Evaluation of simplified methods of analysis for structures with triple friction pendulum isolators

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2010; 39:5–22Published online 8 June 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.930

Evaluation of simplified methods of analysis for structureswith triple friction pendulum isolators

Fabio Fadi‡ and Michael C. Constantinou∗,†,§

Department of Civil, Structural and Environmental Engineering, University at Buffalo, The State Universityof New York, 132 Ketter Hall, Buffalo, NY 14260, U.S.A.

SUMMARY

Triple friction pendulum isolators, that exhibit behavior with amplitude-dependent strength and instanta-neous stiffness, represent a new development in seismic isolation. The application of simplified methodsof analysis for this type of seismically isolated structures requires development of tools of simplifiedanalysis and demonstration of their accuracy. This paper describes these tools and presents validationstudies based on a large number of nonlinear response history analysis results. It is shown that simplifiedmethods of analysis systematically provide good and often conservative estimates of isolator displacementdemands and good estimates of isolator peak velocities. Copyright q 2009 John Wiley & Sons, Ltd.

Received 5 July 2008; Revised 12 April 2009; Accepted 17 April 2009

KEY WORDS: seismic isolation; triple friction pendulum bearing; simplified analysis

1. INTRODUCTION

The triple friction pendulum isolator exhibits multiple changes in stiffness and strength withincreasing amplitude of displacement. (The name ‘triple’ describes a behavior in which slidingoccurs in no more than three sliding surfaces regardless of geometry and frictional values.)When properly configured, these isolators may result in reduced isolator displacement demandsand/or reduced demands in force and acceleration in the primary and secondary structural systemsin comparison with conventional friction pendulum isolators. Previous work on these isolators[1–4]) established the principles of operation, presented mathematical descriptions of the behavior,

∗Correspondence to: Michael C. Constantinou, Department of Civil, Structural and Environmental Engineering,University at Buffalo, The State University of New York, 132 Ketter Hall, Buffalo, NY 14260, U.S.A.

†E-mail: [email protected]‡Doctoral Student, University of Udine, Italy.§Professor.

Contract/grant sponsor: ERDISU (Ente Regionale per il Diritto allo Studio Universitario)

Copyright q 2009 John Wiley & Sons, Ltd.

6 F. FADI AND M. C. CONSTANTINOU

presented experimental data that validate the models and described models for analysis in commer-cial computer software. The application of these isolators requires also that simplified methodsof analysis are developed and validated. Simplified methods of analysis are typically prescribedin applicable codes and specifications (e.g. ASCE [5] and AASHTO [6]) with the primary intentof serving as gauge or criteria for the results of nonlinear response history analysis that is mostoften performed for seismically isolated structures. The accuracy and limitations of the simplifiedanalysis procedures are therefore of much importance.

Simplified methods of analysis employ a single-degree-of-freedom system with equivalentlinear elastic and viscous behavior to predict the displacement demand directly from prescribedsmooth response spectra. Numerous equivalent linear models have been proposed and evaluated(e.g. [7–16]). Also, related studies evaluated damping reduction factors to obtain the displace-ment response of systems with high effective damping with the use of the 5% damped spectra(e.g. Newmark and Rosenblueth [17]; Ramirez et al. [18]; Lin and Chang [19]).

The simplest method of linearization is based on the effective stiffness (or secant stiffness)and effective damping at the maximum displacement and has been systematically used in allapplicable codes and specifications for seismically isolated and damped structures in the UnitedStates since the first drafts in the late 1980s. As all simplified methods of analysis, the effectivestiffness/damping method is an approximate method of analysis that requires validation on thebasis of comparison of results of nonlinear response history analysis to predictions of the methodfor a wide range of nonlinear system behavior. This paper presents an evaluation of the capabilityof this simplified method of analysis to predict the displacement demand and peak velocity ofisolation systems that consist of triple friction pendulum bearings.

