Research ArticleFragility-Based Improvement of System Seismic Performance forLong-Span Suspension Bridges
Guanya Lu 1 Kehai Wang 12 and Wenhua Qiu1
1School of Transportation Southeast University Nanjing 210096 China2Research Institute of Highway Ministry of Transport Beijing 100088 China
Correspondence should be addressed to Kehai Wang khwangriohcn
Received 31 January 2020 Revised 7 June 2020 Accepted 27 August 2020 Published 10 September 2020
Academic Editor Antonio Formisano
Copyright copy 2020 Guanya Lu et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
In this study a procedure is developed to evaluate and improve the seismic performance of long-span suspension bridges based on theperformance objectives under the fragility function framework A common type of suspension bridge in China was utilized in theproposed procedure considering its approach structures according to earthquake damage experience and fortification criteriaComponent-level fragility curves were derived by probabilistic seismic demand models (PSDMs) using a set of nonlinear time-historyanalyses that incorporated the related uncertainties such as earthquake motions and structural properties In addition one step that wascovered was to pinpoint the capacity limit states of critical components including bearings pylons and columnse stepwise improvedseismic designs were proposed in terms of the component fragility results of the as-built design Results of the comparison of improveddesigns showed that the retrofit measure of the suspension span should be selected based on two attributes ie displacement and forceand the restraint system of the approach bridges was a key factor affecting the reasonable damage sequence Necessarily from thecomparison of different system vulnerabilitymodels themean values of earthquake intensity of system-level fragility function developedby the composite damage state indices were used to assess the overall seismic performance of the suspension bridgee results showedthat compared to the absolutely serial and serial-parallel assumptions the defined composite damage indices incorporating the thoughtof component classification and structural relative importance between the main bridge and approach structures were necessary andwere able to derive a good indicator of seismic performance assessment hence validating the point that the different damage states weredominated by the seismic demands of different structures for the retrofitted bridges
1 Introduction
Long-span suspension bridges serve as the highly importanttransportation links during their service lives and there are noexpectations for these bridges to be retired in the foreseeablefuture One of the greatest challenges of structural design can beattributed to the large displacements low damping andcomplex vibration modes resulting from the strong vibrationscaused by earthquakes wind and traffic loadings However theseismic performances of such bridges must be of great concernparticularly in high-intensity earthquake regions
In the past studies related to suspension bridges re-searchers were mainly concerned about the vibrationalcharacteristics and dynamic analysis of suspension bridgesinvolving the use of the finite element method [1] and
continuum approach [2] which were the bases of the seismicanalysis Although the performance-based seismic evalua-tion framework has been widely applied to structural en-gineering and conventional highway bridges only a smallnumber of studies have investigated the seismic perfor-mance of suspension bridges by employing the perfor-mance-based evaluation methods Arzoumanidis et al [3]set the seismic performance objectives of the new TacomaNarrows Bridge and validated the design through the ex-tensive analysis including the demand analysis of globalstructure and the capacity analysis of components In ad-dition in order to account for the uncertainties of certainfactors related to the ground motion input Sgambi et al [4]conducted a seismic analysis of long-span suspensionbridges through the Monte Carlo simulations e mean
HindawiAdvances in Civil EngineeringVolume 2020 Article ID 8693729 21 pageshttpsdoiorg10115520208693729
value and variance of displacements and stresses were usedto describe the seismic responses of structures Karmakaret al [5] studied the seismic performance of the Vincentomas suspension bridge retrofitted with viscoelasticdampers and obtained the ductility demands of towers interms of fragility probability However the studies men-tioned above solely considered the component-level fragil-ities and focused on the damage of the towers Additionallyit must be noted that the approach span of the San Fran-cisco-Oakland Bay Bridge had fallen-off during an earth-quake event which caused the closure of transportation link[6] erefore it can be concluded that leaving out the effecton approach structures may skew the results in the systemseismic evaluation of long-span bridges Due to the differ-ence and connection in structural dynamic behavior anddesign strategy between the suspension span and approachstructures the optimization of system seismic performanceand a reasonable indicator that can reflect the overall seismicperformance for long-span bridges will be appealing
Previous studies have indicated that fragility functionsare effective tools by which to assess the damaging potentialof small-to-medium-span highway bridges under earth-quakes Fragility functions can connect the given damagestates with earthquake intensity Both component-level andsystem-level fragilities have been developed for a class ofbridges in a certain region such as Central America [7 8]California USA [9] Turkey [10] Greece [11] and Italy [12]Barnawi and Dyke [13] and Zhong et al [14 15] evaluatedthe seismic fragility of a cable-stayed bridge and compareddifferent retrofit measures using fragility techniques Al-though their studies noted the necessity of derivation ofsystem-level fragility of piers (towers) and bearings a serialconnection was used in estimating the system fragility ekey point of their methods depends on a relatively con-servative way to quantify correlations between criticalcomponents One is full correlation while the other is partialcorrelation obtained from joint probabilistic seismic de-mand models [8] It is worth discussing whether it is rea-sonable to apply these correlations to the fragilitydevelopment of overall long-span bridge system In additionthe application of fragility curves has been proficient forcomparing and selecting retrofit measures and isolationstrategies for conventional highway bridges [16ndash18]
In the context of the aforesaid review the key purposes ofthis study are to develop a procedure of the performance-basedseismic assessment for long-span suspension bridges and toexamine the applicability of the proposed indicator for evalu-ating the overall bridge system seismic performance A typicallong-span suspension bridge in China was selected A series ofdetailed finite element models were built using OpenSEES [19]software that can perform nonlinear time-history analysis eprocedure incorporated the uncertainties of ground motionsand structural properties e backbone of the proposed pro-cedure is fragility-based improvement of seismic performancefor long-span suspension bridges including the definition ofengineering demand parameters (EDPs) and the capacity limitstate thresholds associated with the case-dependent compo-nents Component fragility is used to pinpoint the vulnerablecomponents of the overall bridge under earthquakes and served
as the basis of stepwise retrofit strategies e retrofit candidatesare determined by earthquake damage experiences relatedcomponent seismic experiments and expert opinions Finallythe indicators appropriate for characterizing the seismic per-formance of overall system for long-span suspension bridges areproposed and discussed Finally the effectiveness of the sug-gested index in the system seismic performance evaluation isfurther identified by comparing the indicators of the retrofitteddesigns with those of the as-built design
2 Procedure Description
is study used PSDM generated by cloud approach andcapacity limit state models based on multiperformance ob-jectives to develop the fragility curves of bridge componentsrooted in the response data obtained from a set of nonlineartime-history analyses e seismic fragility could be defined asthe conditional probability of a structure seismic demand Sd
reaching or exceeding its capacity Sc at a given earthquakeintensity measure (IM) which could be expressed as follows
It is well known that the PSDM built by cloud approachis based on three assumptions (1) lognormal distribution ofseismic demand and structural capacity (2) constant dis-persion assumption for all IM ranges and (3) a power modelbetween seismic demand and IM erefore the conditionalprobability under a given IM can be written as follows
P Sd ge Sc
1113868111386811138681113868 IM1113872 1113873 1 minus Φln Sc( 1113857 minus ln aIMb
1113872 1113873
β2d | IM + β2c1113969⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠ (2)
where Sd is the mean value of seismic demand conditioned onan IM Sc is the limit state threshold of structure correspondingto each defined damage state (DS) βd|IM and βc indicate thedispersion values of demand and capacity respectively andΦ(middot) is the standard normal cumulative distribution functionBoth a and b are regression coefficients which can linearlyregress in log-transformed spaces
e dispersion of seismic demand βd|IM can be estimatedas follows
βd|IM
1113936 ln di( 1113857( 1113857 minus (ln a + b ln(IM))( 11138572
N minus 2
1113971
(3)
where di is the ith demand value obtained from the analyticalsamples and N denotes the sample sizes
e definitions of PSDM and capacity limit states areimperative in the formation of component fragility curvesComponent fragility curves facilitate to highlight the weak linksin the overall bridge system and develop the retrofit and repairdecisions However if the comprehensive seismic evaluation ofcomplicated structures with multicomponents is conductedthen the correlations of various components must be con-sidered According to the objective of this study the highlightsof the proposed procedure are outlined as follows
(1) e propagation of uncertainties is beneficial to thereflection of their statistical significance which include
2 Advances in Civil Engineering
the ground motions that meet the site conditionsrepresenting a broad range of intensities and variousmodeling parameters such as material strengthstructural properties and damping ratio e generalpractice is taking the uncertainty sources as randomvariables to formN finite elementmodels of suspensionbridges in the as-built design using a Latin hypercubesampling (LHS) technique and then pair these modelswith a set of N ground motions in order to carry out aseries of nonlinear time-history analyses
(2) e final step to derive bridge component fragilitycurves is to pinpoint the vulnerable componentsbased on the recorded peak response data in non-linear time-history analyses by which to define theircorresponding limit state thresholds at variousdamage states In addition the application of optimalIM is also the key link to ensure the reliability of theseismic vulnerability assessment
(3) e component-level fragility curves are calculatedusing equation (2) For the purpose of reducing thevarious damage potential of critical bridge componentsin the initial design different seismic designs areproposed based on earthquake damage experiencesrelated component seismic experiments and expertopinions Next the effectiveness of seismic measures isevaluated using the critical component-level fragility
(4) e optimum structural seismic performance is se-lected in this study taking the bridge system-levelfragility as the objective function according to thedefinition of damage states and fragility results ofvarious components obtained from steps 2 and 3Considering the vulnerability degree of each com-ponent and postearthquake repair cost it is helpfulto evaluate the seismic performance of the bridgesystem by combining the critical components in areasonable manner For suspension bridges cablesystems and pylons are primary load-carryingcomponents and pylons can be only allowed to be oflimited damage e damage of bearings can beallowed to compensate by the effective seismicmeasures For approach bridges with the small-to-medium spans the postearthquake serviceability ofbridges depends on the damage stage of columnsabutments and bearings A popular seismic designof this type of bridges is fusing actions of bearings asforce-limiting connections to reduce the damage ofsubstructures A bridge structure consists of multiplemembers with the same function but in variouslocations e working principle of pylons and
columns is similar to that of the series system due tothe fact that the damage or collapse of one of thepylons or columns will directly affect the post-earthquake availability of the bridge system emultiple bearings form a parallel system to jointlysupport superstructures and a single damage has lessinfluence on the overall bridge system Earth pres-sures on abutments are further increased as a resultof the pounding between the deck and abutmentbackwall for the seat abutments which leads to anincrease in the abutment deformation However thedifficulty of repairing abutments is relatively smalland the effect on emergency traffic is slight A parallelsystem can also be adopted to simulate an abutmentsystem For a global bridge structure once any typeof component system is damaged it will have asignificant impact on the overall bridge system eoverall bridge system is close to the series combi-nation of different component systems which can becalled a bridge system based on the series-parallelsystem
Zhong et al [14] summarized the method of componentclassification appropriate for long-span bridges and held thatthe components of long-span bridges can be classified asprimary secondary and accessory based on the relativeimportance of load-carrying capacity and repairing costserefore the application of a serial assumption betweencomponent systemsmay bias the seismic fragility assessmentof the bridge system Zhang and Huo [18] assigned aweighting ratio of 075 and 025 to columns and bearingsrespectively for ordinary highway bridges in USA andproposed the composite damage state for the bridge systeme repair-related strategy variables including repair costand time are considered based on the direct and indirecteconomic losses that are caused by the damage of bridgecomponents e damage to primary and secondary com-ponents at any level will account for a majority of repair costincluding repairing most components and a long closure oftraffic e replacement costs of accessory componentsprimarily rest on their extensive damage including rein-stallation of bearings and restoration of girder that might beresulted from pounding and unseating which interruptstraffic at a short time Combined with the reality of bridgeconstruction in China and the repair-related variables of thecritical components incorporated in the bridge system thisstudy separately proposes the composite damage state in-dices of the long-span and conventional bridge system asfollows
It should be noted that the aforementioned developmentof damage indices for the bridge system is only for a singlebridge system and is closely related to the definition ofdamage states of the components For long-span suspensionbridges composed of a main bridge and several approachbridges a reasonable and comprehensive seismic vulnera-bility evaluation is of great significance to the construction ofproject and postearthquake repairing cost A bridge systemwith the similar structural properties could be regarded as aseries system However the differences in structural im-portance and repairing costs cannot be neglected because ofthe structural differences between the long-span bridges andsmall-to-medium-span bridges e weighting ratios of aand (1 minus a) are assigned to the main bridge system andapproach bridge system respectively which have the abilityto capture the contributions of the main bridge and ap-proach bridges to the vulnerability of the overall bridgesystem erefore the rational range of system fragilitycould be accurately determined as follows
DSsystem round a middot DSmain bridge +(1 minus a) middot DSapproach bridge1113960 1113961
(5)
where the round-off principle can be used for the ldquoroundrdquoe composite damage state in equations (4) and (5) can
be incorporated in the PSDM when the correlation betweenconcerned components with the use of covariance matrix isknown In this case Monte Carlo simulation is used tocompare the demands obtained from the joint probabilisticseismic demand models [8] and multidimensional compo-nent capacities in theM (105) random samples at a range ofthe IM values e basic principle of developing fragilitycurves for the long-span suspension bridge system is shownin Figure 1
3 Case Study
31 Bridge Description and Simulation is study selectedthe Taoyuan Bridge spanning the Jinsha River as the studycase e bridge is characterized by the geometric andmechanical features of a suspension bridge which is themost common type of bridge construction in China ebridge has a total length of 731m and consists of a single-span simply supported suspension bridge measuring 636mand a continuous steel box-girder approach bridge withthree spans measuring 30m 35m and 30m respectivelye longitudinal elevation of the bridge is illustrated inFigure 2
e main bridge has a thin-wall steel girder along thelongitudinal direction with an overall width of 31m fromcurb to curb and a height of 3m accommodating four lanesof traffic e gate-type towers include two RC box-sectionlegs with overall heights of 75m and 70m on its two sidesrespectively Also there is one crossbeam installed along theheight of pylons in order to enhance the lateral stiffness ofstructures At the base the tower legs are anchored to the 6m-thick concrete footings e pedestals for the placingbearings are designed between the concrete footings due tothe shorter distance between the deck and tower base e
bored piles serving as the foundations of pylons and locatedat the leg of each tower have a diameter of 22me areas ofeach main cable and suspender are 01626 m2 and 0001316m2 respectively Gravity anchorage has been applied to themain cable anchorage on both sides of the bridge eapproach bridges are divided into left and right bridges withthe identical configurations on the north side For eachapproach bridge there is a three-span continuous steel box-girder bridge with a pile-bent abutment of which the middlespan is supported by two double-column bents with integralpile-shaft e width and depth of the superstructure are1275m and 153m respectively e circular sections ofcolumns with a diameter of 18m have a height of 7m andthe cast-in-drilled-hole piles measure 19m in diameter epiles of abutment are provided by a circular section of 15min diameter and the height of the backwall is 188m
e description of this studyrsquos analytical modelingprocedure is outlined using OpenSEES software [19] for theprototype bridge in the as-built design A beam-columnelement with distributed plasticity fiber was used to modelthe pylons and columns in order to account for the materialnonlinearity In addition the axial force-moment interac-tions of pylons were captured in the model Each fiber wassimulated via a reasonable stress-strain relationship whichwas dependent on unconfined concrete confined concretedefined by the Mander theoretical model and longitudinalreinforcements that considered the Bauschinger effects asillustrated in Figure 3 e corresponding mechanical in-dices such as material strength axial compression ratio(ACR) and reinforcement ratios of Sections I and II all metthe Chinese code requirements [20]
e main cable element was tension-only which wasmodeled as a finite number of large-displacement truss el-ements using the Ernst method accounting for the sag effect[21] Each suspender element was identical to the simulationof the main cable but there was no sag effect e initialstress of the cable system was also considered in the modelas shown in Figure 4(a) An elastic beam element was uti-lized to simulate the crossbeam of the pylons and stiffeninggirder as well as the girders and cap beams of the approachbridges e foundations were modeled by six spring ele-ments which indicated that in the future they are notexpected to be damaged under earthquake shaking
It should be noted that pot bearings were applied to theas-built design which consequently became the significantfactors in the overall responses and functionality of theapproach bridges e longitudinal response of sliding potbearings detailed in Figure 4(b) was simulated using a bi-linear element [20] e full-scale experiments of Steelmanet al [22] found that properly proportioned steel for fixedpot bearings can achieve a reliable dry friction responsebetween the bearing component and superstructure orsubstructure after the rupture of the anchor bolt Corre-spondingly the behavior of fixed pot bearings was modeledas shown in Figure 4(c) e fixed pot bearings weredesigned to be located at the C2 column as shown inFigure 1 Finally the pounding stiffness between the adjacentdecks (Figure 4(d)) was estimated by the sum of their axialstiffnesses [23]
4 Advances in Civil Engineering
Approach bridge
Composite damage state (4)
Main bridge
Composite damagestate (4)
Composite damage state (5)
Bearing 1 Bearing iBearing 2
Bearing 1 Bearing iBearing 2
Bearing 1 Bearing iBearing 2
Bearing 1 Bearing iBearing 2
Column 1
Column 2
Column j
Abutment 1
Abutment 2
Bearing 1 Bearing iBearing 2
Bearing system
Column 1 Column iColumn 2
Column system
Bearing 1 Bearing iBearing 2
Bearing 1 Bearing iBearing 2
Pylon 1
Pylon 2
Bearing system
Pylon 1 Pylon 2
Pylon system
Damage state definitionsand component-level
fragility results
Abutment 1
Abutment 2
Abutmentsystem
Figure 1 Bridge system based on the series-parallel system and composite damage state
C2C1
4000
7000
6000
7500 63
60
14000
3500
30001600063600
73100
3000South
North
Figure 2 Configuration of the Taoyuan Suspension Bridge (cm)
SouthH = 750m
North H = 700m
H
Sec I
Pylon
Pedestal
(a)
Confined(Mander model)
εu εcu
fc
Kfc
ε
σ
Unconfined
Concrete0002
fcufu
εc0
(b)
σy
εy ε
σ0005E0
E0
Bar
(c)
Figure 3 Continued
Advances in Civil Engineering 5
e shear keys contribute to the constraint of transversemovement between the girder and pier Xu and Li [24]proposed a modified model based on the structural prop-erties of shear keys in China which was used to build theforce-displacement relation of shear keys by separating thecontributions of steel and concrete in parallel as shown inFigure 5 e critical parameters of the shear keys can becalculated using Xursquos method according to the actual designwhich represents the yield (Δ1Y) nominal (Δ1n) degraded(Δ1d) and ultimate (Δ1u) deformations respectively esimulation can accurately reflect the mechanical behavior ofthe shear keys
e seat type abutment is shown in Figure 6 e lon-gitudinal resistance to seismic forces is provided jointly bythe passive action of the backfill soil and active action of thepiles [25] Meanwhile the transverse response is resisted bythe pilese passive and active responses of abutments weresimulated using hyperbolic soil material [26] and trilinearforce-displacement relationship respectively which wereconnected in parallel to capture the abutment responses Inaddition the pounding effect between the deck and abut-ment was simulated using a linear model without energydissipation e stiffness of pounding spring between thesuperstructure and abutment was proposed to be ten timesthat of the backfill soil stiffness [27]
32 Uncertainty Treatment in the FE Model e LHS ap-proach is usually used to account for the uncertainties infragility analysis such as structural geometry materialstrength component properties deck mass and damping in
the modeling which is considered to be a variance reductionsampling technique It is noted that the uncertainty ofstructural geometry is not taken into account in this studyfor the development of bridge-specific fragility curves In theforegoing approach the probability distributions were as-sumed for each parameter and each random variable wasdivided into the several equal intervals which correspond to5 to 95 of the cumulative probability e consideringmodeling parameters and their probability distributions arepresented in Table 1 Statistically significant yet nominallyidentical 3D bridges were built using the LHS so as toaccount for the abovementioned uncertainties then thebridges were paired randomly with the selected groundmotions
33 Ground Motion Input and Intensity Measure e ne-cessity to carry out a large number of nonlinear time-historyanalyses depends on a significant number of earthquakerecords selected for the PSDM In this study a data set of 80recorded ground motions and 20 Los Angeles-pertinentunscaled ground motions were extracted from the PEERstrong motion database and the SAC project database re-spectively [29] ese ground motions can represent thecharacteristics of the bridge site ey were placed into fivemagnitude-distance bins for the purpose of covering therational and broad ground motion intensity e first fourbins included combinations of low and high magnitudesand large and small fault distances e magnitudes variedbetween 57 and 70 while the fault distances ranged from130 to 600 km e characteristic of the ground motions in
Stress
Initialstress
K1
Strain
K2 = 0005K1
(a)
K1
Δy = 2mm
Fy
ΔΔy
K2 = 000001K1
F
(b)
Δu
F
Fd
Δy
K
Rh
ΔΔy = 1mm
Δu = 20mm
(c)
K
Gap
F
Δ
(d)
Δy
FΔult
K
Δ
Fult
Fult2
Δy= 154mmΔult = 1504mm
Per width (m)of abutment
(e)
Δ1 = 762hmmΔ2 = 254hmm
h height of backwall
Keff
K2
F2F1
Δ1
K1
F
ΔΔ2
Per pileKeff = 14kNmm
(f)
Figure 4 Constitutive relations of the various components (a) cable and suspender (b) sliding pot bearing (c) fixed pot bearing(d) pounding (e) abutment backfill (f ) abutment pile
Bar (HRB500 376)
z
y
Confined concrete (stirrup HRB400
1256)
Sec I
Unconfined concrete (C55)
ACR 0159
(d)
Sec II
h =
70m
Double-column bent
(e)
Confined concrete(stirrup HRB335
08)
Unconfined concrete (C35)
Bar(HRB335 1163)
z
y
Sec II
ACR 0037
(f )
Figure 3 Sketch and material model of the pylons and bents of the examined suspension bridge
6 Advances in Civil Engineering
the fifth bin was that the fault distance was less than 15 kme ground motions were given in two orthogonal com-ponents and the acceleration values adopted were computedas the geometric mean values (Sa
SaL middot SaT
1113968) e peak
ground acceleration ranged from 0043 g to 112 g Figure 7shows the distributions of peak ground accelerations (PGA)peak ground velocities (PGV) earthquake magnitudes (M)and fault distances (R) of 100 ground motion records as well
as the acceleration spectra with a damping ratio of 002 etwo orthogonal components of the selected earthquakerecords were randomly input along the longitudinal andtransverse directions of the bridge
PGA PGV PGD spectral acceleration or Arias In-tensity of earthquake can be selected as the IM candidateswhich are used to characterize earthquake intensity elinear relations with the interesting EDPs from probabilistic
Δ
V
Δ1n Δ1d
Vcp
Δ
V
Δ1y Δ1u
Vsp
i
j
Gap
Hysteretic property of concrete Hysteretic property of steel
Vcp = 7624kN
Δ1n = 106mmΔ1d = 422mm
Vsp = 10143kN
Δ1y = 11mmΔ1u = 633mm
Figure 5 Analytical model of shear keys
Abutment
Deck
Figure 4 (b)
Figure 4 (d)
Figure 4 (c)Figure 5 Figure 4 (e)
Figure 4 (f)
Figure 6 Pile-bent abutment simulations
Table 1 Uncertainty parameters incorporated in the modeling design
Modeling parameter Probability distribution Mean COV Unit Source
Concrete compressive strength C55 Normal 466 0149 MPa
[20]C35 321 0164
Steel yield strengthHRB500
Lognormal5696
00743 MPaHRB400 4557HRB335 3816
Friction coefficient Sliding pot bearing Uniform 003 033 mdash [14]Fixed pot bearing Lognormal 02 0002 mdash [7]
Deck-abutment 80Deck-shear key Uniform 120 029 mm [16]
Damping ratio Uniform 1 01 mdash [7]Deck mass Normal 002 015 mdash [28]COV coefficient of variation
Advances in Civil Engineering 7
seismic demand analysis in the logarithmic reference framecan be adopted to determine the optimal IM In the previousstudies PGA and spectral acceleration were selected as theoptimal IMs for highway bridges [9 30] when applying themultiple measures containing efficiency practicality profi-ciency and sufficiency Mackie et al [31] suggested both PGAand PGV are good choices of IM in related to the EDPmeasures in highway overpass bridges Zhong et al [14]pointed out that PGA and PGV are themost efficient practicaland proficient IMs for long-span cable-stayed bridges yet PGVtends to be the optimal IM in terms of sufficiency ereforePGV can be utilized as the IM in this study
34 Engineering Demand Parameters e peak demands ofcritical components are adopted as EDPs for the developmentof PSDMs and the choice of optimal IM Table 2 lists thecritical EDPs for the examined bridge which have beenconsidered in the fragility evaluation of the suspension bridge
35 Capacity Limit State Models In the present study acapacity model was used to measure component damagewhich was described by a damage index as a function of the
selected EDPs ese models are usually discrete andcharacterized by the onset of various damage states based onthe experimental data Similarly the uncertainties of ca-pacity models are also expressed by median Sc and dis-persion βc e definition of damage states must be matchedwith postearthquake bridge functionality and repair strat-egies in the bridge seismic fragility assessments Founda-tions decks and anchorages are typically identified asnondamaged components in the simulations e damagestate values of bearings were determined so as to comparewith two factors including the pounding gaps between theadjacent components and seating width which can reflectthe possible pounding and unseating as shown in Figure 8 inaddition to the damages to bearings themselves that is thedeformation capacity of the bearings Table 3 lists the slight(DS1) moderate (DS2) extensive (DS3) and collapsed(DS4) damage states defined in the initial design for potbearings at various locations based on their allowable de-formation and structural dimensions
e columns of conventional highway bridges are oftenforced into the state of nonlinearity under strong earth-quakes of which the criteria of damage states were studied[8 9 16] It is found that there are major differences in
0
10
20
30
40
50N
umbe
r of r
ecor
ds
08
01
03
05
09
00
07
04
02
10
06
PGA (g)
(a)
0
10
20
30
40
50
Num
ber o
f rec
ords
20 806040 100
180
140
160
2000
220
120
PGV (cms)
(b)
20 40 600Fault distance (R) (km)
55
60
65
70
75
Mag
nitu
de (M
)
(c)
Mean
0
1
2
3
4
5
S a (g
)
2 4 6 8 100Period (s)
(d)
Figure 7 Characteristics of the selected ground motions (a) PGA (b) PGV (c) M and R (d) acceleration response spectra
8 Advances in Civil Engineering
geometric sizes material strengths and ACR of sectionsbetween the pylons of suspension bridges and columns ofconventional highway bridges erefore it is necessary todevelop a modal pushover analysis to define the damagestates of critical cross-sections of pylons In the analyticalprocesses of this study the material strain and nonlinearmoment-curvature relationship of critical cross-sectionswere characterized to measure the damage levels of thesuspension bridge pylons
First the reasonable pylon models were required to bedeveloped using fiber beam elements in longitudinal andtransverse directions as detailed in Figure 3 An elastic
spring which was equivalent to the restraint of main cablewas attached to the top of the pylon taking into account theinfluence of the main cable stiffness in the longitudinaldirection models e determining process of the equivalentstiffness can be referred to in the Ernst formula and theresults reported by Kiureghian and Sackman [21]
Second as illustrated in Figure 9 the inertial forcedistributions were back-calculated from the multimodespectrum analysis using a 3D bridge model as shown inequations (6)ndash(8) which included the contributions of highvibration modes and reflected the properties of inertial forcedistributions as follows
DS3
DS4
DeckDeck
Pedestal
PedestalBearing
Bearing
Gap
(a)
DS2
DS4
Gap
Abutmentnonlinearity
Deck
AbutmentBearing
(b)
GapShear key
degradationDS4
DS2
Bearing
(c)
Figure 8 Damage control and unseating prevention damage states for bearing displacement (a) between decks (b) between deck andabutment (c) between deck and shear key
Table 2 e critical EDPs for the examined bridge
Engineering demand parameters Abbreviation DirectionCurvature ductility of pylon cross-section (south north) PCD(S N) Longitudinal and transverseDisplacement of bearings of the main bridge MBD Longitudinal and transverseAxial force relative to yield value of main cables MAF mdashAxial force relative to yield value of suspenders SAF mdashCurvature ductility of column cross-section at columns (C1 C2) CCD(C1 C2) Longitudinal and transverseDisplacement of bearings of the approach bridge at column (C) and abutment (A) ABD(C A) Longitudinal and transverseDeformation of backfill passive pressure at abutment BPDA LongitudinalDeformation of piles at abutment PDA Longitudinal and transverseDeformation of expansion joint between decks(D) and at abutment (A) EJD(D A) LongitudinalDeformation of shear key at columns (C) and at abutments (A) SKD(C A) Transverse
Table 3 Limit state thresholds for pot bearings (m)
DS1 DS2 DS3 DS4
Criteria Allowable slippingdisplacements
Spherical skateboard radiusbody separation of bearings
Possible pounding betweenadjacent decks or deck and pylon Possible unseating
MBDL 025 031 076 1555MBDT 002 031 045 mdash
Criteria Allowable slippingdisplacements
Possible pounding entering of abutmentsnonlinearity or shear key degradation
Spherical skateboard radiusbody separation of bearings Possible unseating
where Sj and cj are the spectral acceleration and participationfactor associated with the jth mode respectively Fij φij andQij represent the seismic load mode displacement and shearforce at the i-node associated with the jth mode respectivelymi Qj and Pi indicate the mass shear force and inertial forceat the i-node respectively andN denotes the number of nodes
Next the inertial forces were loaded to the nodes ofpylon models Both the material and geometric nonlinear-ities were incorporated into the pushover analyses whichwere used to investigate the potential failure mechanismsand assess the ductility capacities of pylons
e seismic designs of pylons can develop inelastic be-haviors which were controlled by the material strain limitsFigure 9 details the curvature distributions along the pylonheights with the increase of δ where δ was the loading dis-placement at the control point It was observed that plastichinge zones typically occur at the bases of pylons and above andbelow the horizontal crossbeams Also plastic hinges mayprogress to the intermediate height in the longitudinal direc-tion e development of plastic hinges at the bases of pylonscan be much severer and also occur earlier than in other zoneserefore the curvatures at the bottoms of pylons may ef-fectively provide measurements of the damages to pylons
e mechanical indices of pylon cross-sections whichcorrespond to the four damage state characteristics
following the damages are detailed in Figure 10 e cur-vature ductility can be easily determined by the pushoveranalysis results in addition to that of the DS2e numericaldata can be transformed to a bilinear curve that has a value ofDS2 based on the equal energy rule per original curveSimilarly another pushover analysis was applied to thedouble-column bend of the approach bridge based on thespecifications [20] e four limit state thresholds of thecross-sections at the bottom of pylon and columns in boththe longitudinal and transverse directions are listed in Ta-ble 4 It was observed that the seismic capacity of longitu-dinal direction was superior to that of transverse directionand a good ductility capacity existed in both directions
In the present study the definitions of damage statethresholds of main cables and suspenders were dependent onthe steel strand characteristics of the strength and deformationrates A total of 85 of the ultimate strength (nominal yieldstrength) was conservatively acted as the failure index Namelythe corresponding ratio of force to yield value was 10 It is acommon viewpoint that the accidental loss of one suspenderwill not cause immediate failures of a bridge with the exceptionof the loss of the main cables erefore DS3 and DS4 wereonly defined based on the classification of suspenders andmaincables respectively e corresponding elastic and ultimatedeformation of abutments and shear keys were respectivelyspecified the thresholds of DS1 and DS2 with the recom-mendations of Caltrans [25] e damage of expansion jointsonly defined as the DS1 due to the lack of a higher-leveldamage modelis was objectively logical since the joints canbe replaced with the new once damaged thereby suggesting therationality for just one threshold of component damage Table 5summarizes the damage state thresholds of the above-mentioned components
e assignments of dispersions of capacity models ofcomponents appeared to be the most difficult tasks for theresearchers According to the previous research studies 035was adopted in the abutment and RC components including
Figure 9 Inertial force and curvature distributions (a) longitudinal (b) transverse
10 Advances in Civil Engineering
pylons and columns [9] e COV of main cables andsuspenders was 01 taking the low variability of highstrength steel strands into account [16] e descriptivedispersions of the remaining components were calculated tobe 025 for the two lower damage states (DS1 and DS2) and047 for the higher damage states (DS3 and DS4) [8]
36 Assessment of Vulnerable Components e componentfragility curves were derived by steps 1ndash3 detailed in Section 2to identify vulnerable components and regions as shown inFigures 11 and 12e pot bearings were shown to be the mostvulnerable components for the main bridge which inferred thecomplete damages to pot bearings themselves and reflected ahigher probability of pounding between the adjacent decks andunseating in the longitudinal direction e transverse re-sponses of bearings did not cause obvious damage compared tothe longitudinal responses e pylons were the primarycomponents for load carrying where inelasticmaterial responsecould potentially occur e transverse vulnerability was moreobvious than the longitudinal vulnerability for the lowerdamage states However the probability of extensive damagewas extremely low or even zeroWith respect to themain cablesand suspenders the increases in their forces which couldpotentially occur during rare earthquakes were confined toabout 15 (for themean values) of the forces resulting from thedead loads as illustrated in Figure 12 which also indicated thatthe maximum values of main cable stress had occurred at thesaddles of pylons and in the side spans e stress values of
suspenders achieved the maximum at the middle and end Insummary the cable system was found to be secured againstdamages in which the stress values were much lower than theyield values even under strong earthquake shaking It wasnoted that the large deflection and cable angle changes betweenthe main span and side spans would lead to the strong shear inthe longitudinal direction between the cables and saddles andthe slipping damages to the saddles at the top of the pylonsduring earthquakes were identified as high-risk
For the approach bridges the columns were determinedto be the most fragile components in both the longitudinaland transverse directions e application of the fixed potbearings resulted in a large force and deformation of thecolumns the complete damage of sliding pot bearings andapparent pounding between the decks and abutments eseactions subsequently caused the inelastic responses ofabutments but the shear keys were protected from failuretransversely Although the ductility mechanism of approachbridges was designed it was considered to be a defectiveseismic system with irrational sequences of componentdamage occurrences in high-seismicity regions
41 ImprovedDesign ofMainBridge As results shown in theevaluation of vulnerable components a large deck dis-placement of the suspension span would occur duringearthquakes which caused pounding and even unseating
Original curveBilinear curve
MprimeyMy
φs or φccφucφprimeyφy
DS4 reinforcing bar rupture core concrete crushing
DS3 major spalling of cover concrete exposed core concrete
DS2 formation of stable plastic hinge in the pylon (column)
Mom
ent
Curvature
DS1 narrow cracking in concrete first yielding of reinforcing bar
Figure 10 e damage state definitions of the pylon and column cross-sections
Table 4 Limit state thresholds of pylon and column cross-sections
Figure 11 Component fragility curves for the suspension bridge (a) main bridge (b) approach bridge
200 400 600 800 10000Location (m)
00
02
04
06
08
10
MA
F
MaximumMinimumMean
Dead loadYield valueHigh stress region
(a)
00
02
04
06
08
10
SAF
200 400 600 800 10000Location (m)
MaximumMinimumMean High stress region
Dead loadYield value
(b)
Figure 12 Force distributions of the (a) main cables and (b) suspenders
12 Advances in Civil Engineering
when there were no additional restraints in the longitudinaldirection erefore it was consistent to suggest that sup-plementary devices should be equipped to mitigate the deckmovements to some extente reduction of deck or bearingdisplacements can be taken as the retrofitted objective for thesuspension span Fragility analyses were conducted on theretrofitted bridges that covered the general strategies asfollows
Case 1 the analysis model of the bridge used a fluidviscous damper equipped between the deck and eachpedestal on both sides of the bridgeCase 2 the analysis model of the bridge used an elasticcable installed between the deck and each pedestal onboth sides of the bridgeCase 3 the analysis model of the bridge used three pairsof flexible central buckles located in the middle sectionof the span
e main bridge was assumed to have identical me-chanical parameters at each location in various designschemes e dampers and elastic cables were respectivelymodeled using zero-length elements with Maxwell andtension-only materials and the flexible central buckles weresimulated as the same as the suspenders elements inOpenSEES
e experimental testing of fluid viscous dampers hasshown that the force-velocity with fractional velocity powerlaw relation can describe the suitable mechanical behavior
fD CD| _u|αsgn( _u) (9)
where α is a real positive exponent of which value dependson the piston head orifice CD is viscous damping coefficientwith units of force per velocity raised to the α power kNmiddot(ms)-α _u indicates the velocity of damper and sgn(middot) is thesignum function Research by Li et al [32] indicated thatviscous damping coefficients range from 2500 kN (ms)-α to5000 kN (ms)-α and can be simulated with a uniform dis-tribution e velocity exponents can be assumed to varyuniformly from 03 to 05
e axial stiffness of elastic cables and flexible centralbuckles is regarded with the material length and cross-section area It was assumed that the yield strength of steelstrands of fy 1420MPa was the mean value and followed alognormal distribution with a COV 01 e stiffnessvalues of single elastic cable based on past work employed auniform distribution ranging from 125times105 kNm to25 times105 kNm In addition the other mechanical prop-erties such as length and area can be determined by thefollowing relations
LgtEΔfy
1113888 1113889
kE EA
L1113874 1113875
(10)
where L E and fy are the length Youngrsquos modulus andyield stress of elastic cables respectively and Δ represents the
controlled deck displacement which was available for 06min the examined bridge
According to expert opinion three pairs of flexiblecentral buckles were the most common type of centralbuckle of which the cross-section areas were the same asthose of the suspenders
Figures 13(a) and 13(b) illustrate bearing displacementdistributions are formed for the three retrofit measures at thesame ground motion intensity levels e mechanical be-haviors of measures strongly affect the effectiveness It can beconcluded that owing to the similar EDP distributions ofdampers and elastic cables the corresponding groundmotion intensities were similar when generating the samepounding probability e flexible central buckles wereextremely susceptible to yielding failure thereby causing lowefficiency in the displacement restrictions ereforepounding probabilities can be directly observed by condi-tioning on the median responses of dampers and elasticcables instead (Figure 13(c)) e control force produced bydevices is another noteworthy consideration in retrofitdesign e more force the devices require the larger andmore expensive they usually are [13] To take both defor-mation and force into account Figures 13(d) and 13(e)provide the contours of the pounding probability ecomparison showed that the damper deformations featuredslightly larger slopes in the horizontal direction yet thedifference in effect on the deformations of the dampers andelastic cables was not significant for reducing the objectivepounding probability Conversely damper force was lesssensitive to the pounding probability than cable force eelastic cable forces reached on average up to e32 (asymp245)times those of the damper under the same poundingprobabilities ese findings implied that difficult anchoragechallenges existed in the case of guaranteeing the efficiencyof elastic cables and the bridge decks possibly experiencedhigher stress In addition the difference in force was sub-stantial when controlling the displacement which showedthat dampers can contribute additional damping and stiff-ness to the structural system while elastic cables completelytransferred the inertial forces erefore the coordinationand balance of displacements and forces should be simul-taneously considered in the vibration mitigation of large-span bridges In this study the application of fluid viscousdampers was identified as the preferred retrofit in thelongitudinal direction for the main bridge e additionaldamping ratio provided by the fluid viscous dampers can becalculated by the following equation
ε CD
2mωeff (11)
where CD is damping coefficient of linear fluid damperskNmiddot(sm) which is equivalent to nonlinear dampers basedon equal energy dissipation per hysteresis loop [17] ωeff issystem natural frequency corresponding to the vibrationalong the axis of bridge andm is system mass It was shownthat an additional damping of 206ndash318 can offer aneffective controlling mechanism for the reduction ofpounding between the adjacent decks
Advances in Civil Engineering 13
Figure 14 depicts the change of components vulnera-bilities before and after the application of dampers It can beobserved that the retrofit measure played limited roles in thefragility reduction of the self-damaged bearings in long-spanbridges Since the additional longitudinal rigidity to girderwas provided by the dampers the transverse responses suchas girder displacement and pylon curvature would increaseas ground motion intensity increased e influences on thecable system responses and the longitudinal responses ofpylons were found to be negligible Another phenomenonwas that the longitudinal fragility of the abutments signif-icantly decreased with the reduction in the poundingprobabilities between the decks and abutments of the ap-proach bridges It was inferred that the pounding betweenthe adjacent decks was likely to affect movement of theapproach bridges and that the pounding effect would betransmitted along the girder
42 Improved Design of Approach Bridges e damagepotential of each component for the approach bridges washigh due to the high collapse risks posed by the usage of thefixed piers e structural designs of small-to-medium-spanhighway bridges in China show the fact that the super-structures are usually placed onto laminated elastomericbearings (LEBs) which tend to come into direct contact withthe bridge girders and substructures without anchoring etypical types of seismic damages of these bridges consisted of
large girder movements bearings sliding failures of shearkeys and expansion joints and abutment cracking duringthe Wenchuan and Yushu earthquake events in ChinaMeanwhile the bridge columns displayed relatively lowdamage ratios under earthquakes [33] e design can bereferred to as a quasi-isolation system which is primarilyrealized by strategically employing bearings sliding as force-limiting connections between girders and substructures Inaddition the isolation strategies of lead-rubber bearings(LRBs) are usually employed in the high-seismicity regionswhich can be taken as a comparison of the quasi-isolationsystem e two restraint systems consist of bearings shearkeys transversely located at the cap beams and restrainercables longitudinally anchored to the abutments (or ped-estals) In order to calculate the seismic responses of theapproach bridge the related experiments were completedand summarized and applied to the force-displacementrelations of the restraint system components
e detailed experiment of LEB consulted previousacademic achievements [34] An example is shown inFigure 15(a) where the feature points marked by rectangularand circular represented initial walking and obvious slidingresponses which illustrated that the mechanical property ofbearings would degrade as the displacements increased afterentering the friction sliding state until it remained nearlyconstant When there was no specific test the horizontalforce Fd in the stable friction stage can be 07 Fmax whereFmax was the corresponding maximum static friction at the
Bear
ing
disp
lace
men
t (m
)
Logarithmic spacendash5
ndash4
ndash3
ndash2
ndash1
0
1
2 4 6PGV (cms)
Case 1Case 2Case 3
(a)Po
undi
ng p
roba
bilit
y
00
02
04
06
08
05 10 15 2000PGV (ms)
Case 1Case 2Case 3
(b)
Poun
ding
pro
babi
lity
00
02
04
06
02 04 06 08 1000Deformation (m)
Case 1Case 2
(c)
ln [f
orce
(kN
)]
5
10
15
15
20
20
25
25
30
3030
35
3535
40
40
4550
7
72
74
76
78
8
033 041 049 057 065025Deformation (m)
(d)
ln [f
orce
(kN
)]5
10
15
15 20
20 25
25
30
3030
35
35
40
40
45
50
102
104
106
108
11
112
114
0155 0205 0255 0305 03550105Deformation (m)
(e)
Figure 13 Choice of the three retrofitmeasures based on deformations and forces (a) EDP-IM relation (b) pounding probability conditioned onIM (c) pounding probability conditioned on deformation contours of pounding probability for (d) dampers and (e) elastic cables
14 Advances in Civil Engineering
initial walking e displacement value Δ2 can be of 250effective shear strain (ESS) e initial stiffness K0 is cal-culated by MOTC [20] erefore the coefficients can becalculated by the following sequence FdK0⟶ Fmax⟶Δ1⟶Δ2 e built analytical relation ofthe LEB is shown in Figure 16(a) which can correctlycapture mechanical behavior under earthquakes A log-normal distribution [8] with a logarithmic standard devi-ation of 01 and a normal distribution with aCOV 016MPa were used to model the randomness offriction coefficient and shear modulus respectively For theirrespective mean values 02 and 12MPa were taken
Figure 15(b) shows a cyclic loading experiment of LRBwhich reveals that the degradation of bearings stiffness willoccur with the increase of loading displacements e elasticstiffness can reach uniformly 3 to 7 times the postyieldingstiffness according to the experimental results e
mechanical behavior of LRB can be modeled by a bilinearrelation as seen in Figure 16(b)
e cable restrainers were modeled as nonlinear tension-only elements characterized by an initial slack as shown inFigure 16(c) e parameters affecting the functionality ofcable restrainers were the cable slack length and yieldstrength according to the research of Padgett and DesRoches[16] e first two parameters were assumed to follow theuniform distributions and the strength can be modeled bythe lognormal distribution An adopted range of cablestiffness values was from 30times105 kNm to 50times105 kNmuniformly
e key parameters in analytical models of Figure 16 canbe determined according to the structural design of therealistic bridge e damage states of two restraint systemsare listed in Table 6 in which ESS is used as the damageindex
Dam
age p
roba
bilit
y
05 10 15 2000PGV (ms)
00
02
04
06
08
10
Case 0Case 1
MBDLMBDT
(a)
Dam
age p
roba
bilit
y
00
01
02
03
04
05
05 10 15 2000PGV (ms)
Case 0Case 1
PCDNLPCDNT
(b)
Dam
age p
roba
bilit
y
00
02
04
06
08
10
05 10 15 2000PGV (ms)
Case 0Case 1
BPDAEJDA
(c)
Figure 14 Influence of dampers on vulnerabilties of (a) bearing responses for DS2 (b) pylon responses for DS2 and (c) abutment andpounding responses
Advances in Civil Engineering 15
Figure 17 illustrates the probabilities of critical damage stateoccurrence sequences of the bridge components in Cases 1 4and 5 when PGV is 10ms which is equal to E2 earthquakeintensity in the seismic assessment reports e componentvulnerabilities of initial design (Case 0) were taken as the ref-erence values which were marked by different colored strips inFigure 17 e desirable sequence preferred Cases 4 or 5 for theapproach bridgese initial design of approach bridges resultedin unreasonable damage sequences In the strategies pertainingto quasi-isolation and isolation systems multiple tiers of seismicstructural redundancy can be normally employed to preventstructural collapse during strong earthquake events Tier 1 in-dicated that the bearings were weakly fused which was evidentin the high fragility of bearing sliding behaviors (ABDAL andABDC2T) Another advantage was that the extensive dam-ages of bearing would be reduced when compared with the self-damage of the pot bearings Tier 2 indicated that the designedbearings had sufficient seat widths for adapting to the sliding ofsuperstructures However the devices had a certain damageprobability (EJDA and SKDC2) and dissipated a portion of
the seismic energy when the displacement of the superstructuresbecame larger than the allowable values of constraint devicesTier 3 redundancy was designed as a potential plastic hinge inthe substructures (CCDC2) including a certain yielding of thebackfill at the abutments (BPDA) e response of theabutments in the transverse direction (PDAT) was slightlyincreased due to the pounding effect between the girders andshear keys caused by bearing sliding e reasonable damagesequence of two restraint systems can be reflected through thecorresponding damage probabilities of bearings seismic mea-sures and substructures respectively as indicated by the red andgreen lines in Figure 17 erefore the rational sequence ofcritical limit state occurrences for the approach bridges isbearings⟶ expansion joints (or restrainer cables) shearkeys⟶ columns abutments
43 Comprehensive Evaluation of Bridge System Padgett andDesRoches [16] pointed out the necessity to derive the bridgesystem vulnerabilities when assessing retrofitted bridges
Figure 15 Experimental results of (a) LEB and (b) LRB
ΔΔ1
F
Δ2
Fmax
Fd
KdK0
(a)
Δy Δ
F
K1 = (3~7) K2
K1
K2
(b)
Slack
F
K
Δ
Fy
(c)
Figure 16 Analytical models of (a) LEB (b) LRB and (c) cable restrainers
16 Advances in Civil Engineering
because different components may have a different responsewith a given retrofit measure Also Zhang and Huo [18]studied the optimal parameters of isolation bearings byderiving the intensity median value corresponding to sys-tem-level fragility function which can effectively avoid theimpacts of different isolation parameters on differentcomponents As can be seen from Figures 14 and 17 theabove analysis results were also very apparent in this studyFor example the longitudinal deck displacement was sig-nificantly reduced by using dampers yet increased thetransverse demands of stiffening girder and pylons e useof LEB or LRB reduced the demands placed on columns yetled to an increase of the transverse deformations of abut-ments which was due to the use of shear keys limiting theexcessive sliding displacement of bearings erefore acomprehensive evaluation of retrofit strategies on bridgesparticularly long-span suspension bridges that have multiplecomponents should be based on system-level fragility in-stead of component-level
Taking the findings that no damages had occurred to thecable system and the definition of bearing damage states intoaccount the derivation of system-level fragility of the wholebridge depended on critical components including pylonscolumns bearings and abutments e fragility data byemploying the absolutely serial and serial-parallel assump-tions of bridge system are plotted in Figure 18 and comparedto the system fragility obtained from the proposed com-posite damage state e absolutely serial assumption
generated serious bias for fragility curves and may result inmisdirection if they were served as a design guide eabsolutely serial model overestimated the potential damageof the bridge system since the fragility curve directlydepended on the most vulnerable components In otherwords if the difference in location and performance of eachcomponent was neglected then the seismic performanceevaluation would be achieved by attaching equal importanceto the strength of pylons approach columns and bearingsis method contradicted the philosophy of capacity designin which the long-span structure can dissipate energythrough a reasonable swinging of stiffening girder and thebearings were designed as fuse elements in the small-to-medium-span bridgese serial-parallel model qualitativelyincorporated the differences of components and was verysensitive to the seismic design of the structure as seen fromthe comparisons among the cases in Figure 18 e modeldistinguished the serial components and parallel compo-nents yet still overestimated the bridge system-level fragilitywhen bearings became the vulnerable components espe-cially for long-span bridges is special situation such asthe serial assumption may also lead to misunderstandingOnce the relative importance between the components wasconsidered in the serial-composite model the damage po-tential was clearly reduced within the range of earthquakeintensity of interest based on the case studies is alsoreflected the necessity of the classification idea of bridgecomponents which can be expressed by the numerical value
Table 6 Damage states of the quasi-isolation and isolation bearings (m)
Figure 17 Probabilities of critical damage states occurrence sequences for the approach bridges when PGV is 1ms
Advances in Civil Engineering 17
of system-level fragility erefore the application ofcomposite damage state (equation (4)) considering com-ponent difference was instead more appropriate than theaforementioned two models
e damage probability distribution represented byblack circles in Figure 18 can be obtained from the definedcomposite damage state (equation (5)) where the parametera is 06 It was found that system fragility result was a moreaccurate assessment with further taking the relative im-portance between themain bridge and approach bridges intoaccount is is due to the principle that the seismic designperformance objective of long-span bridges generally has nocollapse for the main body structures and is necessarily opento emergency vehicle service In addition attention must bepaid to the fact that the differences in repair cost and time ofthe main and approach bridges are significant when thesame damage state is achieved e results of compositedamage state may match the performance objectives of theseismic design In order to investigate the influence ofimproved strategies and structural relative importance ondamage potential of bridges the mean values of the fragility
functions can be employed to evaluate the seismic designe smooth fragility curves for the specific damage states canbe fitted with the lognormal cumulative probability distri-bution function by employing the least-squares technique
P DSgtDSi | PGV1113858 1113859 Φln(pgv) minus λPGV
ζPGV1113888 1113889 (12)
where λPGV and ζPGV are the logarithmic mean and dis-persion values of earthquake intensity (PGV) for a specifieddamage level e larger mean values represent the smallerdamage potential to reach the specified level
Figure 19 indicates the mean values of earthquakeintensity at the specific damage states as a function of theimproved strategies and parameter a Each step of theretrofit measures would enhance the seismic performanceof the overall bridge system at all levels e improvementof the restraint system of the approach bridge played acritical role of the overall bridge system For instance themean values of the earthquake intensity featured mutationof slopes along the measure direction which reflected the
Figure 18 System fragility of the original design (a) Case 0 and improved design including (b) Case 1 and (c) Case 5
18 Advances in Civil Engineering
fact that the effectiveness of improvement for the ap-proach bridges was clearer than that of the main bridgee fragility of the highway network would be under-estimated if only the damage potential of the suspensionspan was adopted erefore when the structures such asapproach bridges transition piers and auxiliary pierswere ignored the damage identification and repair de-cisions would be misleading in the seismic evaluation oflong-span bridges Conversely the damaging probabilityof the overall bridge system will be amplified regardless ofthe higher importance of the main bridge When respectthe fact that the importance of suspension span is higherthan that of approach bridge in transportation networks aparameter a of 06ndash08 may be a better choice in thepresent study
e seismic fragility of conventionally designedbridges was mainly governed by the approach bridges at alldamage levels However for the retrofitted bridges thatpossess reasonable seismic performance different damagestates were controlled by the earthquake demands ofvarious structures as shown by the trend of the curvesalong parameter a in Case 4 and Case 5 e damagingpotential of the overall bridge from the low-to-moderatelimit states was dominated by the approach bridge eprimary reasons were the sliding of bearings and thetransfer of limited forces from the girder to substructure
by the damage of restriction devices of the approachbridges e extensive damage was controlled by thedisplacement of stiffening girder of the main bridge whichinferred the effect of dampers will be conspicuous understrong earthquake shaking At the same time the overalllong-span bridges with the reasonable seismic system willnot collapse during a strong earthquake event and canmeet emergency traffic service It can be seen that the meanvalues of fragility functions developed by compositedamage indices (equations (4) and (5)) are good indicatorsto evaluate seismic performance of long-span bridges andcan be consistent with the established seismic performanceobjectives
5 Conclusions
In the present study a procedure for investigating thereasonable seismic performance of long-span suspensionbridges was developed under the fragility function frame-work e seismic performance of a long-span suspensionbridge was examined and improved using the proposedprocedure from which the results revealed distinguishableconclusions as follows
(1) e definition of damage states for bearings con-sidered the interactions between the adjacent
014036059081103125
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(a)
020051081112142173
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
0002
0406
0810
CaseParameter a
(b)
130251371492612733
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(c)
Mea
n va
lue
exp
(λ) (
ms
)
Parameter a Case0 1 2 3 4 5
0001
0203
0405
131305478652826
1000
(d)
Figure 19 Influence of relative importance and improved designs on mean values of earthquake intensity for different damage states(a) slight (b) moderate (c) extensive (d) collapse
Advances in Civil Engineering 19
structures and self-damage to bearings in accor-dance with the actual design and experimentalconclusions e damage states of pylons in terms ofcurvature ductility were defined by developingmodal pushover analysis and idealized bilinear fit-ting methods in the equivalent pylon fiber models ofthe suspension bridge
(2) When investigating the component fragility curvesof the initial design it was found that the large deckdisplacement of the suspension span was the mostserious damage present and a slight inelastic re-sponse may have occurred only in the pylons Al-though the cable system was able to maintainelasticity under earthquakes the saddles at the py-lons were identified with a high-risk region indi-cating the damaging potential For the approachbridges the ductility mechanism did not realize arational damage sequence e columns were proneto collapse because of the application of fixed potbearings which in turn caused the damage to thebearings and abutments
(3) e stepwise improved seismic designs were pro-posed in terms of the component fragility results ofthe initial design Among the candidates fluid vis-cous dampers were proven to be a good choice forlong-span bridges based on the two attributes ofdisplacement and force when the similar efficiency ofdisplacement control was achieved e restraintsystem had a critical effect on the seismic responsesof the approach bridges e quasi-isolation orisolation can achieve a reasonable seismic damagesequence of the approach bridges
(4) e defined composite damage indices showed thatthe incorporate component classification conceptand structural relative importance were necessary toderive a good indicator of performance evaluationdue to the fact that the damage indices were based oneither an absolutely serial or serial-parallel systemmodel may significantly overestimate the damagepotential of the overall bridge system e meanvalues of earthquake intensity at the specific damagestates can serve as a function of the improvedstrategies and relative importance to further evaluatethe overall seismic performance of the suspensionbridge It can be concluded that the damaging po-tential of the overall bridge from the low-to-mod-erate limit states was governed by the approachbridges in the retrofitted bridges with a favorableseismic performance whereas the displacementdemand of the suspension span dominated thefragility probability of the extensive damage
Data Availability
e nature of the data is the analytical data of finite elementmodels of the case bridge e data used to support thefindings of this study are available from the correspondingauthor upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the Basic Research ServiceProject of Central-Level Public Welfare Research Institute(grant nos 2016-9018 and 2019-0034) and Science andTechnology Project of Communicationsrsquo Construction inWestern China MOC (grant no 2009318223094)
References
[1] L W David A Kartoum K Chang and I Roy ldquoSuspensionreport seismic performance and design considerations of longspan suspension bridgesrdquo Report CSMIP00-03 CaliforniaDepartment of Conservation Division of Mines And GeologyOffice of Strong Motion Studies Sacramento CA USA 2000
[2] S-G Gwon and D-H Choi ldquoImproved continuummodel forfree vibration analysis of suspension bridgesrdquo Journal ofEngineering Mechanics vol 143 no 7 Article ID 040170382017
[3] S Arzoumanidis A Shama and F Ostadan ldquoPerformance-based seismic analysis and design of suspension bridgesrdquoEarthquake Engineering and Structural Dynamics vol 34no 4-5 pp 349ndash367 2005
[4] L Sgambi E Garavaglia N Basso and F Bontempi ldquoMonteCarlo simulation for seismic analysis of a long span sus-pension bridgerdquo Engineering Structures vol 78 pp 100ndash1112014
[5] D Karmakar S Ray-Chaudhuri and M Shinozuka ldquoFiniteelement model development validation and probabilisticseismic performance evaluation of Vincent omas suspen-sion bridgerdquo Structure and Infrastructure Engineering vol 11no 2 pp 223ndash237 2015
[6] W-F Chen and L Duan Bridge Engineering HandbookSeismic Design CRC Press Boca Raton FL USA 2nd edition2014
[7] E Choi ldquoSeismic analysis and retrofit of mid-americabridgesrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2002
[8] B G Nielson and R DesRoches ldquoSeismic fragility method-ology for highway bridges using a component level approachrdquoEarthquake Engineering amp Structural Dynamics vol 36 no 6pp 823ndash839 2007
[9] K N Ramanathan ldquoNext generation seismic fragility curvesfor california bridges incorporating the evolution in seismicdesignrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2012
[10] O Avsar A Yakut and A Caner ldquoAnalytical fragility curvesfor ordinary highway bridges in Turkeyrdquo Earthquake Spectravol 27 no 4 pp 971ndash996 2011
[11] S P Stefanidou and A J Kappos ldquoMethodology for thedevelopment of bridge-specific fragility curvesrdquo EarthquakeEngineering amp Structural Dynamics vol 46 no 1 pp 73ndash932016
[12] B Borzi P Ceresa P Franchin F Noto G M Calvi andP E Pinto ldquoSeismic vulnerability of the Italian roadwaybridge stockrdquo Earthquake Spectra vol 31 no 4pp 2137ndash2161 2015
[13] W T Barnawi and S J Dyke ldquoSeismic fragility relationshipsof a cable-stayed bridge equipped with response modification
20 Advances in Civil Engineering
systemsrdquo Journal of Bridge Engineering vol 19 no 8 ArticleID A4013003 2014
[14] J Zhong Y Pang J Jong-Su J-S Jeon R DesRoches andW Yuan ldquoSeismic fragility assessment of long-span cable-stayed bridges in Chinardquo Advances in Structural Engineeringvol 19 no 11 pp 1797ndash1812 2016
[15] J Zhong Z Hu W Yuan and L Chen ldquoSystem-basedprobabilistic optimization of fluid viscous dampers equippedin cable-stayed bridgesrdquo Advances in Structural Engineeringvol 21 no 12 pp 1815ndash1825 2018
[16] J E Padgett and R DesRoches ldquoMethodology for the de-velopment of analytical fragility curves for retrofitted bridgesfitted bridgesrdquo Earthquake Engineering amp Structural Dy-namics vol 37 no 8 pp 1157ndash1174 2008
[17] Y Xie and J Zhang ldquoOptimal design of seismic protectivedevices for highway bridges using performance-basedmethodology and multiobjective genetic optimizationrdquoJournal of Bridge Engineering vol 22 no 3 Article ID04016129 2017
[18] J Zhang and Y Huo ldquoEvaluating effectiveness and optimumdesign of isolation devices for highway bridges using thefragility function methodrdquo Engineering Structures vol 31no 8 pp 1648ndash1660 2009
[19] F McKenna M H Scott and G L Fenves ldquoNonlinear finite-element analysis software architecture using object compo-sitionrdquo Journal of Computing in Civil Engineering vol 24no 1 pp 95ndash107 2010
[20] MOTC ldquoGuidelines for seismic design of highway bridgesrdquoChina Ministry of Transportation and CommunicationsPress Beijing China JTGT B02-01-2008 2008
[21] A D Kiureghian and J L Sackman ldquoTangent geometricstiffness of inclined cables under self-weightrdquo Journal ofStructural Engineering vol 131 no 6 pp 941ndash945 2005
[22] J S Steelman E T Filipov L A Fahnestock et al ldquoEx-perimental behavior of steel fixed bearings and implicationsfor seismic bridge responserdquo Journal of Bridge Engineeringvol 19 no 8 Article ID A4014007 2014
[23] B Madani F Behnamfar and H Tajmir Riahi ldquoDynamicresponse of structures subjected to pounding and structure-soil-structure interactionrdquo Soil Dynamics and EarthquakeEngineering vol 78 pp 46ndash60 2015
[24] L Xu and J Li ldquoDesign and experimental investigation of anew type sliding retainer and its efficacy in seismic fortifi-cationrdquo Engineering Mechanics vol 33 no 2 pp 111ndash1182016 in Chinese
[25] Caltrans Caltrans Seismic Design Criteria California De-partment of Transportation Sacramento CA USA 2013
[26] A Shamsabadi K M Rollins and M Kapuskar ldquoNonlinearsoil-abutment-bridge structure interaction for seismic per-formance-based designrdquo Journal of Geotechnical and Geo-environmental Engineering vol 133 no 6 pp 707ndash720 2007
[27] X Li Z Sun D Wang and Y Shi ldquoLongitudinal seismicpounding effect of bridges abutment and backfilling damagerdquoJournal of Changrsquoan University vol 35 no 4 pp 76ndash82 2015in Chinese
[28] MOTC ldquoSpecification for design of highway suspensionbridgerdquo China Ministry of Transportation and Communi-cations Press Beijing China JTGT D65-05-2015 2015
[29] A Shafieezadeh K Ramanathan J E Padgett andR DesRoches ldquoFractional order intensity measures forprobabilistic seismic demand modeling applied to highwaybridgesrdquo Earthquake Engineering amp Structural Dynamicsvol 41 no 3 pp 391ndash409 2012
[30] J E Padgett B G Nielson and R DesRoches ldquoSelection ofoptimal intensity measures in probabilistic seismic demandmodels of highway bridge portfoliosrdquo Earthquake Engineeringamp Structural Dynamics vol 37 no 5 pp 711ndash725 2008
[31] K R Mackie and B Stojadinovic ldquoFragility basis for californiahighway overpass bridge seismic decision makingrdquo PEERReport 200502 Pacific Earthquake Engineering ResearchCenter University of California Berkeley Berkeley CA USA2005
[32] J Li T Peng Ji Lin L Zhou and YuL Deng Seismic Design ofMulti-Pylon Multi-Span Suspension Bridge China Commu-nications Press Beijing China 2013 in Chinese
[33] L Chen W Zhuang H Zhao and Z Wan ldquoReport onHighwaysrsquo Damage in the Wenchuan Earthquake-BridgeChina Communications Press Beijing China 2012 inChinese
[34] G Wu K Wang G Lu and P Zhang ldquoAn experimentalinvestigation of unbonded laminated elastomeric bearingsand the seismic evaluations of highway bridges with testedbearing componentsrdquo Shock and Vibration vol 2018 ArticleID 8439321 18 pages 2018
Advances in Civil Engineering 21
value and variance of displacements and stresses were usedto describe the seismic responses of structures Karmakaret al [5] studied the seismic performance of the Vincentomas suspension bridge retrofitted with viscoelasticdampers and obtained the ductility demands of towers interms of fragility probability However the studies men-tioned above solely considered the component-level fragil-ities and focused on the damage of the towers Additionallyit must be noted that the approach span of the San Fran-cisco-Oakland Bay Bridge had fallen-off during an earth-quake event which caused the closure of transportation link[6] erefore it can be concluded that leaving out the effecton approach structures may skew the results in the systemseismic evaluation of long-span bridges Due to the differ-ence and connection in structural dynamic behavior anddesign strategy between the suspension span and approachstructures the optimization of system seismic performanceand a reasonable indicator that can reflect the overall seismicperformance for long-span bridges will be appealing
Previous studies have indicated that fragility functionsare effective tools by which to assess the damaging potentialof small-to-medium-span highway bridges under earth-quakes Fragility functions can connect the given damagestates with earthquake intensity Both component-level andsystem-level fragilities have been developed for a class ofbridges in a certain region such as Central America [7 8]California USA [9] Turkey [10] Greece [11] and Italy [12]Barnawi and Dyke [13] and Zhong et al [14 15] evaluatedthe seismic fragility of a cable-stayed bridge and compareddifferent retrofit measures using fragility techniques Al-though their studies noted the necessity of derivation ofsystem-level fragility of piers (towers) and bearings a serialconnection was used in estimating the system fragility ekey point of their methods depends on a relatively con-servative way to quantify correlations between criticalcomponents One is full correlation while the other is partialcorrelation obtained from joint probabilistic seismic de-mand models [8] It is worth discussing whether it is rea-sonable to apply these correlations to the fragilitydevelopment of overall long-span bridge system In additionthe application of fragility curves has been proficient forcomparing and selecting retrofit measures and isolationstrategies for conventional highway bridges [16ndash18]
In the context of the aforesaid review the key purposes ofthis study are to develop a procedure of the performance-basedseismic assessment for long-span suspension bridges and toexamine the applicability of the proposed indicator for evalu-ating the overall bridge system seismic performance A typicallong-span suspension bridge in China was selected A series ofdetailed finite element models were built using OpenSEES [19]software that can perform nonlinear time-history analysis eprocedure incorporated the uncertainties of ground motionsand structural properties e backbone of the proposed pro-cedure is fragility-based improvement of seismic performancefor long-span suspension bridges including the definition ofengineering demand parameters (EDPs) and the capacity limitstate thresholds associated with the case-dependent compo-nents Component fragility is used to pinpoint the vulnerablecomponents of the overall bridge under earthquakes and served
as the basis of stepwise retrofit strategies e retrofit candidatesare determined by earthquake damage experiences relatedcomponent seismic experiments and expert opinions Finallythe indicators appropriate for characterizing the seismic per-formance of overall system for long-span suspension bridges areproposed and discussed Finally the effectiveness of the sug-gested index in the system seismic performance evaluation isfurther identified by comparing the indicators of the retrofitteddesigns with those of the as-built design
2 Procedure Description
is study used PSDM generated by cloud approach andcapacity limit state models based on multiperformance ob-jectives to develop the fragility curves of bridge componentsrooted in the response data obtained from a set of nonlineartime-history analyses e seismic fragility could be defined asthe conditional probability of a structure seismic demand Sd
reaching or exceeding its capacity Sc at a given earthquakeintensity measure (IM) which could be expressed as follows
It is well known that the PSDM built by cloud approachis based on three assumptions (1) lognormal distribution ofseismic demand and structural capacity (2) constant dis-persion assumption for all IM ranges and (3) a power modelbetween seismic demand and IM erefore the conditionalprobability under a given IM can be written as follows
P Sd ge Sc
1113868111386811138681113868 IM1113872 1113873 1 minus Φln Sc( 1113857 minus ln aIMb
1113872 1113873
β2d | IM + β2c1113969⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠ (2)
where Sd is the mean value of seismic demand conditioned onan IM Sc is the limit state threshold of structure correspondingto each defined damage state (DS) βd|IM and βc indicate thedispersion values of demand and capacity respectively andΦ(middot) is the standard normal cumulative distribution functionBoth a and b are regression coefficients which can linearlyregress in log-transformed spaces
e dispersion of seismic demand βd|IM can be estimatedas follows
βd|IM
1113936 ln di( 1113857( 1113857 minus (ln a + b ln(IM))( 11138572
N minus 2
1113971
(3)
where di is the ith demand value obtained from the analyticalsamples and N denotes the sample sizes
e definitions of PSDM and capacity limit states areimperative in the formation of component fragility curvesComponent fragility curves facilitate to highlight the weak linksin the overall bridge system and develop the retrofit and repairdecisions However if the comprehensive seismic evaluation ofcomplicated structures with multicomponents is conductedthen the correlations of various components must be con-sidered According to the objective of this study the highlightsof the proposed procedure are outlined as follows
(1) e propagation of uncertainties is beneficial to thereflection of their statistical significance which include
2 Advances in Civil Engineering
the ground motions that meet the site conditionsrepresenting a broad range of intensities and variousmodeling parameters such as material strengthstructural properties and damping ratio e generalpractice is taking the uncertainty sources as randomvariables to formN finite elementmodels of suspensionbridges in the as-built design using a Latin hypercubesampling (LHS) technique and then pair these modelswith a set of N ground motions in order to carry out aseries of nonlinear time-history analyses
(2) e final step to derive bridge component fragilitycurves is to pinpoint the vulnerable componentsbased on the recorded peak response data in non-linear time-history analyses by which to define theircorresponding limit state thresholds at variousdamage states In addition the application of optimalIM is also the key link to ensure the reliability of theseismic vulnerability assessment
(3) e component-level fragility curves are calculatedusing equation (2) For the purpose of reducing thevarious damage potential of critical bridge componentsin the initial design different seismic designs areproposed based on earthquake damage experiencesrelated component seismic experiments and expertopinions Next the effectiveness of seismic measures isevaluated using the critical component-level fragility
(4) e optimum structural seismic performance is se-lected in this study taking the bridge system-levelfragility as the objective function according to thedefinition of damage states and fragility results ofvarious components obtained from steps 2 and 3Considering the vulnerability degree of each com-ponent and postearthquake repair cost it is helpfulto evaluate the seismic performance of the bridgesystem by combining the critical components in areasonable manner For suspension bridges cablesystems and pylons are primary load-carryingcomponents and pylons can be only allowed to be oflimited damage e damage of bearings can beallowed to compensate by the effective seismicmeasures For approach bridges with the small-to-medium spans the postearthquake serviceability ofbridges depends on the damage stage of columnsabutments and bearings A popular seismic designof this type of bridges is fusing actions of bearings asforce-limiting connections to reduce the damage ofsubstructures A bridge structure consists of multiplemembers with the same function but in variouslocations e working principle of pylons and
columns is similar to that of the series system due tothe fact that the damage or collapse of one of thepylons or columns will directly affect the post-earthquake availability of the bridge system emultiple bearings form a parallel system to jointlysupport superstructures and a single damage has lessinfluence on the overall bridge system Earth pres-sures on abutments are further increased as a resultof the pounding between the deck and abutmentbackwall for the seat abutments which leads to anincrease in the abutment deformation However thedifficulty of repairing abutments is relatively smalland the effect on emergency traffic is slight A parallelsystem can also be adopted to simulate an abutmentsystem For a global bridge structure once any typeof component system is damaged it will have asignificant impact on the overall bridge system eoverall bridge system is close to the series combi-nation of different component systems which can becalled a bridge system based on the series-parallelsystem
Zhong et al [14] summarized the method of componentclassification appropriate for long-span bridges and held thatthe components of long-span bridges can be classified asprimary secondary and accessory based on the relativeimportance of load-carrying capacity and repairing costserefore the application of a serial assumption betweencomponent systemsmay bias the seismic fragility assessmentof the bridge system Zhang and Huo [18] assigned aweighting ratio of 075 and 025 to columns and bearingsrespectively for ordinary highway bridges in USA andproposed the composite damage state for the bridge systeme repair-related strategy variables including repair costand time are considered based on the direct and indirecteconomic losses that are caused by the damage of bridgecomponents e damage to primary and secondary com-ponents at any level will account for a majority of repair costincluding repairing most components and a long closure oftraffic e replacement costs of accessory componentsprimarily rest on their extensive damage including rein-stallation of bearings and restoration of girder that might beresulted from pounding and unseating which interruptstraffic at a short time Combined with the reality of bridgeconstruction in China and the repair-related variables of thecritical components incorporated in the bridge system thisstudy separately proposes the composite damage state in-dices of the long-span and conventional bridge system asfollows
It should be noted that the aforementioned developmentof damage indices for the bridge system is only for a singlebridge system and is closely related to the definition ofdamage states of the components For long-span suspensionbridges composed of a main bridge and several approachbridges a reasonable and comprehensive seismic vulnera-bility evaluation is of great significance to the construction ofproject and postearthquake repairing cost A bridge systemwith the similar structural properties could be regarded as aseries system However the differences in structural im-portance and repairing costs cannot be neglected because ofthe structural differences between the long-span bridges andsmall-to-medium-span bridges e weighting ratios of aand (1 minus a) are assigned to the main bridge system andapproach bridge system respectively which have the abilityto capture the contributions of the main bridge and ap-proach bridges to the vulnerability of the overall bridgesystem erefore the rational range of system fragilitycould be accurately determined as follows
DSsystem round a middot DSmain bridge +(1 minus a) middot DSapproach bridge1113960 1113961
(5)
where the round-off principle can be used for the ldquoroundrdquoe composite damage state in equations (4) and (5) can
be incorporated in the PSDM when the correlation betweenconcerned components with the use of covariance matrix isknown In this case Monte Carlo simulation is used tocompare the demands obtained from the joint probabilisticseismic demand models [8] and multidimensional compo-nent capacities in theM (105) random samples at a range ofthe IM values e basic principle of developing fragilitycurves for the long-span suspension bridge system is shownin Figure 1
3 Case Study
31 Bridge Description and Simulation is study selectedthe Taoyuan Bridge spanning the Jinsha River as the studycase e bridge is characterized by the geometric andmechanical features of a suspension bridge which is themost common type of bridge construction in China ebridge has a total length of 731m and consists of a single-span simply supported suspension bridge measuring 636mand a continuous steel box-girder approach bridge withthree spans measuring 30m 35m and 30m respectivelye longitudinal elevation of the bridge is illustrated inFigure 2
e main bridge has a thin-wall steel girder along thelongitudinal direction with an overall width of 31m fromcurb to curb and a height of 3m accommodating four lanesof traffic e gate-type towers include two RC box-sectionlegs with overall heights of 75m and 70m on its two sidesrespectively Also there is one crossbeam installed along theheight of pylons in order to enhance the lateral stiffness ofstructures At the base the tower legs are anchored to the 6m-thick concrete footings e pedestals for the placingbearings are designed between the concrete footings due tothe shorter distance between the deck and tower base e
bored piles serving as the foundations of pylons and locatedat the leg of each tower have a diameter of 22me areas ofeach main cable and suspender are 01626 m2 and 0001316m2 respectively Gravity anchorage has been applied to themain cable anchorage on both sides of the bridge eapproach bridges are divided into left and right bridges withthe identical configurations on the north side For eachapproach bridge there is a three-span continuous steel box-girder bridge with a pile-bent abutment of which the middlespan is supported by two double-column bents with integralpile-shaft e width and depth of the superstructure are1275m and 153m respectively e circular sections ofcolumns with a diameter of 18m have a height of 7m andthe cast-in-drilled-hole piles measure 19m in diameter epiles of abutment are provided by a circular section of 15min diameter and the height of the backwall is 188m
e description of this studyrsquos analytical modelingprocedure is outlined using OpenSEES software [19] for theprototype bridge in the as-built design A beam-columnelement with distributed plasticity fiber was used to modelthe pylons and columns in order to account for the materialnonlinearity In addition the axial force-moment interac-tions of pylons were captured in the model Each fiber wassimulated via a reasonable stress-strain relationship whichwas dependent on unconfined concrete confined concretedefined by the Mander theoretical model and longitudinalreinforcements that considered the Bauschinger effects asillustrated in Figure 3 e corresponding mechanical in-dices such as material strength axial compression ratio(ACR) and reinforcement ratios of Sections I and II all metthe Chinese code requirements [20]
e main cable element was tension-only which wasmodeled as a finite number of large-displacement truss el-ements using the Ernst method accounting for the sag effect[21] Each suspender element was identical to the simulationof the main cable but there was no sag effect e initialstress of the cable system was also considered in the modelas shown in Figure 4(a) An elastic beam element was uti-lized to simulate the crossbeam of the pylons and stiffeninggirder as well as the girders and cap beams of the approachbridges e foundations were modeled by six spring ele-ments which indicated that in the future they are notexpected to be damaged under earthquake shaking
It should be noted that pot bearings were applied to theas-built design which consequently became the significantfactors in the overall responses and functionality of theapproach bridges e longitudinal response of sliding potbearings detailed in Figure 4(b) was simulated using a bi-linear element [20] e full-scale experiments of Steelmanet al [22] found that properly proportioned steel for fixedpot bearings can achieve a reliable dry friction responsebetween the bearing component and superstructure orsubstructure after the rupture of the anchor bolt Corre-spondingly the behavior of fixed pot bearings was modeledas shown in Figure 4(c) e fixed pot bearings weredesigned to be located at the C2 column as shown inFigure 1 Finally the pounding stiffness between the adjacentdecks (Figure 4(d)) was estimated by the sum of their axialstiffnesses [23]
4 Advances in Civil Engineering
Approach bridge
Composite damage state (4)
Main bridge
Composite damagestate (4)
Composite damage state (5)
Bearing 1 Bearing iBearing 2
Bearing 1 Bearing iBearing 2
Bearing 1 Bearing iBearing 2
Bearing 1 Bearing iBearing 2
Column 1
Column 2
Column j
Abutment 1
Abutment 2
Bearing 1 Bearing iBearing 2
Bearing system
Column 1 Column iColumn 2
Column system
Bearing 1 Bearing iBearing 2
Bearing 1 Bearing iBearing 2
Pylon 1
Pylon 2
Bearing system
Pylon 1 Pylon 2
Pylon system
Damage state definitionsand component-level
fragility results
Abutment 1
Abutment 2
Abutmentsystem
Figure 1 Bridge system based on the series-parallel system and composite damage state
C2C1
4000
7000
6000
7500 63
60
14000
3500
30001600063600
73100
3000South
North
Figure 2 Configuration of the Taoyuan Suspension Bridge (cm)
SouthH = 750m
North H = 700m
H
Sec I
Pylon
Pedestal
(a)
Confined(Mander model)
εu εcu
fc
Kfc
ε
σ
Unconfined
Concrete0002
fcufu
εc0
(b)
σy
εy ε
σ0005E0
E0
Bar
(c)
Figure 3 Continued
Advances in Civil Engineering 5
e shear keys contribute to the constraint of transversemovement between the girder and pier Xu and Li [24]proposed a modified model based on the structural prop-erties of shear keys in China which was used to build theforce-displacement relation of shear keys by separating thecontributions of steel and concrete in parallel as shown inFigure 5 e critical parameters of the shear keys can becalculated using Xursquos method according to the actual designwhich represents the yield (Δ1Y) nominal (Δ1n) degraded(Δ1d) and ultimate (Δ1u) deformations respectively esimulation can accurately reflect the mechanical behavior ofthe shear keys
e seat type abutment is shown in Figure 6 e lon-gitudinal resistance to seismic forces is provided jointly bythe passive action of the backfill soil and active action of thepiles [25] Meanwhile the transverse response is resisted bythe pilese passive and active responses of abutments weresimulated using hyperbolic soil material [26] and trilinearforce-displacement relationship respectively which wereconnected in parallel to capture the abutment responses Inaddition the pounding effect between the deck and abut-ment was simulated using a linear model without energydissipation e stiffness of pounding spring between thesuperstructure and abutment was proposed to be ten timesthat of the backfill soil stiffness [27]
32 Uncertainty Treatment in the FE Model e LHS ap-proach is usually used to account for the uncertainties infragility analysis such as structural geometry materialstrength component properties deck mass and damping in
the modeling which is considered to be a variance reductionsampling technique It is noted that the uncertainty ofstructural geometry is not taken into account in this studyfor the development of bridge-specific fragility curves In theforegoing approach the probability distributions were as-sumed for each parameter and each random variable wasdivided into the several equal intervals which correspond to5 to 95 of the cumulative probability e consideringmodeling parameters and their probability distributions arepresented in Table 1 Statistically significant yet nominallyidentical 3D bridges were built using the LHS so as toaccount for the abovementioned uncertainties then thebridges were paired randomly with the selected groundmotions
33 Ground Motion Input and Intensity Measure e ne-cessity to carry out a large number of nonlinear time-historyanalyses depends on a significant number of earthquakerecords selected for the PSDM In this study a data set of 80recorded ground motions and 20 Los Angeles-pertinentunscaled ground motions were extracted from the PEERstrong motion database and the SAC project database re-spectively [29] ese ground motions can represent thecharacteristics of the bridge site ey were placed into fivemagnitude-distance bins for the purpose of covering therational and broad ground motion intensity e first fourbins included combinations of low and high magnitudesand large and small fault distances e magnitudes variedbetween 57 and 70 while the fault distances ranged from130 to 600 km e characteristic of the ground motions in
Stress
Initialstress
K1
Strain
K2 = 0005K1
(a)
K1
Δy = 2mm
Fy
ΔΔy
K2 = 000001K1
F
(b)
Δu
F
Fd
Δy
K
Rh
ΔΔy = 1mm
Δu = 20mm
(c)
K
Gap
F
Δ
(d)
Δy
FΔult
K
Δ
Fult
Fult2
Δy= 154mmΔult = 1504mm
Per width (m)of abutment
(e)
Δ1 = 762hmmΔ2 = 254hmm
h height of backwall
Keff
K2
F2F1
Δ1
K1
F
ΔΔ2
Per pileKeff = 14kNmm
(f)
Figure 4 Constitutive relations of the various components (a) cable and suspender (b) sliding pot bearing (c) fixed pot bearing(d) pounding (e) abutment backfill (f ) abutment pile
Bar (HRB500 376)
z
y
Confined concrete (stirrup HRB400
1256)
Sec I
Unconfined concrete (C55)
ACR 0159
(d)
Sec II
h =
70m
Double-column bent
(e)
Confined concrete(stirrup HRB335
08)
Unconfined concrete (C35)
Bar(HRB335 1163)
z
y
Sec II
ACR 0037
(f )
Figure 3 Sketch and material model of the pylons and bents of the examined suspension bridge
6 Advances in Civil Engineering
the fifth bin was that the fault distance was less than 15 kme ground motions were given in two orthogonal com-ponents and the acceleration values adopted were computedas the geometric mean values (Sa
SaL middot SaT
1113968) e peak
ground acceleration ranged from 0043 g to 112 g Figure 7shows the distributions of peak ground accelerations (PGA)peak ground velocities (PGV) earthquake magnitudes (M)and fault distances (R) of 100 ground motion records as well
as the acceleration spectra with a damping ratio of 002 etwo orthogonal components of the selected earthquakerecords were randomly input along the longitudinal andtransverse directions of the bridge
PGA PGV PGD spectral acceleration or Arias In-tensity of earthquake can be selected as the IM candidateswhich are used to characterize earthquake intensity elinear relations with the interesting EDPs from probabilistic
Δ
V
Δ1n Δ1d
Vcp
Δ
V
Δ1y Δ1u
Vsp
i
j
Gap
Hysteretic property of concrete Hysteretic property of steel
Vcp = 7624kN
Δ1n = 106mmΔ1d = 422mm
Vsp = 10143kN
Δ1y = 11mmΔ1u = 633mm
Figure 5 Analytical model of shear keys
Abutment
Deck
Figure 4 (b)
Figure 4 (d)
Figure 4 (c)Figure 5 Figure 4 (e)
