Research ArticleApplicability of a Simplified SDOF Method in LongitudinalDeck-Pier Poundings of Simply Supported Girder Bridges

Tianbo Peng12 and Ning Guo12

1State Key Laboratory of Disaster Reduction in Civil Engineering Tongji University Shanghai 200092 China2College of Civil Engineering Tongji University Shanghai 200092 China

Correspondence should be addressed to Tianbo Peng ptbtongjieducn

Received 2 January 2016 Accepted 20 March 2016

Academic Editor Stefano Sorace

Copyright copy 2016 T Peng and N GuoThis 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

The pounding issue between decks in the earthquake has been a great concern of many researchers but the research on the deck-pier pounding issue was inadequate In this paper a simplified SDOFmethod was proposed to study the issue for simply supportedgirder bridges Theoretical analysis shaking table test and finite element analysis were conducted to study the applicability of thesimplified SDOF method in longitudinal deck-pier poundings A whole structural model and a SDOF model for the longitudinalpounding issue were also established to study influences of the pier stiffness and the pier mass on longitudinal pounding responsesIt is shown that the simplified SDOF method can estimate the pounding force and deck displacement fairly accurately for almostall cases The pier mass has little effect on pounding responses except for bridges with very rigid piers but the pier stiffness has agreat influence The larger the pier stiffness is the higher the peak pounding force is

1 Introduction

The pounding issue of bridges under earthquakes has beena research focus studied by a lot of scholars in recent yearsSome kinds of classical methods were proposed for the issueFor example stereo mechanical approach [1 2] which isa theoretical approach based on momentum conservationtheorem of two particles was used by several researchersIn stereo mechanical approach the coefficient of restitution(COR) was used to represent the ratio of speeds after andbefore an impact Pairs of particles with COR = 1 collideelastically while particles with COR lt 1 collide inelasticallyFor aCOR=0 the particles effectively ldquostoprdquo at the poundingThe concept of the approach is simple and complicatedcalculations are avoidedHowever amain shortcoming is thatthe pounding time must be very short to neglect effects ofother forces in the pounding process

Compared with the stereo mechanical approach severalkinds of contact element methods were more often usedfor their convenience for commercial finite element software[3] In these methods the impact of two pounding objectsis simulated by a contact element inserted between the

two objects These methods have been improved a lot inorder to simulate different types of poundings Contactelement methods include several kinds of models such aslinear spring model Kelvin model Hertz model and Hertzdamping model These models have been widely used toinvestigate pounding responses of buildings and bridges [4ndash13] Contact elements are built by springs dampers andgaps by series-parallel connections The concept of contactelement method is clear and different types of poundings canbe simulated using a variety of models Pounding force canbe calculated using finite element software easily

Goldsmith suggested a one-dimensional wave methodto study the pounding issue theoretically [1] Based onthe method Watanabe and Kawashima proposed a contactelement method to simulate the coaxial pounding issuebetween adjacent decks [14] Parametric analyses of meshdensity and stiffness of the contact element were conductedand it was found that results obtained by FEM and by theone-dimensional wave method would be very close if thestiffness of the contact element was close to the axial stiffnessbetween two adjacent particles aftermeshing of the deck Andit was also found that the finer the meshing of the deck was

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 3569674 13 pageshttpdxdoiorg10115520163569674

2 Shock and Vibration

Deck Block

Pier

(a) Pounding between decks and blocks

Higher deck

Pier

(b) Pounding between the higher deck and the pier top

Figure 1 Examples related to the deck-pier pounding issue

o Deck Stationary pier

(a) Deck-pier pounding issue

Pier stiffness

o

Nonpounding end

Pounding end

(b) A one-dimensional wave theoretical model

Figure 2 A one-dimensional wave theoretical model for the deck-pier pounding issue

the closer the results obtained by FEM were to the resultsobtained by the one-dimensional wave method

Up to now many researches were focused on poundingsbetween the bridge deck and the abutment Dimitrakopoulosproposed a novel nonsmooth rigid body approach to analyzeseismic responses of pounding skew bridges which involvedoblique frictional multicontact phenomena [15] Li et alevaluated influences of abutment excitations on the poundingbehavior of a system consisting of a single bridge spanwith various fundamental frequencies and its abutmentsexperimentally [16] Wang and Shih presented a case studyon the sliding of bridge decks of a multispan concrete bridgeand pounding at abutment-backfill under the 1999 Chi-Chiearthquake [17] Huo and Zhang studied effects of poundingand skewness on the seismic behaviors of typical three-spanbridges in California [18]

There were also many researches on poundings betweenbridge decks Bi andHao performednumerical simulations ofpounding damage between bridge decks based on a detailed3D finite element model [19] Abdel Raheem developedan analytical model of expansion joints considering theinteraction between adjacent bridge segments and the effectof impact and restrainers [20] Li and Chouw studied theinelastic response of a two-segment bridge due to spatiallyvarying ground motion including pounding using shaketables [21] Kim and Shinozuka studied the effect of poundingat expansion joints on concrete bridge response to earthquakeground motions [22]

However there is little research on the deck-pier pound-ing issue but this kind of phenomenon was far from rareFor example decks and unseating prevention blocks onpiers would collide with each other in the longitudinal andtransverse directions in earthquakes as shown in Figure 1(a)[23]

Another example is related to a pier supporting two deckswith different heights The higher deck will hit the pier top inthe longitudinal direction as shown in Figure 1(b)

Because of the flexibility of piers the deck-pier poundingissue is different from the pounding issue between decks orthat between the bridge deck and the abutment so poundingresponses should receive adequate attention

Taking a simply supported girder bridge as an examplethe deck-pier pounding issue in the longitudinal directionis studied using two theoretical methods at first in thispaper According to the findings a simplified single degree offreedom (SDOF)model is established and verified by shakingtable test method At last influences of the pier stiffness andthe pier mass on pounding responses of a simply supportedgirder bridge were studied by finite element method

2 Theoretical Analysis of the Deck-PierPounding Issue

Two theoreticalmodels are used in this section to simulate thepounding between a deck with an initial velocity and the topof a stationary pier as shown in Figure 2(a)The first model is

Shock and Vibration 3

V2c

minusV2c

minusV2c

f998400+g998400

c(f998400minusg998400 )

f998400

f998400

f998400

g998400 g998400

g998400

V V

0

t = 0

minusP(t)EA

t = L12c t = 2L1c

V2c minus P(t)EA

V minus cP(t)EA

Figure 3 Waveforms of the first-order derivatives of 119891 and 119892

based on the one-dimensional wave theory which had beensuccessfully applied on the deck-deck pounding issue [1]Theinfluence of the piermass on pounding responses is neglectedfor the possibility of solving differential equations So thetheory is not applicable for very tall or heavy piers In the one-dimensional wave theory the deck mass and pier flexibilityare considered The model is shown in Figure 2(b) Supposethat piers are not damaged during the pounding the pierstiffness is a constant

Physical meanings of parameters in the theoretical anal-ysis are as follows

119864 119860 119871 and 120588 are elastic modulus cross-sectionalarea length and density of the deck respectively119881 is the initial velocity of the deck before pounding119875(119905) is the time history of the pounding force120573 is the ratio of the horizontal stiffness of bridge pierto the axial stiffness of the deck119896119901 = 120573119864119860119871 is the horizontal stiffness of bridge pier119906(119909 119905) is the displacement of the particle which is at adistance of 119909 from the nonpounding end of the deckat time 119905 According to the one-dimensional wavetheory 119906(119909 119905) = 119891(119909 + 119888119905) + 119892(119909 minus 119888119905) And 119891 and119892 are displacement waves propagating in the positiveand negative directions respectively1198911015840 and 1198921015840 are thefirst-order derivatives of 119891 and 119892 119888 = radic119864120588 is thepropagation velocity of the displacement wave

The model shown in Figure 2(b) is analyzed theoreticallyusing the one-dimensional wave theory According to initialconditions that the initial velocity is119881 and the initial strain iszero when 119905 = 0 we have

120597119906 (119909 119905)

120597119905

= 119888 (1198911015840minus 1198921015840) = 119881

120597119906 (119909 119905)

120597119909

= 1198911015840+ 1198921015840= 0

997904rArr 1198911015840= minus1198921015840=

119881

2119888

(1)

According to the boundary condition that the strain iszero at 119909 = 0 we have

1198911015840= minus1198921015840 (2)

According to the boundary condition that the strain isminus119875(119905)119864119860 at 119909 = 119871 we have

1198911015840= minus

119875 (119905)

119864119860

minus 1198921015840 (3)

According to the one-dimensional wave theory the first-order derivatives of displacement wave equations can beshown in Figure 3 The wave equations of the nonpoundingend and the pounding end are determined based on (2) and(3)

For 119905 isin [0 2119871119888] the velocity time history at 119909 = 119871 is asfollows

120597119906 (119871 119905)

120597119905

= minus119888119891 (119871 119905) + 119888119892 (119871 119905) = 119881 minus

119888

119864119860

119875 (119905)

= 119881 minus

120573119888

119871

119906 (119871 119905)

(4)

Solving the differential equations we have

119906 (119871 119905) =

119881119871

120573119888

[1 minus exp(minus120573119888119905

119871

)] (5)

And the time history of the pounding force is

119875 (119905) =

119864119860119881

119888

[1 minus exp(minus120573119888119905

119871

)] (6)

For 119905 isin [2119871119888 4119871119888] the velocity time history at 119909 = 119871 isas follows

120597119906 (119871 119905)

120597119905

= 119881 minus

119888

119864119860

119875 (119905) minus

2119888

119864119860

119875(119905 minus

2119871

119888

)

= 119881 minus

120573119888

119871

119906 (119871 119905)

minus 2119881[1 minus exp(minus120573119888119905

119871

+ 2120573)]

(7)

4 Shock and Vibration

0 1 2 3 4

00

01

02

03

04

05

120573 = 001

120573 = 002

120573 = 005

120573 = 01120573 = 02

120591

R(120591)

(a)

0 1 2 3 4

00

02

04

06

08

10

120573 = 05120573 = 10120573 = 20

120573 = 50

120573 = 100

120591

R(120591)

(b)

Figure 4 Time histories of the deck-pier pounding force

Solving the differential equations according to a continu-ous condition namely 119906(119871 2119871119888) = (119881119871120573119888)[1 minus exp(minus2120573)]we have

119906 (119871 119905)

=

119881119871

120573119888

[2 exp(minus120573119888119905

119871

+ 2120573) minus exp(minus120573119888119905

119871

) minus 1]

minus

4119881119871

119888

exp(minus120573119888119905

119871

+ 2120573)

+ 2119881119905 exp(minus120573119888119905

119871

+ 2120573)

(8)

And the time history of the pounding force is

119875 (119905)

=

119864119860119881

119888

[2 exp(minus120573119888119905

119871

+ 2120573) minus exp(minus120573119888119905

119871

) minus 1]

minus

4120573119864119860119881

119888

exp(minus120573119888119905

119871

+ 2120573)

+

2120573119864119860119881

119871

119905 exp(minus120573119888119905

119871

+ 2120573)

(9)

Introducing dimensionless pounding force 119877(119905) =

119875(119905)119888119864119860119881 and dimensionless time coordinate 120591 = 119888119905119871 thetime history of the dimensionless pounding force using theone-dimensional wave theory is as follows

119877 (120591) =

1 minus exp (minus120573120591) 0 le 120591 le 2

2 exp (minus120573120591 + 2120573) minus 4120573 exp (minus120573120591 + 2120573) + 2120573120591 exp (minus120573120591 + 2120573) minus exp (minus120573120591) minus 1 2 lt 120591 le 4

(10)

According to (10) time histories of the deck-pier pound-ing force for different 120573 can be drawn in Figure 4

As shown in Figure 4 for minor 120573 the deck-pier pound-ing force curve is close to a sinusoidal curve And formajor 120573the deck-pier pounding force curve is close to a rectangularcurve

The secondmodel is based on a simplified SDOFmethodIn the method the axial stiffness of the deck is assumed to befar larger than the pier stiffness which is usually valid andpractical Therefore the deck is regarded as a rigid body and

simulated by a particle with the deck mass and the pier issimulated by a spring as shown in Figure 5 119898 and 119896 are thedeck mass and the pier stiffness in the figure

For the simplified SDOF method the circular frequencyof the time history of the deck-pier pounding force is

120596 = radic120573119864119860119871

120588119860119871

=

119888

119871

radic120573 (11)

Shock and Vibration 5

Deck mass

Pier stiffness

Figure 5 A simplified SDOF theoretical model for the deck-pier pounding issue

0 1 2 3 4

00

01

02

03

04

05

120573 = 005

120573 = 01

120573 = 02

120591

R(120591)

(a)

0 1 2 3 4

00

02

04

06

08

10

120573 = 05120573 = 10

120573 = 04

120591

R(120591)

(b)

Figure 6 Comparisons of pounding force time histories between two theoretical models

Then time histories of the pounding force and the dimen-sionless pounding force are as follows

119875 (119905) = radic120573

119864119860119881

119888

sin(119888119905119871

radic120573) (12)

119875max = radic120573119864119860119881

119888

=radic119896119901119871

119864119860

119864119860119881

radic119864120588

= 119881radic120588119860119871 sdot 119896119901

= 119881radic119898 sdot 119896119901

(13)

119877 (120591) = radic120573 sin(radic120573120591) (14)

Time histories of the pounding force obtained using twotheoretical methods are compared in Figure 6 Solid linesindicate results of the one-dimensional wave theory andbroken lines indicate those of the simplified SDOF methodAs shown forminor120573 results of the simplified SDOFmethodagree well with those of the one-dimensional wave theoryHowever for major 120573 such as 120573 ge 04 differences betweenthe results are remarkable Therefore the deck should not beregarded as a rigid body when the pier stiffness is close to theaxial stiffness of the deck

Also as shown in Figure 6 as the axial flexibility of thedeck is considered in the one-dimensional wave theory thepounding duration will be extended and the peak poundingforce will be reduced

It should be noted that the simplified SDOF theoreticalmodel is linear and no energy dissipation is consideredTherefore if the pier is damaged heavily or much energy is

dissipated during the pounding process the simplified SDOFmethod is no more applicable

3 Experimental Study of Deck-Pier Pounding

Shaking table tests were conducted in the State Key Labora-tory of Disaster Reduction in Civil Engineering Tongji Uni-versity to verify the simplified SDOF method for the deck-pier pounding issue Two shaking tables named Table B andTable C were included to simulate ground motions

A simply supported bridge was used as the test specimenand consisted of two piers four bearings and a deck as shownin Figure 7 The two piers were installed on Table B andTable C respectively Each pier consisted of two columnsand one crossbeam The columns were fixed on the tableand the crossbeam was bolted on tops of columns Fourload transducers were placed on the two crossbeams Oneach load transducer one Double Spherical Seismic Isolationbearing (DSSI bearing) was placed The DSSI bearing wasdeveloped by substituting the flat sliding surface of anordinary spherical sliding bearing with a spherical slidingsurface Its working mechanism in earthquakes was similarto a Friction Pendulum Sliding (FPS) bearing The deck wassupported on the four bearings and steel plates were boltedon the deck to achieve an additional mass of 100 tons

Steel blocks were installed on the connection plates ofbearings on Table B to restrict excessive bearing displace-ments as shown in Figure 8 A certain horizontal distance wasdesigned between the block and the side of the upper bearingplate which was named Clearance in this paper

6 Shock and Vibration

Deck

Pier 1

Table B Table C

Load transducer

Pier 2

DSSI bearing

Steel block

Additional mass

Figure 7 The test setup and specimen

Lower bearing plateSteel block

Connection plate

SliderUpper bearing plate

Clearance

Figure 8 Configuration of a bearing with shear pins used as a fixed bearing

A record of Wenchuan Earthquake as shown in Figure 9was input in the cases The motion was recorded in the 119864-119882 direction in Wolong Sichuan Province The peak groundacceleration is 948 gal

Four shaking table test cases were arranged and listed inTable 1 As shown the peak ground acceleration was scaleddown in order not to damage load transducers

In Cases 1 and 3 a rubber plate was inserted in eachClearance to study the applicability of the method for dif-ferent pounding stiffness so pounding stiffness of these twocases was lower

Responses of the bridge model were measured by 33transducers collecting data from the bridge model with asampling frequency of 512Hz Bearing displacements were

Table 1 Test cases

Cases Clearance Peak acceleration1 10mm 02 g2 20mm 02 g3 50mm 05 g4 60mm 05 g

measured by 8 displacement transducers bearing forces weremeasured by 4 load transducers and acceleration for thetables decks and piers was measured by 21 accelerometers

In Figures 10ndash12 time histories of displacements forcesand relative velocities of bearings on Table B in all the four

Shock and Vibration 7

0 10 20 30 40 50minus800

minus600

minus400

minus200

0

200

400

600

800

Acce

lera

tion

(gal

)

Time (s)

(a) A ground acceleration record of Wenchuan Earthquake

0 1 2 3 4 5

0

500

1000

1500

2000

2500

3000

Pseu

doac

cele

ratio

n (g

al)

Period (s)

(b) Pseudoacceleration response spectra (5 damping)

Figure 9 A record of Wenchuan Earthquake used in the cases

10 20 30 40minus20

minus15

minus10

minus5

0

5

10

15

20

Disp

lace

men

t (m

m)

Time (t)

(a) Case 1

10 20 30 40minus30

minus20

minus10

0

10

20

30

Disp

lace

men

t (m

m)

Time (t)

(b) Case 2

10 20 30 40minus80

minus60

minus40

minus20

0

20

40

60

80

Disp

lace

men

t (m

m)

Time (t)

(c) Case 3

10 20 30 40minus80

minus60

minus40

minus20

0

20

40

60

80

Disp

lace

men

t (m

m)

Time (t)

(d) Case 4

Figure 10 Time histories of bearing displacements on Table B in all the cases

8 Shock and Vibration

minus80

minus60

minus40

minus20

0

20

40

60

80

10 20 30 40

Forc

e (kN

)

Time (t)

(a) Case 1

minus80

minus60

minus40

minus20

0

20

40

60

80

10 20 30 40

Forc

e (kN

)

Time (t)

(b) Case 2

minus200

minus150

minus100

minus50

0

50

100

150

200

10 20 30 40

Forc

e (kN

)

Time (t)

(c) Case 3

10 20 30 40minus200

minus150

minus100

minus50

0

50

100

150

200

Forc

e (kN

)

Time (t)

(d) Case 4

Figure 11 Time histories of bearing forces on Table B in all the cases

cases are shown The bearing displacement is the averagevalue of two bearings the bearing force is the resultant forceof two bearings and the relative velocity is the derivative ofthe bearing displacement

In order to verify the simplified SDOF method (13) isused to calculate the theoretical pounding force for eachpounding and then the theoretical pounding force 119865theoreticalis compared with the test pounding force 119865test

The theoretical pounding force is calculated and listedin Table 2 119898 is the pounding mass and equals 100 tonsfor all the cases 119896 is the pounding stiffness and equals thepounding force divided by the displacement measured in thetest Once the pounding happens bearing relative velocitydrops rapidly The bearing relative velocity just before themoment of pounding is extracted as the pounding velocity119881

As shown errors between theoretical pounding forcesand test pounding forces are less than 20 so the simplifiedSDOF method is verified By the comparison it is found thatthe deck can be simulated as a particle and the simplified

SDOF method can be used to study the deck-pier poundingissue

Because no energy dissipation is considered in themodelthe simplified SDOF method may overestimate the actualpounding force and the theoretical pounding force is alwayslarger than the test pounding force Therefore the simplifiedSDOF method can provide a reasonable upper limit ofpounding forcesThe actual pounding force is also influencedby energy dissipation and damage condition during thepounding process

4 FEM Analysis of the Deck-PierPounding Issue

In order to study pounding responses under longitudinalearthquakes and the applicability of the simplified SDOFmethod a typical simply supported girder bridge is studiedThe span is 16m and piers are both 5m high The left end ofthe deck is supported on two same expansion bearings and

Shock and Vibration 9

10 20 30 40minus150

minus100

minus50

0

50

100

150

Velo

city

(mm

s)

Time (t)

(a) Case 1

10 20 30 40minus150

minus100

minus50

0

50

100

150

Velo

city

(mm

s)

Time (t)

(b) Case 2

10 20 30 40minus400

minus300

minus200

minus100

0

100

200

300

400

Velo

city

(mm

s)

Time (t)

(c) Case 3

10 20 30 40minus400

minus300

minus200

minus100

0

100

200

300

400

Velo

city

(mm

s)

Time (t)

(d) Case 4

Figure 12 Time histories of bearing velocities on Table B in all the cases

Table 2 Comparisons of theoretical pounding forces and test pounding forces

Case Pounding number 119898 (ton) 119896 (kNmm) 119881 (mms) 119865theoretical (kN) 119865test (kN) Error ()1 1 100 1077 minus5861 minus6081 minus5533 9911 2 100 1265 5708 6420 6124 4831 3 100 1133 minus5581 minus5942 minus5020 18381 4 100 1391 minus5540 minus6533 minus6217 5081 5 100 1231 6463 7170 6708 6892 1 100 5025 minus3089 minus6924 minus6096 13583 1 100 1001 17079 17092 16657 2613 2 100 848 minus14378 minus13243 minus11373 16443 3 100 1415 18700 22247 19786 12444 1 100 4589 10447 22378 19045 17504 2 100 6546 minus8090 minus20698 minus17905 15604 3 100 5195 9835 22418 19196 16784 4 100 5071 minus5128 minus11548 minus10224 12954 5 100 5514 minus4639 minus10894 minus9116 1950

10 Shock and Vibration

Seismic isolation bearing Expansion bearing

Steel block

Clearance

Steel block

Clearance

Deck

Pier

(a) The whole structural model

Gap

Hook

Expansion bearing Seismic isolation bearing

Plastic Wen Plastic Wen

Gap

Hook

mdkpkp

(b) The SDOF model

Figure 13 Models of a simply supported girder bridge

Table 3 Parameters of a simply supported girder bridge model

Components ParametersDeck Elasticity modulus is 345 times 107 kPa density is 25 tonm3 and cross-sectional area is 54m2

Pier Elasticity modulus is 345 times 107 kPa density is 25 tonm3 and cross-sectional area is 20m2

Expansion bearing Horizontal stiffness before yielding is 432 times 104 kNm yield force is 432 kN and horizontal stiffness after yieldingis 0

Seismic isolationbearing

Horizontal stiffness before yielding is 432 times 104 kNm yield force is 432 kN and horizontal stiffness after yieldingis 216 kNm

Steel block Stiffness of both gap and hook link elements is 10 times 1010 kNm and Clearances are 01m

the right end of the deck is supported on two same seismicisolation bearings Steel blocks for unseating prevention areinstalled on both sides of each bearing Clearances are allset to be 01m for all the steel blocks Deck-pier poundingswill happen between the deck and steel blocks under longi-tudinal earthquakes until bearing displacements exceed thepreserved Clearances

41 Analysis Models and Seismic Input Two models of thesimply supported girder bridge are established with the FEMsoftware SAP2000 One is a whole structural model and theother one is a SDOF model For the whole structural model

all the components are simulated as shown in Figure 13(a)The deck and piers are simulated with frame elementsBearings are simulated with Plastic Wen link elements Steelblocks are simulated with gap and hook link elements All theelements and their parameters are listed in Table 3 For theSDOF model pier masses are neglected and the deck massis concentrated on a particle as shown in Figure 13(b) 119896119901represents the horizontal stiffness of piers and the deck mass119898119889 is 216 tons

Sinusoidal acceleration wave is input as the seismic exci-tation for simplicity Peak acceleration is 4ms2 frequency is1Hz duration is 5 s and integration step is 1ms

Shock and Vibration 11

minus3000minus2000minus1000

0100020003000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(a) 120573 = 00034

minus4000minus2000

020004000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(b) 120573 = 00138

minus6000minus4000minus2000

0200040006000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(c) 120573 = 0034

minus10000minus5000

05000

10000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(d) 120573 = 0103

minus20000minus15000minus10000minus5000

05000

100001500020000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(e) 120573 = 0345

0 1 2 3 4 5minus30000minus20000minus10000

0100002000030000

Bear

ing

forc

e (kN

)

Time (s)

(f) 120573 = 10

Figure 14 Force time histories of the expansion bearing

42 Applicability of the Simplified SDOFMethod To evaluatethe applicability of the simplified SDOF method results ofthe SDOF model for different pier stiffness are comparedwith those of the whole structural model The influenceof the pier stiffness on pounding response is studied byaltering the elastic modulus of piers The horizontal stiffnessis designed to be 40 times 104 kNm 16 times 105 kNm 40times 105 kNm 12 times 106 kNm 40 times 106 kNm and 116 times107 kNm respectively The axial stiffness of the deck is 116 times107 kNm So stiffness ratios are 00034 00138 0034 01030345 and 10 respectively

Force time histories of the expansion bearing of the twomodels are shown in Figure 14 The expansion bearing forceis composed of the friction force and the pounding force andthe latter is far larger than the former In the figure solid linesindicate results of the whole structural model and brokenlines indicate results of the SDOF model

Every spike in Figure 14 indicates one pounding betweenthe deck and the pier As shown along with the increase ofthe pier stiffness the pounding duration becomes shorter andthe pounding force becomes larger Differences between thetwo models are very small except for 120573 = 10 The poundingforce and the pounding time of the two models are almostthe same for the first five cases Therefore the SDOF modelcan estimate the pounding force fairly accurately for 120573 lt

0345 at least All the peak pounding forces calculated by theSDOF model are larger than those of the whole structural

model which is similar to the theoretical results It needs tobe noted that 120573 = 0345 is already very big pier stiffness andapplicability of the simplified SDOF method is verified foralmost all the bridges

For 120573 = 10 differences of the pounding force and thepounding time between the twomodels are much larger thanother cases So the SDOF model is not applicable anymoreHowever because the horizontal stiffness of bridge pier isalways much less than the axial stiffness of the deck 120573 = 10is a very rare condition

43 Influences of the Pier Stiffness and the Pier Mass onPounding Responses The influence of the pier horizontalstiffness on pounding responses is studied by altering theelastic modulus of piers as mentioned in Section 42 Theinfluence of the pier mass on pounding responses is studiedby altering the density of piers Pier masses are designed tobe 5 tons 10 tons 25 tons 50 tons and 125 tons respectivelywhich are named as M1 to M5

The maximum pounding responses of the whole struc-tural model are shown in Figure 15

As shown the pounding force will increase and thedeck displacement will decrease with the increase of thepier stiffness The influence of the pier mass on the deckdisplacement is very small

For minor pier stiffness the influence of the pier mass onthe pounding force is also not too much so the pier mass can

12 Shock and Vibration

3 10 100 10000

5000

10000

15000

20000

25000

30000Po

undi

ng fo

rce (

kN)

M1M2M3

M4M5

Pier stiffness (104 kNm)

(a) Maximum pounding forces

3 10 100 1000010

012

014

016

018

M1M2M3

M4M5

Gird

er d

ispla

cem

ent (

m)

Pier stiffness (104 kNm)

(b) Maximum deck displacements

Figure 15 Maximum pounding responses

be neglected and the SDOFmodel is applicable However theinfluence of the pier mass on the pounding force is bigger formajor pier stiffness and so the pier mass is not negligible andthe SDOF model is no more applicable

5 Conclusions

In this paper theoretical analysis shaking table test andfinite element analysis were conducted to study the deck-pierpounding issue the following conclusions are drawn

(1) Based on theoretical results and test results it wasfound that the longitudinal deck-pier pounding issuecan be simulated by the simplified SDOF method ifenergy dissipation is negligible and the pier stiffnessis much less than the axial stiffness of the deck

