research article evaluation on impact interaction between...

Research ArticleEvaluation on Impact Interaction between Abutmentand Steel Girder Subjected to Nonuniform Seismic Excitation

Yue Zheng,1 Xiang Xiao,2 Lunhai Zhi,3 and Guobo Wang3

1 Department of Civil and Environmental Engineering, Hong Kong Polytechnic University, Kowloon, Hong Kong2 School of Transportation, Wuhan University of Technology, Wuhan 430070, China3 School of Civil Engineering and Architecture, Wuhan University of Technology, Wuhan 430070, China

Correspondence should be addressed to Xiang Xiao; [email protected]

Received 1 September 2014; Accepted 27 September 2014

Academic Editor: Bo Chen

Copyright © 2015 Yue Zheng et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper aims to evaluate the impact interaction between the abutment and the girder subjected to nonuniform seismic excitation.An impact model based on tests is presented by taking material properties of the backfill of the abutment into consideration. Theconditional simulation is performed to investigate the spatial variation of earthquake groundmotions. A two-span continuous steelbox girder bridge is taken as the example to analyze and assess the pounding interaction between the abutment and the girder. Thedetailed nonlinear finite element (FE) model is established and the steel girder and the reinforced concrete piers are modeled bynonlinear fiber elements. The pounding element of the abutment is simulated by using a trilinear compression gap element. Theelastic-perfectly plastic element is used to model the nonlinear rubber bearings.The comparisons of the pounding forces, the shearforces of the nonlinear bearings, the moments of reinforced concrete piers, and the axial pounding stresses of the steel girder arestudied. The made observations indicate that the nonuniform excitation for multisupport bridge is imperative in the analysis andevaluation of the pounding effects of the bridges.

1. Introduction

Long span bridges are subjected to environmental loadingsand dynamic excitations due to the interaction between thebridges and the surrounding environment [1–3]. Externalexcitations acting on bridges, such as earthquakes, mayinduce the structural deformation and stresses due to theindeterminacy, which may cause the damage events of thestructural components [4, 5]. The out-of-phase oscillation inadjacent structures due to the difference in dynamic char-acteristics and the spatial variations of earthquake groundmotion may result in the collision if their structural separa-tion is not enough to accommodate the relative displacement.Reconnaissance shows that the pounding scenario is one ofthe most critical factors resulting in damages and failures ofhighway bridges [6]. The collision between bridge decks andabutments commonly results in the extensive damages to thehighway bridges with seat abutment in many earthquakes,such as the 1971 San Fernando earthquake. Desroches andFenves [7] reported that the frame stiffness ratios, earthquake

loading, hinge gap, frame yield strength, and restrainer stiff-ness are the major factors in quantizing the pounding effectsin multiple-frame bridges. Maragakis et al. [8] investigatedthe effects of energy losses when the bridge deck collideswith the abutment. Various damper elements were placed onthe abutment to account for the energy dissipation duringthe pounding action. Many parameters, such as coefficientof restitution, abutment and deck stiffness, gap, and deckto abutment mass ratio, have been studied to prove thatthe pounding effect plays an important disadvantageous rolein the seismic action on bridges with flexural abutments.Guo et al. [9] presented an experimental and analytical studyon the pounding reduction of highway bridges subjected toseismic excitations by using magnetorheological dampers.However, the aforementioned references do not consider theeffects of spatial variation of the earthquake ground motionson the impact between the girder and the abutment. Biet al. [10] found that the minimum gap should be pro-vided to avoid the pounding at the abutments and betweenthe girders. Nevertheless, if the expansion joint between

Hindawi Publishing CorporationShock and VibrationVolume 2015, Article ID 981804, 14 pageshttp://dx.doi.org/10.1155/2015/981804

2 Shock and Vibration

the abutment and the girder is too large, it may disturb thecomfortable experience of drivers and passengers. Therefore,the pounding interaction between the abutment and thegirder subjected to the uniform and nonuniform excitationsshould be systematically investigated.

Prior to conducting the impact analysis, a rational impactmodel is essential to reflect the physical collision scenario.There exist many pounding models which have been pre-sented to account for the impact effect between adjacentstructures in the past several decades. Muthukumar andDesRoches [11] gave the cogency of various impact modelsin capturing the impact response of neighboring structures.Four impact models, namely, the contact force-based linearspring model, the Kelvin model, the Hertz model, and therestitution-based stereomechnical model, have been widelyutilized in the pounding analysis. However, the four impactmodels do not well model the performance response ofthe collision between the girder and the abutment whenthe bridge is subjected to intensive earthquakes. It is worthnoting that the axial stiffness between the girder and theabutment, which is a critical parameter in the impact model,is associated with the material properties of the backfill. Anapproximate stiffness is proposed based on many large-scaleabutment tests [12, 13].

