seismic reliability of a cable-stayed bridge retrofitted with hysteretic devices

8/17/2019 Seismic Reliability of a Cable-stayed Bridge Retrofitted With Hysteretic Devices

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Seismic reliability of a cable-stayed bridge retrofittedwith hysteretic devices

Fabio Casciati a,*, Gian Paolo Cimellaro b, Marco Domaneschi a

a Department of Structural Mechanics, University of Pavia, Pavia, Italyb Department of Civil, Structural and Environmental Engineering, University at Buffalo, The State University of New York, Buffalo, NY 14260, USA

Received 18 May 2007; accepted 24 January 2008Available online 21 March 2008

Abstract

This study evaluates the fragility curves of a cable-stayed bridge by an analytical approach based on time history analysis. The pre-sentation of the vulnerability information in the form of fragility curves is a widely practiced approach when several uncertain sources areinvolved. The ASCE benchmark problem of a cable-stayed bridge is considered as a case study. A passive control technique is adoptedand results are compared in term of fragility curves. In order to consider the uncertainties related to the ground motion 24 groundmotion time history are considered, corresponding to four different hazard levels, while the uncertainties in the structural characteristicsare introduced by defining the different performance thresholds as random variables. The fragility evaluation shows how important acorrect estimation of the limit state is for the comparison of different retrofit techniques. 2008 Elsevier Ltd. All rights reserved.

Keywords: Benchmark; Bridge control; Cable-stayed bridges; Fragility; Passive control; Random variables

1. Introduction

In the past few decades, cable-stayed bridges have foundwide application throughout the world [1]. The increasingpopularity of contemporary cable-stayed bridges amongbridges engineers can be attributed to (i) the appealingesthetics; (ii) the full and efficient utilization of structuralmaterials; (iii) the increased stiffness over suspensionbridges; (iv) the efficient and fast mode of construction;

(v) the relatively small size of the bridge elements.The central span length of cable-stayed bridges has nowreach length of 1000 m or longer, leading to very longstayed cables. Long-span cables due to their flexibility,small mass and very low inherent damping [2], are suscep-tible to vibration with large amplitude under earthquake,wind, traffic and rain loadings. Therefore, cable-stayed

bridges due to the peculiarity of their structural behaviorare vulnerable to dynamic loadings such as earthquakes,strong wind loads and traffic loads.

It is a widely practiced approach to develop vulnerabil-ity information in the form of fragility curves (FC) [3] forbridges [4], since they are structures of relevant importance.A large number of methods have been proposed to com-pute fragility functions in the last 20 years, ranging fromexpert judgment, to data analysis on observed damage

[5], to fully analytical approaches [6]. Karim and Yamakazi[7] developed an analytical approach to construct fragilitycurves for the piers of specific bridges, using the dynamicresponse of an equivalent nonlinear single degree of free-dom system.

Other authors have used records of damage resultingfrom past earthquakes to develop empirical fragilitycurves. Bazon and Kiremidjian [8], using observationson bridge damage after the Northridge earthquake,developed empirical fragility curves by logistic regression.Shinozuka et al. [9] developed empirical fragility curves

0045-7949/$ - see front matter 2008 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruc.2008.01.012

* Corresponding author.E-mail addresses: [email protected] (F. Casciati), gpc2@buffalo.

edu (G.P. Cimellaro), [email protected] (M. Domaneschi).

www.elsevier.com/locate/compstruc

Available online at www.sciencedirect.com

Computers and Structures 86 (2008) 1769–1781

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assuming that the fragility curves can be expressed in theform of two-parameter lognormal distribution functions.The location and scale parameters of the distributionhave been estimated by the maximum likelihood methodusing the observed damage data from the 1995 Hyogo-ken Nambu (Kobe) earthquake. Tanaka et al. [10] used

a two-parameter normal distribution function instead of using a cumulative lognormal distribution to fit the fra-gility curve. 3683 bridges are grouped into five classesand the damage is ranked into five levels. These fielddata were used to determine the two unknown parame-ters of the distribution.

Many other works can be found in literature and this lit-erature survey is by no mean comprehensive, it is presentedhere to highlight several distinct techniques.

Disadvantages related to all these approaches are thattheir fragility curves are related to specific structural mod-els and cannot be used to assess the fragility of other struc-tures of the same type, unless in a very crude way. Another

disadvantage is that these models cannot be verified by lab-oratory testing, since this requires multiple physical modelsbrought to failure, which are too expensive and time con-suming. In addition, not all these models consider all theuncertainties involved in fragility analyses. For example,the performance limit threshold is usually considered as acrisp quantity, whereas it should be modeled as a randomvariable as well.

This study develops analytical fragility curves of acable-stayed bridge using Monte Carlo simulations. Theperformance threshold is modelled as a random variableassuming a lognormal distribution. The fragility curves

evaluated for the passive control techniques are comparedwith the fragility curves estimated for the uncontrolledcase.

2. Description of the methodology

The methodology described in this section aims at com-puting fragility functions of cable-stayed bridge modellingthe performance threshold as a random variable. Fragilitycurves are functions that represents the probability that theresponse R(x, I , t)={R1, . . ., Rn} of a specific structure (orfamily of structures) exceeds a given threshold rlim(-x, I , t)={rlim1, . . ., rlimn}, associated with a given limit state,conditional on earthquake intensity parameter I (Pga, Pgv,Return period, Sa, Sd, MMI, etc.). Response R andresponse thresholds rlim are functions of the structuralproperties of the system x, the ground motion intensity I and the time t. However, in this formulation it is assumedthat the response threshold r lim(x) does not depend on theground motion history and so it does not depend on time,while the response R±(x, I , t) of the generic i th componentis replaced by its maximum value over the duration of theresponse history Ri (x, I ). In the following, the dependenceof the response R(x, I ) on x and I , and the dependence of the response threshold rlim(x) on x will be omitted for con-

venience. With these assumptions, the general definition of

fragility can be written in the following form when thenumber of parameters to be checked is n:

F ¼ P [ni¼1

ð Ri P r limiÞ

( )

¼Xi P

[ni¼1

ð Ri P r limiÞ= I ¼ i( )

P ð I ¼ iÞ ð1Þ

where the conditional probability of exceeding P is givenby the following equation:

P [ni¼1

ð RiP r limiÞ= I

( )

