paraskeva et al-2006-earthquake engineering & structural dynamics

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2006; 35:12691293Published online 19 June 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.582

Extension of modal pushover analysis toseismic assessment of bridges

T. S. Paraskeva1;, A. J. Kappos1;;; and A. G. Sextos2;

1Laboratory of Concrete and Masonry Structures; Department of Civil Engineering;Aristotle University of Thessaloniki; Greece

2Department of Civil Engineering; Division of Structures; Aristotle University of Thessaloniki; Greece

SUMMARY

Nonlinear static (pushover) analysis has become a popular tool during the last decade for the seismicassessment of buildings. Nevertheless, its main advantage of lower computational cost compared to non-linear dynamic time-history analysis (THA) is counter-balanced by its inherent restriction to structureswherein the fundamental mode dominates the response. Extension of the pushover approach to considerhigher modes eects has attracted attention, but such work has hitherto focused mainly on buildings,while corresponding work on bridges has been very limited. Hence, the aim of this study is to adapt themodal pushover analysis procedure for the assessment of bridges, and investigate its applicability in thecase of an existing, long and curved, bridge, designed according to current seismic codes; this bridgeis assessed using three nonlinear static analysis methods, as well as THA. Comparative evaluation ofthe calculated response of the bridge illustrates the applicability and potential of the modal pushovermethod for bridges, and quanties its relative accuracy compared to that obtained through other inelasticmethods. Copyright ? 2006 John Wiley & Sons, Ltd.

KEY WORDS: bridges; seismic design; modal pushover

INTRODUCTION

Although elastic analysis provides a useful estimate of the expected dynamic response of abridge, as a rule it cannot predict the failure mechanisms or the redistribution of forces thatfollow plastic hinge development during strong ground shaking. Nonlinear static (pushover)

Correspondence to: A. J. Kappos, Laboratory of Concrete and Masonry Structures, Department of Civil Engineering,Aristotle University of Thessaloniki, Greece.

E-mail: [email protected] Student.Professor.Lecturer.

Contract=grant sponsor: General Secretariat of Research and Technology (GGET)

Received 26 September 2005Revised 10 March 2006

Copyright ? 2006 John Wiley & Sons, Ltd. Accepted 14 March 2006

1270 T. S. PARASKEVA, A. J. KAPPOS AND A. G. SEXTOS

analysis on the other hand, is a widely used analytical tool for the evaluation of the structuralbehaviour in the inelastic range and the identication of the locations of structural weaknessesas well as of failure mechanisms [1, 2]. Nevertheless, the method is limited by the assump-tion that the response of the structure is controlled by its fundamental mode. In particular,the structure is subjected to monotonically increasing lateral forces with an invariant spatialdistribution until a predetermined target displacement is reached at a monitoring point. As aresult, both the invariant (fundamental mode based) force distributions and the target displace-ment, do not account for higher mode contribution, which can aect both, particularly in theinelastic range, thus limiting the application of the approach to cases where the fundamentalmode is dominant.Extension of the standard pushover analysis (SPA) to consider higher modes eects has

attracted attention, the eort being to match as closely as possible the results of nonlineartime-history analysis (NL-THA). In an early eort [3], the multi-mode pushover procedure wasused to identify the eects of higher modes in pushover analysis of buildings by appropriatelyextending the capacity spectrum method (CSM), which directly compares building capacity toearthquake demand; separate pushover curves were derived for each mode, without an attemptto combine modal responses. A series of adaptive multi-mode pushover analysis methodsfollowed References [46], involving redenition of the loading pattern, which is determinedby modal combination rules (e.g. SRSS of modal loads), at each stage of the response duringwhich the dynamic characteristics of the structure change (usually at each step when a newplastic hinge forms).While in the aforementioned adaptive methods modal superposition is carried out at the

level of loading, in the modal pushover analysis (MPA) proposed by Chopra and Goel [7],subsequently improved by the same authors [8, 9], pushover analyses are carried out separatelyfor each signicant mode, and the contributions from individual modes to calculated responsequantities (displacements, drifts, etc.) are combined using an appropriate combination rule(SRSS or CQC). Although the rule of superposition of modal responses does not apply inthe inelastic range of the response (modes are not uncoupled anymore), Goel and Chopra [8]have shown that the error, taking the results of inelastic THA as the benchmark, is typicallysmaller than in the case that superposition is carried out at the level of loading (with xedloading pattern), as recommended in the FEMA356 Guidelines [10]; these guidelines adoptthe nonlinear static procedure (NSP), i.e. pushover analysis, carried out with two dierentloading patterns, one based on rst mode loading (triangular distribution) and one withmodal distribution (SRSS combination of elastic modal loads).In another recent development, Aydinoglu [11] has proposed the so-called incremental

response spectrum analysis (IRSA), wherein each time a new hinge forms in a structure,elastic modal spectrum analysis is performed, taking into account the changes in the dynamicproperties of the structure.So far, most of the work performed in the direction of extending the applicability of

pushover analysis to structures with more complex dynamic characteristics focused on build-ings. Bridges, on the other hand, are structures wherein higher modes usually play a morecritical role than in buildings; hence developing a modal pushover procedure for such struc-tures is even more of a challenge than in the case of buildings. From the previouslymentioned studies attempting to account for higher modes in pushover analysis, only that ofAydinoglu [11], which focuses mainly on buildings, includes an application to a bridge struc-ture; the IRSA procedure is used, taking one or eight modes into account, without detailed

Copyright ? 2006 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2006; 35:12691293

