Cable Loss Analyses and Collapse Behavior of Cable Stayed Bridges (Wolff-Starossek)

7/30/2019 Cable Loss Analyses and Collapse Behavior of Cable Stayed Bridges (Wolff-Starossek)

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IABMAS2010, The Fifth International Conference on Bridge Maintenance, Safety and Management, July 11-15, 2010, Philadelphia, USA

1 INTRODUCTION

The failure of one structural element can lead to the

failure of further structural elements and thus to thecollapse of large structural sections or the entirestructure. In many cases, the initial triggering eventand the resulting damage are disproportionate. Suchcollapses have frequently been discussed andinvestigated in recent years and are generallysummarized under the term progressive collapse.But work in this field refers mainly to buildings.

Collapse resistance means insensitivity toaccidental circumstances. This can be achieved byensuring a high level of safety against local failureor by using a design which allows for local failure.The structure's property of being insensitive to localfailure is termed robustness (Starossek 2009). Forcable-stayed bridges, collapse resistance is primarilyachieved by increasing the robustness. The loss ofcables must be considered as a possible local failuresince the cross sections of cables have usually a lowresistance against accidental lateral loads stemmingfrom vehicle impact or malicious actions. The lossof cables can lead to overloading and rupture ofadjacent cables. A collapse progressing in such away is called a zipper-type collapse (Starossek

2007). Because the bridge girder is in compression,the loss of cables, which leads to a reduction of

bracing, increases the risk of buckling. To createrobust structures, it is necessary to know the collapse

behavior of a structure. With this knowledge,

structural properties can be identified which areresponsible for collapse propagation.

This paper examines the dynamic response of a

cable-stayed bridge to the loss of one or morecables. Such analyses require a great amount ofexpertise and modeling effort. When designing forthe loss of a cable, only the maximum responses areof interest. Therefore, quasi-static analyses usingdynamic amplification factors to account for thedynamic effects can be conducted instead(PTI 2007). In the first part of this paper, dynamicamplification factors are determined, information isgiven on how to determine these factors and thelimits of the quasi-static approach for cable-stayed

bridges are outlined. In the second part of this paper,the collapse behavior of a cable-stayed bridge afterthe loss of cables is analyses. To trace the collapse

progression following the rupture of one or morecables, geometric and material nonlinear dynamicanalyses in the time domain are conducted. Hereby,critical elements are identified and the prevailingcollapse type is described. Finally, recommendationsfor the design of robust cable-stayed bridges aregiven.

2 INVESTIGATED BRIDGE SYSTEM AND ITSMODELING

The cable-stayed bridge being considered and whichwas the basis for a number of parameter variations is

Cable-loss analyses and collapse behavior of cable-stayed bridges

M. WolffGrassl Engineering Consultans, Hamburg, Germany

U. StarossekStructural Analysis and Steel Structures Institute, Hamburg University of Technology, Hamburg, Germany

ABSTRACT: The general aim in designing structures, where the consequences of a collapse are high, must becollapse resistance. This means that no structural damage should develop that is disproportionate to thetriggering event. Generally, structures can be made collapse resistant by ensuring a high level of safetyagainst local failure or by designing for the failure of elements and thus increasing the robustness. Increasing

the robustness of cable-stayed bridges is achieved by means of designing for the loss of cables. For this,quasi-static analyses using a dynamic amplification factor are recommended by guidelines. This paper showsthe possibilities and limits of such an approach for cable-stayed bridges. Furthermore, collapse analyses of acable-stayed bridge are conducted. With this, structural properties are identified which are responsible forcollapse propagation. The prevailing collapse type is described and recommendations for the design of robustcable-stayed bridges are given.


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shown in Figure 1. Two cable planes are placed in amodified fan arrangement with 80 cables in eachvertical plane and a cable spacing of 15 m at decklevel, apart from the closely spaced outermost back-stay cables. The pylons consist of reinforcedconcrete. The bridge girder consists of a 21.60 mwide orthotropic steel deck, two 2.6 m deeplongitudinal steel girders and cross girders spaced at

3.75 m apart. In the longitudinal direction, the bridgegirder is only restrained by the cables.The numeric investigation is conducted using a

three-dimensional model of the bridge. The pylonand bridge girder (longitudinal and cross girders) aremodeled with beam elements. For investigationswith nonlinear material behavior, a combination ofshell and beam elements are used for the deck. Thecables are modeled with a series of truss elementswith distributed mass and self weight. The influenceof cable sag and transverse cable vibration on thedynamic response could therefore be investigated. It

shows that for the loss of one cable, this detailedmodeling of cables leads to smaller dynamicresponses (Wolff & Starossek 2009).

