Effect of Frame-Restoring Force Characteristics on MF Bridges Muthukumar and DesRoches Nov 06

Effect of Frame-Restoring ForceCharacteristics on the Pounding Responseof Multiple-Frame Bridges

Susendar Muthukumar,a… M.EERI, and Reginald DesRoches,b… M.EERI

This study examines the effect of column hysteretic behavior on the impactresponse of adjacent frames in multiple-frame bridges. A simplified planaranalytical bridge model is developed including inelastic frame action,nonlinear hinge behavior, and abutment effects. Pounding is simulated using astereomechanical approach. The frame hysteretic models considered includethe elasto-plastic and bilinear �traditional�, Q-Hyst �stiffness-degrading�, andpivot hysteresis �strength-degrading� models. Analytical studies conducted onadjacent bridge frames reveal that the traditional models underestimate thestiff frame displacement amplification due to pounding, and overestimate theflexible frame displacement amplification, when compared with otherhysteretic models. A stiffness-degrading model is recommended to accuratelyestimate the pounding response of bridge frames subjected to far-field groundmotion. The use of a strength-degrading model increases the stiff framedisplacement amplification by 125% when compared to the stiffness-degrading model for highly out-of-phase frames, and is recommended in thepresence of near-field ground motions. �DOI: 10.1193/1.2103107�

INTRODUCTION

Bridges are the lifeline of a highway transportation network and past earthquakeshave illustrated that they are vulnerable to severe damage and/or collapse during mod-erate to strong ground motion. Among the possible structural damages, seismic-inducedpounding has commonly been observed in several recent earthquakes. The 1994Northridge earthquake revealed substantial impact damage at the expansion hinges andabutments of standing portions of the connectors at the Interstate 5/State Road 14 inter-change that were located at close proximity to the epicenter �EERI 1995a�. Reconnais-sance reports from the 1995 Kobe earthquake identify pounding as a major cause offracture of the bearing supports and potential contributor to the collapse of the bridgedecks �EERI 1995b�. Hammering at the expansion joints in some bridges resulted indamage to shear keys, bearings, and anchor bolts during the 1999 Chi-Chi earthquake inTaiwan �EERI 2001a�. Cracking and spalling at expansion joints of concrete bridgeswere observed during the 2001 Nisqually, Washington, earthquake �EERI 2001b�. Morerecently, pounding of adjacent simply supported spans resulting in failure of girder endsand bearing damage was observed during the 2001 Bhuj earthquake in Gujarat, India�EERI 2002�.

a� Browder + LeGuizamon & Associates, 174 West Wieuca Road NE, Atlanta, GA 30342-3220b�
School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0355
1113Earthquake Spectra, Volume 21, No. 4, pages 1113–1135, November 2005; © 2005, Earthquake Engineering Research Institute

1114 S. MUTHUKUMAR AND R. DESROCHES

The multiple-frame bridge and the multi-span simply supported bridge shown in Fig-ure 1 are most susceptible to pounding damage due to numerous independent compo-nents and lack of continuity in the structure. In a multiple-frame bridge, the interactionbetween adjacent frames can result in pounding at the intermediate hinge locations or atthe abutments. Pounding of girder ends at the pier locations and end abutments can oc-cur in a multi-span simply supported bridge. Based on observations from past earth-quakes, seismic pounding can lead to local crushing and spalling of concrete, result indamage to column bents, abutments, shear keys, bearing pads, and restrainers, and pos-sibly contribute to the collapse of deck spans.

Past research into seismic pounding has focused primarily on determining the factorsaffecting pounding �DesRoches and Muthukumar 2002�, modeling the impact phenom-enon �Jankowski et al. 1998, Malhotra 1998�, studying the effects of pounding on thebridge response �Kim and Shinozuka 2003�, and developing mitigation strategies forpounding hazard reduction �Kawashima and Shoji 2000, Shinozuka et al. 2000�. Typi-cally, the participating structural systems have been modeled using bilinear or stiffness-degrading models. However, experimental tests on concrete columns have shown thatstrength degradation occurs under increased cycles of loading, which is accelerated un-der the presence of axial compressive loads �Saatcioglu and Ozcebe 1989�. To the au-thors’ knowledge, no study has yet considered the effects of strength-degrading columnson the pounding response of bridges.

This paper investigates the influence of frame hysteretic characteristics, such as stiff-ness degradation, strength deterioration, and pinching on the pounding response of atypical multiple-frame bridge. Several analytical models are considered for the framehysteretic behavior, including the bilinear, Q-Hyst �stiffness-degrading�, and pivot hys-teresis �strength-degrading� models. A simplified planar nonlinear analytical model of amultiple-frame bridge is developed. Inelastic frame action, nonlinear hinge behavior,and the effects of abutments are modeled. Pounding is simulated using the stereome-

Figure 1. Types of bridges vulnerable to seismic pounding: �a� multiple-frame bridge, and �b�multi-span simply supported bridge.

