part II bending moment acting on the pylons andstress on the staysrdquo Engineering Structures vol 32 no 10pp 3301ndash3312 2010

[6] Y Zhou and S Chen ldquoTime-progressive dynamic assessmentof abrupt cable-breakage events on cable-stayed bridgesrdquoJournal of Bridge Engineering vol 19 no 2 pp 159ndash171 2014

[7] Y Park U Starossek H-M Koh J F Choo H Kim andS W Lee ldquoEffect of cable loss in cable stayed bridgesmdashfocuson dynamic amplificationrdquo IABSE Symposium Report vol 93no 5 pp 9ndash16 2007

[8] J-g Cai Y-x Xu L-p Zhuang J Feng and J ZhangldquoComparison of various procedures for progressive collapseanalysis of cable-stayed bridgesrdquo Journal of Zhejiang Uni-versity Science A vol 13 no 5 pp 323ndash334 2012

[9] Y Xiang Z Chen Y Yang H Lin and S Zhu ldquoDynamicresponse analysis for submerged floating tunnel with anchor-cables subjected to sudden cable breakagerdquoMarine Structuresvol 59 pp 179ndash191 2018

[10] Y F Zhou and S R Chen ldquoFramework of nonlinear dynamicsimulation of long-span cable-stayed bridge and traffic systemsubjected to cable-loss incidentsrdquo Journal of Structural En-gineering vol 142 no 3 2016

[11] W L Qiu M Jiang and C L Huang ldquoParametric study onresponses of a self-anchored suspension bridge to suddenbreakage of a hangerrdquo Scientific World Journal vol 2014Article ID 512120 10 pages 2014

[12] W Qiu M Jiang and Z Zhang ldquoResponses of self-anchoredsuspension bridge to sudden breakage of hangersrdquo StructuralEngineering and Mechanics vol 50 no 2 pp 241ndash255 2014

[13] C M Mozos and A C Aparicio ldquoNumerical and experi-mental study on the interaction cable structure during thefailure of a stay in a cable stayed bridgerdquo EngineeringStructures vol 33 no 8 pp 2330ndash2341 2011

[14] V Hoang O Kiyomiya and T An ldquoExperimental and nu-merical study of lateral cable rupture in cable-stayed bridgescase studyrdquo Journal of Bridge Engineering vol 23 no 6 2018

[15] M Domaneschi G P Cimellaro and G Scutiero ldquoDispro-portionate collapse of a cable-stayed bridgerdquo Proceedings ofthe Institution of Civil EngineersmdashBridge Engineering vol 172no 1 pp 13ndash26 2019

[16] M Wolff and U Starossek ldquoCable loss and progressivecollapse in cable-stayed bridgesrdquo Bridge Structures vol 5no 1 pp 17ndash28 2009

[17] G Wu W Qiu and T Wu ldquoNonlinear dynamic analysis ofthe self-anchored suspension bridge subjected to suddenbreakage of a hangerrdquo Engineering Failure Analysis vol 97pp 701ndash717 2019

[18] M Morgese F Ansari M Domaneschi and G P CimellaroldquoPost-collapse analysis of morandirsquos polcevera viaduct ingenoa Italyrdquo Journal of Civil Structural Health Monitoringvol 10 no 1 pp 69ndash85 2020

[19] M Shoghijavan and U Starossek ldquoStructural robustness oflong-span cable-supported bridges in a cable-loss scenariordquoJournal of Bridge Engineering vol 23 no 2 2018

[20] M Shoghijavan and U Starossek ldquoAn analytical study on thebending moment acting on the girder of a long-span cable-

supported bridge suffering from cable failurerdquo EngineeringStructures vol 167 pp 166ndash174 2018

[21] M Domaneschi C Pellecchia E De Iuliis et al ldquoCollapseanalysis of the Polcevera viaduct by the applied elementmethodrdquo Engineering Structures vol 214 p 110659 2020

[22] O Olamigoke Structural Response of Cable-Stayed Bridges toCable Loss University of Surrey Guildford UK 2018

[23] J Xu H H Sun and S Y Cai ldquoEffect of symmetrical brokenwires damage on mechanical characteristics of stay cablerdquoJournal of Sound and Vibration vol 461 2019

14 Shock and Vibration

Td 12Ts + 14Tv + 075 Tp + Tt1113872 1113873

Tf 12Ts + 075 Tv + Tp + Tt1113872 1113873 + Tc(6)

where Ts Tv Tp and Tt are the tension of the dead loadvehicle load pedestrian load and temperature load re-spectively Tc is the net dynamic effect of the cable breakwhich is equal to the maximum tension of the dynamicresponse minus the cable tension under the dead load

Figure 18 shows the comparison of the design limitcombination Td and the accidental combination Tf of each

cable It is observed that the values of cable fracture ac-cidental combination are all higher than the carryingcapability limit state combination (is means that thecable breakage accidental combination is the most un-favorable situation and should not be ignored in thedesign stage

(e maximum value of the accidental combination Tf is1130 kN and the minimum safety factor of the cablecompared with the ultimate bearing capability fu is 31 Itshows that the cable system has good robustness under acable breakage accidental combination and a single cable

MLTMTMRT

MLBMBMRB

0

100

200

300

400

500M

omen

t of a

rch

rib (k

Nmiddotm

)

10 20 30 40 500Time (s)

(a)

MLTMTMRT

MLBMBMRB

10 20 30 40 500Time(s)

0

100

200

300

400

500

Mom

ent o

f arc

h rib

(kN

middotm)

(b)

Figure 12 Bending moment time-history curves of the arch rib (a) cable S29 break (b) cable S30 break

Midspan

MaxMin

ndash100

0

100

200

300

Mom

ent o

f upp

er ch

ords

(kN

middotm)

5 10 15 20 25 300Cable number

(a)

MaxMin

Midspan

ndash100

0

100

200

300

400

500

Mom

ent o

f low

er ch

ords

(kN

middotm)

5 10 15 20 25 300Cable number

(b)

Figure 13 Envelope of the arch rib bending moment subjected to a single cable break (a) upper chords (b) lower chords

10 Shock and Vibration

break will not trigger a progressive breakage of adjacentcables

52 Robustness of the Deck Different from cable-stayedbridges that have a strong longitudinal stiffening girdercable-supported arch bridge deck systems always usetransverse girders as the main load-bearing members Toevaluate the damage to the deck system the capability of thedeck was obtained by a pushover analysis

(e pushover model was composed of three transversegirders with its flange plate An elastic support was used tosimulate the rigidity of the cables at the anchoring positions(e horizontal side displacement of the deck was notallowed by fixing the left and right ends of the deck system toavoid any rigid movement (e vertical pushover load wasapplied to the anchored position of the broken cable on the

right side of the middle girder as indicated by the arrow inFigure 19 (e relative displacements between the transversegirders were chosen as the capability index to represent thebearing capacity of the deck system

