a study on potential progressive collapse responses of

1. INTRODUCTIONStays of cable-stayed bridges are critical structuralelements which are subjected to corrosion, abrasion,wind, vehicle impact and malicious actions and theseextreme scenarios may lead to severe damage and lossof cable(s) (Walther 1999; Åkesson 2008; Yang et al.2011). Such cable loss scenarios would lead to highimpulsive dynamic loads in the structure (Starossek2011) that can potentially trigger a “zipper-type”progressive collapse of the entire bridge. Accordingly,cable-stayed bridges should be designed for potentialcable loss scenarios as recommended by someguidelines (PTI 2007).

The progressive collapse of structures triggered byextreme events typically has a dynamic nature, however,in practice for progressive collapse assessment ofstructures a static analysis is considered to be adequateprovided a dynamic amplification factor (DAF) isapplied. The value of DAF widely adopted by differentstandards and guidelines varies between 1.5 to 2 (PTI

Advances in Structural Engineering Vol. 16 No. 4 2013 689

A Study on Potential Progressive Collapse Responses

of Cable-Stayed Bridges

Yukari Aoki1,*, Hamid Valipour2, Bijan Samali1 and Ali Saleh1

1School of Civil and Environmental, Centre for Built Infrastructure Research, University of Technology, Sydney, Australia2School of Civil and Environmental, Centre for Infrastructure Engineering and Safety, The University of New South Wales, Sydney

(Received: 24 February 2012; Received revised form: 26 September 2012; Accepted: 17 October 2012)

Abstract: In this paper, a finite element (FE) model for a cable-stayed bridge designedaccording to Australian standards is developed and analysed statically anddynamically with and without geometrical nonlinearities. The dynamic amplificationfactor (DAF) and demand-to-capacity ratio (DCR) in different structural componentsincluding cables, towers and the deck are calculated and it is shown that DCR usuallyremains below one (no material nonlinearity occurs) in the scenarios studied for thebridge under investigation, however, DAF can take values larger than two. Moreover,effects of location, duration and number of cable(s) loss as well as effect of dampinglevel on the progressive collapse resistance of the bridge are studied and importanceof each factor on the potential progressive collapse response of the bridgeisinvestigated.

Key words: cable-stayed bridge, DAF, demand-to-capacity ratio, structural damping.

2007; ARUP 2010). However, application of a DAF = 2(obtained for a single degree of freedom system) hasbeen questioned by some researchers (Ruiz-Teran andAparicio 2007; Wolff and Starossek 2009; Mozos andAparicio 2010). Wolff and Staroussek (2010) found thatapplying a single DAF for different structuralcomponents in a cable stayed structure cannotadequately capture the dynamic effects, because thevalue of DAF for each structural component depends onlocation of the ruptured cable and type of the statevariable (i.e. deflection, shear force, bending moment)under consideration. Mozos and Aparicio (2010) did aparametric study on the response of a series ofhypothetical cable stayed bridges with different deck sizeand two different types of cable configuration (i.e. fanand harp) and they found that fan type and stiffer decklead to a larger DAF for deck and pylons.

At the global level, most of the research on thepotential progressive collapse response of cable stayedbridges take advantage of linear elastic FE models to

*Corresponding author. Email address: [email protected]; Fax: +61-2-9514-7803; Tel: +61-2-9514-8337.Associate Editor: S.S. Law.

determine the DAF for different structural componentsand investigate the contributing factors (such as bridgegeometry, location of ruptured cable, load cases,duration of cable loss and damping ratio) on themagnitude of DAF (Zoli and Woodward 2005; Ruiz-Teran and Aparicio 2007; Wolff and Starossek 2009;Mozos and Aparicio 2010; Wolff and Starossek 2010;Starossek 2011). On the other hand, some researchershave used the sophisticated continuum-based FE modelsand hydro-codes for investigating the local behaviour ofcable stayed bridges subjected to abnormal loads such asblast and fire (Hao and Tang 2010; Tang and Hao 2010;Son and Lee 2011; Wang 2011). The nonlinear FEmodels can provide valuable insight into localbehaviour of structural elements; however, applicationof such models for collapse analysis of a cable stayedbridge can be computationally expensive and timeconsuming. Accordingly, in this study the demand-to-capacity ratio (DCR) is introduced as the ratio ofexisting stress to the yield stress of the materials DCRvalues are employed to determine whether the materialnonlinearities have occurred or still the structuralcomponents are within their elastic range(Saadeghvaziri and Yazdani-Motlagh 2008). The DCRindices calculated for cables, deck and pylons provide arational basis to determine whether considering materialnonlinearity at global level is required or it can beignored.

In this paper a two-dimensional FE model of a cablestayed bridge with a steel box deck, designed accordingto Australian Standards AS5100.6 (2004) is developed.Using the developed FE model, a comprehensiveparametric study is undertaken and the effects oflocation, duration and number of lost cables, as well asapplied load case and the structural damping ratio on theDAF and DCR values are investigated. Moreover, thevalue of DAF and its distribution within differentstructural elements is determined and the correlationbetween DCR and DAF values for the cases underconsideration is discussed. The developed FE modelsare analysed with and without geometricalnonlinearities to demonstrate the effect of geometricalnonlinearities on the global response of the bridgeduring progressive collapse analysis.

