International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Integrated multi-hazard framework for the fragility analysis of roadway bridges Gehl, Pierre; D’Ayala, Dina 2015-07

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Integrated Multi-Hazard Framework for the Fragility Analysis ofRoadway BridgesPierre GehlResearch Associate, EPICentre, Dept. of Civil Engineering, University College London,London, UKDina D’AyalaProfessor, EPICentre, Dept. of Civil Engineering, University College London, London,UKABSTRACT: This paper presents a method for the development of bridge fragility functions that areable to account for the cumulated impact of different hazard types, namely earthquakes, ground failuresand fluvial floods. After identifying which loading mechanisms are affecting which bridge components,specific damage-dependent component fragility curves are derived. The definition of the global damagestates at system level through a fault-tree analysis is coupled with a Bayesian Network formulation inorder to account for the correlation structure between failure events. Fragility functions for four sys-tem damage states are finally derived as a function of flow discharge Q (for floods) and peak groundacceleration PGA (for earthquakes and ground failures): the results are able to represent specific failureconfigurations that can be linked to functionality levels or repair durations.1. INTRODUCTIONSpatially distributed infrastructure networks maybe exposed to a wide range of natural hazards,whose impacts have to be integrated and homog-enized in order to assess the reliability of the sys-tem over the design lifetime of its components (e.g.roadway bridges). While previous studies haveproposed probabilistic frameworks for multi-hazardassessment, either for joint independent events orcascading events (Marzocchi et al., 2012), there re-mains a lack of fragility models that are able toaccount for hazard interactions at the vulnerabil-ity level. Even though some vulnerability analyseshave addressed the combined effects of earthquakesand scour (Alipour and Shafei, 2012) or earth-quakes, scour and truck traffic on bridges (Liangand Lee, 2013), integrated fragility functions withrespect to the various intensity measures that rep-resent the hazard types are needed in order to becoupled to the probabilistic hazard distributions.To this end, a component-based approach is pro-posed in this paper: the bridge system is decom-posed into its various components, so that thehazard-specific loading mechanisms and their de-mand can be straightforwardly quantified at thecomponent level. In the case where some damagedcomponents influence the response of other compo-nents with respect to another hazard types, a set ofdamage-dependent component fragility curves haveto be derived in order to account for all possibledamage configurations. Finally, the use of proba-bilistic tools such as Bayesian Networks can facili-tate the assembly of the component fragility curvesat system level, while accounting for the possiblecorrelations between the component failure events.Multi-variate fragility functions are then expectedto be derived, where each input variable repre-sents a hazard-specific intensity measure, so thatthe damage of the bridge system can be assessedfor joint hazard events as well as single events.2. MULTI-HAZARD SCENARIOSThree main hazard types are considered in thepresent study, namely earthquakes (EQ), fluvialfloods (FL) and earthquake-triggered ground fail-112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015ures (GF). They have been chosen because theymay affect a wide range of bridge components andthey are among the main causes of bridge failures(Sharma and Mohan, 2011).Following the classification proposed by Lee andSternberg (2008), different types of multi-hazardscenarios have to be considered:• Simple events: for instance, EQ, FL or GFevents taken separately (Scenario 1).• Combined multi-hazard events (i.e. a sin-gle event triggering multiple loading mecha-nisms): a GF event triggered by an EQ event(Scenario 2).• Subsequent multi-hazard events (i.e. unrelatedsingle events separated in time): it could bea FL event followed by an EQ, which in turntriggers a GF event (Scenario 3).