The spatial variations include three aspects [14, 15]: (1)local site effects caused from the variation by filtering effectsof overlying soil columns; (2) wave passage, where thenonvertical waves arrive at different points of the groundsurface at different times; (3) geometric incoherence resultsin the scattering in the heterogeneous ground.The numericalsimulation of spatial variations is imperative because theasynchronous ground motions can significantly affect thepounding responses of the abutment. The nonstationaryconditional simulation approach presented by Vanmarckeet al. [16] is widely used by many researchers due to itssimplicity. Therefore, this approach is utilized in this studyto generate the earthquake ground motions at each supportof the continuous bridge.

The primary objective of this study is to investigate thepounding interaction between the abutment and the steelgirder subjected to the uniform and nonuniform excitationswith different wave propagation speeds. An elastic-perfectlyplastic element is used to simulate the nonlinear rubberbearings. A trilinear impact model based on experimentaldata is adopted to simulate the pounding effects betweenthe abutment and the girder. The FE model is establishedin the package Open System for Earthquake EngineeringSimulation (OpenSees). The steel girder and the reinforcedconcrete piers are modeled by nonlinear displacement-basedfiber elements. The nonstationary conditional simulation isperformed to generate the asynchronous ground motions. Areal two-span continuous steel box girder bridge is taken asthe example to analyze and assess the pounding interactionbetween the abutment and the girder. The comparisons ofthe pounding forces, the shear forces of nonlinear bearings,themoments of reinforced concrete piers, and axial poundingstresses of the steel girder are studied. The made observa-tions indicate that the nonuniform excitation for multisup-port bridge is imperative in the analysis and evaluation of

Force

Displacement

Pbw

Kabut

Δgap

Figure 1: Force-deflection relationship of the abutments.

the pounding effects of the highway bridges for it alwaysresults in disadvantageous load case.

2. Impact Interactions betweenAbutment and Steel Girder

2.1. Impact Model. Several impact models have been pre-sented by many researchers [17]. However, the impact mech-anism between the abutment and the girder is not similarto collision between two rigid or elastic bodies completely.Therefore, Clatrans [18] recommended a trilinear approx-imation of the force-deformation relationship to simulatethe longitudinal response of seat abutment; see Figure 1. Thetrilinear model consists of three segments: (i) a zero stiffnesssegment, which accounts for the expansion gap; (ii) a realisticstiffness segment of the embankment fill response; and (iii)a yielding stage segment with ultimate longitudinal forcecapacity 𝑃bw. The stiffness of the abutment 𝐾abut (kN/m) canbe given by

𝐾abut = 𝐾𝑖 × 𝑤bw × (ℎbw1.7

) , 𝐾𝑖≈ 28.7. (1)

The passive pressure force resisting the moment at theabutment is expressed as

𝑃bw = 𝐴𝑒 × 239 × (ℎbw1.7

) , (2)

where 𝑤bw and ℎbw are projected width and height of theback wall for seat abutment, respectively. The area of a seatabutment is

𝐴𝑒= ℎbw × 𝑤bw. (3)

It is noted that the initial stiffness 𝐾𝑖for embankment

fill material should meet the requirements of the CaltransStandard Specifications. For clarity, the contact force basedon the bilinear model can be formulated as𝐹𝑐= 0 𝑢

𝑖− 𝑢𝑗− 𝑔𝑝≤ 0,

𝐹𝑐= 𝐾abut × (𝑢𝑖 − 𝑢𝑗 − 𝑔𝑝) 0 < 𝑢

𝑖− 𝑢𝑗− 𝑔𝑝≤𝑃bw𝐾abut

,

𝐹𝑐= 𝑃bw 𝑢

𝑖− 𝑢𝑗− 𝑔𝑝>𝑃bw𝐾abut

,

(4)

Shock and Vibration 3

where 𝐹𝑐is contact force; 𝑢

𝑖and 𝑢

𝑗are displacements of

pounding nodes 𝑖 and 𝑗, respectively; 𝑔𝑝is the distance of the

gap.

2.2. Simulation on Ground Motion. The out-of-phase effecton the pounding of adjacent girder and the abutment ismainly investigated in this paper.Therefore, as one of the pri-mary factors resulting in the inhomogeneous phenomenon atdifferent supports of the bridge, the spatial variation of seis-mic waves is an imperative issue required to be solved. Threefactors may affect the spatial variations of seismic groundmotion, the wave passage effect, the incoherence effect, andthe local site effect [19]. Many researchers have presentedseveral solutions for this critical issue [16, 20, 21]. Amongthese approaches, the conditional simulation of a nonuniformseismic ground motion field based on nonstationary theorydeveloped by Vanmarcke et al. [16] is adopted in this studydue to its simplicity. This critical principle of the method isconcisely introduced in the following paragraph.

