Bridge seismic response as a function of the Friction Pendulum System (FPS)modeling assumptionsMurat Eröz a,∗,1, Reginald DesRoches b
a Forell/Elsesser Engineers, Inc. 160 Pine Street Suite 600, San Francisco, CA 94111, United Statesb School of Civil and Environmental Engineering, Georgia Institute of Technology, 790 Atlantic Drive N.W., Atlanta, GA 30332, United States
a r t i c l e i n f o
Article history:Received 9 July 2007Received in revised form29 March 2008Accepted 27 April 2008Available online 9 June 2008Keywords:Seismic isolationBridgeNonlinear analysisComparative modelingFriction pendulum system
a b s t r a c t
Friction Pendulum Systems (FPS) are seismic isolation bearings that have been used as a means of bridgeretrofit in numerous cases around the world. To assess their impact on bridge performance, models areneeded to capture the intricate behavior of these highly nonlinear elements. In this paper, a new model forthe FPS that can represent the variation of the normal force and friction coefficient, bi-directional couplingand large deformation effects during nonlinear dynamic analyses is presented. The paper also examinesthe effect of modeling parameters on the response of a three dimensional multi-span continuous (MSC)steel girder bridge model seismically isolated with the FPS.
Seismic isolation can be an effective tool for the earthquakeresistant design of bridges that can be used in both newconstruction and retrofit. The Friction Pendulum System (FPS) is aseismic isolation bearing, with a mechanism based on its concavegeometry and surface friction properties (Fig. 1) [7]. The supportedstructure is administered into a pendulum motion as the housingplate simultaneously glides on the concave dish and dissipateshysteretic energy via friction [15].
A detailed inventory analysis of existing highway bridges inthe Central and Southeastern US (CSUS) shows that multi-spancontinuous (MSC) and multi-span simply supported (MSSS) SteelGirder bridges are among the most common classes of bridgesfound in the CSUS inventory [2,16,17]. Previous research identifiedsignificant vulnerabilities of the steel fixed and rocker bearingsemployed in these bridges to seismic loads [10]. Seismic isolationof these bridges via replacing the existing steel bearings withthe Friction Pendulum System (FPS) may be an effective tool forimproving the earthquake performance [5,22].
Characteristics of the FPS pertaining to durability under severeenvironmental conditions, reduced height, and insensitivity to
Fig. 1. Components of the Friction Pendulum System (FPS).
the frequency content of the ground motions, make it a viableoption for bridge seismic isolation [1,8,24]. The behavior of theFPS is strongly nonlinear and involves the coupling of multiplecomponents of the dynamic response, posing challenges for thoseattempting to model their response. The main modeling aspectsof the response of the FPS are: (1) the normal force (N); (2)the coefficient of friction (µ); (3) the in-plane bi-directionalsliding interaction; and (4) large deformation effects (P–∆) [14].The response of the FPS is typically modeled by a simplified
M. Eröz, R. DesRoches / Engineering Structures 30 (2008) 3204–3212 3205
bilinear force–deformation relationship. Ample theoretical andexperimental research findings are available in the literature forseparately representing each of these modeling aspects. However,the required level of accuracy for modeling the FPS in seismicallyisolated bridges has not received notable attention.
Previous research considering the effects of different aspectsof nonlinearities in the response of the FPS showed that theremight be a significant divergence from a bilinear idealization.Experiments showed that N fluctuations were consequentialin the response of individual isolators incorporating a slidingmechanism [14,20]. Dicleli [6] showed that varying dead loadson FPS along a seismically isolated bridge (SIB) might lead to anon-uniform transverse response that could result in excessivedisplacements. Nakajima et al. [18] tested combined rubber andfriction bearings, and concluded that omitting the variation of theµ in the modeling process caused considerable error. Jangid [9]performed a parametric study on a SIB and showed that the µ ofthe bearings is influential in optimizing the seismic response ofthe bridge. Warn and Whittaker [23] showed that the neglect ofthe bi-directional coupling of isolation bearing models in bridgesresulted in discrepancies in the force–deformation histories, andunderestimation of displacements of the bearings. Almazan and Dela Llera [1] showed that the exact kinematics and P–∆ momentsmight be considerable in estimating the peak bending momentsof columns in SIBs. The authors highlighted that the upward ordownward positioning of the FPS can be used to control the flowof the P–∆ moments.