EFFECT OF FRAME-RESTORING FORCE CHARACTERISTICS ON THE POUNDING RESPONSE 1115

chanical approach. Effects of skew, curvilinear bridge geometry, and multi-support ex-citation are not considered. Two adjacent frames of the bridge are isolated and parameterstudies are conducted using a suite of far-field ground motion records to evaluate theeffect of various hysteretic models on the frame pounding responses. The pounding re-sponse of the various hysteretic models in the presence of near-field earthquakes is alsoassessed. Finally, the differences in the global responses of a multiple-frame bridge withrestrainers, bearings, and abutments when various hysteretic models are used, are high-lighted through an example.

HYSTERETIC MODELS FOR REINFORCED CONCRETEBRIDGE COLUMNS

Reinforced concrete bridge columns can develop inelastic deformations and exhibitnonlinear behavior under moderate to strong base excitation. In the past, elasto-plasticand bilinear models were used due to their simplicity in concept and numerical imple-mentation. Stiffness degradation in concrete was first accounted for with the introduc-tion of a degrading stiffness approach �Clough and Johnston 1966�. Subsequent experi-mental tests on both small-scale and full-scale column specimens have shown that cyclicbehavior of reinforced concrete is characterized by constantly changing stiffness,strength degradation, and a reduction in energy absorption capacity �Takeda et al. 1970,Saatcioglu and Ozcebe 1989, Dowell et al. 1998�.

A typical lateral load-deflection hysteretic relationship for a reinforced concrete col-umn is shown in Figure 2. The general hysteretic characteristics can be summarized asfollows:

• Reduction in stiffness occurs with increased displacements, which can be attrib-uted to the flexural cracking in concrete and the Bauschinger effect in steel.

Figure 2. Lateral load-deflection relation for a reinforced concrete column obtained throughexperiment �Saatcioglu and Ozcebe 1989�.


• The peak strength attained in each cycle decreases with increased loading cycles.This strength degradation is a result of the disintegration of core concrete.

• The hysteretic loop exhibits pronounced pinching effects, which can be attributedto high shear stress reversals and slippage of the longitudinal reinforcementwithin the anchorage area.

• The hysteretic characteristics of reinforced concrete are dependent on the loadinghistory.

Several hysteretic models have been developed to capture the nonlinear dynamic re-sponse of reinforced concrete columns subjected to base excitation. These range fromrelatively simplistic models such as the elasto-plastic and bilinear models, to more rig-orous models such as the Takeda �Takeda et al. 1970�, Park �Kunnath et al. 1990�, andpivot hysteresis models �Dowell et al. 1998�. Other models such as the Clough model�Clough and Johnston 1966� and the Q-Hyst model �Saiidi and Sozen 1979� have alsobeen popular. A brief discussion of the hysteretic models considered in this study is pro-vided below.

ELASTO-PLASTIC MODEL

This is a simple model defined by three rules. The backbone curve is defined by anelastic stiffness �k� that represents cracked-section behavior and a post-yield portionwith zero stiffness, as shown in Figure 3a. The unloading stiffness is taken to be thesame as the elastic loading stiffness. This model is a very poor representation of the hys-teretic behavior of concrete as it does not represent stiffness deterioration with increas-ing displacement amplitude reversals. However, it has been extensively used because ofits simplicity in modeling.

BILINEAR MODEL

This is very similar to the elasto-plastic model, but it also accounts for the strainhardening effect in steel using non-zero post-yield stiffness, as shown in Figure 3b. Stiff-ness and strength degradation effects cannot be represented. Both the elasto-plastic and

Figure 3. �a� Elasto-plastic model; �b� bilinear model.


bilinear models do not consider hysteretic energy dissipation for small displacements.Many studies evaluating the effects of pounding have used bilinear models to representthe behavior of adjacent structures �Anagnostopoulos 1988, Pantelides and Ma 1998,Kim et al. 2000�.

Q-HYST MODEL

The Q-hyst model �Saiidi and Sozen 1979� is defined by four rules and closely rep-resents the response from a Takeda model, which is a more realistic representation of thecyclic behavior of reinforced concrete columns. The backbone curve used is bilinearwith strain hardening as shown in Figure 4. Stiffness degradation is accounted for at un-loading and load reversal. The unloading stiffness is defined by Kq=K�Dy /D�0.5, whereK is the initial elastic slope, D is the largest absolute deformation, and Dy is the yielddeformation. The reloading stiffness Kp is defined as the slope of the line connecting theintersection of the latest unloading branch with the displacement axis �point A� to themaximum absolute displacement �point B�, as shown in Figure 4. The Q-Hyst model ismuch simpler than the Takeda model. Neither the Q-Hyst and Takeda models, however,account for the effect of column axial loads and strength degradation in concrete.