Figure 4 (f)
Figure 6 Pile-bent abutment simulations
Table 1 Uncertainty parameters incorporated in the modeling design
Modeling parameter Probability distribution Mean COV Unit Source
Concrete compressive strength C55 Normal 466 0149 MPa
[20]C35 321 0164
Steel yield strengthHRB500
Lognormal5696
00743 MPaHRB400 4557HRB335 3816
Friction coefficient Sliding pot bearing Uniform 003 033 mdash [14]Fixed pot bearing Lognormal 02 0002 mdash [7]
Deck-abutment 80Deck-shear key Uniform 120 029 mm [16]
Damping ratio Uniform 1 01 mdash [7]Deck mass Normal 002 015 mdash [28]COV coefficient of variation
Advances in Civil Engineering 7
seismic demand analysis in the logarithmic reference framecan be adopted to determine the optimal IM In the previousstudies PGA and spectral acceleration were selected as theoptimal IMs for highway bridges [9 30] when applying themultiple measures containing efficiency practicality profi-ciency and sufficiency Mackie et al [31] suggested both PGAand PGV are good choices of IM in related to the EDPmeasures in highway overpass bridges Zhong et al [14]pointed out that PGA and PGV are themost efficient practicaland proficient IMs for long-span cable-stayed bridges yet PGVtends to be the optimal IM in terms of sufficiency ereforePGV can be utilized as the IM in this study
34 Engineering Demand Parameters e peak demands ofcritical components are adopted as EDPs for the developmentof PSDMs and the choice of optimal IM Table 2 lists thecritical EDPs for the examined bridge which have beenconsidered in the fragility evaluation of the suspension bridge
35 Capacity Limit State Models In the present study acapacity model was used to measure component damagewhich was described by a damage index as a function of the
selected EDPs ese models are usually discrete andcharacterized by the onset of various damage states based onthe experimental data Similarly the uncertainties of ca-pacity models are also expressed by median Sc and dis-persion βc e definition of damage states must be matchedwith postearthquake bridge functionality and repair strat-egies in the bridge seismic fragility assessments Founda-tions decks and anchorages are typically identified asnondamaged components in the simulations e damagestate values of bearings were determined so as to comparewith two factors including the pounding gaps between theadjacent components and seating width which can reflectthe possible pounding and unseating as shown in Figure 8 inaddition to the damages to bearings themselves that is thedeformation capacity of the bearings Table 3 lists the slight(DS1) moderate (DS2) extensive (DS3) and collapsed(DS4) damage states defined in the initial design for potbearings at various locations based on their allowable de-formation and structural dimensions
e columns of conventional highway bridges are oftenforced into the state of nonlinearity under strong earth-quakes of which the criteria of damage states were studied[8 9 16] It is found that there are major differences in
0
10
20
30
40
50N
umbe
r of r
ecor
ds
08
01
03
05
09
00
07
04
02
10
06
PGA (g)
(a)
0
10
20
30
40
50
Num
ber o
f rec
ords
20 806040 100
180
140
160
2000
220
120
PGV (cms)
(b)
20 40 600Fault distance (R) (km)
55
60
65
70
75
Mag
nitu
de (M
)
(c)
Mean
0
1
2
3
4
5
S a (g
)
2 4 6 8 100Period (s)
(d)
Figure 7 Characteristics of the selected ground motions (a) PGA (b) PGV (c) M and R (d) acceleration response spectra
8 Advances in Civil Engineering
geometric sizes material strengths and ACR of sectionsbetween the pylons of suspension bridges and columns ofconventional highway bridges erefore it is necessary todevelop a modal pushover analysis to define the damagestates of critical cross-sections of pylons In the analyticalprocesses of this study the material strain and nonlinearmoment-curvature relationship of critical cross-sectionswere characterized to measure the damage levels of thesuspension bridge pylons
First the reasonable pylon models were required to bedeveloped using fiber beam elements in longitudinal andtransverse directions as detailed in Figure 3 An elastic
spring which was equivalent to the restraint of main cablewas attached to the top of the pylon taking into account theinfluence of the main cable stiffness in the longitudinaldirection models e determining process of the equivalentstiffness can be referred to in the Ernst formula and theresults reported by Kiureghian and Sackman [21]
Second as illustrated in Figure 9 the inertial forcedistributions were back-calculated from the multimodespectrum analysis using a 3D bridge model as shown inequations (6)ndash(8) which included the contributions of highvibration modes and reflected the properties of inertial forcedistributions as follows
DS3
DS4
DeckDeck
Pedestal
PedestalBearing
Bearing
Gap
(a)
DS2
DS4
Gap
Abutmentnonlinearity
Deck
AbutmentBearing
(b)
GapShear key
degradationDS4
DS2
Bearing
(c)
Figure 8 Damage control and unseating prevention damage states for bearing displacement (a) between decks (b) between deck andabutment (c) between deck and shear key
Table 2 e critical EDPs for the examined bridge
Engineering demand parameters Abbreviation DirectionCurvature ductility of pylon cross-section (south north) PCD(S N) Longitudinal and transverseDisplacement of bearings of the main bridge MBD Longitudinal and transverseAxial force relative to yield value of main cables MAF mdashAxial force relative to yield value of suspenders SAF mdashCurvature ductility of column cross-section at columns (C1 C2) CCD(C1 C2) Longitudinal and transverseDisplacement of bearings of the approach bridge at column (C) and abutment (A) ABD(C A) Longitudinal and transverseDeformation of backfill passive pressure at abutment BPDA LongitudinalDeformation of piles at abutment PDA Longitudinal and transverseDeformation of expansion joint between decks(D) and at abutment (A) EJD(D A) LongitudinalDeformation of shear key at columns (C) and at abutments (A) SKD(C A) Transverse
Table 3 Limit state thresholds for pot bearings (m)
DS1 DS2 DS3 DS4
Criteria Allowable slippingdisplacements
Spherical skateboard radiusbody separation of bearings
Possible pounding betweenadjacent decks or deck and pylon Possible unseating
MBDL 025 031 076 1555MBDT 002 031 045 mdash
Criteria Allowable slippingdisplacements
Possible pounding entering of abutmentsnonlinearity or shear key degradation
Spherical skateboard radiusbody separation of bearings Possible unseating
where Sj and cj are the spectral acceleration and participationfactor associated with the jth mode respectively Fij φij andQij represent the seismic load mode displacement and shearforce at the i-node associated with the jth mode respectivelymi Qj and Pi indicate the mass shear force and inertial forceat the i-node respectively andN denotes the number of nodes
Next the inertial forces were loaded to the nodes ofpylon models Both the material and geometric nonlinear-ities were incorporated into the pushover analyses whichwere used to investigate the potential failure mechanismsand assess the ductility capacities of pylons
e seismic designs of pylons can develop inelastic be-haviors which were controlled by the material strain limitsFigure 9 details the curvature distributions along the pylonheights with the increase of δ where δ was the loading dis-placement at the control point It was observed that plastichinge zones typically occur at the bases of pylons and above andbelow the horizontal crossbeams Also plastic hinges mayprogress to the intermediate height in the longitudinal direc-tion e development of plastic hinges at the bases of pylonscan be much severer and also occur earlier than in other zoneserefore the curvatures at the bottoms of pylons may ef-fectively provide measurements of the damages to pylons
e mechanical indices of pylon cross-sections whichcorrespond to the four damage state characteristics
following the damages are detailed in Figure 10 e cur-vature ductility can be easily determined by the pushoveranalysis results in addition to that of the DS2e numericaldata can be transformed to a bilinear curve that has a value ofDS2 based on the equal energy rule per original curveSimilarly another pushover analysis was applied to thedouble-column bend of the approach bridge based on thespecifications [20] e four limit state thresholds of thecross-sections at the bottom of pylon and columns in boththe longitudinal and transverse directions are listed in Ta-ble 4 It was observed that the seismic capacity of longitu-dinal direction was superior to that of transverse directionand a good ductility capacity existed in both directions
In the present study the definitions of damage statethresholds of main cables and suspenders were dependent onthe steel strand characteristics of the strength and deformationrates A total of 85 of the ultimate strength (nominal yieldstrength) was conservatively acted as the failure index Namelythe corresponding ratio of force to yield value was 10 It is acommon viewpoint that the accidental loss of one suspenderwill not cause immediate failures of a bridge with the exceptionof the loss of the main cables erefore DS3 and DS4 wereonly defined based on the classification of suspenders andmaincables respectively e corresponding elastic and ultimatedeformation of abutments and shear keys were respectivelyspecified the thresholds of DS1 and DS2 with the recom-mendations of Caltrans [25] e damage of expansion jointsonly defined as the DS1 due to the lack of a higher-leveldamage modelis was objectively logical since the joints canbe replaced with the new once damaged thereby suggesting therationality for just one threshold of component damage Table 5summarizes the damage state thresholds of the above-mentioned components
e assignments of dispersions of capacity models ofcomponents appeared to be the most difficult tasks for theresearchers According to the previous research studies 035was adopted in the abutment and RC components including
Figure 9 Inertial force and curvature distributions (a) longitudinal (b) transverse
10 Advances in Civil Engineering
pylons and columns [9] e COV of main cables andsuspenders was 01 taking the low variability of highstrength steel strands into account [16] e descriptivedispersions of the remaining components were calculated tobe 025 for the two lower damage states (DS1 and DS2) and047 for the higher damage states (DS3 and DS4) [8]
36 Assessment of Vulnerable Components e componentfragility curves were derived by steps 1ndash3 detailed in Section 2to identify vulnerable components and regions as shown inFigures 11 and 12e pot bearings were shown to be the mostvulnerable components for the main bridge which inferred thecomplete damages to pot bearings themselves and reflected ahigher probability of pounding between the adjacent decks andunseating in the longitudinal direction e transverse re-sponses of bearings did not cause obvious damage compared tothe longitudinal responses e pylons were the primarycomponents for load carrying where inelasticmaterial responsecould potentially occur e transverse vulnerability was moreobvious than the longitudinal vulnerability for the lowerdamage states However the probability of extensive damagewas extremely low or even zeroWith respect to themain cablesand suspenders the increases in their forces which couldpotentially occur during rare earthquakes were confined toabout 15 (for themean values) of the forces resulting from thedead loads as illustrated in Figure 12 which also indicated thatthe maximum values of main cable stress had occurred at thesaddles of pylons and in the side spans e stress values of
suspenders achieved the maximum at the middle and end Insummary the cable system was found to be secured againstdamages in which the stress values were much lower than theyield values even under strong earthquake shaking It wasnoted that the large deflection and cable angle changes betweenthe main span and side spans would lead to the strong shear inthe longitudinal direction between the cables and saddles andthe slipping damages to the saddles at the top of the pylonsduring earthquakes were identified as high-risk
For the approach bridges the columns were determinedto be the most fragile components in both the longitudinaland transverse directions e application of the fixed potbearings resulted in a large force and deformation of thecolumns the complete damage of sliding pot bearings andapparent pounding between the decks and abutments eseactions subsequently caused the inelastic responses ofabutments but the shear keys were protected from failuretransversely Although the ductility mechanism of approachbridges was designed it was considered to be a defectiveseismic system with irrational sequences of componentdamage occurrences in high-seismicity regions
41 ImprovedDesign ofMainBridge As results shown in theevaluation of vulnerable components a large deck dis-placement of the suspension span would occur duringearthquakes which caused pounding and even unseating
Original curveBilinear curve
MprimeyMy
φs or φccφucφprimeyφy
DS4 reinforcing bar rupture core concrete crushing
DS3 major spalling of cover concrete exposed core concrete
DS2 formation of stable plastic hinge in the pylon (column)
Mom
ent
Curvature
DS1 narrow cracking in concrete first yielding of reinforcing bar
Figure 10 e damage state definitions of the pylon and column cross-sections
Table 4 Limit state thresholds of pylon and column cross-sections
Figure 11 Component fragility curves for the suspension bridge (a) main bridge (b) approach bridge
200 400 600 800 10000Location (m)
00
02
04
06
08
10
MA
F
MaximumMinimumMean
Dead loadYield valueHigh stress region
(a)
00
02
04
06
08
10
SAF
200 400 600 800 10000Location (m)
MaximumMinimumMean High stress region
Dead loadYield value
(b)
Figure 12 Force distributions of the (a) main cables and (b) suspenders
12 Advances in Civil Engineering
when there were no additional restraints in the longitudinaldirection erefore it was consistent to suggest that sup-plementary devices should be equipped to mitigate the deckmovements to some extente reduction of deck or bearingdisplacements can be taken as the retrofitted objective for thesuspension span Fragility analyses were conducted on theretrofitted bridges that covered the general strategies asfollows
Case 1 the analysis model of the bridge used a fluidviscous damper equipped between the deck and eachpedestal on both sides of the bridgeCase 2 the analysis model of the bridge used an elasticcable installed between the deck and each pedestal onboth sides of the bridgeCase 3 the analysis model of the bridge used three pairsof flexible central buckles located in the middle sectionof the span
e main bridge was assumed to have identical me-chanical parameters at each location in various designschemes e dampers and elastic cables were respectivelymodeled using zero-length elements with Maxwell andtension-only materials and the flexible central buckles weresimulated as the same as the suspenders elements inOpenSEES
e experimental testing of fluid viscous dampers hasshown that the force-velocity with fractional velocity powerlaw relation can describe the suitable mechanical behavior
fD CD| _u|αsgn( _u) (9)
where α is a real positive exponent of which value dependson the piston head orifice CD is viscous damping coefficientwith units of force per velocity raised to the α power kNmiddot(ms)-α _u indicates the velocity of damper and sgn(middot) is thesignum function Research by Li et al [32] indicated thatviscous damping coefficients range from 2500 kN (ms)-α to5000 kN (ms)-α and can be simulated with a uniform dis-tribution e velocity exponents can be assumed to varyuniformly from 03 to 05
e axial stiffness of elastic cables and flexible centralbuckles is regarded with the material length and cross-section area It was assumed that the yield strength of steelstrands of fy 1420MPa was the mean value and followed alognormal distribution with a COV 01 e stiffnessvalues of single elastic cable based on past work employed auniform distribution ranging from 125times105 kNm to25 times105 kNm In addition the other mechanical prop-erties such as length and area can be determined by thefollowing relations
LgtEΔfy
1113888 1113889
kE EA
L1113874 1113875
(10)
where L E and fy are the length Youngrsquos modulus andyield stress of elastic cables respectively and Δ represents the
controlled deck displacement which was available for 06min the examined bridge
According to expert opinion three pairs of flexiblecentral buckles were the most common type of centralbuckle of which the cross-section areas were the same asthose of the suspenders
Figures 13(a) and 13(b) illustrate bearing displacementdistributions are formed for the three retrofit measures at thesame ground motion intensity levels e mechanical be-haviors of measures strongly affect the effectiveness It can beconcluded that owing to the similar EDP distributions ofdampers and elastic cables the corresponding groundmotion intensities were similar when generating the samepounding probability e flexible central buckles wereextremely susceptible to yielding failure thereby causing lowefficiency in the displacement restrictions ereforepounding probabilities can be directly observed by condi-tioning on the median responses of dampers and elasticcables instead (Figure 13(c)) e control force produced bydevices is another noteworthy consideration in retrofitdesign e more force the devices require the larger andmore expensive they usually are [13] To take both defor-mation and force into account Figures 13(d) and 13(e)provide the contours of the pounding probability ecomparison showed that the damper deformations featuredslightly larger slopes in the horizontal direction yet thedifference in effect on the deformations of the dampers andelastic cables was not significant for reducing the objectivepounding probability Conversely damper force was lesssensitive to the pounding probability than cable force eelastic cable forces reached on average up to e32 (asymp245)times those of the damper under the same poundingprobabilities ese findings implied that difficult anchoragechallenges existed in the case of guaranteeing the efficiencyof elastic cables and the bridge decks possibly experiencedhigher stress In addition the difference in force was sub-stantial when controlling the displacement which showedthat dampers can contribute additional damping and stiff-ness to the structural system while elastic cables completelytransferred the inertial forces erefore the coordinationand balance of displacements and forces should be simul-taneously considered in the vibration mitigation of large-span bridges In this study the application of fluid viscousdampers was identified as the preferred retrofit in thelongitudinal direction for the main bridge e additionaldamping ratio provided by the fluid viscous dampers can becalculated by the following equation
ε CD
2mωeff (11)
where CD is damping coefficient of linear fluid damperskNmiddot(sm) which is equivalent to nonlinear dampers basedon equal energy dissipation per hysteresis loop [17] ωeff issystem natural frequency corresponding to the vibrationalong the axis of bridge andm is system mass It was shownthat an additional damping of 206ndash318 can offer aneffective controlling mechanism for the reduction ofpounding between the adjacent decks
Advances in Civil Engineering 13
Figure 14 depicts the change of components vulnera-bilities before and after the application of dampers It can beobserved that the retrofit measure played limited roles in thefragility reduction of the self-damaged bearings in long-spanbridges Since the additional longitudinal rigidity to girderwas provided by the dampers the transverse responses suchas girder displacement and pylon curvature would increaseas ground motion intensity increased e influences on thecable system responses and the longitudinal responses ofpylons were found to be negligible Another phenomenonwas that the longitudinal fragility of the abutments signif-icantly decreased with the reduction in the poundingprobabilities between the decks and abutments of the ap-proach bridges It was inferred that the pounding betweenthe adjacent decks was likely to affect movement of theapproach bridges and that the pounding effect would betransmitted along the girder
42 Improved Design of Approach Bridges e damagepotential of each component for the approach bridges washigh due to the high collapse risks posed by the usage of thefixed piers e structural designs of small-to-medium-spanhighway bridges in China show the fact that the super-structures are usually placed onto laminated elastomericbearings (LEBs) which tend to come into direct contact withthe bridge girders and substructures without anchoring etypical types of seismic damages of these bridges consisted of
large girder movements bearings sliding failures of shearkeys and expansion joints and abutment cracking duringthe Wenchuan and Yushu earthquake events in ChinaMeanwhile the bridge columns displayed relatively lowdamage ratios under earthquakes [33] e design can bereferred to as a quasi-isolation system which is primarilyrealized by strategically employing bearings sliding as force-limiting connections between girders and substructures Inaddition the isolation strategies of lead-rubber bearings(LRBs) are usually employed in the high-seismicity regionswhich can be taken as a comparison of the quasi-isolationsystem e two restraint systems consist of bearings shearkeys transversely located at the cap beams and restrainercables longitudinally anchored to the abutments (or ped-estals) In order to calculate the seismic responses of theapproach bridge the related experiments were completedand summarized and applied to the force-displacementrelations of the restraint system components
e detailed experiment of LEB consulted previousacademic achievements [34] An example is shown inFigure 15(a) where the feature points marked by rectangularand circular represented initial walking and obvious slidingresponses which illustrated that the mechanical property ofbearings would degrade as the displacements increased afterentering the friction sliding state until it remained nearlyconstant When there was no specific test the horizontalforce Fd in the stable friction stage can be 07 Fmax whereFmax was the corresponding maximum static friction at the
Bear
ing
disp
lace
men
t (m
)
Logarithmic spacendash5
ndash4
ndash3
ndash2
ndash1
0
1
2 4 6PGV (cms)
Case 1Case 2Case 3
(a)Po
undi
ng p
roba
bilit
y
00
02
04
06
08
05 10 15 2000PGV (ms)
Case 1Case 2Case 3
(b)
Poun
ding
pro
babi
lity
00
02
04
06
02 04 06 08 1000Deformation (m)
Case 1Case 2
(c)
ln [f
orce
(kN
)]
5
10
15
15
20
20
25
25
30
3030
35
3535
40
40
4550
7
72
74
76
78
8
033 041 049 057 065025Deformation (m)
(d)
ln [f
orce
(kN
)]5
10
15
15 20
20 25
25
30
3030
35
35
40
40
45
50
102
104
106
108
11
112
114
0155 0205 0255 0305 03550105Deformation (m)
(e)
Figure 13 Choice of the three retrofitmeasures based on deformations and forces (a) EDP-IM relation (b) pounding probability conditioned onIM (c) pounding probability conditioned on deformation contours of pounding probability for (d) dampers and (e) elastic cables
14 Advances in Civil Engineering
initial walking e displacement value Δ2 can be of 250effective shear strain (ESS) e initial stiffness K0 is cal-culated by MOTC [20] erefore the coefficients can becalculated by the following sequence FdK0⟶ Fmax⟶Δ1⟶Δ2 e built analytical relation ofthe LEB is shown in Figure 16(a) which can correctlycapture mechanical behavior under earthquakes A log-normal distribution [8] with a logarithmic standard devi-ation of 01 and a normal distribution with aCOV 016MPa were used to model the randomness offriction coefficient and shear modulus respectively For theirrespective mean values 02 and 12MPa were taken
Figure 15(b) shows a cyclic loading experiment of LRBwhich reveals that the degradation of bearings stiffness willoccur with the increase of loading displacements e elasticstiffness can reach uniformly 3 to 7 times the postyieldingstiffness according to the experimental results e
mechanical behavior of LRB can be modeled by a bilinearrelation as seen in Figure 16(b)
e cable restrainers were modeled as nonlinear tension-only elements characterized by an initial slack as shown inFigure 16(c) e parameters affecting the functionality ofcable restrainers were the cable slack length and yieldstrength according to the research of Padgett and DesRoches[16] e first two parameters were assumed to follow theuniform distributions and the strength can be modeled bythe lognormal distribution An adopted range of cablestiffness values was from 30times105 kNm to 50times105 kNmuniformly
e key parameters in analytical models of Figure 16 canbe determined according to the structural design of therealistic bridge e damage states of two restraint systemsare listed in Table 6 in which ESS is used as the damageindex
Dam
age p
roba
bilit
y
05 10 15 2000PGV (ms)
00
02
04
06
08
10
Case 0Case 1
MBDLMBDT
(a)
Dam
age p
roba
bilit
y
00
01
02
03
04
05
05 10 15 2000PGV (ms)
Case 0Case 1
PCDNLPCDNT
(b)
Dam
age p
roba
bilit
y
00
02
04
06
08
10
05 10 15 2000PGV (ms)
Case 0Case 1
BPDAEJDA
(c)
Figure 14 Influence of dampers on vulnerabilties of (a) bearing responses for DS2 (b) pylon responses for DS2 and (c) abutment andpounding responses
Advances in Civil Engineering 15
Figure 17 illustrates the probabilities of critical damage stateoccurrence sequences of the bridge components in Cases 1 4and 5 when PGV is 10ms which is equal to E2 earthquakeintensity in the seismic assessment reports e componentvulnerabilities of initial design (Case 0) were taken as the ref-erence values which were marked by different colored strips inFigure 17 e desirable sequence preferred Cases 4 or 5 for theapproach bridgese initial design of approach bridges resultedin unreasonable damage sequences In the strategies pertainingto quasi-isolation and isolation systems multiple tiers of seismicstructural redundancy can be normally employed to preventstructural collapse during strong earthquake events Tier 1 in-dicated that the bearings were weakly fused which was evidentin the high fragility of bearing sliding behaviors (ABDAL andABDC2T) Another advantage was that the extensive dam-ages of bearing would be reduced when compared with the self-damage of the pot bearings Tier 2 indicated that the designedbearings had sufficient seat widths for adapting to the sliding ofsuperstructures However the devices had a certain damageprobability (EJDA and SKDC2) and dissipated a portion of
the seismic energy when the displacement of the superstructuresbecame larger than the allowable values of constraint devicesTier 3 redundancy was designed as a potential plastic hinge inthe substructures (CCDC2) including a certain yielding of thebackfill at the abutments (BPDA) e response of theabutments in the transverse direction (PDAT) was slightlyincreased due to the pounding effect between the girders andshear keys caused by bearing sliding e reasonable damagesequence of two restraint systems can be reflected through thecorresponding damage probabilities of bearings seismic mea-sures and substructures respectively as indicated by the red andgreen lines in Figure 17 erefore the rational sequence ofcritical limit state occurrences for the approach bridges isbearings⟶ expansion joints (or restrainer cables) shearkeys⟶ columns abutments
43 Comprehensive Evaluation of Bridge System Padgett andDesRoches [16] pointed out the necessity to derive the bridgesystem vulnerabilities when assessing retrofitted bridges
Figure 15 Experimental results of (a) LEB and (b) LRB
ΔΔ1
F
Δ2
Fmax
Fd
KdK0
(a)
Δy Δ
F
K1 = (3~7) K2
K1
K2
(b)
Slack
F
K
Δ
Fy
(c)
Figure 16 Analytical models of (a) LEB (b) LRB and (c) cable restrainers
16 Advances in Civil Engineering
because different components may have a different responsewith a given retrofit measure Also Zhang and Huo [18]studied the optimal parameters of isolation bearings byderiving the intensity median value corresponding to sys-tem-level fragility function which can effectively avoid theimpacts of different isolation parameters on differentcomponents As can be seen from Figures 14 and 17 theabove analysis results were also very apparent in this studyFor example the longitudinal deck displacement was sig-nificantly reduced by using dampers yet increased thetransverse demands of stiffening girder and pylons e useof LEB or LRB reduced the demands placed on columns yetled to an increase of the transverse deformations of abut-ments which was due to the use of shear keys limiting theexcessive sliding displacement of bearings erefore acomprehensive evaluation of retrofit strategies on bridgesparticularly long-span suspension bridges that have multiplecomponents should be based on system-level fragility in-stead of component-level
Taking the findings that no damages had occurred to thecable system and the definition of bearing damage states intoaccount the derivation of system-level fragility of the wholebridge depended on critical components including pylonscolumns bearings and abutments e fragility data byemploying the absolutely serial and serial-parallel assump-tions of bridge system are plotted in Figure 18 and comparedto the system fragility obtained from the proposed com-posite damage state e absolutely serial assumption
generated serious bias for fragility curves and may result inmisdirection if they were served as a design guide eabsolutely serial model overestimated the potential damageof the bridge system since the fragility curve directlydepended on the most vulnerable components In otherwords if the difference in location and performance of eachcomponent was neglected then the seismic performanceevaluation would be achieved by attaching equal importanceto the strength of pylons approach columns and bearingsis method contradicted the philosophy of capacity designin which the long-span structure can dissipate energythrough a reasonable swinging of stiffening girder and thebearings were designed as fuse elements in the small-to-medium-span bridgese serial-parallel model qualitativelyincorporated the differences of components and was verysensitive to the seismic design of the structure as seen fromthe comparisons among the cases in Figure 18 e modeldistinguished the serial components and parallel compo-nents yet still overestimated the bridge system-level fragilitywhen bearings became the vulnerable components espe-cially for long-span bridges is special situation such asthe serial assumption may also lead to misunderstandingOnce the relative importance between the components wasconsidered in the serial-composite model the damage po-tential was clearly reduced within the range of earthquakeintensity of interest based on the case studies is alsoreflected the necessity of the classification idea of bridgecomponents which can be expressed by the numerical value
Table 6 Damage states of the quasi-isolation and isolation bearings (m)
Figure 17 Probabilities of critical damage states occurrence sequences for the approach bridges when PGV is 1ms
Advances in Civil Engineering 17
of system-level fragility erefore the application ofcomposite damage state (equation (4)) considering com-ponent difference was instead more appropriate than theaforementioned two models
e damage probability distribution represented byblack circles in Figure 18 can be obtained from the definedcomposite damage state (equation (5)) where the parametera is 06 It was found that system fragility result was a moreaccurate assessment with further taking the relative im-portance between themain bridge and approach bridges intoaccount is is due to the principle that the seismic designperformance objective of long-span bridges generally has nocollapse for the main body structures and is necessarily opento emergency vehicle service In addition attention must bepaid to the fact that the differences in repair cost and time ofthe main and approach bridges are significant when thesame damage state is achieved e results of compositedamage state may match the performance objectives of theseismic design In order to investigate the influence ofimproved strategies and structural relative importance ondamage potential of bridges the mean values of the fragility
functions can be employed to evaluate the seismic designe smooth fragility curves for the specific damage states canbe fitted with the lognormal cumulative probability distri-bution function by employing the least-squares technique
P DSgtDSi | PGV1113858 1113859 Φln(pgv) minus λPGV
ζPGV1113888 1113889 (12)
where λPGV and ζPGV are the logarithmic mean and dis-persion values of earthquake intensity (PGV) for a specifieddamage level e larger mean values represent the smallerdamage potential to reach the specified level
Figure 19 indicates the mean values of earthquakeintensity at the specific damage states as a function of theimproved strategies and parameter a Each step of theretrofit measures would enhance the seismic performanceof the overall bridge system at all levels e improvementof the restraint system of the approach bridge played acritical role of the overall bridge system For instance themean values of the earthquake intensity featured mutationof slopes along the measure direction which reflected the
Figure 18 System fragility of the original design (a) Case 0 and improved design including (b) Case 1 and (c) Case 5
18 Advances in Civil Engineering
fact that the effectiveness of improvement for the ap-proach bridges was clearer than that of the main bridgee fragility of the highway network would be under-estimated if only the damage potential of the suspensionspan was adopted erefore when the structures such asapproach bridges transition piers and auxiliary pierswere ignored the damage identification and repair de-cisions would be misleading in the seismic evaluation oflong-span bridges Conversely the damaging probabilityof the overall bridge system will be amplified regardless ofthe higher importance of the main bridge When respectthe fact that the importance of suspension span is higherthan that of approach bridge in transportation networks aparameter a of 06ndash08 may be a better choice in thepresent study
e seismic fragility of conventionally designedbridges was mainly governed by the approach bridges at alldamage levels However for the retrofitted bridges thatpossess reasonable seismic performance different damagestates were controlled by the earthquake demands ofvarious structures as shown by the trend of the curvesalong parameter a in Case 4 and Case 5 e damagingpotential of the overall bridge from the low-to-moderatelimit states was dominated by the approach bridge eprimary reasons were the sliding of bearings and thetransfer of limited forces from the girder to substructure
by the damage of restriction devices of the approachbridges e extensive damage was controlled by thedisplacement of stiffening girder of the main bridge whichinferred the effect of dampers will be conspicuous understrong earthquake shaking At the same time the overalllong-span bridges with the reasonable seismic system willnot collapse during a strong earthquake event and canmeet emergency traffic service It can be seen that the meanvalues of fragility functions developed by compositedamage indices (equations (4) and (5)) are good indicatorsto evaluate seismic performance of long-span bridges andcan be consistent with the established seismic performanceobjectives
5 Conclusions
In the present study a procedure for investigating thereasonable seismic performance of long-span suspensionbridges was developed under the fragility function frame-work e seismic performance of a long-span suspensionbridge was examined and improved using the proposedprocedure from which the results revealed distinguishableconclusions as follows
(1) e definition of damage states for bearings con-sidered the interactions between the adjacent
014036059081103125
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(a)
020051081112142173
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
0002
0406
0810
CaseParameter a
(b)
130251371492612733
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(c)
Mea
n va
lue
exp
(λ) (
ms
)
Parameter a Case0 1 2 3 4 5
0001
0203
0405
131305478652826
1000
(d)
Figure 19 Influence of relative importance and improved designs on mean values of earthquake intensity for different damage states(a) slight (b) moderate (c) extensive (d) collapse
Advances in Civil Engineering 19
structures and self-damage to bearings in accor-dance with the actual design and experimentalconclusions e damage states of pylons in terms ofcurvature ductility were defined by developingmodal pushover analysis and idealized bilinear fit-ting methods in the equivalent pylon fiber models ofthe suspension bridge
(2) When investigating the component fragility curvesof the initial design it was found that the large deckdisplacement of the suspension span was the mostserious damage present and a slight inelastic re-sponse may have occurred only in the pylons Al-though the cable system was able to maintainelasticity under earthquakes the saddles at the py-lons were identified with a high-risk region indi-cating the damaging potential For the approachbridges the ductility mechanism did not realize arational damage sequence e columns were proneto collapse because of the application of fixed potbearings which in turn caused the damage to thebearings and abutments
(3) e stepwise improved seismic designs were pro-posed in terms of the component fragility results ofthe initial design Among the candidates fluid vis-cous dampers were proven to be a good choice forlong-span bridges based on the two attributes ofdisplacement and force when the similar efficiency ofdisplacement control was achieved e restraintsystem had a critical effect on the seismic responsesof the approach bridges e quasi-isolation orisolation can achieve a reasonable seismic damagesequence of the approach bridges
(4) e defined composite damage indices showed thatthe incorporate component classification conceptand structural relative importance were necessary toderive a good indicator of performance evaluationdue to the fact that the damage indices were based oneither an absolutely serial or serial-parallel systemmodel may significantly overestimate the damagepotential of the overall bridge system e meanvalues of earthquake intensity at the specific damagestates can serve as a function of the improvedstrategies and relative importance to further evaluatethe overall seismic performance of the suspensionbridge It can be concluded that the damaging po-tential of the overall bridge from the low-to-mod-erate limit states was governed by the approachbridges in the retrofitted bridges with a favorableseismic performance whereas the displacementdemand of the suspension span dominated thefragility probability of the extensive damage
Data Availability
e nature of the data is the analytical data of finite elementmodels of the case bridge e data used to support thefindings of this study are available from the correspondingauthor upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the Basic Research ServiceProject of Central-Level Public Welfare Research Institute(grant nos 2016-9018 and 2019-0034) and Science andTechnology Project of Communicationsrsquo Construction inWestern China MOC (grant no 2009318223094)
References
[1] L W David A Kartoum K Chang and I Roy ldquoSuspensionreport seismic performance and design considerations of longspan suspension bridgesrdquo Report CSMIP00-03 CaliforniaDepartment of Conservation Division of Mines And GeologyOffice of Strong Motion Studies Sacramento CA USA 2000
[2] S-G Gwon and D-H Choi ldquoImproved continuummodel forfree vibration analysis of suspension bridgesrdquo Journal ofEngineering Mechanics vol 143 no 7 Article ID 040170382017
[3] S Arzoumanidis A Shama and F Ostadan ldquoPerformance-based seismic analysis and design of suspension bridgesrdquoEarthquake Engineering and Structural Dynamics vol 34no 4-5 pp 349ndash367 2005
[4] L Sgambi E Garavaglia N Basso and F Bontempi ldquoMonteCarlo simulation for seismic analysis of a long span sus-pension bridgerdquo Engineering Structures vol 78 pp 100ndash1112014
[5] D Karmakar S Ray-Chaudhuri and M Shinozuka ldquoFiniteelement model development validation and probabilisticseismic performance evaluation of Vincent omas suspen-sion bridgerdquo Structure and Infrastructure Engineering vol 11no 2 pp 223ndash237 2015
[6] W-F Chen and L Duan Bridge Engineering HandbookSeismic Design CRC Press Boca Raton FL USA 2nd edition2014
[7] E Choi ldquoSeismic analysis and retrofit of mid-americabridgesrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2002
[8] B G Nielson and R DesRoches ldquoSeismic fragility method-ology for highway bridges using a component level approachrdquoEarthquake Engineering amp Structural Dynamics vol 36 no 6pp 823ndash839 2007
[9] K N Ramanathan ldquoNext generation seismic fragility curvesfor california bridges incorporating the evolution in seismicdesignrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2012
[10] O Avsar A Yakut and A Caner ldquoAnalytical fragility curvesfor ordinary highway bridges in Turkeyrdquo Earthquake Spectravol 27 no 4 pp 971ndash996 2011
[11] S P Stefanidou and A J Kappos ldquoMethodology for thedevelopment of bridge-specific fragility curvesrdquo EarthquakeEngineering amp Structural Dynamics vol 46 no 1 pp 73ndash932016
[12] B Borzi P Ceresa P Franchin F Noto G M Calvi andP E Pinto ldquoSeismic vulnerability of the Italian roadwaybridge stockrdquo Earthquake Spectra vol 31 no 4pp 2137ndash2161 2015
[13] W T Barnawi and S J Dyke ldquoSeismic fragility relationshipsof a cable-stayed bridge equipped with response modification
20 Advances in Civil Engineering
systemsrdquo Journal of Bridge Engineering vol 19 no 8 ArticleID A4013003 2014
[14] J Zhong Y Pang J Jong-Su J-S Jeon R DesRoches andW Yuan ldquoSeismic fragility assessment of long-span cable-stayed bridges in Chinardquo Advances in Structural Engineeringvol 19 no 11 pp 1797ndash1812 2016
[15] J Zhong Z Hu W Yuan and L Chen ldquoSystem-basedprobabilistic optimization of fluid viscous dampers equippedin cable-stayed bridgesrdquo Advances in Structural Engineeringvol 21 no 12 pp 1815ndash1825 2018
[16] J E Padgett and R DesRoches ldquoMethodology for the de-velopment of analytical fragility curves for retrofitted bridgesfitted bridgesrdquo Earthquake Engineering amp Structural Dy-namics vol 37 no 8 pp 1157ndash1174 2008
[17] Y Xie and J Zhang ldquoOptimal design of seismic protectivedevices for highway bridges using performance-basedmethodology and multiobjective genetic optimizationrdquoJournal of Bridge Engineering vol 22 no 3 Article ID04016129 2017
[18] J Zhang and Y Huo ldquoEvaluating effectiveness and optimumdesign of isolation devices for highway bridges using thefragility function methodrdquo Engineering Structures vol 31no 8 pp 1648ndash1660 2009
[19] F McKenna M H Scott and G L Fenves ldquoNonlinear finite-element analysis software architecture using object compo-sitionrdquo Journal of Computing in Civil Engineering vol 24no 1 pp 95ndash107 2010
[20] MOTC ldquoGuidelines for seismic design of highway bridgesrdquoChina Ministry of Transportation and CommunicationsPress Beijing China JTGT B02-01-2008 2008
[21] A D Kiureghian and J L Sackman ldquoTangent geometricstiffness of inclined cables under self-weightrdquo Journal ofStructural Engineering vol 131 no 6 pp 941ndash945 2005
[22] J S Steelman E T Filipov L A Fahnestock et al ldquoEx-perimental behavior of steel fixed bearings and implicationsfor seismic bridge responserdquo Journal of Bridge Engineeringvol 19 no 8 Article ID A4014007 2014
[23] B Madani F Behnamfar and H Tajmir Riahi ldquoDynamicresponse of structures subjected to pounding and structure-soil-structure interactionrdquo Soil Dynamics and EarthquakeEngineering vol 78 pp 46ndash60 2015
[24] L Xu and J Li ldquoDesign and experimental investigation of anew type sliding retainer and its efficacy in seismic fortifi-cationrdquo Engineering Mechanics vol 33 no 2 pp 111ndash1182016 in Chinese
[25] Caltrans Caltrans Seismic Design Criteria California De-partment of Transportation Sacramento CA USA 2013
[26] A Shamsabadi K M Rollins and M Kapuskar ldquoNonlinearsoil-abutment-bridge structure interaction for seismic per-formance-based designrdquo Journal of Geotechnical and Geo-environmental Engineering vol 133 no 6 pp 707ndash720 2007
[27] X Li Z Sun D Wang and Y Shi ldquoLongitudinal seismicpounding effect of bridges abutment and backfilling damagerdquoJournal of Changrsquoan University vol 35 no 4 pp 76ndash82 2015in Chinese
[28] MOTC ldquoSpecification for design of highway suspensionbridgerdquo China Ministry of Transportation and Communi-cations Press Beijing China JTGT D65-05-2015 2015
[29] A Shafieezadeh K Ramanathan J E Padgett andR DesRoches ldquoFractional order intensity measures forprobabilistic seismic demand modeling applied to highwaybridgesrdquo Earthquake Engineering amp Structural Dynamicsvol 41 no 3 pp 391ndash409 2012
[30] J E Padgett B G Nielson and R DesRoches ldquoSelection ofoptimal intensity measures in probabilistic seismic demandmodels of highway bridge portfoliosrdquo Earthquake Engineeringamp Structural Dynamics vol 37 no 5 pp 711ndash725 2008
[31] K R Mackie and B Stojadinovic ldquoFragility basis for californiahighway overpass bridge seismic decision makingrdquo PEERReport 200502 Pacific Earthquake Engineering ResearchCenter University of California Berkeley Berkeley CA USA2005
[32] J Li T Peng Ji Lin L Zhou and YuL Deng Seismic Design ofMulti-Pylon Multi-Span Suspension Bridge China Commu-nications Press Beijing China 2013 in Chinese
[33] L Chen W Zhuang H Zhao and Z Wan ldquoReport onHighwaysrsquo Damage in the Wenchuan Earthquake-BridgeChina Communications Press Beijing China 2012 inChinese
[34] G Wu K Wang G Lu and P Zhang ldquoAn experimentalinvestigation of unbonded laminated elastomeric bearingsand the seismic evaluations of highway bridges with testedbearing componentsrdquo Shock and Vibration vol 2018 ArticleID 8439321 18 pages 2018
Advances in Civil Engineering 21
the ground motions that meet the site conditionsrepresenting a broad range of intensities and variousmodeling parameters such as material strengthstructural properties and damping ratio e generalpractice is taking the uncertainty sources as randomvariables to formN finite elementmodels of suspensionbridges in the as-built design using a Latin hypercubesampling (LHS) technique and then pair these modelswith a set of N ground motions in order to carry out aseries of nonlinear time-history analyses
(2) e final step to derive bridge component fragilitycurves is to pinpoint the vulnerable componentsbased on the recorded peak response data in non-linear time-history analyses by which to define theircorresponding limit state thresholds at variousdamage states In addition the application of optimalIM is also the key link to ensure the reliability of theseismic vulnerability assessment
(3) e component-level fragility curves are calculatedusing equation (2) For the purpose of reducing thevarious damage potential of critical bridge componentsin the initial design different seismic designs areproposed based on earthquake damage experiencesrelated component seismic experiments and expertopinions Next the effectiveness of seismic measures isevaluated using the critical component-level fragility
(4) e optimum structural seismic performance is se-lected in this study taking the bridge system-levelfragility as the objective function according to thedefinition of damage states and fragility results ofvarious components obtained from steps 2 and 3Considering the vulnerability degree of each com-ponent and postearthquake repair cost it is helpfulto evaluate the seismic performance of the bridgesystem by combining the critical components in areasonable manner For suspension bridges cablesystems and pylons are primary load-carryingcomponents and pylons can be only allowed to be oflimited damage e damage of bearings can beallowed to compensate by the effective seismicmeasures For approach bridges with the small-to-medium spans the postearthquake serviceability ofbridges depends on the damage stage of columnsabutments and bearings A popular seismic designof this type of bridges is fusing actions of bearings asforce-limiting connections to reduce the damage ofsubstructures A bridge structure consists of multiplemembers with the same function but in variouslocations e working principle of pylons and
columns is similar to that of the series system due tothe fact that the damage or collapse of one of thepylons or columns will directly affect the post-earthquake availability of the bridge system emultiple bearings form a parallel system to jointlysupport superstructures and a single damage has lessinfluence on the overall bridge system Earth pres-sures on abutments are further increased as a resultof the pounding between the deck and abutmentbackwall for the seat abutments which leads to anincrease in the abutment deformation However thedifficulty of repairing abutments is relatively smalland the effect on emergency traffic is slight A parallelsystem can also be adopted to simulate an abutmentsystem For a global bridge structure once any typeof component system is damaged it will have asignificant impact on the overall bridge system eoverall bridge system is close to the series combi-nation of different component systems which can becalled a bridge system based on the series-parallelsystem
Zhong et al [14] summarized the method of componentclassification appropriate for long-span bridges and held thatthe components of long-span bridges can be classified asprimary secondary and accessory based on the relativeimportance of load-carrying capacity and repairing costserefore the application of a serial assumption betweencomponent systemsmay bias the seismic fragility assessmentof the bridge system Zhang and Huo [18] assigned aweighting ratio of 075 and 025 to columns and bearingsrespectively for ordinary highway bridges in USA andproposed the composite damage state for the bridge systeme repair-related strategy variables including repair costand time are considered based on the direct and indirecteconomic losses that are caused by the damage of bridgecomponents e damage to primary and secondary com-ponents at any level will account for a majority of repair costincluding repairing most components and a long closure oftraffic e replacement costs of accessory componentsprimarily rest on their extensive damage including rein-stallation of bearings and restoration of girder that might beresulted from pounding and unseating which interruptstraffic at a short time Combined with the reality of bridgeconstruction in China and the repair-related variables of thecritical components incorporated in the bridge system thisstudy separately proposes the composite damage state in-dices of the long-span and conventional bridge system asfollows
It should be noted that the aforementioned developmentof damage indices for the bridge system is only for a singlebridge system and is closely related to the definition ofdamage states of the components For long-span suspensionbridges composed of a main bridge and several approachbridges a reasonable and comprehensive seismic vulnera-bility evaluation is of great significance to the construction ofproject and postearthquake repairing cost A bridge systemwith the similar structural properties could be regarded as aseries system However the differences in structural im-portance and repairing costs cannot be neglected because ofthe structural differences between the long-span bridges andsmall-to-medium-span bridges e weighting ratios of aand (1 minus a) are assigned to the main bridge system andapproach bridge system respectively which have the abilityto capture the contributions of the main bridge and ap-proach bridges to the vulnerability of the overall bridgesystem erefore the rational range of system fragilitycould be accurately determined as follows
DSsystem round a middot DSmain bridge +(1 minus a) middot DSapproach bridge1113960 1113961
(5)
where the round-off principle can be used for the ldquoroundrdquoe composite damage state in equations (4) and (5) can
be incorporated in the PSDM when the correlation betweenconcerned components with the use of covariance matrix isknown In this case Monte Carlo simulation is used tocompare the demands obtained from the joint probabilisticseismic demand models [8] and multidimensional compo-nent capacities in theM (105) random samples at a range ofthe IM values e basic principle of developing fragilitycurves for the long-span suspension bridge system is shownin Figure 1
3 Case Study
31 Bridge Description and Simulation is study selectedthe Taoyuan Bridge spanning the Jinsha River as the studycase e bridge is characterized by the geometric andmechanical features of a suspension bridge which is themost common type of bridge construction in China ebridge has a total length of 731m and consists of a single-span simply supported suspension bridge measuring 636mand a continuous steel box-girder approach bridge withthree spans measuring 30m 35m and 30m respectivelye longitudinal elevation of the bridge is illustrated inFigure 2
e main bridge has a thin-wall steel girder along thelongitudinal direction with an overall width of 31m fromcurb to curb and a height of 3m accommodating four lanesof traffic e gate-type towers include two RC box-sectionlegs with overall heights of 75m and 70m on its two sidesrespectively Also there is one crossbeam installed along theheight of pylons in order to enhance the lateral stiffness ofstructures At the base the tower legs are anchored to the 6m-thick concrete footings e pedestals for the placingbearings are designed between the concrete footings due tothe shorter distance between the deck and tower base e
bored piles serving as the foundations of pylons and locatedat the leg of each tower have a diameter of 22me areas ofeach main cable and suspender are 01626 m2 and 0001316m2 respectively Gravity anchorage has been applied to themain cable anchorage on both sides of the bridge eapproach bridges are divided into left and right bridges withthe identical configurations on the north side For eachapproach bridge there is a three-span continuous steel box-girder bridge with a pile-bent abutment of which the middlespan is supported by two double-column bents with integralpile-shaft e width and depth of the superstructure are1275m and 153m respectively e circular sections ofcolumns with a diameter of 18m have a height of 7m andthe cast-in-drilled-hole piles measure 19m in diameter epiles of abutment are provided by a circular section of 15min diameter and the height of the backwall is 188m
e description of this studyrsquos analytical modelingprocedure is outlined using OpenSEES software [19] for theprototype bridge in the as-built design A beam-columnelement with distributed plasticity fiber was used to modelthe pylons and columns in order to account for the materialnonlinearity In addition the axial force-moment interac-tions of pylons were captured in the model Each fiber wassimulated via a reasonable stress-strain relationship whichwas dependent on unconfined concrete confined concretedefined by the Mander theoretical model and longitudinalreinforcements that considered the Bauschinger effects asillustrated in Figure 3 e corresponding mechanical in-dices such as material strength axial compression ratio(ACR) and reinforcement ratios of Sections I and II all metthe Chinese code requirements [20]
e main cable element was tension-only which wasmodeled as a finite number of large-displacement truss el-ements using the Ernst method accounting for the sag effect[21] Each suspender element was identical to the simulationof the main cable but there was no sag effect e initialstress of the cable system was also considered in the modelas shown in Figure 4(a) An elastic beam element was uti-lized to simulate the crossbeam of the pylons and stiffeninggirder as well as the girders and cap beams of the approachbridges e foundations were modeled by six spring ele-ments which indicated that in the future they are notexpected to be damaged under earthquake shaking
It should be noted that pot bearings were applied to theas-built design which consequently became the significantfactors in the overall responses and functionality of theapproach bridges e longitudinal response of sliding potbearings detailed in Figure 4(b) was simulated using a bi-linear element [20] e full-scale experiments of Steelmanet al [22] found that properly proportioned steel for fixedpot bearings can achieve a reliable dry friction responsebetween the bearing component and superstructure orsubstructure after the rupture of the anchor bolt Corre-spondingly the behavior of fixed pot bearings was modeledas shown in Figure 4(c) e fixed pot bearings weredesigned to be located at the C2 column as shown inFigure 1 Finally the pounding stiffness between the adjacentdecks (Figure 4(d)) was estimated by the sum of their axialstiffnesses [23]
4 Advances in Civil Engineering
Approach bridge
Composite damage state (4)
Main bridge
Composite damagestate (4)
Composite damage state (5)
Bearing 1 Bearing iBearing 2
Bearing 1 Bearing iBearing 2
Bearing 1 Bearing iBearing 2
Bearing 1 Bearing iBearing 2
Column 1
Column 2
Column j
Abutment 1
Abutment 2
Bearing 1 Bearing iBearing 2
Bearing system
Column 1 Column iColumn 2
Column system
Bearing 1 Bearing iBearing 2
Bearing 1 Bearing iBearing 2
Pylon 1
Pylon 2
Bearing system
Pylon 1 Pylon 2
Pylon system
Damage state definitionsand component-level
fragility results
Abutment 1
Abutment 2
Abutmentsystem
Figure 1 Bridge system based on the series-parallel system and composite damage state
C2C1
4000
7000
6000
7500 63
60
14000
3500
30001600063600
73100
3000South
North
Figure 2 Configuration of the Taoyuan Suspension Bridge (cm)
SouthH = 750m
North H = 700m
H
Sec I
Pylon
Pedestal
(a)
Confined(Mander model)
εu εcu
fc
Kfc
ε
σ
Unconfined
Concrete0002
fcufu
εc0
(b)
σy
εy ε
σ0005E0
E0
Bar
(c)
Figure 3 Continued
Advances in Civil Engineering 5
e shear keys contribute to the constraint of transversemovement between the girder and pier Xu and Li [24]proposed a modified model based on the structural prop-erties of shear keys in China which was used to build theforce-displacement relation of shear keys by separating thecontributions of steel and concrete in parallel as shown inFigure 5 e critical parameters of the shear keys can becalculated using Xursquos method according to the actual designwhich represents the yield (Δ1Y) nominal (Δ1n) degraded(Δ1d) and ultimate (Δ1u) deformations respectively esimulation can accurately reflect the mechanical behavior ofthe shear keys
e seat type abutment is shown in Figure 6 e lon-gitudinal resistance to seismic forces is provided jointly bythe passive action of the backfill soil and active action of thepiles [25] Meanwhile the transverse response is resisted bythe pilese passive and active responses of abutments weresimulated using hyperbolic soil material [26] and trilinearforce-displacement relationship respectively which wereconnected in parallel to capture the abutment responses Inaddition the pounding effect between the deck and abut-ment was simulated using a linear model without energydissipation e stiffness of pounding spring between thesuperstructure and abutment was proposed to be ten timesthat of the backfill soil stiffness [27]
32 Uncertainty Treatment in the FE Model e LHS ap-proach is usually used to account for the uncertainties infragility analysis such as structural geometry materialstrength component properties deck mass and damping in
the modeling which is considered to be a variance reductionsampling technique It is noted that the uncertainty ofstructural geometry is not taken into account in this studyfor the development of bridge-specific fragility curves In theforegoing approach the probability distributions were as-sumed for each parameter and each random variable wasdivided into the several equal intervals which correspond to5 to 95 of the cumulative probability e consideringmodeling parameters and their probability distributions arepresented in Table 1 Statistically significant yet nominallyidentical 3D bridges were built using the LHS so as toaccount for the abovementioned uncertainties then thebridges were paired randomly with the selected groundmotions
33 Ground Motion Input and Intensity Measure e ne-cessity to carry out a large number of nonlinear time-historyanalyses depends on a significant number of earthquakerecords selected for the PSDM In this study a data set of 80recorded ground motions and 20 Los Angeles-pertinentunscaled ground motions were extracted from the PEERstrong motion database and the SAC project database re-spectively [29] ese ground motions can represent thecharacteristics of the bridge site ey were placed into fivemagnitude-distance bins for the purpose of covering therational and broad ground motion intensity e first fourbins included combinations of low and high magnitudesand large and small fault distances e magnitudes variedbetween 57 and 70 while the fault distances ranged from130 to 600 km e characteristic of the ground motions in
Stress
Initialstress
K1
Strain
K2 = 0005K1
(a)
K1
Δy = 2mm
Fy
ΔΔy
K2 = 000001K1
F
(b)
Δu
F
Fd
Δy
K
Rh
ΔΔy = 1mm
Δu = 20mm
(c)
K
Gap
F
Δ
(d)
Δy
FΔult
K
Δ
Fult
Fult2
Δy= 154mmΔult = 1504mm
Per width (m)of abutment
(e)
Δ1 = 762hmmΔ2 = 254hmm
h height of backwall
Keff
K2
F2F1
Δ1
K1
F
ΔΔ2
Per pileKeff = 14kNmm
(f)
Figure 4 Constitutive relations of the various components (a) cable and suspender (b) sliding pot bearing (c) fixed pot bearing(d) pounding (e) abutment backfill (f ) abutment pile
Bar (HRB500 376)
z
y
Confined concrete (stirrup HRB400
1256)
Sec I
Unconfined concrete (C55)
ACR 0159
(d)
Sec II
h =
70m
Double-column bent
(e)
Confined concrete(stirrup HRB335
08)
Unconfined concrete (C35)
Bar(HRB335 1163)
z
y
Sec II
ACR 0037
(f )
Figure 3 Sketch and material model of the pylons and bents of the examined suspension bridge
6 Advances in Civil Engineering
the fifth bin was that the fault distance was less than 15 kme ground motions were given in two orthogonal com-ponents and the acceleration values adopted were computedas the geometric mean values (Sa
SaL middot SaT
1113968) e peak
ground acceleration ranged from 0043 g to 112 g Figure 7shows the distributions of peak ground accelerations (PGA)peak ground velocities (PGV) earthquake magnitudes (M)and fault distances (R) of 100 ground motion records as well
as the acceleration spectra with a damping ratio of 002 etwo orthogonal components of the selected earthquakerecords were randomly input along the longitudinal andtransverse directions of the bridge
PGA PGV PGD spectral acceleration or Arias In-tensity of earthquake can be selected as the IM candidateswhich are used to characterize earthquake intensity elinear relations with the interesting EDPs from probabilistic
Δ
V
Δ1n Δ1d
Vcp
Δ
V
Δ1y Δ1u
Vsp
i
j
Gap
Hysteretic property of concrete Hysteretic property of steel
Vcp = 7624kN
Δ1n = 106mmΔ1d = 422mm
Vsp = 10143kN
Δ1y = 11mmΔ1u = 633mm
Figure 5 Analytical model of shear keys
Abutment
Deck
Figure 4 (b)
Figure 4 (d)
Figure 4 (c)Figure 5 Figure 4 (e)
Figure 4 (f)
Figure 6 Pile-bent abutment simulations
Table 1 Uncertainty parameters incorporated in the modeling design
Modeling parameter Probability distribution Mean COV Unit Source
Concrete compressive strength C55 Normal 466 0149 MPa
[20]C35 321 0164
Steel yield strengthHRB500
Lognormal5696
00743 MPaHRB400 4557HRB335 3816
Friction coefficient Sliding pot bearing Uniform 003 033 mdash [14]Fixed pot bearing Lognormal 02 0002 mdash [7]
Deck-abutment 80Deck-shear key Uniform 120 029 mm [16]
Damping ratio Uniform 1 01 mdash [7]Deck mass Normal 002 015 mdash [28]COV coefficient of variation
Advances in Civil Engineering 7
seismic demand analysis in the logarithmic reference framecan be adopted to determine the optimal IM In the previousstudies PGA and spectral acceleration were selected as theoptimal IMs for highway bridges [9 30] when applying themultiple measures containing efficiency practicality profi-ciency and sufficiency Mackie et al [31] suggested both PGAand PGV are good choices of IM in related to the EDPmeasures in highway overpass bridges Zhong et al [14]pointed out that PGA and PGV are themost efficient practicaland proficient IMs for long-span cable-stayed bridges yet PGVtends to be the optimal IM in terms of sufficiency ereforePGV can be utilized as the IM in this study
34 Engineering Demand Parameters e peak demands ofcritical components are adopted as EDPs for the developmentof PSDMs and the choice of optimal IM Table 2 lists thecritical EDPs for the examined bridge which have beenconsidered in the fragility evaluation of the suspension bridge
35 Capacity Limit State Models In the present study acapacity model was used to measure component damagewhich was described by a damage index as a function of the
selected EDPs ese models are usually discrete andcharacterized by the onset of various damage states based onthe experimental data Similarly the uncertainties of ca-pacity models are also expressed by median Sc and dis-persion βc e definition of damage states must be matchedwith postearthquake bridge functionality and repair strat-egies in the bridge seismic fragility assessments Founda-tions decks and anchorages are typically identified asnondamaged components in the simulations e damagestate values of bearings were determined so as to comparewith two factors including the pounding gaps between theadjacent components and seating width which can reflectthe possible pounding and unseating as shown in Figure 8 inaddition to the damages to bearings themselves that is thedeformation capacity of the bearings Table 3 lists the slight(DS1) moderate (DS2) extensive (DS3) and collapsed(DS4) damage states defined in the initial design for potbearings at various locations based on their allowable de-formation and structural dimensions
e columns of conventional highway bridges are oftenforced into the state of nonlinearity under strong earth-quakes of which the criteria of damage states were studied[8 9 16] It is found that there are major differences in
0
10
20
30
40
50N
umbe
r of r
ecor
ds
08
01
03
05
09
00
07
04
02
10
06
PGA (g)
(a)
0
10
20
30
40
50
Num
ber o
f rec
ords
20 806040 100
180
140
160
2000
220
120
PGV (cms)
(b)
20 40 600Fault distance (R) (km)
55
60
65
70
75
Mag
nitu
de (M
)
(c)
Mean
0
1
2
3
4
5
S a (g
)
2 4 6 8 100Period (s)
(d)
Figure 7 Characteristics of the selected ground motions (a) PGA (b) PGV (c) M and R (d) acceleration response spectra
8 Advances in Civil Engineering
geometric sizes material strengths and ACR of sectionsbetween the pylons of suspension bridges and columns ofconventional highway bridges erefore it is necessary todevelop a modal pushover analysis to define the damagestates of critical cross-sections of pylons In the analyticalprocesses of this study the material strain and nonlinearmoment-curvature relationship of critical cross-sectionswere characterized to measure the damage levels of thesuspension bridge pylons
First the reasonable pylon models were required to bedeveloped using fiber beam elements in longitudinal andtransverse directions as detailed in Figure 3 An elastic
spring which was equivalent to the restraint of main cablewas attached to the top of the pylon taking into account theinfluence of the main cable stiffness in the longitudinaldirection models e determining process of the equivalentstiffness can be referred to in the Ernst formula and theresults reported by Kiureghian and Sackman [21]
Second as illustrated in Figure 9 the inertial forcedistributions were back-calculated from the multimodespectrum analysis using a 3D bridge model as shown inequations (6)ndash(8) which included the contributions of highvibration modes and reflected the properties of inertial forcedistributions as follows
DS3
DS4
DeckDeck
Pedestal
PedestalBearing
Bearing
Gap
(a)
DS2
DS4
Gap
Abutmentnonlinearity
Deck
AbutmentBearing
(b)
GapShear key
degradationDS4
DS2
Bearing
(c)
Figure 8 Damage control and unseating prevention damage states for bearing displacement (a) between decks (b) between deck andabutment (c) between deck and shear key
Table 2 e critical EDPs for the examined bridge
Engineering demand parameters Abbreviation DirectionCurvature ductility of pylon cross-section (south north) PCD(S N) Longitudinal and transverseDisplacement of bearings of the main bridge MBD Longitudinal and transverseAxial force relative to yield value of main cables MAF mdashAxial force relative to yield value of suspenders SAF mdashCurvature ductility of column cross-section at columns (C1 C2) CCD(C1 C2) Longitudinal and transverseDisplacement of bearings of the approach bridge at column (C) and abutment (A) ABD(C A) Longitudinal and transverseDeformation of backfill passive pressure at abutment BPDA LongitudinalDeformation of piles at abutment PDA Longitudinal and transverseDeformation of expansion joint between decks(D) and at abutment (A) EJD(D A) LongitudinalDeformation of shear key at columns (C) and at abutments (A) SKD(C A) Transverse
Table 3 Limit state thresholds for pot bearings (m)
DS1 DS2 DS3 DS4
Criteria Allowable slippingdisplacements
Spherical skateboard radiusbody separation of bearings
Possible pounding betweenadjacent decks or deck and pylon Possible unseating
MBDL 025 031 076 1555MBDT 002 031 045 mdash
Criteria Allowable slippingdisplacements
Possible pounding entering of abutmentsnonlinearity or shear key degradation
Spherical skateboard radiusbody separation of bearings Possible unseating
where Sj and cj are the spectral acceleration and participationfactor associated with the jth mode respectively Fij φij andQij represent the seismic load mode displacement and shearforce at the i-node associated with the jth mode respectivelymi Qj and Pi indicate the mass shear force and inertial forceat the i-node respectively andN denotes the number of nodes
Next the inertial forces were loaded to the nodes ofpylon models Both the material and geometric nonlinear-ities were incorporated into the pushover analyses whichwere used to investigate the potential failure mechanismsand assess the ductility capacities of pylons
e seismic designs of pylons can develop inelastic be-haviors which were controlled by the material strain limitsFigure 9 details the curvature distributions along the pylonheights with the increase of δ where δ was the loading dis-placement at the control point It was observed that plastichinge zones typically occur at the bases of pylons and above andbelow the horizontal crossbeams Also plastic hinges mayprogress to the intermediate height in the longitudinal direc-tion e development of plastic hinges at the bases of pylonscan be much severer and also occur earlier than in other zoneserefore the curvatures at the bottoms of pylons may ef-fectively provide measurements of the damages to pylons
e mechanical indices of pylon cross-sections whichcorrespond to the four damage state characteristics
following the damages are detailed in Figure 10 e cur-vature ductility can be easily determined by the pushoveranalysis results in addition to that of the DS2e numericaldata can be transformed to a bilinear curve that has a value ofDS2 based on the equal energy rule per original curveSimilarly another pushover analysis was applied to thedouble-column bend of the approach bridge based on thespecifications [20] e four limit state thresholds of thecross-sections at the bottom of pylon and columns in boththe longitudinal and transverse directions are listed in Ta-ble 4 It was observed that the seismic capacity of longitu-dinal direction was superior to that of transverse directionand a good ductility capacity existed in both directions
In the present study the definitions of damage statethresholds of main cables and suspenders were dependent onthe steel strand characteristics of the strength and deformationrates A total of 85 of the ultimate strength (nominal yieldstrength) was conservatively acted as the failure index Namelythe corresponding ratio of force to yield value was 10 It is acommon viewpoint that the accidental loss of one suspenderwill not cause immediate failures of a bridge with the exceptionof the loss of the main cables erefore DS3 and DS4 wereonly defined based on the classification of suspenders andmaincables respectively e corresponding elastic and ultimatedeformation of abutments and shear keys were respectivelyspecified the thresholds of DS1 and DS2 with the recom-mendations of Caltrans [25] e damage of expansion jointsonly defined as the DS1 due to the lack of a higher-leveldamage modelis was objectively logical since the joints canbe replaced with the new once damaged thereby suggesting therationality for just one threshold of component damage Table 5summarizes the damage state thresholds of the above-mentioned components
e assignments of dispersions of capacity models ofcomponents appeared to be the most difficult tasks for theresearchers According to the previous research studies 035was adopted in the abutment and RC components including
Figure 9 Inertial force and curvature distributions (a) longitudinal (b) transverse
10 Advances in Civil Engineering
pylons and columns [9] e COV of main cables andsuspenders was 01 taking the low variability of highstrength steel strands into account [16] e descriptivedispersions of the remaining components were calculated tobe 025 for the two lower damage states (DS1 and DS2) and047 for the higher damage states (DS3 and DS4) [8]
36 Assessment of Vulnerable Components e componentfragility curves were derived by steps 1ndash3 detailed in Section 2to identify vulnerable components and regions as shown inFigures 11 and 12e pot bearings were shown to be the mostvulnerable components for the main bridge which inferred thecomplete damages to pot bearings themselves and reflected ahigher probability of pounding between the adjacent decks andunseating in the longitudinal direction e transverse re-sponses of bearings did not cause obvious damage compared tothe longitudinal responses e pylons were the primarycomponents for load carrying where inelasticmaterial responsecould potentially occur e transverse vulnerability was moreobvious than the longitudinal vulnerability for the lowerdamage states However the probability of extensive damagewas extremely low or even zeroWith respect to themain cablesand suspenders the increases in their forces which couldpotentially occur during rare earthquakes were confined toabout 15 (for themean values) of the forces resulting from thedead loads as illustrated in Figure 12 which also indicated thatthe maximum values of main cable stress had occurred at thesaddles of pylons and in the side spans e stress values of
suspenders achieved the maximum at the middle and end Insummary the cable system was found to be secured againstdamages in which the stress values were much lower than theyield values even under strong earthquake shaking It wasnoted that the large deflection and cable angle changes betweenthe main span and side spans would lead to the strong shear inthe longitudinal direction between the cables and saddles andthe slipping damages to the saddles at the top of the pylonsduring earthquakes were identified as high-risk
For the approach bridges the columns were determinedto be the most fragile components in both the longitudinaland transverse directions e application of the fixed potbearings resulted in a large force and deformation of thecolumns the complete damage of sliding pot bearings andapparent pounding between the decks and abutments eseactions subsequently caused the inelastic responses ofabutments but the shear keys were protected from failuretransversely Although the ductility mechanism of approachbridges was designed it was considered to be a defectiveseismic system with irrational sequences of componentdamage occurrences in high-seismicity regions
41 ImprovedDesign ofMainBridge As results shown in theevaluation of vulnerable components a large deck dis-placement of the suspension span would occur duringearthquakes which caused pounding and even unseating
Original curveBilinear curve
MprimeyMy
φs or φccφucφprimeyφy
DS4 reinforcing bar rupture core concrete crushing
DS3 major spalling of cover concrete exposed core concrete
DS2 formation of stable plastic hinge in the pylon (column)
Mom
ent
Curvature
DS1 narrow cracking in concrete first yielding of reinforcing bar
Figure 10 e damage state definitions of the pylon and column cross-sections
Table 4 Limit state thresholds of pylon and column cross-sections
Figure 11 Component fragility curves for the suspension bridge (a) main bridge (b) approach bridge
200 400 600 800 10000Location (m)
00
02
04
06
08
10
MA
F
MaximumMinimumMean
Dead loadYield valueHigh stress region
(a)
00
02
04
06
08
10
SAF
200 400 600 800 10000Location (m)
MaximumMinimumMean High stress region
Dead loadYield value
(b)
Figure 12 Force distributions of the (a) main cables and (b) suspenders
12 Advances in Civil Engineering
when there were no additional restraints in the longitudinaldirection erefore it was consistent to suggest that sup-plementary devices should be equipped to mitigate the deckmovements to some extente reduction of deck or bearingdisplacements can be taken as the retrofitted objective for thesuspension span Fragility analyses were conducted on theretrofitted bridges that covered the general strategies asfollows
Case 1 the analysis model of the bridge used a fluidviscous damper equipped between the deck and eachpedestal on both sides of the bridgeCase 2 the analysis model of the bridge used an elasticcable installed between the deck and each pedestal onboth sides of the bridgeCase 3 the analysis model of the bridge used three pairsof flexible central buckles located in the middle sectionof the span
e main bridge was assumed to have identical me-chanical parameters at each location in various designschemes e dampers and elastic cables were respectivelymodeled using zero-length elements with Maxwell andtension-only materials and the flexible central buckles weresimulated as the same as the suspenders elements inOpenSEES
e experimental testing of fluid viscous dampers hasshown that the force-velocity with fractional velocity powerlaw relation can describe the suitable mechanical behavior
fD CD| _u|αsgn( _u) (9)
where α is a real positive exponent of which value dependson the piston head orifice CD is viscous damping coefficientwith units of force per velocity raised to the α power kNmiddot(ms)-α _u indicates the velocity of damper and sgn(middot) is thesignum function Research by Li et al [32] indicated thatviscous damping coefficients range from 2500 kN (ms)-α to5000 kN (ms)-α and can be simulated with a uniform dis-tribution e velocity exponents can be assumed to varyuniformly from 03 to 05
e axial stiffness of elastic cables and flexible centralbuckles is regarded with the material length and cross-section area It was assumed that the yield strength of steelstrands of fy 1420MPa was the mean value and followed alognormal distribution with a COV 01 e stiffnessvalues of single elastic cable based on past work employed auniform distribution ranging from 125times105 kNm to25 times105 kNm In addition the other mechanical prop-erties such as length and area can be determined by thefollowing relations
LgtEΔfy
1113888 1113889
kE EA
L1113874 1113875
(10)
where L E and fy are the length Youngrsquos modulus andyield stress of elastic cables respectively and Δ represents the
controlled deck displacement which was available for 06min the examined bridge
According to expert opinion three pairs of flexiblecentral buckles were the most common type of centralbuckle of which the cross-section areas were the same asthose of the suspenders
Figures 13(a) and 13(b) illustrate bearing displacementdistributions are formed for the three retrofit measures at thesame ground motion intensity levels e mechanical be-haviors of measures strongly affect the effectiveness It can beconcluded that owing to the similar EDP distributions ofdampers and elastic cables the corresponding groundmotion intensities were similar when generating the samepounding probability e flexible central buckles wereextremely susceptible to yielding failure thereby causing lowefficiency in the displacement restrictions ereforepounding probabilities can be directly observed by condi-tioning on the median responses of dampers and elasticcables instead (Figure 13(c)) e control force produced bydevices is another noteworthy consideration in retrofitdesign e more force the devices require the larger andmore expensive they usually are [13] To take both defor-mation and force into account Figures 13(d) and 13(e)provide the contours of the pounding probability ecomparison showed that the damper deformations featuredslightly larger slopes in the horizontal direction yet thedifference in effect on the deformations of the dampers andelastic cables was not significant for reducing the objectivepounding probability Conversely damper force was lesssensitive to the pounding probability than cable force eelastic cable forces reached on average up to e32 (asymp245)times those of the damper under the same poundingprobabilities ese findings implied that difficult anchoragechallenges existed in the case of guaranteeing the efficiencyof elastic cables and the bridge decks possibly experiencedhigher stress In addition the difference in force was sub-stantial when controlling the displacement which showedthat dampers can contribute additional damping and stiff-ness to the structural system while elastic cables completelytransferred the inertial forces erefore the coordinationand balance of displacements and forces should be simul-taneously considered in the vibration mitigation of large-span bridges In this study the application of fluid viscousdampers was identified as the preferred retrofit in thelongitudinal direction for the main bridge e additionaldamping ratio provided by the fluid viscous dampers can becalculated by the following equation
ε CD
2mωeff (11)
where CD is damping coefficient of linear fluid damperskNmiddot(sm) which is equivalent to nonlinear dampers basedon equal energy dissipation per hysteresis loop [17] ωeff issystem natural frequency corresponding to the vibrationalong the axis of bridge andm is system mass It was shownthat an additional damping of 206ndash318 can offer aneffective controlling mechanism for the reduction ofpounding between the adjacent decks
Advances in Civil Engineering 13
Figure 14 depicts the change of components vulnera-bilities before and after the application of dampers It can beobserved that the retrofit measure played limited roles in thefragility reduction of the self-damaged bearings in long-spanbridges Since the additional longitudinal rigidity to girderwas provided by the dampers the transverse responses suchas girder displacement and pylon curvature would increaseas ground motion intensity increased e influences on thecable system responses and the longitudinal responses ofpylons were found to be negligible Another phenomenonwas that the longitudinal fragility of the abutments signif-icantly decreased with the reduction in the poundingprobabilities between the decks and abutments of the ap-proach bridges It was inferred that the pounding betweenthe adjacent decks was likely to affect movement of theapproach bridges and that the pounding effect would betransmitted along the girder
42 Improved Design of Approach Bridges e damagepotential of each component for the approach bridges washigh due to the high collapse risks posed by the usage of thefixed piers e structural designs of small-to-medium-spanhighway bridges in China show the fact that the super-structures are usually placed onto laminated elastomericbearings (LEBs) which tend to come into direct contact withthe bridge girders and substructures without anchoring etypical types of seismic damages of these bridges consisted of
large girder movements bearings sliding failures of shearkeys and expansion joints and abutment cracking duringthe Wenchuan and Yushu earthquake events in ChinaMeanwhile the bridge columns displayed relatively lowdamage ratios under earthquakes [33] e design can bereferred to as a quasi-isolation system which is primarilyrealized by strategically employing bearings sliding as force-limiting connections between girders and substructures Inaddition the isolation strategies of lead-rubber bearings(LRBs) are usually employed in the high-seismicity regionswhich can be taken as a comparison of the quasi-isolationsystem e two restraint systems consist of bearings shearkeys transversely located at the cap beams and restrainercables longitudinally anchored to the abutments (or ped-estals) In order to calculate the seismic responses of theapproach bridge the related experiments were completedand summarized and applied to the force-displacementrelations of the restraint system components
e detailed experiment of LEB consulted previousacademic achievements [34] An example is shown inFigure 15(a) where the feature points marked by rectangularand circular represented initial walking and obvious slidingresponses which illustrated that the mechanical property ofbearings would degrade as the displacements increased afterentering the friction sliding state until it remained nearlyconstant When there was no specific test the horizontalforce Fd in the stable friction stage can be 07 Fmax whereFmax was the corresponding maximum static friction at the
Bear
ing
disp
lace
men
t (m
)
Logarithmic spacendash5
ndash4
ndash3
ndash2
ndash1
0
1
2 4 6PGV (cms)
Case 1Case 2Case 3
(a)Po
undi
ng p
roba
bilit
y
00
02
04
06
08
05 10 15 2000PGV (ms)
Case 1Case 2Case 3
(b)
Poun
ding
pro
babi
lity
00
02
04
06
02 04 06 08 1000Deformation (m)
Case 1Case 2
(c)
ln [f
orce
(kN
)]
5
10
15
15
20
20
25
25
30
3030
35
3535
40
40
4550
7
72
74
76
78
8
033 041 049 057 065025Deformation (m)
(d)
ln [f
orce
(kN
)]5
10
15
15 20
20 25
25
30
3030
35
35
40
40
45
50
102
104
106
108
11
112
114
0155 0205 0255 0305 03550105Deformation (m)
(e)
Figure 13 Choice of the three retrofitmeasures based on deformations and forces (a) EDP-IM relation (b) pounding probability conditioned onIM (c) pounding probability conditioned on deformation contours of pounding probability for (d) dampers and (e) elastic cables
14 Advances in Civil Engineering
initial walking e displacement value Δ2 can be of 250effective shear strain (ESS) e initial stiffness K0 is cal-culated by MOTC [20] erefore the coefficients can becalculated by the following sequence FdK0⟶ Fmax⟶Δ1⟶Δ2 e built analytical relation ofthe LEB is shown in Figure 16(a) which can correctlycapture mechanical behavior under earthquakes A log-normal distribution [8] with a logarithmic standard devi-ation of 01 and a normal distribution with aCOV 016MPa were used to model the randomness offriction coefficient and shear modulus respectively For theirrespective mean values 02 and 12MPa were taken
Figure 15(b) shows a cyclic loading experiment of LRBwhich reveals that the degradation of bearings stiffness willoccur with the increase of loading displacements e elasticstiffness can reach uniformly 3 to 7 times the postyieldingstiffness according to the experimental results e
mechanical behavior of LRB can be modeled by a bilinearrelation as seen in Figure 16(b)
e cable restrainers were modeled as nonlinear tension-only elements characterized by an initial slack as shown inFigure 16(c) e parameters affecting the functionality ofcable restrainers were the cable slack length and yieldstrength according to the research of Padgett and DesRoches[16] e first two parameters were assumed to follow theuniform distributions and the strength can be modeled bythe lognormal distribution An adopted range of cablestiffness values was from 30times105 kNm to 50times105 kNmuniformly
e key parameters in analytical models of Figure 16 canbe determined according to the structural design of therealistic bridge e damage states of two restraint systemsare listed in Table 6 in which ESS is used as the damageindex
Dam
age p
roba
bilit
y
05 10 15 2000PGV (ms)
00
02
04
06
08
10
Case 0Case 1
MBDLMBDT
(a)
Dam
age p
roba
bilit
y
00
01
02
03
04
05
05 10 15 2000PGV (ms)
Case 0Case 1
PCDNLPCDNT
(b)
Dam
age p
roba
bilit
y
00
02
04
06
08
10
05 10 15 2000PGV (ms)
Case 0Case 1
BPDAEJDA
(c)
Figure 14 Influence of dampers on vulnerabilties of (a) bearing responses for DS2 (b) pylon responses for DS2 and (c) abutment andpounding responses
Advances in Civil Engineering 15
Figure 17 illustrates the probabilities of critical damage stateoccurrence sequences of the bridge components in Cases 1 4and 5 when PGV is 10ms which is equal to E2 earthquakeintensity in the seismic assessment reports e componentvulnerabilities of initial design (Case 0) were taken as the ref-erence values which were marked by different colored strips inFigure 17 e desirable sequence preferred Cases 4 or 5 for theapproach bridgese initial design of approach bridges resultedin unreasonable damage sequences In the strategies pertainingto quasi-isolation and isolation systems multiple tiers of seismicstructural redundancy can be normally employed to preventstructural collapse during strong earthquake events Tier 1 in-dicated that the bearings were weakly fused which was evidentin the high fragility of bearing sliding behaviors (ABDAL andABDC2T) Another advantage was that the extensive dam-ages of bearing would be reduced when compared with the self-damage of the pot bearings Tier 2 indicated that the designedbearings had sufficient seat widths for adapting to the sliding ofsuperstructures However the devices had a certain damageprobability (EJDA and SKDC2) and dissipated a portion of
the seismic energy when the displacement of the superstructuresbecame larger than the allowable values of constraint devicesTier 3 redundancy was designed as a potential plastic hinge inthe substructures (CCDC2) including a certain yielding of thebackfill at the abutments (BPDA) e response of theabutments in the transverse direction (PDAT) was slightlyincreased due to the pounding effect between the girders andshear keys caused by bearing sliding e reasonable damagesequence of two restraint systems can be reflected through thecorresponding damage probabilities of bearings seismic mea-sures and substructures respectively as indicated by the red andgreen lines in Figure 17 erefore the rational sequence ofcritical limit state occurrences for the approach bridges isbearings⟶ expansion joints (or restrainer cables) shearkeys⟶ columns abutments
43 Comprehensive Evaluation of Bridge System Padgett andDesRoches [16] pointed out the necessity to derive the bridgesystem vulnerabilities when assessing retrofitted bridges
Figure 15 Experimental results of (a) LEB and (b) LRB
ΔΔ1
F
Δ2
Fmax
Fd
KdK0
(a)
Δy Δ
F
K1 = (3~7) K2
K1
K2
(b)
Slack
F
K
Δ
Fy
(c)
Figure 16 Analytical models of (a) LEB (b) LRB and (c) cable restrainers
16 Advances in Civil Engineering
because different components may have a different responsewith a given retrofit measure Also Zhang and Huo [18]studied the optimal parameters of isolation bearings byderiving the intensity median value corresponding to sys-tem-level fragility function which can effectively avoid theimpacts of different isolation parameters on differentcomponents As can be seen from Figures 14 and 17 theabove analysis results were also very apparent in this studyFor example the longitudinal deck displacement was sig-nificantly reduced by using dampers yet increased thetransverse demands of stiffening girder and pylons e useof LEB or LRB reduced the demands placed on columns yetled to an increase of the transverse deformations of abut-ments which was due to the use of shear keys limiting theexcessive sliding displacement of bearings erefore acomprehensive evaluation of retrofit strategies on bridgesparticularly long-span suspension bridges that have multiplecomponents should be based on system-level fragility in-stead of component-level
Taking the findings that no damages had occurred to thecable system and the definition of bearing damage states intoaccount the derivation of system-level fragility of the wholebridge depended on critical components including pylonscolumns bearings and abutments e fragility data byemploying the absolutely serial and serial-parallel assump-tions of bridge system are plotted in Figure 18 and comparedto the system fragility obtained from the proposed com-posite damage state e absolutely serial assumption
generated serious bias for fragility curves and may result inmisdirection if they were served as a design guide eabsolutely serial model overestimated the potential damageof the bridge system since the fragility curve directlydepended on the most vulnerable components In otherwords if the difference in location and performance of eachcomponent was neglected then the seismic performanceevaluation would be achieved by attaching equal importanceto the strength of pylons approach columns and bearingsis method contradicted the philosophy of capacity designin which the long-span structure can dissipate energythrough a reasonable swinging of stiffening girder and thebearings were designed as fuse elements in the small-to-medium-span bridgese serial-parallel model qualitativelyincorporated the differences of components and was verysensitive to the seismic design of the structure as seen fromthe comparisons among the cases in Figure 18 e modeldistinguished the serial components and parallel compo-nents yet still overestimated the bridge system-level fragilitywhen bearings became the vulnerable components espe-cially for long-span bridges is special situation such asthe serial assumption may also lead to misunderstandingOnce the relative importance between the components wasconsidered in the serial-composite model the damage po-tential was clearly reduced within the range of earthquakeintensity of interest based on the case studies is alsoreflected the necessity of the classification idea of bridgecomponents which can be expressed by the numerical value
Table 6 Damage states of the quasi-isolation and isolation bearings (m)
Figure 17 Probabilities of critical damage states occurrence sequences for the approach bridges when PGV is 1ms
Advances in Civil Engineering 17
of system-level fragility erefore the application ofcomposite damage state (equation (4)) considering com-ponent difference was instead more appropriate than theaforementioned two models
e damage probability distribution represented byblack circles in Figure 18 can be obtained from the definedcomposite damage state (equation (5)) where the parametera is 06 It was found that system fragility result was a moreaccurate assessment with further taking the relative im-portance between themain bridge and approach bridges intoaccount is is due to the principle that the seismic designperformance objective of long-span bridges generally has nocollapse for the main body structures and is necessarily opento emergency vehicle service In addition attention must bepaid to the fact that the differences in repair cost and time ofthe main and approach bridges are significant when thesame damage state is achieved e results of compositedamage state may match the performance objectives of theseismic design In order to investigate the influence ofimproved strategies and structural relative importance ondamage potential of bridges the mean values of the fragility
functions can be employed to evaluate the seismic designe smooth fragility curves for the specific damage states canbe fitted with the lognormal cumulative probability distri-bution function by employing the least-squares technique
P DSgtDSi | PGV1113858 1113859 Φln(pgv) minus λPGV
ζPGV1113888 1113889 (12)
where λPGV and ζPGV are the logarithmic mean and dis-persion values of earthquake intensity (PGV) for a specifieddamage level e larger mean values represent the smallerdamage potential to reach the specified level
Figure 19 indicates the mean values of earthquakeintensity at the specific damage states as a function of theimproved strategies and parameter a Each step of theretrofit measures would enhance the seismic performanceof the overall bridge system at all levels e improvementof the restraint system of the approach bridge played acritical role of the overall bridge system For instance themean values of the earthquake intensity featured mutationof slopes along the measure direction which reflected the
Figure 18 System fragility of the original design (a) Case 0 and improved design including (b) Case 1 and (c) Case 5
18 Advances in Civil Engineering