2. DESCRIPTION OF SIMPLIFIED METHOD OF ANALYSIS

The simplified method evaluated herein is the effective stiffness and effective damping methodthat is described in codes and specifications for seismically isolated structures (e.g. ASCE [5]and AASHTO [6]). While not necessarily the most accurate method of simplified analysis, themethod is conceptually simple and uncomplicated to apply. The method is based on the followingsteps: (a) represent the isolated structure by a single-degree-of-freedom system, (b) assume thepeak isolator displacement, (c) construct the isolation system force–displacement loop at theassumed displacement, (d) calculate the effective stiffness and effective damping on the basis ofthe constructed loop (the latter requires calculation of the energy enclosed by the hysteresis loop),(e) calculate the spectral displacement from the 5% damped response spectrum for the periodcorresponding to the effective stiffness, (f) calculate the displacement demand as the spectraldisplacement divided by the damping factor corresponding to the calculated effective damping,and (g) repeat the process of steps (b) to (f) until the assumed and calculated displacements aresufficiently close. Upon calculation of the displacement demand, the maximum isolation systemforce is obtained directly from the force–displacement loop. The isolation system force (afterdivision by an appropriate response modification factor) is then distributed over the height of thestructure as inertia forces to calculate story shear forces for design.

In the case of triple friction pendulum isolators, the construction of the force–displacementloop is complex as it may contain several transition points that depend on the assumed properties.Figure 1 shows the geometry of a triple friction pendulum bearing. Its behavior is characterized byradii R1, R2, R3 and R4 (typically R1= R4 and R2= R3), heights h1, h2, h3 and h4, displacement

Copyright q 2009 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2010; 39:5–22DOI: 10.1002/eqe

EVALUATION OF SIMPLIFIED METHODS OF ANALYSIS FOR STRUCTURES 7

Figure 1. Cross section of the triple friction pendulum bearing.

Figure 2. Force–displacement behavior of triple friction pendulum isolation system.

capacities d1, d2, d3 and d4 (typically d2=d3 and d1=d4) and friction coefficients �1, �2, �3 and�4 (typically �2=�3). Herein we consider that all isolators are of the same geometry (typical of allbuildings and for most bridges) and that the coefficients of friction represent the weighted averagevalues for the entire isolation system (defined as the sum of friction forces from all isolatorsdivided by weight carried by the isolators).

The lateral force–displacement relation of the isolation system is illustrated in Figure 2. Fivedifferent loops are shown in Figure 2, each one valid in one of five different regimes of displacement.The parameters in the loops relate to the geometry of the bearing, the friction coefficient values andtotal weight W carried by the isolation system as described in Fenz and Constantinou [2]. Triplefriction pendulum isolators are designed to operate in regimes I to IV, whereas regime V should,in the opinion of the authors, be reserved for exceptional conditions to provide for a displacementrestrainer in earthquakes beyond the maximum considered. In regime V the isolator has consumedits displacement capacities d1 and d4 and only slides on surfaces 2 and 3 (see Figure 1).



The effective stiffness, effective period and effective damping are, respectively, defined asfollows:

Keff = Fmax

D(1)

Teff = 2�

(W

gKeff

)1/2

(2)

�eff =EDC

2�KeffD2(3)

where Fmax is the lateral force at displacement D and EDC is the energy dissipated in a cycleof harmonic motion at displacement amplitude D. Evaluation of EDC by analytical means iscomplex and it is best performed by first constructing the force–displacement relationship and thennumerically evaluating the area enclosed by the loop.

Upon calculation of the displacement in the simplified method of analysis, the peak velocityVmax in the isolation system may be calculated as the pseudo-velocity multiplied by the velocitycorrection factor CFV as described in Pekcan et al. [20] and Ramirez et al. [14]

Vmax=(2�

Teff

)·D ·CFV (4)

Factor CFV depends on the effective period and the effective damping of the analyzed system.Although the factor was established by analyzing bilinear hysteretic systems [14] or linear systems[20], the values in the two studies are nearly the same. Herein the values reported by Ramirezet al. [14] are used as they apply to a wider range of values of effective period and damping.