Let us consider a segment of the ground motion at apoint 𝑥

𝑖and assume it can be represented by a nonergodic,

zero-mean, homogeneous, mean-square continuous spacetime process 𝑍

𝑖(𝑡). The process 𝑍

𝑖(𝑡) can be expressed as a

sum of independent frequency-specific spatial processes inconsecutive constant-size frequency intervals as follows:

𝑍𝑖 (𝑡) =

𝐾

∑

𝑘=1

[𝐴𝑖𝑘cos (𝜔

𝑘𝑡) + 𝐵

𝑖𝑘sin (𝜔

𝑘𝑡)] ,

𝜔𝑘= (𝑘 − 1) Δ𝜔 (𝑘 = 1, 2, . . . 𝐾) ,

Δ𝜔 =2𝜋

𝑡𝑓+ Δ𝑡

,

(5)

where the coefficients 𝐴𝑖𝑘and 𝐵

𝑖𝑘are zero-mean random

variables; 𝑡𝑓is the length of the process. For a discrete-time

process, defined at times𝑡𝑗= (𝑗 − 1) Δ𝑡 (𝑘 = 1, 2, . . . 𝐾) ,

Δ𝑡 =

𝑡𝑓

𝐾 − 1,

(6)

the coefficients 𝐴𝑖𝑘and 𝐵

𝑖𝑘are related to 𝑍

𝑖(𝑡𝑗) through the

discrete Fourier Transform,

𝐴𝑖𝑘=1

𝐾

𝐾

∑

𝑗=1

𝑍𝑖(𝑡𝑗) cos(

2𝜋 (𝑘 − 1) (𝑗 − 1)

𝐾) ,

𝐵𝑖𝑘=1

𝐾

𝐾

∑

𝑗=1

𝑍𝑖(𝑡𝑗) sin(

2𝜋 (𝑘 − 1) (𝑗 − 1)

𝐾) .

(7)

The following symmetry conditions about the Nyquist fre-quency, 𝜔

1+𝐾/2= 𝜋/Δ𝑡, apply to the Fourier coefficients 𝐴

𝑖𝑘

and 𝐵𝑖𝑘when the process 𝑍

𝑖(𝑡) is real:

𝐴𝑖𝑘= 𝐴𝑖(𝐾−𝑘+2)

,

𝐵𝑖𝑘= − 𝐵

𝑖(𝐾−𝑘+2),

𝑘 = 2, 3, . . . , 1 +𝐾

2.

(8)

It is shown that the covariance between coefficients at points𝑥𝑖and 𝑥

𝑗, 𝐶𝑖𝑗(𝜔𝑘), can be written as [22]

𝐶𝑖𝑗(𝜔𝑘)

= 𝐸 (𝐴𝑖𝑘𝐴𝑗𝑘)

=

{{{{{{{{{

{{{{{{{{{

{

1

2𝜌𝜔𝑘(𝛾𝑖𝑗)𝐺 (𝜔

𝑘) Δ𝜔, for 𝑘 = 1,

1

4{𝜌𝜔𝑘(𝛾𝑖𝑗)𝐺 (𝜔

𝑘)

+𝜌𝜔𝐾−𝑘+2

(𝛾𝑖𝑗)𝐺 (𝜔

𝐾−𝑘+2)} Δ𝜔, for 𝑘 = 2, . . . , 𝐾

2,

𝜌𝜔𝑘(𝛾𝑖𝑗)𝐺 (𝜔

𝑘) Δ𝜔, for 𝑘 = 𝐾

2+ 1,

𝛾𝑖𝑗= 𝑥𝑖− 𝑥𝑗,

(9)

where 𝛾𝑖𝑗is the relative position vector, 𝐺(𝜔) is the one-

sided “point” spectral density function, and 𝜌𝜔𝑘(𝛾𝑖𝑗) is the

frequency-dependent spatial correlation function. Note that

𝐸 (𝐴𝑖𝑘𝐴𝑗𝑘) = 𝐸 (𝐵

𝑖𝑘𝐵𝑗𝑘) = 𝐶

𝑖𝑗(𝜔𝑘) for 𝑘 = 2, . . . , 𝐾

2.

(10)

The spatially correlated groundmotions can then be obtainedby the following steps: (i) generate sets of Fourier coefficients𝐴𝑖𝑘and 𝐵

𝑖𝑘, for each frequency 𝜔

𝑘(𝑘 = 1, 2, . . . 1 + 𝐾/2); (ii)

obtain the remainder using the symmetry conditions in (3);and (iii) use the Fast Fourier Transform algorithm to performthe inverse discrete transform.

Consider the simulation of seismic ground motions at aset of𝑚 target points 𝑥

𝛽, given that some motions have been

recorded at a set of 𝑛 = 𝑁 − 𝑚 measurement points, where𝑁 is the total number of points. The covariance matrix, 𝐶

𝑘=

[𝐶𝑖𝑗(𝜔𝑘)] (𝑖, 𝑗 = 1, 2, . . . , 𝑛 + 𝑚), for each Fourier frequency

𝜔𝑘(𝑘 = 1, 2, . . . , 1 + 𝐾/2) can be assembled and expressed as

𝐶𝑘= [

𝐶𝛼𝛼

𝐶𝛼𝛽

𝐶𝑇

𝛼𝛽𝐶𝛽𝛽

] . (11)

