seismicstudyofskewbridgesupportedon laminated-rubberbearings

Research ArticleSeismic Study of Skew Bridge Supported onLaminated-Rubber Bearings

Xueshan Liu1 Wei Guo 2 Jianzhong Li1 and Hua Zhang1

1State Key Lab for Disaster Reduction in Civil Engineering Tongji University Shanghai 200092 China2TYLin International Engineering Consulting (China) Co Ltd Chongqing 401120 China

Correspondence should be addressed to Wei Guo guoweitylincomcn

Received 25 May 2020 Revised 31 October 2020 Accepted 4 November 2020 Published 19 November 2020

Academic Editor Daniele Baraldi

Copyright copy 2020 Xueshan Liu 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

Skew bridges consisting of simply supported girders continuous decks and laminated-rubber bearings are widely used in westernChina however they are highly vulnerable to strong earthquakes To investigate the seismic performance of skew bridgesconsidering the sliding behavior of laminated-rubber bearings the Duxiufeng Bridge located in Sichuan China was used as aprototype bridge -is bridge is a skew bridge that suffered seismic damage during the 2008 Wenchuan earthquake -e possibleseismic response of this skew bridge under theWenchuan earthquake was simulated and the postearthquake repair methods wereanalyzed considering the effects of bearing types and cable restrainers Parametric studies using the finite element method werealso performed to investigate the effects of the skew angle and friction coefficient of the bearings on the seismic response of theskew bridge -e results indicate that pin-free bearings could effectively control the seismic displacement of the bridge and thecable restrainers with an appropriate stiffness could significantly reduce the longitudinal residual displacements-e effect of skewangles is less significant on skew bridges with laminated-rubber bearings than on rigid-frame skew bridges because of the slidingbetween the girders and bearings -e residual displacements of the bearings were more sensitive to the variation in the frictioncoefficient between the laminated-rubber bearings and the girders compared to the maximum seismic displacements

1 Introduction

Highway bridges which are a crucial component of thetransportation system with a significant role in the rescueoperations after earthquakes have been extensively damagedto different degrees during major earthquakes in the past-e 1994 Northridge earthquake [1] 1995 Kobe earthquake[2] 1999 Chi-Chi earthquake [3] and 2008 Wenchuanearthquake [4] have demonstrated that highway bridges arehighly vulnerable to earthquakes and the collapse of a bridgemight directly cut off the trunk roads hindering the post-earthquake rescue efforts to a large extent

Laminated-rubber bearings featuring alternate layers ofrubber and steel sheets as shown in Figure 1 have beenextensively used in small-to-medium-span highway bridgesin China in recent decades owing to their low cost andsimple configurations -e bearings possess the capacity tosustain large vertical loads and accommodate the thermal

movements of the superstructure with little or no mainte-nance requirements Laminated-rubber bearings are used tosupport the superstructures and the friction force is the onlyhorizontal resisting force provided at the supports [5 6]Once the seismic force exceeds the friction force slidingleads to potentially large displacements Large residualdisplacements may be generated hindering the usage of thebridge after the earthquake It is also possible that the bridgeunseats and collapses as a result of large relative displace-ments between the superstructure and substructure (egMiaoziping Bridge [7]) -e seismic behavior of laminated-rubber bearings has been investigated by many researchers[8ndash10] To obtain an analytical model of laminated-rubberbearings under earthquakes Xiang and Li [10] performed anexperiment on laminated-rubber bearings placed betweentwo steel plates An analytical model considering the slidingresponse of the bearings was developed and calibrated -eseismic behavior of the bearings before an evident sliding

HindawiAdvances in Civil EngineeringVolume 2020 Article ID 8899693 17 pageshttpsdoiorg10115520208899693

could be approximated as a linear elastic response with aneffective shear modulus in the range of 610ndash1100 kPa -esliding coefficients of friction were observed to be inverselyrelated to the normal force and positively related to thesliding velocity

Considering the potential seismic damage of small-to-medium-span highway bridges with laminated-rubberbearings some restraining devices have been proposed toimprove the seismic performance of these bridges such asyielding steel dampers [11 12] using which the seismicgirder displacement can be effectively controlled while thebridges can achieve a reasonable balance between thebearing displacements and substructure seismic demandsAdditionally some studies [13ndash16] showed that cable re-strainers could reduce the residual displacement of bridgebearings under earthquake loadings Won et al [14] studiedthe seismic response of a multispan simply supported bridgewith cable restrainers -eir results showed that the relativedisplacement between the abutment and the next pier waseffectively controlled thus preventing the span from col-lapsing However few studies have investigated the effect oflaminated-rubber bearings associated with restrainers on theseismic performance of skew bridges

-e design of skew bridges is often dictated by theorientation of the roadway that includes the bridge relativeto the orientation of natural (rivers) and man-made (otherroads) objects Numerous studies [17ndash23] were conductedon the seismic response of skew bridges after the 1971 SanFernando 1994 Northridge and 2010 Chile earthquakesSkewed bridges exhibit a significantly more complex seismicresponse than do the straight bridges It was found that anin-plane rotation of the decks in skew bridges might causelarge torques in the columns thus making them morevulnerable Moreover a large skew is likely to increase therelative displacements between the girders and abutmentsthus making skew bridges more vulnerable to unseating andcollapse To understand the unseating mechanism of a skewbridge with seat-type abutments Wu et al [24 25] per-formed shake table tests of four single-span bridge modelswith skew angles of 0 (straight) 30 45 and 60 Based on thetest results on the unseating mechanism it was furtherconcluded [26] that the obtuse corner of the superstructure

of a skew bridge engaged the adjacent back wall and thesuperstructure then rotated about this corner -e impact ofthe span against the back wall was then followed immedi-ately by a rebound away from the wall and the span con-tinued to rotate in the same direction in free vibration aboutthe center of the stiffness of the substructure Most of theseaforementioned studies only focused on the seismic be-havior of single-span skew bridges or two-span shew bridgeswith rigid girder-deck connections Rasouli and Mahmoodi[27] assessed the effect of skewness and number of spans onthe seismic behavior of multispan skew bridges with rigidgirder-deck connections using a modal pushover analysis Intheir study an increase in the number of spans decreased thedeck rotation responses at the abutments and the adverseeffect of skewness

For a multispan skew girder bridge with laminated-rubber bearings bearing sliding could occur under strongearthquakes which might change the seismic response of thebridge and lead to a seismic damage However little researchhas been carried out on this type of bridges -is studypresents an investigation of the seismic performance of amultispan skew bridge with laminated-rubber bearingswhich suffered seismic damage during the 2008 Wenchuanearthquake To understand the seismic response of thebridge a nonlinear analysis was carried out using the finiteelement method In addition parametric studies includingbearing types skew angle friction coefficient and cablerestrainers were conducted and measures to improve theseismic performance of the bridge were also identified

2 Background and Ground Motions

21 Prototype Bridge To achieve the objectives of this studythe Duxiufeng Bridge located in Wenchuan China wasadopted as the research object -is bridge was approxi-mately 153 km from the epicenter and 57 km from the faultin the 2008 Wenchuan earthquake -e bridge has 6 spanswith a total length of 18806m and each span is 30m inlength Figure 2 shows the elevation and plan view of thebridge -e superstructure is supported on five bents eachconsisting of two 15m diameter columns Figure 3 shows atypical bent section-e heights of the columns from the topof the footing to the bottom of the cap are 830 1149 13011429 and 1428m for bents 1 to 5 respectively Eachcolumn is an extension of a 18m diameter pile -e top ofthe piles in each bent is linked by a transverse beam with across-section of 13times15m-e lengths of the piles are 43254130 3800 2700 and 2730m for bents 1 to 5 respectively-e height and width of the bent cap are 15m and 18mrespectively -e site consists mostly of gravel and sand andthe site class of the bridge is Type II according to the ChineseGuidelines for seismic design of highway bridges [28] Forthe superstructure four identical simply supported I-Shapedgirders are located in each span and two continuous deckswith a width of 85m each are placed on the three left andright spans Expansion joints with a length of 80 cm are usedat bent 3 and the abutments -e three left spans are slightlycurved with a radius of 200m while the three right spans arestraight -e skew angle is 47deg Polytetrafluoroethylene

Concrete shear key

Concrete substructure

Supporting pad

Superstructure

Laminated rubber bearingSteel plate

Steel plate

Figure 1 Layout of laminated-rubber bearings

2 Advances in Civil Engineering

(PTFE) sliding bearings with a small friction coefficient areused at the abutments and bent 3 so that a thermalmovement of the girder is allowed Laminated-rubberbearings support the girders directly at the other bentswithout any other connections All the bearings allow slidingonce the friction forces between the girders and bearings areexceeded by any horizontal forces such as seismic forces-is bearing placement is a conventional design practice forsmall-to-medium-span highway bridges in western Chinaconsidering the low project costs and construction conve-nience -ese bearings will fail when the relative displace-ment between the girder and the bearing is large enough tocause the girder to fall off from the bearing To prevent thegirder from falling off the Duxiufeng bridge caused by a

failure of the bearing the distances from the edges of thegirder to the edge of the bearing in the longitudinal andtransverse directions are 85 and 75 cm respectively thesecalculations were made based on the Chinese Guidelines forseismic design of highway bridges [28]

Figures 4(a) and 4(b) show an overview of the DuxiufengBridge after the 2008 Wenchuan earthquake Figures 4(c)and 4(d) show the transverse residual displacement betweentwo adjacent decks after the earthquake In addition theexpansion joints at the abutment and bent 3 were damagedand had to be replaced To this end this bridge was selectedas the prototype bridge in this study to investigate theseismic response of skew bridges with laminated-rubberbearings

18806Dujiangyan Wenchuan

3 times 30 3 times 30

0 1 2 3 4 5

6

Figure 2 Elevation and plan view of Duxiufeng Bridge (unit m)

50cos43deg

Laminated-rubberbearing

375cos43deg

2075 20753125

150

180

Center line

3125

375cos43deg850cos43deg

50cos43deg

8515

065

Figure 3 Typical bent section (unit cm)