PIVOT HYSTERESIS MODEL

The pivot model �Dowell et al. 1998� is governed by three simple rules and has theability to capture the dominant nonlinear characteristics of concrete under cyclic load.The backbone curve used for positive and negative loading is shown in Figure 5. Thefirst and second branches of the strength envelope represent cracked-section stiffnessand strain-hardening stiffness, respectively. Strength degradation from shear failure orconfinement failure is represented by the third branch. The fourth branch allows for alinearly decreasing residual strength.

Figure 4. Q-Hyst model for reinforced concrete.


Primary pivot points P1 through P4 control the amount of softening expected withincreasing displacement, using parameters �1, �2 as shown in Figure 5. Pinching pivotpoints PP2 and PP4 fix the degree of pinching following a load reversal, through param-eters �1

*, �2*. The response follows the strength envelope as long as no displacement re-

versal occurs. Once the yield displacement is exceeded in either direction, a modifiedstrength envelope is defined by the lines joining PP4 to S1 and PP2 to S2, as illustrated inFigure 5. The pinching pivot points start moving toward the origin of the force-deformation relation, once strength degradation occurs. The pinching parameters, �1

*,�2

*, are given by the following equations:

�i* = �i;dimax

� dti�1�

�i* =

Fimax

Fti

�i;dimax� dti

�2�

where �1, �2 define the degree of pinching for a ductile flexural response before strengthdegradation occurs. Displacements d and d represent the maximum displacement

Figure 5. Strength envelope for the pivot hysteresis model.

imax ti


and strength degradation displacement, respectively, in the ith direction of loading. Fimax

and Fti represent the force levels corresponding to dimax and dti, respectively.

The primary advantage of the pivot hysteresis model when compared to the othermodels is its ability to represent effects of cyclic axial load due to frame action, unsym-metrical sections, and strength degradation. Cyclic actual load effects are achieved withpredefined parameters, and the more general variation in axial load due to vertical ac-celerations is not included. Unlike other models, the pivot model recognizes that yield-ing in one direction does not soften the member in the opposite loading direction. Forinstance, if the yield strength is exceeded in the positive loading direction, unloadingoccurs and the member reloads in quadrant Q2 towards PP2, the response will follow theinitial elastic loading line if yielding has not yet occurred in the negative loading direc-tion. This study assumes symmetric sections for the bridge columns and only the effectsof strength degradation will be considered.

ANALYTICAL BRIDGE MODEL

A simplified planar nonlinear analytical model of an n-frame bridge is developed, asshown in Figure 6. Each bridge frame is idealized as a single-degree-of-freedom�SDOF� yielding element with mass mi, initial stiffness ki, and a viscous damping coef-ficient ci. For the frame force-deformation relation, any of the hysteretic models de-scribed in the previous section can be used. The elastomeric bearings at the hinge loca-tions and abutments are modeled using bilinear spring elements, following Kelly’s model�Naeim and Kelly 1999�. Cable restrainers at the intermediate hinges are modeled usingtension-only bilinear elements �5% strain hardening� with a slack. Abutments are mod-eled using linearized springs with different properties in active and passive action. Theequations of motion for the bridge system subjected to horizontal ground motion are

Figure 6. Analytical model of multiple-frame bridge used in study.


�m1

.

.

mn

��u1

.

.

un

� + �c1

.

.

cn

��u1

.

.

un

� +�FF1

�u1�

.

.

FFn�un�

�− �

FR1�u2 − u1�

FR2�u3 − u2� − FR1

�u2 − u1�

.

− FRn−1�un − un−1�

� − �FB1

�u2 − u1� − FB0�u1 − u0�

FB2�u3 − u2� − FB1

�u2 − u1�

.

FBn�un+1 − un� − FBn−1

�un − un−1��

+ �FI1

�u2 − u1�

FI2�u3 − u2� − FI1

�u2 − u1�

.

− FIn−1�un − un−1�

� = − �m1

.

.

mn

��1

.

.

1�ug �3a�

FA1�u0�

FA2�− un+1�

= FB0�u1 − u0�

FBn�un+1 − un�

�3b�

where FFi is the inelastic restoring force for each frame based on the hysteretic relationchosen, FRi is the force from restrainer Ri, FBi is the force in bearing Bi, FIi is the forcedue to impact between frames i and i+1, and FAi is the force in abutment Ai; ui, ui, andui �i=1 to n� represent the frame acceleration, velocity, and displacement relative to theground; u0, un+1 are abutment displacements obtained using static condensation; and ug

represents the horizontal ground motion applied to the bridge. The solution of Equation3 is obtained numerically using the fourth-order Runge-Kutta method �Kreyzig 1999�.

Seismic pounding occurs when the relative displacement between adjacent frames�ui−ui+1� exceeds the hinge gap �gp�. Pounding is accounted for using the stereome-chanical approach �Goldsmith 1960�, which assumes that impact is instantaneous. Themomentum balance principle is used along with the coefficient of restitution to modelimpact. The force due to impact is taken as zero. However, the velocities of the collidingmasses are modified at the instant of impact as

v1� = v1 − �1 + e�m2�v1 − v2�

m1 + m2�4a�

v2� = v2 + �1 + e�m1�v1 − v2�

m1 + m2�4b�

where vi� and vi+1� are the velocities of adjacent frames after impact; vi, vi+1 are theframe velocities before impact; and e is the coefficient of restitution, which is assumedas 0.8 in this study.