The previous studies, among others, have been invaluable inacquiring an understanding pertaining to the response of the FPS.However, the influence of one or more of the four modelingaspects associated with the FPS response has been previouslyneglected. The lack of guidance on the relative influence of themodeling aspects of the FPS on bridge response has promptedconservative attitudes among the engineering community andlimited the use of this technology [11]. There is a need todevelop a better understanding of the influence of the modelingassumptions and the required level of accuracy for the FPS in three-dimensional (3-D) bridge models. In this paper, a 3-D zero-lengthnonlinear finite element (FE) model of the FPS that can includevarious of the above modeling aspects in any desired combination,is developed in OpenSees [12] to ascertain the importance ofthese assumptions. The modeling aspects mentioned above areaddressed and important physical phenomenon involved in theresponse are presented. These aspects are encapsulated into theformulation of a nonlinear zero-length element. A summary of theimplementation of the mathematical formulation into OpenSeesis presented. A detailed 3-D model of a seismically isolated multi-span continuous (MSC) bridge incorporating the FPS is developed.The influence of the variations in N and µ, in-plane bi-directionalsliding interaction, P–∆ effects, and the orientation of the FPS arehighlighted.
2. Modeling aspects
The two components of the intrinsic forces of the FPS consist ofthe pendulum motion of the mass, fR, and the friction between themass and the sliding surface, fµ. Assuming small deformations, theunidirectional force–deformation response of the FPS is:
f =
fµ︷ ︸︸ ︷Nµ sgn(δ̇)+
fR︷︸︸︷N
Rδ (1)
where N is the normal force acting on the sliding surface, R is theradius of the concave surface, δ is the sliding deformation, δ̇ is thesliding velocity, and sgn(δ̇) is the signum function, i.e., equal to+1or−1 depending on whether δ̇ is negative or positive, respectively.
Fig. 2. Force-deformation characteristics of the unidirectional rigid–plasticresponse of the FPS.
This relationship corresponds to a unidirectional rigid–plastichysteretic model given in Fig. 2. If the yield displacementsof steel–Teflon sliding surfaces reported are on the order on0.06–0.13 cm by Constantinou et al. [4] for conditions relevantto the FPS are considered, Fig. 2 takes up the characteristics of abilinear model. This model is based on the assumptions that: (1) Nis constant; (2) µ is constant; (3) the response is uncoupled in theorthogonal directions; and (4) deformations are small and planar.
The normal force, N, acting on the FPS is inherent in bothresisting force components, fµ and fR, of the response. The increasein the magnitude of N is indicative of a higher yield force, whichmay delay the mobilization of the FPS under dynamic and ahigher post-yield stiffness, which may reduce the flexibility of theFPS. Additionally, N changes the magnitude of µ, however thisrelationship is discussed subsequently. The conventional FPS doesnot have resistance in tension and it is approximately rigid incompression. This behavior closely matches the response of a zero-length gap element defined with a force:
fg ={kgδg if δg 6 0.00.0 if δg > 0.0 (2)
where kg is a high compression stiffness and δg is the deformation.Modeling the vertical response of the FPS with a gap elementallows simultaneously the monitoring of the variations in theN andcapturing the effects of uplift and impact in the FPS.
The coefficient of friction, µ, in addition to the materialproperties of the surface, were found to be primarily a function ofδ̇ and N [13]. Accurate mathematical models have been developedby Constantinou et al. [4] to capture the value of µ for a range ofδ̇ and N that is of interest to the response of the FPS. The influenceof δ̇ on theµwas approximated via the aid of experimental resultsas:
µ = fmax − Df e(−a|δ̇|) (3)
where, fmax and fmin are the values of coefficient of friction atlarge and small sliding velocities, respectively, Df is the differencebetween fmax and fmin, and a is a constant, having units of timeper unit length, that controls the variation of the coefficient offriction with velocity. Only the dependency of fmax to pressure, P, isconsidered in this study as the influence of pressure on fmin and awere shown to be negligible by Tsopelas et al. [21]. The term fmaxas a function of P was given as:
fmax = fmax,0 − Dfmax tanh(εP) (4)
where fmax,0 and fmax,p are the values of fmax at very low andhigh pressures respectively, Dfmax is the difference between fmax,0and fmax,p, and ε is a constant that controls the variation of fmaxbetween very low and very high pressures. Typical values of µ
3206 M. Eröz, R. DesRoches / Engineering Structures 30 (2008) 3204–3212
Fig. 3. Deformed shape of the seismic isolator between the superstructure and the substructure with concave dish at the (a) bottom (b) top.
range between 0.05 to 0.12 [7]. The parameters used in definingEqs. (3) and (4) are a function of surface properties and are obtainedfrom full-scale tests of the isolators prior to installation. Samplevalues may be found elsewhere [3,4,13,14].