Advances in Civil Engineering 3

22 Ground Motions from the 2008 Wenchuan EarthquakeTo simulate as accurately as possible the actual seismicresponse of the Duxiufeng Bridge under the 2008Wenchuanearthquake and acceleration time histories (Figure 5)recorded at the Wolong station approximately 30 km fromthe bridge were used to conduct a nonlinear analysis of thebridge -e peak ground acceleration (PGA) was the largestamong all the records in the 2008Wenchuan earthquake [7]-e PGAs recorded in the EW NS and UD directions were0957 g 0653 g and 0948 g respectively 01 and 25Hzbandpasses were applied to filter the original records -epart of the record used in the analysis was from 20 to 70 s-e three records were applied simultaneously -e corre-sponding acceleration response spectra are shown inFigure 6

-e longitudinal direction of the bridge is NW 10deg andthe global coordinate system of the model was taken asfollows the X- Y- and Z- axes were along the longitudinaltransverse and vertical directions of the bridge respectively-erefore the NS record with an angle of 10deg to the X-axiswas taken as the longitudinal input and the EW record withan angle of 10deg to the Y-axis was taken as the transverseinput -e vertical motion was also applied

3 Finite Element Model

31 Modelling of the Superstructure To understand theseismic response of the bridge during the earthquake adetailed finite element model was developed using a

SAP2000 software [29] From a comparative study Mengand [20] concluded that the effect of the flexibility of thesuperstructure was significant and should not be ignored inthe dynamic analysis because the use of the rigid deck orstick model in the dynamic analysis of skew bridges withlarge skew angles would lead to inaccurate axial forces in thecolumns during an earthquake -erefore the superstruc-ture was modelled using a grid system that represented thebeamrsquos flexibility In each span four elastic beams were usedto model the I-Shaped girders -e stiffness of the deck wasincluded by enlarging the flange of the girders Rigidtransverse elements were assigned to the nodes to simulatethe high in-plane stiffness of the deck

32 Modelling of the Bents Plastic hinges were assigned tothe top and bottom of the columns -e plastic hinge sec-tions consisted of three types of fibers confined concreteunconfined concrete and longitudinal bars -e Mander[30] model was used to model the unconfined and confinedconcrete -e hinge length (Lp) was calculated as followsbased on [28]

Lp min 008H + 0022fyds ge 0044fyds23

b1113874 1113875 (1)

where H is the distance from the top or bottom to the pointwhere the moment is zero b is the minimum cross-sectionaldimension or the diameter of the circular section fy (MPa)is the specified steel bar yield strength and ds is the bardiameter -e concrete grade in the bents was C30 with a

(a) (b)

(c) (d)

Figure 4 Photographs of the Duxiufeng Bridge after the 2008 Wenchuan earthquake (a) Photographs of the bents (b) Photographs of thedeck (c) Transverse residual displacement at abutment 0 (d) Transverse residual displacement at bent 3


EW

ndash10

ndash05

00

05

10

Acc

eler

atio

n (g

)

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 1600Time (s)

NS

ndash10

ndash05

00

05

10

Acc

eler

atio

n (g

)

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 1600Time (s)

UD

ndash10

ndash05

00

05

10

Acc

eler

atio

n (g

)

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 1600Time (s)

Figure 5 Acceleration time histories of Wolong records

(064 058)

(033 163)

NS

00

05

10

15

20

25

30

35

Acc

eler

atio

n (g

)

1 2 3 4 5 60Period (s)

(064 075)

(033 273)

EW

1 2 3 4 5 60Period (s)

00

05

10

15

20

25

30

35

Acc

eler

atio

n (g

)

UD

00

05

10

15

20

25

30

35

Acc

eler

atio

n (g

)

1 2 3 4 5 60Period (s)

Figure 6 Acceleration response spectra of Wolong records


specified compressive strength of 201MPa [31] -e steelgrades of the longitudinal and transverse bars were HRB335and R235 respectively and the yield strengths of these steelgrades were 335 and 235MPa respectively [31] -e loca-tions of the hinges were defined in the middle of the plastichinge length -e bent caps were modelled as elasticelements

33 Modelling of the Bearings As stated before the linksbetween girders and bents are laminated-rubber bearingswithout any other connections -erefore friction forces arethe only horizontal forces No tension force can be trans-mitted through the bearings -e friction isolator link inSAP2000 was used to simulate this mechanical behaviorFriction factors were assigned in the horizontal direction tocalculate the horizontal forces based on the normal force onthe bearing -e normal force (P) is always nonlinear

P kd if dlt 0

0 otherwise1113896 (2)

where k is the compressive stiffness of the laminated-rubberbearing and d is the normal deformation of the bearing

To simulate the sliding behavior of elastomeric bearingsan analytical model considering the effect of the normalpressure is shown in Figure 7 -e model incorporates twocomponents elastic shear stiffness (Ke) and sliding coeffi-cient of friction (μ)

-e value of μ between the laminated-rubber bearingsand concrete girders was taken as 03 from the experimentalresults of Huang [32] while that between the PTFE slidingbearings (at the abutments and bent 3) and concrete girderswas taken as 002 -e elastic shear stiffness (Ke) of elas-tomeric bearings can be calculated using the followingequation

Ke GA

1113936 t (3)

where G is the shear modulus of the rubber A is the area ofthe bearing and 1113936 t is the total thickness of the rubber layersin the elastomeric bearings

34 Modelling of the Expansion Joints Gap elements wereused to model the pounding effect between the super-structure segments at bent 3 and between the superstructureand the abutments -e initial gap was set as 8 cm which isthe standard expansion joint gap length in bridges -egirder axial stiffness was used as the gap element com-pressive stiffness

35 Modelling of the Foundations and the AbutmentsLinear rotational and translational springs were used tomodel the bent foundation to account for the soil-pile in-teraction -e stiffnesses of the bent foundations were cal-culated by considering the arrangement of piles and soilinformation at the bridge site based on the ChineseGuidelines for seismic design of highway bridges [28]

Longitudinal linear springs were used to model theabutment to account for the abutmentndashsuperstructure in-teraction upon the gap closure -e transverse springs werenot considered because there were no concrete shear keys atthe abutments -e stiffnesses of the abutments were de-termined using the California Department of Transportation(Caltrans) guidelines [33] -e initial stiffness yield forceand yield displacement for the abutment spring were13366 kNmm 6168 kN and 46mm respectively -e finiteelement model is shown in Figure 8

4 Response Analysis

41 Dynamic Characteristics -e dynamic response of astructure is generally based on its natural vibration modes-erefore the natural periods and mode shapes for the firstseven modes of the bridge were calculated (see Table 1) -eresults indicate that the fundamental mode of the bridge wasthe in-plane rotation of the superstructure which wasmainly because of the large skewness of the bridge andunsymmetrical column heights Under excitations withpredominant periods close to the in-plane rotation periodsthe rotation response would be significant and cause rela-tively large displacements compared with nonskew bridgesAs the heights of the columns of the three right bents werelarger than those of the two left bents the natural periods ofthe former were longer than those of the three left spans Forthe 7th mode the longitudinal vibration of bent 3 wassimilar to that of a cantilever beam because the PTFE slidingbearings had small friction forces and provided minimalconstraint to the bent

42 Results of Nonlinear Analysis First a nonlinear staticpushover analysis was performed for each bent consideringthe dead load effect of the bridge -e pushover results arelisted in Table 2 -e displacement ductility capacity is theratio of the ultimate displacement to the yield displacementIt can be seen that the yield displacements increased as thebent heights increase while the horizontal load capacitydecreased -e longitudinal ductility and displacement ca-pacities were larger than those in the transverse direction

Force

Displ

Ke Ke

Nμ

Nμ

Figure 7 Analytical model of the sliding of elastomeric bearings


which meant that the bents were more vulnerable to theloadings in the transverse direction

A nonlinear analysis was conducted to study the seismicresponse of the skew bridge using the finite element methodAs representative results the displacement time histories ofthe bent tops for bents 2 and 4 in the two horizontal di-rections are illustrated in Figure 9 It can be seen that thedisplacement of bent 2 (a relatively short bent) in thetransverse direction was smaller than the longitudinal dis-placement However this trend was reversed in the tallerbents (eg bent 4) -e yield displacements in the longi-tudinal and transverse directions were 60 and 15mm re-spectively for bent 2 and 100 and 25mm respectively forbent 4 -e results also show that in the longitudinal di-rection the bents remained elastic however they yielded in

the transverse direction with displacement ductility de-mands of 197 and 291 for bents 2 and 4 respectively -eductility demands were smaller than the correspondingcalculated capacities for bents 2 and 4 As shown in Figure 9there was no significant residual displacement in eitherdirection

Figure 10 compares the displacement ductility demands andductility capacities of the bents μD is the displacement ductilitydemand and μC is the calculated displacement ductility capacity(See Table 2)-e displacement ductility demand μD is the ratioof the maximum seismic displacement of the bent to thecorresponding yield displacement -e bent remains elastic ifμle 1 It can be seen that all the bents remained elastic in thelongitudinal direction except bent 1 while for the transversedirection all the bents yielded however the ductility demands

XYZ

Figure 8 Finite element model

Table 1 First seven natural modes of the bridge

Mode no Periods (s) Frequency (Hz) Mode description1 2462 0406 In-plane rotation (right three spans)2 2087 0479 In-plane rotation (left three spans)3 1754 0570 Longitudinal vibration (right three spans)4 1430 0699 Longitudinal vibration (left three spans)5 1329 0752 Transverse vibration (right three spans)6 1108 0902 Transverse vibration (left three spans)7 0645 1551 Longitudinal vibration of bent 3

Table 2 Pushover results of the bents

Bent noYield displacement (mm) Displacement ductility capacities Maximum horizontal force (kN)