2. DYNAMIC AMPLIFICATION FACTOR(DAF ) AND DEMAND-TO-CAPACITYRATIO (DCR )

The focus of this paper is on DAF and DCR valueswhich are calculated for the most critical sections in thedeck, pylons and cables.

The dynamic amplification factor (DAF) adopted byWolff and Starossek (2009) is defined as

(1)

and the equivalent DAF (Zoli and Woodward 2005) dueto sudden loss of cable(s) is calculated from

(2)

where Sdyn(t) is the maximum/minimum value of theresponse at instant time t of the dynamic response,Shealthy is the response obtained from the static analysisof the healthy bridge [Figure 1(a)] and SF0 and SF1 arethe responses obtained from the static analysis of caseF0 and F1 shown in Figures 1(b) and 1(c), respectively.

Before removing the cable(s), the healthy bridge isanalysed statically and the internal force of the members,Shealthy, are obtained from this analysis [Figure 1(a)]. In thenext stage the cable is removed [Figure 1(b)] and then theinternal force of the removed cable with opposite sign,Finit, is applied on the deck and tower [Figure 1(c)] asrecommended by Zoli and Woodward (2005) and PTI(2007). The SF1 values are taken from the results of thisrecent analysis [Figure 1(c)].

In this study, demand-to-capacity ratio (DCR) isdefined as

(3)

where σ is the existing stress from dynamic analysis andσy is the yield stress.

The existing stress σ at the most critical section iscalculated from

(4)

where N and M are the axial force and bending moment,respectively, and y is distance from the neutral axis. A isthe cross-sectional area and I is the moment of inertia ofthe section. The location of critical sections, wheremaximum σ occurs, is determined from the envelope ofdynamic analysis and DCR values are always calculatedfor combination of axial force and bending moment(obtained from the envelope of dynamic analysis) andtypically the maximum DCR values are calculated at themost critical sections where either maximum bendingmoment or maximum axial force has occurred. Thematerial is assumed to be elastic-perfect plastic (seeFigure 2) and the DCR values greater than 100% is theindicator of material yielding (material nonlinearity).

σ = ± +N

A

M y

I

DCRy

= ×σσ

100

Equivalent DAFS S

S Sdyn t healthy

F healthy=

−−

( )

1

DAFS S

S Sdyn t healthy

F healthy=

−−

( )

0

690 Advances in Structural Engineering Vol. 16 No. 4 2013

A Study on Potential Progressive Collapse Responses of Cable-Stayed Bridges

3. DESCRIPTION OF MATERIALS,GEOMETRY AND LOADS

3.1. Material Properties and Geometry of the

Bridge

The dimensions of the bridge and geometry of sectionsas well as the configuration of cables for the bridgeconsidered in this paper are shown in Figure 3. Thebridge is symmetrical and has a main-span length of

600 m supported by two 140 m tall towers and 60 pairsof cables. All cables, except the first and the last fourback stays (i.e. cables no. 1–4 and 57–60), are regularlyspaced (20 m apart) along the deck (Figure 3a).Regarding the back stays no. 1–4 and 57–60 [see Figure3(b)], a 2.5 m spacing along the deck was considered.Further, all cables are regularly spaced (4 m apart) overthe pylons height over a distance of 56 m from the topof the pylon. The bridge deck is 25.6 m wide (8 trafficlanes according to AS5100.2 (2004) and 2 m deep andmade of a multi-cell steel box girder which is depictedin Figure 3(c). This bridge deck is restrained by a pinsupport at the far left end and by a roller support at thefar right end [Figure 3(a)], and there is no directconnection between the deck and the pylons. The crosssection of pylons of each leg which is fully fixed at theground level is shown in Figure 3(d).

The modulus of elasticity, E, and the yield stress ofsteel, σy, as well as the geometrical properties of thedeck including second moment of area, I, and crosssectional area , A, are given in Table 1. Three differentsizes of stays are used as given in Table 1. Further, theaxial force and bending moment corresponding to thefirst yield of material for the cables, deck and towers arecalculated and summarized in Table 1.


Yukari Aoki, Hamid Valipour, Bijan Samali and Ali Saleh

(a) Healthy structure

(b) Case F0

(c) Case F1

Finit

Figure 1. Different cases considered in conjunction with DAF or equivalent DAF

500

400

300

200

100

00 0.02 0.04 0.06

Strain

Str

ess

(Mpa

)

0.08 0.1 0.12

Figure 2. Adopted elastic-perfect plastic stress-strain relationship

for steel



No. 15No. 30 No. 31

No. 60

120

m

140

m

No. 1

A

B

C D

DCBE E10 m

20 m10 mA

10@ 20 m = 200 m 14@ 20 m = 280 m 14@ 20 m = 280 m

14@

4 m

= 5

6 m

230 m 600 m

2@ 2.5 m5 m

10 m

8 lanes × 3.2 m = 25.6

0.02 m

8 m

0.06 m

6.5

m

0.06

m

0.45 m

0.02

5 m

2 m

(a) Bridge configuration and location of critical sections

(b) Cable spacing at the support

(c) Cross section of the deck

(d) Cross section of the tower

230 m

Figure 3. Bridge elevation and principal dimensions

3.2. Design Loads

The bridge considered in this paper has been designed forgravity loads (i.e. permanent and traffic loads) as well asthe post-tensioning forces determined according toAustralian Standard AS5100.2 (2004). The permanentload includes the self-weight of the structural elements aswell as the wearing surface of the road [see Figure 3(c)].The adopted traffic load is S1600 stationary traffic loadwhich contains a uniform distributed load and heavytruck loads as defined in AS5100.2 (2004) and shown inFigure 4. The post-tensioning force in the cables wascalculated based on serviceability design requirementsand maximum mid-span deflection due to traffic loadwhich is allowed by AS5100.2 (2004). Further, the post-tensioning forces in the cables were adjusted according to‘zero displacement method’ to achieve the desired profilefor the bridge deck (Wang et al. 1993).