Figure 20 depicts the load-displacement curve of thedeck It can be noticed that the failure process of the deck canbe divided into three stages elastic stage cracked stage andfailure stage When the relative displacement is less than235mm the deck is in the elastic stage When the relativedisplacement increases from 235mm to 315mm cracksgradually increase and the deck works with the cracks Whenthe relative displacement exceeds 315mm the longitudinalreinforcement yields and the displacement increases rapidly

(e robustness of the deck was evaluated by comparingthe relative displacement of the cable breakage with that ofthe load-displacement curve As displayed in Figure 21 themaximum relative displacement of the deck is between 8mmand 15mm subjected to a single cable break which increases

Midspan

Upper chordsLower chords

0

100

200

300

Mom

ent v

aria

tion

(kN

middotm)

5 10 15 20 25 300Cable number

Figure 14 Moment variation of arch rib chords caused by a singlecable break

0 5 10 15 20 25 3010

12

14

16

18

DA

F of

adja

cent

cabl

e ten

sion

Cable number

Figure 15 DAF of the cable tension caused by a single cable break

10

12

14

16

18

DA

F of

dec

k di

spla

cem

ent

5 10 15 20 25 300Cable number

Figure 16 DAF of the deck displacement caused by a single cablebreak

10

12

14

16

18

20

22

24D

AF

of ar

ch ri

b be

ndin

g m

omen

t

5 10 15 20 25 300Cable number

Figure 17 DAF of the arch rib moment caused by a single cablebreak

Shock and Vibration 11

with the length of the broken cable Compared with that ofthe load-displacement curve in Figure 20 the relative dis-placement of the deck subjected to a single cable breakage iswithin the cracked stage and will not lead to a deck collapseUnder the worst conditions the deck system has a safety

factor of approximately 20 which indicates that the decksystem has relatively lower robustness than the cable system

53 Robustness of theArch Rib To evaluate the robustness ofthe arch rib which is an eccentric compression member anM-N curve was drawn to evaluate its bearing capacityFigures 22 and 23 show the axial force and bending momentcurves of the arch rib subjected to S1 and S30 cable breakrespectively (e curves in the circle are the internal forcecurves of the arch rib which are enlarged at the upper rightof the figure It is observed that the arch ribs are mainlyunder compression (e axial force is between 14000 kNand 17500 kN and the bending moment is between minus150and 450 kNmiddotm (e impact effect caused by the cable break

TdTf

400

600

800

1000

1200

1400Te

nsio

n of

cabl

es (k

N)

5 10 15 20 25 300Cable number

Figure 18 Comparison of the limit state combination Td andaccidental combination Tf of each cable

Figure 19 Pushover model of the deck system

0

500

1000

1500

2000

Push

forc

e (kN

)

10 20 30 400Relative displacement of anchorage end (mm)

Figure 20 Load-displacement curve of the deck

5 10 15 20 25 300Cable number

8

10

12

14

16

Max

imum

relat

ive d

ispla

cem

ent (

mm

)

Figure 21 Maximum relative deck displacements caused by asingle cable break

ndash200 ndash100 0 100 200 30014000

15000

16000

17000

MLTMLBMT

MBMRTMRB

Axi

al fo

rce (

kN)

Moments (kNmiddotm)

ndash50000

0

50000

100000

150000A

xial

forc

e (kN

)

0 10000 20000 30000ndash10000Moments (kNmiddotm)

Figure 22 Internal force change curves of the arch rib due to cableS1 break within its M-N curves

12 Shock and Vibration

was within 20 of the arch rib bearing capability indicatingthat the failure of a single cable does not cause failure of thearch rib (e safety factor of the arch rib is greater than 50which reveals that the arch rib has sufficient robustnesssubjected to single cable breakage

6 Conclusions

Cable breaking is common damage in cable-supported archbridges as a result of aging and erosion caused by the harshenvironment To evaluate the dynamic response and robust-ness of cable-supported arch bridges subjected to cablebreaking a nonlinear dynamic simulation procedure and ro-bustness evaluation method are developed in this study Aprototype arch bridge with a single cable break was analyzed asan example Several critical factors for cable breakage simu-lation such as the cable break mode cable break duration anddamping are studied (e dynamic responses of the cablesystem deck and rib are evaluated and the DAF of differentcomponents of the cable-supported arch bridge is also dis-cussed Robustness evaluation indexes of the cable systemdeck and arch rib are calculated and their robustness levels areevaluated (e main conclusions are as follows

(1) Using an appropriate simulation method is essentialto the dynamic analysis of a cable breaking processDirectly removing the failed cable in an appropriatetime step of a transient analysis can effectivelysimulate the cable breaking effect When the cablebreak duration is sufficiently short the maximumimpact effect can be fully attained regardless of thetype of the break simulation mode used (e Ray-leigh damping coefficient calculated by a higher-order frequency can avoid the suppression effect of alocal vibration and better reflect the local vibrationcaused by a cable break

(2) (e closest cable to the broken cable receives thelargest impact effect (e impact effect of the cable

tension caused by the break of a short cable is greaterthan that of a long cable (e deck displacement andbending moment increase with the length of thefailure cable(e breaking of a downstream cable haslittle effect on the upstream cables

(3) (e DAF of each component is different(e DAF ofcable tension is between 14 and 16 while that of thedeck displacement is between 13 and 165 (e archrib bending moment has the largest DAF that isbetween 15 and 24 which is greater than the PTIrecommended value of 20 in some cases Using aunified DAF of 20 cannot satisfy all the componentsof cable-supported arch bridges

(4) (e deck system is the least robust component ofcable-supported arch bridges (e cable system has asafety factor of 31 under the cable breakage acci-dental combination a single cable break will nottrigger progressive damage (e arch rib has suffi-cient robustness subjected to a single cable breakagethough it has the largest DAF

(e results obtained by this analysis are conducive to thedesign and robustness evaluation of cable-supported archbridges Although only the breakage of a single cable wasdemonstrated the proposed methodology can be applied tomultiple cable breakage scenario of cable-supported bridgesFuture research can be carried out in different types of cable-supported arch bridges subjected to multiple cablebreakages

Data Availability

(e data that support the findings of this study are availablefrom the corresponding author upon reasonable request

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(e authors would like to acknowledge Professor RuinianJiang of NMSU for providing help during the preparation ofthe manuscript and the authors also gratefully acknowledgethe support of Taizhou Highway Administration (is re-search was funded by the Zhejiang Provincial Transportationand Transportation Office Technology Funding Project(2012H47)

References

[1] PTI Recommendations for Stay Cable Design Testing andInstallation Post-Tensioning Institute Farmington Hills MIUSA 2012

[2] T Zoli and R Woodward ldquoDesign of long span bridges forcable lossrdquo IABSE Symposium Report vol 90 no 9 pp 17ndash252005