Assume a set of simulated Fourier coefficients

𝐴𝑠= {𝐴𝑠𝛼, 𝐴𝑠𝛽} , (12)

where 𝐴𝑠𝛼

and 𝐴𝑠𝛽

denote the coefficients at recordingand target points, respectively. A set of simulated Fouriercoefficients 𝐵

𝑠can be defined in the same way. To simulate𝐴

𝑠

and 𝐵𝑠, the covariance matrix 𝐶

𝑘is evaluated for frequencies

up to 𝜔1+𝐾/2

= 𝜋/Δ𝑡. For admissible spatial correlation andspectral density functions, 𝐶

𝑘is positive definite and can be

expressed as the product of a nonsingular lower triangularmatrix, 𝐿

𝑘, and its transpose by means of the Cholesky

decomposition

𝐶𝑘= 𝐿𝑘𝐿𝑇

𝑘. (13)


In case the limited-duration segment of the ground motioncan be modelled as a Gaussian process, two sets of indepen-dent standard normal random variables can be simulated foreach frequency:

𝑈𝑘= {𝑈1𝑘, 𝑈2𝑘, . . . 𝑈

𝑁𝑘} ,

𝑉𝑘= {𝑉1𝑘, 𝑉2𝑘, . . . 𝑉𝑁𝑘} .

(14)

Sets 𝐴𝑠and 𝐵

𝑠are then generated by

𝐴𝑠= 𝐿𝑘𝑈𝑘,

𝐵𝑠= 𝐿𝑘𝑉𝑘.

(15)

It is easy to see that 𝐴𝑠and 𝐵

𝑠have the proper covariance

structure. Based on the subsets of simulation values atthe recording points, 𝐴

𝑠𝛼and 𝐵

𝑠𝛼, the linear prediction

estimators at the target points are given by

𝐴∗

𝑠𝛽= 𝐶𝑇

𝛼𝛽𝐶−1

𝛼𝛼𝐴𝑠𝛼,

𝐵∗

𝑠𝛽= 𝐶𝑇

𝛼𝛽𝐶−1

𝛼𝛼𝐵𝑠𝛼.

(16)

The conditional simulation now involves generating for eachfrequency 𝜔

𝑘sets of Fourier coefficients 𝐴

𝑠𝑐,𝑘and 𝐵

𝑠𝑐,𝑘at

target points 𝑥𝛽, according to the conditional simulation

algorithm

𝐴𝑠𝑐,𝑘

= {𝐴∗

𝛽+ 𝐴𝑠𝛽− 𝐴∗

𝑠𝛽}(𝑘),

𝐵𝑠𝑐,𝑘

= {𝐵∗

𝛽+ 𝐵𝑠𝛽− 𝐵∗

𝑠𝛽}(𝑘),

(17)

where 𝐴∗𝛽and 𝐵∗

𝛽are the linear prediction estimators based

on the observed Fourier coefficients at the recording points𝐴𝛼and 𝐵

𝛼:

𝐴∗

𝛽= 𝐶𝑇

𝛼𝛽𝐶−1

𝛼𝛼𝐴𝛼,

𝐵∗

𝛽= 𝐶𝑇

𝛼𝛽𝐶−1

𝛼𝛼𝐵𝛼.

(18)

The remaining Fourier coefficients at frequency 𝜔𝑘(𝑘 =

2 + 𝐾/2, . . . , 𝐾) are then obtained using the symmetryconditions. Once coefficients have been generated for theentire frequency range, an inverse FFT can be applied to yielda set of ground motion time histories at the target points.The conditional simulation of seismic ground motions isdisplayed in Figure 2.

2.3. Equation of Motion for Pounding Action. For a lumpedmass system, the dynamic equilibrium equation in terms ofthe nonuniform excitation can be written as

[M𝑠𝑠

00 M

𝑏𝑏

]{

u𝑡𝑠

u𝑡𝑏

} + [C𝑠𝑠

C𝑠𝑏

C𝑏𝑠

C𝑏𝑏

]{

u𝑡𝑠

u𝑡𝑏

}

+ [K𝑠𝑠

K𝑠𝑏

K𝑏𝑠

K𝑏𝑏

]{u𝑡𝑠

u𝑡𝑏

} + [K𝑝𝑠𝑠

K𝑝𝑠𝑏

K𝑝𝑏𝑠

K𝑝𝑏𝑏

]{d𝑝𝑠

d𝑝𝑏

} = {00} ,

(19)

Stop

Start

At point i

At point j

No

Segment counter

Yes

No

Yes

sn = 1

sn = sn + 1

k = k + 1

Yisn(t) = f(𝜔k)

Get Aik and Bik

Get Ck

Get As and Bs

Get A∗s𝛽 and B∗s𝛽

Get Asc,k and Bsc,k

k = K

Yjsn(t) by inverse FFT

Yj(t)

Segment: sn

sn = NT

Figure 2: Flowchart of conditional simulation.