Longitudinal Transverse Longitudinal Transverse Longitudinal Transverse1 40 10 690 267 1022 10252 60 15 700 305 765 7673 80 20 748 317 653 6544 100 25 588 279 619 6195 100 25 640 288 608 608


were smaller than the corresponding ductility capacities for allthe bents except bent 4 It is interesting to note that even thoughthe friction coefficient of the bearings on bent 3 was small andhence only a little friction force could be transferred from thesuperstructure to the bent through the bearing the displace-ment ductility demands of bent 3 in both the directions werealmost the same as the those of the other bents -is result canbe explained by the response spectra (Figure 6) As stated abovethere was almost no constraint at the top of bent 3 and it couldvibrate as a cantilever system -e first two natural modes ofbent 3 were longitudinal and transverse modes with periods of064 and 033 s respectively From the response spectra it isevident that the corresponding accelerations at these periodswere relatively large as shown in Figure 5 -e other bents areconstrained to the superstructure through the friction of thebearings and the natural vibration periods (the transverserotation periods of the bridge as shown in Table 1) were rel-atively large Furthermore the spectral accelerations were muchsmaller during these periods -erefore displacements gener-ated with large accelerations and small masses (bent 3) were

approximately the same as those generated with small accel-erations and large masses (the remaining bents)

-e maximum residual displacements of the bearings inthe longitudinal and transverse directions were 859 and1555mm respectively -e maximum axial deformation ofthe gap elements was 845mm and the corresponding forcewas 1546 kN Pounding occurred at the left side of bent 3 dueto the in-plane rotation of the decks -e calculated residualrelative displacements between two adjacent decks are listedin Table 3 It can be seen that the maximum residual relativedisplacement in Table 3 was 1104mm appearing at bent 3which was much smaller than the measured value Althoughthe error between the calculated maximum residual dis-placement and the real value was relatively large the cal-culated maximum residual displacement between the twoadjacent decks also happened at bent 3 of the DuxiufengBridge which reflected the real seismic behavior of this skewbridge under the 2008 Wenchuan Earthquake that the re-sidual displacement between span 3 and span 4 was thelargest and may cause severe seismic damage -e possibleexplanation for the error is that the ground motion used inthe work was not the real ground motion at the bridge siteand the recorded station of the ground motion was ap-proximately 30 km far from the bridge site -e character-istics of the ground motion might be influenced by somefactors such as the landform of the bridge site that wasdifferent from the recorded site

(1) In the longitudinal direction the fixed-pin bearingswere used at bents 2 and 4 and roller bearings whichallow the movement of the girder were used at otherbents In the transverse direction the girder wasrestrained by bearings at the other bents

(2) In the longitudinal direction the situation of thebearings was the same as in the first case however inthe transverse direction the movement of the girderwas free at the other bents

43 Postearthquake Repair

431 Effect of Bearing Types From the postearthquakedisaster investigation and the results from the numericalsimulation it can be seen that the Duxiufeng Bridge under

6080

40

Disp

(m

m)

200

Time (s)0 10 20 30 40 50

ndash20ndash40ndash60

LongitudinalTransverse

(a)

Time (s)0 10 20 30 40 50

6080

40

Disp

(m

m)

200

ndash20ndash40ndash60ndash80


(b)

Figure 9 Displacement time histories of the bent tops (a) Bent 2 (b) Bent 4

μD (longitudinal)μD (transverse)

μ = 1μC (transverse)

2 3 4 51Bent No

00

05

10

15

20

25

30

35

40

45

μ

Figure 10 Ductility demands of bents


the Wenchuan earthquake suffered severe seismic damageincluding a large residual displacement girder pounding atbent 3 and yielding of rebars in bents which hampered thepostearthquake functioning of the bridge To repair thisbridge all the damaged bearings needed to be replacedTaking this into consideration a comparison of differentbearing types was analyzed to obtain a feasible plan toimprove the seismic performance of the bridge at a lowproject cost In this analysis the other two common bearingtypes were selected based on the current design practice ofsmall-to-medium-span bridges in China as described below

To facilitate the discussion the bridge model with thefirst bearing combination was labelled as the ldquopin-rollerbearingrdquo and the one with the second type was labelled asthe ldquopin-free bearingrdquo -e model with the actual bearingplacement used in the Duxiufeng Bridge was referred to as aldquorubber bearingrdquo

Figure 11 shows the displacement ductility demandsof the bents for different bearing types -e pin-rollerbearing experienced the largest bent displacement duc-tility demands in the longitudinal direction None of thebents in the other two models yielded in the longitudinaldirection -e in-plane rotation of the superstructure wasrestrained because the transverse movement was re-strained It is seen that the transverse restraint led to largebent forces in both directions which increased the dis-placement ductility demands in the bents -e differencesin the displacement ductility demands in the longitudinaldirection between the pin-free and rubber bearings wererelatively small -e pin-roller bearing also had thelargest displacement ductility demands in the transversedirection of all the bents except for bents 4 and 5 Incontrast the displacements of the pin-free bearing wereslightly larger than those of the rubber bearing at thefixed-pin bents (bents 2 and 4) -erefore the pin-freeand rubber bearing models were preferred for controllingthe damage in the bents under the Wolong records

-e maximum seismic displacements of the bearings fordifferent bearing types are listed in Table 4 -ese resultsindicate that the longitudinal and transverse maximumdisplacements for the bridge with laminated-rubbers werelarger than those of the others -e results for the pin-freebearing model were approximately one-half of the results forthe rubber bearing model For the pin-roller bearing modelthe longitudinal displacement was slightly larger than thatfor the rubber bearing model and the transverse displace-ment was zero because the bearings were fixed in thetransverse direction -e results indicate a significant re-duction in both the longitudinal and transverse maximum

displacements for the pin-free model compared with thoseof the rubber bearing model -erefore the pin-free bearingwas recommended for the Duxiufeng Bridge to control theseismic displacements

432 Effect of Cable Restrainers -e skew bridge withlaminated rubber bearings suffered from the possibility ofthe girder falling off under strong earthquakes thereforeseismic devices are needed to improve its seismic perfor-mance Considering the feasibility and the budgetary con-straints of the postearthquake repair for the DuxiufengBridge cable restrainers with elastic behavior were selectedas the seismic control device -e cable restrainers wereassumed to be installed at the expansion joints at abutments0 and 6 and bent 3 -e layout of the cable restrainers isshown in Figure 12 Two cable restrainers were symmetri-cally arranged at each abutment expansion joint attachingthe girders to the abutment Four more cable restrainerswere symmetrically arranged at the middle expansion jointattaching spans 3 and 4 to bent 3 -e cable restrainerstiffness was assumed to be the same at different locationsAlthough the minimum restrainer stiffness suggested was25 kNmm according to a study by Saiidi et al [13] stiffnessvalues of 125 25 5 and 10 kNmm were employed toperform a parametric study Additionally the elastic be-havior of the cable restrainers is also shown in Figure 12-einitial slack of the cable restrainers was taken as zero toaccount for the extremely low temperature and the mostcritical condition for restrainer loading

Figure 13 compares the maximum residual dis-placements of the bearings considering the cable re-strainers with different stiffnesses positions 0 and 6represent abutments 0 and 6 while positions 1 to 5represent bents 1 to 5 In Figure 13(a) it can be seen thatthe longitudinal maximum residual displacements of thebearings without the cable restrainers were larger thanthose with the cable restrainers at positions 0 to 6 Ingeneral even a nominal cable restrainer (with a relativelysmall stiffness) could reduce these residual displacementsby approximately 50 in most positions With an in-crease in the stiffness significantly complicated varia-tions of the residual displacements were presentedAdditionally Table 5 shows the percentages of the de-crease in the maximum residual displacements withdifferent stiffnesses at positions 0 3 and 6 because theseismic damage led by the residual displacements couldappear at positions other than the above -e percentageof decrease η can be calculated using

Table 3 Calculated residual relative displacement between two adjacent decks

Position Longitudinal (mm) Transverse (mm)

Abutment 0 Right side 647 248Left side 834 437

Bent 3 Right side 20 1104Left side 223 870



η D minus D

cable

D (4)

where D is the residual displacement of the bearing withoutthe cable restrainers while D

cableis the residual displacement

of the bearing with the cable restrainers It can be seen that

the residual displacement at position 6 was significantlyaffected by the cable restrainers as the minimum value of ηwas larger than 60 When the stiffness was less than 5 kNmm the values of η at 0 and 3 were smaller than 50 Basedon these results the stiffness of the cable restrainers used inthe Duxiufeng Bridge should be equal or greater than 5 kN

RubberPin-freePin-roller

00

05

10

15

20

μ

2 3 4 51Bent no

(a)


2 3 4 51Bent no

0

2

4

6

8

μ

(b)

Figure 11 Displacement ductility demands of bents with different bearing types (a) Longitudinal direction (b) Transverse direction

Table 4 Maximum seismic displacement of bearings for different bearing types

Bearing type Longitudinal (mm) Transverse (mm)Rubber 119 229Pin-free 68 118Pin-roller 121 0

Bent 3 Abutment 6

Abutment 0

Cable restrainers

Stiffness

Disp

Force

Elastic behavior of cable restrainers

Span 4 Span 5 Span 6

Span 1

Span 2Span 3

Force

Laminated-rubber bearings(not shown in the figure)

Ke

Disp

Figure 12 Layout of the cable restrainers for the Duxiufeng Bridge


mm In addition for a given cable restrainer stiffness theaverage value of η was also calculated With an increase inthe stiffness the average value increased indicating that thecable restrainer with a larger stiffness could achieve a betterperformance for controlling the residual displacements ofthe bearings

Figure 13(b) illustrates that the longitudinal cable re-strainers had an insignificant effect on limiting the maxi-mum transverse residual displacements of the bearings andthey might increase the displacements at some positionsTaking the cable restrainers with a stiffness of 5 kNmm as anexample the longitudinal residual displacement at bent 3decreased while the transverse displacement increased -iswas expected because the cable restrainers acted mainly inthe longitudinal direction Meanwhile the biaxial horizontalinput motion and rotation of the superstructure resulted incoupled longitudinal and transverse displacements of thesuperstructure When the longitudinal movement wasconstrained the girders tended to move transversely whichtended to amplify the rotation of the superstructure