[3] A M Ruiz-Teran and A C Aparicio ldquoResponse of under-deck cable-stayed bridges to the accidental breakage of stay

MLTMLBMT

MBMRTMRB

0 100 200 300 400 50013000

14000

15000

16000

17000

18000

Axi

al fo

rce (

kN)

Moments (kNmiddotm)

ndash50000

0

50000

100000

150000A

xial

forc

e (kN

)

0 10000 20000 30000ndash10000Moments (kNmiddotm)

Figure 23 Internal force change curves of the arch rib due to cableS30 break within its M-N curves

Shock and Vibration 13

cablesrdquo Engineering Structures vol 31 no 7 pp 1425ndash14342009

[4] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part I bending moment acting on the deckrdquo En-gineering Structures vol 32 no 10 pp 3288ndash3300 2010

[5] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part II bending moment acting on the pylons andstress on the staysrdquo Engineering Structures vol 32 no 10pp 3301ndash3312 2010

[6] Y Zhou and S Chen ldquoTime-progressive dynamic assessmentof abrupt cable-breakage events on cable-stayed bridgesrdquoJournal of Bridge Engineering vol 19 no 2 pp 159ndash171 2014

[7] Y Park U Starossek H-M Koh J F Choo H Kim andS W Lee ldquoEffect of cable loss in cable stayed bridgesmdashfocuson dynamic amplificationrdquo IABSE Symposium Report vol 93no 5 pp 9ndash16 2007

[8] J-g Cai Y-x Xu L-p Zhuang J Feng and J ZhangldquoComparison of various procedures for progressive collapseanalysis of cable-stayed bridgesrdquo Journal of Zhejiang Uni-versity Science A vol 13 no 5 pp 323ndash334 2012

[9] Y Xiang Z Chen Y Yang H Lin and S Zhu ldquoDynamicresponse analysis for submerged floating tunnel with anchor-cables subjected to sudden cable breakagerdquoMarine Structuresvol 59 pp 179ndash191 2018

[10] Y F Zhou and S R Chen ldquoFramework of nonlinear dynamicsimulation of long-span cable-stayed bridge and traffic systemsubjected to cable-loss incidentsrdquo Journal of Structural En-gineering vol 142 no 3 2016

[11] W L Qiu M Jiang and C L Huang ldquoParametric study onresponses of a self-anchored suspension bridge to suddenbreakage of a hangerrdquo Scientific World Journal vol 2014Article ID 512120 10 pages 2014

[12] W Qiu M Jiang and Z Zhang ldquoResponses of self-anchoredsuspension bridge to sudden breakage of hangersrdquo StructuralEngineering and Mechanics vol 50 no 2 pp 241ndash255 2014

[13] C M Mozos and A C Aparicio ldquoNumerical and experi-mental study on the interaction cable structure during thefailure of a stay in a cable stayed bridgerdquo EngineeringStructures vol 33 no 8 pp 2330ndash2341 2011

[14] V Hoang O Kiyomiya and T An ldquoExperimental and nu-merical study of lateral cable rupture in cable-stayed bridgescase studyrdquo Journal of Bridge Engineering vol 23 no 6 2018

[15] M Domaneschi G P Cimellaro and G Scutiero ldquoDispro-portionate collapse of a cable-stayed bridgerdquo Proceedings ofthe Institution of Civil EngineersmdashBridge Engineering vol 172no 1 pp 13ndash26 2019

[16] M Wolff and U Starossek ldquoCable loss and progressivecollapse in cable-stayed bridgesrdquo Bridge Structures vol 5no 1 pp 17ndash28 2009

[17] G Wu W Qiu and T Wu ldquoNonlinear dynamic analysis ofthe self-anchored suspension bridge subjected to suddenbreakage of a hangerrdquo Engineering Failure Analysis vol 97pp 701ndash717 2019

[18] M Morgese F Ansari M Domaneschi and G P CimellaroldquoPost-collapse analysis of morandirsquos polcevera viaduct ingenoa Italyrdquo Journal of Civil Structural Health Monitoringvol 10 no 1 pp 69ndash85 2020

[19] M Shoghijavan and U Starossek ldquoStructural robustness oflong-span cable-supported bridges in a cable-loss scenariordquoJournal of Bridge Engineering vol 23 no 2 2018

[20] M Shoghijavan and U Starossek ldquoAn analytical study on thebending moment acting on the girder of a long-span cable-

supported bridge suffering from cable failurerdquo EngineeringStructures vol 167 pp 166ndash174 2018

[21] M Domaneschi C Pellecchia E De Iuliis et al ldquoCollapseanalysis of the Polcevera viaduct by the applied elementmethodrdquo Engineering Structures vol 214 p 110659 2020

[22] O Olamigoke Structural Response of Cable-Stayed Bridges toCable Loss University of Surrey Guildford UK 2018

[23] J Xu H H Sun and S Y Cai ldquoEffect of symmetrical brokenwires damage on mechanical characteristics of stay cablerdquoJournal of Sound and Vibration vol 461 2019

14 Shock and Vibration

break will not trigger a progressive breakage of adjacentcables

52 Robustness of the Deck Different from cable-stayedbridges that have a strong longitudinal stiffening girdercable-supported arch bridge deck systems always usetransverse girders as the main load-bearing members Toevaluate the damage to the deck system the capability of thedeck was obtained by a pushover analysis

(e pushover model was composed of three transversegirders with its flange plate An elastic support was used tosimulate the rigidity of the cables at the anchoring positions(e horizontal side displacement of the deck was notallowed by fixing the left and right ends of the deck system toavoid any rigid movement (e vertical pushover load wasapplied to the anchored position of the broken cable on the

right side of the middle girder as indicated by the arrow inFigure 19 (e relative displacements between the transversegirders were chosen as the capability index to represent thebearing capacity of the deck system

Figure 20 depicts the load-displacement curve of thedeck It can be noticed that the failure process of the deck canbe divided into three stages elastic stage cracked stage andfailure stage When the relative displacement is less than235mm the deck is in the elastic stage When the relativedisplacement increases from 235mm to 315mm cracksgradually increase and the deck works with the cracks Whenthe relative displacement exceeds 315mm the longitudinalreinforcement yields and the displacement increases rapidly

(e robustness of the deck was evaluated by comparingthe relative displacement of the cable breakage with that ofthe load-displacement curve As displayed in Figure 21 themaximum relative displacement of the deck is between 8mmand 15mm subjected to a single cable break which increases

Midspan

Upper chordsLower chords

0

100

200

300

Mom

ent v

aria

tion

(kN

middotm)

