a nonlinear static procedure for the seismic design of...

Research ArticleA Nonlinear Static Procedure for the Seismic Design ofSymmetrical Irregular Bridges

Shanshan Li1 Ping Xiang1 Biao Wei 1 Lu Yan1 and Ye Xia2

1School of Civil Engineering Central South University 22 Shaoshan South Road Changsha 410075 China2Department of Bridge Engineering Tongji University 1239 Siping Road Shanghai 200092 China

Correspondence should be addressed to Biao Wei weibiaocsueducn

Received 1 May 2020 Revised 4 August 2020 Accepted 14 September 2020 Published 26 September 2020

Academic Editor Marco Lepidi

Copyright copy 2020 Shanshan Li 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

Displacement-based seismic design methods support the performance-based seismic design philosophy known to be the mostadvanced seismic design theoryis paper explores one common type of irregular-continuous bridges and studies the predictionof their elastoplastic displacement demand based on a new nonlinear static procedure is benefits to achieve the operation ofdisplacement-based seismic design ree irregular-continuous bridges are analyzed to advance the equivalent SDOF systembuild the capacity spectrum and the inelastic spectrum and generate the new nonlinear static analysis e proposed approach isused to simplify the prediction of elastoplastic displacement demand and is validated by parametric analysis e new nonlinearstatic procedure is also used to achieve the displacement-based seismic design procedure It is tested by an example to obtainresults which show that after several combinations of the capacity spectrum (obtained by a pushover analysis) and the inelasticdemand spectrum the simplified displacement-based seismic design of the common irregular-continuous bridges can beachieved By this design the seismic damage on structures is effectively controlled

1 Introduction

In recent years displacement-based seismic design methodshave rapidly developed and the prediction of elastoplasticdisplacement demand on structures under seismic actionhas become a crucial issue [1] Although inelastic timehistory analysis (ITHA) can calculate the elastoplastic dis-placement demand on structures recent studies have con-centrated on developing simplified methods for relatedquestions due to the complexity of ITHA In the presenttime however the mature simplified methods are primarilysuitable for regular-continuous bridges eg the pushoveranalysis method under uniform load in AASHTO [1] Whencomparing irregular-continuous bridges [2] influenced bymodes to the regular-continuous bridges the simplifiedprediction methods of elastoplastic displacement demandrequire further study [3] Although various of methodolo-gies such as the modal adaptive nonlinear static procedure(MANSP) [4ndash6] the modal pushover analysis (MPA) [7ndash10]and the incremental response spectral analysis (IRSA) [11]

have been proposed further investigation on their calcu-lation accuracy application scope and the degree of sim-plification is needed

In previous studies some extremely irregular-continu-ous bridges were selected to validate the calculation accuracy[12 13] ose bridges with asymmetry obviously unequalpier length and other irregular properties at the same timeare common cases in mountain area or other similar areasRegarding practical continuous bridges in general areascases with many regular factors and only few irregularfactors [14] leading to the obvious influence of high modesare referred to as the common irregular-continuous bridgese irregular behavior comes from the dynamic response ofthe bridges under investigation despite these bridges havinga rather regular geometric layout In terms of common ir-regular-continuous bridges the many rules in their seismicresponse benefit in finding a proper simplified predictionmethod of elastoplastic displacement demand

In addition to the pushover-based analysis methodsmentioned above many other methods have been

HindawiShock and VibrationVolume 2020 Article ID 8899705 16 pageshttpsdoiorg10115520208899705

established for the simplified prediction of elastoplasticdisplacement demand in irregular-continuous bridges Forinstance Kowalsky proposed a displacement-based seismicdesign method grounded in the ideas of equivalent systemand equivalent damping ratio etc [15ndash18] is proposedmethod involves the advantages of easy operation KapposGidaris and Gkatzogias further improved some aspects ofthe proposed method [19 20] such as how to combine thedamping ratios of structural components to form thedamping ratio of bridge systems e ideas introduced inthese methods are worth studying in the research of a newnonlinear static procedure analysis methods

By taking one common type of irregular-continuousbridges in transverse direction as the object of study thispaper proposes a simplified prediction method of seismicdisplacement demand Based on their seismic responsecharacteristics (the equivalent system concept and the basicidea of pushover analysis) this paper also proposes thecorresponding displacement-based seismic designprocedure

2 A Common Type of Irregular-Continuous Bridges

Regular-continuous bridges are generally defined as bridgesthat can be simplified as single-degree-of-freedom (SDOF)systems or as those with dynamic responses controlled byonly a decisive fundamental mode To carry out the sim-plified seismic design safely the AASHTO [1] and ChinarsquosGuidelines for Seismic Design of Highway Bridges (2008)define regular-continuous bridges according to structuralcharacteristics However transverse dynamic responses ofmany actual continuous bridges are controlled by two ormore modes When this occurs they are referred to as ir-regular-continuous bridges Figure 1 shows one kind of themost popular irregular-continuous bridges containing manyregular aspects eg nearly symmetric distribution of pierlength nearly equivalent distribution of span length andmany other aspects which satisfy the structural requirementsof regular-continuous bridges However the transversedynamic response of these bridges is still controlled by twoor more modes is is due to the influence of the followingirregular factors (1) different pier lengths and a compara-tively small stiffness ratio of girder to pier (2) in order toprevent excessive internal forces and deformations atabutments under earthquakes bidirectional sliding bearingsare set on all abutments as shown in Figure 1 Meanwhile ifthe stiffness ratio of girder to pier is comparatively smallerthe mass of superstructure endured by each pier will bedifferent under transverse seismic actions even if the pierheights are nearly the same In Figure 1 the modal shapes forthe first and second modes are nearly identical in shape forthe three different bridges e reason is that the bidirec-tional sliding bearings are set on all abutments and thestiffness ratio of girder to pier is small as described abovewhich control the first and second modes of the three dif-ferent bridges However there are some differences in thebending degree of the corresponding vibration modes of thethree bridges since these bridges have different pier

distributions It is noted that shear keys are considered anddesigned in the transverse direction of abutments to meetthe requirements under normal loading and their failure isonly permitted under severe earthquakes [13]

e irregular-continuous bridges with relatively regulargeometry are the study object of this paper ree 4times 40mtypical common irregular-continuous bridges in Figure 1have been selected for analysis e friction coefficient ofsliding bearings is equal to 002 e section properties ofgirders and columns are shown in Table 1 Earthquake loadadopts the elastic response spectrum for soil profile III inChinese criteria (JTJ 004-89) as shown in Figure 2(a)According to requirements of this paper it is transformed inthe following two ways (i) it is converted into the inelasticdemand spectrum which will be used for the simplifiedprediction method of seismic displacement demand edetailed procedure will be discussed in the following sec-tions (ii) To transform the ground motion input of ITHAseven accelerograms are generated by the Simqke procedure[21] Results of ITHA are regarded as the benchmark forcomparison using the simplified prediction methodFigure 2(b) gives the first accelerogram while other figuresare omitted due to similarity

3 Characteristics of Seismic Displacement

e seismic displacement of girder is nearly symmetric forthe common irregular-continuous bridges and when in-duced by the gradual increase of peak ground accelerations(PGAs) its shape changes as the pierrsquos yielding degree in-creases [22] Furthermore the shape will be relatively un-changed after the pierrsquos yielding degree arrives at a givenvalue In this section a concept of the equivalent system isused to decompose the girderrsquos seismic displacement intothe displacement of the equivalent SDOF system e co-efficient of displacement shape is applied to the foregoingthree irregular-continuous bridges to study their charac-teristics of seismic displacement

31 Concept of Equivalent SDOF System To operate thepushover analysis and to carry out displacement-basedseismic design procedure it is necessary to first analyze howto transform a multi-degree-of-freedom (MDOF) system ofa continuous bridge into a SDOF system ie how to de-compose the seismic displacement of the bridge into thedisplacement equivalent SDOF system and the coefficient ofdisplacement shape

In the finite element model (FEM) mi Δi Fi and ai aredefined to be the ith structure nodersquos mass displacementinertia force [23] and acceleration respectively e cor-responding values of the equivalent SDOF system aredenoted by meq Δeq Feq and aeq respectively e rela-tionship between Δi and Δeq and ai and aeq is supposed as

Δi ciΔeq (1)

ai ciaeq (2)

where ci is the coefficient of displacement shape

2 Shock and Vibration

e inertia force of the equivalent system is to be equal tothe resultant inertia force of the original system hence

Feq 1113944n

i1Fi 1113944

n

i1miai aeq 1113944

n

i1mici (3)

And thus the mass of the equivalent system meq isdenoted by

meq 1113944n

i1mici (4)

According to (2) and (3) it is obtained by

Fi miai miciaeq mici

1113936ni1mici

Feq (5)

Substitute equation (1) into equation (5) hence

Fi miΔi

1113936ni1miΔi

Feq (6)

Suppose the inertia forces of the equivalent system andthe original system to be equal as follows

Table 1 Section properties of girder and piers

Components Area(m2)

Moment of inertia(m4)

Polar moment of inertia(m4)

Concretetype

Longitudinal reinforcement steel and arearatio

Girder 7 40 14 C50 mdashColumn 225 0422 0722 C30 HRB335 066

5m

0 1 2 3 4Girder point

Mode 1 21167 0473727 248

ndash006ndash004ndash002

0002004006008

01


0002004006008

01


0002004006008

01


Mode 1Mode 2

Mode 1Mode 2

Mode 1Mode 2

Mode 1 2Period(s) 0798 0564Meff() 233 741


Mode 1 20755 0429488 494

Period (s)Meff ()

Period(s)Meff()

0 43215m10m5m

0 43215m 10m10m

051005 bridge

100510 bridge

0 43215m5m

050505 bridge

Arrangement of bearing of all bridges

ϕ nϕ n

ϕ n

Figure 1 One type of most popular irregular-continuous bridges

Shock and Vibration 3

FeqΔeq 1113944n

i1FiΔi (7)

Substituting equation (6) into equation (7) obtains

Δeq 1113944

n

i1miΔ2i

1113944n

i1miΔi

(8)

Substitute equation (8) into equation (1) and then

ci Δi

Δeq

1113944n

i1miΔi

1113944n

i1miΔ2i

Δi (9)


meq 1113944

n

i1miΔi1113872 11138732

1113944n

i1miΔ2i

(10)

erefore the relationship MDOF system beingequivalent to the SDOF system is developed with the fol-lowing characteristics

(1) When a bridge structure is under elastic state pa-rameters meq ci and Δeq are only related to the shapeof the elastic displacement vector Δ which isequivalent to mode vectorΦn Compared to a certainmode in multimode pushover analysis [8] meq inequation (10) similar to mode participation mass ci

in equation (9) is similar to the product of the modeparticipation factor Γn and the corresponding value

ϕin of the mode vector Φn and Δeq in equation (8) issimilar to the response spectrum displacement sd of acertain single mode

(2) When the bridge is under plastic state meq ci andΔeq are still related to the shape of the displacementvector Δ