In design, it was assumed that local buckling wouldnot occur and the ultimate loading capacity of membersis governed by yielding of the material according toAS5100.6 (2004) requirements. Furthermore, themaximum stress induced in the deck and towers due toservice load of (permanent action + S1600 traffic load)plus the post-tensioning forces is always less than 0.5 σy

and the maximum stress in the cables is limited to0.45σy which is consistent with design requirementsaccording to JSCE (1990).

3.3. Cable Loss Scenarios

With regard to existing guidelines (PTI 2007) forprogressive collapse design of cable stayed bridges, onlyconsidering the scenarios associated with loss of a singlecable is adequate, however, some researchers believe thatscenarios in which more than one cable is lost should notbe ignored (Wolff and Starossek 2010). Accordingly, inthis study, three different scenarios corresponding to lossof only one cable and simultaneous loss of two and threecables are considered. In the first scenario (only one cableis lost) four different cases which are potentially the mostcritical ones are investigated (Wolff and Starossek 2010;Aoki et al. 2011). In case-1 and -4 the loss of cable no. 1and 60, respectively, connected to the end supports[Figure 3(a)] are studied, in the case-2 loss of the shortestcable adjacent to the left tower (cable no. 15) isconsidered and in case-3 longest cable connected to mid-span (cable no. 30) is lost. In the second scenario (loss ofa pair of cables) two different cases are analysed. In case-1, the longest cables connected to the pin support (cables



Table 1. Material and geometrical properties of the deck, towers and cables

E A I σy Yield moment Yield force

(GPa) (m2) (m4) (MPa) My* (MN.m) Ny* (MN)

Girder 200 1.43 1.21 450 470 363.9Tower 200 4.75 45.7 450 1,457

Cable No. 1–5, 26–35, 56–60 200 0.0327 1,860 60.86–10, 21–25, 36–40, 51–55 200 0.0183 1,860 34.0

11–20, 41–50 200 0.0165 1,860 30.7

*My and Ny is the value corresponds to the first yield of material by absence of interaction between axial force and moment.

3.2 m standarddesign

240 kN

1.25 1.25 3.75

0.4 0.6

0.2 2.0

5.0 0.6Varies 6.25 min.1.25 1.25 1.25 1.25 1.25 1.25

240 kN 240 kN24 kN/m

Elevation

Plan

240 kN

Figure 4. S1600 stationary traffic load according to AS5100.2 (2004)

no. 1 and 2) are lost, and loss of the cables connected tothe mid span (cable no. 30 and 31) is considered in case-2. In the third scenario only one case is considered inwhich the three cables connected to the pin support(cables no. 1, 2 and 3) are lost. The scenarios consideredin conjunction with different loading pattern are listed inTable 2. LC-1 and LC-2 in this table are different loadcases which are explained in section 3.5.

3.4. Finite Element Model

A 2D finite element model of the bridge is developed inANSYS (2009). The structural elements are modelledby frame elements and the cables are treated as onlytension members with limited tensile capacity. Thematerial is assumed to be linear elastic-perfect plastic(see Figure 2), however, effects of large displacementsand geometrical nonlinearities are taken into account byactivating the required solution algorithm of theANSYS software (2009). The demand-to-capacity ratio

(DCR) calculated at the most critical cross sections ofthe deck, pylons and cables are employed to verify theadequacy of the adopt linear elastic material behaviour[see Figure 3(a)].

3.5. Load Combinations Adopted for

Progressive Collapse Assessment

In conjunction with cable loss scenarios, two differentload combinations are considered as shown in Figure 5.In load combination LC-1 [Figure 5(a)] the distributedcomponent of the traffic load along with dead load isapplied over the entire bridge length whereas in LC-2[Figure 5(b)] the distributed component of the trafficload as well as the heavy truck loads are only placed onthe middle span and the back spans are only underpermanent action (dead load). The preliminaryinvestigations based on influence line theory showedthat LC-2 is more critical than LC-1.

For the cable loss scenarios, adopted load factors areas recommended by Post-tensioning Institute (PTI 2007),

(4)

where, DC = dead load of structural components andnon-structural attachments, DW = dead load of wearingsurfaces and utilities, LL = full vehicular live loadplaced in actual stripped lanes, IM = vehicular dynamicload allowance taken equal to zero in this paper andCLDF = cable loss dynamic forces. In the exampleconsidered in this paper, the exact value of cableinternal forces are available, and accordingly the loadfactor 1.1 on the cable loss dynamic forces that accountsfor a variation of final cable force in construction hasbeen ignored.