Other approaches to model pounding include the contact force-based linear springelement �Maison and Kasai 1992�, Hertz nonlinear spring �Davis 1992�, and the Kelvinelement �Jankowski et al. 1998�. The high impact stiffness used in the contact force-based approaches produces acceleration spikes at the instances of impact, which appearin the dynamic response. The stereomechanical approach does not produce these accel-eration spikes. Furthermore, the stereomechanical method is momentum-based, and thusthe uncertainty associated with the value of impact spring stiffness is eliminated. Thestereomechanical approach is no longer valid, however, if the impact duration is largeenough so that significant changes occur in the configuration of the colliding bodies. Re-cently, a study has been conducted to address the validity of the various approaches tomodel pounding �Muthukumar and DesRoches 2004�. The results indicate that the ste-reomechanical and contact force-based approaches produce similar displacement re-sponses for a given coefficient of restitution. The accelerations are much smaller for thestereomechanical approach.

Abutments play an important role in the seismic response of a bridge, as they attracta large portion of the earthquake loads and many design guidelines require their inclu-sion as equivalent linear springs. Impact between the bridge deck and end abutmentswill induce high passive pressures, and hence the effect of deck impact is indirectly con-sidered by including a high passive stiffness in the abutment model. The passive stiffnessof the abutment is determined using the Caltrans procedure �Caltrans 1999�. The stiff-ness is calculated to be around 2600 kips/ in for a backwall height of eight feet and anacceptable deformation of one inch.

COMPARISON OF HYSTERETIC MODEL RESPONSES

An SDOF system shown in Figure 7 is considered to study the differences in frameresponse when various hysteretic models are used. The system has an initial stiffness,

Figure 7. Single-degree-of-freedom system used to compare hysteretic model responses.


K=295 kips/ in, damping ratio, �=5%, and a period, T=1 second. The Saratoga-AlohaAvenue record with a peak ground acceleration �PGA� of 0.51 g, from the 1989 LomaPrieta earthquake is chosen for analysis. The yield strength of the system is selectedsuch that the target ductility is µ=4 when the Q-Hyst model is used as the frame force-deformation relation. Five percent strain hardening is used wherever applicable.

The hysteretic parameters of the Q-Hyst model �unloading stiffness KqQH, reloadingstiffness KpQH� and the pivot model without strength degradation �pivot parameters�1 ,�2, and the pinching pivot parameters �1 ,�2� need to be correlated so that their re-sponses are as close to each other as possible. Consider the unloading stiffnesses of thetwo models, as illustrated in Figure 8. From Figure 8a, KqQH can be expressed as

KqQH= K�Dy

Dm=

K�µ

�5�

where µ is the ductility ratio from the Q-Hyst model. From Figure 8b, the unloadingstiffness for the pivot model, KqPH can be written as

Figure 8. Unloading stiffnesses: �a� Q-Hyst model, and �b� pivot model.


KqPH=

K

µ

�1 − �* + �*µ + ��

�1 +�

µ �6�

where �1=�2=� �pivot parameter�. Equating Equations 5 and 6, an expression for thepivot parameter � can be found in terms of the strain hardening ratio �* and the ductilityratio µ of the Q-Hyst model, as given below.

� = �µ�1 − �*�1 + �µ�� 7�

Figure 9 sketches the reloading stiffnesses for the two models. For the Q-Hystmodel, the reloading stiffness KpQH can be written as

KpQH=

K

µ

�1 − �1 − µ��*�

�2 − �*�µ −�1 − �*�

�µ� �8�

Assuming �1=�2=�, the pinching pivot parameter can be expressed as

� =− KpPH

Xr

Fy�1 −Kp

K �9�

where Xr is x-coordinate of point A in Figure 9b and can be expressed as

Xr =Fy

K��1 − �*�µ�µ − �µ�1 − µ�� 10�

where Fy is the yield strength of the system and K is the initial elastic stiffness. SettingKpQH=KpPH and using Equations 8 and 10, the pinching pivot parameter � can be sim-plified to

Figure 9. Reloading stiffnesses: �a� Q-Hyst model, and �b� pivot model.


� =

�µ�1 − �* − �µ + 2�*µ + �*µ3/2�1 − �*�µ

�* − 1 �

1 − µ + �µ + µ�1 − �*�µ

�* − 1 �11�

It should be noted that the pivot model assumes that yielding in one direction doesnot soften the member in the opposite direction. Thus the response proceeds towardspoint C from point PP4, if the yield deformation has not been exceeded in the negativeloading direction, in Figure 9b. On the other hand, the response from the Q-Hyst modelwill proceed toward point B, the largest absolute displacement. This implies that in mostcases, the maximum response from the pivot model will either be equal to or smallerthan the maximum Q-Hyst model response.