5 10 15 20 25 300Cable number

Figure 14 Moment variation of arch rib chords caused by a singlecable break

0 5 10 15 20 25 3010

12

14

16

18

DA

F of

adja

cent

cabl

e ten

sion

Cable number

Figure 15 DAF of the cable tension caused by a single cable break

10

12

14

16

18

DA

F of

dec

k di

spla

cem

ent

5 10 15 20 25 300Cable number

Figure 16 DAF of the deck displacement caused by a single cablebreak

10

12

14

16

18

20

22

24D

AF

of ar

ch ri

b be

ndin

g m

omen

t

5 10 15 20 25 300Cable number

Figure 17 DAF of the arch rib moment caused by a single cablebreak

Shock and Vibration 11

with the length of the broken cable Compared with that ofthe load-displacement curve in Figure 20 the relative dis-placement of the deck subjected to a single cable breakage iswithin the cracked stage and will not lead to a deck collapseUnder the worst conditions the deck system has a safety

factor of approximately 20 which indicates that the decksystem has relatively lower robustness than the cable system

53 Robustness of theArch Rib To evaluate the robustness ofthe arch rib which is an eccentric compression member anM-N curve was drawn to evaluate its bearing capacityFigures 22 and 23 show the axial force and bending momentcurves of the arch rib subjected to S1 and S30 cable breakrespectively (e curves in the circle are the internal forcecurves of the arch rib which are enlarged at the upper rightof the figure It is observed that the arch ribs are mainlyunder compression (e axial force is between 14000 kNand 17500 kN and the bending moment is between minus150and 450 kNmiddotm (e impact effect caused by the cable break

TdTf

400

600

800

1000

1200

1400Te

nsio

n of

cabl

es (k

N)

5 10 15 20 25 300Cable number

Figure 18 Comparison of the limit state combination Td andaccidental combination Tf of each cable

Figure 19 Pushover model of the deck system

0

500

1000

1500

2000

Push

forc

e (kN

)

10 20 30 400Relative displacement of anchorage end (mm)

Figure 20 Load-displacement curve of the deck

5 10 15 20 25 300Cable number

8

10

12

14

16

Max

imum

relat

ive d

ispla

cem

ent (

mm

)

Figure 21 Maximum relative deck displacements caused by asingle cable break

ndash200 ndash100 0 100 200 30014000

15000

16000

17000

MLTMLBMT

MBMRTMRB

Axi

al fo

rce (

kN)

Moments (kNmiddotm)

ndash50000

0

50000

100000

150000A

xial

forc

e (kN

)

0 10000 20000 30000ndash10000Moments (kNmiddotm)

Figure 22 Internal force change curves of the arch rib due to cableS1 break within its M-N curves

12 Shock and Vibration

was within 20 of the arch rib bearing capability indicatingthat the failure of a single cable does not cause failure of thearch rib (e safety factor of the arch rib is greater than 50which reveals that the arch rib has sufficient robustnesssubjected to single cable breakage

6 Conclusions

Cable breaking is common damage in cable-supported archbridges as a result of aging and erosion caused by the harshenvironment To evaluate the dynamic response and robust-ness of cable-supported arch bridges subjected to cablebreaking a nonlinear dynamic simulation procedure and ro-bustness evaluation method are developed in this study Aprototype arch bridge with a single cable break was analyzed asan example Several critical factors for cable breakage simu-lation such as the cable break mode cable break duration anddamping are studied (e dynamic responses of the cablesystem deck and rib are evaluated and the DAF of differentcomponents of the cable-supported arch bridge is also dis-cussed Robustness evaluation indexes of the cable systemdeck and arch rib are calculated and their robustness levels areevaluated (e main conclusions are as follows

(1) Using an appropriate simulation method is essentialto the dynamic analysis of a cable breaking processDirectly removing the failed cable in an appropriatetime step of a transient analysis can effectivelysimulate the cable breaking effect When the cablebreak duration is sufficiently short the maximumimpact effect can be fully attained regardless of thetype of the break simulation mode used (e Ray-leigh damping coefficient calculated by a higher-order frequency can avoid the suppression effect of alocal vibration and better reflect the local vibrationcaused by a cable break

(2) (e closest cable to the broken cable receives thelargest impact effect (e impact effect of the cable

tension caused by the break of a short cable is greaterthan that of a long cable (e deck displacement andbending moment increase with the length of thefailure cable(e breaking of a downstream cable haslittle effect on the upstream cables

(3) (e DAF of each component is different(e DAF ofcable tension is between 14 and 16 while that of thedeck displacement is between 13 and 165 (e archrib bending moment has the largest DAF that isbetween 15 and 24 which is greater than the PTIrecommended value of 20 in some cases Using aunified DAF of 20 cannot satisfy all the componentsof cable-supported arch bridges

(4) (e deck system is the least robust component ofcable-supported arch bridges (e cable system has asafety factor of 31 under the cable breakage acci-dental combination a single cable break will nottrigger progressive damage (e arch rib has suffi-cient robustness subjected to a single cable breakagethough it has the largest DAF

(e results obtained by this analysis are conducive to thedesign and robustness evaluation of cable-supported archbridges Although only the breakage of a single cable wasdemonstrated the proposed methodology can be applied tomultiple cable breakage scenario of cable-supported bridgesFuture research can be carried out in different types of cable-supported arch bridges subjected to multiple cablebreakages

Data Availability

(e data that support the findings of this study are availablefrom the corresponding author upon reasonable request

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(e authors would like to acknowledge Professor RuinianJiang of NMSU for providing help during the preparation ofthe manuscript and the authors also gratefully acknowledgethe support of Taizhou Highway Administration (is re-search was funded by the Zhejiang Provincial Transportationand Transportation Office Technology Funding Project(2012H47)

References

[1] PTI Recommendations for Stay Cable Design Testing andInstallation Post-Tensioning Institute Farmington Hills MIUSA 2012

[2] T Zoli and R Woodward ldquoDesign of long span bridges forcable lossrdquo IABSE Symposium Report vol 90 no 9 pp 17ndash252005

[3] A M Ruiz-Teran and A C Aparicio ldquoResponse of under-deck cable-stayed bridges to the accidental breakage of stay

MLTMLBMT

MBMRTMRB

0 100 200 300 400 50013000

14000

15000

16000

17000

18000

Axi

al fo

rce (

kN)

Moments (kNmiddotm)

ndash50000

0

50000

100000

150000A

xial

forc

e (kN

)

0 10000 20000 30000ndash10000Moments (kNmiddotm)

Figure 23 Internal force change curves of the arch rib due to cableS30 break within its M-N curves

Shock and Vibration 13

cablesrdquo Engineering Structures vol 31 no 7 pp 1425ndash14342009

[4] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part I bending moment acting on the deckrdquo En-gineering Structures vol 32 no 10 pp 3288ndash3300 2010

[5] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part II bending moment acting on the pylons andstress on the staysrdquo Engineering Structures vol 32 no 10pp 3301ndash3312 2010