1 1 1 35 0 75 1 1. . . ( ) .DC DW LL IM CLDF+ + + +



Table 2. Scenarios considered in this study

Scenario Lost cable Load case

name no. (Figure 5)

scenario -1/LC-1, case-1 1 LC-1scenario -1/LC-1, case-2 15scenario -1/LC-1, case-3 30scenario -1/LC-1, case-4 60

scenario -1/LC-2, case-1 1 LC-2scenario -1/LC-2, case-2 30

scenario -2/LC-2, case-1 1&2scenario -2/LC-2, case-2 30 & 31

scenario -3/LC-2, case-1 1 & 2 & 3

Dead + live load

Dead + live load

Dead load

Truck loads

(a) LC-1

(b) LC-2

Dead load

Figure 5. Gravity load cases considered for progressive collapse analysis

3.6. Cable Removal Method and Type of

Analysis

In progressive collapse assessment of structures/bridgesbased on alternate load path (ALP), the time over whichthe critical members (i.e. column or cable) are removedcan significantly affect the response of the structures. Ifthe member is removed over a longer period of time(typically longer than the first natural period ofstructure) a static analysis will be adequate, however,removing the members (i.e. columns or cables) over ashorter period of time warrants a dynamic analysis. Foranalysis of cable stayed bridges against progressivecollapse, the cables can be removed through a static ordynamic procedure and the FE model should beanalysed accordingly. In this paper, for dynamicanalysis a consistent mass matrix with a proportionaldamping is adopted (Bathe 1996) and Newmarkconstant acceleration method which is unconditionallystable is used for time integration.

4. PARAMETRIC STUDIES ANDDISCUSSIONS

A parametric study is undertaken and parametersinfluencing DAF and DCR such as the time over whichthe cable is removed, damping ratio and geometrical

nonlinearities (large displacements) are studied.Moreover, different cable loss scenarios are examinedand the most critical ones which lead to the maximumDAF and DCR are identified.

4.1. Time Step over which the Cable is

Removed (Cable Removal Time Step)

For dynamic simulation of a cable loss scenario, thebridge without the lost cable is modelled first and theinitial force in the lost cable, Finit, is gradually applied onthe tower and deck [see Figure 6(a)], then bridge isallowed to reach the equilibrium state which is basicallyequivalent to the healthy bridge. After the bridge reachesthe equilibrium state, the cable loss scenario is simulatedby reducing the force, Finit, down to zero over a time stepof ∆tf [see Figure 6(b)]. Alternatively, the cable lossscenario can be achieved by activating a special solutionprocedure in ANSYS by applying the “EKILL”command. It is noteworthy that the EKILL commanddeactivates the lost cable element over an integration timestep (∆t = 0.01 sec) whereby the element contributes anear-zero stiffness value to the overall matrix and nothingto the overall mass matrix. Any solution-dependent statevariables, such as the corresponding cable forces, are setto zero at the end of the time step.



Finit

(a) Cable loss scenario considered fo sensitivity analysis with respect to removal time step ∆tf

Steady state

Loading time (t)

0.1 sec

1 sec

For

ce

8 sec

0 5 10 15 20

Unloading duration (0.1 sec to 8 sec)

*Ekill (cable removed within 0.01 sec)

0 5 10 15 20

Fo

rce

8 sec 1 sec

0.1 sec

*Ekill (cable removed within 0.01 sec)

Loading time (t)

Steady state Unloading duration (0.1 sec to 8 sec)

(b) Loading curve

Time (sec)

Figure 6. Load direction and curve for dynamic analysis of cable removal

Since the time step over which the cable is removed,∆tf, can affect the dynamic response of the bridge, in thispart of the parametric study sensitivity of dynamicresponse in relation to ∆tf is examined.

Four different cases corresponding to removal ofcable no. 30 [see Figure 6(a)] over ∆tf = 8 sec, 1 sec,0.1 sec and application of EKILL command areconsidered [Figure 6(b)]. The time history of bendingmoment in scenario-1, case-3 at section C-C [Figure3(a)] for four different cases (i.e. ∆tf = 8 sec, 1 sec, 0.1sec and EKILL) are shown in Figure 7 and it isobserved that dynamic responses (particularly themaximum and minimum value of the responses)provided by different ∆tf, more or less have the sameaccuracy except for the case in which the cable isremoved over 8 sec. The maximum DCR andcorresponding DAF values calculated for one, two andthree cable loss scenarios are given in Tables 3 to 5,respectively. With regard to Tables 3 to 5, it isobserved that DAF and DCR values are quite sensitiveto ∆tf (particularly when two cables are lost) and foraccurate estimation of DAF and DCR a fairly small ∆tf

(about 0.001-0.1 sec) is required, which is consistentwith the recommendations of some guidelines forprogressive collapse analysis of framed structures(GSA 2003; DoD 2005). It should be noted thatremoving the cable over 8 sec or 1 sec in this examplecannot adequately simulate the sudden loss of cableand such a slow removal of cables produces a quasi-static scenario rather than a dynamic one.

4.2. Structural Damping

Mozos and Aparisio (2010) identified the structuraldamping ratio as one of the factors that can influencethe progressive collapse response of the bridge andDAF values, however, they do not provide any detailsin this regard. Accordingly, the importance of dampingratio and its impact on DAF and DCR values areinvestigated in this section. For parametric study, threedifferent ratios of critical damping, i.e. 0.5%, 1% and2% within the acceptable range for steel structures areadopted (Clough and Penzien 1993). For the triplecable loss scenario-3, the time history of bendingmoment at Section A-A [refer to Figure 3(a)] for threedifferent levels of critical damping ratio (i.e. 0.5%, 1%and 2%) are shown in Figure 8 and it is observable thatthe time history and the maximum/minimum valuesare very similar for all adopted damping ratios.Further, the maximum DCR and corresponding DAFvalues calculated for the scenarios with three cablelosses are given in Table 5. The results provided inTable 5 clearly show that DAF and DCR values are notsensitive to the level of damping ratio adopted fordynamic analysis. Further, it can be concluded that thedamping ratio within the range considered (i.e. 0–2%),has a minor impact on the potential progressivecollapse response of the cable stayed bridges.