• Simultaneous multi-hazard events: they repre-sent the most unlikely scenarios and are notaddressed here.These different scenarios are then consideredthroughout the rest of the paper, as the objectiveis to generate multi-hazard fragility curves that areable to account for different combinations of load-ing mechanisms.3. BRIDGE COMPONENTS AND ASSOCI-ATED FAILURE MECHANISMSA multi-span concrete bridge with simply-supported independent decks is considered in thepresent study. It is composed of two seat-typeabutments and two piers with three cylindricalRC columns. Deck displacement is restrained byelastomeric bearings (i.e. alternation of expansionand fixed devices) in the longitudinal directionand shear keys in the transversal direction. Ateach bridge extremity, an embankment approach isadded in order to simulate the transition betweenthe plain roadway segment and the bridge (see halfbridge system in Figure 1).The hazard types that may potentially affect thevarious bridge components are detailed in Table 1:• Earthquakes affect all structural componentsof the bridge (i.e. abutments, piers, bearings,shear keys).• Fluvial floods may excavate pier foundationsdue to scour, while in extreme cases decks may------Abutment approach (embankment)AbutmentAbutment foundationS hear keyBearingP iercolumnP ier foundationFigure 1: Sketch of the studied bridge system and dislodged by hydraulic forces, after shearkeys have been damaged (Padgett et al., 2008).• Earthquake-triggered ground failures arelikely to affect the approach embankment, asthere is usually a vertical settlement betweenthe embankment and the bridge, due to thedifference in foundation depth (Puppala et al.,2009). Deep-seated landslides may also affectthe slope on which the abutment is built.Table 1: Bridge components and associated hazardtypes.Component EQ FL GFPier foundation XPier XBearing XDeck XAbutment foundation XAbutment XAbutment approach XShear key X X4. COMPONENT FRAGILITY CURVESFragility models are derived to quantify the damageprobability of each component to their associatedhazard types.4.1. EarthquakeThe layout, dimensions and constitutive modelsof the different bridge components are directly212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015taken from the multi-span simply-supported con-crete (MSSSC) girder bridge described in the studyby Nielson (2005). However two modifications arebrought to the bridge system:1. Pier foundations are assumed to be anchoredup to a depth of 8 m: the group of pile foun-dations, as described by Nielson (2005), isapproximated by an equivalent elastic beam,which is connected to the ground through a setof Winkler p-y springs in order to model thesoil resistance. The bending stiffness of theequivalent pile and the parameters of the p-ycurves are based on the configuration of thegroup piles and the soil properties (Prasad andBanerjee, 2013).2. External shear keys are added on the pier capsin order to restrain the displacement of the in-dependent decks in the transversal direction.The shear keys are modelled according to asliding friction shear mechanism: first, thedeck can slide on the pier cap (i.e. frictionCoulomb law) until the gap with the shear keyis closed. Then, the capacity of the shear key isengaged until it ruptures through a shear mech-anism. Once the shear key has failed, it is as-sumed that the deck can slide again until un-seating.The bridge is modelled through a simplified sys-tem of connectors whose stiffness and hystereticmodels represent the behaviour of the differentbridge components, such as piers, bearings or abut-ments (Gehl and D’Ayala, 2014). For instance,each three-column pier is fully modelled with a fi-nite element program and its cyclic pushover curvein both longitudinal and transversal directions isthen used to model an hysteretic material spring inthe OpenSees platform.Following the multi-hazard scenarios defined inthe previous section, the seismic fragility curvesshould be derived for various configurations:• Intact bridge;• Pier foundations excavated by scour;• Shear keys damaged by hydraulic forces (i.e.fluvial flood);• Both pier foundations and shear keys affectedby a fluvial flood.Therefore the bridge response has to be re-assessed for each of the identified configurations,leading to a series of damage-dependent compo-nent fragility functions. The effect of scour is in-troduced by removing the Winkler springs that areexcavated by the scour depth (Alipour et al., 2013).If damaged, shear keys are simply removed and thetransversal deck movement is only restrained by afriction law.Fragility curves for each