¼Xn j¼1

P ðð R jP r lim jÞ= I Þ

Xn

i¼1 Xn

j¼2

P ðð RiP r limiÞð R jP r lim jÞ= I Þ

þXni¼1

Xn j¼2

Xnk ¼3

P ðð RiP r limiÞð R jP r lim jÞð Rk P r limk Þ= I Þ þ

þ ð1Þn P ðð R1P r lim1Þð R2P r lim2Þ . . . ð RnP r limnÞÞ ð2Þ

where (Ri P rlimi ), i = 1,2, . . . , n, are arbitrary events. Ri isthe response measure that can be either a deformationquantity like deck displacements, or a force quantity likebending moment or shear forces in bridge members orany other measure of damage for which adequate capacitymodels exist. ri ,lim is the response threshold related to a cer-tain quantity that is correlated with damage and I is theearthquake intensity. P (I = i ) is the probability that the

seismic input experienced is ‘‘level” i . The evaluation of fra-gility necessitates the definition of the performance thresh-old that is given through a generalized formula that allowsconsidering in the same formulation multiple limit statesrelated to different quantities. Details about the generaldescription of the methodology can be found in Cimellaroet al. [11,12]. The relation among different limit states canbe determined through an opportune choice of the param-eters that allows different shapes of performance threshold,and they can be estimated using probabilistic analysis andengineering judgment. In this framework, limit states canbe linear or nonlinear, dependent or independent, random

or deterministic. All these options can be formulated asparticular cases of the main general one with a suitablechoice of the parameters involved.

The generalized formula for multidimensional perfor-mance limit threshold can be formulated as:

LðrlimÞ ¼Xni¼1

r i lim

r i lim;0

Ni 1 ð3Þ

where ri lim is the dependent response threshold parameter(deformation, force, velocity, etc.), that is correlated withdamage, ri lim,0 is the independent response thresholdparameter and N i are the interaction factors determining

the shape of n-dimensional surface.

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The development of the analytical fragility curves of thecable-stayed bridge are illustrated in the following steps:

Step 1: Compute the maximum pier’s shear forces andmoment, deck’s displacement and cable tensionfor each earthquake record consistent with a given

hazard level.Step 2: Estimate the probability density function (PDS) of the response distribution – assumed lognormal – from the values of the mean and the standarddeviation.

Step 3: Generate a larger ensemble of random numberslognormally distributed.

Step 4: Count the number of times N f that the maximumresponse R exceeds a given performance thresholdrlim and divide by the total number of trials N related to the respective hazard level, so the prob-ability P f of reaching or exceeding the perfor-mance limit state is P f = N f /N . As N approaches

infinity, P f approaches the true probability of reaching or exceeding a limit state.

Step 5: The probability P f of exceeding the limit state isevaluated for different hazard levels (four in thecase study considered), so in the plane probabilityof exceedance vs. earthquake intensity as manypoints as the number of considered hazard levelscan be located;

Step 6 : Fragility curves are plotted showing the probabil-ity of reaching or exceeding a limit state vs. thecorresponding return period. Experimental datapoints corresponding to the different hazard levels

in the probability of exceedance vs. earthquakeintensity plane are transformed into a fragilitycurve by assuming that the response of the struc-tural system is lognormally distributed. Fragilitycurves in a more compact form are described bythe following equation:

F Y ð y Þ ¼ U 1

b lnð y =hY Þ

y P 0 ð4Þ

where U is the standardized cumulative normal dis-tribution function, h y is the median of y, and b isthe standard deviation of the natural logarithmof y [13]. A straightforward optimization algorithmbased on chi-squared v2 goodness-of-fit test ob-tains the estimation of the optimal parameters of the lognormal distribution (h y and b) [11].

Step7 : The procedure is repeated for every selected mem-ber of the cable-stayed bridge and fragility curvesare constructed. Then, performance levels of dif-ferent bridge elements are combined using Eq.(3) and the global fragility curve of the cable-stayed bridge is evaluated.

The advantage of this proposed methodology is that itallows considering dependences among different limit states

and the uncertainties of different limits, but it also allows

combining fragility curves of different members to obtainthe global fragility curve of the bridge.

2.1. Uncertainties of the limit states and failure modes

Engineers usually make their recommendations using

fragility curves based on deterministic performance limitstates obtained from design provisions, public policies doc-uments, engineering judgment, experimentation, etc. Thedefinition of performance limit states (PLSs) plays animportant role in the construction of fragility curvesbecause their values have a direct effect.

Therefore, a crisp description of PLSs can be inappro-priate. Instead, PLSs should be modeled as random vari-ables although often are described by crisp quantities(because the uncertainty in the earthquake load is con-siderably larger than the uncertainty in the PLSsthemselves).

In this study, the performance limit states are considered

as random variables, and are defined in terms of deck’s dis-placement, shear forces and moment at the base and decklevel of the tower and cable tensions, since they governthe behavior and failure modes of the cable-stayed bridge.

One damage state threshold has been considered thatcorresponds at the elastic limit state of bridge members(e.g. piers or towers). Bridge piers are designed to remainin the elastic range even during strong earthquakes, sothe possibility that the piers exhibit plastic behavior hasbeen assumed like bridge failure. A cable-stayed bridgeusually has many possible ways to reach collapse and manyfailures modes that are dependent each other. In fragility

analysis, it is essential to specify the failure modes for thestructural system considered, because the purpose of fragil-ity curves is to predict which component will fail and how,and with what probability. A refined fragility analysisrequires a good description of failure modes and theiruncertainties.

There are many uncertainties that should be consideredin fragility analysis of a cable-stayed bridge to obtain accu-rate results. However, in this paper a balance is foundbetween cost and accuracy of the analysis, so dependencybetween failure modes has been neglected. Uncertaintiesof the structural model are here considered through a ran-dom description of performance limit states. The choice of an ensemble of cable-stayed bridges with different struc-tural characteristics could have been done, but in this case,the model should have been simplified reducing the compu-tational time of each model, but this may not guarantee theaccuracy of the final analysis.

3. The structural problem

To verify the effectiveness of the fragility analysisdescribed above, the ASCE benchmark problem has beenselected [14–17]. The bridge is the Bill Emerson MemorialBridge, located in the USA, spanning the Missisipi River

near Cape Girardeau. It is a fan-type, medium-span,

F. Casciati et al. / Computers and Structures 86 (2008) 1769–1781 1771


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cable-stayed bridge (the main span length is 350.6 m). Thebridge is composed of two towers, 128 cables, and 12 addi-tional piers. It has a total length of 1205.8 m. The sidespans are 142.7 m in length, and the approach on the otherside is 570 m. The 128 cables are made of high-strength,low-relaxation steel. The smallest cable area is 28.5 cm2

and the largest cable area is 76.3 cm2

. The damping matrixis developed by assigning 3% of critical damping to eachmode. Sixteen 6.67 MN shock transmission devices areinstalled at the connection between towers and the deckin the uncontrolled configuration. The devices allow theelongation of the deck due to temperature change, but rig-idly connect the tower and the deck under a strong motion.