It should be noted that velocity Vmax is the peak velocity of the top of the isolator with respectto its bottom and is not the peak sliding velocity at any of the sliding interfaces of the isolator.The peak velocity may be partitioned among the four sliding interfaces of the isolator as describedin Fenz and Constantinou [2]. The value of Vmax is useful in establishing testing protocols forisolators, whereas values of the sliding velocities (after partitioning) are useful in estimating thetemperature at the sliding interfaces and its effects on friction and wear [21].

3. GROUND MOTIONS USED IN RESPONSE HISTORY ANALYSIS

The evaluation of the accuracy of simplified methods of analysis is based on comparison of theresults of nonlinear response history analysis with results of the simplified analysis. The responsehistory analysis is performed for a large number of ground motions that collectively (in an averagesense) represent specific response spectra. Two different response spectra with corresponding twodifferent sets of ground motions are used in this study. These spectra and sets of motion werepreviously used in studies of structures with damping systems by Tsopelas et al. [12] and Pavlouand Constantinou [15].

The first set of ground motions consists of the scaled horizontal components of 10 earthquakemotions recorded on soft rock or stiff soil. Each of these earthquakes was selected to have amagnitude larger than 6.5, an epicentral distance between 10 and 20 km and a site conditionaccording with the definition of Site Class C or D per ASCE [5]. The ground motions were scaled



Table I. Soft rock to stiff soil ground motions (SR).

Earthquake Station Component Scale factor

W. Washington (1949) 325 (USGS) N04W 2.74N86E 2.74

Eureka (1954) 022 (USGS) N11W 1.74N79E 1.74

San Fernando (1971) 241 (USGS) N00W 1.96N90W 1.96

458 (USGS) S00W 2.22S90W 2.22

Loma Prieta (1989) Gilroy 2 90 1.070 1.07

Hollister 90 1.460 1.46

Landers (1992) Yermo 360 1.28270 1.28

Joshua 90 1.480 1.48

Northridge (1994) Moorpark 180 2.6190 2.61

Century 90 2.27360 2.27

following the procedure described in Tsopelas et al. [12] to represent in an average sense a designresponse spectrum per ASCE [5] with parameters SD1=0.6, SDS =1.0 and Ts =0.6 s. Table Ipresents the ground motions components and the scale factors used. This set of motions is referredto as the SR (stiff soil or rock) motions.

The second set of ground motions consists of the scaled horizontal components of seven soft-soilearthquake histories selected to have a site condition in accordance with the definition of SiteClass E in ASCE [5]. The selected motions were scaled to represent in an average sense a designresponse spectrum per ASCE [5] with parameters SD1=0.75, SDS =0.8 and Ts =0.94 s. Table IIpresents the ground motions components and the scale factors used. This set of motions is referredto as the SS (soft soil) motions. Figure 3 presents a comparison of the average 5% damped spectraof the scaled ground motions and the target design spectrum. It should be noted that the averageof the scaled motions represent well the target design spectra for periods generally larger than 1 sand up to about 2.8 s for the SR set and up to at least 4 s for the SS set of motions.

The two sets of scaled motions are characterized by average values of spectral acceleration atperiod of 1 s equal to 0.6g (SR set) and 0.75g (SS set). Analyses were performed by further scalingthe amplitude of the motions so that the one-second average 5% damped spectral accelerationvalues were in the range of 0.2 to 1.6g with increments of 0.2g. This allowed analysis of structureswith triple friction pendulum isolators that operated in all possible regimes of displacement asshown in Figure 2 and provided a more comprehensive set of data for evaluation. It should benoted that the scaled SR and SS sets have the same target spectral shapes in the long period rangefor the same value of spectral acceleration at 1 s. However, the two sets of scaled motions differ



Table II. Soft-soil ground motions (SS).