fact that the effectiveness of improvement for the ap-proach bridges was clearer than that of the main bridgee fragility of the highway network would be under-estimated if only the damage potential of the suspensionspan was adopted erefore when the structures such asapproach bridges transition piers and auxiliary pierswere ignored the damage identification and repair de-cisions would be misleading in the seismic evaluation oflong-span bridges Conversely the damaging probabilityof the overall bridge system will be amplified regardless ofthe higher importance of the main bridge When respectthe fact that the importance of suspension span is higherthan that of approach bridge in transportation networks aparameter a of 06ndash08 may be a better choice in thepresent study
e seismic fragility of conventionally designedbridges was mainly governed by the approach bridges at alldamage levels However for the retrofitted bridges thatpossess reasonable seismic performance different damagestates were controlled by the earthquake demands ofvarious structures as shown by the trend of the curvesalong parameter a in Case 4 and Case 5 e damagingpotential of the overall bridge from the low-to-moderatelimit states was dominated by the approach bridge eprimary reasons were the sliding of bearings and thetransfer of limited forces from the girder to substructure
by the damage of restriction devices of the approachbridges e extensive damage was controlled by thedisplacement of stiffening girder of the main bridge whichinferred the effect of dampers will be conspicuous understrong earthquake shaking At the same time the overalllong-span bridges with the reasonable seismic system willnot collapse during a strong earthquake event and canmeet emergency traffic service It can be seen that the meanvalues of fragility functions developed by compositedamage indices (equations (4) and (5)) are good indicatorsto evaluate seismic performance of long-span bridges andcan be consistent with the established seismic performanceobjectives
5 Conclusions
In the present study a procedure for investigating thereasonable seismic performance of long-span suspensionbridges was developed under the fragility function frame-work e seismic performance of a long-span suspensionbridge was examined and improved using the proposedprocedure from which the results revealed distinguishableconclusions as follows
(1) e definition of damage states for bearings con-sidered the interactions between the adjacent
014036059081103125
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(a)
020051081112142173
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
0002
0406
0810
CaseParameter a
(b)
130251371492612733
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(c)
Mea
n va
lue
exp
(λ) (
ms
)
Parameter a Case0 1 2 3 4 5
0001
0203
0405
131305478652826
1000
(d)
Figure 19 Influence of relative importance and improved designs on mean values of earthquake intensity for different damage states(a) slight (b) moderate (c) extensive (d) collapse
Advances in Civil Engineering 19
structures and self-damage to bearings in accor-dance with the actual design and experimentalconclusions e damage states of pylons in terms ofcurvature ductility were defined by developingmodal pushover analysis and idealized bilinear fit-ting methods in the equivalent pylon fiber models ofthe suspension bridge
(2) When investigating the component fragility curvesof the initial design it was found that the large deckdisplacement of the suspension span was the mostserious damage present and a slight inelastic re-sponse may have occurred only in the pylons Al-though the cable system was able to maintainelasticity under earthquakes the saddles at the py-lons were identified with a high-risk region indi-cating the damaging potential For the approachbridges the ductility mechanism did not realize arational damage sequence e columns were proneto collapse because of the application of fixed potbearings which in turn caused the damage to thebearings and abutments
(3) e stepwise improved seismic designs were pro-posed in terms of the component fragility results ofthe initial design Among the candidates fluid vis-cous dampers were proven to be a good choice forlong-span bridges based on the two attributes ofdisplacement and force when the similar efficiency ofdisplacement control was achieved e restraintsystem had a critical effect on the seismic responsesof the approach bridges e quasi-isolation orisolation can achieve a reasonable seismic damagesequence of the approach bridges
(4) e defined composite damage indices showed thatthe incorporate component classification conceptand structural relative importance were necessary toderive a good indicator of performance evaluationdue to the fact that the damage indices were based oneither an absolutely serial or serial-parallel systemmodel may significantly overestimate the damagepotential of the overall bridge system e meanvalues of earthquake intensity at the specific damagestates can serve as a function of the improvedstrategies and relative importance to further evaluatethe overall seismic performance of the suspensionbridge It can be concluded that the damaging po-tential of the overall bridge from the low-to-mod-erate limit states was governed by the approachbridges in the retrofitted bridges with a favorableseismic performance whereas the displacementdemand of the suspension span dominated thefragility probability of the extensive damage
Data Availability
e nature of the data is the analytical data of finite elementmodels of the case bridge e data used to support thefindings of this study are available from the correspondingauthor upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the Basic Research ServiceProject of Central-Level Public Welfare Research Institute(grant nos 2016-9018 and 2019-0034) and Science andTechnology Project of Communicationsrsquo Construction inWestern China MOC (grant no 2009318223094)
References
[1] L W David A Kartoum K Chang and I Roy ldquoSuspensionreport seismic performance and design considerations of longspan suspension bridgesrdquo Report CSMIP00-03 CaliforniaDepartment of Conservation Division of Mines And GeologyOffice of Strong Motion Studies Sacramento CA USA 2000
[2] S-G Gwon and D-H Choi ldquoImproved continuummodel forfree vibration analysis of suspension bridgesrdquo Journal ofEngineering Mechanics vol 143 no 7 Article ID 040170382017
[3] S Arzoumanidis A Shama and F Ostadan ldquoPerformance-based seismic analysis and design of suspension bridgesrdquoEarthquake Engineering and Structural Dynamics vol 34no 4-5 pp 349ndash367 2005
[4] L Sgambi E Garavaglia N Basso and F Bontempi ldquoMonteCarlo simulation for seismic analysis of a long span sus-pension bridgerdquo Engineering Structures vol 78 pp 100ndash1112014
[5] D Karmakar S Ray-Chaudhuri and M Shinozuka ldquoFiniteelement model development validation and probabilisticseismic performance evaluation of Vincent omas suspen-sion bridgerdquo Structure and Infrastructure Engineering vol 11no 2 pp 223ndash237 2015
[6] W-F Chen and L Duan Bridge Engineering HandbookSeismic Design CRC Press Boca Raton FL USA 2nd edition2014
[7] E Choi ldquoSeismic analysis and retrofit of mid-americabridgesrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2002
[8] B G Nielson and R DesRoches ldquoSeismic fragility method-ology for highway bridges using a component level approachrdquoEarthquake Engineering amp Structural Dynamics vol 36 no 6pp 823ndash839 2007
[9] K N Ramanathan ldquoNext generation seismic fragility curvesfor california bridges incorporating the evolution in seismicdesignrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2012
[10] O Avsar A Yakut and A Caner ldquoAnalytical fragility curvesfor ordinary highway bridges in Turkeyrdquo Earthquake Spectravol 27 no 4 pp 971ndash996 2011
[11] S P Stefanidou and A J Kappos ldquoMethodology for thedevelopment of bridge-specific fragility curvesrdquo EarthquakeEngineering amp Structural Dynamics vol 46 no 1 pp 73ndash932016
[12] B Borzi P Ceresa P Franchin F Noto G M Calvi andP E Pinto ldquoSeismic vulnerability of the Italian roadwaybridge stockrdquo Earthquake Spectra vol 31 no 4pp 2137ndash2161 2015
[13] W T Barnawi and S J Dyke ldquoSeismic fragility relationshipsof a cable-stayed bridge equipped with response modification
20 Advances in Civil Engineering
systemsrdquo Journal of Bridge Engineering vol 19 no 8 ArticleID A4013003 2014
[14] J Zhong Y Pang J Jong-Su J-S Jeon R DesRoches andW Yuan ldquoSeismic fragility assessment of long-span cable-stayed bridges in Chinardquo Advances in Structural Engineeringvol 19 no 11 pp 1797ndash1812 2016
[15] J Zhong Z Hu W Yuan and L Chen ldquoSystem-basedprobabilistic optimization of fluid viscous dampers equippedin cable-stayed bridgesrdquo Advances in Structural Engineeringvol 21 no 12 pp 1815ndash1825 2018
[16] J E Padgett and R DesRoches ldquoMethodology for the de-velopment of analytical fragility curves for retrofitted bridgesfitted bridgesrdquo Earthquake Engineering amp Structural Dy-namics vol 37 no 8 pp 1157ndash1174 2008
[17] Y Xie and J Zhang ldquoOptimal design of seismic protectivedevices for highway bridges using performance-basedmethodology and multiobjective genetic optimizationrdquoJournal of Bridge Engineering vol 22 no 3 Article ID04016129 2017
[18] J Zhang and Y Huo ldquoEvaluating effectiveness and optimumdesign of isolation devices for highway bridges using thefragility function methodrdquo Engineering Structures vol 31no 8 pp 1648ndash1660 2009
[19] F McKenna M H Scott and G L Fenves ldquoNonlinear finite-element analysis software architecture using object compo-sitionrdquo Journal of Computing in Civil Engineering vol 24no 1 pp 95ndash107 2010
[20] MOTC ldquoGuidelines for seismic design of highway bridgesrdquoChina Ministry of Transportation and CommunicationsPress Beijing China JTGT B02-01-2008 2008
[21] A D Kiureghian and J L Sackman ldquoTangent geometricstiffness of inclined cables under self-weightrdquo Journal ofStructural Engineering vol 131 no 6 pp 941ndash945 2005
[22] J S Steelman E T Filipov L A Fahnestock et al ldquoEx-perimental behavior of steel fixed bearings and implicationsfor seismic bridge responserdquo Journal of Bridge Engineeringvol 19 no 8 Article ID A4014007 2014
[23] B Madani F Behnamfar and H Tajmir Riahi ldquoDynamicresponse of structures subjected to pounding and structure-soil-structure interactionrdquo Soil Dynamics and EarthquakeEngineering vol 78 pp 46ndash60 2015
[24] L Xu and J Li ldquoDesign and experimental investigation of anew type sliding retainer and its efficacy in seismic fortifi-cationrdquo Engineering Mechanics vol 33 no 2 pp 111ndash1182016 in Chinese
[25] Caltrans Caltrans Seismic Design Criteria California De-partment of Transportation Sacramento CA USA 2013
[26] A Shamsabadi K M Rollins and M Kapuskar ldquoNonlinearsoil-abutment-bridge structure interaction for seismic per-formance-based designrdquo Journal of Geotechnical and Geo-environmental Engineering vol 133 no 6 pp 707ndash720 2007
[27] X Li Z Sun D Wang and Y Shi ldquoLongitudinal seismicpounding effect of bridges abutment and backfilling damagerdquoJournal of Changrsquoan University vol 35 no 4 pp 76ndash82 2015in Chinese
[28] MOTC ldquoSpecification for design of highway suspensionbridgerdquo China Ministry of Transportation and Communi-cations Press Beijing China JTGT D65-05-2015 2015
[29] A Shafieezadeh K Ramanathan J E Padgett andR DesRoches ldquoFractional order intensity measures forprobabilistic seismic demand modeling applied to highwaybridgesrdquo Earthquake Engineering amp Structural Dynamicsvol 41 no 3 pp 391ndash409 2012
[30] J E Padgett B G Nielson and R DesRoches ldquoSelection ofoptimal intensity measures in probabilistic seismic demandmodels of highway bridge portfoliosrdquo Earthquake Engineeringamp Structural Dynamics vol 37 no 5 pp 711ndash725 2008
[31] K R Mackie and B Stojadinovic ldquoFragility basis for californiahighway overpass bridge seismic decision makingrdquo PEERReport 200502 Pacific Earthquake Engineering ResearchCenter University of California Berkeley Berkeley CA USA2005
[32] J Li T Peng Ji Lin L Zhou and YuL Deng Seismic Design ofMulti-Pylon Multi-Span Suspension Bridge China Commu-nications Press Beijing China 2013 in Chinese
[33] L Chen W Zhuang H Zhao and Z Wan ldquoReport onHighwaysrsquo Damage in the Wenchuan Earthquake-BridgeChina Communications Press Beijing China 2012 inChinese
[34] G Wu K Wang G Lu and P Zhang ldquoAn experimentalinvestigation of unbonded laminated elastomeric bearingsand the seismic evaluations of highway bridges with testedbearing componentsrdquo Shock and Vibration vol 2018 ArticleID 8439321 18 pages 2018
Advances in Civil Engineering 21
It should be noted that the aforementioned developmentof damage indices for the bridge system is only for a singlebridge system and is closely related to the definition ofdamage states of the components For long-span suspensionbridges composed of a main bridge and several approachbridges a reasonable and comprehensive seismic vulnera-bility evaluation is of great significance to the construction ofproject and postearthquake repairing cost A bridge systemwith the similar structural properties could be regarded as aseries system However the differences in structural im-portance and repairing costs cannot be neglected because ofthe structural differences between the long-span bridges andsmall-to-medium-span bridges e weighting ratios of aand (1 minus a) are assigned to the main bridge system andapproach bridge system respectively which have the abilityto capture the contributions of the main bridge and ap-proach bridges to the vulnerability of the overall bridgesystem erefore the rational range of system fragilitycould be accurately determined as follows
DSsystem round a middot DSmain bridge +(1 minus a) middot DSapproach bridge1113960 1113961
(5)
where the round-off principle can be used for the ldquoroundrdquoe composite damage state in equations (4) and (5) can
be incorporated in the PSDM when the correlation betweenconcerned components with the use of covariance matrix isknown In this case Monte Carlo simulation is used tocompare the demands obtained from the joint probabilisticseismic demand models [8] and multidimensional compo-nent capacities in theM (105) random samples at a range ofthe IM values e basic principle of developing fragilitycurves for the long-span suspension bridge system is shownin Figure 1
3 Case Study
31 Bridge Description and Simulation is study selectedthe Taoyuan Bridge spanning the Jinsha River as the studycase e bridge is characterized by the geometric andmechanical features of a suspension bridge which is themost common type of bridge construction in China ebridge has a total length of 731m and consists of a single-span simply supported suspension bridge measuring 636mand a continuous steel box-girder approach bridge withthree spans measuring 30m 35m and 30m respectivelye longitudinal elevation of the bridge is illustrated inFigure 2
e main bridge has a thin-wall steel girder along thelongitudinal direction with an overall width of 31m fromcurb to curb and a height of 3m accommodating four lanesof traffic e gate-type towers include two RC box-sectionlegs with overall heights of 75m and 70m on its two sidesrespectively Also there is one crossbeam installed along theheight of pylons in order to enhance the lateral stiffness ofstructures At the base the tower legs are anchored to the 6m-thick concrete footings e pedestals for the placingbearings are designed between the concrete footings due tothe shorter distance between the deck and tower base e
bored piles serving as the foundations of pylons and locatedat the leg of each tower have a diameter of 22me areas ofeach main cable and suspender are 01626 m2 and 0001316m2 respectively Gravity anchorage has been applied to themain cable anchorage on both sides of the bridge eapproach bridges are divided into left and right bridges withthe identical configurations on the north side For eachapproach bridge there is a three-span continuous steel box-girder bridge with a pile-bent abutment of which the middlespan is supported by two double-column bents with integralpile-shaft e width and depth of the superstructure are1275m and 153m respectively e circular sections ofcolumns with a diameter of 18m have a height of 7m andthe cast-in-drilled-hole piles measure 19m in diameter epiles of abutment are provided by a circular section of 15min diameter and the height of the backwall is 188m
e description of this studyrsquos analytical modelingprocedure is outlined using OpenSEES software [19] for theprototype bridge in the as-built design A beam-columnelement with distributed plasticity fiber was used to modelthe pylons and columns in order to account for the materialnonlinearity In addition the axial force-moment interac-tions of pylons were captured in the model Each fiber wassimulated via a reasonable stress-strain relationship whichwas dependent on unconfined concrete confined concretedefined by the Mander theoretical model and longitudinalreinforcements that considered the Bauschinger effects asillustrated in Figure 3 e corresponding mechanical in-dices such as material strength axial compression ratio(ACR) and reinforcement ratios of Sections I and II all metthe Chinese code requirements [20]
e main cable element was tension-only which wasmodeled as a finite number of large-displacement truss el-ements using the Ernst method accounting for the sag effect[21] Each suspender element was identical to the simulationof the main cable but there was no sag effect e initialstress of the cable system was also considered in the modelas shown in Figure 4(a) An elastic beam element was uti-lized to simulate the crossbeam of the pylons and stiffeninggirder as well as the girders and cap beams of the approachbridges e foundations were modeled by six spring ele-ments which indicated that in the future they are notexpected to be damaged under earthquake shaking
It should be noted that pot bearings were applied to theas-built design which consequently became the significantfactors in the overall responses and functionality of theapproach bridges e longitudinal response of sliding potbearings detailed in Figure 4(b) was simulated using a bi-linear element [20] e full-scale experiments of Steelmanet al [22] found that properly proportioned steel for fixedpot bearings can achieve a reliable dry friction responsebetween the bearing component and superstructure orsubstructure after the rupture of the anchor bolt Corre-spondingly the behavior of fixed pot bearings was modeledas shown in Figure 4(c) e fixed pot bearings weredesigned to be located at the C2 column as shown inFigure 1 Finally the pounding stiffness between the adjacentdecks (Figure 4(d)) was estimated by the sum of their axialstiffnesses [23]
4 Advances in Civil Engineering
Approach bridge
Composite damage state (4)
Main bridge
Composite damagestate (4)
Composite damage state (5)
Bearing 1 Bearing iBearing 2
Bearing 1 Bearing iBearing 2
Bearing 1 Bearing iBearing 2
Bearing 1 Bearing iBearing 2
Column 1
Column 2
Column j
Abutment 1
Abutment 2
Bearing 1 Bearing iBearing 2
Bearing system
Column 1 Column iColumn 2
Column system
Bearing 1 Bearing iBearing 2
Bearing 1 Bearing iBearing 2
Pylon 1
Pylon 2
Bearing system
Pylon 1 Pylon 2
Pylon system
Damage state definitionsand component-level
fragility results
Abutment 1
Abutment 2
Abutmentsystem
Figure 1 Bridge system based on the series-parallel system and composite damage state
C2C1
4000
7000
6000
7500 63
60
14000
3500
30001600063600
73100
3000South
North
Figure 2 Configuration of the Taoyuan Suspension Bridge (cm)
SouthH = 750m
North H = 700m
H
Sec I
Pylon
Pedestal
(a)
Confined(Mander model)
εu εcu
fc
Kfc
ε
σ
Unconfined
Concrete0002
fcufu
εc0
(b)
σy
εy ε
σ0005E0
E0
Bar
(c)
Figure 3 Continued
Advances in Civil Engineering 5
e shear keys contribute to the constraint of transversemovement between the girder and pier Xu and Li [24]proposed a modified model based on the structural prop-erties of shear keys in China which was used to build theforce-displacement relation of shear keys by separating thecontributions of steel and concrete in parallel as shown inFigure 5 e critical parameters of the shear keys can becalculated using Xursquos method according to the actual designwhich represents the yield (Δ1Y) nominal (Δ1n) degraded(Δ1d) and ultimate (Δ1u) deformations respectively esimulation can accurately reflect the mechanical behavior ofthe shear keys
e seat type abutment is shown in Figure 6 e lon-gitudinal resistance to seismic forces is provided jointly bythe passive action of the backfill soil and active action of thepiles [25] Meanwhile the transverse response is resisted bythe pilese passive and active responses of abutments weresimulated using hyperbolic soil material [26] and trilinearforce-displacement relationship respectively which wereconnected in parallel to capture the abutment responses Inaddition the pounding effect between the deck and abut-ment was simulated using a linear model without energydissipation e stiffness of pounding spring between thesuperstructure and abutment was proposed to be ten timesthat of the backfill soil stiffness [27]
32 Uncertainty Treatment in the FE Model e LHS ap-proach is usually used to account for the uncertainties infragility analysis such as structural geometry materialstrength component properties deck mass and damping in
the modeling which is considered to be a variance reductionsampling technique It is noted that the uncertainty ofstructural geometry is not taken into account in this studyfor the development of bridge-specific fragility curves In theforegoing approach the probability distributions were as-sumed for each parameter and each random variable wasdivided into the several equal intervals which correspond to5 to 95 of the cumulative probability e consideringmodeling parameters and their probability distributions arepresented in Table 1 Statistically significant yet nominallyidentical 3D bridges were built using the LHS so as toaccount for the abovementioned uncertainties then thebridges were paired randomly with the selected groundmotions
33 Ground Motion Input and Intensity Measure e ne-cessity to carry out a large number of nonlinear time-historyanalyses depends on a significant number of earthquakerecords selected for the PSDM In this study a data set of 80recorded ground motions and 20 Los Angeles-pertinentunscaled ground motions were extracted from the PEERstrong motion database and the SAC project database re-spectively [29] ese ground motions can represent thecharacteristics of the bridge site ey were placed into fivemagnitude-distance bins for the purpose of covering therational and broad ground motion intensity e first fourbins included combinations of low and high magnitudesand large and small fault distances e magnitudes variedbetween 57 and 70 while the fault distances ranged from130 to 600 km e characteristic of the ground motions in
Stress
Initialstress
K1
Strain
K2 = 0005K1
(a)
K1
Δy = 2mm
Fy
ΔΔy
K2 = 000001K1
F
(b)
Δu
F
Fd
Δy
K
Rh
ΔΔy = 1mm
Δu = 20mm
(c)
K
Gap
F
Δ
(d)
Δy
FΔult
K
Δ
Fult
Fult2
Δy= 154mmΔult = 1504mm
Per width (m)of abutment
(e)
Δ1 = 762hmmΔ2 = 254hmm
h height of backwall
Keff
K2
F2F1
Δ1
K1
F
ΔΔ2
Per pileKeff = 14kNmm
(f)
Figure 4 Constitutive relations of the various components (a) cable and suspender (b) sliding pot bearing (c) fixed pot bearing(d) pounding (e) abutment backfill (f ) abutment pile
Bar (HRB500 376)
z
y
Confined concrete (stirrup HRB400
1256)
Sec I
Unconfined concrete (C55)
ACR 0159
(d)
Sec II
h =
70m
Double-column bent
(e)
Confined concrete(stirrup HRB335
08)
Unconfined concrete (C35)
Bar(HRB335 1163)
z
y
Sec II
ACR 0037
(f )
Figure 3 Sketch and material model of the pylons and bents of the examined suspension bridge
6 Advances in Civil Engineering
the fifth bin was that the fault distance was less than 15 kme ground motions were given in two orthogonal com-ponents and the acceleration values adopted were computedas the geometric mean values (Sa
SaL middot SaT
1113968) e peak
ground acceleration ranged from 0043 g to 112 g Figure 7shows the distributions of peak ground accelerations (PGA)peak ground velocities (PGV) earthquake magnitudes (M)and fault distances (R) of 100 ground motion records as well
as the acceleration spectra with a damping ratio of 002 etwo orthogonal components of the selected earthquakerecords were randomly input along the longitudinal andtransverse directions of the bridge
PGA PGV PGD spectral acceleration or Arias In-tensity of earthquake can be selected as the IM candidateswhich are used to characterize earthquake intensity elinear relations with the interesting EDPs from probabilistic
Δ
V
Δ1n Δ1d
Vcp
Δ
V
Δ1y Δ1u
Vsp
i
j
Gap
Hysteretic property of concrete Hysteretic property of steel
Vcp = 7624kN
Δ1n = 106mmΔ1d = 422mm
Vsp = 10143kN
Δ1y = 11mmΔ1u = 633mm
Figure 5 Analytical model of shear keys
Abutment
Deck
Figure 4 (b)
Figure 4 (d)
Figure 4 (c)Figure 5 Figure 4 (e)
Figure 4 (f)
Figure 6 Pile-bent abutment simulations
Table 1 Uncertainty parameters incorporated in the modeling design
Modeling parameter Probability distribution Mean COV Unit Source
Concrete compressive strength C55 Normal 466 0149 MPa
[20]C35 321 0164
Steel yield strengthHRB500
Lognormal5696
00743 MPaHRB400 4557HRB335 3816
Friction coefficient Sliding pot bearing Uniform 003 033 mdash [14]Fixed pot bearing Lognormal 02 0002 mdash [7]
Deck-abutment 80Deck-shear key Uniform 120 029 mm [16]
Damping ratio Uniform 1 01 mdash [7]Deck mass Normal 002 015 mdash [28]COV coefficient of variation
Advances in Civil Engineering 7
seismic demand analysis in the logarithmic reference framecan be adopted to determine the optimal IM In the previousstudies PGA and spectral acceleration were selected as theoptimal IMs for highway bridges [9 30] when applying themultiple measures containing efficiency practicality profi-ciency and sufficiency Mackie et al [31] suggested both PGAand PGV are good choices of IM in related to the EDPmeasures in highway overpass bridges Zhong et al [14]pointed out that PGA and PGV are themost efficient practicaland proficient IMs for long-span cable-stayed bridges yet PGVtends to be the optimal IM in terms of sufficiency ereforePGV can be utilized as the IM in this study
34 Engineering Demand Parameters e peak demands ofcritical components are adopted as EDPs for the developmentof PSDMs and the choice of optimal IM Table 2 lists thecritical EDPs for the examined bridge which have beenconsidered in the fragility evaluation of the suspension bridge
35 Capacity Limit State Models In the present study acapacity model was used to measure component damagewhich was described by a damage index as a function of the
selected EDPs ese models are usually discrete andcharacterized by the onset of various damage states based onthe experimental data Similarly the uncertainties of ca-pacity models are also expressed by median Sc and dis-persion βc e definition of damage states must be matchedwith postearthquake bridge functionality and repair strat-egies in the bridge seismic fragility assessments Founda-tions decks and anchorages are typically identified asnondamaged components in the simulations e damagestate values of bearings were determined so as to comparewith two factors including the pounding gaps between theadjacent components and seating width which can reflectthe possible pounding and unseating as shown in Figure 8 inaddition to the damages to bearings themselves that is thedeformation capacity of the bearings Table 3 lists the slight(DS1) moderate (DS2) extensive (DS3) and collapsed(DS4) damage states defined in the initial design for potbearings at various locations based on their allowable de-formation and structural dimensions
e columns of conventional highway bridges are oftenforced into the state of nonlinearity under strong earth-quakes of which the criteria of damage states were studied[8 9 16] It is found that there are major differences in
0
10
20
30
40
50N
umbe
r of r
ecor
ds
08
01
03
05
09
00
07
04
02
10
06
PGA (g)
(a)
0
10
20
30
40
50
Num
ber o
f rec
ords
20 806040 100
180
140
160
2000
220
120
PGV (cms)
(b)
20 40 600Fault distance (R) (km)
55
60
65
70
75
Mag
nitu
de (M
)
(c)
Mean
0
1
2
3
4
5
S a (g
)
2 4 6 8 100Period (s)
(d)
Figure 7 Characteristics of the selected ground motions (a) PGA (b) PGV (c) M and R (d) acceleration response spectra
8 Advances in Civil Engineering
geometric sizes material strengths and ACR of sectionsbetween the pylons of suspension bridges and columns ofconventional highway bridges erefore it is necessary todevelop a modal pushover analysis to define the damagestates of critical cross-sections of pylons In the analyticalprocesses of this study the material strain and nonlinearmoment-curvature relationship of critical cross-sectionswere characterized to measure the damage levels of thesuspension bridge pylons
First the reasonable pylon models were required to bedeveloped using fiber beam elements in longitudinal andtransverse directions as detailed in Figure 3 An elastic
spring which was equivalent to the restraint of main cablewas attached to the top of the pylon taking into account theinfluence of the main cable stiffness in the longitudinaldirection models e determining process of the equivalentstiffness can be referred to in the Ernst formula and theresults reported by Kiureghian and Sackman [21]
Second as illustrated in Figure 9 the inertial forcedistributions were back-calculated from the multimodespectrum analysis using a 3D bridge model as shown inequations (6)ndash(8) which included the contributions of highvibration modes and reflected the properties of inertial forcedistributions as follows
DS3
DS4
DeckDeck
Pedestal
PedestalBearing
Bearing
Gap
(a)
DS2
DS4
Gap
Abutmentnonlinearity
Deck
AbutmentBearing
(b)
GapShear key
degradationDS4
DS2
Bearing
(c)
Figure 8 Damage control and unseating prevention damage states for bearing displacement (a) between decks (b) between deck andabutment (c) between deck and shear key
Table 2 e critical EDPs for the examined bridge
Engineering demand parameters Abbreviation DirectionCurvature ductility of pylon cross-section (south north) PCD(S N) Longitudinal and transverseDisplacement of bearings of the main bridge MBD Longitudinal and transverseAxial force relative to yield value of main cables MAF mdashAxial force relative to yield value of suspenders SAF mdashCurvature ductility of column cross-section at columns (C1 C2) CCD(C1 C2) Longitudinal and transverseDisplacement of bearings of the approach bridge at column (C) and abutment (A) ABD(C A) Longitudinal and transverseDeformation of backfill passive pressure at abutment BPDA LongitudinalDeformation of piles at abutment PDA Longitudinal and transverseDeformation of expansion joint between decks(D) and at abutment (A) EJD(D A) LongitudinalDeformation of shear key at columns (C) and at abutments (A) SKD(C A) Transverse
Table 3 Limit state thresholds for pot bearings (m)
DS1 DS2 DS3 DS4
Criteria Allowable slippingdisplacements
Spherical skateboard radiusbody separation of bearings
Possible pounding betweenadjacent decks or deck and pylon Possible unseating
MBDL 025 031 076 1555MBDT 002 031 045 mdash
Criteria Allowable slippingdisplacements
Possible pounding entering of abutmentsnonlinearity or shear key degradation
Spherical skateboard radiusbody separation of bearings Possible unseating
where Sj and cj are the spectral acceleration and participationfactor associated with the jth mode respectively Fij φij andQij represent the seismic load mode displacement and shearforce at the i-node associated with the jth mode respectivelymi Qj and Pi indicate the mass shear force and inertial forceat the i-node respectively andN denotes the number of nodes
Next the inertial forces were loaded to the nodes ofpylon models Both the material and geometric nonlinear-ities were incorporated into the pushover analyses whichwere used to investigate the potential failure mechanismsand assess the ductility capacities of pylons
e seismic designs of pylons can develop inelastic be-haviors which were controlled by the material strain limitsFigure 9 details the curvature distributions along the pylonheights with the increase of δ where δ was the loading dis-placement at the control point It was observed that plastichinge zones typically occur at the bases of pylons and above andbelow the horizontal crossbeams Also plastic hinges mayprogress to the intermediate height in the longitudinal direc-tion e development of plastic hinges at the bases of pylonscan be much severer and also occur earlier than in other zoneserefore the curvatures at the bottoms of pylons may ef-fectively provide measurements of the damages to pylons
e mechanical indices of pylon cross-sections whichcorrespond to the four damage state characteristics
following the damages are detailed in Figure 10 e cur-vature ductility can be easily determined by the pushoveranalysis results in addition to that of the DS2e numericaldata can be transformed to a bilinear curve that has a value ofDS2 based on the equal energy rule per original curveSimilarly another pushover analysis was applied to thedouble-column bend of the approach bridge based on thespecifications [20] e four limit state thresholds of thecross-sections at the bottom of pylon and columns in boththe longitudinal and transverse directions are listed in Ta-ble 4 It was observed that the seismic capacity of longitu-dinal direction was superior to that of transverse directionand a good ductility capacity existed in both directions
In the present study the definitions of damage statethresholds of main cables and suspenders were dependent onthe steel strand characteristics of the strength and deformationrates A total of 85 of the ultimate strength (nominal yieldstrength) was conservatively acted as the failure index Namelythe corresponding ratio of force to yield value was 10 It is acommon viewpoint that the accidental loss of one suspenderwill not cause immediate failures of a bridge with the exceptionof the loss of the main cables erefore DS3 and DS4 wereonly defined based on the classification of suspenders andmaincables respectively e corresponding elastic and ultimatedeformation of abutments and shear keys were respectivelyspecified the thresholds of DS1 and DS2 with the recom-mendations of Caltrans [25] e damage of expansion jointsonly defined as the DS1 due to the lack of a higher-leveldamage modelis was objectively logical since the joints canbe replaced with the new once damaged thereby suggesting therationality for just one threshold of component damage Table 5summarizes the damage state thresholds of the above-mentioned components
e assignments of dispersions of capacity models ofcomponents appeared to be the most difficult tasks for theresearchers According to the previous research studies 035was adopted in the abutment and RC components including
Figure 9 Inertial force and curvature distributions (a) longitudinal (b) transverse
10 Advances in Civil Engineering
pylons and columns [9] e COV of main cables andsuspenders was 01 taking the low variability of highstrength steel strands into account [16] e descriptivedispersions of the remaining components were calculated tobe 025 for the two lower damage states (DS1 and DS2) and047 for the higher damage states (DS3 and DS4) [8]
36 Assessment of Vulnerable Components e componentfragility curves were derived by steps 1ndash3 detailed in Section 2to identify vulnerable components and regions as shown inFigures 11 and 12e pot bearings were shown to be the mostvulnerable components for the main bridge which inferred thecomplete damages to pot bearings themselves and reflected ahigher probability of pounding between the adjacent decks andunseating in the longitudinal direction e transverse re-sponses of bearings did not cause obvious damage compared tothe longitudinal responses e pylons were the primarycomponents for load carrying where inelasticmaterial responsecould potentially occur e transverse vulnerability was moreobvious than the longitudinal vulnerability for the lowerdamage states However the probability of extensive damagewas extremely low or even zeroWith respect to themain cablesand suspenders the increases in their forces which couldpotentially occur during rare earthquakes were confined toabout 15 (for themean values) of the forces resulting from thedead loads as illustrated in Figure 12 which also indicated thatthe maximum values of main cable stress had occurred at thesaddles of pylons and in the side spans e stress values of
suspenders achieved the maximum at the middle and end Insummary the cable system was found to be secured againstdamages in which the stress values were much lower than theyield values even under strong earthquake shaking It wasnoted that the large deflection and cable angle changes betweenthe main span and side spans would lead to the strong shear inthe longitudinal direction between the cables and saddles andthe slipping damages to the saddles at the top of the pylonsduring earthquakes were identified as high-risk
For the approach bridges the columns were determinedto be the most fragile components in both the longitudinaland transverse directions e application of the fixed potbearings resulted in a large force and deformation of thecolumns the complete damage of sliding pot bearings andapparent pounding between the decks and abutments eseactions subsequently caused the inelastic responses ofabutments but the shear keys were protected from failuretransversely Although the ductility mechanism of approachbridges was designed it was considered to be a defectiveseismic system with irrational sequences of componentdamage occurrences in high-seismicity regions
41 ImprovedDesign ofMainBridge As results shown in theevaluation of vulnerable components a large deck dis-placement of the suspension span would occur duringearthquakes which caused pounding and even unseating
Original curveBilinear curve
MprimeyMy
φs or φccφucφprimeyφy
DS4 reinforcing bar rupture core concrete crushing
DS3 major spalling of cover concrete exposed core concrete
DS2 formation of stable plastic hinge in the pylon (column)
Mom
ent
Curvature
DS1 narrow cracking in concrete first yielding of reinforcing bar
Figure 10 e damage state definitions of the pylon and column cross-sections
Table 4 Limit state thresholds of pylon and column cross-sections
Figure 11 Component fragility curves for the suspension bridge (a) main bridge (b) approach bridge
200 400 600 800 10000Location (m)
00
02
04
06
08
10
MA
F
MaximumMinimumMean
Dead loadYield valueHigh stress region
(a)
00
02
04
06
08
10
SAF
200 400 600 800 10000Location (m)
MaximumMinimumMean High stress region
Dead loadYield value
(b)
Figure 12 Force distributions of the (a) main cables and (b) suspenders
12 Advances in Civil Engineering
when there were no additional restraints in the longitudinaldirection erefore it was consistent to suggest that sup-plementary devices should be equipped to mitigate the deckmovements to some extente reduction of deck or bearingdisplacements can be taken as the retrofitted objective for thesuspension span Fragility analyses were conducted on theretrofitted bridges that covered the general strategies asfollows
Case 1 the analysis model of the bridge used a fluidviscous damper equipped between the deck and eachpedestal on both sides of the bridgeCase 2 the analysis model of the bridge used an elasticcable installed between the deck and each pedestal onboth sides of the bridgeCase 3 the analysis model of the bridge used three pairsof flexible central buckles located in the middle sectionof the span
e main bridge was assumed to have identical me-chanical parameters at each location in various designschemes e dampers and elastic cables were respectivelymodeled using zero-length elements with Maxwell andtension-only materials and the flexible central buckles weresimulated as the same as the suspenders elements inOpenSEES
e experimental testing of fluid viscous dampers hasshown that the force-velocity with fractional velocity powerlaw relation can describe the suitable mechanical behavior
fD CD| _u|αsgn( _u) (9)
where α is a real positive exponent of which value dependson the piston head orifice CD is viscous damping coefficientwith units of force per velocity raised to the α power kNmiddot(ms)-α _u indicates the velocity of damper and sgn(middot) is thesignum function Research by Li et al [32] indicated thatviscous damping coefficients range from 2500 kN (ms)-α to5000 kN (ms)-α and can be simulated with a uniform dis-tribution e velocity exponents can be assumed to varyuniformly from 03 to 05
e axial stiffness of elastic cables and flexible centralbuckles is regarded with the material length and cross-section area It was assumed that the yield strength of steelstrands of fy 1420MPa was the mean value and followed alognormal distribution with a COV 01 e stiffnessvalues of single elastic cable based on past work employed auniform distribution ranging from 125times105 kNm to25 times105 kNm In addition the other mechanical prop-erties such as length and area can be determined by thefollowing relations
LgtEΔfy
1113888 1113889
kE EA
L1113874 1113875
(10)
where L E and fy are the length Youngrsquos modulus andyield stress of elastic cables respectively and Δ represents the
controlled deck displacement which was available for 06min the examined bridge
According to expert opinion three pairs of flexiblecentral buckles were the most common type of centralbuckle of which the cross-section areas were the same asthose of the suspenders
Figures 13(a) and 13(b) illustrate bearing displacementdistributions are formed for the three retrofit measures at thesame ground motion intensity levels e mechanical be-haviors of measures strongly affect the effectiveness It can beconcluded that owing to the similar EDP distributions ofdampers and elastic cables the corresponding groundmotion intensities were similar when generating the samepounding probability e flexible central buckles wereextremely susceptible to yielding failure thereby causing lowefficiency in the displacement restrictions ereforepounding probabilities can be directly observed by condi-tioning on the median responses of dampers and elasticcables instead (Figure 13(c)) e control force produced bydevices is another noteworthy consideration in retrofitdesign e more force the devices require the larger andmore expensive they usually are [13] To take both defor-mation and force into account Figures 13(d) and 13(e)provide the contours of the pounding probability ecomparison showed that the damper deformations featuredslightly larger slopes in the horizontal direction yet thedifference in effect on the deformations of the dampers andelastic cables was not significant for reducing the objectivepounding probability Conversely damper force was lesssensitive to the pounding probability than cable force eelastic cable forces reached on average up to e32 (asymp245)times those of the damper under the same poundingprobabilities ese findings implied that difficult anchoragechallenges existed in the case of guaranteeing the efficiencyof elastic cables and the bridge decks possibly experiencedhigher stress In addition the difference in force was sub-stantial when controlling the displacement which showedthat dampers can contribute additional damping and stiff-ness to the structural system while elastic cables completelytransferred the inertial forces erefore the coordinationand balance of displacements and forces should be simul-taneously considered in the vibration mitigation of large-span bridges In this study the application of fluid viscousdampers was identified as the preferred retrofit in thelongitudinal direction for the main bridge e additionaldamping ratio provided by the fluid viscous dampers can becalculated by the following equation
ε CD
2mωeff (11)
where CD is damping coefficient of linear fluid damperskNmiddot(sm) which is equivalent to nonlinear dampers basedon equal energy dissipation per hysteresis loop [17] ωeff issystem natural frequency corresponding to the vibrationalong the axis of bridge andm is system mass It was shownthat an additional damping of 206ndash318 can offer aneffective controlling mechanism for the reduction ofpounding between the adjacent decks
Advances in Civil Engineering 13
Figure 14 depicts the change of components vulnera-bilities before and after the application of dampers It can beobserved that the retrofit measure played limited roles in thefragility reduction of the self-damaged bearings in long-spanbridges Since the additional longitudinal rigidity to girderwas provided by the dampers the transverse responses suchas girder displacement and pylon curvature would increaseas ground motion intensity increased e influences on thecable system responses and the longitudinal responses ofpylons were found to be negligible Another phenomenonwas that the longitudinal fragility of the abutments signif-icantly decreased with the reduction in the poundingprobabilities between the decks and abutments of the ap-proach bridges It was inferred that the pounding betweenthe adjacent decks was likely to affect movement of theapproach bridges and that the pounding effect would betransmitted along the girder
42 Improved Design of Approach Bridges e damagepotential of each component for the approach bridges washigh due to the high collapse risks posed by the usage of thefixed piers e structural designs of small-to-medium-spanhighway bridges in China show the fact that the super-structures are usually placed onto laminated elastomericbearings (LEBs) which tend to come into direct contact withthe bridge girders and substructures without anchoring etypical types of seismic damages of these bridges consisted of
large girder movements bearings sliding failures of shearkeys and expansion joints and abutment cracking duringthe Wenchuan and Yushu earthquake events in ChinaMeanwhile the bridge columns displayed relatively lowdamage ratios under earthquakes [33] e design can bereferred to as a quasi-isolation system which is primarilyrealized by strategically employing bearings sliding as force-limiting connections between girders and substructures Inaddition the isolation strategies of lead-rubber bearings(LRBs) are usually employed in the high-seismicity regionswhich can be taken as a comparison of the quasi-isolationsystem e two restraint systems consist of bearings shearkeys transversely located at the cap beams and restrainercables longitudinally anchored to the abutments (or ped-estals) In order to calculate the seismic responses of theapproach bridge the related experiments were completedand summarized and applied to the force-displacementrelations of the restraint system components
e detailed experiment of LEB consulted previousacademic achievements [34] An example is shown inFigure 15(a) where the feature points marked by rectangularand circular represented initial walking and obvious slidingresponses which illustrated that the mechanical property ofbearings would degrade as the displacements increased afterentering the friction sliding state until it remained nearlyconstant When there was no specific test the horizontalforce Fd in the stable friction stage can be 07 Fmax whereFmax was the corresponding maximum static friction at the
Bear
ing
disp
lace
men
t (m
)
Logarithmic spacendash5
ndash4
ndash3
ndash2
ndash1
0
1
2 4 6PGV (cms)
Case 1Case 2Case 3
(a)Po
undi
ng p
roba
bilit
y
00
02
04
06
08
05 10 15 2000PGV (ms)
Case 1Case 2Case 3
(b)
Poun
ding
pro
babi
lity
00
02
04
06
02 04 06 08 1000Deformation (m)
Case 1Case 2
(c)
ln [f
orce
(kN
)]
5
10
15
15
20
20
25
25
30
3030
35
3535
40
40
4550
7
72
74
76
78
8
033 041 049 057 065025Deformation (m)
(d)
ln [f
orce
(kN
)]5
10
15
15 20
20 25
25
30
3030
35
35
40
40
45
50
102
104
106
108
11
112
114
0155 0205 0255 0305 03550105Deformation (m)
(e)
Figure 13 Choice of the three retrofitmeasures based on deformations and forces (a) EDP-IM relation (b) pounding probability conditioned onIM (c) pounding probability conditioned on deformation contours of pounding probability for (d) dampers and (e) elastic cables
14 Advances in Civil Engineering
initial walking e displacement value Δ2 can be of 250effective shear strain (ESS) e initial stiffness K0 is cal-culated by MOTC [20] erefore the coefficients can becalculated by the following sequence FdK0⟶ Fmax⟶Δ1⟶Δ2 e built analytical relation ofthe LEB is shown in Figure 16(a) which can correctlycapture mechanical behavior under earthquakes A log-normal distribution [8] with a logarithmic standard devi-ation of 01 and a normal distribution with aCOV 016MPa were used to model the randomness offriction coefficient and shear modulus respectively For theirrespective mean values 02 and 12MPa were taken
Figure 15(b) shows a cyclic loading experiment of LRBwhich reveals that the degradation of bearings stiffness willoccur with the increase of loading displacements e elasticstiffness can reach uniformly 3 to 7 times the postyieldingstiffness according to the experimental results e
mechanical behavior of LRB can be modeled by a bilinearrelation as seen in Figure 16(b)
e cable restrainers were modeled as nonlinear tension-only elements characterized by an initial slack as shown inFigure 16(c) e parameters affecting the functionality ofcable restrainers were the cable slack length and yieldstrength according to the research of Padgett and DesRoches[16] e first two parameters were assumed to follow theuniform distributions and the strength can be modeled bythe lognormal distribution An adopted range of cablestiffness values was from 30times105 kNm to 50times105 kNmuniformly
e key parameters in analytical models of Figure 16 canbe determined according to the structural design of therealistic bridge e damage states of two restraint systemsare listed in Table 6 in which ESS is used as the damageindex
Dam
age p
roba
bilit
y
05 10 15 2000PGV (ms)
00
02
04
06
08
10
Case 0Case 1
MBDLMBDT
(a)
Dam
age p
roba
bilit
y
00
01
02
03
04
05
05 10 15 2000PGV (ms)
Case 0Case 1
PCDNLPCDNT
(b)
Dam
age p
roba
bilit
y
00
02
04
06
08
10
05 10 15 2000PGV (ms)
Case 0Case 1
BPDAEJDA
(c)
Figure 14 Influence of dampers on vulnerabilties of (a) bearing responses for DS2 (b) pylon responses for DS2 and (c) abutment andpounding responses
Advances in Civil Engineering 15
Figure 17 illustrates the probabilities of critical damage stateoccurrence sequences of the bridge components in Cases 1 4and 5 when PGV is 10ms which is equal to E2 earthquakeintensity in the seismic assessment reports e componentvulnerabilities of initial design (Case 0) were taken as the ref-erence values which were marked by different colored strips inFigure 17 e desirable sequence preferred Cases 4 or 5 for theapproach bridgese initial design of approach bridges resultedin unreasonable damage sequences In the strategies pertainingto quasi-isolation and isolation systems multiple tiers of seismicstructural redundancy can be normally employed to preventstructural collapse during strong earthquake events Tier 1 in-dicated that the bearings were weakly fused which was evidentin the high fragility of bearing sliding behaviors (ABDAL andABDC2T) Another advantage was that the extensive dam-ages of bearing would be reduced when compared with the self-damage of the pot bearings Tier 2 indicated that the designedbearings had sufficient seat widths for adapting to the sliding ofsuperstructures However the devices had a certain damageprobability (EJDA and SKDC2) and dissipated a portion of
the seismic energy when the displacement of the superstructuresbecame larger than the allowable values of constraint devicesTier 3 redundancy was designed as a potential plastic hinge inthe substructures (CCDC2) including a certain yielding of thebackfill at the abutments (BPDA) e response of theabutments in the transverse direction (PDAT) was slightlyincreased due to the pounding effect between the girders andshear keys caused by bearing sliding e reasonable damagesequence of two restraint systems can be reflected through thecorresponding damage probabilities of bearings seismic mea-sures and substructures respectively as indicated by the red andgreen lines in Figure 17 erefore the rational sequence ofcritical limit state occurrences for the approach bridges isbearings⟶ expansion joints (or restrainer cables) shearkeys⟶ columns abutments
43 Comprehensive Evaluation of Bridge System Padgett andDesRoches [16] pointed out the necessity to derive the bridgesystem vulnerabilities when assessing retrofitted bridges
Figure 15 Experimental results of (a) LEB and (b) LRB
ΔΔ1
F
Δ2
Fmax
Fd
KdK0
(a)
Δy Δ
F
K1 = (3~7) K2
K1
K2
(b)
Slack
F
K
Δ
Fy
(c)
Figure 16 Analytical models of (a) LEB (b) LRB and (c) cable restrainers
16 Advances in Civil Engineering
because different components may have a different responsewith a given retrofit measure Also Zhang and Huo [18]studied the optimal parameters of isolation bearings byderiving the intensity median value corresponding to sys-tem-level fragility function which can effectively avoid theimpacts of different isolation parameters on differentcomponents As can be seen from Figures 14 and 17 theabove analysis results were also very apparent in this studyFor example the longitudinal deck displacement was sig-nificantly reduced by using dampers yet increased thetransverse demands of stiffening girder and pylons e useof LEB or LRB reduced the demands placed on columns yetled to an increase of the transverse deformations of abut-ments which was due to the use of shear keys limiting theexcessive sliding displacement of bearings erefore acomprehensive evaluation of retrofit strategies on bridgesparticularly long-span suspension bridges that have multiplecomponents should be based on system-level fragility in-stead of component-level
Taking the findings that no damages had occurred to thecable system and the definition of bearing damage states intoaccount the derivation of system-level fragility of the wholebridge depended on critical components including pylonscolumns bearings and abutments e fragility data byemploying the absolutely serial and serial-parallel assump-tions of bridge system are plotted in Figure 18 and comparedto the system fragility obtained from the proposed com-posite damage state e absolutely serial assumption
generated serious bias for fragility curves and may result inmisdirection if they were served as a design guide eabsolutely serial model overestimated the potential damageof the bridge system since the fragility curve directlydepended on the most vulnerable components In otherwords if the difference in location and performance of eachcomponent was neglected then the seismic performanceevaluation would be achieved by attaching equal importanceto the strength of pylons approach columns and bearingsis method contradicted the philosophy of capacity designin which the long-span structure can dissipate energythrough a reasonable swinging of stiffening girder and thebearings were designed as fuse elements in the small-to-medium-span bridgese serial-parallel model qualitativelyincorporated the differences of components and was verysensitive to the seismic design of the structure as seen fromthe comparisons among the cases in Figure 18 e modeldistinguished the serial components and parallel compo-nents yet still overestimated the bridge system-level fragilitywhen bearings became the vulnerable components espe-cially for long-span bridges is special situation such asthe serial assumption may also lead to misunderstandingOnce the relative importance between the components wasconsidered in the serial-composite model the damage po-tential was clearly reduced within the range of earthquakeintensity of interest based on the case studies is alsoreflected the necessity of the classification idea of bridgecomponents which can be expressed by the numerical value
Table 6 Damage states of the quasi-isolation and isolation bearings (m)
Figure 17 Probabilities of critical damage states occurrence sequences for the approach bridges when PGV is 1ms
Advances in Civil Engineering 17
of system-level fragility erefore the application ofcomposite damage state (equation (4)) considering com-ponent difference was instead more appropriate than theaforementioned two models
e damage probability distribution represented byblack circles in Figure 18 can be obtained from the definedcomposite damage state (equation (5)) where the parametera is 06 It was found that system fragility result was a moreaccurate assessment with further taking the relative im-portance between themain bridge and approach bridges intoaccount is is due to the principle that the seismic designperformance objective of long-span bridges generally has nocollapse for the main body structures and is necessarily opento emergency vehicle service In addition attention must bepaid to the fact that the differences in repair cost and time ofthe main and approach bridges are significant when thesame damage state is achieved e results of compositedamage state may match the performance objectives of theseismic design In order to investigate the influence ofimproved strategies and structural relative importance ondamage potential of bridges the mean values of the fragility
functions can be employed to evaluate the seismic designe smooth fragility curves for the specific damage states canbe fitted with the lognormal cumulative probability distri-bution function by employing the least-squares technique
P DSgtDSi | PGV1113858 1113859 Φln(pgv) minus λPGV
ζPGV1113888 1113889 (12)
where λPGV and ζPGV are the logarithmic mean and dis-persion values of earthquake intensity (PGV) for a specifieddamage level e larger mean values represent the smallerdamage potential to reach the specified level
Figure 19 indicates the mean values of earthquakeintensity at the specific damage states as a function of theimproved strategies and parameter a Each step of theretrofit measures would enhance the seismic performanceof the overall bridge system at all levels e improvementof the restraint system of the approach bridge played acritical role of the overall bridge system For instance themean values of the earthquake intensity featured mutationof slopes along the measure direction which reflected the
Figure 18 System fragility of the original design (a) Case 0 and improved design including (b) Case 1 and (c) Case 5
18 Advances in Civil Engineering
fact that the effectiveness of improvement for the ap-proach bridges was clearer than that of the main bridgee fragility of the highway network would be under-estimated if only the damage potential of the suspensionspan was adopted erefore when the structures such asapproach bridges transition piers and auxiliary pierswere ignored the damage identification and repair de-cisions would be misleading in the seismic evaluation oflong-span bridges Conversely the damaging probabilityof the overall bridge system will be amplified regardless ofthe higher importance of the main bridge When respectthe fact that the importance of suspension span is higherthan that of approach bridge in transportation networks aparameter a of 06ndash08 may be a better choice in thepresent study
e seismic fragility of conventionally designedbridges was mainly governed by the approach bridges at alldamage levels However for the retrofitted bridges thatpossess reasonable seismic performance different damagestates were controlled by the earthquake demands ofvarious structures as shown by the trend of the curvesalong parameter a in Case 4 and Case 5 e damagingpotential of the overall bridge from the low-to-moderatelimit states was dominated by the approach bridge eprimary reasons were the sliding of bearings and thetransfer of limited forces from the girder to substructure
by the damage of restriction devices of the approachbridges e extensive damage was controlled by thedisplacement of stiffening girder of the main bridge whichinferred the effect of dampers will be conspicuous understrong earthquake shaking At the same time the overalllong-span bridges with the reasonable seismic system willnot collapse during a strong earthquake event and canmeet emergency traffic service It can be seen that the meanvalues of fragility functions developed by compositedamage indices (equations (4) and (5)) are good indicatorsto evaluate seismic performance of long-span bridges andcan be consistent with the established seismic performanceobjectives
5 Conclusions
In the present study a procedure for investigating thereasonable seismic performance of long-span suspensionbridges was developed under the fragility function frame-work e seismic performance of a long-span suspensionbridge was examined and improved using the proposedprocedure from which the results revealed distinguishableconclusions as follows
(1) e definition of damage states for bearings con-sidered the interactions between the adjacent
014036059081103125
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(a)
020051081112142173
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
0002
0406
0810
CaseParameter a
(b)
130251371492612733
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(c)
Mea
n va
lue
exp
(λ) (
ms
)
Parameter a Case0 1 2 3 4 5
0001
0203
0405
131305478652826
1000
(d)
Figure 19 Influence of relative importance and improved designs on mean values of earthquake intensity for different damage states(a) slight (b) moderate (c) extensive (d) collapse
Advances in Civil Engineering 19
structures and self-damage to bearings in accor-dance with the actual design and experimentalconclusions e damage states of pylons in terms ofcurvature ductility were defined by developingmodal pushover analysis and idealized bilinear fit-ting methods in the equivalent pylon fiber models ofthe suspension bridge
(2) When investigating the component fragility curvesof the initial design it was found that the large deckdisplacement of the suspension span was the mostserious damage present and a slight inelastic re-sponse may have occurred only in the pylons Al-though the cable system was able to maintainelasticity under earthquakes the saddles at the py-lons were identified with a high-risk region indi-cating the damaging potential For the approachbridges the ductility mechanism did not realize arational damage sequence e columns were proneto collapse because of the application of fixed potbearings which in turn caused the damage to thebearings and abutments
(3) e stepwise improved seismic designs were pro-posed in terms of the component fragility results ofthe initial design Among the candidates fluid vis-cous dampers were proven to be a good choice forlong-span bridges based on the two attributes ofdisplacement and force when the similar efficiency ofdisplacement control was achieved e restraintsystem had a critical effect on the seismic responsesof the approach bridges e quasi-isolation orisolation can achieve a reasonable seismic damagesequence of the approach bridges
(4) e defined composite damage indices showed thatthe incorporate component classification conceptand structural relative importance were necessary toderive a good indicator of performance evaluationdue to the fact that the damage indices were based oneither an absolutely serial or serial-parallel systemmodel may significantly overestimate the damagepotential of the overall bridge system e meanvalues of earthquake intensity at the specific damagestates can serve as a function of the improvedstrategies and relative importance to further evaluatethe overall seismic performance of the suspensionbridge It can be concluded that the damaging po-tential of the overall bridge from the low-to-mod-erate limit states was governed by the approachbridges in the retrofitted bridges with a favorableseismic performance whereas the displacementdemand of the suspension span dominated thefragility probability of the extensive damage
Data Availability
e nature of the data is the analytical data of finite elementmodels of the case bridge e data used to support thefindings of this study are available from the correspondingauthor upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the Basic Research ServiceProject of Central-Level Public Welfare Research Institute(grant nos 2016-9018 and 2019-0034) and Science andTechnology Project of Communicationsrsquo Construction inWestern China MOC (grant no 2009318223094)
References
[1] L W David A Kartoum K Chang and I Roy ldquoSuspensionreport seismic performance and design considerations of longspan suspension bridgesrdquo Report CSMIP00-03 CaliforniaDepartment of Conservation Division of Mines And GeologyOffice of Strong Motion Studies Sacramento CA USA 2000
[2] S-G Gwon and D-H Choi ldquoImproved continuummodel forfree vibration analysis of suspension bridgesrdquo Journal ofEngineering Mechanics vol 143 no 7 Article ID 040170382017
[3] S Arzoumanidis A Shama and F Ostadan ldquoPerformance-based seismic analysis and design of suspension bridgesrdquoEarthquake Engineering and Structural Dynamics vol 34no 4-5 pp 349ndash367 2005
[4] L Sgambi E Garavaglia N Basso and F Bontempi ldquoMonteCarlo simulation for seismic analysis of a long span sus-pension bridgerdquo Engineering Structures vol 78 pp 100ndash1112014
[5] D Karmakar S Ray-Chaudhuri and M Shinozuka ldquoFiniteelement model development validation and probabilisticseismic performance evaluation of Vincent omas suspen-sion bridgerdquo Structure and Infrastructure Engineering vol 11no 2 pp 223ndash237 2015
[6] W-F Chen and L Duan Bridge Engineering HandbookSeismic Design CRC Press Boca Raton FL USA 2nd edition2014
[7] E Choi ldquoSeismic analysis and retrofit of mid-americabridgesrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2002
[8] B G Nielson and R DesRoches ldquoSeismic fragility method-ology for highway bridges using a component level approachrdquoEarthquake Engineering amp Structural Dynamics vol 36 no 6pp 823ndash839 2007
[9] K N Ramanathan ldquoNext generation seismic fragility curvesfor california bridges incorporating the evolution in seismicdesignrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2012
[10] O Avsar A Yakut and A Caner ldquoAnalytical fragility curvesfor ordinary highway bridges in Turkeyrdquo Earthquake Spectravol 27 no 4 pp 971ndash996 2011
[11] S P Stefanidou and A J Kappos ldquoMethodology for thedevelopment of bridge-specific fragility curvesrdquo EarthquakeEngineering amp Structural Dynamics vol 46 no 1 pp 73ndash932016
[12] B Borzi P Ceresa P Franchin F Noto G M Calvi andP E Pinto ldquoSeismic vulnerability of the Italian roadwaybridge stockrdquo Earthquake Spectra vol 31 no 4pp 2137ndash2161 2015
[13] W T Barnawi and S J Dyke ldquoSeismic fragility relationshipsof a cable-stayed bridge equipped with response modification
20 Advances in Civil Engineering
systemsrdquo Journal of Bridge Engineering vol 19 no 8 ArticleID A4013003 2014
[14] J Zhong Y Pang J Jong-Su J-S Jeon R DesRoches andW Yuan ldquoSeismic fragility assessment of long-span cable-stayed bridges in Chinardquo Advances in Structural Engineeringvol 19 no 11 pp 1797ndash1812 2016
[15] J Zhong Z Hu W Yuan and L Chen ldquoSystem-basedprobabilistic optimization of fluid viscous dampers equippedin cable-stayed bridgesrdquo Advances in Structural Engineeringvol 21 no 12 pp 1815ndash1825 2018
[16] J E Padgett and R DesRoches ldquoMethodology for the de-velopment of analytical fragility curves for retrofitted bridgesfitted bridgesrdquo Earthquake Engineering amp Structural Dy-namics vol 37 no 8 pp 1157ndash1174 2008
[17] Y Xie and J Zhang ldquoOptimal design of seismic protectivedevices for highway bridges using performance-basedmethodology and multiobjective genetic optimizationrdquoJournal of Bridge Engineering vol 22 no 3 Article ID04016129 2017
[18] J Zhang and Y Huo ldquoEvaluating effectiveness and optimumdesign of isolation devices for highway bridges using thefragility function methodrdquo Engineering Structures vol 31no 8 pp 1648ndash1660 2009
[19] F McKenna M H Scott and G L Fenves ldquoNonlinear finite-element analysis software architecture using object compo-sitionrdquo Journal of Computing in Civil Engineering vol 24no 1 pp 95ndash107 2010
[20] MOTC ldquoGuidelines for seismic design of highway bridgesrdquoChina Ministry of Transportation and CommunicationsPress Beijing China JTGT B02-01-2008 2008
[21] A D Kiureghian and J L Sackman ldquoTangent geometricstiffness of inclined cables under self-weightrdquo Journal ofStructural Engineering vol 131 no 6 pp 941ndash945 2005
[22] J S Steelman E T Filipov L A Fahnestock et al ldquoEx-perimental behavior of steel fixed bearings and implicationsfor seismic bridge responserdquo Journal of Bridge Engineeringvol 19 no 8 Article ID A4014007 2014
[23] B Madani F Behnamfar and H Tajmir Riahi ldquoDynamicresponse of structures subjected to pounding and structure-soil-structure interactionrdquo Soil Dynamics and EarthquakeEngineering vol 78 pp 46ndash60 2015
[24] L Xu and J Li ldquoDesign and experimental investigation of anew type sliding retainer and its efficacy in seismic fortifi-cationrdquo Engineering Mechanics vol 33 no 2 pp 111ndash1182016 in Chinese
[25] Caltrans Caltrans Seismic Design Criteria California De-partment of Transportation Sacramento CA USA 2013
[26] A Shamsabadi K M Rollins and M Kapuskar ldquoNonlinearsoil-abutment-bridge structure interaction for seismic per-formance-based designrdquo Journal of Geotechnical and Geo-environmental Engineering vol 133 no 6 pp 707ndash720 2007
[27] X Li Z Sun D Wang and Y Shi ldquoLongitudinal seismicpounding effect of bridges abutment and backfilling damagerdquoJournal of Changrsquoan University vol 35 no 4 pp 76ndash82 2015in Chinese
[28] MOTC ldquoSpecification for design of highway suspensionbridgerdquo China Ministry of Transportation and Communi-cations Press Beijing China JTGT D65-05-2015 2015
[29] A Shafieezadeh K Ramanathan J E Padgett andR DesRoches ldquoFractional order intensity measures forprobabilistic seismic demand modeling applied to highwaybridgesrdquo Earthquake Engineering amp Structural Dynamicsvol 41 no 3 pp 391ndash409 2012
[30] J E Padgett B G Nielson and R DesRoches ldquoSelection ofoptimal intensity measures in probabilistic seismic demandmodels of highway bridge portfoliosrdquo Earthquake Engineeringamp Structural Dynamics vol 37 no 5 pp 711ndash725 2008
[31] K R Mackie and B Stojadinovic ldquoFragility basis for californiahighway overpass bridge seismic decision makingrdquo PEERReport 200502 Pacific Earthquake Engineering ResearchCenter University of California Berkeley Berkeley CA USA2005
[32] J Li T Peng Ji Lin L Zhou and YuL Deng Seismic Design ofMulti-Pylon Multi-Span Suspension Bridge China Commu-nications Press Beijing China 2013 in Chinese
[33] L Chen W Zhuang H Zhao and Z Wan ldquoReport onHighwaysrsquo Damage in the Wenchuan Earthquake-BridgeChina Communications Press Beijing China 2012 inChinese
[34] G Wu K Wang G Lu and P Zhang ldquoAn experimentalinvestigation of unbonded laminated elastomeric bearingsand the seismic evaluations of highway bridges with testedbearing componentsrdquo Shock and Vibration vol 2018 ArticleID 8439321 18 pages 2018
Advances in Civil Engineering 21
Approach bridge
Composite damage state (4)
Main bridge
Composite damagestate (4)
Composite damage state (5)
Bearing 1 Bearing iBearing 2
Bearing 1 Bearing iBearing 2
Bearing 1 Bearing iBearing 2
Bearing 1 Bearing iBearing 2
Column 1
Column 2
Column j
Abutment 1
Abutment 2
Bearing 1 Bearing iBearing 2
Bearing system
Column 1 Column iColumn 2
Column system
Bearing 1 Bearing iBearing 2
Bearing 1 Bearing iBearing 2
Pylon 1
Pylon 2
Bearing system
Pylon 1 Pylon 2
Pylon system
Damage state definitionsand component-level
fragility results
Abutment 1
Abutment 2
Abutmentsystem
Figure 1 Bridge system based on the series-parallel system and composite damage state
C2C1
4000
7000
6000
7500 63
60
14000
3500
30001600063600
73100
3000South
North
Figure 2 Configuration of the Taoyuan Suspension Bridge (cm)
SouthH = 750m
North H = 700m
H
Sec I
Pylon
Pedestal
(a)
Confined(Mander model)
εu εcu
fc
Kfc
ε
σ
Unconfined
Concrete0002
fcufu
εc0
(b)
σy
εy ε
σ0005E0
E0
Bar
(c)
Figure 3 Continued
Advances in Civil Engineering 5
e shear keys contribute to the constraint of transversemovement between the girder and pier Xu and Li [24]proposed a modified model based on the structural prop-erties of shear keys in China which was used to build theforce-displacement relation of shear keys by separating thecontributions of steel and concrete in parallel as shown inFigure 5 e critical parameters of the shear keys can becalculated using Xursquos method according to the actual designwhich represents the yield (Δ1Y) nominal (Δ1n) degraded(Δ1d) and ultimate (Δ1u) deformations respectively esimulation can accurately reflect the mechanical behavior ofthe shear keys
e seat type abutment is shown in Figure 6 e lon-gitudinal resistance to seismic forces is provided jointly bythe passive action of the backfill soil and active action of thepiles [25] Meanwhile the transverse response is resisted bythe pilese passive and active responses of abutments weresimulated using hyperbolic soil material [26] and trilinearforce-displacement relationship respectively which wereconnected in parallel to capture the abutment responses Inaddition the pounding effect between the deck and abut-ment was simulated using a linear model without energydissipation e stiffness of pounding spring between thesuperstructure and abutment was proposed to be ten timesthat of the backfill soil stiffness [27]
32 Uncertainty Treatment in the FE Model e LHS ap-proach is usually used to account for the uncertainties infragility analysis such as structural geometry materialstrength component properties deck mass and damping in
the modeling which is considered to be a variance reductionsampling technique It is noted that the uncertainty ofstructural geometry is not taken into account in this studyfor the development of bridge-specific fragility curves In theforegoing approach the probability distributions were as-sumed for each parameter and each random variable wasdivided into the several equal intervals which correspond to5 to 95 of the cumulative probability e consideringmodeling parameters and their probability distributions arepresented in Table 1 Statistically significant yet nominallyidentical 3D bridges were built using the LHS so as toaccount for the abovementioned uncertainties then thebridges were paired randomly with the selected groundmotions
33 Ground Motion Input and Intensity Measure e ne-cessity to carry out a large number of nonlinear time-historyanalyses depends on a significant number of earthquakerecords selected for the PSDM In this study a data set of 80recorded ground motions and 20 Los Angeles-pertinentunscaled ground motions were extracted from the PEERstrong motion database and the SAC project database re-spectively [29] ese ground motions can represent thecharacteristics of the bridge site ey were placed into fivemagnitude-distance bins for the purpose of covering therational and broad ground motion intensity e first fourbins included combinations of low and high magnitudesand large and small fault distances e magnitudes variedbetween 57 and 70 while the fault distances ranged from130 to 600 km e characteristic of the ground motions in
Stress
Initialstress
K1
Strain
K2 = 0005K1
(a)
K1
Δy = 2mm
Fy
ΔΔy
K2 = 000001K1
F
(b)
Δu
F
Fd
Δy
K
Rh
ΔΔy = 1mm
Δu = 20mm
(c)
K
Gap
F
Δ
(d)
Δy
FΔult
K
Δ
Fult
Fult2
Δy= 154mmΔult = 1504mm
Per width (m)of abutment
(e)
Δ1 = 762hmmΔ2 = 254hmm
h height of backwall
Keff
K2
F2F1
Δ1
K1
F
ΔΔ2
Per pileKeff = 14kNmm
(f)
Figure 4 Constitutive relations of the various components (a) cable and suspender (b) sliding pot bearing (c) fixed pot bearing(d) pounding (e) abutment backfill (f ) abutment pile
Bar (HRB500 376)
z
y
Confined concrete (stirrup HRB400
1256)
Sec I
Unconfined concrete (C55)
ACR 0159
(d)
Sec II
h =
70m
Double-column bent
(e)
Confined concrete(stirrup HRB335
08)
Unconfined concrete (C35)
Bar(HRB335 1163)
z
y
Sec II
ACR 0037
(f )
Figure 3 Sketch and material model of the pylons and bents of the examined suspension bridge
6 Advances in Civil Engineering
the fifth bin was that the fault distance was less than 15 kme ground motions were given in two orthogonal com-ponents and the acceleration values adopted were computedas the geometric mean values (Sa
SaL middot SaT
1113968) e peak
ground acceleration ranged from 0043 g to 112 g Figure 7shows the distributions of peak ground accelerations (PGA)peak ground velocities (PGV) earthquake magnitudes (M)and fault distances (R) of 100 ground motion records as well
as the acceleration spectra with a damping ratio of 002 etwo orthogonal components of the selected earthquakerecords were randomly input along the longitudinal andtransverse directions of the bridge
PGA PGV PGD spectral acceleration or Arias In-tensity of earthquake can be selected as the IM candidateswhich are used to characterize earthquake intensity elinear relations with the interesting EDPs from probabilistic
Δ
V
Δ1n Δ1d
Vcp
Δ
V
Δ1y Δ1u
Vsp
i
j
Gap
Hysteretic property of concrete Hysteretic property of steel
Vcp = 7624kN
Δ1n = 106mmΔ1d = 422mm
Vsp = 10143kN
Δ1y = 11mmΔ1u = 633mm
Figure 5 Analytical model of shear keys
Abutment
Deck
Figure 4 (b)
Figure 4 (d)
Figure 4 (c)Figure 5 Figure 4 (e)
Figure 4 (f)
Figure 6 Pile-bent abutment simulations
Table 1 Uncertainty parameters incorporated in the modeling design
Modeling parameter Probability distribution Mean COV Unit Source
Concrete compressive strength C55 Normal 466 0149 MPa
[20]C35 321 0164
Steel yield strengthHRB500
Lognormal5696
00743 MPaHRB400 4557HRB335 3816
Friction coefficient Sliding pot bearing Uniform 003 033 mdash [14]Fixed pot bearing Lognormal 02 0002 mdash [7]
Deck-abutment 80Deck-shear key Uniform 120 029 mm [16]
Damping ratio Uniform 1 01 mdash [7]Deck mass Normal 002 015 mdash [28]COV coefficient of variation
Advances in Civil Engineering 7
seismic demand analysis in the logarithmic reference framecan be adopted to determine the optimal IM In the previousstudies PGA and spectral acceleration were selected as theoptimal IMs for highway bridges [9 30] when applying themultiple measures containing efficiency practicality profi-ciency and sufficiency Mackie et al [31] suggested both PGAand PGV are good choices of IM in related to the EDPmeasures in highway overpass bridges Zhong et al [14]pointed out that PGA and PGV are themost efficient practicaland proficient IMs for long-span cable-stayed bridges yet PGVtends to be the optimal IM in terms of sufficiency ereforePGV can be utilized as the IM in this study
34 Engineering Demand Parameters e peak demands ofcritical components are adopted as EDPs for the developmentof PSDMs and the choice of optimal IM Table 2 lists thecritical EDPs for the examined bridge which have beenconsidered in the fragility evaluation of the suspension bridge
35 Capacity Limit State Models In the present study acapacity model was used to measure component damagewhich was described by a damage index as a function of the
selected EDPs ese models are usually discrete andcharacterized by the onset of various damage states based onthe experimental data Similarly the uncertainties of ca-pacity models are also expressed by median Sc and dis-persion βc e definition of damage states must be matchedwith postearthquake bridge functionality and repair strat-egies in the bridge seismic fragility assessments Founda-tions decks and anchorages are typically identified asnondamaged components in the simulations e damagestate values of bearings were determined so as to comparewith two factors including the pounding gaps between theadjacent components and seating width which can reflectthe possible pounding and unseating as shown in Figure 8 inaddition to the damages to bearings themselves that is thedeformation capacity of the bearings Table 3 lists the slight(DS1) moderate (DS2) extensive (DS3) and collapsed(DS4) damage states defined in the initial design for potbearings at various locations based on their allowable de-formation and structural dimensions
e columns of conventional highway bridges are oftenforced into the state of nonlinearity under strong earth-quakes of which the criteria of damage states were studied[8 9 16] It is found that there are major differences in
0
10
20
30
40
50N
umbe
r of r
ecor
ds
08
01
03
05
09
00
07
04
02
10
06
PGA (g)
(a)
0
10
20
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Num
ber o
f rec
ords
20 806040 100
180
140
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2000
220
120
PGV (cms)
(b)
20 40 600Fault distance (R) (km)
55
60
65
70
75
Mag
nitu
de (M
)
(c)
Mean
0
1
2
3
4
5
S a (g
)
2 4 6 8 100Period (s)
(d)
Figure 7 Characteristics of the selected ground motions (a) PGA (b) PGV (c) M and R (d) acceleration response spectra
8 Advances in Civil Engineering
geometric sizes material strengths and ACR of sectionsbetween the pylons of suspension bridges and columns ofconventional highway bridges erefore it is necessary todevelop a modal pushover analysis to define the damagestates of critical cross-sections of pylons In the analyticalprocesses of this study the material strain and nonlinearmoment-curvature relationship of critical cross-sectionswere characterized to measure the damage levels of thesuspension bridge pylons
First the reasonable pylon models were required to bedeveloped using fiber beam elements in longitudinal andtransverse directions as detailed in Figure 3 An elastic
spring which was equivalent to the restraint of main cablewas attached to the top of the pylon taking into account theinfluence of the main cable stiffness in the longitudinaldirection models e determining process of the equivalentstiffness can be referred to in the Ernst formula and theresults reported by Kiureghian and Sackman [21]
Second as illustrated in Figure 9 the inertial forcedistributions were back-calculated from the multimodespectrum analysis using a 3D bridge model as shown inequations (6)ndash(8) which included the contributions of highvibration modes and reflected the properties of inertial forcedistributions as follows
DS3
DS4
DeckDeck
Pedestal
PedestalBearing
Bearing
Gap
(a)
DS2
DS4
Gap
Abutmentnonlinearity
Deck
AbutmentBearing
(b)
GapShear key
degradationDS4
DS2
Bearing
(c)
Figure 8 Damage control and unseating prevention damage states for bearing displacement (a) between decks (b) between deck andabutment (c) between deck and shear key
Table 2 e critical EDPs for the examined bridge
Engineering demand parameters Abbreviation DirectionCurvature ductility of pylon cross-section (south north) PCD(S N) Longitudinal and transverseDisplacement of bearings of the main bridge MBD Longitudinal and transverseAxial force relative to yield value of main cables MAF mdashAxial force relative to yield value of suspenders SAF mdashCurvature ductility of column cross-section at columns (C1 C2) CCD(C1 C2) Longitudinal and transverseDisplacement of bearings of the approach bridge at column (C) and abutment (A) ABD(C A) Longitudinal and transverseDeformation of backfill passive pressure at abutment BPDA LongitudinalDeformation of piles at abutment PDA Longitudinal and transverseDeformation of expansion joint between decks(D) and at abutment (A) EJD(D A) LongitudinalDeformation of shear key at columns (C) and at abutments (A) SKD(C A) Transverse
Table 3 Limit state thresholds for pot bearings (m)
DS1 DS2 DS3 DS4
Criteria Allowable slippingdisplacements
Spherical skateboard radiusbody separation of bearings
Possible pounding betweenadjacent decks or deck and pylon Possible unseating
MBDL 025 031 076 1555MBDT 002 031 045 mdash
Criteria Allowable slippingdisplacements
Possible pounding entering of abutmentsnonlinearity or shear key degradation
Spherical skateboard radiusbody separation of bearings Possible unseating
where Sj and cj are the spectral acceleration and participationfactor associated with the jth mode respectively Fij φij andQij represent the seismic load mode displacement and shearforce at the i-node associated with the jth mode respectivelymi Qj and Pi indicate the mass shear force and inertial forceat the i-node respectively andN denotes the number of nodes
Next the inertial forces were loaded to the nodes ofpylon models Both the material and geometric nonlinear-ities were incorporated into the pushover analyses whichwere used to investigate the potential failure mechanismsand assess the ductility capacities of pylons
e seismic designs of pylons can develop inelastic be-haviors which were controlled by the material strain limitsFigure 9 details the curvature distributions along the pylonheights with the increase of δ where δ was the loading dis-placement at the control point It was observed that plastichinge zones typically occur at the bases of pylons and above andbelow the horizontal crossbeams Also plastic hinges mayprogress to the intermediate height in the longitudinal direc-tion e development of plastic hinges at the bases of pylonscan be much severer and also occur earlier than in other zoneserefore the curvatures at the bottoms of pylons may ef-fectively provide measurements of the damages to pylons
e mechanical indices of pylon cross-sections whichcorrespond to the four damage state characteristics
following the damages are detailed in Figure 10 e cur-vature ductility can be easily determined by the pushoveranalysis results in addition to that of the DS2e numericaldata can be transformed to a bilinear curve that has a value ofDS2 based on the equal energy rule per original curveSimilarly another pushover analysis was applied to thedouble-column bend of the approach bridge based on thespecifications [20] e four limit state thresholds of thecross-sections at the bottom of pylon and columns in boththe longitudinal and transverse directions are listed in Ta-ble 4 It was observed that the seismic capacity of longitu-dinal direction was superior to that of transverse directionand a good ductility capacity existed in both directions
In the present study the definitions of damage statethresholds of main cables and suspenders were dependent onthe steel strand characteristics of the strength and deformationrates A total of 85 of the ultimate strength (nominal yieldstrength) was conservatively acted as the failure index Namelythe corresponding ratio of force to yield value was 10 It is acommon viewpoint that the accidental loss of one suspenderwill not cause immediate failures of a bridge with the exceptionof the loss of the main cables erefore DS3 and DS4 wereonly defined based on the classification of suspenders andmaincables respectively e corresponding elastic and ultimatedeformation of abutments and shear keys were respectivelyspecified the thresholds of DS1 and DS2 with the recom-mendations of Caltrans [25] e damage of expansion jointsonly defined as the DS1 due to the lack of a higher-leveldamage modelis was objectively logical since the joints canbe replaced with the new once damaged thereby suggesting therationality for just one threshold of component damage Table 5summarizes the damage state thresholds of the above-mentioned components
e assignments of dispersions of capacity models ofcomponents appeared to be the most difficult tasks for theresearchers According to the previous research studies 035was adopted in the abutment and RC components including
Figure 9 Inertial force and curvature distributions (a) longitudinal (b) transverse
10 Advances in Civil Engineering
pylons and columns [9] e COV of main cables andsuspenders was 01 taking the low variability of highstrength steel strands into account [16] e descriptivedispersions of the remaining components were calculated tobe 025 for the two lower damage states (DS1 and DS2) and047 for the higher damage states (DS3 and DS4) [8]
36 Assessment of Vulnerable Components e componentfragility curves were derived by steps 1ndash3 detailed in Section 2to identify vulnerable components and regions as shown inFigures 11 and 12e pot bearings were shown to be the mostvulnerable components for the main bridge which inferred thecomplete damages to pot bearings themselves and reflected ahigher probability of pounding between the adjacent decks andunseating in the longitudinal direction e transverse re-sponses of bearings did not cause obvious damage compared tothe longitudinal responses e pylons were the primarycomponents for load carrying where inelasticmaterial responsecould potentially occur e transverse vulnerability was moreobvious than the longitudinal vulnerability for the lowerdamage states However the probability of extensive damagewas extremely low or even zeroWith respect to themain cablesand suspenders the increases in their forces which couldpotentially occur during rare earthquakes were confined toabout 15 (for themean values) of the forces resulting from thedead loads as illustrated in Figure 12 which also indicated thatthe maximum values of main cable stress had occurred at thesaddles of pylons and in the side spans e stress values of
suspenders achieved the maximum at the middle and end Insummary the cable system was found to be secured againstdamages in which the stress values were much lower than theyield values even under strong earthquake shaking It wasnoted that the large deflection and cable angle changes betweenthe main span and side spans would lead to the strong shear inthe longitudinal direction between the cables and saddles andthe slipping damages to the saddles at the top of the pylonsduring earthquakes were identified as high-risk
For the approach bridges the columns were determinedto be the most fragile components in both the longitudinaland transverse directions e application of the fixed potbearings resulted in a large force and deformation of thecolumns the complete damage of sliding pot bearings andapparent pounding between the decks and abutments eseactions subsequently caused the inelastic responses ofabutments but the shear keys were protected from failuretransversely Although the ductility mechanism of approachbridges was designed it was considered to be a defectiveseismic system with irrational sequences of componentdamage occurrences in high-seismicity regions
41 ImprovedDesign ofMainBridge As results shown in theevaluation of vulnerable components a large deck dis-placement of the suspension span would occur duringearthquakes which caused pounding and even unseating
Original curveBilinear curve
MprimeyMy
φs or φccφucφprimeyφy
DS4 reinforcing bar rupture core concrete crushing
DS3 major spalling of cover concrete exposed core concrete
DS2 formation of stable plastic hinge in the pylon (column)
Mom
ent
Curvature
DS1 narrow cracking in concrete first yielding of reinforcing bar
Figure 10 e damage state definitions of the pylon and column cross-sections
Table 4 Limit state thresholds of pylon and column cross-sections
Figure 11 Component fragility curves for the suspension bridge (a) main bridge (b) approach bridge
200 400 600 800 10000Location (m)
00
02
04
06
08
10
MA
F
MaximumMinimumMean
Dead loadYield valueHigh stress region
(a)
00
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10
SAF
200 400 600 800 10000Location (m)
MaximumMinimumMean High stress region
Dead loadYield value
(b)
Figure 12 Force distributions of the (a) main cables and (b) suspenders
12 Advances in Civil Engineering
when there were no additional restraints in the longitudinaldirection erefore it was consistent to suggest that sup-plementary devices should be equipped to mitigate the deckmovements to some extente reduction of deck or bearingdisplacements can be taken as the retrofitted objective for thesuspension span Fragility analyses were conducted on theretrofitted bridges that covered the general strategies asfollows
Case 1 the analysis model of the bridge used a fluidviscous damper equipped between the deck and eachpedestal on both sides of the bridgeCase 2 the analysis model of the bridge used an elasticcable installed between the deck and each pedestal onboth sides of the bridgeCase 3 the analysis model of the bridge used three pairsof flexible central buckles located in the middle sectionof the span
e main bridge was assumed to have identical me-chanical parameters at each location in various designschemes e dampers and elastic cables were respectivelymodeled using zero-length elements with Maxwell andtension-only materials and the flexible central buckles weresimulated as the same as the suspenders elements inOpenSEES
e experimental testing of fluid viscous dampers hasshown that the force-velocity with fractional velocity powerlaw relation can describe the suitable mechanical behavior
fD CD| _u|αsgn( _u) (9)
where α is a real positive exponent of which value dependson the piston head orifice CD is viscous damping coefficientwith units of force per velocity raised to the α power kNmiddot(ms)-α _u indicates the velocity of damper and sgn(middot) is thesignum function Research by Li et al [32] indicated thatviscous damping coefficients range from 2500 kN (ms)-α to5000 kN (ms)-α and can be simulated with a uniform dis-tribution e velocity exponents can be assumed to varyuniformly from 03 to 05
e axial stiffness of elastic cables and flexible centralbuckles is regarded with the material length and cross-section area It was assumed that the yield strength of steelstrands of fy 1420MPa was the mean value and followed alognormal distribution with a COV 01 e stiffnessvalues of single elastic cable based on past work employed auniform distribution ranging from 125times105 kNm to25 times105 kNm In addition the other mechanical prop-erties such as length and area can be determined by thefollowing relations
LgtEΔfy
1113888 1113889
kE EA
L1113874 1113875
(10)
where L E and fy are the length Youngrsquos modulus andyield stress of elastic cables respectively and Δ represents the
controlled deck displacement which was available for 06min the examined bridge
According to expert opinion three pairs of flexiblecentral buckles were the most common type of centralbuckle of which the cross-section areas were the same asthose of the suspenders
Figures 13(a) and 13(b) illustrate bearing displacementdistributions are formed for the three retrofit measures at thesame ground motion intensity levels e mechanical be-haviors of measures strongly affect the effectiveness It can beconcluded that owing to the similar EDP distributions ofdampers and elastic cables the corresponding groundmotion intensities were similar when generating the samepounding probability e flexible central buckles wereextremely susceptible to yielding failure thereby causing lowefficiency in the displacement restrictions ereforepounding probabilities can be directly observed by condi-tioning on the median responses of dampers and elasticcables instead (Figure 13(c)) e control force produced bydevices is another noteworthy consideration in retrofitdesign e more force the devices require the larger andmore expensive they usually are [13] To take both defor-mation and force into account Figures 13(d) and 13(e)provide the contours of the pounding probability ecomparison showed that the damper deformations featuredslightly larger slopes in the horizontal direction yet thedifference in effect on the deformations of the dampers andelastic cables was not significant for reducing the objectivepounding probability Conversely damper force was lesssensitive to the pounding probability than cable force eelastic cable forces reached on average up to e32 (asymp245)times those of the damper under the same poundingprobabilities ese findings implied that difficult anchoragechallenges existed in the case of guaranteeing the efficiencyof elastic cables and the bridge decks possibly experiencedhigher stress In addition the difference in force was sub-stantial when controlling the displacement which showedthat dampers can contribute additional damping and stiff-ness to the structural system while elastic cables completelytransferred the inertial forces erefore the coordinationand balance of displacements and forces should be simul-taneously considered in the vibration mitigation of large-span bridges In this study the application of fluid viscousdampers was identified as the preferred retrofit in thelongitudinal direction for the main bridge e additionaldamping ratio provided by the fluid viscous dampers can becalculated by the following equation
ε CD
2mωeff (11)
where CD is damping coefficient of linear fluid damperskNmiddot(sm) which is equivalent to nonlinear dampers basedon equal energy dissipation per hysteresis loop [17] ωeff issystem natural frequency corresponding to the vibrationalong the axis of bridge andm is system mass It was shownthat an additional damping of 206ndash318 can offer aneffective controlling mechanism for the reduction ofpounding between the adjacent decks
Advances in Civil Engineering 13
Figure 14 depicts the change of components vulnera-bilities before and after the application of dampers It can beobserved that the retrofit measure played limited roles in thefragility reduction of the self-damaged bearings in long-spanbridges Since the additional longitudinal rigidity to girderwas provided by the dampers the transverse responses suchas girder displacement and pylon curvature would increaseas ground motion intensity increased e influences on thecable system responses and the longitudinal responses ofpylons were found to be negligible Another phenomenonwas that the longitudinal fragility of the abutments signif-icantly decreased with the reduction in the poundingprobabilities between the decks and abutments of the ap-proach bridges It was inferred that the pounding betweenthe adjacent decks was likely to affect movement of theapproach bridges and that the pounding effect would betransmitted along the girder