A three-dimensional finite element model of the bridgewas developed in MATLAB (1997) [14], that allows toinclude the torsional effects and the torsional mode of vibration. The model of the structure is pseudo-linear,because the stiffness matrix used in this model is deter-mined through a nonlinear static analysis corresponding

to the deformed state of the bridge with dead loads. Thisis a standard procedure [18] applied also from otherresearchers [19] in different bridges in China.

The nonlinear static analysis was performed using thecommercial finite element program ABAQUS, giving themodel tangent stiffness matrix at the (deformed) equilib-rium position. The B31 beam element was used for thestructural beam element, and the element T3D2 (it is a3D truss element with two nodes) was used for the cableelements. Rigid links were used to connect the cables tothe tower. Then, the element mass and stiffness matricesobtained with ABAQUS are output to MATLAB for

assembly. Subsequently, the constraints are applied, anda reduction is performed to reduce the size of the modelto something more manageable. Finally, the first 10undamped frequencies of the evaluation model are0.2899, 0.3699, 0.4683, 0.5158, 0.5812, 0.6490, 0.6687,0.6970, 0.7102, and 0.7203 Hz. Phase II of the benchmark[16] considers the horizontal directions for the seismic exci-tation and its coupling with the snow action, respect tobenchmark phase I. Since the bridge is supported by bed-rock, the soil–structure interaction effects are neglected[17]. The ground motion is assumed identical at each sup-port, but it is not applied simultaneously. It is assumed thatwhen the first support undergoes a specified groundmotion, the motion at the other three supports is onlydelayed proportionally to the ratio of the distance betweenadjacent supports and the speed of the L-wave (3 km/s).More details about the description of the structural modeland the structural members can be found in Caceido et al.[17].

3.1. Nonlinear considerations and safety evaluation

An important feature of cable-stayed bridges is the effectof the dead load that may contribute to 80–90% to total

bridge loads. Dead loads are usually applied before the

earthquake, so that the seismic analysis should start fromthe deformed equilibrium configuration due to dead loads.

In a cable-stayed bridge, Ren and Obota [19], but alsoFleming and Egeseli [20] and Nazmy and Abdel-Ghaffar[21], found that there is only small difference between linearand geometrically nonlinear seismic time-history analysis

under strong ground motions, when the analysis startsfrom the deformed equilibrium configuration due to deadloads. Ren and Obota also found that the elasto-plasticeffects tend to reduce the seismic responses of the bridgethat depend highly on the characteristic of the earthquakerecords and the dynamic properties of the bridge. Based onthis considerations, it has been used a pseudo-linear struc-ture in order to avoid the huge computational afford neces-sary to run a nonlinear finite element model with a muchbigger DOFs, that will make the fragility analysis basedon Monte Carlo simulation very long and not practical.Therefore, towers and cables are assumed with a linearelastic behavior, while the girder is connected to the bent

and the piers by orthogonal independent passive deviceswhich are modeled using the Bouc-Wen model.

3.2. Ultimate load-carrying capacity and safety evaluation

The yield moment, yield shear and yield curvatures of piers are obtained using a fiber section analysis (Kent-Parkmodel is used for concrete and a bilinear model for steel).

In Table 1 are shown the values of the yielding curvatureand ultimate curvature together with their respective forcesboth in the longitudinal and transversal direction for thetwo cross sections of the piers at the base level (sectionE) and at the deck level (section D). Performance thresh-olds are modeled as random variables using a coefficientof variation of 50% for the piers while a coefficient of var-iation of 10% is assumed for the yielding forces of cables.This value of the coefficient of variation for steel is in agree-ment with the results of Ellingwood et al. [22] that sug-gested coefficient of variation for steel between 10% and30%. Indeed, the random strength-constant stiffness modeladopted for steel cables appears to be a reasonable model,because the elastic modulus of steel that determines thestiffness have little uncertain, whereas steel strength is lesscertain. The lower bound of the values of the coefficient

of variation proposed by Ellingwood has been chosen(10%), because cables are high-strength, low-relaxation

Table 1Yield and ultimate curvatures and moment of the piers sections

M y (kN m) H y[1/1000]

M u (kN m) Hu[1/1000]

(1) (2) (3) (4)

Section E Trans. 2160 4.13 2370 61.74Base level Long. 63,447 5.29 63,447 65.17Section D Trans. 43,000 0.41 43,500 6.17Deck level Long. 6330 0.76 6920 5.58



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steel tested in laboratory, so it is reasonable to assume areduced uncertainty.

3.3. Cables sag effects

There are three main sources of geometrically nonlinear

behavior of cable-stayed bridges: (i) the beam-columneffect; (ii) the large displacements (known as P-D) effect;(iii) the cable sag effect [23]. It is generally accepted thatthe latter is the most relevant of those and, consequently,even in this simplified model adopted to evaluate fragilityof the case study considered the sag effect has beenincluded.

In the last decades, several finite elements have been pro-posed for cable modelling [24,25]. Gambhir and Batchelor[26] developed a two-node curved finite element, usingcubic polynomial interpolation functions, and used forthe static and dynamic analysis of three-dimensional (3-D) prestressed cable nets. Ozdemir [27] developed another

two-node curved finite element using Lagrangian functionsfor the interpolation of element geometry. Ali and Ghaffar[28] used a four-node iso-parametric cable element formodelling cables in cable-stayed bridges.

Despite the complexity and accuracy of all these models,the Ernst method [29] or modified elastic modulus methodhas been used in this analysis, given its capability toaccount for the sag effect and the easy of use. The methodassumes a parabolic instead of a catenary shape for thecable, which is acceptable only for moderate curvatures,typical of highly tensioned cables.

The cable element is modeled as a large-displacement

truss element that has a modified modulus of elasticity,E eq, given by

E eq ¼ E c

1 þ ðwL xÞ2 Ac E c

12T 3c

h i ð5Þwhere Ac is area of the cross-section, T c is the tension in thecable, w is its unit weight, Lx is the projected length in theX – Z plane, and E c is the modulus of elasticity of thematerial.

The cable elements are modeled as truss elements inABAQUS, and their equivalent elastic modulus are usedin the nonlinear static analysis. The cable stiffness contribu-

tion to the global stiffness matrix is only applied when thecable is under tension and is omitted otherwise:

K c ¼ E eq A

L ; ui > 0

0; ui < 0

( ð6Þ

It has been shown [24] that the equivalent modulus ap-proach results in softer cable response as it accounts forthe sag effect, but does not account for the stiffening effectdue to large displacements.