(3) Δ can be decomposed into the product of Δeq and ci

no matter what status the bridge is under eg elasticstate or plastic state

32 Study Case Displacement vector Δ of a bridge can bedecomposed into the product of Δeq and ci according to theforegoing concept of the equivalent system It is used tostudy the seismic displacement characteristics of three ir-regular-continuous bridges in Figure 1

FEM for each bridge is developed by OpenSeesprogram [24] e girders piers and bearings are sim-ulated by elastic beam fiber and zero-length link ele-ments respectively e cross section of piers is dividedinto three parts including cover concrete core concreteand longitudinal bars e concrete is simulated byconcrete07 and the longitudinal bars are simulated byreinforcing steel material with the low-cycle fatigueparameters e displacement-based fiber elements withadequate integral points are used to calculate the seismicresponses Different zero-length link elements are usedto simulate the fixed and sliding bearings respectivelyAs for the fixed bearings the zero-length link element isan elastic link element with a large stiffness and an

01 045 1 150

1

2

225

Structure period T (s)

Dyn

amic

mag

nific

atio

n fa

ctor

β

Spectrum for soil profile III in ChineseSpectrum 1~7 generated by simkqeSpectrum 1~7 generated by simkqeSpectrum 1~7 generated by simkqeSpectrum 1~7 generated by simkqeSpectrum 1~7 generated by simkqeSpectrum 1~7 generated by simkqeSpectrum 1~7 generated by simkqe

β = 225 times (045T)095

(a)

Acce

lera

tion

(ms

2 )

Time (s)

15

10

5

0

ndash5

ndash10

ndash150 10 20 30 40

(b)

Figure 2 Earthquake input (a) response spectra for soil profile III in Chinese criteria (JTJ 004-89) and generated by Simqke and (b) oneaccelerogram corresponding to (a)


assumed large force which is unyielding forever In termsof the middle and side sliding bearings the zero-lengthlink element is an elastoplastic link element with ayielding force of 75 kN and 375 kN respectively

e accelerograms corresponding to the responsespectrum of soil type III are designated for seismic input 59levels of PGA are investigated ranging from 002 g to 06 gwith an interval of 001 g [25 26] e seismic displacementis calculated by ITHA Parameters meq Δeq and ci aredetermined by the concept of the equivalent system inSection 31

e mass meq is computed by equation (10) and shownin Figure 3(a) As the total mass of each bridge is 30293058 and 3000 tons respectively Figure 3(a) shows thefollowing

(1) e ratios of mass meq to bridge total mass are948sim967 854sim981 and 865sim981 re-spectively erefore with the inclusion of theparticipation masses of each mode the mass meq isnearly the same as the bridge total mass

(2) e ratio of mass meq to bridge total mass for eachbridge increases gradually as PGA increases thusmaking meq closer to the bridge total mass

e displacement Δeq is then computed by equation(8) and displayed in Figure 3(b) It shows that the dis-placement Δeq for each bridge gradually increases as PGAincreases Figure 3(b) also shows that Δeq of the 051005bridge and 050505 bridge almost coincides with eachother since both trends increase at similar rates

e coefficient of displacement shape ci is then com-puted by equation (9) ci of the girder points at 0 1 and 2of each bridge in Figure 1 is shown in Figures 3(c)ndash3(e)illustrating how ci changes for each bridge along with theincrease of PGA as follows

(1) When PGA is small and the bridge is in an elasticstate the value of ci is stable as PGA increases

(2) When PGA is larger and the bridge begins to yield atdifferent degrees the value of ci changes rapidly asPGA increases

(3) When PGA is noticeably larger than case (2) thevalue of ci changes little and tends to stabilize as PGAincreases

(4) e changing range of ci at the node 0 of the girderend ie girder point 0 in Figure 1 is relativelysmaller when compared to the corresponding valueci in its elastic state

4 Procedure of SimplifiedPrediction of SeismicDisplacement Demand

is section gives a simplified prediction procedure ofseismic displacement demand e principle of the proce-dure is to combine the structural capacity spectrum and theinelastic demand spectrum to estimate the seismic

displacement response of structure e following willdiscuss each part of the simplified prediction procedure

41 Capacity Spectrum e transformation from seismicdynamic loading to static loading and the transformationfrom theMDOF system to the SDOF systemmust be studiedin order to estimate the seismic displacement of the con-tinuous bridge In regard to studying the transformationfrom the MDOF system to the SDOF system two mainmethods exist One solution is the same as the multimodepushover analysis method in which mode decomposition isexecuted and each mode refers to a single SDOF system Itcan directly use the pushover analysis in theory Becauseeach important mode is used to determine the distributionof forces for the pushover analysis separately this method iscomplex in practice It also requires several pushover pro-cesses e alternative method treats a continuous bridge asapproximately a single SDOF system It is pushed by rea-sonable distribution of forces which have been indirectlyadopted in the equivalent linear methodese forces will beused to build the capacity spectrum of irregular-continuousbridge in this section is alternative method is simplerthan the previous solution

e relationship between the MDOF system and itsequivalent SDOF system can be linked by the concept of theequivalent system according to the discussion in Section 3Based on the above analysis the following steps are used toobtain the capacity spectrum

(1) e FEM of a bridge is analyzed by the responsespectrum analysis to obtain the elastic displacementvector Δ

(2) e bridge is pushed to a certain plastic state underthe distribution of forces mΔ and the Vb minus ur curveis obtained whereVb is the summation of shear forceat the bottom of each pier and ur is the displacementof reference point and m is the mass matrix

(3) e Vb minus ur curve is then transformed into theSa minus Sd curve by assigning Sa Vbmeq andSd urci

is process of the pushover analysis method is referredto as the pushover analysis method based on responsespectrum loads For short it is referenced to as RSP Its basicidea comes from the N2 method [27] and the FEMApushover method [8] and some similar methods have beenused for bridge structures [28 29]

When the bridge is pushed by the distribution offorces mΔ the position of displacement reference pointrequires further discussion in this method When thebridge is under elastic state the displacement shape ob-tained by pushover analysis is nearly consistent with theshape of elastic displacement vector Δ Sd urci of dif-ferent displacement reference points is nearly the samewith each other and the corresponding Sa minus Sd curve isirrelevant to the position of the displacement referencepoint However when the bridge enters into plastic state


the displacement shape (obtained by pushover analysis)and the shape of elastic displacement vector Δ (obtainedby response spectrum analysis) become more and moreinconsistent erefore Sd urci of different displace-ment reference points is not the same e correspondingSa minus Sd curve is also different for various positions ofdisplacement reference points

Based on the concept of the equivalent system in Section3 vector Δ can be decomposed into the product of Δeq andci In pushover analysis ur can be expressed as ur ciSd inwhich Sd is corresponding to Δeq of an equivalent system

from a physics concept If the displacement vector Δ ob-tained by pushover analysis is required to be equal to theresults from ITHA when Sd Δeq the coefficient of dis-placement shape ci of the two methods must be the same Inthe pushover analysis ci is constantly changing creatingdifficulty in tracking the complexity of its transformationus simplified measures are needed

According to the case analysis in Section 3 the changingrange of ci at the point 0 of the girder end is relatively smallerwhen compared to the corresponding value ci in its elasticstate Hence the change of ci at the point 0 of the girder end

2500

2700

2900

3100

051005 bridge100510 bridge050505 bridge

meq

(t)

01 02 03 04 05 060PGA (g)

(a)


∆ eq

(m)

0002004006008

01012014

01 02 03 04 05 060PGA (g)

(b)

Girder point 0Girder point 1Girder point 2

01 02 03 04 05 060PGA (g)

06

08

1

12

14

16

18

ci

(c)


0

03

06

09

12

15

18

ci

01 02 03 04 05 060PGA (g)

(d)


01 02 03 04 05 060PGA (g)

0

04

08

12

16

2

c i

(e)

Figure 3 Equivalent SDOF system (a) mass (b) displacement (c) coefficient of displacement shape of the 051005 bridge (d) coefficient ofdisplacement shape of the 100510 bridge and (e) coefficient of displacement shape of the 050505 bridge


under seismic actions is omitted and the corresponding ci isassumed and set to be always equal to the value of elasticstate erefore the girder point 0 is chosen as the dis-placement reference point and the Sa minus Sd curve of bridgestructure can be obtained through the formula Sd urci inwhich ur and ci are all the corresponding values of the girderpoint 0

42 Inelastic Demand Spectrum Based on Section 2 theelastic response spectrum should be converted into theinelastic demand spectrum used by the simplified predictionmethod of seismic displacement demand e conversioncan use C the ratio of displacement demand of the elas-toplastic model to that of its elastic counterpart for oneSDOF system subjected to the same earthquake Many re-searchers have investigated C to simplify the estimation ofseismic displacement demand of a structure [30 31] and C

used here adopts Mirandarsquos equation shown as follows [32]

C 1 +1μ

minus 11113888 1113889 middot exp minus12Tμminus 081113872 11138731113890 1113891

minus 1

(11)

where T is the period of SDOF and μ is its displacementductility demand

e aforementioned elastic response spectrum is con-verted as follows

Say CSa

μ

Su CSd CSaT

2

4π2

(12)

where Sd and Sa are respectively the displacement value andacceleration value of the elastic response spectrum Su andSay are respectively the displacement value and accelerationvalue of the inelastic response spectrum

Figure 4 shows how to construct the inelastic demandspectrum based on the aforementioned equations ere-fore the inelastic demand spectrum and the aforementionedcapacity spectrum can be applied to the Sa minus Sd coordinatesystem to obtain the modal displacement response Sd [33]

43 Prediction of Seismic Displacement e inelastic de-mand spectrum and the capacity spectrum are drawn in thesame figure e capacity spectrum will intersect with dif-ferent demand spectrums corresponding to different μvalues which are the displacement ductility demand factorDifferent Sd of the intersection points will then also beobtained Denote μprime as SdSdy where Sdy is the spectrumvalue of yield-point displacement and μprime 1 when Sd is in theelastic regione Sd of the intersection point correspondingto μ asymp μprime where μ is the displacement ductility demand inFigure 4 and μprime SdSdy in the capacity spectrum is theseismic displacement demand of the equivalent SDOFsystem Sd is equivalent to Δeq in equation (8)

Seismic displacement demand Δi of each node in itsoriginal structure needs to be reversely solved by usingequation (1) after obtaining the Sd or Δeq of the equivalent

SDOF system in theory As to further simply the predictionof displacement demand in practice it adopts the actualpushover displacement vector u corresponding to Sd as theseismic displacement demand Δi of each node in the bridgesystem

5 Verification Case of SimplifiedPrediction Procedure

Results show that the seismic displacement response of ir-regular-continuous bridges has two characteristics as PGAincreases ① the displacement Δeq of the equivalent SDOFsystem increases gradually and ② the coefficient ci of dis-placement shape is constantly changing e two charac-teristics above should be reflected when judging if asimplified prediction method can correctly predict theseismic displacement response of irregular-continuousbridges In this part RSP is applied to three irregular-continuous bridges in Figure 1 to verify the effectiveness ofthe simplified prediction method proposed in Section 4