42 Improved Design of Approach Bridges e damagepotential of each component for the approach bridges washigh due to the high collapse risks posed by the usage of thefixed piers e structural designs of small-to-medium-spanhighway bridges in China show the fact that the super-structures are usually placed onto laminated elastomericbearings (LEBs) which tend to come into direct contact withthe bridge girders and substructures without anchoring etypical types of seismic damages of these bridges consisted of
large girder movements bearings sliding failures of shearkeys and expansion joints and abutment cracking duringthe Wenchuan and Yushu earthquake events in ChinaMeanwhile the bridge columns displayed relatively lowdamage ratios under earthquakes [33] e design can bereferred to as a quasi-isolation system which is primarilyrealized by strategically employing bearings sliding as force-limiting connections between girders and substructures Inaddition the isolation strategies of lead-rubber bearings(LRBs) are usually employed in the high-seismicity regionswhich can be taken as a comparison of the quasi-isolationsystem e two restraint systems consist of bearings shearkeys transversely located at the cap beams and restrainercables longitudinally anchored to the abutments (or ped-estals) In order to calculate the seismic responses of theapproach bridge the related experiments were completedand summarized and applied to the force-displacementrelations of the restraint system components
e detailed experiment of LEB consulted previousacademic achievements [34] An example is shown inFigure 15(a) where the feature points marked by rectangularand circular represented initial walking and obvious slidingresponses which illustrated that the mechanical property ofbearings would degrade as the displacements increased afterentering the friction sliding state until it remained nearlyconstant When there was no specific test the horizontalforce Fd in the stable friction stage can be 07 Fmax whereFmax was the corresponding maximum static friction at the
Bear
ing
disp
lace
men
t (m
)
Logarithmic spacendash5
ndash4
ndash3
ndash2
ndash1
0
1
2 4 6PGV (cms)
Case 1Case 2Case 3
(a)Po
undi
ng p
roba
bilit
y
00
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05 10 15 2000PGV (ms)
Case 1Case 2Case 3
(b)
Poun
ding
pro
babi
lity
00
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02 04 06 08 1000Deformation (m)
Case 1Case 2
(c)
ln [f
orce
(kN
)]
5
10
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3030
35
3535
40
40
4550
7
72
74
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78
8
033 041 049 057 065025Deformation (m)
(d)
ln [f
orce
(kN
)]5
10
15
15 20
20 25
25
30
3030
35
35
40
40
45
50
102
104
106
108
11
112
114
0155 0205 0255 0305 03550105Deformation (m)
(e)
Figure 13 Choice of the three retrofitmeasures based on deformations and forces (a) EDP-IM relation (b) pounding probability conditioned onIM (c) pounding probability conditioned on deformation contours of pounding probability for (d) dampers and (e) elastic cables
14 Advances in Civil Engineering
initial walking e displacement value Δ2 can be of 250effective shear strain (ESS) e initial stiffness K0 is cal-culated by MOTC [20] erefore the coefficients can becalculated by the following sequence FdK0⟶ Fmax⟶Δ1⟶Δ2 e built analytical relation ofthe LEB is shown in Figure 16(a) which can correctlycapture mechanical behavior under earthquakes A log-normal distribution [8] with a logarithmic standard devi-ation of 01 and a normal distribution with aCOV 016MPa were used to model the randomness offriction coefficient and shear modulus respectively For theirrespective mean values 02 and 12MPa were taken
Figure 15(b) shows a cyclic loading experiment of LRBwhich reveals that the degradation of bearings stiffness willoccur with the increase of loading displacements e elasticstiffness can reach uniformly 3 to 7 times the postyieldingstiffness according to the experimental results e
mechanical behavior of LRB can be modeled by a bilinearrelation as seen in Figure 16(b)
e cable restrainers were modeled as nonlinear tension-only elements characterized by an initial slack as shown inFigure 16(c) e parameters affecting the functionality ofcable restrainers were the cable slack length and yieldstrength according to the research of Padgett and DesRoches[16] e first two parameters were assumed to follow theuniform distributions and the strength can be modeled bythe lognormal distribution An adopted range of cablestiffness values was from 30times105 kNm to 50times105 kNmuniformly
e key parameters in analytical models of Figure 16 canbe determined according to the structural design of therealistic bridge e damage states of two restraint systemsare listed in Table 6 in which ESS is used as the damageindex
Dam
age p
roba
bilit
y
05 10 15 2000PGV (ms)
00
02
04
06
08
10
Case 0Case 1
MBDLMBDT
(a)
Dam
age p
roba
bilit
y
00
01
02
03
04
05
05 10 15 2000PGV (ms)
Case 0Case 1
PCDNLPCDNT
(b)
Dam
age p
roba
bilit
y
00
02
04
06
08
10
05 10 15 2000PGV (ms)
Case 0Case 1
BPDAEJDA
(c)
Figure 14 Influence of dampers on vulnerabilties of (a) bearing responses for DS2 (b) pylon responses for DS2 and (c) abutment andpounding responses
Advances in Civil Engineering 15
Figure 17 illustrates the probabilities of critical damage stateoccurrence sequences of the bridge components in Cases 1 4and 5 when PGV is 10ms which is equal to E2 earthquakeintensity in the seismic assessment reports e componentvulnerabilities of initial design (Case 0) were taken as the ref-erence values which were marked by different colored strips inFigure 17 e desirable sequence preferred Cases 4 or 5 for theapproach bridgese initial design of approach bridges resultedin unreasonable damage sequences In the strategies pertainingto quasi-isolation and isolation systems multiple tiers of seismicstructural redundancy can be normally employed to preventstructural collapse during strong earthquake events Tier 1 in-dicated that the bearings were weakly fused which was evidentin the high fragility of bearing sliding behaviors (ABDAL andABDC2T) Another advantage was that the extensive dam-ages of bearing would be reduced when compared with the self-damage of the pot bearings Tier 2 indicated that the designedbearings had sufficient seat widths for adapting to the sliding ofsuperstructures However the devices had a certain damageprobability (EJDA and SKDC2) and dissipated a portion of
the seismic energy when the displacement of the superstructuresbecame larger than the allowable values of constraint devicesTier 3 redundancy was designed as a potential plastic hinge inthe substructures (CCDC2) including a certain yielding of thebackfill at the abutments (BPDA) e response of theabutments in the transverse direction (PDAT) was slightlyincreased due to the pounding effect between the girders andshear keys caused by bearing sliding e reasonable damagesequence of two restraint systems can be reflected through thecorresponding damage probabilities of bearings seismic mea-sures and substructures respectively as indicated by the red andgreen lines in Figure 17 erefore the rational sequence ofcritical limit state occurrences for the approach bridges isbearings⟶ expansion joints (or restrainer cables) shearkeys⟶ columns abutments
43 Comprehensive Evaluation of Bridge System Padgett andDesRoches [16] pointed out the necessity to derive the bridgesystem vulnerabilities when assessing retrofitted bridges
Figure 15 Experimental results of (a) LEB and (b) LRB
ΔΔ1
F
Δ2
Fmax
Fd
KdK0
(a)
Δy Δ
F
K1 = (3~7) K2
K1
K2
(b)
Slack
F
K
Δ
Fy
(c)
Figure 16 Analytical models of (a) LEB (b) LRB and (c) cable restrainers
16 Advances in Civil Engineering
because different components may have a different responsewith a given retrofit measure Also Zhang and Huo [18]studied the optimal parameters of isolation bearings byderiving the intensity median value corresponding to sys-tem-level fragility function which can effectively avoid theimpacts of different isolation parameters on differentcomponents As can be seen from Figures 14 and 17 theabove analysis results were also very apparent in this studyFor example the longitudinal deck displacement was sig-nificantly reduced by using dampers yet increased thetransverse demands of stiffening girder and pylons e useof LEB or LRB reduced the demands placed on columns yetled to an increase of the transverse deformations of abut-ments which was due to the use of shear keys limiting theexcessive sliding displacement of bearings erefore acomprehensive evaluation of retrofit strategies on bridgesparticularly long-span suspension bridges that have multiplecomponents should be based on system-level fragility in-stead of component-level
Taking the findings that no damages had occurred to thecable system and the definition of bearing damage states intoaccount the derivation of system-level fragility of the wholebridge depended on critical components including pylonscolumns bearings and abutments e fragility data byemploying the absolutely serial and serial-parallel assump-tions of bridge system are plotted in Figure 18 and comparedto the system fragility obtained from the proposed com-posite damage state e absolutely serial assumption
generated serious bias for fragility curves and may result inmisdirection if they were served as a design guide eabsolutely serial model overestimated the potential damageof the bridge system since the fragility curve directlydepended on the most vulnerable components In otherwords if the difference in location and performance of eachcomponent was neglected then the seismic performanceevaluation would be achieved by attaching equal importanceto the strength of pylons approach columns and bearingsis method contradicted the philosophy of capacity designin which the long-span structure can dissipate energythrough a reasonable swinging of stiffening girder and thebearings were designed as fuse elements in the small-to-medium-span bridgese serial-parallel model qualitativelyincorporated the differences of components and was verysensitive to the seismic design of the structure as seen fromthe comparisons among the cases in Figure 18 e modeldistinguished the serial components and parallel compo-nents yet still overestimated the bridge system-level fragilitywhen bearings became the vulnerable components espe-cially for long-span bridges is special situation such asthe serial assumption may also lead to misunderstandingOnce the relative importance between the components wasconsidered in the serial-composite model the damage po-tential was clearly reduced within the range of earthquakeintensity of interest based on the case studies is alsoreflected the necessity of the classification idea of bridgecomponents which can be expressed by the numerical value
Table 6 Damage states of the quasi-isolation and isolation bearings (m)
Figure 17 Probabilities of critical damage states occurrence sequences for the approach bridges when PGV is 1ms
Advances in Civil Engineering 17
of system-level fragility erefore the application ofcomposite damage state (equation (4)) considering com-ponent difference was instead more appropriate than theaforementioned two models
e damage probability distribution represented byblack circles in Figure 18 can be obtained from the definedcomposite damage state (equation (5)) where the parametera is 06 It was found that system fragility result was a moreaccurate assessment with further taking the relative im-portance between themain bridge and approach bridges intoaccount is is due to the principle that the seismic designperformance objective of long-span bridges generally has nocollapse for the main body structures and is necessarily opento emergency vehicle service In addition attention must bepaid to the fact that the differences in repair cost and time ofthe main and approach bridges are significant when thesame damage state is achieved e results of compositedamage state may match the performance objectives of theseismic design In order to investigate the influence ofimproved strategies and structural relative importance ondamage potential of bridges the mean values of the fragility
functions can be employed to evaluate the seismic designe smooth fragility curves for the specific damage states canbe fitted with the lognormal cumulative probability distri-bution function by employing the least-squares technique
P DSgtDSi | PGV1113858 1113859 Φln(pgv) minus λPGV
ζPGV1113888 1113889 (12)
where λPGV and ζPGV are the logarithmic mean and dis-persion values of earthquake intensity (PGV) for a specifieddamage level e larger mean values represent the smallerdamage potential to reach the specified level
Figure 19 indicates the mean values of earthquakeintensity at the specific damage states as a function of theimproved strategies and parameter a Each step of theretrofit measures would enhance the seismic performanceof the overall bridge system at all levels e improvementof the restraint system of the approach bridge played acritical role of the overall bridge system For instance themean values of the earthquake intensity featured mutationof slopes along the measure direction which reflected the
Figure 18 System fragility of the original design (a) Case 0 and improved design including (b) Case 1 and (c) Case 5
18 Advances in Civil Engineering
fact that the effectiveness of improvement for the ap-proach bridges was clearer than that of the main bridgee fragility of the highway network would be under-estimated if only the damage potential of the suspensionspan was adopted erefore when the structures such asapproach bridges transition piers and auxiliary pierswere ignored the damage identification and repair de-cisions would be misleading in the seismic evaluation oflong-span bridges Conversely the damaging probabilityof the overall bridge system will be amplified regardless ofthe higher importance of the main bridge When respectthe fact that the importance of suspension span is higherthan that of approach bridge in transportation networks aparameter a of 06ndash08 may be a better choice in thepresent study
e seismic fragility of conventionally designedbridges was mainly governed by the approach bridges at alldamage levels However for the retrofitted bridges thatpossess reasonable seismic performance different damagestates were controlled by the earthquake demands ofvarious structures as shown by the trend of the curvesalong parameter a in Case 4 and Case 5 e damagingpotential of the overall bridge from the low-to-moderatelimit states was dominated by the approach bridge eprimary reasons were the sliding of bearings and thetransfer of limited forces from the girder to substructure
by the damage of restriction devices of the approachbridges e extensive damage was controlled by thedisplacement of stiffening girder of the main bridge whichinferred the effect of dampers will be conspicuous understrong earthquake shaking At the same time the overalllong-span bridges with the reasonable seismic system willnot collapse during a strong earthquake event and canmeet emergency traffic service It can be seen that the meanvalues of fragility functions developed by compositedamage indices (equations (4) and (5)) are good indicatorsto evaluate seismic performance of long-span bridges andcan be consistent with the established seismic performanceobjectives
5 Conclusions
In the present study a procedure for investigating thereasonable seismic performance of long-span suspensionbridges was developed under the fragility function frame-work e seismic performance of a long-span suspensionbridge was examined and improved using the proposedprocedure from which the results revealed distinguishableconclusions as follows
(1) e definition of damage states for bearings con-sidered the interactions between the adjacent
014036059081103125
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(a)
020051081112142173
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
0002
0406
0810
CaseParameter a
(b)
130251371492612733
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(c)
Mea
n va
lue
exp
(λ) (
ms
)
Parameter a Case0 1 2 3 4 5
0001
0203
0405
131305478652826
1000
(d)
Figure 19 Influence of relative importance and improved designs on mean values of earthquake intensity for different damage states(a) slight (b) moderate (c) extensive (d) collapse
Advances in Civil Engineering 19
structures and self-damage to bearings in accor-dance with the actual design and experimentalconclusions e damage states of pylons in terms ofcurvature ductility were defined by developingmodal pushover analysis and idealized bilinear fit-ting methods in the equivalent pylon fiber models ofthe suspension bridge
(2) When investigating the component fragility curvesof the initial design it was found that the large deckdisplacement of the suspension span was the mostserious damage present and a slight inelastic re-sponse may have occurred only in the pylons Al-though the cable system was able to maintainelasticity under earthquakes the saddles at the py-lons were identified with a high-risk region indi-cating the damaging potential For the approachbridges the ductility mechanism did not realize arational damage sequence e columns were proneto collapse because of the application of fixed potbearings which in turn caused the damage to thebearings and abutments
(3) e stepwise improved seismic designs were pro-posed in terms of the component fragility results ofthe initial design Among the candidates fluid vis-cous dampers were proven to be a good choice forlong-span bridges based on the two attributes ofdisplacement and force when the similar efficiency ofdisplacement control was achieved e restraintsystem had a critical effect on the seismic responsesof the approach bridges e quasi-isolation orisolation can achieve a reasonable seismic damagesequence of the approach bridges
(4) e defined composite damage indices showed thatthe incorporate component classification conceptand structural relative importance were necessary toderive a good indicator of performance evaluationdue to the fact that the damage indices were based oneither an absolutely serial or serial-parallel systemmodel may significantly overestimate the damagepotential of the overall bridge system e meanvalues of earthquake intensity at the specific damagestates can serve as a function of the improvedstrategies and relative importance to further evaluatethe overall seismic performance of the suspensionbridge It can be concluded that the damaging po-tential of the overall bridge from the low-to-mod-erate limit states was governed by the approachbridges in the retrofitted bridges with a favorableseismic performance whereas the displacementdemand of the suspension span dominated thefragility probability of the extensive damage
Data Availability
e nature of the data is the analytical data of finite elementmodels of the case bridge e data used to support thefindings of this study are available from the correspondingauthor upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the Basic Research ServiceProject of Central-Level Public Welfare Research Institute(grant nos 2016-9018 and 2019-0034) and Science andTechnology Project of Communicationsrsquo Construction inWestern China MOC (grant no 2009318223094)
References
[1] L W David A Kartoum K Chang and I Roy ldquoSuspensionreport seismic performance and design considerations of longspan suspension bridgesrdquo Report CSMIP00-03 CaliforniaDepartment of Conservation Division of Mines And GeologyOffice of Strong Motion Studies Sacramento CA USA 2000
[2] S-G Gwon and D-H Choi ldquoImproved continuummodel forfree vibration analysis of suspension bridgesrdquo Journal ofEngineering Mechanics vol 143 no 7 Article ID 040170382017
[3] S Arzoumanidis A Shama and F Ostadan ldquoPerformance-based seismic analysis and design of suspension bridgesrdquoEarthquake Engineering and Structural Dynamics vol 34no 4-5 pp 349ndash367 2005
[4] L Sgambi E Garavaglia N Basso and F Bontempi ldquoMonteCarlo simulation for seismic analysis of a long span sus-pension bridgerdquo Engineering Structures vol 78 pp 100ndash1112014
[5] D Karmakar S Ray-Chaudhuri and M Shinozuka ldquoFiniteelement model development validation and probabilisticseismic performance evaluation of Vincent omas suspen-sion bridgerdquo Structure and Infrastructure Engineering vol 11no 2 pp 223ndash237 2015
[6] W-F Chen and L Duan Bridge Engineering HandbookSeismic Design CRC Press Boca Raton FL USA 2nd edition2014
[7] E Choi ldquoSeismic analysis and retrofit of mid-americabridgesrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2002
[8] B G Nielson and R DesRoches ldquoSeismic fragility method-ology for highway bridges using a component level approachrdquoEarthquake Engineering amp Structural Dynamics vol 36 no 6pp 823ndash839 2007
[9] K N Ramanathan ldquoNext generation seismic fragility curvesfor california bridges incorporating the evolution in seismicdesignrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2012
[10] O Avsar A Yakut and A Caner ldquoAnalytical fragility curvesfor ordinary highway bridges in Turkeyrdquo Earthquake Spectravol 27 no 4 pp 971ndash996 2011
[11] S P Stefanidou and A J Kappos ldquoMethodology for thedevelopment of bridge-specific fragility curvesrdquo EarthquakeEngineering amp Structural Dynamics vol 46 no 1 pp 73ndash932016
[12] B Borzi P Ceresa P Franchin F Noto G M Calvi andP E Pinto ldquoSeismic vulnerability of the Italian roadwaybridge stockrdquo Earthquake Spectra vol 31 no 4pp 2137ndash2161 2015
[13] W T Barnawi and S J Dyke ldquoSeismic fragility relationshipsof a cable-stayed bridge equipped with response modification
20 Advances in Civil Engineering
systemsrdquo Journal of Bridge Engineering vol 19 no 8 ArticleID A4013003 2014
[14] J Zhong Y Pang J Jong-Su J-S Jeon R DesRoches andW Yuan ldquoSeismic fragility assessment of long-span cable-stayed bridges in Chinardquo Advances in Structural Engineeringvol 19 no 11 pp 1797ndash1812 2016
[15] J Zhong Z Hu W Yuan and L Chen ldquoSystem-basedprobabilistic optimization of fluid viscous dampers equippedin cable-stayed bridgesrdquo Advances in Structural Engineeringvol 21 no 12 pp 1815ndash1825 2018
[16] J E Padgett and R DesRoches ldquoMethodology for the de-velopment of analytical fragility curves for retrofitted bridgesfitted bridgesrdquo Earthquake Engineering amp Structural Dy-namics vol 37 no 8 pp 1157ndash1174 2008
[17] Y Xie and J Zhang ldquoOptimal design of seismic protectivedevices for highway bridges using performance-basedmethodology and multiobjective genetic optimizationrdquoJournal of Bridge Engineering vol 22 no 3 Article ID04016129 2017
[18] J Zhang and Y Huo ldquoEvaluating effectiveness and optimumdesign of isolation devices for highway bridges using thefragility function methodrdquo Engineering Structures vol 31no 8 pp 1648ndash1660 2009
[19] F McKenna M H Scott and G L Fenves ldquoNonlinear finite-element analysis software architecture using object compo-sitionrdquo Journal of Computing in Civil Engineering vol 24no 1 pp 95ndash107 2010
[20] MOTC ldquoGuidelines for seismic design of highway bridgesrdquoChina Ministry of Transportation and CommunicationsPress Beijing China JTGT B02-01-2008 2008
[21] A D Kiureghian and J L Sackman ldquoTangent geometricstiffness of inclined cables under self-weightrdquo Journal ofStructural Engineering vol 131 no 6 pp 941ndash945 2005
[22] J S Steelman E T Filipov L A Fahnestock et al ldquoEx-perimental behavior of steel fixed bearings and implicationsfor seismic bridge responserdquo Journal of Bridge Engineeringvol 19 no 8 Article ID A4014007 2014
[23] B Madani F Behnamfar and H Tajmir Riahi ldquoDynamicresponse of structures subjected to pounding and structure-soil-structure interactionrdquo Soil Dynamics and EarthquakeEngineering vol 78 pp 46ndash60 2015
[24] L Xu and J Li ldquoDesign and experimental investigation of anew type sliding retainer and its efficacy in seismic fortifi-cationrdquo Engineering Mechanics vol 33 no 2 pp 111ndash1182016 in Chinese
[25] Caltrans Caltrans Seismic Design Criteria California De-partment of Transportation Sacramento CA USA 2013
[26] A Shamsabadi K M Rollins and M Kapuskar ldquoNonlinearsoil-abutment-bridge structure interaction for seismic per-formance-based designrdquo Journal of Geotechnical and Geo-environmental Engineering vol 133 no 6 pp 707ndash720 2007
[27] X Li Z Sun D Wang and Y Shi ldquoLongitudinal seismicpounding effect of bridges abutment and backfilling damagerdquoJournal of Changrsquoan University vol 35 no 4 pp 76ndash82 2015in Chinese
[28] MOTC ldquoSpecification for design of highway suspensionbridgerdquo China Ministry of Transportation and Communi-cations Press Beijing China JTGT D65-05-2015 2015
[29] A Shafieezadeh K Ramanathan J E Padgett andR DesRoches ldquoFractional order intensity measures forprobabilistic seismic demand modeling applied to highwaybridgesrdquo Earthquake Engineering amp Structural Dynamicsvol 41 no 3 pp 391ndash409 2012
[30] J E Padgett B G Nielson and R DesRoches ldquoSelection ofoptimal intensity measures in probabilistic seismic demandmodels of highway bridge portfoliosrdquo Earthquake Engineeringamp Structural Dynamics vol 37 no 5 pp 711ndash725 2008
[31] K R Mackie and B Stojadinovic ldquoFragility basis for californiahighway overpass bridge seismic decision makingrdquo PEERReport 200502 Pacific Earthquake Engineering ResearchCenter University of California Berkeley Berkeley CA USA2005
[32] J Li T Peng Ji Lin L Zhou and YuL Deng Seismic Design ofMulti-Pylon Multi-Span Suspension Bridge China Commu-nications Press Beijing China 2013 in Chinese
[33] L Chen W Zhuang H Zhao and Z Wan ldquoReport onHighwaysrsquo Damage in the Wenchuan Earthquake-BridgeChina Communications Press Beijing China 2012 inChinese
[34] G Wu K Wang G Lu and P Zhang ldquoAn experimentalinvestigation of unbonded laminated elastomeric bearingsand the seismic evaluations of highway bridges with testedbearing componentsrdquo Shock and Vibration vol 2018 ArticleID 8439321 18 pages 2018
Advances in Civil Engineering 21
e shear keys contribute to the constraint of transversemovement between the girder and pier Xu and Li [24]proposed a modified model based on the structural prop-erties of shear keys in China which was used to build theforce-displacement relation of shear keys by separating thecontributions of steel and concrete in parallel as shown inFigure 5 e critical parameters of the shear keys can becalculated using Xursquos method according to the actual designwhich represents the yield (Δ1Y) nominal (Δ1n) degraded(Δ1d) and ultimate (Δ1u) deformations respectively esimulation can accurately reflect the mechanical behavior ofthe shear keys
e seat type abutment is shown in Figure 6 e lon-gitudinal resistance to seismic forces is provided jointly bythe passive action of the backfill soil and active action of thepiles [25] Meanwhile the transverse response is resisted bythe pilese passive and active responses of abutments weresimulated using hyperbolic soil material [26] and trilinearforce-displacement relationship respectively which wereconnected in parallel to capture the abutment responses Inaddition the pounding effect between the deck and abut-ment was simulated using a linear model without energydissipation e stiffness of pounding spring between thesuperstructure and abutment was proposed to be ten timesthat of the backfill soil stiffness [27]
32 Uncertainty Treatment in the FE Model e LHS ap-proach is usually used to account for the uncertainties infragility analysis such as structural geometry materialstrength component properties deck mass and damping in
the modeling which is considered to be a variance reductionsampling technique It is noted that the uncertainty ofstructural geometry is not taken into account in this studyfor the development of bridge-specific fragility curves In theforegoing approach the probability distributions were as-sumed for each parameter and each random variable wasdivided into the several equal intervals which correspond to5 to 95 of the cumulative probability e consideringmodeling parameters and their probability distributions arepresented in Table 1 Statistically significant yet nominallyidentical 3D bridges were built using the LHS so as toaccount for the abovementioned uncertainties then thebridges were paired randomly with the selected groundmotions
33 Ground Motion Input and Intensity Measure e ne-cessity to carry out a large number of nonlinear time-historyanalyses depends on a significant number of earthquakerecords selected for the PSDM In this study a data set of 80recorded ground motions and 20 Los Angeles-pertinentunscaled ground motions were extracted from the PEERstrong motion database and the SAC project database re-spectively [29] ese ground motions can represent thecharacteristics of the bridge site ey were placed into fivemagnitude-distance bins for the purpose of covering therational and broad ground motion intensity e first fourbins included combinations of low and high magnitudesand large and small fault distances e magnitudes variedbetween 57 and 70 while the fault distances ranged from130 to 600 km e characteristic of the ground motions in
Stress
Initialstress
K1
Strain
K2 = 0005K1
(a)
K1
Δy = 2mm
Fy
ΔΔy
K2 = 000001K1
F
(b)
Δu
F
Fd
Δy
K
Rh
ΔΔy = 1mm
Δu = 20mm
(c)
K
Gap
F
Δ
(d)
Δy
FΔult
K
Δ
Fult
Fult2
Δy= 154mmΔult = 1504mm
Per width (m)of abutment
(e)
Δ1 = 762hmmΔ2 = 254hmm
h height of backwall
Keff
K2
F2F1
Δ1
K1
F
ΔΔ2
Per pileKeff = 14kNmm
(f)
Figure 4 Constitutive relations of the various components (a) cable and suspender (b) sliding pot bearing (c) fixed pot bearing(d) pounding (e) abutment backfill (f ) abutment pile
Bar (HRB500 376)
z
y
Confined concrete (stirrup HRB400
1256)
Sec I
Unconfined concrete (C55)
ACR 0159
(d)
Sec II
h =
70m
Double-column bent
(e)
Confined concrete(stirrup HRB335
08)
Unconfined concrete (C35)
Bar(HRB335 1163)
z
y
Sec II
ACR 0037
(f )
Figure 3 Sketch and material model of the pylons and bents of the examined suspension bridge
6 Advances in Civil Engineering
the fifth bin was that the fault distance was less than 15 kme ground motions were given in two orthogonal com-ponents and the acceleration values adopted were computedas the geometric mean values (Sa
SaL middot SaT
1113968) e peak
ground acceleration ranged from 0043 g to 112 g Figure 7shows the distributions of peak ground accelerations (PGA)peak ground velocities (PGV) earthquake magnitudes (M)and fault distances (R) of 100 ground motion records as well
as the acceleration spectra with a damping ratio of 002 etwo orthogonal components of the selected earthquakerecords were randomly input along the longitudinal andtransverse directions of the bridge
PGA PGV PGD spectral acceleration or Arias In-tensity of earthquake can be selected as the IM candidateswhich are used to characterize earthquake intensity elinear relations with the interesting EDPs from probabilistic
Δ
V
Δ1n Δ1d
Vcp
Δ
V
Δ1y Δ1u
Vsp
i
j
Gap
Hysteretic property of concrete Hysteretic property of steel
Vcp = 7624kN
Δ1n = 106mmΔ1d = 422mm
Vsp = 10143kN
Δ1y = 11mmΔ1u = 633mm
Figure 5 Analytical model of shear keys
Abutment
Deck
Figure 4 (b)
Figure 4 (d)
Figure 4 (c)Figure 5 Figure 4 (e)
Figure 4 (f)
Figure 6 Pile-bent abutment simulations
Table 1 Uncertainty parameters incorporated in the modeling design
Modeling parameter Probability distribution Mean COV Unit Source
Concrete compressive strength C55 Normal 466 0149 MPa
[20]C35 321 0164
Steel yield strengthHRB500
Lognormal5696
00743 MPaHRB400 4557HRB335 3816
Friction coefficient Sliding pot bearing Uniform 003 033 mdash [14]Fixed pot bearing Lognormal 02 0002 mdash [7]
Deck-abutment 80Deck-shear key Uniform 120 029 mm [16]
Damping ratio Uniform 1 01 mdash [7]Deck mass Normal 002 015 mdash [28]COV coefficient of variation
Advances in Civil Engineering 7
seismic demand analysis in the logarithmic reference framecan be adopted to determine the optimal IM In the previousstudies PGA and spectral acceleration were selected as theoptimal IMs for highway bridges [9 30] when applying themultiple measures containing efficiency practicality profi-ciency and sufficiency Mackie et al [31] suggested both PGAand PGV are good choices of IM in related to the EDPmeasures in highway overpass bridges Zhong et al [14]pointed out that PGA and PGV are themost efficient practicaland proficient IMs for long-span cable-stayed bridges yet PGVtends to be the optimal IM in terms of sufficiency ereforePGV can be utilized as the IM in this study
34 Engineering Demand Parameters e peak demands ofcritical components are adopted as EDPs for the developmentof PSDMs and the choice of optimal IM Table 2 lists thecritical EDPs for the examined bridge which have beenconsidered in the fragility evaluation of the suspension bridge
35 Capacity Limit State Models In the present study acapacity model was used to measure component damagewhich was described by a damage index as a function of the
selected EDPs ese models are usually discrete andcharacterized by the onset of various damage states based onthe experimental data Similarly the uncertainties of ca-pacity models are also expressed by median Sc and dis-persion βc e definition of damage states must be matchedwith postearthquake bridge functionality and repair strat-egies in the bridge seismic fragility assessments Founda-tions decks and anchorages are typically identified asnondamaged components in the simulations e damagestate values of bearings were determined so as to comparewith two factors including the pounding gaps between theadjacent components and seating width which can reflectthe possible pounding and unseating as shown in Figure 8 inaddition to the damages to bearings themselves that is thedeformation capacity of the bearings Table 3 lists the slight(DS1) moderate (DS2) extensive (DS3) and collapsed(DS4) damage states defined in the initial design for potbearings at various locations based on their allowable de-formation and structural dimensions
e columns of conventional highway bridges are oftenforced into the state of nonlinearity under strong earth-quakes of which the criteria of damage states were studied[8 9 16] It is found that there are major differences in
0
10
20
30
40
50N
umbe
r of r
ecor
ds
08
01
03
05
09
00
07
04
02
10
06
PGA (g)
(a)
0
10
20
30
40
50
Num
ber o
f rec
ords
20 806040 100
180
140
160
2000
220
120
PGV (cms)
(b)
20 40 600Fault distance (R) (km)
55
60
65
70
75
Mag
nitu
de (M
)
(c)
Mean
0
1
2
3
4
5
S a (g
)
2 4 6 8 100Period (s)
(d)
Figure 7 Characteristics of the selected ground motions (a) PGA (b) PGV (c) M and R (d) acceleration response spectra
8 Advances in Civil Engineering
geometric sizes material strengths and ACR of sectionsbetween the pylons of suspension bridges and columns ofconventional highway bridges erefore it is necessary todevelop a modal pushover analysis to define the damagestates of critical cross-sections of pylons In the analyticalprocesses of this study the material strain and nonlinearmoment-curvature relationship of critical cross-sectionswere characterized to measure the damage levels of thesuspension bridge pylons
First the reasonable pylon models were required to bedeveloped using fiber beam elements in longitudinal andtransverse directions as detailed in Figure 3 An elastic
spring which was equivalent to the restraint of main cablewas attached to the top of the pylon taking into account theinfluence of the main cable stiffness in the longitudinaldirection models e determining process of the equivalentstiffness can be referred to in the Ernst formula and theresults reported by Kiureghian and Sackman [21]
Second as illustrated in Figure 9 the inertial forcedistributions were back-calculated from the multimodespectrum analysis using a 3D bridge model as shown inequations (6)ndash(8) which included the contributions of highvibration modes and reflected the properties of inertial forcedistributions as follows
DS3
DS4
DeckDeck
Pedestal
PedestalBearing
Bearing
Gap
(a)
DS2
DS4
Gap
Abutmentnonlinearity
Deck
AbutmentBearing
(b)
GapShear key
degradationDS4
DS2
Bearing
(c)
Figure 8 Damage control and unseating prevention damage states for bearing displacement (a) between decks (b) between deck andabutment (c) between deck and shear key
Table 2 e critical EDPs for the examined bridge
Engineering demand parameters Abbreviation DirectionCurvature ductility of pylon cross-section (south north) PCD(S N) Longitudinal and transverseDisplacement of bearings of the main bridge MBD Longitudinal and transverseAxial force relative to yield value of main cables MAF mdashAxial force relative to yield value of suspenders SAF mdashCurvature ductility of column cross-section at columns (C1 C2) CCD(C1 C2) Longitudinal and transverseDisplacement of bearings of the approach bridge at column (C) and abutment (A) ABD(C A) Longitudinal and transverseDeformation of backfill passive pressure at abutment BPDA LongitudinalDeformation of piles at abutment PDA Longitudinal and transverseDeformation of expansion joint between decks(D) and at abutment (A) EJD(D A) LongitudinalDeformation of shear key at columns (C) and at abutments (A) SKD(C A) Transverse
Table 3 Limit state thresholds for pot bearings (m)
DS1 DS2 DS3 DS4
Criteria Allowable slippingdisplacements
Spherical skateboard radiusbody separation of bearings
Possible pounding betweenadjacent decks or deck and pylon Possible unseating
MBDL 025 031 076 1555MBDT 002 031 045 mdash
Criteria Allowable slippingdisplacements
Possible pounding entering of abutmentsnonlinearity or shear key degradation
Spherical skateboard radiusbody separation of bearings Possible unseating
where Sj and cj are the spectral acceleration and participationfactor associated with the jth mode respectively Fij φij andQij represent the seismic load mode displacement and shearforce at the i-node associated with the jth mode respectivelymi Qj and Pi indicate the mass shear force and inertial forceat the i-node respectively andN denotes the number of nodes
Next the inertial forces were loaded to the nodes ofpylon models Both the material and geometric nonlinear-ities were incorporated into the pushover analyses whichwere used to investigate the potential failure mechanismsand assess the ductility capacities of pylons
e seismic designs of pylons can develop inelastic be-haviors which were controlled by the material strain limitsFigure 9 details the curvature distributions along the pylonheights with the increase of δ where δ was the loading dis-placement at the control point It was observed that plastichinge zones typically occur at the bases of pylons and above andbelow the horizontal crossbeams Also plastic hinges mayprogress to the intermediate height in the longitudinal direc-tion e development of plastic hinges at the bases of pylonscan be much severer and also occur earlier than in other zoneserefore the curvatures at the bottoms of pylons may ef-fectively provide measurements of the damages to pylons
e mechanical indices of pylon cross-sections whichcorrespond to the four damage state characteristics
following the damages are detailed in Figure 10 e cur-vature ductility can be easily determined by the pushoveranalysis results in addition to that of the DS2e numericaldata can be transformed to a bilinear curve that has a value ofDS2 based on the equal energy rule per original curveSimilarly another pushover analysis was applied to thedouble-column bend of the approach bridge based on thespecifications [20] e four limit state thresholds of thecross-sections at the bottom of pylon and columns in boththe longitudinal and transverse directions are listed in Ta-ble 4 It was observed that the seismic capacity of longitu-dinal direction was superior to that of transverse directionand a good ductility capacity existed in both directions
In the present study the definitions of damage statethresholds of main cables and suspenders were dependent onthe steel strand characteristics of the strength and deformationrates A total of 85 of the ultimate strength (nominal yieldstrength) was conservatively acted as the failure index Namelythe corresponding ratio of force to yield value was 10 It is acommon viewpoint that the accidental loss of one suspenderwill not cause immediate failures of a bridge with the exceptionof the loss of the main cables erefore DS3 and DS4 wereonly defined based on the classification of suspenders andmaincables respectively e corresponding elastic and ultimatedeformation of abutments and shear keys were respectivelyspecified the thresholds of DS1 and DS2 with the recom-mendations of Caltrans [25] e damage of expansion jointsonly defined as the DS1 due to the lack of a higher-leveldamage modelis was objectively logical since the joints canbe replaced with the new once damaged thereby suggesting therationality for just one threshold of component damage Table 5summarizes the damage state thresholds of the above-mentioned components
e assignments of dispersions of capacity models ofcomponents appeared to be the most difficult tasks for theresearchers According to the previous research studies 035was adopted in the abutment and RC components including
Figure 9 Inertial force and curvature distributions (a) longitudinal (b) transverse
10 Advances in Civil Engineering
pylons and columns [9] e COV of main cables andsuspenders was 01 taking the low variability of highstrength steel strands into account [16] e descriptivedispersions of the remaining components were calculated tobe 025 for the two lower damage states (DS1 and DS2) and047 for the higher damage states (DS3 and DS4) [8]
36 Assessment of Vulnerable Components e componentfragility curves were derived by steps 1ndash3 detailed in Section 2to identify vulnerable components and regions as shown inFigures 11 and 12e pot bearings were shown to be the mostvulnerable components for the main bridge which inferred thecomplete damages to pot bearings themselves and reflected ahigher probability of pounding between the adjacent decks andunseating in the longitudinal direction e transverse re-sponses of bearings did not cause obvious damage compared tothe longitudinal responses e pylons were the primarycomponents for load carrying where inelasticmaterial responsecould potentially occur e transverse vulnerability was moreobvious than the longitudinal vulnerability for the lowerdamage states However the probability of extensive damagewas extremely low or even zeroWith respect to themain cablesand suspenders the increases in their forces which couldpotentially occur during rare earthquakes were confined toabout 15 (for themean values) of the forces resulting from thedead loads as illustrated in Figure 12 which also indicated thatthe maximum values of main cable stress had occurred at thesaddles of pylons and in the side spans e stress values of
suspenders achieved the maximum at the middle and end Insummary the cable system was found to be secured againstdamages in which the stress values were much lower than theyield values even under strong earthquake shaking It wasnoted that the large deflection and cable angle changes betweenthe main span and side spans would lead to the strong shear inthe longitudinal direction between the cables and saddles andthe slipping damages to the saddles at the top of the pylonsduring earthquakes were identified as high-risk
For the approach bridges the columns were determinedto be the most fragile components in both the longitudinaland transverse directions e application of the fixed potbearings resulted in a large force and deformation of thecolumns the complete damage of sliding pot bearings andapparent pounding between the decks and abutments eseactions subsequently caused the inelastic responses ofabutments but the shear keys were protected from failuretransversely Although the ductility mechanism of approachbridges was designed it was considered to be a defectiveseismic system with irrational sequences of componentdamage occurrences in high-seismicity regions
41 ImprovedDesign ofMainBridge As results shown in theevaluation of vulnerable components a large deck dis-placement of the suspension span would occur duringearthquakes which caused pounding and even unseating
Original curveBilinear curve
MprimeyMy
φs or φccφucφprimeyφy
DS4 reinforcing bar rupture core concrete crushing
DS3 major spalling of cover concrete exposed core concrete
DS2 formation of stable plastic hinge in the pylon (column)
Mom
ent
Curvature
DS1 narrow cracking in concrete first yielding of reinforcing bar
Figure 10 e damage state definitions of the pylon and column cross-sections
Table 4 Limit state thresholds of pylon and column cross-sections
Figure 11 Component fragility curves for the suspension bridge (a) main bridge (b) approach bridge
200 400 600 800 10000Location (m)
00
02
04
06
08
10
MA
F
MaximumMinimumMean
Dead loadYield valueHigh stress region
(a)
00
02
04
06
08
10
SAF
200 400 600 800 10000Location (m)
MaximumMinimumMean High stress region
Dead loadYield value
(b)
Figure 12 Force distributions of the (a) main cables and (b) suspenders
12 Advances in Civil Engineering
when there were no additional restraints in the longitudinaldirection erefore it was consistent to suggest that sup-plementary devices should be equipped to mitigate the deckmovements to some extente reduction of deck or bearingdisplacements can be taken as the retrofitted objective for thesuspension span Fragility analyses were conducted on theretrofitted bridges that covered the general strategies asfollows
Case 1 the analysis model of the bridge used a fluidviscous damper equipped between the deck and eachpedestal on both sides of the bridgeCase 2 the analysis model of the bridge used an elasticcable installed between the deck and each pedestal onboth sides of the bridgeCase 3 the analysis model of the bridge used three pairsof flexible central buckles located in the middle sectionof the span
e main bridge was assumed to have identical me-chanical parameters at each location in various designschemes e dampers and elastic cables were respectivelymodeled using zero-length elements with Maxwell andtension-only materials and the flexible central buckles weresimulated as the same as the suspenders elements inOpenSEES
e experimental testing of fluid viscous dampers hasshown that the force-velocity with fractional velocity powerlaw relation can describe the suitable mechanical behavior
fD CD| _u|αsgn( _u) (9)
where α is a real positive exponent of which value dependson the piston head orifice CD is viscous damping coefficientwith units of force per velocity raised to the α power kNmiddot(ms)-α _u indicates the velocity of damper and sgn(middot) is thesignum function Research by Li et al [32] indicated thatviscous damping coefficients range from 2500 kN (ms)-α to5000 kN (ms)-α and can be simulated with a uniform dis-tribution e velocity exponents can be assumed to varyuniformly from 03 to 05
e axial stiffness of elastic cables and flexible centralbuckles is regarded with the material length and cross-section area It was assumed that the yield strength of steelstrands of fy 1420MPa was the mean value and followed alognormal distribution with a COV 01 e stiffnessvalues of single elastic cable based on past work employed auniform distribution ranging from 125times105 kNm to25 times105 kNm In addition the other mechanical prop-erties such as length and area can be determined by thefollowing relations
LgtEΔfy
1113888 1113889
kE EA
L1113874 1113875
(10)
where L E and fy are the length Youngrsquos modulus andyield stress of elastic cables respectively and Δ represents the
controlled deck displacement which was available for 06min the examined bridge
According to expert opinion three pairs of flexiblecentral buckles were the most common type of centralbuckle of which the cross-section areas were the same asthose of the suspenders
Figures 13(a) and 13(b) illustrate bearing displacementdistributions are formed for the three retrofit measures at thesame ground motion intensity levels e mechanical be-haviors of measures strongly affect the effectiveness It can beconcluded that owing to the similar EDP distributions ofdampers and elastic cables the corresponding groundmotion intensities were similar when generating the samepounding probability e flexible central buckles wereextremely susceptible to yielding failure thereby causing lowefficiency in the displacement restrictions ereforepounding probabilities can be directly observed by condi-tioning on the median responses of dampers and elasticcables instead (Figure 13(c)) e control force produced bydevices is another noteworthy consideration in retrofitdesign e more force the devices require the larger andmore expensive they usually are [13] To take both defor-mation and force into account Figures 13(d) and 13(e)provide the contours of the pounding probability ecomparison showed that the damper deformations featuredslightly larger slopes in the horizontal direction yet thedifference in effect on the deformations of the dampers andelastic cables was not significant for reducing the objectivepounding probability Conversely damper force was lesssensitive to the pounding probability than cable force eelastic cable forces reached on average up to e32 (asymp245)times those of the damper under the same poundingprobabilities ese findings implied that difficult anchoragechallenges existed in the case of guaranteeing the efficiencyof elastic cables and the bridge decks possibly experiencedhigher stress In addition the difference in force was sub-stantial when controlling the displacement which showedthat dampers can contribute additional damping and stiff-ness to the structural system while elastic cables completelytransferred the inertial forces erefore the coordinationand balance of displacements and forces should be simul-taneously considered in the vibration mitigation of large-span bridges In this study the application of fluid viscousdampers was identified as the preferred retrofit in thelongitudinal direction for the main bridge e additionaldamping ratio provided by the fluid viscous dampers can becalculated by the following equation
ε CD
2mωeff (11)
where CD is damping coefficient of linear fluid damperskNmiddot(sm) which is equivalent to nonlinear dampers basedon equal energy dissipation per hysteresis loop [17] ωeff issystem natural frequency corresponding to the vibrationalong the axis of bridge andm is system mass It was shownthat an additional damping of 206ndash318 can offer aneffective controlling mechanism for the reduction ofpounding between the adjacent decks
Advances in Civil Engineering 13
Figure 14 depicts the change of components vulnera-bilities before and after the application of dampers It can beobserved that the retrofit measure played limited roles in thefragility reduction of the self-damaged bearings in long-spanbridges Since the additional longitudinal rigidity to girderwas provided by the dampers the transverse responses suchas girder displacement and pylon curvature would increaseas ground motion intensity increased e influences on thecable system responses and the longitudinal responses ofpylons were found to be negligible Another phenomenonwas that the longitudinal fragility of the abutments signif-icantly decreased with the reduction in the poundingprobabilities between the decks and abutments of the ap-proach bridges It was inferred that the pounding betweenthe adjacent decks was likely to affect movement of theapproach bridges and that the pounding effect would betransmitted along the girder
42 Improved Design of Approach Bridges e damagepotential of each component for the approach bridges washigh due to the high collapse risks posed by the usage of thefixed piers e structural designs of small-to-medium-spanhighway bridges in China show the fact that the super-structures are usually placed onto laminated elastomericbearings (LEBs) which tend to come into direct contact withthe bridge girders and substructures without anchoring etypical types of seismic damages of these bridges consisted of
large girder movements bearings sliding failures of shearkeys and expansion joints and abutment cracking duringthe Wenchuan and Yushu earthquake events in ChinaMeanwhile the bridge columns displayed relatively lowdamage ratios under earthquakes [33] e design can bereferred to as a quasi-isolation system which is primarilyrealized by strategically employing bearings sliding as force-limiting connections between girders and substructures Inaddition the isolation strategies of lead-rubber bearings(LRBs) are usually employed in the high-seismicity regionswhich can be taken as a comparison of the quasi-isolationsystem e two restraint systems consist of bearings shearkeys transversely located at the cap beams and restrainercables longitudinally anchored to the abutments (or ped-estals) In order to calculate the seismic responses of theapproach bridge the related experiments were completedand summarized and applied to the force-displacementrelations of the restraint system components
e detailed experiment of LEB consulted previousacademic achievements [34] An example is shown inFigure 15(a) where the feature points marked by rectangularand circular represented initial walking and obvious slidingresponses which illustrated that the mechanical property ofbearings would degrade as the displacements increased afterentering the friction sliding state until it remained nearlyconstant When there was no specific test the horizontalforce Fd in the stable friction stage can be 07 Fmax whereFmax was the corresponding maximum static friction at the
Bear
ing
disp
lace
men
t (m
)
Logarithmic spacendash5
ndash4
ndash3
ndash2
ndash1
0
1
2 4 6PGV (cms)
Case 1Case 2Case 3
(a)Po
undi
ng p
roba
bilit
y
00
02
04
06
08
05 10 15 2000PGV (ms)
Case 1Case 2Case 3
(b)
Poun
ding
pro
babi
lity
00
02
04
06
02 04 06 08 1000Deformation (m)
Case 1Case 2
(c)
ln [f
orce
(kN
)]
5
10
15
15
20
20
25
25
30
3030
35
3535
40
40
4550
7
72
74
76
78
8
033 041 049 057 065025Deformation (m)
(d)
ln [f
orce
(kN
)]5
10
15
15 20
20 25
25
30
3030
35
35
40
40
45
50
102
104
106
108
11
112
114
0155 0205 0255 0305 03550105Deformation (m)
(e)
Figure 13 Choice of the three retrofitmeasures based on deformations and forces (a) EDP-IM relation (b) pounding probability conditioned onIM (c) pounding probability conditioned on deformation contours of pounding probability for (d) dampers and (e) elastic cables
14 Advances in Civil Engineering
initial walking e displacement value Δ2 can be of 250effective shear strain (ESS) e initial stiffness K0 is cal-culated by MOTC [20] erefore the coefficients can becalculated by the following sequence FdK0⟶ Fmax⟶Δ1⟶Δ2 e built analytical relation ofthe LEB is shown in Figure 16(a) which can correctlycapture mechanical behavior under earthquakes A log-normal distribution [8] with a logarithmic standard devi-ation of 01 and a normal distribution with aCOV 016MPa were used to model the randomness offriction coefficient and shear modulus respectively For theirrespective mean values 02 and 12MPa were taken
Figure 15(b) shows a cyclic loading experiment of LRBwhich reveals that the degradation of bearings stiffness willoccur with the increase of loading displacements e elasticstiffness can reach uniformly 3 to 7 times the postyieldingstiffness according to the experimental results e
mechanical behavior of LRB can be modeled by a bilinearrelation as seen in Figure 16(b)
e cable restrainers were modeled as nonlinear tension-only elements characterized by an initial slack as shown inFigure 16(c) e parameters affecting the functionality ofcable restrainers were the cable slack length and yieldstrength according to the research of Padgett and DesRoches[16] e first two parameters were assumed to follow theuniform distributions and the strength can be modeled bythe lognormal distribution An adopted range of cablestiffness values was from 30times105 kNm to 50times105 kNmuniformly
e key parameters in analytical models of Figure 16 canbe determined according to the structural design of therealistic bridge e damage states of two restraint systemsare listed in Table 6 in which ESS is used as the damageindex
Dam
age p
roba
bilit
y
05 10 15 2000PGV (ms)
00
02
04
06
08
10
Case 0Case 1
MBDLMBDT
(a)
Dam
age p
roba
bilit
y
00
01
02
03
04
05
05 10 15 2000PGV (ms)
Case 0Case 1
PCDNLPCDNT
(b)
Dam
age p
roba
bilit
y
00
02
04
06
08
10
05 10 15 2000PGV (ms)
Case 0Case 1
BPDAEJDA
(c)
Figure 14 Influence of dampers on vulnerabilties of (a) bearing responses for DS2 (b) pylon responses for DS2 and (c) abutment andpounding responses
Advances in Civil Engineering 15
Figure 17 illustrates the probabilities of critical damage stateoccurrence sequences of the bridge components in Cases 1 4and 5 when PGV is 10ms which is equal to E2 earthquakeintensity in the seismic assessment reports e componentvulnerabilities of initial design (Case 0) were taken as the ref-erence values which were marked by different colored strips inFigure 17 e desirable sequence preferred Cases 4 or 5 for theapproach bridgese initial design of approach bridges resultedin unreasonable damage sequences In the strategies pertainingto quasi-isolation and isolation systems multiple tiers of seismicstructural redundancy can be normally employed to preventstructural collapse during strong earthquake events Tier 1 in-dicated that the bearings were weakly fused which was evidentin the high fragility of bearing sliding behaviors (ABDAL andABDC2T) Another advantage was that the extensive dam-ages of bearing would be reduced when compared with the self-damage of the pot bearings Tier 2 indicated that the designedbearings had sufficient seat widths for adapting to the sliding ofsuperstructures However the devices had a certain damageprobability (EJDA and SKDC2) and dissipated a portion of
the seismic energy when the displacement of the superstructuresbecame larger than the allowable values of constraint devicesTier 3 redundancy was designed as a potential plastic hinge inthe substructures (CCDC2) including a certain yielding of thebackfill at the abutments (BPDA) e response of theabutments in the transverse direction (PDAT) was slightlyincreased due to the pounding effect between the girders andshear keys caused by bearing sliding e reasonable damagesequence of two restraint systems can be reflected through thecorresponding damage probabilities of bearings seismic mea-sures and substructures respectively as indicated by the red andgreen lines in Figure 17 erefore the rational sequence ofcritical limit state occurrences for the approach bridges isbearings⟶ expansion joints (or restrainer cables) shearkeys⟶ columns abutments
43 Comprehensive Evaluation of Bridge System Padgett andDesRoches [16] pointed out the necessity to derive the bridgesystem vulnerabilities when assessing retrofitted bridges
Figure 15 Experimental results of (a) LEB and (b) LRB
ΔΔ1
F
Δ2
Fmax
Fd
KdK0
(a)
Δy Δ
F
K1 = (3~7) K2
K1
K2
(b)
Slack
F
K
Δ
Fy
(c)
Figure 16 Analytical models of (a) LEB (b) LRB and (c) cable restrainers
16 Advances in Civil Engineering
because different components may have a different responsewith a given retrofit measure Also Zhang and Huo [18]studied the optimal parameters of isolation bearings byderiving the intensity median value corresponding to sys-tem-level fragility function which can effectively avoid theimpacts of different isolation parameters on differentcomponents As can be seen from Figures 14 and 17 theabove analysis results were also very apparent in this studyFor example the longitudinal deck displacement was sig-nificantly reduced by using dampers yet increased thetransverse demands of stiffening girder and pylons e useof LEB or LRB reduced the demands placed on columns yetled to an increase of the transverse deformations of abut-ments which was due to the use of shear keys limiting theexcessive sliding displacement of bearings erefore acomprehensive evaluation of retrofit strategies on bridgesparticularly long-span suspension bridges that have multiplecomponents should be based on system-level fragility in-stead of component-level
Taking the findings that no damages had occurred to thecable system and the definition of bearing damage states intoaccount the derivation of system-level fragility of the wholebridge depended on critical components including pylonscolumns bearings and abutments e fragility data byemploying the absolutely serial and serial-parallel assump-tions of bridge system are plotted in Figure 18 and comparedto the system fragility obtained from the proposed com-posite damage state e absolutely serial assumption
generated serious bias for fragility curves and may result inmisdirection if they were served as a design guide eabsolutely serial model overestimated the potential damageof the bridge system since the fragility curve directlydepended on the most vulnerable components In otherwords if the difference in location and performance of eachcomponent was neglected then the seismic performanceevaluation would be achieved by attaching equal importanceto the strength of pylons approach columns and bearingsis method contradicted the philosophy of capacity designin which the long-span structure can dissipate energythrough a reasonable swinging of stiffening girder and thebearings were designed as fuse elements in the small-to-medium-span bridgese serial-parallel model qualitativelyincorporated the differences of components and was verysensitive to the seismic design of the structure as seen fromthe comparisons among the cases in Figure 18 e modeldistinguished the serial components and parallel compo-nents yet still overestimated the bridge system-level fragilitywhen bearings became the vulnerable components espe-cially for long-span bridges is special situation such asthe serial assumption may also lead to misunderstandingOnce the relative importance between the components wasconsidered in the serial-composite model the damage po-tential was clearly reduced within the range of earthquakeintensity of interest based on the case studies is alsoreflected the necessity of the classification idea of bridgecomponents which can be expressed by the numerical value
Table 6 Damage states of the quasi-isolation and isolation bearings (m)
Figure 17 Probabilities of critical damage states occurrence sequences for the approach bridges when PGV is 1ms
Advances in Civil Engineering 17
of system-level fragility erefore the application ofcomposite damage state (equation (4)) considering com-ponent difference was instead more appropriate than theaforementioned two models
e damage probability distribution represented byblack circles in Figure 18 can be obtained from the definedcomposite damage state (equation (5)) where the parametera is 06 It was found that system fragility result was a moreaccurate assessment with further taking the relative im-portance between themain bridge and approach bridges intoaccount is is due to the principle that the seismic designperformance objective of long-span bridges generally has nocollapse for the main body structures and is necessarily opento emergency vehicle service In addition attention must bepaid to the fact that the differences in repair cost and time ofthe main and approach bridges are significant when thesame damage state is achieved e results of compositedamage state may match the performance objectives of theseismic design In order to investigate the influence ofimproved strategies and structural relative importance ondamage potential of bridges the mean values of the fragility
functions can be employed to evaluate the seismic designe smooth fragility curves for the specific damage states canbe fitted with the lognormal cumulative probability distri-bution function by employing the least-squares technique
P DSgtDSi | PGV1113858 1113859 Φln(pgv) minus λPGV
ζPGV1113888 1113889 (12)
where λPGV and ζPGV are the logarithmic mean and dis-persion values of earthquake intensity (PGV) for a specifieddamage level e larger mean values represent the smallerdamage potential to reach the specified level
Figure 19 indicates the mean values of earthquakeintensity at the specific damage states as a function of theimproved strategies and parameter a Each step of theretrofit measures would enhance the seismic performanceof the overall bridge system at all levels e improvementof the restraint system of the approach bridge played acritical role of the overall bridge system For instance themean values of the earthquake intensity featured mutationof slopes along the measure direction which reflected the
Figure 18 System fragility of the original design (a) Case 0 and improved design including (b) Case 1 and (c) Case 5
18 Advances in Civil Engineering
fact that the effectiveness of improvement for the ap-proach bridges was clearer than that of the main bridgee fragility of the highway network would be under-estimated if only the damage potential of the suspensionspan was adopted erefore when the structures such asapproach bridges transition piers and auxiliary pierswere ignored the damage identification and repair de-cisions would be misleading in the seismic evaluation oflong-span bridges Conversely the damaging probabilityof the overall bridge system will be amplified regardless ofthe higher importance of the main bridge When respectthe fact that the importance of suspension span is higherthan that of approach bridge in transportation networks aparameter a of 06ndash08 may be a better choice in thepresent study
e seismic fragility of conventionally designedbridges was mainly governed by the approach bridges at alldamage levels However for the retrofitted bridges thatpossess reasonable seismic performance different damagestates were controlled by the earthquake demands ofvarious structures as shown by the trend of the curvesalong parameter a in Case 4 and Case 5 e damagingpotential of the overall bridge from the low-to-moderatelimit states was dominated by the approach bridge eprimary reasons were the sliding of bearings and thetransfer of limited forces from the girder to substructure
by the damage of restriction devices of the approachbridges e extensive damage was controlled by thedisplacement of stiffening girder of the main bridge whichinferred the effect of dampers will be conspicuous understrong earthquake shaking At the same time the overalllong-span bridges with the reasonable seismic system willnot collapse during a strong earthquake event and canmeet emergency traffic service It can be seen that the meanvalues of fragility functions developed by compositedamage indices (equations (4) and (5)) are good indicatorsto evaluate seismic performance of long-span bridges andcan be consistent with the established seismic performanceobjectives
5 Conclusions
In the present study a procedure for investigating thereasonable seismic performance of long-span suspensionbridges was developed under the fragility function frame-work e seismic performance of a long-span suspensionbridge was examined and improved using the proposedprocedure from which the results revealed distinguishableconclusions as follows
(1) e definition of damage states for bearings con-sidered the interactions between the adjacent
014036059081103125
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(a)
020051081112142173
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
0002
0406
0810
CaseParameter a
(b)
130251371492612733
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(c)
Mea
n va
lue
exp
(λ) (
ms
)
Parameter a Case0 1 2 3 4 5
0001
0203
0405
131305478652826
1000
(d)
Figure 19 Influence of relative importance and improved designs on mean values of earthquake intensity for different damage states(a) slight (b) moderate (c) extensive (d) collapse
Advances in Civil Engineering 19
structures and self-damage to bearings in accor-dance with the actual design and experimentalconclusions e damage states of pylons in terms ofcurvature ductility were defined by developingmodal pushover analysis and idealized bilinear fit-ting methods in the equivalent pylon fiber models ofthe suspension bridge
(2) When investigating the component fragility curvesof the initial design it was found that the large deckdisplacement of the suspension span was the mostserious damage present and a slight inelastic re-sponse may have occurred only in the pylons Al-though the cable system was able to maintainelasticity under earthquakes the saddles at the py-lons were identified with a high-risk region indi-cating the damaging potential For the approachbridges the ductility mechanism did not realize arational damage sequence e columns were proneto collapse because of the application of fixed potbearings which in turn caused the damage to thebearings and abutments
(3) e stepwise improved seismic designs were pro-posed in terms of the component fragility results ofthe initial design Among the candidates fluid vis-cous dampers were proven to be a good choice forlong-span bridges based on the two attributes ofdisplacement and force when the similar efficiency ofdisplacement control was achieved e restraintsystem had a critical effect on the seismic responsesof the approach bridges e quasi-isolation orisolation can achieve a reasonable seismic damagesequence of the approach bridges
(4) e defined composite damage indices showed thatthe incorporate component classification conceptand structural relative importance were necessary toderive a good indicator of performance evaluationdue to the fact that the damage indices were based oneither an absolutely serial or serial-parallel systemmodel may significantly overestimate the damagepotential of the overall bridge system e meanvalues of earthquake intensity at the specific damagestates can serve as a function of the improvedstrategies and relative importance to further evaluatethe overall seismic performance of the suspensionbridge It can be concluded that the damaging po-tential of the overall bridge from the low-to-mod-erate limit states was governed by the approachbridges in the retrofitted bridges with a favorableseismic performance whereas the displacementdemand of the suspension span dominated thefragility probability of the extensive damage
Data Availability
e nature of the data is the analytical data of finite elementmodels of the case bridge e data used to support thefindings of this study are available from the correspondingauthor upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the Basic Research ServiceProject of Central-Level Public Welfare Research Institute(grant nos 2016-9018 and 2019-0034) and Science andTechnology Project of Communicationsrsquo Construction inWestern China MOC (grant no 2009318223094)
References
[1] L W David A Kartoum K Chang and I Roy ldquoSuspensionreport seismic performance and design considerations of longspan suspension bridgesrdquo Report CSMIP00-03 CaliforniaDepartment of Conservation Division of Mines And GeologyOffice of Strong Motion Studies Sacramento CA USA 2000
[2] S-G Gwon and D-H Choi ldquoImproved continuummodel forfree vibration analysis of suspension bridgesrdquo Journal ofEngineering Mechanics vol 143 no 7 Article ID 040170382017
[3] S Arzoumanidis A Shama and F Ostadan ldquoPerformance-based seismic analysis and design of suspension bridgesrdquoEarthquake Engineering and Structural Dynamics vol 34no 4-5 pp 349ndash367 2005
[4] L Sgambi E Garavaglia N Basso and F Bontempi ldquoMonteCarlo simulation for seismic analysis of a long span sus-pension bridgerdquo Engineering Structures vol 78 pp 100ndash1112014
[5] D Karmakar S Ray-Chaudhuri and M Shinozuka ldquoFiniteelement model development validation and probabilisticseismic performance evaluation of Vincent omas suspen-sion bridgerdquo Structure and Infrastructure Engineering vol 11no 2 pp 223ndash237 2015
[6] W-F Chen and L Duan Bridge Engineering HandbookSeismic Design CRC Press Boca Raton FL USA 2nd edition2014
[7] E Choi ldquoSeismic analysis and retrofit of mid-americabridgesrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2002
[8] B G Nielson and R DesRoches ldquoSeismic fragility method-ology for highway bridges using a component level approachrdquoEarthquake Engineering amp Structural Dynamics vol 36 no 6pp 823ndash839 2007
[9] K N Ramanathan ldquoNext generation seismic fragility curvesfor california bridges incorporating the evolution in seismicdesignrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2012
[10] O Avsar A Yakut and A Caner ldquoAnalytical fragility curvesfor ordinary highway bridges in Turkeyrdquo Earthquake Spectravol 27 no 4 pp 971ndash996 2011
[11] S P Stefanidou and A J Kappos ldquoMethodology for thedevelopment of bridge-specific fragility curvesrdquo EarthquakeEngineering amp Structural Dynamics vol 46 no 1 pp 73ndash932016
[12] B Borzi P Ceresa P Franchin F Noto G M Calvi andP E Pinto ldquoSeismic vulnerability of the Italian roadwaybridge stockrdquo Earthquake Spectra vol 31 no 4pp 2137ndash2161 2015
[13] W T Barnawi and S J Dyke ldquoSeismic fragility relationshipsof a cable-stayed bridge equipped with response modification
20 Advances in Civil Engineering
systemsrdquo Journal of Bridge Engineering vol 19 no 8 ArticleID A4013003 2014
[14] J Zhong Y Pang J Jong-Su J-S Jeon R DesRoches andW Yuan ldquoSeismic fragility assessment of long-span cable-stayed bridges in Chinardquo Advances in Structural Engineeringvol 19 no 11 pp 1797ndash1812 2016
[15] J Zhong Z Hu W Yuan and L Chen ldquoSystem-basedprobabilistic optimization of fluid viscous dampers equippedin cable-stayed bridgesrdquo Advances in Structural Engineeringvol 21 no 12 pp 1815ndash1825 2018
[16] J E Padgett and R DesRoches ldquoMethodology for the de-velopment of analytical fragility curves for retrofitted bridgesfitted bridgesrdquo Earthquake Engineering amp Structural Dy-namics vol 37 no 8 pp 1157ndash1174 2008
[17] Y Xie and J Zhang ldquoOptimal design of seismic protectivedevices for highway bridges using performance-basedmethodology and multiobjective genetic optimizationrdquoJournal of Bridge Engineering vol 22 no 3 Article ID04016129 2017
[18] J Zhang and Y Huo ldquoEvaluating effectiveness and optimumdesign of isolation devices for highway bridges using thefragility function methodrdquo Engineering Structures vol 31no 8 pp 1648ndash1660 2009
[19] F McKenna M H Scott and G L Fenves ldquoNonlinear finite-element analysis software architecture using object compo-sitionrdquo Journal of Computing in Civil Engineering vol 24no 1 pp 95ndash107 2010
[20] MOTC ldquoGuidelines for seismic design of highway bridgesrdquoChina Ministry of Transportation and CommunicationsPress Beijing China JTGT B02-01-2008 2008
[21] A D Kiureghian and J L Sackman ldquoTangent geometricstiffness of inclined cables under self-weightrdquo Journal ofStructural Engineering vol 131 no 6 pp 941ndash945 2005
[22] J S Steelman E T Filipov L A Fahnestock et al ldquoEx-perimental behavior of steel fixed bearings and implicationsfor seismic bridge responserdquo Journal of Bridge Engineeringvol 19 no 8 Article ID A4014007 2014
[23] B Madani F Behnamfar and H Tajmir Riahi ldquoDynamicresponse of structures subjected to pounding and structure-soil-structure interactionrdquo Soil Dynamics and EarthquakeEngineering vol 78 pp 46ndash60 2015
[24] L Xu and J Li ldquoDesign and experimental investigation of anew type sliding retainer and its efficacy in seismic fortifi-cationrdquo Engineering Mechanics vol 33 no 2 pp 111ndash1182016 in Chinese
[25] Caltrans Caltrans Seismic Design Criteria California De-partment of Transportation Sacramento CA USA 2013
[26] A Shamsabadi K M Rollins and M Kapuskar ldquoNonlinearsoil-abutment-bridge structure interaction for seismic per-formance-based designrdquo Journal of Geotechnical and Geo-environmental Engineering vol 133 no 6 pp 707ndash720 2007
[27] X Li Z Sun D Wang and Y Shi ldquoLongitudinal seismicpounding effect of bridges abutment and backfilling damagerdquoJournal of Changrsquoan University vol 35 no 4 pp 76ndash82 2015in Chinese
[28] MOTC ldquoSpecification for design of highway suspensionbridgerdquo China Ministry of Transportation and Communi-cations Press Beijing China JTGT D65-05-2015 2015
[29] A Shafieezadeh K Ramanathan J E Padgett andR DesRoches ldquoFractional order intensity measures forprobabilistic seismic demand modeling applied to highwaybridgesrdquo Earthquake Engineering amp Structural Dynamicsvol 41 no 3 pp 391ndash409 2012
[30] J E Padgett B G Nielson and R DesRoches ldquoSelection ofoptimal intensity measures in probabilistic seismic demandmodels of highway bridge portfoliosrdquo Earthquake Engineeringamp Structural Dynamics vol 37 no 5 pp 711ndash725 2008
[31] K R Mackie and B Stojadinovic ldquoFragility basis for californiahighway overpass bridge seismic decision makingrdquo PEERReport 200502 Pacific Earthquake Engineering ResearchCenter University of California Berkeley Berkeley CA USA2005
[32] J Li T Peng Ji Lin L Zhou and YuL Deng Seismic Design ofMulti-Pylon Multi-Span Suspension Bridge China Commu-nications Press Beijing China 2013 in Chinese
[33] L Chen W Zhuang H Zhao and Z Wan ldquoReport onHighwaysrsquo Damage in the Wenchuan Earthquake-BridgeChina Communications Press Beijing China 2012 inChinese
[34] G Wu K Wang G Lu and P Zhang ldquoAn experimentalinvestigation of unbonded laminated elastomeric bearingsand the seismic evaluations of highway bridges with testedbearing componentsrdquo Shock and Vibration vol 2018 ArticleID 8439321 18 pages 2018
Advances in Civil Engineering 21
the fifth bin was that the fault distance was less than 15 kme ground motions were given in two orthogonal com-ponents and the acceleration values adopted were computedas the geometric mean values (Sa
SaL middot SaT
1113968) e peak
ground acceleration ranged from 0043 g to 112 g Figure 7shows the distributions of peak ground accelerations (PGA)peak ground velocities (PGV) earthquake magnitudes (M)and fault distances (R) of 100 ground motion records as well
as the acceleration spectra with a damping ratio of 002 etwo orthogonal components of the selected earthquakerecords were randomly input along the longitudinal andtransverse directions of the bridge
PGA PGV PGD spectral acceleration or Arias In-tensity of earthquake can be selected as the IM candidateswhich are used to characterize earthquake intensity elinear relations with the interesting EDPs from probabilistic
Δ
V
Δ1n Δ1d
Vcp
Δ
V
Δ1y Δ1u
Vsp
i
j
Gap
Hysteretic property of concrete Hysteretic property of steel
Vcp = 7624kN
Δ1n = 106mmΔ1d = 422mm
Vsp = 10143kN
Δ1y = 11mmΔ1u = 633mm
Figure 5 Analytical model of shear keys
Abutment
Deck
Figure 4 (b)
Figure 4 (d)
Figure 4 (c)Figure 5 Figure 4 (e)
Figure 4 (f)
Figure 6 Pile-bent abutment simulations
Table 1 Uncertainty parameters incorporated in the modeling design
Modeling parameter Probability distribution Mean COV Unit Source
Concrete compressive strength C55 Normal 466 0149 MPa
[20]C35 321 0164
Steel yield strengthHRB500
Lognormal5696
00743 MPaHRB400 4557HRB335 3816
Friction coefficient Sliding pot bearing Uniform 003 033 mdash [14]Fixed pot bearing Lognormal 02 0002 mdash [7]
Deck-abutment 80Deck-shear key Uniform 120 029 mm [16]
Damping ratio Uniform 1 01 mdash [7]Deck mass Normal 002 015 mdash [28]COV coefficient of variation
Advances in Civil Engineering 7
seismic demand analysis in the logarithmic reference framecan be adopted to determine the optimal IM In the previousstudies PGA and spectral acceleration were selected as theoptimal IMs for highway bridges [9 30] when applying themultiple measures containing efficiency practicality profi-ciency and sufficiency Mackie et al [31] suggested both PGAand PGV are good choices of IM in related to the EDPmeasures in highway overpass bridges Zhong et al [14]pointed out that PGA and PGV are themost efficient practicaland proficient IMs for long-span cable-stayed bridges yet PGVtends to be the optimal IM in terms of sufficiency ereforePGV can be utilized as the IM in this study
34 Engineering Demand Parameters e peak demands ofcritical components are adopted as EDPs for the developmentof PSDMs and the choice of optimal IM Table 2 lists thecritical EDPs for the examined bridge which have beenconsidered in the fragility evaluation of the suspension bridge
35 Capacity Limit State Models In the present study acapacity model was used to measure component damagewhich was described by a damage index as a function of the
selected EDPs ese models are usually discrete andcharacterized by the onset of various damage states based onthe experimental data Similarly the uncertainties of ca-pacity models are also expressed by median Sc and dis-persion βc e definition of damage states must be matchedwith postearthquake bridge functionality and repair strat-egies in the bridge seismic fragility assessments Founda-tions decks and anchorages are typically identified asnondamaged components in the simulations e damagestate values of bearings were determined so as to comparewith two factors including the pounding gaps between theadjacent components and seating width which can reflectthe possible pounding and unseating as shown in Figure 8 inaddition to the damages to bearings themselves that is thedeformation capacity of the bearings Table 3 lists the slight(DS1) moderate (DS2) extensive (DS3) and collapsed(DS4) damage states defined in the initial design for potbearings at various locations based on their allowable de-formation and structural dimensions
e columns of conventional highway bridges are oftenforced into the state of nonlinearity under strong earth-quakes of which the criteria of damage states were studied[8 9 16] It is found that there are major differences in
0
10
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50N
umbe
r of r
ecor
ds
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ber o
f rec
ords
20 806040 100
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PGV (cms)
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nitu
de (M
)
(c)
Mean
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)
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(d)
Figure 7 Characteristics of the selected ground motions (a) PGA (b) PGV (c) M and R (d) acceleration response spectra
8 Advances in Civil Engineering
geometric sizes material strengths and ACR of sectionsbetween the pylons of suspension bridges and columns ofconventional highway bridges erefore it is necessary todevelop a modal pushover analysis to define the damagestates of critical cross-sections of pylons In the analyticalprocesses of this study the material strain and nonlinearmoment-curvature relationship of critical cross-sectionswere characterized to measure the damage levels of thesuspension bridge pylons
First the reasonable pylon models were required to bedeveloped using fiber beam elements in longitudinal andtransverse directions as detailed in Figure 3 An elastic
spring which was equivalent to the restraint of main cablewas attached to the top of the pylon taking into account theinfluence of the main cable stiffness in the longitudinaldirection models e determining process of the equivalentstiffness can be referred to in the Ernst formula and theresults reported by Kiureghian and Sackman [21]
Second as illustrated in Figure 9 the inertial forcedistributions were back-calculated from the multimodespectrum analysis using a 3D bridge model as shown inequations (6)ndash(8) which included the contributions of highvibration modes and reflected the properties of inertial forcedistributions as follows
DS3
DS4
DeckDeck
Pedestal
PedestalBearing
Bearing
Gap
(a)
DS2
DS4
Gap
Abutmentnonlinearity
Deck
AbutmentBearing
(b)
GapShear key
degradationDS4
DS2
Bearing
(c)
Figure 8 Damage control and unseating prevention damage states for bearing displacement (a) between decks (b) between deck andabutment (c) between deck and shear key
Table 2 e critical EDPs for the examined bridge
Engineering demand parameters Abbreviation DirectionCurvature ductility of pylon cross-section (south north) PCD(S N) Longitudinal and transverseDisplacement of bearings of the main bridge MBD Longitudinal and transverseAxial force relative to yield value of main cables MAF mdashAxial force relative to yield value of suspenders SAF mdashCurvature ductility of column cross-section at columns (C1 C2) CCD(C1 C2) Longitudinal and transverseDisplacement of bearings of the approach bridge at column (C) and abutment (A) ABD(C A) Longitudinal and transverseDeformation of backfill passive pressure at abutment BPDA LongitudinalDeformation of piles at abutment PDA Longitudinal and transverseDeformation of expansion joint between decks(D) and at abutment (A) EJD(D A) LongitudinalDeformation of shear key at columns (C) and at abutments (A) SKD(C A) Transverse
Table 3 Limit state thresholds for pot bearings (m)
DS1 DS2 DS3 DS4
Criteria Allowable slippingdisplacements
Spherical skateboard radiusbody separation of bearings
Possible pounding betweenadjacent decks or deck and pylon Possible unseating
MBDL 025 031 076 1555MBDT 002 031 045 mdash
Criteria Allowable slippingdisplacements
Possible pounding entering of abutmentsnonlinearity or shear key degradation
Spherical skateboard radiusbody separation of bearings Possible unseating
where Sj and cj are the spectral acceleration and participationfactor associated with the jth mode respectively Fij φij andQij represent the seismic load mode displacement and shearforce at the i-node associated with the jth mode respectivelymi Qj and Pi indicate the mass shear force and inertial forceat the i-node respectively andN denotes the number of nodes
Next the inertial forces were loaded to the nodes ofpylon models Both the material and geometric nonlinear-ities were incorporated into the pushover analyses whichwere used to investigate the potential failure mechanismsand assess the ductility capacities of pylons
e seismic designs of pylons can develop inelastic be-haviors which were controlled by the material strain limitsFigure 9 details the curvature distributions along the pylonheights with the increase of δ where δ was the loading dis-placement at the control point It was observed that plastichinge zones typically occur at the bases of pylons and above andbelow the horizontal crossbeams Also plastic hinges mayprogress to the intermediate height in the longitudinal direc-tion e development of plastic hinges at the bases of pylonscan be much severer and also occur earlier than in other zoneserefore the curvatures at the bottoms of pylons may ef-fectively provide measurements of the damages to pylons
e mechanical indices of pylon cross-sections whichcorrespond to the four damage state characteristics
following the damages are detailed in Figure 10 e cur-vature ductility can be easily determined by the pushoveranalysis results in addition to that of the DS2e numericaldata can be transformed to a bilinear curve that has a value ofDS2 based on the equal energy rule per original curveSimilarly another pushover analysis was applied to thedouble-column bend of the approach bridge based on thespecifications [20] e four limit state thresholds of thecross-sections at the bottom of pylon and columns in boththe longitudinal and transverse directions are listed in Ta-ble 4 It was observed that the seismic capacity of longitu-dinal direction was superior to that of transverse directionand a good ductility capacity existed in both directions
In the present study the definitions of damage statethresholds of main cables and suspenders were dependent onthe steel strand characteristics of the strength and deformationrates A total of 85 of the ultimate strength (nominal yieldstrength) was conservatively acted as the failure index Namelythe corresponding ratio of force to yield value was 10 It is acommon viewpoint that the accidental loss of one suspenderwill not cause immediate failures of a bridge with the exceptionof the loss of the main cables erefore DS3 and DS4 wereonly defined based on the classification of suspenders andmaincables respectively e corresponding elastic and ultimatedeformation of abutments and shear keys were respectivelyspecified the thresholds of DS1 and DS2 with the recom-mendations of Caltrans [25] e damage of expansion jointsonly defined as the DS1 due to the lack of a higher-leveldamage modelis was objectively logical since the joints canbe replaced with the new once damaged thereby suggesting therationality for just one threshold of component damage Table 5summarizes the damage state thresholds of the above-mentioned components
e assignments of dispersions of capacity models ofcomponents appeared to be the most difficult tasks for theresearchers According to the previous research studies 035was adopted in the abutment and RC components including
Figure 9 Inertial force and curvature distributions (a) longitudinal (b) transverse
10 Advances in Civil Engineering
pylons and columns [9] e COV of main cables andsuspenders was 01 taking the low variability of highstrength steel strands into account [16] e descriptivedispersions of the remaining components were calculated tobe 025 for the two lower damage states (DS1 and DS2) and047 for the higher damage states (DS3 and DS4) [8]
36 Assessment of Vulnerable Components e componentfragility curves were derived by steps 1ndash3 detailed in Section 2to identify vulnerable components and regions as shown inFigures 11 and 12e pot bearings were shown to be the mostvulnerable components for the main bridge which inferred thecomplete damages to pot bearings themselves and reflected ahigher probability of pounding between the adjacent decks andunseating in the longitudinal direction e transverse re-sponses of bearings did not cause obvious damage compared tothe longitudinal responses e pylons were the primarycomponents for load carrying where inelasticmaterial responsecould potentially occur e transverse vulnerability was moreobvious than the longitudinal vulnerability for the lowerdamage states However the probability of extensive damagewas extremely low or even zeroWith respect to themain cablesand suspenders the increases in their forces which couldpotentially occur during rare earthquakes were confined toabout 15 (for themean values) of the forces resulting from thedead loads as illustrated in Figure 12 which also indicated thatthe maximum values of main cable stress had occurred at thesaddles of pylons and in the side spans e stress values of
suspenders achieved the maximum at the middle and end Insummary the cable system was found to be secured againstdamages in which the stress values were much lower than theyield values even under strong earthquake shaking It wasnoted that the large deflection and cable angle changes betweenthe main span and side spans would lead to the strong shear inthe longitudinal direction between the cables and saddles andthe slipping damages to the saddles at the top of the pylonsduring earthquakes were identified as high-risk
For the approach bridges the columns were determinedto be the most fragile components in both the longitudinaland transverse directions e application of the fixed potbearings resulted in a large force and deformation of thecolumns the complete damage of sliding pot bearings andapparent pounding between the decks and abutments eseactions subsequently caused the inelastic responses ofabutments but the shear keys were protected from failuretransversely Although the ductility mechanism of approachbridges was designed it was considered to be a defectiveseismic system with irrational sequences of componentdamage occurrences in high-seismicity regions
41 ImprovedDesign ofMainBridge As results shown in theevaluation of vulnerable components a large deck dis-placement of the suspension span would occur duringearthquakes which caused pounding and even unseating
Original curveBilinear curve
MprimeyMy
φs or φccφucφprimeyφy
DS4 reinforcing bar rupture core concrete crushing
DS3 major spalling of cover concrete exposed core concrete
DS2 formation of stable plastic hinge in the pylon (column)
Mom
ent
Curvature
DS1 narrow cracking in concrete first yielding of reinforcing bar
Figure 10 e damage state definitions of the pylon and column cross-sections
Table 4 Limit state thresholds of pylon and column cross-sections
Figure 11 Component fragility curves for the suspension bridge (a) main bridge (b) approach bridge
200 400 600 800 10000Location (m)
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F
MaximumMinimumMean
Dead loadYield valueHigh stress region
(a)
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MaximumMinimumMean High stress region
Dead loadYield value
(b)
Figure 12 Force distributions of the (a) main cables and (b) suspenders
12 Advances in Civil Engineering
when there were no additional restraints in the longitudinaldirection erefore it was consistent to suggest that sup-plementary devices should be equipped to mitigate the deckmovements to some extente reduction of deck or bearingdisplacements can be taken as the retrofitted objective for thesuspension span Fragility analyses were conducted on theretrofitted bridges that covered the general strategies asfollows
Case 1 the analysis model of the bridge used a fluidviscous damper equipped between the deck and eachpedestal on both sides of the bridgeCase 2 the analysis model of the bridge used an elasticcable installed between the deck and each pedestal onboth sides of the bridgeCase 3 the analysis model of the bridge used three pairsof flexible central buckles located in the middle sectionof the span
e main bridge was assumed to have identical me-chanical parameters at each location in various designschemes e dampers and elastic cables were respectivelymodeled using zero-length elements with Maxwell andtension-only materials and the flexible central buckles weresimulated as the same as the suspenders elements inOpenSEES
e experimental testing of fluid viscous dampers hasshown that the force-velocity with fractional velocity powerlaw relation can describe the suitable mechanical behavior
fD CD| _u|αsgn( _u) (9)
where α is a real positive exponent of which value dependson the piston head orifice CD is viscous damping coefficientwith units of force per velocity raised to the α power kNmiddot(ms)-α _u indicates the velocity of damper and sgn(middot) is thesignum function Research by Li et al [32] indicated thatviscous damping coefficients range from 2500 kN (ms)-α to5000 kN (ms)-α and can be simulated with a uniform dis-tribution e velocity exponents can be assumed to varyuniformly from 03 to 05
e axial stiffness of elastic cables and flexible centralbuckles is regarded with the material length and cross-section area It was assumed that the yield strength of steelstrands of fy 1420MPa was the mean value and followed alognormal distribution with a COV 01 e stiffnessvalues of single elastic cable based on past work employed auniform distribution ranging from 125times105 kNm to25 times105 kNm In addition the other mechanical prop-erties such as length and area can be determined by thefollowing relations
LgtEΔfy
1113888 1113889
kE EA
L1113874 1113875
(10)
where L E and fy are the length Youngrsquos modulus andyield stress of elastic cables respectively and Δ represents the
controlled deck displacement which was available for 06min the examined bridge
According to expert opinion three pairs of flexiblecentral buckles were the most common type of centralbuckle of which the cross-section areas were the same asthose of the suspenders
Figures 13(a) and 13(b) illustrate bearing displacementdistributions are formed for the three retrofit measures at thesame ground motion intensity levels e mechanical be-haviors of measures strongly affect the effectiveness It can beconcluded that owing to the similar EDP distributions ofdampers and elastic cables the corresponding groundmotion intensities were similar when generating the samepounding probability e flexible central buckles wereextremely susceptible to yielding failure thereby causing lowefficiency in the displacement restrictions ereforepounding probabilities can be directly observed by condi-tioning on the median responses of dampers and elasticcables instead (Figure 13(c)) e control force produced bydevices is another noteworthy consideration in retrofitdesign e more force the devices require the larger andmore expensive they usually are [13] To take both defor-mation and force into account Figures 13(d) and 13(e)provide the contours of the pounding probability ecomparison showed that the damper deformations featuredslightly larger slopes in the horizontal direction yet thedifference in effect on the deformations of the dampers andelastic cables was not significant for reducing the objectivepounding probability Conversely damper force was lesssensitive to the pounding probability than cable force eelastic cable forces reached on average up to e32 (asymp245)times those of the damper under the same poundingprobabilities ese findings implied that difficult anchoragechallenges existed in the case of guaranteeing the efficiencyof elastic cables and the bridge decks possibly experiencedhigher stress In addition the difference in force was sub-stantial when controlling the displacement which showedthat dampers can contribute additional damping and stiff-ness to the structural system while elastic cables completelytransferred the inertial forces erefore the coordinationand balance of displacements and forces should be simul-taneously considered in the vibration mitigation of large-span bridges In this study the application of fluid viscousdampers was identified as the preferred retrofit in thelongitudinal direction for the main bridge e additionaldamping ratio provided by the fluid viscous dampers can becalculated by the following equation