(2) It was found that the pounding duration becomesshorter and the pounding force becomes larger alongwith the increase of the pier stiffness

(3) The pier mass has little effect on pounding responsesexcept for bridges with very rigid piers

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of China (no 51278372) and the Ministry ofScience and Technology of China Grant no SLDRCE 14-B-15

References

[1] W Goldsmith Impact Dover New York NY USA 2001[2] G Cole R Dhakal A Carr and D Bull ldquoAn investigation

of the effects of mass distribution on pounding structuresrdquoEarthquake Engineering and Structural Dynamics vol 40 no6 pp 641ndash659 2011

[3] H Hao K Bi N Chouw and W-X Ren ldquoState-of-the-artreview on seismic induced pounding response of bridge struc-turesrdquo Journal of Earthquake and Tsunami vol 7 no 3 ArticleID 1350019 2013

[4] B F Maison and K Kasai ldquoAnalysis for a type of structuralpoundingrdquo Journal of Structural Engineering vol 116 no 4 pp957ndash977 1990

[5] B F Maison and K Kasai ldquoDynamics of pounding whentwo buildings colliderdquo Earthquake Engineering and StructuralDynamics vol 21 no 9 pp 771ndash786 1992

[6] R Jankowski KWilde and Y Fujino ldquoPounding of superstruc-ture segments in isolated elevated bridge during earthquakesrdquoEarthquake Engineering and Structural Dynamics vol 27 no 5pp 487ndash502 1998

[7] K T Chau and X X Wei ldquoPounding of structures modelled asnon-linear impacts of two oscillatorsrdquo Earthquake Engineeringand Structural Dynamics vol 30 no 5 pp 633ndash651 2001

[8] S Muthukumar and R DesRoches ldquoA Hertz contact modelwith non-linear damping for pounding simulationrdquo EarthquakeEngineering and Structural Dynamics vol 35 no 7 pp 811ndash8282006

[9] P Komodromos P C Polycarpou L Papaloizou and M CPhocas ldquoResponse of seismically isolated buildings consideringpoundingsrdquo Earthquake Engineering and Structural Dynamicsvol 36 no 12 pp 1605ndash1622 2007

[10] K Ye L Li and H Zhu ldquoA note on the Hertz contact modelwith nonlinear damping for pounding simulationrdquo EarthquakeEngineering and Structural Dynamics vol 38 no 9 pp 1135ndash1142 2009

Shock and Vibration 13

[11] C C Patel and R S Jangid ldquoDynamic response of adjacentstructures connected by friction damperrdquo Earthquake andStructures vol 2 no 2 pp 149ndash169 2011

[12] R Jankowski ldquoNon-linear FEM analysis of pounding-involvedresponse of buildings under non-uniform earthquake excita-tionrdquo Engineering Structures vol 37 pp 99ndash105 2012

[13] D G Weng C Zhang X L Lu S Zeng and S M Zhang ldquoAsimplified design procedure for seismic retrofit of earthquake-damaged RC frames with viscous dampersrdquo Structural Engi-neering and Mechanics vol 44 no 5 pp 611ndash631 2012

[14] G Watanabe and K Kawashima ldquoNumerical simulation ofpounding of bridge decksrdquo in Proceedings of the 13thWorld Con-ference on Earthquake Engineering Paper No 884 VancouverCanada August 2004

[15] E G Dimitrakopoulos ldquoSeismic response analysis of skewbridges with pounding deck-abutment jointsrdquo EngineeringStructures vol 33 no 3 pp 813ndash826 2011

[16] B Li K Bi N Chouw J W Butterworth and H Hao ldquoEffectof abutment excitation on bridge poundingrdquo Engineering Struc-tures vol 54 pp 57ndash68 2013

[17] C-J Wang and M-H Shih ldquoPerformance study of a bridgeinvolving sliding decks and pounded abutment during a violentearthquakerdquo Engineering Structures vol 29 no 5 pp 802ndash8122007

[18] Y Huo and J Zhang ldquoEffects of pounding and skewness onseismic responses of typical multispan highway bridges usingthe fragility function methodrdquo Journal of Bridge Engineeringvol 18 no 6 pp 499ndash515 2013

[19] K Bi and H Hao ldquoNumerical simulation of pounding damageto bridge structures under spatially varying ground motionsrdquoEngineering Structures vol 46 pp 62ndash76 2013

[20] S E Abdel Raheem ldquoPounding mitigation and unseatingprevention at expansion joints of isolated multi-span bridgesrdquoEngineering Structures vol 31 no 10 pp 2345ndash2356 2009

[21] B Li and N Chouw ldquoExperimental investigation of inelasticbridge response under spatially varying excitationswith pound-ingrdquo Engineering Structures vol 79 pp 106ndash116 2014

[22] S-H Kim and M Shinozuka ldquoEffects of seismically inducedpounding at expansion joints of concrete bridgesrdquo Journal ofEngineering Mechanics vol 129 no 11 pp 1225ndash1234 2003

[23] M J N Priestley F Seible and G M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

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2 Shock and Vibration

Deck Block

Pier

(a) Pounding between decks and blocks

Higher deck

Pier

(b) Pounding between the higher deck and the pier top

Figure 1 Examples related to the deck-pier pounding issue

o Deck Stationary pier

(a) Deck-pier pounding issue

Pier stiffness

o

Nonpounding end

Pounding end

(b) A one-dimensional wave theoretical model

Figure 2 A one-dimensional wave theoretical model for the deck-pier pounding issue

the closer the results obtained by FEM were to the resultsobtained by the one-dimensional wave method

Up to now many researches were focused on poundingsbetween the bridge deck and the abutment Dimitrakopoulosproposed a novel nonsmooth rigid body approach to analyzeseismic responses of pounding skew bridges which involvedoblique frictional multicontact phenomena [15] Li et alevaluated influences of abutment excitations on the poundingbehavior of a system consisting of a single bridge spanwith various fundamental frequencies and its abutmentsexperimentally [16] Wang and Shih presented a case studyon the sliding of bridge decks of a multispan concrete bridgeand pounding at abutment-backfill under the 1999 Chi-Chiearthquake [17] Huo and Zhang studied effects of poundingand skewness on the seismic behaviors of typical three-spanbridges in California [18]

There were also many researches on poundings betweenbridge decks Bi andHao performednumerical simulations ofpounding damage between bridge decks based on a detailed3D finite element model [19] Abdel Raheem developedan analytical model of expansion joints considering theinteraction between adjacent bridge segments and the effectof impact and restrainers [20] Li and Chouw studied theinelastic response of a two-segment bridge due to spatiallyvarying ground motion including pounding using shaketables [21] Kim and Shinozuka studied the effect of poundingat expansion joints on concrete bridge response to earthquakeground motions [22]

However there is little research on the deck-pier pound-ing issue but this kind of phenomenon was far from rareFor example decks and unseating prevention blocks onpiers would collide with each other in the longitudinal andtransverse directions in earthquakes as shown in Figure 1(a)[23]

Another example is related to a pier supporting two deckswith different heights The higher deck will hit the pier top inthe longitudinal direction as shown in Figure 1(b)

Because of the flexibility of piers the deck-pier poundingissue is different from the pounding issue between decks orthat between the bridge deck and the abutment so poundingresponses should receive adequate attention

Taking a simply supported girder bridge as an examplethe deck-pier pounding issue in the longitudinal directionis studied using two theoretical methods at first in thispaper According to the findings a simplified single degree offreedom (SDOF)model is established and verified by shakingtable test method At last influences of the pier stiffness andthe pier mass on pounding responses of a simply supportedgirder bridge were studied by finite element method

2 Theoretical Analysis of the Deck-PierPounding Issue

Two theoreticalmodels are used in this section to simulate thepounding between a deck with an initial velocity and the topof a stationary pier as shown in Figure 2(a)The first model is

Shock and Vibration 3

V2c

minusV2c

minusV2c

f998400+g998400

c(f998400minusg998400 )

f998400

f998400

f998400

g998400 g998400

g998400

V V

0

t = 0

minusP(t)EA

t = L12c t = 2L1c

V2c minus P(t)EA

V minus cP(t)EA

Figure 3 Waveforms of the first-order derivatives of 119891 and 119892

based on the one-dimensional wave theory which had beensuccessfully applied on the deck-deck pounding issue [1]Theinfluence of the piermass on pounding responses is neglectedfor the possibility of solving differential equations So thetheory is not applicable for very tall or heavy piers In the one-dimensional wave theory the deck mass and pier flexibilityare considered The model is shown in Figure 2(b) Supposethat piers are not damaged during the pounding the pierstiffness is a constant

Physical meanings of parameters in the theoretical anal-ysis are as follows

119864 119860 119871 and 120588 are elastic modulus cross-sectionalarea length and density of the deck respectively119881 is the initial velocity of the deck before pounding119875(119905) is the time history of the pounding force120573 is the ratio of the horizontal stiffness of bridge pierto the axial stiffness of the deck119896119901 = 120573119864119860119871 is the horizontal stiffness of bridge pier119906(119909 119905) is the displacement of the particle which is at adistance of 119909 from the nonpounding end of the deckat time 119905 According to the one-dimensional wavetheory 119906(119909 119905) = 119891(119909 + 119888119905) + 119892(119909 minus 119888119905) And 119891 and119892 are displacement waves propagating in the positiveand negative directions respectively1198911015840 and 1198921015840 are thefirst-order derivatives of 119891 and 119892 119888 = radic119864120588 is thepropagation velocity of the displacement wave

The model shown in Figure 2(b) is analyzed theoreticallyusing the one-dimensional wave theory According to initialconditions that the initial velocity is119881 and the initial strain iszero when 119905 = 0 we have

120597119906 (119909 119905)

120597119905

= 119888 (1198911015840minus 1198921015840) = 119881

120597119906 (119909 119905)

120597119909

= 1198911015840+ 1198921015840= 0

997904rArr 1198911015840= minus1198921015840=

119881

2119888

(1)

According to the boundary condition that the strain iszero at 119909 = 0 we have

1198911015840= minus1198921015840 (2)

According to the boundary condition that the strain isminus119875(119905)119864119860 at 119909 = 119871 we have

1198911015840= minus

119875 (119905)

119864119860

minus 1198921015840 (3)

According to the one-dimensional wave theory the first-order derivatives of displacement wave equations can beshown in Figure 3 The wave equations of the nonpoundingend and the pounding end are determined based on (2) and(3)

For 119905 isin [0 2119871119888] the velocity time history at 119909 = 119871 is asfollows

120597119906 (119871 119905)

120597119905

= minus119888119891 (119871 119905) + 119888119892 (119871 119905) = 119881 minus

119888

119864119860

119875 (119905)

= 119881 minus

120573119888

119871

119906 (119871 119905)

(4)

Solving the differential equations we have

119906 (119871 119905) =

119881119871

120573119888

[1 minus exp(minus120573119888119905

119871

)] (5)

And the time history of the pounding force is

119875 (119905) =

119864119860119881

119888

[1 minus exp(minus120573119888119905

119871

)] (6)

For 119905 isin [2119871119888 4119871119888] the velocity time history at 119909 = 119871 isas follows

120597119906 (119871 119905)

120597119905

= 119881 minus

119888

119864119860

119875 (119905) minus

2119888

119864119860

119875(119905 minus

2119871

119888

)

= 119881 minus

120573119888

119871

119906 (119871 119905)

minus 2119881[1 minus exp(minus120573119888119905

119871

+ 2120573)]

(7)

4 Shock and Vibration

0 1 2 3 4

00

01

02

03

04

05

120573 = 001

120573 = 002

120573 = 005

120573 = 01120573 = 02

120591

R(120591)

(a)

0 1 2 3 4

00

02

04

06

08

10

120573 = 05120573 = 10120573 = 20

120573 = 50

120573 = 100

120591

R(120591)

(b)

Figure 4 Time histories of the deck-pier pounding force

Solving the differential equations according to a continu-ous condition namely 119906(119871 2119871119888) = (119881119871120573119888)[1 minus exp(minus2120573)]we have

119906 (119871 119905)

=

119881119871

120573119888

[2 exp(minus120573119888119905

119871

+ 2120573) minus exp(minus120573119888119905

119871

) minus 1]

minus

4119881119871

119888

exp(minus120573119888119905

119871

+ 2120573)

+ 2119881119905 exp(minus120573119888119905

119871

+ 2120573)

(8)

And the time history of the pounding force is

119875 (119905)

=

119864119860119881

119888

[2 exp(minus120573119888119905

119871

+ 2120573) minus exp(minus120573119888119905

119871

) minus 1]

minus

4120573119864119860119881

119888

exp(minus120573119888119905

119871

+ 2120573)

+

2120573119864119860119881

119871

119905 exp(minus120573119888119905

119871

+ 2120573)

(9)

Introducing dimensionless pounding force 119877(119905) =

119875(119905)119888119864119860119881 and dimensionless time coordinate 120591 = 119888119905119871 thetime history of the dimensionless pounding force using theone-dimensional wave theory is as follows

119877 (120591) =

1 minus exp (minus120573120591) 0 le 120591 le 2

2 exp (minus120573120591 + 2120573) minus 4120573 exp (minus120573120591 + 2120573) + 2120573120591 exp (minus120573120591 + 2120573) minus exp (minus120573120591) minus 1 2 lt 120591 le 4

(10)

According to (10) time histories of the deck-pier pound-ing force for different 120573 can be drawn in Figure 4

As shown in Figure 4 for minor 120573 the deck-pier pound-ing force curve is close to a sinusoidal curve And formajor 120573the deck-pier pounding force curve is close to a rectangularcurve

The secondmodel is based on a simplified SDOFmethodIn the method the axial stiffness of the deck is assumed to befar larger than the pier stiffness which is usually valid andpractical Therefore the deck is regarded as a rigid body and

simulated by a particle with the deck mass and the pier issimulated by a spring as shown in Figure 5 119898 and 119896 are thedeck mass and the pier stiffness in the figure

For the simplified SDOF method the circular frequencyof the time history of the deck-pier pounding force is

120596 = radic120573119864119860119871

120588119860119871

=

119888

119871

radic120573 (11)

Shock and Vibration 5

Deck mass

Pier stiffness

Figure 5 A simplified SDOF theoretical model for the deck-pier pounding issue

0 1 2 3 4

00

01

02

03

04

05

120573 = 005

120573 = 01

120573 = 02

120591

R(120591)

(a)

0 1 2 3 4

00

02

04

06

08

10

120573 = 05120573 = 10

120573 = 04

120591

R(120591)

(b)

Figure 6 Comparisons of pounding force time histories between two theoretical models

Then time histories of the pounding force and the dimen-sionless pounding force are as follows

119875 (119905) = radic120573

119864119860119881

119888

sin(119888119905119871

radic120573) (12)

119875max = radic120573119864119860119881

119888

=radic119896119901119871

119864119860

119864119860119881

radic119864120588

= 119881radic120588119860119871 sdot 119896119901

= 119881radic119898 sdot 119896119901

(13)

119877 (120591) = radic120573 sin(radic120573120591) (14)

Time histories of the pounding force obtained using twotheoretical methods are compared in Figure 6 Solid linesindicate results of the one-dimensional wave theory andbroken lines indicate those of the simplified SDOF methodAs shown forminor120573 results of the simplified SDOFmethodagree well with those of the one-dimensional wave theoryHowever for major 120573 such as 120573 ge 04 differences betweenthe results are remarkable Therefore the deck should not beregarded as a rigid body when the pier stiffness is close to theaxial stiffness of the deck

Also as shown in Figure 6 as the axial flexibility of thedeck is considered in the one-dimensional wave theory thepounding duration will be extended and the peak poundingforce will be reduced

It should be noted that the simplified SDOF theoreticalmodel is linear and no energy dissipation is consideredTherefore if the pier is damaged heavily or much energy is

dissipated during the pounding process the simplified SDOFmethod is no more applicable

3 Experimental Study of Deck-Pier Pounding

Shaking table tests were conducted in the State Key Labora-tory of Disaster Reduction in Civil Engineering Tongji Uni-versity to verify the simplified SDOF method for the deck-pier pounding issue Two shaking tables named Table B andTable C were included to simulate ground motions

A simply supported bridge was used as the test specimenand consisted of two piers four bearings and a deck as shownin Figure 7 The two piers were installed on Table B andTable C respectively Each pier consisted of two columnsand one crossbeam The columns were fixed on the tableand the crossbeam was bolted on tops of columns Fourload transducers were placed on the two crossbeams Oneach load transducer one Double Spherical Seismic Isolationbearing (DSSI bearing) was placed The DSSI bearing wasdeveloped by substituting the flat sliding surface of anordinary spherical sliding bearing with a spherical slidingsurface Its working mechanism in earthquakes was similarto a Friction Pendulum Sliding (FPS) bearing The deck wassupported on the four bearings and steel plates were boltedon the deck to achieve an additional mass of 100 tons

Steel blocks were installed on the connection plates ofbearings on Table B to restrict excessive bearing displace-ments as shown in Figure 8 A certain horizontal distance wasdesigned between the block and the side of the upper bearingplate which was named Clearance in this paper

6 Shock and Vibration

Deck

Pier 1

Table B Table C

Load transducer

Pier 2

DSSI bearing

Steel block

Additional mass

Figure 7 The test setup and specimen

Lower bearing plateSteel block

Connection plate

SliderUpper bearing plate

Clearance

Figure 8 Configuration of a bearing with shear pins used as a fixed bearing

A record of Wenchuan Earthquake as shown in Figure 9was input in the cases The motion was recorded in the 119864-119882 direction in Wolong Sichuan Province The peak groundacceleration is 948 gal

Four shaking table test cases were arranged and listed inTable 1 As shown the peak ground acceleration was scaleddown in order not to damage load transducers

In Cases 1 and 3 a rubber plate was inserted in eachClearance to study the applicability of the method for dif-ferent pounding stiffness so pounding stiffness of these twocases was lower

Responses of the bridge model were measured by 33transducers collecting data from the bridge model with asampling frequency of 512Hz Bearing displacements were

Table 1 Test cases

Cases Clearance Peak acceleration1 10mm 02 g2 20mm 02 g3 50mm 05 g4 60mm 05 g

measured by 8 displacement transducers bearing forces weremeasured by 4 load transducers and acceleration for thetables decks and piers was measured by 21 accelerometers

In Figures 10ndash12 time histories of displacements forcesand relative velocities of bearings on Table B in all the four

Shock and Vibration 7

0 10 20 30 40 50minus800

minus600

minus400

minus200

0

200

400

600

800

Acce

lera

tion

(gal

)

Time (s)

(a) A ground acceleration record of Wenchuan Earthquake

0 1 2 3 4 5

0

500

1000

1500

2000

2500

3000

Pseu

doac

cele

ratio

n (g

al)

Period (s)

(b) Pseudoacceleration response spectra (5 damping)

Figure 9 A record of Wenchuan Earthquake used in the cases

10 20 30 40minus20

minus15

minus10

minus5

0

5

10

15

20

Disp

lace

men

t (m

m)

Time (t)

(a) Case 1

10 20 30 40minus30

minus20

minus10

0

10

20

30

Disp

lace

men

t (m

m)

Time (t)

(b) Case 2

10 20 30 40minus80

minus60

minus40

minus20

0

20

40

60

80

Disp

lace

men

t (m

m)

Time (t)

(c) Case 3

10 20 30 40minus80

minus60

minus40

minus20

0

20

40

60

80

Disp

lace

men

t (m

m)

Time (t)

(d) Case 4

Figure 10 Time histories of bearing displacements on Table B in all the cases

8 Shock and Vibration

minus80

minus60

minus40

minus20

0

20

40

60

80

10 20 30 40

Forc

e (kN

)

Time (t)

(a) Case 1

minus80

minus60

minus40

minus20

0

20

40

60

80

10 20 30 40

Forc

e (kN

)

Time (t)

(b) Case 2

minus200

minus150

minus100

minus50

0

50

100

150

200

10 20 30 40

Forc

e (kN

)

Time (t)

(c) Case 3

10 20 30 40minus200

minus150

minus100

minus50

0

50

100

150

200

Forc

e (kN

)

Time (t)

(d) Case 4

Figure 11 Time histories of bearing forces on Table B in all the cases

cases are shown The bearing displacement is the averagevalue of two bearings the bearing force is the resultant forceof two bearings and the relative velocity is the derivative ofthe bearing displacement

In order to verify the simplified SDOF method (13) isused to calculate the theoretical pounding force for eachpounding and then the theoretical pounding force 119865theoreticalis compared with the test pounding force 119865test

The theoretical pounding force is calculated and listedin Table 2 119898 is the pounding mass and equals 100 tonsfor all the cases 119896 is the pounding stiffness and equals thepounding force divided by the displacement measured in thetest Once the pounding happens bearing relative velocitydrops rapidly The bearing relative velocity just before themoment of pounding is extracted as the pounding velocity119881

As shown errors between theoretical pounding forcesand test pounding forces are less than 20 so the simplifiedSDOF method is verified By the comparison it is found thatthe deck can be simulated as a particle and the simplified

SDOF method can be used to study the deck-pier poundingissue

Because no energy dissipation is considered in themodelthe simplified SDOF method may overestimate the actualpounding force and the theoretical pounding force is alwayslarger than the test pounding force Therefore the simplifiedSDOF method can provide a reasonable upper limit ofpounding forcesThe actual pounding force is also influencedby energy dissipation and damage condition during thepounding process

4 FEM Analysis of the Deck-PierPounding Issue

In order to study pounding responses under longitudinalearthquakes and the applicability of the simplified SDOFmethod a typical simply supported girder bridge is studiedThe span is 16m and piers are both 5m high The left end ofthe deck is supported on two same expansion bearings and

Shock and Vibration 9

10 20 30 40minus150

minus100

minus50

0

50

100

150

Velo

city

(mm

s)

Time (t)

(a) Case 1

10 20 30 40minus150

minus100

minus50

0

50

100

150

Velo

city

(mm

s)

Time (t)

(b) Case 2

10 20 30 40minus400

minus300

minus200

minus100

0

100

200

300

400

Velo

city

(mm

s)

Time (t)

(c) Case 3

10 20 30 40minus400

minus300

minus200

minus100

0

100

200

300

400

Velo

city

(mm

s)

Time (t)

(d) Case 4

Figure 12 Time histories of bearing velocities on Table B in all the cases

Table 2 Comparisons of theoretical pounding forces and test pounding forces

Case Pounding number 119898 (ton) 119896 (kNmm) 119881 (mms) 119865theoretical (kN) 119865test (kN) Error ()1 1 100 1077 minus5861 minus6081 minus5533 9911 2 100 1265 5708 6420 6124 4831 3 100 1133 minus5581 minus5942 minus5020 18381 4 100 1391 minus5540 minus6533 minus6217 5081 5 100 1231 6463 7170 6708 6892 1 100 5025 minus3089 minus6924 minus6096 13583 1 100 1001 17079 17092 16657 2613 2 100 848 minus14378 minus13243 minus11373 16443 3 100 1415 18700 22247 19786 12444 1 100 4589 10447 22378 19045 17504 2 100 6546 minus8090 minus20698 minus17905 15604 3 100 5195 9835 22418 19196 16784 4 100 5071 minus5128 minus11548 minus10224 12954 5 100 5514 minus4639 minus10894 minus9116 1950

10 Shock and Vibration

Seismic isolation bearing Expansion bearing

Steel block

Clearance

Steel block

Clearance

Deck

Pier

(a) The whole structural model

Gap

Hook

Expansion bearing Seismic isolation bearing

Plastic Wen Plastic Wen

Gap

Hook

mdkpkp

(b) The SDOF model

Figure 13 Models of a simply supported girder bridge

Table 3 Parameters of a simply supported girder bridge model

Components ParametersDeck Elasticity modulus is 345 times 107 kPa density is 25 tonm3 and cross-sectional area is 54m2

Pier Elasticity modulus is 345 times 107 kPa density is 25 tonm3 and cross-sectional area is 20m2

Expansion bearing Horizontal stiffness before yielding is 432 times 104 kNm yield force is 432 kN and horizontal stiffness after yieldingis 0

Seismic isolationbearing

Horizontal stiffness before yielding is 432 times 104 kNm yield force is 432 kN and horizontal stiffness after yieldingis 216 kNm

Steel block Stiffness of both gap and hook link elements is 10 times 1010 kNm and Clearances are 01m

the right end of the deck is supported on two same seismicisolation bearings Steel blocks for unseating prevention areinstalled on both sides of each bearing Clearances are allset to be 01m for all the steel blocks Deck-pier poundingswill happen between the deck and steel blocks under longi-tudinal earthquakes until bearing displacements exceed thepreserved Clearances

41 Analysis Models and Seismic Input Two models of thesimply supported girder bridge are established with the FEMsoftware SAP2000 One is a whole structural model and theother one is a SDOF model For the whole structural model

all the components are simulated as shown in Figure 13(a)The deck and piers are simulated with frame elementsBearings are simulated with Plastic Wen link elements Steelblocks are simulated with gap and hook link elements All theelements and their parameters are listed in Table 3 For theSDOF model pier masses are neglected and the deck massis concentrated on a particle as shown in Figure 13(b) 119896119901represents the horizontal stiffness of piers and the deck mass119898119889 is 216 tons

Sinusoidal acceleration wave is input as the seismic exci-tation for simplicity Peak acceleration is 4ms2 frequency is1Hz duration is 5 s and integration step is 1ms

Shock and Vibration 11

minus3000minus2000minus1000

0100020003000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(a) 120573 = 00034

minus4000minus2000

020004000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(b) 120573 = 00138

minus6000minus4000minus2000

0200040006000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(c) 120573 = 0034

minus10000minus5000

05000

10000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(d) 120573 = 0103

minus20000minus15000minus10000minus5000

05000

100001500020000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(e) 120573 = 0345

0 1 2 3 4 5minus30000minus20000minus10000

0100002000030000

Bear

ing

forc

e (kN

)

Time (s)

(f) 120573 = 10

Figure 14 Force time histories of the expansion bearing

42 Applicability of the Simplified SDOFMethod To evaluatethe applicability of the simplified SDOF method results ofthe SDOF model for different pier stiffness are comparedwith those of the whole structural model The influenceof the pier stiffness on pounding response is studied byaltering the elastic modulus of piers The horizontal stiffnessis designed to be 40 times 104 kNm 16 times 105 kNm 40times 105 kNm 12 times 106 kNm 40 times 106 kNm and 116 times107 kNm respectively The axial stiffness of the deck is 116 times107 kNm So stiffness ratios are 00034 00138 0034 01030345 and 10 respectively

Force time histories of the expansion bearing of the twomodels are shown in Figure 14 The expansion bearing forceis composed of the friction force and the pounding force andthe latter is far larger than the former In the figure solid linesindicate results of the whole structural model and brokenlines indicate results of the SDOF model

Every spike in Figure 14 indicates one pounding betweenthe deck and the pier As shown along with the increase ofthe pier stiffness the pounding duration becomes shorter andthe pounding force becomes larger Differences between thetwo models are very small except for 120573 = 10 The poundingforce and the pounding time of the two models are almostthe same for the first five cases Therefore the SDOF modelcan estimate the pounding force fairly accurately for 120573 lt

0345 at least All the peak pounding forces calculated by theSDOF model are larger than those of the whole structural

model which is similar to the theoretical results It needs tobe noted that 120573 = 0345 is already very big pier stiffness andapplicability of the simplified SDOF method is verified foralmost all the bridges