51 Characteristics of RSP Based on the concept of theequivalent system the displacement vector Δ can bedecomposed into the product of Δeq and ci If RSP correctlypredicts the seismic displacement response of irregular-continuous bridges it must have the followingcharacteristics

(1) Sd fromRSPmust be almost consistent withΔeq fromITHA

(2) Displacement shape from RSP must reflect thechanges of ci from ITHA

Taking irregular-continuous bridges in Figure 1 as anexample the seismic displacement is solved by RSP andITHA respectively ey are compared with each other toverify RSPrsquos validity e detailed processes are as follows

(1) FEM of each bridge is built in OpenSees program inwhich elastic beam element fiber element andnonlinear link element are used to simulate thegirder the piers and the bearings e Chinese re-sponse spectrum of soil type III in Figure 2(a) andthe corresponding accelerograms in Figure 2(b) arechosen as the earthquake input PGA is divided into59 levels from 002 g to 06 g by intervals of 001 g

(2) e seismic displacement for each seismic level iscalculated by ITHA and the corresponding displace-mentΔeq of the equivalent SDOF system is obtained byequation (8)

(3) Sd of the equivalent SDOF system is calculated byRSP for each seismic level and the correspondingpushover displacement vector u is adopted as theseismic displacement of the bridge

(4) Sd from RSP and Δeq from ITHA are compared asshown in Figure 5

(5) Seismic displacements from RSP and ITHA for thesame Sd or Δeq are compared as shown in Figure 6


According to Figure 5 some conclusions are obtained asfollows

(1) As a whole Sd calculated by RSP is close to Δeq byITHA

(2) e difference between Sd and Δeq becomes moreand more obvious as PGA increases and Sd cal-culated by RSP is larger

Based on Figure 6 some conclusions are obtained as follows

(1) In general as for the same displacement of theequivalent SDOF system seismic displacement fromRSP is close to the one from ITHA is indirectlyshows that the displacement shape from RSP canreflect the changes of ci from ITHA based onequation (9)

(2) e difference between seismic displacement fromRSP and that from ITHA becomes more obvious as awhole as PGA increases

Results from Figures 5 and 6 show that the simplifiedprediction method proposed in Section 4 can be used to

predict seismic displacement for the irregular-continuousbridges of the case study

As to evaluate the prediction errors of the simplifiedprediction method in detail the Chinese response spectrumof soil type III in Figure 2(a) and the correspondingaccelerograms in Figure 2(b) are chosen as the earthquakeinput for the irregular bridges in Figure 1 and PGA adopts01 g 02 g 04 g 08 g and 16 g respectively e corre-sponding results are shown in the following sections

52 Case 1 051005 Bridge As for the 051005 bridge takingPGA of a 02 g for example the procedure for seismic dis-placement prediction is described in detail shown in Figure 7

In Figure 7 the girder end point 0 is chosen as the dis-placement reference point e capacity spectrum is obtainedby pushing the bridge under the response spectrum loaddistribution in which the spectrum value of yield-point dis-placement is Sdy 0021m e values of Sd for the threeintersection points of the capacity spectrum curve and threedemand spectrum curves with μ 10 15 and 20 are 0055m0032m and 0029m respectivelye corresponding values of

T

Su Sd

S ay

S ay

S a

S aT

micro = 2 micro = 2

micro = 1

micro = 1

micro = 2

micro = 1

micro = 2

micro = 1

S uT

Figure 4 Generation procedure of the inelastic demand spectrum


μprime SdSdy are 262 153 and 138 respectively Note thatμ asymp μprime is only satisfied when μ 15 hence Sd of theequivalent SDOF system is 0032m According to the result ofpushover analysis the values of seismic displacement of thegirder points of 0 1 23 and 4 corresponding to Sd

0032m are 0045m 0018m 0024m 0018m and 0045mrespectively e corresponding values of ITHA are 0042m0016m 0021m 0016m and 0042m respectivelye resultsshow that the seismic displacement of the simplified predictionmethod is close to that of ITHA

e comparison of seismic displacement calculated bythe simplified prediction method using RSP and that byITHA under five PGA levels of a 01 g 02 g 04 g 08 gand 16 g is shown in Figure 8(a) For each PGA level theseismic displacement calculated by the simplified predictionmethod using RSP is close to that of ITHA Even for the PGAlevel of a 16 g the maximum relative error of the seismicdisplacement of the simplified prediction method using RSPis only 16 when compared to that of ITHAis can satisfythe engineering application It is meaningless for the PGAlevel of a 16 g since most bridges will not suffer such astrong earthquake Such a case is only used to identify theaccuracy of the simplified prediction method using RSP

53 Case 2 100510 Bridge As for the 100510 bridge thecomparison of seismic displacement calculated by the

simplified prediction method using RSP and that by ITHA isshown in Figure 8(b) In terms of the PGA level of a 01 g02 g 04 g and 08 g the ratio of seismic displacement of thesimplified prediction method using RSP to that of ITHAranges from 85 to 118 which can meet the requirementof the engineering application At a PGA level of a 16 g theratio of seismic displacement of the simplified predictionmethod using RSP to that of ITHA ranges from 75 to 130which shows that as PGA increases the relative error of theseismic displacement of the simplified prediction methodusing RSP increases when compared to that of ITHA

54 Case 3 050505 Bridge As for the 050505 bridge thecomparison of seismic displacement calculated by thesimplified prediction method using RSP and that by ITHA isshown in Figure 8(c) In terms of the PGA level of a 01 g02 g 04 g and 08 g the ratio of seismic displacement of thesimplified prediction method using RSP to that of ITHAranges from 85 to 119 which can meet the requirementof engineering application At a PGA level of a 16 g theratio of seismic displacement of the simplified predictionmethod using RSP to that of ITHA ranges from 98 to130 which shows that as PGA increases the relative errorof the seismic displacement of the simplified predictionmethod using RSP increases when compared to that ofITHA

ITHARSP

Δ eq (

m)

01 02 03 04 05 060PGA (g)

0002004006008

01012

(a)

ITHARSP

Δ eq (

m)

01 02 03 04 05 060PGA(g)

0

005

01

015

02

(b)

ITHARSP

Δ eq (

m)

01 02 03 04 05 060PGA(g)

0

002

004

006

008

01

(c)

Figure 5 Comparison of the equivalent SDOF systemrsquos displacement by ITHA and RSP (a) 051005 bridge (b) 100510 bridge and (c)050505 bridge


6 Parametric Analysis of CalculationAccuracy of Simplified Prediction Procedure

e results from the foregoing three cases show that thesimplified prediction method using RSP is a good predictor

of the seismic displacement of irregular-continuous bridgesHowever just like other simplified methods it still is asemitheoretical and semiempirical method Some assump-tions are adopted in the theoretical analysis therefore it isnot enough to verify the efficiency of the simplified pre-diction method using RSP based on only three cases Car-rying out more parametric analyses is necessary to ensurethe validity of the simplified prediction method using RSPbefore applying its theories to simplified displacement-basedseismic design of irregular-continuous bridges

61 Bridge Structure and Seismic Input ree cases ofcontinuous bridges are identified as the reference of analysiswhose geometry shapes and section properties of girders andpiers are shown in Figure 1 and Table 1 respectively Basedon the three cases some parameters are changed to producemore combinations as shown in Table 2 e combinationrule changes one parameter by keeping the other parametersthe same As the three cases are the simplified model of thetrue bridges the new models of Table 2 obtained bychanging only one parameter are reasonable to includemany practical bridges ey can be used for numericalsimulation

0

002

004

006

008

01

012

Seism

ic d

ispla

cem

ent (

m)

002 004 006 008 010∆eq (m)

ITHA girder point 0ITHA girder point 1ITHA girder point 2

RSP girder point 0RSP girder point 1RSP girder point 2

(a)

002 004 006 008 010∆eq (m)

0

002

004

006

008

01

012

014

Seism

ic d

ispla

cem

ent (

m)



(b)

0 002 004 006 008∆eq (m)

0

002

004

006

008

01

012

Seism

ic d

ispla

cem

ent (

m)



(c)

Figure 6 Comparison of seismic displacement by ITHA and RSP (a) 051005 bridge (b) 100510 bridge and (c) 050505 bridge

0

1

2

3

4

5

0 01 02 03 04 05

S a

Sd

00550032

0029

Capacity spectrum

Demand spectrum

μ = 10

μ = 15μ = 20

Figure 7 Analysis process of the simplified prediction methodcorresponding to PGA 02 g


Based on Table 2 and to satisfy the study requirements ofthis paper a majority of cases are obtained with 69 sym-metrical bridges selected as the study object of the para-metric analysis

When earthquake load is concerned the simplifiedprediction method using RSP and ITHA adopt the inelasticdemand spectrum and seven accelerograms respectivelywhich are all corresponding to the elastic response spectrumas shown in Figure 2(a) and PGA adopts 01 g 02 g 04 gand 08 g respectively

62 Numerical Results As for each bridge model thesimplified prediction method using RSP and ITHA are usedto calculate its seismic displacement respectively e ratiosof the displacement values of the girder points 0 1 2 3 and4 in Figure 1 calculated from RSP to that of ITHA are shownin Figure 9

According to Figure 9 when compared to the results ofITHA the simplified prediction method using RSP can

obtain the reasonable and conservative seismic displace-ment e average values of these ratios are 103 105 109and 115 when PGA 01 g 02 g 04 g and 08 g respec-tively e relative error of the simplified prediction methodusing RSP increases as PGA increases

7 Procedure of Simplified Displacement-BasedSeismic Design

e displacement is the soul in the whole procedure of thedisplacement-based seismic design method to keep thebalance between target displacement and seismic displace-ment demand is can effectively control the structurersquosseismic damage is procedure has been achieved by usingan ITHA method but consumes too long computing time[34] e simplified prediction method using RSP simplifiesthe calculation of seismic displacement demand of bridgesand saves the computing time is section will discuss howto apply the simplified prediction method using RSP to the

Girder point

04

03

02

Seism

ic d

ispla

cem

ent (

m)

01

00 1 2 3 4

α = 16g

α = 08g

α = 04gα = 02gα = 01g

ITHARSP

(a)

Girder point

04

05

06

07

03

02

01

00 1 2 3 4

α = 16g

α = 08g

α = 04g

α = 02gα = 01g

ITHARSP

Seism

ic d

ispla

cem

ent (

m)

(b)

04

05

06

03

02

01

00 1 2 3 4

α = 16g

α = 08g

α = 04gα = 02gα = 01g

Girder point

ITHARSP

Seism

ic d

ispla

cem

ent (

m)

(c)

Figure 8 Comparison of seismic displacement by the ITHA and simplified prediction method using RSP (a) 051005 bridge (b) 100510bridge and (c) 050505 bridge


displacement-based seismic design of the irregular-contin-uous bridges especially for equilibrium iteration of targetdisplacement and seismic displacement demand (Figure 10)

71 Target Displacement Irregular-continuous bridges canbe designed according to two design levels of E1 and E2

(1) As for the design level of small earthquake E1 mainparts of the structure only require little damage iethe maximum section curvature φE1 of main ductilemembers should be less than the corresponding yieldcurvature φy e force-based seismic design canthen be applied but this is not the topic of this paper