Furthermore, the effect of dynamic interaction with deckand towers is also important in the seismic response of cable-stayed bridge [30] and cables play an important role

in amplification or attenuation of structural response. In

particular amplification was noticed when a narrow bandexcitation is applied, that can excite the higher cable modesand induce contribution to higher-order global mode of thestructure, causing high frequency vibration to occur [31].However to take in account this effect, it is necessary tomodel each cable with multiple nodes [32,33] including a

large number of DOFs, that increase the computationaltime of the analysis based on Monte Carlo simulations.Still, for some cases, e.g. for short and medium span

cable-stayed bridges, linear analysis utilizing the equivalentmodulus approach is often sufficient [34,23], so cables hasbeen model in this way in order to find a balance betweenaccuracy and computational time.

3.4. Characterization of input ground motion

The bridge is located in the New Madrid seismic zone(Cape Girardeau: Lat. 37.2971N Long. 89.5163W) forwhich no strong motion records from similar historical

earthquakes exists. Hence, synthetic accelerograms areused. The synthetic accelerograms are generated by theprogram SMSIM_TD version 2.10 [35]. SMSIM_TD usesa stochastic method and assumes a point source. Sixtyaccelerograms corresponding to a given return period areobtained from the most likely (magnitude M , distance R)event (modal event) that is determined from the seismichazard deaggregation maps of US Geological Survey[36]. The modal (R, M ) pair corresponds to an earthquakethat is more likely than any other pair to produce a givenground-motion exceedance in the future. The deaggrega-tion maps are discrete surfaces that show the probability

of occurrence of given event expressed in term of magni-tude M and distance R for different return periods. Fourdifferent hazard levels are considered in this study: 20%in 50 years (return period T r = 224 years), 10% in 50 years(T r = 475 years), 5% in 50 years (T r = 975 years) and 2% in50 years (T r = 2475 years). The uniform hazard responsespectrum is derived according to FEMA 356 [37] and usingthe information provided by U.S.G.S [36]. Each of therecords is scaled so that the ground motion expressed interm of pseudo spectra acceleration (SA) with 5% dampingat the period of 2 s is equal to the SA of the target spectrumrelated to a given probability of exceedance (PE). Amongthe 60 accelerograms corresponding to a given return per-iod, a subset of 6 accelerograms has been selected whoseresponse spectra most closely match an approximate uni-form hazard response spectrum over most of its domain.The sampling frequency is 200 Hz and the duration is81.9 s. In Fig. 1, six response spectra corresponding totwo return periods (475 years and 2475 years) are plottedand compared with the corresponding uniform hazardspectrum. The original benchmark problem uses threerecorded accelerograms with different characteristics tocompare the various control strategies, but it does notallow considering the problem using a probabilisticapproach, as it should be due to the uncertainties of the

input ground motion.



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In Fig. 2, the two uniform hazard spectra correspondingto the return period of 475 years and 2475 years are shownand compared with the response spectra of the benchmark.It is observed a close matching of the response spectrum atlarger periods, in particular when it is considered the firstnatural mode of the bridge (T 0 = 3.44 s). On the contrary,there are big discrepancies at the higher frequency region.Therefore, it is also possible to consider the effects of thehigher modes of the bridge, using the proposed set of syn-thetic accelerograms. Besides, using 24 accelerogramsinstead of three, it is possible to approach the problem ina statistic way.

4. Retrofit strategy

Passive control has been implemented in bridges world-wide since 1970s [38–40]. In particular, the seismic isola-tion, with limited increase of a natural period to limitdisplacement, which is known as ‘‘the Menshin Design”,has been widely accepted in highway bridges in Japan afterthe 1995 Hyogo-ken Nanbu Earthquake. A passive controlsystem based on elastomeric lead–rubber bearings has beenadopted as retrofit strategy, because it limits the transfer of the input seismic energy to the structure. In this way, most

of the displacements occur across the device, while the

superstructure deforms pretty much as a rigid body. Thepassive control strategy for the mitigation of seismic effectsimplemented by the present work was developed using thebenchmark bridge model in literature [39]. Different typol-ogies of elasto-plastic passive devices are located in eightpositions along the bridge, symmetric respect the longitudi-nal axis, between deck, piers and bents. They are able todissipate energy on the horizontal plane in longitudinaland transversal direction (Fig. 3).

In the benchmark model, the structure is analyzed onlyfor two horizontal components of ground motion. Undersuch seismic excitation, which neglects the vertical compo-

nent, the response of the bridge deck was investigated andit results as a rigid body motion in the horizontal planewith unimportant vertical displacements (Fig. 4). For thereasons above, the vertical dissipative contribution of thecontrol devices was not implemented.

The devices behaviour is modeled in each orthogonaldirection using the Bouc-Wen endochronic hystereticmodel [41,42] in the form:

_ z ¼ A _ x b _ xj z jn

cj _ xj z j z jn1 ð7Þ

V ¼ ð1 aÞkz þ akx þ c _ x ð8Þ

where z is the auxiliary variable controlling the hysteretic

behavior. V is the lateral force expressed by the sum of

Response Spectrum (10 %PE; Tr=475 yrs) matching point: T=2sec

t (sec)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

a (

g )

0.0

0.2

0.4

0.6

0.8

1.0 A1-A6 [R=41.7km; M=7.7]

UHS 2002 - Target Spectrum

Median spectrum

Response Spectrum (2%PE; Tr=2475 yrs) matching point: T=2sec

t (sec)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

a ( g )

0

1

2

3

4

5

A19-A24 [R=16.7km;M=7.7]

UHS 2002 - Target Spectrum

Median Spectrum

a

b

Fig. 1. Comparison among response spectra of synthetic accelerograms and uniform hazard response spectrum for two return periods: (a) 475 years; (b)2475 years.



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three terms acting in parallel; a = k 2/k , the ratio betweenpost and pre-yielding stiffness; A, b, c, n are time invariantparameters defining the amplitude and the shape of the cy-cles, c is the damping and x and _ x are the relative displace-ment and velocity. The viscous damping component isoften very small, so, in this formulation the term ðc_ xÞ isneglected.