component, for bothloading directions, and for two damage states (i.e.yield and collapse for piers and abutments; restraintfailure and unseating for bearings and shear keys)are then derived using the limit states defined inNielson (2005). Non-linear dynamic analyses onthe simplified bridge model are carried out with 288synthetic records for an appropriate range of magni-tude and epicentral distance, based on the seismo-tectonic context of the area where the bridge hasbeen modelled (i.e. Central Southern United States,see Nielson (2005)). The synthetic signals havebeen generated using a stochastic procedure devel-oped by Pousse et al. (2006) and they are succes-sively applied to the bridge system along the lon-gitudinal and transversal directions. In the longitu-dinal direction, 10 components are considered (i.e.piers P1 and P2, abutments Ab1 and Ab2, fixed andexpansion bearings B1 to B6), as well as in thetransversal direction, except that the bearings arethen replaced by the shear keys.Fragility curves have been derived for differentscour depths (i.e. from 0 to 5.1 m): based on theevolution of the bridge response and the fragilityparameters, three threshold levels of scour depthhave been identified: starting from 1 m, somechanges in the bridge response can be observed (i.e.scour damage state D1); from 3.6 m, the behaviourof the pier changes dramatically and the effect ofscour is much more noticeable (D2); finally, it is as-sumed that, after 5.1 m, the pier has almost a pinnedconnection to the soil and the stability of the systemcannot be guaranteed (D3).From Figure 2 it can be observed that scour hasmainly a detrimental effect on the bearings, whichtend to fail earlier as scour depth increases. Sincethe pier connection is relaxed at their base, they ex-312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20150 5 10 15P1P2Ab1B1B2B3B4B5B6Ab2D1 (EQ) : Yield / Restraint failureα [m/s2]0 5 10 15P1P2Ab1B1B2B3B4B5B6Ab2D2 (EQ): Collapse / Unseatingα [m/s2]scour D0scour D1scour D2Figure 2: Mean fragility parameter α in the longitu-dinal direction, for the different components and theirtwo damage states, for different initial states of scourdamage.perience lower bending moments and their failureprobability decreases for higher scour levels (how-ever shear failure is not currently modelled). Fi-nally, the response of abutments seems to remainstable across the different scour depth. Similar ob-servations have been made when fragility curvesare computed in the transversal direction.The effect of previous shear key failure is onlyobserved for fragility curves in the transversal di-rection: the influence of shear keys is not signif-icant for lower damage states (i.e. yield), howeverthe absence of shear keys leads to a much earlier oc-currence of heavier damage states, especially deckunseating.4.2. Ground failureAs discussed in the previous section, earthquake-triggered ground failures are more likely to affectthe components that are located at the bridge ex-tremities, namely the approach embankment andthe abutment foundations. First, the fragility of ap-proach embankments with respect to lateral spread-ing of the supporting soil is taken from Kaynia et al.(2012). Using the first two damage states (i.e. slightdamage D1 with permanent ground displacementof 3 cm and moderate D2 with permanent grounddisplacement of 15 cm), fragility parameters forD1 (respectively D2) are the mean α1 = 1.96 m/s2(resp. α2 = 4.12 m/s2) and the standard deviationβ1 = 0.70 (resp. β2 = 0.70), for a European soiltype D and an embankment height of 2 m.Regarding the abutments, their deep foundationsprevent the occurrence of damage due to superfi-cial landslides with a planar sliding surface: how-ever the damage due to a deep-seated circular land-slide that occurs below the foundations may haveto be considered. In this case, the factor of safetyFS is estimated with the limit equilibrium method(i.e. Bishop’s simplified method), assuming a cir-cular slip surface (see Figure 3). The surface issubdivided into a number n of vertical slices andthe factor of safety FS is then expressed as the ratioof resisting versus destabilizing moments of all theslices.0 5 10 15 20 25−2024681012X [m]Z [m]RMoment arm LiθiPore pressure u1 = 0 kPaUnit weight ρ1 = 18 kN/m3Friction angle φ1 ~ N(30°,6)Cohesion c1 ~ N(50kPa,20)Cohesion c2 ~ N(75kPa,30)Friction angle φ2 ~ N(34°,6.8)Weight WiWidth biPore pressure u2 = 0 