-e maximum elongations of the cable restrainers were84 77 70 and 68mm corresponding to stiffnesses of 12525 5 and 10 kNmm respectively -ese values weresmaller than those of the yield displacement (107mm) of the

widely used cable restrainers [13] indicating that the cablerestrainers could remain elastic during the earthquake

5 Parametric Analysis

51 Ground Motions for Parametric Analysis To furtherstudy the seismic performance of skew bridges with lami-nated-rubber bearings the effects of the skew angle andfriction coefficient were investigated in this study through anumerical analysis Ten real earthquake records with thesame seismic soil classification of the Duxiufeng Bridge wereused in the analysis including theWolong records describedin the previous section as shown in Table 6 -e other nineearthquake records were selected from the Pacific Earth-quake Engineering Research Centre (PEER) ground motiondatabase For each record all the three components wereincluded in this analysis and the input direction was thesame as that specified in the previous section Figure 14shows the acceleration response spectra of the largestcomponents of the ten records In these analyses the largestPGA of the three components was scaled to 0957 g which isthe same as the PGA of the Wolong records -e samescaling factor was also used for the other two components ofeach earthquake record -e seismic responses of the bridge

Kr = 5kNmmKr = 10kNmm

No restrainerKr = 125kNmmKr = 25kNmm

0

30

60

90

120

150M

ax b

earin

g re

sidua

l disp

(m

m)

1 2 3 4 5 60Position

(a)



0

50

100

150

200

250

Max

bea

ring

resid

ual d

isp (

mm

)


(b)

Figure 13 Comparison of maximum residual displacements of bearings (a) Longitudinal direction (b) Transverse direction

Table 5 Percentages of decrease in the maximum residual displacements of bearings

PositionStiffness of cable restrainers

125 kNmm () 25 kNmm () 5 kNmm () 10 kNmm ()Abutment 0 313 275 575 850Bent 3 301 289 765 916Abutment 6 664 896 828 888Average value 426 487 723 885


under these records were averaged to ensure that the con-clusions were independent of the input

52 Effect of Skew Angle -e skew angle of the DuxiufengBridge was 47deg To study the effect of the skew angle on thedynamic responses of the bridge different models with thesame properties but different skew angles (15 30 47 and60deg) were developed and analyzed -e dynamic charac-teristics and seismic responses were calculated to study theinfluence of the skew angle

521 Dynamic Characteristics -e effect of the skew angleon the modal periods is shown in Figure 15 -e resultsindicate that with an increase in the skew angle the periodsof the first four modes were enlarged -is is because boththe torsional stiffness of the bents and the in-plane rotationalinertia of the superstructure increased with the skew anglehowever the increase in the torsional stiffness was smallerthan that of the rotational inertia

522 Seismic Response Figure 16 shows a comparison ofthe maximum and residual seismic displacements of thebearings with different skew angles Avg (L) is the averagevalue for all the bearings in the longitudinal direction Avg(T) is the average value for all the bearings in the transversedirection Max (L) is the maximum value for all the bearingsin the longitudinal direction and Max (T) is the maximumvalue for all the bearings in the transverse direction It can beseen that with an increase in the skew angle the maximumand average values of the longitudinal maximum seismicdisplacements varied slightly while the same in the trans-verse direction were significantly affected by the skew angleFor the residual displacement the maximum and averagevalues reached a minimum in both the longitudinal andtransverse directions for a skew angle of 47deg Previous studies[20 33] have pointed out that an increase in the skew angleincreases the girder displacement response of the bridge-is conclusion was based on the investigations and nu-merical analyses on rigid-frame bridges However for themodels analyzed in this study sliding occurred between thebearings and girders thus decreasing the in-plane rotation-erefore the effect of the skew angle on the seismic dis-placement response of the bridge with laminated-rubberbearings was less significant than that of a rigid-framebridge

-e effect of skew angle on the displacement ductilitydemands of bents is shown in Figure 17 Generally theskew angle has an insignificant influence on the ductilitydemands in the longitudinal direction if the skew angle isless than 47deg -e ductility demands in bents 2 4 and 5 inthe model with a skew angle of 60deg were larger than thosein the other models indicating that taller bents and bentswith laminated-rubber bearings were sensitive to a largeskew angle compared with the shorter bents and bentswith PTFE sliding bearings (bent 3) Sliding occurred inbent 3 thus reducing the effect of the skew angle For the

WenchuanNorthridgeLoma PrietaImperial ValleyChi-Chi

Kobe-KakogawaKobe-KJMASan FernandoParkfieldKern County

0

1

2

3

4

PGA

(g)

1 2 3 40Period (s)

Figure 14 Acceleration response spectra of selected records

Table 6 Earthquake records for seismic analysis

Earthquake Year Station name Scale factorWenchuan 2008 Wolong 10Northridge 1994 Sylmar hospital 1136Loma prieta 1989 SF airport 2911Imperial valley 1940 El centro 3058Chi-Chi 1999 CHY025 5921Kobe 1995 Kakogawa 2952Kobe 1995 KJMA 1147San Fernando 1971 Castaic old ridge route 2802Parkfield 1966 Cholame - Shandon array 8 3866Kern county 1952 Taft Lincoln school 5308

Skew angle = 15degSkew angle = 30deg


00

05

10

15

20

25

30

Perio

d (s

)

1 2 3 4 5 6 7 80Mode number

Figure 15 Comparison of modal periods with different skewangles


transverse direction the ductility demands might in-crease with an increase in the skew angle

53 Effect of Friction Coefficient -e sliding between thebearings and girders was significantly affected by the frictioncoefficient Caltrans [29] specified a dynamic friction co-efficient of 04 between concrete and neoprene and 035between neoprene and steel -e friction coefficient mightdepend on the sliding velocity and temperature -e sen-sitivity of the response was determined using two sets offriction coefficients -e coefficient between the laminated-

rubber bearings and the girders (CR) was varied from 02 to05 while the coefficient between the PTFE sliding bearingsand girders (CT) was kept constant at 002 for the first groupIn the second group CR was constant at 03 but CT wasvaried from 0005 to 002

-e numerical results with differentCR values are comparedin Figures 18 and 19 As shown in Figure 18 the maximum andaverage values of the maximum displacements of all thebearings are not sensitive to CR values except for the maximumvalues in the transverse direction which increase with the in-crease of CR values On the other hand the residual displace-ments of the bearings are significantly affected by the variations

Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)

30 45 6015Skew angle (deg)

(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

100

200

300

400

Disp

(m

m)

(b)

Figure 16 Comparison of seismic displacements of bearings with different skew angles (a) Maximum displacement (b) Residualdisplacement



15

20

25

30

35

40

45

μ

2 3 4 51Bent no

(a)



20

25

30

35

40

45

50

55

60

μ

2 3 4 51Bent no

(b)

Figure 17 Displacement ductility demands of bridgemodels with different skew angles (a) Longitudinal direction (b) Transverse direction


of CR values -e influence of CR on the residual displacements(in Figure 18(b)) is erratic and no clear trends can be identifiedWhen the CR is 03 the average and maximum values of theresidual displacements of all the bearings reach theminimum inthe transverse and longitudinal directions

In Figure 19 it can be seen that except for bent 3 thelongitudinal and transverse displacement ductility de-mands for the bents increased with an increase in CR -isis because when the friction coefficient increased theshear force transmitted from the bearings to the bentsincreased resulting in larger displacement ductility de-mands Because the bearings on bent 3 were PTFE slidingbearings with a constant friction coefficient of 002 theresponse of bent 3 was not significantly affected by CR

-e effect of CT on the seismic response is shown inFigures 20 and 21 -e effect of CT on the seismic dis-placements of bearings was approximately the same as thatof CR -e residual displacements were more sensitive to CTcompared to themaximum displacements An increase inCTdecreased the average and maximum values of the longi-tudinal residual displacements however it did not neces-sarily lead to a reduction in the transverse residualdisplacements

As shown in Figure 21 the displacement ductility de-mands of the bents appeared to be insensitive to a variationin CT -e possible reason is that CT was very small com-pared with CR -erefore with a variation in CT the totalforce on the bents changed only by a small amount and

Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700

Disp

(m

m)

02 03 04 05 0601Friction coefficient

(a)

Avg(L)Avg(T)

Max(L)Max(T)

0

100

200

300

400

500

Disp

(m

m)


(b)

Figure 18 Effect of CR on the bearing displacements (a) Maximum displacement (b) Residual displacement

CR = 02CR = 03

CR = 04CR = 05

15

20

25

30

35

40

45

50

μ

2 3 4 51Bent no

(a)

CR = 02CR = 03

CR = 04CR = 05

20

25

30

35

40

45

50

55

60

65

70

μ

2 3 4 51Bent no

(b)

Figure 19 Effect of CR on the displacement ductility demands (a) Longitudinal direction (b) Transverse direction


hence the displacement ductility demands of the bents werenot significantly affected

6 Conclusions

From this study the following conclusions could be drawn

(1) Large sliding displacements occurred between thelaminated-rubber bearings and the girders for theDuxiufeng Bridge under the 2008 Wenchuanearthquake

(2) Compared with the bridge with a pin-free bearing orrubber bearing the displacement ductility demandsof the bents were relatively large in the longitudinal

and transverse directions for the skew bridge with apin-roller bearing

(3) -e cable restrainers with an appropriate stiffnesscould effectively reduce the longitudinal residualdisplacements of the bearings

(4) -e effect of the skew angle was less significant forthe bridge with the bearings than for a rigid-framebridge because of the sliding between the girders andthe laminated-rubber bearings

(5) -e residual displacements were more sensitive tothe variation in the friction coefficient between thelaminated-rubber bearings and the girders comparedwith the maximum seismic displacements However

Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)

0010 0015 00200005Friction coefficient

(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

50

100

150

200

250

300

350

Disp

(m

m)

(b)

Figure 20 Effect of CT on the seismic displacements of bearings (a) Maximum displacement (b) Residual displacement

CT = 0005CT = 0010

CT = 0015CT = 0020

0

1

2

3

4

5

μ

2 3 4 51Bent no

(a)