Bi-directional motion may commence in the FPS subject tomultidirectional excitations. The orthogonal components of theplanar friction force vector, fµ = [ fµx fµy ]
T , defined in theplanar x and y directions as fµx and fµy respectively, are coupled.The interaction surface of fµx and fµx is circular and the resultantsliding friction force magnitude, ||fµ|| =
√f 2µx + f 2
µy, is equal to Nµ
regardless of the planar sliding direction. If each of the componentsof fµ is modeled uncoupled as in Eq. (1), the magnitude of ||fµ||during sliding ranges from Nµ, if sliding along the x or y axes, and√
2Nµ, if sliding along a path along the 45 degree direction.Constantinou [4] extended the work of Park et al. [19] and
presented bi-dimensional hysteretic parameters, η = [ ηx ηy ]T,
to evaluate fµ in planar steel–Teflon sliding interfaces. Thefrictional force vector in this case is defined as:
fµ = Nµη. (5)
The parameters of η evolve according to the following coupled setof differential equations:
η =
[η̇x
η̇y
]=
1∆s
[A− axη
2x −ayηxηy
−axηxηy A− axη2x
] [δ̇xδ̇y
](6)
where δ̇x and δ̇y, are the components of the sliding velocity in thex and y directions, respectively, ax = β + γ sgn(δ̇xηx), ay = β +
γ sgn(δ̇yηy), ∆s = max(µk/N,∆smin) is the deformation at whichsliding occurs with k the stiffness during the sticking phase and∆smin the minimum value of ∆s used to avoid instability problemsin the integration procedure when N tends to zero; and A, β, γ aredimensionless constants that control the shape of the hystereticloops. Constantinou [4] showed that for A/(β+γ) = 1, the solutionof Eq. (6): (1) describes a circular interaction curve; (2) for slidingconditions the hysteretic parameters are ηx = cos θ and ηy =
sin θ, where θ = arctan(δ̇y/δ̇x); and (3) for sticking conditions‖η‖ < 1.0. A value of A = 1, β = 0.1 and γ = 0.9 was foundto produce satisfactory accordance of the analytical response withthose obtained from the experiments in surfaces similar to theFPS [4].
The orientation of the FPS controls whether the P–∆ momentsoccur at the structural members below or above the bearing [7].This unique feature of the FPS does not have implications on thein-plane force–deformation response and allows for diverting P–∆
moments from weak elements of the structure [1]. Fig. 3 is a
schematic of the displaced shape of an FPS between a simplifiedbridge superstructure and the column. The normal force, N, istransmitted through the slider to the concave dish. Assuming thatthe rotations at the superstructure and the top of the columnare negligible, the displaced configuration of the FPS results in aninternal moment M = Nδ. This internal moment, M, is balancedat the tip of the column if the concave dish is at the bottom and bythe superstructure if the concave dish is at the top. Almazan and Dela Llera [1] presented a nonlinear transformation matrix for theirFPS model to account for this aspect, which is elaborated in thesubsequent section.
3. Mathematical model
The exact 3-D kinematics equations considering large defor-mation effects of the FPS were developed by Almazan and Dela Llera [1]. The zero-length element developed in OpenSees isconstructed based on these principles by assuming no nodal ro-tations. The authors do not claim any innovation for implement-ing this simplification and refer the reader to Almazan and De laLlera [1] for a more detailed presentation of these principles. A no-table modification made to the model used by Almazan and De laLlera [1] is the inclusion of the variations in the of theµ via Eqs. (3)and (4) at each integration time step. Here, only a brief summary ofthe mathematical formulation is presented with similar nomencla-ture as the original equations. The zero-length element is 3-D with6 degrees of freedom (DOF) per node (Fig. 4). This model accountsfor the variations of the N via an inherent gap element describedby Eq. (2). The coupling of the sliding forces are included via thehysteretic parameters, η = [ ηx ηy ]
T , as described in the previ-ous sections. The P–∆ moments are transferred to the nodes of themodel via a nonlinear transformation matrix, which is describedsubsequently.
Assuming that the nodal rotations are negligible, the localslider and global coordinates coincide. The nodal deformationsof the element are defined as u = [u(J)u(I)
]T , where u(J)
=
[u(J)x u(J)
y u(J)z r(J)x r(J)y r(J)z ]
T and u(I)= [u(I)
x u(I)y u(I)
z r(I)x r(I)y r(I)z ]T define the
motions of nodes J and I, respectively. The instantaneous positionand velocity of the slider is defined by the vectors δ =
[δx δy δz
]and δ̇ =
[δ̇x δ̇y δ̇z
], respectively (Fig. 4). The slider’s motion is
bounded by the spherical surface of the concave dish defined as:
G = δ2x + δ
2y + (δz − R)2
= 0. (7)The unitary vectors in the outward normal direction and thetangential to the trajectory of the slider are:
n =∇G
‖∇G‖=
1R
[δx δy (δz − R)
]T (8)
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Fig. 4. Schematic view of the model.