(2) In terms of the design level of large earthquake E2the structure can have severe damage without col-lapsing or causing other fatal damage ie themaximum section curvature φE2 of main ductilemembers should be larger than the correspondingyield curvature φy and not exceed the permitted limitcurvature φu e displacement-based seismic designcan then be used and this is the topic of this paper

Under the design level of large earthquake E2 the dis-placement-based seismic design using a nonlinear static methodwill be proposed on the irregular-continuous bridges in thissection and the following sections First how to obtain the targetdisplacement of the irregular-continuous bridges is listed asfollows

(1) FEM of the bridge is built with experience-guidedpier size and reinforcement arrangement which isalso achieved by the force-based seismic designunder the design level of small earthquake E1 It isseen as the preliminary scheme of the design level oflarge earthquake E2 which will be continuouslyoptimized in the following process e FEM is usedto obtain the response spectrum load distributionand carry out the following pushover analysis

(2) e structure is pushed by the response spectrumload distribution and the curvature of the mostdangerous section of the first yielding pier is mon-itored e general displacement ur yielding dis-placement Δy and ultimate displacement Δu of thewhole bridge system represented by the girder point0 in Figure 1 are obtained when the monitoredcurvature reaches φy and φu respectively

(3) e corresponding general displacement Sd yielddisplacement Sdy and limit displacement Sdu of thecapacity spectrum are calculated according toSd urci Sdy Δyci and Sdu Δuci respectivelyci refers to the coefficient of displacement shapeusing the girder point 0 in Figure 1 and equation (9)for the elastic state of the bridge

72 Check of Preliminary Scheme e capacity coefficient μc

is calculated by μc SduSdy e coefficient μdE 2 corre-sponding to the inelastic demand spectrum of E2 designlevel is set to be μdE 2 μc When the capacity spectrum andthe inelastic demand spectrum are drawn in the same figureas shown in Figure 11 the actual seismic displacement of E2design level situates at Sd2 sim Sdu On this basis there are twopossibilities

(1) If Sdu asymp Sd2 the preliminary scheme will be satis-factory for E2 design level

(2) Under other conditions a new scheme should be chosen

73NewScheme e bridge pier should be redesigned if theformer scheme is not satisfactory ie the case (2) in Section72 Specify San San2 + (Sau minus Sa2) in which all the piersyield when Sau is arrived since the structure is pushed untilSa does not dramatically increase In fact all the piers willnot yield at the same time under a special ground motion if

Table 2 Changing parameters of girder and piers

Membertype Variables Parameter values

Girder

Lateral moment of inertia (m4) 20 40 80 and 160Polar moment of inertia (m4) 7 14 28 and 56

Section area (m2) 35 7 14 and 28Single span length (m) 20 40 80 and 160

Pier

Section area (m2) 10mtimes 10m 15mtimes 15m 20mtimes 20m and 25mtimes 25mArea ratio of longitudinal

reinforcement 04 08 12 and 16

Height distribution of piers Pier2 varies as 5m 10m and 15m while pier1 equals to pier 3 and varies as 5m 10m15m and 20m synchronously

06

08

1

12

14

16

RSP

ITH

A

02 04 06 080PGA (g)

Figure 9 Ratio of seismic displacement of the simplified predictionmethod using RSP to that of ITHA


the piers have different length However when the groundmotion continuously increases the different piers willgradually enter the yield state such as the capacity spectrumin Figure 11 Finally all the piers yield if the ground motionis large enough and this state corresponds to Sau on thecapacity spectrum in Figure 11 e state that all the piersyield can help to distribute the resultant force to each pier inthe following process

erefore the total inertial force of the new scheme afterall the piers yield is Fg mgSan Fg includes all of the shearforce at piers and abutments e sum of shear force at eachpier can be denoted by Fp Fg minus Fa and Fp is expressed byequation (13) where Fa refers to the sum of shear force atabutments and Fpn is the shear force of the n pier

In many cases bridge piers are often designed with thesame cross section and the same reinforcement ratio Aprinciple of the same yield bending moment of each pier canbe followed to distribute Fp and calculate the yield bendingmoment My of each pier as shown in the former expressionof equation (14) where hn is the length of the n pier If thebridge piers are designed with different cross sections ordifferent reinforcement ratios other special but simple re-lations can be written as shown in the latter expression ofequation (14) e yield bending moment My calculated byequation (14) can be used to design the new cross section andreinforcement of piers

Fp1 + Fp2 + Fp3 + middot middot middot + Fpn Fp (13)

My Fp1h1 Fp2h2 Fp3h3 middot middot middot Fpnhn (14)

or other special relations

74 Final Scheme e sections above are repeated escheme that satisfies the requirement of Sdu asymp Sd2 is the finalscheme because the limit displacement Sdu of the capacityspectrum line and the inelastic demand spectrum line hasthe same ductility coefficient and the two lines just intersectat the point of Sdu After the piers are designed based onequations (13) and (14) other detailed designs of the stirrup

of piers the foundation and the bearing can then be exe-cuted under the principle of capacity protection which is notthe topic of this paper

8 Verification Case of the SimplifiedDisplacement-Based SeismicDesign Procedure

As to better describe the procedure of the foregoing dis-placement-based seismic design a relatively simple irregu-lar-continuous bridge is selected to carry out thedisplacement-based seismic design It is then furtherchecked by ITHA

81 Introduction of Case e known conditions are asfollows

(1) e first bridge with a total mass 2912t of the su-perstructure in Figure 1 is selected as the design case

(2) Earthquake load adopts the response spectrum forsoil profile III in Chinese criteria (JTJ 004-89) asshown in Figure 2(a) and PGA of E2 design leveladopts 04 g

Determination of earthquake levels E1 and E2

Conceptual and force-based design (not the topic of this paper) under E1 earthquake determine structural system and design pier

Elastic response spectrum analysis determine system displacement shapeDetermine the new pier size and reinforcement

Determination of new yield moment of pier

Determine new schemeNo Check draw the capacity spectrum and demand spectrum in the same diagram to judge

whether the target displacement meets the reqrirements of E2 level displacementYes

Detail design

The elastic shape is used as the lateral force mode for pushover analysis determinetarget displacement capacity spectrum and demand spectrum under E2 earthquake

Figure 10 Displacement-based seismic design process using a nonlinear static method

S aS a

n2S a

2S a

u

Py

Pu

Sdy Sd2

Sd

Sdu

Demand spectrum

Capacity spectrum

Figure 11 Check of design scheme


Note that the pier cross section and the reinforcementare unknown and need further design based on the dis-placement-based seismic design procedure

82 Design Procedure e preliminary pier scheme can beobtained by the conceptual design the experience-guideddesign or the force-based seismic design under the designlevel of small earthquake E1 however this is not the topic ofthis paper In this section the cross section of the prelim-inary pier is assumed and given by 12mtimes 12m with alongitudinal reinforcement ratio of 12

FEM of the above bridge is the preliminary scheme builtin OpenSees program According to the material straincapacity the curvature information of the pier section isφy 000273 radm and φu 00394radm and the latter ofwhich corresponds to a collapse prevention state but has asafety factor of 20 according to Chinese criteria When thewhole bridge structure is pushed by the response spectrumload distribution the curvature of the most dangeroussection of the first yield 5m pier and the displacement of thegirder point 0 in Figure 1 are monitored e displacementof Δy and Δu of the whole bridge system represented by the

displacement of the girder point 0 in Figure 1 is obtainedwhen the monitored curvature reaches φy and φu respec-tively e corresponding displacement information of thecapacity spectrum is Sdy Δyci 00213mSdu Δuci 01077m and μc SduSdy 506 e ca-pacity spectrum represents the global measures of ductilitybecause it is obtained by pushing the whole bridge structureIt also represents the local measures of ductility because itmonitors the most strained 5m pier and puts the corre-sponding indexes Sdy and Sdu in Figure 12

e demand spectrum of the E2 design level is builtbased on the assumption of μdE 2 μc and it corresponds toa collapse prevention state of the global measures of ductilitycontrolled by the 5m pier e combination of the capacityspectrum and demand spectrum is shown in Figure 12(a)Because Sdu gt Sd2 in Figure 12(a) being as well as that inFigure 11 the preliminary scheme is so safe that it needs todecrease the pier cross section or the longitudinal rein-forcement ratio

From Figure 12(a) San2 + (Sau minus Sa2)

08759 + (18671 minus 11560) 15869ms2 ieSan 15869ms2 for the new scheme and the corre-sponding total inertia force of the new scheme is

Py

Pu

Demand spectrum

Capacity spectrum

0

1

2

3S a

01 02 03 04 050Sd

(a)

0

1

2

3

S a

Py

Pu

Demand spectrum

Capacity spectrum

01 02 03 04 050Sd

(b)

Figure 12 Check of design scheme (a) preliminary scheme and (b) new scheme

Girder point0

016

012

008

0041 2 3 4

Seism

ic d

ispla

cem

ent (

m)

ITHARSP

(a)

Sect

ion

curv

atur

e (ra

dm

)

Curvature from ITHALimit curvature

Girder point0 1 2 3 4

008007006005004003002001

0

(b)

Figure 13 Check of design result (a) seismic displacement calculated by RSP and ITHA and (b) curvature of the pier base section


Fg 2912 times 15869 4621 kN Note that piers almostsupport the total inertia force since the abutment bearing isbidirectional sliding only taking a small amount of inertiaforce According to the equal yield moment principle theshear force Fpn of three piers is 1852183 kN 916635 kN and1852183 kN respectively based on equations (13) and (14)and the yield moment My of each pier is 5171033 kNmiddotmerefore in the new scheme in Figure 12(b) the crosssection of the pier remains unchanged and the longitudinalreinforcement ratio decreases to 0866 based on the pieryield moment of My 5171033 kN middot m

e combination of the capacity spectrum and the de-mand spectrum of the new scheme is shown in Figure 12(b)e result shows Sdu asymp Sd2 which implies that the capacityspectrum line and the demand spectrum line just intersect atthe point of Sdu and satisfies the requirement of seismicdesign Consequently the scheme can be chosen as the finalone

83 Check of Design Result To check the validity of thedesign result the final scheme is calculated by ITHA eaccelerograms in Section 2 are chosen as the seismic inputand PGA adopts 04 g e seismic displacement calculatedby the simplified prediction method using RSP and ITHA isshown in Figure 13(a) e check of target curvature of thepier base section is shown in Figure 13(b)

Figure 13(a) shows that the seismic displacement fromthe simplified prediction method using RSP is close to thatfrom ITHA

Figure 13(b) shows that the base section curvatures of thetwo short piers reach the limit value and the base sectioncurvature of the long pier is much less than the limit valueerefore the seismic design of the final scheme is con-trolled by the short pierrsquos deformation capacity of E2 designlevel

e check results show that the seismic design result isproper and correct

9 Conclusion

By taking one common type of irregular-continuous bridgeswith quasi-regular geometry the building procedures of thecapacity spectrum and the demand spectrum are discussedAs a result the simplified displacement-based seismic designprocedure is advanced us conclusions include thefollowing