kPaUnit weight ρ2 = 18 kN/m3Figure 3: Slice equilibrium method for the estimationof the factor of safety. The black shape is a simplifiedview of the studied bridge and its foundations.Fragility functions for slope instability (D1) arethen derived following the method proposed byWu (2014): for each increasing value of PGA,the reliability index of lnFS > 0 is estimatedusing a Mean-Value First-Order Second Moment(MFOSM) method. The input random variablesare the cohesion and friction angle of each soillayer, while a correlation factor of −0.4 is assumedbetween the cohesion and the friction angle (Wu,2013). The search algorithm for the probabilis-tic critical surfaces proposed by Hassan and Wolff(1999) is used in order to ensure that the minimumreliability index is found for each combination of412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015the soil parameters, which is different that findingthe surface with the smallest factor of safety. Anadditional constraint is introduced by the locationof the abutment and pier foundations, since the crit-ical surface is unlikely to generate any failures ifit intersects with the bridge foundations. Finally,the reliability index is converted into the probabil-ity of failure Pf and the points PGA−Pf are fittedinto a fragility curve with a lognormal cumulativedistribution function: a mean of 12.62 m/s2 and astandard deviation of 1.03 have been found.4.3. ScourFor the purpose of the demonstration, only localscour at piers is considered, since it could be as-sumed that there is no contraction of the river bedcross-section. Also, general scour due to riverbed degradation is usually neglected with respectto local scour. Empirical equations from HEC-18(Richardson and Davis, 1995) are used to quantifythe excavated depth ys due to local scour at piers:ys = 2 ·K1 ·K2 ·K3 ·K4 · y ·(Dy)0.65·F0.43 (1)where y represents the flow height, D is the pierwidth, F is the Froude number and the Ki param-eters are corrective coefficients (see Table 2). Fi-nally, the flow discharge Q can be expressed as afunction of velocity v and height y (Alipour et al.,2013), with river section width b:Q = b · y · v = b · yn·(b · yb+2y)2/3·S1/20 (2)The Manning’s roughness coefficient n and slopegrade S0 are specified in Table 2.Based on the probabilistic distribution of the in-put parameters, a Monte Carlo scheme is used withsampled values of flow height y in order to gener-ate around 10,000 couples of points Q− ys, whichare related by the combination of Equations 1 and2. This data set is then used to generate fragilitycurves with respect to flow discharge Q, using thethree scour depth thresholds that have been previ-ously defined. A comparative analysis of differ-ent statistical models has shown that the lognormalTable 2: Values used in the scour equations. Some ofthe parameter distributions are taken from Alipour andShafei (2012).Variable Description DistributionK1 Factor for pier K1 = 1nose shapeK2 Factor for flow U (1,1.5)angle of attackK3 Factor for bed N(1.1,0.0552)conditionK4 Factor for bed K4 = 1material sizen Manning’s roughness lnN(0.025,0.2752)coefficientS0 Slope grade lnN(0.02,0.52)cumulative distribution function is the best fit forthese scour fragility curves. The derived fragilityparameters for damage states D1, D2 and D3 aremeans α1 = 2.74 m3/s, α2 = 285.77 m3/s, α3 =847.62 m3/s and standard deviations β1 = 0.55,β2 = 0.57 and β3 = SubmersionWhile several studies have dealt with the vulner-ability of bridges due to hurricanes and relatedstorm surges (Kameshwar and Padgett, 2014; Pad-gett et al., 2008), the impact of fluvial floods onbridge superstructures remains difficult to quantify.Therefore, as a very raw approximation and for thesake of the demonstration, the fragility curve fromKameshwar and Padgett (2014) for bridge failuredue to storm surge is taken in order to represent theprobability of deck unseating (D2), while keepingthe coefficient related to wave height equal to 0.Regarding damage to shear keys (D1), a conser-vative assumption could consist in considering fail-ure as soon as the flow height reaches the top of thepier cap. These threshold values can then be con-verted in terms of flow discharge using Equation 2.5. GLOBAL DAMAGE STATESGlobal damage states should be defined so that theyare consistent in terms of degrees of damage andconsequences on the bridge functionality. There-fore a rationale is proposed here in order