SEISMIC ASSESSMENT OF BRIDGES 1271

discussion of the resulting dierences. At the same time as Aydinoglu [11], two more studiesinvolving higher mode eects in pushover analysis of bridges appeared [12, 13]; the study byKappos et al. [12] applies a multi-modal pushover procedure generally similar to that ofChopra and Goel [7] to an actual curved bridge (studied in more detail herein) considering itsrst three transverse modes, and compares the resulting displacements with those of single-mode pushover and of THA for spectrum-compatible records. In the studies by Fischingeret al. [13] and Isakovic and Fischinger [14] slightly dierent versions of these three methods,as well as IRSA, are used for the analysis of hypothetical irregular, torsionally sensitive bridges(an interesting, albeit extreme, case, appropriate for testing the limitations of the method),and results are compared. Very recently Pinho et al. [15] applied a number of existingpushover procedures (standard and adaptive), as well as a new version of adaptive pushover(called displacement-based adaptive pushover) to a number of idealized bridges (regular andirregular), and compared with results from incremental inelastic dynamic analysis. The needfor further work is pointed out in all these few studies on bridges.In view of the previous considerations, the present study rst attempts to extend the MPA

procedure [79], which was found to provide good results for buildings and (unlike theadaptive procedure) can be implemented using standard software tools, to the case of bridges.It then proceeds to quantify the relative accuracy of three inelastic analysis methods, i.e. SPA(with single-mode or multi-mode load patterns), MPA, and NL-THA, by focusing on therealistic case of a long and curved-in-plan, actual bridge, analysed with the aid of a three-dimensional model. This structure was deemed to be the most appropriate one (for reasonsexplained in more detail in a later section) for testing the pushover methods, among a numberof actual bridges in Greece that were studied by the authors, as part of an ongoing researchproject (see Acknowledgements).

PROPOSED EXTENSION OF MPA TO BRIDGES

According to the MPA procedure [79], standard pushover analysis is performed for eachmode independently, wherein the elastic modal forces are applied as invariant seismic loadpatterns. Modal pushover curves are then plotted and can be converted to capacity diagramsusing modal conversion parameters [3, 16]. Response quantities are separately estimated foreach individual mode, and then superimposed using an appropriate modal combination rule.The basic steps of the method have been presented by Chopra and Goel [7], but a set ofadditional assumptions and decisions regarding alternative procedures that can be used areneeded in order to apply the method in the case of bridges; a key issue is the selection of anappropriate point for monitoring the displacement demand (and also for drawing the pushovercurve for each mode). Other issues include the way a pushover curve is bilinearized beforebeing transformed into a capacity curve, the use of the capacity spectrum for dening theearthquake demand for each mode and then combining modal responses, and the number ofmodes that should be considered in the case of bridges. For the sake of completeness (andthe benet of the reader) all steps of the procedure (including those that are the same as inthe Chopra and Goel method) are summarized in the following.

Step 1: Compute the natural periods, Tn and modes n, for linearly elastic vibration ofthe structure. It is noted that in the case of bridges, the number of modes that have to beconsidered is signicantly higher than in the case of buildings; in fact, in order to capture all



Vbn

urn Sd1

Vbny

urny

1

Sdn

Sa

Idealized curve

Actual curve

1ankn

kn

an4 2

n2

Demand diagram

4 2

n2

1

Figure 1. Idealized pushover curve of the nth mode of the MDOF system, and correspondingcapacity curve for the nth mode of the equivalent inelastic SDOF system.

modes whose masses contribute to at least 90% of the total mass of a complex bridge structure(a criterion commonly used in seismic codes) one might need up to a few hundred modes; thisissue is further discussed under Step 7. In reinforced concrete bridges, the elastic periodsshould correspond to the eective stiness of the members, which for piers that respondclose to their yield point or beyond it, could be taken as the secant stiness at yield of thecritical section, with a possible correction for tension stiening, for instance as suggested inEurocode 8 [17].Step 2: Carry out separate pushover analyses for force distribution, sn =mn, where m is

the mass matrix of the structure, for each signicant mode of the bridge, and construct thebase shear vs displacement of the monitoring point (Vbnurn) pushover curve for each mode.The selection of an appropriate monitoring point for bridges (in buildings it is typically theroof) is further discussed in the following (Step 5). Gravity loads are applied before eachMPA, and P- eects are included, if signicant (e.g. bridges with tall piers). It is notedthat the value of the lateral deck displacement due to gravity loads, urg, is negligible for abridge with nearly symmetrically distributed gravity loading.Step 3: A critical issue in MPA is the way that response quantities individually calculated

for each mode are superimposed, in the sense that modal contributions should correspond to thesame earthquake intensity. Most of the currently available procedures [10, 16, 17] developedfor SPA require that the pushover curve be idealized as a bilinear curve (Figure 1left),so that a yield point and ductility factor can be dened and then be used to appropriatelyreduce the elastic response spectra representing the seismic action considered for assessment.This idealization can be done in a number of ways, some more involved than others; it issuggested to do this once (as opposed, for instance, to the ATC [16] procedure) using thefull pushover curve (i.e. analysis up to failure of the structure, dened by a drop in peakstrength of about 20%) and the equal energy absorption rule (equal areas under the actualand the bilinear curve). It is noted that the remaining steps of the MPA procedure can beapplied even if a dierent method for producing a bilinear curve is used.Step 4: Several procedures are available [7, 10, 16, 17] for dening the earthquake dis-

placement demand associated with each of the pushover curves derived in Step 3. In thisstudy the CSM [3, 16] is used for dening the displacement demand for a given earthquakeintensity, hence Step 4 consists in converting the idealized Vbn urn pushover curve of themulti-degree-of-freedom (MDOF) system (calculated in Step 3) to a capacity diagram, as



shown in Figure 1 (right). The base shear forces and the corresponding displacements in eachpushover curve are converted to spectral accelerations (Sa) and spectral displacements (Sd),respectively, of an equivalent single-degree-of-freedom (SDOF) system, using the relationships[7, 16]:

Sa =VbnM n

(1)

Sd =urnnrn

(2)

wherein rn is the value of n at the reference (or monitoring) point, M n =Lnn is theeective modal mass, Ln=Tnm1 , n=Ln=Mn, and Mn=