4.3. Geometrical Nonlinearities

Effect of geometrical nonlinearity should be takeninto account when the deflections of the structure are



–1000 5 10 15 20 25 30

–120

–140

–160

–180

Ekill

F1

F0

0.1 s

1s

8sHealthy (–136 MN.m)

–200

Mom

ent (

MN

.m)

Time (sec)

Figure 7. Time history of bending moment in scenario-1, case-3 at section C-C of the deck after removing cable no. 30



Table 3. Maximum DCR and corresponding DAF values forscenario-1/LC-1 in which one cable is lost(Gravity

load case 1 is applied and critical damping ratio is taken as 0.5%)

Geometrical DAF**

Lost cables no. Duration ∆tf (sec) nonlinearity (equivalent DAF) DCR*

1 8 On 1.1 (0.5) 40%(scenario-1/LC-1, Off 1.1 (0.5) 40%case-1, Section A-A) 1 On 1.7 (0.8) 46%

Off 3.8 (1.8) 46%0.1 On 1.8 (0.8) 48%

Off 7.9 (3.8) 48%Ekill On 1.8 (0.8) 48%

15 8 On 1.7 (0.8) 28%(scenario-1/LC-1, case-2, Off 2.4 (1.0) 28%Section B-B) 1 On 9.4 (4.1) 28%

Off 9.7 (4.3) 28%0.1 On 60 (26) 29%

Off 78 (35) 29%Ekill On 83 (36) 30%

30 8 On 1.1 (0.6) 27%(scenario-1/LC-1, Off 1.0 (0.5) 26%case-3, Section C-C ) 1 On 2.4 (1.2) 30%

Off 2.0 (1.1) 29%0.1 On 2.7 (1.3) 31%

Off 2.3 (1.4) 31%Ekill On 2.6 (1.3) 31%

60 8 On 1.1 (0.6) 34%(scenario-1/LC-1, Off 1.1 (0.5) 33%case-4, Section D-D ) 1 On 1.6 (0.8) 39%

Off 1.6 (0.7) 37%0.1 On 1.6 (0.9) 40%

Off 1.6 (0.7) 37%Ekill On 1.7 (0.9) 40%

*DCR values are calculated based on combination of axial force and bending moment at different critical sections (with maximum bending moment) along thedeck [see Figure 3(a) for location of critical sections].**DAF values were calculated based on maximum bending moment.

Table 4. Maximum DCR and corresponding DAF values forscenario-2/LC-2 in which two cables are lost

(Gravity load case 2 is applied and critical damping ratio is taken as 0.5%)

Geometrical DAF**


1&2 8 On 1.2 (0.6) 69%(scenario-2/LC-2, Off 1.1 (0.5) 64%case-1, Section A-A) 1 On 2.7 (1.3) 84%

Off 14 (7.1) 74%0.1 On 5.1 (2.4) 90%

Off 4.0 (2.0) 83%Ekill On 5.7 (2.7) 91%

30&31 8 On 1.9 (0.7) 45%(scenario-2/LC-2, Off 1.0 (1.0) 44%case-2, Section C-C) 1 On 20# (7.5) 60%

Off 1.1 (1.1) 59%0.1 On 35# (13) 64%

Off 1.3 (1.3) 64%Ekill On 36# (14) 63%

*DCR values are calculated based on combination of axial force and bending moment at different critical sections (with maximum bending moment) along thedeck [see Figure 3(a) for location of critical sections].**DAF values were calculated based on maximum bending moment except the ones with # right superscript.#DAF value was calculated based on maximum axial force.

large enough compared with the size of structuralmembers. Furthermore, for compressive structuralmembers such as columns or pylons the second order

P–∆ effects can reduce the stiffness and loadingcapacity of the members, particularly in cable stayedbridges the P–∆ effects can be quite significant for



Table 5. Maximum DCR and corresponding DAF values for scenario-3/LC-2 in which three cables are lost

(Gravity load case 2 is applied)

Geometrical DAF**

Lost cables no. Duration ∆tf (sec) nonlinearity Damping (equivalent DAF) DCR*

1&2&3 8 On 0.5% 1.2 (0.7) 81%(scenario-3/LC-2, Off 1.1 (0.6) 78%case-1, 1 On 1.9 (1.1) 108%Section A-A) Off 1.8 (0.9) 104%

0.1 On 3.1 (1.4) 117%Off 2.4 (1.2) 111%

Ekill On 3.2 (1.5) 120%

8 On 1.0% 1.2 (0.7) 81%Off 1.1 (0.6) 78%

1 On 1.9 (1.1) 107%Off 1.8 (0.9) 104%

0.1 On 2.5 (1.2) 114%Off 1.9 (1.0) 109%

Ekill On 2.7 (1.2) 116%

8 On 2.0% 1.2 (0.7) 80%Off 1.1 (0.6) 78%

1 On 1.9 (1.1) 106%Off 1.8 (0.9) 102%

0.1 On 2.0 (1.2) 110%Off 1.9 (1.0) 106%

Ekill On 2.0 (1.2) 112%

*DCR values are calculated based on combination of axial force and bending moment at different critical sections (with maximum bending moment) along thedeck [see Figure 3(a) for location of critical sections].**DAF values were calculated based on maximum bending moment within the deck.