(1) e pushover curve resulted from a pushoveranalysis can be selected as the capacity spectrum ofone common type of irregular-continuous bridgesIn the pushover analysis the girder end point 0 isselected as the displacement reference point and thedisplacement shape from the response spectrumanalysis is used to determine the load distribution

(2) By combining the capacity spectrum and the in-elastic demand spectrum the seismic displacementdemand can be properly predicted for one commontype of irregular-continuous bridges

(3) After several iterations of the combination of thecapacity spectrum and the inelastic demand spec-trum the simplified displacement-based seismicdesign of one common type of irregular-continuousbridges can be achieved

It is noted that the above proposed nonlinear staticprocedure is only applicable for the common irregular-continuous bridges with similar characteristics of those usedin the case study and those used for the parametric analysisose bridges have many regular factors and only few ir-regular factors leading to the obvious influence of highmodes And the higher mode effects are mild for the four-span bridges which improves the accuracy of the conven-tional force-based single-load pattern pushover analysis Itneeds further investigation whether the above proposednonlinear static procedure extends beyond to what waspresented for the designed bridge in this paper [35 36]Furthermore the above proposed nonlinear static procedureis a little complex such as using a FEM model to helpanalysis It needs investigation about how to further simplythe proposed nonlinear static procedure in the future

Data Availability

e data used to support the findings of this study are in-cluded within the article e data include the structuralparameters ground motion inputs calculation methodsand calculation results

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is paper was supported by the National Natural ScienceFoundation of China under grant nos 51778635 and51778630 the Natural Science Foundation of HunanProvince under grant no 2019JJ40386 and the Innovation-Driven Plan in Central South University under grant no20200017050004 ese financial supports are gratefullyacknowledged

References

[1] AASHTO AASHTO LRFD Bridge Design SpecificationsAASHTO Washington DC USA 4th edition 2007

[2] H R Ahmadi N Namdari M S Cao and M Bayat ldquoSeismicinvestigation of pushover methods for concrete piers ofcurved bridges in planrdquo Computers and Concrete vol 23no 1 pp 1ndash10 2019

[3] T S Paraskeva A J Kappos and A G Sextos ldquoExtension ofmodal pushover analysis to seismic assessment of bridgesrdquoEarthquake Engineering amp Structural Dynamics vol 35no 10 pp 1269ndash1293 2006

[4] M Jafari and M Soltani ldquoA stochastic adaptive pushoverprocedure for seismic assessment of buildingsrdquo Earthquakesand Structures vol 14 no 5 pp 477ndash492 2018

[5] J Luo L A Fahnestock and J M LaFave ldquoNonlinear staticpushover and eigenvalue modal analyses of quasi-isolated


highway bridges with seat-type abutmentsrdquo Structuresvol 12 pp 145ndash167 2017

[6] A M Reinhorn ldquoInelastic analysis techniques in seismicevaluationsrdquo in Proceedings of the International Workshop onSeismic Design Methodologies for the Next Generation ofCodes pp 277ndash287 Bled Slovenia 1997

[7] A R Ghotbi ldquoModal pushover analysis of skewed bridges-case study of jack tone road on-ramp overcrossingrdquo KSCEJournal of Civil Engineering vol 20 no 5 pp 1948ndash19572016

[8] A K Chopra and R K Goel ldquoEvaluation of modal and FEMApushover analyses SAC Buildingsrdquo Earthquake Spectravol 20 pp 225ndash254 2004

[9] K Shakeri K Tarbali and M Mohebbi ldquoAn adaptive modalpushover procedure for asymmetric-plan buildingsrdquo Engi-neering Structures vol 36 pp 160ndash172 2012

[10] T S Paraskeva and A J Kappos ldquoFurther development of amultimodal pushover analysis procedure for seismic assess-ment of bridgesrdquo Earthquake Engineering and StructureDynamics vol 39 pp 211ndash222 2010

[11] M N Aydinoglu ldquoAn improved pushover procedure forengineering practice incremental response spectrum analysisIRSArdquo in Proceedings of the International Workshop Perfor-mance-Based Seismic Design Concepts and Implementationno 5 pp 345ndash356 Bled Slovenia 2004

[12] T Isakovic and M Fischinger ldquoHigher modes in simplifiedinelastic seismic analysis of single column bent viaductsrdquoEarthquake Engineering and Structure Dynamics vol 35pp 95ndash114 2006

[13] B Wei ldquoStudy of the applicability of modal pushover analysison irregular continuous bridgesrdquo Structural Engineering In-ternational vol 21 no 2 pp 233ndash237 2011

[14] M R Falamarz-Sheikhabadi and A Zerva ldquoEffect of nu-merical soil-foundation-structure modeling on the seismicresponse of a tall bridge pier via pushover analysisrdquo SoilDynamics and Earthquake Engineering vol 90 pp 52ndash732016

[15] M J Kowalsky M J N Priestley and G A Macrae ldquoDis-placement-based design of RC bridge columns in seismicregionsrdquo Earthquake Engineering amp Structural Dynamicsvol 24 no 12 pp 1623ndash1643 1995

[16] M J Kowalsky ldquoDirect displacement-based design a seismicdesign methodology and its application to concrete bridgesrdquoDoctoral dissertation University of California at San DiegoSan Diego CA USA 1997

[17] M J Kowalsky ldquoA displacement-based approach for theseismic design of continuous concrete bridgesrdquo EarthquakeEngineering amp Structural Dynamics vol 31 no 3 pp 719ndash747 2002

[18] M J N Priestley G M Calvi and M J Kowalsky Dis-placement-Based Seismic Design of Structures IUSS PressVienna Austria 2007

[19] A J Kappos I G Gidaris and K I Gkatzogias ldquoProblemsassociated with direct displacement-based design of concretebridges with single-column piers and some suggested im-provementsrdquo Bulletin of Earthquake Engineering vol 10no 4 pp 1237ndash1266 2012

[20] A J Kappos K I Gkatzogias and I G Gidaris ldquoExtension ofdirect displacement-based design methodology for bridges toaccount for higher mode effectsrdquo Earthquake Engineering ampStructural Dynamics vol 42 no 4 pp 581ndash602 2013

[21] Y Fahjan and Z Ozdemir ldquoScaling of earthquake accelero-grams for non-linear dynamic analysis to match the

earthquake design spectrardquo in Proceedings of the 14th WorldConference on Earthquake Engineering Beijing China 2008

[22] H Dwairi and M Kowalsky ldquoInelastic displacement patternsin support of displacement-based design for multi-spanbridgesrdquo in Proceedings of the 13th World Conference onEarthquake Engineering Vancouver Canada 2004

[23] C Perdomo R Monteiro and H Sucuoglu ldquoGeneralizedforce vectors for multi-mode pushover analysis of bridgesrdquoBulletin of Earthquake Engineering vol 15 no 12pp 5247ndash5280 2017

[24] S Mazzoni F McKenna and M H Scott OpenSees Com-mand Language Manual Pacific Earthquake EngineeringResearch University of California Oakland CA USA 2007

[25] B Wei Z L Hu X H He and L Z Jiang ldquoEvaluation ofoptimal ground motion intensity measures and seismic vul-nerability analysis of multi-pylon cable-stayed bridge withsuper-high piers in mountainous areasrdquo Soil Dynamics andEarthquake Engineering vol 129 2020

[26] B Wei C Li and X He ldquoe applicability of differentearthquake intensity measures to the seismic vulnerability of ahigh-speed railway continuous bridgerdquo International Journalof Civil Engineering vol 17 no 7 pp 981ndash997 2019

[27] P Fajfar ldquoA nonlinear analysis method for performance-based seismic designrdquo Earthquake Spectra vol 16 no 3pp 573ndash592 2000

[28] M Kohrangi R Bento and M Lopes ldquoSeismic performanceof irregular bridges-comparison of different nonlinear staticproceduresrdquo Structure and Infrastructure Engineering vol 11no 12 pp 1632ndash1650 2015

[29] R Pinho R Monteiro C Casarotti and R Delgado ldquoAs-sessment of continuous span bridges through nonlinear staticproceduresrdquo Earthquake Spectra vol 25 no 1 pp 143ndash1592009

[30] N M Newmark and W J Hall ldquoSeismic design criteria fornuclear reactor facilitiesrdquo in Proceedings of the 4th WorldConference on Earthquake Engineering Santiago Chile 1969

[31] G H Cui C G Liu X X Tao and X M Chen ldquoSeismicsafety evaluation of bridge structures based on inelasticspectrum methodrdquo in Proceedings of the 14th World Con-ference on Earthquake Engineering Beijing China 2008

[32] E Miranda ldquoInelastic displacement ratios for structures onfirm sitesrdquo Journal of Structural Engineering vol 126 no 10pp 1150ndash1159 2000

[33] M Ozgenoglu and Y Arıcı ldquoComparison of ASCESEIStandard and modal pushover-based ground motion scalingprocedures for pre-tensioned concrete bridgesrdquo Structure andInfrastructure Engineering vol 13 no 12 pp 1609ndash16232017

[34] A J Kappos and A Manafpour ldquoSeismic design of RCbuildings with the aid of advanced analytical techniquesrdquoEngineering Structures vol 23 no 4 pp 319ndash332 2001

[35] Y Xia L M Chen H Y Ma and D Su ldquoExperimental andnumerical study on shear studs connecting steel girder andprecast concrete deckrdquo Structural Engineering and Mechanicsvol 71 no 4 pp 433ndash444 2019

[36] S Wu ldquoUnseating mechanism of a skew bridge with seat-typeabutments and a simplified method for estimating its supportlength requirementrdquo Engineering Structures vol 191pp 194ndash205 2019


established for the simplified prediction of elastoplasticdisplacement demand in irregular-continuous bridges Forinstance Kowalsky proposed a displacement-based seismicdesign method grounded in the ideas of equivalent systemand equivalent damping ratio etc [15ndash18] is proposedmethod involves the advantages of easy operation KapposGidaris and Gkatzogias further improved some aspects ofthe proposed method [19 20] such as how to combine thedamping ratios of structural components to form thedamping ratio of bridge systems e ideas introduced inthese methods are worth studying in the research of a newnonlinear static procedure analysis methods

By taking one common type of irregular-continuousbridges in transverse direction as the object of study thispaper proposes a simplified prediction method of seismicdisplacement demand Based on their seismic responsecharacteristics (the equivalent system concept and the basicidea of pushover analysis) this paper also proposes thecorresponding displacement-based seismic designprocedure