and
s =δ̇∥∥∥δ̇∥∥∥ =
[ ηx
‖η‖cosα
ηy
‖η‖cosα sinα
]T(9)
respectively (Fig. 5), where,
α = arctan(δx
ηx‖η‖+ δy
ηy‖η‖
R− δz
)(10)
denotes the angle between the frictional force component and thex–y plane. The local slider restoring forces for all phases is:
f =[fx fy fz
]T= Nr (11)
where, r, is the restoring force orientation vector that is constitutedfrom the normal and tangential components of the slider as:
r =normal component︷︸︸︷
n +
tangential component︷ ︸︸ ︷µ ‖η‖ s . (12)
The concavity and friction based components of the isolator arefR = nN and fµ = Nµ ‖η‖ s, respectively. The normal force in theisolator is:
N =fgrz
(13)
where rz is the axial component of the vector r. The nodal forcevector F = [F(J)F(I)
]T , where F(J)
= [F(J)x F(J)
y F(J)z M(J)
x M(J)y M(J)
z ]T and
F(I)= [F(I)
x F(I)y F(I)
z M(I)x M(I)
y M(I)z ]
T define the forces at nodes J and I,respectively, is:
F = L̂Tf (14)
where L̂T is the transform of the nonlinear transformation matrixdefined as:
The top and bottom signs in Eq. (15) are used to differentiatebetween the downward and upward positions of the FPS,respectively. The transformation matrix, L̂, depends exclusively ongeometry and is nonlinear to account for the variation of the P–∆
moments. The exclusion of the vertical rise in the concave dishcorresponds to δz = 0, α = 0, and N = fg . In this case Eq. (11)becomes:
f =[fx fy fz
]T= N
[δx
R+ ηxµ
δy
R+ ηyµ −1
]T. (16)
Fig. 5. Deflections and forces acting on the slider.
4. Evaluation platform
The Open System for Earthquake Engineering Simulation(OpenSees) is an open source software framework for simu-lating the earthquake response of structural and geotechnicalsystems [12]. OpenSees has an open source object-oriented ar-chitecture in the C++ programming language that maximizes itsmodularity, thus making it a viable choice for research purposes.Material and element models describing the hysteretic response ofnew structural members can be developed as C++ classes in anobject-oriented platform and inserted into the existing library ofOpenSees without the need for performing changes in the existingsolution algorithms. Although OpenSees provides a variety of hys-teretic uniaxial force–deformation response models, none of themwas found to be capable of adequately representing the mathemat-ical model described previously. Consequently, a 3-D zero-lengthelement class and a complimentary material class that can have theoption to include/exclude the modeling aspects of the FPS were im-plemented into the OpenSees library.
5. Bridge modeling
The bridge type selected for the nonlinear time history (NLTH)analysis is an MSC Steel Girder Bridge seismically isolated withFPS isolators. The 3-D SIB model was developed in OpenSees.This model includes material and geometric nonlinearities. Thegeometry and modeling approach for the bridge is illustrated inFigs. 6 and 7. The bridge has three spans and a continuous slab-on-girder deck with a total of eight steel girders. The seismicisolation of the bridge is achieved by placing FPS bearings undereach of the eight girders above the piers and abutments. The FPSisolation bearings are selected to achieve approximately a 2.0–2.5s fundamental period. The FPS bearings have R = 99 cm withan in-plane displacement capacity of 23 cm and positioned as theconcave dish at the top. The slider diameter has 7.7 cm to ensurepressures below 310 MPa under gravity, live and seismic loads in
3208 M. Eröz, R. DesRoches / Engineering Structures 30 (2008) 3204–3212
Fig. 6. Multi-span continuous (MSC) steel girder bridge general elevation and modeling details.
Fig. 7. Pier configuration and bent and column discretization.
accordance with the recommendations of the manufacturer. Thecharacteristic properties of the µ are selected as: a = 59.1 s/m,∆s = 0.025 cm, ε = 0.012 MPa−1, fmax,0 = 0.12, fmax,p = 0.07 andfmin = 0.05 [3,14]. The bi-directional coupling is simulated withthe solution of Eq. (6) based on A = 1, β = 0.1 and γ = 0.9 [4].