The lateral yielding force V y for value A = 1 and a close

to zero assumes the following form:

V y ¼ k

ðc þ bÞ1=n

ð9Þ

Literature is rich of devices modeled by the Bouc-Wenmodel [42], like magnetorheological dampers [43], metallicdampers [44,45], rubber bearings [46], electroinductive de-vices [47], etc. No formal optimization algorithm has beenadopted, so parametric analysis has been performed in or-

der to obtain the optimal values of the yielding forces and

Uniform hazard spectrum vs. benchmark accelerograms response spectra

t (sec)0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

a ( g )

0.0

0.5

1.0

1.5

2.0

2.5

El Centro

Gebze

Mexico

UHS 2002 (Tr=475yrs)

UHS 2002 (Tr=2475 yrs)

t (sec)

3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

a ( g )

0.0

0.1

0.2

0.3

0.4

El Centro

Gebze

Mexico

UHS 2002 (Tr=475 years)

UHS 2002 (Tr=2475 years) f i r s t m o d e - T 0 = 3 . 4 4 s e c

Fig. 2. Comparison between UHS and benchmark response spectra.

Fig. 3. Position of the passive devices.



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the post-yield stiffness k 2 by considering as objective func-tion the moments of the piers and the displacement of thedeck. Table 2 summarizes the adopted parameters, with theinitial stiffness k assigned, different values of the yieldingforce V y and the post-yield stiffness k 2 has been consideredin order to optimize the response reduction.

5. Numerical results of the bridge response

Dynamic analyses were performed using 24 syntheticearthquakes and fragility curves were developed beforeand after retrofitting. The adopted passive control solu-tions (Table 2) were analyzed and results compared in termof fragility curves with the uncontrolled case. Note that the

reference structure used for comparison of the passive con-trol strategy is the bridge with shock transmission devicesinstalled. In each case considered, the following time-his-tory seismic responses are recorded: (i) shear and momentat the base of the piers and at the deck level; (ii) displace-ment of the deck; (iii) cable tension time history response.All results are obtained starting from the deformed equilib-rium configuration due to dead loads. The reference vari-ables in building the fragility curves are the moments andthe shears at the base of the piers and at the deck levelrespectively; the tension in the cables and the displacementsof the deck, however only part of the results are shown in

this paper. Fig. 5 shows details of the time response for thebase moment and base shear of pier 2 for the uncontrolledand controlled case, but also the tension in one of thecables and the displacement response at bent 1. The passivecontrol adopted is leading to a reduction in term of forcesin the structural members, with a small increase of deck

displacements that is within the acceptable limits. In fact,the expansion joints of the deck level are constrained inthe relative vertical movements, while they allow horizontalmovements. The maximum allowed relative movementsbetween adjacent decks are 15 cm.

Fig. 6 depicts the hysteretic loops in the transversal andlongitudinal direction at pier 2. It is worth noting the differ-ence in the energy dissipated by the two orthogonal inde-pendent devices, the transversal one seems unnecessary.

Fig. 7 shows the Fourier transforms of the moment atthe deck level and the base shear of pier 2 in the longitudi-nal direction, for the uncontrolled and passive case. In bothsignals it is clear the first natural frequency of the bridge in

the uncontrolled case ( f = 0.2899 Hz).In Table 3 the mean of the maximum values of the base

shear at pier 2 in the longitudinal direction are tabulatedfor three different values of the yielding force and four dif-ferent hazard levels, assuming an angle of incidence u of the seismic action of 15. This pier has been selectedbecause it receives more forces respect to the other piers,while the angle of 15 has been selected because it is themore demanding for the pier in question. This conclusionhas been addressed after a sensitivity analysis observingdifferent angle of incidence of the seismic action.

Tables 4 and 5 are similar to Table 3, but related to the

base moment of pier 2 and to displacement of bent 1respectively. Type 8 in Table 1, shows better reduction interm of base shear and moment compared to the othertwo types, but the displacements of the deck compared totype 1 are bigger. Type 2 is able to reduce the displacementsof the deck for higher hazard levels, but definitely, it per-forms worse in term of displacement at the lower hazardlevel (Fig. 8). Besides in term of shear and moment the type1 performs better compared to type 2 (Tables 3 and 4). InFig. 9 the fragility curves for pier 2 are plotted consideringas reference variable the base moment. The graphs want toshow the influence of the uncertainties in the limit thresh-old using the coefficient of variation m = r/l defined asratio between the standard deviation and the mean[11,12]. The retrofit technique adopted is very effective interm of fragility curves as shown from the shift of the fra-gility curves to the right (Fig. 7). Fragility curves are verysensitive to the uncertainties of the performance thresholdexpressed using the coefficient of variation m. The uncer-tainties of the performance threshold increase the probabil-ity of exceedance; however even with this increase, theretrofit technique chosen is still very effective.

It is interesting to mention that instead when the uncer-tainties of the limit threshold of the deck displacement areconsidered the fragility curves of type 1 are not sensitive to

this dispersion.

Time (sec)

0 20 40 60 80 100 120 140 160 180 200

D e c k D i s p l a c e m e n t ( 1 0 - 3 m )

-2

-1

0

1

2

Fig. 4. Vertical displacement of the deck to the pier 2. The seismic recordhas a return period of 2475 years.

Table 2Device parameters

A n a V y (kN) b = c k (kN/m)

(1) (2) (3) (4) (5) (6)

Type 1 1 1 0.02 1000 40 80,000Type 2 1 1 0.02 2000 20 80,000Type 3 1 1 0.02 5000 8 80,000Type 4 1 1 1E5 5000 8 80,000Type 5 1 1 0.02 250 160 80,000Type 6 1 1 0.02 1 40,000 80,000Type 7 1 1 1E5 1 40,000 80,000Type 8 1 1 0.02 500 80 80,000

Type 9 1 1 0.02 4000 10 80,000



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6. Directivity effects of seismic action on fragility curves

Synthetic earthquake accelerograms have two horizon-tal components X (longitudinal) and Z (transverse). Thearrival times to the piers are [0 0.05 0.16 0.20] s for anangle of incidence of 15, while for an angle of incidenceof 45 the arrival times are [0 0.03 0.12 0.15] s. The inci-dence angle is defined between the longitudinal directionof the bridge and the N–S wave of the earthquake. The

earthquakes with the angle of incidence of the seismic

action inclined of 15 are more demanding than theearthquakes with angle of incidence of 45. This can beseen by looking at the fragility curves for the uncon-trolled and controlled cases for base shear and momentof pier 2 (Fig. 10). In fact, the probability of exceedingthe performance threshold is higher for the angle of inci-dence of 15. In addition, deck displacements with anangle of incidence of 45 in the longitudinal directionare one order of magnitude lower than for the angle of

incidence of 15 (Table 6).

displacement (m)

-0.4 -0.2 0.0 0.2 0.4

F o r c e ( 1 0 3 k

N

m )

-2

-1

0

1

2

displacement (10-3 m)

-4 -2 0 2 4

F o r c e ( 1 0 3 k

N )

-0.2

-0.1

0.0

0.1

0.2

Fig. 6. Hysteretic cycles of the transversal device (a) and the longitudinal one (b). The seismic record has a return period of 2475 years. The devices arelocated at the pier 2. Different scale resolution.