[6] Y Zhou and S Chen ldquoTime-progressive dynamic assessmentof abrupt cable-breakage events on cable-stayed bridgesrdquoJournal of Bridge Engineering vol 19 no 2 pp 159ndash171 2014

[7] Y Park U Starossek H-M Koh J F Choo H Kim andS W Lee ldquoEffect of cable loss in cable stayed bridgesmdashfocuson dynamic amplificationrdquo IABSE Symposium Report vol 93no 5 pp 9ndash16 2007

[8] J-g Cai Y-x Xu L-p Zhuang J Feng and J ZhangldquoComparison of various procedures for progressive collapseanalysis of cable-stayed bridgesrdquo Journal of Zhejiang Uni-versity Science A vol 13 no 5 pp 323ndash334 2012

[9] Y Xiang Z Chen Y Yang H Lin and S Zhu ldquoDynamicresponse analysis for submerged floating tunnel with anchor-cables subjected to sudden cable breakagerdquoMarine Structuresvol 59 pp 179ndash191 2018

[10] Y F Zhou and S R Chen ldquoFramework of nonlinear dynamicsimulation of long-span cable-stayed bridge and traffic systemsubjected to cable-loss incidentsrdquo Journal of Structural En-gineering vol 142 no 3 2016

[11] W L Qiu M Jiang and C L Huang ldquoParametric study onresponses of a self-anchored suspension bridge to suddenbreakage of a hangerrdquo Scientific World Journal vol 2014Article ID 512120 10 pages 2014

[12] W Qiu M Jiang and Z Zhang ldquoResponses of self-anchoredsuspension bridge to sudden breakage of hangersrdquo StructuralEngineering and Mechanics vol 50 no 2 pp 241ndash255 2014

[13] C M Mozos and A C Aparicio ldquoNumerical and experi-mental study on the interaction cable structure during thefailure of a stay in a cable stayed bridgerdquo EngineeringStructures vol 33 no 8 pp 2330ndash2341 2011

[14] V Hoang O Kiyomiya and T An ldquoExperimental and nu-merical study of lateral cable rupture in cable-stayed bridgescase studyrdquo Journal of Bridge Engineering vol 23 no 6 2018

[15] M Domaneschi G P Cimellaro and G Scutiero ldquoDispro-portionate collapse of a cable-stayed bridgerdquo Proceedings ofthe Institution of Civil EngineersmdashBridge Engineering vol 172no 1 pp 13ndash26 2019

[16] M Wolff and U Starossek ldquoCable loss and progressivecollapse in cable-stayed bridgesrdquo Bridge Structures vol 5no 1 pp 17ndash28 2009

[17] G Wu W Qiu and T Wu ldquoNonlinear dynamic analysis ofthe self-anchored suspension bridge subjected to suddenbreakage of a hangerrdquo Engineering Failure Analysis vol 97pp 701ndash717 2019

[18] M Morgese F Ansari M Domaneschi and G P CimellaroldquoPost-collapse analysis of morandirsquos polcevera viaduct ingenoa Italyrdquo Journal of Civil Structural Health Monitoringvol 10 no 1 pp 69ndash85 2020

[19] M Shoghijavan and U Starossek ldquoStructural robustness oflong-span cable-supported bridges in a cable-loss scenariordquoJournal of Bridge Engineering vol 23 no 2 2018

[20] M Shoghijavan and U Starossek ldquoAn analytical study on thebending moment acting on the girder of a long-span cable-

supported bridge suffering from cable failurerdquo EngineeringStructures vol 167 pp 166ndash174 2018

[21] M Domaneschi C Pellecchia E De Iuliis et al ldquoCollapseanalysis of the Polcevera viaduct by the applied elementmethodrdquo Engineering Structures vol 214 p 110659 2020

[22] O Olamigoke Structural Response of Cable-Stayed Bridges toCable Loss University of Surrey Guildford UK 2018

[23] J Xu H H Sun and S Y Cai ldquoEffect of symmetrical brokenwires damage on mechanical characteristics of stay cablerdquoJournal of Sound and Vibration vol 461 2019

14 Shock and Vibration

with the length of the broken cable Compared with that ofthe load-displacement curve in Figure 20 the relative dis-placement of the deck subjected to a single cable breakage iswithin the cracked stage and will not lead to a deck collapseUnder the worst conditions the deck system has a safety

factor of approximately 20 which indicates that the decksystem has relatively lower robustness than the cable system

53 Robustness of theArch Rib To evaluate the robustness ofthe arch rib which is an eccentric compression member anM-N curve was drawn to evaluate its bearing capacityFigures 22 and 23 show the axial force and bending momentcurves of the arch rib subjected to S1 and S30 cable breakrespectively (e curves in the circle are the internal forcecurves of the arch rib which are enlarged at the upper rightof the figure It is observed that the arch ribs are mainlyunder compression (e axial force is between 14000 kNand 17500 kN and the bending moment is between minus150and 450 kNmiddotm (e impact effect caused by the cable break

TdTf

400

600

800

1000

1200

1400Te

nsio

n of

cabl

es (k

N)

5 10 15 20 25 300Cable number

Figure 18 Comparison of the limit state combination Td andaccidental combination Tf of each cable

Figure 19 Pushover model of the deck system

0

500

1000

1500

2000

Push

forc

e (kN

)

10 20 30 400Relative displacement of anchorage end (mm)

Figure 20 Load-displacement curve of the deck

5 10 15 20 25 300Cable number

8

10

12

14

16

Max

imum

relat

ive d

ispla

cem

ent (

mm

)

Figure 21 Maximum relative deck displacements caused by asingle cable break

ndash200 ndash100 0 100 200 30014000

15000

16000

17000

MLTMLBMT

MBMRTMRB

Axi

al fo

rce (

kN)

Moments (kNmiddotm)

ndash50000

0

50000

100000

150000A

xial

forc

e (kN

)

0 10000 20000 30000ndash10000Moments (kNmiddotm)

Figure 22 Internal force change curves of the arch rib due to cableS1 break within its M-N curves

12 Shock and Vibration

was within 20 of the arch rib bearing capability indicatingthat the failure of a single cable does not cause failure of thearch rib (e safety factor of the arch rib is greater than 50which reveals that the arch rib has sufficient robustnesssubjected to single cable breakage

6 Conclusions

Cable breaking is common damage in cable-supported archbridges as a result of aging and erosion caused by the harshenvironment To evaluate the dynamic response and robust-ness of cable-supported arch bridges subjected to cablebreaking a nonlinear dynamic simulation procedure and ro-bustness evaluation method are developed in this study Aprototype arch bridge with a single cable break was analyzed asan example Several critical factors for cable breakage simu-lation such as the cable break mode cable break duration anddamping are studied (e dynamic responses of the cablesystem deck and rib are evaluated and the DAF of differentcomponents of the cable-supported arch bridge is also dis-cussed Robustness evaluation indexes of the cable systemdeck and arch rib are calculated and their robustness levels areevaluated (e main conclusions are as follows