For 120573 = 10 differences of the pounding force and thepounding time between the twomodels are much larger thanother cases So the SDOF model is not applicable anymoreHowever because the horizontal stiffness of bridge pier isalways much less than the axial stiffness of the deck 120573 = 10is a very rare condition

43 Influences of the Pier Stiffness and the Pier Mass onPounding Responses The influence of the pier horizontalstiffness on pounding responses is studied by altering theelastic modulus of piers as mentioned in Section 42 Theinfluence of the pier mass on pounding responses is studiedby altering the density of piers Pier masses are designed tobe 5 tons 10 tons 25 tons 50 tons and 125 tons respectivelywhich are named as M1 to M5

The maximum pounding responses of the whole struc-tural model are shown in Figure 15

As shown the pounding force will increase and thedeck displacement will decrease with the increase of thepier stiffness The influence of the pier mass on the deckdisplacement is very small

For minor pier stiffness the influence of the pier mass onthe pounding force is also not too much so the pier mass can

12 Shock and Vibration

3 10 100 10000

5000

10000

15000

20000

25000

30000Po

undi

ng fo

rce (

kN)

M1M2M3

M4M5

Pier stiffness (104 kNm)

(a) Maximum pounding forces

3 10 100 1000010

012

014

016

018

M1M2M3

M4M5

Gird

er d

ispla

cem

ent (

m)

Pier stiffness (104 kNm)

(b) Maximum deck displacements

Figure 15 Maximum pounding responses

be neglected and the SDOFmodel is applicable However theinfluence of the pier mass on the pounding force is bigger formajor pier stiffness and so the pier mass is not negligible andthe SDOF model is no more applicable

5 Conclusions

In this paper theoretical analysis shaking table test andfinite element analysis were conducted to study the deck-pierpounding issue the following conclusions are drawn

(1) Based on theoretical results and test results it wasfound that the longitudinal deck-pier pounding issuecan be simulated by the simplified SDOF method ifenergy dissipation is negligible and the pier stiffnessis much less than the axial stiffness of the deck

(2) It was found that the pounding duration becomesshorter and the pounding force becomes larger alongwith the increase of the pier stiffness

(3) The pier mass has little effect on pounding responsesexcept for bridges with very rigid piers

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of China (no 51278372) and the Ministry ofScience and Technology of China Grant no SLDRCE 14-B-15

References

[1] W Goldsmith Impact Dover New York NY USA 2001[2] G Cole R Dhakal A Carr and D Bull ldquoAn investigation

of the effects of mass distribution on pounding structuresrdquoEarthquake Engineering and Structural Dynamics vol 40 no6 pp 641ndash659 2011

[3] H Hao K Bi N Chouw and W-X Ren ldquoState-of-the-artreview on seismic induced pounding response of bridge struc-turesrdquo Journal of Earthquake and Tsunami vol 7 no 3 ArticleID 1350019 2013

[4] B F Maison and K Kasai ldquoAnalysis for a type of structuralpoundingrdquo Journal of Structural Engineering vol 116 no 4 pp957ndash977 1990

[5] B F Maison and K Kasai ldquoDynamics of pounding whentwo buildings colliderdquo Earthquake Engineering and StructuralDynamics vol 21 no 9 pp 771ndash786 1992

[6] R Jankowski KWilde and Y Fujino ldquoPounding of superstruc-ture segments in isolated elevated bridge during earthquakesrdquoEarthquake Engineering and Structural Dynamics vol 27 no 5pp 487ndash502 1998

[7] K T Chau and X X Wei ldquoPounding of structures modelled asnon-linear impacts of two oscillatorsrdquo Earthquake Engineeringand Structural Dynamics vol 30 no 5 pp 633ndash651 2001

[8] S Muthukumar and R DesRoches ldquoA Hertz contact modelwith non-linear damping for pounding simulationrdquo EarthquakeEngineering and Structural Dynamics vol 35 no 7 pp 811ndash8282006

[9] P Komodromos P C Polycarpou L Papaloizou and M CPhocas ldquoResponse of seismically isolated buildings consideringpoundingsrdquo Earthquake Engineering and Structural Dynamicsvol 36 no 12 pp 1605ndash1622 2007

[10] K Ye L Li and H Zhu ldquoA note on the Hertz contact modelwith nonlinear damping for pounding simulationrdquo EarthquakeEngineering and Structural Dynamics vol 38 no 9 pp 1135ndash1142 2009

Shock and Vibration 13

[11] C C Patel and R S Jangid ldquoDynamic response of adjacentstructures connected by friction damperrdquo Earthquake andStructures vol 2 no 2 pp 149ndash169 2011

[12] R Jankowski ldquoNon-linear FEM analysis of pounding-involvedresponse of buildings under non-uniform earthquake excita-tionrdquo Engineering Structures vol 37 pp 99ndash105 2012

[13] D G Weng C Zhang X L Lu S Zeng and S M Zhang ldquoAsimplified design procedure for seismic retrofit of earthquake-damaged RC frames with viscous dampersrdquo Structural Engi-neering and Mechanics vol 44 no 5 pp 611ndash631 2012

[14] G Watanabe and K Kawashima ldquoNumerical simulation ofpounding of bridge decksrdquo in Proceedings of the 13thWorld Con-ference on Earthquake Engineering Paper No 884 VancouverCanada August 2004

[15] E G Dimitrakopoulos ldquoSeismic response analysis of skewbridges with pounding deck-abutment jointsrdquo EngineeringStructures vol 33 no 3 pp 813ndash826 2011

[16] B Li K Bi N Chouw J W Butterworth and H Hao ldquoEffectof abutment excitation on bridge poundingrdquo Engineering Struc-tures vol 54 pp 57ndash68 2013

[17] C-J Wang and M-H Shih ldquoPerformance study of a bridgeinvolving sliding decks and pounded abutment during a violentearthquakerdquo Engineering Structures vol 29 no 5 pp 802ndash8122007

[18] Y Huo and J Zhang ldquoEffects of pounding and skewness onseismic responses of typical multispan highway bridges usingthe fragility function methodrdquo Journal of Bridge Engineeringvol 18 no 6 pp 499ndash515 2013

[19] K Bi and H Hao ldquoNumerical simulation of pounding damageto bridge structures under spatially varying ground motionsrdquoEngineering Structures vol 46 pp 62ndash76 2013

[20] S E Abdel Raheem ldquoPounding mitigation and unseatingprevention at expansion joints of isolated multi-span bridgesrdquoEngineering Structures vol 31 no 10 pp 2345ndash2356 2009

[21] B Li and N Chouw ldquoExperimental investigation of inelasticbridge response under spatially varying excitationswith pound-ingrdquo Engineering Structures vol 79 pp 106ndash116 2014

[22] S-H Kim and M Shinozuka ldquoEffects of seismically inducedpounding at expansion joints of concrete bridgesrdquo Journal ofEngineering Mechanics vol 129 no 11 pp 1225ndash1234 2003

[23] M J N Priestley F Seible and G M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

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Shock and Vibration

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International Journal of

Shock and Vibration 3

V2c

minusV2c

minusV2c

f998400+g998400

c(f998400minusg998400 )

f998400

f998400

f998400

g998400 g998400

g998400

V V

0

t = 0

minusP(t)EA

t = L12c t = 2L1c

V2c minus P(t)EA

V minus cP(t)EA

Figure 3 Waveforms of the first-order derivatives of 119891 and 119892

based on the one-dimensional wave theory which had beensuccessfully applied on the deck-deck pounding issue [1]Theinfluence of the piermass on pounding responses is neglectedfor the possibility of solving differential equations So thetheory is not applicable for very tall or heavy piers In the one-dimensional wave theory the deck mass and pier flexibilityare considered The model is shown in Figure 2(b) Supposethat piers are not damaged during the pounding the pierstiffness is a constant

Physical meanings of parameters in the theoretical anal-ysis are as follows

119864 119860 119871 and 120588 are elastic modulus cross-sectionalarea length and density of the deck respectively119881 is the initial velocity of the deck before pounding119875(119905) is the time history of the pounding force120573 is the ratio of the horizontal stiffness of bridge pierto the axial stiffness of the deck119896119901 = 120573119864119860119871 is the horizontal stiffness of bridge pier119906(119909 119905) is the displacement of the particle which is at adistance of 119909 from the nonpounding end of the deckat time 119905 According to the one-dimensional wavetheory 119906(119909 119905) = 119891(119909 + 119888119905) + 119892(119909 minus 119888119905) And 119891 and119892 are displacement waves propagating in the positiveand negative directions respectively1198911015840 and 1198921015840 are thefirst-order derivatives of 119891 and 119892 119888 = radic119864120588 is thepropagation velocity of the displacement wave

The model shown in Figure 2(b) is analyzed theoreticallyusing the one-dimensional wave theory According to initialconditions that the initial velocity is119881 and the initial strain iszero when 119905 = 0 we have

120597119906 (119909 119905)

120597119905

= 119888 (1198911015840minus 1198921015840) = 119881

120597119906 (119909 119905)

120597119909

= 1198911015840+ 1198921015840= 0

997904rArr 1198911015840= minus1198921015840=

119881

2119888

(1)

According to the boundary condition that the strain iszero at 119909 = 0 we have

1198911015840= minus1198921015840 (2)

According to the boundary condition that the strain isminus119875(119905)119864119860 at 119909 = 119871 we have

1198911015840= minus

119875 (119905)

119864119860

minus 1198921015840 (3)

According to the one-dimensional wave theory the first-order derivatives of displacement wave equations can beshown in Figure 3 The wave equations of the nonpoundingend and the pounding end are determined based on (2) and(3)

For 119905 isin [0 2119871119888] the velocity time history at 119909 = 119871 is asfollows

120597119906 (119871 119905)

120597119905

= minus119888119891 (119871 119905) + 119888119892 (119871 119905) = 119881 minus

119888

119864119860

119875 (119905)

= 119881 minus

120573119888

119871

119906 (119871 119905)

(4)

Solving the differential equations we have

119906 (119871 119905) =

119881119871

120573119888

[1 minus exp(minus120573119888119905

119871

)] (5)

And the time history of the pounding force is

119875 (119905) =

119864119860119881

119888

[1 minus exp(minus120573119888119905

119871

)] (6)

For 119905 isin [2119871119888 4119871119888] the velocity time history at 119909 = 119871 isas follows

120597119906 (119871 119905)

120597119905

= 119881 minus

119888

119864119860

119875 (119905) minus

2119888

119864119860

119875(119905 minus

2119871

119888

)

= 119881 minus

120573119888

119871

119906 (119871 119905)

minus 2119881[1 minus exp(minus120573119888119905

119871

+ 2120573)]

(7)

4 Shock and Vibration

0 1 2 3 4

00

01

02

03

04

05

120573 = 001

120573 = 002

120573 = 005

120573 = 01120573 = 02

120591

R(120591)

(a)

0 1 2 3 4

00

02

04

06

08

10

120573 = 05120573 = 10120573 = 20

120573 = 50

120573 = 100

120591

R(120591)

(b)

Figure 4 Time histories of the deck-pier pounding force

Solving the differential equations according to a continu-ous condition namely 119906(119871 2119871119888) = (119881119871120573119888)[1 minus exp(minus2120573)]we have

119906 (119871 119905)

=

119881119871

120573119888

[2 exp(minus120573119888119905

119871

+ 2120573) minus exp(minus120573119888119905

119871

) minus 1]

minus

4119881119871

119888

exp(minus120573119888119905

119871

+ 2120573)

+ 2119881119905 exp(minus120573119888119905

119871

+ 2120573)

(8)

And the time history of the pounding force is

119875 (119905)

=

119864119860119881

119888

[2 exp(minus120573119888119905

119871

+ 2120573) minus exp(minus120573119888119905

119871

) minus 1]

minus

4120573119864119860119881

119888

exp(minus120573119888119905

119871

+ 2120573)

+

2120573119864119860119881

119871

119905 exp(minus120573119888119905

119871

+ 2120573)

(9)

Introducing dimensionless pounding force 119877(119905) =

119875(119905)119888119864119860119881 and dimensionless time coordinate 120591 = 119888119905119871 thetime history of the dimensionless pounding force using theone-dimensional wave theory is as follows

119877 (120591) =

1 minus exp (minus120573120591) 0 le 120591 le 2

2 exp (minus120573120591 + 2120573) minus 4120573 exp (minus120573120591 + 2120573) + 2120573120591 exp (minus120573120591 + 2120573) minus exp (minus120573120591) minus 1 2 lt 120591 le 4

(10)

According to (10) time histories of the deck-pier pound-ing force for different 120573 can be drawn in Figure 4

As shown in Figure 4 for minor 120573 the deck-pier pound-ing force curve is close to a sinusoidal curve And formajor 120573the deck-pier pounding force curve is close to a rectangularcurve

The secondmodel is based on a simplified SDOFmethodIn the method the axial stiffness of the deck is assumed to befar larger than the pier stiffness which is usually valid andpractical Therefore the deck is regarded as a rigid body and

simulated by a particle with the deck mass and the pier issimulated by a spring as shown in Figure 5 119898 and 119896 are thedeck mass and the pier stiffness in the figure

For the simplified SDOF method the circular frequencyof the time history of the deck-pier pounding force is

120596 = radic120573119864119860119871

120588119860119871

=

119888

119871

radic120573 (11)

Shock and Vibration 5

Deck mass

Pier stiffness

Figure 5 A simplified SDOF theoretical model for the deck-pier pounding issue

0 1 2 3 4

00

01

02

03

04

05

120573 = 005

120573 = 01

120573 = 02

120591

R(120591)

(a)

0 1 2 3 4

00

02

04

06

08

10

120573 = 05120573 = 10

120573 = 04

120591

R(120591)

(b)

Figure 6 Comparisons of pounding force time histories between two theoretical models

Then time histories of the pounding force and the dimen-sionless pounding force are as follows

119875 (119905) = radic120573

119864119860119881

119888

sin(119888119905119871

radic120573) (12)

119875max = radic120573119864119860119881

119888

=radic119896119901119871

119864119860

119864119860119881

radic119864120588

= 119881radic120588119860119871 sdot 119896119901

= 119881radic119898 sdot 119896119901

(13)

119877 (120591) = radic120573 sin(radic120573120591) (14)

Time histories of the pounding force obtained using twotheoretical methods are compared in Figure 6 Solid linesindicate results of the one-dimensional wave theory andbroken lines indicate those of the simplified SDOF methodAs shown forminor120573 results of the simplified SDOFmethodagree well with those of the one-dimensional wave theoryHowever for major 120573 such as 120573 ge 04 differences betweenthe results are remarkable Therefore the deck should not beregarded as a rigid body when the pier stiffness is close to theaxial stiffness of the deck

Also as shown in Figure 6 as the axial flexibility of thedeck is considered in the one-dimensional wave theory thepounding duration will be extended and the peak poundingforce will be reduced

It should be noted that the simplified SDOF theoreticalmodel is linear and no energy dissipation is consideredTherefore if the pier is damaged heavily or much energy is

dissipated during the pounding process the simplified SDOFmethod is no more applicable

3 Experimental Study of Deck-Pier Pounding

Shaking table tests were conducted in the State Key Labora-tory of Disaster Reduction in Civil Engineering Tongji Uni-versity to verify the simplified SDOF method for the deck-pier pounding issue Two shaking tables named Table B andTable C were included to simulate ground motions

A simply supported bridge was used as the test specimenand consisted of two piers four bearings and a deck as shownin Figure 7 The two piers were installed on Table B andTable C respectively Each pier consisted of two columnsand one crossbeam The columns were fixed on the tableand the crossbeam was bolted on tops of columns Fourload transducers were placed on the two crossbeams Oneach load transducer one Double Spherical Seismic Isolationbearing (DSSI bearing) was placed The DSSI bearing wasdeveloped by substituting the flat sliding surface of anordinary spherical sliding bearing with a spherical slidingsurface Its working mechanism in earthquakes was similarto a Friction Pendulum Sliding (FPS) bearing The deck wassupported on the four bearings and steel plates were boltedon the deck to achieve an additional mass of 100 tons

Steel blocks were installed on the connection plates ofbearings on Table B to restrict excessive bearing displace-ments as shown in Figure 8 A certain horizontal distance wasdesigned between the block and the side of the upper bearingplate which was named Clearance in this paper

6 Shock and Vibration

Deck

Pier 1

Table B Table C

Load transducer

Pier 2

DSSI bearing

Steel block

Additional mass

Figure 7 The test setup and specimen

Lower bearing plateSteel block

Connection plate

SliderUpper bearing plate

Clearance

Figure 8 Configuration of a bearing with shear pins used as a fixed bearing

A record of Wenchuan Earthquake as shown in Figure 9was input in the cases The motion was recorded in the 119864-119882 direction in Wolong Sichuan Province The peak groundacceleration is 948 gal

Four shaking table test cases were arranged and listed inTable 1 As shown the peak ground acceleration was scaleddown in order not to damage load transducers

In Cases 1 and 3 a rubber plate was inserted in eachClearance to study the applicability of the method for dif-ferent pounding stiffness so pounding stiffness of these twocases was lower

Responses of the bridge model were measured by 33transducers collecting data from the bridge model with asampling frequency of 512Hz Bearing displacements were

Table 1 Test cases

Cases Clearance Peak acceleration1 10mm 02 g2 20mm 02 g3 50mm 05 g4 60mm 05 g

measured by 8 displacement transducers bearing forces weremeasured by 4 load transducers and acceleration for thetables decks and piers was measured by 21 accelerometers

In Figures 10ndash12 time histories of displacements forcesand relative velocities of bearings on Table B in all the four

Shock and Vibration 7

0 10 20 30 40 50minus800

minus600

minus400

minus200

0

200

400

600

800

Acce

lera

tion

(gal

)

Time (s)

(a) A ground acceleration record of Wenchuan Earthquake

0 1 2 3 4 5

0

500

1000

1500

2000

2500

3000

Pseu

doac

cele

ratio

n (g

al)

Period (s)

(b) Pseudoacceleration response spectra (5 damping)

Figure 9 A record of Wenchuan Earthquake used in the cases

10 20 30 40minus20

minus15

minus10

minus5

0

5

10

15

20

Disp

lace

men

t (m

m)

Time (t)

(a) Case 1

10 20 30 40minus30

minus20

minus10

0

10

20

30

Disp

lace

men

t (m

m)

Time (t)

(b) Case 2

10 20 30 40minus80

minus60

minus40

minus20

0

20

40

60

80

Disp

lace

men

t (m

m)

Time (t)

(c) Case 3

10 20 30 40minus80

minus60

minus40

minus20

0

20

40

60

80

Disp

lace

men

t (m

m)

Time (t)

(d) Case 4

Figure 10 Time histories of bearing displacements on Table B in all the cases

8 Shock and Vibration

minus80

minus60

minus40

minus20

0

20

40

60

80

10 20 30 40

Forc

e (kN

)

Time (t)

(a) Case 1

minus80

minus60

minus40

minus20

0

20

40

60

80

10 20 30 40

Forc

e (kN

)

Time (t)

(b) Case 2

minus200

minus150

minus100

minus50

0

50

100

150

200

10 20 30 40

Forc

e (kN

)

Time (t)

(c) Case 3

10 20 30 40minus200

minus150

minus100

minus50

0

50

100

150

200

Forc

e (kN

)

Time (t)

(d) Case 4

Figure 11 Time histories of bearing forces on Table B in all the cases

cases are shown The bearing displacement is the averagevalue of two bearings the bearing force is the resultant forceof two bearings and the relative velocity is the derivative ofthe bearing displacement

In order to verify the simplified SDOF method (13) isused to calculate the theoretical pounding force for eachpounding and then the theoretical pounding force 119865theoreticalis compared with the test pounding force 119865test

The theoretical pounding force is calculated and listedin Table 2 119898 is the pounding mass and equals 100 tonsfor all the cases 119896 is the pounding stiffness and equals thepounding force divided by the displacement measured in thetest Once the pounding happens bearing relative velocitydrops rapidly The bearing relative velocity just before themoment of pounding is extracted as the pounding velocity119881

As shown errors between theoretical pounding forcesand test pounding forces are less than 20 so the simplifiedSDOF method is verified By the comparison it is found thatthe deck can be simulated as a particle and the simplified

SDOF method can be used to study the deck-pier poundingissue

Because no energy dissipation is considered in themodelthe simplified SDOF method may overestimate the actualpounding force and the theoretical pounding force is alwayslarger than the test pounding force Therefore the simplifiedSDOF method can provide a reasonable upper limit ofpounding forcesThe actual pounding force is also influencedby energy dissipation and damage condition during thepounding process

4 FEM Analysis of the Deck-PierPounding Issue

In order to study pounding responses under longitudinalearthquakes and the applicability of the simplified SDOFmethod a typical simply supported girder bridge is studiedThe span is 16m and piers are both 5m high The left end ofthe deck is supported on two same expansion bearings and

Shock and Vibration 9

10 20 30 40minus150

minus100

minus50

0

50

100

150

Velo

city

(mm

s)

Time (t)

(a) Case 1

10 20 30 40minus150

minus100

minus50

0

50

100

150

Velo

city

(mm

s)

Time (t)

(b) Case 2

10 20 30 40minus400

minus300

minus200

minus100

0

100

200

300

400

Velo

city

(mm

s)

Time (t)

(c) Case 3

10 20 30 40minus400

minus300

minus200

minus100

0

100

200

300

400

Velo

city

(mm

s)

Time (t)

(d) Case 4

Figure 12 Time histories of bearing velocities on Table B in all the cases

Table 2 Comparisons of theoretical pounding forces and test pounding forces

Case Pounding number 119898 (ton) 119896 (kNmm) 119881 (mms) 119865theoretical (kN) 119865test (kN) Error ()1 1 100 1077 minus5861 minus6081 minus5533 9911 2 100 1265 5708 6420 6124 4831 3 100 1133 minus5581 minus5942 minus5020 18381 4 100 1391 minus5540 minus6533 minus6217 5081 5 100 1231 6463 7170 6708 6892 1 100 5025 minus3089 minus6924 minus6096 13583 1 100 1001 17079 17092 16657 2613 2 100 848 minus14378 minus13243 minus11373 16443 3 100 1415 18700 22247 19786 12444 1 100 4589 10447 22378 19045 17504 2 100 6546 minus8090 minus20698 minus17905 15604 3 100 5195 9835 22418 19196 16784 4 100 5071 minus5128 minus11548 minus10224 12954 5 100 5514 minus4639 minus10894 minus9116 1950

10 Shock and Vibration

Seismic isolation bearing Expansion bearing

Steel block

Clearance

Steel block

Clearance

Deck

Pier

(a) The whole structural model

Gap

Hook

Expansion bearing Seismic isolation bearing

Plastic Wen Plastic Wen

Gap

Hook

mdkpkp

(b) The SDOF model

Figure 13 Models of a simply supported girder bridge

Table 3 Parameters of a simply supported girder bridge model

Components ParametersDeck Elasticity modulus is 345 times 107 kPa density is 25 tonm3 and cross-sectional area is 54m2

Pier Elasticity modulus is 345 times 107 kPa density is 25 tonm3 and cross-sectional area is 20m2

Expansion bearing Horizontal stiffness before yielding is 432 times 104 kNm yield force is 432 kN and horizontal stiffness after yieldingis 0

Seismic isolationbearing

Horizontal stiffness before yielding is 432 times 104 kNm yield force is 432 kN and horizontal stiffness after yieldingis 216 kNm

Steel block Stiffness of both gap and hook link elements is 10 times 1010 kNm and Clearances are 01m

the right end of the deck is supported on two same seismicisolation bearings Steel blocks for unseating prevention areinstalled on both sides of each bearing Clearances are allset to be 01m for all the steel blocks Deck-pier poundingswill happen between the deck and steel blocks under longi-tudinal earthquakes until bearing displacements exceed thepreserved Clearances

41 Analysis Models and Seismic Input Two models of thesimply supported girder bridge are established with the FEMsoftware SAP2000 One is a whole structural model and theother one is a SDOF model For the whole structural model

all the components are simulated as shown in Figure 13(a)The deck and piers are simulated with frame elementsBearings are simulated with Plastic Wen link elements Steelblocks are simulated with gap and hook link elements All theelements and their parameters are listed in Table 3 For theSDOF model pier masses are neglected and the deck massis concentrated on a particle as shown in Figure 13(b) 119896119901represents the horizontal stiffness of piers and the deck mass119898119889 is 216 tons

Sinusoidal acceleration wave is input as the seismic exci-tation for simplicity Peak acceleration is 4ms2 frequency is1Hz duration is 5 s and integration step is 1ms

Shock and Vibration 11

minus3000minus2000minus1000

0100020003000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(a) 120573 = 00034

minus4000minus2000

020004000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(b) 120573 = 00138

minus6000minus4000minus2000

0200040006000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(c) 120573 = 0034

minus10000minus5000

05000

10000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(d) 120573 = 0103

minus20000minus15000minus10000minus5000

05000

100001500020000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(e) 120573 = 0345

0 1 2 3 4 5minus30000minus20000minus10000

0100002000030000

Bear

ing

forc

e (kN

)

Time (s)

(f) 120573 = 10

Figure 14 Force time histories of the expansion bearing

42 Applicability of the Simplified SDOFMethod To evaluatethe applicability of the simplified SDOF method results ofthe SDOF model for different pier stiffness are comparedwith those of the whole structural model The influenceof the pier stiffness on pounding response is studied byaltering the elastic modulus of piers The horizontal stiffnessis designed to be 40 times 104 kNm 16 times 105 kNm 40times 105 kNm 12 times 106 kNm 40 times 106 kNm and 116 times107 kNm respectively The axial stiffness of the deck is 116 times107 kNm So stiffness ratios are 00034 00138 0034 01030345 and 10 respectively

Force time histories of the expansion bearing of the twomodels are shown in Figure 14 The expansion bearing forceis composed of the friction force and the pounding force andthe latter is far larger than the former In the figure solid linesindicate results of the whole structural model and brokenlines indicate results of the SDOF model

Every spike in Figure 14 indicates one pounding betweenthe deck and the pier As shown along with the increase ofthe pier stiffness the pounding duration becomes shorter andthe pounding force becomes larger Differences between thetwo models are very small except for 120573 = 10 The poundingforce and the pounding time of the two models are almostthe same for the first five cases Therefore the SDOF modelcan estimate the pounding force fairly accurately for 120573 lt

0345 at least All the peak pounding forces calculated by theSDOF model are larger than those of the whole structural

model which is similar to the theoretical results It needs tobe noted that 120573 = 0345 is already very big pier stiffness andapplicability of the simplified SDOF method is verified foralmost all the bridges

For 120573 = 10 differences of the pounding force and thepounding time between the twomodels are much larger thanother cases So the SDOF model is not applicable anymoreHowever because the horizontal stiffness of bridge pier isalways much less than the axial stiffness of the deck 120573 = 10is a very rare condition

43 Influences of the Pier Stiffness and the Pier Mass onPounding Responses The influence of the pier horizontalstiffness on pounding responses is studied by altering theelastic modulus of piers as mentioned in Section 42 Theinfluence of the pier mass on pounding responses is studiedby altering the density of piers Pier masses are designed tobe 5 tons 10 tons 25 tons 50 tons and 125 tons respectivelywhich are named as M1 to M5

The maximum pounding responses of the whole struc-tural model are shown in Figure 15

As shown the pounding force will increase and thedeck displacement will decrease with the increase of thepier stiffness The influence of the pier mass on the deckdisplacement is very small

For minor pier stiffness the influence of the pier mass onthe pounding force is also not too much so the pier mass can

12 Shock and Vibration

3 10 100 10000

5000

10000

15000

20000

25000

30000Po

undi

ng fo

rce (

kN)