The superstructure is expected to remain within the linearelastic range; thus, the deck elements are modeled using elasticbeam column elements, using the composite section properties.Despite that the substructure of a seismically isolated bridgeshould be designed for elastic response, the piers are modeledusing nonlinear properties to monitor if the isolator modelingassumptions result in yielding of the substructure. The sectionproperties for the columns and the bent beams are created usingfiber elements with appropriate constitutive models for boththe concrete and the steel reinforcement. The reinforcing steelis modeled as a bilinear material with a yield strength, fys =414 MPa, and an elastic modulus, Es = 200 GPa. A strain-hardening ratio of 0.018 is used for this material. The unconfinedand confined concrete behavior is modeled via the Kent-Scott-Park model, which utilizes degraded linear uploading/reloadingstiffness and a residual stress. The concrete compressive strength,fc, and associated strain, εc, are 27.6 MPa and 2 × 10−3 forthe unconfined case and 28.5 MPa and (2.062)10−3 for theconfined case, respectively (Fig. 7). The bridge has footings, which
are 2.44 m square and use eight piles. The horizontal, kt , androtational, kr , stiffnesses of the foundation are 130.5 kN/mm and(6.06)105 kN m/rad, respectively. It is assumed that the bridgeends can tolerate maximum isolator movements in the plane ofthe bridge deck and that pounding between the deck and theabutments do not occur. Structural damping is assumed to be 5%.
6. FPS Modeling
Seven SIB models are generated with the above propertieswhere the only difference is in the FPS modeling assumptions. Thefirst model is theoretically exact, i.e., accounts for the variationsof the N and µ, has bi-directional coupling of the sliding forcesand incorporates P–∆ effects. The second model is a simplifiedbilinear model that is insensitive to the variations in N and µ,with uncoupled bi-directional sliding forces and small deformationassumptions. In Model 2, the constant value of N is taken as thecorresponding value after gravity load analysis and µ as 0.07.The third model is developed to monitor the influence of notaccounting for the variations of N on the response of the FPS. Itis the same as Model 1 with the only difference of assuming aconstant N of the corresponding value after gravity load analysis.The fourth model is developed to identify the influence of the bi-directional coupling in estimating the response of the FPS. It is
M. Eröz, R. DesRoches / Engineering Structures 30 (2008) 3204–3212 3209
Table 1Summary of model properties
Modeling aspect Modela
1 2 3 4 5 6b 7c
Normal force x x x x xBi-directional coupling x x x x xLarge deformations x x x x xFiction coefficient x x x x x
a ‘x’ denotes exact modeling.b Concave dish of the FPS at the bottom.c Seven models with value ofµ ranging from 0.05 to 0.12 with increments of 0.01.
Fig. 8. Mode shapes of the deck.
the same model as Model 1 with the only difference of assumingthe orthogonal sliding forces of the FPS isolators to be uncoupled.This is achieved by assuming η =
[sgn(δ̇x) sgn(δ̇y)
]T. The fifth
model is developed to monitor the influence of not accounting forthe inclination due to the concavity in the FPS. This is achieved bycomputing f with Eq. (16). The sixth model is generated to identifythe influence of the FPS orientation. Model 6 is same as Model 1with the only difference being that the FPS isolators are positionedwith the concave dish at the bottom, which is accommodated asthe corresponding sign shift in the L̂ in Eq. (15). The seventh modelis developed to monitor the influence of the assumptions on thevalue of µ. Model 7 is established with the same principles asModel 1 with the only difference of having a µ that is constant,i.e. insensitive to variations in pressure and sliding velocity. Model7 is discussed separately from the other models and analyzed for aconstant value of µ ranging from 0.05 to 0.12 with increments of0.01. The properties of the models are summarized in Table 1.
7. Dynamic analyses
The modal properties of the SIB in Model 1 are established byassigning linear effective stiffness to the FPS isolators. The firstthree modes of vibration are those involving the isolation system,which shows that the characteristics and the design of the FPSisolators govern the dynamic response of the bridge (Fig. 8). Thefirst three modal periods of the SIB are 2.22 s, 2.15 s, and 1.93 s,respectively. The first mode is longitudinal, the second mode istransverse and the third mode is torsional.
An important recommendation by the bridge engineeringcommunity is the use of design earthquakes that have a 2%
Fig. 9. Response spectrums for the suite of ground motions.
probability of exceedance in 50 years (an earthquake with a meanrecurrence interval of 2475 years) (FEMA 1997). A suite of tenearthquake records from rock sites is used in the NLTH analysisof the bridges (Table 2). The geometric mean of the longitudinaland transverse component of each record is scaled to match thespectral value of 0.118g at a period of 2.22 s corresponding to a 2%probability of exceedance in 50 years hazard level earthquake inMemphis, TN. The response spectra of the scaled ground motionrecords for 5% damping, ξ, and their median are given in Fig. 9.
The in-plane orthogonal components of the earthquakes areoriented to result in the maximum demands on the columns forall cases. The SIB models were first analyzed for gravity loads andsequentially subjected to NLTH analyses using simultaneously thelongitudinal, transverse and vertical acceleration records of thegiven earthquake. It is found from the gravity load analysis thateach isolator above the pier and the abutments carry a gravity load,No, of approximately 125 kN and 258 kN, respectively (neglectingthe normal load variation between the isolators at the exterior andthe interior ends at the same pier and abutment).