The loss of a cable is investigated by nonlineardynamic analyses in the time domain, taking intoaccount large deformations. Firstly, the static initialstate of the structure with the considered static loadcases is calculated. The cable to be considered forfailure is eliminated from the structural model andthe corresponding cable forces are applied to theanchorage nodes of that cable as static loads. The

time-history analysis is begun on this modified andloaded system at rest. To model the sudden loss ofthe cable, a step loading of the same size as the staticcable force but acting in opposite directions isapplied to both anchorage nodes. The numericcalculations are done using the finite elementanalysis program ANSYS.

3 DYNAMIC AMPLIFICATION FACTORS

In the design, only the maximum dynamic responsesto cable loss are of interest. Therefore, guidelinessuggest a quasi-static approach which accounts forthe dynamic effects by a dynamic amplificationfactor (DAF). For single-degree-of-freedom

systems, this factor is 2.0.According to the PTI Recommendations (2007), a

force which is the cable force multiplied by thisfactor of 2.0 acting in the opposite direction must beapplied to calculate the maximum responses due tocable loss. The EC 3 (2006) stipulates that the

bending moments and forces due to static removal ofa cable be multiplied by a factor of 1.5. This smaller

value might consider the fact that a sudden failure ofa whole cable this causes a higher response than agradual reduction is unlikely. But the failure of acable anchorage at the Cycle Arc bridge in Glasgow,the rupture of the main cable of a cable car inCavalese due to a jet plane impact or the rupture of acable at the Rion-Antirion Bridge due to a lightningstrike tell a different story.

The PTI Recommendations additionally allow thedetermination of a dynamic amplification factor in anonlinear dynamic analysis, because it is assumedthat, in general, smaller factors can be chosen for

cable stayed-bridges. But how this factor is to bedetermined or which assumptions are to be made arenot described.

Investigations as to realistic ranges of dynamicamplification factors are rare. Single values arecalculated for an arch bridge in Zoli & Woodward(2005), in a simplified manner in Hyttinen et al.(1994), and in Park et al. (2007).

In the following, dynamic amplification factorsare calculated separately for all state variables in allstructural elements of the described bridge. The aim

is to give advice on how to determine this factor, toprove if the use of a uniform amplification factor isvalid and if reductions are generally possible.

The method by which the DAF is determined isdescribed in (Wolff & Starossek 2008). It wasconcluded that a unique dynamic amplificationfactor cannot be specified. Instead, the value isdependent on the location of the ruptured cable aswell as the type and location of the state variable

being considered. Very different dynamicamplification factors result if the rupture of onecable is considered (Wolff & Starossek 2009).Amplification factors at locations further away fromthe ruptured cable are high. At these locations, staticresponses are small. While the static removal of acable mainly causes local deflections and bending

Figure 1. Structural system.


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moments, the sudden removal of a cable excitesnatural modes with deflections and moments overthe whole girder length. Thus, the mainly excitednatural modes are not affine to the static deflectioncurve.

The dynamic responses at locations further awayfrom the ruptured cable are, however, irrelevantwhen considering all cable loss load cases, because

only responses in the vicinity of the ruptured cableare design governing. Concerning the positivevertical deflections, a dynamic amplification factorof between 1.5 and 1.8 results at these locations. Theamplification factor for the positive bendingmoments in proximity to the ruptured cable lies

between 1.3 and 1.6, while that for the negativebending moments between 1.4 and 2.7. In Figure 2,the envelopes of extreme bending moments from allload cases are shown, together with thecorresponding amplification factors.

The reason for amplification factors smaller than

2.0 is that the maxima of all superposed eigenmodesdo not occur at the same time in the considered time

period. Figure 3 shows the contribution of the first300 eigenmodes in time-history to the total bendingmoment at the location of a ruptured cable. For thesake of clarity, the contributions are not sketchedindividually but grouped. The sum of the maxima ofthese groups (encircled) is My= 18.3 MN which ishigher than the maximum of the total response inthis time range. The theoretical maximum developsat t= 300 s provided no damping is present.