Tnmn is the generalized mass, for

the nth natural mode. For inelastic behaviour, the procedure used in the present study forestimating the displacement demand at the monitoring point is based on the use of inelasticspectra; this is equally simple, more consistent, and more accurate than the ATC [16] proce-dure based on reducing the elastic spectra with ductility-dependent damping factors, as shownin a number of studies [18, 19]. Several options are available for deriving inelastic spectra bydividing the elastic spectra with the reduction factor R (ratio of the elastic force demand toyield strength). In this paper, the formula proposed by Miranda and Bertero [20], was used

R= 1

+ 11 (3a)

where , is a function of the ductility factor , period T , and soil conditions at the site, andis given by

=1 +1

10T T 12T

exp

[32

(ln T 3

5

)2](3b)

for rock sites, while similar expressions apply for other soil conditions. It is recalled thatin the CSM the relevant intersection point (of the demand and the capacity diagrams) is theone for which the ductility factor calculated from the capacity diagram matches the ductilityvalue associated with the intersecting demand curve (inelastic spectrum). In many bridges, thepredominant modes correspond to long fundamental natural periods, and for those the equaldisplacement rule applies, hence there is no need for iterating to arrive at inelastic spectraconsistent with the ductility demand; however, in general some iteration is required, at leastfor higher modes. An alternative, computationally more demanding, procedure was used (forbuildings) by Chopra and Goel [7], wherein for each SDOF system with known period Tnand damping ratio n, the displacement demand was calculated from an NL-THA for thegiven motion; as a simpler alternative, they recommended using inelastic spectra (as detailedin Reference [18]).Step 5: Since the displacement demand calculated in Step 4 (for each mode) refers to SDOF

systems with periods equal to those of the corresponding modes, the next step is to correlatethese displacements to those of the actual bridge. Hence, Step 5 consists in converting thedisplacement demand of the nth mode inelastic SDOF system to the peak displacement ofthe monitoring point, urm of the bridge, using Equation (2). The selection of this point is acritical issue for MPA of bridges and is discussed in the following.Natural choices for the monitoring point in a bridge are the deck mass centre [17], or

the top of the nearest to it pier, if the displacement of the two is practically the same,



i.e. for monolithic or hinged pier-to-deck connections, but not for sliding or exible connec-tions (e.g. through pot bearings or elastomeric bearings). By analogy to building structures[18], it can also be selected as the point of the deck that corresponds to the location (xn ) alongthe longitudinal axis of the bridge of an equivalent SDOF system, dened by the location ofthe resultant of the modal load pattern of Step 2, which can be calculated from the propertiesof the MDOF system using the following relationship:

xn =

Nj=1xjmjjnNj=1mjjn

(4)

in which, xj is the distance of the jth mass from a (selected) point of the MDOF system (ina bridge, the left abutment is a natural choice), and jn is the value of n at the jth mass;xn is essentially independent of the way the mode is normalized. It is noted that whereasin buildings locating the SDOF system to a height above the ground dened by (4) ensuresthat the overturning moment at its base is the same as that resulting in the MDOF structurefrom the application of the modal load pattern of Step 2, in bridges it simply ensures that themoment at the abutments resulting from applying the base shear at a distance xn is the sameas that resulting from the modal loads applied on the actual (MDOF) bridge, which is a lessimportant condition than that regarding the overturning moment in buildings.Another proposal [13, 14] for the monitoring point of the bridge is at the point of the

deck where the (current) displacement is maximum, while in the present study the top ofthe pier that exhibits the most critical plastic rotation (again, for identical pier and deckdisplacements), which does not have to be the same for all individual analyses (i.e. for allmodes) was also used. An initial analysis of the structure for each mode is required in thelatter case, to dene the most critical location that will be used for constructing the rele-vant pushover curve (Figure 1); even this extra eort is not always enough when multipleearthquake intensities are considered, since the location of the critical point might changeas the bridge enters the inelastic range and the relative contribution of each mode possiblychanges. On the other hand, if the current maximum displacement point (of the deck) isselected, appropriate software should be used, or post-processing of the results is necessary,which in the case of large-scale 3D structures (such as the bridge studied in the next sec-tion) involves rather substantial eort. As will be made clear from the case study presentedlater, the selection of the monitoring point aects the shape of the pushover curve (as alsonoted in an earlier study by the writers [21]), as well as the second branch of the capacitycurve; however it does not aect the initial branch of the capacity curve, whose slope isequal to !2n (i.e. the square of the circular frequency of the considered mode) if the initialslope of the corresponding pushover curve reects the elastic (eective) stiness properties ofthe bridge.Step 6: The response quantities of interest (displacements, plastic hinge rotations, forces in

the piers) are evaluated by extracting from the database of the individual pushover analysesthe values of the desired responses rn, due to the combined eects of gravity and lateral loadsfor the analysis step at which the displacement at the reference point is equal to urn (seeEquation (2)).Step 7: Steps 36 are repeated for as many modes as required for sucient accuracy.

Judging the required number of modes is far from straightforward in the case of bridges. Asmentioned in Step 1, capturing all modes whose masses add up to 90% of the total mass of a



long and=or complex bridge structure might need considering up to a few hundred modes (asan example, modal spectral analysis of the 1036m long, 8-span, Arachthos bridge, currentlyunder construction in Heperus, Greece, involved consideration of 450 modes, to capture 90%of the total mass in all directions, including the vertical one). On the other hand, work carriedout within the present study (partly reported herein) has shown that there is little merit inadding modes whose participation factor is very low (say less than 1%), and less rigid rulesthan the 90% one (calibrated only for buildings) could be adopted.Step 8: The total value for any desired response quantity (and each level of earthquake

intensity considered) can be determined by combining the peak modal responses, rno usingan appropriate modal combination rule, e.g. the SRSS combination rule, or the CQC rule.This simple procedure was used for both displacements and plastic hinge rotations in thepresent study, which were the main quantities used for assessing the bridges analysed (whoseresponse to service gravity loading was, of course, elastic).If member (e.g. pier) forces have to be determined accurately (in an inelastic procedure

this is equivalent to determining the percentage by which yield strength of the members isexceeded), a more involved procedure of combining modal responses should be used. Sucha procedure was suggested by Goel and Chopra [8] for buildings, consisting essentially incorrecting the bending moments at member ends (whenever yield values were exceeded) onthe basis of the relevant momentrotation diagram and the value of the calculated plastichinge rotation. This procedure, which blends well with the capabilities of currently availablesoftware, has also been used in the case study presented in the next section.