M1M2M3

M4M5

Pier stiffness (104 kNm)

(a) Maximum pounding forces

3 10 100 1000010

012

014

016

018

M1M2M3

M4M5

Gird

er d

ispla

cem

ent (

m)

Pier stiffness (104 kNm)

(b) Maximum deck displacements

Figure 15 Maximum pounding responses

be neglected and the SDOFmodel is applicable However theinfluence of the pier mass on the pounding force is bigger formajor pier stiffness and so the pier mass is not negligible andthe SDOF model is no more applicable

5 Conclusions

In this paper theoretical analysis shaking table test andfinite element analysis were conducted to study the deck-pierpounding issue the following conclusions are drawn

(1) Based on theoretical results and test results it wasfound that the longitudinal deck-pier pounding issuecan be simulated by the simplified SDOF method ifenergy dissipation is negligible and the pier stiffnessis much less than the axial stiffness of the deck

(2) It was found that the pounding duration becomesshorter and the pounding force becomes larger alongwith the increase of the pier stiffness

(3) The pier mass has little effect on pounding responsesexcept for bridges with very rigid piers

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of China (no 51278372) and the Ministry ofScience and Technology of China Grant no SLDRCE 14-B-15

References

[1] W Goldsmith Impact Dover New York NY USA 2001[2] G Cole R Dhakal A Carr and D Bull ldquoAn investigation

of the effects of mass distribution on pounding structuresrdquoEarthquake Engineering and Structural Dynamics vol 40 no6 pp 641ndash659 2011

[3] H Hao K Bi N Chouw and W-X Ren ldquoState-of-the-artreview on seismic induced pounding response of bridge struc-turesrdquo Journal of Earthquake and Tsunami vol 7 no 3 ArticleID 1350019 2013

[4] B F Maison and K Kasai ldquoAnalysis for a type of structuralpoundingrdquo Journal of Structural Engineering vol 116 no 4 pp957ndash977 1990

[5] B F Maison and K Kasai ldquoDynamics of pounding whentwo buildings colliderdquo Earthquake Engineering and StructuralDynamics vol 21 no 9 pp 771ndash786 1992

[6] R Jankowski KWilde and Y Fujino ldquoPounding of superstruc-ture segments in isolated elevated bridge during earthquakesrdquoEarthquake Engineering and Structural Dynamics vol 27 no 5pp 487ndash502 1998

[7] K T Chau and X X Wei ldquoPounding of structures modelled asnon-linear impacts of two oscillatorsrdquo Earthquake Engineeringand Structural Dynamics vol 30 no 5 pp 633ndash651 2001

[8] S Muthukumar and R DesRoches ldquoA Hertz contact modelwith non-linear damping for pounding simulationrdquo EarthquakeEngineering and Structural Dynamics vol 35 no 7 pp 811ndash8282006

[9] P Komodromos P C Polycarpou L Papaloizou and M CPhocas ldquoResponse of seismically isolated buildings consideringpoundingsrdquo Earthquake Engineering and Structural Dynamicsvol 36 no 12 pp 1605ndash1622 2007

[10] K Ye L Li and H Zhu ldquoA note on the Hertz contact modelwith nonlinear damping for pounding simulationrdquo EarthquakeEngineering and Structural Dynamics vol 38 no 9 pp 1135ndash1142 2009

Shock and Vibration 13

[11] C C Patel and R S Jangid ldquoDynamic response of adjacentstructures connected by friction damperrdquo Earthquake andStructures vol 2 no 2 pp 149ndash169 2011

[12] R Jankowski ldquoNon-linear FEM analysis of pounding-involvedresponse of buildings under non-uniform earthquake excita-tionrdquo Engineering Structures vol 37 pp 99ndash105 2012

[13] D G Weng C Zhang X L Lu S Zeng and S M Zhang ldquoAsimplified design procedure for seismic retrofit of earthquake-damaged RC frames with viscous dampersrdquo Structural Engi-neering and Mechanics vol 44 no 5 pp 611ndash631 2012

[14] G Watanabe and K Kawashima ldquoNumerical simulation ofpounding of bridge decksrdquo in Proceedings of the 13thWorld Con-ference on Earthquake Engineering Paper No 884 VancouverCanada August 2004

[15] E G Dimitrakopoulos ldquoSeismic response analysis of skewbridges with pounding deck-abutment jointsrdquo EngineeringStructures vol 33 no 3 pp 813ndash826 2011

[16] B Li K Bi N Chouw J W Butterworth and H Hao ldquoEffectof abutment excitation on bridge poundingrdquo Engineering Struc-tures vol 54 pp 57ndash68 2013

[17] C-J Wang and M-H Shih ldquoPerformance study of a bridgeinvolving sliding decks and pounded abutment during a violentearthquakerdquo Engineering Structures vol 29 no 5 pp 802ndash8122007

[18] Y Huo and J Zhang ldquoEffects of pounding and skewness onseismic responses of typical multispan highway bridges usingthe fragility function methodrdquo Journal of Bridge Engineeringvol 18 no 6 pp 499ndash515 2013

[19] K Bi and H Hao ldquoNumerical simulation of pounding damageto bridge structures under spatially varying ground motionsrdquoEngineering Structures vol 46 pp 62ndash76 2013

[20] S E Abdel Raheem ldquoPounding mitigation and unseatingprevention at expansion joints of isolated multi-span bridgesrdquoEngineering Structures vol 31 no 10 pp 2345ndash2356 2009

[21] B Li and N Chouw ldquoExperimental investigation of inelasticbridge response under spatially varying excitationswith pound-ingrdquo Engineering Structures vol 79 pp 106ndash116 2014

[22] S-H Kim and M Shinozuka ldquoEffects of seismically inducedpounding at expansion joints of concrete bridgesrdquo Journal ofEngineering Mechanics vol 129 no 11 pp 1225ndash1234 2003

[23] M J N Priestley F Seible and G M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

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Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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DistributedSensor Networks

International Journal of

4 Shock and Vibration

0 1 2 3 4

00

01

02

03

04

05

120573 = 001

120573 = 002

120573 = 005

120573 = 01120573 = 02

120591

R(120591)

(a)

0 1 2 3 4

00

02

04

06

08

10

120573 = 05120573 = 10120573 = 20

120573 = 50

120573 = 100

120591

R(120591)

(b)

Figure 4 Time histories of the deck-pier pounding force

Solving the differential equations according to a continu-ous condition namely 119906(119871 2119871119888) = (119881119871120573119888)[1 minus exp(minus2120573)]we have

119906 (119871 119905)

=

119881119871

120573119888

[2 exp(minus120573119888119905

119871

+ 2120573) minus exp(minus120573119888119905

119871

) minus 1]

minus

4119881119871

119888

exp(minus120573119888119905

119871

+ 2120573)

+ 2119881119905 exp(minus120573119888119905

119871

+ 2120573)

(8)

And the time history of the pounding force is

119875 (119905)

=

119864119860119881

119888

[2 exp(minus120573119888119905

119871

+ 2120573) minus exp(minus120573119888119905

119871

) minus 1]

minus

4120573119864119860119881

119888

exp(minus120573119888119905

119871

+ 2120573)

+

2120573119864119860119881

119871

119905 exp(minus120573119888119905

119871

+ 2120573)

(9)

Introducing dimensionless pounding force 119877(119905) =

119875(119905)119888119864119860119881 and dimensionless time coordinate 120591 = 119888119905119871 thetime history of the dimensionless pounding force using theone-dimensional wave theory is as follows

119877 (120591) =

1 minus exp (minus120573120591) 0 le 120591 le 2

2 exp (minus120573120591 + 2120573) minus 4120573 exp (minus120573120591 + 2120573) + 2120573120591 exp (minus120573120591 + 2120573) minus exp (minus120573120591) minus 1 2 lt 120591 le 4

(10)

According to (10) time histories of the deck-pier pound-ing force for different 120573 can be drawn in Figure 4

As shown in Figure 4 for minor 120573 the deck-pier pound-ing force curve is close to a sinusoidal curve And formajor 120573the deck-pier pounding force curve is close to a rectangularcurve

The secondmodel is based on a simplified SDOFmethodIn the method the axial stiffness of the deck is assumed to befar larger than the pier stiffness which is usually valid andpractical Therefore the deck is regarded as a rigid body and

simulated by a particle with the deck mass and the pier issimulated by a spring as shown in Figure 5 119898 and 119896 are thedeck mass and the pier stiffness in the figure

For the simplified SDOF method the circular frequencyof the time history of the deck-pier pounding force is

120596 = radic120573119864119860119871

120588119860119871

=

119888

119871

radic120573 (11)

Shock and Vibration 5

Deck mass

Pier stiffness

Figure 5 A simplified SDOF theoretical model for the deck-pier pounding issue

0 1 2 3 4

00

01

02

03

04

05

120573 = 005

120573 = 01

120573 = 02

120591

R(120591)

(a)

0 1 2 3 4

00

02

04

06

08

10

120573 = 05120573 = 10

120573 = 04

120591

R(120591)

(b)

Figure 6 Comparisons of pounding force time histories between two theoretical models

Then time histories of the pounding force and the dimen-sionless pounding force are as follows

119875 (119905) = radic120573

119864119860119881

119888

sin(119888119905119871

radic120573) (12)

119875max = radic120573119864119860119881

119888

=radic119896119901119871

119864119860

119864119860119881

radic119864120588

= 119881radic120588119860119871 sdot 119896119901

= 119881radic119898 sdot 119896119901

(13)

119877 (120591) = radic120573 sin(radic120573120591) (14)

Time histories of the pounding force obtained using twotheoretical methods are compared in Figure 6 Solid linesindicate results of the one-dimensional wave theory andbroken lines indicate those of the simplified SDOF methodAs shown forminor120573 results of the simplified SDOFmethodagree well with those of the one-dimensional wave theoryHowever for major 120573 such as 120573 ge 04 differences betweenthe results are remarkable Therefore the deck should not beregarded as a rigid body when the pier stiffness is close to theaxial stiffness of the deck

Also as shown in Figure 6 as the axial flexibility of thedeck is considered in the one-dimensional wave theory thepounding duration will be extended and the peak poundingforce will be reduced

It should be noted that the simplified SDOF theoreticalmodel is linear and no energy dissipation is consideredTherefore if the pier is damaged heavily or much energy is

dissipated during the pounding process the simplified SDOFmethod is no more applicable

3 Experimental Study of Deck-Pier Pounding

Shaking table tests were conducted in the State Key Labora-tory of Disaster Reduction in Civil Engineering Tongji Uni-versity to verify the simplified SDOF method for the deck-pier pounding issue Two shaking tables named Table B andTable C were included to simulate ground motions

A simply supported bridge was used as the test specimenand consisted of two piers four bearings and a deck as shownin Figure 7 The two piers were installed on Table B andTable C respectively Each pier consisted of two columnsand one crossbeam The columns were fixed on the tableand the crossbeam was bolted on tops of columns Fourload transducers were placed on the two crossbeams Oneach load transducer one Double Spherical Seismic Isolationbearing (DSSI bearing) was placed The DSSI bearing wasdeveloped by substituting the flat sliding surface of anordinary spherical sliding bearing with a spherical slidingsurface Its working mechanism in earthquakes was similarto a Friction Pendulum Sliding (FPS) bearing The deck wassupported on the four bearings and steel plates were boltedon the deck to achieve an additional mass of 100 tons

Steel blocks were installed on the connection plates ofbearings on Table B to restrict excessive bearing displace-ments as shown in Figure 8 A certain horizontal distance wasdesigned between the block and the side of the upper bearingplate which was named Clearance in this paper

6 Shock and Vibration

Deck

Pier 1

Table B Table C

Load transducer

Pier 2

DSSI bearing

Steel block

Additional mass

Figure 7 The test setup and specimen

Lower bearing plateSteel block

Connection plate

SliderUpper bearing plate

Clearance

Figure 8 Configuration of a bearing with shear pins used as a fixed bearing

A record of Wenchuan Earthquake as shown in Figure 9was input in the cases The motion was recorded in the 119864-119882 direction in Wolong Sichuan Province The peak groundacceleration is 948 gal

Four shaking table test cases were arranged and listed inTable 1 As shown the peak ground acceleration was scaleddown in order not to damage load transducers

In Cases 1 and 3 a rubber plate was inserted in eachClearance to study the applicability of the method for dif-ferent pounding stiffness so pounding stiffness of these twocases was lower

Responses of the bridge model were measured by 33transducers collecting data from the bridge model with asampling frequency of 512Hz Bearing displacements were

Table 1 Test cases

Cases Clearance Peak acceleration1 10mm 02 g2 20mm 02 g3 50mm 05 g4 60mm 05 g

measured by 8 displacement transducers bearing forces weremeasured by 4 load transducers and acceleration for thetables decks and piers was measured by 21 accelerometers

In Figures 10ndash12 time histories of displacements forcesand relative velocities of bearings on Table B in all the four

Shock and Vibration 7

0 10 20 30 40 50minus800

minus600

minus400

minus200

0

200

400

600

800

Acce

lera

tion

(gal

)

Time (s)

(a) A ground acceleration record of Wenchuan Earthquake

0 1 2 3 4 5

0

500

1000

1500

2000

2500

3000

Pseu

doac

cele

ratio

n (g

al)

Period (s)

(b) Pseudoacceleration response spectra (5 damping)

Figure 9 A record of Wenchuan Earthquake used in the cases

10 20 30 40minus20

minus15

minus10

minus5

0

5

10

15

20

Disp

lace

men

t (m

m)

Time (t)

(a) Case 1

10 20 30 40minus30

minus20

minus10

0

10

20

30

Disp

lace

men

t (m

m)

Time (t)

(b) Case 2

10 20 30 40minus80

minus60

minus40

minus20

0

20

40

60

80

Disp

lace

men

t (m

m)

Time (t)

(c) Case 3

10 20 30 40minus80

minus60

minus40

minus20

0

20

40

60

80

Disp

lace

men

t (m

m)

Time (t)

(d) Case 4

Figure 10 Time histories of bearing displacements on Table B in all the cases

8 Shock and Vibration

minus80

minus60

minus40

minus20

0

20

40

60

80

10 20 30 40

Forc

e (kN

)

Time (t)

(a) Case 1

minus80

minus60

minus40

minus20

0

20

40

60

80

10 20 30 40

Forc

e (kN

)

Time (t)

(b) Case 2

minus200

minus150

minus100

minus50

0

50

100

150

200

10 20 30 40

Forc

e (kN

)

Time (t)

(c) Case 3

10 20 30 40minus200

minus150

minus100

minus50

0

50

100

150

200

Forc

e (kN

)

Time (t)

(d) Case 4

Figure 11 Time histories of bearing forces on Table B in all the cases

cases are shown The bearing displacement is the averagevalue of two bearings the bearing force is the resultant forceof two bearings and the relative velocity is the derivative ofthe bearing displacement

In order to verify the simplified SDOF method (13) isused to calculate the theoretical pounding force for eachpounding and then the theoretical pounding force 119865theoreticalis compared with the test pounding force 119865test

The theoretical pounding force is calculated and listedin Table 2 119898 is the pounding mass and equals 100 tonsfor all the cases 119896 is the pounding stiffness and equals thepounding force divided by the displacement measured in thetest Once the pounding happens bearing relative velocitydrops rapidly The bearing relative velocity just before themoment of pounding is extracted as the pounding velocity119881

As shown errors between theoretical pounding forcesand test pounding forces are less than 20 so the simplifiedSDOF method is verified By the comparison it is found thatthe deck can be simulated as a particle and the simplified

SDOF method can be used to study the deck-pier poundingissue

Because no energy dissipation is considered in themodelthe simplified SDOF method may overestimate the actualpounding force and the theoretical pounding force is alwayslarger than the test pounding force Therefore the simplifiedSDOF method can provide a reasonable upper limit ofpounding forcesThe actual pounding force is also influencedby energy dissipation and damage condition during thepounding process

4 FEM Analysis of the Deck-PierPounding Issue

In order to study pounding responses under longitudinalearthquakes and the applicability of the simplified SDOFmethod a typical simply supported girder bridge is studiedThe span is 16m and piers are both 5m high The left end ofthe deck is supported on two same expansion bearings and

Shock and Vibration 9

10 20 30 40minus150

minus100

minus50

0

50

100

150

Velo

city

(mm

s)

Time (t)

(a) Case 1

10 20 30 40minus150

minus100

minus50

0

50

100

150

Velo

city

(mm

s)

Time (t)

(b) Case 2

10 20 30 40minus400

minus300

minus200

minus100

0

100

200

300

400

Velo

city

(mm

s)

Time (t)

(c) Case 3

10 20 30 40minus400

minus300

minus200

minus100

0

100

200

300

400

Velo

city

(mm

s)

Time (t)

(d) Case 4

Figure 12 Time histories of bearing velocities on Table B in all the cases

Table 2 Comparisons of theoretical pounding forces and test pounding forces

Case Pounding number 119898 (ton) 119896 (kNmm) 119881 (mms) 119865theoretical (kN) 119865test (kN) Error ()1 1 100 1077 minus5861 minus6081 minus5533 9911 2 100 1265 5708 6420 6124 4831 3 100 1133 minus5581 minus5942 minus5020 18381 4 100 1391 minus5540 minus6533 minus6217 5081 5 100 1231 6463 7170 6708 6892 1 100 5025 minus3089 minus6924 minus6096 13583 1 100 1001 17079 17092 16657 2613 2 100 848 minus14378 minus13243 minus11373 16443 3 100 1415 18700 22247 19786 12444 1 100 4589 10447 22378 19045 17504 2 100 6546 minus8090 minus20698 minus17905 15604 3 100 5195 9835 22418 19196 16784 4 100 5071 minus5128 minus11548 minus10224 12954 5 100 5514 minus4639 minus10894 minus9116 1950

10 Shock and Vibration

Seismic isolation bearing Expansion bearing

Steel block

Clearance

Steel block

Clearance

Deck

Pier

(a) The whole structural model

Gap

Hook

Expansion bearing Seismic isolation bearing

Plastic Wen Plastic Wen

Gap

Hook

mdkpkp

(b) The SDOF model

Figure 13 Models of a simply supported girder bridge

Table 3 Parameters of a simply supported girder bridge model

Components ParametersDeck Elasticity modulus is 345 times 107 kPa density is 25 tonm3 and cross-sectional area is 54m2

Pier Elasticity modulus is 345 times 107 kPa density is 25 tonm3 and cross-sectional area is 20m2

Expansion bearing Horizontal stiffness before yielding is 432 times 104 kNm yield force is 432 kN and horizontal stiffness after yieldingis 0

Seismic isolationbearing

Horizontal stiffness before yielding is 432 times 104 kNm yield force is 432 kN and horizontal stiffness after yieldingis 216 kNm

Steel block Stiffness of both gap and hook link elements is 10 times 1010 kNm and Clearances are 01m

the right end of the deck is supported on two same seismicisolation bearings Steel blocks for unseating prevention areinstalled on both sides of each bearing Clearances are allset to be 01m for all the steel blocks Deck-pier poundingswill happen between the deck and steel blocks under longi-tudinal earthquakes until bearing displacements exceed thepreserved Clearances

41 Analysis Models and Seismic Input Two models of thesimply supported girder bridge are established with the FEMsoftware SAP2000 One is a whole structural model and theother one is a SDOF model For the whole structural model

all the components are simulated as shown in Figure 13(a)The deck and piers are simulated with frame elementsBearings are simulated with Plastic Wen link elements Steelblocks are simulated with gap and hook link elements All theelements and their parameters are listed in Table 3 For theSDOF model pier masses are neglected and the deck massis concentrated on a particle as shown in Figure 13(b) 119896119901represents the horizontal stiffness of piers and the deck mass119898119889 is 216 tons

Sinusoidal acceleration wave is input as the seismic exci-tation for simplicity Peak acceleration is 4ms2 frequency is1Hz duration is 5 s and integration step is 1ms

Shock and Vibration 11

minus3000minus2000minus1000

0100020003000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(a) 120573 = 00034

minus4000minus2000

020004000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(b) 120573 = 00138

minus6000minus4000minus2000

0200040006000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(c) 120573 = 0034

minus10000minus5000

05000

10000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(d) 120573 = 0103

minus20000minus15000minus10000minus5000

05000

100001500020000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(e) 120573 = 0345

0 1 2 3 4 5minus30000minus20000minus10000

0100002000030000

Bear

ing

forc

e (kN

)

Time (s)

(f) 120573 = 10

Figure 14 Force time histories of the expansion bearing

42 Applicability of the Simplified SDOFMethod To evaluatethe applicability of the simplified SDOF method results ofthe SDOF model for different pier stiffness are comparedwith those of the whole structural model The influenceof the pier stiffness on pounding response is studied byaltering the elastic modulus of piers The horizontal stiffnessis designed to be 40 times 104 kNm 16 times 105 kNm 40times 105 kNm 12 times 106 kNm 40 times 106 kNm and 116 times107 kNm respectively The axial stiffness of the deck is 116 times107 kNm So stiffness ratios are 00034 00138 0034 01030345 and 10 respectively

Force time histories of the expansion bearing of the twomodels are shown in Figure 14 The expansion bearing forceis composed of the friction force and the pounding force andthe latter is far larger than the former In the figure solid linesindicate results of the whole structural model and brokenlines indicate results of the SDOF model

Every spike in Figure 14 indicates one pounding betweenthe deck and the pier As shown along with the increase ofthe pier stiffness the pounding duration becomes shorter andthe pounding force becomes larger Differences between thetwo models are very small except for 120573 = 10 The poundingforce and the pounding time of the two models are almostthe same for the first five cases Therefore the SDOF modelcan estimate the pounding force fairly accurately for 120573 lt

0345 at least All the peak pounding forces calculated by theSDOF model are larger than those of the whole structural

model which is similar to the theoretical results It needs tobe noted that 120573 = 0345 is already very big pier stiffness andapplicability of the simplified SDOF method is verified foralmost all the bridges

For 120573 = 10 differences of the pounding force and thepounding time between the twomodels are much larger thanother cases So the SDOF model is not applicable anymoreHowever because the horizontal stiffness of bridge pier isalways much less than the axial stiffness of the deck 120573 = 10is a very rare condition

43 Influences of the Pier Stiffness and the Pier Mass onPounding Responses The influence of the pier horizontalstiffness on pounding responses is studied by altering theelastic modulus of piers as mentioned in Section 42 Theinfluence of the pier mass on pounding responses is studiedby altering the density of piers Pier masses are designed tobe 5 tons 10 tons 25 tons 50 tons and 125 tons respectivelywhich are named as M1 to M5

The maximum pounding responses of the whole struc-tural model are shown in Figure 15

As shown the pounding force will increase and thedeck displacement will decrease with the increase of thepier stiffness The influence of the pier mass on the deckdisplacement is very small

For minor pier stiffness the influence of the pier mass onthe pounding force is also not too much so the pier mass can

12 Shock and Vibration

3 10 100 10000

5000

10000

15000

20000

25000

30000Po

undi

ng fo

rce (

kN)

M1M2M3

M4M5

Pier stiffness (104 kNm)

(a) Maximum pounding forces

3 10 100 1000010

012

014

016

018

M1M2M3

M4M5

Gird

er d

ispla

cem

ent (

m)

Pier stiffness (104 kNm)

(b) Maximum deck displacements

Figure 15 Maximum pounding responses

be neglected and the SDOFmodel is applicable However theinfluence of the pier mass on the pounding force is bigger formajor pier stiffness and so the pier mass is not negligible andthe SDOF model is no more applicable

5 Conclusions

In this paper theoretical analysis shaking table test andfinite element analysis were conducted to study the deck-pierpounding issue the following conclusions are drawn

(1) Based on theoretical results and test results it wasfound that the longitudinal deck-pier pounding issuecan be simulated by the simplified SDOF method ifenergy dissipation is negligible and the pier stiffnessis much less than the axial stiffness of the deck

(2) It was found that the pounding duration becomesshorter and the pounding force becomes larger alongwith the increase of the pier stiffness

(3) The pier mass has little effect on pounding responsesexcept for bridges with very rigid piers

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of China (no 51278372) and the Ministry ofScience and Technology of China Grant no SLDRCE 14-B-15

References

[1] W Goldsmith Impact Dover New York NY USA 2001[2] G Cole R Dhakal A Carr and D Bull ldquoAn investigation

of the effects of mass distribution on pounding structuresrdquoEarthquake Engineering and Structural Dynamics vol 40 no6 pp 641ndash659 2011

[3] H Hao K Bi N Chouw and W-X Ren ldquoState-of-the-artreview on seismic induced pounding response of bridge struc-turesrdquo Journal of Earthquake and Tsunami vol 7 no 3 ArticleID 1350019 2013

[4] B F Maison and K Kasai ldquoAnalysis for a type of structuralpoundingrdquo Journal of Structural Engineering vol 116 no 4 pp957ndash977 1990

[5] B F Maison and K Kasai ldquoDynamics of pounding whentwo buildings colliderdquo Earthquake Engineering and StructuralDynamics vol 21 no 9 pp 771ndash786 1992

[6] R Jankowski KWilde and Y Fujino ldquoPounding of superstruc-ture segments in isolated elevated bridge during earthquakesrdquoEarthquake Engineering and Structural Dynamics vol 27 no 5pp 487ndash502 1998

[7] K T Chau and X X Wei ldquoPounding of structures modelled asnon-linear impacts of two oscillatorsrdquo Earthquake Engineeringand Structural Dynamics vol 30 no 5 pp 633ndash651 2001

[8] S Muthukumar and R DesRoches ldquoA Hertz contact modelwith non-linear damping for pounding simulationrdquo EarthquakeEngineering and Structural Dynamics vol 35 no 7 pp 811ndash8282006

[9] P Komodromos P C Polycarpou L Papaloizou and M CPhocas ldquoResponse of seismically isolated buildings consideringpoundingsrdquo Earthquake Engineering and Structural Dynamicsvol 36 no 12 pp 1605ndash1622 2007

[10] K Ye L Li and H Zhu ldquoA note on the Hertz contact modelwith nonlinear damping for pounding simulationrdquo EarthquakeEngineering and Structural Dynamics vol 38 no 9 pp 1135ndash1142 2009

Shock and Vibration 13

[11] C C Patel and R S Jangid ldquoDynamic response of adjacentstructures connected by friction damperrdquo Earthquake andStructures vol 2 no 2 pp 149ndash169 2011

[12] R Jankowski ldquoNon-linear FEM analysis of pounding-involvedresponse of buildings under non-uniform earthquake excita-tionrdquo Engineering Structures vol 37 pp 99ndash105 2012

[13] D G Weng C Zhang X L Lu S Zeng and S M Zhang ldquoAsimplified design procedure for seismic retrofit of earthquake-damaged RC frames with viscous dampersrdquo Structural Engi-neering and Mechanics vol 44 no 5 pp 611ndash631 2012

[14] G Watanabe and K Kawashima ldquoNumerical simulation ofpounding of bridge decksrdquo in Proceedings of the 13thWorld Con-ference on Earthquake Engineering Paper No 884 VancouverCanada August 2004

[15] E G Dimitrakopoulos ldquoSeismic response analysis of skewbridges with pounding deck-abutment jointsrdquo EngineeringStructures vol 33 no 3 pp 813ndash826 2011

[16] B Li K Bi N Chouw J W Butterworth and H Hao ldquoEffectof abutment excitation on bridge poundingrdquo Engineering Struc-tures vol 54 pp 57ndash68 2013

[17] C-J Wang and M-H Shih ldquoPerformance study of a bridgeinvolving sliding decks and pounded abutment during a violentearthquakerdquo Engineering Structures vol 29 no 5 pp 802ndash8122007