The dynamic amplification factor for the designgoverning dynamic axial forces in the bridge girdercan be high (Wolff & Starossek 2009). But theincrease in the girders total stresses due to thenormal forces resulting from cable loss is smallcompared to the increase due to bending moments:

For the investigated bridge, the value is 2 %.The dynamic amplification factors for the design

relevant cable forces which develop in the cablesadjacent to the lost cable are between 1.35 and 2.0,depending on the lost cable being considered.

Special attention is necessary for the bendingmoments in the pylons. The dynamic amplificationfactor for the bending moments over the whole

pylon height and for all cable losses is significantlyhigher than 2.0. At the pylon base, values of about30 for negative and about 8 for positive momentsoccur. The static moments are small. However, thedynamic bending moments in the pylon aresignificant (Fig. 4). Here too, higher modes whichare not affine to the static deflection curve areexcited. Furthermore, the pylon is not only excited

by the step loading of the failing cable, but each ofthe cables induces irregular forces which arecomposed of the redistributed loads from the failedcable plus the inertia forces from the vibrating

bridge deck. (The anchorage points are evenlydistributed over a length of 26 m at the pylon head.)These forces cause a complex structural responsewhich cannot be simplified as described above.

The results show that in the present case, dynamicamplification factors smaller than 2.0 are only

possible for the bending moments in the bridgegirder, since over wide parts, the values are smaller.Here, an explicit calculation of the DAF can be

beneficial. For the safe design of the cables, adynamic amplification factor of 2.0 is necessary. For

the bending moments in the pylons, high dynamicforces occur which cannot be safely accounted for

by a quasi-static analysis using amplification factors.In particular, when the static removal causes adecrease in responses while the dynamic removalcauses an increase, quasi-static analyses cannot yield

Figure 2. a) Dynamic amplification factor (DAF) for positive () and negative () bending moments; (b) extreme bendingmoments in longitudinal girder in the plane of cable rupture due to permanent loads and cable losses (envelope)


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Figure 4. Extreme bending moments in one pylon leg due to

permanent loads and cable losses (envelope)

correct results. Dynamic analyses seem vital here.The results presented in this paragraph are for the

undamped system under self-weight without liveloads. The dynamic amplification factors aredetermined in nonlinear analyses. However, it turnsout that even if nonlinearities cannot be neglected incalculating the maximum responses, the dynamic

amplification is nearly independent of the type ofcalculation. Therefore, it is also independent of thetype of additional loading. Thus, different live load

positions need not be considered.In (Wolff & Starossek 2008), the influence of

damping on the structural response after cable loss isdescribed. The effect of damping depends on theoccurrence of the maximum responses in time-history. As stated before, the theoretical maximum

responses occur very late in time history when evena very small damping has stopped any vibration. Theconsidered time-range for calculating the dynamicamplification factor was chosen in such a way thatdamping reduces the amplitudes by at least 25 % bythe end of this time range. Design relevantmaximum deflections and bending moments of the

bridge girder and maximum cable forces developearly in this time range. Therefore, a damping hasonly a small effect on these maximum responses andthe impact on reducing the dynamic amplificationfactor is small. The bending moments in the pylons,however, are significantly reduced, even by a smalldamping ratio of = 0.2 %. However, the dynamicamplification factors are still much higher than 2.0.

4 IMPORTANT AND CRITICAL ELEMENTS

The effort of determining maximum responses dueto cable loss can be reduced by minimizing thenumber of investigated cable-loss load cases. This is

possible if the important elements of the structure

are known, which are defined as those elementswhose failure cause the highest responses.

For the investigated bridge, the increase in cablestresses is highest when a short cable fails. Thedesign governing stresses in the bridge girder are

Figure 3. Contribution of first 300 eigenmodes in time-history to total bending moment at the location of a ruptured cable

(= maximum bending moment)


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due to the loss of a long cable near the center of thebridge. These results might be applicable to othercable-stayed bridges with closely spaced cables

because the reason for this is the bridge girderselastic support by the cables: in the range of shortcables, the support is stiffer because of the largerinclination and stiffer force-deformation behavior ofshort stay-cables. Thus, the loads are transferred to

few cables. The soft support in the range of longcables causes a load distribution to more cables butat the expense of higher bending moments in the

bridge girder. Details are given in (Wolff 2010)For the maximum bending moments in the

pylons, the loss of long cables has to be analyzed,because the induced horizontal forces are highest.But this cannot be generalized, because resonanceeffects also play a role.