2 A Common Type of Irregular-Continuous Bridges

Regular-continuous bridges are generally defined as bridgesthat can be simplified as single-degree-of-freedom (SDOF)systems or as those with dynamic responses controlled byonly a decisive fundamental mode To carry out the sim-plified seismic design safely the AASHTO [1] and ChinarsquosGuidelines for Seismic Design of Highway Bridges (2008)define regular-continuous bridges according to structuralcharacteristics However transverse dynamic responses ofmany actual continuous bridges are controlled by two ormore modes When this occurs they are referred to as ir-regular-continuous bridges Figure 1 shows one kind of themost popular irregular-continuous bridges containing manyregular aspects eg nearly symmetric distribution of pierlength nearly equivalent distribution of span length andmany other aspects which satisfy the structural requirementsof regular-continuous bridges However the transversedynamic response of these bridges is still controlled by twoor more modes is is due to the influence of the followingirregular factors (1) different pier lengths and a compara-tively small stiffness ratio of girder to pier (2) in order toprevent excessive internal forces and deformations atabutments under earthquakes bidirectional sliding bearingsare set on all abutments as shown in Figure 1 Meanwhile ifthe stiffness ratio of girder to pier is comparatively smallerthe mass of superstructure endured by each pier will bedifferent under transverse seismic actions even if the pierheights are nearly the same In Figure 1 the modal shapes forthe first and second modes are nearly identical in shape forthe three different bridges e reason is that the bidirec-tional sliding bearings are set on all abutments and thestiffness ratio of girder to pier is small as described abovewhich control the first and second modes of the three dif-ferent bridges However there are some differences in thebending degree of the corresponding vibration modes of thethree bridges since these bridges have different pier

distributions It is noted that shear keys are considered anddesigned in the transverse direction of abutments to meetthe requirements under normal loading and their failure isonly permitted under severe earthquakes [13]

e irregular-continuous bridges with relatively regulargeometry are the study object of this paper ree 4times 40mtypical common irregular-continuous bridges in Figure 1have been selected for analysis e friction coefficient ofsliding bearings is equal to 002 e section properties ofgirders and columns are shown in Table 1 Earthquake loadadopts the elastic response spectrum for soil profile III inChinese criteria (JTJ 004-89) as shown in Figure 2(a)According to requirements of this paper it is transformed inthe following two ways (i) it is converted into the inelasticdemand spectrum which will be used for the simplifiedprediction method of seismic displacement demand edetailed procedure will be discussed in the following sec-tions (ii) To transform the ground motion input of ITHAseven accelerograms are generated by the Simqke procedure[21] Results of ITHA are regarded as the benchmark forcomparison using the simplified prediction methodFigure 2(b) gives the first accelerogram while other figuresare omitted due to similarity

3 Characteristics of Seismic Displacement

e seismic displacement of girder is nearly symmetric forthe common irregular-continuous bridges and when in-duced by the gradual increase of peak ground accelerations(PGAs) its shape changes as the pierrsquos yielding degree in-creases [22] Furthermore the shape will be relatively un-changed after the pierrsquos yielding degree arrives at a givenvalue In this section a concept of the equivalent system isused to decompose the girderrsquos seismic displacement intothe displacement of the equivalent SDOF system e co-efficient of displacement shape is applied to the foregoingthree irregular-continuous bridges to study their charac-teristics of seismic displacement

31 Concept of Equivalent SDOF System To operate thepushover analysis and to carry out displacement-basedseismic design procedure it is necessary to first analyze howto transform a multi-degree-of-freedom (MDOF) system ofa continuous bridge into a SDOF system ie how to de-compose the seismic displacement of the bridge into thedisplacement equivalent SDOF system and the coefficient ofdisplacement shape

In the finite element model (FEM) mi Δi Fi and ai aredefined to be the ith structure nodersquos mass displacementinertia force [23] and acceleration respectively e cor-responding values of the equivalent SDOF system aredenoted by meq Δeq Feq and aeq respectively e rela-tionship between Δi and Δeq and ai and aeq is supposed as

Δi ciΔeq (1)

ai ciaeq (2)

where ci is the coefficient of displacement shape



Feq 1113944n

i1Fi 1113944

n

i1miai aeq 1113944

n

i1mici (3)


meq 1113944n

i1mici (4)



1113936ni1mici

Feq (5)


Fi miΔi

1113936ni1miΔi

Feq (6)



Components Area(m2)



Concretetype



5m


Mode 1 21167 0473727 248


0002004006008

01


0002004006008

01


0002004006008

01


Mode 1Mode 2

Mode 1Mode 2

Mode 1Mode 2



Mode 1 20755 0429488 494

Period (s)Meff ()

Period(s)Meff()

0 43215m10m5m

0 43215m 10m10m

051005 bridge

100510 bridge

0 43215m5m

050505 bridge


ϕ nϕ n

ϕ n



FeqΔeq 1113944n

i1FiΔi (7)


Δeq 1113944

n

i1miΔ2i

1113944n

i1miΔi

(8)


ci Δi

Δeq

1113944n

i1miΔi

1113944n

i1miΔ2i

Δi (9)


meq 1113944

n

i1miΔi1113872 11138732

1113944n

i1miΔ2i

(10)










01 045 1 150

1

2

225


Dyn

amic

mag

nific

atio

n fa

ctor

β


β = 225 times (045T)095

(a)

Acce

lera

tion

(ms

2 )

Time (s)

15

10

5

0

ndash5

ndash10

ndash150 10 20 30 40

(b)





























2500

2700

2900

3100


meq

(t)

01 02 03 04 05 060PGA (g)

(a)


∆ eq

(m)

0002004006008

01012014

01 02 03 04 05 060PGA (g)

(b)


01 02 03 04 05 060PGA (g)

06

08

1

12

14

16

18

ci

(c)


0

03

06

09

12

15

18

ci

01 02 03 04 05 060PGA (g)

(d)


01 02 03 04 05 060PGA (g)

0

04

08

12

16

2

c i

(e)






C 1 +1μ


minus 1

(11)



Say CSa

μ

Su CSd CSaT

2

4π2

(12)





























T

Su Sd

S ay

S ay

S a

S aT

micro = 2 micro = 2

micro = 1

micro = 1

micro = 2

micro = 1

micro = 2

micro = 1

S uT









ITHARSP

Δ eq (

m)

01 02 03 04 05 060PGA (g)

0002004006008

01012

(a)

ITHARSP

Δ eq (

m)

01 02 03 04 05 060PGA(g)

0

005

01

015

02

(b)

ITHARSP

Δ eq (

m)

01 02 03 04 05 060PGA(g)

0

002

004

006

008

01

(c)







0

002

004

006

008

01

012

Seism

ic d

ispla

cem

ent (

m)

002 004 006 008 010∆eq (m)



(a)

002 004 006 008 010∆eq (m)

0

002

004

006

008

01

012

014

Seism

ic d

ispla

cem

ent (

m)



(b)

0 002 004 006 008∆eq (m)

0

002

004

006

008

01

012

Seism

ic d

ispla

cem

ent (

m)



(c)


0

1

2

3

4

5

0 01 02 03 04 05

S a

Sd

00550032

0029

Capacity spectrum

Demand spectrum

μ = 10

μ = 15μ = 20










Girder point

04

03

02

Seism

ic d

ispla

cem

ent (

m)

01

00 1 2 3 4

α = 16g

α = 08g

α = 04gα = 02gα = 01g

ITHARSP

(a)

Girder point

04

05

06

07

03

02

01

00 1 2 3 4

α = 16g

α = 08g

α = 04g

α = 02gα = 01g

ITHARSP

Seism

ic d

ispla

cem

ent (

m)

(b)

04

05

06

03

02

01

00 1 2 3 4

α = 16g

α = 08g

α = 04gα = 02gα = 01g

Girder point

ITHARSP

Seism

ic d

ispla

cem

ent (

m)

(c)


















Girder



Pier




06

08

1

12

14

16

RSP

ITH

A

02 04 06 080PGA (g)






















Detail design



S aS a

n2S a

2S a

u

Py

Pu

Sdy Sd2

Sd

Sdu

Demand spectrum

Capacity spectrum










Py

Pu

Demand spectrum

Capacity spectrum

0

1

2

3S a

01 02 03 04 050Sd

(a)

0

1

2

3

S a

Py

Pu

Demand spectrum

Capacity spectrum

01 02 03 04 050Sd

(b)


Girder point0

016

012

008

0041 2 3 4

Seism

ic d

ispla

cem

ent (

m)

ITHARSP

(a)

Sect

ion

curv

atur

e (ra

dm

)



008007006005004003002001

0

(b)









9 Conclusion






Data Availability




Acknowledgments


References










































Feq 1113944n

i1Fi 1113944

n

i1miai aeq 1113944

n

i1mici (3)


meq 1113944n

i1mici (4)



1113936ni1mici

Feq (5)


Fi miΔi

1113936ni1miΔi

Feq (6)



Components Area(m2)



Concretetype



5m


Mode 1 21167 0473727 248


0002004006008

01


0002004006008

01


0002004006008

01


Mode 1Mode 2

Mode 1Mode 2

Mode 1Mode 2



Mode 1 20755 0429488 494

Period (s)Meff ()

Period(s)Meff()

0 43215m10m5m

0 43215m 10m10m

051005 bridge

100510 bridge

0 43215m5m

050505 bridge


ϕ nϕ n

ϕ n



FeqΔeq 1113944n

i1FiΔi (7)


Δeq 1113944

n

i1miΔ2i

1113944n

i1miΔi

(8)


ci Δi

Δeq

1113944n

i1miΔi

1113944n

i1miΔ2i

Δi (9)


meq 1113944

n

i1miΔi1113872 11138732

1113944n

i1miΔ2i

(10)










01 045 1 150

1

2

225


Dyn

amic

mag

nific

atio

n fa

ctor

β


β = 225 times (045T)095

(a)

Acce

lera

tion

(ms

2 )

Time (s)

15

10

5

0

ndash5

ndash10

ndash150 10 20 30 40

(b)





























2500

2700

2900

3100


meq

(t)

01 02 03 04 05 060PGA (g)

(a)


∆ eq

(m)

0002004006008

01012014

01 02 03 04 05 060PGA (g)

(b)


01 02 03 04 05 060PGA (g)

06

08

1

12

14

16

18

ci

(c)


0

03

06

09

12

15

18

ci

01 02 03 04 05 060PGA (g)

(d)


01 02 03 04 05 060PGA (g)

0

04

08

12

16

2

c i

(e)






C 1 +1μ


minus 1

(11)



Say CSa

μ

Su CSd CSaT

2

4π2

(12)





























T

Su Sd

S ay

S ay

S a

S aT

micro = 2 micro = 2

micro = 1

micro = 1

micro = 2

micro = 1

micro = 2

micro = 1

S uT









ITHARSP

Δ eq (

m)

01 02 03 04 05 060PGA (g)

0002004006008

01012

(a)

ITHARSP

Δ eq (

m)

01 02 03 04 05 060PGA(g)

0

005

01

015

02

(b)

ITHARSP

Δ eq (

m)

01 02 03 04 05 060PGA(g)

0

002

004

006

008

01

(c)







0

002

004

006

008

01

012

Seism

ic d

ispla

cem

ent (

m)

002 004 006 008 010∆eq (m)



(a)

002 004 006 008 010∆eq (m)

0

002

004

006

008

01

012

014

Seism

ic d

ispla

cem

ent (

m)



(b)

0 002 004 006 008∆eq (m)

0

002

004

006

008

01

012

Seism

ic d

ispla

cem

ent (

m)



(c)