For a target ductility, µ=4, and a strain hardening ratio, �*=5%, parameters � and �can be determined as 1.70 and 0.43, respectively. The responses from the various hys-teretic models when the SDOF system is subjected to the Saratoga-Aloha Avenue recordare presented in Figure 10. The time-history responses from all the models are identicalfor the first 6.2 sec, as nonlinear deformations have not yet occurred. However, once theyield force has been exceeded, the elasto-plastic and bilinear model responses exhibitmore permanent deformations, with a pronounced shift in the equilibrium position. Thisis because neither of the two models considers hysteretic energy dissipation for smalldisplacements. Thus, should a load reversal occur immediately after the reloading stage,there will be no energy dissipation from the elasto-plastic and bilinear models. TheQ-Hyst and pivot models will have some energy dissipation, however, since their un-loading and reloading stiffnesses are generally different.

The absolute maximum displacement from the Q-Hyst model is larger than the cor-responding bilinear response, as stiffness degradation in the Q-Hyst model produces lessdamping per cycle. However, despite major differences in the force-deformation rela-

Figure 10. Responses of SDOF system with various hysteretic models subjected to the 1989Saratoga-Aloha Avenue record: �a� time history of displacements, and �b� hysteresis loops.


tions, the absolute peak displacements from the Q-Hyst and elasto-plastic models areidentical �6.3 inches�. Typically, the Q-Hyst response is expected to be larger than theelasto-plastic response due to a smaller hysteretic loop for the Q-Hyst model. But forthis particular ground motion, the elasto-plastic model shows large excursions along thepost-yielding branch, which could account for the peak displacements being identical.The lack of strain hardening in the elasto-plastic model could be a factor as well.

The hysteretic loops from the pivot �without strength degradation� and Q-Hyst mod-els are very similar, with only a 15% difference in the maximum displacement response,for this particular ground motion record. The disparity can be attributed to the assump-tion in the pivot model that yielding in one direction does not soften the member in theopposite loading direction. Preliminary studies indicate that the maximum displace-ments from both the models can also be identical depending on the ground motionrecord. Thus, for the purposes of comparing the maximum displacement response, thecorrelation of hysteretic parameters between the Q-Hyst and the pivot models given byEquations 7 and 11 appears to be satisfactory and will be utilized through the rest of thestudy.

PARAMETER STUDY TO COMPARE THE IMPACT RESPONSE OF VARIOUSHYSTERETIC MODELS

To investigate the influence of column hysteretic characteristics on the pounding re-sponse of multiple-frame bridges, two adjacent frames are isolated and a parameterstudy is conducted with the two-degree-of-freedom idealized system shown in Figure11. The structure on the left �represented by a thick line� is taken as the stiff frame andthe structure on the right is assumed to be the flexible frame. The fundamental periods of

Figure 11. Two-degree-of-freedom model of adjacent frames used for hysteretic model param-eter study. Fundamental periods, T1=2��m1 /k1; T2=2��m2 /k2.


the stiff and flexible frames are T1, and T2, respectively. Restrainers, bearings, and abut-ments are not included so that differences in frame pounding responses can be directlyattributed to differences in hysteretic model characteristics.

A previous study by the authors identified the frame period ratio �T1/T2� as the pri-mary factor affecting the seismic pounding response �DesRoches and Muthukumar2002�. Smaller period ratios �T1/T2�0.3� represent highly out-of-phase frames and pe-riod ratios closer to one �T1/T2�0.7� indicate that the frames are more in-phase. Basedon a normalization of the governing equations of motion, the ground motion effectiveperiod ratio �T2eff

/Tg=T2�µ /Tg� was identified as another parameter affecting the

pounding response. The characteristic period of ground motion �Tg� is defined as theperiod at which the input energy of a 5% damped linear elastic system is a maximum.The results of the study showed that the frame displacement amplification due to pound-ing can be classified into three zones depending on the ground motion effective periodratio, T2eff /Tg. Impact was found to be most detrimental in Zone I �T2eff /Tg�1�. Thus,in the following study, only Zone I responses from the various hysteretic models are con-sidered, with three values for the frame period ratio: T1/T2=0.3 �highly out-of-phaseframes�, T1 /T2=0.5 �moderately out-of-phase frames�, and T1/T2=0.7 �essentially in-phase frames�.

The flexible frame period is fixed at 0.40 second and the stiff frame period is variedto get the desired period ratio. The yield strength of the system is selected such that thetarget ductility is µ=4 when the Q-Hyst model is used. Five percent strain hardening isassumed, wherever applicable. The effect of pounding is expressed in terms of framedisplacement amplification ��, which is the ratio of the maximum pounding displace-ment to the maximum frame displacement if pounding does not occur. The hinge gap isset very large for the no-pounding analysis, and is assumed as 1

2 inch for the poundinganalysis.