In addition to the important elements, criticalelements are defined which are elements whosefailure leads to a disproportionate damage. Thus,

they are dependent on the definition of the termdisproportionate. Furthermore, the additionalloading is important. Depending on their location inthe structure, the number or size of critical elementscan be different. For the investigated bridge, the lossof three adjacent short cables is critical, as explainedin Section 5.

5 COLLAPSE BEHAVIOR

To trace the collapse progression after an initialfailure of one or more cables, geometrically andmaterially nonlinear dynamic analyses arenecessary.

The material behavior of the bridge girder and thecables is assumed to be elasto-plastic. In the collapsestate, cable stresses can be significantly higher thanin the initial state but also a slacking is possible. Forthese high stress variations, the cables force-deformation behavior is highly nonlinear. Takinginto account cable sag and transverse cable vibrationtherefore becomes crucial. To account for the exactstress distribution in the bridge girder, a combinationof shell and beam elements is used. The element sizeis adapted according to the strain gradients.

For the investigated bridge, the failure of onesingle cable does not lead to collapse progression.(Unfactored live loads are placed in the mostunfavorable position.) Only local plastificationsdevelop and deflections are not significant. Also, thecable tensions remain comparatively small. Liveloads can be increased by a factor of three untilultimate state is reached.

The minimum number of cables which have tofail for total collapse of the bridge are three adjacentshort cables. In Figure 5, the initial state with theintact critical cables is depicted. After the suddenloss of these cables, vertical deformations with

plastic regions first begin to develop in thelongitudinal girder of the damaged cable plane (inthe front part of the figure). Thereby, the normalforces acting on the whole section of the bridgegirder are transferred to the longitudinal girder of theintact cable plane (at the rear of the figure) wherevertical deflections are small. Although this girder iscontinuously supported by the cables, it cannot resist

these high normal forces and begins to buckle in thevertical direction. From this moment on, verticaldeflections grow strongly and cannot be arrestedsince the bridge deck is not restrained by fixsupports in the longitudinal direction. Ultimatestresses in the bridge girder are exceeded. Duringthis process, the upward deflections and the missingrestraint in the longitudinal direction cause aslacking of some cables which can lead to themdisengaging from their anchors. The downwarddeflection of the longitudinal girder of the intactcable plane finally causes the rupture of the cable at

this location. This state of the bridge deck is shownin Figure 6. The collapse state is sketched in Figure7: due to the longitudinal motion of the bridge decktowards the damaged region, normal forces aretransferred to the other, intact half of the bridge withthe second pylon (not shown in the figures). Thisleads to high unbalanced cable forces in both bridge

parts which in turn results in high bending momentsin both pylons. The continuity of the bridge girderthus causes the failure of both pylons whichultimately leads to the total collapse of the bridge.

The collapse can only be arrested by providing analternate load path for the normal forces in the

bridge girder. Therefore, an alternative system wasinvestigated where the bridge deck was horizontallyfixed at the ends of the bridge or alternatively at the

pylons. But the results show that high forces of40 MN develop indicating that horizontal bearingsare not a good measure for increasing the robustness.

Figure 5: Von Mises stresses in the bridge girder due topermanent and live loads prior to loss of cables (one half of thebridge)


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Figure 6: Von Mises stresses in the bridge girder at collapse state due to loss of three cables, permanent and live loads(for a better illustration, the pylon is omitted), -- slack cables, ruptured cable

Figure 8: Deformed state after the loss of 10 long cables (von Mises stresses)

max w = 3220 cm

Figure 7: Final collapse state for the failure of critical elements, scheme


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The structure is more robust regarding the loss oflong cables. This is due to the fact that at theanchorage points of the longer cables, the bridgegirder exhibits smaller normal forces and thereforethe increase in bending moments due to secondorder effects is small. Furthermore, the support ofthe bridge girder at the bridge center is softer than atthe pylons. Thus, the forces from the failing cables

are distributed to more cables. Figure 8 shows thedeformed state after the loss of five cables in eachcable plane. The bridge will finally collapse due tohigh bending moments in both pylons, sinceadditionally to the high vertical deformations, highhorizontal deflections towards the bridge centeroccur.