Earthquake Station Component Scale factor

Northridge (1994) Sylmar 90 0.99360 0.59

Loma Prieta (1989) Gilroy 3 90 1.740 2.20

Hollister and Pine 90 2.460 1.21

Landers (1992) Barstow-Vineyard 90 3.030 3.46

Amboy 90 4.260 3.79

Hot Springs 90 4.010 3.65

Southern Alaska (1979) Yakutat 09 3.20279 2.45

(a) in the seed motions used to obtain the scaled motions and (b) in how well they match the targetspectra in the long period range. As noted earlier, the SS set better matches the long period rangeof the target spectra.

4. DESCRIPTION OF ISOLATION SYSTEM

The isolation system considered in this study had three different geometrical configurations andthree different sets of frictional properties. The three geometrical configurations are presented inTable III. The three configurations primarily differ in the displacement capacities, which respec-tively are 0.48, 1.05 and 1.52m, and the effective radii of curvature. The effective radius isdefined as the difference between the radius of curvature of the relevant spherical surface and thecorresponding portion of the height of the slider, Reffi = Ri −hi [2].

Three different sets of friction coefficients are assumed for each of the three geometricalconfigurations. The first and second sets are characterized by unequal friction values on the twomain sliding interfaces and by either low values of friction (referred to as LF) or high values(referred to as HF). The third set is characterized by equal and low values of friction on the twomain sliding interfaces (referred to as EF). All three sets have the same friction coefficient for thecontact surfaces between the rigid slider and the internal plates. Table IV presents the frictionalproperties considered in this study.

Figure 4 presents force–displacements loops for the nine configurations considered in this study.The loops for each of the five possible regimes of the isolation systems are shown (note that for theEF configurations, the five regimes collapse to three). Table V presents values of displacement andlateral force (normalized by weight) at each of the transition points between regimes (I to II, II toIII, III to IV and IV to V). The nomenclature used for forces and displacements at the transitionpoints and the calculation of their values was based on Fenz and Constantinou [2]. It should benoted that the loops in Figure 4 are shown to extend into displacement regime V although triplefriction pendulum isolators are not typically designed to operate in that regime.



Figure 3. Average 5% damped spectral acceleration of SR and SS ground motions and design spectra.

Table III. Dimensions of triple friction pendulum isolators.

Displacement capacities Effective radii

Designation d2=d3 (m) d1=d4 (m) dTOT (m) Reff2= Reff3 (m) Reff1= Reff4 (m)

GC1 0.064 0.178 0.484 0.356 1.499GC2 0.125 0.400 1.050 0.300 2.085GC3 0.178 0.584 1.524 0.902 3.823



Table IV. Friction coefficient values.

Designation Surface 1 Surface 2 Surface 3 Surface 4

LF 0.04 0.02 0.02 0.10HF 0.07 0.02 0.02 0.14EF 0.07 0.02 0.02 0.07

Figure 4. Force–displacement loops (five displacement regimes) of studied systems.

5. COMPARISON OF SIMPLIFIED AND RESPONSE HISTORY ANALYSIS RESULTS

Dynamic response history analysis was conducted in computer code SAP2000 [22] in which thetriple friction pendulum isolators were modeled using the model described in Fenz and Constantinou[4]. In this model, each isolator is modeled as three single friction pendulum isolators in seriestogether with gap elements. The model is capable of capturing the behavior of triple friction



Table V. Displacements and lateral forces at transition points between regimes.

Transition displacement (m) Horizontal force/Weight

Configuration u∗ u∗∗ udr1 udr4 F∗/W F∗∗/W Fdr1/W Fdr4/W

GC1-LF 0.014 0.123 0.302 0.413 0.04 0.10 0.16 0.22GC1-HF 0.036 0.166 0.312 0.441 0.07 0.14 0.19 0.26GC1-EF 0.036 0.036 0.392 0.392 0.07 0.07 0.19 0.19GC2-LF 0.012 0.155 0.705 0.848 0.04 0.10 0.23 0.29GC2-HF 0.030 0.197 0.705 0.872 0.07 0.14 0.26 0.33GC2-EF 0.030 0.030 0.830 0.830 0.07 0.07 0.26 0.26GC3-LF 0.036 0.320 1.029 1.312 0.04 0.10 0.19 0.25GC3-HF 0.090 0.421 1.054 1.385 0.07 0.14 0.22 0.29GC3-EF 0.090 0.090 1.258 1.258 0.07 0.07 0.22 0.22