The parameters of the pivot model are selected such that strength degradation doesnot occur during the no-pounding analysis. The following values are used for the variousparameters: pivot parameters, �1 ,�2=1.70, pinching pivot parameters, �1 ,�2=0.43,strength degradation ductility, µt

+ , �µt−�=4.0, residual strength ratio, Fdr

+ , �Fdr−�=0.7, re-

sidual strength reduction ductility, µd+ , �µd

−�=8, failure ductility, µf+ , �µf

−�=100. The corre-lation of the � and � parameters with the Q-Hyst model parameters ensures that theductility of each frame when the pivot model is used and when no pounding occurs isclose to 4. Thus differences in displacement amplifications between the Q-Hyst andpivot models can be directly related to the effects of strength degradation.

Ten far-field ground motions recorded on medium soil �Tg=0.6–1.2 sec� are se-lected for analysis from the PEER Strong Motion Database �http://peer.berkeley.edu/smcat/�. The characteristic motions include the Pasadena record from the 1971 SanFernando earthquake, the Waho record from the 1989 Loma Prieta earthquake, and theWonderland Avenue record from the 1994 Northridge earthquake. The pseudo accelera-tion response spectra and record names are presented in Figure 12a. Each record isscaled such that the spectral acceleration at fundamental period equals the mean spectralacceleration of the suite of records at the fundamental period of the system �T


=0.40 sec�. The yield strengths of the system for each ground motion record at the vari-ous frame periods are determined using an iterative scheme. The characteristic periodsof the records �Tg� ensure that the ground motion effective period ratio, T2eff /Tg, lies inZone I.

Figure 13 presents the mean plus one standard deviation of the displacement ampli-fication due to pounding for the various hysteretic models as a function of the frameperiod ratio �T1/T2�, for ground motion effective period ratios in Zone I �T2eff /Tg�1�.In general, the elasto-plastic and bilinear models �traditional models� underestimate thestiff frame amplification and overestimate the flexible frame amplification, when com-pared to the Q-Hyst and pivot models �advanced models�. For instance, at T1/T2=0.3,the stiff frame displacement amplification predicted by the advanced models is 30%more than that predicted by the traditional models. The traditional models underestimate

Figure 12. Pseudo acceleration spectra of records used in analysis: �a� far field, and �b� nearfield.

Figure 13. Mean plus one standard deviation of displacement amplification due to pounding
from various hysteresis models—10 far-field ground motion records; Zone I �T2eff /Tg�1�.


the flexible frame displacement amplification by 20%, when T1/T2=0.3. The differencesbecome smaller with increasing period ratio. At T1/T2=0.5, the differences between thetraditional and advanced model responses are 20% for the stiff frame and 10% for theflexible frame. For essentially in-phase frames �T1/T2=0.7�, the deviations are only 5%and 2% for the stiff and flexible frames, respectively.

While comparing the Q-Hyst and pivot models, the strength degradation effect im-poses no additional demands on the response of the system. The pivot model only showsa 7% increase in the mean displacement amplification when compared to the stiffness-degrading �Q-Hyst� model, for T1/T2=0.3. In fact, at T1 /T2=0.5, the stiff frame ampli-fication from the pivot model is smaller than the Q-Hyst model response by approxi-mately 12%. All of the hysteretic models predict similar displacement amplificationswhen the frames are essentially in-phase �T1/T2=0.7�. The coefficient of variation�COV�, defined as the ratio of the standard deviation to the mean, ranges from 55% atlow-period ratios �T1/T2=0.3� to 14% at high-period ratios, for the pivot hysteresismodel. The COVs for the Q-Hyst model range from 34% to 14%, for low to high frameperiod ratios.

The results indicate that the effects of pounding are highly dependent on the framehysteretic model, especially for out-of-phase frames. The selection of traditional modelslike elasto-plastic and bilinear models can result in an underestimation of the impact am-plifications for the stiff frame and predict higher amplifications for the flexible framewhen compared with more advanced models. The effects of strength degradation in pre-dicting the pounding response of adjacent frames are not significant as long as stiffness-degradation is modeled. A parameter study is presented in the following subsection, tostudy the effects of strength degradation and pounding in the presence of near-fieldground motions.

EFFECT OF NEAR-SOURCE GROUND MOTIONS

Near-field earthquake motions are characterized by high peak ground accelerationsand velocity pulses with a long-period component �Yang and Agrawal 2002�. Such char-acteristics may greatly amplify the dynamic response of multiple-frame bridges, result-ing in severe damage. Such recent earthquakes as the 1994 Northridge, 1995 Kobe, 1999Kocaeli, and 1999 Chi-Chi earthquakes have demonstrated the damage that can becaused by near-field ground motions. To study the effects of near-field ground motion onthe pounding response of strength-degrading bridge frames, ten near-source records�Tg=0.6–1.2 sec� are selected for analysis from the PEER Strong Motion Database�http://peer.berkeley.edu/smcat/�. The characteristic ground motions considered includethe Bonds Corner record from the 1979 Imperial Valley earthquake, the Sylmar recordfrom the 1994 Northridge earthquake, and the TCU129N record from the 1999 Chi-Chiearthquake in Taiwan. The two-degree-of-freedom system considered in the earlier sec-tion is adopted for analysis. The frame yield strengths at various periods for each groundmotion record are obtained such that the frame ductility demands equal four when theQ-Hyst model is used. All records are scaled to the mean spectral acceleration at the


fundamental period of the system. Figure 12b presents the pseudo acceleration responsespectra and the record names for the near-field ground motions. The mean spectral ac-celeration at the fundamental period �T=0.40s� is 0.83 g.