6 COLLAPSE TYPES

According to Starossek (2007), six types of

structural collapse can be distinguished: five purecollapse types and one mixed-type collapse. Eachcollapse type can be characterized by a propagatingaction which, after the failure of one element, leadsto the failure of the next element. The study of

propagating actions can give insights into thestructural properties which promote collapse

propagation.The collapse propagation for the failure of the

critical elements, described above, does exhibitfeatures of at least two of the pure collapse types and

must therefore be categorized as mixed-typecollapse. At the beginning, there is the initial failureof elements which are responsible although not

primarily - for the stabilization of the bridge girderin compression. Lack of bracing then leads to anincrease in vertical deflections and high stresses dueto second order effects, firstly in the longitudinalgirder of the affected cable plane and then in thesecond longitudinal girder (a more detailed analysisis given in Wolff (2010)). These are features of theinstability-type collapse although the failure is not a

pure buckling. The compression force is thusresponsible for the onset of collapse propagation andthe failure of the bridge deck.

After the failure of the bridge deck, the normalforces in the bridge deck, resulting from thehorizontal cable forces, are redirected to the intact

bridge part utilizing the decks tension resistance(Figure 7). The forces are transferred to the pylons

by the main spans cables (Figure 7). The pylons arepulled towards the main span of the bridge and failin bending. This process exhibits features of thedomino-type collapse: the horizontal forces cause an

overturning and thus failure of these two individualstructures with mainly vertical load bearing capacity.They would not develop if the bridge deck was madeof concrete or is not continuous.

A rupture of adjacent cables as a directconsequence of the initial failing cables which isthe main example for the zipper-type collapse and isoften connected to the collapse of cable-stayed

bridges does not occur.

7 ROBUSTNESS

From the observations made regarding the collapseanalysis, conclusions can be drawn as to therobustness of the investigated cable-stayed bridge,which has a slenderness ratio of 1/230 and a cablespacing of 15 m. In general, robustness can bedefined as insensitivity to local failure (Starossek2009). Robustness is therefore always related to thesize of the initial failure and to the accepted amountof damage to the remaining structure. Both itemshave to be quantified as design aims (Starossek &Wolff 2006). Recommendations for cable-stayed

bridges require the analysis for the failure of onesingle cable; other authors propose the failure ofcables within a 10 m range (Starossek 2009).

If the design aim is that no or only smallplastifications are allowed, the investigated bridgeacts only robust to the loss of one cable. Here,alternate paths can develop. Loads are transferredthrough the bridge girder to adjacent cables. Onlylocal plastifications develop in the bridge girder andthe cables remain elastic. If only total collapseshould be avoided, the bridge can be termed robust

regarding the loss of two adjacent cables plus oneadjacent cable in the second cable plane. The failureof more cables is not possible without seriousdamage or total collapse. The bridges robustness istherefore limited to the failure of one to three cablesdepending on the predefined design criteria.

Although the quantitative description ofrobustness will be different, the qualitativedescription will be analogous to similar cable-stayed

bridges. The reason for the relatively robustbehavior of the bridge lies in some general featuresof cable-stayed bridges with close cable spacing.The bridge decks elastic support by the cablesallows a load distribution to many cables. Togetherwith the low stresses in the serviceability state and ahigh deformation capacity due to their length, afailure of cables due to overloading is not verylikely.

For the investigated bridge, the bridge girder isthe critical structural element. Its resistance againstinstability is crucial for the robustness of thestructure in case of the loss of short cables. Theresistance can either be increased by increasing the

stiffness or by reducing the unsupported length,which means closer cable spacing. In the wholecontext of avoiding disproportionate collapse, theformer alternative should be chosen: if the cables are

placed close together, the probability that more


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cables fail due to the same event is higher (ShankarNair 2006) and the consequence to the girderremains the same. Additionally to a high cross-sectional stiffness, the resistance to lateral bucklingof the longitudinal girders should be high. Here,close cross girders with the same height as thelongitudinal girders are useful.

In case of the loss of long cables, the bridge

decks deformation capacity is important. Thus, asteel deck is preferable. Another advantage of a steeldeck is that tension forces can be transferred to theintact bridge part. Although the loss of the criticalelements leads to the progression of collapse to theintact bridge half, it increases the robustness for theloss of long cables.