pendulum isolators in all five regimes of displacement. The analysis included only horizontalcomponents of seismic excitation because experimental studies with vertical excitation reportedby Morgan [1] and Fenz and Constantinou [23] concluded that the vertical excitation does nothave effects that warrant consideration in the prediction of the isolator displacement response ofstructures with triple friction pendulum isolators. However, the vertical excitation may have effectson the isolation system forces that warrant consideration.

Figures 5–7 present a comparison of peak isolator displacement as calculated by simplifiedanalysis and by nonlinear response history analysis (the latter as average of the peak displacementcalculated for each of the motions). For the simplified analysis, the average spectra of the scaledmotions were utilized so that direct comparisons to results of response history analysis could bemade. Each graph in Figures 5–7 corresponds to one of the nine configurations analyzed, theforce–displacement characteristics of which are described in Figure 4 and Table V. Also, eachgraph identifies the transition displacements (see Table V) so that the reader can easily identify theregime of displacement in which the isolators operated. The results are presented with the averageof nonlinear response history analysis plotted on the vertical axis and the results of the simplifiedanalysis plotted on the horizontal axis. Each graph includes the results of the analysis for the SRand SS sets of motions as scaled to represent a range of values of spectral acceleration at a periodof 1 s. It should be noted that (a) the results of the response history analysis differ between thetwo cases of motions due to differences in the details of the scaled motions, and (b) the results ofthe simplified analysis differ between the two cases of motions due to differences in the averageresponse spectra (whereas the target spectra are identical for the two cases and for the same spectralacceleration at 1 second). In order to discuss the results of analysis, Table VI was prepared. Itpresents values of the effective period and effective damping calculated in the simplified analysis.Note that in this table the range of values of effective period and damping shown correspond tothe various levels of seismic excitation considered. The first value in each range corresponds tothe lowest level of excitation and the second value corresponds to the highest level of excitation.In all cases but two, the effective period increases as the excitation level increases. The two casesidentified in the table correspond to operation of the isolators well within the displacement regimeV , where significant stiffening occurred.

The results of Figure 5–7 demonstrate a systematically good and often conservative estimationof displacement demand by the simplified method of analysis. The most conservative estimation of



Figure 5. Comparison of response history (RHA) and simplified analysis (SA) resultson displacements for system GC1.









Table VI. Effective period and effective damping calculated in simplified analysis.

Effective period (s) Effective damping (%)

Configuration Motions SR Motions SS Motions SR Motions SS

GC1-LF 1.87–2.76 1.86–2.70 30–17 31–14GC1-HF 2.05–2.62 2.07–2.53 23–18 23–15GC1-EF 1.67–2.84 1.60–2.69 27–20 25–15GC2-LF 1.84–3.45 2.97–2.80∗ 23–14 23–7GC2-HF 2.47–3.21 2.50–3.27 25–21 25–18GC2-EF 2.75–3.54 2.90–2.79∗ 32–16 29–7GC3-LF 3.55–4.59 3.70–4.60 21–17 21–15GC3-HF 3.06–4.09 3.27–4.35 25–22 23–20GC3-EF 3.21–5.54 3.49–4.71 31–20 31–17

∗Cases of operation in displacement regime V.

displacement demand by the simplified method occurred in cases in which the isolators operatedwell within the displacement regime V. Specifically, in two out of 90 analyzed cases (system GC2,LF and EF, motions SS) there was approximately 35% overestimation of displacement by thesimplified method. The overestimation was the result of underestimation of the effective damping(7%—see Table VI) due to operation in the stiffening range of regime V. It should be noted thatin these two cases the response history analysis method did not predict operation in regime V.