[18] Y Huo and J Zhang ldquoEffects of pounding and skewness onseismic responses of typical multispan highway bridges usingthe fragility function methodrdquo Journal of Bridge Engineeringvol 18 no 6 pp 499ndash515 2013

[19] K Bi and H Hao ldquoNumerical simulation of pounding damageto bridge structures under spatially varying ground motionsrdquoEngineering Structures vol 46 pp 62ndash76 2013

[20] S E Abdel Raheem ldquoPounding mitigation and unseatingprevention at expansion joints of isolated multi-span bridgesrdquoEngineering Structures vol 31 no 10 pp 2345ndash2356 2009

[21] B Li and N Chouw ldquoExperimental investigation of inelasticbridge response under spatially varying excitationswith pound-ingrdquo Engineering Structures vol 79 pp 106ndash116 2014

[22] S-H Kim and M Shinozuka ldquoEffects of seismically inducedpounding at expansion joints of concrete bridgesrdquo Journal ofEngineering Mechanics vol 129 no 11 pp 1225ndash1234 2003

[23] M J N Priestley F Seible and G M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

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Shock and Vibration

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International Journal of

Shock and Vibration 5

Deck mass

Pier stiffness

Figure 5 A simplified SDOF theoretical model for the deck-pier pounding issue

0 1 2 3 4

00

01

02

03

04

05

120573 = 005

120573 = 01

120573 = 02

120591

R(120591)

(a)

0 1 2 3 4

00

02

04

06

08

10

120573 = 05120573 = 10

120573 = 04

120591

R(120591)

(b)

Figure 6 Comparisons of pounding force time histories between two theoretical models

Then time histories of the pounding force and the dimen-sionless pounding force are as follows

119875 (119905) = radic120573

119864119860119881

119888

sin(119888119905119871

radic120573) (12)

119875max = radic120573119864119860119881

119888

=radic119896119901119871

119864119860

119864119860119881

radic119864120588

= 119881radic120588119860119871 sdot 119896119901

= 119881radic119898 sdot 119896119901

(13)

119877 (120591) = radic120573 sin(radic120573120591) (14)

Time histories of the pounding force obtained using twotheoretical methods are compared in Figure 6 Solid linesindicate results of the one-dimensional wave theory andbroken lines indicate those of the simplified SDOF methodAs shown forminor120573 results of the simplified SDOFmethodagree well with those of the one-dimensional wave theoryHowever for major 120573 such as 120573 ge 04 differences betweenthe results are remarkable Therefore the deck should not beregarded as a rigid body when the pier stiffness is close to theaxial stiffness of the deck

Also as shown in Figure 6 as the axial flexibility of thedeck is considered in the one-dimensional wave theory thepounding duration will be extended and the peak poundingforce will be reduced

It should be noted that the simplified SDOF theoreticalmodel is linear and no energy dissipation is consideredTherefore if the pier is damaged heavily or much energy is

dissipated during the pounding process the simplified SDOFmethod is no more applicable

3 Experimental Study of Deck-Pier Pounding

Shaking table tests were conducted in the State Key Labora-tory of Disaster Reduction in Civil Engineering Tongji Uni-versity to verify the simplified SDOF method for the deck-pier pounding issue Two shaking tables named Table B andTable C were included to simulate ground motions

A simply supported bridge was used as the test specimenand consisted of two piers four bearings and a deck as shownin Figure 7 The two piers were installed on Table B andTable C respectively Each pier consisted of two columnsand one crossbeam The columns were fixed on the tableand the crossbeam was bolted on tops of columns Fourload transducers were placed on the two crossbeams Oneach load transducer one Double Spherical Seismic Isolationbearing (DSSI bearing) was placed The DSSI bearing wasdeveloped by substituting the flat sliding surface of anordinary spherical sliding bearing with a spherical slidingsurface Its working mechanism in earthquakes was similarto a Friction Pendulum Sliding (FPS) bearing The deck wassupported on the four bearings and steel plates were boltedon the deck to achieve an additional mass of 100 tons

Steel blocks were installed on the connection plates ofbearings on Table B to restrict excessive bearing displace-ments as shown in Figure 8 A certain horizontal distance wasdesigned between the block and the side of the upper bearingplate which was named Clearance in this paper

6 Shock and Vibration

Deck

Pier 1

Table B Table C

Load transducer

Pier 2

DSSI bearing

Steel block

Additional mass

Figure 7 The test setup and specimen

Lower bearing plateSteel block

Connection plate

SliderUpper bearing plate

Clearance

Figure 8 Configuration of a bearing with shear pins used as a fixed bearing

A record of Wenchuan Earthquake as shown in Figure 9was input in the cases The motion was recorded in the 119864-119882 direction in Wolong Sichuan Province The peak groundacceleration is 948 gal

Four shaking table test cases were arranged and listed inTable 1 As shown the peak ground acceleration was scaleddown in order not to damage load transducers

In Cases 1 and 3 a rubber plate was inserted in eachClearance to study the applicability of the method for dif-ferent pounding stiffness so pounding stiffness of these twocases was lower

Responses of the bridge model were measured by 33transducers collecting data from the bridge model with asampling frequency of 512Hz Bearing displacements were

Table 1 Test cases

Cases Clearance Peak acceleration1 10mm 02 g2 20mm 02 g3 50mm 05 g4 60mm 05 g

measured by 8 displacement transducers bearing forces weremeasured by 4 load transducers and acceleration for thetables decks and piers was measured by 21 accelerometers

In Figures 10ndash12 time histories of displacements forcesand relative velocities of bearings on Table B in all the four

Shock and Vibration 7

0 10 20 30 40 50minus800

minus600

minus400

minus200

0

200

400

600

800

Acce

lera

tion

(gal

)

Time (s)

(a) A ground acceleration record of Wenchuan Earthquake

0 1 2 3 4 5

0

500

1000

1500

2000

2500

3000

Pseu

doac

cele

ratio

n (g

al)

Period (s)

(b) Pseudoacceleration response spectra (5 damping)

Figure 9 A record of Wenchuan Earthquake used in the cases

10 20 30 40minus20

minus15

minus10

minus5

0

5

10

15

20

Disp

lace

men

t (m

m)

Time (t)

(a) Case 1

10 20 30 40minus30

minus20

minus10

0

10

20

30

Disp

lace

men

t (m

m)

Time (t)

(b) Case 2

10 20 30 40minus80

minus60

minus40

minus20

0

20

40

60

80

Disp

lace

men

t (m

m)

Time (t)

(c) Case 3

10 20 30 40minus80

minus60

minus40

minus20

0

20

40

60

80

Disp

lace

men

t (m

m)

Time (t)

(d) Case 4

Figure 10 Time histories of bearing displacements on Table B in all the cases

8 Shock and Vibration

minus80

minus60

minus40

minus20

0

20

40

60

80

10 20 30 40

Forc

e (kN

)

Time (t)

(a) Case 1

minus80

minus60

minus40

minus20

0

20

40

60

80

10 20 30 40

Forc

e (kN

)

Time (t)

(b) Case 2

minus200

minus150

minus100

minus50

0

50

100

150

200

10 20 30 40

Forc

e (kN

)

Time (t)

(c) Case 3

10 20 30 40minus200

minus150

minus100

minus50

0

50

100

150

200

Forc

e (kN

)

Time (t)

(d) Case 4

Figure 11 Time histories of bearing forces on Table B in all the cases

cases are shown The bearing displacement is the averagevalue of two bearings the bearing force is the resultant forceof two bearings and the relative velocity is the derivative ofthe bearing displacement

In order to verify the simplified SDOF method (13) isused to calculate the theoretical pounding force for eachpounding and then the theoretical pounding force 119865theoreticalis compared with the test pounding force 119865test

The theoretical pounding force is calculated and listedin Table 2 119898 is the pounding mass and equals 100 tonsfor all the cases 119896 is the pounding stiffness and equals thepounding force divided by the displacement measured in thetest Once the pounding happens bearing relative velocitydrops rapidly The bearing relative velocity just before themoment of pounding is extracted as the pounding velocity119881

As shown errors between theoretical pounding forcesand test pounding forces are less than 20 so the simplifiedSDOF method is verified By the comparison it is found thatthe deck can be simulated as a particle and the simplified

SDOF method can be used to study the deck-pier poundingissue

Because no energy dissipation is considered in themodelthe simplified SDOF method may overestimate the actualpounding force and the theoretical pounding force is alwayslarger than the test pounding force Therefore the simplifiedSDOF method can provide a reasonable upper limit ofpounding forcesThe actual pounding force is also influencedby energy dissipation and damage condition during thepounding process

4 FEM Analysis of the Deck-PierPounding Issue

In order to study pounding responses under longitudinalearthquakes and the applicability of the simplified SDOFmethod a typical simply supported girder bridge is studiedThe span is 16m and piers are both 5m high The left end ofthe deck is supported on two same expansion bearings and

Shock and Vibration 9

10 20 30 40minus150

minus100

minus50

0

50

100

150

Velo

city

(mm

s)

Time (t)

(a) Case 1

10 20 30 40minus150

minus100

minus50

0

50

100

150

Velo

city

(mm

s)

Time (t)

(b) Case 2

10 20 30 40minus400

minus300

minus200

minus100

0

100

200

300

400

Velo

city

(mm

s)

Time (t)

(c) Case 3

10 20 30 40minus400

minus300

minus200

minus100

0

100

200

300

400

Velo

city

(mm

s)

Time (t)

(d) Case 4

Figure 12 Time histories of bearing velocities on Table B in all the cases

Table 2 Comparisons of theoretical pounding forces and test pounding forces

Case Pounding number 119898 (ton) 119896 (kNmm) 119881 (mms) 119865theoretical (kN) 119865test (kN) Error ()1 1 100 1077 minus5861 minus6081 minus5533 9911 2 100 1265 5708 6420 6124 4831 3 100 1133 minus5581 minus5942 minus5020 18381 4 100 1391 minus5540 minus6533 minus6217 5081 5 100 1231 6463 7170 6708 6892 1 100 5025 minus3089 minus6924 minus6096 13583 1 100 1001 17079 17092 16657 2613 2 100 848 minus14378 minus13243 minus11373 16443 3 100 1415 18700 22247 19786 12444 1 100 4589 10447 22378 19045 17504 2 100 6546 minus8090 minus20698 minus17905 15604 3 100 5195 9835 22418 19196 16784 4 100 5071 minus5128 minus11548 minus10224 12954 5 100 5514 minus4639 minus10894 minus9116 1950

10 Shock and Vibration

Seismic isolation bearing Expansion bearing

Steel block

Clearance

Steel block

Clearance

Deck

Pier

(a) The whole structural model

Gap

Hook

Expansion bearing Seismic isolation bearing

Plastic Wen Plastic Wen

Gap

Hook

mdkpkp

(b) The SDOF model

Figure 13 Models of a simply supported girder bridge

Table 3 Parameters of a simply supported girder bridge model

Components ParametersDeck Elasticity modulus is 345 times 107 kPa density is 25 tonm3 and cross-sectional area is 54m2

Pier Elasticity modulus is 345 times 107 kPa density is 25 tonm3 and cross-sectional area is 20m2

Expansion bearing Horizontal stiffness before yielding is 432 times 104 kNm yield force is 432 kN and horizontal stiffness after yieldingis 0

Seismic isolationbearing

Horizontal stiffness before yielding is 432 times 104 kNm yield force is 432 kN and horizontal stiffness after yieldingis 216 kNm

Steel block Stiffness of both gap and hook link elements is 10 times 1010 kNm and Clearances are 01m

the right end of the deck is supported on two same seismicisolation bearings Steel blocks for unseating prevention areinstalled on both sides of each bearing Clearances are allset to be 01m for all the steel blocks Deck-pier poundingswill happen between the deck and steel blocks under longi-tudinal earthquakes until bearing displacements exceed thepreserved Clearances

41 Analysis Models and Seismic Input Two models of thesimply supported girder bridge are established with the FEMsoftware SAP2000 One is a whole structural model and theother one is a SDOF model For the whole structural model

all the components are simulated as shown in Figure 13(a)The deck and piers are simulated with frame elementsBearings are simulated with Plastic Wen link elements Steelblocks are simulated with gap and hook link elements All theelements and their parameters are listed in Table 3 For theSDOF model pier masses are neglected and the deck massis concentrated on a particle as shown in Figure 13(b) 119896119901represents the horizontal stiffness of piers and the deck mass119898119889 is 216 tons

Sinusoidal acceleration wave is input as the seismic exci-tation for simplicity Peak acceleration is 4ms2 frequency is1Hz duration is 5 s and integration step is 1ms

Shock and Vibration 11

minus3000minus2000minus1000

0100020003000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(a) 120573 = 00034

minus4000minus2000

020004000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(b) 120573 = 00138

minus6000minus4000minus2000

0200040006000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(c) 120573 = 0034

minus10000minus5000

05000

10000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(d) 120573 = 0103

minus20000minus15000minus10000minus5000

05000

100001500020000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(e) 120573 = 0345

0 1 2 3 4 5minus30000minus20000minus10000

0100002000030000

Bear

ing

forc

e (kN

)

Time (s)

(f) 120573 = 10

Figure 14 Force time histories of the expansion bearing

42 Applicability of the Simplified SDOFMethod To evaluatethe applicability of the simplified SDOF method results ofthe SDOF model for different pier stiffness are comparedwith those of the whole structural model The influenceof the pier stiffness on pounding response is studied byaltering the elastic modulus of piers The horizontal stiffnessis designed to be 40 times 104 kNm 16 times 105 kNm 40times 105 kNm 12 times 106 kNm 40 times 106 kNm and 116 times107 kNm respectively The axial stiffness of the deck is 116 times107 kNm So stiffness ratios are 00034 00138 0034 01030345 and 10 respectively

Force time histories of the expansion bearing of the twomodels are shown in Figure 14 The expansion bearing forceis composed of the friction force and the pounding force andthe latter is far larger than the former In the figure solid linesindicate results of the whole structural model and brokenlines indicate results of the SDOF model

Every spike in Figure 14 indicates one pounding betweenthe deck and the pier As shown along with the increase ofthe pier stiffness the pounding duration becomes shorter andthe pounding force becomes larger Differences between thetwo models are very small except for 120573 = 10 The poundingforce and the pounding time of the two models are almostthe same for the first five cases Therefore the SDOF modelcan estimate the pounding force fairly accurately for 120573 lt

0345 at least All the peak pounding forces calculated by theSDOF model are larger than those of the whole structural

model which is similar to the theoretical results It needs tobe noted that 120573 = 0345 is already very big pier stiffness andapplicability of the simplified SDOF method is verified foralmost all the bridges

For 120573 = 10 differences of the pounding force and thepounding time between the twomodels are much larger thanother cases So the SDOF model is not applicable anymoreHowever because the horizontal stiffness of bridge pier isalways much less than the axial stiffness of the deck 120573 = 10is a very rare condition

43 Influences of the Pier Stiffness and the Pier Mass onPounding Responses The influence of the pier horizontalstiffness on pounding responses is studied by altering theelastic modulus of piers as mentioned in Section 42 Theinfluence of the pier mass on pounding responses is studiedby altering the density of piers Pier masses are designed tobe 5 tons 10 tons 25 tons 50 tons and 125 tons respectivelywhich are named as M1 to M5

The maximum pounding responses of the whole struc-tural model are shown in Figure 15

As shown the pounding force will increase and thedeck displacement will decrease with the increase of thepier stiffness The influence of the pier mass on the deckdisplacement is very small

For minor pier stiffness the influence of the pier mass onthe pounding force is also not too much so the pier mass can

12 Shock and Vibration

3 10 100 10000

5000

10000

15000

20000

25000

30000Po

undi

ng fo

rce (

kN)

M1M2M3

M4M5

Pier stiffness (104 kNm)

(a) Maximum pounding forces

3 10 100 1000010

012

014

016

018

M1M2M3

M4M5

Gird

er d

ispla

cem

ent (

m)

Pier stiffness (104 kNm)

(b) Maximum deck displacements

Figure 15 Maximum pounding responses

be neglected and the SDOFmodel is applicable However theinfluence of the pier mass on the pounding force is bigger formajor pier stiffness and so the pier mass is not negligible andthe SDOF model is no more applicable

5 Conclusions

In this paper theoretical analysis shaking table test andfinite element analysis were conducted to study the deck-pierpounding issue the following conclusions are drawn

(1) Based on theoretical results and test results it wasfound that the longitudinal deck-pier pounding issuecan be simulated by the simplified SDOF method ifenergy dissipation is negligible and the pier stiffnessis much less than the axial stiffness of the deck

(2) It was found that the pounding duration becomesshorter and the pounding force becomes larger alongwith the increase of the pier stiffness

(3) The pier mass has little effect on pounding responsesexcept for bridges with very rigid piers

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of China (no 51278372) and the Ministry ofScience and Technology of China Grant no SLDRCE 14-B-15

References

[1] W Goldsmith Impact Dover New York NY USA 2001[2] G Cole R Dhakal A Carr and D Bull ldquoAn investigation

of the effects of mass distribution on pounding structuresrdquoEarthquake Engineering and Structural Dynamics vol 40 no6 pp 641ndash659 2011

[3] H Hao K Bi N Chouw and W-X Ren ldquoState-of-the-artreview on seismic induced pounding response of bridge struc-turesrdquo Journal of Earthquake and Tsunami vol 7 no 3 ArticleID 1350019 2013

[4] B F Maison and K Kasai ldquoAnalysis for a type of structuralpoundingrdquo Journal of Structural Engineering vol 116 no 4 pp957ndash977 1990

[5] B F Maison and K Kasai ldquoDynamics of pounding whentwo buildings colliderdquo Earthquake Engineering and StructuralDynamics vol 21 no 9 pp 771ndash786 1992

[6] R Jankowski KWilde and Y Fujino ldquoPounding of superstruc-ture segments in isolated elevated bridge during earthquakesrdquoEarthquake Engineering and Structural Dynamics vol 27 no 5pp 487ndash502 1998

[7] K T Chau and X X Wei ldquoPounding of structures modelled asnon-linear impacts of two oscillatorsrdquo Earthquake Engineeringand Structural Dynamics vol 30 no 5 pp 633ndash651 2001

[8] S Muthukumar and R DesRoches ldquoA Hertz contact modelwith non-linear damping for pounding simulationrdquo EarthquakeEngineering and Structural Dynamics vol 35 no 7 pp 811ndash8282006

[9] P Komodromos P C Polycarpou L Papaloizou and M CPhocas ldquoResponse of seismically isolated buildings consideringpoundingsrdquo Earthquake Engineering and Structural Dynamicsvol 36 no 12 pp 1605ndash1622 2007

[10] K Ye L Li and H Zhu ldquoA note on the Hertz contact modelwith nonlinear damping for pounding simulationrdquo EarthquakeEngineering and Structural Dynamics vol 38 no 9 pp 1135ndash1142 2009

Shock and Vibration 13

[11] C C Patel and R S Jangid ldquoDynamic response of adjacentstructures connected by friction damperrdquo Earthquake andStructures vol 2 no 2 pp 149ndash169 2011

[12] R Jankowski ldquoNon-linear FEM analysis of pounding-involvedresponse of buildings under non-uniform earthquake excita-tionrdquo Engineering Structures vol 37 pp 99ndash105 2012

[13] D G Weng C Zhang X L Lu S Zeng and S M Zhang ldquoAsimplified design procedure for seismic retrofit of earthquake-damaged RC frames with viscous dampersrdquo Structural Engi-neering and Mechanics vol 44 no 5 pp 611ndash631 2012

[14] G Watanabe and K Kawashima ldquoNumerical simulation ofpounding of bridge decksrdquo in Proceedings of the 13thWorld Con-ference on Earthquake Engineering Paper No 884 VancouverCanada August 2004

[15] E G Dimitrakopoulos ldquoSeismic response analysis of skewbridges with pounding deck-abutment jointsrdquo EngineeringStructures vol 33 no 3 pp 813ndash826 2011

[16] B Li K Bi N Chouw J W Butterworth and H Hao ldquoEffectof abutment excitation on bridge poundingrdquo Engineering Struc-tures vol 54 pp 57ndash68 2013

[17] C-J Wang and M-H Shih ldquoPerformance study of a bridgeinvolving sliding decks and pounded abutment during a violentearthquakerdquo Engineering Structures vol 29 no 5 pp 802ndash8122007

[18] Y Huo and J Zhang ldquoEffects of pounding and skewness onseismic responses of typical multispan highway bridges usingthe fragility function methodrdquo Journal of Bridge Engineeringvol 18 no 6 pp 499ndash515 2013

[19] K Bi and H Hao ldquoNumerical simulation of pounding damageto bridge structures under spatially varying ground motionsrdquoEngineering Structures vol 46 pp 62ndash76 2013

[20] S E Abdel Raheem ldquoPounding mitigation and unseatingprevention at expansion joints of isolated multi-span bridgesrdquoEngineering Structures vol 31 no 10 pp 2345ndash2356 2009

[21] B Li and N Chouw ldquoExperimental investigation of inelasticbridge response under spatially varying excitationswith pound-ingrdquo Engineering Structures vol 79 pp 106ndash116 2014

[22] S-H Kim and M Shinozuka ldquoEffects of seismically inducedpounding at expansion joints of concrete bridgesrdquo Journal ofEngineering Mechanics vol 129 no 11 pp 1225ndash1234 2003

[23] M J N Priestley F Seible and G M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Shock and Vibration

Deck

Pier 1

Table B Table C

Load transducer

Pier 2

DSSI bearing

Steel block

Additional mass

Figure 7 The test setup and specimen

Lower bearing plateSteel block

Connection plate

SliderUpper bearing plate

Clearance

Figure 8 Configuration of a bearing with shear pins used as a fixed bearing

A record of Wenchuan Earthquake as shown in Figure 9was input in the cases The motion was recorded in the 119864-119882 direction in Wolong Sichuan Province The peak groundacceleration is 948 gal

Four shaking table test cases were arranged and listed inTable 1 As shown the peak ground acceleration was scaleddown in order not to damage load transducers

In Cases 1 and 3 a rubber plate was inserted in eachClearance to study the applicability of the method for dif-ferent pounding stiffness so pounding stiffness of these twocases was lower

Responses of the bridge model were measured by 33transducers collecting data from the bridge model with asampling frequency of 512Hz Bearing displacements were

Table 1 Test cases

Cases Clearance Peak acceleration1 10mm 02 g2 20mm 02 g3 50mm 05 g4 60mm 05 g

measured by 8 displacement transducers bearing forces weremeasured by 4 load transducers and acceleration for thetables decks and piers was measured by 21 accelerometers

In Figures 10ndash12 time histories of displacements forcesand relative velocities of bearings on Table B in all the four

Shock and Vibration 7

0 10 20 30 40 50minus800

minus600

minus400

minus200

0

200

400

600

800

Acce

lera

tion

(gal

)

Time (s)

(a) A ground acceleration record of Wenchuan Earthquake

0 1 2 3 4 5

0

500

1000

1500

2000

2500

3000

Pseu

doac

cele

ratio

n (g

al)

Period (s)

(b) Pseudoacceleration response spectra (5 damping)

Figure 9 A record of Wenchuan Earthquake used in the cases

10 20 30 40minus20

minus15

minus10

minus5

0

5

10

15

20

Disp

lace

men

t (m

m)

Time (t)

(a) Case 1

10 20 30 40minus30

minus20

minus10

0

10

20

30

Disp

lace

men

t (m

m)

Time (t)

(b) Case 2

10 20 30 40minus80

minus60

minus40

minus20

0

20

40

60

80

Disp

lace

men

t (m

m)

Time (t)

(c) Case 3

10 20 30 40minus80

minus60

minus40

minus20

0

20

40

60

80

Disp

lace

men

t (m

m)

Time (t)

(d) Case 4

Figure 10 Time histories of bearing displacements on Table B in all the cases

8 Shock and Vibration

minus80

minus60

minus40

minus20

0

20

40

60

80

10 20 30 40

Forc

e (kN

)

Time (t)

(a) Case 1

minus80

minus60

minus40

minus20

0

20

40

60

80

10 20 30 40

Forc

e (kN

)

Time (t)

(b) Case 2

minus200

minus150

minus100

minus50

0

50

100

150

200

10 20 30 40

Forc

e (kN

)

Time (t)

(c) Case 3

10 20 30 40minus200

minus150

minus100

minus50

0

50

100

150

200

Forc

e (kN

)

Time (t)

(d) Case 4

Figure 11 Time histories of bearing forces on Table B in all the cases

cases are shown The bearing displacement is the averagevalue of two bearings the bearing force is the resultant forceof two bearings and the relative velocity is the derivative ofthe bearing displacement

In order to verify the simplified SDOF method (13) isused to calculate the theoretical pounding force for eachpounding and then the theoretical pounding force 119865theoreticalis compared with the test pounding force 119865test

The theoretical pounding force is calculated and listedin Table 2 119898 is the pounding mass and equals 100 tonsfor all the cases 119896 is the pounding stiffness and equals thepounding force divided by the displacement measured in thetest Once the pounding happens bearing relative velocitydrops rapidly The bearing relative velocity just before themoment of pounding is extracted as the pounding velocity119881

As shown errors between theoretical pounding forcesand test pounding forces are less than 20 so the simplifiedSDOF method is verified By the comparison it is found thatthe deck can be simulated as a particle and the simplified

SDOF method can be used to study the deck-pier poundingissue

Because no energy dissipation is considered in themodelthe simplified SDOF method may overestimate the actualpounding force and the theoretical pounding force is alwayslarger than the test pounding force Therefore the simplifiedSDOF method can provide a reasonable upper limit ofpounding forcesThe actual pounding force is also influencedby energy dissipation and damage condition during thepounding process

4 FEM Analysis of the Deck-PierPounding Issue

In order to study pounding responses under longitudinalearthquakes and the applicability of the simplified SDOFmethod a typical simply supported girder bridge is studiedThe span is 16m and piers are both 5m high The left end ofthe deck is supported on two same expansion bearings and

Shock and Vibration 9

10 20 30 40minus150

minus100

minus50

0

50

100

150

Velo

city

(mm

s)

Time (t)

(a) Case 1

10 20 30 40minus150

minus100

minus50

0

50

100

150

Velo

city

(mm

s)

Time (t)

(b) Case 2

10 20 30 40minus400

minus300

minus200

minus100

0

100

200

300

400

Velo

city

(mm

s)

Time (t)

(c) Case 3

10 20 30 40minus400

minus300

minus200

minus100

0

100

200

300

400

Velo

city

(mm

s)

Time (t)

(d) Case 4

Figure 12 Time histories of bearing velocities on Table B in all the cases

Table 2 Comparisons of theoretical pounding forces and test pounding forces

Case Pounding number 119898 (ton) 119896 (kNmm) 119881 (mms) 119865theoretical (kN) 119865test (kN) Error ()1 1 100 1077 minus5861 minus6081 minus5533 9911 2 100 1265 5708 6420 6124 4831 3 100 1133 minus5581 minus5942 minus5020 18381 4 100 1391 minus5540 minus6533 minus6217 5081 5 100 1231 6463 7170 6708 6892 1 100 5025 minus3089 minus6924 minus6096 13583 1 100 1001 17079 17092 16657 2613 2 100 848 minus14378 minus13243 minus11373 16443 3 100 1415 18700 22247 19786 12444 1 100 4589 10447 22378 19045 17504 2 100 6546 minus8090 minus20698 minus17905 15604 3 100 5195 9835 22418 19196 16784 4 100 5071 minus5128 minus11548 minus10224 12954 5 100 5514 minus4639 minus10894 minus9116 1950