EVALUATION OF ALTERNATIVE INELASTIC ANALYSIS PROCEDURESIN THE CASE OF AN EXISTING BRIDGE

Description of studied bridge

To investigate the accuracy, eciency, and also the practicality, of the proposed procedure itwas deemed appropriate to apply it on an actual bridge structure, whose length, curvature inplan, and complex support conditions resulted in an increased contribution of higher modes.The selected Krystallopigi Bridge is a 12-span structure of 638m total length (Figure 2) thatcrosses a valley in northern Greece, and is characterized by a large curvature in plan (radiusequal to 488m). The deck consists of a 13m wide prestressed concrete box girder section (seeinsert in Figure 2left). Piers are rectangular hollow reinforced concrete members, a typicalchoice in modern European motorway bridges. As is common in valley-crossing viaducts, thepier height varies along the length. In particular, the structure is supported on 11 piers ofheights between 11 and 27m. The support of the deck on the piers follows a rather complex,albeit not uncommon in modern bridges in seismic areas like Greece, pattern wherein for theouter piers 1, 2, 3, 9, 10, and 11 (see Figure 2) a bearing-type pier-to-deck connection isadopted, allowing movement in the longitudinal (tangential, due to the curved conguration)direction, but restricting movement in the transverse (radial) direction, while the inner (taller)piers 48 are monolithically connected to the deck. This complex articulation scheme assures arather favourable seismic response (without excessive concentration of demand in the squaterpiers), while it also permits avoiding completely expansion joints in the deck (except, ofcourse, at the abutments). It is also noted that for practical reasons (i.e. anchorage of the



Figure 2. Layout of the bridge conguration and nite element modelling.

prestressing cables) the initial 0:50 0:20m pier section is widened to 0:70 0:20m at thepier top. The piers are supported on pile groups of length and conguration that dier betweensupports due to the change of the soil prole along the bridge axis.The Greek Seismic Code (EAK2000) design spectrum [22] for Zone II (PGA of 0:24g) was

the basis for seismic design; it corresponded to soil conditions category B of the Code, whichcan be deemed equivalent to subsoil class B of older drafts of Eurocode 8-2. Moreover, animportance factor equal to 1.30 was adopted (as prescribed for bridges of high importance),hence the bridge was nally designed for a spectrum scaled to a PGA of 0:31g. A behaviour(or force reduction) factor of 3.0 was adopted for design, i.e. the bridge was designed as aductile structure (plastic hinges expected to form in the piers).The bridge was assessed using inelastic standard and modal pushover analysis (the

demand spectrum for both pushover analyses was the design one, or multiples of it), pushoveranalysis for an SRSS combination of modal loads (one of the methods recommended byASCE-FEMA 356 [10]), as well as NL-THA, for articial records closely matching thedemand spectrum. Analyses were carried out using the SAP2000 program [23]; additionalverication of results was made with extra analyses using the general purpose FE programANSYS [24]. The reference nite element model involved 220 nonprismatic 3D beam ele-ments, while appropriate nonlinear links and plastic hinges were utilized for time-history andstatic inelastic analyses, respectively. For the piers connected to the deck through bearings,the movement along the longitudinal axis, as well as the rotation around both the longitudinaland transverse axis was unrestrained. On the contrary, the presence of shear keys (Figure 2)resulted in the restraint of transverse (radial) displacements, as well as displacement along,and rotation about, the vertical axis. For the pushover analyses, the inelastic behaviour wassimulated through software built-in plastic hinges, whereas for the case of THA, a compatible



lumped plasticity model (i.e. nonlinear links) was used. Elastic parts of the piers were mod-elled with cracked stiness properties allowing for moderate tension stiening, as per theEurocode 8 [17] recommendations.In the analyses presented in the following, the focus is on the transverse response of the

bridge, as it is well known (e.g. Reference [13]) that this is the response most aectedby higher modes; additional analyses in the longitudinal direction are also briey presented.Soilstructure interaction (SSI) was accounted for in previous studies by the writers, throughthe use of an appropriate foundation (pile group) stiness matrix [25]. Due to the relativelysti soil formations underneath the studied bridge, SSI was found [21] to little aect theresponse (no more than 15%), hence it was considered that the relative evaluation of the threetypes of nonlinear analysis is not aected by SSI eects, which were subsequently ignoredherein.

SPA

A fundamental mode-based (standard) pushover analysis was rst performed, to serve asthe reference (i.e. the least involved procedure) for assessing the inelastic response of thebridge studied. It is worth noting that unlike the case of buildings, wherein the pushovercurve is generally dened in terms of base shear vs top displacement (in the direction underconsideration), in bridges the shape of the pushover curve depends on the location of themonitoring point (particularly when piers are of unequal height, as in the bridge studied). Thedisplacement of the monitoring point is used not only as a parameter of the capacity curve,but also to establish the seismic demand along the structure at the estimated peak displacement(Step 5). In the case of Krystallopigi Bridge, the monitoring point was initially selected as thedeck displacement at the location of the middle pier 6 of the bridge (monolithically connectedto the deck), which practically coincided with the centre of mass of the structure (which isthe recommended location in Eurocode 8).A typical pushover curve calculated by applying the modal load pattern of the rst mode in

the transverse direction of the bridge is shown in Figure 3 (referring to the deck displacement

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045

Transverse direction

Longitudinal direction

Plastic hinge development

V/W

Figure 3. Base shear vs deck displacement at central pier location, and sequence of plastic hingeformation in both directions of the bridge.