The mean plus one standard deviation of the displacement amplification due topounding for the various hysteretic models is presented in Figure 14. As observed for thefar-field records, the traditional models �elasto-plastic and bilinear models� underesti-mate the stiff frame amplification and overestimate the flexible frame amplification whencompared to the more advanced models �Q-Hyst and pivot models�. However, when us-ing near-source records, the differences between the traditional and advanced modelspersist even when the frames get more in-phase �T1/T2=0.5,0.7�, unlike the case withfar-field ground motions. At T1/T2=0.7, the traditional models underestimate the stiffframe displacement amplification by 20% and overestimate the flexible frame amplifi-cation by 15% when compared to the more rigorous models.

The biggest difference in using near-field records is that strength degradation andpounding significantly affect the frame response, especially when the frame is highlyout-of-phase. The pivot model results in a mean stiff frame displacement amplificationof 5.8 as opposed to 2.6 for the Q-Hyst model. Large amplifications are observed forseveral ground motions, including the 1984 Halls Valley, 1994 Sylmar, and 1999TCU076N records. Thus accounting for strength degradation increases the stiff framedemand by 125%. The corresponding increase for the case with far-field records is only7%. In the presence of near-field ground motions, a bilinear or stiffness-degradingmodel for the column will grossly underestimate the pounding demands when comparedwith a strength-degrading model, for highly out-of-phase frames.

The earlier study using far-field ground motions indicated that the frame amplifica-tions get closer to unity as the period ratio becomes higher. The pivot model showed astiff frame amplification of 1.07 and a flexible frame amplification of 0.95, at T1/T2

Figure 14. Mean plus one standard deviation of displacement amplification due to poundingfrom various hysteresis models—10 near-field ground motion records; Zone I �T2eff /Tg�1�.


=0.7. However, for near-field ground motions, the frame amplifications show greaterdiscrepancy from unity. In the latter case, the corresponding pivot model amplificationsare 1.3 and 0.85 for the stiff and flexible frames, respectively.

EFFECTS OF HYSTERETIC MODEL ON THE GLOBALPOUNDING RESPONSE

In this section, the differences in the global responses of a multiple-frame bridgesystem, due to various hysteretic frame models are investigated. The bridge consideredconsists of four frames connected at three intermediate hinges. The hinge gap is taken as12 in. at all intermediate hinge locations. The simplified bridge model, as shown in Fig-ure 6 is developed with frame weights of 2880 k, 7080 k, 7080 k, and 2880 k, forframes 1 through 4, respectively. The damping ratio for each frame is taken as 5%. Theproperties of various elements used in the model are listed in Table 1. The hystereticmodels discussed earlier, namely, the elasto-plastic, bilinear, Q-Hyst, and pivot hyster-esis models are used to describe the frame behavior, with all models having the sameinitial stiffness and yield strength. The bilinear and Q-Hyst models assume a strain hard-ening ratio of 5%. The pivot hysteresis model is assumed to have the same properties inboth loading directions with dt=2*dy, dd=4*dy, and df=6*dy.

The restrainers are designed according to the procedure recommended by DesRo-ches and Fenves �DesRoches and Fenves 2001�. The restrainer slack is assumed as 1

2 in.The properties for the elastomeric bearings at the hinge locations are calculated based onthe bearing dimensions �12 in.8 in.4 in.,LWH�. The bearings at the abutmentlocations are designed to have a stiffness proportional to the passive stiffness of the abut-

Table 1. Properties of various bridge components used in study

Element ComponentInitial stiffness

�kips/in�Yield strength

�kips� Period �s�

Frame F1,F4 1333 877 0.47F2 ,F3 577 750 1.12

Element ComponentInitial stiffness

�kips/in�Yield strength

�kips�Strain

hardening �%�

Restrainer R1 ,R3 200 840 5R2 100 420 5

Bearing B1,B2 ,B3 6 2.4 33B0,B4 2600 1560 33

Element ComponentActive stiffness

�kips/in� Passive stiffness �kips/in�

Abutment A1,A2 10 2600


ment. Since the abutment and the bearing at the abutment are springs in series, the activestiffness of the abutment is taken proportional to the stiffness of the hinge bearing. Thebridge is subjected to horizontal ground motion from the 1989 Loma Prieta earthquake.The Saratoga record is used, which has a peak ground acceleration of 0.5 g, and a char-acteristic period �Tg� of 1.8 second. To study the effect of pounding on the bridge re-sponse, two cases are considered; Case 1, where the hinge gap is set very large so thatpounding does not occur, and Case 2, where the hinge gap is set at 1

2 inch and poundingoccurs. Figure 15 presents the displacement time history of the stiff frame �frame 1� forthe various hysteretic models. The corresponding hysteresis loops for the pounding andno-pounding cases are shown in Figure 16.