where the subscript 𝑠 and 𝑏 denote the superstructureand base foundation, respectively; M

𝑠𝑠and M

𝑏𝑏are the

lumped mass matrices of the superstructure and foundation,respectively; C

𝑠𝑠, C𝑠𝑏, C𝑏𝑠, and C

𝑏𝑏are damping matrices,

respectively, in terms of damping of superstructure andfoundation; similarly, K

𝑠𝑠, K𝑠𝑏, K𝑏𝑠, and K

𝑏𝑏are the matrices

with respect to stiffness of the superstructure and the foun-dation, respectively; I is the earthquake influence coefficientvector; u𝑡

𝑠and u𝑡

𝑏are the absolute acceleration vector of

the superstructure and the foundation, respectively; u𝑡𝑠and

u𝑡𝑏are the absolute velocity vectors of the superstructure

and the foundation, respectively; u𝑡𝑠and u𝑡

𝑏are the absolute

displacement matrices of the superstructure and the founda-tion, respectively; K𝑝

𝑠𝑠are the stiffness matrix of the partial

superstructure where the impact occurs; K𝑝𝑠𝑏and K𝑝

𝑏𝑠are the

stiffness matrices of the coupled part of the superstructureand the foundation where the pounding occurs; K𝑝

𝑏𝑏are the

stiffness matrix of foundation at which the impact occurs;d𝑝𝑠and d𝑝

𝑏are the relative displacement matrices between


200

550

325 10 5000

PlaneA

7000B

325

550

10B

220

200 188

B

A

XY

Z

250

500

300

150

B

(a) Profile of a two-span steel continuous girder bridge (unit: cm)

1000

1000

1050

486

150

150200

975

250

500

CC

200

192354 163

626

A–A

B–B

(b) Detailed cross section of the pier and abutment (unit: cm)

Pier

1001300

1500

1500

100

(c) Detailed cross section of a pier (unit:mm)

Figure 3: Profile of a steel continuous girder bridge.

adjacent components located in the superstructure and thefoundation, respectively.

It is noted that the pounding scenario which generateshighmagnitude and short duration acceleration pulse duringan earthquake will make the numerical convergence difficult.Therefore, a variable time stepping procedure is utilized todetermine the impact time and solve the equation of motion.

3. Pounding Effect of the Bridge

3.1. Structural Description. To assess the pounding interac-tion between the girder and the abutment subjected to thenonuniform seismic excitations, a real two-span continuoussteel girder bridge constructed in China is selected as anexample.The bridge is supported on reinforced concrete piersand the two spans are 50mand 70m, respectively, as shown inFigure 3(a). The detailed profile of the pier and the abutment

0 5 10 15 20 25 30 35 40 45 50

0

5

Time (s)

El CentroAcc.

(m/s2)

−5

Figure 4: Original time histories of El Centro (NS).

is displayed in Figure 3(b) and the reinforcement details ofa pier section are shown in Figure 3(c). The longitudinal,transversal, and vertical directions are denoted by 𝑋, 𝑌,and 𝑍, respectively. The bridge pier consists of single squarecolumn and the cross-sectional area is 2.25m2. The clearheight of the column is 9.75m.The rubber bearings are placed


0 5 10 15 20 25 30 35 40 45 50 55 60

0

0.02

0.04

0.06

0.08

Time (s)

Displacement

Acceleration

−0.02

−0.04

−0.06

−0.08

Disp

lace

men

t (m

)

Velocity

0

2

4

6

8

10

−10

−8

−6

−4

−2

Acce

lera

tion

(m/s2)

0 5 10 15 20 25 30 35 40 45 50 55 60Time (s)

0 5 10 15 20 25 30 35 40 45 50 55 60Time (s)

Velo

city

(m/s

)

0

0.2

0.4

0.6

0.8

−0.2

−0.4

−0.6

−0.8

(a) Left abutment

0

0.03

0.06

0.09

0.12

0.15

Disp

lace

men

t (m

)Ve

loci

ty (m

/s)

Velocity

−0.15

−0.12

−0.09

−0.06

−0.03

0 5 10 15 20 25 30 35 40 45 50 55 60Time (s)

Displacement

0

0.2

0.4

0.6

0.8

−0.2

−0.4

−0.6

−0.80 5 10 15 20 25 30 35 40 45 50 55 60

Time (s)

0

2

4

6

8

10

Acceleration

−10

−8

−6

−4

−2

Acce

lera

tion

(m/s2)

0 5 10 15 20 25 30 35 40 45 50 55 60Time (s)

(b) Right abutment

Figure 5: Dynamic responses of abutments (𝑉𝑤= 400m/s).