00 5 10

Healthy (–150MN.m)

2% (–473 MN.m)

1% (–487 MN.m)

0.5% (–499 MN.m)

15 20 25 30

–100

–200

–300

–400

–500

–600

Mom

ent (

MN

.m)

Time (sec)

Figure 8. Time history of the bending moment in scenario 3 at section A-A of the deck obtained from the dynamic analysis with different

critical damping ratios

pylons/towers subjected to lateral air blast pressure(Son and Lee 2011). The general understanding is thatduring cable loss scenarios, geometrical nonlinearitiescan affect the response of the cable stayed bridges andthey should be taken into account (Mozos andAparicio 2010; Wolff and Starossek 2010), however,there is no comprehensive study to clearlydemonstrate the contribution of the geometricalnonlinearities in DAF values and the progressivecollapse response of the cable stayed bridges. Themaximum DAF and DCR values calculated fordifferent cable loss scenarios with and withoutgeometrical nonlinearities included are summarised inTables 3 to 5. It is observed that for the bridge underconsideration, the effect of geometrical nonlinearitieson DAF is considerable and the major contribution ofgeometrical nonlinearity in DAF comes from thedynamic analysis Sdyn(t); in particular, inclusion ofgeometrical nonlinearities for scenario-1, case-1 inwhich cable No. 1 is lost has led to a 200% to 400%increase in DAF. Among different cable loss scenariosinvestigated in this study, the scenarios involved withloss of cable No. 1 (back stay connected to the hingesupport) are the most critical ones in terms ofsignificance of geometrical nonlinearity. In addition,with regard to Tables 3 to 5 it is concluded thatmaximum and minimum DCR values calculated fortowers, deck and cables are not really sensitive to

geometrical nonlinearity. In other words, geometricalnonlinearities have a minor role (for this example lessthan 7%) in driving the structural members towardstheir ultimate state during cable loss scenarios.

4.4. Cable Removal Scenarios

In this part, dynamic progressive collapse response ofthe bridge due to different cable loss scenarios isinvestigated. The configuration of gravity load casesapplied on the deck is shown in Figure 5 and a constantcritical damping ratio of 0.5% is adopted for dynamicanalyses.

4.4.1. Single cable loss

For the sake of comparison, in the single cable lossscenarios two different gravity load cases (i.e. LC-1 andLC-2) are considered (Figure 5).

(a) LC-1Figure 9 shows the time history of the bending

moment at section E-E [see Figure 3(a)] for differentcases within scenario-1/LC-1, in which cables no. 1,15, 30 and 60 are removed. It is observed that loss ofcables no. 1, 30 and 60 can significantly affect themagnitude of bending moment at the bottom of lefttower, whereas loss of cable no.15 has minor effects.Among four different cases considered underscenario-1/LC-1, loss of cable no.1 (the longest backstays) is the most critical in terms of maximum



-800

-600

-400

-200

0

200

4000 5 10 15 20 25 30

Mom

ent (

MN

.m)

Time (sec)

case-1 (no.1)

56MN.m (healthy)

case-3 (no.30) case-4 (no.60)

case-2 (no.15)

Figure 9. Time history of bending moment at the bottom of the left tower for different cable loss cases under scenario-1/LC-1 (only one

cable is lost)

dynamic bending moment induced in the bottom ofthe towers. For scenario-1/LC-1, the value of DAF andDCR calculated based on maximum bending momentat the most critical sections along the deck [see Figure3(a)] are given in Table 3. Furthermore, the maximumDCR values and corresponding DAF observed in thedeck, towers and cables for scenario-1/LC-1 are givenin Table 6. With regard to Table 6, it can be concludedthat DAF can take values higher than 2.0, however theDCR values are still well below 100% (no yielding hasoccurred). Further, for all structural components, it isconcluded that loss of cables no. 1 and 30 are morecritical than the other cases under scenario-1/LC-1

and accordingly for scenario-1/LC-2 only loss of thesecables are considered.

(b) LC-2The time history of the bending moment at the

bottom of the left tower for different cases withinscenario-1/LC-2 is given in Figure 10 that clearlyshows that loss of cable no. 1 is more critical thanother cases in terms of maximum dynamic bendingmoment induced in the bottom of the towers. Forscenario-1/LC-2, the value of DAF and DCRcalculated at the most critical sections along the deck(section with maximum bending moment) are given inTable 7. Furthermore, the maximum DCR values andcorresponding DAF observed in the deck, towers andcables for scenario-1/LC-2 are given in Table 8 thatshows DAF can take values higher than 2.0, howeverthe DCR values are still well below 100% (no yieldinghas occurred).

4.4.2. Loss of two cables

Among different possible scenarios in which twocables are simultaneously lost, the most critical onesare related to simultaneous loss of cables no. 1 and 2(case-1) and cables no. 30 and 31 (caes-2) which areinvestigated in this part of the parametric studies. Thetime history of bending moment at section A-A forcase-1 and -2 within scenario-2 are shown in Figure 11. It is observable that scenario-2/case-1corresponding to loss of two back stays (cables no. 1and 2) is the most critical one in terms of maximumdynamic bending moment induced in the deck and,therefore, the maximum DCR values andcorresponding DAF observed at different locations inthe deck, towers and cables for scenario-2/case-1 aregiven in Figure 12. Moreover, for scenario-2/LC-2,DAF and DCR values determined at the most criticalsections (based on maximum bending moment) alongthe deck are given in Table 4. With regard to theresults obtained for scenario-2, it is concluded thatalike scenario-1, DAF can take values higher than 2.0,however the DCR values are still below 100% (noyielding has occurred) and accordingly no potentialprogressive collapse due to material nonlinearity isexpected.