8. Results
The structural response of the isolators and columns alongthe same transverse axis were essentially the same. Therefore,the results are presented for one of the isolators on top ofthe piers and the abutments and one of the columns. Themain response quantities monitored for the FPS isolators arethe maximum normalized force, MNF = max(
√f 2L + f 2
T /No),where fL is the longitudinal and fT is the transverse isolatorforce, respectively, and the maximum normalized displacement,MND=max(
√δ2L + δ
2T/R), where δL is the longitudinal and δT is the
transverse isolator displacement, respectively, for the FPS abovethe pier at a given earthquake. Maximum column drifts, dmax, at agiven earthquake are selected as the response quantity to monitorthe structural demands on the SIB.
It was observed from the NLTH analyses of Model 1 that upliftoccurred between the sliding surfaces of the FPS isolators in the
Table 2Ground motion suite
No. Earthquake record Component PGA (g) ScaleLongitudinal Transverse Vertical
3210 M. Eröz, R. DesRoches / Engineering Structures 30 (2008) 3204–3212
Fig. 10. Time history of the N/No for the FPS during the Nahanni earthquake NLTHanalysis.
vertical direction for all of the records except for the Loma Prieta,Helena and Landers. The time-history of the N/No of the Model 1FPS isolator for the Nahanni earthquake is given in Fig. 10. Themaximum allowable N is limited by the allowable pressure of 310MPa on the slider, which corresponds to N/No = 5.4. This ratio wasnot exceeded during any of the NLTH analyses, however, during theNahanni earthquake a peak value of N/No = 3.51 were reached.This substantial increase is indicative of a proportional increase inthe post-yield stiffness and yield force of the isolator. It is observedfrom Fig. 10 that the contact between the two sliding surfaces waslost at least once which resulted in N/No = 0. This uplift causedinstantaneous yet complete loss of stiffness of the isolators duringthe earthquakes. However, due to the indeterminacy of the modelthere was no instability.
Fig. 11 shows the normalized force–deformation (NF-ND)histories of an FPS isolator on top of a pier among Models 1 to
4 in the longitudinal direction of the bridge during the N. PalmSprings record. Model 1 can capture the abrupt changes in isolatorforce and instances of uplift in the vertical direction. These twoaspects of the isolator response could not be observed in Model2. Additionally, Model 2 underestimated both the MNF and theMND in comparison to Model 1. These differences between Model1 and Model 2 NF-ND histories can be explained via the responseobserved in Models 3 and 4. Model 3 was unable to capturepeak isolator forces indicating that the normal components of theground motion were influential in this response quantity. AlthoughModel 4 was able to account for the significant variations in isolatorforces, the peak isolator force was overestimated and the peakisolator deformation was underestimated. This characteristic ofModel 4 indicates an overestimation of isolator stiffness when thebi-directional effects are neglected.
The influence of the isolator modeling parameters on MNF,MND and dmax for the suite of ground motions is illustrated viabox plots given in Fig. 12. Box plots are a useful way of presentingthe graphical description of variability of data. This informationprovides an overview of the expected demands on the isolatorsand the structural system as well as the scatter in the results. Thestatistical interpretation of the results is presented with numericalvalues of the median and plots of the 10th, 25th, 10th, 75th, and90th percentile cumulative probabilities.
It is observed that Model 2 underestimated the median ofthe MNF by 20% and the peak MNF as 44% of Model 1. Similarresults were observed for Model 3, which indicated that the normalcomponents of the force are influential in design level isolatorforces. It was observed that not including the influence of thevariations in the normal forces acting on the isolators results inloss in the variability of the MNF results. The underestimation ofMNF in Models 2 and 3 also lead to the underestimation of peakdmax by approximately 33%. Peak MND values for Models 2 to 6 hadnegligible difference with Model 1. However, there was a notable
Fig. 11. Force-deformation history of the FPS in the longitudinal directions on top of the pier for the N. Palm Springs earthquake record with (a) Model 1 (b) Model 2 (c)Model 3 and; (d) Model 4.
M. Eröz, R. DesRoches / Engineering Structures 30 (2008) 3204–3212 3211
Fig. 12. The influence of modeling assumptions on (a) MNF; (b) MND; and (c) dmax .