0

0.04

0.08

0.12

0.16

0.2

0

0.3

0.6

0.9

1.2

1.5

Velo

city

(m/s

)

0 5 10 15 20 25 30 35 40 45 50 55 60Time (s)

Displacement

−0.04

−0.08

−0.12

−0.3

−0.6

−0.9

−1.2

−1.5

−0.16

−0.2

Disp

lace

men

t (m

)

0

4

8

12

16

Acceleration

−16

−12

−8

−4

Acce

lera

tion

(m/s2)

0 5 10 15 20 25 30 35 40 45 50 55 60Time (s)

Velocity

0 5 10 15 20 25 30 35 40 45 50 55 60Time (s)

Figure 6: Dynamic responses of the middle foundation (𝑉𝑤= 400m/s).

on the bent cap and the abutments. The space between tworectangular octagonal hoops is 0.4m. The densities of thesteel and the reinforced concrete are 7850 and 2500 kg/m3,respectively. The diameters of longitudinal and transversalreinforcing bars are 32mm and 16mm, respectively. Theyield strength 𝑓

𝑦of the reinforcing steel is 280MPa. Young’s

modulus of the steel 𝐸𝑠and the concrete 𝐸

𝑐are 2.0 × 105MPa

and 3.0 × 104MPa, respectively. The parameters about therubber bearing are tabulated in Table 1. The shear modulusand the height of all the bearings are 1.0 GPa and 0.15m,respectively.The bearings at left and right abutments have thesame size of which the area of cross section and the shearstiffness (𝐾

ℎ) are 0.03m2 and 209.0 kN/m, respectively. The

counterparts of the bearing at middle bent cap are 0.126m2and 838.0 kN/m, respectively. The yielding strengths of twotypes of bearing are 20.9 kN and 83.8 kN, respectively.

The FEmodel of this bridge is established with the aids ofthe packageOpenSees.The steel box girder and the reinforcedconcrete pier are modeled by using nonlinear fiber elements.The rubber bearings aremodeled by using an elastic-perfectly

plastic zero-length element. A trilinear zero-length element isused to simulate the impact effects between the abutment andthe girder. The dynamic equilibrium equation involved mul-tisupport excitation which can be solved using theNewmark-beta method with a self-adaptive integration time step from1.0× 10−4 s to 2.0× 10−2 s to ensure the numerical convergenceat the end of each step. The Rayleigh damping assumption isadopted to construct the structural damping matrix, and thedamping ratio of the bridge is set as 0.03 [22–25].

3.2. Conditional Simulation of Ground Motion at Three Sup-port Points. To evaluate the pounding interaction betweenthe girder and the abutment of the continuous steel girderbridge, the El Centro NS (1940) ground motion is selected asinputs to the example bridge structure.The time history of theseismic record is shown in Figure 4.The original peak groundacceleration (PGA) of the seismic record is 3.417m/s2. Theoriginal time history of the seismic record is scaled to 8.6m/s2to perform the pounding analysis. Three scenarios are takeninto consideration to investigate the pounding effects under


0 5 10 15 20 25 30 35 40 45 50 55 60

0

0.02

0.04

0.06

0.08

0.1

Time (s)

0

3

6

9

12

−0.1

−0.08

−0.06

−0.04

−0.02

Disp

lace

men

t (m

)

Displacement

Velo

city

(m/s

)

Velocity

0

0.2

0.4

0.6

0.8

−0.2

−0.4

−0.6

−0.80 5 10 15 20 25 30 35 40 45 50 55 60

Time (s)

Velo

city

(m/s

)

Velocity

0

0.2

0.4

0.6

0.8

−0.2

−0.4

−0.6

−0.80 5 10 15 20 25 30 35 40 45 50 55 60

Time (s)

0 5 10 15 20 25 30 35 40 45 50 55 60Time (s)

Acceleration

−3

−6

−9

−12

Acce

lera

tion

(m/s2)

Displacement

Acceleration

0 5 10 15 20 25 30 35 40 45 50 55 60

0

0.02

0.04

0.06

0.08

Time (s)

−0.02

−0.04

−0.06

−0.08

Disp

lace

men

t (m

)

0

2

4

6

8

10

−10

−8

−6

−4

−2

Acce

lera

tion

(m/s2)

0 5 10 15 20 25 30 35 40 45 50Time (s)

Figure 7: Dynamic responses of abutments (𝑉𝑤= 200m/s).


0 5 10 15 20 25 30 35 40 45 50 55 60Time (s)

0 5 10 15 20 25 30 35 40 45 50 55 60Time (s)

0

0.04

0.08

0.12

0.16

0.2

Displacement

−0.04

−0.08

−0.12

−0.16

−0.2

Disp

lace

men

t (m

)

0

0.3

0.6

0.9

1.2

1.5

Velo

city

(m/s

)

−0.3

−0.6

−0.9

−1.2

−1.5

0

4

8

12

16

Acceleration

−16

−12

−8

−4

Acce

lera

tion

(m/s2)

0 5 10 15 20 25 30 35 40 45 50 55 60Time (s)

Velocity

Figure 8: Dynamic responses of the middle foundation (𝑉𝑤= 200m/s).