CT = 0005CT = 0010

CT = 0015CT = 0020

1

2

3

4

5

6

μ

2 3 4 51Bent no

(b)

Figure 21 Effect of CT on the displacement ductility demands of bents (a) Longitudinal direction (b) Transverse direction


the displacement ductility demands of the bentsincreased with an increase in the friction coefficient-e friction coefficient between the PTFE slidingbearings and the girders had an insignificant impacton the seismic response of the bridge

As the current study considered one specific type of skewbridges without considering the effect of geometric varia-tions such as bent height and span length a further studyusing different skew bridges with various configurationsshould be conducted for a better understanding of theseismic response of such irregular bridges -is study in-vestigated the seismic performance of a laminated-rubberbearing-supported skew bridge considering bearing slidingonly under horizontal earthquakes In addition to the cablerestrainers the effectiveness of other possible restrainerssuch as concrete shear keys and yielding steel dampersshould be studied for better seismic retrofitting of skewbridges

Data Availability

All data and models generated or used during the study areavailable from the corresponding author upon request (datafor figures and finite element models)

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is study was financially supported by the Scientific andTechnical Project on Western Transportation Construction(no 2008318200098) for the project ldquoSeismic DamageEvaluation Mechanism Analysis and Evaluation of SeismicFortification Criterion on Highway during WenchuanEarthquakerdquo and the Postdoctoral Science Fund Project ofChongqing Natural Science Foundation (no cstc2020jcyj-bshX0086) -e supports are gratefully acknowledged

References

[1] F Seible and M J N Priestley ldquoLessons learned from bridgeperformance during Northridge earthquakerdquo Special Publi-cation vol 187 pp 29ndash56 1999

[2] S Hashimoto Y Fujino and M Abe ldquoDamage analysis ofHanshin Expressway viaducts during 1995 Kobe earthquakeII damage mode of single reinforced concrete piersrdquo Journalof Bridge Engineering vol 10 no 1 pp 54ndash60 2005

[3] Y T Hsu and C C Fu ldquoSeismic effect on highway bridges inChi Chi earthquakerdquo Journal of Performance of ConstructedFacilities vol 18 no 1 pp 47ndash53 2004

[4] Q Han X Du J Liu Z Li L Li and J Zhao ldquoSeismic damageof highway bridges during the 2008 Wenchuan earthquakerdquoEarthquake Engineering and Engineering Vibration vol 8no 2 pp 263ndash273 2009

[5] N Xiang and J Li ldquoSeismic performance of highway bridgeswith different transverse unseating-prevention devicesrdquoJournal of Bridge Engineering vol 21 no 9 Article ID04016045 2016

[6] J Li N Xiang H Tang and Z Guan ldquoShake-table tests andnumerical simulation of an innovative isolation system forhighway bridgesrdquo Soil Dynamics and Earthquake Engineeringvol 86 pp 55ndash70 2016

[7] J Li T Peng and Y Xu ldquoDamage investigation of girderbridges under the Wenchuan earthquake and correspondingseismic design recommendationsrdquo Earthquake Engineeringand Engineering Vibration vol 7 no 4 pp 337ndash344 2008

[8] D Konstantinidis J M Kelly and N Makris ldquoExperimentalinvestigation on the seismic response of bridge bearingsrdquoEarthquake Engineering Research Center University ofCalifornia Berkeley CA USA UCBEERC-200802 2008

[9] J S Steelman L A Fahnestock E T Filipov et al ldquoShear andfriction response of nonseismic laminated elastomeric bridgebearings subject to seismic demandsrdquo Journal of Bridge En-gineering vol 18 no 7 pp 612ndash623 2012

[10] N Xiang and J Li ldquoExperimental and numerical study onseismic sliding mechanism of laminated-rubber bearingsrdquoEngineering Structures vol 141 pp 159ndash174 2017

[11] N Xiang M S Alam and J Li ldquoShake table studies of ahighway bridge model by allowing the sliding of laminated-rubber bearings with and without restraining devicesrdquo En-gineering Structures vol 171 pp 583ndash601 2018

[12] N Xiang M S Alam and J Li ldquoYielding steel dampers asrestraining devices to control seismic sliding of laminatedrubber bearings for highway bridges analytical and experi-mental studyrdquo Journal of Bridge Engineering vol 24 no 11Article ID 04019103 2019

[13] M Saiidi M Randall E Maragakis and T Isakovic ldquoSeismicrestrainer design methods for simply supported bridgesrdquoJournal of Bridge Engineering vol 6 no 5 pp 307ndash315 2001

[14] J-H Won H-S Mha K-I Cho and S-H Kim ldquoEffects ofthe restrainer upon bridge motions under seismic excita-tionsrdquo Engineering Structures vol 30 no 12 pp 3532ndash35442008

[15] J-q Wang S Li F Hedayati Dezfuli and M S AlamldquoSensitivity analysis and multi-criteria optimization of SMAcable restrainers for longitudinal seismic protection of iso-lated simply supported highway bridgesrdquo EngineeringStructures vol 189 pp 509ndash522 2019

[16] S Li F H Dezfuli J-q Wang and M S Alam ldquoLongitudinalseismic response control of long-span cable-stayed bridgesusing shape memory alloy wire-based lead rubber bearingsunder near-fault recordsrdquo Journal of Intelligent MaterialSystems and Structures vol 29 no 5 pp 703ndash728 2018

[17] A A Ghobarah and W K Tso ldquoSeismic analysis of skewedhighway bridges with intermediate supportsrdquo EarthquakeEngineering amp Structural Dynamics vol 2 no 3 pp 235ndash2481973

[18] R S Srinivasan and K Munaswamy ldquoDynamic response ofskew bridge decksrdquo Earthquake Engineering amp StructuralDynamics vol 6 no 2 pp 139ndash156 1978

[19] E A Maragakis and P C Jennings ldquoAnalytical models for therigid body motions of skew bridgesrdquo Earthquake Engineeringamp Structural Dynamics vol 15 no 8 pp 923ndash944 1987

[20] J Y Meng and E M Lui ldquoSeismic analysis and assessment ofa skew highway bridgerdquo Engineering Structures vol 22 no 11pp 1433ndash1452 2000

[21] J Chen Q Han X Liang and X Du ldquoEffect of pounding onnonlinear seismic response of skewed highway bridgesrdquo SoilDynamics and Earthquake Engineering vol 103 pp 151ndash1652017

[22] A Abdel-Mohti and G Pekcan ldquoEffect of skew angle onseismic vulnerability of RC box-girder highway bridgesrdquo


International Journal of Structural Stability and Dynamicsvol 13 no 6 Article ID 1350013 2013

[23] S P Deepu K Prajapat and S Ray-Chaudhuri ldquoSeismicvulnerability of skew bridges under bi-directional groundmotionsrdquo Engineering Structures vol 71 pp 150ndash160 2014

[24] SWu I G Buckle A M Itani et al ldquoExperimental studies onseismic response of skew bridges with seat-type abutments Ishake table experimentsrdquo Journal of Bridge Engineeringvol 24 no 10 Article ID 04019096 2019

[25] SWu I G Buckle A M Itani et al ldquoExperimental studies onseismic response of skew bridges with seat-type AbutmentsII resultsrdquo Journal of Bridge Engineering vol 24 no 10Article ID 04019097 2019

[26] 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

[27] S M Rasouli and M Mahmoodi ldquoAssessment the effect ofskewness and number of spans in seismic behavior of bridgeswith continuous multiple spans using MPArdquo KSCE Journal ofCivil Engineering vol 22 no 4 pp 1328ndash1335 2018

[28] Ministry of Transport of the Peoplersquos Republic of ChinaldquoGuidelines for seismic design of highway bridgesrdquo Ministryof Transport of the Peoplersquos Republic of China Beijing ChinaJTGT B02-01-2008 2008

[29] E L Wilson Bree Dimensional Static and Dynamic Analysisof Structures Computers and Structures Berkeley CA USA2000

[30] J B Mander M J N Priestley and R Park ldquo-eoreticalstress-strain model for confined concreterdquo Journal of Struc-tural Engineering vol 114 no 8 pp 1804ndash1826 1988

[31] Ministry of Transport of the Peoplersquos Republic of ChinaldquoCode for Design of Highway Reinforced Concrete andPrestressed Concrete Bridges and Culvertsrdquo Ministry ofTransport of the Peoplersquos Republic of China Beijing ChinaJTG D62-2004 2004

[32] X Huang and J Li Experimental and Analytical Study onUnseating Prevention System for Continuous Bridge TongjiUniversity Shanghai China 2009 In Chinese

[33] S D C Caltrans Caltrans Seismic Design Criteria Version 16California Department of Transportation Sacramento CAUSA 2010


could be approximated as a linear elastic response with aneffective shear modulus in the range of 610ndash1100 kPa -esliding coefficients of friction were observed to be inverselyrelated to the normal force and positively related to thesliding velocity

Considering the potential seismic damage of small-to-medium-span highway bridges with laminated-rubberbearings some restraining devices have been proposed toimprove the seismic performance of these bridges such asyielding steel dampers [11 12] using which the seismicgirder displacement can be effectively controlled while thebridges can achieve a reasonable balance between thebearing displacements and substructure seismic demandsAdditionally some studies [13ndash16] showed that cable re-strainers could reduce the residual displacement of bridgebearings under earthquake loadings Won et al [14] studiedthe seismic response of a multispan simply supported bridgewith cable restrainers -eir results showed that the relativedisplacement between the abutment and the next pier waseffectively controlled thus preventing the span from col-lapsing However few studies have investigated the effect oflaminated-rubber bearings associated with restrainers on theseismic performance of skew bridges