variation in the median values of the MND as a function of themagnitude of the constant value of µ in Model 7. This effect iselaborated in the subsequent section. Model 2 overestimated themedian of the dmax by 12% and underestimated the peak dmax as 69%of Model 1. The absence of the variability in the dmax with Model 2stems from the inability of this model to account for normal forcevariations on the isolators. On the other hand, Model 2 attained ageneral increase in the median of the dmax, which is attributed tothe overestimation of the stiffness caused by uncoupled responsein the orthogonal directions of the isolator lateral motion. Anotherfactor that contributed to the increase in the median of the dmaxin Model 2 was not accounting for the variation of µ. The peakvalue of MNF was overestimated by 7% in Model 4 in comparison toModel 1, however, the median of this response quantity remainedessentially the same. This is reflected as a 10% overestimation ofthe median dmax in Model 4 in comparison to Model 1. Models 5and 6 predicted the MNF, MND and dmax approximately the sameas Model 1 for the entire suite of ground motion records. Theexclusion of the exact kinematics pertaining to the concavity ofthe FPS in Model 6 was insignificant since the MND was limitedto 0.19 for the suite of ground motions. It is concluded that largedeformation effects associated with the orientation and exactkinematics were not significant in the response where averageMND was smaller than 0.20.
The influence of different magnitudes of µ in Model 7 on MNF,MND and dmax for the suite of ground motions is illustrated viabox plots given in Fig. 13. It is observed that the median of theMNF for the suite of ground motions increase consistently withincreasing values of µ. The values of the peak MNF has a similartrend with an exception atµ = 0.05 and 0.06, where the response
Fig. 13. The influence of constant value of µ assumptions on (a) MNF; (b) MND;and (c) dmax .
is underestimated by less than 5% in comparison to Model 1. Themedian MND is overestimated by 28% and underestimated by 11%in comparison to Model 1 when µ = 0.05 and 0.12, respectively,and generally decreases with the decreasing values ofµ. However,the peak MND remains essentially the same for all values of µ inModel 7. The median of the dmax was overestimated for all values ofµ by Model 7 in comparison to Model 1. The largest overestimationof the median dmax was for µ = 0.12 with approximately 15%.
Smaller No developed in the FPS isolators at the abutments thanat the piers, due to the difference in the corresponding tributarymass of the superstructure. Fig. 14 shows the comparison of thetotal MNF transferred to the pier, ΣMNFpier, and the total MNFtransferred to the abutment, ΣMNFabutment; and MND on top of thepier, MNDpier, and abutments, MNDabutment for the suite of groundmotions. It is observed that the isolators transferred 2.17 times theforce to the piers in comparison to the abutments on the median.The ratio between median MNDpier and MNDabutment was 0.97,which indicates that the difference between the ΣMNFabutment andΣMNFpier is associated with isolator stiffness characteristics ratherthan maximum deformations. Abutments, which are typically thestructurally stronger components of bridges compared to the piers,may be further engaged into resisting earthquake-induced loads inSIB by designing the FPS isolators by designing for a more uniformstiffness distribution among isolators throughout the bridge.
9. Conclusion
In this paper, the modeling of a typical highway bridgeseismically isolated with the FPS has been presented. The influenceof FPS modeling assumptions on normal force, N, and friction
3212 M. Eröz, R. DesRoches / Engineering Structures 30 (2008) 3204–3212
Fig. 14. Comparison of the (a) total MNF transferred to the pier, ΣMNFpier , and thetotal MNF transferred to the abutment, ΣMNFabutment; (b) MND on top of the pier,MNDpier , and abutments, MNDabutments .
coefficient, µ, orthogonal coupling and large deformation, P–∆,effects in a seismically isolated multi-span continuous (MSC) steelgirder bridge has been highlighted via nonlinear time-history(NLTH) analyses. The following conclusions are drawn:
(1) The most influential modeling aspects of the FPS in this bridgeapplication were found to be the variation of the normal forceand friction coefficient, and bi-directional coupling. The incor-poration of the modeling effects of orientation and the exactconcave geometry of the FPS has negligible affects if maximumnormalized displacements (MND) remain under 0.20.
(2) A simplified bilinear force–deformation model of the FPS with-out the coupling of the orthogonal directions was unable tocapture the variability of the response quantities. This simpli-fication led to the underestimation of the maximum columndrifts (dmax) by up to 31%. This was mainly a result of not ac-counting for the effects of vertical components of ground mo-tions, bi-directional coupling and the variable magnitude of thefriction coefficient.
(3) The uplift and subsequent pounding of the deck in the verticaldirection had notable effects on the maximum forces of the FPSthat in one case caused an increase of up to 3.51 times in theinitial gravity load acting on the isolators (No). Not accountingfor the variations in the N of the isolators led to underestimat-ing peak isolator maximum normalized forces (MNF) and dmaxup to 44% and 33%, respectively.
(4) Excluding the bi-directional coupling of the FPS isolators re-sulted in an increase of MNF values and consequently the over-estimation of the median dmax by 10%. However, the values ofMND remained essentially the same.
(5) The peak and median values of MND of the isolators had theleast amount of variation in comparison among all the mod-eling assumptions for the suite of ground motion evaluated.However, the median MND was influenced by the assumptionsin the magnitude ofµ and varied between 73% overestimationto 11% underestimation based on the assumed value of µ.