Base Moment (Mz) at Pier 2 in long. dir. X - A7

Time (sec)

25 30 35

B a s e M o m e n t ( 1 0 3

k N

m )

-300

-200

-100

0

100

200

300

Uncontrolled

Controlled

Base Shear Force (Vz) at Pier 2 in long. dir. X - A7

Time (sec)

25 30 35

B a s e S h e a r F o

r c e ( 1 0 3

k N )

-15

-10

-5

0

5

10

15

Uncontrolled

Controlled

Displacement at Bent 1 in long. dir. X - A7

Time (sec)

20 25 30 35 40 45 50

D e c k D i s p l a c e m e n t ( 1 0 - 3 m

)

-60

-40

-20

0

20

40

60

Uncontrolled

Controlled

Tension Force in cable 81 - A7

Time (sec)

25 30 35

T e n s i o n F o r c e

( k N )

1300

1350

1400

1450

1500

1550

1600

Uncontrolled

Controlled

Fig. 5. Time history responses of the bridge: (a) base moment; (b) base shear; (c) cable tension; (d) displacement of bent 1.



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7. Fragility curves for cables

One of the important aspects of cable-stayed bridge isthe determination of tension force in the cable, which isdirectly related to forces in the tower and the girder. Con-trol on the tension force in cables is critical. In most cases,cables are tensioned to about 40% of their ultimate strength

under permanent load condition.

The passive control technique adopted is able to keepthe tension in the cable within the limits for all the fourhazard levels chosen, when the angle of incidence of theseismic action is of 15. However using an angle of inci-dence for the seismic action of 45 the tension in some cableis not able to remain within the acceptable limits for thehighest risk level (T r = 2475 years). This is shown by the

fragility curves for two cables (81 and 114) for the uncon-

Table 3Mean of the maximum base shear at pier 2

Max. base shear pier 2 (kN) T r = 224 years T r = 475 years T r = 975 years T r = 2475 years

(1) (2) (3) (4) (5) (6) (7) (8)

Uncontrolled 3087.0 % 10362.0 % 25320.0 % 52793.0 %Type 1 (V y = 1000 kN) 2511.0 18.6 6525.4 37.0 14427.0 43.0 28783.0 45.5Type 2 (V y = 2000 kN) 2941.8 4.7 7200.6 30.5 15189.0 40.0 29220.0 44.6Type 8 (V y = 500 kN) 2186.1 29.2 6226.7 39.9 13749.0 45.7 28543.0 45.9

Longitudinal direction; u = 15; no snow.

Table 4Mean of the maximum base moment at pier 2

Max. base moment pier 2 (kN m) T r = 224 years T r = 475 years T r = 975 years T r = 2475 years

(1) (2) (3) (4) (5) (6) (7) (8)Uncontrolled 59458.0 % 190600.0 % 466830.0 % 973060.0 %Type 1 (V y = 1000 kN) 48184.0 18.9 111520.0 41.5 247590.0 46.9 501690.0 48.4Type 2 (V y = 2000 kN) 61659.0 -3.7 136600.0 28.3 255640.0 45.2 482010.0 50.5Type 8 (V y = 500 kN) 36622.0 38.4 107660.0 43.5 261920.0 43.9 517410.0 46.8


Table 5Mean of the maximum displacement at bent 1

Max. displ. bent 1 (m) T r = 224 years T r = 475 years T r = 975 years T r = 2475 years

(1) (2) (3) (4) (5) (6) (7) (8)

Type 1 (V y = 1000 kN) 0.0234 % 0.0645 % 0.1723 % 0.3679 %Type 2 (V y = 2000 kN) 0.0293 25.2 0.0671 4 0.1539 10.6 0.2602 29.2Type 8 (V y = 500 kN) 0.0202 13.4 0.0740 14.8 0.2104 22.1 0.4781 29.9


FT Moment (Mz) at deck level Pier 2 in long. dir. X - A7

Frequency (Hz)

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

M a g n i t u d e ( 1 0 3 k

N

m t

)

0.0

5.0e+3

1.0e+4

1.5e+4

2.0e+4

2.5e+4

Uncontrolled

Passive

FT Base Shear Vz Pier 2 in long. dir. X - A7

Frequency (Hz)

M a g n i t u d e ( 1 0 3

k N

s e c )

0

200

400

600

800

1000

1200

Uncontrolled

Passive

Fig. 7. Representative Fourier transform of the bridge: (a) longitudinal ground acceleration to moment at the deck level before and after retrofit; (b)longitudinal ground acceleration to base shear before and after retrofit.



11/13

My=444915 kN*m

PIER 2 BASE SHEAR [ ν=50%]

Return Period (yrs)

0 500 1000 1500 2000 2500

P e x c e e d

Uncontrolled (θy=996 yrs β=0.05)

Passive - Type 1 (θy=2130 yrs β=0.37)



PE in 50 years20% 10% 2%5%

DISPLACEMENT BENT 1

Return Period (yrs)

0 500 1000 1500 2000 2500

P e x c e e d

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0




PE in 50 years20% 10% 2%5%

a b

Fig. 8. Comparison between the best passive devices in term of fragility using uncertainties of the performance threshold using a coefficient of variation of 50%.

PIER 2 BASE MOMENT

Return Period (yrs)

0 500 1000 1500 2000 2500

P e x c e e d

0.0

0.2

0.4

0.6

0.8

1.0

Uncontrolled ν=100% (θy=667 yrs β=0.57)

Passive type 1 ν=100% (θy=1241 yrs β=0.77)

Uncontrolled ν=50% (θy=783 yrs β=0.22)

Passive type 1 ν=50% (θy=1610 yrs β=0.39)

Uncontrolled ν=0% det. (θy=515 yrs β=0.22)

Passive type 1 ν=0%

det. (θy=1675

yrs β=0.38)

PE in 50 years20% 10% 2%5%

Fig. 9. Comparison between different values of dispersions of the performance threshold and the passive retrofit technique adopted.