0 20 40 60

0

Left abutment

Poun

ding

forc

e (kN

)

Time (s)

−1000

−2000

−3000

Uniform400m/s)Nonuniform (

Nonuniform (200m/s)

(a) Left abutment

Right abutment

0 20 40 60

0

Poun

ding

forc

e (kN

)

Time (s)

−1000

−2000

−3000


Nonuniform (200m/s)

(b) Right abutment

Figure 9: Time history of pounding force at abutments.


0 20 40 60

0

20

40

60

Left bearing

Shea

r for

ce (k

N)

Time (s)

−20

−40

−60


Nonuniform (200m/s)

(a) Left bearing

Right bearing0 20 40 60

0

20

40

60

Shea

r for

ce (k

N)

Time (s)

−20

−40

−60


Nonuniform (200m/s)

(b) Right bearing

050

100150200

Middle bearing−200

−150

−100

−50

Shea

r for

ce (k

N)

0 20 40 60Time (s)


Nonuniform (200m/s)

(c) Middle bearing

Figure 10: Time history of shear force of bearings.

Table 1: Bearing properties.

Location of bearings 𝐺 (kPa) 𝐴 (m2) ℎ (m) 𝐾ℎ= GA/ℎ (kN/m) Yielding strength (kN)

Left abutment 1.0 × 103 0.03 0.15 209.0 20.9Middle pier cap 1.0 × 103 0.126 0.15 838.0 83.8Right abutment 1.0 × 103 0.03 0.15 209.0 20.9

both uniform and nonuniform excitations with two wavepropagation velocities 𝑉

𝑤, namely, 200m/s and 400m/s. The

simulated acceleration time histories at three supports of thebridge are indicated in Figures 5, 6, 7, and 8, respectively, forthe two wave propagation velocities. In the simulation of theground motion, the frequency-dependent spatial correlationfunction of phase-aligned ground acceleration presented byHarichandran and Vanmarcke [26] is adopted:

𝜌𝜔𝑘(𝛾𝑖𝑗) = 0.736𝑒

−0.536(|𝛾𝑖𝑗|/𝜃𝜔) + 0.264𝑒−0.744(|𝛾𝑖𝑗|/𝜃𝜔),

𝜃𝜔= 5210{1 + (

𝜔

2.17𝜋)

2.78

}

−1/2

,

(20)

where 𝜃𝜔is the frequency-dependent scale of the fluctuation.

In addition, it is noted that the ground motion used in thecomputation is divided into five segments with the time

duration of 1.48 s, 4.32 s, 20.8 s, 8.4 s, and 15 s, respectively.Each of the five segments is assumed as a stationary process.Then the five segments are assembled together through thelinear interpolation and the wave propagation phase delaysare postprocessed. Finally, the velocity and displacement timehistories at each support can be calculated by integrating theacceleration time histories with respect to time, respectively.

4. Seismic Response Analyses

4.1. Response of Abutment. As discussed above, three sce-narios are considered to evaluate the dynamic responses ofpounding interaction between the abutment and the steelgirder. Figure 9 shows the time histories of pounding forcesat the abutment and the peak responses of the poundingforce are listed in Table 2. It is seen from Figure 9 and Table 2


Table 2: Peak response of pounding force.

Location Uniform Nonuniform (400m/s) Nonuniform (200m/s)Time (s) Force (kN) Time (s) Force (kN) Time (s) Force (kN)

Left abutment 15.46 2.79 × 103 7.96 2.79 × 103 12.12 1.57 × 103

Right abutment 14.88 2.79 × 103 7.52 2.79 × 103 12.62 2.79 × 103

Table 3: Peak responses of shear forces.

Location Uniform Nonuniform (400m/s) Nonuniform (200m/s)Time (s) Force (kN) Time (s) Force (kN) Time (s) Force (kN)

Left bearing 3.74 41.9 4.7 41.9 8.2 41.9Middle bearing 2.34 168.0 3.02 168.0 4.02 168.0Right bearing 3.74 41.9 5.08 41.9 7.10 41.9

0 20 40 60

0

5000

10000

Time (s)

−5000

−100000

Mom

ent (

kN·m

)


Nonuniform (200m/s)

Figure 11: Time histories of moment force at the bottom of RC pier.

that the right abutment experienced yielding state underboth uniform and nonuniform excitations. The nonuniformexcitation with the wave propagation speed of 400m/s isthe most critical loading case for the right abutment. Theright abutment is the latest one to trigger yielding state whensubjected to uniform excitation. However, the left abutmentdoes not reach yielding state in the case of nonuniformexcitation with the propagation speed of 200m/s. In this case,the peak pounding force is 1.57 × 103 kN which is much lessthan that of the right abutment. The comparison betweentwo abutments indicates that the spatial variations of theground motion should be taken into consideration in theassessment on the pounding effects because it provides thedisadvantageous loading excitations.

4.2. Response of Rubber Bearing. Figure 10 displays the timehistories of the shear forces of the bearings at the leftabutment, the middle bent cap, and the right abutment forthe concerned three scenarios. Listed in Table 3 are the peakresponses and the time occurrence of shear forces of bearingat three locations. It can be observed that the bearings at thethree locations experience the yielding stages for all the threeloading cases. As far as one rubber bearing among them isconcerned, it yields at first when subjected to the uniformexcitation in comparisonwith the case under the nonuniform

Table 4: Peak response of moment force and curvature.