The no-pounding responses for the various models are very similar because there arenot too many excursions into the nonlinear range and the displacement ductility �µ� issmall �µ�2�. However, for Case 2, seismic pounding amplifies the displacement re-sponse of frame 1 by 100% to 173%, depending on the hysteretic model. The maximumdisplacement from the Q-Hyst and bilinear models is around 2.0 in, for the poundingcase, while the pivot hysteresis model response is 3.0 in.

Figure 15. Time history of displacement—stiff frame �frame 1�; 1989 Saratoga record �PGA=0.5g�.


Figure 16. Hysteresis loops for stiff frame �frame 1�—1989 Saratoga record �PGA=0.5g�.


The results indicate that strength degradation in bridge columns has a significant in-fluence on the pounding response of the stiff frame �frame 1�. Strength degradation withincreased loading cycles combined with the interaction of adjacent frames increases thestiff frame displacement demand by 50% when compared to other hysteretic models.This example serves to highlight the importance of correct hysteretic modeling in cap-turing the pounding response of multiple-frame bridges. The use of traditional modelslike the elasto-plastic and bilinear models can underestimate the severity of the pound-ing effect on the global frame response.

CONCLUSIONS

Recent earthquakes have illustrated that multiple-frame and multi-span simply sup-ported bridges are vulnerable to damage from seismic pounding. Previous studies havetypically used bilinear or stiffness-degrading models to represent the response of bridgecolumns while analyzing the effects of pounding. This paper investigates the influence ofcolumn hysteretic characteristics such as stiffness degradation, strength deterioration,and pinching on the impact response of adjacent frames in a multiple-frame bridge. Tra-ditional analytical models such as the elasto-plastic and bilinear models, and more ad-vanced models such as the Q-Hyst �stiffness-degrading� and pivot �strength-degrading�models are considered for representing the response of bridge columns. A simplified pla-nar nonlinear analytical model of a typical multiple-frame bridge is developed includinginelastic frame action, nonlinear hinge behavior, and abutment effects.

Parameter studies conducted on two adjacent bridge frames subject to ten far-fieldearthquake records show that the traditional models underestimate the stiff frame dis-placement amplification due to pounding, and overestimate the flexible frame amplifi-cation, when compared to the advanced models, for moderately to highly out-of-phaseframes. At T1/T2=0.3, the traditional models underestimate the stiff frame pounding re-sponse by 30% and overestimate the flexible frame response by 20%. The effect ofpounding is not significant for in-phase frames �T1/T2=0.7�, irrespective of the hyster-etic model considered. The strength degradation effect imposes no additional demandson the pounding response as long as stiffness degradation is modeled. However, in thepresence of near-field records, strength degradation increases the stiff frame displace-ment demand by 125% when compared to stiffness-degrading effect, for highly out-of-phase frames. Moreover, at T1/T2=0.7, the displacement amplifications due to poundingshow greater discrepancy from unity for near-field ground motions, with a stiff frameamplification of 1.3 and a flexible frame deamplification of 0.85.

A case study conducted on a four-frame bridge with the 1989 Saratoga record�PGA=0.5 g� indicates that strength degradation in bridge columns combined withpounding can increase the stiff frame displacement response by 50%, when compared toother hysteretic models. The traditional models underestimate the stiff frame displace-ment, in good agreement with findings from the earlier parameter study.

In conclusion, the authors recommend the use of a stiffness-degrading model to ac-curately estimate the frame pounding response in the presence of far-field ground mo-tion records. The expected value of stiff frame displacement amplification for ground


motion effective period ratios in Zone I �T2eff /Tg�1� is 1.7 �with a standard deviationof 0.6� for highly out-of-phase frames �T1/T2=0.3�, and 1.4 �standard deviation of 0.4�for moderately out-of-phase frames �T1/T2=0.5�. The above results are applicable forframe ductility demands around 4. In the case of near-field ground motions, a strength-degrading model is suggested for the bridge frame, to accurately estimate the poundingresponse. However, it is difficult to recommend a range for the frame displacement am-plification factors, since the values are dependent on the strength degradation parametersselected and are highly sensitive to near-source records used in analysis.

ACKNOWLEDGMENTS

This study has been supported primarily by the Earthquake Engineering ResearchCenters Program of the National Science Foundation under Award Number EEC-9701785 and the CAREER Program of the National Science Foundation under Grant0093868.

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�Received 21 April 2004; accepted 31 January 2005�

Effect of Frame-Restoring Force Characteristics on MF Bridges Muthukumar and DesRoches Nov 06

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