4.4.3. Loss of three cables

With regard to the results obtained for the previousscenarios, it can be concluded that scenariosassociated with loss of back stays connected to the pinsupport are the most critical ones. Accordingly,simultaneous loss of cables no. 1, 2 and 3 is only



Table 6. Maximum DCR and corresponding DAFvalues in towers, deck and cables under scenario-

1/LC-1 (Gravity load case 1 is applied and critical

damping ratio is taken as 0.5%)

Lost DAFcables no. Component (equivalent DAF) DCR

1 Left tower(scenario-1/LC-1, (bottom) 2.3 (1.1) 25%case-1) Right tower

(bottom) 6.8 (3.0) 25%Deck

(section A-A) 1.8 (0.8) 48%Cable

(Cable no.2) 1.7 (0.8) 36%


(bottom) 122# (30#) 14%Deck

(section B-B) 83# (36#) 30%Cable

(cable no.14) 1.9 (1.0) 40%


(bottom) 2.7 (1.3) 18%Deck

(section C-C) 2.7 (1.3) 31%Cable

(cable no.31) 1.5 (0.7) 33%


(bottom) 1.2 (0.6) 17%Deck

(section D-D) 1.7 (0.9) 40%Cable

(cable no.59) 1.6 (0.8) 35%

# DAF value was calculated based on maximum axial force.

considered in this part. For scenario-3/LC-2, DAF andDCR values determined based on maximum bendingmoment at the most critical sections along the deckare given in Table 5. In addition, the maximum DCR

and corresponding DAF values observed in the deck,towers and cables for scenario-3/LC-2 are given inFigure 13. The maximum DAF calculated within thisscenario is about 4.5 and related to the maximum



Table 7. Maximum DCR and corresponding DAF values for scenario-1/LC-2 in which one cable is lost (Gravity

load case 2 is applied and critical damping ratio is taken as 0.5%)

Geometrical DAF**


1 8 On 1.1 (0.6) 55%(scenario-1/LC-2, Off 1.0 (0.5) 52%case-1, Section A-A) 1 On 4.0 (2.0) 62%

Off 3.0 (1.6) 57%0.1 On 7.1 (3.7) 63%

Off 6.1 (3.2) 60%Ekill On 8.2 (4.2) 64%

30 8 On 1.1 (0.6) 43%(scenario-1/LC-2, Off 1.1 (0.5) 40%case-2, Section C-C) 1 On 2.4 (1.2) 47%

Off 2.3 (1.0) 44%0.1 On 2.8 (1.3) 49%

Off 3.0 (1.3) 46%Ekill On 2.7 (1.3) 49%

00 5 10 15 20 25 30

–200

–400

–600

–800

–1000

–1200

–1400

Mom

ent (

MN

.m)

Time (sec)

Case – 2 (no. 30)

Case – 1 (no. 1)

510 MN.m (Healthy)

Figure 10. Time history of bending moment at the bottom of the left tower for different cable loss cases under scenario-1/LC-2 (only one

cable is lost)

moment in the right tower. In scenario-3, DCR takesvalues greater than 100% (yielding may occur due tocombination of bending moment and axial load in thedeck, see Table 5), however, even in the event offormation of a plastic hinge at one location of thedeck, the redundancy provided by the support through

the remainder of the cables will guard againstprogressive collapse of the bridge.

5. DISCUSSION ON THE ADEQUACY OFTHE PROPOSED 2D MODEL

Due to lack of experimental data, the proposed 2Dmodels in this study cannot be validated directly;however, the adequacy of the proposed FE models andadopted constitutive law for steel can be verified againstmore complex 3D models. A 2D FE model of the bridgewas built in Microstran software and the results obtainedfrom this model for the healthy bridge under LC-1 werecompared with ANSYS model (see Figure 14). It isobservable that the response (internal force in cables,bending moment at the base of tower and the maximumdeflection at the mid-span) obtained from ANSYS andMicrostran model correlates well. The differencebetween Microstran and ANSYS model predictions isattributed to the method used for post-tensioning cablesand applying traffic loads. In ANSYS model, alldistributed loads treated as point loads and in Microstranthe cables were post-tensioned by applying atemperature gradient.