(1) Using an appropriate simulation method is essentialto the dynamic analysis of a cable breaking processDirectly removing the failed cable in an appropriatetime step of a transient analysis can effectivelysimulate the cable breaking effect When the cablebreak duration is sufficiently short the maximumimpact effect can be fully attained regardless of thetype of the break simulation mode used (e Ray-leigh damping coefficient calculated by a higher-order frequency can avoid the suppression effect of alocal vibration and better reflect the local vibrationcaused by a cable break

(2) (e closest cable to the broken cable receives thelargest impact effect (e impact effect of the cable

tension caused by the break of a short cable is greaterthan that of a long cable (e deck displacement andbending moment increase with the length of thefailure cable(e breaking of a downstream cable haslittle effect on the upstream cables

(3) (e DAF of each component is different(e DAF ofcable tension is between 14 and 16 while that of thedeck displacement is between 13 and 165 (e archrib bending moment has the largest DAF that isbetween 15 and 24 which is greater than the PTIrecommended value of 20 in some cases Using aunified DAF of 20 cannot satisfy all the componentsof cable-supported arch bridges

(4) (e deck system is the least robust component ofcable-supported arch bridges (e cable system has asafety factor of 31 under the cable breakage acci-dental combination a single cable break will nottrigger progressive damage (e arch rib has suffi-cient robustness subjected to a single cable breakagethough it has the largest DAF

(e results obtained by this analysis are conducive to thedesign and robustness evaluation of cable-supported archbridges Although only the breakage of a single cable wasdemonstrated the proposed methodology can be applied tomultiple cable breakage scenario of cable-supported bridgesFuture research can be carried out in different types of cable-supported arch bridges subjected to multiple cablebreakages

Data Availability

(e data that support the findings of this study are availablefrom the corresponding author upon reasonable request

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(e authors would like to acknowledge Professor RuinianJiang of NMSU for providing help during the preparation ofthe manuscript and the authors also gratefully acknowledgethe support of Taizhou Highway Administration (is re-search was funded by the Zhejiang Provincial Transportationand Transportation Office Technology Funding Project(2012H47)

References

[1] PTI Recommendations for Stay Cable Design Testing andInstallation Post-Tensioning Institute Farmington Hills MIUSA 2012

[2] T Zoli and R Woodward ldquoDesign of long span bridges forcable lossrdquo IABSE Symposium Report vol 90 no 9 pp 17ndash252005

[3] A M Ruiz-Teran and A C Aparicio ldquoResponse of under-deck cable-stayed bridges to the accidental breakage of stay

MLTMLBMT

MBMRTMRB

0 100 200 300 400 50013000

14000

15000

16000

17000

18000

Axi

al fo

rce (

kN)

Moments (kNmiddotm)

ndash50000

0

50000

100000

150000A

xial

forc

e (kN

)

0 10000 20000 30000ndash10000Moments (kNmiddotm)

Figure 23 Internal force change curves of the arch rib due to cableS30 break within its M-N curves

Shock and Vibration 13

cablesrdquo Engineering Structures vol 31 no 7 pp 1425ndash14342009

[4] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part I bending moment acting on the deckrdquo En-gineering Structures vol 32 no 10 pp 3288ndash3300 2010

[5] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part II bending moment acting on the pylons andstress on the staysrdquo Engineering Structures vol 32 no 10pp 3301ndash3312 2010

[6] Y Zhou and S Chen ldquoTime-progressive dynamic assessmentof abrupt cable-breakage events on cable-stayed bridgesrdquoJournal of Bridge Engineering vol 19 no 2 pp 159ndash171 2014

[7] Y Park U Starossek H-M Koh J F Choo H Kim andS W Lee ldquoEffect of cable loss in cable stayed bridgesmdashfocuson dynamic amplificationrdquo IABSE Symposium Report vol 93no 5 pp 9ndash16 2007

[8] J-g Cai Y-x Xu L-p Zhuang J Feng and J ZhangldquoComparison of various procedures for progressive collapseanalysis of cable-stayed bridgesrdquo Journal of Zhejiang Uni-versity Science A vol 13 no 5 pp 323ndash334 2012

[9] Y Xiang Z Chen Y Yang H Lin and S Zhu ldquoDynamicresponse analysis for submerged floating tunnel with anchor-cables subjected to sudden cable breakagerdquoMarine Structuresvol 59 pp 179ndash191 2018

[10] Y F Zhou and S R Chen ldquoFramework of nonlinear dynamicsimulation of long-span cable-stayed bridge and traffic systemsubjected to cable-loss incidentsrdquo Journal of Structural En-gineering vol 142 no 3 2016

[11] W L Qiu M Jiang and C L Huang ldquoParametric study onresponses of a self-anchored suspension bridge to suddenbreakage of a hangerrdquo Scientific World Journal vol 2014Article ID 512120 10 pages 2014

[12] W Qiu M Jiang and Z Zhang ldquoResponses of self-anchoredsuspension bridge to sudden breakage of hangersrdquo StructuralEngineering and Mechanics vol 50 no 2 pp 241ndash255 2014

[13] C M Mozos and A C Aparicio ldquoNumerical and experi-mental study on the interaction cable structure during thefailure of a stay in a cable stayed bridgerdquo EngineeringStructures vol 33 no 8 pp 2330ndash2341 2011

[14] V Hoang O Kiyomiya and T An ldquoExperimental and nu-merical study of lateral cable rupture in cable-stayed bridgescase studyrdquo Journal of Bridge Engineering vol 23 no 6 2018

[15] M Domaneschi G P Cimellaro and G Scutiero ldquoDispro-portionate collapse of a cable-stayed bridgerdquo Proceedings ofthe Institution of Civil EngineersmdashBridge Engineering vol 172no 1 pp 13ndash26 2019

[16] M Wolff and U Starossek ldquoCable loss and progressivecollapse in cable-stayed bridgesrdquo Bridge Structures vol 5no 1 pp 17ndash28 2009

[17] G Wu W Qiu and T Wu ldquoNonlinear dynamic analysis ofthe self-anchored suspension bridge subjected to suddenbreakage of a hangerrdquo Engineering Failure Analysis vol 97pp 701ndash717 2019

[18] M Morgese F Ansari M Domaneschi and G P CimellaroldquoPost-collapse analysis of morandirsquos polcevera viaduct ingenoa Italyrdquo Journal of Civil Structural Health Monitoringvol 10 no 1 pp 69ndash85 2020

[19] M Shoghijavan and U Starossek ldquoStructural robustness oflong-span cable-supported bridges in a cable-loss scenariordquoJournal of Bridge Engineering vol 23 no 2 2018