ε CD
2mωeff (11)
where CD is damping coefficient of linear fluid damperskNmiddot(sm) which is equivalent to nonlinear dampers basedon equal energy dissipation per hysteresis loop [17] ωeff issystem natural frequency corresponding to the vibrationalong the axis of bridge andm is system mass It was shownthat an additional damping of 206ndash318 can offer aneffective controlling mechanism for the reduction ofpounding between the adjacent decks
Advances in Civil Engineering 13
Figure 14 depicts the change of components vulnera-bilities before and after the application of dampers It can beobserved that the retrofit measure played limited roles in thefragility reduction of the self-damaged bearings in long-spanbridges Since the additional longitudinal rigidity to girderwas provided by the dampers the transverse responses suchas girder displacement and pylon curvature would increaseas ground motion intensity increased e influences on thecable system responses and the longitudinal responses ofpylons were found to be negligible Another phenomenonwas that the longitudinal fragility of the abutments signif-icantly decreased with the reduction in the poundingprobabilities between the decks and abutments of the ap-proach bridges It was inferred that the pounding betweenthe adjacent decks was likely to affect movement of theapproach bridges and that the pounding effect would betransmitted along the girder
42 Improved Design of Approach Bridges e damagepotential of each component for the approach bridges washigh due to the high collapse risks posed by the usage of thefixed piers e structural designs of small-to-medium-spanhighway bridges in China show the fact that the super-structures are usually placed onto laminated elastomericbearings (LEBs) which tend to come into direct contact withthe bridge girders and substructures without anchoring etypical types of seismic damages of these bridges consisted of
large girder movements bearings sliding failures of shearkeys and expansion joints and abutment cracking duringthe Wenchuan and Yushu earthquake events in ChinaMeanwhile the bridge columns displayed relatively lowdamage ratios under earthquakes [33] e design can bereferred to as a quasi-isolation system which is primarilyrealized by strategically employing bearings sliding as force-limiting connections between girders and substructures Inaddition the isolation strategies of lead-rubber bearings(LRBs) are usually employed in the high-seismicity regionswhich can be taken as a comparison of the quasi-isolationsystem e two restraint systems consist of bearings shearkeys transversely located at the cap beams and restrainercables longitudinally anchored to the abutments (or ped-estals) In order to calculate the seismic responses of theapproach bridge the related experiments were completedand summarized and applied to the force-displacementrelations of the restraint system components
e detailed experiment of LEB consulted previousacademic achievements [34] An example is shown inFigure 15(a) where the feature points marked by rectangularand circular represented initial walking and obvious slidingresponses which illustrated that the mechanical property ofbearings would degrade as the displacements increased afterentering the friction sliding state until it remained nearlyconstant When there was no specific test the horizontalforce Fd in the stable friction stage can be 07 Fmax whereFmax was the corresponding maximum static friction at the
Bear
ing
disp
lace
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t (m
)
Logarithmic spacendash5
ndash4
ndash3
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ndash1
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Case 1Case 2Case 3
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undi
ng p
roba
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Case 1Case 2Case 3
(b)
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Case 1Case 2
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)]
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033 041 049 057 065025Deformation (m)
(d)
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orce
(kN
)]5
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35
35
40
40
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50
102
104
106
108
11
112
114
0155 0205 0255 0305 03550105Deformation (m)
(e)
Figure 13 Choice of the three retrofitmeasures based on deformations and forces (a) EDP-IM relation (b) pounding probability conditioned onIM (c) pounding probability conditioned on deformation contours of pounding probability for (d) dampers and (e) elastic cables
14 Advances in Civil Engineering
initial walking e displacement value Δ2 can be of 250effective shear strain (ESS) e initial stiffness K0 is cal-culated by MOTC [20] erefore the coefficients can becalculated by the following sequence FdK0⟶ Fmax⟶Δ1⟶Δ2 e built analytical relation ofthe LEB is shown in Figure 16(a) which can correctlycapture mechanical behavior under earthquakes A log-normal distribution [8] with a logarithmic standard devi-ation of 01 and a normal distribution with aCOV 016MPa were used to model the randomness offriction coefficient and shear modulus respectively For theirrespective mean values 02 and 12MPa were taken
Figure 15(b) shows a cyclic loading experiment of LRBwhich reveals that the degradation of bearings stiffness willoccur with the increase of loading displacements e elasticstiffness can reach uniformly 3 to 7 times the postyieldingstiffness according to the experimental results e
mechanical behavior of LRB can be modeled by a bilinearrelation as seen in Figure 16(b)
e cable restrainers were modeled as nonlinear tension-only elements characterized by an initial slack as shown inFigure 16(c) e parameters affecting the functionality ofcable restrainers were the cable slack length and yieldstrength according to the research of Padgett and DesRoches[16] e first two parameters were assumed to follow theuniform distributions and the strength can be modeled bythe lognormal distribution An adopted range of cablestiffness values was from 30times105 kNm to 50times105 kNmuniformly
e key parameters in analytical models of Figure 16 canbe determined according to the structural design of therealistic bridge e damage states of two restraint systemsare listed in Table 6 in which ESS is used as the damageindex
Dam
age p
roba
bilit
y
05 10 15 2000PGV (ms)
00
02
04
06
08
10
Case 0Case 1
MBDLMBDT
(a)
Dam
age p
roba
bilit
y
00
01
02
03
04
05
05 10 15 2000PGV (ms)
Case 0Case 1
PCDNLPCDNT
(b)
Dam
age p
roba
bilit
y
00
02
04
06
08
10
05 10 15 2000PGV (ms)
Case 0Case 1
BPDAEJDA
(c)
Figure 14 Influence of dampers on vulnerabilties of (a) bearing responses for DS2 (b) pylon responses for DS2 and (c) abutment andpounding responses
Advances in Civil Engineering 15
Figure 17 illustrates the probabilities of critical damage stateoccurrence sequences of the bridge components in Cases 1 4and 5 when PGV is 10ms which is equal to E2 earthquakeintensity in the seismic assessment reports e componentvulnerabilities of initial design (Case 0) were taken as the ref-erence values which were marked by different colored strips inFigure 17 e desirable sequence preferred Cases 4 or 5 for theapproach bridgese initial design of approach bridges resultedin unreasonable damage sequences In the strategies pertainingto quasi-isolation and isolation systems multiple tiers of seismicstructural redundancy can be normally employed to preventstructural collapse during strong earthquake events Tier 1 in-dicated that the bearings were weakly fused which was evidentin the high fragility of bearing sliding behaviors (ABDAL andABDC2T) Another advantage was that the extensive dam-ages of bearing would be reduced when compared with the self-damage of the pot bearings Tier 2 indicated that the designedbearings had sufficient seat widths for adapting to the sliding ofsuperstructures However the devices had a certain damageprobability (EJDA and SKDC2) and dissipated a portion of
the seismic energy when the displacement of the superstructuresbecame larger than the allowable values of constraint devicesTier 3 redundancy was designed as a potential plastic hinge inthe substructures (CCDC2) including a certain yielding of thebackfill at the abutments (BPDA) e response of theabutments in the transverse direction (PDAT) was slightlyincreased due to the pounding effect between the girders andshear keys caused by bearing sliding e reasonable damagesequence of two restraint systems can be reflected through thecorresponding damage probabilities of bearings seismic mea-sures and substructures respectively as indicated by the red andgreen lines in Figure 17 erefore the rational sequence ofcritical limit state occurrences for the approach bridges isbearings⟶ expansion joints (or restrainer cables) shearkeys⟶ columns abutments
43 Comprehensive Evaluation of Bridge System Padgett andDesRoches [16] pointed out the necessity to derive the bridgesystem vulnerabilities when assessing retrofitted bridges
Figure 15 Experimental results of (a) LEB and (b) LRB
ΔΔ1
F
Δ2
Fmax
Fd
KdK0
(a)
Δy Δ
F
K1 = (3~7) K2
K1
K2
(b)
Slack
F
K
Δ
Fy
(c)
Figure 16 Analytical models of (a) LEB (b) LRB and (c) cable restrainers
16 Advances in Civil Engineering
because different components may have a different responsewith a given retrofit measure Also Zhang and Huo [18]studied the optimal parameters of isolation bearings byderiving the intensity median value corresponding to sys-tem-level fragility function which can effectively avoid theimpacts of different isolation parameters on differentcomponents As can be seen from Figures 14 and 17 theabove analysis results were also very apparent in this studyFor example the longitudinal deck displacement was sig-nificantly reduced by using dampers yet increased thetransverse demands of stiffening girder and pylons e useof LEB or LRB reduced the demands placed on columns yetled to an increase of the transverse deformations of abut-ments which was due to the use of shear keys limiting theexcessive sliding displacement of bearings erefore acomprehensive evaluation of retrofit strategies on bridgesparticularly long-span suspension bridges that have multiplecomponents should be based on system-level fragility in-stead of component-level
Taking the findings that no damages had occurred to thecable system and the definition of bearing damage states intoaccount the derivation of system-level fragility of the wholebridge depended on critical components including pylonscolumns bearings and abutments e fragility data byemploying the absolutely serial and serial-parallel assump-tions of bridge system are plotted in Figure 18 and comparedto the system fragility obtained from the proposed com-posite damage state e absolutely serial assumption
generated serious bias for fragility curves and may result inmisdirection if they were served as a design guide eabsolutely serial model overestimated the potential damageof the bridge system since the fragility curve directlydepended on the most vulnerable components In otherwords if the difference in location and performance of eachcomponent was neglected then the seismic performanceevaluation would be achieved by attaching equal importanceto the strength of pylons approach columns and bearingsis method contradicted the philosophy of capacity designin which the long-span structure can dissipate energythrough a reasonable swinging of stiffening girder and thebearings were designed as fuse elements in the small-to-medium-span bridgese serial-parallel model qualitativelyincorporated the differences of components and was verysensitive to the seismic design of the structure as seen fromthe comparisons among the cases in Figure 18 e modeldistinguished the serial components and parallel compo-nents yet still overestimated the bridge system-level fragilitywhen bearings became the vulnerable components espe-cially for long-span bridges is special situation such asthe serial assumption may also lead to misunderstandingOnce the relative importance between the components wasconsidered in the serial-composite model the damage po-tential was clearly reduced within the range of earthquakeintensity of interest based on the case studies is alsoreflected the necessity of the classification idea of bridgecomponents which can be expressed by the numerical value
Table 6 Damage states of the quasi-isolation and isolation bearings (m)
Figure 17 Probabilities of critical damage states occurrence sequences for the approach bridges when PGV is 1ms
Advances in Civil Engineering 17
of system-level fragility erefore the application ofcomposite damage state (equation (4)) considering com-ponent difference was instead more appropriate than theaforementioned two models
e damage probability distribution represented byblack circles in Figure 18 can be obtained from the definedcomposite damage state (equation (5)) where the parametera is 06 It was found that system fragility result was a moreaccurate assessment with further taking the relative im-portance between themain bridge and approach bridges intoaccount is is due to the principle that the seismic designperformance objective of long-span bridges generally has nocollapse for the main body structures and is necessarily opento emergency vehicle service In addition attention must bepaid to the fact that the differences in repair cost and time ofthe main and approach bridges are significant when thesame damage state is achieved e results of compositedamage state may match the performance objectives of theseismic design In order to investigate the influence ofimproved strategies and structural relative importance ondamage potential of bridges the mean values of the fragility
functions can be employed to evaluate the seismic designe smooth fragility curves for the specific damage states canbe fitted with the lognormal cumulative probability distri-bution function by employing the least-squares technique
P DSgtDSi | PGV1113858 1113859 Φln(pgv) minus λPGV
ζPGV1113888 1113889 (12)
where λPGV and ζPGV are the logarithmic mean and dis-persion values of earthquake intensity (PGV) for a specifieddamage level e larger mean values represent the smallerdamage potential to reach the specified level
Figure 19 indicates the mean values of earthquakeintensity at the specific damage states as a function of theimproved strategies and parameter a Each step of theretrofit measures would enhance the seismic performanceof the overall bridge system at all levels e improvementof the restraint system of the approach bridge played acritical role of the overall bridge system For instance themean values of the earthquake intensity featured mutationof slopes along the measure direction which reflected the
Figure 18 System fragility of the original design (a) Case 0 and improved design including (b) Case 1 and (c) Case 5
18 Advances in Civil Engineering
fact that the effectiveness of improvement for the ap-proach bridges was clearer than that of the main bridgee fragility of the highway network would be under-estimated if only the damage potential of the suspensionspan was adopted erefore when the structures such asapproach bridges transition piers and auxiliary pierswere ignored the damage identification and repair de-cisions would be misleading in the seismic evaluation oflong-span bridges Conversely the damaging probabilityof the overall bridge system will be amplified regardless ofthe higher importance of the main bridge When respectthe fact that the importance of suspension span is higherthan that of approach bridge in transportation networks aparameter a of 06ndash08 may be a better choice in thepresent study
e seismic fragility of conventionally designedbridges was mainly governed by the approach bridges at alldamage levels However for the retrofitted bridges thatpossess reasonable seismic performance different damagestates were controlled by the earthquake demands ofvarious structures as shown by the trend of the curvesalong parameter a in Case 4 and Case 5 e damagingpotential of the overall bridge from the low-to-moderatelimit states was dominated by the approach bridge eprimary reasons were the sliding of bearings and thetransfer of limited forces from the girder to substructure
by the damage of restriction devices of the approachbridges e extensive damage was controlled by thedisplacement of stiffening girder of the main bridge whichinferred the effect of dampers will be conspicuous understrong earthquake shaking At the same time the overalllong-span bridges with the reasonable seismic system willnot collapse during a strong earthquake event and canmeet emergency traffic service It can be seen that the meanvalues of fragility functions developed by compositedamage indices (equations (4) and (5)) are good indicatorsto evaluate seismic performance of long-span bridges andcan be consistent with the established seismic performanceobjectives
5 Conclusions
In the present study a procedure for investigating thereasonable seismic performance of long-span suspensionbridges was developed under the fragility function frame-work e seismic performance of a long-span suspensionbridge was examined and improved using the proposedprocedure from which the results revealed distinguishableconclusions as follows
(1) e definition of damage states for bearings con-sidered the interactions between the adjacent
014036059081103125
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(a)
020051081112142173
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
0002
0406
0810
CaseParameter a
(b)
130251371492612733
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(c)
Mea
n va
lue
exp
(λ) (
ms
)
Parameter a Case0 1 2 3 4 5
0001
0203
0405
131305478652826
1000
(d)
Figure 19 Influence of relative importance and improved designs on mean values of earthquake intensity for different damage states(a) slight (b) moderate (c) extensive (d) collapse
Advances in Civil Engineering 19
structures and self-damage to bearings in accor-dance with the actual design and experimentalconclusions e damage states of pylons in terms ofcurvature ductility were defined by developingmodal pushover analysis and idealized bilinear fit-ting methods in the equivalent pylon fiber models ofthe suspension bridge
(2) When investigating the component fragility curvesof the initial design it was found that the large deckdisplacement of the suspension span was the mostserious damage present and a slight inelastic re-sponse may have occurred only in the pylons Al-though the cable system was able to maintainelasticity under earthquakes the saddles at the py-lons were identified with a high-risk region indi-cating the damaging potential For the approachbridges the ductility mechanism did not realize arational damage sequence e columns were proneto collapse because of the application of fixed potbearings which in turn caused the damage to thebearings and abutments
(3) e stepwise improved seismic designs were pro-posed in terms of the component fragility results ofthe initial design Among the candidates fluid vis-cous dampers were proven to be a good choice forlong-span bridges based on the two attributes ofdisplacement and force when the similar efficiency ofdisplacement control was achieved e restraintsystem had a critical effect on the seismic responsesof the approach bridges e quasi-isolation orisolation can achieve a reasonable seismic damagesequence of the approach bridges
(4) e defined composite damage indices showed thatthe incorporate component classification conceptand structural relative importance were necessary toderive a good indicator of performance evaluationdue to the fact that the damage indices were based oneither an absolutely serial or serial-parallel systemmodel may significantly overestimate the damagepotential of the overall bridge system e meanvalues of earthquake intensity at the specific damagestates can serve as a function of the improvedstrategies and relative importance to further evaluatethe overall seismic performance of the suspensionbridge It can be concluded that the damaging po-tential of the overall bridge from the low-to-mod-erate limit states was governed by the approachbridges in the retrofitted bridges with a favorableseismic performance whereas the displacementdemand of the suspension span dominated thefragility probability of the extensive damage
Data Availability
e nature of the data is the analytical data of finite elementmodels of the case bridge e data used to support thefindings of this study are available from the correspondingauthor upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the Basic Research ServiceProject of Central-Level Public Welfare Research Institute(grant nos 2016-9018 and 2019-0034) and Science andTechnology Project of Communicationsrsquo Construction inWestern China MOC (grant no 2009318223094)
References
[1] L W David A Kartoum K Chang and I Roy ldquoSuspensionreport seismic performance and design considerations of longspan suspension bridgesrdquo Report CSMIP00-03 CaliforniaDepartment of Conservation Division of Mines And GeologyOffice of Strong Motion Studies Sacramento CA USA 2000
[2] S-G Gwon and D-H Choi ldquoImproved continuummodel forfree vibration analysis of suspension bridgesrdquo Journal ofEngineering Mechanics vol 143 no 7 Article ID 040170382017
[3] S Arzoumanidis A Shama and F Ostadan ldquoPerformance-based seismic analysis and design of suspension bridgesrdquoEarthquake Engineering and Structural Dynamics vol 34no 4-5 pp 349ndash367 2005
[4] L Sgambi E Garavaglia N Basso and F Bontempi ldquoMonteCarlo simulation for seismic analysis of a long span sus-pension bridgerdquo Engineering Structures vol 78 pp 100ndash1112014
[5] D Karmakar S Ray-Chaudhuri and M Shinozuka ldquoFiniteelement model development validation and probabilisticseismic performance evaluation of Vincent omas suspen-sion bridgerdquo Structure and Infrastructure Engineering vol 11no 2 pp 223ndash237 2015
[6] W-F Chen and L Duan Bridge Engineering HandbookSeismic Design CRC Press Boca Raton FL USA 2nd edition2014
[7] E Choi ldquoSeismic analysis and retrofit of mid-americabridgesrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2002
[8] B G Nielson and R DesRoches ldquoSeismic fragility method-ology for highway bridges using a component level approachrdquoEarthquake Engineering amp Structural Dynamics vol 36 no 6pp 823ndash839 2007
[9] K N Ramanathan ldquoNext generation seismic fragility curvesfor california bridges incorporating the evolution in seismicdesignrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2012
[10] O Avsar A Yakut and A Caner ldquoAnalytical fragility curvesfor ordinary highway bridges in Turkeyrdquo Earthquake Spectravol 27 no 4 pp 971ndash996 2011
[11] S P Stefanidou and A J Kappos ldquoMethodology for thedevelopment of bridge-specific fragility curvesrdquo EarthquakeEngineering amp Structural Dynamics vol 46 no 1 pp 73ndash932016
[12] B Borzi P Ceresa P Franchin F Noto G M Calvi andP E Pinto ldquoSeismic vulnerability of the Italian roadwaybridge stockrdquo Earthquake Spectra vol 31 no 4pp 2137ndash2161 2015
[13] W T Barnawi and S J Dyke ldquoSeismic fragility relationshipsof a cable-stayed bridge equipped with response modification
20 Advances in Civil Engineering
systemsrdquo Journal of Bridge Engineering vol 19 no 8 ArticleID A4013003 2014
[14] J Zhong Y Pang J Jong-Su J-S Jeon R DesRoches andW Yuan ldquoSeismic fragility assessment of long-span cable-stayed bridges in Chinardquo Advances in Structural Engineeringvol 19 no 11 pp 1797ndash1812 2016
[15] J Zhong Z Hu W Yuan and L Chen ldquoSystem-basedprobabilistic optimization of fluid viscous dampers equippedin cable-stayed bridgesrdquo Advances in Structural Engineeringvol 21 no 12 pp 1815ndash1825 2018
[16] J E Padgett and R DesRoches ldquoMethodology for the de-velopment of analytical fragility curves for retrofitted bridgesfitted bridgesrdquo Earthquake Engineering amp Structural Dy-namics vol 37 no 8 pp 1157ndash1174 2008
[17] Y Xie and J Zhang ldquoOptimal design of seismic protectivedevices for highway bridges using performance-basedmethodology and multiobjective genetic optimizationrdquoJournal of Bridge Engineering vol 22 no 3 Article ID04016129 2017
[18] J Zhang and Y Huo ldquoEvaluating effectiveness and optimumdesign of isolation devices for highway bridges using thefragility function methodrdquo Engineering Structures vol 31no 8 pp 1648ndash1660 2009
[19] F McKenna M H Scott and G L Fenves ldquoNonlinear finite-element analysis software architecture using object compo-sitionrdquo Journal of Computing in Civil Engineering vol 24no 1 pp 95ndash107 2010
[20] MOTC ldquoGuidelines for seismic design of highway bridgesrdquoChina Ministry of Transportation and CommunicationsPress Beijing China JTGT B02-01-2008 2008
[21] A D Kiureghian and J L Sackman ldquoTangent geometricstiffness of inclined cables under self-weightrdquo Journal ofStructural Engineering vol 131 no 6 pp 941ndash945 2005
[22] J S Steelman E T Filipov L A Fahnestock et al ldquoEx-perimental behavior of steel fixed bearings and implicationsfor seismic bridge responserdquo Journal of Bridge Engineeringvol 19 no 8 Article ID A4014007 2014
[23] B Madani F Behnamfar and H Tajmir Riahi ldquoDynamicresponse of structures subjected to pounding and structure-soil-structure interactionrdquo Soil Dynamics and EarthquakeEngineering vol 78 pp 46ndash60 2015
[24] L Xu and J Li ldquoDesign and experimental investigation of anew type sliding retainer and its efficacy in seismic fortifi-cationrdquo Engineering Mechanics vol 33 no 2 pp 111ndash1182016 in Chinese
[25] Caltrans Caltrans Seismic Design Criteria California De-partment of Transportation Sacramento CA USA 2013
[26] A Shamsabadi K M Rollins and M Kapuskar ldquoNonlinearsoil-abutment-bridge structure interaction for seismic per-formance-based designrdquo Journal of Geotechnical and Geo-environmental Engineering vol 133 no 6 pp 707ndash720 2007
[27] X Li Z Sun D Wang and Y Shi ldquoLongitudinal seismicpounding effect of bridges abutment and backfilling damagerdquoJournal of Changrsquoan University vol 35 no 4 pp 76ndash82 2015in Chinese
[28] MOTC ldquoSpecification for design of highway suspensionbridgerdquo China Ministry of Transportation and Communi-cations Press Beijing China JTGT D65-05-2015 2015
[29] A Shafieezadeh K Ramanathan J E Padgett andR DesRoches ldquoFractional order intensity measures forprobabilistic seismic demand modeling applied to highwaybridgesrdquo Earthquake Engineering amp Structural Dynamicsvol 41 no 3 pp 391ndash409 2012
[30] J E Padgett B G Nielson and R DesRoches ldquoSelection ofoptimal intensity measures in probabilistic seismic demandmodels of highway bridge portfoliosrdquo Earthquake Engineeringamp Structural Dynamics vol 37 no 5 pp 711ndash725 2008
[31] K R Mackie and B Stojadinovic ldquoFragility basis for californiahighway overpass bridge seismic decision makingrdquo PEERReport 200502 Pacific Earthquake Engineering ResearchCenter University of California Berkeley Berkeley CA USA2005
[32] J Li T Peng Ji Lin L Zhou and YuL Deng Seismic Design ofMulti-Pylon Multi-Span Suspension Bridge China Commu-nications Press Beijing China 2013 in Chinese
[33] L Chen W Zhuang H Zhao and Z Wan ldquoReport onHighwaysrsquo Damage in the Wenchuan Earthquake-BridgeChina Communications Press Beijing China 2012 inChinese
[34] G Wu K Wang G Lu and P Zhang ldquoAn experimentalinvestigation of unbonded laminated elastomeric bearingsand the seismic evaluations of highway bridges with testedbearing componentsrdquo Shock and Vibration vol 2018 ArticleID 8439321 18 pages 2018
Advances in Civil Engineering 21
seismic demand analysis in the logarithmic reference framecan be adopted to determine the optimal IM In the previousstudies PGA and spectral acceleration were selected as theoptimal IMs for highway bridges [9 30] when applying themultiple measures containing efficiency practicality profi-ciency and sufficiency Mackie et al [31] suggested both PGAand PGV are good choices of IM in related to the EDPmeasures in highway overpass bridges Zhong et al [14]pointed out that PGA and PGV are themost efficient practicaland proficient IMs for long-span cable-stayed bridges yet PGVtends to be the optimal IM in terms of sufficiency ereforePGV can be utilized as the IM in this study
34 Engineering Demand Parameters e peak demands ofcritical components are adopted as EDPs for the developmentof PSDMs and the choice of optimal IM Table 2 lists thecritical EDPs for the examined bridge which have beenconsidered in the fragility evaluation of the suspension bridge
35 Capacity Limit State Models In the present study acapacity model was used to measure component damagewhich was described by a damage index as a function of the
selected EDPs ese models are usually discrete andcharacterized by the onset of various damage states based onthe experimental data Similarly the uncertainties of ca-pacity models are also expressed by median Sc and dis-persion βc e definition of damage states must be matchedwith postearthquake bridge functionality and repair strat-egies in the bridge seismic fragility assessments Founda-tions decks and anchorages are typically identified asnondamaged components in the simulations e damagestate values of bearings were determined so as to comparewith two factors including the pounding gaps between theadjacent components and seating width which can reflectthe possible pounding and unseating as shown in Figure 8 inaddition to the damages to bearings themselves that is thedeformation capacity of the bearings Table 3 lists the slight(DS1) moderate (DS2) extensive (DS3) and collapsed(DS4) damage states defined in the initial design for potbearings at various locations based on their allowable de-formation and structural dimensions
e columns of conventional highway bridges are oftenforced into the state of nonlinearity under strong earth-quakes of which the criteria of damage states were studied[8 9 16] It is found that there are major differences in
0
10
20
30
40
50N
umbe
r of r
ecor
ds
08
01
03
05
09
00
07
04
02
10
06
PGA (g)
(a)
0
10
20
30
40
50
Num
ber o
f rec
ords
20 806040 100
180
140
160
2000
220
120
PGV (cms)
(b)
20 40 600Fault distance (R) (km)
55
60
65
70
75
Mag
nitu
de (M
)
(c)
Mean
0
1
2
3
4
5
S a (g
)
2 4 6 8 100Period (s)
(d)
Figure 7 Characteristics of the selected ground motions (a) PGA (b) PGV (c) M and R (d) acceleration response spectra
8 Advances in Civil Engineering
geometric sizes material strengths and ACR of sectionsbetween the pylons of suspension bridges and columns ofconventional highway bridges erefore it is necessary todevelop a modal pushover analysis to define the damagestates of critical cross-sections of pylons In the analyticalprocesses of this study the material strain and nonlinearmoment-curvature relationship of critical cross-sectionswere characterized to measure the damage levels of thesuspension bridge pylons
First the reasonable pylon models were required to bedeveloped using fiber beam elements in longitudinal andtransverse directions as detailed in Figure 3 An elastic
spring which was equivalent to the restraint of main cablewas attached to the top of the pylon taking into account theinfluence of the main cable stiffness in the longitudinaldirection models e determining process of the equivalentstiffness can be referred to in the Ernst formula and theresults reported by Kiureghian and Sackman [21]
Second as illustrated in Figure 9 the inertial forcedistributions were back-calculated from the multimodespectrum analysis using a 3D bridge model as shown inequations (6)ndash(8) which included the contributions of highvibration modes and reflected the properties of inertial forcedistributions as follows
DS3
DS4
DeckDeck
Pedestal
PedestalBearing
Bearing
Gap
(a)
DS2
DS4
Gap
Abutmentnonlinearity
Deck
AbutmentBearing
(b)
GapShear key
degradationDS4
DS2
Bearing
(c)
Figure 8 Damage control and unseating prevention damage states for bearing displacement (a) between decks (b) between deck andabutment (c) between deck and shear key
Table 2 e critical EDPs for the examined bridge
Engineering demand parameters Abbreviation DirectionCurvature ductility of pylon cross-section (south north) PCD(S N) Longitudinal and transverseDisplacement of bearings of the main bridge MBD Longitudinal and transverseAxial force relative to yield value of main cables MAF mdashAxial force relative to yield value of suspenders SAF mdashCurvature ductility of column cross-section at columns (C1 C2) CCD(C1 C2) Longitudinal and transverseDisplacement of bearings of the approach bridge at column (C) and abutment (A) ABD(C A) Longitudinal and transverseDeformation of backfill passive pressure at abutment BPDA LongitudinalDeformation of piles at abutment PDA Longitudinal and transverseDeformation of expansion joint between decks(D) and at abutment (A) EJD(D A) LongitudinalDeformation of shear key at columns (C) and at abutments (A) SKD(C A) Transverse
Table 3 Limit state thresholds for pot bearings (m)
DS1 DS2 DS3 DS4
Criteria Allowable slippingdisplacements
Spherical skateboard radiusbody separation of bearings
Possible pounding betweenadjacent decks or deck and pylon Possible unseating
MBDL 025 031 076 1555MBDT 002 031 045 mdash
Criteria Allowable slippingdisplacements
Possible pounding entering of abutmentsnonlinearity or shear key degradation
Spherical skateboard radiusbody separation of bearings Possible unseating
where Sj and cj are the spectral acceleration and participationfactor associated with the jth mode respectively Fij φij andQij represent the seismic load mode displacement and shearforce at the i-node associated with the jth mode respectivelymi Qj and Pi indicate the mass shear force and inertial forceat the i-node respectively andN denotes the number of nodes
Next the inertial forces were loaded to the nodes ofpylon models Both the material and geometric nonlinear-ities were incorporated into the pushover analyses whichwere used to investigate the potential failure mechanismsand assess the ductility capacities of pylons
e seismic designs of pylons can develop inelastic be-haviors which were controlled by the material strain limitsFigure 9 details the curvature distributions along the pylonheights with the increase of δ where δ was the loading dis-placement at the control point It was observed that plastichinge zones typically occur at the bases of pylons and above andbelow the horizontal crossbeams Also plastic hinges mayprogress to the intermediate height in the longitudinal direc-tion e development of plastic hinges at the bases of pylonscan be much severer and also occur earlier than in other zoneserefore the curvatures at the bottoms of pylons may ef-fectively provide measurements of the damages to pylons
e mechanical indices of pylon cross-sections whichcorrespond to the four damage state characteristics
following the damages are detailed in Figure 10 e cur-vature ductility can be easily determined by the pushoveranalysis results in addition to that of the DS2e numericaldata can be transformed to a bilinear curve that has a value ofDS2 based on the equal energy rule per original curveSimilarly another pushover analysis was applied to thedouble-column bend of the approach bridge based on thespecifications [20] e four limit state thresholds of thecross-sections at the bottom of pylon and columns in boththe longitudinal and transverse directions are listed in Ta-ble 4 It was observed that the seismic capacity of longitu-dinal direction was superior to that of transverse directionand a good ductility capacity existed in both directions
In the present study the definitions of damage statethresholds of main cables and suspenders were dependent onthe steel strand characteristics of the strength and deformationrates A total of 85 of the ultimate strength (nominal yieldstrength) was conservatively acted as the failure index Namelythe corresponding ratio of force to yield value was 10 It is acommon viewpoint that the accidental loss of one suspenderwill not cause immediate failures of a bridge with the exceptionof the loss of the main cables erefore DS3 and DS4 wereonly defined based on the classification of suspenders andmaincables respectively e corresponding elastic and ultimatedeformation of abutments and shear keys were respectivelyspecified the thresholds of DS1 and DS2 with the recom-mendations of Caltrans [25] e damage of expansion jointsonly defined as the DS1 due to the lack of a higher-leveldamage modelis was objectively logical since the joints canbe replaced with the new once damaged thereby suggesting therationality for just one threshold of component damage Table 5summarizes the damage state thresholds of the above-mentioned components
e assignments of dispersions of capacity models ofcomponents appeared to be the most difficult tasks for theresearchers According to the previous research studies 035was adopted in the abutment and RC components including
Figure 9 Inertial force and curvature distributions (a) longitudinal (b) transverse
10 Advances in Civil Engineering
pylons and columns [9] e COV of main cables andsuspenders was 01 taking the low variability of highstrength steel strands into account [16] e descriptivedispersions of the remaining components were calculated tobe 025 for the two lower damage states (DS1 and DS2) and047 for the higher damage states (DS3 and DS4) [8]
36 Assessment of Vulnerable Components e componentfragility curves were derived by steps 1ndash3 detailed in Section 2to identify vulnerable components and regions as shown inFigures 11 and 12e pot bearings were shown to be the mostvulnerable components for the main bridge which inferred thecomplete damages to pot bearings themselves and reflected ahigher probability of pounding between the adjacent decks andunseating in the longitudinal direction e transverse re-sponses of bearings did not cause obvious damage compared tothe longitudinal responses e pylons were the primarycomponents for load carrying where inelasticmaterial responsecould potentially occur e transverse vulnerability was moreobvious than the longitudinal vulnerability for the lowerdamage states However the probability of extensive damagewas extremely low or even zeroWith respect to themain cablesand suspenders the increases in their forces which couldpotentially occur during rare earthquakes were confined toabout 15 (for themean values) of the forces resulting from thedead loads as illustrated in Figure 12 which also indicated thatthe maximum values of main cable stress had occurred at thesaddles of pylons and in the side spans e stress values of
suspenders achieved the maximum at the middle and end Insummary the cable system was found to be secured againstdamages in which the stress values were much lower than theyield values even under strong earthquake shaking It wasnoted that the large deflection and cable angle changes betweenthe main span and side spans would lead to the strong shear inthe longitudinal direction between the cables and saddles andthe slipping damages to the saddles at the top of the pylonsduring earthquakes were identified as high-risk
For the approach bridges the columns were determinedto be the most fragile components in both the longitudinaland transverse directions e application of the fixed potbearings resulted in a large force and deformation of thecolumns the complete damage of sliding pot bearings andapparent pounding between the decks and abutments eseactions subsequently caused the inelastic responses ofabutments but the shear keys were protected from failuretransversely Although the ductility mechanism of approachbridges was designed it was considered to be a defectiveseismic system with irrational sequences of componentdamage occurrences in high-seismicity regions
41 ImprovedDesign ofMainBridge As results shown in theevaluation of vulnerable components a large deck dis-placement of the suspension span would occur duringearthquakes which caused pounding and even unseating
Original curveBilinear curve
MprimeyMy
φs or φccφucφprimeyφy
DS4 reinforcing bar rupture core concrete crushing
DS3 major spalling of cover concrete exposed core concrete
DS2 formation of stable plastic hinge in the pylon (column)
Mom
ent
Curvature
DS1 narrow cracking in concrete first yielding of reinforcing bar
Figure 10 e damage state definitions of the pylon and column cross-sections
Table 4 Limit state thresholds of pylon and column cross-sections
Figure 11 Component fragility curves for the suspension bridge (a) main bridge (b) approach bridge
200 400 600 800 10000Location (m)
00
02
04
06
08
10
MA
F
MaximumMinimumMean
Dead loadYield valueHigh stress region
(a)
00
02
04
06
08
10
SAF
200 400 600 800 10000Location (m)
MaximumMinimumMean High stress region
Dead loadYield value
(b)
Figure 12 Force distributions of the (a) main cables and (b) suspenders
12 Advances in Civil Engineering
when there were no additional restraints in the longitudinaldirection erefore it was consistent to suggest that sup-plementary devices should be equipped to mitigate the deckmovements to some extente reduction of deck or bearingdisplacements can be taken as the retrofitted objective for thesuspension span Fragility analyses were conducted on theretrofitted bridges that covered the general strategies asfollows
Case 1 the analysis model of the bridge used a fluidviscous damper equipped between the deck and eachpedestal on both sides of the bridgeCase 2 the analysis model of the bridge used an elasticcable installed between the deck and each pedestal onboth sides of the bridgeCase 3 the analysis model of the bridge used three pairsof flexible central buckles located in the middle sectionof the span
e main bridge was assumed to have identical me-chanical parameters at each location in various designschemes e dampers and elastic cables were respectivelymodeled using zero-length elements with Maxwell andtension-only materials and the flexible central buckles weresimulated as the same as the suspenders elements inOpenSEES
e experimental testing of fluid viscous dampers hasshown that the force-velocity with fractional velocity powerlaw relation can describe the suitable mechanical behavior
fD CD| _u|αsgn( _u) (9)
where α is a real positive exponent of which value dependson the piston head orifice CD is viscous damping coefficientwith units of force per velocity raised to the α power kNmiddot(ms)-α _u indicates the velocity of damper and sgn(middot) is thesignum function Research by Li et al [32] indicated thatviscous damping coefficients range from 2500 kN (ms)-α to5000 kN (ms)-α and can be simulated with a uniform dis-tribution e velocity exponents can be assumed to varyuniformly from 03 to 05
e axial stiffness of elastic cables and flexible centralbuckles is regarded with the material length and cross-section area It was assumed that the yield strength of steelstrands of fy 1420MPa was the mean value and followed alognormal distribution with a COV 01 e stiffnessvalues of single elastic cable based on past work employed auniform distribution ranging from 125times105 kNm to25 times105 kNm In addition the other mechanical prop-erties such as length and area can be determined by thefollowing relations
LgtEΔfy
1113888 1113889
kE EA
L1113874 1113875
(10)
where L E and fy are the length Youngrsquos modulus andyield stress of elastic cables respectively and Δ represents the
controlled deck displacement which was available for 06min the examined bridge
According to expert opinion three pairs of flexiblecentral buckles were the most common type of centralbuckle of which the cross-section areas were the same asthose of the suspenders
Figures 13(a) and 13(b) illustrate bearing displacementdistributions are formed for the three retrofit measures at thesame ground motion intensity levels e mechanical be-haviors of measures strongly affect the effectiveness It can beconcluded that owing to the similar EDP distributions ofdampers and elastic cables the corresponding groundmotion intensities were similar when generating the samepounding probability e flexible central buckles wereextremely susceptible to yielding failure thereby causing lowefficiency in the displacement restrictions ereforepounding probabilities can be directly observed by condi-tioning on the median responses of dampers and elasticcables instead (Figure 13(c)) e control force produced bydevices is another noteworthy consideration in retrofitdesign e more force the devices require the larger andmore expensive they usually are [13] To take both defor-mation and force into account Figures 13(d) and 13(e)provide the contours of the pounding probability ecomparison showed that the damper deformations featuredslightly larger slopes in the horizontal direction yet thedifference in effect on the deformations of the dampers andelastic cables was not significant for reducing the objectivepounding probability Conversely damper force was lesssensitive to the pounding probability than cable force eelastic cable forces reached on average up to e32 (asymp245)times those of the damper under the same poundingprobabilities ese findings implied that difficult anchoragechallenges existed in the case of guaranteeing the efficiencyof elastic cables and the bridge decks possibly experiencedhigher stress In addition the difference in force was sub-stantial when controlling the displacement which showedthat dampers can contribute additional damping and stiff-ness to the structural system while elastic cables completelytransferred the inertial forces erefore the coordinationand balance of displacements and forces should be simul-taneously considered in the vibration mitigation of large-span bridges In this study the application of fluid viscousdampers was identified as the preferred retrofit in thelongitudinal direction for the main bridge e additionaldamping ratio provided by the fluid viscous dampers can becalculated by the following equation
ε CD
2mωeff (11)
where CD is damping coefficient of linear fluid damperskNmiddot(sm) which is equivalent to nonlinear dampers basedon equal energy dissipation per hysteresis loop [17] ωeff issystem natural frequency corresponding to the vibrationalong the axis of bridge andm is system mass It was shownthat an additional damping of 206ndash318 can offer aneffective controlling mechanism for the reduction ofpounding between the adjacent decks
Advances in Civil Engineering 13
Figure 14 depicts the change of components vulnera-bilities before and after the application of dampers It can beobserved that the retrofit measure played limited roles in thefragility reduction of the self-damaged bearings in long-spanbridges Since the additional longitudinal rigidity to girderwas provided by the dampers the transverse responses suchas girder displacement and pylon curvature would increaseas ground motion intensity increased e influences on thecable system responses and the longitudinal responses ofpylons were found to be negligible Another phenomenonwas that the longitudinal fragility of the abutments signif-icantly decreased with the reduction in the poundingprobabilities between the decks and abutments of the ap-proach bridges It was inferred that the pounding betweenthe adjacent decks was likely to affect movement of theapproach bridges and that the pounding effect would betransmitted along the girder
42 Improved Design of Approach Bridges e damagepotential of each component for the approach bridges washigh due to the high collapse risks posed by the usage of thefixed piers e structural designs of small-to-medium-spanhighway bridges in China show the fact that the super-structures are usually placed onto laminated elastomericbearings (LEBs) which tend to come into direct contact withthe bridge girders and substructures without anchoring etypical types of seismic damages of these bridges consisted of
large girder movements bearings sliding failures of shearkeys and expansion joints and abutment cracking duringthe Wenchuan and Yushu earthquake events in ChinaMeanwhile the bridge columns displayed relatively lowdamage ratios under earthquakes [33] e design can bereferred to as a quasi-isolation system which is primarilyrealized by strategically employing bearings sliding as force-limiting connections between girders and substructures Inaddition the isolation strategies of lead-rubber bearings(LRBs) are usually employed in the high-seismicity regionswhich can be taken as a comparison of the quasi-isolationsystem e two restraint systems consist of bearings shearkeys transversely located at the cap beams and restrainercables longitudinally anchored to the abutments (or ped-estals) In order to calculate the seismic responses of theapproach bridge the related experiments were completedand summarized and applied to the force-displacementrelations of the restraint system components
e detailed experiment of LEB consulted previousacademic achievements [34] An example is shown inFigure 15(a) where the feature points marked by rectangularand circular represented initial walking and obvious slidingresponses which illustrated that the mechanical property ofbearings would degrade as the displacements increased afterentering the friction sliding state until it remained nearlyconstant When there was no specific test the horizontalforce Fd in the stable friction stage can be 07 Fmax whereFmax was the corresponding maximum static friction at the
Bear
ing
disp
lace
men
t (m
)
Logarithmic spacendash5
ndash4
ndash3
ndash2
ndash1
0
1
2 4 6PGV (cms)
Case 1Case 2Case 3
(a)Po
undi
ng p
roba
bilit
y
00
02
04
06
08
05 10 15 2000PGV (ms)
Case 1Case 2Case 3
(b)
Poun
ding
pro
babi
lity
00
02
04
06
02 04 06 08 1000Deformation (m)
Case 1Case 2
(c)
ln [f
orce
(kN
)]
5
10
15
15
20
20
25
25
30
3030
35
3535
40
40
4550
7
72
74
76
78
8
033 041 049 057 065025Deformation (m)
(d)
ln [f
orce
(kN
)]5
10
15
15 20
20 25
25
30
3030
35
35
40
40
45
50
102
104
106
108
11
112
114
0155 0205 0255 0305 03550105Deformation (m)
(e)
Figure 13 Choice of the three retrofitmeasures based on deformations and forces (a) EDP-IM relation (b) pounding probability conditioned onIM (c) pounding probability conditioned on deformation contours of pounding probability for (d) dampers and (e) elastic cables
14 Advances in Civil Engineering
initial walking e displacement value Δ2 can be of 250effective shear strain (ESS) e initial stiffness K0 is cal-culated by MOTC [20] erefore the coefficients can becalculated by the following sequence FdK0⟶ Fmax⟶Δ1⟶Δ2 e built analytical relation ofthe LEB is shown in Figure 16(a) which can correctlycapture mechanical behavior under earthquakes A log-normal distribution [8] with a logarithmic standard devi-ation of 01 and a normal distribution with aCOV 016MPa were used to model the randomness offriction coefficient and shear modulus respectively For theirrespective mean values 02 and 12MPa were taken
Figure 15(b) shows a cyclic loading experiment of LRBwhich reveals that the degradation of bearings stiffness willoccur with the increase of loading displacements e elasticstiffness can reach uniformly 3 to 7 times the postyieldingstiffness according to the experimental results e
mechanical behavior of LRB can be modeled by a bilinearrelation as seen in Figure 16(b)
e cable restrainers were modeled as nonlinear tension-only elements characterized by an initial slack as shown inFigure 16(c) e parameters affecting the functionality ofcable restrainers were the cable slack length and yieldstrength according to the research of Padgett and DesRoches[16] e first two parameters were assumed to follow theuniform distributions and the strength can be modeled bythe lognormal distribution An adopted range of cablestiffness values was from 30times105 kNm to 50times105 kNmuniformly
e key parameters in analytical models of Figure 16 canbe determined according to the structural design of therealistic bridge e damage states of two restraint systemsare listed in Table 6 in which ESS is used as the damageindex
Dam
age p
roba
bilit
y
05 10 15 2000PGV (ms)
00
02
04
06
08
10
Case 0Case 1
MBDLMBDT
(a)
Dam
age p
roba
bilit
y
00
01
02
03
04
05
05 10 15 2000PGV (ms)
Case 0Case 1
PCDNLPCDNT
(b)
Dam
age p
roba
bilit
y
00
02
04
06
08
10
05 10 15 2000PGV (ms)
Case 0Case 1
BPDAEJDA
(c)
Figure 14 Influence of dampers on vulnerabilties of (a) bearing responses for DS2 (b) pylon responses for DS2 and (c) abutment andpounding responses
Advances in Civil Engineering 15
Figure 17 illustrates the probabilities of critical damage stateoccurrence sequences of the bridge components in Cases 1 4and 5 when PGV is 10ms which is equal to E2 earthquakeintensity in the seismic assessment reports e componentvulnerabilities of initial design (Case 0) were taken as the ref-erence values which were marked by different colored strips inFigure 17 e desirable sequence preferred Cases 4 or 5 for theapproach bridgese initial design of approach bridges resultedin unreasonable damage sequences In the strategies pertainingto quasi-isolation and isolation systems multiple tiers of seismicstructural redundancy can be normally employed to preventstructural collapse during strong earthquake events Tier 1 in-dicated that the bearings were weakly fused which was evidentin the high fragility of bearing sliding behaviors (ABDAL andABDC2T) Another advantage was that the extensive dam-ages of bearing would be reduced when compared with the self-damage of the pot bearings Tier 2 indicated that the designedbearings had sufficient seat widths for adapting to the sliding ofsuperstructures However the devices had a certain damageprobability (EJDA and SKDC2) and dissipated a portion of
the seismic energy when the displacement of the superstructuresbecame larger than the allowable values of constraint devicesTier 3 redundancy was designed as a potential plastic hinge inthe substructures (CCDC2) including a certain yielding of thebackfill at the abutments (BPDA) e response of theabutments in the transverse direction (PDAT) was slightlyincreased due to the pounding effect between the girders andshear keys caused by bearing sliding e reasonable damagesequence of two restraint systems can be reflected through thecorresponding damage probabilities of bearings seismic mea-sures and substructures respectively as indicated by the red andgreen lines in Figure 17 erefore the rational sequence ofcritical limit state occurrences for the approach bridges isbearings⟶ expansion joints (or restrainer cables) shearkeys⟶ columns abutments
43 Comprehensive Evaluation of Bridge System Padgett andDesRoches [16] pointed out the necessity to derive the bridgesystem vulnerabilities when assessing retrofitted bridges
Figure 15 Experimental results of (a) LEB and (b) LRB
ΔΔ1
F
Δ2
Fmax
Fd
KdK0
(a)
Δy Δ
F
K1 = (3~7) K2
K1
K2
(b)
Slack
F
K
Δ
Fy
(c)
Figure 16 Analytical models of (a) LEB (b) LRB and (c) cable restrainers
16 Advances in Civil Engineering
because different components may have a different responsewith a given retrofit measure Also Zhang and Huo [18]studied the optimal parameters of isolation bearings byderiving the intensity median value corresponding to sys-tem-level fragility function which can effectively avoid theimpacts of different isolation parameters on differentcomponents As can be seen from Figures 14 and 17 theabove analysis results were also very apparent in this studyFor example the longitudinal deck displacement was sig-nificantly reduced by using dampers yet increased thetransverse demands of stiffening girder and pylons e useof LEB or LRB reduced the demands placed on columns yetled to an increase of the transverse deformations of abut-ments which was due to the use of shear keys limiting theexcessive sliding displacement of bearings erefore acomprehensive evaluation of retrofit strategies on bridgesparticularly long-span suspension bridges that have multiplecomponents should be based on system-level fragility in-stead of component-level
Taking the findings that no damages had occurred to thecable system and the definition of bearing damage states intoaccount the derivation of system-level fragility of the wholebridge depended on critical components including pylonscolumns bearings and abutments e fragility data byemploying the absolutely serial and serial-parallel assump-tions of bridge system are plotted in Figure 18 and comparedto the system fragility obtained from the proposed com-posite damage state e absolutely serial assumption
generated serious bias for fragility curves and may result inmisdirection if they were served as a design guide eabsolutely serial model overestimated the potential damageof the bridge system since the fragility curve directlydepended on the most vulnerable components In otherwords if the difference in location and performance of eachcomponent was neglected then the seismic performanceevaluation would be achieved by attaching equal importanceto the strength of pylons approach columns and bearingsis method contradicted the philosophy of capacity designin which the long-span structure can dissipate energythrough a reasonable swinging of stiffening girder and thebearings were designed as fuse elements in the small-to-medium-span bridgese serial-parallel model qualitativelyincorporated the differences of components and was verysensitive to the seismic design of the structure as seen fromthe comparisons among the cases in Figure 18 e modeldistinguished the serial components and parallel compo-nents yet still overestimated the bridge system-level fragilitywhen bearings became the vulnerable components espe-cially for long-span bridges is special situation such asthe serial assumption may also lead to misunderstandingOnce the relative importance between the components wasconsidered in the serial-composite model the damage po-tential was clearly reduced within the range of earthquakeintensity of interest based on the case studies is alsoreflected the necessity of the classification idea of bridgecomponents which can be expressed by the numerical value
Table 6 Damage states of the quasi-isolation and isolation bearings (m)
Figure 17 Probabilities of critical damage states occurrence sequences for the approach bridges when PGV is 1ms
Advances in Civil Engineering 17
of system-level fragility erefore the application ofcomposite damage state (equation (4)) considering com-ponent difference was instead more appropriate than theaforementioned two models
e damage probability distribution represented byblack circles in Figure 18 can be obtained from the definedcomposite damage state (equation (5)) where the parametera is 06 It was found that system fragility result was a moreaccurate assessment with further taking the relative im-portance between themain bridge and approach bridges intoaccount is is due to the principle that the seismic designperformance objective of long-span bridges generally has nocollapse for the main body structures and is necessarily opento emergency vehicle service In addition attention must bepaid to the fact that the differences in repair cost and time ofthe main and approach bridges are significant when thesame damage state is achieved e results of compositedamage state may match the performance objectives of theseismic design In order to investigate the influence ofimproved strategies and structural relative importance ondamage potential of bridges the mean values of the fragility
functions can be employed to evaluate the seismic designe smooth fragility curves for the specific damage states canbe fitted with the lognormal cumulative probability distri-bution function by employing the least-squares technique
P DSgtDSi | PGV1113858 1113859 Φln(pgv) minus λPGV
ζPGV1113888 1113889 (12)
where λPGV and ζPGV are the logarithmic mean and dis-persion values of earthquake intensity (PGV) for a specifieddamage level e larger mean values represent the smallerdamage potential to reach the specified level
Figure 19 indicates the mean values of earthquakeintensity at the specific damage states as a function of theimproved strategies and parameter a Each step of theretrofit measures would enhance the seismic performanceof the overall bridge system at all levels e improvementof the restraint system of the approach bridge played acritical role of the overall bridge system For instance themean values of the earthquake intensity featured mutationof slopes along the measure direction which reflected the
Figure 18 System fragility of the original design (a) Case 0 and improved design including (b) Case 1 and (c) Case 5
18 Advances in Civil Engineering
fact that the effectiveness of improvement for the ap-proach bridges was clearer than that of the main bridgee fragility of the highway network would be under-estimated if only the damage potential of the suspensionspan was adopted erefore when the structures such asapproach bridges transition piers and auxiliary pierswere ignored the damage identification and repair de-cisions would be misleading in the seismic evaluation oflong-span bridges Conversely the damaging probabilityof the overall bridge system will be amplified regardless ofthe higher importance of the main bridge When respectthe fact that the importance of suspension span is higherthan that of approach bridge in transportation networks aparameter a of 06ndash08 may be a better choice in thepresent study
e seismic fragility of conventionally designedbridges was mainly governed by the approach bridges at alldamage levels However for the retrofitted bridges thatpossess reasonable seismic performance different damagestates were controlled by the earthquake demands ofvarious structures as shown by the trend of the curvesalong parameter a in Case 4 and Case 5 e damagingpotential of the overall bridge from the low-to-moderatelimit states was dominated by the approach bridge eprimary reasons were the sliding of bearings and thetransfer of limited forces from the girder to substructure
by the damage of restriction devices of the approachbridges e extensive damage was controlled by thedisplacement of stiffening girder of the main bridge whichinferred the effect of dampers will be conspicuous understrong earthquake shaking At the same time the overalllong-span bridges with the reasonable seismic system willnot collapse during a strong earthquake event and canmeet emergency traffic service It can be seen that the meanvalues of fragility functions developed by compositedamage indices (equations (4) and (5)) are good indicatorsto evaluate seismic performance of long-span bridges andcan be consistent with the established seismic performanceobjectives
5 Conclusions
In the present study a procedure for investigating thereasonable seismic performance of long-span suspensionbridges was developed under the fragility function frame-work e seismic performance of a long-span suspensionbridge was examined and improved using the proposedprocedure from which the results revealed distinguishableconclusions as follows
(1) e definition of damage states for bearings con-sidered the interactions between the adjacent
014036059081103125
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(a)
020051081112142173
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
0002
0406
0810
CaseParameter a
(b)
130251371492612733
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(c)
Mea
n va
lue
exp
(λ) (
ms
)
Parameter a Case0 1 2 3 4 5
0001
0203
0405
131305478652826
1000
(d)
Figure 19 Influence of relative importance and improved designs on mean values of earthquake intensity for different damage states(a) slight (b) moderate (c) extensive (d) collapse
Advances in Civil Engineering 19
structures and self-damage to bearings in accor-dance with the actual design and experimentalconclusions e damage states of pylons in terms ofcurvature ductility were defined by developingmodal pushover analysis and idealized bilinear fit-ting methods in the equivalent pylon fiber models ofthe suspension bridge
(2) When investigating the component fragility curvesof the initial design it was found that the large deckdisplacement of the suspension span was the mostserious damage present and a slight inelastic re-sponse may have occurred only in the pylons Al-though the cable system was able to maintainelasticity under earthquakes the saddles at the py-lons were identified with a high-risk region indi-cating the damaging potential For the approachbridges the ductility mechanism did not realize arational damage sequence e columns were proneto collapse because of the application of fixed potbearings which in turn caused the damage to thebearings and abutments
(3) e stepwise improved seismic designs were pro-posed in terms of the component fragility results ofthe initial design Among the candidates fluid vis-cous dampers were proven to be a good choice forlong-span bridges based on the two attributes ofdisplacement and force when the similar efficiency ofdisplacement control was achieved e restraintsystem had a critical effect on the seismic responsesof the approach bridges e quasi-isolation orisolation can achieve a reasonable seismic damagesequence of the approach bridges
(4) e defined composite damage indices showed thatthe incorporate component classification conceptand structural relative importance were necessary toderive a good indicator of performance evaluationdue to the fact that the damage indices were based oneither an absolutely serial or serial-parallel systemmodel may significantly overestimate the damagepotential of the overall bridge system e meanvalues of earthquake intensity at the specific damagestates can serve as a function of the improvedstrategies and relative importance to further evaluatethe overall seismic performance of the suspensionbridge It can be concluded that the damaging po-tential of the overall bridge from the low-to-mod-erate limit states was governed by the approachbridges in the retrofitted bridges with a favorableseismic performance whereas the displacementdemand of the suspension span dominated thefragility probability of the extensive damage
Data Availability
e nature of the data is the analytical data of finite elementmodels of the case bridge e data used to support thefindings of this study are available from the correspondingauthor upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the Basic Research ServiceProject of Central-Level Public Welfare Research Institute(grant nos 2016-9018 and 2019-0034) and Science andTechnology Project of Communicationsrsquo Construction inWestern China MOC (grant no 2009318223094)
References
[1] L W David A Kartoum K Chang and I Roy ldquoSuspensionreport seismic performance and design considerations of longspan suspension bridgesrdquo Report CSMIP00-03 CaliforniaDepartment of Conservation Division of Mines And GeologyOffice of Strong Motion Studies Sacramento CA USA 2000
[2] S-G Gwon and D-H Choi ldquoImproved continuummodel forfree vibration analysis of suspension bridgesrdquo Journal ofEngineering Mechanics vol 143 no 7 Article ID 040170382017
[3] S Arzoumanidis A Shama and F Ostadan ldquoPerformance-based seismic analysis and design of suspension bridgesrdquoEarthquake Engineering and Structural Dynamics vol 34no 4-5 pp 349ndash367 2005
[4] L Sgambi E Garavaglia N Basso and F Bontempi ldquoMonteCarlo simulation for seismic analysis of a long span sus-pension bridgerdquo Engineering Structures vol 78 pp 100ndash1112014
[5] D Karmakar S Ray-Chaudhuri and M Shinozuka ldquoFiniteelement model development validation and probabilisticseismic performance evaluation of Vincent omas suspen-sion bridgerdquo Structure and Infrastructure Engineering vol 11no 2 pp 223ndash237 2015
[6] W-F Chen and L Duan Bridge Engineering HandbookSeismic Design CRC Press Boca Raton FL USA 2nd edition2014
[7] E Choi ldquoSeismic analysis and retrofit of mid-americabridgesrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2002
[8] B G Nielson and R DesRoches ldquoSeismic fragility method-ology for highway bridges using a component level approachrdquoEarthquake Engineering amp Structural Dynamics vol 36 no 6pp 823ndash839 2007
[9] K N Ramanathan ldquoNext generation seismic fragility curvesfor california bridges incorporating the evolution in seismicdesignrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2012
[10] O Avsar A Yakut and A Caner ldquoAnalytical fragility curvesfor ordinary highway bridges in Turkeyrdquo Earthquake Spectravol 27 no 4 pp 971ndash996 2011
[11] S P Stefanidou and A J Kappos ldquoMethodology for thedevelopment of bridge-specific fragility curvesrdquo EarthquakeEngineering amp Structural Dynamics vol 46 no 1 pp 73ndash932016
[12] B Borzi P Ceresa P Franchin F Noto G M Calvi andP E Pinto ldquoSeismic vulnerability of the Italian roadwaybridge stockrdquo Earthquake Spectra vol 31 no 4pp 2137ndash2161 2015
[13] W T Barnawi and S J Dyke ldquoSeismic fragility relationshipsof a cable-stayed bridge equipped with response modification
20 Advances in Civil Engineering
systemsrdquo Journal of Bridge Engineering vol 19 no 8 ArticleID A4013003 2014
[14] J Zhong Y Pang J Jong-Su J-S Jeon R DesRoches andW Yuan ldquoSeismic fragility assessment of long-span cable-stayed bridges in Chinardquo Advances in Structural Engineeringvol 19 no 11 pp 1797ndash1812 2016
[15] J Zhong Z Hu W Yuan and L Chen ldquoSystem-basedprobabilistic optimization of fluid viscous dampers equippedin cable-stayed bridgesrdquo Advances in Structural Engineeringvol 21 no 12 pp 1815ndash1825 2018
[16] J E Padgett and R DesRoches ldquoMethodology for the de-velopment of analytical fragility curves for retrofitted bridgesfitted bridgesrdquo Earthquake Engineering amp Structural Dy-namics vol 37 no 8 pp 1157ndash1174 2008
[17] Y Xie and J Zhang ldquoOptimal design of seismic protectivedevices for highway bridges using performance-basedmethodology and multiobjective genetic optimizationrdquoJournal of Bridge Engineering vol 22 no 3 Article ID04016129 2017
[18] J Zhang and Y Huo ldquoEvaluating effectiveness and optimumdesign of isolation devices for highway bridges using thefragility function methodrdquo Engineering Structures vol 31no 8 pp 1648ndash1660 2009
[19] F McKenna M H Scott and G L Fenves ldquoNonlinear finite-element analysis software architecture using object compo-sitionrdquo Journal of Computing in Civil Engineering vol 24no 1 pp 95ndash107 2010
[20] MOTC ldquoGuidelines for seismic design of highway bridgesrdquoChina Ministry of Transportation and CommunicationsPress Beijing China JTGT B02-01-2008 2008
[21] A D Kiureghian and J L Sackman ldquoTangent geometricstiffness of inclined cables under self-weightrdquo Journal ofStructural Engineering vol 131 no 6 pp 941ndash945 2005
[22] J S Steelman E T Filipov L A Fahnestock et al ldquoEx-perimental behavior of steel fixed bearings and implicationsfor seismic bridge responserdquo Journal of Bridge Engineeringvol 19 no 8 Article ID A4014007 2014
[23] B Madani F Behnamfar and H Tajmir Riahi ldquoDynamicresponse of structures subjected to pounding and structure-soil-structure interactionrdquo Soil Dynamics and EarthquakeEngineering vol 78 pp 46ndash60 2015
[24] L Xu and J Li ldquoDesign and experimental investigation of anew type sliding retainer and its efficacy in seismic fortifi-cationrdquo Engineering Mechanics vol 33 no 2 pp 111ndash1182016 in Chinese
[25] Caltrans Caltrans Seismic Design Criteria California De-partment of Transportation Sacramento CA USA 2013
[26] A Shamsabadi K M Rollins and M Kapuskar ldquoNonlinearsoil-abutment-bridge structure interaction for seismic per-formance-based designrdquo Journal of Geotechnical and Geo-environmental Engineering vol 133 no 6 pp 707ndash720 2007
[27] X Li Z Sun D Wang and Y Shi ldquoLongitudinal seismicpounding effect of bridges abutment and backfilling damagerdquoJournal of Changrsquoan University vol 35 no 4 pp 76ndash82 2015in Chinese
[28] MOTC ldquoSpecification for design of highway suspensionbridgerdquo China Ministry of Transportation and Communi-cations Press Beijing China JTGT D65-05-2015 2015
[29] A Shafieezadeh K Ramanathan J E Padgett andR DesRoches ldquoFractional order intensity measures forprobabilistic seismic demand modeling applied to highwaybridgesrdquo Earthquake Engineering amp Structural Dynamicsvol 41 no 3 pp 391ndash409 2012
[30] J E Padgett B G Nielson and R DesRoches ldquoSelection ofoptimal intensity measures in probabilistic seismic demandmodels of highway bridge portfoliosrdquo Earthquake Engineeringamp Structural Dynamics vol 37 no 5 pp 711ndash725 2008
[31] K R Mackie and B Stojadinovic ldquoFragility basis for californiahighway overpass bridge seismic decision makingrdquo PEERReport 200502 Pacific Earthquake Engineering ResearchCenter University of California Berkeley Berkeley CA USA2005
[32] J Li T Peng Ji Lin L Zhou and YuL Deng Seismic Design ofMulti-Pylon Multi-Span Suspension Bridge China Commu-nications Press Beijing China 2013 in Chinese
[33] L Chen W Zhuang H Zhao and Z Wan ldquoReport onHighwaysrsquo Damage in the Wenchuan Earthquake-BridgeChina Communications Press Beijing China 2012 inChinese
[34] G Wu K Wang G Lu and P Zhang ldquoAn experimentalinvestigation of unbonded laminated elastomeric bearingsand the seismic evaluations of highway bridges with testedbearing componentsrdquo Shock and Vibration vol 2018 ArticleID 8439321 18 pages 2018
Advances in Civil Engineering 21
geometric sizes material strengths and ACR of sectionsbetween the pylons of suspension bridges and columns ofconventional highway bridges erefore it is necessary todevelop a modal pushover analysis to define the damagestates of critical cross-sections of pylons In the analyticalprocesses of this study the material strain and nonlinearmoment-curvature relationship of critical cross-sectionswere characterized to measure the damage levels of thesuspension bridge pylons
First the reasonable pylon models were required to bedeveloped using fiber beam elements in longitudinal andtransverse directions as detailed in Figure 3 An elastic
spring which was equivalent to the restraint of main cablewas attached to the top of the pylon taking into account theinfluence of the main cable stiffness in the longitudinaldirection models e determining process of the equivalentstiffness can be referred to in the Ernst formula and theresults reported by Kiureghian and Sackman [21]
Second as illustrated in Figure 9 the inertial forcedistributions were back-calculated from the multimodespectrum analysis using a 3D bridge model as shown inequations (6)ndash(8) which included the contributions of highvibration modes and reflected the properties of inertial forcedistributions as follows
DS3
DS4
DeckDeck
Pedestal
PedestalBearing
Bearing
Gap
(a)
DS2
DS4
Gap
Abutmentnonlinearity
Deck
AbutmentBearing
(b)
GapShear key
degradationDS4
DS2
Bearing
(c)
Figure 8 Damage control and unseating prevention damage states for bearing displacement (a) between decks (b) between deck andabutment (c) between deck and shear key
Table 2 e critical EDPs for the examined bridge
Engineering demand parameters Abbreviation DirectionCurvature ductility of pylon cross-section (south north) PCD(S N) Longitudinal and transverseDisplacement of bearings of the main bridge MBD Longitudinal and transverseAxial force relative to yield value of main cables MAF mdashAxial force relative to yield value of suspenders SAF mdashCurvature ductility of column cross-section at columns (C1 C2) CCD(C1 C2) Longitudinal and transverseDisplacement of bearings of the approach bridge at column (C) and abutment (A) ABD(C A) Longitudinal and transverseDeformation of backfill passive pressure at abutment BPDA LongitudinalDeformation of piles at abutment PDA Longitudinal and transverseDeformation of expansion joint between decks(D) and at abutment (A) EJD(D A) LongitudinalDeformation of shear key at columns (C) and at abutments (A) SKD(C A) Transverse
Table 3 Limit state thresholds for pot bearings (m)
DS1 DS2 DS3 DS4
Criteria Allowable slippingdisplacements
Spherical skateboard radiusbody separation of bearings
Possible pounding betweenadjacent decks or deck and pylon Possible unseating
MBDL 025 031 076 1555MBDT 002 031 045 mdash
Criteria Allowable slippingdisplacements
Possible pounding entering of abutmentsnonlinearity or shear key degradation
Spherical skateboard radiusbody separation of bearings Possible unseating
where Sj and cj are the spectral acceleration and participationfactor associated with the jth mode respectively Fij φij andQij represent the seismic load mode displacement and shearforce at the i-node associated with the jth mode respectivelymi Qj and Pi indicate the mass shear force and inertial forceat the i-node respectively andN denotes the number of nodes
Next the inertial forces were loaded to the nodes ofpylon models Both the material and geometric nonlinear-ities were incorporated into the pushover analyses whichwere used to investigate the potential failure mechanismsand assess the ductility capacities of pylons
e seismic designs of pylons can develop inelastic be-haviors which were controlled by the material strain limitsFigure 9 details the curvature distributions along the pylonheights with the increase of δ where δ was the loading dis-placement at the control point It was observed that plastichinge zones typically occur at the bases of pylons and above andbelow the horizontal crossbeams Also plastic hinges mayprogress to the intermediate height in the longitudinal direc-tion e development of plastic hinges at the bases of pylonscan be much severer and also occur earlier than in other zoneserefore the curvatures at the bottoms of pylons may ef-fectively provide measurements of the damages to pylons
e mechanical indices of pylon cross-sections whichcorrespond to the four damage state characteristics
following the damages are detailed in Figure 10 e cur-vature ductility can be easily determined by the pushoveranalysis results in addition to that of the DS2e numericaldata can be transformed to a bilinear curve that has a value ofDS2 based on the equal energy rule per original curveSimilarly another pushover analysis was applied to thedouble-column bend of the approach bridge based on thespecifications [20] e four limit state thresholds of thecross-sections at the bottom of pylon and columns in boththe longitudinal and transverse directions are listed in Ta-ble 4 It was observed that the seismic capacity of longitu-dinal direction was superior to that of transverse directionand a good ductility capacity existed in both directions
In the present study the definitions of damage statethresholds of main cables and suspenders were dependent onthe steel strand characteristics of the strength and deformationrates A total of 85 of the ultimate strength (nominal yieldstrength) was conservatively acted as the failure index Namelythe corresponding ratio of force to yield value was 10 It is acommon viewpoint that the accidental loss of one suspenderwill not cause immediate failures of a bridge with the exceptionof the loss of the main cables erefore DS3 and DS4 wereonly defined based on the classification of suspenders andmaincables respectively e corresponding elastic and ultimatedeformation of abutments and shear keys were respectivelyspecified the thresholds of DS1 and DS2 with the recom-mendations of Caltrans [25] e damage of expansion jointsonly defined as the DS1 due to the lack of a higher-leveldamage modelis was objectively logical since the joints canbe replaced with the new once damaged thereby suggesting therationality for just one threshold of component damage Table 5summarizes the damage state thresholds of the above-mentioned components
e assignments of dispersions of capacity models ofcomponents appeared to be the most difficult tasks for theresearchers According to the previous research studies 035was adopted in the abutment and RC components including
Figure 9 Inertial force and curvature distributions (a) longitudinal (b) transverse
10 Advances in Civil Engineering
pylons and columns [9] e COV of main cables andsuspenders was 01 taking the low variability of highstrength steel strands into account [16] e descriptivedispersions of the remaining components were calculated tobe 025 for the two lower damage states (DS1 and DS2) and047 for the higher damage states (DS3 and DS4) [8]
36 Assessment of Vulnerable Components e componentfragility curves were derived by steps 1ndash3 detailed in Section 2to identify vulnerable components and regions as shown inFigures 11 and 12e pot bearings were shown to be the mostvulnerable components for the main bridge which inferred thecomplete damages to pot bearings themselves and reflected ahigher probability of pounding between the adjacent decks andunseating in the longitudinal direction e transverse re-sponses of bearings did not cause obvious damage compared tothe longitudinal responses e pylons were the primarycomponents for load carrying where inelasticmaterial responsecould potentially occur e transverse vulnerability was moreobvious than the longitudinal vulnerability for the lowerdamage states However the probability of extensive damagewas extremely low or even zeroWith respect to themain cablesand suspenders the increases in their forces which couldpotentially occur during rare earthquakes were confined toabout 15 (for themean values) of the forces resulting from thedead loads as illustrated in Figure 12 which also indicated thatthe maximum values of main cable stress had occurred at thesaddles of pylons and in the side spans e stress values of
suspenders achieved the maximum at the middle and end Insummary the cable system was found to be secured againstdamages in which the stress values were much lower than theyield values even under strong earthquake shaking It wasnoted that the large deflection and cable angle changes betweenthe main span and side spans would lead to the strong shear inthe longitudinal direction between the cables and saddles andthe slipping damages to the saddles at the top of the pylonsduring earthquakes were identified as high-risk
For the approach bridges the columns were determinedto be the most fragile components in both the longitudinaland transverse directions e application of the fixed potbearings resulted in a large force and deformation of thecolumns the complete damage of sliding pot bearings andapparent pounding between the decks and abutments eseactions subsequently caused the inelastic responses ofabutments but the shear keys were protected from failuretransversely Although the ductility mechanism of approachbridges was designed it was considered to be a defectiveseismic system with irrational sequences of componentdamage occurrences in high-seismicity regions
41 ImprovedDesign ofMainBridge As results shown in theevaluation of vulnerable components a large deck dis-placement of the suspension span would occur duringearthquakes which caused pounding and even unseating
Original curveBilinear curve
MprimeyMy
φs or φccφucφprimeyφy
DS4 reinforcing bar rupture core concrete crushing
DS3 major spalling of cover concrete exposed core concrete
DS2 formation of stable plastic hinge in the pylon (column)
Mom
ent
Curvature
DS1 narrow cracking in concrete first yielding of reinforcing bar
Figure 10 e damage state definitions of the pylon and column cross-sections
Table 4 Limit state thresholds of pylon and column cross-sections
Figure 11 Component fragility curves for the suspension bridge (a) main bridge (b) approach bridge
200 400 600 800 10000Location (m)
00
02
04
06
08
10
MA
F
MaximumMinimumMean
Dead loadYield valueHigh stress region
(a)
00
02
04
06
08
10
SAF
200 400 600 800 10000Location (m)
MaximumMinimumMean High stress region
Dead loadYield value
(b)
Figure 12 Force distributions of the (a) main cables and (b) suspenders
12 Advances in Civil Engineering
when there were no additional restraints in the longitudinaldirection erefore it was consistent to suggest that sup-plementary devices should be equipped to mitigate the deckmovements to some extente reduction of deck or bearingdisplacements can be taken as the retrofitted objective for thesuspension span Fragility analyses were conducted on theretrofitted bridges that covered the general strategies asfollows
Case 1 the analysis model of the bridge used a fluidviscous damper equipped between the deck and eachpedestal on both sides of the bridgeCase 2 the analysis model of the bridge used an elasticcable installed between the deck and each pedestal onboth sides of the bridgeCase 3 the analysis model of the bridge used three pairsof flexible central buckles located in the middle sectionof the span
e main bridge was assumed to have identical me-chanical parameters at each location in various designschemes e dampers and elastic cables were respectivelymodeled using zero-length elements with Maxwell andtension-only materials and the flexible central buckles weresimulated as the same as the suspenders elements inOpenSEES
e experimental testing of fluid viscous dampers hasshown that the force-velocity with fractional velocity powerlaw relation can describe the suitable mechanical behavior
fD CD| _u|αsgn( _u) (9)
where α is a real positive exponent of which value dependson the piston head orifice CD is viscous damping coefficientwith units of force per velocity raised to the α power kNmiddot(ms)-α _u indicates the velocity of damper and sgn(middot) is thesignum function Research by Li et al [32] indicated thatviscous damping coefficients range from 2500 kN (ms)-α to5000 kN (ms)-α and can be simulated with a uniform dis-tribution e velocity exponents can be assumed to varyuniformly from 03 to 05
e axial stiffness of elastic cables and flexible centralbuckles is regarded with the material length and cross-section area It was assumed that the yield strength of steelstrands of fy 1420MPa was the mean value and followed alognormal distribution with a COV 01 e stiffnessvalues of single elastic cable based on past work employed auniform distribution ranging from 125times105 kNm to25 times105 kNm In addition the other mechanical prop-erties such as length and area can be determined by thefollowing relations
LgtEΔfy
1113888 1113889
kE EA
L1113874 1113875
(10)
where L E and fy are the length Youngrsquos modulus andyield stress of elastic cables respectively and Δ represents the
controlled deck displacement which was available for 06min the examined bridge
According to expert opinion three pairs of flexiblecentral buckles were the most common type of centralbuckle of which the cross-section areas were the same asthose of the suspenders
Figures 13(a) and 13(b) illustrate bearing displacementdistributions are formed for the three retrofit measures at thesame ground motion intensity levels e mechanical be-haviors of measures strongly affect the effectiveness It can beconcluded that owing to the similar EDP distributions ofdampers and elastic cables the corresponding groundmotion intensities were similar when generating the samepounding probability e flexible central buckles wereextremely susceptible to yielding failure thereby causing lowefficiency in the displacement restrictions ereforepounding probabilities can be directly observed by condi-tioning on the median responses of dampers and elasticcables instead (Figure 13(c)) e control force produced bydevices is another noteworthy consideration in retrofitdesign e more force the devices require the larger andmore expensive they usually are [13] To take both defor-mation and force into account Figures 13(d) and 13(e)provide the contours of the pounding probability ecomparison showed that the damper deformations featuredslightly larger slopes in the horizontal direction yet thedifference in effect on the deformations of the dampers andelastic cables was not significant for reducing the objectivepounding probability Conversely damper force was lesssensitive to the pounding probability than cable force eelastic cable forces reached on average up to e32 (asymp245)times those of the damper under the same poundingprobabilities ese findings implied that difficult anchoragechallenges existed in the case of guaranteeing the efficiencyof elastic cables and the bridge decks possibly experiencedhigher stress In addition the difference in force was sub-stantial when controlling the displacement which showedthat dampers can contribute additional damping and stiff-ness to the structural system while elastic cables completelytransferred the inertial forces erefore the coordinationand balance of displacements and forces should be simul-taneously considered in the vibration mitigation of large-span bridges In this study the application of fluid viscousdampers was identified as the preferred retrofit in thelongitudinal direction for the main bridge e additionaldamping ratio provided by the fluid viscous dampers can becalculated by the following equation
ε CD
2mωeff (11)
where CD is damping coefficient of linear fluid damperskNmiddot(sm) which is equivalent to nonlinear dampers basedon equal energy dissipation per hysteresis loop [17] ωeff issystem natural frequency corresponding to the vibrationalong the axis of bridge andm is system mass It was shownthat an additional damping of 206ndash318 can offer aneffective controlling mechanism for the reduction ofpounding between the adjacent decks
Advances in Civil Engineering 13
Figure 14 depicts the change of components vulnera-bilities before and after the application of dampers It can beobserved that the retrofit measure played limited roles in thefragility reduction of the self-damaged bearings in long-spanbridges Since the additional longitudinal rigidity to girderwas provided by the dampers the transverse responses suchas girder displacement and pylon curvature would increaseas ground motion intensity increased e influences on thecable system responses and the longitudinal responses ofpylons were found to be negligible Another phenomenonwas that the longitudinal fragility of the abutments signif-icantly decreased with the reduction in the poundingprobabilities between the decks and abutments of the ap-proach bridges It was inferred that the pounding betweenthe adjacent decks was likely to affect movement of theapproach bridges and that the pounding effect would betransmitted along the girder
42 Improved Design of Approach Bridges e damagepotential of each component for the approach bridges washigh due to the high collapse risks posed by the usage of thefixed piers e structural designs of small-to-medium-spanhighway bridges in China show the fact that the super-structures are usually placed onto laminated elastomericbearings (LEBs) which tend to come into direct contact withthe bridge girders and substructures without anchoring etypical types of seismic damages of these bridges consisted of
large girder movements bearings sliding failures of shearkeys and expansion joints and abutment cracking duringthe Wenchuan and Yushu earthquake events in ChinaMeanwhile the bridge columns displayed relatively lowdamage ratios under earthquakes [33] e design can bereferred to as a quasi-isolation system which is primarilyrealized by strategically employing bearings sliding as force-limiting connections between girders and substructures Inaddition the isolation strategies of lead-rubber bearings(LRBs) are usually employed in the high-seismicity regionswhich can be taken as a comparison of the quasi-isolationsystem e two restraint systems consist of bearings shearkeys transversely located at the cap beams and restrainercables longitudinally anchored to the abutments (or ped-estals) In order to calculate the seismic responses of theapproach bridge the related experiments were completedand summarized and applied to the force-displacementrelations of the restraint system components
e detailed experiment of LEB consulted previousacademic achievements [34] An example is shown inFigure 15(a) where the feature points marked by rectangularand circular represented initial walking and obvious slidingresponses which illustrated that the mechanical property ofbearings would degrade as the displacements increased afterentering the friction sliding state until it remained nearlyconstant When there was no specific test the horizontalforce Fd in the stable friction stage can be 07 Fmax whereFmax was the corresponding maximum static friction at the
Bear
ing
disp
lace
men
t (m
)
Logarithmic spacendash5
ndash4
ndash3
ndash2
ndash1
0
1
2 4 6PGV (cms)
Case 1Case 2Case 3
(a)Po
undi
ng p
roba
bilit
y
00
02
04
06
08
05 10 15 2000PGV (ms)
Case 1Case 2Case 3
(b)
Poun
ding
pro
babi
lity
00
02
04
06
02 04 06 08 1000Deformation (m)
Case 1Case 2
(c)
ln [f
orce
(kN
)]
5
10
15
15
20
20
25
25
30
3030
35
3535
40
40
4550
7
72
74
76
78
8
033 041 049 057 065025Deformation (m)
(d)
ln [f
orce
(kN
)]5
10
15
15 20
20 25
25
30
3030
35
35
40
40
45
50
102
104
106
108
11
112
114
0155 0205 0255 0305 03550105Deformation (m)
(e)
Figure 13 Choice of the three retrofitmeasures based on deformations and forces (a) EDP-IM relation (b) pounding probability conditioned onIM (c) pounding probability conditioned on deformation contours of pounding probability for (d) dampers and (e) elastic cables
14 Advances in Civil Engineering
initial walking e displacement value Δ2 can be of 250effective shear strain (ESS) e initial stiffness K0 is cal-culated by MOTC [20] erefore the coefficients can becalculated by the following sequence FdK0⟶ Fmax⟶Δ1⟶Δ2 e built analytical relation ofthe LEB is shown in Figure 16(a) which can correctlycapture mechanical behavior under earthquakes A log-normal distribution [8] with a logarithmic standard devi-ation of 01 and a normal distribution with aCOV 016MPa were used to model the randomness offriction coefficient and shear modulus respectively For theirrespective mean values 02 and 12MPa were taken
Figure 15(b) shows a cyclic loading experiment of LRBwhich reveals that the degradation of bearings stiffness willoccur with the increase of loading displacements e elasticstiffness can reach uniformly 3 to 7 times the postyieldingstiffness according to the experimental results e
mechanical behavior of LRB can be modeled by a bilinearrelation as seen in Figure 16(b)
e cable restrainers were modeled as nonlinear tension-only elements characterized by an initial slack as shown inFigure 16(c) e parameters affecting the functionality ofcable restrainers were the cable slack length and yieldstrength according to the research of Padgett and DesRoches[16] e first two parameters were assumed to follow theuniform distributions and the strength can be modeled bythe lognormal distribution An adopted range of cablestiffness values was from 30times105 kNm to 50times105 kNmuniformly
e key parameters in analytical models of Figure 16 canbe determined according to the structural design of therealistic bridge e damage states of two restraint systemsare listed in Table 6 in which ESS is used as the damageindex
Dam
age p
roba
bilit
y
05 10 15 2000PGV (ms)
00
02
04
06
08
10
Case 0Case 1
MBDLMBDT
(a)
Dam
age p
roba
bilit
y
00
01
02
03
04
05
05 10 15 2000PGV (ms)
Case 0Case 1
PCDNLPCDNT
(b)
Dam
age p
roba
bilit
y
00
02
04
06
08
10
05 10 15 2000PGV (ms)
Case 0Case 1
BPDAEJDA
(c)
Figure 14 Influence of dampers on vulnerabilties of (a) bearing responses for DS2 (b) pylon responses for DS2 and (c) abutment andpounding responses
Advances in Civil Engineering 15
Figure 17 illustrates the probabilities of critical damage stateoccurrence sequences of the bridge components in Cases 1 4and 5 when PGV is 10ms which is equal to E2 earthquakeintensity in the seismic assessment reports e componentvulnerabilities of initial design (Case 0) were taken as the ref-erence values which were marked by different colored strips inFigure 17 e desirable sequence preferred Cases 4 or 5 for theapproach bridgese initial design of approach bridges resultedin unreasonable damage sequences In the strategies pertainingto quasi-isolation and isolation systems multiple tiers of seismicstructural redundancy can be normally employed to preventstructural collapse during strong earthquake events Tier 1 in-dicated that the bearings were weakly fused which was evidentin the high fragility of bearing sliding behaviors (ABDAL andABDC2T) Another advantage was that the extensive dam-ages of bearing would be reduced when compared with the self-damage of the pot bearings Tier 2 indicated that the designedbearings had sufficient seat widths for adapting to the sliding ofsuperstructures However the devices had a certain damageprobability (EJDA and SKDC2) and dissipated a portion of