In addition, a 3D FE model of the bridge was developedusing shell elements in ANSYS (Aoki et al. 2012). In the3D model all material and geometrical nonlinearities areconsidered and element size varies between 0.3-1.0 m.Outline of the 3D FE mesh is shown in Figure 15(a).Regarding material nonlinearity, Von Mises yield



Table 8. Maximum DCR and corresponding DAFvalues in towers, deck and cables under scenario-

1/LC-2 (Gravity load case 2 is applied and critical

damping ratio is taken as 0.5%)

Lost DAFCables No. Component (equivalent DAF) DCR


(bottom) 0.6 (0.3) 37%Deck

(section A-A) 8.2 (4.2) 64%Cable

(cable no.2) 1.1 (0.6) 38%


(bottom) 0.6 (1.2) 17%Deck

(section C-C) 2.8 (1.3) 49%Cable

(cable no.31) 2.3 (1.1) 36%

00 5 10 15 20 25 30

–50

–100

–150

–200

–250

–300

–350

–400

Mom

ent (

MN

.m)

Time (sec)

Case – 2(no. 30 & 31 are lost)

Case – 1(no.1 & 2 are lost)

150 MN.m (healthy)

Figure 11. Time history of the bending moment at section A-A of the deck under scenario-2/LC-2 (two cables are lost)



Left tower

DAF

100%

DC

R

50%

0%0 1 2 3 4

max moment

min moment

max axial force

mini axial force

5

Right tower

DAF

100%

DC

R

50%

0%0 1 2 3 4

max momentmin momentmax axial forcemini axial force

5

Deck

DAF

100%

DC

R

50%

0%0 1 2 3 4


5

Cables

DAF

100%

DC

R50%

0%0 1 2 3 4

max axial force

mini axial force

5

Figure 12. DAF (from Eqn 1) versus DCR values for scneaio-2, case-1, under LC-2 (loss of cables no. 1 and 2)

Left tower

DAF

100%

DC

R

50%

0%0 1 2 3 4


5

Right tower

DAF

100%

DC

R

50%

0%0 1 2 3 4

max moment

min momentmax axial forcemini axial force

5

Deck

DAF

100%

DC

R

50%

0%0 1 2 3 4


5

Cables

DAF

100%

DC

R

50%

0%0 1 2 3 4

max axial force

mini axial force

5

Figure 13. DAF (from Eqn 1) versus DCR values for scneaio-3 under LC-2 (loss of cables no. 1, 2 and 3).



Ncable (1) = 20 (MN) Ncable (30) = 18 (MN)

(a)

Mtower = 700 (MN.m)

δmax = –0.75 (m)

MN

MN


(b)

Mtower = 670 (MN.m)δmax = –0.75 (m)

Figure 14. Comparison of results obtained from 2D models developed in (a) ANSYS and (b) Microstran for healthy bridge under LC-1


(b)

(a)

δmax = –0.75 (m)

MN

MN

MN


(c)

δmax = –0.67 (m)

Figure 15. (a) Outline of the mesh for the 3D FE model and internal force; and deflections predicted by (b) linear-elastic 2D models (c) 3D

model with material and geometrical nonlinearities included

criterion with perfect plastic behaviour (i.e. no strainhardening) was adopted (Aoki et al. 2012). Comparisonbetween 3D and 2D model predictions for the healthybridge under LC-1 is shown in Figures 15(b) and (c),which show very good correlations. Further, comparisonof 2D and 3D model predictions for the mid-span timehistory due to loss of cable no. 1 (under LC-1 load case)are shown in Figure 16, which clearly demonstratesadequacy of the proposed 2D model for predicting theglobal response of the cable-stayed bridges.

6. CONCLUSIONSA numerical study on the potential progressive collapseof cable stayed bridges due to different cable lossscenarios is carried out in this paper. A comprehensiveparametric study is undertaken and effect of location,duration and number of lost cables as well as appliedload case and the structural damping ratio on thedynamic amplification factor (DAF) and demand-to-capacity ratio (DCR) in different structural members(towers, deck and cables) are investigated. With regardto the parametric studies undertaken in this paper, thefollowing conclusions are drawn;

(1) Among different cable loss scenarios consideredfor potential progressive collapse of the cablestayed bridge been considered, the onesassociated with loss of the longest back stays arethe most critical ones (the highest DCR valueswere observed for all structural components).

(2) The DAF for the bending moment and axialforce at different sections along the deck, towers

and cables can take values much higher than two(typically adopted by different guidelines),however, the DCR value is usually less than100%. In other words, material nonlinearity hasminor effect on the global progressive collapseresponse of the bridge due to cable loss, and it isnot un-conservative to allow a DAF of greatertwo as long as DCR remains below one.

(3) The DAF values alone, do not provide anyinformation about the progressive collapseresponse of the cable stayed bridge under study,whereas DCR values at different locations of thestructure can be used as an indicator of materialnonlinearity and formation of plastic hinges.

(4) Damping ratio has a minor impact on thedynamic progressive collapse response of thecable stayed bridges as long as the adoptedvalue of the critical damping ratio is within theacceptable range (less than 2% of critical forthe bridge considered in this study).

(5) The developed FE model for the bridgeisanalysed with and without geometricalnonlinearities included to demonstrate the effectof geometrical nonlinearities on the globalprogressive collapse response of the bridge. It isshown that for the cases studied in this paper, theeffect of geometrical nonlinearities on theprogressive collapse response (due to cableloss), for a properly designed bridge is limited(less than 7% for the cases considered in thispaper).



00

–0.4

–0.8

–1.2

–1.6

–2

5 10

Time (sec)

Def

lect

ion

(m)

15 20

3D

2D

Figure 16. Time history of mid-span deflection predicted by 2D and 3D FE model for the bridge subject to loss of cable no. 1 and LC-1

(6) The value of DAF is sensitive to the time stepover which the cables are removed.

(7) The value of DAF for bending moment and axialforce in the towers and deck is larger than theDAF that should be applied for axial force in thecables.

(8) For all cases and scenarios studied in this paper,values of DAF for axial force in the cables aregenerally less than 2.0.

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