8 CONCLUSION

This paper investigates the loss of any one cable

by nonlinear dynamic analysis of a three-dimensional model of a cable-stayed bridge. Dyna-mic amplification factors for quasi-static analysesare determined. The aim of these analyses is to giveadvice on how to determine this dynamicamplification factor, to prove if the use of a uniformamplification factor to calculate the maximumresponses to cable loss is valid and if reductions ofthis factor are generally possible.

The results show that a unique dynamicamplification factor cannot be specified. Instead, the

dynamic amplification factor depends on thelocation of the ruptured cable and on the type andlocation of the state variable being examined. Usinga factor smaller than 2.0 is only possible for the

bending moments in the bridge girder. Here, anexplicit calculation of the DAF can be beneficial. Adynamic amplification factor of 2.0 is necessary forthe safe design of the cables. Regarding the bendingmoments in the pylons, large dynamic amplificationfactors result because of large dynamic responsesresulting from a complex excitation. Dynamic time-history analyses are therefore recommended, at leastfor the cable loss cases which yield the highestresponses. These are generally the longer cables butnot necessarily the longest cable.

The dynamic amplification factor can bedetermined in linear dynamic analyses. Differentlive load positions do not have to be considered.

Additionally to having appropriate analysis toolsfor creating robust structures, it is important to knowwhich structural properties are important to increasestructural robustness. These properties are identified

by investigating the collapse behavior of a cable-

stayed bridge after the loss of cables. The resultsshow that the normal forces in the bridge girder isthe collapse promoting attribute of a cable-stayed

bridge. For this reason, self-anchored cable-stayedreact less robust to the loss of short cables where the

normal force in the bridge deck is highest. For theinvestigated cable-stayed bridge, two adjacent shortcables in one cable plane plus one cable in thesecond cable plane can fail without disproportionatecollapse. In case of the failure of three adjacentcables, the bridge collapses due to instability failureof the bridge deck. The robustness of the bridge can

be increased by preventing instability. This is

possible by increasing the stiffness of the bridgegirder or by reducing the unsupported length, whichmeans closer cable spacing. The former isrecommended here due to a higher failure

probability of closely spaced cables.

ACKNOWLEDGMENTS

This research was funded by the DeutscheForschungsgemeinschaft DFG (German ResearchFoundation) which is gratefully acknowledged.

REFERENCES

Eurocode 3. 2006. Design of steel structures, Design ofstructures with tension components. EN 1993-1-11, 2006

Fib (International Federation for Structural Concrete) 2005.Acceptance of stay cable systems using prestressing steels.Lausanne: fib.

Hyttinen, E. & Vlimki, J. & Jrvenp, E. 1994. Cable-stayed bridges effect, of breaking of a cable. In Cable-stayed and suspension bridges, Proceedings AFPCConference, October 12-15, 1994.

Park, Y.S., Starossek, U., Koh, H.M., Choo, J.F., Kim, H.K. &Lee, S.W. 2007. Effect of cable loss in cable stayedbridgesFocus on dynamic amplification. Improvinginfrastructure worldwide; Report IABSE symposium,Weimar, Germany, 19-21 September 2007. Zurich: IABSE.

Post-Tensioning Institute (PTI) 2007. Recommendations forstay cable design, testing and installation. 5th edition,Cable-Stayed Bridges Committee. Phoenix: PTI.

Shankar Nair, R. 2006. Preventing Dispoportionate Collapse.Jounal of Performance of Cnstructed Facilities, 20(4), 309-314, November 2006.

Starossek U. 2007. Typology of progressive collapse.Engineering Structures, 29(9), 2302-2307, September 2007.

www.starossek.de.Starossek, U. (2009). Progressive Collapse of Structures.

Thomas Telford Publishing, London, July 2009.www.starossek.de

Wolff, M. & Starossek, U. 2008. Robustness assessment of acable-stayed bridge. Proceedings, 4th InternationalConference on Bridge Maintenance, Safety, andManagement (IABMAS08), Seoul, Korea, July 13-17,2008.

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Wolff, M. 2010. Seilausfall bei Schrgseilbrcken undprogressiver Kollaps (Cable loss in cable-stayed bridgesand progressive collapse). Phd-Thesis, Structural Analysisand Steel Struct. Inst.. Hamburg University of Technology.

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Cable Loss Analyses and Collapse Behavior of Cable Stayed Bridges (Wolff-Starossek)

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