the seismic energy when the displacement of the superstructuresbecame larger than the allowable values of constraint devicesTier 3 redundancy was designed as a potential plastic hinge inthe substructures (CCDC2) including a certain yielding of thebackfill at the abutments (BPDA) e response of theabutments in the transverse direction (PDAT) was slightlyincreased due to the pounding effect between the girders andshear keys caused by bearing sliding e reasonable damagesequence of two restraint systems can be reflected through thecorresponding damage probabilities of bearings seismic mea-sures and substructures respectively as indicated by the red andgreen lines in Figure 17 erefore the rational sequence ofcritical limit state occurrences for the approach bridges isbearings⟶ expansion joints (or restrainer cables) shearkeys⟶ columns abutments
43 Comprehensive Evaluation of Bridge System Padgett andDesRoches [16] pointed out the necessity to derive the bridgesystem vulnerabilities when assessing retrofitted bridges
Figure 15 Experimental results of (a) LEB and (b) LRB
ΔΔ1
F
Δ2
Fmax
Fd
KdK0
(a)
Δy Δ
F
K1 = (3~7) K2
K1
K2
(b)
Slack
F
K
Δ
Fy
(c)
Figure 16 Analytical models of (a) LEB (b) LRB and (c) cable restrainers
16 Advances in Civil Engineering
because different components may have a different responsewith a given retrofit measure Also Zhang and Huo [18]studied the optimal parameters of isolation bearings byderiving the intensity median value corresponding to sys-tem-level fragility function which can effectively avoid theimpacts of different isolation parameters on differentcomponents As can be seen from Figures 14 and 17 theabove analysis results were also very apparent in this studyFor example the longitudinal deck displacement was sig-nificantly reduced by using dampers yet increased thetransverse demands of stiffening girder and pylons e useof LEB or LRB reduced the demands placed on columns yetled to an increase of the transverse deformations of abut-ments which was due to the use of shear keys limiting theexcessive sliding displacement of bearings erefore acomprehensive evaluation of retrofit strategies on bridgesparticularly long-span suspension bridges that have multiplecomponents should be based on system-level fragility in-stead of component-level
Taking the findings that no damages had occurred to thecable system and the definition of bearing damage states intoaccount the derivation of system-level fragility of the wholebridge depended on critical components including pylonscolumns bearings and abutments e fragility data byemploying the absolutely serial and serial-parallel assump-tions of bridge system are plotted in Figure 18 and comparedto the system fragility obtained from the proposed com-posite damage state e absolutely serial assumption
generated serious bias for fragility curves and may result inmisdirection if they were served as a design guide eabsolutely serial model overestimated the potential damageof the bridge system since the fragility curve directlydepended on the most vulnerable components In otherwords if the difference in location and performance of eachcomponent was neglected then the seismic performanceevaluation would be achieved by attaching equal importanceto the strength of pylons approach columns and bearingsis method contradicted the philosophy of capacity designin which the long-span structure can dissipate energythrough a reasonable swinging of stiffening girder and thebearings were designed as fuse elements in the small-to-medium-span bridgese serial-parallel model qualitativelyincorporated the differences of components and was verysensitive to the seismic design of the structure as seen fromthe comparisons among the cases in Figure 18 e modeldistinguished the serial components and parallel compo-nents yet still overestimated the bridge system-level fragilitywhen bearings became the vulnerable components espe-cially for long-span bridges is special situation such asthe serial assumption may also lead to misunderstandingOnce the relative importance between the components wasconsidered in the serial-composite model the damage po-tential was clearly reduced within the range of earthquakeintensity of interest based on the case studies is alsoreflected the necessity of the classification idea of bridgecomponents which can be expressed by the numerical value
Table 6 Damage states of the quasi-isolation and isolation bearings (m)
Figure 17 Probabilities of critical damage states occurrence sequences for the approach bridges when PGV is 1ms
Advances in Civil Engineering 17
of system-level fragility erefore the application ofcomposite damage state (equation (4)) considering com-ponent difference was instead more appropriate than theaforementioned two models
e damage probability distribution represented byblack circles in Figure 18 can be obtained from the definedcomposite damage state (equation (5)) where the parametera is 06 It was found that system fragility result was a moreaccurate assessment with further taking the relative im-portance between themain bridge and approach bridges intoaccount is is due to the principle that the seismic designperformance objective of long-span bridges generally has nocollapse for the main body structures and is necessarily opento emergency vehicle service In addition attention must bepaid to the fact that the differences in repair cost and time ofthe main and approach bridges are significant when thesame damage state is achieved e results of compositedamage state may match the performance objectives of theseismic design In order to investigate the influence ofimproved strategies and structural relative importance ondamage potential of bridges the mean values of the fragility
functions can be employed to evaluate the seismic designe smooth fragility curves for the specific damage states canbe fitted with the lognormal cumulative probability distri-bution function by employing the least-squares technique
P DSgtDSi | PGV1113858 1113859 Φln(pgv) minus λPGV
ζPGV1113888 1113889 (12)
where λPGV and ζPGV are the logarithmic mean and dis-persion values of earthquake intensity (PGV) for a specifieddamage level e larger mean values represent the smallerdamage potential to reach the specified level
Figure 19 indicates the mean values of earthquakeintensity at the specific damage states as a function of theimproved strategies and parameter a Each step of theretrofit measures would enhance the seismic performanceof the overall bridge system at all levels e improvementof the restraint system of the approach bridge played acritical role of the overall bridge system For instance themean values of the earthquake intensity featured mutationof slopes along the measure direction which reflected the
Figure 18 System fragility of the original design (a) Case 0 and improved design including (b) Case 1 and (c) Case 5
18 Advances in Civil Engineering
fact that the effectiveness of improvement for the ap-proach bridges was clearer than that of the main bridgee fragility of the highway network would be under-estimated if only the damage potential of the suspensionspan was adopted erefore when the structures such asapproach bridges transition piers and auxiliary pierswere ignored the damage identification and repair de-cisions would be misleading in the seismic evaluation oflong-span bridges Conversely the damaging probabilityof the overall bridge system will be amplified regardless ofthe higher importance of the main bridge When respectthe fact that the importance of suspension span is higherthan that of approach bridge in transportation networks aparameter a of 06ndash08 may be a better choice in thepresent study
e seismic fragility of conventionally designedbridges was mainly governed by the approach bridges at alldamage levels However for the retrofitted bridges thatpossess reasonable seismic performance different damagestates were controlled by the earthquake demands ofvarious structures as shown by the trend of the curvesalong parameter a in Case 4 and Case 5 e damagingpotential of the overall bridge from the low-to-moderatelimit states was dominated by the approach bridge eprimary reasons were the sliding of bearings and thetransfer of limited forces from the girder to substructure
by the damage of restriction devices of the approachbridges e extensive damage was controlled by thedisplacement of stiffening girder of the main bridge whichinferred the effect of dampers will be conspicuous understrong earthquake shaking At the same time the overalllong-span bridges with the reasonable seismic system willnot collapse during a strong earthquake event and canmeet emergency traffic service It can be seen that the meanvalues of fragility functions developed by compositedamage indices (equations (4) and (5)) are good indicatorsto evaluate seismic performance of long-span bridges andcan be consistent with the established seismic performanceobjectives
5 Conclusions
In the present study a procedure for investigating thereasonable seismic performance of long-span suspensionbridges was developed under the fragility function frame-work e seismic performance of a long-span suspensionbridge was examined and improved using the proposedprocedure from which the results revealed distinguishableconclusions as follows
(1) e definition of damage states for bearings con-sidered the interactions between the adjacent
014036059081103125
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(a)
020051081112142173
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
0002
0406
0810
CaseParameter a
(b)
130251371492612733
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(c)
Mea
n va
lue
exp
(λ) (
ms
)
Parameter a Case0 1 2 3 4 5
0001
0203
0405
131305478652826
1000
(d)
Figure 19 Influence of relative importance and improved designs on mean values of earthquake intensity for different damage states(a) slight (b) moderate (c) extensive (d) collapse
Advances in Civil Engineering 19
structures and self-damage to bearings in accor-dance with the actual design and experimentalconclusions e damage states of pylons in terms ofcurvature ductility were defined by developingmodal pushover analysis and idealized bilinear fit-ting methods in the equivalent pylon fiber models ofthe suspension bridge
(2) When investigating the component fragility curvesof the initial design it was found that the large deckdisplacement of the suspension span was the mostserious damage present and a slight inelastic re-sponse may have occurred only in the pylons Al-though the cable system was able to maintainelasticity under earthquakes the saddles at the py-lons were identified with a high-risk region indi-cating the damaging potential For the approachbridges the ductility mechanism did not realize arational damage sequence e columns were proneto collapse because of the application of fixed potbearings which in turn caused the damage to thebearings and abutments
(3) e stepwise improved seismic designs were pro-posed in terms of the component fragility results ofthe initial design Among the candidates fluid vis-cous dampers were proven to be a good choice forlong-span bridges based on the two attributes ofdisplacement and force when the similar efficiency ofdisplacement control was achieved e restraintsystem had a critical effect on the seismic responsesof the approach bridges e quasi-isolation orisolation can achieve a reasonable seismic damagesequence of the approach bridges
(4) e defined composite damage indices showed thatthe incorporate component classification conceptand structural relative importance were necessary toderive a good indicator of performance evaluationdue to the fact that the damage indices were based oneither an absolutely serial or serial-parallel systemmodel may significantly overestimate the damagepotential of the overall bridge system e meanvalues of earthquake intensity at the specific damagestates can serve as a function of the improvedstrategies and relative importance to further evaluatethe overall seismic performance of the suspensionbridge It can be concluded that the damaging po-tential of the overall bridge from the low-to-mod-erate limit states was governed by the approachbridges in the retrofitted bridges with a favorableseismic performance whereas the displacementdemand of the suspension span dominated thefragility probability of the extensive damage
Data Availability
e nature of the data is the analytical data of finite elementmodels of the case bridge e data used to support thefindings of this study are available from the correspondingauthor upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the Basic Research ServiceProject of Central-Level Public Welfare Research Institute(grant nos 2016-9018 and 2019-0034) and Science andTechnology Project of Communicationsrsquo Construction inWestern China MOC (grant no 2009318223094)
References
[1] L W David A Kartoum K Chang and I Roy ldquoSuspensionreport seismic performance and design considerations of longspan suspension bridgesrdquo Report CSMIP00-03 CaliforniaDepartment of Conservation Division of Mines And GeologyOffice of Strong Motion Studies Sacramento CA USA 2000
[2] S-G Gwon and D-H Choi ldquoImproved continuummodel forfree vibration analysis of suspension bridgesrdquo Journal ofEngineering Mechanics vol 143 no 7 Article ID 040170382017
[3] S Arzoumanidis A Shama and F Ostadan ldquoPerformance-based seismic analysis and design of suspension bridgesrdquoEarthquake Engineering and Structural Dynamics vol 34no 4-5 pp 349ndash367 2005
[4] L Sgambi E Garavaglia N Basso and F Bontempi ldquoMonteCarlo simulation for seismic analysis of a long span sus-pension bridgerdquo Engineering Structures vol 78 pp 100ndash1112014
[5] D Karmakar S Ray-Chaudhuri and M Shinozuka ldquoFiniteelement model development validation and probabilisticseismic performance evaluation of Vincent omas suspen-sion bridgerdquo Structure and Infrastructure Engineering vol 11no 2 pp 223ndash237 2015
[6] W-F Chen and L Duan Bridge Engineering HandbookSeismic Design CRC Press Boca Raton FL USA 2nd edition2014
[7] E Choi ldquoSeismic analysis and retrofit of mid-americabridgesrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2002
[8] B G Nielson and R DesRoches ldquoSeismic fragility method-ology for highway bridges using a component level approachrdquoEarthquake Engineering amp Structural Dynamics vol 36 no 6pp 823ndash839 2007
[9] K N Ramanathan ldquoNext generation seismic fragility curvesfor california bridges incorporating the evolution in seismicdesignrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2012
[10] O Avsar A Yakut and A Caner ldquoAnalytical fragility curvesfor ordinary highway bridges in Turkeyrdquo Earthquake Spectravol 27 no 4 pp 971ndash996 2011
[11] S P Stefanidou and A J Kappos ldquoMethodology for thedevelopment of bridge-specific fragility curvesrdquo EarthquakeEngineering amp Structural Dynamics vol 46 no 1 pp 73ndash932016
[12] B Borzi P Ceresa P Franchin F Noto G M Calvi andP E Pinto ldquoSeismic vulnerability of the Italian roadwaybridge stockrdquo Earthquake Spectra vol 31 no 4pp 2137ndash2161 2015
[13] W T Barnawi and S J Dyke ldquoSeismic fragility relationshipsof a cable-stayed bridge equipped with response modification
20 Advances in Civil Engineering
systemsrdquo Journal of Bridge Engineering vol 19 no 8 ArticleID A4013003 2014
[14] J Zhong Y Pang J Jong-Su J-S Jeon R DesRoches andW Yuan ldquoSeismic fragility assessment of long-span cable-stayed bridges in Chinardquo Advances in Structural Engineeringvol 19 no 11 pp 1797ndash1812 2016
[15] J Zhong Z Hu W Yuan and L Chen ldquoSystem-basedprobabilistic optimization of fluid viscous dampers equippedin cable-stayed bridgesrdquo Advances in Structural Engineeringvol 21 no 12 pp 1815ndash1825 2018
[16] J E Padgett and R DesRoches ldquoMethodology for the de-velopment of analytical fragility curves for retrofitted bridgesfitted bridgesrdquo Earthquake Engineering amp Structural Dy-namics vol 37 no 8 pp 1157ndash1174 2008
[17] Y Xie and J Zhang ldquoOptimal design of seismic protectivedevices for highway bridges using performance-basedmethodology and multiobjective genetic optimizationrdquoJournal of Bridge Engineering vol 22 no 3 Article ID04016129 2017
[18] J Zhang and Y Huo ldquoEvaluating effectiveness and optimumdesign of isolation devices for highway bridges using thefragility function methodrdquo Engineering Structures vol 31no 8 pp 1648ndash1660 2009
[19] F McKenna M H Scott and G L Fenves ldquoNonlinear finite-element analysis software architecture using object compo-sitionrdquo Journal of Computing in Civil Engineering vol 24no 1 pp 95ndash107 2010
[20] MOTC ldquoGuidelines for seismic design of highway bridgesrdquoChina Ministry of Transportation and CommunicationsPress Beijing China JTGT B02-01-2008 2008
[21] A D Kiureghian and J L Sackman ldquoTangent geometricstiffness of inclined cables under self-weightrdquo Journal ofStructural Engineering vol 131 no 6 pp 941ndash945 2005
[22] J S Steelman E T Filipov L A Fahnestock et al ldquoEx-perimental behavior of steel fixed bearings and implicationsfor seismic bridge responserdquo Journal of Bridge Engineeringvol 19 no 8 Article ID A4014007 2014
[23] B Madani F Behnamfar and H Tajmir Riahi ldquoDynamicresponse of structures subjected to pounding and structure-soil-structure interactionrdquo Soil Dynamics and EarthquakeEngineering vol 78 pp 46ndash60 2015
[24] L Xu and J Li ldquoDesign and experimental investigation of anew type sliding retainer and its efficacy in seismic fortifi-cationrdquo Engineering Mechanics vol 33 no 2 pp 111ndash1182016 in Chinese
[25] Caltrans Caltrans Seismic Design Criteria California De-partment of Transportation Sacramento CA USA 2013
[26] A Shamsabadi K M Rollins and M Kapuskar ldquoNonlinearsoil-abutment-bridge structure interaction for seismic per-formance-based designrdquo Journal of Geotechnical and Geo-environmental Engineering vol 133 no 6 pp 707ndash720 2007
[27] X Li Z Sun D Wang and Y Shi ldquoLongitudinal seismicpounding effect of bridges abutment and backfilling damagerdquoJournal of Changrsquoan University vol 35 no 4 pp 76ndash82 2015in Chinese
[28] MOTC ldquoSpecification for design of highway suspensionbridgerdquo China Ministry of Transportation and Communi-cations Press Beijing China JTGT D65-05-2015 2015
[29] A Shafieezadeh K Ramanathan J E Padgett andR DesRoches ldquoFractional order intensity measures forprobabilistic seismic demand modeling applied to highwaybridgesrdquo Earthquake Engineering amp Structural Dynamicsvol 41 no 3 pp 391ndash409 2012
[30] J E Padgett B G Nielson and R DesRoches ldquoSelection ofoptimal intensity measures in probabilistic seismic demandmodels of highway bridge portfoliosrdquo Earthquake Engineeringamp Structural Dynamics vol 37 no 5 pp 711ndash725 2008
[31] K R Mackie and B Stojadinovic ldquoFragility basis for californiahighway overpass bridge seismic decision makingrdquo PEERReport 200502 Pacific Earthquake Engineering ResearchCenter University of California Berkeley Berkeley CA USA2005
[32] J Li T Peng Ji Lin L Zhou and YuL Deng Seismic Design ofMulti-Pylon Multi-Span Suspension Bridge China Commu-nications Press Beijing China 2013 in Chinese
[33] L Chen W Zhuang H Zhao and Z Wan ldquoReport onHighwaysrsquo Damage in the Wenchuan Earthquake-BridgeChina Communications Press Beijing China 2012 inChinese
[34] G Wu K Wang G Lu and P Zhang ldquoAn experimentalinvestigation of unbonded laminated elastomeric bearingsand the seismic evaluations of highway bridges with testedbearing componentsrdquo Shock and Vibration vol 2018 ArticleID 8439321 18 pages 2018
Advances in Civil Engineering 21
Fij Sjcjϕijmi (6)
Qij 1113944N
mi
Fmj
Qi
1113944
N
j1Q
2ij
11139741113972
(7)
Pi Qi minus Qi+1 (8)
where Sj and cj are the spectral acceleration and participationfactor associated with the jth mode respectively Fij φij andQij represent the seismic load mode displacement and shearforce at the i-node associated with the jth mode respectivelymi Qj and Pi indicate the mass shear force and inertial forceat the i-node respectively andN denotes the number of nodes
Next the inertial forces were loaded to the nodes ofpylon models Both the material and geometric nonlinear-ities were incorporated into the pushover analyses whichwere used to investigate the potential failure mechanismsand assess the ductility capacities of pylons
e seismic designs of pylons can develop inelastic be-haviors which were controlled by the material strain limitsFigure 9 details the curvature distributions along the pylonheights with the increase of δ where δ was the loading dis-placement at the control point It was observed that plastichinge zones typically occur at the bases of pylons and above andbelow the horizontal crossbeams Also plastic hinges mayprogress to the intermediate height in the longitudinal direc-tion e development of plastic hinges at the bases of pylonscan be much severer and also occur earlier than in other zoneserefore the curvatures at the bottoms of pylons may ef-fectively provide measurements of the damages to pylons
e mechanical indices of pylon cross-sections whichcorrespond to the four damage state characteristics
following the damages are detailed in Figure 10 e cur-vature ductility can be easily determined by the pushoveranalysis results in addition to that of the DS2e numericaldata can be transformed to a bilinear curve that has a value ofDS2 based on the equal energy rule per original curveSimilarly another pushover analysis was applied to thedouble-column bend of the approach bridge based on thespecifications [20] e four limit state thresholds of thecross-sections at the bottom of pylon and columns in boththe longitudinal and transverse directions are listed in Ta-ble 4 It was observed that the seismic capacity of longitu-dinal direction was superior to that of transverse directionand a good ductility capacity existed in both directions
In the present study the definitions of damage statethresholds of main cables and suspenders were dependent onthe steel strand characteristics of the strength and deformationrates A total of 85 of the ultimate strength (nominal yieldstrength) was conservatively acted as the failure index Namelythe corresponding ratio of force to yield value was 10 It is acommon viewpoint that the accidental loss of one suspenderwill not cause immediate failures of a bridge with the exceptionof the loss of the main cables erefore DS3 and DS4 wereonly defined based on the classification of suspenders andmaincables respectively e corresponding elastic and ultimatedeformation of abutments and shear keys were respectivelyspecified the thresholds of DS1 and DS2 with the recom-mendations of Caltrans [25] e damage of expansion jointsonly defined as the DS1 due to the lack of a higher-leveldamage modelis was objectively logical since the joints canbe replaced with the new once damaged thereby suggesting therationality for just one threshold of component damage Table 5summarizes the damage state thresholds of the above-mentioned components
e assignments of dispersions of capacity models ofcomponents appeared to be the most difficult tasks for theresearchers According to the previous research studies 035was adopted in the abutment and RC components including
Figure 9 Inertial force and curvature distributions (a) longitudinal (b) transverse
10 Advances in Civil Engineering
pylons and columns [9] e COV of main cables andsuspenders was 01 taking the low variability of highstrength steel strands into account [16] e descriptivedispersions of the remaining components were calculated tobe 025 for the two lower damage states (DS1 and DS2) and047 for the higher damage states (DS3 and DS4) [8]
36 Assessment of Vulnerable Components e componentfragility curves were derived by steps 1ndash3 detailed in Section 2to identify vulnerable components and regions as shown inFigures 11 and 12e pot bearings were shown to be the mostvulnerable components for the main bridge which inferred thecomplete damages to pot bearings themselves and reflected ahigher probability of pounding between the adjacent decks andunseating in the longitudinal direction e transverse re-sponses of bearings did not cause obvious damage compared tothe longitudinal responses e pylons were the primarycomponents for load carrying where inelasticmaterial responsecould potentially occur e transverse vulnerability was moreobvious than the longitudinal vulnerability for the lowerdamage states However the probability of extensive damagewas extremely low or even zeroWith respect to themain cablesand suspenders the increases in their forces which couldpotentially occur during rare earthquakes were confined toabout 15 (for themean values) of the forces resulting from thedead loads as illustrated in Figure 12 which also indicated thatthe maximum values of main cable stress had occurred at thesaddles of pylons and in the side spans e stress values of
suspenders achieved the maximum at the middle and end Insummary the cable system was found to be secured againstdamages in which the stress values were much lower than theyield values even under strong earthquake shaking It wasnoted that the large deflection and cable angle changes betweenthe main span and side spans would lead to the strong shear inthe longitudinal direction between the cables and saddles andthe slipping damages to the saddles at the top of the pylonsduring earthquakes were identified as high-risk
For the approach bridges the columns were determinedto be the most fragile components in both the longitudinaland transverse directions e application of the fixed potbearings resulted in a large force and deformation of thecolumns the complete damage of sliding pot bearings andapparent pounding between the decks and abutments eseactions subsequently caused the inelastic responses ofabutments but the shear keys were protected from failuretransversely Although the ductility mechanism of approachbridges was designed it was considered to be a defectiveseismic system with irrational sequences of componentdamage occurrences in high-seismicity regions
41 ImprovedDesign ofMainBridge As results shown in theevaluation of vulnerable components a large deck dis-placement of the suspension span would occur duringearthquakes which caused pounding and even unseating
Original curveBilinear curve
MprimeyMy
φs or φccφucφprimeyφy
DS4 reinforcing bar rupture core concrete crushing
DS3 major spalling of cover concrete exposed core concrete
DS2 formation of stable plastic hinge in the pylon (column)
Mom
ent
Curvature
DS1 narrow cracking in concrete first yielding of reinforcing bar
Figure 10 e damage state definitions of the pylon and column cross-sections
Table 4 Limit state thresholds of pylon and column cross-sections
Figure 11 Component fragility curves for the suspension bridge (a) main bridge (b) approach bridge
200 400 600 800 10000Location (m)
00
02
04
06
08
10
MA
F
MaximumMinimumMean
Dead loadYield valueHigh stress region
(a)
00
02
04
06
08
10
SAF
200 400 600 800 10000Location (m)
MaximumMinimumMean High stress region
Dead loadYield value
(b)
Figure 12 Force distributions of the (a) main cables and (b) suspenders
12 Advances in Civil Engineering
when there were no additional restraints in the longitudinaldirection erefore it was consistent to suggest that sup-plementary devices should be equipped to mitigate the deckmovements to some extente reduction of deck or bearingdisplacements can be taken as the retrofitted objective for thesuspension span Fragility analyses were conducted on theretrofitted bridges that covered the general strategies asfollows
Case 1 the analysis model of the bridge used a fluidviscous damper equipped between the deck and eachpedestal on both sides of the bridgeCase 2 the analysis model of the bridge used an elasticcable installed between the deck and each pedestal onboth sides of the bridgeCase 3 the analysis model of the bridge used three pairsof flexible central buckles located in the middle sectionof the span
e main bridge was assumed to have identical me-chanical parameters at each location in various designschemes e dampers and elastic cables were respectivelymodeled using zero-length elements with Maxwell andtension-only materials and the flexible central buckles weresimulated as the same as the suspenders elements inOpenSEES
e experimental testing of fluid viscous dampers hasshown that the force-velocity with fractional velocity powerlaw relation can describe the suitable mechanical behavior
fD CD| _u|αsgn( _u) (9)
where α is a real positive exponent of which value dependson the piston head orifice CD is viscous damping coefficientwith units of force per velocity raised to the α power kNmiddot(ms)-α _u indicates the velocity of damper and sgn(middot) is thesignum function Research by Li et al [32] indicated thatviscous damping coefficients range from 2500 kN (ms)-α to5000 kN (ms)-α and can be simulated with a uniform dis-tribution e velocity exponents can be assumed to varyuniformly from 03 to 05
e axial stiffness of elastic cables and flexible centralbuckles is regarded with the material length and cross-section area It was assumed that the yield strength of steelstrands of fy 1420MPa was the mean value and followed alognormal distribution with a COV 01 e stiffnessvalues of single elastic cable based on past work employed auniform distribution ranging from 125times105 kNm to25 times105 kNm In addition the other mechanical prop-erties such as length and area can be determined by thefollowing relations
LgtEΔfy
1113888 1113889
kE EA
L1113874 1113875
(10)
where L E and fy are the length Youngrsquos modulus andyield stress of elastic cables respectively and Δ represents the
controlled deck displacement which was available for 06min the examined bridge
According to expert opinion three pairs of flexiblecentral buckles were the most common type of centralbuckle of which the cross-section areas were the same asthose of the suspenders
Figures 13(a) and 13(b) illustrate bearing displacementdistributions are formed for the three retrofit measures at thesame ground motion intensity levels e mechanical be-haviors of measures strongly affect the effectiveness It can beconcluded that owing to the similar EDP distributions ofdampers and elastic cables the corresponding groundmotion intensities were similar when generating the samepounding probability e flexible central buckles wereextremely susceptible to yielding failure thereby causing lowefficiency in the displacement restrictions ereforepounding probabilities can be directly observed by condi-tioning on the median responses of dampers and elasticcables instead (Figure 13(c)) e control force produced bydevices is another noteworthy consideration in retrofitdesign e more force the devices require the larger andmore expensive they usually are [13] To take both defor-mation and force into account Figures 13(d) and 13(e)provide the contours of the pounding probability ecomparison showed that the damper deformations featuredslightly larger slopes in the horizontal direction yet thedifference in effect on the deformations of the dampers andelastic cables was not significant for reducing the objectivepounding probability Conversely damper force was lesssensitive to the pounding probability than cable force eelastic cable forces reached on average up to e32 (asymp245)times those of the damper under the same poundingprobabilities ese findings implied that difficult anchoragechallenges existed in the case of guaranteeing the efficiencyof elastic cables and the bridge decks possibly experiencedhigher stress In addition the difference in force was sub-stantial when controlling the displacement which showedthat dampers can contribute additional damping and stiff-ness to the structural system while elastic cables completelytransferred the inertial forces erefore the coordinationand balance of displacements and forces should be simul-taneously considered in the vibration mitigation of large-span bridges In this study the application of fluid viscousdampers was identified as the preferred retrofit in thelongitudinal direction for the main bridge e additionaldamping ratio provided by the fluid viscous dampers can becalculated by the following equation
ε CD
2mωeff (11)
where CD is damping coefficient of linear fluid damperskNmiddot(sm) which is equivalent to nonlinear dampers basedon equal energy dissipation per hysteresis loop [17] ωeff issystem natural frequency corresponding to the vibrationalong the axis of bridge andm is system mass It was shownthat an additional damping of 206ndash318 can offer aneffective controlling mechanism for the reduction ofpounding between the adjacent decks
Advances in Civil Engineering 13
Figure 14 depicts the change of components vulnera-bilities before and after the application of dampers It can beobserved that the retrofit measure played limited roles in thefragility reduction of the self-damaged bearings in long-spanbridges Since the additional longitudinal rigidity to girderwas provided by the dampers the transverse responses suchas girder displacement and pylon curvature would increaseas ground motion intensity increased e influences on thecable system responses and the longitudinal responses ofpylons were found to be negligible Another phenomenonwas that the longitudinal fragility of the abutments signif-icantly decreased with the reduction in the poundingprobabilities between the decks and abutments of the ap-proach bridges It was inferred that the pounding betweenthe adjacent decks was likely to affect movement of theapproach bridges and that the pounding effect would betransmitted along the girder
42 Improved Design of Approach Bridges e damagepotential of each component for the approach bridges washigh due to the high collapse risks posed by the usage of thefixed piers e structural designs of small-to-medium-spanhighway bridges in China show the fact that the super-structures are usually placed onto laminated elastomericbearings (LEBs) which tend to come into direct contact withthe bridge girders and substructures without anchoring etypical types of seismic damages of these bridges consisted of
large girder movements bearings sliding failures of shearkeys and expansion joints and abutment cracking duringthe Wenchuan and Yushu earthquake events in ChinaMeanwhile the bridge columns displayed relatively lowdamage ratios under earthquakes [33] e design can bereferred to as a quasi-isolation system which is primarilyrealized by strategically employing bearings sliding as force-limiting connections between girders and substructures Inaddition the isolation strategies of lead-rubber bearings(LRBs) are usually employed in the high-seismicity regionswhich can be taken as a comparison of the quasi-isolationsystem e two restraint systems consist of bearings shearkeys transversely located at the cap beams and restrainercables longitudinally anchored to the abutments (or ped-estals) In order to calculate the seismic responses of theapproach bridge the related experiments were completedand summarized and applied to the force-displacementrelations of the restraint system components
e detailed experiment of LEB consulted previousacademic achievements [34] An example is shown inFigure 15(a) where the feature points marked by rectangularand circular represented initial walking and obvious slidingresponses which illustrated that the mechanical property ofbearings would degrade as the displacements increased afterentering the friction sliding state until it remained nearlyconstant When there was no specific test the horizontalforce Fd in the stable friction stage can be 07 Fmax whereFmax was the corresponding maximum static friction at the
Bear
ing
disp
lace
men
t (m
)
Logarithmic spacendash5
ndash4
ndash3
ndash2
ndash1
0
1
2 4 6PGV (cms)
Case 1Case 2Case 3
(a)Po
undi
ng p
roba
bilit
y
00
02
04
06
08
05 10 15 2000PGV (ms)
Case 1Case 2Case 3
(b)
Poun
ding
pro
babi
lity
00
02
04
06
02 04 06 08 1000Deformation (m)
Case 1Case 2
(c)
ln [f
orce
(kN
)]
5
10
15
15
20
20
25
25
30
3030
35
3535
40
40
4550
7
72
74
76
78
8
033 041 049 057 065025Deformation (m)
(d)
ln [f
orce
(kN
)]5
10
15
15 20
20 25
25
30
3030
35
35
40
40
45
50
102
104
106
108
11
112
114
0155 0205 0255 0305 03550105Deformation (m)
(e)
Figure 13 Choice of the three retrofitmeasures based on deformations and forces (a) EDP-IM relation (b) pounding probability conditioned onIM (c) pounding probability conditioned on deformation contours of pounding probability for (d) dampers and (e) elastic cables
14 Advances in Civil Engineering
initial walking e displacement value Δ2 can be of 250effective shear strain (ESS) e initial stiffness K0 is cal-culated by MOTC [20] erefore the coefficients can becalculated by the following sequence FdK0⟶ Fmax⟶Δ1⟶Δ2 e built analytical relation ofthe LEB is shown in Figure 16(a) which can correctlycapture mechanical behavior under earthquakes A log-normal distribution [8] with a logarithmic standard devi-ation of 01 and a normal distribution with aCOV 016MPa were used to model the randomness offriction coefficient and shear modulus respectively For theirrespective mean values 02 and 12MPa were taken
Figure 15(b) shows a cyclic loading experiment of LRBwhich reveals that the degradation of bearings stiffness willoccur with the increase of loading displacements e elasticstiffness can reach uniformly 3 to 7 times the postyieldingstiffness according to the experimental results e
mechanical behavior of LRB can be modeled by a bilinearrelation as seen in Figure 16(b)
e cable restrainers were modeled as nonlinear tension-only elements characterized by an initial slack as shown inFigure 16(c) e parameters affecting the functionality ofcable restrainers were the cable slack length and yieldstrength according to the research of Padgett and DesRoches[16] e first two parameters were assumed to follow theuniform distributions and the strength can be modeled bythe lognormal distribution An adopted range of cablestiffness values was from 30times105 kNm to 50times105 kNmuniformly
e key parameters in analytical models of Figure 16 canbe determined according to the structural design of therealistic bridge e damage states of two restraint systemsare listed in Table 6 in which ESS is used as the damageindex
Dam
age p
roba
bilit
y
05 10 15 2000PGV (ms)
00
02
04
06
08
10
Case 0Case 1
MBDLMBDT
(a)
Dam
age p
roba
bilit
y
00
01
02
03
04
05
05 10 15 2000PGV (ms)
Case 0Case 1
PCDNLPCDNT
(b)
Dam
age p
roba
bilit
y
00
02
04
06
08
10
05 10 15 2000PGV (ms)
Case 0Case 1
BPDAEJDA
(c)
Figure 14 Influence of dampers on vulnerabilties of (a) bearing responses for DS2 (b) pylon responses for DS2 and (c) abutment andpounding responses
Advances in Civil Engineering 15
Figure 17 illustrates the probabilities of critical damage stateoccurrence sequences of the bridge components in Cases 1 4and 5 when PGV is 10ms which is equal to E2 earthquakeintensity in the seismic assessment reports e componentvulnerabilities of initial design (Case 0) were taken as the ref-erence values which were marked by different colored strips inFigure 17 e desirable sequence preferred Cases 4 or 5 for theapproach bridgese initial design of approach bridges resultedin unreasonable damage sequences In the strategies pertainingto quasi-isolation and isolation systems multiple tiers of seismicstructural redundancy can be normally employed to preventstructural collapse during strong earthquake events Tier 1 in-dicated that the bearings were weakly fused which was evidentin the high fragility of bearing sliding behaviors (ABDAL andABDC2T) Another advantage was that the extensive dam-ages of bearing would be reduced when compared with the self-damage of the pot bearings Tier 2 indicated that the designedbearings had sufficient seat widths for adapting to the sliding ofsuperstructures However the devices had a certain damageprobability (EJDA and SKDC2) and dissipated a portion of
the seismic energy when the displacement of the superstructuresbecame larger than the allowable values of constraint devicesTier 3 redundancy was designed as a potential plastic hinge inthe substructures (CCDC2) including a certain yielding of thebackfill at the abutments (BPDA) e response of theabutments in the transverse direction (PDAT) was slightlyincreased due to the pounding effect between the girders andshear keys caused by bearing sliding e reasonable damagesequence of two restraint systems can be reflected through thecorresponding damage probabilities of bearings seismic mea-sures and substructures respectively as indicated by the red andgreen lines in Figure 17 erefore the rational sequence ofcritical limit state occurrences for the approach bridges isbearings⟶ expansion joints (or restrainer cables) shearkeys⟶ columns abutments
43 Comprehensive Evaluation of Bridge System Padgett andDesRoches [16] pointed out the necessity to derive the bridgesystem vulnerabilities when assessing retrofitted bridges
Figure 15 Experimental results of (a) LEB and (b) LRB
ΔΔ1
F
Δ2
Fmax
Fd
KdK0
(a)
Δy Δ
F
K1 = (3~7) K2
K1
K2
(b)
Slack
F
K
Δ
Fy
(c)
Figure 16 Analytical models of (a) LEB (b) LRB and (c) cable restrainers
16 Advances in Civil Engineering
because different components may have a different responsewith a given retrofit measure Also Zhang and Huo [18]studied the optimal parameters of isolation bearings byderiving the intensity median value corresponding to sys-tem-level fragility function which can effectively avoid theimpacts of different isolation parameters on differentcomponents As can be seen from Figures 14 and 17 theabove analysis results were also very apparent in this studyFor example the longitudinal deck displacement was sig-nificantly reduced by using dampers yet increased thetransverse demands of stiffening girder and pylons e useof LEB or LRB reduced the demands placed on columns yetled to an increase of the transverse deformations of abut-ments which was due to the use of shear keys limiting theexcessive sliding displacement of bearings erefore acomprehensive evaluation of retrofit strategies on bridgesparticularly long-span suspension bridges that have multiplecomponents should be based on system-level fragility in-stead of component-level
Taking the findings that no damages had occurred to thecable system and the definition of bearing damage states intoaccount the derivation of system-level fragility of the wholebridge depended on critical components including pylonscolumns bearings and abutments e fragility data byemploying the absolutely serial and serial-parallel assump-tions of bridge system are plotted in Figure 18 and comparedto the system fragility obtained from the proposed com-posite damage state e absolutely serial assumption
generated serious bias for fragility curves and may result inmisdirection if they were served as a design guide eabsolutely serial model overestimated the potential damageof the bridge system since the fragility curve directlydepended on the most vulnerable components In otherwords if the difference in location and performance of eachcomponent was neglected then the seismic performanceevaluation would be achieved by attaching equal importanceto the strength of pylons approach columns and bearingsis method contradicted the philosophy of capacity designin which the long-span structure can dissipate energythrough a reasonable swinging of stiffening girder and thebearings were designed as fuse elements in the small-to-medium-span bridgese serial-parallel model qualitativelyincorporated the differences of components and was verysensitive to the seismic design of the structure as seen fromthe comparisons among the cases in Figure 18 e modeldistinguished the serial components and parallel compo-nents yet still overestimated the bridge system-level fragilitywhen bearings became the vulnerable components espe-cially for long-span bridges is special situation such asthe serial assumption may also lead to misunderstandingOnce the relative importance between the components wasconsidered in the serial-composite model the damage po-tential was clearly reduced within the range of earthquakeintensity of interest based on the case studies is alsoreflected the necessity of the classification idea of bridgecomponents which can be expressed by the numerical value
Table 6 Damage states of the quasi-isolation and isolation bearings (m)
Figure 17 Probabilities of critical damage states occurrence sequences for the approach bridges when PGV is 1ms
Advances in Civil Engineering 17
of system-level fragility erefore the application ofcomposite damage state (equation (4)) considering com-ponent difference was instead more appropriate than theaforementioned two models
e damage probability distribution represented byblack circles in Figure 18 can be obtained from the definedcomposite damage state (equation (5)) where the parametera is 06 It was found that system fragility result was a moreaccurate assessment with further taking the relative im-portance between themain bridge and approach bridges intoaccount is is due to the principle that the seismic designperformance objective of long-span bridges generally has nocollapse for the main body structures and is necessarily opento emergency vehicle service In addition attention must bepaid to the fact that the differences in repair cost and time ofthe main and approach bridges are significant when thesame damage state is achieved e results of compositedamage state may match the performance objectives of theseismic design In order to investigate the influence ofimproved strategies and structural relative importance ondamage potential of bridges the mean values of the fragility
functions can be employed to evaluate the seismic designe smooth fragility curves for the specific damage states canbe fitted with the lognormal cumulative probability distri-bution function by employing the least-squares technique
P DSgtDSi | PGV1113858 1113859 Φln(pgv) minus λPGV
ζPGV1113888 1113889 (12)
where λPGV and ζPGV are the logarithmic mean and dis-persion values of earthquake intensity (PGV) for a specifieddamage level e larger mean values represent the smallerdamage potential to reach the specified level
Figure 19 indicates the mean values of earthquakeintensity at the specific damage states as a function of theimproved strategies and parameter a Each step of theretrofit measures would enhance the seismic performanceof the overall bridge system at all levels e improvementof the restraint system of the approach bridge played acritical role of the overall bridge system For instance themean values of the earthquake intensity featured mutationof slopes along the measure direction which reflected the
Figure 18 System fragility of the original design (a) Case 0 and improved design including (b) Case 1 and (c) Case 5
18 Advances in Civil Engineering
fact that the effectiveness of improvement for the ap-proach bridges was clearer than that of the main bridgee fragility of the highway network would be under-estimated if only the damage potential of the suspensionspan was adopted erefore when the structures such asapproach bridges transition piers and auxiliary pierswere ignored the damage identification and repair de-cisions would be misleading in the seismic evaluation oflong-span bridges Conversely the damaging probabilityof the overall bridge system will be amplified regardless ofthe higher importance of the main bridge When respectthe fact that the importance of suspension span is higherthan that of approach bridge in transportation networks aparameter a of 06ndash08 may be a better choice in thepresent study
e seismic fragility of conventionally designedbridges was mainly governed by the approach bridges at alldamage levels However for the retrofitted bridges thatpossess reasonable seismic performance different damagestates were controlled by the earthquake demands ofvarious structures as shown by the trend of the curvesalong parameter a in Case 4 and Case 5 e damagingpotential of the overall bridge from the low-to-moderatelimit states was dominated by the approach bridge eprimary reasons were the sliding of bearings and thetransfer of limited forces from the girder to substructure
by the damage of restriction devices of the approachbridges e extensive damage was controlled by thedisplacement of stiffening girder of the main bridge whichinferred the effect of dampers will be conspicuous understrong earthquake shaking At the same time the overalllong-span bridges with the reasonable seismic system willnot collapse during a strong earthquake event and canmeet emergency traffic service It can be seen that the meanvalues of fragility functions developed by compositedamage indices (equations (4) and (5)) are good indicatorsto evaluate seismic performance of long-span bridges andcan be consistent with the established seismic performanceobjectives
5 Conclusions
In the present study a procedure for investigating thereasonable seismic performance of long-span suspensionbridges was developed under the fragility function frame-work e seismic performance of a long-span suspensionbridge was examined and improved using the proposedprocedure from which the results revealed distinguishableconclusions as follows
(1) e definition of damage states for bearings con-sidered the interactions between the adjacent
014036059081103125
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(a)
020051081112142173
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
0002
0406
0810
CaseParameter a
(b)
130251371492612733
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(c)
Mea
n va
lue
exp
(λ) (
ms
)
Parameter a Case0 1 2 3 4 5
0001
0203
0405
131305478652826
1000
(d)
Figure 19 Influence of relative importance and improved designs on mean values of earthquake intensity for different damage states(a) slight (b) moderate (c) extensive (d) collapse
Advances in Civil Engineering 19
structures and self-damage to bearings in accor-dance with the actual design and experimentalconclusions e damage states of pylons in terms ofcurvature ductility were defined by developingmodal pushover analysis and idealized bilinear fit-ting methods in the equivalent pylon fiber models ofthe suspension bridge
(2) When investigating the component fragility curvesof the initial design it was found that the large deckdisplacement of the suspension span was the mostserious damage present and a slight inelastic re-sponse may have occurred only in the pylons Al-though the cable system was able to maintainelasticity under earthquakes the saddles at the py-lons were identified with a high-risk region indi-cating the damaging potential For the approachbridges the ductility mechanism did not realize arational damage sequence e columns were proneto collapse because of the application of fixed potbearings which in turn caused the damage to thebearings and abutments
(3) e stepwise improved seismic designs were pro-posed in terms of the component fragility results ofthe initial design Among the candidates fluid vis-cous dampers were proven to be a good choice forlong-span bridges based on the two attributes ofdisplacement and force when the similar efficiency ofdisplacement control was achieved e restraintsystem had a critical effect on the seismic responsesof the approach bridges e quasi-isolation orisolation can achieve a reasonable seismic damagesequence of the approach bridges
(4) e defined composite damage indices showed thatthe incorporate component classification conceptand structural relative importance were necessary toderive a good indicator of performance evaluationdue to the fact that the damage indices were based oneither an absolutely serial or serial-parallel systemmodel may significantly overestimate the damagepotential of the overall bridge system e meanvalues of earthquake intensity at the specific damagestates can serve as a function of the improvedstrategies and relative importance to further evaluatethe overall seismic performance of the suspensionbridge It can be concluded that the damaging po-tential of the overall bridge from the low-to-mod-erate limit states was governed by the approachbridges in the retrofitted bridges with a favorableseismic performance whereas the displacementdemand of the suspension span dominated thefragility probability of the extensive damage
Data Availability
e nature of the data is the analytical data of finite elementmodels of the case bridge e data used to support thefindings of this study are available from the correspondingauthor upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the Basic Research ServiceProject of Central-Level Public Welfare Research Institute(grant nos 2016-9018 and 2019-0034) and Science andTechnology Project of Communicationsrsquo Construction inWestern China MOC (grant no 2009318223094)
References
[1] L W David A Kartoum K Chang and I Roy ldquoSuspensionreport seismic performance and design considerations of longspan suspension bridgesrdquo Report CSMIP00-03 CaliforniaDepartment of Conservation Division of Mines And GeologyOffice of Strong Motion Studies Sacramento CA USA 2000
[2] S-G Gwon and D-H Choi ldquoImproved continuummodel forfree vibration analysis of suspension bridgesrdquo Journal ofEngineering Mechanics vol 143 no 7 Article ID 040170382017
[3] S Arzoumanidis A Shama and F Ostadan ldquoPerformance-based seismic analysis and design of suspension bridgesrdquoEarthquake Engineering and Structural Dynamics vol 34no 4-5 pp 349ndash367 2005
[4] L Sgambi E Garavaglia N Basso and F Bontempi ldquoMonteCarlo simulation for seismic analysis of a long span sus-pension bridgerdquo Engineering Structures vol 78 pp 100ndash1112014
[5] D Karmakar S Ray-Chaudhuri and M Shinozuka ldquoFiniteelement model development validation and probabilisticseismic performance evaluation of Vincent omas suspen-sion bridgerdquo Structure and Infrastructure Engineering vol 11no 2 pp 223ndash237 2015
[6] W-F Chen and L Duan Bridge Engineering HandbookSeismic Design CRC Press Boca Raton FL USA 2nd edition2014
[7] E Choi ldquoSeismic analysis and retrofit of mid-americabridgesrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2002
[8] B G Nielson and R DesRoches ldquoSeismic fragility method-ology for highway bridges using a component level approachrdquoEarthquake Engineering amp Structural Dynamics vol 36 no 6pp 823ndash839 2007
[9] K N Ramanathan ldquoNext generation seismic fragility curvesfor california bridges incorporating the evolution in seismicdesignrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2012
[10] O Avsar A Yakut and A Caner ldquoAnalytical fragility curvesfor ordinary highway bridges in Turkeyrdquo Earthquake Spectravol 27 no 4 pp 971ndash996 2011
[11] S P Stefanidou and A J Kappos ldquoMethodology for thedevelopment of bridge-specific fragility curvesrdquo EarthquakeEngineering amp Structural Dynamics vol 46 no 1 pp 73ndash932016
[12] B Borzi P Ceresa P Franchin F Noto G M Calvi andP E Pinto ldquoSeismic vulnerability of the Italian roadwaybridge stockrdquo Earthquake Spectra vol 31 no 4pp 2137ndash2161 2015
[13] W T Barnawi and S J Dyke ldquoSeismic fragility relationshipsof a cable-stayed bridge equipped with response modification
20 Advances in Civil Engineering
systemsrdquo Journal of Bridge Engineering vol 19 no 8 ArticleID A4013003 2014
[14] J Zhong Y Pang J Jong-Su J-S Jeon R DesRoches andW Yuan ldquoSeismic fragility assessment of long-span cable-stayed bridges in Chinardquo Advances in Structural Engineeringvol 19 no 11 pp 1797ndash1812 2016
[15] J Zhong Z Hu W Yuan and L Chen ldquoSystem-basedprobabilistic optimization of fluid viscous dampers equippedin cable-stayed bridgesrdquo Advances in Structural Engineeringvol 21 no 12 pp 1815ndash1825 2018
[16] J E Padgett and R DesRoches ldquoMethodology for the de-velopment of analytical fragility curves for retrofitted bridgesfitted bridgesrdquo Earthquake Engineering amp Structural Dy-namics vol 37 no 8 pp 1157ndash1174 2008
[17] Y Xie and J Zhang ldquoOptimal design of seismic protectivedevices for highway bridges using performance-basedmethodology and multiobjective genetic optimizationrdquoJournal of Bridge Engineering vol 22 no 3 Article ID04016129 2017
[18] J Zhang and Y Huo ldquoEvaluating effectiveness and optimumdesign of isolation devices for highway bridges using thefragility function methodrdquo Engineering Structures vol 31no 8 pp 1648ndash1660 2009
[19] F McKenna M H Scott and G L Fenves ldquoNonlinear finite-element analysis software architecture using object compo-sitionrdquo Journal of Computing in Civil Engineering vol 24no 1 pp 95ndash107 2010
[20] MOTC ldquoGuidelines for seismic design of highway bridgesrdquoChina Ministry of Transportation and CommunicationsPress Beijing China JTGT B02-01-2008 2008
[21] A D Kiureghian and J L Sackman ldquoTangent geometricstiffness of inclined cables under self-weightrdquo Journal ofStructural Engineering vol 131 no 6 pp 941ndash945 2005
[22] J S Steelman E T Filipov L A Fahnestock et al ldquoEx-perimental behavior of steel fixed bearings and implicationsfor seismic bridge responserdquo Journal of Bridge Engineeringvol 19 no 8 Article ID A4014007 2014
[23] B Madani F Behnamfar and H Tajmir Riahi ldquoDynamicresponse of structures subjected to pounding and structure-soil-structure interactionrdquo Soil Dynamics and EarthquakeEngineering vol 78 pp 46ndash60 2015
[24] L Xu and J Li ldquoDesign and experimental investigation of anew type sliding retainer and its efficacy in seismic fortifi-cationrdquo Engineering Mechanics vol 33 no 2 pp 111ndash1182016 in Chinese
[25] Caltrans Caltrans Seismic Design Criteria California De-partment of Transportation Sacramento CA USA 2013
[26] A Shamsabadi K M Rollins and M Kapuskar ldquoNonlinearsoil-abutment-bridge structure interaction for seismic per-formance-based designrdquo Journal of Geotechnical and Geo-environmental Engineering vol 133 no 6 pp 707ndash720 2007
[27] X Li Z Sun D Wang and Y Shi ldquoLongitudinal seismicpounding effect of bridges abutment and backfilling damagerdquoJournal of Changrsquoan University vol 35 no 4 pp 76ndash82 2015in Chinese
[28] MOTC ldquoSpecification for design of highway suspensionbridgerdquo China Ministry of Transportation and Communi-cations Press Beijing China JTGT D65-05-2015 2015
[29] A Shafieezadeh K Ramanathan J E Padgett andR DesRoches ldquoFractional order intensity measures forprobabilistic seismic demand modeling applied to highwaybridgesrdquo Earthquake Engineering amp Structural Dynamicsvol 41 no 3 pp 391ndash409 2012
[30] J E Padgett B G Nielson and R DesRoches ldquoSelection ofoptimal intensity measures in probabilistic seismic demandmodels of highway bridge portfoliosrdquo Earthquake Engineeringamp Structural Dynamics vol 37 no 5 pp 711ndash725 2008
[31] K R Mackie and B Stojadinovic ldquoFragility basis for californiahighway overpass bridge seismic decision makingrdquo PEERReport 200502 Pacific Earthquake Engineering ResearchCenter University of California Berkeley Berkeley CA USA2005
[32] J Li T Peng Ji Lin L Zhou and YuL Deng Seismic Design ofMulti-Pylon Multi-Span Suspension Bridge China Commu-nications Press Beijing China 2013 in Chinese
[33] L Chen W Zhuang H Zhao and Z Wan ldquoReport onHighwaysrsquo Damage in the Wenchuan Earthquake-BridgeChina Communications Press Beijing China 2012 inChinese
[34] G Wu K Wang G Lu and P Zhang ldquoAn experimentalinvestigation of unbonded laminated elastomeric bearingsand the seismic evaluations of highway bridges with testedbearing componentsrdquo Shock and Vibration vol 2018 ArticleID 8439321 18 pages 2018
Advances in Civil Engineering 21
pylons and columns [9] e COV of main cables andsuspenders was 01 taking the low variability of highstrength steel strands into account [16] e descriptivedispersions of the remaining components were calculated tobe 025 for the two lower damage states (DS1 and DS2) and047 for the higher damage states (DS3 and DS4) [8]
36 Assessment of Vulnerable Components e componentfragility curves were derived by steps 1ndash3 detailed in Section 2to identify vulnerable components and regions as shown inFigures 11 and 12e pot bearings were shown to be the mostvulnerable components for the main bridge which inferred thecomplete damages to pot bearings themselves and reflected ahigher probability of pounding between the adjacent decks andunseating in the longitudinal direction e transverse re-sponses of bearings did not cause obvious damage compared tothe longitudinal responses e pylons were the primarycomponents for load carrying where inelasticmaterial responsecould potentially occur e transverse vulnerability was moreobvious than the longitudinal vulnerability for the lowerdamage states However the probability of extensive damagewas extremely low or even zeroWith respect to themain cablesand suspenders the increases in their forces which couldpotentially occur during rare earthquakes were confined toabout 15 (for themean values) of the forces resulting from thedead loads as illustrated in Figure 12 which also indicated thatthe maximum values of main cable stress had occurred at thesaddles of pylons and in the side spans e stress values of
suspenders achieved the maximum at the middle and end Insummary the cable system was found to be secured againstdamages in which the stress values were much lower than theyield values even under strong earthquake shaking It wasnoted that the large deflection and cable angle changes betweenthe main span and side spans would lead to the strong shear inthe longitudinal direction between the cables and saddles andthe slipping damages to the saddles at the top of the pylonsduring earthquakes were identified as high-risk
For the approach bridges the columns were determinedto be the most fragile components in both the longitudinaland transverse directions e application of the fixed potbearings resulted in a large force and deformation of thecolumns the complete damage of sliding pot bearings andapparent pounding between the decks and abutments eseactions subsequently caused the inelastic responses ofabutments but the shear keys were protected from failuretransversely Although the ductility mechanism of approachbridges was designed it was considered to be a defectiveseismic system with irrational sequences of componentdamage occurrences in high-seismicity regions
41 ImprovedDesign ofMainBridge As results shown in theevaluation of vulnerable components a large deck dis-placement of the suspension span would occur duringearthquakes which caused pounding and even unseating
Original curveBilinear curve
MprimeyMy
φs or φccφucφprimeyφy
DS4 reinforcing bar rupture core concrete crushing
DS3 major spalling of cover concrete exposed core concrete
DS2 formation of stable plastic hinge in the pylon (column)
Mom
ent
Curvature
DS1 narrow cracking in concrete first yielding of reinforcing bar
Figure 10 e damage state definitions of the pylon and column cross-sections
Table 4 Limit state thresholds of pylon and column cross-sections
Figure 11 Component fragility curves for the suspension bridge (a) main bridge (b) approach bridge
200 400 600 800 10000Location (m)
00
02
04
06
08
10
MA
F
MaximumMinimumMean
Dead loadYield valueHigh stress region
(a)
00
02
04
06
08
10
SAF
200 400 600 800 10000Location (m)
MaximumMinimumMean High stress region
Dead loadYield value
(b)
Figure 12 Force distributions of the (a) main cables and (b) suspenders
12 Advances in Civil Engineering
when there were no additional restraints in the longitudinaldirection erefore it was consistent to suggest that sup-plementary devices should be equipped to mitigate the deckmovements to some extente reduction of deck or bearingdisplacements can be taken as the retrofitted objective for thesuspension span Fragility analyses were conducted on theretrofitted bridges that covered the general strategies asfollows
Case 1 the analysis model of the bridge used a fluidviscous damper equipped between the deck and eachpedestal on both sides of the bridgeCase 2 the analysis model of the bridge used an elasticcable installed between the deck and each pedestal onboth sides of the bridgeCase 3 the analysis model of the bridge used three pairsof flexible central buckles located in the middle sectionof the span
e main bridge was assumed to have identical me-chanical parameters at each location in various designschemes e dampers and elastic cables were respectivelymodeled using zero-length elements with Maxwell andtension-only materials and the flexible central buckles weresimulated as the same as the suspenders elements inOpenSEES
e experimental testing of fluid viscous dampers hasshown that the force-velocity with fractional velocity powerlaw relation can describe the suitable mechanical behavior
fD CD| _u|αsgn( _u) (9)
where α is a real positive exponent of which value dependson the piston head orifice CD is viscous damping coefficientwith units of force per velocity raised to the α power kNmiddot(ms)-α _u indicates the velocity of damper and sgn(middot) is thesignum function Research by Li et al [32] indicated thatviscous damping coefficients range from 2500 kN (ms)-α to5000 kN (ms)-α and can be simulated with a uniform dis-tribution e velocity exponents can be assumed to varyuniformly from 03 to 05
e axial stiffness of elastic cables and flexible centralbuckles is regarded with the material length and cross-section area It was assumed that the yield strength of steelstrands of fy 1420MPa was the mean value and followed alognormal distribution with a COV 01 e stiffnessvalues of single elastic cable based on past work employed auniform distribution ranging from 125times105 kNm to25 times105 kNm In addition the other mechanical prop-erties such as length and area can be determined by thefollowing relations
LgtEΔfy
1113888 1113889
kE EA
L1113874 1113875
(10)
where L E and fy are the length Youngrsquos modulus andyield stress of elastic cables respectively and Δ represents the
controlled deck displacement which was available for 06min the examined bridge
According to expert opinion three pairs of flexiblecentral buckles were the most common type of centralbuckle of which the cross-section areas were the same asthose of the suspenders
Figures 13(a) and 13(b) illustrate bearing displacementdistributions are formed for the three retrofit measures at thesame ground motion intensity levels e mechanical be-haviors of measures strongly affect the effectiveness It can beconcluded that owing to the similar EDP distributions ofdampers and elastic cables the corresponding groundmotion intensities were similar when generating the samepounding probability e flexible central buckles wereextremely susceptible to yielding failure thereby causing lowefficiency in the displacement restrictions ereforepounding probabilities can be directly observed by condi-tioning on the median responses of dampers and elasticcables instead (Figure 13(c)) e control force produced bydevices is another noteworthy consideration in retrofitdesign e more force the devices require the larger andmore expensive they usually are [13] To take both defor-mation and force into account Figures 13(d) and 13(e)provide the contours of the pounding probability ecomparison showed that the damper deformations featuredslightly larger slopes in the horizontal direction yet thedifference in effect on the deformations of the dampers andelastic cables was not significant for reducing the objectivepounding probability Conversely damper force was lesssensitive to the pounding probability than cable force eelastic cable forces reached on average up to e32 (asymp245)times those of the damper under the same poundingprobabilities ese findings implied that difficult anchoragechallenges existed in the case of guaranteeing the efficiencyof elastic cables and the bridge decks possibly experiencedhigher stress In addition the difference in force was sub-stantial when controlling the displacement which showedthat dampers can contribute additional damping and stiff-ness to the structural system while elastic cables completelytransferred the inertial forces erefore the coordinationand balance of displacements and forces should be simul-taneously considered in the vibration mitigation of large-span bridges In this study the application of fluid viscousdampers was identified as the preferred retrofit in thelongitudinal direction for the main bridge e additionaldamping ratio provided by the fluid viscous dampers can becalculated by the following equation
ε CD
2mωeff (11)
where CD is damping coefficient of linear fluid damperskNmiddot(sm) which is equivalent to nonlinear dampers basedon equal energy dissipation per hysteresis loop [17] ωeff issystem natural frequency corresponding to the vibrationalong the axis of bridge andm is system mass It was shownthat an additional damping of 206ndash318 can offer aneffective controlling mechanism for the reduction ofpounding between the adjacent decks
Advances in Civil Engineering 13
Figure 14 depicts the change of components vulnera-bilities before and after the application of dampers It can beobserved that the retrofit measure played limited roles in thefragility reduction of the self-damaged bearings in long-spanbridges Since the additional longitudinal rigidity to girderwas provided by the dampers the transverse responses suchas girder displacement and pylon curvature would increaseas ground motion intensity increased e influences on thecable system responses and the longitudinal responses ofpylons were found to be negligible Another phenomenonwas that the longitudinal fragility of the abutments signif-icantly decreased with the reduction in the poundingprobabilities between the decks and abutments of the ap-proach bridges It was inferred that the pounding betweenthe adjacent decks was likely to affect movement of theapproach bridges and that the pounding effect would betransmitted along the girder
42 Improved Design of Approach Bridges e damagepotential of each component for the approach bridges washigh due to the high collapse risks posed by the usage of thefixed piers e structural designs of small-to-medium-spanhighway bridges in China show the fact that the super-structures are usually placed onto laminated elastomericbearings (LEBs) which tend to come into direct contact withthe bridge girders and substructures without anchoring etypical types of seismic damages of these bridges consisted of
large girder movements bearings sliding failures of shearkeys and expansion joints and abutment cracking duringthe Wenchuan and Yushu earthquake events in ChinaMeanwhile the bridge columns displayed relatively lowdamage ratios under earthquakes [33] e design can bereferred to as a quasi-isolation system which is primarilyrealized by strategically employing bearings sliding as force-limiting connections between girders and substructures Inaddition the isolation strategies of lead-rubber bearings(LRBs) are usually employed in the high-seismicity regionswhich can be taken as a comparison of the quasi-isolationsystem e two restraint systems consist of bearings shearkeys transversely located at the cap beams and restrainercables longitudinally anchored to the abutments (or ped-estals) In order to calculate the seismic responses of theapproach bridge the related experiments were completedand summarized and applied to the force-displacementrelations of the restraint system components
e detailed experiment of LEB consulted previousacademic achievements [34] An example is shown inFigure 15(a) where the feature points marked by rectangularand circular represented initial walking and obvious slidingresponses which illustrated that the mechanical property ofbearings would degrade as the displacements increased afterentering the friction sliding state until it remained nearlyconstant When there was no specific test the horizontalforce Fd in the stable friction stage can be 07 Fmax whereFmax was the corresponding maximum static friction at the
Bear
ing
disp
lace
men
t (m
)
Logarithmic spacendash5
ndash4
ndash3
ndash2
ndash1
0
1
2 4 6PGV (cms)
Case 1Case 2Case 3
(a)Po
undi
ng p
roba
bilit
y
00
02
04
06
08
05 10 15 2000PGV (ms)
Case 1Case 2Case 3
(b)
Poun
ding
pro
babi
lity
00
02
04
06
02 04 06 08 1000Deformation (m)
Case 1Case 2
(c)
ln [f
orce
(kN
)]
5
10
15
15
20
20
25
25
30
3030
35
3535
40
40
4550
7
72
74
76
78
8
033 041 049 057 065025Deformation (m)
(d)
ln [f
orce
(kN
)]5
10
15
15 20
20 25
25
30
3030
35
35
40
40
45
50
102
104
106
108
11
112
114
0155 0205 0255 0305 03550105Deformation (m)
(e)
Figure 13 Choice of the three retrofitmeasures based on deformations and forces (a) EDP-IM relation (b) pounding probability conditioned onIM (c) pounding probability conditioned on deformation contours of pounding probability for (d) dampers and (e) elastic cables
14 Advances in Civil Engineering
initial walking e displacement value Δ2 can be of 250effective shear strain (ESS) e initial stiffness K0 is cal-culated by MOTC [20] erefore the coefficients can becalculated by the following sequence FdK0⟶ Fmax⟶Δ1⟶Δ2 e built analytical relation ofthe LEB is shown in Figure 16(a) which can correctlycapture mechanical behavior under earthquakes A log-normal distribution [8] with a logarithmic standard devi-ation of 01 and a normal distribution with aCOV 016MPa were used to model the randomness offriction coefficient and shear modulus respectively For theirrespective mean values 02 and 12MPa were taken
Figure 15(b) shows a cyclic loading experiment of LRBwhich reveals that the degradation of bearings stiffness willoccur with the increase of loading displacements e elasticstiffness can reach uniformly 3 to 7 times the postyieldingstiffness according to the experimental results e
mechanical behavior of LRB can be modeled by a bilinearrelation as seen in Figure 16(b)
e cable restrainers were modeled as nonlinear tension-only elements characterized by an initial slack as shown inFigure 16(c) e parameters affecting the functionality ofcable restrainers were the cable slack length and yieldstrength according to the research of Padgett and DesRoches[16] e first two parameters were assumed to follow theuniform distributions and the strength can be modeled bythe lognormal distribution An adopted range of cablestiffness values was from 30times105 kNm to 50times105 kNmuniformly
e key parameters in analytical models of Figure 16 canbe determined according to the structural design of therealistic bridge e damage states of two restraint systemsare listed in Table 6 in which ESS is used as the damageindex
Dam
age p
roba
bilit
y
05 10 15 2000PGV (ms)
00
02
04
06
08
10
Case 0Case 1
MBDLMBDT
(a)
Dam
age p
roba
bilit
y
00
01
02
03
04
05
05 10 15 2000PGV (ms)
Case 0Case 1
PCDNLPCDNT
(b)
Dam
age p
roba
bilit
y
00
02
04
06
08
10
05 10 15 2000PGV (ms)
Case 0Case 1
BPDAEJDA
(c)
Figure 14 Influence of dampers on vulnerabilties of (a) bearing responses for DS2 (b) pylon responses for DS2 and (c) abutment andpounding responses
Advances in Civil Engineering 15
Figure 17 illustrates the probabilities of critical damage stateoccurrence sequences of the bridge components in Cases 1 4and 5 when PGV is 10ms which is equal to E2 earthquakeintensity in the seismic assessment reports e componentvulnerabilities of initial design (Case 0) were taken as the ref-erence values which were marked by different colored strips inFigure 17 e desirable sequence preferred Cases 4 or 5 for theapproach bridgese initial design of approach bridges resultedin unreasonable damage sequences In the strategies pertainingto quasi-isolation and isolation systems multiple tiers of seismicstructural redundancy can be normally employed to preventstructural collapse during strong earthquake events Tier 1 in-dicated that the bearings were weakly fused which was evidentin the high fragility of bearing sliding behaviors (ABDAL andABDC2T) Another advantage was that the extensive dam-ages of bearing would be reduced when compared with the self-damage of the pot bearings Tier 2 indicated that the designedbearings had sufficient seat widths for adapting to the sliding ofsuperstructures However the devices had a certain damageprobability (EJDA and SKDC2) and dissipated a portion of
the seismic energy when the displacement of the superstructuresbecame larger than the allowable values of constraint devicesTier 3 redundancy was designed as a potential plastic hinge inthe substructures (CCDC2) including a certain yielding of thebackfill at the abutments (BPDA) e response of theabutments in the transverse direction (PDAT) was slightlyincreased due to the pounding effect between the girders andshear keys caused by bearing sliding e reasonable damagesequence of two restraint systems can be reflected through thecorresponding damage probabilities of bearings seismic mea-sures and substructures respectively as indicated by the red andgreen lines in Figure 17 erefore the rational sequence ofcritical limit state occurrences for the approach bridges isbearings⟶ expansion joints (or restrainer cables) shearkeys⟶ columns abutments
43 Comprehensive Evaluation of Bridge System Padgett andDesRoches [16] pointed out the necessity to derive the bridgesystem vulnerabilities when assessing retrofitted bridges
Figure 15 Experimental results of (a) LEB and (b) LRB
ΔΔ1
F
Δ2
Fmax
Fd
KdK0
(a)
Δy Δ
F
K1 = (3~7) K2
K1
K2
(b)
Slack
F
K
Δ
Fy
(c)
Figure 16 Analytical models of (a) LEB (b) LRB and (c) cable restrainers
16 Advances in Civil Engineering
because different components may have a different responsewith a given retrofit measure Also Zhang and Huo [18]studied the optimal parameters of isolation bearings byderiving the intensity median value corresponding to sys-tem-level fragility function which can effectively avoid theimpacts of different isolation parameters on differentcomponents As can be seen from Figures 14 and 17 theabove analysis results were also very apparent in this studyFor example the longitudinal deck displacement was sig-nificantly reduced by using dampers yet increased thetransverse demands of stiffening girder and pylons e useof LEB or LRB reduced the demands placed on columns yetled to an increase of the transverse deformations of abut-ments which was due to the use of shear keys limiting theexcessive sliding displacement of bearings erefore acomprehensive evaluation of retrofit strategies on bridgesparticularly long-span suspension bridges that have multiplecomponents should be based on system-level fragility in-stead of component-level
Taking the findings that no damages had occurred to thecable system and the definition of bearing damage states intoaccount the derivation of system-level fragility of the wholebridge depended on critical components including pylonscolumns bearings and abutments e fragility data byemploying the absolutely serial and serial-parallel assump-tions of bridge system are plotted in Figure 18 and comparedto the system fragility obtained from the proposed com-posite damage state e absolutely serial assumption
generated serious bias for fragility curves and may result inmisdirection if they were served as a design guide eabsolutely serial model overestimated the potential damageof the bridge system since the fragility curve directlydepended on the most vulnerable components In otherwords if the difference in location and performance of eachcomponent was neglected then the seismic performanceevaluation would be achieved by attaching equal importanceto the strength of pylons approach columns and bearingsis method contradicted the philosophy of capacity designin which the long-span structure can dissipate energythrough a reasonable swinging of stiffening girder and thebearings were designed as fuse elements in the small-to-medium-span bridgese serial-parallel model qualitativelyincorporated the differences of components and was verysensitive to the seismic design of the structure as seen fromthe comparisons among the cases in Figure 18 e modeldistinguished the serial components and parallel compo-nents yet still overestimated the bridge system-level fragilitywhen bearings became the vulnerable components espe-cially for long-span bridges is special situation such asthe serial assumption may also lead to misunderstandingOnce the relative importance between the components wasconsidered in the serial-composite model the damage po-tential was clearly reduced within the range of earthquakeintensity of interest based on the case studies is alsoreflected the necessity of the classification idea of bridgecomponents which can be expressed by the numerical value
Table 6 Damage states of the quasi-isolation and isolation bearings (m)
Figure 17 Probabilities of critical damage states occurrence sequences for the approach bridges when PGV is 1ms
Advances in Civil Engineering 17
of system-level fragility erefore the application ofcomposite damage state (equation (4)) considering com-ponent difference was instead more appropriate than theaforementioned two models
e damage probability distribution represented byblack circles in Figure 18 can be obtained from the definedcomposite damage state (equation (5)) where the parametera is 06 It was found that system fragility result was a moreaccurate assessment with further taking the relative im-portance between themain bridge and approach bridges intoaccount is is due to the principle that the seismic designperformance objective of long-span bridges generally has nocollapse for the main body structures and is necessarily opento emergency vehicle service In addition attention must bepaid to the fact that the differences in repair cost and time ofthe main and approach bridges are significant when thesame damage state is achieved e results of compositedamage state may match the performance objectives of theseismic design In order to investigate the influence ofimproved strategies and structural relative importance ondamage potential of bridges the mean values of the fragility
functions can be employed to evaluate the seismic designe smooth fragility curves for the specific damage states canbe fitted with the lognormal cumulative probability distri-bution function by employing the least-squares technique
P DSgtDSi | PGV1113858 1113859 Φln(pgv) minus λPGV
ζPGV1113888 1113889 (12)
where λPGV and ζPGV are the logarithmic mean and dis-persion values of earthquake intensity (PGV) for a specifieddamage level e larger mean values represent the smallerdamage potential to reach the specified level
Figure 19 indicates the mean values of earthquakeintensity at the specific damage states as a function of theimproved strategies and parameter a Each step of theretrofit measures would enhance the seismic performanceof the overall bridge system at all levels e improvementof the restraint system of the approach bridge played acritical role of the overall bridge system For instance themean values of the earthquake intensity featured mutationof slopes along the measure direction which reflected the
Figure 18 System fragility of the original design (a) Case 0 and improved design including (b) Case 1 and (c) Case 5
18 Advances in Civil Engineering
fact that the effectiveness of improvement for the ap-proach bridges was clearer than that of the main bridgee fragility of the highway network would be under-estimated if only the damage potential of the suspensionspan was adopted erefore when the structures such asapproach bridges transition piers and auxiliary pierswere ignored the damage identification and repair de-cisions would be misleading in the seismic evaluation oflong-span bridges Conversely the damaging probabilityof the overall bridge system will be amplified regardless ofthe higher importance of the main bridge When respectthe fact that the importance of suspension span is higherthan that of approach bridge in transportation networks aparameter a of 06ndash08 may be a better choice in thepresent study
e seismic fragility of conventionally designedbridges was mainly governed by the approach bridges at alldamage levels However for the retrofitted bridges thatpossess reasonable seismic performance different damagestates were controlled by the earthquake demands ofvarious structures as shown by the trend of the curvesalong parameter a in Case 4 and Case 5 e damagingpotential of the overall bridge from the low-to-moderatelimit states was dominated by the approach bridge eprimary reasons were the sliding of bearings and thetransfer of limited forces from the girder to substructure
by the damage of restriction devices of the approachbridges e extensive damage was controlled by thedisplacement of stiffening girder of the main bridge whichinferred the effect of dampers will be conspicuous understrong earthquake shaking At the same time the overalllong-span bridges with the reasonable seismic system willnot collapse during a strong earthquake event and canmeet emergency traffic service It can be seen that the meanvalues of fragility functions developed by compositedamage indices (equations (4) and (5)) are good indicatorsto evaluate seismic performance of long-span bridges andcan be consistent with the established seismic performanceobjectives
5 Conclusions
In the present study a procedure for investigating thereasonable seismic performance of long-span suspensionbridges was developed under the fragility function frame-work e seismic performance of a long-span suspensionbridge was examined and improved using the proposedprocedure from which the results revealed distinguishableconclusions as follows
(1) e definition of damage states for bearings con-sidered the interactions between the adjacent
014036059081103125
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(a)
020051081112142173
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
0002
0406
0810
CaseParameter a
(b)
130251371492612733
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(c)
Mea
n va
lue
exp
(λ) (
ms
)
Parameter a Case0 1 2 3 4 5
0001
0203
0405
131305478652826
1000
(d)
Figure 19 Influence of relative importance and improved designs on mean values of earthquake intensity for different damage states(a) slight (b) moderate (c) extensive (d) collapse
Advances in Civil Engineering 19
structures and self-damage to bearings in accor-dance with the actual design and experimentalconclusions e damage states of pylons in terms ofcurvature ductility were defined by developingmodal pushover analysis and idealized bilinear fit-ting methods in the equivalent pylon fiber models ofthe suspension bridge
(2) When investigating the component fragility curvesof the initial design it was found that the large deckdisplacement of the suspension span was the mostserious damage present and a slight inelastic re-sponse may have occurred only in the pylons Al-though the cable system was able to maintainelasticity under earthquakes the saddles at the py-lons were identified with a high-risk region indi-cating the damaging potential For the approachbridges the ductility mechanism did not realize arational damage sequence e columns were proneto collapse because of the application of fixed potbearings which in turn caused the damage to thebearings and abutments
(3) e stepwise improved seismic designs were pro-posed in terms of the component fragility results ofthe initial design Among the candidates fluid vis-cous dampers were proven to be a good choice forlong-span bridges based on the two attributes ofdisplacement and force when the similar efficiency ofdisplacement control was achieved e restraintsystem had a critical effect on the seismic responsesof the approach bridges e quasi-isolation orisolation can achieve a reasonable seismic damagesequence of the approach bridges
(4) e defined composite damage indices showed thatthe incorporate component classification conceptand structural relative importance were necessary toderive a good indicator of performance evaluationdue to the fact that the damage indices were based oneither an absolutely serial or serial-parallel systemmodel may significantly overestimate the damagepotential of the overall bridge system e meanvalues of earthquake intensity at the specific damagestates can serve as a function of the improvedstrategies and relative importance to further evaluatethe overall seismic performance of the suspensionbridge It can be concluded that the damaging po-tential of the overall bridge from the low-to-mod-erate limit states was governed by the approachbridges in the retrofitted bridges with a favorableseismic performance whereas the displacementdemand of the suspension span dominated thefragility probability of the extensive damage
Data Availability
e nature of the data is the analytical data of finite elementmodels of the case bridge e data used to support thefindings of this study are available from the correspondingauthor upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the Basic Research ServiceProject of Central-Level Public Welfare Research Institute(grant nos 2016-9018 and 2019-0034) and Science andTechnology Project of Communicationsrsquo Construction inWestern China MOC (grant no 2009318223094)
References
[1] L W David A Kartoum K Chang and I Roy ldquoSuspensionreport seismic performance and design considerations of longspan suspension bridgesrdquo Report CSMIP00-03 CaliforniaDepartment of Conservation Division of Mines And GeologyOffice of Strong Motion Studies Sacramento CA USA 2000
[2] S-G Gwon and D-H Choi ldquoImproved continuummodel forfree vibration analysis of suspension bridgesrdquo Journal ofEngineering Mechanics vol 143 no 7 Article ID 040170382017
[3] S Arzoumanidis A Shama and F Ostadan ldquoPerformance-based seismic analysis and design of suspension bridgesrdquoEarthquake Engineering and Structural Dynamics vol 34no 4-5 pp 349ndash367 2005
[4] L Sgambi E Garavaglia N Basso and F Bontempi ldquoMonteCarlo simulation for seismic analysis of a long span sus-pension bridgerdquo Engineering Structures vol 78 pp 100ndash1112014
[5] D Karmakar S Ray-Chaudhuri and M Shinozuka ldquoFiniteelement model development validation and probabilisticseismic performance evaluation of Vincent omas suspen-sion bridgerdquo Structure and Infrastructure Engineering vol 11no 2 pp 223ndash237 2015
[6] W-F Chen and L Duan Bridge Engineering HandbookSeismic Design CRC Press Boca Raton FL USA 2nd edition2014
[7] E Choi ldquoSeismic analysis and retrofit of mid-americabridgesrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2002
[8] B G Nielson and R DesRoches ldquoSeismic fragility method-ology for highway bridges using a component level approachrdquoEarthquake Engineering amp Structural Dynamics vol 36 no 6pp 823ndash839 2007
[9] K N Ramanathan ldquoNext generation seismic fragility curvesfor california bridges incorporating the evolution in seismicdesignrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2012
[10] O Avsar A Yakut and A Caner ldquoAnalytical fragility curvesfor ordinary highway bridges in Turkeyrdquo Earthquake Spectravol 27 no 4 pp 971ndash996 2011
[11] S P Stefanidou and A J Kappos ldquoMethodology for thedevelopment of bridge-specific fragility curvesrdquo EarthquakeEngineering amp Structural Dynamics vol 46 no 1 pp 73ndash932016
[12] B Borzi P Ceresa P Franchin F Noto G M Calvi andP E Pinto ldquoSeismic vulnerability of the Italian roadwaybridge stockrdquo Earthquake Spectra vol 31 no 4pp 2137ndash2161 2015
[13] W T Barnawi and S J Dyke ldquoSeismic fragility relationshipsof a cable-stayed bridge equipped with response modification
20 Advances in Civil Engineering
systemsrdquo Journal of Bridge Engineering vol 19 no 8 ArticleID A4013003 2014
[14] J Zhong Y Pang J Jong-Su J-S Jeon R DesRoches andW Yuan ldquoSeismic fragility assessment of long-span cable-stayed bridges in Chinardquo Advances in Structural Engineeringvol 19 no 11 pp 1797ndash1812 2016
[15] J Zhong Z Hu W Yuan and L Chen ldquoSystem-basedprobabilistic optimization of fluid viscous dampers equippedin cable-stayed bridgesrdquo Advances in Structural Engineeringvol 21 no 12 pp 1815ndash1825 2018
[16] J E Padgett and R DesRoches ldquoMethodology for the de-velopment of analytical fragility curves for retrofitted bridgesfitted bridgesrdquo Earthquake Engineering amp Structural Dy-namics vol 37 no 8 pp 1157ndash1174 2008
[17] Y Xie and J Zhang ldquoOptimal design of seismic protectivedevices for highway bridges using performance-basedmethodology and multiobjective genetic optimizationrdquoJournal of Bridge Engineering vol 22 no 3 Article ID04016129 2017
[18] J Zhang and Y Huo ldquoEvaluating effectiveness and optimumdesign of isolation devices for highway bridges using thefragility function methodrdquo Engineering Structures vol 31no 8 pp 1648ndash1660 2009
[19] F McKenna M H Scott and G L Fenves ldquoNonlinear finite-element analysis software architecture using object compo-sitionrdquo Journal of Computing in Civil Engineering vol 24no 1 pp 95ndash107 2010
[20] MOTC ldquoGuidelines for seismic design of highway bridgesrdquoChina Ministry of Transportation and CommunicationsPress Beijing China JTGT B02-01-2008 2008
[21] A D Kiureghian and J L Sackman ldquoTangent geometricstiffness of inclined cables under self-weightrdquo Journal ofStructural Engineering vol 131 no 6 pp 941ndash945 2005
[22] J S Steelman E T Filipov L A Fahnestock et al ldquoEx-perimental behavior of steel fixed bearings and implicationsfor seismic bridge responserdquo Journal of Bridge Engineeringvol 19 no 8 Article ID A4014007 2014
[23] B Madani F Behnamfar and H Tajmir Riahi ldquoDynamicresponse of structures subjected to pounding and structure-soil-structure interactionrdquo Soil Dynamics and EarthquakeEngineering vol 78 pp 46ndash60 2015
[24] L Xu and J Li ldquoDesign and experimental investigation of anew type sliding retainer and its efficacy in seismic fortifi-cationrdquo Engineering Mechanics vol 33 no 2 pp 111ndash1182016 in Chinese
[25] Caltrans Caltrans Seismic Design Criteria California De-partment of Transportation Sacramento CA USA 2013
[26] A Shamsabadi K M Rollins and M Kapuskar ldquoNonlinearsoil-abutment-bridge structure interaction for seismic per-formance-based designrdquo Journal of Geotechnical and Geo-environmental Engineering vol 133 no 6 pp 707ndash720 2007
[27] X Li Z Sun D Wang and Y Shi ldquoLongitudinal seismicpounding effect of bridges abutment and backfilling damagerdquoJournal of Changrsquoan University vol 35 no 4 pp 76ndash82 2015in Chinese
[28] MOTC ldquoSpecification for design of highway suspensionbridgerdquo China Ministry of Transportation and Communi-cations Press Beijing China JTGT D65-05-2015 2015
[29] A Shafieezadeh K Ramanathan J E Padgett andR DesRoches ldquoFractional order intensity measures forprobabilistic seismic demand modeling applied to highwaybridgesrdquo Earthquake Engineering amp Structural Dynamicsvol 41 no 3 pp 391ndash409 2012
[30] J E Padgett B G Nielson and R DesRoches ldquoSelection ofoptimal intensity measures in probabilistic seismic demandmodels of highway bridge portfoliosrdquo Earthquake Engineeringamp Structural Dynamics vol 37 no 5 pp 711ndash725 2008
[31] K R Mackie and B Stojadinovic ldquoFragility basis for californiahighway overpass bridge seismic decision makingrdquo PEERReport 200502 Pacific Earthquake Engineering ResearchCenter University of California Berkeley Berkeley CA USA2005
[32] J Li T Peng Ji Lin L Zhou and YuL Deng Seismic Design ofMulti-Pylon Multi-Span Suspension Bridge China Commu-nications Press Beijing China 2013 in Chinese
[33] L Chen W Zhuang H Zhao and Z Wan ldquoReport onHighwaysrsquo Damage in the Wenchuan Earthquake-BridgeChina Communications Press Beijing China 2012 inChinese
[34] G Wu K Wang G Lu and P Zhang ldquoAn experimentalinvestigation of unbonded laminated elastomeric bearingsand the seismic evaluations of highway bridges with testedbearing componentsrdquo Shock and Vibration vol 2018 ArticleID 8439321 18 pages 2018
Advances in Civil Engineering 21
Table 5 Limit state thresholds of cable system abutment shear key and expansion joint
Figure 11 Component fragility curves for the suspension bridge (a) main bridge (b) approach bridge
200 400 600 800 10000Location (m)
00
02
04
06
08
10
MA
F
MaximumMinimumMean
Dead loadYield valueHigh stress region
(a)
00
02
04
06
08
10
SAF
200 400 600 800 10000Location (m)
MaximumMinimumMean High stress region
Dead loadYield value
(b)
Figure 12 Force distributions of the (a) main cables and (b) suspenders
12 Advances in Civil Engineering
when there were no additional restraints in the longitudinaldirection erefore it was consistent to suggest that sup-plementary devices should be equipped to mitigate the deckmovements to some extente reduction of deck or bearingdisplacements can be taken as the retrofitted objective for thesuspension span Fragility analyses were conducted on theretrofitted bridges that covered the general strategies asfollows
Case 1 the analysis model of the bridge used a fluidviscous damper equipped between the deck and eachpedestal on both sides of the bridgeCase 2 the analysis model of the bridge used an elasticcable installed between the deck and each pedestal onboth sides of the bridgeCase 3 the analysis model of the bridge used three pairsof flexible central buckles located in the middle sectionof the span
e main bridge was assumed to have identical me-chanical parameters at each location in various designschemes e dampers and elastic cables were respectivelymodeled using zero-length elements with Maxwell andtension-only materials and the flexible central buckles weresimulated as the same as the suspenders elements inOpenSEES