0

1

2

3

4

5

0 01 02 03 04 05

S a

Sd

00550032

0029

Capacity spectrum

Demand spectrum

μ = 10

μ = 15μ = 20










Girder point

04

03

02

Seism

ic d

ispla

cem

ent (

m)

01

00 1 2 3 4

α = 16g

α = 08g

α = 04gα = 02gα = 01g

ITHARSP

(a)

Girder point

04

05

06

07

03

02

01

00 1 2 3 4

α = 16g

α = 08g

α = 04g

α = 02gα = 01g

ITHARSP

Seism

ic d

ispla

cem

ent (

m)

(b)

04

05

06

03

02

01

00 1 2 3 4

α = 16g

α = 08g

α = 04gα = 02gα = 01g

Girder point

ITHARSP

Seism

ic d

ispla

cem

ent (

m)

(c)


















Girder



Pier




06

08

1

12

14

16

RSP

ITH

A

02 04 06 080PGA (g)






















Detail design



S aS a

n2S a

2S a

u

Py

Pu

Sdy Sd2

Sd

Sdu

Demand spectrum

Capacity spectrum










Py

Pu

Demand spectrum

Capacity spectrum

0

1

2

3S a

01 02 03 04 050Sd

(a)

0

1

2

3

S a

Py

Pu

Demand spectrum

Capacity spectrum

01 02 03 04 050Sd

(b)


Girder point0

016

012

008

0041 2 3 4

Seism

ic d

ispla

cem

ent (

m)

ITHARSP

(a)

Sect

ion

curv

atur

e (ra

dm

)



008007006005004003002001

0

(b)









9 Conclusion






Data Availability




Acknowledgments


References









































FeqΔeq 1113944n

i1FiΔi (7)


Δeq 1113944

n

i1miΔ2i

1113944n

i1miΔi

(8)


ci Δi

Δeq

1113944n

i1miΔi

1113944n

i1miΔ2i

Δi (9)


meq 1113944

n

i1miΔi1113872 11138732

1113944n

i1miΔ2i

(10)










01 045 1 150

1

2

225


Dyn

amic

mag

nific

atio

n fa

ctor

β


β = 225 times (045T)095

(a)

Acce

lera

tion

(ms

2 )

Time (s)

15

10

5

0

ndash5

ndash10

ndash150 10 20 30 40

(b)





























2500

2700

2900

3100


meq

(t)

01 02 03 04 05 060PGA (g)

(a)


∆ eq

(m)

0002004006008

01012014

01 02 03 04 05 060PGA (g)

(b)


01 02 03 04 05 060PGA (g)

06

08

1

12

14

16

18

ci

(c)


0

03

06

09

12

15

18

ci

01 02 03 04 05 060PGA (g)

(d)


01 02 03 04 05 060PGA (g)

0

04

08

12

16

2

c i

(e)






C 1 +1μ


minus 1

(11)



Say CSa

μ

Su CSd CSaT

2

4π2

(12)





























T

Su Sd

S ay

S ay

S a

S aT

micro = 2 micro = 2

micro = 1

micro = 1

micro = 2

micro = 1

micro = 2

micro = 1

S uT









ITHARSP

Δ eq (

m)

01 02 03 04 05 060PGA (g)

0002004006008

01012

(a)

ITHARSP

Δ eq (

m)

01 02 03 04 05 060PGA(g)

0

005

01

015

02

(b)

ITHARSP

Δ eq (

m)

01 02 03 04 05 060PGA(g)

0

002

004

006

008

01

(c)







0

002

004

006

008

01

012

Seism

ic d

ispla

cem

ent (

m)

002 004 006 008 010∆eq (m)



(a)

002 004 006 008 010∆eq (m)

0

002

004

006

008

01

012

014

Seism

ic d

ispla

cem

ent (

m)



(b)

0 002 004 006 008∆eq (m)

0

002

004

006

008

01

012

Seism

ic d

ispla

cem

ent (

m)



(c)


0

1

2

3

4

5

0 01 02 03 04 05

S a

Sd

00550032

0029

Capacity spectrum

Demand spectrum

μ = 10

μ = 15μ = 20










Girder point

04

03

02

Seism

ic d

ispla

cem

ent (

m)

01

00 1 2 3 4

α = 16g

α = 08g

α = 04gα = 02gα = 01g

ITHARSP

(a)

Girder point

04

05

06

07

03

02

01

00 1 2 3 4

α = 16g

α = 08g

α = 04g

α = 02gα = 01g

ITHARSP

Seism

ic d

ispla

cem

ent (

m)

(b)

04

05

06

03

02

01

00 1 2 3 4

α = 16g

α = 08g

α = 04gα = 02gα = 01g

Girder point

ITHARSP

Seism

ic d

ispla

cem

ent (

m)

(c)


















Girder



Pier




06

08

1

12

14

16

RSP

ITH

A

02 04 06 080PGA (g)






















Detail design



S aS a

n2S a

2S a

u

Py

Pu

Sdy Sd2

Sd

Sdu

Demand spectrum

Capacity spectrum










Py

Pu

Demand spectrum

Capacity spectrum

0

1

2

3S a

01 02 03 04 050Sd

(a)

0

1

2

3

S a

Py

Pu

Demand spectrum

Capacity spectrum

01 02 03 04 050Sd

(b)


Girder point0

016

012

008

0041 2 3 4

Seism

ic d

ispla

cem

ent (

m)

ITHARSP

(a)

Sect

ion

curv

atur

e (ra

dm

)



008007006005004003002001

0

(b)









9 Conclusion






Data Availability




Acknowledgments


References



































































2500

2700

2900

3100


meq

(t)

01 02 03 04 05 060PGA (g)

(a)


∆ eq

(m)

0002004006008

01012014

01 02 03 04 05 060PGA (g)

(b)


01 02 03 04 05 060PGA (g)

06

08

1

12

14

16

18

ci

(c)


0

03

06

09

12

15

18

ci

01 02 03 04 05 060PGA (g)

(d)


01 02 03 04 05 060PGA (g)

0

04

08

12

16

2

c i

(e)






C 1 +1μ


minus 1

(11)



Say CSa

μ

Su CSd CSaT

2

4π2

(12)





























T

Su Sd

S ay

S ay

S a

S aT

micro = 2 micro = 2

micro = 1

micro = 1

micro = 2

micro = 1

micro = 2

micro = 1

S uT









ITHARSP

Δ eq (

m)

01 02 03 04 05 060PGA (g)

0002004006008

01012

(a)

ITHARSP

Δ eq (

m)

01 02 03 04 05 060PGA(g)

0

005

01

015

02

(b)

ITHARSP

Δ eq (

m)

01 02 03 04 05 060PGA(g)

0

002

004

006

008

01

(c)







0

002

004

006

008

01

012

Seism

ic d

ispla

cem

ent (

m)

002 004 006 008 010∆eq (m)



(a)

002 004 006 008 010∆eq (m)

0

002

004

006

008

01

012

014

Seism

ic d

ispla

cem

ent (

m)



(b)

0 002 004 006 008∆eq (m)

0

002

004

006

008

01

012

Seism

ic d

ispla

cem

ent (

m)



(c)


0

1

2

3

4

5

0 01 02 03 04 05

S a

Sd

00550032

0029

Capacity spectrum

Demand spectrum

μ = 10

μ = 15μ = 20










Girder point

04

03

02

Seism

ic d

ispla

cem

ent (

m)

01

00 1 2 3 4

α = 16g

α = 08g

α = 04gα = 02gα = 01g

ITHARSP

(a)

Girder point

04

05

06

07

03

02

01

00 1 2 3 4

α = 16g

α = 08g

α = 04g

α = 02gα = 01g

ITHARSP

Seism

ic d

ispla

cem

ent (

m)

(b)

04

05

06

03

02

01

00 1 2 3 4

α = 16g

α = 08g

α = 04gα = 02gα = 01g

Girder point

ITHARSP

Seism

ic d

ispla

cem

ent (

m)

(c)


















Girder



Pier




06

08

1

12

14

16

RSP

ITH

A

02 04 06 080PGA (g)






















Detail design



S aS a

n2S a

2S a

u

Py

Pu

Sdy Sd2

Sd

Sdu

Demand spectrum

Capacity spectrum










Py

Pu

Demand spectrum

Capacity spectrum

0

1

2

3S a

01 02 03 04 050Sd

(a)

0

1

2

3

S a

Py

Pu

Demand spectrum

Capacity spectrum

01 02 03 04 050Sd

(b)


Girder point0

016

012

008

0041 2 3 4

Seism

ic d

ispla

cem

ent (

m)

ITHARSP

(a)

Sect

ion

curv

atur

e (ra

dm

)



008007006005004003002001

0

(b)









9 Conclusion






Data Availability




Acknowledgments


References












































C 1 +1μ


minus 1

(11)



Say CSa

μ

Su CSd CSaT

2

4π2

(12)





























T

Su Sd

S ay

S ay

S a

S aT

micro = 2 micro = 2

micro = 1

micro = 1

micro = 2

micro = 1

micro = 2

micro = 1

S uT









ITHARSP

Δ eq (

m)

01 02 03 04 05 060PGA (g)

0002004006008

01012

(a)

ITHARSP

Δ eq (

m)

01 02 03 04 05 060PGA(g)

0

005

01

015

02

(b)

ITHARSP

Δ eq (

m)

01 02 03 04 05 060PGA(g)

0

002

004

006

008

01

(c)







0

002

004

006

008

01

012

Seism

ic d

ispla

cem

ent (

m)

002 004 006 008 010∆eq (m)



(a)

002 004 006 008 010∆eq (m)

0

002

004

006

008

01

012

014

Seism

ic d

ispla

cem

ent (

m)



(b)

0 002 004 006 008∆eq (m)

0

002

004

006

008

01

012

Seism

ic d

ispla

cem

ent (

m)



(c)


0

1

2

3

4

5

0 01 02 03 04 05

S a

Sd

00550032

0029

Capacity spectrum

Demand spectrum

μ = 10

μ = 15μ = 20










Girder point

04

03

02

Seism

ic d

ispla

cem

ent (

m)

01

00 1 2 3 4

α = 16g

α = 08g

α = 04gα = 02gα = 01g

ITHARSP

(a)

Girder point

04

05

06

07

03

02

01

00 1 2 3 4

α = 16g

α = 08g

α = 04g

α = 02gα = 01g

ITHARSP

Seism

ic d

ispla

cem

ent (

m)

(b)

04

05

06

03

02

01

00 1 2 3 4

α = 16g

α = 08g

α = 04gα = 02gα = 01g

Girder point

ITHARSP

Seism

ic d

ispla

cem

ent (

m)

(c)


















Girder



Pier




06

08

1

12

14

16

RSP

ITH

A

02 04 06 080PGA (g)






















Detail design



S aS a

n2S a

2S a

u

Py

Pu

Sdy Sd2

Sd

Sdu

Demand spectrum

Capacity spectrum










Py

Pu

Demand spectrum

Capacity spectrum

0

1

2

3S a

01 02 03 04 050Sd

(a)

0

1

2

3

S a

Py

Pu

Demand spectrum

Capacity spectrum

01 02 03 04 050Sd

(b)