at the location of pier 6); a similar curve was also derived for the longitudinal direction.As discussed in more detail elsewhere [21], the overall performance of the bridge was verysatisfactory, since neither local nor global failure was predicted, even under seismic actionsthat far exceed the design level. The sequence of plastic hinge formation (along with thenormalized force (V=W )displacement (=H) curves), was also derived for both the longitudi-nal and transverse direction of the bridge. It is noted (Figure 3) that plastic hinge formation ismuch closer to being simultaneous in the longitudinal direction in which the bridge behavesmuch more like an SDOF system, than in the transverse one, wherein hinging is also aectedby higher modes and takes places at distinctly dierent stages of the response, somethingwhich is also manifested by the much steeper slope of the yielding branch of the bilinearcurve (strain-hardening) in this direction.In addition to the rst mode loading pattern, wherein the force at each node is proportional

to the modal displacement and the corresponding nodal concentrated mass (and acts in thedirection of the modal displacement), for comparison purposes, the alternative pattern ofuniform loading was also used in SPA; according to this pattern, the force acting at eachnode is proportional to the nodal mass in the direction of the control displacement. Thispattern is usually required by codes for pushover analysis of buildings [10], mainly as a(rough) means of identifying critical combinations of shear and exure (V=M).Results indicated that the adoption of a particular loading model for the SPA plays indeed

an important role with respect to the inelastic response of the bridge. In particular, when theuniform loading pattern was applied, the overall strength of the system appeared to be higher(i.e. yielding occurred at a higher level of base shear); this trend was found to be clearerin the transverse direction. The more pronounced eect in the transverse direction should beattributed to the fact that due the shape of the rst transverse mode, the largest displacementcorresponds to the middle pier (P6) location, hence the modal force is higher at that particularpier compared to the rest, whereas in the uniform pattern, forces at all piers are about thesame (since masses are similar). As a result, for the same displacement demand, higherforces are developed in the latter case. It is interesting to note that a similar overestimationof strength is found in pushover analysis of buildings, but in that case the key reason isthe distribution of the lateral loading along the height of the structure (which results inhigher overturning moment at the base in the case of modal loading). The above observationis an indication that consideration of higher modes in the MPA procedure could possiblyhighlight aspects of the inelastic response of the bridge that would be otherwise hiddenin SPA.In terms of the overall assessment obtained through the reference approach (SPA) presented

herein, it can be also concluded that the available displacement ductility was found to be highin both principal directions (i.e. 6.1 in the longitudinal and 4.6 in the transverse direction)due to the signicant available curvature ductility at the critical locations of potential plastichinging; these local ductilities resulted from the application of provisions that are relatedto the prevention of longitudinal bar buckling and the capacity design against shear of themiddle piers. Moreover, a signicant overstrength of the bridge was found due not only tothe partial safety factors applied during design, but also to the use of a minimum amountof reinforcement even when the strength demand is less (code minimum requirements), andto force redistribution after yielding. It has to be noted that the observed overstrength washigher in the transverse direction, since the longitudinal direction was the critical one for thedesign of pier reinforcement.



MPA

The dynamic characteristics required within the context of the MPA approach, were determinedusing standard eigenvalue analysis. Figure 4 illustrates the rst four transverse mode shapesof the bridge (note that the lowest mode is a longitudinal one, with T =1:46 s), togetherwith the corresponding participation factors and mass ratios, as well as the locations (fromEquation (4)) of the equivalent SDOF system for each mode. It is seen that the modal massparticipation factors of higher transverse modes are much lower than that of the fundamentaltransverse mode. Moreover, even the lowest antisymmetric mode (with T =0:73 s), has amodal mass participation factor equal to less than 0.1% in the transverse direction, and istherefore ignored in the results presented subsequently; this mode is the one with the highestparticipation factor (15%) in torsion (rotation of the deck around a vertical axis passing fromits mass centre). Both of the previous characteristics should be attributed to the curvature inplan of the bridge, and the restraint of the transverse movement at the abutments. Considerationof the four modes of Figure 4 assures that more than 90% of the total mass in the transversedirection is considered.Applying the modal load pattern of the nth mode in the transverse direction of the bridge,

the corresponding pushover curve, involving the deck displacement at the central pier location,as well as the position of the corresponding equivalent SDOF system (Equation (4)), wasconstructed and then idealized as a bilinear curve. The way the bilinearization of the curveis made is important, since it aects the eective period of the bridge in each mode. If therst branch of the bilinear curve is drawn based on the part of the actual pushover curvethat corresponds to elastic response of the bridge, the eective period remains the sameas that of the pertinent elastic mode; in this case the resulting capacity curve is the sameregardless of the selection of monitoring point. If the bilinearization is carried out using equalenergy absorption or similar concepts, without specically intending to dene the rst branch

Figure 4. Force distribution, sn =mn, location of the equivalent SDOF systems, and modal parametersfor the main transverse modes of the bridge.



0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

EAK2000 mode1 (a),(b) mode2 (a),(b) mode3 (a)mode3 (b) mode4 (a) mode4 (b)

Sa

S d (m)

(m/s2

)

Figure 5. Capacity curves derived with respect to the deck displacement: (a) at the location of the deckmass centre; and (b) at the location of the resultant of the modal forces, for the four transverse modes

(the elastic spectrum of the design earthquake is also shown).