Figures 8–10 present a comparison of peak velocity calculated by the simplified procedure tothe average results of response history analysis. Evidently, the simplified method underpredicts thepeak velocity. Among the 90 cases analyzed, the greatest under-estimation of velocity occurred forsystem GC3 and the cases of strongest excitation where the simplified method under-predicted thepeak velocity by as much as 35%. In general, the percentage amount of under-estimation of thepeak velocity by the simplified method increased with increasing effective period of the isolationsystem. In the case of the aforementioned systems GC3 with greatest error in the prediction ofvelocity, the effective period exceeded 4 s.

It should be noted that codes and standards such as ASCE [5] would not allow the use ofsimplified methods of analysis under the conditions in which either overestimation of displacementoccurred (significant stiffening of isolators, very strong excitation) or when underestimation ofvelocity occurred (effective period exceeding 3 s). Rather, nonlinear response history analysis wouldbe required but subject to limits on the calculated response that are related to the results obtained insimplified analysis. Therefore, the results of simplified analysis are important even when responsehistory analysis is used and the aforementioned limits on response should be revisited in the lightof the degree of conservatism, or lack of, in simplified methods of analysis.

6. CONCLUSION

This paper investigated the validity and accuracy of simplified methods of analysis for structureswith triple friction pendulum seismic isolators, a system that exhibits complex behavior withamplitude dependent stiffness and strength. The simplified analysis method investigated is the onetypically described in codes and specifications. The method is based on modeling the isolatedstructure as a single-degree-of-freedom system with linear stiffness and linear viscous damping



Figure 8. Comparison of response history (RHA) and simplified analysis (SA) resultson velocities for system GC1.









equal, respectively, to the effective or secant stiffness and effective damping of the isolation system.The study compared the results of nonlinear dynamic response history analysis with results ofsimplified analysis using two sets of ground motions that were scaled to match design spectra in thelong period range and then scaled to represent earthquakes of different intensities. Nine differentconfigurations were analyzed, each one of which operated in all possible displacement regimes ofthe triple friction pendulum isolator. The study led to the conclusion that the simplified methodof analysis (a) provides good and often conservative estimates of peak isolator displacement and(b) generally under-predicts the peak velocity by as much as 35%. The percentage amount ofunderestimation of the peak velocity increased with increasing effective period of the isolationsystem. The greatest overestimation of displacement by the simplified method occurred in casesof operation of the isolators in regime V, where significant stiffening occurs. When the simplifiedmethod significantly overestimated (by about 35%) the displacement or significantly underestimated(by about 35%) the velocity, the conditions were such that the simplified methods of analysiswould not be allowed by applicable standards such as ASCE [5] (significant stiffening of isolators,very strong seismic excitation or effective period exceeding 3 s).

ACKNOWLEDGEMENTS

The authors are thankful to ERDISU (Ente Regionale per il Diritto allo Studio Universitario) of Udine,Italy for sponsoring the studies of Fabio Fadi at the University at Buffalo.

REFERENCES

1. Morgan TA. The use of innovative base isolation systems to achieve complex seismic performance objectives.Ph.D. Dissertation, Department of Civil and Environmental Engineering, University of California, Berkeley, 2007.

2. Fenz DM, Constantinou MC. Spherical sliding isolation bearings with adaptive behavior: theory. EarthquakeEngineering and Structural Dynamics 2008; 37(2):163–183.

3. Fenz DM, Constantinou MC. Spherical sliding isolation bearings with adaptive behavior: experimental verification.Earthquake Engineering and Structural Dynamics 2008; 37(2):185–205.

4. Fenz DM, Constantinou MC. Modeling triple friction pendulum bearings for response history analysis. EarthquakeSpectra 2008; 24(4):1011–1028.