PIER 2 BASE SHEAR [ ν=100%]

Return Period (yrs)

0 500 1000 1500 2000 2500

P e x c e e d

ϕ=150 Uncontr. (θy=800yrs β=0.58)

ϕ=450 Uncontr. (θy=1966yrs β=0.74)

ϕ=150 Type 1 (θy=1486yrs β=0.77)

ϕ=450

Type 1 (θy=2602yrs β=0.03)

PE in 50 years20% 10% 2%5%

PIER 2 BASE MOMENT [ ν=100%]

Return Period (yrs)

0 500 1000 1500 2000 2500

P e x c e e d

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

ϕ=150 Uncontr. (θy=668 yrs β= 0.57)

ϕ=450 Uncontr. (θy=1744 yrs β= 0.75)

ϕ=150 Type 1 (θy= 1241yrs β= 0.77)

ϕ=450

Type 1 (θy=2602 yrs β= 0.03)

My=444915kN*m

PE in 50 years20% 10% 2%5%

a b

Fig. 10. Influence of the angle of incidence u of the seismic action on fragility curves.



12/13

trolled and controlled cases (Fig. 11). The uncertainty of the performance threshold in this case is chosen equal to

m = 10% because the ultimate strength of the cables aremore accurately defined than the ultimate load of the piersections made of concrete, for which uncertainties equalto m = 100% have been chosen.

8. Remark and conclusions

This study evaluates the fragility curves of a cable-stayed bridge using an analytical approach based on timehistory analysis. The bridge is located in USA spammingthe Mississippi river near Cape Girardeau, and is a partof the benchmark problem phase II that, with respect tophase I, includes the bi-directional effects of the seismicaction. Passive control devices are modelled using theBouc-Wen hysteretic model. The passive control is com-pared with the uncontrolled case in term of fragility curves.The uncertainty related to the ground motion has beenconsidered using four different hazard levels, while theuncertainty of the structural model has been consideredby modelling the different performance thresholds as ran-dom variables. Fragility curves were developed at the baseof the pier and at the deck level, considering both momentsand shears, but they also were developed in the cables con-sidering the tension and in the deck considering the dis-

placements. The paper shows how important is a correct

evaluation of the performance threshold of structuralmembers, because they can affect significantly the results

in term of fragility curves. Control strategy should be cho-sen in order to overcome the effect of uncertainties.

Acknowledgements

The second author wants to acknowledge his advisorProfessor A.M. Reinhorn and Professor T.T. Soong fortheir discussion and teaching in the field of structural con-trol, along with their guidance and encouragement that hasbeen essential in the completion of this work and are grate-fully acknowledged. The second author wants also toacknowledge the support of MCEER, which is supported

by the National Science Foundation. A research grantfrom the University of Pavia is acknowledged to supportthe contribution of the other two authors.

References

[1] Evangelista L, Petrangeli MP, Traini G. The cable-stayed bridge overthe PO river. In: IABSE symposium on structures for high-speedrailway transportation, Antwerp, August 2003.

[2] Pacheco BM, Fujino Y, Sulekh A. Estimation curves for modaldamping in stay cables with viscous damper. J Struct Eng, ASCE1993;119(6):1961–79.

[3] Casciati F, Faravelli L. Fragility analysis of complex structuralsystems. Taunton, Somerset, England: Research Studies Press Ltd;

1991.

Table 6Mean of the maximum displacement at bent 1

Max. displ. bent 1 (m) T r = 224 years T r = 475 years T r = 975 years T r = 2475 years

Type1 u = 15 0.0234 0.0645 0.1723 0.3679Type1 u = 45 0.0019 0.0058 0.0142 0.0338

Longitudinal direction; u = 15/45; no snow.

CABLE 81 ν=10% ϕ=450

Return Period (yrs)

0 500 1000 1500 2000 2500

PE in 50 years

P e x c e e d

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Uncontrolled (θy=1037yrs β=0.03)

Passive Type 1 (θy=1610yrs β=0.28)

20% 10% 2%5%

CABLE 114 ν=10% ϕ=450

Return Period (yrs)

0 500 1000 1500 2000 2500

P e x c e e d

Uncontrolled (θy=1710yrs β=0.38)

Passive Type 1 (θy=2602yrs β=0.05)

PE in 50 years20% 10% 2%5%

a b

Fig. 11. Fragility curves of cable 81 and 114 for an angle of incidence u = 45 and uncertainties of the limit threshold of m = 10%.



13/13

[4] Shinozuka M, Feng MQ, Lee J, Naganuma T. Statistical analysis of fragility curves. J Eng Mech, ASCE 2000;126(12):1224–31.

[5] Singhal A, Kiremidjian A. Bayesian updating of fragilities withapplication to RC frames. J Struct Eng, ASCE 1998;124(8):922–9.

[6] Cornell AC, Jalayer F, Hamburger R, Foutch D. Probabilistic basisfor the 2000 SAC/FEMA steel moment frame guide-lines. J EngMech, ASCE 2002;128(4):526–33.

[7] Karim KR, Yamazaki F. Effect of earthquake ground motions onfragility curves of highway bridge piers based on numerical simula-tion. Earthquake Eng Struct Dynam 2001;30(12):1839–56.

[8] Basoz N, Kiremidjian AS. Risk assessment of bridges and highwaysystems from the Northridge earthquake. In: Proceedings of thenational seismic conference on bridges and highways: progress inresearch and practice. Sacramento (CA); 1997. p. 65–79.

[9] Shinozuka M, Feng MQ, Kim H, Kim S. Non linear static procedurefor fragility curve development. J Eng Mech, ASCE2000;126(12):1287–95.

[10] Tanaka S, Kameda H, Nojima N, Ohnishi S. Evaluation of seismicfragility for highway transportation systems. In: Proceedings of the12th world conference on earthquake engineering. Upper Hutt, NewZealand. Paper No. 0546; 2000.

[11] Cimellaro GP, Reinhorn, AM, Bruneau M, Rutenberg A. Multidi-mensional fragility of structures: formulation and evaluation.MCEER Technical Report – MCEER-06-0002, University at Buffalo,The State University of New York, Buffalo, NY; 2006.

[12] Cimellaro GP, Kafali C. Fragility based life estimation methodologyfor critical facilities. In: Topping BHV, editor. Proceedings of thetenth international conference on civil, structural and environmentalengineering computing. Stirling, United Kingdom: Civil-Comp Press;2005 [paper 223].

[13] Soong TT. Fundamental of probability and statistics for engi-neers. John Wiley & Sons; 2004.

[14] Dyke SJ, Turan G, Caceido JM, Bergman LA, Hauge S. Benchmarkcontrol problem for seismic response of cable-stayed bridges. Wash-ington University in St. Louis; 2000.

[15] Bontempi F, Casciati F, Giudici M. Seismic Response of a cable-stayed bridge: active and passive control systems (Benchmarkproblem). J Struct Contr 2003;10(3–4):169–85.