on the basis of the linear elastic part of the response, the eective period of the correspond-ing mode will be longer than the elastic one, and the resulting capacity curve will depend(to a certain extent) on the selection of monitoring point, as will be made clear in thefollowing.The bilinearized pushover curves for the four transverse modes were converted to the

capacity curves shown in Figure 5, using the procedure described earlier in Step 4; curveswere drawn both with respect to the mass centre of the deck and the location of the resultantof the modal forces. It is noted that these curves are not necessarily representative of theactual response of all structural members of the bridge. For instance, the capacity curvecorresponding to modes 3 and 4 is purely linear, hence conveying the impression that thebridge does not enter the inelastic range when subjected to the third or fourth mode loadpattern, even for very high accelerations. In reality, it is only the central pier that respondselastically in those cases, whereas the edge piers do enter the inelastic range; this is clearlydue to the form of the load pattern of these two higher modes (see lower row of Figure 4),which is obviously not critical for the central pier. It is worth pointing out here that unrealisticshapes of pushover curves for higher modes have also been noted in the case of buildings;as discussed in a recent paper by Goel and Chopra [26], even reversal in a higher-modepushover curve might occur after formation of a plastic mechanism, if the resultant forceabove the bottom of the mechanism is in the direction that moves the roof in a directionopposite to that prior to formation of the mechanism. It is important, though, that despite therather misleading form of the aforementioned curves for some higher modes, correct localquantities like plastic hinge rotations can be extracted from the pushover analysis results, solong as the displacement demand is the correct one.To explore other possibilities that might produce more representative pictures of the overall

behaviour of the bridge, alternative pushover and capacity curves were derived with respectto the deck displacement at the location of: (a) the most critical pier (in terms of maximumplastic rotation) for each individual modal load pattern; (b) the maximum deck displacement,



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Figure 6. Capacity curves derived with respect to the deck displacement: (a) at the location of themost critical pier for each mode; and (b) at the location of the current maximum displacement.

which generally changes during the analysis procedure. It is clear that to identify the criticalpier in each case (in order to construct the pushover curve with respect to its location), apreliminary pushover analysis of the structure is required. After carrying out such analyses,it was decided to draw the capacity curve of the rst transverse mode in terms of the deckdisplacement at the central pier location, that of the second mode in terms of the deckdisplacement at pier 10 location, that of the third mode in terms of the deck displacementat pier 3 location, and that of the fourth mode in terms of the deck displacement at pier2 location (Figure 6). On the other hand, the second option based on the current point ofmax displacement could not be implemented with currently available software [23, 24], and anad hoc post-processor had to be written, to handle the large volume of results produced bythe 3D pushover analysis of this long and curved bridge.Notwithstanding the aforementioned increase in computational eort, the capacity curves

produced using the alternative procedures are clearly more representative of the actualbehaviour of the bridge, since they indicate (correctly) that at some stage of the responseand for an earthquake intensity that is higher (in most cases much higher) than the designone, one or more piers of the structure yield. While it is clear that yielding of the structurewill initiate from its response to the fundamental mode (the corresponding capacity curve hasthe lowest ordinate), one cannot predict a priori which the second most critical mode willbe; for instance, in the studied bridge, the capacity curves of Figure 6 indicate that yieldingdue to higher mode eects will initiate due to third mode response (at the outer piers). Fromthe practical application point of view, two remarks are in order: (a) With regard to the fun-damental transverse mode, the pushover analysis can be carried out as suggested by currentcodes such as the Eurocode 8-2 [17], i.e. drawing the pushover curve with respect to the dis-placement at the mass centre of the deck. (b) The capacity curves for higher modes drawn onthe basis of the most critical pier and those drawn on the basis of the current point ofmaximum deck displacement are generally very close to each other (Figure 6), and for



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Figure 7. Response to the design earthquake (transverse direction): (a) deck displacements at pierlocations; and (b) plastic rotation at the base of the piers.

practical purposes either one can be used (especially considering the other uncertainties in-volved in the procedure). In this case, the choice will depend on whether the analyst willprefer to carry out two analyses (the rst one for identifying the most critical pier), or developa post-processor for storing the values of base shear and maximum displacement (which,for transverse response of curved bridges should be the radial, rather than the global Ydisplacement) at each step of the analysis.Another important aspect of the MPA procedure is the estimation of the displacement

demand with the aid of the CSM, according to Step 4 of the methodology presentedearlier. Depending on the reference point selected for drawing the pushover curve for eachmode, Equation (2) will give a dierent value of urn, whereas the spectral displacement Sd isindependent of the selection of monitoring point if the elastic branch of the pushover curvecorresponds to the elastic period of the pertinent mode of the bridge, as discussedearlier.The peak modal responses rno, each determined by a pushover analysis, are then combined

using an appropriate modal combination rule, to obtain an estimate of the peak value, ro, ofthe total response. Figures 7(a) , 8(a) and 9(a), illustrate the radial (i.e. perpendicular to the



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(b)Figure 8. Response to 1.5 times the design earthquake (transverse direction): (a) deck displacements at

pier locations; and (b) plastic rotation at the base of the piers.

tangent to the deck axis) displacements of the deck at the location of each pier for the foursignicant modal patterns along the transverse direction, as well as the total displacementsas derived by applying the SRSS combination rule to the quantities corresponding to thedisplacement demand calculated for each mode using the CSM. To investigate the eect of thelevel of inelasticity on the calculated response, dierent levels of excitation were considered,i.e. the design earthquake was multiplied by a factor of 1.0, 1.5 and 2.0; Figures 7(b), 8(b)and 9(b), present the respective plastic rotations, developed at the basis of the piers for eachexcitation level, and their SRSS combinations (MPA).From the displacement distributions shown in Figures 7(a), 8(a) and 9(a), it is noted that

the contribution of the rst transverse mode to the overall response is signicant for the caseof the design earthquake, but it is reduced as the excitation level increases, since higher modesare participating more actively, particularly towards the ends of the bridge. Consequently, thedierence between the displacement prole calculated from SPA and that from MPA becomesmore substantial as the induced level of inelasticity increases. For the design earthquake level,plastic hinges primarily form at the base of the central piers due to the dominant participationof the rst mode (Figure 7(b)), while additional plastic hinges form at P3, P9 and P10 for thecase of 1.5 times the design earthquake (Figure 8(b)) and P3, P4, P9 and P10 for the case



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Figure 9. Response to twice the design earthquake (transverse direction): (a) deck displacements at pierlocations; and (b) plastic rotation at the base of the piers.

of twice the design earthquake (Figure 9(b)). This is a strong indication that the necessity ofimplementing the MPA is closely related to the magnitude of earthquake forces, as well asto the characteristics of the structure itself.