10 Shock and Vibration

Seismic isolation bearing Expansion bearing

Steel block

Clearance

Steel block

Clearance

Deck

Pier

(a) The whole structural model

Gap

Hook

Expansion bearing Seismic isolation bearing

Plastic Wen Plastic Wen

Gap

Hook

mdkpkp

(b) The SDOF model

Figure 13 Models of a simply supported girder bridge

Table 3 Parameters of a simply supported girder bridge model

Components ParametersDeck Elasticity modulus is 345 times 107 kPa density is 25 tonm3 and cross-sectional area is 54m2

Pier Elasticity modulus is 345 times 107 kPa density is 25 tonm3 and cross-sectional area is 20m2

Expansion bearing Horizontal stiffness before yielding is 432 times 104 kNm yield force is 432 kN and horizontal stiffness after yieldingis 0

Seismic isolationbearing

Horizontal stiffness before yielding is 432 times 104 kNm yield force is 432 kN and horizontal stiffness after yieldingis 216 kNm

Steel block Stiffness of both gap and hook link elements is 10 times 1010 kNm and Clearances are 01m

the right end of the deck is supported on two same seismicisolation bearings Steel blocks for unseating prevention areinstalled on both sides of each bearing Clearances are allset to be 01m for all the steel blocks Deck-pier poundingswill happen between the deck and steel blocks under longi-tudinal earthquakes until bearing displacements exceed thepreserved Clearances

41 Analysis Models and Seismic Input Two models of thesimply supported girder bridge are established with the FEMsoftware SAP2000 One is a whole structural model and theother one is a SDOF model For the whole structural model

all the components are simulated as shown in Figure 13(a)The deck and piers are simulated with frame elementsBearings are simulated with Plastic Wen link elements Steelblocks are simulated with gap and hook link elements All theelements and their parameters are listed in Table 3 For theSDOF model pier masses are neglected and the deck massis concentrated on a particle as shown in Figure 13(b) 119896119901represents the horizontal stiffness of piers and the deck mass119898119889 is 216 tons

Sinusoidal acceleration wave is input as the seismic exci-tation for simplicity Peak acceleration is 4ms2 frequency is1Hz duration is 5 s and integration step is 1ms

Shock and Vibration 11

minus3000minus2000minus1000

0100020003000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(a) 120573 = 00034

minus4000minus2000

020004000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(b) 120573 = 00138

minus6000minus4000minus2000

0200040006000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(c) 120573 = 0034

minus10000minus5000

05000

10000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(d) 120573 = 0103

minus20000minus15000minus10000minus5000

05000

100001500020000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(e) 120573 = 0345

0 1 2 3 4 5minus30000minus20000minus10000

0100002000030000

Bear

ing

forc

e (kN

)

Time (s)

(f) 120573 = 10

Figure 14 Force time histories of the expansion bearing

42 Applicability of the Simplified SDOFMethod To evaluatethe applicability of the simplified SDOF method results ofthe SDOF model for different pier stiffness are comparedwith those of the whole structural model The influenceof the pier stiffness on pounding response is studied byaltering the elastic modulus of piers The horizontal stiffnessis designed to be 40 times 104 kNm 16 times 105 kNm 40times 105 kNm 12 times 106 kNm 40 times 106 kNm and 116 times107 kNm respectively The axial stiffness of the deck is 116 times107 kNm So stiffness ratios are 00034 00138 0034 01030345 and 10 respectively

Force time histories of the expansion bearing of the twomodels are shown in Figure 14 The expansion bearing forceis composed of the friction force and the pounding force andthe latter is far larger than the former In the figure solid linesindicate results of the whole structural model and brokenlines indicate results of the SDOF model

Every spike in Figure 14 indicates one pounding betweenthe deck and the pier As shown along with the increase ofthe pier stiffness the pounding duration becomes shorter andthe pounding force becomes larger Differences between thetwo models are very small except for 120573 = 10 The poundingforce and the pounding time of the two models are almostthe same for the first five cases Therefore the SDOF modelcan estimate the pounding force fairly accurately for 120573 lt

0345 at least All the peak pounding forces calculated by theSDOF model are larger than those of the whole structural

model which is similar to the theoretical results It needs tobe noted that 120573 = 0345 is already very big pier stiffness andapplicability of the simplified SDOF method is verified foralmost all the bridges

For 120573 = 10 differences of the pounding force and thepounding time between the twomodels are much larger thanother cases So the SDOF model is not applicable anymoreHowever because the horizontal stiffness of bridge pier isalways much less than the axial stiffness of the deck 120573 = 10is a very rare condition

43 Influences of the Pier Stiffness and the Pier Mass onPounding Responses The influence of the pier horizontalstiffness on pounding responses is studied by altering theelastic modulus of piers as mentioned in Section 42 Theinfluence of the pier mass on pounding responses is studiedby altering the density of piers Pier masses are designed tobe 5 tons 10 tons 25 tons 50 tons and 125 tons respectivelywhich are named as M1 to M5

The maximum pounding responses of the whole struc-tural model are shown in Figure 15

As shown the pounding force will increase and thedeck displacement will decrease with the increase of thepier stiffness The influence of the pier mass on the deckdisplacement is very small

For minor pier stiffness the influence of the pier mass onthe pounding force is also not too much so the pier mass can

12 Shock and Vibration

3 10 100 10000

5000

10000

15000

20000

25000

30000Po

undi

ng fo

rce (

kN)

M1M2M3

M4M5

Pier stiffness (104 kNm)

(a) Maximum pounding forces

3 10 100 1000010

012

014

016

018

M1M2M3

M4M5

Gird

er d

ispla

cem

ent (

m)

Pier stiffness (104 kNm)

(b) Maximum deck displacements

Figure 15 Maximum pounding responses

be neglected and the SDOFmodel is applicable However theinfluence of the pier mass on the pounding force is bigger formajor pier stiffness and so the pier mass is not negligible andthe SDOF model is no more applicable

5 Conclusions

In this paper theoretical analysis shaking table test andfinite element analysis were conducted to study the deck-pierpounding issue the following conclusions are drawn

(1) Based on theoretical results and test results it wasfound that the longitudinal deck-pier pounding issuecan be simulated by the simplified SDOF method ifenergy dissipation is negligible and the pier stiffnessis much less than the axial stiffness of the deck

(2) It was found that the pounding duration becomesshorter and the pounding force becomes larger alongwith the increase of the pier stiffness

(3) The pier mass has little effect on pounding responsesexcept for bridges with very rigid piers

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of China (no 51278372) and the Ministry ofScience and Technology of China Grant no SLDRCE 14-B-15

References

[1] W Goldsmith Impact Dover New York NY USA 2001[2] G Cole R Dhakal A Carr and D Bull ldquoAn investigation

of the effects of mass distribution on pounding structuresrdquoEarthquake Engineering and Structural Dynamics vol 40 no6 pp 641ndash659 2011

[3] H Hao K Bi N Chouw and W-X Ren ldquoState-of-the-artreview on seismic induced pounding response of bridge struc-turesrdquo Journal of Earthquake and Tsunami vol 7 no 3 ArticleID 1350019 2013

[4] B F Maison and K Kasai ldquoAnalysis for a type of structuralpoundingrdquo Journal of Structural Engineering vol 116 no 4 pp957ndash977 1990

[5] B F Maison and K Kasai ldquoDynamics of pounding whentwo buildings colliderdquo Earthquake Engineering and StructuralDynamics vol 21 no 9 pp 771ndash786 1992

[6] R Jankowski KWilde and Y Fujino ldquoPounding of superstruc-ture segments in isolated elevated bridge during earthquakesrdquoEarthquake Engineering and Structural Dynamics vol 27 no 5pp 487ndash502 1998

[7] K T Chau and X X Wei ldquoPounding of structures modelled asnon-linear impacts of two oscillatorsrdquo Earthquake Engineeringand Structural Dynamics vol 30 no 5 pp 633ndash651 2001

[8] S Muthukumar and R DesRoches ldquoA Hertz contact modelwith non-linear damping for pounding simulationrdquo EarthquakeEngineering and Structural Dynamics vol 35 no 7 pp 811ndash8282006

[9] P Komodromos P C Polycarpou L Papaloizou and M CPhocas ldquoResponse of seismically isolated buildings consideringpoundingsrdquo Earthquake Engineering and Structural Dynamicsvol 36 no 12 pp 1605ndash1622 2007

[10] K Ye L Li and H Zhu ldquoA note on the Hertz contact modelwith nonlinear damping for pounding simulationrdquo EarthquakeEngineering and Structural Dynamics vol 38 no 9 pp 1135ndash1142 2009

Shock and Vibration 13

[11] C C Patel and R S Jangid ldquoDynamic response of adjacentstructures connected by friction damperrdquo Earthquake andStructures vol 2 no 2 pp 149ndash169 2011

[12] R Jankowski ldquoNon-linear FEM analysis of pounding-involvedresponse of buildings under non-uniform earthquake excita-tionrdquo Engineering Structures vol 37 pp 99ndash105 2012

[13] D G Weng C Zhang X L Lu S Zeng and S M Zhang ldquoAsimplified design procedure for seismic retrofit of earthquake-damaged RC frames with viscous dampersrdquo Structural Engi-neering and Mechanics vol 44 no 5 pp 611ndash631 2012

[14] G Watanabe and K Kawashima ldquoNumerical simulation ofpounding of bridge decksrdquo in Proceedings of the 13thWorld Con-ference on Earthquake Engineering Paper No 884 VancouverCanada August 2004

[15] E G Dimitrakopoulos ldquoSeismic response analysis of skewbridges with pounding deck-abutment jointsrdquo EngineeringStructures vol 33 no 3 pp 813ndash826 2011

[16] B Li K Bi N Chouw J W Butterworth and H Hao ldquoEffectof abutment excitation on bridge poundingrdquo Engineering Struc-tures vol 54 pp 57ndash68 2013

[17] C-J Wang and M-H Shih ldquoPerformance study of a bridgeinvolving sliding decks and pounded abutment during a violentearthquakerdquo Engineering Structures vol 29 no 5 pp 802ndash8122007

[18] Y Huo and J Zhang ldquoEffects of pounding and skewness onseismic responses of typical multispan highway bridges usingthe fragility function methodrdquo Journal of Bridge Engineeringvol 18 no 6 pp 499ndash515 2013

[19] K Bi and H Hao ldquoNumerical simulation of pounding damageto bridge structures under spatially varying ground motionsrdquoEngineering Structures vol 46 pp 62ndash76 2013

[20] S E Abdel Raheem ldquoPounding mitigation and unseatingprevention at expansion joints of isolated multi-span bridgesrdquoEngineering Structures vol 31 no 10 pp 2345ndash2356 2009

[21] B Li and N Chouw ldquoExperimental investigation of inelasticbridge response under spatially varying excitationswith pound-ingrdquo Engineering Structures vol 79 pp 106ndash116 2014

[22] S-H Kim and M Shinozuka ldquoEffects of seismically inducedpounding at expansion joints of concrete bridgesrdquo Journal ofEngineering Mechanics vol 129 no 11 pp 1225ndash1234 2003

[23] M J N Priestley F Seible and G M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 7

0 10 20 30 40 50minus800

minus600

minus400

minus200

0

200

400

600

800

Acce

lera

tion

(gal

)

Time (s)

(a) A ground acceleration record of Wenchuan Earthquake

0 1 2 3 4 5

0

500

1000

1500

2000

2500

3000

Pseu

doac

cele

ratio

n (g

al)

Period (s)

(b) Pseudoacceleration response spectra (5 damping)

Figure 9 A record of Wenchuan Earthquake used in the cases

10 20 30 40minus20

minus15

minus10

minus5

0

5

10

15

20

Disp

lace

men

t (m

m)

Time (t)

(a) Case 1

10 20 30 40minus30

minus20

minus10

0

10

20

30

Disp

lace

men

t (m

m)

Time (t)

(b) Case 2

10 20 30 40minus80

minus60

minus40

minus20

0

20

40

60

80

Disp

lace

men

t (m

m)

Time (t)

(c) Case 3

10 20 30 40minus80

minus60

minus40

minus20

0

20

40

60

80

Disp

lace

men

t (m

m)

Time (t)

(d) Case 4

Figure 10 Time histories of bearing displacements on Table B in all the cases

8 Shock and Vibration

minus80

minus60

minus40

minus20

0

20

40

60

80

10 20 30 40

Forc

e (kN

)

Time (t)

(a) Case 1

minus80

minus60

minus40

minus20

0

20

40

60

80

10 20 30 40

Forc

e (kN

)

Time (t)

(b) Case 2

minus200

minus150

minus100

minus50

0

50

100

150

200

10 20 30 40

Forc

e (kN

)

Time (t)

(c) Case 3

10 20 30 40minus200

minus150

minus100

minus50

0

50

100

150

200

Forc

e (kN

)

Time (t)

(d) Case 4

Figure 11 Time histories of bearing forces on Table B in all the cases

cases are shown The bearing displacement is the averagevalue of two bearings the bearing force is the resultant forceof two bearings and the relative velocity is the derivative ofthe bearing displacement

In order to verify the simplified SDOF method (13) isused to calculate the theoretical pounding force for eachpounding and then the theoretical pounding force 119865theoreticalis compared with the test pounding force 119865test

The theoretical pounding force is calculated and listedin Table 2 119898 is the pounding mass and equals 100 tonsfor all the cases 119896 is the pounding stiffness and equals thepounding force divided by the displacement measured in thetest Once the pounding happens bearing relative velocitydrops rapidly The bearing relative velocity just before themoment of pounding is extracted as the pounding velocity119881

As shown errors between theoretical pounding forcesand test pounding forces are less than 20 so the simplifiedSDOF method is verified By the comparison it is found thatthe deck can be simulated as a particle and the simplified

SDOF method can be used to study the deck-pier poundingissue

Because no energy dissipation is considered in themodelthe simplified SDOF method may overestimate the actualpounding force and the theoretical pounding force is alwayslarger than the test pounding force Therefore the simplifiedSDOF method can provide a reasonable upper limit ofpounding forcesThe actual pounding force is also influencedby energy dissipation and damage condition during thepounding process

4 FEM Analysis of the Deck-PierPounding Issue

In order to study pounding responses under longitudinalearthquakes and the applicability of the simplified SDOFmethod a typical simply supported girder bridge is studiedThe span is 16m and piers are both 5m high The left end ofthe deck is supported on two same expansion bearings and

Shock and Vibration 9

10 20 30 40minus150

minus100

minus50

0

50

100

150

Velo

city

(mm

s)

Time (t)

(a) Case 1

10 20 30 40minus150

minus100

minus50

0

50

100

150

Velo

city

(mm

s)

Time (t)

(b) Case 2

10 20 30 40minus400

minus300

minus200

minus100

0

100

200

300

400

Velo

city

(mm

s)

Time (t)

(c) Case 3

10 20 30 40minus400

minus300

minus200

minus100

0

100

200

300

400

Velo

city

(mm

s)

Time (t)

(d) Case 4

Figure 12 Time histories of bearing velocities on Table B in all the cases

Table 2 Comparisons of theoretical pounding forces and test pounding forces

Case Pounding number 119898 (ton) 119896 (kNmm) 119881 (mms) 119865theoretical (kN) 119865test (kN) Error ()1 1 100 1077 minus5861 minus6081 minus5533 9911 2 100 1265 5708 6420 6124 4831 3 100 1133 minus5581 minus5942 minus5020 18381 4 100 1391 minus5540 minus6533 minus6217 5081 5 100 1231 6463 7170 6708 6892 1 100 5025 minus3089 minus6924 minus6096 13583 1 100 1001 17079 17092 16657 2613 2 100 848 minus14378 minus13243 minus11373 16443 3 100 1415 18700 22247 19786 12444 1 100 4589 10447 22378 19045 17504 2 100 6546 minus8090 minus20698 minus17905 15604 3 100 5195 9835 22418 19196 16784 4 100 5071 minus5128 minus11548 minus10224 12954 5 100 5514 minus4639 minus10894 minus9116 1950

10 Shock and Vibration

Seismic isolation bearing Expansion bearing

Steel block

Clearance

Steel block

Clearance

Deck

Pier

(a) The whole structural model

Gap

Hook

Expansion bearing Seismic isolation bearing

Plastic Wen Plastic Wen

Gap

Hook

mdkpkp

(b) The SDOF model

Figure 13 Models of a simply supported girder bridge

Table 3 Parameters of a simply supported girder bridge model

Components ParametersDeck Elasticity modulus is 345 times 107 kPa density is 25 tonm3 and cross-sectional area is 54m2

Pier Elasticity modulus is 345 times 107 kPa density is 25 tonm3 and cross-sectional area is 20m2

Expansion bearing Horizontal stiffness before yielding is 432 times 104 kNm yield force is 432 kN and horizontal stiffness after yieldingis 0

Seismic isolationbearing

Horizontal stiffness before yielding is 432 times 104 kNm yield force is 432 kN and horizontal stiffness after yieldingis 216 kNm

Steel block Stiffness of both gap and hook link elements is 10 times 1010 kNm and Clearances are 01m

the right end of the deck is supported on two same seismicisolation bearings Steel blocks for unseating prevention areinstalled on both sides of each bearing Clearances are allset to be 01m for all the steel blocks Deck-pier poundingswill happen between the deck and steel blocks under longi-tudinal earthquakes until bearing displacements exceed thepreserved Clearances

41 Analysis Models and Seismic Input Two models of thesimply supported girder bridge are established with the FEMsoftware SAP2000 One is a whole structural model and theother one is a SDOF model For the whole structural model

all the components are simulated as shown in Figure 13(a)The deck and piers are simulated with frame elementsBearings are simulated with Plastic Wen link elements Steelblocks are simulated with gap and hook link elements All theelements and their parameters are listed in Table 3 For theSDOF model pier masses are neglected and the deck massis concentrated on a particle as shown in Figure 13(b) 119896119901represents the horizontal stiffness of piers and the deck mass119898119889 is 216 tons

Sinusoidal acceleration wave is input as the seismic exci-tation for simplicity Peak acceleration is 4ms2 frequency is1Hz duration is 5 s and integration step is 1ms

Shock and Vibration 11

minus3000minus2000minus1000

0100020003000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(a) 120573 = 00034

minus4000minus2000

020004000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(b) 120573 = 00138

minus6000minus4000minus2000

0200040006000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(c) 120573 = 0034

minus10000minus5000

05000

10000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(d) 120573 = 0103

minus20000minus15000minus10000minus5000

05000

100001500020000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(e) 120573 = 0345

0 1 2 3 4 5minus30000minus20000minus10000

0100002000030000

Bear

ing

forc

e (kN

)

Time (s)

(f) 120573 = 10

Figure 14 Force time histories of the expansion bearing

42 Applicability of the Simplified SDOFMethod To evaluatethe applicability of the simplified SDOF method results ofthe SDOF model for different pier stiffness are comparedwith those of the whole structural model The influenceof the pier stiffness on pounding response is studied byaltering the elastic modulus of piers The horizontal stiffnessis designed to be 40 times 104 kNm 16 times 105 kNm 40times 105 kNm 12 times 106 kNm 40 times 106 kNm and 116 times107 kNm respectively The axial stiffness of the deck is 116 times107 kNm So stiffness ratios are 00034 00138 0034 01030345 and 10 respectively

Force time histories of the expansion bearing of the twomodels are shown in Figure 14 The expansion bearing forceis composed of the friction force and the pounding force andthe latter is far larger than the former In the figure solid linesindicate results of the whole structural model and brokenlines indicate results of the SDOF model

Every spike in Figure 14 indicates one pounding betweenthe deck and the pier As shown along with the increase ofthe pier stiffness the pounding duration becomes shorter andthe pounding force becomes larger Differences between thetwo models are very small except for 120573 = 10 The poundingforce and the pounding time of the two models are almostthe same for the first five cases Therefore the SDOF modelcan estimate the pounding force fairly accurately for 120573 lt

0345 at least All the peak pounding forces calculated by theSDOF model are larger than those of the whole structural

model which is similar to the theoretical results It needs tobe noted that 120573 = 0345 is already very big pier stiffness andapplicability of the simplified SDOF method is verified foralmost all the bridges

For 120573 = 10 differences of the pounding force and thepounding time between the twomodels are much larger thanother cases So the SDOF model is not applicable anymoreHowever because the horizontal stiffness of bridge pier isalways much less than the axial stiffness of the deck 120573 = 10is a very rare condition

43 Influences of the Pier Stiffness and the Pier Mass onPounding Responses The influence of the pier horizontalstiffness on pounding responses is studied by altering theelastic modulus of piers as mentioned in Section 42 Theinfluence of the pier mass on pounding responses is studiedby altering the density of piers Pier masses are designed tobe 5 tons 10 tons 25 tons 50 tons and 125 tons respectivelywhich are named as M1 to M5

The maximum pounding responses of the whole struc-tural model are shown in Figure 15

As shown the pounding force will increase and thedeck displacement will decrease with the increase of thepier stiffness The influence of the pier mass on the deckdisplacement is very small

For minor pier stiffness the influence of the pier mass onthe pounding force is also not too much so the pier mass can

12 Shock and Vibration

3 10 100 10000

5000

10000

15000

20000

25000

30000Po

undi

ng fo

rce (

kN)

M1M2M3

M4M5

Pier stiffness (104 kNm)

(a) Maximum pounding forces

3 10 100 1000010

012

014

016

018

M1M2M3

M4M5

Gird

er d

ispla

cem

ent (

m)

Pier stiffness (104 kNm)

(b) Maximum deck displacements

Figure 15 Maximum pounding responses

be neglected and the SDOFmodel is applicable However theinfluence of the pier mass on the pounding force is bigger formajor pier stiffness and so the pier mass is not negligible andthe SDOF model is no more applicable

5 Conclusions

In this paper theoretical analysis shaking table test andfinite element analysis were conducted to study the deck-pierpounding issue the following conclusions are drawn

(1) Based on theoretical results and test results it wasfound that the longitudinal deck-pier pounding issuecan be simulated by the simplified SDOF method ifenergy dissipation is negligible and the pier stiffnessis much less than the axial stiffness of the deck

(2) It was found that the pounding duration becomesshorter and the pounding force becomes larger alongwith the increase of the pier stiffness

(3) The pier mass has little effect on pounding responsesexcept for bridges with very rigid piers

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of China (no 51278372) and the Ministry ofScience and Technology of China Grant no SLDRCE 14-B-15

References

[1] W Goldsmith Impact Dover New York NY USA 2001[2] G Cole R Dhakal A Carr and D Bull ldquoAn investigation

of the effects of mass distribution on pounding structuresrdquoEarthquake Engineering and Structural Dynamics vol 40 no6 pp 641ndash659 2011

[3] H Hao K Bi N Chouw and W-X Ren ldquoState-of-the-artreview on seismic induced pounding response of bridge struc-turesrdquo Journal of Earthquake and Tsunami vol 7 no 3 ArticleID 1350019 2013

[4] B F Maison and K Kasai ldquoAnalysis for a type of structuralpoundingrdquo Journal of Structural Engineering vol 116 no 4 pp957ndash977 1990

[5] B F Maison and K Kasai ldquoDynamics of pounding whentwo buildings colliderdquo Earthquake Engineering and StructuralDynamics vol 21 no 9 pp 771ndash786 1992

[6] R Jankowski KWilde and Y Fujino ldquoPounding of superstruc-ture segments in isolated elevated bridge during earthquakesrdquoEarthquake Engineering and Structural Dynamics vol 27 no 5pp 487ndash502 1998

[7] K T Chau and X X Wei ldquoPounding of structures modelled asnon-linear impacts of two oscillatorsrdquo Earthquake Engineeringand Structural Dynamics vol 30 no 5 pp 633ndash651 2001

[8] S Muthukumar and R DesRoches ldquoA Hertz contact modelwith non-linear damping for pounding simulationrdquo EarthquakeEngineering and Structural Dynamics vol 35 no 7 pp 811ndash8282006

[9] P Komodromos P C Polycarpou L Papaloizou and M CPhocas ldquoResponse of seismically isolated buildings consideringpoundingsrdquo Earthquake Engineering and Structural Dynamicsvol 36 no 12 pp 1605ndash1622 2007

[10] K Ye L Li and H Zhu ldquoA note on the Hertz contact modelwith nonlinear damping for pounding simulationrdquo EarthquakeEngineering and Structural Dynamics vol 38 no 9 pp 1135ndash1142 2009

Shock and Vibration 13

[11] C C Patel and R S Jangid ldquoDynamic response of adjacentstructures connected by friction damperrdquo Earthquake andStructures vol 2 no 2 pp 149ndash169 2011

[12] R Jankowski ldquoNon-linear FEM analysis of pounding-involvedresponse of buildings under non-uniform earthquake excita-tionrdquo Engineering Structures vol 37 pp 99ndash105 2012

[13] D G Weng C Zhang X L Lu S Zeng and S M Zhang ldquoAsimplified design procedure for seismic retrofit of earthquake-damaged RC frames with viscous dampersrdquo Structural Engi-neering and Mechanics vol 44 no 5 pp 611ndash631 2012

[14] G Watanabe and K Kawashima ldquoNumerical simulation ofpounding of bridge decksrdquo in Proceedings of the 13thWorld Con-ference on Earthquake Engineering Paper No 884 VancouverCanada August 2004

[15] E G Dimitrakopoulos ldquoSeismic response analysis of skewbridges with pounding deck-abutment jointsrdquo EngineeringStructures vol 33 no 3 pp 813ndash826 2011

[16] B Li K Bi N Chouw J W Butterworth and H Hao ldquoEffectof abutment excitation on bridge poundingrdquo Engineering Struc-tures vol 54 pp 57ndash68 2013

[17] C-J Wang and M-H Shih ldquoPerformance study of a bridgeinvolving sliding decks and pounded abutment during a violentearthquakerdquo Engineering Structures vol 29 no 5 pp 802ndash8122007

[18] Y Huo and J Zhang ldquoEffects of pounding and skewness onseismic responses of typical multispan highway bridges usingthe fragility function methodrdquo Journal of Bridge Engineeringvol 18 no 6 pp 499ndash515 2013

[19] K Bi and H Hao ldquoNumerical simulation of pounding damageto bridge structures under spatially varying ground motionsrdquoEngineering Structures vol 46 pp 62ndash76 2013

[20] S E Abdel Raheem ldquoPounding mitigation and unseatingprevention at expansion joints of isolated multi-span bridgesrdquoEngineering Structures vol 31 no 10 pp 2345ndash2356 2009

[21] B Li and N Chouw ldquoExperimental investigation of inelasticbridge response under spatially varying excitationswith pound-ingrdquo Engineering Structures vol 79 pp 106ndash116 2014

[22] S-H Kim and M Shinozuka ldquoEffects of seismically inducedpounding at expansion joints of concrete bridgesrdquo Journal ofEngineering Mechanics vol 129 no 11 pp 1225ndash1234 2003

[23] M J N Priestley F Seible and G M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Shock and Vibration

minus80

minus60

minus40

minus20

0

20

40

60

80

10 20 30 40

Forc

e (kN

)

Time (t)

(a) Case 1

minus80

minus60

minus40

minus20

0

20

40

60

80

10 20 30 40

Forc

e (kN

)

Time (t)

(b) Case 2

minus200

minus150

minus100

minus50

0

50

100

150

200

10 20 30 40

Forc

e (kN

)

Time (t)

(c) Case 3

10 20 30 40minus200

minus150

minus100

minus50

0

50

100

150

200

Forc

e (kN

)

Time (t)