Case Curvature (1/m) Moment (kN⋅m)Uniform 0.0026 7.51 × 103

Nonuniform (400m/s) 0.0051 9.28 × 103

Nonuniform (200m/s) 0.0032 8.06 × 103

excitation. The three bearings reach the yielding state latestwhen they are subjected to nonuniform excitation with thewave propagation speed of 200m/s.

4.3. Response of Reinforced Concrete Pier. Figure 11 displaysthe time histories of moment of the pier and Figure 12indicates themoment versus the curvature relationship of thepier.The peak moment is listed in Table 4.The curvature andthe moment responses at the bottom of the pier are variablewith the types of the excitation, of which the nonuniformexcitation with wave propagation speed of 400m/s is themost critical case. The most critical peak curvature andmoment are 0.0051 1/m and 9.28× 103 kN⋅m, respectively.Themoment versus the curvature hysteretic loop under nonuni-form excitation with wave propagation speed of 400m/s isplumper than the other two cases, which indicates that thepier dissipates much energy of the earthquake input.The caseof uniform excitation is the one that experienced the smallestdamage.

4.4. Response of Steel Box Girder. To be the primary com-ponent of the bridge, the steel girder bears the momentsinduced by various loadings, such as dead load, live load,and temperature load. If the pounding interaction betweenthe girder and the abutment occurs during an earthquake,the axial stress in the girder may increase sharply to inducedamage events of the steel girder. The time histories of axialstress at both ends of the girder are computed and displayedin Figure 13. The peak axial stresses are tabulated in Table 5.

The axial stress responses of the steel girder indicate thatthe most dangerous loading case for both ends of the steelgirder is the nonuniform excitation with the wave propaga-tion speed of 400m/s. However, the nonuniform excitation


0 1 2 3

0

2000

4000

6000

8000

Curvature (/m)

Uniform

−2000

−4000

−6000

−8000−3 −2 −1

×10−3

Mom

ent f

orce

(kN·m

)

(a)

0 1 2 3 4 5 6

0

0.2

0.4

0.6

0.8

1

Curvature (/m)

×104

−0.2

−0.4

−0.6

−0.8

−1−1−2−3−4−5

×10−3

Mom

ent f

orce

(kN·m

)

400m/s)Nonuniform (

(b)

0 1 2 3

0

2000

4000

6000

8000

Curvature (/m)

−2000

−4000

−6000

−8000

−10000−1−2−3−4

×10−3

Mom

ent f

orce

(kN·m

)

400m/s)Nonuniform (

(c)

Figure 12: Moment versus curvature relationship of piers.

Table 5: Peak response of axial stress of steel girder.

Location Uniform Nonuniform (400m/s) Nonuniform (200m/s)Time (s) Stress (kPa) Time (s) Stress (kPa) Time (s) Stress (kPa)

Left end 15.28 8.26 × 103 7.78 8.39 × 103 11.92 4.44 × 103

Right end 16.0 8.05 × 103 7.34 8.96 × 103 12.44 7.97 × 103

with the wave propagation speed of 200m/s presents thesmallest seismic responses.

5. Concluding Remarks

The paper investigates the pounding effects between theabutment and the steel girder by considering the uniform

and nonuniform excitation with different wave propagationspeeds. A detailed nonlinear FE model is established andtwo types of zero-length nonlinear elements are used tomodel the pounding interaction between the girder and theabutment and the rubber bearings, respectively. In addition,two types of nonlinear fiber displacement-based elements areused tomodel the steel box girder and the reinforced concretepier, respectively. The dynamic analyses on impact effects are


0

2000

Axi

al st

ress

(kP) −2000

−4000

−6000

−8000

−100000 20 40 60

Time (s)


Nonuniform (200m/s)

(a) Left end of girder

Axi

al st

ress

(kP) 0

2000

4000

−2000

−4000

−6000

−8000

−100000 20 40 60

Time (s)


Nonuniform (200m/s)

(b) Right end of girder

Figure 13: Time history response of axial stress in steel girder.

carried out under the uniform and nonuniform excitationswith two wave propagation speeds. The comparison of theseismic responses, such as the pounding force between theabutment and the girder, the shear forces of the rubberbearings, the moments of reinforced concrete pier, and theaxial pounding stresses in the steel girder is performed.

The consequence of the comparison indicates that thenonuniform excitation always results in the disadvantageousresponses in the seismic analysis of the bridge under impactaction.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The authors are grateful for the financial support from theNational Natural Science Foundation of China (51178366),the Technological Project of the Chinese Southern PowerGrid Co. Ltd. (K-GD2013-0222), the Fok Ying-Tong Educa-tion Foundation (131072), and the Natural Science Founda-tion of Hubei Province (2014CFA026).

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