Girder point0

016

012

008

0041 2 3 4

Seism

ic d

ispla

cem

ent (

m)

ITHARSP

(a)

Sect

ion

curv

atur

e (ra

dm

)



008007006005004003002001

0

(b)









9 Conclusion






Data Availability




Acknowledgments


References




















































T

Su Sd

S ay

S ay

S a

S aT

micro = 2 micro = 2

micro = 1

micro = 1

micro = 2

micro = 1

micro = 2

micro = 1

S uT









ITHARSP

Δ eq (

m)

01 02 03 04 05 060PGA (g)

0002004006008

01012

(a)

ITHARSP

Δ eq (

m)

01 02 03 04 05 060PGA(g)

0

005

01

015

02

(b)

ITHARSP

Δ eq (

m)

01 02 03 04 05 060PGA(g)

0

002

004

006

008

01

(c)







0

002

004

006

008

01

012

Seism

ic d

ispla

cem

ent (

m)

002 004 006 008 010∆eq (m)



(a)

002 004 006 008 010∆eq (m)

0

002

004

006

008

01

012

014

Seism

ic d

ispla

cem

ent (

m)



(b)

0 002 004 006 008∆eq (m)

0

002

004

006

008

01

012

Seism

ic d

ispla

cem

ent (

m)



(c)


0

1

2

3

4

5

0 01 02 03 04 05

S a

Sd

00550032

0029

Capacity spectrum

Demand spectrum

μ = 10

μ = 15μ = 20










Girder point

04

03

02

Seism

ic d

ispla

cem

ent (

m)

01

00 1 2 3 4

α = 16g

α = 08g

α = 04gα = 02gα = 01g

ITHARSP

(a)

Girder point

04

05

06

07

03

02

01

00 1 2 3 4

α = 16g

α = 08g

α = 04g

α = 02gα = 01g

ITHARSP

Seism

ic d

ispla

cem

ent (

m)

(b)

04

05

06

03

02

01

00 1 2 3 4

α = 16g

α = 08g

α = 04gα = 02gα = 01g

Girder point

ITHARSP

Seism

ic d

ispla

cem

ent (

m)

(c)


















Girder



Pier




06

08

1

12

14

16

RSP

ITH

A

02 04 06 080PGA (g)






















Detail design



S aS a

n2S a

2S a

u

Py

Pu

Sdy Sd2

Sd

Sdu

Demand spectrum

Capacity spectrum










Py

Pu

Demand spectrum

Capacity spectrum

0

1

2

3S a

01 02 03 04 050Sd

(a)

0

1

2

3

S a

Py

Pu

Demand spectrum

Capacity spectrum

01 02 03 04 050Sd

(b)


Girder point0

016

012

008

0041 2 3 4

Seism

ic d

ispla

cem

ent (

m)

ITHARSP

(a)

Sect

ion

curv

atur

e (ra

dm

)



008007006005004003002001

0

(b)









9 Conclusion






Data Availability




Acknowledgments


References















































ITHARSP

Δ eq (

m)

01 02 03 04 05 060PGA (g)

0002004006008

01012

(a)

ITHARSP

Δ eq (

m)

01 02 03 04 05 060PGA(g)

0

005

01

015

02

(b)

ITHARSP

Δ eq (

m)

01 02 03 04 05 060PGA(g)

0

002

004

006

008

01

(c)







0

002

004

006

008

01

012

Seism

ic d

ispla

cem

ent (

m)

002 004 006 008 010∆eq (m)



(a)

002 004 006 008 010∆eq (m)

0

002

004

006

008

01

012

014

Seism

ic d

ispla

cem

ent (

m)



(b)

0 002 004 006 008∆eq (m)

0

002

004

006

008

01

012

Seism

ic d

ispla

cem

ent (

m)



(c)


0

1

2

3

4

5

0 01 02 03 04 05

S a

Sd

00550032

0029

Capacity spectrum

Demand spectrum

μ = 10

μ = 15μ = 20










Girder point

04

03

02

Seism

ic d

ispla

cem

ent (

m)

01

00 1 2 3 4

α = 16g

α = 08g

α = 04gα = 02gα = 01g

ITHARSP

(a)

Girder point

04

05

06

07

03

02

01

00 1 2 3 4

α = 16g

α = 08g

α = 04g

α = 02gα = 01g

ITHARSP

Seism

ic d

ispla

cem

ent (

m)

(b)

04

05

06

03

02

01

00 1 2 3 4

α = 16g

α = 08g

α = 04gα = 02gα = 01g

Girder point

ITHARSP

Seism

ic d

ispla

cem

ent (

m)

(c)


















Girder



Pier




06

08

1

12

14

16

RSP

ITH

A

02 04 06 080PGA (g)






















Detail design



S aS a

n2S a

2S a

u

Py

Pu

Sdy Sd2

Sd

Sdu

Demand spectrum

Capacity spectrum










Py

Pu

Demand spectrum

Capacity spectrum

0

1

2

3S a

01 02 03 04 050Sd

(a)

0

1

2

3

S a

Py

Pu

Demand spectrum

Capacity spectrum

01 02 03 04 050Sd

(b)


Girder point0

016

012

008

0041 2 3 4

Seism

ic d

ispla

cem

ent (

m)

ITHARSP

(a)

Sect

ion

curv

atur

e (ra

dm

)



008007006005004003002001

0

(b)









9 Conclusion






Data Availability




Acknowledgments


References













































0

002

004

006

008

01

012

Seism

ic d

ispla

cem

ent (

m)

002 004 006 008 010∆eq (m)



(a)

002 004 006 008 010∆eq (m)

0

002

004

006

008

01

012

014

Seism

ic d

ispla

cem

ent (

m)



(b)

0 002 004 006 008∆eq (m)

0

002

004

006

008

01

012

Seism

ic d

ispla

cem

ent (

m)



(c)


0

1

2

3

4

5

0 01 02 03 04 05

S a

Sd

00550032

0029

Capacity spectrum

Demand spectrum

μ = 10

μ = 15μ = 20










Girder point

04

03

02

Seism

ic d

ispla

cem

ent (

m)

01

00 1 2 3 4

α = 16g

α = 08g

α = 04gα = 02gα = 01g

ITHARSP

(a)

Girder point

04

05

06

07

03

02

01

00 1 2 3 4

α = 16g

α = 08g

α = 04g

α = 02gα = 01g

ITHARSP

Seism

ic d

ispla

cem

ent (

m)

(b)

04

05

06

03

02

01

00 1 2 3 4

α = 16g

α = 08g

α = 04gα = 02gα = 01g

Girder point

ITHARSP

Seism

ic d

ispla

cem

ent (

m)

(c)


















Girder



Pier




06

08

1

12

14

16

RSP

ITH

A

02 04 06 080PGA (g)






















Detail design



S aS a

n2S a

2S a

u

Py

Pu

Sdy Sd2

Sd

Sdu

Demand spectrum

Capacity spectrum










Py

Pu

Demand spectrum

Capacity spectrum

0

1

2

3S a

01 02 03 04 050Sd

(a)

0

1

2

3

S a

Py

Pu

Demand spectrum

Capacity spectrum

01 02 03 04 050Sd

(b)


Girder point0

016

012

008

0041 2 3 4

Seism

ic d

ispla

cem

ent (

m)

ITHARSP

(a)

Sect

ion

curv

atur

e (ra

dm

)



008007006005004003002001

0

(b)









9 Conclusion






Data Availability




Acknowledgments


References
















































Girder point

04

03

02

Seism

ic d

ispla

cem

ent (

m)

01

00 1 2 3 4

α = 16g

α = 08g

α = 04gα = 02gα = 01g

ITHARSP

(a)

Girder point

04

05

06

07

03

02

01

00 1 2 3 4

α = 16g

α = 08g

α = 04g

α = 02gα = 01g

ITHARSP

Seism

ic d

ispla

cem

ent (

m)

(b)

04

05

06

03

02

01

00 1 2 3 4

α = 16g

α = 08g

α = 04gα = 02gα = 01g

Girder point

ITHARSP

Seism

ic d

ispla

cem

ent (

m)

(c)


















Girder



Pier




06

08

1

12

14

16

RSP

ITH

A

02 04 06 080PGA (g)






















Detail design



S aS a

n2S a

2S a

u

Py

Pu

Sdy Sd2

Sd

Sdu

Demand spectrum

Capacity spectrum










Py

Pu

Demand spectrum

Capacity spectrum

0

1

2

3S a

01 02 03 04 050Sd

(a)

0

1

2

3

S a

Py

Pu

Demand spectrum

Capacity spectrum

01 02 03 04 050Sd

(b)


Girder point0

016

012

008

0041 2 3 4

Seism

ic d

ispla

cem

ent (

m)

ITHARSP

(a)

Sect

ion

curv

atur

e (ra

dm

)



008007006005004003002001

0

(b)









9 Conclusion






Data Availability




Acknowledgments


References
























































Girder



Pier




06

08

1

12

14

16

RSP

ITH

A

02 04 06 080PGA (g)






















Detail design



S aS a

n2S a

2S a

u

Py

Pu

Sdy Sd2

Sd

Sdu

Demand spectrum

Capacity spectrum










Py

Pu

Demand spectrum

Capacity spectrum

0

1

2

3S a

01 02 03 04 050Sd

(a)

0

1

2

3

S a

Py

Pu

Demand spectrum

Capacity spectrum

01 02 03 04 050Sd

(b)


Girder point0

016

012

008

0041 2 3 4

Seism

ic d

ispla

cem

ent (

m)

ITHARSP

(a)

Sect

ion

curv

atur

e (ra

dm

)



008007006005004003002001

0

(b)









9 Conclusion






Data Availability




Acknowledgments


References




























































Detail design



S aS a

n2S a

2S a

u

Py

Pu

Sdy Sd2

Sd

Sdu

Demand spectrum

Capacity spectrum










Py

Pu

Demand spectrum

Capacity spectrum

0

1

2

3S a

01 02 03 04 050Sd

(a)

0

1

2

3

S a

Py

Pu

Demand spectrum

Capacity spectrum

01 02 03 04 050Sd

(b)


Girder point0

016

012

008

0041 2 3 4

Seism

ic d

ispla

cem

ent (

m)

ITHARSP

(a)

Sect

ion

curv

atur

e (ra

dm

)



008007006005004003002001

0

(b)









9 Conclusion






Data Availability




Acknowledgments


References
















































Py

Pu

Demand spectrum

Capacity spectrum

0

1

2

3S a

01 02 03 04 050Sd

(a)

0

1

2

3

S a

Py

Pu

Demand spectrum

Capacity spectrum

01 02 03 04 050Sd

(b)


Girder point0

016

012

008

0041 2 3 4

Seism

ic d

ispla

cem

ent (

m)

ITHARSP

(a)

Sect

ion

curv

atur

e (ra

dm

)



008007006005004003002001

0

(b)









9 Conclusion






Data Availability




Acknowledgments


References















































9 Conclusion






Data Availability




Acknowledgments


References









































a nonlinear static procedure for the seismic design of...

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