NL-THA

In line with most previous similar studies, it was deemed necessary to compare results ofthe standard and modal inelastic pushover approaches with those from NL-THA, the latterconsidered to be the most rigorous procedure to compute seismic demand. To this eect, a setof NL-THAs was performed using ve articial records compatible with the EAK2000 elasticspectrum and generated with the use of the computer code ASING [27]. The Newmark =1=2,=1=4 integration method was used, with time step t=0:002 s and a total of 10 000 steps(20 s of input). A uniform damping value of 3.5% was assumed for all modes of vibration;as discussed in Reference [21], this is a value lying between that typically used for reinforcedconcrete structures (5%) and that for prestressed concrete structures (2%).It has to be noted that plastic hinging in the piers had to be modelled slightly dierently

in the NL-THA and the pushover analysis, due to limitations of the software used [23]. In



particular, nonlinear rotational spring elements were used in the FE models used in NL-THA,while the built-in beam hinge feature of SAP2000 was implemented in the models set upfor pushover analysis. In both cases, though, the same momentrotation (M) relationshipwas used (i.e. bilinear with 26% hardening, depending on the calculated ultimate moment),with input parameters dened from bre analysis performed for each particular pier section,utilising the computer program RCCOLA-90 [28].

Evaluation of dierent procedures

When assessing the feasibility of MPA, it has to be noted that the procedure is based on twoprincipal approximations: (a) coupling among modal coordinates associated with the modesof the corresponding linear system, arising from the yielding of the system, is essentiallyneglected, and (b) the estimate of the total response is obtained by combining the peakmodal responses using a statistical combination rule.In order to investigate the potential implications of each of these approximations, the

bridge was rst analysed elastically, using the SPA, MPA and the THA procedure, assumingelastic response in all cases. The peak deck displacements (radial direction) at pier locationscalculated from each analysis are shown in Figure 10; note that in this and all subsequentgures, the displacement demand is estimated independently in static and dynamic (time-history) inelastic analysis (unlike some previous studies wherein comparisons of displacementproles are made assuming the same maximum displacement in both cases). It is observed that,consistently with the mode shapes depicted in Figure 4, the main dierence between displace-ments calculated from THA and those from the two static methods is towards the abutmentsof the bridge, with dierences diminishing in the area of the central piers (an area dominatedby the rst mode). MPA which accounts for the other three transverse modes is much closerto THA at the end areas of the bridge, but some dierences persist, possibly indicating a biasin the MPA procedure due to the estimation of the total response by using the modal combi-nation rule (SRSS); as pointed out by Fischinger et al. [13], other rules that dully account for

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Figure 10. Deck displacements at pier locations calculated from SPA,MPA and THA, for elastic behaviour.



the sign of the response quantities (not only their magnitude) might be more appropriate inthis case.The displacements determined by the SPA and MPA procedures were compared to those

from NL-THA for increasing levels of earthquake excitation, as shown in Figures 11(a),12(a) and 13(a). It is noted that the deck displacements shown in the gures as the NL-THA case are the average of the maximum displacements recorded in the structure during theve THAs.From Figure 11(a) it is observed that MPA predicts very well (i.e. matches closely the

values from the NL-THA approach) the maximum transverse displacement of the bridge(150mm, compared to the 156mm predicted by NL-THA). On the other hand, both pushoveranalysis procedures underestimate the displacements of piers P3, P4, P8, P9 and P10 comparedto the more rened NL-THA approach. This can be primarily attributed to the fact that duringNL-THA, plastic hinges additional to those predicted from MPA develop at the base of thesepiers (Figure 11(c)), leading to relatively larger pier top displacements, and to a lesser extentto the fact that relatively higher values of plastic rotations were derived through NL-THA atthe locations where plastic hinges develop, as seen in Figure 11(b). In any case, it is notedthat plastic rotation demands for the design earthquake are very modest for all piers (between0.001 and 0.002 rad), another indication of the sound seismic design of the bridge, and pierswherein no plastic hinges formed, were found in MPA to be very close to yielding; this willbecome clearer for higher earthquake intensities, as discussed in the following.As the level of excitation increases and higher mode contributions become more signicant

(without substantially altering the shape of the modes), the displacement prole derived bythe MPA method tends to match that obtained by the NL-THA, whereas SPAs predictionsbecome less accurate as the level of inelasticity increases. Especially, for twice the designearthquake intensity case presented in Figure 13, consideration of the higher modes with theproposed MPA scheme, signicantly improves the accuracy of the predicted displacements,although its predictions are rather poor (but still better than those from SPA) in the areasclose to the piers 5 and 8. Another signicant advantage of the MPA method is that it isable to capture the plastic hinge development at P2, P3, P4, P9 and P10, something the SPAfailed to do, hence, the overall degree of agreement between MPA and NL-THA is deemedquite satisfactory. SPAs eectiveness is similar to that of the MPA method only in the caseof the design earthquake where both methods capture well the inelastic behaviour of thecentral piers.Although in seismic assessment of structures the emphasis is mainly on displacement and

deformation quantities, the magnitude of forces developed in critical members (like the piers inbridges) is also of some interest. Figures 11(d), 12(d) and 13(d) show the maximum momentsat the base of the piers, calculated as described in Step 8 of the proposed procedure. It isclear that SPA strongly underpredicts moments (and corresponding shear forces) in the outerpiers of the bridge, whereas moments predicted from MPA are much closer to those resultingfrom THA. Not surprisingly, the dierences between MPA and NL-THA become smallerwhen higher intensities are considered, and all piers yield (Figure 13(d)), hence their bendingmoment is controlled by their exural capacity. Since, as discussed previously, SPA failedto identify hinge formation in all outer piers, it is only natural that corresponding momentpredictions are also very poor, particularly in P1, P2 and P11. It is worth pointing out that inthe strongly curved-in-plan bridge analysed herein, the shape of the rst mode (Figure 4) ischaracterized by very small transverse amplitudes towards the edges, which are also due to



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Figure 11. Response to the design earthquake calculated from SPA, MPA and NL-THA: (a) deckdisplacements at pier locations; (b) plastic rotation at the base of the piers; (c) plastic hinge pattern;

and (d) bending moment at the base of the piers.