-e design of skew bridges is often dictated by theorientation of the roadway that includes the bridge relativeto the orientation of natural (rivers) and man-made (otherroads) objects Numerous studies [17ndash23] were conductedon the seismic response of skew bridges after the 1971 SanFernando 1994 Northridge and 2010 Chile earthquakesSkewed bridges exhibit a significantly more complex seismicresponse than do the straight bridges It was found that anin-plane rotation of the decks in skew bridges might causelarge torques in the columns thus making them morevulnerable Moreover a large skew is likely to increase therelative displacements between the girders and abutmentsthus making skew bridges more vulnerable to unseating andcollapse To understand the unseating mechanism of a skewbridge with seat-type abutments Wu et al [24 25] per-formed shake table tests of four single-span bridge modelswith skew angles of 0 (straight) 30 45 and 60 Based on thetest results on the unseating mechanism it was furtherconcluded [26] that the obtuse corner of the superstructure

of a skew bridge engaged the adjacent back wall and thesuperstructure then rotated about this corner -e impact ofthe span against the back wall was then followed immedi-ately by a rebound away from the wall and the span con-tinued to rotate in the same direction in free vibration aboutthe center of the stiffness of the substructure Most of theseaforementioned studies only focused on the seismic be-havior of single-span skew bridges or two-span shew bridgeswith rigid girder-deck connections Rasouli and Mahmoodi[27] assessed the effect of skewness and number of spans onthe seismic behavior of multispan skew bridges with rigidgirder-deck connections using a modal pushover analysis Intheir study an increase in the number of spans decreased thedeck rotation responses at the abutments and the adverseeffect of skewness

For a multispan skew girder bridge with laminated-rubber bearings bearing sliding could occur under strongearthquakes which might change the seismic response of thebridge and lead to a seismic damage However little researchhas been carried out on this type of bridges -is studypresents an investigation of the seismic performance of amultispan skew bridge with laminated-rubber bearingswhich suffered seismic damage during the 2008 Wenchuanearthquake To understand the seismic response of thebridge a nonlinear analysis was carried out using the finiteelement method In addition parametric studies includingbearing types skew angle friction coefficient and cablerestrainers were conducted and measures to improve theseismic performance of the bridge were also identified

2 Background and Ground Motions

21 Prototype Bridge To achieve the objectives of this studythe Duxiufeng Bridge located in Wenchuan China wasadopted as the research object -is bridge was approxi-mately 153 km from the epicenter and 57 km from the faultin the 2008 Wenchuan earthquake -e bridge has 6 spanswith a total length of 18806m and each span is 30m inlength Figure 2 shows the elevation and plan view of thebridge -e superstructure is supported on five bents eachconsisting of two 15m diameter columns Figure 3 shows atypical bent section-e heights of the columns from the topof the footing to the bottom of the cap are 830 1149 13011429 and 1428m for bents 1 to 5 respectively Eachcolumn is an extension of a 18m diameter pile -e top ofthe piles in each bent is linked by a transverse beam with across-section of 13times15m-e lengths of the piles are 43254130 3800 2700 and 2730m for bents 1 to 5 respectively-e height and width of the bent cap are 15m and 18mrespectively -e site consists mostly of gravel and sand andthe site class of the bridge is Type II according to the ChineseGuidelines for seismic design of highway bridges [28] Forthe superstructure four identical simply supported I-Shapedgirders are located in each span and two continuous deckswith a width of 85m each are placed on the three left andright spans Expansion joints with a length of 80 cm are usedat bent 3 and the abutments -e three left spans are slightlycurved with a radius of 200m while the three right spans arestraight -e skew angle is 47deg Polytetrafluoroethylene

Concrete shear key

Concrete substructure

Supporting pad

Superstructure

Laminated rubber bearingSteel plate

Steel plate

Figure 1 Layout of laminated-rubber bearings







0 1 2 3 4 5

6


50cos43deg


375cos43deg

2075 20753125

150

180

Center line

3125


50cos43deg

8515

065










b1113874 1113875 (1)


(a) (b)

(c) (d)



EW

ndash10

ndash05

00

05

10

Acc

eler

atio

n (g

)

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 1600Time (s)

NS

ndash10

ndash05

00

05

10

Acc

eler

atio

n (g

)

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 1600Time (s)

UD

ndash10

ndash05

00

05

10

Acc

eler

atio

n (g

)

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 1600Time (s)


(064 058)

(033 163)

NS

00

05

10

15

20

25

30

35

Acc

eler

atio

n (g

)

1 2 3 4 5 60Period (s)

(064 075)

(033 273)

EW

1 2 3 4 5 60Period (s)

00

05

10

15

20

25

30

35

Acc

eler

atio

n (g

)

UD

00

05

10

15

20

25

30

35

Acc

eler

atio

n (g

)

1 2 3 4 5 60Period (s)





P kd if dlt 0





Ke GA

1113936 t (3)





4 Response Analysis



Force

Displ

Ke Ke

Nμ

Nμ







XYZ















6080

40

Disp

(m

m)

200

Time (s)0 10 20 30 40 50



(a)

Time (s)0 10 20 30 40 50

6080

40

Disp

(m

m)

200



(b)




2 3 4 51Bent No

00

05

10

15

20

25

30

35

40

45

μ
















η D minus D

cable

D (4)






00

05

10

15

20

μ

2 3 4 51Bent no

(a)


2 3 4 51Bent no

0

2

4

6

8

μ

(b)




Bent 3 Abutment 6

Abutment 0

Cable restrainers

Stiffness

Disp

Force



Span 1

Span 2Span 3

Force


Ke

Disp











0

30

60

90

120

150M

ax b

earin

g re

sidua

l disp

(m

m)


(a)



0

50

100

150

200

250

Max

bea

ring

resid

ual d

isp (

mm

)


(b)













0

1

2

3

4

PGA

(g)

1 2 3 40Period (s)






00

05

10

15

20

25

30

Perio

d (s

)

1 2 3 4 5 6 7 80Mode number







Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)


(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

100

200

300

400

Disp

(m

m)

(b)




15

20

25

30

35

40

45

μ

2 3 4 51Bent no

(a)



20

25

30

35

40

45

50

55

60

μ

2 3 4 51Bent no

(b)







Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700

Disp

(m

m)


(a)

Avg(L)Avg(T)

Max(L)Max(T)

0

100

200

300

400

500

Disp

(m

m)


(b)


CR = 02CR = 03

CR = 04CR = 05

15

20

25

30

35

40

45

50

μ

2 3 4 51Bent no

(a)

CR = 02CR = 03

CR = 04CR = 05

20

25

30

35

40

45

50

55

60

65

70

μ

2 3 4 51Bent no

(b)




6 Conclusions








Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)


(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

50

100

150

200

250

300

350

Disp

(m

m)

(b)


CT = 0005CT = 0010

CT = 0015CT = 0020

0

1

2

3

4

5

μ

2 3 4 51Bent no

(a)

CT = 0005CT = 0010

CT = 0015CT = 0020

1

2

3

4

5

6

μ

2 3 4 51Bent no

(b)





Data Availability




Acknowledgments


References










































0 1 2 3 4 5

6


50cos43deg


375cos43deg

2075 20753125

150

180

Center line

3125


50cos43deg

8515

065










b1113874 1113875 (1)


(a) (b)

(c) (d)



EW

ndash10

ndash05

00

05

10

Acc

eler

atio

n (g

)

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 1600Time (s)

NS

ndash10

ndash05

00

05

10

Acc

eler

atio

n (g

)

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 1600Time (s)

UD

ndash10

ndash05

00

05

10

Acc

eler

atio

n (g

)

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 1600Time (s)


(064 058)

(033 163)

NS

00

05

10

15

20

25

30

35

Acc

eler

atio

n (g

)

1 2 3 4 5 60Period (s)

(064 075)

(033 273)

EW

1 2 3 4 5 60Period (s)

00

05

10

15

20

25

30

35

Acc

eler

atio

n (g

)

UD

00

05

10

15

20

25

30

35

Acc

eler

atio

n (g

)

1 2 3 4 5 60Period (s)





P kd if dlt 0





Ke GA

1113936 t (3)





4 Response Analysis



Force

Displ

Ke Ke

Nμ

Nμ







XYZ















6080

40

Disp

(m

m)

200

Time (s)0 10 20 30 40 50



(a)

Time (s)0 10 20 30 40 50

6080

40

Disp

(m

m)

200



(b)




2 3 4 51Bent No

00

05

10

15

20

25

30

35

40

45

μ
















η D minus D

cable

D (4)






00

05

10

15

20

μ

2 3 4 51Bent no

(a)


2 3 4 51Bent no

0

2

4

6

8

μ

(b)




Bent 3 Abutment 6

Abutment 0

Cable restrainers

Stiffness

Disp

Force



Span 1

Span 2Span 3

Force


Ke

Disp











0

30

60

90

120

150M

ax b

earin

g re

sidua

l disp

(m

m)


(a)



0

50

100

150

200

250

Max

bea

ring

resid

ual d

isp (

mm

)


(b)













0

1

2

3

4

PGA

(g)

1 2 3 40Period (s)






00

05

10

15

20

25

30

Perio

d (s

)

1 2 3 4 5 6 7 80Mode number







Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)


(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

100

200

300

400

Disp

(m

m)

(b)




15

20

25

30

35

40

45

μ

2 3 4 51Bent no

(a)



20

25

30

35

40

45

50

55

60

μ

2 3 4 51Bent no

(b)







Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700

Disp

(m

m)


(a)

Avg(L)Avg(T)

Max(L)Max(T)

0

100

200

300

400

500

Disp

(m

m)


(b)


CR = 02CR = 03

CR = 04CR = 05

15

20

25

30

35

40

45

50

μ

2 3 4 51Bent no

(a)

CR = 02CR = 03

CR = 04CR = 05

20

25

30

35

40

45

50

55

60

65

70

μ

2 3 4 51Bent no

(b)




6 Conclusions








Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)


(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

50

100

150

200

250

300

350

Disp

(m

m)

(b)


CT = 0005CT = 0010

CT = 0015CT = 0020

0

1

2

3

4

5

μ

2 3 4 51Bent no

(a)

CT = 0005CT = 0010

CT = 0015CT = 0020

1

2

3

4

5

6

μ

2 3 4 51Bent no

(b)





Data Availability




Acknowledgments


References












































b1113874 1113875 (1)


(a) (b)

(c) (d)



EW

ndash10

ndash05

00

05

10

Acc

eler

atio

n (g

)