to iden-tify homogeneous system damage states (SDSs) de-512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015pending on all possible combinations of componentdamage states, based on the functionality level ofthe bridge and the repair operations required:• SDS1 (slight repairs and no closing time):slight damage to approach embankments (D1)without a significant impact on the bridgefunctionality, even though repair operationsare eventually necessary;• SDS2 (moderate repairs and short closingtime): structural damage to bridge components(i.e. piers, abutments, bearings and shear keysin damage state D1 due to earthquake, ap-proach embankments in damage state D2 dueto ground failure, damaged shear keys D1 dueto flood);• SDS3 (extensive repairs and closing time):deck unseating that induces long term closureof the bridge, even though temporary deckspans could be installed if the substructurecomponents have not collapsed (i.e. deck un-seating D2 due to flood, bearings and shearkeys in damage state D2 due to earthquake);• SDS4 (irreparable with full collapse): sub-structure components have collapsed and in-duce the total failure of the bridge system (i.e.piers and abutments in damage state D2 dueto earthquake, scour damage state D3 at pierfoundations, slope failure D1 beneath abut-ment foundations).These system damage states may be representedthrough a fault-tree analysis, where the cascadingfailures leading to the global failure event can bedetailed. In this case, the top event would be one ofthe system damage states, while the root events arethe potential hazard loadings and the intermediateevents the various component damages. One of themajor issues of fault tree analyses is the treatmentof common cause failures (e.g. earthquake load-ing potentially leading to multiple component fail-ures): each hazard event has to be represented mul-tiple times in the fault tree and the statistical depen-dency between component failures is not properlyexplicated, unless a common cause failure node isadded.Therefore a Bayesian Network representation ispreferred here, since it leads to a more compact de-scription of the problem and to a more rigoroustreatment of statistical dependencies, as it is ex-plained in the next section.6. MULTI-HAZARD FRAGILITY FUNC-TIONSOnce the component fragility curves and the sys-tem damage states have been fully described, itis possible to derive system-level fragility func-tions that are able to account for the joint effectof the different loading mechanisms: since hazard-specific fragility curves from previous sections areexpressed with either flow discharge Q (i.e. fluvialflood) or PGA (i.e. earthquake and ground failure),then it is possible to derive a fragility surface thatexpresses the damage failure with respect to twostatistically independent intensity measures.The series of events leading to deck unseating(i.e. SDS3) may be translated into a Bayesian Net-work formulation (see Figure 4): this enables toclearly identify the potential common cause fail-ures induced by the hazard types. The BayesianNetwork has been developed according to the fol-lowing sequence of events:• Deck unseating occurs if one the bearings (orshear keys) exceeds a given deformation levelin the longitudinal (or transversal) direction, orif the deck is directly upset by the fluvial flood;• Bearing (or shear key) deformation is trig-gered by seismic loading and may be modifiedby the state of the pier foundations;• Piers foundations are altered by scour due tofluvial flood;• The seismic response of shear keys in thetransversal direction is also influenced by thestate of the shear keys due to fluvial flood.Correlations between failure events can then beapproximated with a Dunnett-Sobel class of Gaus-sian random variables, as proposed by Song andKang (2009): for each component i, the standard-ized safety factor Zi (i.e. ratio of demand over ca-pacity) can be expressed as a combination of ran-dom variables such as Zi =√1− t2 ·Vi + ti ·U ,where the tis have to be optimized by fitting the cor-relation matrix of Zi. Numerical seismic analysisof the bridge system enables to obtain a straight-forward correlation matrix of the component re-612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Figure 4: Bayesian Network for deck unseating (SDS3).Dunnett-Sobel variables are represented in red.sponses, however this is not the case for floods andground failures. It can however