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Figure 12. Response to 1.5 times the design earthquake calculated from SPA, MPA and NL-THA:(a) deck displacements at pier locations; (b) plastic rotation at the base of the piers; (c) plastic

hinge pattern; and (d) bending moment at the base of the piers.



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Figure 13. Response to twice the design earthquake calculated from SPA, MPA and NL-THA: (a) deckdisplacements at pier location; (b) plastic rotation at the base of the piers; (c) plastic hinge pattern;

and (d) bending moment at the base of the piers.



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Figure 14. Response to 1.5 times the design earthquake calculated from SPA, SRSS of modalloads (fno), MPA (CQC or SRSS rule) and NL-THA: deck displacements at pier locations.

the articulation scheme adopted by the designers of the bridge (free sliding in the tangentialdirection in piers 13 and 911); as a result of the curvature and the articulation scheme,radial displacements, and hence modal loads, are very small in these areas. In straight bridgesanalysed by the writers (not reported herein due to space limitations) predictions by SPA ofplastic rotations and moments in edge piers were generally better, but again results from MPAwere closer to those from THA.As a further comparison, pushover analysis using an alternative pattern of loading was

also conducted; according to this pattern, which is denoted as SRSS(fno) and is one of therecommended procedures in FEMA 356 [10], the force acting at each node is computed bycombining the modal loads from an appropriate number of modes (the rst four transversemodes here), using the SRSS rule. From Figure 14, it is seen that when the SRSS(fno) isapplied, only a minor improvement of the results obtained through SPA is achieved. It isalso noticeable that in this case, the results lie in-between those derived from the MPA andSPA procedures, generally being closer to the latter. Also shown in the same gure is anothercomparison performed for the cases that the MPA method is applied, assuming the standard(SRSS) or complete quadratic (CQC) combination rules. It is observed that the application ofthe CQC rule instead of the commonly used SRSS one, hardly aects the calculated response.A nal remark regarding the feasibility of matching the results of the two methods (pushover

and time-history) is that in NL-THA all structural modes (transverse, longitudinal, andvertical) are automatically included for the evaluation of the nonlinear response, whereasonly four transverse modes were used in the MPA method, and just a single mode in thestandard pushover analysis. This observation should be especially relevant in the case ofcurved bridges, like the one considered here, wherein transverse displacements are alsoaected by longitudinal response.

CONCLUSIONS

An existing methodology for MPA was extended here to the case of bridges, and its feasibilityand accuracy were evaluated by applying it to an actual long and curved-in-plan bridge,



designed to modern seismic practice. The key issue in this extension was the proper deni-tion of the monitoring point for estimating the earthquake displacement demand in pushoveranalysis, while additional issues addressed included the way the pushover curve is bilinearizedbefore being transformed into a capacity curve, the use of the capacity spectrum for deningthe earthquake demand for each mode, and the number of modes that should be consideredin the case of bridges.By applying inelastic SPA and MPA, as well NL-THA, to the aforementioned bridge, which

is strongly curved and with complex articulation, but properly designed according to moderncodes, it was concluded that:

All three methods yielded similar values of maximum inelastic deck displacement, how-ever the variation of displacement along the bridge was rather dierent. The SPA methodpredicted well the displacements only in the central, rst mode dominated, area of thebridge. On the contrary, MPA provided a signicantly improved estimate with respect tothe maximum displacement pattern, reasonably matching the results of the more renedNL-THA analysis, even for increasing levels of earthquake loading that trigger increasedcontribution of higher modes.

The MPA also provided a good estimate of the plastic hinge distribution (similar oreven identical to that indicated by the NL-THA) for earthquake intensities exceedingthe design one. SPA failed to identify plastic hinging in the outer piers of the bridge(even for twice the design earthquake intensity), and it strongly underpredicted bendingmoments in these piers.

Previous studies have indicated that SPA generally works reasonably well when appliedto bridges of regular conguration (as opposed to irregular ones, mainly those aectedby torsion). The present study revealed that a single mode-based load pattern should notbe used in bridges with strong curvature in plan, even when they qualify as regular(i.e. dominated by symmetric modes), especially when combined with articulationschemes that lead to very low modal amplitudes (and hence to very low modal loads)towards the edges of the bridge.

Carrying out pushover analysis based on a load pattern resulting from statistical com-bination of modal loads, improves slightly the results; however, they generally remaincloser to those of SPA than MPA (which, in turn is closer to NL-THA results). It isinteresting to note that such observations were also made in previous studies involvingpushover analysis of bridges.

On the basis of the results obtained for the studied bridge, MPA seems to be a promisingapproach that yields more accurate results compared to the standard pushover, with-out requiring the higher modelling eort and computational cost, as well as the othercomplications involved in NL-THA (like the selection and scaling of natural records, orthe generation of synthetic ones), or of other proposals (which have, of course, theirown merits) involving multiple eigenvalue analyses of the structure to dene improvedloading patterns in the inelastic range.

More work is clearly required, to further investigate the eectiveness of MPA by apply-ing it to bridge structures with dierent conguration, degree of irregularity, and dynamiccharacteristics (in terms of higher mode signicance, in particular bridges with impor-tant anti-symmetric modes), since MPA is expected to be even more valuable for theassessment of the actual inelastic response of bridges with signicant higher modes.



ACKNOWLEDGEMENTS

This work has been performed within the framework of the research project ASProGe: SeismicProtection of Bridges, funded by the General Secretariat of Research and Technology (GGET)of Greece.

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