(6) The structural demands transferred by the isolators to theabutments and the piers were significantly different. The
median abutment forces from the isolators were approxi-mately twice as large as those at the piers. This is a result ofthe uneven distribution of isolator stiffness properties alongthe bridge as a function of deck tributary mass. However, iso-lators acquired similar MND on the median at the abutmentsand piers.
References
[1] Almazan J, De la Llera J. Physical model for dynamic analysis of structures withFPS isolators. Earthquake Eng Struct Dynam 2003;32:1157–84.
[2] Choi E. Seismic analysis and retrofit of Mid-America bridges. Ph.D. thesis.School of Civil and Environmental Engineering, Georgia Institute of Technol-ogy; 2002.
[3] Constantinou MC, Tsopelas P, Kim Y-S, Okamoto S. NCEER-Taisei corpora-tion research program on sliding seismic isolation system for bridges: Exper-imental and analytical study of a frictional pendulum system (FPS). TechnicalReport NCEER-93-0020. Buffalo (New York): National Center for EarthquakeEngineering Research; 1993.
[4] Constantinou MC, Mokha A, Reinhorn A. Teflon bearings in base isolation. II:Modeling. J Struct Eng 1990;116(2):455–74.
[5] Dicleli M, Mansour MY. Seismic retrofitting of highway bridges in Illinois usingfriction pendulum seismic isolation bearings and modeling procedures. EngStruct 2003;25(9):1139–56.
[6] Dicleli M. Seismic design of lifeline bridge using hybrid seismic isolation.J Bridge Eng 2002;7(2):94–103.
[8] HITEC (Highway Innovative Technology Evaluation Center). Technical evalu-ation report: Evaluation findings for earthquake protection systems, inc. fric-tion pendulum bearings. USA. 1998.
[9] Jangid RS. Optimum friction pendulum system for near-fault motions. EngStruct 2004;27(3):349–59.
[10] Mander JB, Kim DK, Chem SS, Premus GJ. Response of steel bridge bearings tothe reversed cyclic loading. Report no. NCEER 96-0014. Buffalo (New York):National Center for Earthquake Engineering Research; 1996.
[11] Mayes R. Impediments to the implementation of seismic isolation. ATCseminar on response modification technologies for performance-basedseismic design. 2002.
[12] Mazzoni S, McKenna F, Scott MH, Fenves GL, Jeremic B. Open System forEarthquake Engineering Simulation (Opensees). Berkeley, California. 2006.
[13] Mokha A, Constantinou M, Reinhorn A. Teflon bearings in base isolation. I:Testing. J Struct Eng 1990;116(2):438–54.
[14] Mosqueda G, Whittaker AS, Fenves GL. Characterization and modeling offriction pendulum bearings subjected to multiple components of excitation.J Struct Eng 2004;130(3):433–42.
[15] Naeim F, Kelly J. Design of seismic isolated structures. 1st ed. New York: Wiley;1996.
[16] Nielson B, DesRoches R. Influence of modeling assumptions on the seismicresponse of multi-span simply supported steel girder bridges in moderateseismic zones. Eng Struct 2006;28(8):1083–92.
[17] Nielson B. Analytical fragility curves for highway bridges in moderate seismiczones. Ph.D. thesis. School of Civil and Environmental Engineering, GeorgiaInstitute of Technology; 2005.
[18] Nakajima K, Iemura H, Takahashi Y, Ogawa K. Pseudo dynamic tests andimplementation of sliding bridge isolators with vertical motion. In: 12thWorld Conference on Earthquake Engineering. 2000.
[19] Park YJ, Wen YK, Ang AHS. Random vibration of hysteretic systems under bi-directional motions. Earthquake Eng Struct Dynam 1986;14(4):543–60.
[20] Takahashi Y, Iemura H, Yanagawa S, Hibi M. Shaking table test for frictionalisolated bridges and tribological numerical model of frictional isolator. In: 13thWorld conference on earthquake engineering. 2004.
[21] Tsopelas PC, Constantinou MC, Reinhorn AM. 3D-BASIS-ME: Computerprogram for nonlinear dynamic analysis of seismically isolated single andmultiple structures and liquid storage tanks. Technical report MCEER-94-0010. Buffalo (New York): Multidisciplinary Center for EarthquakeEngineering Research; 1994.
[23] Warn PW, Whittaker AS. Performance estimates in seismically isolated bridgestructures. Eng Struct 2004;26(9):1261–78.
[24] Zayas VA, Low S. Seismic isolation for extreme cold temperatures. In: 8thCanadian conference on earthquake engineering. Vancouver: CanadianAssociation for Earthquake Engineering; 1999.