[20] M Shoghijavan and U Starossek ldquoAn analytical study on thebending moment acting on the girder of a long-span cable-

supported bridge suffering from cable failurerdquo EngineeringStructures vol 167 pp 166ndash174 2018

[21] M Domaneschi C Pellecchia E De Iuliis et al ldquoCollapseanalysis of the Polcevera viaduct by the applied elementmethodrdquo Engineering Structures vol 214 p 110659 2020

[22] O Olamigoke Structural Response of Cable-Stayed Bridges toCable Loss University of Surrey Guildford UK 2018

[23] J Xu H H Sun and S Y Cai ldquoEffect of symmetrical brokenwires damage on mechanical characteristics of stay cablerdquoJournal of Sound and Vibration vol 461 2019

14 Shock and Vibration

was within 20 of the arch rib bearing capability indicatingthat the failure of a single cable does not cause failure of thearch rib (e safety factor of the arch rib is greater than 50which reveals that the arch rib has sufficient robustnesssubjected to single cable breakage

6 Conclusions

Cable breaking is common damage in cable-supported archbridges as a result of aging and erosion caused by the harshenvironment To evaluate the dynamic response and robust-ness of cable-supported arch bridges subjected to cablebreaking a nonlinear dynamic simulation procedure and ro-bustness evaluation method are developed in this study Aprototype arch bridge with a single cable break was analyzed asan example Several critical factors for cable breakage simu-lation such as the cable break mode cable break duration anddamping are studied (e dynamic responses of the cablesystem deck and rib are evaluated and the DAF of differentcomponents of the cable-supported arch bridge is also dis-cussed Robustness evaluation indexes of the cable systemdeck and arch rib are calculated and their robustness levels areevaluated (e main conclusions are as follows

(1) Using an appropriate simulation method is essentialto the dynamic analysis of a cable breaking processDirectly removing the failed cable in an appropriatetime step of a transient analysis can effectivelysimulate the cable breaking effect When the cablebreak duration is sufficiently short the maximumimpact effect can be fully attained regardless of thetype of the break simulation mode used (e Ray-leigh damping coefficient calculated by a higher-order frequency can avoid the suppression effect of alocal vibration and better reflect the local vibrationcaused by a cable break

(2) (e closest cable to the broken cable receives thelargest impact effect (e impact effect of the cable

tension caused by the break of a short cable is greaterthan that of a long cable (e deck displacement andbending moment increase with the length of thefailure cable(e breaking of a downstream cable haslittle effect on the upstream cables

(3) (e DAF of each component is different(e DAF ofcable tension is between 14 and 16 while that of thedeck displacement is between 13 and 165 (e archrib bending moment has the largest DAF that isbetween 15 and 24 which is greater than the PTIrecommended value of 20 in some cases Using aunified DAF of 20 cannot satisfy all the componentsof cable-supported arch bridges

(4) (e deck system is the least robust component ofcable-supported arch bridges (e cable system has asafety factor of 31 under the cable breakage acci-dental combination a single cable break will nottrigger progressive damage (e arch rib has suffi-cient robustness subjected to a single cable breakagethough it has the largest DAF

(e results obtained by this analysis are conducive to thedesign and robustness evaluation of cable-supported archbridges Although only the breakage of a single cable wasdemonstrated the proposed methodology can be applied tomultiple cable breakage scenario of cable-supported bridgesFuture research can be carried out in different types of cable-supported arch bridges subjected to multiple cablebreakages

Data Availability

(e data that support the findings of this study are availablefrom the corresponding author upon reasonable request

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(e authors would like to acknowledge Professor RuinianJiang of NMSU for providing help during the preparation ofthe manuscript and the authors also gratefully acknowledgethe support of Taizhou Highway Administration (is re-search was funded by the Zhejiang Provincial Transportationand Transportation Office Technology Funding Project(2012H47)

References

[1] PTI Recommendations for Stay Cable Design Testing andInstallation Post-Tensioning Institute Farmington Hills MIUSA 2012

[2] T Zoli and R Woodward ldquoDesign of long span bridges forcable lossrdquo IABSE Symposium Report vol 90 no 9 pp 17ndash252005

[3] A M Ruiz-Teran and A C Aparicio ldquoResponse of under-deck cable-stayed bridges to the accidental breakage of stay

MLTMLBMT

MBMRTMRB

0 100 200 300 400 50013000

14000

15000

16000

17000

18000

Axi

al fo

rce (

kN)

Moments (kNmiddotm)

ndash50000

0

50000

100000

150000A

xial

forc

e (kN

)

0 10000 20000 30000ndash10000Moments (kNmiddotm)

Figure 23 Internal force change curves of the arch rib due to cableS30 break within its M-N curves

Shock and Vibration 13

cablesrdquo Engineering Structures vol 31 no 7 pp 1425ndash14342009

[4] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part I bending moment acting on the deckrdquo En-gineering Structures vol 32 no 10 pp 3288ndash3300 2010

[5] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part II bending moment acting on the pylons andstress on the staysrdquo Engineering Structures vol 32 no 10pp 3301ndash3312 2010

[6] Y Zhou and S Chen ldquoTime-progressive dynamic assessmentof abrupt cable-breakage events on cable-stayed bridgesrdquoJournal of Bridge Engineering vol 19 no 2 pp 159ndash171 2014

[7] Y Park U Starossek H-M Koh J F Choo H Kim andS W Lee ldquoEffect of cable loss in cable stayed bridgesmdashfocuson dynamic amplificationrdquo IABSE Symposium Report vol 93no 5 pp 9ndash16 2007

[8] J-g Cai Y-x Xu L-p Zhuang J Feng and J ZhangldquoComparison of various procedures for progressive collapseanalysis of cable-stayed bridgesrdquo Journal of Zhejiang Uni-versity Science A vol 13 no 5 pp 323ndash334 2012

[9] Y Xiang Z Chen Y Yang H Lin and S Zhu ldquoDynamicresponse analysis for submerged floating tunnel with anchor-cables subjected to sudden cable breakagerdquoMarine Structuresvol 59 pp 179ndash191 2018

[10] Y F Zhou and S R Chen ldquoFramework of nonlinear dynamicsimulation of long-span cable-stayed bridge and traffic systemsubjected to cable-loss incidentsrdquo Journal of Structural En-gineering vol 142 no 3 2016

[11] W L Qiu M Jiang and C L Huang ldquoParametric study onresponses of a self-anchored suspension bridge to suddenbreakage of a hangerrdquo Scientific World Journal vol 2014Article ID 512120 10 pages 2014

[12] W Qiu M Jiang and Z Zhang ldquoResponses of self-anchoredsuspension bridge to sudden breakage of hangersrdquo StructuralEngineering and Mechanics vol 50 no 2 pp 241ndash255 2014