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 1600Time (s)

NS

ndash10

ndash05

00

05

10

Acc

eler

atio

n (g

)

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 1600Time (s)

UD

ndash10

ndash05

00

05

10

Acc

eler

atio

n (g

)

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 1600Time (s)


(064 058)

(033 163)

NS

00

05

10

15

20

25

30

35

Acc

eler

atio

n (g

)

1 2 3 4 5 60Period (s)

(064 075)

(033 273)

EW

1 2 3 4 5 60Period (s)

00

05

10

15

20

25

30

35

Acc

eler

atio

n (g

)

UD

00

05

10

15

20

25

30

35

Acc

eler

atio

n (g

)

1 2 3 4 5 60Period (s)





P kd if dlt 0





Ke GA

1113936 t (3)





4 Response Analysis



Force

Displ

Ke Ke

Nμ

Nμ







XYZ















6080

40

Disp

(m

m)

200

Time (s)0 10 20 30 40 50



(a)

Time (s)0 10 20 30 40 50

6080

40

Disp

(m

m)

200



(b)




2 3 4 51Bent No

00

05

10

15

20

25

30

35

40

45

μ
















η D minus D

cable

D (4)






00

05

10

15

20

μ

2 3 4 51Bent no

(a)


2 3 4 51Bent no

0

2

4

6

8

μ

(b)




Bent 3 Abutment 6

Abutment 0

Cable restrainers

Stiffness

Disp

Force



Span 1

Span 2Span 3

Force


Ke

Disp











0

30

60

90

120

150M

ax b

earin

g re

sidua

l disp

(m

m)


(a)



0

50

100

150

200

250

Max

bea

ring

resid

ual d

isp (

mm

)


(b)













0

1

2

3

4

PGA

(g)

1 2 3 40Period (s)






00

05

10

15

20

25

30

Perio

d (s

)

1 2 3 4 5 6 7 80Mode number







Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)


(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

100

200

300

400

Disp

(m

m)

(b)




15

20

25

30

35

40

45

μ

2 3 4 51Bent no

(a)



20

25

30

35

40

45

50

55

60

μ

2 3 4 51Bent no

(b)







Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700

Disp

(m

m)


(a)

Avg(L)Avg(T)

Max(L)Max(T)

0

100

200

300

400

500

Disp

(m

m)


(b)


CR = 02CR = 03

CR = 04CR = 05

15

20

25

30

35

40

45

50

μ

2 3 4 51Bent no

(a)

CR = 02CR = 03

CR = 04CR = 05

20

25

30

35

40

45

50

55

60

65

70

μ

2 3 4 51Bent no

(b)




6 Conclusions








Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)


(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

50

100

150

200

250

300

350

Disp

(m

m)

(b)


CT = 0005CT = 0010

CT = 0015CT = 0020

0

1

2

3

4

5

μ

2 3 4 51Bent no

(a)

CT = 0005CT = 0010

CT = 0015CT = 0020

1

2

3

4

5

6

μ

2 3 4 51Bent no

(b)





Data Availability




Acknowledgments


References





































EW

ndash10

ndash05

00

05

10

Acc

eler

atio

n (g

)

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 1600Time (s)

NS

ndash10

ndash05

00

05

10

Acc

eler

atio

n (g

)

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 1600Time (s)

UD

ndash10

ndash05

00

05

10

Acc

eler

atio

n (g

)

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 1600Time (s)


(064 058)

(033 163)

NS

00

05

10

15

20

25

30

35

Acc

eler

atio

n (g

)

1 2 3 4 5 60Period (s)

(064 075)

(033 273)

EW

1 2 3 4 5 60Period (s)

00

05

10

15

20

25

30

35

Acc

eler

atio

n (g

)

UD

00

05

10

15

20

25

30

35

Acc

eler

atio

n (g

)

1 2 3 4 5 60Period (s)





P kd if dlt 0





Ke GA

1113936 t (3)





4 Response Analysis



Force

Displ

Ke Ke

Nμ

Nμ







XYZ















6080

40

Disp

(m

m)

200

Time (s)0 10 20 30 40 50



(a)

Time (s)0 10 20 30 40 50

6080

40

Disp

(m

m)

200



(b)




2 3 4 51Bent No

00

05

10

15

20

25

30

35

40

45

μ
















η D minus D

cable

D (4)






00

05

10

15

20

μ

2 3 4 51Bent no

(a)


2 3 4 51Bent no

0

2

4

6

8

μ

(b)




Bent 3 Abutment 6

Abutment 0

Cable restrainers

Stiffness

Disp

Force



Span 1

Span 2Span 3

Force


Ke

Disp











0

30

60

90

120

150M

ax b

earin

g re

sidua

l disp

(m

m)


(a)



0

50

100

150

200

250

Max

bea

ring

resid

ual d

isp (

mm

)


(b)













0

1

2

3

4

PGA

(g)

1 2 3 40Period (s)






00

05

10

15

20

25

30

Perio

d (s

)

1 2 3 4 5 6 7 80Mode number







Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)


(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

100

200

300

400

Disp

(m

m)

(b)




15

20

25

30

35

40

45

μ

2 3 4 51Bent no

(a)



20

25

30

35

40

45

50

55

60

μ

2 3 4 51Bent no

(b)







Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700

Disp

(m

m)


(a)

Avg(L)Avg(T)

Max(L)Max(T)

0

100

200

300

400

500

Disp

(m

m)


(b)


CR = 02CR = 03

CR = 04CR = 05

15

20

25

30

35

40

45

50

μ

2 3 4 51Bent no

(a)

CR = 02CR = 03

CR = 04CR = 05

20

25

30

35

40

45

50

55

60

65

70

μ

2 3 4 51Bent no

(b)




6 Conclusions








Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)


(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

50

100

150

200

250

300

350

Disp

(m

m)

(b)


CT = 0005CT = 0010

CT = 0015CT = 0020

0

1

2

3

4

5

μ

2 3 4 51Bent no

(a)

CT = 0005CT = 0010

CT = 0015CT = 0020

1

2

3

4

5

6

μ

2 3 4 51Bent no

(b)





Data Availability




Acknowledgments


References







































P kd if dlt 0





Ke GA

1113936 t (3)





4 Response Analysis



Force

Displ

Ke Ke

Nμ

Nμ







XYZ















6080

40

Disp

(m

m)

200

Time (s)0 10 20 30 40 50



(a)

Time (s)0 10 20 30 40 50

6080

40

Disp

(m

m)

200



(b)




2 3 4 51Bent No

00

05

10

15

20

25

30

35

40

45

μ
















η D minus D

cable

D (4)






00

05

10

15

20

μ

2 3 4 51Bent no

(a)


2 3 4 51Bent no

0

2

4

6

8

μ

(b)




Bent 3 Abutment 6

Abutment 0

Cable restrainers

Stiffness

Disp

Force



Span 1

Span 2Span 3

Force


Ke

Disp











0

30

60

90

120

150M

ax b

earin

g re

sidua

l disp

(m

m)


(a)



0

50

100

150

200

250

Max

bea

ring

resid

ual d

isp (

mm

)


(b)













0

1

2

3

4

PGA

(g)

1 2 3 40Period (s)






00

05

10

15

20

25

30

Perio

d (s

)

1 2 3 4 5 6 7 80Mode number







Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)


(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

100

200

300

400

Disp

(m

m)

(b)




15

20

25

30

35

40

45

μ

2 3 4 51Bent no

(a)



20

25

30

35

40

45

50

55

60

μ

2 3 4 51Bent no

(b)







Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700

Disp

(m

m)


(a)

Avg(L)Avg(T)

Max(L)Max(T)

0

100

200

300

400

500

Disp

(m

m)


(b)


CR = 02CR = 03

CR = 04CR = 05

15

20

25

30

35

40

45

50

μ

2 3 4 51Bent no

(a)

CR = 02CR = 03

CR = 04CR = 05

20

25

30

35

40

45

50

55

60

65

70

μ

2 3 4 51Bent no

(b)




6 Conclusions








Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)


(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

50

100

150

200

250

300

350

Disp

(m

m)

(b)


CT = 0005CT = 0010

CT = 0015CT = 0020

0

1

2

3

4

5

μ

2 3 4 51Bent no

(a)

CT = 0005CT = 0010

CT = 0015CT = 0020

1

2

3

4

5

6

μ

2 3 4 51Bent no

(b)





Data Availability




Acknowledgments


References









































XYZ















6080

40

Disp

(m

m)

200

Time (s)0 10 20 30 40 50



(a)

Time (s)0 10 20 30 40 50

6080

40

Disp

(m

m)

200



(b)




2 3 4 51Bent No

00

05

10

15

20

25

30

35

40

45

μ
















η D minus D

cable

D (4)






00

05

10

15

20

μ

2 3 4 51Bent no

(a)


2 3 4 51Bent no

0

2

4

6

8

μ

(b)




Bent 3 Abutment 6

Abutment 0

Cable restrainers

Stiffness

Disp

Force



Span 1

Span 2Span 3

Force


Ke

Disp











0

30

60

90

120

150M

ax b

earin

g re

sidua

l disp

(m

m)


(a)



0

50

100

150

200

250

Max

bea

ring

resid

ual d

isp (

mm

)


(b)













0

1

2

3

4

PGA

(g)

1 2 3 40Period (s)






00

05

10

15

20

25

30

Perio

d (s

)

1 2 3 4 5 6 7 80Mode number







Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)


(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

100

200

300

400

Disp

(m

m)

(b)




15

20

25

30

35

40

45

μ

2 3 4 51Bent no

(a)



20

25

30

35

40

45

50

55

60

μ

2 3 4 51Bent no

(b)







Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700

Disp

(m

m)


(a)

Avg(L)Avg(T)

Max(L)Max(T)

0

100

200

300

400

500

Disp

(m

m)


(b)


CR = 02CR = 03

CR = 04CR = 05

15

20

25

30

35

40

45

50

μ

2 3 4 51Bent no

(a)

CR = 02CR = 03

CR = 04CR = 05

20

25

30

35

40

45

50

55

60

65

70

μ

2 3 4 51Bent no

(b)