(d) Case 4

Figure 11 Time histories of bearing forces on Table B in all the cases

cases are shown The bearing displacement is the averagevalue of two bearings the bearing force is the resultant forceof two bearings and the relative velocity is the derivative ofthe bearing displacement

In order to verify the simplified SDOF method (13) isused to calculate the theoretical pounding force for eachpounding and then the theoretical pounding force 119865theoreticalis compared with the test pounding force 119865test

The theoretical pounding force is calculated and listedin Table 2 119898 is the pounding mass and equals 100 tonsfor all the cases 119896 is the pounding stiffness and equals thepounding force divided by the displacement measured in thetest Once the pounding happens bearing relative velocitydrops rapidly The bearing relative velocity just before themoment of pounding is extracted as the pounding velocity119881

As shown errors between theoretical pounding forcesand test pounding forces are less than 20 so the simplifiedSDOF method is verified By the comparison it is found thatthe deck can be simulated as a particle and the simplified

SDOF method can be used to study the deck-pier poundingissue

Because no energy dissipation is considered in themodelthe simplified SDOF method may overestimate the actualpounding force and the theoretical pounding force is alwayslarger than the test pounding force Therefore the simplifiedSDOF method can provide a reasonable upper limit ofpounding forcesThe actual pounding force is also influencedby energy dissipation and damage condition during thepounding process

4 FEM Analysis of the Deck-PierPounding Issue

In order to study pounding responses under longitudinalearthquakes and the applicability of the simplified SDOFmethod a typical simply supported girder bridge is studiedThe span is 16m and piers are both 5m high The left end ofthe deck is supported on two same expansion bearings and

Shock and Vibration 9

10 20 30 40minus150

minus100

minus50

0

50

100

150

Velo

city

(mm

s)

Time (t)

(a) Case 1

10 20 30 40minus150

minus100

minus50

0

50

100

150

Velo

city

(mm

s)

Time (t)

(b) Case 2

10 20 30 40minus400

minus300

minus200

minus100

0

100

200

300

400

Velo

city

(mm

s)

Time (t)

(c) Case 3

10 20 30 40minus400

minus300

minus200

minus100

0

100

200

300

400

Velo

city

(mm

s)

Time (t)

(d) Case 4

Figure 12 Time histories of bearing velocities on Table B in all the cases

Table 2 Comparisons of theoretical pounding forces and test pounding forces

Case Pounding number 119898 (ton) 119896 (kNmm) 119881 (mms) 119865theoretical (kN) 119865test (kN) Error ()1 1 100 1077 minus5861 minus6081 minus5533 9911 2 100 1265 5708 6420 6124 4831 3 100 1133 minus5581 minus5942 minus5020 18381 4 100 1391 minus5540 minus6533 minus6217 5081 5 100 1231 6463 7170 6708 6892 1 100 5025 minus3089 minus6924 minus6096 13583 1 100 1001 17079 17092 16657 2613 2 100 848 minus14378 minus13243 minus11373 16443 3 100 1415 18700 22247 19786 12444 1 100 4589 10447 22378 19045 17504 2 100 6546 minus8090 minus20698 minus17905 15604 3 100 5195 9835 22418 19196 16784 4 100 5071 minus5128 minus11548 minus10224 12954 5 100 5514 minus4639 minus10894 minus9116 1950

10 Shock and Vibration

Seismic isolation bearing Expansion bearing

Steel block

Clearance

Steel block

Clearance

Deck

Pier

(a) The whole structural model

Gap

Hook

Expansion bearing Seismic isolation bearing

Plastic Wen Plastic Wen

Gap

Hook

mdkpkp

(b) The SDOF model

Figure 13 Models of a simply supported girder bridge

Table 3 Parameters of a simply supported girder bridge model

Components ParametersDeck Elasticity modulus is 345 times 107 kPa density is 25 tonm3 and cross-sectional area is 54m2

Pier Elasticity modulus is 345 times 107 kPa density is 25 tonm3 and cross-sectional area is 20m2

Expansion bearing Horizontal stiffness before yielding is 432 times 104 kNm yield force is 432 kN and horizontal stiffness after yieldingis 0

Seismic isolationbearing

Horizontal stiffness before yielding is 432 times 104 kNm yield force is 432 kN and horizontal stiffness after yieldingis 216 kNm

Steel block Stiffness of both gap and hook link elements is 10 times 1010 kNm and Clearances are 01m

the right end of the deck is supported on two same seismicisolation bearings Steel blocks for unseating prevention areinstalled on both sides of each bearing Clearances are allset to be 01m for all the steel blocks Deck-pier poundingswill happen between the deck and steel blocks under longi-tudinal earthquakes until bearing displacements exceed thepreserved Clearances

41 Analysis Models and Seismic Input Two models of thesimply supported girder bridge are established with the FEMsoftware SAP2000 One is a whole structural model and theother one is a SDOF model For the whole structural model

all the components are simulated as shown in Figure 13(a)The deck and piers are simulated with frame elementsBearings are simulated with Plastic Wen link elements Steelblocks are simulated with gap and hook link elements All theelements and their parameters are listed in Table 3 For theSDOF model pier masses are neglected and the deck massis concentrated on a particle as shown in Figure 13(b) 119896119901represents the horizontal stiffness of piers and the deck mass119898119889 is 216 tons

Sinusoidal acceleration wave is input as the seismic exci-tation for simplicity Peak acceleration is 4ms2 frequency is1Hz duration is 5 s and integration step is 1ms

Shock and Vibration 11

minus3000minus2000minus1000

0100020003000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(a) 120573 = 00034

minus4000minus2000

020004000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(b) 120573 = 00138

minus6000minus4000minus2000

0200040006000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(c) 120573 = 0034

minus10000minus5000

05000

10000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(d) 120573 = 0103

minus20000minus15000minus10000minus5000

05000

100001500020000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(e) 120573 = 0345

0 1 2 3 4 5minus30000minus20000minus10000

0100002000030000

Bear

ing

forc

e (kN

)

Time (s)

(f) 120573 = 10

Figure 14 Force time histories of the expansion bearing

42 Applicability of the Simplified SDOFMethod To evaluatethe applicability of the simplified SDOF method results ofthe SDOF model for different pier stiffness are comparedwith those of the whole structural model The influenceof the pier stiffness on pounding response is studied byaltering the elastic modulus of piers The horizontal stiffnessis designed to be 40 times 104 kNm 16 times 105 kNm 40times 105 kNm 12 times 106 kNm 40 times 106 kNm and 116 times107 kNm respectively The axial stiffness of the deck is 116 times107 kNm So stiffness ratios are 00034 00138 0034 01030345 and 10 respectively

Force time histories of the expansion bearing of the twomodels are shown in Figure 14 The expansion bearing forceis composed of the friction force and the pounding force andthe latter is far larger than the former In the figure solid linesindicate results of the whole structural model and brokenlines indicate results of the SDOF model

Every spike in Figure 14 indicates one pounding betweenthe deck and the pier As shown along with the increase ofthe pier stiffness the pounding duration becomes shorter andthe pounding force becomes larger Differences between thetwo models are very small except for 120573 = 10 The poundingforce and the pounding time of the two models are almostthe same for the first five cases Therefore the SDOF modelcan estimate the pounding force fairly accurately for 120573 lt

0345 at least All the peak pounding forces calculated by theSDOF model are larger than those of the whole structural

model which is similar to the theoretical results It needs tobe noted that 120573 = 0345 is already very big pier stiffness andapplicability of the simplified SDOF method is verified foralmost all the bridges

For 120573 = 10 differences of the pounding force and thepounding time between the twomodels are much larger thanother cases So the SDOF model is not applicable anymoreHowever because the horizontal stiffness of bridge pier isalways much less than the axial stiffness of the deck 120573 = 10is a very rare condition

43 Influences of the Pier Stiffness and the Pier Mass onPounding Responses The influence of the pier horizontalstiffness on pounding responses is studied by altering theelastic modulus of piers as mentioned in Section 42 Theinfluence of the pier mass on pounding responses is studiedby altering the density of piers Pier masses are designed tobe 5 tons 10 tons 25 tons 50 tons and 125 tons respectivelywhich are named as M1 to M5

The maximum pounding responses of the whole struc-tural model are shown in Figure 15

As shown the pounding force will increase and thedeck displacement will decrease with the increase of thepier stiffness The influence of the pier mass on the deckdisplacement is very small

For minor pier stiffness the influence of the pier mass onthe pounding force is also not too much so the pier mass can

12 Shock and Vibration

3 10 100 10000

5000

10000

15000

20000

25000

30000Po

undi

ng fo

rce (

kN)

M1M2M3

M4M5

Pier stiffness (104 kNm)

(a) Maximum pounding forces

3 10 100 1000010

012

014

016

018

M1M2M3

M4M5

Gird

er d

ispla

cem

ent (

m)

Pier stiffness (104 kNm)

(b) Maximum deck displacements

Figure 15 Maximum pounding responses

be neglected and the SDOFmodel is applicable However theinfluence of the pier mass on the pounding force is bigger formajor pier stiffness and so the pier mass is not negligible andthe SDOF model is no more applicable

5 Conclusions

In this paper theoretical analysis shaking table test andfinite element analysis were conducted to study the deck-pierpounding issue the following conclusions are drawn

(1) Based on theoretical results and test results it wasfound that the longitudinal deck-pier pounding issuecan be simulated by the simplified SDOF method ifenergy dissipation is negligible and the pier stiffnessis much less than the axial stiffness of the deck

(2) It was found that the pounding duration becomesshorter and the pounding force becomes larger alongwith the increase of the pier stiffness

(3) The pier mass has little effect on pounding responsesexcept for bridges with very rigid piers

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of China (no 51278372) and the Ministry ofScience and Technology of China Grant no SLDRCE 14-B-15

References

[1] W Goldsmith Impact Dover New York NY USA 2001[2] G Cole R Dhakal A Carr and D Bull ldquoAn investigation

of the effects of mass distribution on pounding structuresrdquoEarthquake Engineering and Structural Dynamics vol 40 no6 pp 641ndash659 2011

[3] H Hao K Bi N Chouw and W-X Ren ldquoState-of-the-artreview on seismic induced pounding response of bridge struc-turesrdquo Journal of Earthquake and Tsunami vol 7 no 3 ArticleID 1350019 2013

[4] B F Maison and K Kasai ldquoAnalysis for a type of structuralpoundingrdquo Journal of Structural Engineering vol 116 no 4 pp957ndash977 1990

[5] B F Maison and K Kasai ldquoDynamics of pounding whentwo buildings colliderdquo Earthquake Engineering and StructuralDynamics vol 21 no 9 pp 771ndash786 1992

[6] R Jankowski KWilde and Y Fujino ldquoPounding of superstruc-ture segments in isolated elevated bridge during earthquakesrdquoEarthquake Engineering and Structural Dynamics vol 27 no 5pp 487ndash502 1998

[7] K T Chau and X X Wei ldquoPounding of structures modelled asnon-linear impacts of two oscillatorsrdquo Earthquake Engineeringand Structural Dynamics vol 30 no 5 pp 633ndash651 2001

[8] S Muthukumar and R DesRoches ldquoA Hertz contact modelwith non-linear damping for pounding simulationrdquo EarthquakeEngineering and Structural Dynamics vol 35 no 7 pp 811ndash8282006

[9] P Komodromos P C Polycarpou L Papaloizou and M CPhocas ldquoResponse of seismically isolated buildings consideringpoundingsrdquo Earthquake Engineering and Structural Dynamicsvol 36 no 12 pp 1605ndash1622 2007

[10] K Ye L Li and H Zhu ldquoA note on the Hertz contact modelwith nonlinear damping for pounding simulationrdquo EarthquakeEngineering and Structural Dynamics vol 38 no 9 pp 1135ndash1142 2009

Shock and Vibration 13

[11] C C Patel and R S Jangid ldquoDynamic response of adjacentstructures connected by friction damperrdquo Earthquake andStructures vol 2 no 2 pp 149ndash169 2011

[12] R Jankowski ldquoNon-linear FEM analysis of pounding-involvedresponse of buildings under non-uniform earthquake excita-tionrdquo Engineering Structures vol 37 pp 99ndash105 2012

[13] D G Weng C Zhang X L Lu S Zeng and S M Zhang ldquoAsimplified design procedure for seismic retrofit of earthquake-damaged RC frames with viscous dampersrdquo Structural Engi-neering and Mechanics vol 44 no 5 pp 611ndash631 2012

[14] G Watanabe and K Kawashima ldquoNumerical simulation ofpounding of bridge decksrdquo in Proceedings of the 13thWorld Con-ference on Earthquake Engineering Paper No 884 VancouverCanada August 2004

[15] E G Dimitrakopoulos ldquoSeismic response analysis of skewbridges with pounding deck-abutment jointsrdquo EngineeringStructures vol 33 no 3 pp 813ndash826 2011

[16] B Li K Bi N Chouw J W Butterworth and H Hao ldquoEffectof abutment excitation on bridge poundingrdquo Engineering Struc-tures vol 54 pp 57ndash68 2013

[17] C-J Wang and M-H Shih ldquoPerformance study of a bridgeinvolving sliding decks and pounded abutment during a violentearthquakerdquo Engineering Structures vol 29 no 5 pp 802ndash8122007

[18] Y Huo and J Zhang ldquoEffects of pounding and skewness onseismic responses of typical multispan highway bridges usingthe fragility function methodrdquo Journal of Bridge Engineeringvol 18 no 6 pp 499ndash515 2013

[19] K Bi and H Hao ldquoNumerical simulation of pounding damageto bridge structures under spatially varying ground motionsrdquoEngineering Structures vol 46 pp 62ndash76 2013

[20] S E Abdel Raheem ldquoPounding mitigation and unseatingprevention at expansion joints of isolated multi-span bridgesrdquoEngineering Structures vol 31 no 10 pp 2345ndash2356 2009

[21] B Li and N Chouw ldquoExperimental investigation of inelasticbridge response under spatially varying excitationswith pound-ingrdquo Engineering Structures vol 79 pp 106ndash116 2014

[22] S-H Kim and M Shinozuka ldquoEffects of seismically inducedpounding at expansion joints of concrete bridgesrdquo Journal ofEngineering Mechanics vol 129 no 11 pp 1225ndash1234 2003

[23] M J N Priestley F Seible and G M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 9

10 20 30 40minus150

minus100

minus50

0

50

100

150

Velo

city

(mm

s)

Time (t)

(a) Case 1

10 20 30 40minus150

minus100

minus50

0

50

100

150

Velo

city

(mm

s)

Time (t)

(b) Case 2

10 20 30 40minus400

minus300

minus200

minus100

0

100

200

300

400

Velo

city

(mm

s)

Time (t)

(c) Case 3

10 20 30 40minus400

minus300

minus200

minus100

0

100

200

300

400

Velo

city

(mm

s)

Time (t)

(d) Case 4

Figure 12 Time histories of bearing velocities on Table B in all the cases

Table 2 Comparisons of theoretical pounding forces and test pounding forces

Case Pounding number 119898 (ton) 119896 (kNmm) 119881 (mms) 119865theoretical (kN) 119865test (kN) Error ()1 1 100 1077 minus5861 minus6081 minus5533 9911 2 100 1265 5708 6420 6124 4831 3 100 1133 minus5581 minus5942 minus5020 18381 4 100 1391 minus5540 minus6533 minus6217 5081 5 100 1231 6463 7170 6708 6892 1 100 5025 minus3089 minus6924 minus6096 13583 1 100 1001 17079 17092 16657 2613 2 100 848 minus14378 minus13243 minus11373 16443 3 100 1415 18700 22247 19786 12444 1 100 4589 10447 22378 19045 17504 2 100 6546 minus8090 minus20698 minus17905 15604 3 100 5195 9835 22418 19196 16784 4 100 5071 minus5128 minus11548 minus10224 12954 5 100 5514 minus4639 minus10894 minus9116 1950

10 Shock and Vibration

Seismic isolation bearing Expansion bearing

Steel block

Clearance

Steel block

Clearance

Deck

Pier

(a) The whole structural model

Gap

Hook

Expansion bearing Seismic isolation bearing

Plastic Wen Plastic Wen

Gap

Hook

mdkpkp

(b) The SDOF model

Figure 13 Models of a simply supported girder bridge

Table 3 Parameters of a simply supported girder bridge model

Components ParametersDeck Elasticity modulus is 345 times 107 kPa density is 25 tonm3 and cross-sectional area is 54m2

Pier Elasticity modulus is 345 times 107 kPa density is 25 tonm3 and cross-sectional area is 20m2

Expansion bearing Horizontal stiffness before yielding is 432 times 104 kNm yield force is 432 kN and horizontal stiffness after yieldingis 0

Seismic isolationbearing

Horizontal stiffness before yielding is 432 times 104 kNm yield force is 432 kN and horizontal stiffness after yieldingis 216 kNm

Steel block Stiffness of both gap and hook link elements is 10 times 1010 kNm and Clearances are 01m

the right end of the deck is supported on two same seismicisolation bearings Steel blocks for unseating prevention areinstalled on both sides of each bearing Clearances are allset to be 01m for all the steel blocks Deck-pier poundingswill happen between the deck and steel blocks under longi-tudinal earthquakes until bearing displacements exceed thepreserved Clearances

41 Analysis Models and Seismic Input Two models of thesimply supported girder bridge are established with the FEMsoftware SAP2000 One is a whole structural model and theother one is a SDOF model For the whole structural model

all the components are simulated as shown in Figure 13(a)The deck and piers are simulated with frame elementsBearings are simulated with Plastic Wen link elements Steelblocks are simulated with gap and hook link elements All theelements and their parameters are listed in Table 3 For theSDOF model pier masses are neglected and the deck massis concentrated on a particle as shown in Figure 13(b) 119896119901represents the horizontal stiffness of piers and the deck mass119898119889 is 216 tons

Sinusoidal acceleration wave is input as the seismic exci-tation for simplicity Peak acceleration is 4ms2 frequency is1Hz duration is 5 s and integration step is 1ms

Shock and Vibration 11

minus3000minus2000minus1000

0100020003000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(a) 120573 = 00034

minus4000minus2000

020004000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(b) 120573 = 00138

minus6000minus4000minus2000

0200040006000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(c) 120573 = 0034

minus10000minus5000

05000

10000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(d) 120573 = 0103

minus20000minus15000minus10000minus5000

05000

100001500020000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(e) 120573 = 0345

0 1 2 3 4 5minus30000minus20000minus10000

0100002000030000

Bear

ing

forc

e (kN

)

Time (s)

(f) 120573 = 10

Figure 14 Force time histories of the expansion bearing

42 Applicability of the Simplified SDOFMethod To evaluatethe applicability of the simplified SDOF method results ofthe SDOF model for different pier stiffness are comparedwith those of the whole structural model The influenceof the pier stiffness on pounding response is studied byaltering the elastic modulus of piers The horizontal stiffnessis designed to be 40 times 104 kNm 16 times 105 kNm 40times 105 kNm 12 times 106 kNm 40 times 106 kNm and 116 times107 kNm respectively The axial stiffness of the deck is 116 times107 kNm So stiffness ratios are 00034 00138 0034 01030345 and 10 respectively

Force time histories of the expansion bearing of the twomodels are shown in Figure 14 The expansion bearing forceis composed of the friction force and the pounding force andthe latter is far larger than the former In the figure solid linesindicate results of the whole structural model and brokenlines indicate results of the SDOF model

Every spike in Figure 14 indicates one pounding betweenthe deck and the pier As shown along with the increase ofthe pier stiffness the pounding duration becomes shorter andthe pounding force becomes larger Differences between thetwo models are very small except for 120573 = 10 The poundingforce and the pounding time of the two models are almostthe same for the first five cases Therefore the SDOF modelcan estimate the pounding force fairly accurately for 120573 lt

0345 at least All the peak pounding forces calculated by theSDOF model are larger than those of the whole structural

model which is similar to the theoretical results It needs tobe noted that 120573 = 0345 is already very big pier stiffness andapplicability of the simplified SDOF method is verified foralmost all the bridges

For 120573 = 10 differences of the pounding force and thepounding time between the twomodels are much larger thanother cases So the SDOF model is not applicable anymoreHowever because the horizontal stiffness of bridge pier isalways much less than the axial stiffness of the deck 120573 = 10is a very rare condition

43 Influences of the Pier Stiffness and the Pier Mass onPounding Responses The influence of the pier horizontalstiffness on pounding responses is studied by altering theelastic modulus of piers as mentioned in Section 42 Theinfluence of the pier mass on pounding responses is studiedby altering the density of piers Pier masses are designed tobe 5 tons 10 tons 25 tons 50 tons and 125 tons respectivelywhich are named as M1 to M5

The maximum pounding responses of the whole struc-tural model are shown in Figure 15

As shown the pounding force will increase and thedeck displacement will decrease with the increase of thepier stiffness The influence of the pier mass on the deckdisplacement is very small

For minor pier stiffness the influence of the pier mass onthe pounding force is also not too much so the pier mass can

12 Shock and Vibration

3 10 100 10000

5000

10000

15000

20000

25000

30000Po

undi

ng fo

rce (

kN)

M1M2M3

M4M5

Pier stiffness (104 kNm)

(a) Maximum pounding forces

3 10 100 1000010

012

014

016

018

M1M2M3

M4M5

Gird

er d

ispla

cem

ent (

m)

Pier stiffness (104 kNm)

(b) Maximum deck displacements

Figure 15 Maximum pounding responses

be neglected and the SDOFmodel is applicable However theinfluence of the pier mass on the pounding force is bigger formajor pier stiffness and so the pier mass is not negligible andthe SDOF model is no more applicable

5 Conclusions

In this paper theoretical analysis shaking table test andfinite element analysis were conducted to study the deck-pierpounding issue the following conclusions are drawn

(1) Based on theoretical results and test results it wasfound that the longitudinal deck-pier pounding issuecan be simulated by the simplified SDOF method ifenergy dissipation is negligible and the pier stiffnessis much less than the axial stiffness of the deck

(2) It was found that the pounding duration becomesshorter and the pounding force becomes larger alongwith the increase of the pier stiffness

(3) The pier mass has little effect on pounding responsesexcept for bridges with very rigid piers

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of China (no 51278372) and the Ministry ofScience and Technology of China Grant no SLDRCE 14-B-15

References

[1] W Goldsmith Impact Dover New York NY USA 2001[2] G Cole R Dhakal A Carr and D Bull ldquoAn investigation

of the effects of mass distribution on pounding structuresrdquoEarthquake Engineering and Structural Dynamics vol 40 no6 pp 641ndash659 2011

[3] H Hao K Bi N Chouw and W-X Ren ldquoState-of-the-artreview on seismic induced pounding response of bridge struc-turesrdquo Journal of Earthquake and Tsunami vol 7 no 3 ArticleID 1350019 2013

[4] B F Maison and K Kasai ldquoAnalysis for a type of structuralpoundingrdquo Journal of Structural Engineering vol 116 no 4 pp957ndash977 1990

[5] B F Maison and K Kasai ldquoDynamics of pounding whentwo buildings colliderdquo Earthquake Engineering and StructuralDynamics vol 21 no 9 pp 771ndash786 1992

[6] R Jankowski KWilde and Y Fujino ldquoPounding of superstruc-ture segments in isolated elevated bridge during earthquakesrdquoEarthquake Engineering and Structural Dynamics vol 27 no 5pp 487ndash502 1998

[7] K T Chau and X X Wei ldquoPounding of structures modelled asnon-linear impacts of two oscillatorsrdquo Earthquake Engineeringand Structural Dynamics vol 30 no 5 pp 633ndash651 2001

[8] S Muthukumar and R DesRoches ldquoA Hertz contact modelwith non-linear damping for pounding simulationrdquo EarthquakeEngineering and Structural Dynamics vol 35 no 7 pp 811ndash8282006

[9] P Komodromos P C Polycarpou L Papaloizou and M CPhocas ldquoResponse of seismically isolated buildings consideringpoundingsrdquo Earthquake Engineering and Structural Dynamicsvol 36 no 12 pp 1605ndash1622 2007

[10] K Ye L Li and H Zhu ldquoA note on the Hertz contact modelwith nonlinear damping for pounding simulationrdquo EarthquakeEngineering and Structural Dynamics vol 38 no 9 pp 1135ndash1142 2009

Shock and Vibration 13

[11] C C Patel and R S Jangid ldquoDynamic response of adjacentstructures connected by friction damperrdquo Earthquake andStructures vol 2 no 2 pp 149ndash169 2011

[12] R Jankowski ldquoNon-linear FEM analysis of pounding-involvedresponse of buildings under non-uniform earthquake excita-tionrdquo Engineering Structures vol 37 pp 99ndash105 2012

[13] D G Weng C Zhang X L Lu S Zeng and S M Zhang ldquoAsimplified design procedure for seismic retrofit of earthquake-damaged RC frames with viscous dampersrdquo Structural Engi-neering and Mechanics vol 44 no 5 pp 611ndash631 2012

[14] G Watanabe and K Kawashima ldquoNumerical simulation ofpounding of bridge decksrdquo in Proceedings of the 13thWorld Con-ference on Earthquake Engineering Paper No 884 VancouverCanada August 2004

[15] E G Dimitrakopoulos ldquoSeismic response analysis of skewbridges with pounding deck-abutment jointsrdquo EngineeringStructures vol 33 no 3 pp 813ndash826 2011

[16] B Li K Bi N Chouw J W Butterworth and H Hao ldquoEffectof abutment excitation on bridge poundingrdquo Engineering Struc-tures vol 54 pp 57ndash68 2013

[17] C-J Wang and M-H Shih ldquoPerformance study of a bridgeinvolving sliding decks and pounded abutment during a violentearthquakerdquo Engineering Structures vol 29 no 5 pp 802ndash8122007

[18] Y Huo and J Zhang ldquoEffects of pounding and skewness onseismic responses of typical multispan highway bridges usingthe fragility function methodrdquo Journal of Bridge Engineeringvol 18 no 6 pp 499ndash515 2013

[19] K Bi and H Hao ldquoNumerical simulation of pounding damageto bridge structures under spatially varying ground motionsrdquoEngineering Structures vol 46 pp 62ndash76 2013

[20] S E Abdel Raheem ldquoPounding mitigation and unseatingprevention at expansion joints of isolated multi-span bridgesrdquoEngineering Structures vol 31 no 10 pp 2345ndash2356 2009

[21] B Li and N Chouw ldquoExperimental investigation of inelasticbridge response under spatially varying excitationswith pound-ingrdquo Engineering Structures vol 79 pp 106ndash116 2014

[22] S-H Kim and M Shinozuka ldquoEffects of seismically inducedpounding at expansion joints of concrete bridgesrdquo Journal ofEngineering Mechanics vol 129 no 11 pp 1225ndash1234 2003

[23] M J N Priestley F Seible and G M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Shock and Vibration

Seismic isolation bearing Expansion bearing

Steel block

Clearance

Steel block

Clearance

Deck

Pier

(a) The whole structural model

Gap

Hook

Expansion bearing Seismic isolation bearing

Plastic Wen Plastic Wen

Gap

Hook

mdkpkp

(b) The SDOF model

Figure 13 Models of a simply supported girder bridge

Table 3 Parameters of a simply supported girder bridge model

Components ParametersDeck Elasticity modulus is 345 times 107 kPa density is 25 tonm3 and cross-sectional area is 54m2

Pier Elasticity modulus is 345 times 107 kPa density is 25 tonm3 and cross-sectional area is 20m2

Expansion bearing Horizontal stiffness before yielding is 432 times 104 kNm yield force is 432 kN and horizontal stiffness after yieldingis 0