5. American Society of Civil Engineers. Minimum design loads for buildings and other structures. Standard ASCE/

SEI 7-05, Reston, VA, 2005.6. American Association of State Highway and Transportation Officials. Guide Specification for Seismic Isolation

Design, Washington, DC, 1999.7. Iwan WD, Gates NC. The effective period ad damping of a class of hysteretic structures. Earthquake Engineering

and Structural Dynamics 1979; 7(3):199–221.8. Iwan WD, Gates NC. Estimating earthquake response of simple hysteretic structures. Journal of the Engineering

Mechanics Division 1979; 105(EM3):391–405.9. Kircher CA, Lashkari B. Statistical evaluation of nonlinear response of seismic isolation systems. Technical

Report No. JBA 109-070, Jack R. Benjamin and Associates, Inc., Mountain View, CA, 1989.10. Theodossiou D, Constantinou MC. Evaluation of SEAOC design requirements for sliding isolated structures.

Technical Report NCEER 91-0015, National Center for Earthquake Engineering Research, University at Buffalo,State University of New York, Buffalo, NY, 1991.

11. Winters CW, Constantinou MC. Evaluation of static and response spectrum analysis procedures of SEAOC/UBCfor seismic isolated structures. Technical Report NCEER 93-0004, National Center for Earthquake EngineeringResearch, University at Buffalo, State University of New York, Buffalo, NY, 1993.

12. Tsopelas P, Constantinou MC, Kircher CA, Whittaker AS. Evaluation of simplified methods of analysis foryielding structures. Technical Report NCEER 97-0012, National Center for Earthquake Engineering Research,University at Buffalo, State University of New York, Buffalo, NY, 1997.



13. Chopra AK, Goel RK. Evaluation of NSP to estimate seismic deformation: SDF systems. Journal of StructuralEngineering (ASCE) 1999; 126(4):482–490.

14. Ramirez OM, Constantinou MC, Whittaker AS, Kircher CA, Chrysostomou CZ. Evaluation of simplified methodsof analysis of yielding structures with damping systems. Earthquake Spectra 2002; 18(3):531–547.

15. Pavlou EA, Constantinou MC. Response of elastic and inelastic structures with damping systems to near-fieldand soft-soil ground motions. Engineering Structures 2004; 26(9):1217–1230.

16. Guyader AC, Iwan WD. Determining equivalent linear parameters for use in a capacity spectrum method ofanalysis. Journal of Structural Engineering (ASCE) 2006; 132(1):59–67.

17. Newmark N, Rosenblueth E. Fundamentals of Earthquake Engineering. Prentice-Hall: London, 1971.18. Ramirez OM, Constantinou MC, Gomez JD, Whittaker AS, Chrysostomou CZ. Elastic and inelastic seismic

response of structure with damping systems. Earthquake Spectra 2002; 18(3):501–530.19. Lin YY, Chang KC. Effect of site classes on damping reduction factors. Journal of Structural Engineering

(ASCE) 2004; 130(11):1667–1675.20. Pekcan G, Mander JB, Chen SS. Design and retrofit methodology for buildings structures with supplemental energy

dissipating systems. Technical Report MCEER-99-0021, Multidisciplinary Center for Earthquake EngineeringResearch, University at Buffalo, State University of New York, Buffalo, NY, 1999.

21. Constantinou MC, Whittaker AS, Kalpakidis Y, Fenz DM, Warn GP. Performance of seismic isolation hardwareunder service and seismic loading. Technical Report MCEER-07-0012, Multidisciplinary Center for EarthquakeEngineering Research, University at Buffalo, State University of New York, Buffalo, NY, 2007.

22. Computers and Structures Inc. SAP2000: Static and Dynamic Finite Element Analysis of Structures (Version11.0.2) Analysis Reference Manual, Berkeley, CA, U.S.A., 2007.

23. Fenz DM, Constantinou MC. Development, implementation and verification of dynamic analysis models formulti-spherical sliding bearings. Technical Report MCEER-08-0018, Multidisciplinary Center for EarthquakeEngineering Research, University at Buffalo, State University of New York, Buffalo, NY, 2008.


Documents

Evaluation of simplified methods of analysis for structures with triple friction pendulum isolators