6 Conclusions








Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)


(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

50

100

150

200

250

300

350

Disp

(m

m)

(b)


CT = 0005CT = 0010

CT = 0015CT = 0020

0

1

2

3

4

5

μ

2 3 4 51Bent no

(a)

CT = 0005CT = 0010

CT = 0015CT = 0020

1

2

3

4

5

6

μ

2 3 4 51Bent no

(b)





Data Availability




Acknowledgments


References












































6080

40

Disp

(m

m)

200

Time (s)0 10 20 30 40 50



(a)

Time (s)0 10 20 30 40 50

6080

40

Disp

(m

m)

200



(b)




2 3 4 51Bent No

00

05

10

15

20

25

30

35

40

45

μ
















η D minus D

cable

D (4)






00

05

10

15

20

μ

2 3 4 51Bent no

(a)


2 3 4 51Bent no

0

2

4

6

8

μ

(b)




Bent 3 Abutment 6

Abutment 0

Cable restrainers

Stiffness

Disp

Force



Span 1

Span 2Span 3

Force


Ke

Disp











0

30

60

90

120

150M

ax b

earin

g re

sidua

l disp

(m

m)


(a)



0

50

100

150

200

250

Max

bea

ring

resid

ual d

isp (

mm

)


(b)













0

1

2

3

4

PGA

(g)

1 2 3 40Period (s)






00

05

10

15

20

25

30

Perio

d (s

)

1 2 3 4 5 6 7 80Mode number







Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)


(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

100

200

300

400

Disp

(m

m)

(b)




15

20

25

30

35

40

45

μ

2 3 4 51Bent no

(a)



20

25

30

35

40

45

50

55

60

μ

2 3 4 51Bent no

(b)







Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700

Disp

(m

m)


(a)

Avg(L)Avg(T)

Max(L)Max(T)

0

100

200

300

400

500

Disp

(m

m)


(b)


CR = 02CR = 03

CR = 04CR = 05

15

20

25

30

35

40

45

50

μ

2 3 4 51Bent no

(a)

CR = 02CR = 03

CR = 04CR = 05

20

25

30

35

40

45

50

55

60

65

70

μ

2 3 4 51Bent no

(b)




6 Conclusions








Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)


(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

50

100

150

200

250

300

350

Disp

(m

m)

(b)


CT = 0005CT = 0010

CT = 0015CT = 0020

0

1

2

3

4

5

μ

2 3 4 51Bent no

(a)

CT = 0005CT = 0010

CT = 0015CT = 0020

1

2

3

4

5

6

μ

2 3 4 51Bent no

(b)





Data Availability




Acknowledgments


References


















































η D minus D

cable

D (4)






00

05

10

15

20

μ

2 3 4 51Bent no

(a)


2 3 4 51Bent no

0

2

4

6

8

μ

(b)




Bent 3 Abutment 6

Abutment 0

Cable restrainers

Stiffness

Disp

Force



Span 1

Span 2Span 3

Force


Ke

Disp











0

30

60

90

120

150M

ax b

earin

g re

sidua

l disp

(m

m)


(a)



0

50

100

150

200

250

Max

bea

ring

resid

ual d

isp (

mm

)


(b)













0

1

2

3

4

PGA

(g)

1 2 3 40Period (s)






00

05

10

15

20

25

30

Perio

d (s

)

1 2 3 4 5 6 7 80Mode number







Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)


(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

100

200

300

400

Disp

(m

m)

(b)




15

20

25

30

35

40

45

μ

2 3 4 51Bent no

(a)



20

25

30

35

40

45

50

55

60

μ

2 3 4 51Bent no

(b)







Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700

Disp

(m

m)


(a)

Avg(L)Avg(T)

Max(L)Max(T)

0

100

200

300

400

500

Disp

(m

m)


(b)


CR = 02CR = 03

CR = 04CR = 05

15

20

25

30

35

40

45

50

μ

2 3 4 51Bent no

(a)

CR = 02CR = 03

CR = 04CR = 05

20

25

30

35

40

45

50

55

60

65

70

μ

2 3 4 51Bent no

(b)




6 Conclusions








Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)


(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

50

100

150

200

250

300

350

Disp

(m

m)

(b)


CT = 0005CT = 0010

CT = 0015CT = 0020

0

1

2

3

4

5

μ

2 3 4 51Bent no

(a)

CT = 0005CT = 0010

CT = 0015CT = 0020

1

2

3

4

5

6

μ

2 3 4 51Bent no

(b)





Data Availability




Acknowledgments


References













































0

30

60

90

120

150M

ax b

earin

g re

sidua

l disp

(m

m)


(a)



0

50

100

150

200

250

Max

bea

ring

resid

ual d

isp (

mm

)


(b)













0

1

2

3

4

PGA

(g)

1 2 3 40Period (s)






00

05

10

15

20

25

30

Perio

d (s

)

1 2 3 4 5 6 7 80Mode number







Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)


(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

100

200

300

400

Disp

(m

m)

(b)




15

20

25

30

35

40

45

μ

2 3 4 51Bent no

(a)



20

25

30

35

40

45

50

55

60

μ

2 3 4 51Bent no

(b)







Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700

Disp

(m

m)


(a)

Avg(L)Avg(T)

Max(L)Max(T)

0

100

200

300

400

500

Disp

(m

m)


(b)


CR = 02CR = 03

CR = 04CR = 05

15

20

25

30

35

40

45

50

μ

2 3 4 51Bent no

(a)

CR = 02CR = 03

CR = 04CR = 05

20

25

30

35

40

45

50

55

60

65

70

μ

2 3 4 51Bent no

(b)




6 Conclusions








Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)


(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

50

100

150

200

250

300

350

Disp

(m

m)

(b)


CT = 0005CT = 0010

CT = 0015CT = 0020

0

1

2

3

4

5

μ

2 3 4 51Bent no

(a)

CT = 0005CT = 0010

CT = 0015CT = 0020

1

2

3

4

5

6

μ

2 3 4 51Bent no

(b)





Data Availability




Acknowledgments


References












































0

1

2

3

4

PGA

(g)

1 2 3 40Period (s)






00

05

10

15

20

25

30

Perio

d (s

)

1 2 3 4 5 6 7 80Mode number







Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)


(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

100

200

300

400

Disp

(m

m)

(b)




15

20

25

30

35

40

45

μ

2 3 4 51Bent no

(a)



20

25

30

35

40

45

50

55

60

μ

2 3 4 51Bent no

(b)







Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700

Disp

(m

m)


(a)

Avg(L)Avg(T)

Max(L)Max(T)

0

100

200

300

400

500

Disp

(m

m)


(b)


CR = 02CR = 03

CR = 04CR = 05

15

20

25

30

35

40

45

50

μ

2 3 4 51Bent no

(a)

CR = 02CR = 03

CR = 04CR = 05

20

25

30

35

40

45

50

55

60

65

70

μ

2 3 4 51Bent no

(b)




6 Conclusions








Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)


(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

50

100

150

200

250

300

350

Disp

(m

m)

(b)


CT = 0005CT = 0010

CT = 0015CT = 0020

0

1

2

3

4

5

μ

2 3 4 51Bent no

(a)

CT = 0005CT = 0010

CT = 0015CT = 0020

1

2

3

4

5

6

μ

2 3 4 51Bent no

(b)





Data Availability




Acknowledgments


References









































Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)


(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

100

200

300

400

Disp

(m

m)

(b)




15

20

25

30

35

40

45

μ

2 3 4 51Bent no

(a)



20

25

30

35

40

45

50

55

60

μ

2 3 4 51Bent no

(b)







Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700

Disp

(m

m)


(a)

Avg(L)Avg(T)

Max(L)Max(T)

0

100

200

300

400

500

Disp

(m

m)


(b)


CR = 02CR = 03

CR = 04CR = 05

15

20

25

30

35

40

45

50

μ

2 3 4 51Bent no

(a)

CR = 02CR = 03

CR = 04CR = 05

20

25

30

35

40

45

50

55

60

65

70

μ

2 3 4 51Bent no

(b)




6 Conclusions








Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)


(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

50

100

150

200

250

300

350

Disp

(m

m)

(b)


CT = 0005CT = 0010

CT = 0015CT = 0020

0

1

2

3

4

5

μ

2 3 4 51Bent no

(a)

CT = 0005CT = 0010

CT = 0015CT = 0020

1

2

3

4

5

6

μ

2 3 4 51Bent no

(b)





Data Availability




Acknowledgments


References









































Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700

Disp

(m

m)


(a)

Avg(L)Avg(T)

Max(L)Max(T)

0

100

200

300

400

500

Disp

(m

m)


(b)


CR = 02CR = 03

CR = 04CR = 05

15

20

25

30

35

40

45

50

μ

2 3 4 51Bent no

(a)

CR = 02CR = 03

CR = 04CR = 05

20

25

30

35

40

45

50

55

60

65

70

μ

2 3 4 51Bent no

(b)




6 Conclusions








Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)


(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

50

100

150

200

250

300

350

Disp

(m

m)

(b)


CT = 0005CT = 0010

CT = 0015CT = 0020

0

1

2

3

4

5

μ

2 3 4 51Bent no

(a)

CT = 0005CT = 0010

CT = 0015CT = 0020

1

2

3

4

5

6

μ

2 3 4 51Bent no

(b)





Data Availability




Acknowledgments


References






































6 Conclusions








Avg(L)Avg(T)

Max(L)Max(T)

100

200

300

400

500

600

700D

isp (

mm

)


(a)

Avg(L)Avg(T)

Max(L)Max(T)


0

50

100

150

200

250

300

350

Disp

(m

m)

(b)


CT = 0005CT = 0010

CT = 0015CT = 0020

0

1

2

3

4

5

μ

2 3 4 51Bent no

(a)

CT = 0005CT = 0010

CT = 0015CT = 0020

1

2

3

4

5

6

μ

2 3 4 51Bent no

(b)





Data Availability




Acknowledgments


References







































Data Availability




Acknowledgments


References





































seismicstudyofskewbridgesupportedon laminated-rubberbearings

Documents