[16] Caceido JM, Dyke SJ, Moon SJ, Bergman LA, Turan G, Hague S.Phase II benchmark control problem for seismic response of cable-stayed bridges. J Struct Contr 2003;10:137–68.

[17] Caceido JM, Dyke SJ, Moon SJ, Bergman L, Turan G, Hague S.Phase II benchmark control problem for seismic response of cable-stayed bridges. http://wusceel.cive.wustl.edu/quake/; 2003.

[18] Wilson J, Gravelle W. Modelling of a cable-stayed bridge fordynamic analysis. Earthquake Eng Struct Dynam 1991;20:707–21.

[19] Ren WX, Obata M. Elastic–plastic behavior of long span cable-stayed bridges. J Bridge Eng, ASCE 1999:194–203.

[20] Fleming JF, Egeseli EA. Dynamic behavior of a cable-stayed bridge.Earthquake Eng Struct Dynam 1980;8:1–16.

[21] Nazmy AS, Abdel-Ghaffar AM. Non-linear earthquake responseanalysis of long-span cable-stayed bridges: theory. Earthquake Eng

Struct Dynam 1990;19:45–62.[22] Ellingwood B, Galambos TV, MacGregor JG, Cornell CA. Devel-opment of a probabilistic-based load criterion for American NationalStandard A58. National Bureau of Standards 1980, Washington(DC). 222pp.

[23] Kanok-Nukulchai W, Yiu PKA. Mathematical modelling of cable-stayed bridges. Struct Eng Int 1992;2:108–13.

[24] Karoumi R. Some modeling aspects in the nonlinear finite elementanalysis of cable supported bridges. Comput Struct 1999;71:397–412.

[25] Freire AMS, Negrao JHO, Lopes AV. Geometric nonlinearities onthe static analysis of highly flexible steel cable-stayed bridges.Comput Struct, doi:10.1016/j.compstruc. 2006.08.047; 2006.

[26] Gambhir ML, Batchelor B. A finite element for 3-D prestressedcablenets. Int J Numer Methods Eng 1977;11:1699–718.

[27] Ozdemir H. A finite element approach for cable problems. Int J

Solids Struct 1979;15:427–37.

[28] Ali HM, Abdel-Ghaffar AM. Modeling the nonlinear seismicbehavior of cable-stayed bridges with passive control bearings.Comput Struct 1995;54:461–92.

[29] Ernst HJ. Der E-Modul von Seilen unter Beruecksich-tigung desDurchhanges. Der Bauingenieur 1965;40:52–5.

[30] Abdel-Ghaffar AM, Khalifa M. Importance of cable vibration indynamics of cable-stayed bridges. J Eng Mech, ASCE1991;117(11):2571–89.

[31] Caetano E, Cunha A, Taylor CA. Investigation of dynamic cable-deck interaction in a physical model of a cable-stayed bridge: Part Imodal analysis; Part II seismic response. Earthquake Eng StructDynam 2000;29(4):481–521.

[32] Loh C-H, Chang C-M. MATLAB-based seismic response control of a cable-stayed bridge: cable vibration. J Struct Contr Health Monitor[in press], published on-line in Wiley InterScience (www.inter-science.wiley.com) doi:10.1002/stc.87.

[33] Loh CH, Chang CM. Vibration control assessment of ASCEbenchmark model of cable-stayed bridge. J Struct Contr HealthMonit; 2006, published online 16 February 2005 in Wiley Inter-Science (www.interscience.wiley.com). doi:10.1002/stc.62.

[34] Karoumi R. Dynamic Response of cable-stayed bridges subjected tomoving vehicles. Licentiate thesis, TRITA-BKN Bulletin 22, Depart-ment of structural Engineering, Royal Institute of Technology,Stockholm; 1996.

[35] Boore D. SMSIM-Fortran programs for simulating groundmotions from earthquakes: version 2.0 – A revision of OFR 96-80-A. U.S. Geological Survey Open-File Report OF 00-509.55p;2000.

[36] United States Geological Survey, National Seismic Hazards MappingProject, version 2002, http://eqint.cr.usgs.gov/eq/html/lookup-2002-interp.html; 2000.

[37] FEMA, Prestandard and commentary for the seismic rehabilitationof buildings, Report No. FEMA 356, Federal Emergency Manage-ment Agency, Washington (DC); 2000.

[38] Buckle I, Mayes R. History and application of seismic isolation tohighway bridges. In: Proceedings of the 1st US–Japan workshop onearthquake protective systems, NCEER 92-4, National Center forEarthquake Engineering Research. University at Buffalo, the StateUniversity of New York, NY, USA; 1992. p. 27–40.

[39] Casciati F, Domaneschi M. 2004. Confinement of vibration appliedto a benchmark problem. In: Proceedings of the second Europeanworkshop on structural health monitoring, Monaco (D), July 7–9;2004. p. 811–8.

[40] Kawashima K. A perspective of Menshin design for highway bridges.In: Proceedings of the 1st US–Japan workshop on earthquakeprotective systems, NCEER 92-4, National Center for EarthquakeEngineering Research. University at Buffalo, The State University of New York, NY, USA; 1992. p. 3–26.

[41] Bouc R. Forced vibration of mechanical systems with hysteresis.In: Proceedings of the 4th conference on nonlinear oscillations;1967.

[42] Ikhouane F, Rodellar J. On the hysteretic Bouc-Wen model: part I

and II. Nonlinear Dynam 2005;42:63–95.[43] Spencer Jr BF, Dyke SJ, Sain MK, Carlson JD. Phenomenologicalmodel for magnetorheological dampers. J Eng Mech1997;123(3):230–8.

[44] Soong TT, Dargush GF. Passive energy dissipation systems instructural engineering. London: Wiley; 1997.

[45] Casciati F, Domaneschi M, Faravelli L. Some remarks on the drift of elasto-plastic – oscillators under stochastic excitation. In: Spence JrBF, Johnson EA, editors. Stochastic structural dynamics. Indiana,USA: Notre Dame; 1998. p. 545–50. August 6–8.

[46] Spencer BF. On the reliability of nonlinear hysteretic structuressubjected to broadband random excitation. In: Brebbia CA, OrszagSA, editors. Lecture notes in engineering, vol. 21. NewYork: Springer; 1996.

[47] Casciati F, Domaneschi M. Semi-active electro-inductive devices:

characterization and modelling. J Vib Contr 2007;13(6):815–38.

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seismic reliability of a cable-stayed bridge retrofitted with hysteretic devices

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