e experimental testing of fluid viscous dampers hasshown that the force-velocity with fractional velocity powerlaw relation can describe the suitable mechanical behavior
fD CD| _u|αsgn( _u) (9)
where α is a real positive exponent of which value dependson the piston head orifice CD is viscous damping coefficientwith units of force per velocity raised to the α power kNmiddot(ms)-α _u indicates the velocity of damper and sgn(middot) is thesignum function Research by Li et al [32] indicated thatviscous damping coefficients range from 2500 kN (ms)-α to5000 kN (ms)-α and can be simulated with a uniform dis-tribution e velocity exponents can be assumed to varyuniformly from 03 to 05
e axial stiffness of elastic cables and flexible centralbuckles is regarded with the material length and cross-section area It was assumed that the yield strength of steelstrands of fy 1420MPa was the mean value and followed alognormal distribution with a COV 01 e stiffnessvalues of single elastic cable based on past work employed auniform distribution ranging from 125times105 kNm to25 times105 kNm In addition the other mechanical prop-erties such as length and area can be determined by thefollowing relations
LgtEΔfy
1113888 1113889
kE EA
L1113874 1113875
(10)
where L E and fy are the length Youngrsquos modulus andyield stress of elastic cables respectively and Δ represents the
controlled deck displacement which was available for 06min the examined bridge
According to expert opinion three pairs of flexiblecentral buckles were the most common type of centralbuckle of which the cross-section areas were the same asthose of the suspenders
Figures 13(a) and 13(b) illustrate bearing displacementdistributions are formed for the three retrofit measures at thesame ground motion intensity levels e mechanical be-haviors of measures strongly affect the effectiveness It can beconcluded that owing to the similar EDP distributions ofdampers and elastic cables the corresponding groundmotion intensities were similar when generating the samepounding probability e flexible central buckles wereextremely susceptible to yielding failure thereby causing lowefficiency in the displacement restrictions ereforepounding probabilities can be directly observed by condi-tioning on the median responses of dampers and elasticcables instead (Figure 13(c)) e control force produced bydevices is another noteworthy consideration in retrofitdesign e more force the devices require the larger andmore expensive they usually are [13] To take both defor-mation and force into account Figures 13(d) and 13(e)provide the contours of the pounding probability ecomparison showed that the damper deformations featuredslightly larger slopes in the horizontal direction yet thedifference in effect on the deformations of the dampers andelastic cables was not significant for reducing the objectivepounding probability Conversely damper force was lesssensitive to the pounding probability than cable force eelastic cable forces reached on average up to e32 (asymp245)times those of the damper under the same poundingprobabilities ese findings implied that difficult anchoragechallenges existed in the case of guaranteeing the efficiencyof elastic cables and the bridge decks possibly experiencedhigher stress In addition the difference in force was sub-stantial when controlling the displacement which showedthat dampers can contribute additional damping and stiff-ness to the structural system while elastic cables completelytransferred the inertial forces erefore the coordinationand balance of displacements and forces should be simul-taneously considered in the vibration mitigation of large-span bridges In this study the application of fluid viscousdampers was identified as the preferred retrofit in thelongitudinal direction for the main bridge e additionaldamping ratio provided by the fluid viscous dampers can becalculated by the following equation
ε CD
2mωeff (11)
where CD is damping coefficient of linear fluid damperskNmiddot(sm) which is equivalent to nonlinear dampers basedon equal energy dissipation per hysteresis loop [17] ωeff issystem natural frequency corresponding to the vibrationalong the axis of bridge andm is system mass It was shownthat an additional damping of 206ndash318 can offer aneffective controlling mechanism for the reduction ofpounding between the adjacent decks
Advances in Civil Engineering 13
Figure 14 depicts the change of components vulnera-bilities before and after the application of dampers It can beobserved that the retrofit measure played limited roles in thefragility reduction of the self-damaged bearings in long-spanbridges Since the additional longitudinal rigidity to girderwas provided by the dampers the transverse responses suchas girder displacement and pylon curvature would increaseas ground motion intensity increased e influences on thecable system responses and the longitudinal responses ofpylons were found to be negligible Another phenomenonwas that the longitudinal fragility of the abutments signif-icantly decreased with the reduction in the poundingprobabilities between the decks and abutments of the ap-proach bridges It was inferred that the pounding betweenthe adjacent decks was likely to affect movement of theapproach bridges and that the pounding effect would betransmitted along the girder
42 Improved Design of Approach Bridges e damagepotential of each component for the approach bridges washigh due to the high collapse risks posed by the usage of thefixed piers e structural designs of small-to-medium-spanhighway bridges in China show the fact that the super-structures are usually placed onto laminated elastomericbearings (LEBs) which tend to come into direct contact withthe bridge girders and substructures without anchoring etypical types of seismic damages of these bridges consisted of
large girder movements bearings sliding failures of shearkeys and expansion joints and abutment cracking duringthe Wenchuan and Yushu earthquake events in ChinaMeanwhile the bridge columns displayed relatively lowdamage ratios under earthquakes [33] e design can bereferred to as a quasi-isolation system which is primarilyrealized by strategically employing bearings sliding as force-limiting connections between girders and substructures Inaddition the isolation strategies of lead-rubber bearings(LRBs) are usually employed in the high-seismicity regionswhich can be taken as a comparison of the quasi-isolationsystem e two restraint systems consist of bearings shearkeys transversely located at the cap beams and restrainercables longitudinally anchored to the abutments (or ped-estals) In order to calculate the seismic responses of theapproach bridge the related experiments were completedand summarized and applied to the force-displacementrelations of the restraint system components
e detailed experiment of LEB consulted previousacademic achievements [34] An example is shown inFigure 15(a) where the feature points marked by rectangularand circular represented initial walking and obvious slidingresponses which illustrated that the mechanical property ofbearings would degrade as the displacements increased afterentering the friction sliding state until it remained nearlyconstant When there was no specific test the horizontalforce Fd in the stable friction stage can be 07 Fmax whereFmax was the corresponding maximum static friction at the
Bear
ing
disp
lace
men
t (m
)
Logarithmic spacendash5
ndash4
ndash3
ndash2
ndash1
0
1
2 4 6PGV (cms)
Case 1Case 2Case 3
(a)Po
undi
ng p
roba
bilit
y
00
02
04
06
08
05 10 15 2000PGV (ms)
Case 1Case 2Case 3
(b)
Poun
ding
pro
babi
lity
00
02
04
06
02 04 06 08 1000Deformation (m)
Case 1Case 2
(c)
ln [f
orce
(kN
)]
5
10
15
15
20
20
25
25
30
3030
35
3535
40
40
4550
7
72
74
76
78
8
033 041 049 057 065025Deformation (m)
(d)
ln [f
orce
(kN
)]5
10
15
15 20
20 25
25
30
3030
35
35
40
40
45
50
102
104
106
108
11
112
114
0155 0205 0255 0305 03550105Deformation (m)
(e)
Figure 13 Choice of the three retrofitmeasures based on deformations and forces (a) EDP-IM relation (b) pounding probability conditioned onIM (c) pounding probability conditioned on deformation contours of pounding probability for (d) dampers and (e) elastic cables
14 Advances in Civil Engineering
initial walking e displacement value Δ2 can be of 250effective shear strain (ESS) e initial stiffness K0 is cal-culated by MOTC [20] erefore the coefficients can becalculated by the following sequence FdK0⟶ Fmax⟶Δ1⟶Δ2 e built analytical relation ofthe LEB is shown in Figure 16(a) which can correctlycapture mechanical behavior under earthquakes A log-normal distribution [8] with a logarithmic standard devi-ation of 01 and a normal distribution with aCOV 016MPa were used to model the randomness offriction coefficient and shear modulus respectively For theirrespective mean values 02 and 12MPa were taken
Figure 15(b) shows a cyclic loading experiment of LRBwhich reveals that the degradation of bearings stiffness willoccur with the increase of loading displacements e elasticstiffness can reach uniformly 3 to 7 times the postyieldingstiffness according to the experimental results e
mechanical behavior of LRB can be modeled by a bilinearrelation as seen in Figure 16(b)
e cable restrainers were modeled as nonlinear tension-only elements characterized by an initial slack as shown inFigure 16(c) e parameters affecting the functionality ofcable restrainers were the cable slack length and yieldstrength according to the research of Padgett and DesRoches[16] e first two parameters were assumed to follow theuniform distributions and the strength can be modeled bythe lognormal distribution An adopted range of cablestiffness values was from 30times105 kNm to 50times105 kNmuniformly
e key parameters in analytical models of Figure 16 canbe determined according to the structural design of therealistic bridge e damage states of two restraint systemsare listed in Table 6 in which ESS is used as the damageindex
Dam
age p
roba
bilit
y
05 10 15 2000PGV (ms)
00
02
04
06
08
10
Case 0Case 1
MBDLMBDT
(a)
Dam
age p
roba
bilit
y
00
01
02
03
04
05
05 10 15 2000PGV (ms)
Case 0Case 1
PCDNLPCDNT
(b)
Dam
age p
roba
bilit
y
00
02
04
06
08
10
05 10 15 2000PGV (ms)
Case 0Case 1
BPDAEJDA
(c)
Figure 14 Influence of dampers on vulnerabilties of (a) bearing responses for DS2 (b) pylon responses for DS2 and (c) abutment andpounding responses
Advances in Civil Engineering 15
Figure 17 illustrates the probabilities of critical damage stateoccurrence sequences of the bridge components in Cases 1 4and 5 when PGV is 10ms which is equal to E2 earthquakeintensity in the seismic assessment reports e componentvulnerabilities of initial design (Case 0) were taken as the ref-erence values which were marked by different colored strips inFigure 17 e desirable sequence preferred Cases 4 or 5 for theapproach bridgese initial design of approach bridges resultedin unreasonable damage sequences In the strategies pertainingto quasi-isolation and isolation systems multiple tiers of seismicstructural redundancy can be normally employed to preventstructural collapse during strong earthquake events Tier 1 in-dicated that the bearings were weakly fused which was evidentin the high fragility of bearing sliding behaviors (ABDAL andABDC2T) Another advantage was that the extensive dam-ages of bearing would be reduced when compared with the self-damage of the pot bearings Tier 2 indicated that the designedbearings had sufficient seat widths for adapting to the sliding ofsuperstructures However the devices had a certain damageprobability (EJDA and SKDC2) and dissipated a portion of
the seismic energy when the displacement of the superstructuresbecame larger than the allowable values of constraint devicesTier 3 redundancy was designed as a potential plastic hinge inthe substructures (CCDC2) including a certain yielding of thebackfill at the abutments (BPDA) e response of theabutments in the transverse direction (PDAT) was slightlyincreased due to the pounding effect between the girders andshear keys caused by bearing sliding e reasonable damagesequence of two restraint systems can be reflected through thecorresponding damage probabilities of bearings seismic mea-sures and substructures respectively as indicated by the red andgreen lines in Figure 17 erefore the rational sequence ofcritical limit state occurrences for the approach bridges isbearings⟶ expansion joints (or restrainer cables) shearkeys⟶ columns abutments
43 Comprehensive Evaluation of Bridge System Padgett andDesRoches [16] pointed out the necessity to derive the bridgesystem vulnerabilities when assessing retrofitted bridges
Figure 15 Experimental results of (a) LEB and (b) LRB
ΔΔ1
F
Δ2
Fmax
Fd
KdK0
(a)
Δy Δ
F
K1 = (3~7) K2
K1
K2
(b)
Slack
F
K
Δ
Fy
(c)
Figure 16 Analytical models of (a) LEB (b) LRB and (c) cable restrainers
16 Advances in Civil Engineering
because different components may have a different responsewith a given retrofit measure Also Zhang and Huo [18]studied the optimal parameters of isolation bearings byderiving the intensity median value corresponding to sys-tem-level fragility function which can effectively avoid theimpacts of different isolation parameters on differentcomponents As can be seen from Figures 14 and 17 theabove analysis results were also very apparent in this studyFor example the longitudinal deck displacement was sig-nificantly reduced by using dampers yet increased thetransverse demands of stiffening girder and pylons e useof LEB or LRB reduced the demands placed on columns yetled to an increase of the transverse deformations of abut-ments which was due to the use of shear keys limiting theexcessive sliding displacement of bearings erefore acomprehensive evaluation of retrofit strategies on bridgesparticularly long-span suspension bridges that have multiplecomponents should be based on system-level fragility in-stead of component-level
Taking the findings that no damages had occurred to thecable system and the definition of bearing damage states intoaccount the derivation of system-level fragility of the wholebridge depended on critical components including pylonscolumns bearings and abutments e fragility data byemploying the absolutely serial and serial-parallel assump-tions of bridge system are plotted in Figure 18 and comparedto the system fragility obtained from the proposed com-posite damage state e absolutely serial assumption
generated serious bias for fragility curves and may result inmisdirection if they were served as a design guide eabsolutely serial model overestimated the potential damageof the bridge system since the fragility curve directlydepended on the most vulnerable components In otherwords if the difference in location and performance of eachcomponent was neglected then the seismic performanceevaluation would be achieved by attaching equal importanceto the strength of pylons approach columns and bearingsis method contradicted the philosophy of capacity designin which the long-span structure can dissipate energythrough a reasonable swinging of stiffening girder and thebearings were designed as fuse elements in the small-to-medium-span bridgese serial-parallel model qualitativelyincorporated the differences of components and was verysensitive to the seismic design of the structure as seen fromthe comparisons among the cases in Figure 18 e modeldistinguished the serial components and parallel compo-nents yet still overestimated the bridge system-level fragilitywhen bearings became the vulnerable components espe-cially for long-span bridges is special situation such asthe serial assumption may also lead to misunderstandingOnce the relative importance between the components wasconsidered in the serial-composite model the damage po-tential was clearly reduced within the range of earthquakeintensity of interest based on the case studies is alsoreflected the necessity of the classification idea of bridgecomponents which can be expressed by the numerical value
Table 6 Damage states of the quasi-isolation and isolation bearings (m)
Figure 17 Probabilities of critical damage states occurrence sequences for the approach bridges when PGV is 1ms
Advances in Civil Engineering 17
of system-level fragility erefore the application ofcomposite damage state (equation (4)) considering com-ponent difference was instead more appropriate than theaforementioned two models
e damage probability distribution represented byblack circles in Figure 18 can be obtained from the definedcomposite damage state (equation (5)) where the parametera is 06 It was found that system fragility result was a moreaccurate assessment with further taking the relative im-portance between themain bridge and approach bridges intoaccount is is due to the principle that the seismic designperformance objective of long-span bridges generally has nocollapse for the main body structures and is necessarily opento emergency vehicle service In addition attention must bepaid to the fact that the differences in repair cost and time ofthe main and approach bridges are significant when thesame damage state is achieved e results of compositedamage state may match the performance objectives of theseismic design In order to investigate the influence ofimproved strategies and structural relative importance ondamage potential of bridges the mean values of the fragility
functions can be employed to evaluate the seismic designe smooth fragility curves for the specific damage states canbe fitted with the lognormal cumulative probability distri-bution function by employing the least-squares technique
P DSgtDSi | PGV1113858 1113859 Φln(pgv) minus λPGV
ζPGV1113888 1113889 (12)
where λPGV and ζPGV are the logarithmic mean and dis-persion values of earthquake intensity (PGV) for a specifieddamage level e larger mean values represent the smallerdamage potential to reach the specified level
Figure 19 indicates the mean values of earthquakeintensity at the specific damage states as a function of theimproved strategies and parameter a Each step of theretrofit measures would enhance the seismic performanceof the overall bridge system at all levels e improvementof the restraint system of the approach bridge played acritical role of the overall bridge system For instance themean values of the earthquake intensity featured mutationof slopes along the measure direction which reflected the
Figure 18 System fragility of the original design (a) Case 0 and improved design including (b) Case 1 and (c) Case 5
18 Advances in Civil Engineering
fact that the effectiveness of improvement for the ap-proach bridges was clearer than that of the main bridgee fragility of the highway network would be under-estimated if only the damage potential of the suspensionspan was adopted erefore when the structures such asapproach bridges transition piers and auxiliary pierswere ignored the damage identification and repair de-cisions would be misleading in the seismic evaluation oflong-span bridges Conversely the damaging probabilityof the overall bridge system will be amplified regardless ofthe higher importance of the main bridge When respectthe fact that the importance of suspension span is higherthan that of approach bridge in transportation networks aparameter a of 06ndash08 may be a better choice in thepresent study
e seismic fragility of conventionally designedbridges was mainly governed by the approach bridges at alldamage levels However for the retrofitted bridges thatpossess reasonable seismic performance different damagestates were controlled by the earthquake demands ofvarious structures as shown by the trend of the curvesalong parameter a in Case 4 and Case 5 e damagingpotential of the overall bridge from the low-to-moderatelimit states was dominated by the approach bridge eprimary reasons were the sliding of bearings and thetransfer of limited forces from the girder to substructure
by the damage of restriction devices of the approachbridges e extensive damage was controlled by thedisplacement of stiffening girder of the main bridge whichinferred the effect of dampers will be conspicuous understrong earthquake shaking At the same time the overalllong-span bridges with the reasonable seismic system willnot collapse during a strong earthquake event and canmeet emergency traffic service It can be seen that the meanvalues of fragility functions developed by compositedamage indices (equations (4) and (5)) are good indicatorsto evaluate seismic performance of long-span bridges andcan be consistent with the established seismic performanceobjectives
5 Conclusions
In the present study a procedure for investigating thereasonable seismic performance of long-span suspensionbridges was developed under the fragility function frame-work e seismic performance of a long-span suspensionbridge was examined and improved using the proposedprocedure from which the results revealed distinguishableconclusions as follows
(1) e definition of damage states for bearings con-sidered the interactions between the adjacent
014036059081103125
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(a)
020051081112142173
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
0002
0406
0810
CaseParameter a
(b)
130251371492612733
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(c)
Mea
n va
lue
exp
(λ) (
ms
)
Parameter a Case0 1 2 3 4 5
0001
0203
0405
131305478652826
1000
(d)
Figure 19 Influence of relative importance and improved designs on mean values of earthquake intensity for different damage states(a) slight (b) moderate (c) extensive (d) collapse
Advances in Civil Engineering 19
structures and self-damage to bearings in accor-dance with the actual design and experimentalconclusions e damage states of pylons in terms ofcurvature ductility were defined by developingmodal pushover analysis and idealized bilinear fit-ting methods in the equivalent pylon fiber models ofthe suspension bridge
(2) When investigating the component fragility curvesof the initial design it was found that the large deckdisplacement of the suspension span was the mostserious damage present and a slight inelastic re-sponse may have occurred only in the pylons Al-though the cable system was able to maintainelasticity under earthquakes the saddles at the py-lons were identified with a high-risk region indi-cating the damaging potential For the approachbridges the ductility mechanism did not realize arational damage sequence e columns were proneto collapse because of the application of fixed potbearings which in turn caused the damage to thebearings and abutments
(3) e stepwise improved seismic designs were pro-posed in terms of the component fragility results ofthe initial design Among the candidates fluid vis-cous dampers were proven to be a good choice forlong-span bridges based on the two attributes ofdisplacement and force when the similar efficiency ofdisplacement control was achieved e restraintsystem had a critical effect on the seismic responsesof the approach bridges e quasi-isolation orisolation can achieve a reasonable seismic damagesequence of the approach bridges
(4) e defined composite damage indices showed thatthe incorporate component classification conceptand structural relative importance were necessary toderive a good indicator of performance evaluationdue to the fact that the damage indices were based oneither an absolutely serial or serial-parallel systemmodel may significantly overestimate the damagepotential of the overall bridge system e meanvalues of earthquake intensity at the specific damagestates can serve as a function of the improvedstrategies and relative importance to further evaluatethe overall seismic performance of the suspensionbridge It can be concluded that the damaging po-tential of the overall bridge from the low-to-mod-erate limit states was governed by the approachbridges in the retrofitted bridges with a favorableseismic performance whereas the displacementdemand of the suspension span dominated thefragility probability of the extensive damage
Data Availability
e nature of the data is the analytical data of finite elementmodels of the case bridge e data used to support thefindings of this study are available from the correspondingauthor upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the Basic Research ServiceProject of Central-Level Public Welfare Research Institute(grant nos 2016-9018 and 2019-0034) and Science andTechnology Project of Communicationsrsquo Construction inWestern China MOC (grant no 2009318223094)
References
[1] L W David A Kartoum K Chang and I Roy ldquoSuspensionreport seismic performance and design considerations of longspan suspension bridgesrdquo Report CSMIP00-03 CaliforniaDepartment of Conservation Division of Mines And GeologyOffice of Strong Motion Studies Sacramento CA USA 2000
[2] S-G Gwon and D-H Choi ldquoImproved continuummodel forfree vibration analysis of suspension bridgesrdquo Journal ofEngineering Mechanics vol 143 no 7 Article ID 040170382017
[3] S Arzoumanidis A Shama and F Ostadan ldquoPerformance-based seismic analysis and design of suspension bridgesrdquoEarthquake Engineering and Structural Dynamics vol 34no 4-5 pp 349ndash367 2005
[4] L Sgambi E Garavaglia N Basso and F Bontempi ldquoMonteCarlo simulation for seismic analysis of a long span sus-pension bridgerdquo Engineering Structures vol 78 pp 100ndash1112014
[5] D Karmakar S Ray-Chaudhuri and M Shinozuka ldquoFiniteelement model development validation and probabilisticseismic performance evaluation of Vincent omas suspen-sion bridgerdquo Structure and Infrastructure Engineering vol 11no 2 pp 223ndash237 2015
[6] W-F Chen and L Duan Bridge Engineering HandbookSeismic Design CRC Press Boca Raton FL USA 2nd edition2014
[7] E Choi ldquoSeismic analysis and retrofit of mid-americabridgesrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2002
[8] B G Nielson and R DesRoches ldquoSeismic fragility method-ology for highway bridges using a component level approachrdquoEarthquake Engineering amp Structural Dynamics vol 36 no 6pp 823ndash839 2007
[9] K N Ramanathan ldquoNext generation seismic fragility curvesfor california bridges incorporating the evolution in seismicdesignrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2012
[10] O Avsar A Yakut and A Caner ldquoAnalytical fragility curvesfor ordinary highway bridges in Turkeyrdquo Earthquake Spectravol 27 no 4 pp 971ndash996 2011
[11] S P Stefanidou and A J Kappos ldquoMethodology for thedevelopment of bridge-specific fragility curvesrdquo EarthquakeEngineering amp Structural Dynamics vol 46 no 1 pp 73ndash932016
[12] B Borzi P Ceresa P Franchin F Noto G M Calvi andP E Pinto ldquoSeismic vulnerability of the Italian roadwaybridge stockrdquo Earthquake Spectra vol 31 no 4pp 2137ndash2161 2015
[13] W T Barnawi and S J Dyke ldquoSeismic fragility relationshipsof a cable-stayed bridge equipped with response modification
20 Advances in Civil Engineering
systemsrdquo Journal of Bridge Engineering vol 19 no 8 ArticleID A4013003 2014
[14] J Zhong Y Pang J Jong-Su J-S Jeon R DesRoches andW Yuan ldquoSeismic fragility assessment of long-span cable-stayed bridges in Chinardquo Advances in Structural Engineeringvol 19 no 11 pp 1797ndash1812 2016
[15] J Zhong Z Hu W Yuan and L Chen ldquoSystem-basedprobabilistic optimization of fluid viscous dampers equippedin cable-stayed bridgesrdquo Advances in Structural Engineeringvol 21 no 12 pp 1815ndash1825 2018
[16] J E Padgett and R DesRoches ldquoMethodology for the de-velopment of analytical fragility curves for retrofitted bridgesfitted bridgesrdquo Earthquake Engineering amp Structural Dy-namics vol 37 no 8 pp 1157ndash1174 2008
[17] Y Xie and J Zhang ldquoOptimal design of seismic protectivedevices for highway bridges using performance-basedmethodology and multiobjective genetic optimizationrdquoJournal of Bridge Engineering vol 22 no 3 Article ID04016129 2017
[18] J Zhang and Y Huo ldquoEvaluating effectiveness and optimumdesign of isolation devices for highway bridges using thefragility function methodrdquo Engineering Structures vol 31no 8 pp 1648ndash1660 2009
[19] F McKenna M H Scott and G L Fenves ldquoNonlinear finite-element analysis software architecture using object compo-sitionrdquo Journal of Computing in Civil Engineering vol 24no 1 pp 95ndash107 2010
[20] MOTC ldquoGuidelines for seismic design of highway bridgesrdquoChina Ministry of Transportation and CommunicationsPress Beijing China JTGT B02-01-2008 2008
[21] A D Kiureghian and J L Sackman ldquoTangent geometricstiffness of inclined cables under self-weightrdquo Journal ofStructural Engineering vol 131 no 6 pp 941ndash945 2005
[22] J S Steelman E T Filipov L A Fahnestock et al ldquoEx-perimental behavior of steel fixed bearings and implicationsfor seismic bridge responserdquo Journal of Bridge Engineeringvol 19 no 8 Article ID A4014007 2014
[23] B Madani F Behnamfar and H Tajmir Riahi ldquoDynamicresponse of structures subjected to pounding and structure-soil-structure interactionrdquo Soil Dynamics and EarthquakeEngineering vol 78 pp 46ndash60 2015
[24] L Xu and J Li ldquoDesign and experimental investigation of anew type sliding retainer and its efficacy in seismic fortifi-cationrdquo Engineering Mechanics vol 33 no 2 pp 111ndash1182016 in Chinese
[25] Caltrans Caltrans Seismic Design Criteria California De-partment of Transportation Sacramento CA USA 2013
[26] A Shamsabadi K M Rollins and M Kapuskar ldquoNonlinearsoil-abutment-bridge structure interaction for seismic per-formance-based designrdquo Journal of Geotechnical and Geo-environmental Engineering vol 133 no 6 pp 707ndash720 2007
[27] X Li Z Sun D Wang and Y Shi ldquoLongitudinal seismicpounding effect of bridges abutment and backfilling damagerdquoJournal of Changrsquoan University vol 35 no 4 pp 76ndash82 2015in Chinese
[28] MOTC ldquoSpecification for design of highway suspensionbridgerdquo China Ministry of Transportation and Communi-cations Press Beijing China JTGT D65-05-2015 2015
[29] A Shafieezadeh K Ramanathan J E Padgett andR DesRoches ldquoFractional order intensity measures forprobabilistic seismic demand modeling applied to highwaybridgesrdquo Earthquake Engineering amp Structural Dynamicsvol 41 no 3 pp 391ndash409 2012
[30] J E Padgett B G Nielson and R DesRoches ldquoSelection ofoptimal intensity measures in probabilistic seismic demandmodels of highway bridge portfoliosrdquo Earthquake Engineeringamp Structural Dynamics vol 37 no 5 pp 711ndash725 2008
[31] K R Mackie and B Stojadinovic ldquoFragility basis for californiahighway overpass bridge seismic decision makingrdquo PEERReport 200502 Pacific Earthquake Engineering ResearchCenter University of California Berkeley Berkeley CA USA2005
[32] J Li T Peng Ji Lin L Zhou and YuL Deng Seismic Design ofMulti-Pylon Multi-Span Suspension Bridge China Commu-nications Press Beijing China 2013 in Chinese
[33] L Chen W Zhuang H Zhao and Z Wan ldquoReport onHighwaysrsquo Damage in the Wenchuan Earthquake-BridgeChina Communications Press Beijing China 2012 inChinese
[34] G Wu K Wang G Lu and P Zhang ldquoAn experimentalinvestigation of unbonded laminated elastomeric bearingsand the seismic evaluations of highway bridges with testedbearing componentsrdquo Shock and Vibration vol 2018 ArticleID 8439321 18 pages 2018
Advances in Civil Engineering 21
when there were no additional restraints in the longitudinaldirection erefore it was consistent to suggest that sup-plementary devices should be equipped to mitigate the deckmovements to some extente reduction of deck or bearingdisplacements can be taken as the retrofitted objective for thesuspension span Fragility analyses were conducted on theretrofitted bridges that covered the general strategies asfollows
Case 1 the analysis model of the bridge used a fluidviscous damper equipped between the deck and eachpedestal on both sides of the bridgeCase 2 the analysis model of the bridge used an elasticcable installed between the deck and each pedestal onboth sides of the bridgeCase 3 the analysis model of the bridge used three pairsof flexible central buckles located in the middle sectionof the span
e main bridge was assumed to have identical me-chanical parameters at each location in various designschemes e dampers and elastic cables were respectivelymodeled using zero-length elements with Maxwell andtension-only materials and the flexible central buckles weresimulated as the same as the suspenders elements inOpenSEES
e experimental testing of fluid viscous dampers hasshown that the force-velocity with fractional velocity powerlaw relation can describe the suitable mechanical behavior
fD CD| _u|αsgn( _u) (9)
where α is a real positive exponent of which value dependson the piston head orifice CD is viscous damping coefficientwith units of force per velocity raised to the α power kNmiddot(ms)-α _u indicates the velocity of damper and sgn(middot) is thesignum function Research by Li et al [32] indicated thatviscous damping coefficients range from 2500 kN (ms)-α to5000 kN (ms)-α and can be simulated with a uniform dis-tribution e velocity exponents can be assumed to varyuniformly from 03 to 05
e axial stiffness of elastic cables and flexible centralbuckles is regarded with the material length and cross-section area It was assumed that the yield strength of steelstrands of fy 1420MPa was the mean value and followed alognormal distribution with a COV 01 e stiffnessvalues of single elastic cable based on past work employed auniform distribution ranging from 125times105 kNm to25 times105 kNm In addition the other mechanical prop-erties such as length and area can be determined by thefollowing relations
LgtEΔfy
1113888 1113889
kE EA
L1113874 1113875
(10)
where L E and fy are the length Youngrsquos modulus andyield stress of elastic cables respectively and Δ represents the
controlled deck displacement which was available for 06min the examined bridge
According to expert opinion three pairs of flexiblecentral buckles were the most common type of centralbuckle of which the cross-section areas were the same asthose of the suspenders
Figures 13(a) and 13(b) illustrate bearing displacementdistributions are formed for the three retrofit measures at thesame ground motion intensity levels e mechanical be-haviors of measures strongly affect the effectiveness It can beconcluded that owing to the similar EDP distributions ofdampers and elastic cables the corresponding groundmotion intensities were similar when generating the samepounding probability e flexible central buckles wereextremely susceptible to yielding failure thereby causing lowefficiency in the displacement restrictions ereforepounding probabilities can be directly observed by condi-tioning on the median responses of dampers and elasticcables instead (Figure 13(c)) e control force produced bydevices is another noteworthy consideration in retrofitdesign e more force the devices require the larger andmore expensive they usually are [13] To take both defor-mation and force into account Figures 13(d) and 13(e)provide the contours of the pounding probability ecomparison showed that the damper deformations featuredslightly larger slopes in the horizontal direction yet thedifference in effect on the deformations of the dampers andelastic cables was not significant for reducing the objectivepounding probability Conversely damper force was lesssensitive to the pounding probability than cable force eelastic cable forces reached on average up to e32 (asymp245)times those of the damper under the same poundingprobabilities ese findings implied that difficult anchoragechallenges existed in the case of guaranteeing the efficiencyof elastic cables and the bridge decks possibly experiencedhigher stress In addition the difference in force was sub-stantial when controlling the displacement which showedthat dampers can contribute additional damping and stiff-ness to the structural system while elastic cables completelytransferred the inertial forces erefore the coordinationand balance of displacements and forces should be simul-taneously considered in the vibration mitigation of large-span bridges In this study the application of fluid viscousdampers was identified as the preferred retrofit in thelongitudinal direction for the main bridge e additionaldamping ratio provided by the fluid viscous dampers can becalculated by the following equation
ε CD
2mωeff (11)
where CD is damping coefficient of linear fluid damperskNmiddot(sm) which is equivalent to nonlinear dampers basedon equal energy dissipation per hysteresis loop [17] ωeff issystem natural frequency corresponding to the vibrationalong the axis of bridge andm is system mass It was shownthat an additional damping of 206ndash318 can offer aneffective controlling mechanism for the reduction ofpounding between the adjacent decks
Advances in Civil Engineering 13
Figure 14 depicts the change of components vulnera-bilities before and after the application of dampers It can beobserved that the retrofit measure played limited roles in thefragility reduction of the self-damaged bearings in long-spanbridges Since the additional longitudinal rigidity to girderwas provided by the dampers the transverse responses suchas girder displacement and pylon curvature would increaseas ground motion intensity increased e influences on thecable system responses and the longitudinal responses ofpylons were found to be negligible Another phenomenonwas that the longitudinal fragility of the abutments signif-icantly decreased with the reduction in the poundingprobabilities between the decks and abutments of the ap-proach bridges It was inferred that the pounding betweenthe adjacent decks was likely to affect movement of theapproach bridges and that the pounding effect would betransmitted along the girder
42 Improved Design of Approach Bridges e damagepotential of each component for the approach bridges washigh due to the high collapse risks posed by the usage of thefixed piers e structural designs of small-to-medium-spanhighway bridges in China show the fact that the super-structures are usually placed onto laminated elastomericbearings (LEBs) which tend to come into direct contact withthe bridge girders and substructures without anchoring etypical types of seismic damages of these bridges consisted of
large girder movements bearings sliding failures of shearkeys and expansion joints and abutment cracking duringthe Wenchuan and Yushu earthquake events in ChinaMeanwhile the bridge columns displayed relatively lowdamage ratios under earthquakes [33] e design can bereferred to as a quasi-isolation system which is primarilyrealized by strategically employing bearings sliding as force-limiting connections between girders and substructures Inaddition the isolation strategies of lead-rubber bearings(LRBs) are usually employed in the high-seismicity regionswhich can be taken as a comparison of the quasi-isolationsystem e two restraint systems consist of bearings shearkeys transversely located at the cap beams and restrainercables longitudinally anchored to the abutments (or ped-estals) In order to calculate the seismic responses of theapproach bridge the related experiments were completedand summarized and applied to the force-displacementrelations of the restraint system components
e detailed experiment of LEB consulted previousacademic achievements [34] An example is shown inFigure 15(a) where the feature points marked by rectangularand circular represented initial walking and obvious slidingresponses which illustrated that the mechanical property ofbearings would degrade as the displacements increased afterentering the friction sliding state until it remained nearlyconstant When there was no specific test the horizontalforce Fd in the stable friction stage can be 07 Fmax whereFmax was the corresponding maximum static friction at the
Bear
ing
disp
lace
men
t (m
)
Logarithmic spacendash5
ndash4
ndash3
ndash2
ndash1
0
1
2 4 6PGV (cms)
Case 1Case 2Case 3
(a)Po
undi
ng p
roba
bilit
y
00
02
04
06
08
05 10 15 2000PGV (ms)
Case 1Case 2Case 3
(b)
Poun
ding
pro
babi
lity
00
02
04
06
02 04 06 08 1000Deformation (m)
Case 1Case 2
(c)
ln [f
orce
(kN
)]
5
10
15
15
20
20
25
25
30
3030
35
3535
40
40
4550
7
72
74
76
78
8
033 041 049 057 065025Deformation (m)
(d)
ln [f
orce
(kN
)]5
10
15
15 20
20 25
25
30
3030
35
35
40
40
45
50
102
104
106
108
11
112
114
0155 0205 0255 0305 03550105Deformation (m)
(e)
Figure 13 Choice of the three retrofitmeasures based on deformations and forces (a) EDP-IM relation (b) pounding probability conditioned onIM (c) pounding probability conditioned on deformation contours of pounding probability for (d) dampers and (e) elastic cables
14 Advances in Civil Engineering
initial walking e displacement value Δ2 can be of 250effective shear strain (ESS) e initial stiffness K0 is cal-culated by MOTC [20] erefore the coefficients can becalculated by the following sequence FdK0⟶ Fmax⟶Δ1⟶Δ2 e built analytical relation ofthe LEB is shown in Figure 16(a) which can correctlycapture mechanical behavior under earthquakes A log-normal distribution [8] with a logarithmic standard devi-ation of 01 and a normal distribution with aCOV 016MPa were used to model the randomness offriction coefficient and shear modulus respectively For theirrespective mean values 02 and 12MPa were taken
Figure 15(b) shows a cyclic loading experiment of LRBwhich reveals that the degradation of bearings stiffness willoccur with the increase of loading displacements e elasticstiffness can reach uniformly 3 to 7 times the postyieldingstiffness according to the experimental results e
mechanical behavior of LRB can be modeled by a bilinearrelation as seen in Figure 16(b)
e cable restrainers were modeled as nonlinear tension-only elements characterized by an initial slack as shown inFigure 16(c) e parameters affecting the functionality ofcable restrainers were the cable slack length and yieldstrength according to the research of Padgett and DesRoches[16] e first two parameters were assumed to follow theuniform distributions and the strength can be modeled bythe lognormal distribution An adopted range of cablestiffness values was from 30times105 kNm to 50times105 kNmuniformly
e key parameters in analytical models of Figure 16 canbe determined according to the structural design of therealistic bridge e damage states of two restraint systemsare listed in Table 6 in which ESS is used as the damageindex
Dam
age p
roba
bilit
y
05 10 15 2000PGV (ms)
00
02
04
06
08
10
Case 0Case 1
MBDLMBDT
(a)
Dam
age p
roba
bilit
y
00
01
02
03
04
05
05 10 15 2000PGV (ms)
Case 0Case 1
PCDNLPCDNT
(b)
Dam
age p
roba
bilit
y
00
02
04
06
08
10
05 10 15 2000PGV (ms)
Case 0Case 1
BPDAEJDA
(c)
Figure 14 Influence of dampers on vulnerabilties of (a) bearing responses for DS2 (b) pylon responses for DS2 and (c) abutment andpounding responses
Advances in Civil Engineering 15
Figure 17 illustrates the probabilities of critical damage stateoccurrence sequences of the bridge components in Cases 1 4and 5 when PGV is 10ms which is equal to E2 earthquakeintensity in the seismic assessment reports e componentvulnerabilities of initial design (Case 0) were taken as the ref-erence values which were marked by different colored strips inFigure 17 e desirable sequence preferred Cases 4 or 5 for theapproach bridgese initial design of approach bridges resultedin unreasonable damage sequences In the strategies pertainingto quasi-isolation and isolation systems multiple tiers of seismicstructural redundancy can be normally employed to preventstructural collapse during strong earthquake events Tier 1 in-dicated that the bearings were weakly fused which was evidentin the high fragility of bearing sliding behaviors (ABDAL andABDC2T) Another advantage was that the extensive dam-ages of bearing would be reduced when compared with the self-damage of the pot bearings Tier 2 indicated that the designedbearings had sufficient seat widths for adapting to the sliding ofsuperstructures However the devices had a certain damageprobability (EJDA and SKDC2) and dissipated a portion of
the seismic energy when the displacement of the superstructuresbecame larger than the allowable values of constraint devicesTier 3 redundancy was designed as a potential plastic hinge inthe substructures (CCDC2) including a certain yielding of thebackfill at the abutments (BPDA) e response of theabutments in the transverse direction (PDAT) was slightlyincreased due to the pounding effect between the girders andshear keys caused by bearing sliding e reasonable damagesequence of two restraint systems can be reflected through thecorresponding damage probabilities of bearings seismic mea-sures and substructures respectively as indicated by the red andgreen lines in Figure 17 erefore the rational sequence ofcritical limit state occurrences for the approach bridges isbearings⟶ expansion joints (or restrainer cables) shearkeys⟶ columns abutments
43 Comprehensive Evaluation of Bridge System Padgett andDesRoches [16] pointed out the necessity to derive the bridgesystem vulnerabilities when assessing retrofitted bridges
Figure 15 Experimental results of (a) LEB and (b) LRB
ΔΔ1
F
Δ2
Fmax
Fd
KdK0
(a)
Δy Δ
F
K1 = (3~7) K2
K1
K2
(b)
Slack
F
K
Δ
Fy
(c)
Figure 16 Analytical models of (a) LEB (b) LRB and (c) cable restrainers
16 Advances in Civil Engineering
because different components may have a different responsewith a given retrofit measure Also Zhang and Huo [18]studied the optimal parameters of isolation bearings byderiving the intensity median value corresponding to sys-tem-level fragility function which can effectively avoid theimpacts of different isolation parameters on differentcomponents As can be seen from Figures 14 and 17 theabove analysis results were also very apparent in this studyFor example the longitudinal deck displacement was sig-nificantly reduced by using dampers yet increased thetransverse demands of stiffening girder and pylons e useof LEB or LRB reduced the demands placed on columns yetled to an increase of the transverse deformations of abut-ments which was due to the use of shear keys limiting theexcessive sliding displacement of bearings erefore acomprehensive evaluation of retrofit strategies on bridgesparticularly long-span suspension bridges that have multiplecomponents should be based on system-level fragility in-stead of component-level
Taking the findings that no damages had occurred to thecable system and the definition of bearing damage states intoaccount the derivation of system-level fragility of the wholebridge depended on critical components including pylonscolumns bearings and abutments e fragility data byemploying the absolutely serial and serial-parallel assump-tions of bridge system are plotted in Figure 18 and comparedto the system fragility obtained from the proposed com-posite damage state e absolutely serial assumption
generated serious bias for fragility curves and may result inmisdirection if they were served as a design guide eabsolutely serial model overestimated the potential damageof the bridge system since the fragility curve directlydepended on the most vulnerable components In otherwords if the difference in location and performance of eachcomponent was neglected then the seismic performanceevaluation would be achieved by attaching equal importanceto the strength of pylons approach columns and bearingsis method contradicted the philosophy of capacity designin which the long-span structure can dissipate energythrough a reasonable swinging of stiffening girder and thebearings were designed as fuse elements in the small-to-medium-span bridgese serial-parallel model qualitativelyincorporated the differences of components and was verysensitive to the seismic design of the structure as seen fromthe comparisons among the cases in Figure 18 e modeldistinguished the serial components and parallel compo-nents yet still overestimated the bridge system-level fragilitywhen bearings became the vulnerable components espe-cially for long-span bridges is special situation such asthe serial assumption may also lead to misunderstandingOnce the relative importance between the components wasconsidered in the serial-composite model the damage po-tential was clearly reduced within the range of earthquakeintensity of interest based on the case studies is alsoreflected the necessity of the classification idea of bridgecomponents which can be expressed by the numerical value
Table 6 Damage states of the quasi-isolation and isolation bearings (m)
Figure 17 Probabilities of critical damage states occurrence sequences for the approach bridges when PGV is 1ms
Advances in Civil Engineering 17
of system-level fragility erefore the application ofcomposite damage state (equation (4)) considering com-ponent difference was instead more appropriate than theaforementioned two models
e damage probability distribution represented byblack circles in Figure 18 can be obtained from the definedcomposite damage state (equation (5)) where the parametera is 06 It was found that system fragility result was a moreaccurate assessment with further taking the relative im-portance between themain bridge and approach bridges intoaccount is is due to the principle that the seismic designperformance objective of long-span bridges generally has nocollapse for the main body structures and is necessarily opento emergency vehicle service In addition attention must bepaid to the fact that the differences in repair cost and time ofthe main and approach bridges are significant when thesame damage state is achieved e results of compositedamage state may match the performance objectives of theseismic design In order to investigate the influence ofimproved strategies and structural relative importance ondamage potential of bridges the mean values of the fragility
functions can be employed to evaluate the seismic designe smooth fragility curves for the specific damage states canbe fitted with the lognormal cumulative probability distri-bution function by employing the least-squares technique
P DSgtDSi | PGV1113858 1113859 Φln(pgv) minus λPGV
ζPGV1113888 1113889 (12)
where λPGV and ζPGV are the logarithmic mean and dis-persion values of earthquake intensity (PGV) for a specifieddamage level e larger mean values represent the smallerdamage potential to reach the specified level
Figure 19 indicates the mean values of earthquakeintensity at the specific damage states as a function of theimproved strategies and parameter a Each step of theretrofit measures would enhance the seismic performanceof the overall bridge system at all levels e improvementof the restraint system of the approach bridge played acritical role of the overall bridge system For instance themean values of the earthquake intensity featured mutationof slopes along the measure direction which reflected the
Figure 18 System fragility of the original design (a) Case 0 and improved design including (b) Case 1 and (c) Case 5
18 Advances in Civil Engineering
fact that the effectiveness of improvement for the ap-proach bridges was clearer than that of the main bridgee fragility of the highway network would be under-estimated if only the damage potential of the suspensionspan was adopted erefore when the structures such asapproach bridges transition piers and auxiliary pierswere ignored the damage identification and repair de-cisions would be misleading in the seismic evaluation oflong-span bridges Conversely the damaging probabilityof the overall bridge system will be amplified regardless ofthe higher importance of the main bridge When respectthe fact that the importance of suspension span is higherthan that of approach bridge in transportation networks aparameter a of 06ndash08 may be a better choice in thepresent study
e seismic fragility of conventionally designedbridges was mainly governed by the approach bridges at alldamage levels However for the retrofitted bridges thatpossess reasonable seismic performance different damagestates were controlled by the earthquake demands ofvarious structures as shown by the trend of the curvesalong parameter a in Case 4 and Case 5 e damagingpotential of the overall bridge from the low-to-moderatelimit states was dominated by the approach bridge eprimary reasons were the sliding of bearings and thetransfer of limited forces from the girder to substructure
by the damage of restriction devices of the approachbridges e extensive damage was controlled by thedisplacement of stiffening girder of the main bridge whichinferred the effect of dampers will be conspicuous understrong earthquake shaking At the same time the overalllong-span bridges with the reasonable seismic system willnot collapse during a strong earthquake event and canmeet emergency traffic service It can be seen that the meanvalues of fragility functions developed by compositedamage indices (equations (4) and (5)) are good indicatorsto evaluate seismic performance of long-span bridges andcan be consistent with the established seismic performanceobjectives
5 Conclusions
In the present study a procedure for investigating thereasonable seismic performance of long-span suspensionbridges was developed under the fragility function frame-work e seismic performance of a long-span suspensionbridge was examined and improved using the proposedprocedure from which the results revealed distinguishableconclusions as follows
(1) e definition of damage states for bearings con-sidered the interactions between the adjacent
014036059081103125
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(a)
020051081112142173
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
0002
0406
0810
CaseParameter a
(b)
130251371492612733
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(c)
Mea
n va
lue
exp
(λ) (
ms
)
Parameter a Case0 1 2 3 4 5
0001
0203
0405
131305478652826
1000
(d)
Figure 19 Influence of relative importance and improved designs on mean values of earthquake intensity for different damage states(a) slight (b) moderate (c) extensive (d) collapse
Advances in Civil Engineering 19
structures and self-damage to bearings in accor-dance with the actual design and experimentalconclusions e damage states of pylons in terms ofcurvature ductility were defined by developingmodal pushover analysis and idealized bilinear fit-ting methods in the equivalent pylon fiber models ofthe suspension bridge
(2) When investigating the component fragility curvesof the initial design it was found that the large deckdisplacement of the suspension span was the mostserious damage present and a slight inelastic re-sponse may have occurred only in the pylons Al-though the cable system was able to maintainelasticity under earthquakes the saddles at the py-lons were identified with a high-risk region indi-cating the damaging potential For the approachbridges the ductility mechanism did not realize arational damage sequence e columns were proneto collapse because of the application of fixed potbearings which in turn caused the damage to thebearings and abutments
(3) e stepwise improved seismic designs were pro-posed in terms of the component fragility results ofthe initial design Among the candidates fluid vis-cous dampers were proven to be a good choice forlong-span bridges based on the two attributes ofdisplacement and force when the similar efficiency ofdisplacement control was achieved e restraintsystem had a critical effect on the seismic responsesof the approach bridges e quasi-isolation orisolation can achieve a reasonable seismic damagesequence of the approach bridges
(4) e defined composite damage indices showed thatthe incorporate component classification conceptand structural relative importance were necessary toderive a good indicator of performance evaluationdue to the fact that the damage indices were based oneither an absolutely serial or serial-parallel systemmodel may significantly overestimate the damagepotential of the overall bridge system e meanvalues of earthquake intensity at the specific damagestates can serve as a function of the improvedstrategies and relative importance to further evaluatethe overall seismic performance of the suspensionbridge It can be concluded that the damaging po-tential of the overall bridge from the low-to-mod-erate limit states was governed by the approachbridges in the retrofitted bridges with a favorableseismic performance whereas the displacementdemand of the suspension span dominated thefragility probability of the extensive damage
Data Availability
e nature of the data is the analytical data of finite elementmodels of the case bridge e data used to support thefindings of this study are available from the correspondingauthor upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the Basic Research ServiceProject of Central-Level Public Welfare Research Institute(grant nos 2016-9018 and 2019-0034) and Science andTechnology Project of Communicationsrsquo Construction inWestern China MOC (grant no 2009318223094)
References
[1] L W David A Kartoum K Chang and I Roy ldquoSuspensionreport seismic performance and design considerations of longspan suspension bridgesrdquo Report CSMIP00-03 CaliforniaDepartment of Conservation Division of Mines And GeologyOffice of Strong Motion Studies Sacramento CA USA 2000
[2] S-G Gwon and D-H Choi ldquoImproved continuummodel forfree vibration analysis of suspension bridgesrdquo Journal ofEngineering Mechanics vol 143 no 7 Article ID 040170382017
[3] S Arzoumanidis A Shama and F Ostadan ldquoPerformance-based seismic analysis and design of suspension bridgesrdquoEarthquake Engineering and Structural Dynamics vol 34no 4-5 pp 349ndash367 2005
[4] L Sgambi E Garavaglia N Basso and F Bontempi ldquoMonteCarlo simulation for seismic analysis of a long span sus-pension bridgerdquo Engineering Structures vol 78 pp 100ndash1112014
[5] D Karmakar S Ray-Chaudhuri and M Shinozuka ldquoFiniteelement model development validation and probabilisticseismic performance evaluation of Vincent omas suspen-sion bridgerdquo Structure and Infrastructure Engineering vol 11no 2 pp 223ndash237 2015
[6] W-F Chen and L Duan Bridge Engineering HandbookSeismic Design CRC Press Boca Raton FL USA 2nd edition2014
[7] E Choi ldquoSeismic analysis and retrofit of mid-americabridgesrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2002
[8] B G Nielson and R DesRoches ldquoSeismic fragility method-ology for highway bridges using a component level approachrdquoEarthquake Engineering amp Structural Dynamics vol 36 no 6pp 823ndash839 2007
[9] K N Ramanathan ldquoNext generation seismic fragility curvesfor california bridges incorporating the evolution in seismicdesignrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2012
[10] O Avsar A Yakut and A Caner ldquoAnalytical fragility curvesfor ordinary highway bridges in Turkeyrdquo Earthquake Spectravol 27 no 4 pp 971ndash996 2011
[11] S P Stefanidou and A J Kappos ldquoMethodology for thedevelopment of bridge-specific fragility curvesrdquo EarthquakeEngineering amp Structural Dynamics vol 46 no 1 pp 73ndash932016
[12] B Borzi P Ceresa P Franchin F Noto G M Calvi andP E Pinto ldquoSeismic vulnerability of the Italian roadwaybridge stockrdquo Earthquake Spectra vol 31 no 4pp 2137ndash2161 2015
[13] W T Barnawi and S J Dyke ldquoSeismic fragility relationshipsof a cable-stayed bridge equipped with response modification
20 Advances in Civil Engineering
systemsrdquo Journal of Bridge Engineering vol 19 no 8 ArticleID A4013003 2014
[14] J Zhong Y Pang J Jong-Su J-S Jeon R DesRoches andW Yuan ldquoSeismic fragility assessment of long-span cable-stayed bridges in Chinardquo Advances in Structural Engineeringvol 19 no 11 pp 1797ndash1812 2016
[15] J Zhong Z Hu W Yuan and L Chen ldquoSystem-basedprobabilistic optimization of fluid viscous dampers equippedin cable-stayed bridgesrdquo Advances in Structural Engineeringvol 21 no 12 pp 1815ndash1825 2018
[16] J E Padgett and R DesRoches ldquoMethodology for the de-velopment of analytical fragility curves for retrofitted bridgesfitted bridgesrdquo Earthquake Engineering amp Structural Dy-namics vol 37 no 8 pp 1157ndash1174 2008
[17] Y Xie and J Zhang ldquoOptimal design of seismic protectivedevices for highway bridges using performance-basedmethodology and multiobjective genetic optimizationrdquoJournal of Bridge Engineering vol 22 no 3 Article ID04016129 2017
[18] J Zhang and Y Huo ldquoEvaluating effectiveness and optimumdesign of isolation devices for highway bridges using thefragility function methodrdquo Engineering Structures vol 31no 8 pp 1648ndash1660 2009
[19] F McKenna M H Scott and G L Fenves ldquoNonlinear finite-element analysis software architecture using object compo-sitionrdquo Journal of Computing in Civil Engineering vol 24no 1 pp 95ndash107 2010
[20] MOTC ldquoGuidelines for seismic design of highway bridgesrdquoChina Ministry of Transportation and CommunicationsPress Beijing China JTGT B02-01-2008 2008
[21] A D Kiureghian and J L Sackman ldquoTangent geometricstiffness of inclined cables under self-weightrdquo Journal ofStructural Engineering vol 131 no 6 pp 941ndash945 2005
[22] J S Steelman E T Filipov L A Fahnestock et al ldquoEx-perimental behavior of steel fixed bearings and implicationsfor seismic bridge responserdquo Journal of Bridge Engineeringvol 19 no 8 Article ID A4014007 2014
[23] B Madani F Behnamfar and H Tajmir Riahi ldquoDynamicresponse of structures subjected to pounding and structure-soil-structure interactionrdquo Soil Dynamics and EarthquakeEngineering vol 78 pp 46ndash60 2015
[24] L Xu and J Li ldquoDesign and experimental investigation of anew type sliding retainer and its efficacy in seismic fortifi-cationrdquo Engineering Mechanics vol 33 no 2 pp 111ndash1182016 in Chinese
[25] Caltrans Caltrans Seismic Design Criteria California De-partment of Transportation Sacramento CA USA 2013
[26] A Shamsabadi K M Rollins and M Kapuskar ldquoNonlinearsoil-abutment-bridge structure interaction for seismic per-formance-based designrdquo Journal of Geotechnical and Geo-environmental Engineering vol 133 no 6 pp 707ndash720 2007
[27] X Li Z Sun D Wang and Y Shi ldquoLongitudinal seismicpounding effect of bridges abutment and backfilling damagerdquoJournal of Changrsquoan University vol 35 no 4 pp 76ndash82 2015in Chinese
[28] MOTC ldquoSpecification for design of highway suspensionbridgerdquo China Ministry of Transportation and Communi-cations Press Beijing China JTGT D65-05-2015 2015
[29] A Shafieezadeh K Ramanathan J E Padgett andR DesRoches ldquoFractional order intensity measures forprobabilistic seismic demand modeling applied to highwaybridgesrdquo Earthquake Engineering amp Structural Dynamicsvol 41 no 3 pp 391ndash409 2012
[30] J E Padgett B G Nielson and R DesRoches ldquoSelection ofoptimal intensity measures in probabilistic seismic demandmodels of highway bridge portfoliosrdquo Earthquake Engineeringamp Structural Dynamics vol 37 no 5 pp 711ndash725 2008
[31] K R Mackie and B Stojadinovic ldquoFragility basis for californiahighway overpass bridge seismic decision makingrdquo PEERReport 200502 Pacific Earthquake Engineering ResearchCenter University of California Berkeley Berkeley CA USA2005
[32] J Li T Peng Ji Lin L Zhou and YuL Deng Seismic Design ofMulti-Pylon Multi-Span Suspension Bridge China Commu-nications Press Beijing China 2013 in Chinese
[33] L Chen W Zhuang H Zhao and Z Wan ldquoReport onHighwaysrsquo Damage in the Wenchuan Earthquake-BridgeChina Communications Press Beijing China 2012 inChinese
[34] G Wu K Wang G Lu and P Zhang ldquoAn experimentalinvestigation of unbonded laminated elastomeric bearingsand the seismic evaluations of highway bridges with testedbearing componentsrdquo Shock and Vibration vol 2018 ArticleID 8439321 18 pages 2018
Advances in Civil Engineering 21
Figure 14 depicts the change of components vulnera-bilities before and after the application of dampers It can beobserved that the retrofit measure played limited roles in thefragility reduction of the self-damaged bearings in long-spanbridges Since the additional longitudinal rigidity to girderwas provided by the dampers the transverse responses suchas girder displacement and pylon curvature would increaseas ground motion intensity increased e influences on thecable system responses and the longitudinal responses ofpylons were found to be negligible Another phenomenonwas that the longitudinal fragility of the abutments signif-icantly decreased with the reduction in the poundingprobabilities between the decks and abutments of the ap-proach bridges It was inferred that the pounding betweenthe adjacent decks was likely to affect movement of theapproach bridges and that the pounding effect would betransmitted along the girder
42 Improved Design of Approach Bridges e damagepotential of each component for the approach bridges washigh due to the high collapse risks posed by the usage of thefixed piers e structural designs of small-to-medium-spanhighway bridges in China show the fact that the super-structures are usually placed onto laminated elastomericbearings (LEBs) which tend to come into direct contact withthe bridge girders and substructures without anchoring etypical types of seismic damages of these bridges consisted of
large girder movements bearings sliding failures of shearkeys and expansion joints and abutment cracking duringthe Wenchuan and Yushu earthquake events in ChinaMeanwhile the bridge columns displayed relatively lowdamage ratios under earthquakes [33] e design can bereferred to as a quasi-isolation system which is primarilyrealized by strategically employing bearings sliding as force-limiting connections between girders and substructures Inaddition the isolation strategies of lead-rubber bearings(LRBs) are usually employed in the high-seismicity regionswhich can be taken as a comparison of the quasi-isolationsystem e two restraint systems consist of bearings shearkeys transversely located at the cap beams and restrainercables longitudinally anchored to the abutments (or ped-estals) In order to calculate the seismic responses of theapproach bridge the related experiments were completedand summarized and applied to the force-displacementrelations of the restraint system components
e detailed experiment of LEB consulted previousacademic achievements [34] An example is shown inFigure 15(a) where the feature points marked by rectangularand circular represented initial walking and obvious slidingresponses which illustrated that the mechanical property ofbearings would degrade as the displacements increased afterentering the friction sliding state until it remained nearlyconstant When there was no specific test the horizontalforce Fd in the stable friction stage can be 07 Fmax whereFmax was the corresponding maximum static friction at the
Bear
ing
disp
lace
men
t (m
)
Logarithmic spacendash5
ndash4
ndash3
ndash2
ndash1
0
1
2 4 6PGV (cms)
Case 1Case 2Case 3
(a)Po
undi
ng p
roba
bilit
y
00
02
04
06
08
05 10 15 2000PGV (ms)
Case 1Case 2Case 3
(b)
Poun
ding
pro
babi
lity
00
02
04
06
02 04 06 08 1000Deformation (m)
Case 1Case 2
(c)
ln [f
orce
(kN
)]
5
10
15
15
20
20
25
25
30
3030
35
3535
40
40
4550
7
72
74
76
78
8
033 041 049 057 065025Deformation (m)
(d)
ln [f
orce
(kN
)]5
10
15
15 20
20 25
25
30
3030
35
35
40
40
45
50
102
104
106
108
11
112
114
0155 0205 0255 0305 03550105Deformation (m)
(e)
Figure 13 Choice of the three retrofitmeasures based on deformations and forces (a) EDP-IM relation (b) pounding probability conditioned onIM (c) pounding probability conditioned on deformation contours of pounding probability for (d) dampers and (e) elastic cables
14 Advances in Civil Engineering
initial walking e displacement value Δ2 can be of 250effective shear strain (ESS) e initial stiffness K0 is cal-culated by MOTC [20] erefore the coefficients can becalculated by the following sequence FdK0⟶ Fmax⟶Δ1⟶Δ2 e built analytical relation ofthe LEB is shown in Figure 16(a) which can correctlycapture mechanical behavior under earthquakes A log-normal distribution [8] with a logarithmic standard devi-ation of 01 and a normal distribution with aCOV 016MPa were used to model the randomness offriction coefficient and shear modulus respectively For theirrespective mean values 02 and 12MPa were taken
Figure 15(b) shows a cyclic loading experiment of LRBwhich reveals that the degradation of bearings stiffness willoccur with the increase of loading displacements e elasticstiffness can reach uniformly 3 to 7 times the postyieldingstiffness according to the experimental results e
mechanical behavior of LRB can be modeled by a bilinearrelation as seen in Figure 16(b)
e cable restrainers were modeled as nonlinear tension-only elements characterized by an initial slack as shown inFigure 16(c) e parameters affecting the functionality ofcable restrainers were the cable slack length and yieldstrength according to the research of Padgett and DesRoches[16] e first two parameters were assumed to follow theuniform distributions and the strength can be modeled bythe lognormal distribution An adopted range of cablestiffness values was from 30times105 kNm to 50times105 kNmuniformly
e key parameters in analytical models of Figure 16 canbe determined according to the structural design of therealistic bridge e damage states of two restraint systemsare listed in Table 6 in which ESS is used as the damageindex
Dam
age p
roba
bilit
y
05 10 15 2000PGV (ms)
00
02
04
06
08
10
Case 0Case 1
MBDLMBDT
(a)
Dam
age p
roba
bilit
y
00
01
02
03
04
05
05 10 15 2000PGV (ms)
Case 0Case 1
PCDNLPCDNT
(b)
Dam
age p
roba
bilit
y
00
02
04
06
08
10
05 10 15 2000PGV (ms)
Case 0Case 1
BPDAEJDA
(c)
Figure 14 Influence of dampers on vulnerabilties of (a) bearing responses for DS2 (b) pylon responses for DS2 and (c) abutment andpounding responses
Advances in Civil Engineering 15
Figure 17 illustrates the probabilities of critical damage stateoccurrence sequences of the bridge components in Cases 1 4and 5 when PGV is 10ms which is equal to E2 earthquakeintensity in the seismic assessment reports e componentvulnerabilities of initial design (Case 0) were taken as the ref-erence values which were marked by different colored strips inFigure 17 e desirable sequence preferred Cases 4 or 5 for theapproach bridgese initial design of approach bridges resultedin unreasonable damage sequences In the strategies pertainingto quasi-isolation and isolation systems multiple tiers of seismicstructural redundancy can be normally employed to preventstructural collapse during strong earthquake events Tier 1 in-dicated that the bearings were weakly fused which was evidentin the high fragility of bearing sliding behaviors (ABDAL andABDC2T) Another advantage was that the extensive dam-ages of bearing would be reduced when compared with the self-damage of the pot bearings Tier 2 indicated that the designedbearings had sufficient seat widths for adapting to the sliding ofsuperstructures However the devices had a certain damageprobability (EJDA and SKDC2) and dissipated a portion of
the seismic energy when the displacement of the superstructuresbecame larger than the allowable values of constraint devicesTier 3 redundancy was designed as a potential plastic hinge inthe substructures (CCDC2) including a certain yielding of thebackfill at the abutments (BPDA) e response of theabutments in the transverse direction (PDAT) was slightlyincreased due to the pounding effect between the girders andshear keys caused by bearing sliding e reasonable damagesequence of two restraint systems can be reflected through thecorresponding damage probabilities of bearings seismic mea-sures and substructures respectively as indicated by the red andgreen lines in Figure 17 erefore the rational sequence ofcritical limit state occurrences for the approach bridges isbearings⟶ expansion joints (or restrainer cables) shearkeys⟶ columns abutments
43 Comprehensive Evaluation of Bridge System Padgett andDesRoches [16] pointed out the necessity to derive the bridgesystem vulnerabilities when assessing retrofitted bridges
Figure 15 Experimental results of (a) LEB and (b) LRB
ΔΔ1
F
Δ2
Fmax
Fd
KdK0
(a)
Δy Δ
F
K1 = (3~7) K2
K1
K2
(b)
Slack
F
K
Δ
Fy
(c)
Figure 16 Analytical models of (a) LEB (b) LRB and (c) cable restrainers
16 Advances in Civil Engineering
because different components may have a different responsewith a given retrofit measure Also Zhang and Huo [18]studied the optimal parameters of isolation bearings byderiving the intensity median value corresponding to sys-tem-level fragility function which can effectively avoid theimpacts of different isolation parameters on differentcomponents As can be seen from Figures 14 and 17 theabove analysis results were also very apparent in this studyFor example the longitudinal deck displacement was sig-nificantly reduced by using dampers yet increased thetransverse demands of stiffening girder and pylons e useof LEB or LRB reduced the demands placed on columns yetled to an increase of the transverse deformations of abut-ments which was due to the use of shear keys limiting theexcessive sliding displacement of bearings erefore acomprehensive evaluation of retrofit strategies on bridgesparticularly long-span suspension bridges that have multiplecomponents should be based on system-level fragility in-stead of component-level
Taking the findings that no damages had occurred to thecable system and the definition of bearing damage states intoaccount the derivation of system-level fragility of the wholebridge depended on critical components including pylonscolumns bearings and abutments e fragility data byemploying the absolutely serial and serial-parallel assump-tions of bridge system are plotted in Figure 18 and comparedto the system fragility obtained from the proposed com-posite damage state e absolutely serial assumption
generated serious bias for fragility curves and may result inmisdirection if they were served as a design guide eabsolutely serial model overestimated the potential damageof the bridge system since the fragility curve directlydepended on the most vulnerable components In otherwords if the difference in location and performance of eachcomponent was neglected then the seismic performanceevaluation would be achieved by attaching equal importanceto the strength of pylons approach columns and bearingsis method contradicted the philosophy of capacity designin which the long-span structure can dissipate energythrough a reasonable swinging of stiffening girder and thebearings were designed as fuse elements in the small-to-medium-span bridgese serial-parallel model qualitativelyincorporated the differences of components and was verysensitive to the seismic design of the structure as seen fromthe comparisons among the cases in Figure 18 e modeldistinguished the serial components and parallel compo-nents yet still overestimated the bridge system-level fragilitywhen bearings became the vulnerable components espe-cially for long-span bridges is special situation such asthe serial assumption may also lead to misunderstandingOnce the relative importance between the components wasconsidered in the serial-composite model the damage po-tential was clearly reduced within the range of earthquakeintensity of interest based on the case studies is alsoreflected the necessity of the classification idea of bridgecomponents which can be expressed by the numerical value
Table 6 Damage states of the quasi-isolation and isolation bearings (m)
Figure 17 Probabilities of critical damage states occurrence sequences for the approach bridges when PGV is 1ms
Advances in Civil Engineering 17
of system-level fragility erefore the application ofcomposite damage state (equation (4)) considering com-ponent difference was instead more appropriate than theaforementioned two models
e damage probability distribution represented byblack circles in Figure 18 can be obtained from the definedcomposite damage state (equation (5)) where the parametera is 06 It was found that system fragility result was a moreaccurate assessment with further taking the relative im-portance between themain bridge and approach bridges intoaccount is is due to the principle that the seismic designperformance objective of long-span bridges generally has nocollapse for the main body structures and is necessarily opento emergency vehicle service In addition attention must bepaid to the fact that the differences in repair cost and time ofthe main and approach bridges are significant when thesame damage state is achieved e results of compositedamage state may match the performance objectives of theseismic design In order to investigate the influence ofimproved strategies and structural relative importance ondamage potential of bridges the mean values of the fragility
functions can be employed to evaluate the seismic designe smooth fragility curves for the specific damage states canbe fitted with the lognormal cumulative probability distri-bution function by employing the least-squares technique
P DSgtDSi | PGV1113858 1113859 Φln(pgv) minus λPGV
ζPGV1113888 1113889 (12)
where λPGV and ζPGV are the logarithmic mean and dis-persion values of earthquake intensity (PGV) for a specifieddamage level e larger mean values represent the smallerdamage potential to reach the specified level
Figure 19 indicates the mean values of earthquakeintensity at the specific damage states as a function of theimproved strategies and parameter a Each step of theretrofit measures would enhance the seismic performanceof the overall bridge system at all levels e improvementof the restraint system of the approach bridge played acritical role of the overall bridge system For instance themean values of the earthquake intensity featured mutationof slopes along the measure direction which reflected the
Figure 18 System fragility of the original design (a) Case 0 and improved design including (b) Case 1 and (c) Case 5
18 Advances in Civil Engineering
fact that the effectiveness of improvement for the ap-proach bridges was clearer than that of the main bridgee fragility of the highway network would be under-estimated if only the damage potential of the suspensionspan was adopted erefore when the structures such asapproach bridges transition piers and auxiliary pierswere ignored the damage identification and repair de-cisions would be misleading in the seismic evaluation oflong-span bridges Conversely the damaging probabilityof the overall bridge system will be amplified regardless ofthe higher importance of the main bridge When respectthe fact that the importance of suspension span is higherthan that of approach bridge in transportation networks aparameter a of 06ndash08 may be a better choice in thepresent study
e seismic fragility of conventionally designedbridges was mainly governed by the approach bridges at alldamage levels However for the retrofitted bridges thatpossess reasonable seismic performance different damagestates were controlled by the earthquake demands ofvarious structures as shown by the trend of the curvesalong parameter a in Case 4 and Case 5 e damagingpotential of the overall bridge from the low-to-moderatelimit states was dominated by the approach bridge eprimary reasons were the sliding of bearings and thetransfer of limited forces from the girder to substructure
by the damage of restriction devices of the approachbridges e extensive damage was controlled by thedisplacement of stiffening girder of the main bridge whichinferred the effect of dampers will be conspicuous understrong earthquake shaking At the same time the overalllong-span bridges with the reasonable seismic system willnot collapse during a strong earthquake event and canmeet emergency traffic service It can be seen that the meanvalues of fragility functions developed by compositedamage indices (equations (4) and (5)) are good indicatorsto evaluate seismic performance of long-span bridges andcan be consistent with the established seismic performanceobjectives
5 Conclusions
In the present study a procedure for investigating thereasonable seismic performance of long-span suspensionbridges was developed under the fragility function frame-work e seismic performance of a long-span suspensionbridge was examined and improved using the proposedprocedure from which the results revealed distinguishableconclusions as follows
(1) e definition of damage states for bearings con-sidered the interactions between the adjacent
014036059081103125
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(a)
020051081112142173
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
0002
0406
0810
CaseParameter a
(b)
130251371492612733
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(c)
Mea
n va
lue
exp
(λ) (
ms
)
Parameter a Case0 1 2 3 4 5
0001
0203
0405
131305478652826
1000
(d)
Figure 19 Influence of relative importance and improved designs on mean values of earthquake intensity for different damage states(a) slight (b) moderate (c) extensive (d) collapse
Advances in Civil Engineering 19
structures and self-damage to bearings in accor-dance with the actual design and experimentalconclusions e damage states of pylons in terms ofcurvature ductility were defined by developingmodal pushover analysis and idealized bilinear fit-ting methods in the equivalent pylon fiber models ofthe suspension bridge
(2) When investigating the component fragility curvesof the initial design it was found that the large deckdisplacement of the suspension span was the mostserious damage present and a slight inelastic re-sponse may have occurred only in the pylons Al-though the cable system was able to maintainelasticity under earthquakes the saddles at the py-lons were identified with a high-risk region indi-cating the damaging potential For the approachbridges the ductility mechanism did not realize arational damage sequence e columns were proneto collapse because of the application of fixed potbearings which in turn caused the damage to thebearings and abutments
(3) e stepwise improved seismic designs were pro-posed in terms of the component fragility results ofthe initial design Among the candidates fluid vis-cous dampers were proven to be a good choice forlong-span bridges based on the two attributes ofdisplacement and force when the similar efficiency ofdisplacement control was achieved e restraintsystem had a critical effect on the seismic responsesof the approach bridges e quasi-isolation orisolation can achieve a reasonable seismic damagesequence of the approach bridges
(4) e defined composite damage indices showed thatthe incorporate component classification conceptand structural relative importance were necessary toderive a good indicator of performance evaluationdue to the fact that the damage indices were based oneither an absolutely serial or serial-parallel systemmodel may significantly overestimate the damagepotential of the overall bridge system e meanvalues of earthquake intensity at the specific damagestates can serve as a function of the improvedstrategies and relative importance to further evaluatethe overall seismic performance of the suspensionbridge It can be concluded that the damaging po-tential of the overall bridge from the low-to-mod-erate limit states was governed by the approachbridges in the retrofitted bridges with a favorableseismic performance whereas the displacementdemand of the suspension span dominated thefragility probability of the extensive damage
Data Availability
e nature of the data is the analytical data of finite elementmodels of the case bridge e data used to support thefindings of this study are available from the correspondingauthor upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the Basic Research ServiceProject of Central-Level Public Welfare Research Institute(grant nos 2016-9018 and 2019-0034) and Science andTechnology Project of Communicationsrsquo Construction inWestern China MOC (grant no 2009318223094)
References
[1] L W David A Kartoum K Chang and I Roy ldquoSuspensionreport seismic performance and design considerations of longspan suspension bridgesrdquo Report CSMIP00-03 CaliforniaDepartment of Conservation Division of Mines And GeologyOffice of Strong Motion Studies Sacramento CA USA 2000
[2] S-G Gwon and D-H Choi ldquoImproved continuummodel forfree vibration analysis of suspension bridgesrdquo Journal ofEngineering Mechanics vol 143 no 7 Article ID 040170382017
[3] S Arzoumanidis A Shama and F Ostadan ldquoPerformance-based seismic analysis and design of suspension bridgesrdquoEarthquake Engineering and Structural Dynamics vol 34no 4-5 pp 349ndash367 2005
[4] L Sgambi E Garavaglia N Basso and F Bontempi ldquoMonteCarlo simulation for seismic analysis of a long span sus-pension bridgerdquo Engineering Structures vol 78 pp 100ndash1112014
[5] D Karmakar S Ray-Chaudhuri and M Shinozuka ldquoFiniteelement model development validation and probabilisticseismic performance evaluation of Vincent omas suspen-sion bridgerdquo Structure and Infrastructure Engineering vol 11no 2 pp 223ndash237 2015
[6] W-F Chen and L Duan Bridge Engineering HandbookSeismic Design CRC Press Boca Raton FL USA 2nd edition2014
[7] E Choi ldquoSeismic analysis and retrofit of mid-americabridgesrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2002
[8] B G Nielson and R DesRoches ldquoSeismic fragility method-ology for highway bridges using a component level approachrdquoEarthquake Engineering amp Structural Dynamics vol 36 no 6pp 823ndash839 2007
[9] K N Ramanathan ldquoNext generation seismic fragility curvesfor california bridges incorporating the evolution in seismicdesignrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2012
[10] O Avsar A Yakut and A Caner ldquoAnalytical fragility curvesfor ordinary highway bridges in Turkeyrdquo Earthquake Spectravol 27 no 4 pp 971ndash996 2011
[11] S P Stefanidou and A J Kappos ldquoMethodology for thedevelopment of bridge-specific fragility curvesrdquo EarthquakeEngineering amp Structural Dynamics vol 46 no 1 pp 73ndash932016
[12] B Borzi P Ceresa P Franchin F Noto G M Calvi andP E Pinto ldquoSeismic vulnerability of the Italian roadwaybridge stockrdquo Earthquake Spectra vol 31 no 4pp 2137ndash2161 2015
[13] W T Barnawi and S J Dyke ldquoSeismic fragility relationshipsof a cable-stayed bridge equipped with response modification
20 Advances in Civil Engineering
systemsrdquo Journal of Bridge Engineering vol 19 no 8 ArticleID A4013003 2014
[14] J Zhong Y Pang J Jong-Su J-S Jeon R DesRoches andW Yuan ldquoSeismic fragility assessment of long-span cable-stayed bridges in Chinardquo Advances in Structural Engineeringvol 19 no 11 pp 1797ndash1812 2016
[15] J Zhong Z Hu W Yuan and L Chen ldquoSystem-basedprobabilistic optimization of fluid viscous dampers equippedin cable-stayed bridgesrdquo Advances in Structural Engineeringvol 21 no 12 pp 1815ndash1825 2018
[16] J E Padgett and R DesRoches ldquoMethodology for the de-velopment of analytical fragility curves for retrofitted bridgesfitted bridgesrdquo Earthquake Engineering amp Structural Dy-namics vol 37 no 8 pp 1157ndash1174 2008
[17] Y Xie and J Zhang ldquoOptimal design of seismic protectivedevices for highway bridges using performance-basedmethodology and multiobjective genetic optimizationrdquoJournal of Bridge Engineering vol 22 no 3 Article ID04016129 2017
[18] J Zhang and Y Huo ldquoEvaluating effectiveness and optimumdesign of isolation devices for highway bridges using thefragility function methodrdquo Engineering Structures vol 31no 8 pp 1648ndash1660 2009
[19] F McKenna M H Scott and G L Fenves ldquoNonlinear finite-element analysis software architecture using object compo-sitionrdquo Journal of Computing in Civil Engineering vol 24no 1 pp 95ndash107 2010
[20] MOTC ldquoGuidelines for seismic design of highway bridgesrdquoChina Ministry of Transportation and CommunicationsPress Beijing China JTGT B02-01-2008 2008
[21] A D Kiureghian and J L Sackman ldquoTangent geometricstiffness of inclined cables under self-weightrdquo Journal ofStructural Engineering vol 131 no 6 pp 941ndash945 2005
[22] J S Steelman E T Filipov L A Fahnestock et al ldquoEx-perimental behavior of steel fixed bearings and implicationsfor seismic bridge responserdquo Journal of Bridge Engineeringvol 19 no 8 Article ID A4014007 2014
[23] B Madani F Behnamfar and H Tajmir Riahi ldquoDynamicresponse of structures subjected to pounding and structure-soil-structure interactionrdquo Soil Dynamics and EarthquakeEngineering vol 78 pp 46ndash60 2015
[24] L Xu and J Li ldquoDesign and experimental investigation of anew type sliding retainer and its efficacy in seismic fortifi-cationrdquo Engineering Mechanics vol 33 no 2 pp 111ndash1182016 in Chinese
[25] Caltrans Caltrans Seismic Design Criteria California De-partment of Transportation Sacramento CA USA 2013
[26] A Shamsabadi K M Rollins and M Kapuskar ldquoNonlinearsoil-abutment-bridge structure interaction for seismic per-formance-based designrdquo Journal of Geotechnical and Geo-environmental Engineering vol 133 no 6 pp 707ndash720 2007
[27] X Li Z Sun D Wang and Y Shi ldquoLongitudinal seismicpounding effect of bridges abutment and backfilling damagerdquoJournal of Changrsquoan University vol 35 no 4 pp 76ndash82 2015in Chinese
[28] MOTC ldquoSpecification for design of highway suspensionbridgerdquo China Ministry of Transportation and Communi-cations Press Beijing China JTGT D65-05-2015 2015
[29] A Shafieezadeh K Ramanathan J E Padgett andR DesRoches ldquoFractional order intensity measures forprobabilistic seismic demand modeling applied to highwaybridgesrdquo Earthquake Engineering amp Structural Dynamicsvol 41 no 3 pp 391ndash409 2012
[30] J E Padgett B G Nielson and R DesRoches ldquoSelection ofoptimal intensity measures in probabilistic seismic demandmodels of highway bridge portfoliosrdquo Earthquake Engineeringamp Structural Dynamics vol 37 no 5 pp 711ndash725 2008
[31] K R Mackie and B Stojadinovic ldquoFragility basis for californiahighway overpass bridge seismic decision makingrdquo PEERReport 200502 Pacific Earthquake Engineering ResearchCenter University of California Berkeley Berkeley CA USA2005
[32] J Li T Peng Ji Lin L Zhou and YuL Deng Seismic Design ofMulti-Pylon Multi-Span Suspension Bridge China Commu-nications Press Beijing China 2013 in Chinese
[33] L Chen W Zhuang H Zhao and Z Wan ldquoReport onHighwaysrsquo Damage in the Wenchuan Earthquake-BridgeChina Communications Press Beijing China 2012 inChinese
[34] G Wu K Wang G Lu and P Zhang ldquoAn experimentalinvestigation of unbonded laminated elastomeric bearingsand the seismic evaluations of highway bridges with testedbearing componentsrdquo Shock and Vibration vol 2018 ArticleID 8439321 18 pages 2018
Advances in Civil Engineering 21
initial walking e displacement value Δ2 can be of 250effective shear strain (ESS) e initial stiffness K0 is cal-culated by MOTC [20] erefore the coefficients can becalculated by the following sequence FdK0⟶ Fmax⟶Δ1⟶Δ2 e built analytical relation ofthe LEB is shown in Figure 16(a) which can correctlycapture mechanical behavior under earthquakes A log-normal distribution [8] with a logarithmic standard devi-ation of 01 and a normal distribution with aCOV 016MPa were used to model the randomness offriction coefficient and shear modulus respectively For theirrespective mean values 02 and 12MPa were taken
Figure 15(b) shows a cyclic loading experiment of LRBwhich reveals that the degradation of bearings stiffness willoccur with the increase of loading displacements e elasticstiffness can reach uniformly 3 to 7 times the postyieldingstiffness according to the experimental results e
mechanical behavior of LRB can be modeled by a bilinearrelation as seen in Figure 16(b)
e cable restrainers were modeled as nonlinear tension-only elements characterized by an initial slack as shown inFigure 16(c) e parameters affecting the functionality ofcable restrainers were the cable slack length and yieldstrength according to the research of Padgett and DesRoches[16] e first two parameters were assumed to follow theuniform distributions and the strength can be modeled bythe lognormal distribution An adopted range of cablestiffness values was from 30times105 kNm to 50times105 kNmuniformly
e key parameters in analytical models of Figure 16 canbe determined according to the structural design of therealistic bridge e damage states of two restraint systemsare listed in Table 6 in which ESS is used as the damageindex
Dam
age p
roba
bilit
y
05 10 15 2000PGV (ms)
00
02
04
06
08
10
Case 0Case 1
MBDLMBDT
(a)
Dam
age p
roba
bilit
y
00
01
02
03
04
05
05 10 15 2000PGV (ms)
Case 0Case 1
PCDNLPCDNT
(b)
Dam
age p
roba
bilit
y
00
02
04
06
08
10
05 10 15 2000PGV (ms)
Case 0Case 1
BPDAEJDA
(c)
Figure 14 Influence of dampers on vulnerabilties of (a) bearing responses for DS2 (b) pylon responses for DS2 and (c) abutment andpounding responses
Advances in Civil Engineering 15
Figure 17 illustrates the probabilities of critical damage stateoccurrence sequences of the bridge components in Cases 1 4and 5 when PGV is 10ms which is equal to E2 earthquakeintensity in the seismic assessment reports e componentvulnerabilities of initial design (Case 0) were taken as the ref-erence values which were marked by different colored strips inFigure 17 e desirable sequence preferred Cases 4 or 5 for theapproach bridgese initial design of approach bridges resultedin unreasonable damage sequences In the strategies pertainingto quasi-isolation and isolation systems multiple tiers of seismicstructural redundancy can be normally employed to preventstructural collapse during strong earthquake events Tier 1 in-dicated that the bearings were weakly fused which was evidentin the high fragility of bearing sliding behaviors (ABDAL andABDC2T) Another advantage was that the extensive dam-ages of bearing would be reduced when compared with the self-damage of the pot bearings Tier 2 indicated that the designedbearings had sufficient seat widths for adapting to the sliding ofsuperstructures However the devices had a certain damageprobability (EJDA and SKDC2) and dissipated a portion of
the seismic energy when the displacement of the superstructuresbecame larger than the allowable values of constraint devicesTier 3 redundancy was designed as a potential plastic hinge inthe substructures (CCDC2) including a certain yielding of thebackfill at the abutments (BPDA) e response of theabutments in the transverse direction (PDAT) was slightlyincreased due to the pounding effect between the girders andshear keys caused by bearing sliding e reasonable damagesequence of two restraint systems can be reflected through thecorresponding damage probabilities of bearings seismic mea-sures and substructures respectively as indicated by the red andgreen lines in Figure 17 erefore the rational sequence ofcritical limit state occurrences for the approach bridges isbearings⟶ expansion joints (or restrainer cables) shearkeys⟶ columns abutments
43 Comprehensive Evaluation of Bridge System Padgett andDesRoches [16] pointed out the necessity to derive the bridgesystem vulnerabilities when assessing retrofitted bridges
Figure 15 Experimental results of (a) LEB and (b) LRB
ΔΔ1
F
Δ2
Fmax
Fd
KdK0
(a)
Δy Δ
F
K1 = (3~7) K2
K1
K2
(b)
Slack
F
K
Δ
Fy
(c)
Figure 16 Analytical models of (a) LEB (b) LRB and (c) cable restrainers
16 Advances in Civil Engineering
because different components may have a different responsewith a given retrofit measure Also Zhang and Huo [18]studied the optimal parameters of isolation bearings byderiving the intensity median value corresponding to sys-tem-level fragility function which can effectively avoid theimpacts of different isolation parameters on differentcomponents As can be seen from Figures 14 and 17 theabove analysis results were also very apparent in this studyFor example the longitudinal deck displacement was sig-nificantly reduced by using dampers yet increased thetransverse demands of stiffening girder and pylons e useof LEB or LRB reduced the demands placed on columns yetled to an increase of the transverse deformations of abut-ments which was due to the use of shear keys limiting theexcessive sliding displacement of bearings erefore acomprehensive evaluation of retrofit strategies on bridgesparticularly long-span suspension bridges that have multiplecomponents should be based on system-level fragility in-stead of component-level
Taking the findings that no damages had occurred to thecable system and the definition of bearing damage states intoaccount the derivation of system-level fragility of the wholebridge depended on critical components including pylonscolumns bearings and abutments e fragility data byemploying the absolutely serial and serial-parallel assump-tions of bridge system are plotted in Figure 18 and comparedto the system fragility obtained from the proposed com-posite damage state e absolutely serial assumption