be assumed thatflood- and earthquake-related failures are statisti-cally independent, therefore a correlation factor of0 is used between the different hazard types.For each global damage state, the correspond-ing Bayesian Network has been implemented intothe Bayes Net toolbox: a junction-tree algorithmis used to estimate the global damage probabilitieswhen the hazard loadings (i.e. PGA and Q) arespecified as evidence. Finally, the fragility surfacesfor SDS1 to 4 are displayed in Figure 5. A cumula-tive representation of the damage probabilities hasbeen chosen, however it can be seen that the def-inition of the four SDSs does not follow a strictlyhierarchical model (e.g. the probability of reach-ing SDS4 is higher than the probability of reachingSDS1 for some combinations of Q and PGA).It can be seen that the effect of fluvial floodis mainly significant for heavier system damagestates, such as deck unseating or full collapse: forthese damage states, it has been observed that theeffect of scour or shear key removal has the most in-fluence on the seismic response of the other bridgecomponents. Finally, it can be checked that themulti-hazard fragility functions are consistent withthe various scenarios defined in Section 2: if theloading history on the system is respected (i.e. aflood followed by an earthquake), the cumulativedamages can be quantified for the various configu-0100500100000.51PGA [m/s2]SDS4Q [m3/s] 0100500100000.51PGA [m/s2]SDS3Q [m3/s]0100500100000.51PGA [m/s2]SDS2Q [m3/s] 0100500100000.51PGA [m/s2]SDS1Q [m3/s]Figure 5: Multi-hazard fragility functions for the foursystem damage states, expressed as a function of Q andPGA.rations.7. CONCLUSIONSThis paper has shown through an elaborate exam-ple that multi-hazard fragility functions can be as-sembled from hazard-specific fragility curves anda corresponding Bayesian Network formulation. Itshould be noted that this framework allows to treatcomponent fragility curves that have been derivedthrough various techniques (e.g. empirical, analyt-ical, judgement-based, with plain Monte Carlo orMFOSM methods), which are usually inherent toeach hazard type. Limited knowledge on a givenquantitative fragility curve could also be addressedby assigning upper and lower bound to the dam-age probabilities and analysing their effects on thesystem-level fragility functions.Finally, this study has assumed that all pier foun-dations or all shear keys are in the same damagestate (i.e. correlation factor of 1) due to fluvialflood, while there could actually be many config-urations where only a few components have failedat the same time. Accounting for this level of de-tail would lead to an almost intractable number ofdamage configurations, while the fragility functionspresented here are already based on a large amountof damage-dependent fragility curves (e.g. assem-bly of 65 component fragility curves for SDS2).712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Such difficulties are mainly due to the nature ofthe seismic response of a structural system, whereany component may have an effect on the behaviourof the other components, and reciprocally. Meta-models or response surfaces could be a reasonablecompromise, since they could be used to adjust thefragility parameters of the components, based onthe damage configuration of the bridge (e.g. num-ber of damaged components, degree of damage).8. ACKNOWLEDGEMENTSThis study has been carried out within the IN-FRARISK project (Novel indicators for identify-ing critical INFRAstructure at RISK from NaturalHazards), which has received funding from the Eu-ropean Union’s Seventh Programme for research,technological development and demonstration un-der grant agreement No 603960.9. REFERENCESAlipour, A. and Shafei, B. (2012). “Performance assess-ment of highway bridges under earthquake and scoureffects.” 15th World Conference on Earthquake Engi-neering, Lisbon, Portugal.Alipour, A., Shafei, B., and Shinozuka, M. (2013).“Reliability-Based Calibration of Load and Resis-tance Factors for Design of RC Bridges under Multi-ple Extreme Events: Scour and Earthquake.” Journalof Bridge Engineering, 18, 362–371.Gehl, P. and D’Ayala, D. (2014). “Developing fragilitycurves for roadway bridges using system reliabilityand support vector machines.” 2nd European Confer-ence on Earthquake Engineering and Seismology, Is-tanbul, Turkey.Hassan, A. 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