[13] C M Mozos and A C Aparicio ldquoNumerical and experi-mental study on the interaction cable structure during thefailure of a stay in a cable stayed bridgerdquo EngineeringStructures vol 33 no 8 pp 2330ndash2341 2011

[14] V Hoang O Kiyomiya and T An ldquoExperimental and nu-merical study of lateral cable rupture in cable-stayed bridgescase studyrdquo Journal of Bridge Engineering vol 23 no 6 2018

[15] M Domaneschi G P Cimellaro and G Scutiero ldquoDispro-portionate collapse of a cable-stayed bridgerdquo Proceedings ofthe Institution of Civil EngineersmdashBridge Engineering vol 172no 1 pp 13ndash26 2019

[16] M Wolff and U Starossek ldquoCable loss and progressivecollapse in cable-stayed bridgesrdquo Bridge Structures vol 5no 1 pp 17ndash28 2009

[17] G Wu W Qiu and T Wu ldquoNonlinear dynamic analysis ofthe self-anchored suspension bridge subjected to suddenbreakage of a hangerrdquo Engineering Failure Analysis vol 97pp 701ndash717 2019

[18] M Morgese F Ansari M Domaneschi and G P CimellaroldquoPost-collapse analysis of morandirsquos polcevera viaduct ingenoa Italyrdquo Journal of Civil Structural Health Monitoringvol 10 no 1 pp 69ndash85 2020

[19] M Shoghijavan and U Starossek ldquoStructural robustness oflong-span cable-supported bridges in a cable-loss scenariordquoJournal of Bridge Engineering vol 23 no 2 2018

[20] M Shoghijavan and U Starossek ldquoAn analytical study on thebending moment acting on the girder of a long-span cable-

supported bridge suffering from cable failurerdquo EngineeringStructures vol 167 pp 166ndash174 2018

[21] M Domaneschi C Pellecchia E De Iuliis et al ldquoCollapseanalysis of the Polcevera viaduct by the applied elementmethodrdquo Engineering Structures vol 214 p 110659 2020

[22] O Olamigoke Structural Response of Cable-Stayed Bridges toCable Loss University of Surrey Guildford UK 2018

[23] J Xu H H Sun and S Y Cai ldquoEffect of symmetrical brokenwires damage on mechanical characteristics of stay cablerdquoJournal of Sound and Vibration vol 461 2019

14 Shock and Vibration

cablesrdquo Engineering Structures vol 31 no 7 pp 1425ndash14342009

[4] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part I bending moment acting on the deckrdquo En-gineering Structures vol 32 no 10 pp 3288ndash3300 2010

[5] C M Mozos and A C Aparicio ldquoParametric study on thedynamic response of cable stayed bridges to the sudden failureof a stay Part II bending moment acting on the pylons andstress on the staysrdquo Engineering Structures vol 32 no 10pp 3301ndash3312 2010

[6] Y Zhou and S Chen ldquoTime-progressive dynamic assessmentof abrupt cable-breakage events on cable-stayed bridgesrdquoJournal of Bridge Engineering vol 19 no 2 pp 159ndash171 2014

[7] Y Park U Starossek H-M Koh J F Choo H Kim andS W Lee ldquoEffect of cable loss in cable stayed bridgesmdashfocuson dynamic amplificationrdquo IABSE Symposium Report vol 93no 5 pp 9ndash16 2007

[8] J-g Cai Y-x Xu L-p Zhuang J Feng and J ZhangldquoComparison of various procedures for progressive collapseanalysis of cable-stayed bridgesrdquo Journal of Zhejiang Uni-versity Science A vol 13 no 5 pp 323ndash334 2012

[9] Y Xiang Z Chen Y Yang H Lin and S Zhu ldquoDynamicresponse analysis for submerged floating tunnel with anchor-cables subjected to sudden cable breakagerdquoMarine Structuresvol 59 pp 179ndash191 2018

[10] Y F Zhou and S R Chen ldquoFramework of nonlinear dynamicsimulation of long-span cable-stayed bridge and traffic systemsubjected to cable-loss incidentsrdquo Journal of Structural En-gineering vol 142 no 3 2016

[11] W L Qiu M Jiang and C L Huang ldquoParametric study onresponses of a self-anchored suspension bridge to suddenbreakage of a hangerrdquo Scientific World Journal vol 2014Article ID 512120 10 pages 2014

[12] W Qiu M Jiang and Z Zhang ldquoResponses of self-anchoredsuspension bridge to sudden breakage of hangersrdquo StructuralEngineering and Mechanics vol 50 no 2 pp 241ndash255 2014

[13] C M Mozos and A C Aparicio ldquoNumerical and experi-mental study on the interaction cable structure during thefailure of a stay in a cable stayed bridgerdquo EngineeringStructures vol 33 no 8 pp 2330ndash2341 2011

[14] V Hoang O Kiyomiya and T An ldquoExperimental and nu-merical study of lateral cable rupture in cable-stayed bridgescase studyrdquo Journal of Bridge Engineering vol 23 no 6 2018

[15] M Domaneschi G P Cimellaro and G Scutiero ldquoDispro-portionate collapse of a cable-stayed bridgerdquo Proceedings ofthe Institution of Civil EngineersmdashBridge Engineering vol 172no 1 pp 13ndash26 2019

[16] M Wolff and U Starossek ldquoCable loss and progressivecollapse in cable-stayed bridgesrdquo Bridge Structures vol 5no 1 pp 17ndash28 2009

[17] G Wu W Qiu and T Wu ldquoNonlinear dynamic analysis ofthe self-anchored suspension bridge subjected to suddenbreakage of a hangerrdquo Engineering Failure Analysis vol 97pp 701ndash717 2019

[18] M Morgese F Ansari M Domaneschi and G P CimellaroldquoPost-collapse analysis of morandirsquos polcevera viaduct ingenoa Italyrdquo Journal of Civil Structural Health Monitoringvol 10 no 1 pp 69ndash85 2020

[19] M Shoghijavan and U Starossek ldquoStructural robustness oflong-span cable-supported bridges in a cable-loss scenariordquoJournal of Bridge Engineering vol 23 no 2 2018

[20] M Shoghijavan and U Starossek ldquoAn analytical study on thebending moment acting on the girder of a long-span cable-

supported bridge suffering from cable failurerdquo EngineeringStructures vol 167 pp 166ndash174 2018

[21] M Domaneschi C Pellecchia E De Iuliis et al ldquoCollapseanalysis of the Polcevera viaduct by the applied elementmethodrdquo Engineering Structures vol 214 p 110659 2020

[22] O Olamigoke Structural Response of Cable-Stayed Bridges toCable Loss University of Surrey Guildford UK 2018

[23] J Xu H H Sun and S Y Cai ldquoEffect of symmetrical brokenwires damage on mechanical characteristics of stay cablerdquoJournal of Sound and Vibration vol 461 2019

14 Shock and Vibration