generated serious bias for fragility curves and may result inmisdirection if they were served as a design guide eabsolutely serial model overestimated the potential damageof the bridge system since the fragility curve directlydepended on the most vulnerable components In otherwords if the difference in location and performance of eachcomponent was neglected then the seismic performanceevaluation would be achieved by attaching equal importanceto the strength of pylons approach columns and bearingsis method contradicted the philosophy of capacity designin which the long-span structure can dissipate energythrough a reasonable swinging of stiffening girder and thebearings were designed as fuse elements in the small-to-medium-span bridgese serial-parallel model qualitativelyincorporated the differences of components and was verysensitive to the seismic design of the structure as seen fromthe comparisons among the cases in Figure 18 e modeldistinguished the serial components and parallel compo-nents yet still overestimated the bridge system-level fragilitywhen bearings became the vulnerable components espe-cially for long-span bridges is special situation such asthe serial assumption may also lead to misunderstandingOnce the relative importance between the components wasconsidered in the serial-composite model the damage po-tential was clearly reduced within the range of earthquakeintensity of interest based on the case studies is alsoreflected the necessity of the classification idea of bridgecomponents which can be expressed by the numerical value
Table 6 Damage states of the quasi-isolation and isolation bearings (m)
Figure 17 Probabilities of critical damage states occurrence sequences for the approach bridges when PGV is 1ms
Advances in Civil Engineering 17
of system-level fragility erefore the application ofcomposite damage state (equation (4)) considering com-ponent difference was instead more appropriate than theaforementioned two models
e damage probability distribution represented byblack circles in Figure 18 can be obtained from the definedcomposite damage state (equation (5)) where the parametera is 06 It was found that system fragility result was a moreaccurate assessment with further taking the relative im-portance between themain bridge and approach bridges intoaccount is is due to the principle that the seismic designperformance objective of long-span bridges generally has nocollapse for the main body structures and is necessarily opento emergency vehicle service In addition attention must bepaid to the fact that the differences in repair cost and time ofthe main and approach bridges are significant when thesame damage state is achieved e results of compositedamage state may match the performance objectives of theseismic design In order to investigate the influence ofimproved strategies and structural relative importance ondamage potential of bridges the mean values of the fragility
functions can be employed to evaluate the seismic designe smooth fragility curves for the specific damage states canbe fitted with the lognormal cumulative probability distri-bution function by employing the least-squares technique
P DSgtDSi | PGV1113858 1113859 Φln(pgv) minus λPGV
ζPGV1113888 1113889 (12)
where λPGV and ζPGV are the logarithmic mean and dis-persion values of earthquake intensity (PGV) for a specifieddamage level e larger mean values represent the smallerdamage potential to reach the specified level
Figure 19 indicates the mean values of earthquakeintensity at the specific damage states as a function of theimproved strategies and parameter a Each step of theretrofit measures would enhance the seismic performanceof the overall bridge system at all levels e improvementof the restraint system of the approach bridge played acritical role of the overall bridge system For instance themean values of the earthquake intensity featured mutationof slopes along the measure direction which reflected the
Figure 18 System fragility of the original design (a) Case 0 and improved design including (b) Case 1 and (c) Case 5
18 Advances in Civil Engineering
fact that the effectiveness of improvement for the ap-proach bridges was clearer than that of the main bridgee fragility of the highway network would be under-estimated if only the damage potential of the suspensionspan was adopted erefore when the structures such asapproach bridges transition piers and auxiliary pierswere ignored the damage identification and repair de-cisions would be misleading in the seismic evaluation oflong-span bridges Conversely the damaging probabilityof the overall bridge system will be amplified regardless ofthe higher importance of the main bridge When respectthe fact that the importance of suspension span is higherthan that of approach bridge in transportation networks aparameter a of 06ndash08 may be a better choice in thepresent study
e seismic fragility of conventionally designedbridges was mainly governed by the approach bridges at alldamage levels However for the retrofitted bridges thatpossess reasonable seismic performance different damagestates were controlled by the earthquake demands ofvarious structures as shown by the trend of the curvesalong parameter a in Case 4 and Case 5 e damagingpotential of the overall bridge from the low-to-moderatelimit states was dominated by the approach bridge eprimary reasons were the sliding of bearings and thetransfer of limited forces from the girder to substructure
by the damage of restriction devices of the approachbridges e extensive damage was controlled by thedisplacement of stiffening girder of the main bridge whichinferred the effect of dampers will be conspicuous understrong earthquake shaking At the same time the overalllong-span bridges with the reasonable seismic system willnot collapse during a strong earthquake event and canmeet emergency traffic service It can be seen that the meanvalues of fragility functions developed by compositedamage indices (equations (4) and (5)) are good indicatorsto evaluate seismic performance of long-span bridges andcan be consistent with the established seismic performanceobjectives
5 Conclusions
In the present study a procedure for investigating thereasonable seismic performance of long-span suspensionbridges was developed under the fragility function frame-work e seismic performance of a long-span suspensionbridge was examined and improved using the proposedprocedure from which the results revealed distinguishableconclusions as follows
(1) e definition of damage states for bearings con-sidered the interactions between the adjacent
014036059081103125
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(a)
020051081112142173
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
0002
0406
0810
CaseParameter a
(b)
130251371492612733
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(c)
Mea
n va
lue
exp
(λ) (
ms
)
Parameter a Case0 1 2 3 4 5
0001
0203
0405
131305478652826
1000
(d)
Figure 19 Influence of relative importance and improved designs on mean values of earthquake intensity for different damage states(a) slight (b) moderate (c) extensive (d) collapse
Advances in Civil Engineering 19
structures and self-damage to bearings in accor-dance with the actual design and experimentalconclusions e damage states of pylons in terms ofcurvature ductility were defined by developingmodal pushover analysis and idealized bilinear fit-ting methods in the equivalent pylon fiber models ofthe suspension bridge
(2) When investigating the component fragility curvesof the initial design it was found that the large deckdisplacement of the suspension span was the mostserious damage present and a slight inelastic re-sponse may have occurred only in the pylons Al-though the cable system was able to maintainelasticity under earthquakes the saddles at the py-lons were identified with a high-risk region indi-cating the damaging potential For the approachbridges the ductility mechanism did not realize arational damage sequence e columns were proneto collapse because of the application of fixed potbearings which in turn caused the damage to thebearings and abutments
(3) e stepwise improved seismic designs were pro-posed in terms of the component fragility results ofthe initial design Among the candidates fluid vis-cous dampers were proven to be a good choice forlong-span bridges based on the two attributes ofdisplacement and force when the similar efficiency ofdisplacement control was achieved e restraintsystem had a critical effect on the seismic responsesof the approach bridges e quasi-isolation orisolation can achieve a reasonable seismic damagesequence of the approach bridges
(4) e defined composite damage indices showed thatthe incorporate component classification conceptand structural relative importance were necessary toderive a good indicator of performance evaluationdue to the fact that the damage indices were based oneither an absolutely serial or serial-parallel systemmodel may significantly overestimate the damagepotential of the overall bridge system e meanvalues of earthquake intensity at the specific damagestates can serve as a function of the improvedstrategies and relative importance to further evaluatethe overall seismic performance of the suspensionbridge It can be concluded that the damaging po-tential of the overall bridge from the low-to-mod-erate limit states was governed by the approachbridges in the retrofitted bridges with a favorableseismic performance whereas the displacementdemand of the suspension span dominated thefragility probability of the extensive damage
Data Availability
e nature of the data is the analytical data of finite elementmodels of the case bridge e data used to support thefindings of this study are available from the correspondingauthor upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the Basic Research ServiceProject of Central-Level Public Welfare Research Institute(grant nos 2016-9018 and 2019-0034) and Science andTechnology Project of Communicationsrsquo Construction inWestern China MOC (grant no 2009318223094)
References
[1] L W David A Kartoum K Chang and I Roy ldquoSuspensionreport seismic performance and design considerations of longspan suspension bridgesrdquo Report CSMIP00-03 CaliforniaDepartment of Conservation Division of Mines And GeologyOffice of Strong Motion Studies Sacramento CA USA 2000
[2] S-G Gwon and D-H Choi ldquoImproved continuummodel forfree vibration analysis of suspension bridgesrdquo Journal ofEngineering Mechanics vol 143 no 7 Article ID 040170382017
[3] S Arzoumanidis A Shama and F Ostadan ldquoPerformance-based seismic analysis and design of suspension bridgesrdquoEarthquake Engineering and Structural Dynamics vol 34no 4-5 pp 349ndash367 2005
[4] L Sgambi E Garavaglia N Basso and F Bontempi ldquoMonteCarlo simulation for seismic analysis of a long span sus-pension bridgerdquo Engineering Structures vol 78 pp 100ndash1112014
[5] D Karmakar S Ray-Chaudhuri and M Shinozuka ldquoFiniteelement model development validation and probabilisticseismic performance evaluation of Vincent omas suspen-sion bridgerdquo Structure and Infrastructure Engineering vol 11no 2 pp 223ndash237 2015
[6] W-F Chen and L Duan Bridge Engineering HandbookSeismic Design CRC Press Boca Raton FL USA 2nd edition2014
[7] E Choi ldquoSeismic analysis and retrofit of mid-americabridgesrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2002
[8] B G Nielson and R DesRoches ldquoSeismic fragility method-ology for highway bridges using a component level approachrdquoEarthquake Engineering amp Structural Dynamics vol 36 no 6pp 823ndash839 2007
[9] K N Ramanathan ldquoNext generation seismic fragility curvesfor california bridges incorporating the evolution in seismicdesignrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2012
[10] O Avsar A Yakut and A Caner ldquoAnalytical fragility curvesfor ordinary highway bridges in Turkeyrdquo Earthquake Spectravol 27 no 4 pp 971ndash996 2011
[11] S P Stefanidou and A J Kappos ldquoMethodology for thedevelopment of bridge-specific fragility curvesrdquo EarthquakeEngineering amp Structural Dynamics vol 46 no 1 pp 73ndash932016
[12] B Borzi P Ceresa P Franchin F Noto G M Calvi andP E Pinto ldquoSeismic vulnerability of the Italian roadwaybridge stockrdquo Earthquake Spectra vol 31 no 4pp 2137ndash2161 2015
[13] W T Barnawi and S J Dyke ldquoSeismic fragility relationshipsof a cable-stayed bridge equipped with response modification
20 Advances in Civil Engineering
systemsrdquo Journal of Bridge Engineering vol 19 no 8 ArticleID A4013003 2014
[14] J Zhong Y Pang J Jong-Su J-S Jeon R DesRoches andW Yuan ldquoSeismic fragility assessment of long-span cable-stayed bridges in Chinardquo Advances in Structural Engineeringvol 19 no 11 pp 1797ndash1812 2016
[15] J Zhong Z Hu W Yuan and L Chen ldquoSystem-basedprobabilistic optimization of fluid viscous dampers equippedin cable-stayed bridgesrdquo Advances in Structural Engineeringvol 21 no 12 pp 1815ndash1825 2018
[16] J E Padgett and R DesRoches ldquoMethodology for the de-velopment of analytical fragility curves for retrofitted bridgesfitted bridgesrdquo Earthquake Engineering amp Structural Dy-namics vol 37 no 8 pp 1157ndash1174 2008
[17] Y Xie and J Zhang ldquoOptimal design of seismic protectivedevices for highway bridges using performance-basedmethodology and multiobjective genetic optimizationrdquoJournal of Bridge Engineering vol 22 no 3 Article ID04016129 2017
[18] J Zhang and Y Huo ldquoEvaluating effectiveness and optimumdesign of isolation devices for highway bridges using thefragility function methodrdquo Engineering Structures vol 31no 8 pp 1648ndash1660 2009
[19] F McKenna M H Scott and G L Fenves ldquoNonlinear finite-element analysis software architecture using object compo-sitionrdquo Journal of Computing in Civil Engineering vol 24no 1 pp 95ndash107 2010
[20] MOTC ldquoGuidelines for seismic design of highway bridgesrdquoChina Ministry of Transportation and CommunicationsPress Beijing China JTGT B02-01-2008 2008
[21] A D Kiureghian and J L Sackman ldquoTangent geometricstiffness of inclined cables under self-weightrdquo Journal ofStructural Engineering vol 131 no 6 pp 941ndash945 2005
[22] J S Steelman E T Filipov L A Fahnestock et al ldquoEx-perimental behavior of steel fixed bearings and implicationsfor seismic bridge responserdquo Journal of Bridge Engineeringvol 19 no 8 Article ID A4014007 2014
[23] B Madani F Behnamfar and H Tajmir Riahi ldquoDynamicresponse of structures subjected to pounding and structure-soil-structure interactionrdquo Soil Dynamics and EarthquakeEngineering vol 78 pp 46ndash60 2015
[24] L Xu and J Li ldquoDesign and experimental investigation of anew type sliding retainer and its efficacy in seismic fortifi-cationrdquo Engineering Mechanics vol 33 no 2 pp 111ndash1182016 in Chinese
[25] Caltrans Caltrans Seismic Design Criteria California De-partment of Transportation Sacramento CA USA 2013
[26] A Shamsabadi K M Rollins and M Kapuskar ldquoNonlinearsoil-abutment-bridge structure interaction for seismic per-formance-based designrdquo Journal of Geotechnical and Geo-environmental Engineering vol 133 no 6 pp 707ndash720 2007
[27] X Li Z Sun D Wang and Y Shi ldquoLongitudinal seismicpounding effect of bridges abutment and backfilling damagerdquoJournal of Changrsquoan University vol 35 no 4 pp 76ndash82 2015in Chinese
[28] MOTC ldquoSpecification for design of highway suspensionbridgerdquo China Ministry of Transportation and Communi-cations Press Beijing China JTGT D65-05-2015 2015
[29] A Shafieezadeh K Ramanathan J E Padgett andR DesRoches ldquoFractional order intensity measures forprobabilistic seismic demand modeling applied to highwaybridgesrdquo Earthquake Engineering amp Structural Dynamicsvol 41 no 3 pp 391ndash409 2012
[30] J E Padgett B G Nielson and R DesRoches ldquoSelection ofoptimal intensity measures in probabilistic seismic demandmodels of highway bridge portfoliosrdquo Earthquake Engineeringamp Structural Dynamics vol 37 no 5 pp 711ndash725 2008
[31] K R Mackie and B Stojadinovic ldquoFragility basis for californiahighway overpass bridge seismic decision makingrdquo PEERReport 200502 Pacific Earthquake Engineering ResearchCenter University of California Berkeley Berkeley CA USA2005
[32] J Li T Peng Ji Lin L Zhou and YuL Deng Seismic Design ofMulti-Pylon Multi-Span Suspension Bridge China Commu-nications Press Beijing China 2013 in Chinese
[33] L Chen W Zhuang H Zhao and Z Wan ldquoReport onHighwaysrsquo Damage in the Wenchuan Earthquake-BridgeChina Communications Press Beijing China 2012 inChinese
[34] G Wu K Wang G Lu and P Zhang ldquoAn experimentalinvestigation of unbonded laminated elastomeric bearingsand the seismic evaluations of highway bridges with testedbearing componentsrdquo Shock and Vibration vol 2018 ArticleID 8439321 18 pages 2018
Advances in Civil Engineering 21
Figure 17 illustrates the probabilities of critical damage stateoccurrence sequences of the bridge components in Cases 1 4and 5 when PGV is 10ms which is equal to E2 earthquakeintensity in the seismic assessment reports e componentvulnerabilities of initial design (Case 0) were taken as the ref-erence values which were marked by different colored strips inFigure 17 e desirable sequence preferred Cases 4 or 5 for theapproach bridgese initial design of approach bridges resultedin unreasonable damage sequences In the strategies pertainingto quasi-isolation and isolation systems multiple tiers of seismicstructural redundancy can be normally employed to preventstructural collapse during strong earthquake events Tier 1 in-dicated that the bearings were weakly fused which was evidentin the high fragility of bearing sliding behaviors (ABDAL andABDC2T) Another advantage was that the extensive dam-ages of bearing would be reduced when compared with the self-damage of the pot bearings Tier 2 indicated that the designedbearings had sufficient seat widths for adapting to the sliding ofsuperstructures However the devices had a certain damageprobability (EJDA and SKDC2) and dissipated a portion of
the seismic energy when the displacement of the superstructuresbecame larger than the allowable values of constraint devicesTier 3 redundancy was designed as a potential plastic hinge inthe substructures (CCDC2) including a certain yielding of thebackfill at the abutments (BPDA) e response of theabutments in the transverse direction (PDAT) was slightlyincreased due to the pounding effect between the girders andshear keys caused by bearing sliding e reasonable damagesequence of two restraint systems can be reflected through thecorresponding damage probabilities of bearings seismic mea-sures and substructures respectively as indicated by the red andgreen lines in Figure 17 erefore the rational sequence ofcritical limit state occurrences for the approach bridges isbearings⟶ expansion joints (or restrainer cables) shearkeys⟶ columns abutments
43 Comprehensive Evaluation of Bridge System Padgett andDesRoches [16] pointed out the necessity to derive the bridgesystem vulnerabilities when assessing retrofitted bridges
Figure 15 Experimental results of (a) LEB and (b) LRB
ΔΔ1
F
Δ2
Fmax
Fd
KdK0
(a)
Δy Δ
F
K1 = (3~7) K2
K1
K2
(b)
Slack
F
K
Δ
Fy
(c)
Figure 16 Analytical models of (a) LEB (b) LRB and (c) cable restrainers
16 Advances in Civil Engineering
because different components may have a different responsewith a given retrofit measure Also Zhang and Huo [18]studied the optimal parameters of isolation bearings byderiving the intensity median value corresponding to sys-tem-level fragility function which can effectively avoid theimpacts of different isolation parameters on differentcomponents As can be seen from Figures 14 and 17 theabove analysis results were also very apparent in this studyFor example the longitudinal deck displacement was sig-nificantly reduced by using dampers yet increased thetransverse demands of stiffening girder and pylons e useof LEB or LRB reduced the demands placed on columns yetled to an increase of the transverse deformations of abut-ments which was due to the use of shear keys limiting theexcessive sliding displacement of bearings erefore acomprehensive evaluation of retrofit strategies on bridgesparticularly long-span suspension bridges that have multiplecomponents should be based on system-level fragility in-stead of component-level
Taking the findings that no damages had occurred to thecable system and the definition of bearing damage states intoaccount the derivation of system-level fragility of the wholebridge depended on critical components including pylonscolumns bearings and abutments e fragility data byemploying the absolutely serial and serial-parallel assump-tions of bridge system are plotted in Figure 18 and comparedto the system fragility obtained from the proposed com-posite damage state e absolutely serial assumption
generated serious bias for fragility curves and may result inmisdirection if they were served as a design guide eabsolutely serial model overestimated the potential damageof the bridge system since the fragility curve directlydepended on the most vulnerable components In otherwords if the difference in location and performance of eachcomponent was neglected then the seismic performanceevaluation would be achieved by attaching equal importanceto the strength of pylons approach columns and bearingsis method contradicted the philosophy of capacity designin which the long-span structure can dissipate energythrough a reasonable swinging of stiffening girder and thebearings were designed as fuse elements in the small-to-medium-span bridgese serial-parallel model qualitativelyincorporated the differences of components and was verysensitive to the seismic design of the structure as seen fromthe comparisons among the cases in Figure 18 e modeldistinguished the serial components and parallel compo-nents yet still overestimated the bridge system-level fragilitywhen bearings became the vulnerable components espe-cially for long-span bridges is special situation such asthe serial assumption may also lead to misunderstandingOnce the relative importance between the components wasconsidered in the serial-composite model the damage po-tential was clearly reduced within the range of earthquakeintensity of interest based on the case studies is alsoreflected the necessity of the classification idea of bridgecomponents which can be expressed by the numerical value
Table 6 Damage states of the quasi-isolation and isolation bearings (m)
Figure 17 Probabilities of critical damage states occurrence sequences for the approach bridges when PGV is 1ms
Advances in Civil Engineering 17
of system-level fragility erefore the application ofcomposite damage state (equation (4)) considering com-ponent difference was instead more appropriate than theaforementioned two models
e damage probability distribution represented byblack circles in Figure 18 can be obtained from the definedcomposite damage state (equation (5)) where the parametera is 06 It was found that system fragility result was a moreaccurate assessment with further taking the relative im-portance between themain bridge and approach bridges intoaccount is is due to the principle that the seismic designperformance objective of long-span bridges generally has nocollapse for the main body structures and is necessarily opento emergency vehicle service In addition attention must bepaid to the fact that the differences in repair cost and time ofthe main and approach bridges are significant when thesame damage state is achieved e results of compositedamage state may match the performance objectives of theseismic design In order to investigate the influence ofimproved strategies and structural relative importance ondamage potential of bridges the mean values of the fragility
functions can be employed to evaluate the seismic designe smooth fragility curves for the specific damage states canbe fitted with the lognormal cumulative probability distri-bution function by employing the least-squares technique
P DSgtDSi | PGV1113858 1113859 Φln(pgv) minus λPGV
ζPGV1113888 1113889 (12)
where λPGV and ζPGV are the logarithmic mean and dis-persion values of earthquake intensity (PGV) for a specifieddamage level e larger mean values represent the smallerdamage potential to reach the specified level
Figure 19 indicates the mean values of earthquakeintensity at the specific damage states as a function of theimproved strategies and parameter a Each step of theretrofit measures would enhance the seismic performanceof the overall bridge system at all levels e improvementof the restraint system of the approach bridge played acritical role of the overall bridge system For instance themean values of the earthquake intensity featured mutationof slopes along the measure direction which reflected the
Figure 18 System fragility of the original design (a) Case 0 and improved design including (b) Case 1 and (c) Case 5
18 Advances in Civil Engineering
fact that the effectiveness of improvement for the ap-proach bridges was clearer than that of the main bridgee fragility of the highway network would be under-estimated if only the damage potential of the suspensionspan was adopted erefore when the structures such asapproach bridges transition piers and auxiliary pierswere ignored the damage identification and repair de-cisions would be misleading in the seismic evaluation oflong-span bridges Conversely the damaging probabilityof the overall bridge system will be amplified regardless ofthe higher importance of the main bridge When respectthe fact that the importance of suspension span is higherthan that of approach bridge in transportation networks aparameter a of 06ndash08 may be a better choice in thepresent study
e seismic fragility of conventionally designedbridges was mainly governed by the approach bridges at alldamage levels However for the retrofitted bridges thatpossess reasonable seismic performance different damagestates were controlled by the earthquake demands ofvarious structures as shown by the trend of the curvesalong parameter a in Case 4 and Case 5 e damagingpotential of the overall bridge from the low-to-moderatelimit states was dominated by the approach bridge eprimary reasons were the sliding of bearings and thetransfer of limited forces from the girder to substructure
by the damage of restriction devices of the approachbridges e extensive damage was controlled by thedisplacement of stiffening girder of the main bridge whichinferred the effect of dampers will be conspicuous understrong earthquake shaking At the same time the overalllong-span bridges with the reasonable seismic system willnot collapse during a strong earthquake event and canmeet emergency traffic service It can be seen that the meanvalues of fragility functions developed by compositedamage indices (equations (4) and (5)) are good indicatorsto evaluate seismic performance of long-span bridges andcan be consistent with the established seismic performanceobjectives
5 Conclusions
In the present study a procedure for investigating thereasonable seismic performance of long-span suspensionbridges was developed under the fragility function frame-work e seismic performance of a long-span suspensionbridge was examined and improved using the proposedprocedure from which the results revealed distinguishableconclusions as follows
(1) e definition of damage states for bearings con-sidered the interactions between the adjacent
014036059081103125
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(a)
020051081112142173
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
0002
0406
0810
CaseParameter a
(b)
130251371492612733
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(c)
Mea
n va
lue
exp
(λ) (
ms
)
Parameter a Case0 1 2 3 4 5
0001
0203
0405
131305478652826
1000
(d)
Figure 19 Influence of relative importance and improved designs on mean values of earthquake intensity for different damage states(a) slight (b) moderate (c) extensive (d) collapse
Advances in Civil Engineering 19
structures and self-damage to bearings in accor-dance with the actual design and experimentalconclusions e damage states of pylons in terms ofcurvature ductility were defined by developingmodal pushover analysis and idealized bilinear fit-ting methods in the equivalent pylon fiber models ofthe suspension bridge
(2) When investigating the component fragility curvesof the initial design it was found that the large deckdisplacement of the suspension span was the mostserious damage present and a slight inelastic re-sponse may have occurred only in the pylons Al-though the cable system was able to maintainelasticity under earthquakes the saddles at the py-lons were identified with a high-risk region indi-cating the damaging potential For the approachbridges the ductility mechanism did not realize arational damage sequence e columns were proneto collapse because of the application of fixed potbearings which in turn caused the damage to thebearings and abutments
(3) e stepwise improved seismic designs were pro-posed in terms of the component fragility results ofthe initial design Among the candidates fluid vis-cous dampers were proven to be a good choice forlong-span bridges based on the two attributes ofdisplacement and force when the similar efficiency ofdisplacement control was achieved e restraintsystem had a critical effect on the seismic responsesof the approach bridges e quasi-isolation orisolation can achieve a reasonable seismic damagesequence of the approach bridges
(4) e defined composite damage indices showed thatthe incorporate component classification conceptand structural relative importance were necessary toderive a good indicator of performance evaluationdue to the fact that the damage indices were based oneither an absolutely serial or serial-parallel systemmodel may significantly overestimate the damagepotential of the overall bridge system e meanvalues of earthquake intensity at the specific damagestates can serve as a function of the improvedstrategies and relative importance to further evaluatethe overall seismic performance of the suspensionbridge It can be concluded that the damaging po-tential of the overall bridge from the low-to-mod-erate limit states was governed by the approachbridges in the retrofitted bridges with a favorableseismic performance whereas the displacementdemand of the suspension span dominated thefragility probability of the extensive damage
Data Availability
e nature of the data is the analytical data of finite elementmodels of the case bridge e data used to support thefindings of this study are available from the correspondingauthor upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the Basic Research ServiceProject of Central-Level Public Welfare Research Institute(grant nos 2016-9018 and 2019-0034) and Science andTechnology Project of Communicationsrsquo Construction inWestern China MOC (grant no 2009318223094)
References
[1] L W David A Kartoum K Chang and I Roy ldquoSuspensionreport seismic performance and design considerations of longspan suspension bridgesrdquo Report CSMIP00-03 CaliforniaDepartment of Conservation Division of Mines And GeologyOffice of Strong Motion Studies Sacramento CA USA 2000
[2] S-G Gwon and D-H Choi ldquoImproved continuummodel forfree vibration analysis of suspension bridgesrdquo Journal ofEngineering Mechanics vol 143 no 7 Article ID 040170382017
[3] S Arzoumanidis A Shama and F Ostadan ldquoPerformance-based seismic analysis and design of suspension bridgesrdquoEarthquake Engineering and Structural Dynamics vol 34no 4-5 pp 349ndash367 2005
[4] L Sgambi E Garavaglia N Basso and F Bontempi ldquoMonteCarlo simulation for seismic analysis of a long span sus-pension bridgerdquo Engineering Structures vol 78 pp 100ndash1112014
[5] D Karmakar S Ray-Chaudhuri and M Shinozuka ldquoFiniteelement model development validation and probabilisticseismic performance evaluation of Vincent omas suspen-sion bridgerdquo Structure and Infrastructure Engineering vol 11no 2 pp 223ndash237 2015
[6] W-F Chen and L Duan Bridge Engineering HandbookSeismic Design CRC Press Boca Raton FL USA 2nd edition2014
[7] E Choi ldquoSeismic analysis and retrofit of mid-americabridgesrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2002
[8] B G Nielson and R DesRoches ldquoSeismic fragility method-ology for highway bridges using a component level approachrdquoEarthquake Engineering amp Structural Dynamics vol 36 no 6pp 823ndash839 2007
[9] K N Ramanathan ldquoNext generation seismic fragility curvesfor california bridges incorporating the evolution in seismicdesignrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2012
[10] O Avsar A Yakut and A Caner ldquoAnalytical fragility curvesfor ordinary highway bridges in Turkeyrdquo Earthquake Spectravol 27 no 4 pp 971ndash996 2011
[11] S P Stefanidou and A J Kappos ldquoMethodology for thedevelopment of bridge-specific fragility curvesrdquo EarthquakeEngineering amp Structural Dynamics vol 46 no 1 pp 73ndash932016
[12] B Borzi P Ceresa P Franchin F Noto G M Calvi andP E Pinto ldquoSeismic vulnerability of the Italian roadwaybridge stockrdquo Earthquake Spectra vol 31 no 4pp 2137ndash2161 2015
[13] W T Barnawi and S J Dyke ldquoSeismic fragility relationshipsof a cable-stayed bridge equipped with response modification
20 Advances in Civil Engineering
systemsrdquo Journal of Bridge Engineering vol 19 no 8 ArticleID A4013003 2014
[14] J Zhong Y Pang J Jong-Su J-S Jeon R DesRoches andW Yuan ldquoSeismic fragility assessment of long-span cable-stayed bridges in Chinardquo Advances in Structural Engineeringvol 19 no 11 pp 1797ndash1812 2016
[15] J Zhong Z Hu W Yuan and L Chen ldquoSystem-basedprobabilistic optimization of fluid viscous dampers equippedin cable-stayed bridgesrdquo Advances in Structural Engineeringvol 21 no 12 pp 1815ndash1825 2018
[16] J E Padgett and R DesRoches ldquoMethodology for the de-velopment of analytical fragility curves for retrofitted bridgesfitted bridgesrdquo Earthquake Engineering amp Structural Dy-namics vol 37 no 8 pp 1157ndash1174 2008
[17] Y Xie and J Zhang ldquoOptimal design of seismic protectivedevices for highway bridges using performance-basedmethodology and multiobjective genetic optimizationrdquoJournal of Bridge Engineering vol 22 no 3 Article ID04016129 2017
[18] J Zhang and Y Huo ldquoEvaluating effectiveness and optimumdesign of isolation devices for highway bridges using thefragility function methodrdquo Engineering Structures vol 31no 8 pp 1648ndash1660 2009
[19] F McKenna M H Scott and G L Fenves ldquoNonlinear finite-element analysis software architecture using object compo-sitionrdquo Journal of Computing in Civil Engineering vol 24no 1 pp 95ndash107 2010
[20] MOTC ldquoGuidelines for seismic design of highway bridgesrdquoChina Ministry of Transportation and CommunicationsPress Beijing China JTGT B02-01-2008 2008
[21] A D Kiureghian and J L Sackman ldquoTangent geometricstiffness of inclined cables under self-weightrdquo Journal ofStructural Engineering vol 131 no 6 pp 941ndash945 2005
[22] J S Steelman E T Filipov L A Fahnestock et al ldquoEx-perimental behavior of steel fixed bearings and implicationsfor seismic bridge responserdquo Journal of Bridge Engineeringvol 19 no 8 Article ID A4014007 2014
[23] B Madani F Behnamfar and H Tajmir Riahi ldquoDynamicresponse of structures subjected to pounding and structure-soil-structure interactionrdquo Soil Dynamics and EarthquakeEngineering vol 78 pp 46ndash60 2015
[24] L Xu and J Li ldquoDesign and experimental investigation of anew type sliding retainer and its efficacy in seismic fortifi-cationrdquo Engineering Mechanics vol 33 no 2 pp 111ndash1182016 in Chinese
[25] Caltrans Caltrans Seismic Design Criteria California De-partment of Transportation Sacramento CA USA 2013
[26] A Shamsabadi K M Rollins and M Kapuskar ldquoNonlinearsoil-abutment-bridge structure interaction for seismic per-formance-based designrdquo Journal of Geotechnical and Geo-environmental Engineering vol 133 no 6 pp 707ndash720 2007
[27] X Li Z Sun D Wang and Y Shi ldquoLongitudinal seismicpounding effect of bridges abutment and backfilling damagerdquoJournal of Changrsquoan University vol 35 no 4 pp 76ndash82 2015in Chinese
[28] MOTC ldquoSpecification for design of highway suspensionbridgerdquo China Ministry of Transportation and Communi-cations Press Beijing China JTGT D65-05-2015 2015
[29] A Shafieezadeh K Ramanathan J E Padgett andR DesRoches ldquoFractional order intensity measures forprobabilistic seismic demand modeling applied to highwaybridgesrdquo Earthquake Engineering amp Structural Dynamicsvol 41 no 3 pp 391ndash409 2012
[30] J E Padgett B G Nielson and R DesRoches ldquoSelection ofoptimal intensity measures in probabilistic seismic demandmodels of highway bridge portfoliosrdquo Earthquake Engineeringamp Structural Dynamics vol 37 no 5 pp 711ndash725 2008
[31] K R Mackie and B Stojadinovic ldquoFragility basis for californiahighway overpass bridge seismic decision makingrdquo PEERReport 200502 Pacific Earthquake Engineering ResearchCenter University of California Berkeley Berkeley CA USA2005
[32] J Li T Peng Ji Lin L Zhou and YuL Deng Seismic Design ofMulti-Pylon Multi-Span Suspension Bridge China Commu-nications Press Beijing China 2013 in Chinese
[33] L Chen W Zhuang H Zhao and Z Wan ldquoReport onHighwaysrsquo Damage in the Wenchuan Earthquake-BridgeChina Communications Press Beijing China 2012 inChinese
[34] G Wu K Wang G Lu and P Zhang ldquoAn experimentalinvestigation of unbonded laminated elastomeric bearingsand the seismic evaluations of highway bridges with testedbearing componentsrdquo Shock and Vibration vol 2018 ArticleID 8439321 18 pages 2018
Advances in Civil Engineering 21
because different components may have a different responsewith a given retrofit measure Also Zhang and Huo [18]studied the optimal parameters of isolation bearings byderiving the intensity median value corresponding to sys-tem-level fragility function which can effectively avoid theimpacts of different isolation parameters on differentcomponents As can be seen from Figures 14 and 17 theabove analysis results were also very apparent in this studyFor example the longitudinal deck displacement was sig-nificantly reduced by using dampers yet increased thetransverse demands of stiffening girder and pylons e useof LEB or LRB reduced the demands placed on columns yetled to an increase of the transverse deformations of abut-ments which was due to the use of shear keys limiting theexcessive sliding displacement of bearings erefore acomprehensive evaluation of retrofit strategies on bridgesparticularly long-span suspension bridges that have multiplecomponents should be based on system-level fragility in-stead of component-level
Taking the findings that no damages had occurred to thecable system and the definition of bearing damage states intoaccount the derivation of system-level fragility of the wholebridge depended on critical components including pylonscolumns bearings and abutments e fragility data byemploying the absolutely serial and serial-parallel assump-tions of bridge system are plotted in Figure 18 and comparedto the system fragility obtained from the proposed com-posite damage state e absolutely serial assumption
generated serious bias for fragility curves and may result inmisdirection if they were served as a design guide eabsolutely serial model overestimated the potential damageof the bridge system since the fragility curve directlydepended on the most vulnerable components In otherwords if the difference in location and performance of eachcomponent was neglected then the seismic performanceevaluation would be achieved by attaching equal importanceto the strength of pylons approach columns and bearingsis method contradicted the philosophy of capacity designin which the long-span structure can dissipate energythrough a reasonable swinging of stiffening girder and thebearings were designed as fuse elements in the small-to-medium-span bridgese serial-parallel model qualitativelyincorporated the differences of components and was verysensitive to the seismic design of the structure as seen fromthe comparisons among the cases in Figure 18 e modeldistinguished the serial components and parallel compo-nents yet still overestimated the bridge system-level fragilitywhen bearings became the vulnerable components espe-cially for long-span bridges is special situation such asthe serial assumption may also lead to misunderstandingOnce the relative importance between the components wasconsidered in the serial-composite model the damage po-tential was clearly reduced within the range of earthquakeintensity of interest based on the case studies is alsoreflected the necessity of the classification idea of bridgecomponents which can be expressed by the numerical value
Table 6 Damage states of the quasi-isolation and isolation bearings (m)
Figure 17 Probabilities of critical damage states occurrence sequences for the approach bridges when PGV is 1ms
Advances in Civil Engineering 17
of system-level fragility erefore the application ofcomposite damage state (equation (4)) considering com-ponent difference was instead more appropriate than theaforementioned two models
e damage probability distribution represented byblack circles in Figure 18 can be obtained from the definedcomposite damage state (equation (5)) where the parametera is 06 It was found that system fragility result was a moreaccurate assessment with further taking the relative im-portance between themain bridge and approach bridges intoaccount is is due to the principle that the seismic designperformance objective of long-span bridges generally has nocollapse for the main body structures and is necessarily opento emergency vehicle service In addition attention must bepaid to the fact that the differences in repair cost and time ofthe main and approach bridges are significant when thesame damage state is achieved e results of compositedamage state may match the performance objectives of theseismic design In order to investigate the influence ofimproved strategies and structural relative importance ondamage potential of bridges the mean values of the fragility
functions can be employed to evaluate the seismic designe smooth fragility curves for the specific damage states canbe fitted with the lognormal cumulative probability distri-bution function by employing the least-squares technique
P DSgtDSi | PGV1113858 1113859 Φln(pgv) minus λPGV
ζPGV1113888 1113889 (12)
where λPGV and ζPGV are the logarithmic mean and dis-persion values of earthquake intensity (PGV) for a specifieddamage level e larger mean values represent the smallerdamage potential to reach the specified level
Figure 19 indicates the mean values of earthquakeintensity at the specific damage states as a function of theimproved strategies and parameter a Each step of theretrofit measures would enhance the seismic performanceof the overall bridge system at all levels e improvementof the restraint system of the approach bridge played acritical role of the overall bridge system For instance themean values of the earthquake intensity featured mutationof slopes along the measure direction which reflected the
Figure 18 System fragility of the original design (a) Case 0 and improved design including (b) Case 1 and (c) Case 5
18 Advances in Civil Engineering
fact that the effectiveness of improvement for the ap-proach bridges was clearer than that of the main bridgee fragility of the highway network would be under-estimated if only the damage potential of the suspensionspan was adopted erefore when the structures such asapproach bridges transition piers and auxiliary pierswere ignored the damage identification and repair de-cisions would be misleading in the seismic evaluation oflong-span bridges Conversely the damaging probabilityof the overall bridge system will be amplified regardless ofthe higher importance of the main bridge When respectthe fact that the importance of suspension span is higherthan that of approach bridge in transportation networks aparameter a of 06ndash08 may be a better choice in thepresent study
e seismic fragility of conventionally designedbridges was mainly governed by the approach bridges at alldamage levels However for the retrofitted bridges thatpossess reasonable seismic performance different damagestates were controlled by the earthquake demands ofvarious structures as shown by the trend of the curvesalong parameter a in Case 4 and Case 5 e damagingpotential of the overall bridge from the low-to-moderatelimit states was dominated by the approach bridge eprimary reasons were the sliding of bearings and thetransfer of limited forces from the girder to substructure
by the damage of restriction devices of the approachbridges e extensive damage was controlled by thedisplacement of stiffening girder of the main bridge whichinferred the effect of dampers will be conspicuous understrong earthquake shaking At the same time the overalllong-span bridges with the reasonable seismic system willnot collapse during a strong earthquake event and canmeet emergency traffic service It can be seen that the meanvalues of fragility functions developed by compositedamage indices (equations (4) and (5)) are good indicatorsto evaluate seismic performance of long-span bridges andcan be consistent with the established seismic performanceobjectives
5 Conclusions
In the present study a procedure for investigating thereasonable seismic performance of long-span suspensionbridges was developed under the fragility function frame-work e seismic performance of a long-span suspensionbridge was examined and improved using the proposedprocedure from which the results revealed distinguishableconclusions as follows
(1) e definition of damage states for bearings con-sidered the interactions between the adjacent
014036059081103125
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(a)
020051081112142173
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
0002
0406
0810
CaseParameter a
(b)
130251371492612733
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(c)
Mea
n va
lue
exp
(λ) (
ms
)
Parameter a Case0 1 2 3 4 5
0001
0203
0405
131305478652826
1000
(d)
Figure 19 Influence of relative importance and improved designs on mean values of earthquake intensity for different damage states(a) slight (b) moderate (c) extensive (d) collapse
Advances in Civil Engineering 19
structures and self-damage to bearings in accor-dance with the actual design and experimentalconclusions e damage states of pylons in terms ofcurvature ductility were defined by developingmodal pushover analysis and idealized bilinear fit-ting methods in the equivalent pylon fiber models ofthe suspension bridge
(2) When investigating the component fragility curvesof the initial design it was found that the large deckdisplacement of the suspension span was the mostserious damage present and a slight inelastic re-sponse may have occurred only in the pylons Al-though the cable system was able to maintainelasticity under earthquakes the saddles at the py-lons were identified with a high-risk region indi-cating the damaging potential For the approachbridges the ductility mechanism did not realize arational damage sequence e columns were proneto collapse because of the application of fixed potbearings which in turn caused the damage to thebearings and abutments
(3) e stepwise improved seismic designs were pro-posed in terms of the component fragility results ofthe initial design Among the candidates fluid vis-cous dampers were proven to be a good choice forlong-span bridges based on the two attributes ofdisplacement and force when the similar efficiency ofdisplacement control was achieved e restraintsystem had a critical effect on the seismic responsesof the approach bridges e quasi-isolation orisolation can achieve a reasonable seismic damagesequence of the approach bridges
(4) e defined composite damage indices showed thatthe incorporate component classification conceptand structural relative importance were necessary toderive a good indicator of performance evaluationdue to the fact that the damage indices were based oneither an absolutely serial or serial-parallel systemmodel may significantly overestimate the damagepotential of the overall bridge system e meanvalues of earthquake intensity at the specific damagestates can serve as a function of the improvedstrategies and relative importance to further evaluatethe overall seismic performance of the suspensionbridge It can be concluded that the damaging po-tential of the overall bridge from the low-to-mod-erate limit states was governed by the approachbridges in the retrofitted bridges with a favorableseismic performance whereas the displacementdemand of the suspension span dominated thefragility probability of the extensive damage
Data Availability
e nature of the data is the analytical data of finite elementmodels of the case bridge e data used to support thefindings of this study are available from the correspondingauthor upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the Basic Research ServiceProject of Central-Level Public Welfare Research Institute(grant nos 2016-9018 and 2019-0034) and Science andTechnology Project of Communicationsrsquo Construction inWestern China MOC (grant no 2009318223094)
References
[1] L W David A Kartoum K Chang and I Roy ldquoSuspensionreport seismic performance and design considerations of longspan suspension bridgesrdquo Report CSMIP00-03 CaliforniaDepartment of Conservation Division of Mines And GeologyOffice of Strong Motion Studies Sacramento CA USA 2000
[2] S-G Gwon and D-H Choi ldquoImproved continuummodel forfree vibration analysis of suspension bridgesrdquo Journal ofEngineering Mechanics vol 143 no 7 Article ID 040170382017
[3] S Arzoumanidis A Shama and F Ostadan ldquoPerformance-based seismic analysis and design of suspension bridgesrdquoEarthquake Engineering and Structural Dynamics vol 34no 4-5 pp 349ndash367 2005
[4] L Sgambi E Garavaglia N Basso and F Bontempi ldquoMonteCarlo simulation for seismic analysis of a long span sus-pension bridgerdquo Engineering Structures vol 78 pp 100ndash1112014
[5] D Karmakar S Ray-Chaudhuri and M Shinozuka ldquoFiniteelement model development validation and probabilisticseismic performance evaluation of Vincent omas suspen-sion bridgerdquo Structure and Infrastructure Engineering vol 11no 2 pp 223ndash237 2015
[6] W-F Chen and L Duan Bridge Engineering HandbookSeismic Design CRC Press Boca Raton FL USA 2nd edition2014
[7] E Choi ldquoSeismic analysis and retrofit of mid-americabridgesrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2002
[8] B G Nielson and R DesRoches ldquoSeismic fragility method-ology for highway bridges using a component level approachrdquoEarthquake Engineering amp Structural Dynamics vol 36 no 6pp 823ndash839 2007
[9] K N Ramanathan ldquoNext generation seismic fragility curvesfor california bridges incorporating the evolution in seismicdesignrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2012
[10] O Avsar A Yakut and A Caner ldquoAnalytical fragility curvesfor ordinary highway bridges in Turkeyrdquo Earthquake Spectravol 27 no 4 pp 971ndash996 2011
[11] S P Stefanidou and A J Kappos ldquoMethodology for thedevelopment of bridge-specific fragility curvesrdquo EarthquakeEngineering amp Structural Dynamics vol 46 no 1 pp 73ndash932016
[12] B Borzi P Ceresa P Franchin F Noto G M Calvi andP E Pinto ldquoSeismic vulnerability of the Italian roadwaybridge stockrdquo Earthquake Spectra vol 31 no 4pp 2137ndash2161 2015
[13] W T Barnawi and S J Dyke ldquoSeismic fragility relationshipsof a cable-stayed bridge equipped with response modification
20 Advances in Civil Engineering
systemsrdquo Journal of Bridge Engineering vol 19 no 8 ArticleID A4013003 2014
[14] J Zhong Y Pang J Jong-Su J-S Jeon R DesRoches andW Yuan ldquoSeismic fragility assessment of long-span cable-stayed bridges in Chinardquo Advances in Structural Engineeringvol 19 no 11 pp 1797ndash1812 2016
[15] J Zhong Z Hu W Yuan and L Chen ldquoSystem-basedprobabilistic optimization of fluid viscous dampers equippedin cable-stayed bridgesrdquo Advances in Structural Engineeringvol 21 no 12 pp 1815ndash1825 2018
[16] J E Padgett and R DesRoches ldquoMethodology for the de-velopment of analytical fragility curves for retrofitted bridgesfitted bridgesrdquo Earthquake Engineering amp Structural Dy-namics vol 37 no 8 pp 1157ndash1174 2008
[17] Y Xie and J Zhang ldquoOptimal design of seismic protectivedevices for highway bridges using performance-basedmethodology and multiobjective genetic optimizationrdquoJournal of Bridge Engineering vol 22 no 3 Article ID04016129 2017
[18] J Zhang and Y Huo ldquoEvaluating effectiveness and optimumdesign of isolation devices for highway bridges using thefragility function methodrdquo Engineering Structures vol 31no 8 pp 1648ndash1660 2009
[19] F McKenna M H Scott and G L Fenves ldquoNonlinear finite-element analysis software architecture using object compo-sitionrdquo Journal of Computing in Civil Engineering vol 24no 1 pp 95ndash107 2010
[20] MOTC ldquoGuidelines for seismic design of highway bridgesrdquoChina Ministry of Transportation and CommunicationsPress Beijing China JTGT B02-01-2008 2008
[21] A D Kiureghian and J L Sackman ldquoTangent geometricstiffness of inclined cables under self-weightrdquo Journal ofStructural Engineering vol 131 no 6 pp 941ndash945 2005
[22] J S Steelman E T Filipov L A Fahnestock et al ldquoEx-perimental behavior of steel fixed bearings and implicationsfor seismic bridge responserdquo Journal of Bridge Engineeringvol 19 no 8 Article ID A4014007 2014
[23] B Madani F Behnamfar and H Tajmir Riahi ldquoDynamicresponse of structures subjected to pounding and structure-soil-structure interactionrdquo Soil Dynamics and EarthquakeEngineering vol 78 pp 46ndash60 2015
[24] L Xu and J Li ldquoDesign and experimental investigation of anew type sliding retainer and its efficacy in seismic fortifi-cationrdquo Engineering Mechanics vol 33 no 2 pp 111ndash1182016 in Chinese
[25] Caltrans Caltrans Seismic Design Criteria California De-partment of Transportation Sacramento CA USA 2013
[26] A Shamsabadi K M Rollins and M Kapuskar ldquoNonlinearsoil-abutment-bridge structure interaction for seismic per-formance-based designrdquo Journal of Geotechnical and Geo-environmental Engineering vol 133 no 6 pp 707ndash720 2007
[27] X Li Z Sun D Wang and Y Shi ldquoLongitudinal seismicpounding effect of bridges abutment and backfilling damagerdquoJournal of Changrsquoan University vol 35 no 4 pp 76ndash82 2015in Chinese
[28] MOTC ldquoSpecification for design of highway suspensionbridgerdquo China Ministry of Transportation and Communi-cations Press Beijing China JTGT D65-05-2015 2015
[29] A Shafieezadeh K Ramanathan J E Padgett andR DesRoches ldquoFractional order intensity measures forprobabilistic seismic demand modeling applied to highwaybridgesrdquo Earthquake Engineering amp Structural Dynamicsvol 41 no 3 pp 391ndash409 2012
[30] J E Padgett B G Nielson and R DesRoches ldquoSelection ofoptimal intensity measures in probabilistic seismic demandmodels of highway bridge portfoliosrdquo Earthquake Engineeringamp Structural Dynamics vol 37 no 5 pp 711ndash725 2008
[31] K R Mackie and B Stojadinovic ldquoFragility basis for californiahighway overpass bridge seismic decision makingrdquo PEERReport 200502 Pacific Earthquake Engineering ResearchCenter University of California Berkeley Berkeley CA USA2005
[32] J Li T Peng Ji Lin L Zhou and YuL Deng Seismic Design ofMulti-Pylon Multi-Span Suspension Bridge China Commu-nications Press Beijing China 2013 in Chinese
[33] L Chen W Zhuang H Zhao and Z Wan ldquoReport onHighwaysrsquo Damage in the Wenchuan Earthquake-BridgeChina Communications Press Beijing China 2012 inChinese
[34] G Wu K Wang G Lu and P Zhang ldquoAn experimentalinvestigation of unbonded laminated elastomeric bearingsand the seismic evaluations of highway bridges with testedbearing componentsrdquo Shock and Vibration vol 2018 ArticleID 8439321 18 pages 2018
Advances in Civil Engineering 21
of system-level fragility erefore the application ofcomposite damage state (equation (4)) considering com-ponent difference was instead more appropriate than theaforementioned two models
e damage probability distribution represented byblack circles in Figure 18 can be obtained from the definedcomposite damage state (equation (5)) where the parametera is 06 It was found that system fragility result was a moreaccurate assessment with further taking the relative im-portance between themain bridge and approach bridges intoaccount is is due to the principle that the seismic designperformance objective of long-span bridges generally has nocollapse for the main body structures and is necessarily opento emergency vehicle service In addition attention must bepaid to the fact that the differences in repair cost and time ofthe main and approach bridges are significant when thesame damage state is achieved e results of compositedamage state may match the performance objectives of theseismic design In order to investigate the influence ofimproved strategies and structural relative importance ondamage potential of bridges the mean values of the fragility
functions can be employed to evaluate the seismic designe smooth fragility curves for the specific damage states canbe fitted with the lognormal cumulative probability distri-bution function by employing the least-squares technique
P DSgtDSi | PGV1113858 1113859 Φln(pgv) minus λPGV
ζPGV1113888 1113889 (12)
where λPGV and ζPGV are the logarithmic mean and dis-persion values of earthquake intensity (PGV) for a specifieddamage level e larger mean values represent the smallerdamage potential to reach the specified level
Figure 19 indicates the mean values of earthquakeintensity at the specific damage states as a function of theimproved strategies and parameter a Each step of theretrofit measures would enhance the seismic performanceof the overall bridge system at all levels e improvementof the restraint system of the approach bridge played acritical role of the overall bridge system For instance themean values of the earthquake intensity featured mutationof slopes along the measure direction which reflected the
Figure 18 System fragility of the original design (a) Case 0 and improved design including (b) Case 1 and (c) Case 5
18 Advances in Civil Engineering
fact that the effectiveness of improvement for the ap-proach bridges was clearer than that of the main bridgee fragility of the highway network would be under-estimated if only the damage potential of the suspensionspan was adopted erefore when the structures such asapproach bridges transition piers and auxiliary pierswere ignored the damage identification and repair de-cisions would be misleading in the seismic evaluation oflong-span bridges Conversely the damaging probabilityof the overall bridge system will be amplified regardless ofthe higher importance of the main bridge When respectthe fact that the importance of suspension span is higherthan that of approach bridge in transportation networks aparameter a of 06ndash08 may be a better choice in thepresent study
e seismic fragility of conventionally designedbridges was mainly governed by the approach bridges at alldamage levels However for the retrofitted bridges thatpossess reasonable seismic performance different damagestates were controlled by the earthquake demands ofvarious structures as shown by the trend of the curvesalong parameter a in Case 4 and Case 5 e damagingpotential of the overall bridge from the low-to-moderatelimit states was dominated by the approach bridge eprimary reasons were the sliding of bearings and thetransfer of limited forces from the girder to substructure
by the damage of restriction devices of the approachbridges e extensive damage was controlled by thedisplacement of stiffening girder of the main bridge whichinferred the effect of dampers will be conspicuous understrong earthquake shaking At the same time the overalllong-span bridges with the reasonable seismic system willnot collapse during a strong earthquake event and canmeet emergency traffic service It can be seen that the meanvalues of fragility functions developed by compositedamage indices (equations (4) and (5)) are good indicatorsto evaluate seismic performance of long-span bridges andcan be consistent with the established seismic performanceobjectives
5 Conclusions
In the present study a procedure for investigating thereasonable seismic performance of long-span suspensionbridges was developed under the fragility function frame-work e seismic performance of a long-span suspensionbridge was examined and improved using the proposedprocedure from which the results revealed distinguishableconclusions as follows
(1) e definition of damage states for bearings con-sidered the interactions between the adjacent
014036059081103125
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(a)
020051081112142173
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
0002
0406
0810
CaseParameter a
(b)
130251371492612733
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(c)
Mea
n va
lue
exp
(λ) (
ms
)
Parameter a Case0 1 2 3 4 5
0001
0203
0405
131305478652826
1000
(d)
Figure 19 Influence of relative importance and improved designs on mean values of earthquake intensity for different damage states(a) slight (b) moderate (c) extensive (d) collapse
Advances in Civil Engineering 19
structures and self-damage to bearings in accor-dance with the actual design and experimentalconclusions e damage states of pylons in terms ofcurvature ductility were defined by developingmodal pushover analysis and idealized bilinear fit-ting methods in the equivalent pylon fiber models ofthe suspension bridge
(2) When investigating the component fragility curvesof the initial design it was found that the large deckdisplacement of the suspension span was the mostserious damage present and a slight inelastic re-sponse may have occurred only in the pylons Al-though the cable system was able to maintainelasticity under earthquakes the saddles at the py-lons were identified with a high-risk region indi-cating the damaging potential For the approachbridges the ductility mechanism did not realize arational damage sequence e columns were proneto collapse because of the application of fixed potbearings which in turn caused the damage to thebearings and abutments
(3) e stepwise improved seismic designs were pro-posed in terms of the component fragility results ofthe initial design Among the candidates fluid vis-cous dampers were proven to be a good choice forlong-span bridges based on the two attributes ofdisplacement and force when the similar efficiency ofdisplacement control was achieved e restraintsystem had a critical effect on the seismic responsesof the approach bridges e quasi-isolation orisolation can achieve a reasonable seismic damagesequence of the approach bridges
(4) e defined composite damage indices showed thatthe incorporate component classification conceptand structural relative importance were necessary toderive a good indicator of performance evaluationdue to the fact that the damage indices were based oneither an absolutely serial or serial-parallel systemmodel may significantly overestimate the damagepotential of the overall bridge system e meanvalues of earthquake intensity at the specific damagestates can serve as a function of the improvedstrategies and relative importance to further evaluatethe overall seismic performance of the suspensionbridge It can be concluded that the damaging po-tential of the overall bridge from the low-to-mod-erate limit states was governed by the approachbridges in the retrofitted bridges with a favorableseismic performance whereas the displacementdemand of the suspension span dominated thefragility probability of the extensive damage
Data Availability
e nature of the data is the analytical data of finite elementmodels of the case bridge e data used to support thefindings of this study are available from the correspondingauthor upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the Basic Research ServiceProject of Central-Level Public Welfare Research Institute(grant nos 2016-9018 and 2019-0034) and Science andTechnology Project of Communicationsrsquo Construction inWestern China MOC (grant no 2009318223094)
References
[1] L W David A Kartoum K Chang and I Roy ldquoSuspensionreport seismic performance and design considerations of longspan suspension bridgesrdquo Report CSMIP00-03 CaliforniaDepartment of Conservation Division of Mines And GeologyOffice of Strong Motion Studies Sacramento CA USA 2000
[2] S-G Gwon and D-H Choi ldquoImproved continuummodel forfree vibration analysis of suspension bridgesrdquo Journal ofEngineering Mechanics vol 143 no 7 Article ID 040170382017
[3] S Arzoumanidis A Shama and F Ostadan ldquoPerformance-based seismic analysis and design of suspension bridgesrdquoEarthquake Engineering and Structural Dynamics vol 34no 4-5 pp 349ndash367 2005
[4] L Sgambi E Garavaglia N Basso and F Bontempi ldquoMonteCarlo simulation for seismic analysis of a long span sus-pension bridgerdquo Engineering Structures vol 78 pp 100ndash1112014
[5] D Karmakar S Ray-Chaudhuri and M Shinozuka ldquoFiniteelement model development validation and probabilisticseismic performance evaluation of Vincent omas suspen-sion bridgerdquo Structure and Infrastructure Engineering vol 11no 2 pp 223ndash237 2015
[6] W-F Chen and L Duan Bridge Engineering HandbookSeismic Design CRC Press Boca Raton FL USA 2nd edition2014
[7] E Choi ldquoSeismic analysis and retrofit of mid-americabridgesrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2002
[8] B G Nielson and R DesRoches ldquoSeismic fragility method-ology for highway bridges using a component level approachrdquoEarthquake Engineering amp Structural Dynamics vol 36 no 6pp 823ndash839 2007
[9] K N Ramanathan ldquoNext generation seismic fragility curvesfor california bridges incorporating the evolution in seismicdesignrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2012
[10] O Avsar A Yakut and A Caner ldquoAnalytical fragility curvesfor ordinary highway bridges in Turkeyrdquo Earthquake Spectravol 27 no 4 pp 971ndash996 2011
[11] S P Stefanidou and A J Kappos ldquoMethodology for thedevelopment of bridge-specific fragility curvesrdquo EarthquakeEngineering amp Structural Dynamics vol 46 no 1 pp 73ndash932016
[12] B Borzi P Ceresa P Franchin F Noto G M Calvi andP E Pinto ldquoSeismic vulnerability of the Italian roadwaybridge stockrdquo Earthquake Spectra vol 31 no 4pp 2137ndash2161 2015
[13] W T Barnawi and S J Dyke ldquoSeismic fragility relationshipsof a cable-stayed bridge equipped with response modification
20 Advances in Civil Engineering
systemsrdquo Journal of Bridge Engineering vol 19 no 8 ArticleID A4013003 2014
[14] J Zhong Y Pang J Jong-Su J-S Jeon R DesRoches andW Yuan ldquoSeismic fragility assessment of long-span cable-stayed bridges in Chinardquo Advances in Structural Engineeringvol 19 no 11 pp 1797ndash1812 2016
[15] J Zhong Z Hu W Yuan and L Chen ldquoSystem-basedprobabilistic optimization of fluid viscous dampers equippedin cable-stayed bridgesrdquo Advances in Structural Engineeringvol 21 no 12 pp 1815ndash1825 2018
[16] J E Padgett and R DesRoches ldquoMethodology for the de-velopment of analytical fragility curves for retrofitted bridgesfitted bridgesrdquo Earthquake Engineering amp Structural Dy-namics vol 37 no 8 pp 1157ndash1174 2008
[17] Y Xie and J Zhang ldquoOptimal design of seismic protectivedevices for highway bridges using performance-basedmethodology and multiobjective genetic optimizationrdquoJournal of Bridge Engineering vol 22 no 3 Article ID04016129 2017
[18] J Zhang and Y Huo ldquoEvaluating effectiveness and optimumdesign of isolation devices for highway bridges using thefragility function methodrdquo Engineering Structures vol 31no 8 pp 1648ndash1660 2009
[19] F McKenna M H Scott and G L Fenves ldquoNonlinear finite-element analysis software architecture using object compo-sitionrdquo Journal of Computing in Civil Engineering vol 24no 1 pp 95ndash107 2010
[20] MOTC ldquoGuidelines for seismic design of highway bridgesrdquoChina Ministry of Transportation and CommunicationsPress Beijing China JTGT B02-01-2008 2008
[21] A D Kiureghian and J L Sackman ldquoTangent geometricstiffness of inclined cables under self-weightrdquo Journal ofStructural Engineering vol 131 no 6 pp 941ndash945 2005
[22] J S Steelman E T Filipov L A Fahnestock et al ldquoEx-perimental behavior of steel fixed bearings and implicationsfor seismic bridge responserdquo Journal of Bridge Engineeringvol 19 no 8 Article ID A4014007 2014
[23] B Madani F Behnamfar and H Tajmir Riahi ldquoDynamicresponse of structures subjected to pounding and structure-soil-structure interactionrdquo Soil Dynamics and EarthquakeEngineering vol 78 pp 46ndash60 2015
[24] L Xu and J Li ldquoDesign and experimental investigation of anew type sliding retainer and its efficacy in seismic fortifi-cationrdquo Engineering Mechanics vol 33 no 2 pp 111ndash1182016 in Chinese
[25] Caltrans Caltrans Seismic Design Criteria California De-partment of Transportation Sacramento CA USA 2013
[26] A Shamsabadi K M Rollins and M Kapuskar ldquoNonlinearsoil-abutment-bridge structure interaction for seismic per-formance-based designrdquo Journal of Geotechnical and Geo-environmental Engineering vol 133 no 6 pp 707ndash720 2007
[27] X Li Z Sun D Wang and Y Shi ldquoLongitudinal seismicpounding effect of bridges abutment and backfilling damagerdquoJournal of Changrsquoan University vol 35 no 4 pp 76ndash82 2015in Chinese
[28] MOTC ldquoSpecification for design of highway suspensionbridgerdquo China Ministry of Transportation and Communi-cations Press Beijing China JTGT D65-05-2015 2015
[29] A Shafieezadeh K Ramanathan J E Padgett andR DesRoches ldquoFractional order intensity measures forprobabilistic seismic demand modeling applied to highwaybridgesrdquo Earthquake Engineering amp Structural Dynamicsvol 41 no 3 pp 391ndash409 2012
[30] J E Padgett B G Nielson and R DesRoches ldquoSelection ofoptimal intensity measures in probabilistic seismic demandmodels of highway bridge portfoliosrdquo Earthquake Engineeringamp Structural Dynamics vol 37 no 5 pp 711ndash725 2008
[31] K R Mackie and B Stojadinovic ldquoFragility basis for californiahighway overpass bridge seismic decision makingrdquo PEERReport 200502 Pacific Earthquake Engineering ResearchCenter University of California Berkeley Berkeley CA USA2005
[32] J Li T Peng Ji Lin L Zhou and YuL Deng Seismic Design ofMulti-Pylon Multi-Span Suspension Bridge China Commu-nications Press Beijing China 2013 in Chinese
[33] L Chen W Zhuang H Zhao and Z Wan ldquoReport onHighwaysrsquo Damage in the Wenchuan Earthquake-BridgeChina Communications Press Beijing China 2012 inChinese
[34] G Wu K Wang G Lu and P Zhang ldquoAn experimentalinvestigation of unbonded laminated elastomeric bearingsand the seismic evaluations of highway bridges with testedbearing componentsrdquo Shock and Vibration vol 2018 ArticleID 8439321 18 pages 2018
Advances in Civil Engineering 21
fact that the effectiveness of improvement for the ap-proach bridges was clearer than that of the main bridgee fragility of the highway network would be under-estimated if only the damage potential of the suspensionspan was adopted erefore when the structures such asapproach bridges transition piers and auxiliary pierswere ignored the damage identification and repair de-cisions would be misleading in the seismic evaluation oflong-span bridges Conversely the damaging probabilityof the overall bridge system will be amplified regardless ofthe higher importance of the main bridge When respectthe fact that the importance of suspension span is higherthan that of approach bridge in transportation networks aparameter a of 06ndash08 may be a better choice in thepresent study
e seismic fragility of conventionally designedbridges was mainly governed by the approach bridges at alldamage levels However for the retrofitted bridges thatpossess reasonable seismic performance different damagestates were controlled by the earthquake demands ofvarious structures as shown by the trend of the curvesalong parameter a in Case 4 and Case 5 e damagingpotential of the overall bridge from the low-to-moderatelimit states was dominated by the approach bridge eprimary reasons were the sliding of bearings and thetransfer of limited forces from the girder to substructure
by the damage of restriction devices of the approachbridges e extensive damage was controlled by thedisplacement of stiffening girder of the main bridge whichinferred the effect of dampers will be conspicuous understrong earthquake shaking At the same time the overalllong-span bridges with the reasonable seismic system willnot collapse during a strong earthquake event and canmeet emergency traffic service It can be seen that the meanvalues of fragility functions developed by compositedamage indices (equations (4) and (5)) are good indicatorsto evaluate seismic performance of long-span bridges andcan be consistent with the established seismic performanceobjectives
5 Conclusions
In the present study a procedure for investigating thereasonable seismic performance of long-span suspensionbridges was developed under the fragility function frame-work e seismic performance of a long-span suspensionbridge was examined and improved using the proposedprocedure from which the results revealed distinguishableconclusions as follows
(1) e definition of damage states for bearings con-sidered the interactions between the adjacent
014036059081103125
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(a)
020051081112142173
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
0002
0406
0810
CaseParameter a
(b)
130251371492612733
Mea
n va
lue
exp
(λ) (
ms
)
01
23
45
002
0406
081
Parameter a Case
(c)
Mea
n va
lue
exp
(λ) (
ms
)
Parameter a Case0 1 2 3 4 5
0001
0203
0405
131305478652826
1000
(d)
Figure 19 Influence of relative importance and improved designs on mean values of earthquake intensity for different damage states(a) slight (b) moderate (c) extensive (d) collapse
Advances in Civil Engineering 19
structures and self-damage to bearings in accor-dance with the actual design and experimentalconclusions e damage states of pylons in terms ofcurvature ductility were defined by developingmodal pushover analysis and idealized bilinear fit-ting methods in the equivalent pylon fiber models ofthe suspension bridge
(2) When investigating the component fragility curvesof the initial design it was found that the large deckdisplacement of the suspension span was the mostserious damage present and a slight inelastic re-sponse may have occurred only in the pylons Al-though the cable system was able to maintainelasticity under earthquakes the saddles at the py-lons were identified with a high-risk region indi-cating the damaging potential For the approachbridges the ductility mechanism did not realize arational damage sequence e columns were proneto collapse because of the application of fixed potbearings which in turn caused the damage to thebearings and abutments
(3) e stepwise improved seismic designs were pro-posed in terms of the component fragility results ofthe initial design Among the candidates fluid vis-cous dampers were proven to be a good choice forlong-span bridges based on the two attributes ofdisplacement and force when the similar efficiency ofdisplacement control was achieved e restraintsystem had a critical effect on the seismic responsesof the approach bridges e quasi-isolation orisolation can achieve a reasonable seismic damagesequence of the approach bridges
(4) e defined composite damage indices showed thatthe incorporate component classification conceptand structural relative importance were necessary toderive a good indicator of performance evaluationdue to the fact that the damage indices were based oneither an absolutely serial or serial-parallel systemmodel may significantly overestimate the damagepotential of the overall bridge system e meanvalues of earthquake intensity at the specific damagestates can serve as a function of the improvedstrategies and relative importance to further evaluatethe overall seismic performance of the suspensionbridge It can be concluded that the damaging po-tential of the overall bridge from the low-to-mod-erate limit states was governed by the approachbridges in the retrofitted bridges with a favorableseismic performance whereas the displacementdemand of the suspension span dominated thefragility probability of the extensive damage
Data Availability
e nature of the data is the analytical data of finite elementmodels of the case bridge e data used to support thefindings of this study are available from the correspondingauthor upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the Basic Research ServiceProject of Central-Level Public Welfare Research Institute(grant nos 2016-9018 and 2019-0034) and Science andTechnology Project of Communicationsrsquo Construction inWestern China MOC (grant no 2009318223094)
References
[1] L W David A Kartoum K Chang and I Roy ldquoSuspensionreport seismic performance and design considerations of longspan suspension bridgesrdquo Report CSMIP00-03 CaliforniaDepartment of Conservation Division of Mines And GeologyOffice of Strong Motion Studies Sacramento CA USA 2000
[2] S-G Gwon and D-H Choi ldquoImproved continuummodel forfree vibration analysis of suspension bridgesrdquo Journal ofEngineering Mechanics vol 143 no 7 Article ID 040170382017
[3] S Arzoumanidis A Shama and F Ostadan ldquoPerformance-based seismic analysis and design of suspension bridgesrdquoEarthquake Engineering and Structural Dynamics vol 34no 4-5 pp 349ndash367 2005
[4] L Sgambi E Garavaglia N Basso and F Bontempi ldquoMonteCarlo simulation for seismic analysis of a long span sus-pension bridgerdquo Engineering Structures vol 78 pp 100ndash1112014
[5] D Karmakar S Ray-Chaudhuri and M Shinozuka ldquoFiniteelement model development validation and probabilisticseismic performance evaluation of Vincent omas suspen-sion bridgerdquo Structure and Infrastructure Engineering vol 11no 2 pp 223ndash237 2015
[6] W-F Chen and L Duan Bridge Engineering HandbookSeismic Design CRC Press Boca Raton FL USA 2nd edition2014
[7] E Choi ldquoSeismic analysis and retrofit of mid-americabridgesrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2002
[8] B G Nielson and R DesRoches ldquoSeismic fragility method-ology for highway bridges using a component level approachrdquoEarthquake Engineering amp Structural Dynamics vol 36 no 6pp 823ndash839 2007
[9] K N Ramanathan ldquoNext generation seismic fragility curvesfor california bridges incorporating the evolution in seismicdesignrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2012
[10] O Avsar A Yakut and A Caner ldquoAnalytical fragility curvesfor ordinary highway bridges in Turkeyrdquo Earthquake Spectravol 27 no 4 pp 971ndash996 2011
[11] S P Stefanidou and A J Kappos ldquoMethodology for thedevelopment of bridge-specific fragility curvesrdquo EarthquakeEngineering amp Structural Dynamics vol 46 no 1 pp 73ndash932016
[12] B Borzi P Ceresa P Franchin F Noto G M Calvi andP E Pinto ldquoSeismic vulnerability of the Italian roadwaybridge stockrdquo Earthquake Spectra vol 31 no 4pp 2137ndash2161 2015
[13] W T Barnawi and S J Dyke ldquoSeismic fragility relationshipsof a cable-stayed bridge equipped with response modification
20 Advances in Civil Engineering
systemsrdquo Journal of Bridge Engineering vol 19 no 8 ArticleID A4013003 2014
[14] J Zhong Y Pang J Jong-Su J-S Jeon R DesRoches andW Yuan ldquoSeismic fragility assessment of long-span cable-stayed bridges in Chinardquo Advances in Structural Engineeringvol 19 no 11 pp 1797ndash1812 2016
[15] J Zhong Z Hu W Yuan and L Chen ldquoSystem-basedprobabilistic optimization of fluid viscous dampers equippedin cable-stayed bridgesrdquo Advances in Structural Engineeringvol 21 no 12 pp 1815ndash1825 2018
[16] J E Padgett and R DesRoches ldquoMethodology for the de-velopment of analytical fragility curves for retrofitted bridgesfitted bridgesrdquo Earthquake Engineering amp Structural Dy-namics vol 37 no 8 pp 1157ndash1174 2008
[17] Y Xie and J Zhang ldquoOptimal design of seismic protectivedevices for highway bridges using performance-basedmethodology and multiobjective genetic optimizationrdquoJournal of Bridge Engineering vol 22 no 3 Article ID04016129 2017
[18] J Zhang and Y Huo ldquoEvaluating effectiveness and optimumdesign of isolation devices for highway bridges using thefragility function methodrdquo Engineering Structures vol 31no 8 pp 1648ndash1660 2009
[19] F McKenna M H Scott and G L Fenves ldquoNonlinear finite-element analysis software architecture using object compo-sitionrdquo Journal of Computing in Civil Engineering vol 24no 1 pp 95ndash107 2010
[20] MOTC ldquoGuidelines for seismic design of highway bridgesrdquoChina Ministry of Transportation and CommunicationsPress Beijing China JTGT B02-01-2008 2008
[21] A D Kiureghian and J L Sackman ldquoTangent geometricstiffness of inclined cables under self-weightrdquo Journal ofStructural Engineering vol 131 no 6 pp 941ndash945 2005
[22] J S Steelman E T Filipov L A Fahnestock et al ldquoEx-perimental behavior of steel fixed bearings and implicationsfor seismic bridge responserdquo Journal of Bridge Engineeringvol 19 no 8 Article ID A4014007 2014
[23] B Madani F Behnamfar and H Tajmir Riahi ldquoDynamicresponse of structures subjected to pounding and structure-soil-structure interactionrdquo Soil Dynamics and EarthquakeEngineering vol 78 pp 46ndash60 2015
[24] L Xu and J Li ldquoDesign and experimental investigation of anew type sliding retainer and its efficacy in seismic fortifi-cationrdquo Engineering Mechanics vol 33 no 2 pp 111ndash1182016 in Chinese
[25] Caltrans Caltrans Seismic Design Criteria California De-partment of Transportation Sacramento CA USA 2013
[26] A Shamsabadi K M Rollins and M Kapuskar ldquoNonlinearsoil-abutment-bridge structure interaction for seismic per-formance-based designrdquo Journal of Geotechnical and Geo-environmental Engineering vol 133 no 6 pp 707ndash720 2007
[27] X Li Z Sun D Wang and Y Shi ldquoLongitudinal seismicpounding effect of bridges abutment and backfilling damagerdquoJournal of Changrsquoan University vol 35 no 4 pp 76ndash82 2015in Chinese
[28] MOTC ldquoSpecification for design of highway suspensionbridgerdquo China Ministry of Transportation and Communi-cations Press Beijing China JTGT D65-05-2015 2015
[29] A Shafieezadeh K Ramanathan J E Padgett andR DesRoches ldquoFractional order intensity measures forprobabilistic seismic demand modeling applied to highwaybridgesrdquo Earthquake Engineering amp Structural Dynamicsvol 41 no 3 pp 391ndash409 2012
[30] J E Padgett B G Nielson and R DesRoches ldquoSelection ofoptimal intensity measures in probabilistic seismic demandmodels of highway bridge portfoliosrdquo Earthquake Engineeringamp Structural Dynamics vol 37 no 5 pp 711ndash725 2008
[31] K R Mackie and B Stojadinovic ldquoFragility basis for californiahighway overpass bridge seismic decision makingrdquo PEERReport 200502 Pacific Earthquake Engineering ResearchCenter University of California Berkeley Berkeley CA USA2005
[32] J Li T Peng Ji Lin L Zhou and YuL Deng Seismic Design ofMulti-Pylon Multi-Span Suspension Bridge China Commu-nications Press Beijing China 2013 in Chinese
[33] L Chen W Zhuang H Zhao and Z Wan ldquoReport onHighwaysrsquo Damage in the Wenchuan Earthquake-BridgeChina Communications Press Beijing China 2012 inChinese
[34] G Wu K Wang G Lu and P Zhang ldquoAn experimentalinvestigation of unbonded laminated elastomeric bearingsand the seismic evaluations of highway bridges with testedbearing componentsrdquo Shock and Vibration vol 2018 ArticleID 8439321 18 pages 2018
Advances in Civil Engineering 21
structures and self-damage to bearings in accor-dance with the actual design and experimentalconclusions e damage states of pylons in terms ofcurvature ductility were defined by developingmodal pushover analysis and idealized bilinear fit-ting methods in the equivalent pylon fiber models ofthe suspension bridge
(2) When investigating the component fragility curvesof the initial design it was found that the large deckdisplacement of the suspension span was the mostserious damage present and a slight inelastic re-sponse may have occurred only in the pylons Al-though the cable system was able to maintainelasticity under earthquakes the saddles at the py-lons were identified with a high-risk region indi-cating the damaging potential For the approachbridges the ductility mechanism did not realize arational damage sequence e columns were proneto collapse because of the application of fixed potbearings which in turn caused the damage to thebearings and abutments
(3) e stepwise improved seismic designs were pro-posed in terms of the component fragility results ofthe initial design Among the candidates fluid vis-cous dampers were proven to be a good choice forlong-span bridges based on the two attributes ofdisplacement and force when the similar efficiency ofdisplacement control was achieved e restraintsystem had a critical effect on the seismic responsesof the approach bridges e quasi-isolation orisolation can achieve a reasonable seismic damagesequence of the approach bridges
(4) e defined composite damage indices showed thatthe incorporate component classification conceptand structural relative importance were necessary toderive a good indicator of performance evaluationdue to the fact that the damage indices were based oneither an absolutely serial or serial-parallel systemmodel may significantly overestimate the damagepotential of the overall bridge system e meanvalues of earthquake intensity at the specific damagestates can serve as a function of the improvedstrategies and relative importance to further evaluatethe overall seismic performance of the suspensionbridge It can be concluded that the damaging po-tential of the overall bridge from the low-to-mod-erate limit states was governed by the approachbridges in the retrofitted bridges with a favorableseismic performance whereas the displacementdemand of the suspension span dominated thefragility probability of the extensive damage
Data Availability
e nature of the data is the analytical data of finite elementmodels of the case bridge e data used to support thefindings of this study are available from the correspondingauthor upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the Basic Research ServiceProject of Central-Level Public Welfare Research Institute(grant nos 2016-9018 and 2019-0034) and Science andTechnology Project of Communicationsrsquo Construction inWestern China MOC (grant no 2009318223094)
References
[1] L W David A Kartoum K Chang and I Roy ldquoSuspensionreport seismic performance and design considerations of longspan suspension bridgesrdquo Report CSMIP00-03 CaliforniaDepartment of Conservation Division of Mines And GeologyOffice of Strong Motion Studies Sacramento CA USA 2000
[2] S-G Gwon and D-H Choi ldquoImproved continuummodel forfree vibration analysis of suspension bridgesrdquo Journal ofEngineering Mechanics vol 143 no 7 Article ID 040170382017
[3] S Arzoumanidis A Shama and F Ostadan ldquoPerformance-based seismic analysis and design of suspension bridgesrdquoEarthquake Engineering and Structural Dynamics vol 34no 4-5 pp 349ndash367 2005
[4] L Sgambi E Garavaglia N Basso and F Bontempi ldquoMonteCarlo simulation for seismic analysis of a long span sus-pension bridgerdquo Engineering Structures vol 78 pp 100ndash1112014
[5] D Karmakar S Ray-Chaudhuri and M Shinozuka ldquoFiniteelement model development validation and probabilisticseismic performance evaluation of Vincent omas suspen-sion bridgerdquo Structure and Infrastructure Engineering vol 11no 2 pp 223ndash237 2015
[6] W-F Chen and L Duan Bridge Engineering HandbookSeismic Design CRC Press Boca Raton FL USA 2nd edition2014
[7] E Choi ldquoSeismic analysis and retrofit of mid-americabridgesrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2002
[8] B G Nielson and R DesRoches ldquoSeismic fragility method-ology for highway bridges using a component level approachrdquoEarthquake Engineering amp Structural Dynamics vol 36 no 6pp 823ndash839 2007
[9] K N Ramanathan ldquoNext generation seismic fragility curvesfor california bridges incorporating the evolution in seismicdesignrdquo Doctoral thesis Georgia Institute of TechnologyAtlanta GA USA 2012
[10] O Avsar A Yakut and A Caner ldquoAnalytical fragility curvesfor ordinary highway bridges in Turkeyrdquo Earthquake Spectravol 27 no 4 pp 971ndash996 2011
[11] S P Stefanidou and A J Kappos ldquoMethodology for thedevelopment of bridge-specific fragility curvesrdquo EarthquakeEngineering amp Structural Dynamics vol 46 no 1 pp 73ndash932016
[12] B Borzi P Ceresa P Franchin F Noto G M Calvi andP E Pinto ldquoSeismic vulnerability of the Italian roadwaybridge stockrdquo Earthquake Spectra vol 31 no 4pp 2137ndash2161 2015
[13] W T Barnawi and S J Dyke ldquoSeismic fragility relationshipsof a cable-stayed bridge equipped with response modification
20 Advances in Civil Engineering
systemsrdquo Journal of Bridge Engineering vol 19 no 8 ArticleID A4013003 2014
[14] J Zhong Y Pang J Jong-Su J-S Jeon R DesRoches andW Yuan ldquoSeismic fragility assessment of long-span cable-stayed bridges in Chinardquo Advances in Structural Engineeringvol 19 no 11 pp 1797ndash1812 2016
[15] J Zhong Z Hu W Yuan and L Chen ldquoSystem-basedprobabilistic optimization of fluid viscous dampers equippedin cable-stayed bridgesrdquo Advances in Structural Engineeringvol 21 no 12 pp 1815ndash1825 2018
[16] J E Padgett and R DesRoches ldquoMethodology for the de-velopment of analytical fragility curves for retrofitted bridgesfitted bridgesrdquo Earthquake Engineering amp Structural Dy-namics vol 37 no 8 pp 1157ndash1174 2008
[17] Y Xie and J Zhang ldquoOptimal design of seismic protectivedevices for highway bridges using performance-basedmethodology and multiobjective genetic optimizationrdquoJournal of Bridge Engineering vol 22 no 3 Article ID04016129 2017
[18] J Zhang and Y Huo ldquoEvaluating effectiveness and optimumdesign of isolation devices for highway bridges using thefragility function methodrdquo Engineering Structures vol 31no 8 pp 1648ndash1660 2009
[19] F McKenna M H Scott and G L Fenves ldquoNonlinear finite-element analysis software architecture using object compo-sitionrdquo Journal of Computing in Civil Engineering vol 24no 1 pp 95ndash107 2010
[20] MOTC ldquoGuidelines for seismic design of highway bridgesrdquoChina Ministry of Transportation and CommunicationsPress Beijing China JTGT B02-01-2008 2008
[21] A D Kiureghian and J L Sackman ldquoTangent geometricstiffness of inclined cables under self-weightrdquo Journal ofStructural Engineering vol 131 no 6 pp 941ndash945 2005
[22] J S Steelman E T Filipov L A Fahnestock et al ldquoEx-perimental behavior of steel fixed bearings and implicationsfor seismic bridge responserdquo Journal of Bridge Engineeringvol 19 no 8 Article ID A4014007 2014
[23] B Madani F Behnamfar and H Tajmir Riahi ldquoDynamicresponse of structures subjected to pounding and structure-soil-structure interactionrdquo Soil Dynamics and EarthquakeEngineering vol 78 pp 46ndash60 2015
[24] L Xu and J Li ldquoDesign and experimental investigation of anew type sliding retainer and its efficacy in seismic fortifi-cationrdquo Engineering Mechanics vol 33 no 2 pp 111ndash1182016 in Chinese
[25] Caltrans Caltrans Seismic Design Criteria California De-partment of Transportation Sacramento CA USA 2013
[26] A Shamsabadi K M Rollins and M Kapuskar ldquoNonlinearsoil-abutment-bridge structure interaction for seismic per-formance-based designrdquo Journal of Geotechnical and Geo-environmental Engineering vol 133 no 6 pp 707ndash720 2007
[27] X Li Z Sun D Wang and Y Shi ldquoLongitudinal seismicpounding effect of bridges abutment and backfilling damagerdquoJournal of Changrsquoan University vol 35 no 4 pp 76ndash82 2015in Chinese
[28] MOTC ldquoSpecification for design of highway suspensionbridgerdquo China Ministry of Transportation and Communi-cations Press Beijing China JTGT D65-05-2015 2015
[29] A Shafieezadeh K Ramanathan J E Padgett andR DesRoches ldquoFractional order intensity measures forprobabilistic seismic demand modeling applied to highwaybridgesrdquo Earthquake Engineering amp Structural Dynamicsvol 41 no 3 pp 391ndash409 2012
[30] J E Padgett B G Nielson and R DesRoches ldquoSelection ofoptimal intensity measures in probabilistic seismic demandmodels of highway bridge portfoliosrdquo Earthquake Engineeringamp Structural Dynamics vol 37 no 5 pp 711ndash725 2008
[31] K R Mackie and B Stojadinovic ldquoFragility basis for californiahighway overpass bridge seismic decision makingrdquo PEERReport 200502 Pacific Earthquake Engineering ResearchCenter University of California Berkeley Berkeley CA USA2005
[32] J Li T Peng Ji Lin L Zhou and YuL Deng Seismic Design ofMulti-Pylon Multi-Span Suspension Bridge China Commu-nications Press Beijing China 2013 in Chinese
[33] L Chen W Zhuang H Zhao and Z Wan ldquoReport onHighwaysrsquo Damage in the Wenchuan Earthquake-BridgeChina Communications Press Beijing China 2012 inChinese
[34] G Wu K Wang G Lu and P Zhang ldquoAn experimentalinvestigation of unbonded laminated elastomeric bearingsand the seismic evaluations of highway bridges with testedbearing componentsrdquo Shock and Vibration vol 2018 ArticleID 8439321 18 pages 2018
Advances in Civil Engineering 21
systemsrdquo Journal of Bridge Engineering vol 19 no 8 ArticleID A4013003 2014
[14] J Zhong Y Pang J Jong-Su J-S Jeon R DesRoches andW Yuan ldquoSeismic fragility assessment of long-span cable-stayed bridges in Chinardquo Advances in Structural Engineeringvol 19 no 11 pp 1797ndash1812 2016
[15] J Zhong Z Hu W Yuan and L Chen ldquoSystem-basedprobabilistic optimization of fluid viscous dampers equippedin cable-stayed bridgesrdquo Advances in Structural Engineeringvol 21 no 12 pp 1815ndash1825 2018
[16] J E Padgett and R DesRoches ldquoMethodology for the de-velopment of analytical fragility curves for retrofitted bridgesfitted bridgesrdquo Earthquake Engineering amp Structural Dy-namics vol 37 no 8 pp 1157ndash1174 2008
[17] Y Xie and J Zhang ldquoOptimal design of seismic protectivedevices for highway bridges using performance-basedmethodology and multiobjective genetic optimizationrdquoJournal of Bridge Engineering vol 22 no 3 Article ID04016129 2017
[18] J Zhang and Y Huo ldquoEvaluating effectiveness and optimumdesign of isolation devices for highway bridges using thefragility function methodrdquo Engineering Structures vol 31no 8 pp 1648ndash1660 2009
[19] F McKenna M H Scott and G L Fenves ldquoNonlinear finite-element analysis software architecture using object compo-sitionrdquo Journal of Computing in Civil Engineering vol 24no 1 pp 95ndash107 2010
[20] MOTC ldquoGuidelines for seismic design of highway bridgesrdquoChina Ministry of Transportation and CommunicationsPress Beijing China JTGT B02-01-2008 2008
[21] A D Kiureghian and J L Sackman ldquoTangent geometricstiffness of inclined cables under self-weightrdquo Journal ofStructural Engineering vol 131 no 6 pp 941ndash945 2005
[22] J S Steelman E T Filipov L A Fahnestock et al ldquoEx-perimental behavior of steel fixed bearings and implicationsfor seismic bridge responserdquo Journal of Bridge Engineeringvol 19 no 8 Article ID A4014007 2014
[23] B Madani F Behnamfar and H Tajmir Riahi ldquoDynamicresponse of structures subjected to pounding and structure-soil-structure interactionrdquo Soil Dynamics and EarthquakeEngineering vol 78 pp 46ndash60 2015
[24] L Xu and J Li ldquoDesign and experimental investigation of anew type sliding retainer and its efficacy in seismic fortifi-cationrdquo Engineering Mechanics vol 33 no 2 pp 111ndash1182016 in Chinese
[25] Caltrans Caltrans Seismic Design Criteria California De-partment of Transportation Sacramento CA USA 2013
[26] A Shamsabadi K M Rollins and M Kapuskar ldquoNonlinearsoil-abutment-bridge structure interaction for seismic per-formance-based designrdquo Journal of Geotechnical and Geo-environmental Engineering vol 133 no 6 pp 707ndash720 2007
[27] X Li Z Sun D Wang and Y Shi ldquoLongitudinal seismicpounding effect of bridges abutment and backfilling damagerdquoJournal of Changrsquoan University vol 35 no 4 pp 76ndash82 2015in Chinese
[28] MOTC ldquoSpecification for design of highway suspensionbridgerdquo China Ministry of Transportation and Communi-cations Press Beijing China JTGT D65-05-2015 2015
[29] A Shafieezadeh K Ramanathan J E Padgett andR DesRoches ldquoFractional order intensity measures forprobabilistic seismic demand modeling applied to highwaybridgesrdquo Earthquake Engineering amp Structural Dynamicsvol 41 no 3 pp 391ndash409 2012
[30] J E Padgett B G Nielson and R DesRoches ldquoSelection ofoptimal intensity measures in probabilistic seismic demandmodels of highway bridge portfoliosrdquo Earthquake Engineeringamp Structural Dynamics vol 37 no 5 pp 711ndash725 2008
[31] K R Mackie and B Stojadinovic ldquoFragility basis for californiahighway overpass bridge seismic decision makingrdquo PEERReport 200502 Pacific Earthquake Engineering ResearchCenter University of California Berkeley Berkeley CA USA2005
[32] J Li T Peng Ji Lin L Zhou and YuL Deng Seismic Design ofMulti-Pylon Multi-Span Suspension Bridge China Commu-nications Press Beijing China 2013 in Chinese
[33] L Chen W Zhuang H Zhao and Z Wan ldquoReport onHighwaysrsquo Damage in the Wenchuan Earthquake-BridgeChina Communications Press Beijing China 2012 inChinese
[34] G Wu K Wang G Lu and P Zhang ldquoAn experimentalinvestigation of unbonded laminated elastomeric bearingsand the seismic evaluations of highway bridges with testedbearing componentsrdquo Shock and Vibration vol 2018 ArticleID 8439321 18 pages 2018