Seismic isolationbearing

Horizontal stiffness before yielding is 432 times 104 kNm yield force is 432 kN and horizontal stiffness after yieldingis 216 kNm

Steel block Stiffness of both gap and hook link elements is 10 times 1010 kNm and Clearances are 01m

the right end of the deck is supported on two same seismicisolation bearings Steel blocks for unseating prevention areinstalled on both sides of each bearing Clearances are allset to be 01m for all the steel blocks Deck-pier poundingswill happen between the deck and steel blocks under longi-tudinal earthquakes until bearing displacements exceed thepreserved Clearances

41 Analysis Models and Seismic Input Two models of thesimply supported girder bridge are established with the FEMsoftware SAP2000 One is a whole structural model and theother one is a SDOF model For the whole structural model

all the components are simulated as shown in Figure 13(a)The deck and piers are simulated with frame elementsBearings are simulated with Plastic Wen link elements Steelblocks are simulated with gap and hook link elements All theelements and their parameters are listed in Table 3 For theSDOF model pier masses are neglected and the deck massis concentrated on a particle as shown in Figure 13(b) 119896119901represents the horizontal stiffness of piers and the deck mass119898119889 is 216 tons

Sinusoidal acceleration wave is input as the seismic exci-tation for simplicity Peak acceleration is 4ms2 frequency is1Hz duration is 5 s and integration step is 1ms

Shock and Vibration 11

minus3000minus2000minus1000

0100020003000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(a) 120573 = 00034

minus4000minus2000

020004000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(b) 120573 = 00138

minus6000minus4000minus2000

0200040006000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(c) 120573 = 0034

minus10000minus5000

05000

10000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(d) 120573 = 0103

minus20000minus15000minus10000minus5000

05000

100001500020000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(e) 120573 = 0345

0 1 2 3 4 5minus30000minus20000minus10000

0100002000030000

Bear

ing

forc

e (kN

)

Time (s)

(f) 120573 = 10

Figure 14 Force time histories of the expansion bearing

42 Applicability of the Simplified SDOFMethod To evaluatethe applicability of the simplified SDOF method results ofthe SDOF model for different pier stiffness are comparedwith those of the whole structural model The influenceof the pier stiffness on pounding response is studied byaltering the elastic modulus of piers The horizontal stiffnessis designed to be 40 times 104 kNm 16 times 105 kNm 40times 105 kNm 12 times 106 kNm 40 times 106 kNm and 116 times107 kNm respectively The axial stiffness of the deck is 116 times107 kNm So stiffness ratios are 00034 00138 0034 01030345 and 10 respectively

Force time histories of the expansion bearing of the twomodels are shown in Figure 14 The expansion bearing forceis composed of the friction force and the pounding force andthe latter is far larger than the former In the figure solid linesindicate results of the whole structural model and brokenlines indicate results of the SDOF model

Every spike in Figure 14 indicates one pounding betweenthe deck and the pier As shown along with the increase ofthe pier stiffness the pounding duration becomes shorter andthe pounding force becomes larger Differences between thetwo models are very small except for 120573 = 10 The poundingforce and the pounding time of the two models are almostthe same for the first five cases Therefore the SDOF modelcan estimate the pounding force fairly accurately for 120573 lt

0345 at least All the peak pounding forces calculated by theSDOF model are larger than those of the whole structural

model which is similar to the theoretical results It needs tobe noted that 120573 = 0345 is already very big pier stiffness andapplicability of the simplified SDOF method is verified foralmost all the bridges

For 120573 = 10 differences of the pounding force and thepounding time between the twomodels are much larger thanother cases So the SDOF model is not applicable anymoreHowever because the horizontal stiffness of bridge pier isalways much less than the axial stiffness of the deck 120573 = 10is a very rare condition

43 Influences of the Pier Stiffness and the Pier Mass onPounding Responses The influence of the pier horizontalstiffness on pounding responses is studied by altering theelastic modulus of piers as mentioned in Section 42 Theinfluence of the pier mass on pounding responses is studiedby altering the density of piers Pier masses are designed tobe 5 tons 10 tons 25 tons 50 tons and 125 tons respectivelywhich are named as M1 to M5

The maximum pounding responses of the whole struc-tural model are shown in Figure 15

As shown the pounding force will increase and thedeck displacement will decrease with the increase of thepier stiffness The influence of the pier mass on the deckdisplacement is very small

For minor pier stiffness the influence of the pier mass onthe pounding force is also not too much so the pier mass can

12 Shock and Vibration

3 10 100 10000

5000

10000

15000

20000

25000

30000Po

undi

ng fo

rce (

kN)

M1M2M3

M4M5

Pier stiffness (104 kNm)

(a) Maximum pounding forces

3 10 100 1000010

012

014

016

018

M1M2M3

M4M5

Gird

er d

ispla

cem

ent (

m)

Pier stiffness (104 kNm)

(b) Maximum deck displacements

Figure 15 Maximum pounding responses

be neglected and the SDOFmodel is applicable However theinfluence of the pier mass on the pounding force is bigger formajor pier stiffness and so the pier mass is not negligible andthe SDOF model is no more applicable

5 Conclusions

In this paper theoretical analysis shaking table test andfinite element analysis were conducted to study the deck-pierpounding issue the following conclusions are drawn

(1) Based on theoretical results and test results it wasfound that the longitudinal deck-pier pounding issuecan be simulated by the simplified SDOF method ifenergy dissipation is negligible and the pier stiffnessis much less than the axial stiffness of the deck

(2) It was found that the pounding duration becomesshorter and the pounding force becomes larger alongwith the increase of the pier stiffness

(3) The pier mass has little effect on pounding responsesexcept for bridges with very rigid piers

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of China (no 51278372) and the Ministry ofScience and Technology of China Grant no SLDRCE 14-B-15

References

[1] W Goldsmith Impact Dover New York NY USA 2001[2] G Cole R Dhakal A Carr and D Bull ldquoAn investigation

of the effects of mass distribution on pounding structuresrdquoEarthquake Engineering and Structural Dynamics vol 40 no6 pp 641ndash659 2011

[3] H Hao K Bi N Chouw and W-X Ren ldquoState-of-the-artreview on seismic induced pounding response of bridge struc-turesrdquo Journal of Earthquake and Tsunami vol 7 no 3 ArticleID 1350019 2013

[4] B F Maison and K Kasai ldquoAnalysis for a type of structuralpoundingrdquo Journal of Structural Engineering vol 116 no 4 pp957ndash977 1990

[5] B F Maison and K Kasai ldquoDynamics of pounding whentwo buildings colliderdquo Earthquake Engineering and StructuralDynamics vol 21 no 9 pp 771ndash786 1992

[6] R Jankowski KWilde and Y Fujino ldquoPounding of superstruc-ture segments in isolated elevated bridge during earthquakesrdquoEarthquake Engineering and Structural Dynamics vol 27 no 5pp 487ndash502 1998

[7] K T Chau and X X Wei ldquoPounding of structures modelled asnon-linear impacts of two oscillatorsrdquo Earthquake Engineeringand Structural Dynamics vol 30 no 5 pp 633ndash651 2001

[8] S Muthukumar and R DesRoches ldquoA Hertz contact modelwith non-linear damping for pounding simulationrdquo EarthquakeEngineering and Structural Dynamics vol 35 no 7 pp 811ndash8282006

[9] P Komodromos P C Polycarpou L Papaloizou and M CPhocas ldquoResponse of seismically isolated buildings consideringpoundingsrdquo Earthquake Engineering and Structural Dynamicsvol 36 no 12 pp 1605ndash1622 2007

[10] K Ye L Li and H Zhu ldquoA note on the Hertz contact modelwith nonlinear damping for pounding simulationrdquo EarthquakeEngineering and Structural Dynamics vol 38 no 9 pp 1135ndash1142 2009

Shock and Vibration 13

[11] C C Patel and R S Jangid ldquoDynamic response of adjacentstructures connected by friction damperrdquo Earthquake andStructures vol 2 no 2 pp 149ndash169 2011

[12] R Jankowski ldquoNon-linear FEM analysis of pounding-involvedresponse of buildings under non-uniform earthquake excita-tionrdquo Engineering Structures vol 37 pp 99ndash105 2012

[13] D G Weng C Zhang X L Lu S Zeng and S M Zhang ldquoAsimplified design procedure for seismic retrofit of earthquake-damaged RC frames with viscous dampersrdquo Structural Engi-neering and Mechanics vol 44 no 5 pp 611ndash631 2012

[14] G Watanabe and K Kawashima ldquoNumerical simulation ofpounding of bridge decksrdquo in Proceedings of the 13thWorld Con-ference on Earthquake Engineering Paper No 884 VancouverCanada August 2004

[15] E G Dimitrakopoulos ldquoSeismic response analysis of skewbridges with pounding deck-abutment jointsrdquo EngineeringStructures vol 33 no 3 pp 813ndash826 2011

[16] B Li K Bi N Chouw J W Butterworth and H Hao ldquoEffectof abutment excitation on bridge poundingrdquo Engineering Struc-tures vol 54 pp 57ndash68 2013

[17] C-J Wang and M-H Shih ldquoPerformance study of a bridgeinvolving sliding decks and pounded abutment during a violentearthquakerdquo Engineering Structures vol 29 no 5 pp 802ndash8122007

[18] Y Huo and J Zhang ldquoEffects of pounding and skewness onseismic responses of typical multispan highway bridges usingthe fragility function methodrdquo Journal of Bridge Engineeringvol 18 no 6 pp 499ndash515 2013

[19] K Bi and H Hao ldquoNumerical simulation of pounding damageto bridge structures under spatially varying ground motionsrdquoEngineering Structures vol 46 pp 62ndash76 2013

[20] S E Abdel Raheem ldquoPounding mitigation and unseatingprevention at expansion joints of isolated multi-span bridgesrdquoEngineering Structures vol 31 no 10 pp 2345ndash2356 2009

[21] B Li and N Chouw ldquoExperimental investigation of inelasticbridge response under spatially varying excitationswith pound-ingrdquo Engineering Structures vol 79 pp 106ndash116 2014

[22] S-H Kim and M Shinozuka ldquoEffects of seismically inducedpounding at expansion joints of concrete bridgesrdquo Journal ofEngineering Mechanics vol 129 no 11 pp 1225ndash1234 2003

[23] M J N Priestley F Seible and G M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 11

minus3000minus2000minus1000

0100020003000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(a) 120573 = 00034

minus4000minus2000

020004000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(b) 120573 = 00138

minus6000minus4000minus2000

0200040006000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(c) 120573 = 0034

minus10000minus5000

05000

10000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(d) 120573 = 0103

minus20000minus15000minus10000minus5000

05000

100001500020000

0 1 2 3 4 5

Bear

ing

forc

e (kN

)

Time (s)

(e) 120573 = 0345

0 1 2 3 4 5minus30000minus20000minus10000

0100002000030000

Bear

ing

forc

e (kN

)

Time (s)

(f) 120573 = 10

Figure 14 Force time histories of the expansion bearing

42 Applicability of the Simplified SDOFMethod To evaluatethe applicability of the simplified SDOF method results ofthe SDOF model for different pier stiffness are comparedwith those of the whole structural model The influenceof the pier stiffness on pounding response is studied byaltering the elastic modulus of piers The horizontal stiffnessis designed to be 40 times 104 kNm 16 times 105 kNm 40times 105 kNm 12 times 106 kNm 40 times 106 kNm and 116 times107 kNm respectively The axial stiffness of the deck is 116 times107 kNm So stiffness ratios are 00034 00138 0034 01030345 and 10 respectively

Force time histories of the expansion bearing of the twomodels are shown in Figure 14 The expansion bearing forceis composed of the friction force and the pounding force andthe latter is far larger than the former In the figure solid linesindicate results of the whole structural model and brokenlines indicate results of the SDOF model

Every spike in Figure 14 indicates one pounding betweenthe deck and the pier As shown along with the increase ofthe pier stiffness the pounding duration becomes shorter andthe pounding force becomes larger Differences between thetwo models are very small except for 120573 = 10 The poundingforce and the pounding time of the two models are almostthe same for the first five cases Therefore the SDOF modelcan estimate the pounding force fairly accurately for 120573 lt

0345 at least All the peak pounding forces calculated by theSDOF model are larger than those of the whole structural

model which is similar to the theoretical results It needs tobe noted that 120573 = 0345 is already very big pier stiffness andapplicability of the simplified SDOF method is verified foralmost all the bridges

For 120573 = 10 differences of the pounding force and thepounding time between the twomodels are much larger thanother cases So the SDOF model is not applicable anymoreHowever because the horizontal stiffness of bridge pier isalways much less than the axial stiffness of the deck 120573 = 10is a very rare condition

43 Influences of the Pier Stiffness and the Pier Mass onPounding Responses The influence of the pier horizontalstiffness on pounding responses is studied by altering theelastic modulus of piers as mentioned in Section 42 Theinfluence of the pier mass on pounding responses is studiedby altering the density of piers Pier masses are designed tobe 5 tons 10 tons 25 tons 50 tons and 125 tons respectivelywhich are named as M1 to M5

The maximum pounding responses of the whole struc-tural model are shown in Figure 15

As shown the pounding force will increase and thedeck displacement will decrease with the increase of thepier stiffness The influence of the pier mass on the deckdisplacement is very small

For minor pier stiffness the influence of the pier mass onthe pounding force is also not too much so the pier mass can

12 Shock and Vibration

3 10 100 10000

5000

10000

15000

20000

25000

30000Po

undi

ng fo

rce (

kN)

M1M2M3

M4M5

Pier stiffness (104 kNm)

(a) Maximum pounding forces

3 10 100 1000010

012

014

016

018

M1M2M3

M4M5

Gird

er d

ispla

cem

ent (

m)

Pier stiffness (104 kNm)

(b) Maximum deck displacements

Figure 15 Maximum pounding responses

be neglected and the SDOFmodel is applicable However theinfluence of the pier mass on the pounding force is bigger formajor pier stiffness and so the pier mass is not negligible andthe SDOF model is no more applicable

5 Conclusions

In this paper theoretical analysis shaking table test andfinite element analysis were conducted to study the deck-pierpounding issue the following conclusions are drawn

(1) Based on theoretical results and test results it wasfound that the longitudinal deck-pier pounding issuecan be simulated by the simplified SDOF method ifenergy dissipation is negligible and the pier stiffnessis much less than the axial stiffness of the deck

(2) It was found that the pounding duration becomesshorter and the pounding force becomes larger alongwith the increase of the pier stiffness

(3) The pier mass has little effect on pounding responsesexcept for bridges with very rigid piers

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of China (no 51278372) and the Ministry ofScience and Technology of China Grant no SLDRCE 14-B-15

References

[1] W Goldsmith Impact Dover New York NY USA 2001[2] G Cole R Dhakal A Carr and D Bull ldquoAn investigation

of the effects of mass distribution on pounding structuresrdquoEarthquake Engineering and Structural Dynamics vol 40 no6 pp 641ndash659 2011

[3] H Hao K Bi N Chouw and W-X Ren ldquoState-of-the-artreview on seismic induced pounding response of bridge struc-turesrdquo Journal of Earthquake and Tsunami vol 7 no 3 ArticleID 1350019 2013

[4] B F Maison and K Kasai ldquoAnalysis for a type of structuralpoundingrdquo Journal of Structural Engineering vol 116 no 4 pp957ndash977 1990

[5] B F Maison and K Kasai ldquoDynamics of pounding whentwo buildings colliderdquo Earthquake Engineering and StructuralDynamics vol 21 no 9 pp 771ndash786 1992

[6] R Jankowski KWilde and Y Fujino ldquoPounding of superstruc-ture segments in isolated elevated bridge during earthquakesrdquoEarthquake Engineering and Structural Dynamics vol 27 no 5pp 487ndash502 1998

[7] K T Chau and X X Wei ldquoPounding of structures modelled asnon-linear impacts of two oscillatorsrdquo Earthquake Engineeringand Structural Dynamics vol 30 no 5 pp 633ndash651 2001

[8] S Muthukumar and R DesRoches ldquoA Hertz contact modelwith non-linear damping for pounding simulationrdquo EarthquakeEngineering and Structural Dynamics vol 35 no 7 pp 811ndash8282006

[9] P Komodromos P C Polycarpou L Papaloizou and M CPhocas ldquoResponse of seismically isolated buildings consideringpoundingsrdquo Earthquake Engineering and Structural Dynamicsvol 36 no 12 pp 1605ndash1622 2007

[10] K Ye L Li and H Zhu ldquoA note on the Hertz contact modelwith nonlinear damping for pounding simulationrdquo EarthquakeEngineering and Structural Dynamics vol 38 no 9 pp 1135ndash1142 2009

Shock and Vibration 13

[11] C C Patel and R S Jangid ldquoDynamic response of adjacentstructures connected by friction damperrdquo Earthquake andStructures vol 2 no 2 pp 149ndash169 2011

[12] R Jankowski ldquoNon-linear FEM analysis of pounding-involvedresponse of buildings under non-uniform earthquake excita-tionrdquo Engineering Structures vol 37 pp 99ndash105 2012

[13] D G Weng C Zhang X L Lu S Zeng and S M Zhang ldquoAsimplified design procedure for seismic retrofit of earthquake-damaged RC frames with viscous dampersrdquo Structural Engi-neering and Mechanics vol 44 no 5 pp 611ndash631 2012

[14] G Watanabe and K Kawashima ldquoNumerical simulation ofpounding of bridge decksrdquo in Proceedings of the 13thWorld Con-ference on Earthquake Engineering Paper No 884 VancouverCanada August 2004

[15] E G Dimitrakopoulos ldquoSeismic response analysis of skewbridges with pounding deck-abutment jointsrdquo EngineeringStructures vol 33 no 3 pp 813ndash826 2011

[16] B Li K Bi N Chouw J W Butterworth and H Hao ldquoEffectof abutment excitation on bridge poundingrdquo Engineering Struc-tures vol 54 pp 57ndash68 2013

[17] C-J Wang and M-H Shih ldquoPerformance study of a bridgeinvolving sliding decks and pounded abutment during a violentearthquakerdquo Engineering Structures vol 29 no 5 pp 802ndash8122007

[18] Y Huo and J Zhang ldquoEffects of pounding and skewness onseismic responses of typical multispan highway bridges usingthe fragility function methodrdquo Journal of Bridge Engineeringvol 18 no 6 pp 499ndash515 2013

[19] K Bi and H Hao ldquoNumerical simulation of pounding damageto bridge structures under spatially varying ground motionsrdquoEngineering Structures vol 46 pp 62ndash76 2013

[20] S E Abdel Raheem ldquoPounding mitigation and unseatingprevention at expansion joints of isolated multi-span bridgesrdquoEngineering Structures vol 31 no 10 pp 2345ndash2356 2009

[21] B Li and N Chouw ldquoExperimental investigation of inelasticbridge response under spatially varying excitationswith pound-ingrdquo Engineering Structures vol 79 pp 106ndash116 2014

[22] S-H Kim and M Shinozuka ldquoEffects of seismically inducedpounding at expansion joints of concrete bridgesrdquo Journal ofEngineering Mechanics vol 129 no 11 pp 1225ndash1234 2003

[23] M J N Priestley F Seible and G M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Shock and Vibration

3 10 100 10000

5000

10000

15000

20000

25000

30000Po

undi

ng fo

rce (

kN)

M1M2M3

M4M5

Pier stiffness (104 kNm)

(a) Maximum pounding forces

3 10 100 1000010

012

014

016

018

M1M2M3

M4M5

Gird

er d

ispla

cem

ent (

m)

Pier stiffness (104 kNm)

(b) Maximum deck displacements

Figure 15 Maximum pounding responses

be neglected and the SDOFmodel is applicable However theinfluence of the pier mass on the pounding force is bigger formajor pier stiffness and so the pier mass is not negligible andthe SDOF model is no more applicable

5 Conclusions

In this paper theoretical analysis shaking table test andfinite element analysis were conducted to study the deck-pierpounding issue the following conclusions are drawn

(1) Based on theoretical results and test results it wasfound that the longitudinal deck-pier pounding issuecan be simulated by the simplified SDOF method ifenergy dissipation is negligible and the pier stiffnessis much less than the axial stiffness of the deck

(2) It was found that the pounding duration becomesshorter and the pounding force becomes larger alongwith the increase of the pier stiffness

(3) The pier mass has little effect on pounding responsesexcept for bridges with very rigid piers

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by National Natural ScienceFoundation of China (no 51278372) and the Ministry ofScience and Technology of China Grant no SLDRCE 14-B-15

References

[1] W Goldsmith Impact Dover New York NY USA 2001[2] G Cole R Dhakal A Carr and D Bull ldquoAn investigation

of the effects of mass distribution on pounding structuresrdquoEarthquake Engineering and Structural Dynamics vol 40 no6 pp 641ndash659 2011

[3] H Hao K Bi N Chouw and W-X Ren ldquoState-of-the-artreview on seismic induced pounding response of bridge struc-turesrdquo Journal of Earthquake and Tsunami vol 7 no 3 ArticleID 1350019 2013

[4] B F Maison and K Kasai ldquoAnalysis for a type of structuralpoundingrdquo Journal of Structural Engineering vol 116 no 4 pp957ndash977 1990

[5] B F Maison and K Kasai ldquoDynamics of pounding whentwo buildings colliderdquo Earthquake Engineering and StructuralDynamics vol 21 no 9 pp 771ndash786 1992

[6] R Jankowski KWilde and Y Fujino ldquoPounding of superstruc-ture segments in isolated elevated bridge during earthquakesrdquoEarthquake Engineering and Structural Dynamics vol 27 no 5pp 487ndash502 1998

[7] K T Chau and X X Wei ldquoPounding of structures modelled asnon-linear impacts of two oscillatorsrdquo Earthquake Engineeringand Structural Dynamics vol 30 no 5 pp 633ndash651 2001

[8] S Muthukumar and R DesRoches ldquoA Hertz contact modelwith non-linear damping for pounding simulationrdquo EarthquakeEngineering and Structural Dynamics vol 35 no 7 pp 811ndash8282006

[9] P Komodromos P C Polycarpou L Papaloizou and M CPhocas ldquoResponse of seismically isolated buildings consideringpoundingsrdquo Earthquake Engineering and Structural Dynamicsvol 36 no 12 pp 1605ndash1622 2007

[10] K Ye L Li and H Zhu ldquoA note on the Hertz contact modelwith nonlinear damping for pounding simulationrdquo EarthquakeEngineering and Structural Dynamics vol 38 no 9 pp 1135ndash1142 2009

Shock and Vibration 13

[11] C C Patel and R S Jangid ldquoDynamic response of adjacentstructures connected by friction damperrdquo Earthquake andStructures vol 2 no 2 pp 149ndash169 2011

[12] R Jankowski ldquoNon-linear FEM analysis of pounding-involvedresponse of buildings under non-uniform earthquake excita-tionrdquo Engineering Structures vol 37 pp 99ndash105 2012

[13] D G Weng C Zhang X L Lu S Zeng and S M Zhang ldquoAsimplified design procedure for seismic retrofit of earthquake-damaged RC frames with viscous dampersrdquo Structural Engi-neering and Mechanics vol 44 no 5 pp 611ndash631 2012

[14] G Watanabe and K Kawashima ldquoNumerical simulation ofpounding of bridge decksrdquo in Proceedings of the 13thWorld Con-ference on Earthquake Engineering Paper No 884 VancouverCanada August 2004

[15] E G Dimitrakopoulos ldquoSeismic response analysis of skewbridges with pounding deck-abutment jointsrdquo EngineeringStructures vol 33 no 3 pp 813ndash826 2011

[16] B Li K Bi N Chouw J W Butterworth and H Hao ldquoEffectof abutment excitation on bridge poundingrdquo Engineering Struc-tures vol 54 pp 57ndash68 2013

[17] C-J Wang and M-H Shih ldquoPerformance study of a bridgeinvolving sliding decks and pounded abutment during a violentearthquakerdquo Engineering Structures vol 29 no 5 pp 802ndash8122007

[18] Y Huo and J Zhang ldquoEffects of pounding and skewness onseismic responses of typical multispan highway bridges usingthe fragility function methodrdquo Journal of Bridge Engineeringvol 18 no 6 pp 499ndash515 2013

[19] K Bi and H Hao ldquoNumerical simulation of pounding damageto bridge structures under spatially varying ground motionsrdquoEngineering Structures vol 46 pp 62ndash76 2013

[20] S E Abdel Raheem ldquoPounding mitigation and unseatingprevention at expansion joints of isolated multi-span bridgesrdquoEngineering Structures vol 31 no 10 pp 2345ndash2356 2009

[21] B Li and N Chouw ldquoExperimental investigation of inelasticbridge response under spatially varying excitationswith pound-ingrdquo Engineering Structures vol 79 pp 106ndash116 2014

[22] S-H Kim and M Shinozuka ldquoEffects of seismically inducedpounding at expansion joints of concrete bridgesrdquo Journal ofEngineering Mechanics vol 129 no 11 pp 1225ndash1234 2003

[23] M J N Priestley F Seible and G M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 13

[11] C C Patel and R S Jangid ldquoDynamic response of adjacentstructures connected by friction damperrdquo Earthquake andStructures vol 2 no 2 pp 149ndash169 2011

[12] R Jankowski ldquoNon-linear FEM analysis of pounding-involvedresponse of buildings under non-uniform earthquake excita-tionrdquo Engineering Structures vol 37 pp 99ndash105 2012

[13] D G Weng C Zhang X L Lu S Zeng and S M Zhang ldquoAsimplified design procedure for seismic retrofit of earthquake-damaged RC frames with viscous dampersrdquo Structural Engi-neering and Mechanics vol 44 no 5 pp 611ndash631 2012

[14] G Watanabe and K Kawashima ldquoNumerical simulation ofpounding of bridge decksrdquo in Proceedings of the 13thWorld Con-ference on Earthquake Engineering Paper No 884 VancouverCanada August 2004

[15] E G Dimitrakopoulos ldquoSeismic response analysis of skewbridges with pounding deck-abutment jointsrdquo EngineeringStructures vol 33 no 3 pp 813ndash826 2011

[16] B Li K Bi N Chouw J W Butterworth and H Hao ldquoEffectof abutment excitation on bridge poundingrdquo Engineering Struc-tures vol 54 pp 57ndash68 2013

[17] C-J Wang and M-H Shih ldquoPerformance study of a bridgeinvolving sliding decks and pounded abutment during a violentearthquakerdquo Engineering Structures vol 29 no 5 pp 802ndash8122007

[18] Y Huo and J Zhang ldquoEffects of pounding and skewness onseismic responses of typical multispan highway bridges usingthe fragility function methodrdquo Journal of Bridge Engineeringvol 18 no 6 pp 499ndash515 2013

[19] K Bi and H Hao ldquoNumerical simulation of pounding damageto bridge structures under spatially varying ground motionsrdquoEngineering Structures vol 46 pp 62ndash76 2013

[20] S E Abdel Raheem ldquoPounding mitigation and unseatingprevention at expansion joints of isolated multi-span bridgesrdquoEngineering Structures vol 31 no 10 pp 2345ndash2356 2009

[21] B Li and N Chouw ldquoExperimental investigation of inelasticbridge response under spatially varying excitationswith pound-ingrdquo Engineering Structures vol 79 pp 106ndash116 2014

[22] S-H Kim and M Shinozuka ldquoEffects of seismically inducedpounding at expansion joints of concrete bridgesrdquo Journal ofEngineering Mechanics vol 129 no 11 pp 1225ndash1234 2003

[23] M J N Priestley F Seible and G M Calvi Seismic Design andRetrofit of Bridges John Wiley amp Sons New York NY USA1996

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of