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

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

Evaluation of simplified methods of analysis for structures with triple friction pendulum isolators Fabio Fadi ‡ and 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.

SUMMARY Triple friction pendulum isolators, that exhibit behavior with amplitude-dependent strength and instantaneous stiffness, represent a new development in seismic isolation. The application of simplified methods of analysis for this type of seismically isolated structures requires development of tools of simplified analysis and demonstration of their accuracy. This paper describes these tools and presents validation studies based on a large number of nonlinear response history analysis results. It is shown that simplified methods of analysis systematically provide good and often conservative estimates of isolator displacement demands 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 with increasing amplitude of displacement. (The name ‘triple’ describes a behavior in which sliding occurs in no more than three sliding surfaces regardless of geometry and frictional values.) When properly configured, these isolators may result in reduced isolator displacement demands and/or reduced demands in force and acceleration in the primary and secondary structural systems in 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

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presented experimental data that validate the models and described models for analysis in commercial computer software. The application of these isolators requires also that simplified methods of analysis are developed and validated. Simplified methods of analysis are typically prescribed in applicable codes and specifications (e.g. ASCE [5] and AASHTO [6]) with the primary intent of serving as gauge or criteria for the results of nonlinear response history analysis that is most often performed for seismically isolated structures. The accuracy and limitations of the simplified analysis procedures are therefore of much importance. Simplified methods of analysis employ a single-degree-of-freedom system with equivalent linear elastic and viscous behavior to predict the displacement demand directly from prescribed smooth 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 displacement 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 all applicable codes and specifications for seismically isolated and damped structures in the United States since the first drafts in the late 1980s. As all simplified methods of analysis, the effective stiffness/damping method is an approximate method of analysis that requires validation on the basis of comparison of results of nonlinear response history analysis to predictions of the method for a wide range of nonlinear system behavior. This paper presents an evaluation of the capability of this simplified method of analysis to predict the displacement demand and peak velocity of isolation 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 method that 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, the method is conceptually simple and uncomplicated to apply. The method is based on the following steps: (a) represent the isolated structure by a single-degree-of-freedom system, (b) assume the peak isolator displacement, (c) construct the isolation system force–displacement loop at the assumed displacement, (d) calculate the effective stiffness and effective damping on the basis of the 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 period corresponding to the effective stiffness, (f) calculate the displacement demand as the spectral displacement 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 are sufficiently close. Upon calculation of the displacement demand, the maximum isolation system force is obtained directly from the force–displacement loop. The isolation system force (after division by an appropriate response modification factor) is then distributed over the height of the structure as inertia forces to calculate story shear forces for design. In the case of triple friction pendulum isolators, the construction of the force–displacement loop 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 by radii R1 , R2 , R3 and R4 (typically R1 = R4 and R2 = R3 ), heights h 1 , h 2 , h 3 and h 4 , displacement Copyright q

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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 all buildings and for most bridges) and that the coefficients of friction represent the weighted average values for the entire isolation system (defined as the sum of friction forces from all isolators divided by weight carried by the isolators). The lateral force–displacement relation of the isolation system is illustrated in Figure 2. Five different 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 and total weight W carried by the isolation system as described in Fenz and Constantinou [2]. Triple friction 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 displacement restrainer in earthquakes beyond the maximum considered. In regime V the isolator has consumed its displacement capacities d1 and d4 and only slides on surfaces 2 and 3 (see Figure 1). Copyright q

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The effective stiffness, effective period and effective damping are, respectively, defined as follows: Fmax D  1/2 W Teff = 2 gK eff

K eff =

eff =

EDC 2K eff D 2

(1) (2) (3)

where Fmax is the lateral force at displacement D and EDC is the energy dissipated in a cycle of harmonic motion at displacement amplitude D. Evaluation of EDC by analytical means is complex and it is best performed by first constructing the force–displacement relationship and then numerically evaluating the area enclosed by the loop. Upon calculation of the displacement in the simplified method of analysis, the peak velocity Vmax in the isolation system may be calculated as the pseudo-velocity multiplied by the velocity correction factor CFV as described in Pekcan et al. [20] and Ramirez et al. [14]   2 Vmax = · D ·CFV (4) Teff 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 Ramirez et 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 respect to 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 described in Fenz and Constantinou [2]. The value of Vmax is useful in establishing testing protocols for isolators, whereas values of the sliding velocities (after partitioning) are useful in estimating the temperature 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 the results of nonlinear response history analysis with results of the simplified analysis. The response history analysis is performed for a large number of ground motions that collectively (in an average sense) represent specific response spectra. Two different response spectra with corresponding two different sets of ground motions are used in this study. These spectra and sets of motion were previously used in studies of structures with damping systems by Tsopelas et al. [12] and Pavlou and Constantinou [15]. The first set of ground motions consists of the scaled horizontal components of 10 earthquake motions recorded on soft rock or stiff soil. Each of these earthquakes was selected to have a magnitude larger than 6.5, an epicentral distance between 10 and 20 km and a site condition according with the definition of Site Class C or D per ASCE [5]. The ground motions were scaled Copyright q

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

Station

Component

Scale factor

W. Washington (1949)

325 (USGS)

N04W N86E

2.74 2.74

Eureka (1954)

022 (USGS)

N11W N79E

1.74 1.74

San Fernando (1971)

241 (USGS)

N00W N90W S00W S90W

1.96 1.96 2.22 2.22

90 0 90 0

1.07 1.07 1.46 1.46

360 270 90 0

1.28 1.28 1.48 1.48

180 90 90 360

2.61 2.61 2.27 2.27

458 (USGS) Loma Prieta (1989)

Gilroy 2 Hollister

Landers (1992)

Yermo Joshua

Northridge (1994)

Moorpark Century

following the procedure described in Tsopelas et al. [12] to represent in an average sense a design response spectrum per ASCE [5] with parameters S D1 = 0.6, S DS = 1.0 and Ts = 0.6 s. Table I presents the ground motions components and the scale factors used. This set of motions is referred to as the SR (stiff soil or rock) motions. The second set of ground motions consists of the scaled horizontal components of seven soft-soil earthquake histories selected to have a site condition in accordance with the definition of Site Class E in ASCE [5]. The selected motions were scaled to represent in an average sense a design response spectrum per ASCE [5] with parameters S D1 = 0.75, S DS = 0.8 and Ts = 0.94 s. Table II presents the ground motions components and the scale factors used. This set of motions is referred to as the SS (soft soil) motions. Figure 3 presents a comparison of the average 5% damped spectra of the scaled ground motions and the target design spectrum. It should be noted that the average of the scaled motions represent well the target design spectra for periods generally larger than 1 s and 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 at period of 1 s equal to 0.6g (SR set) and 0.75g (SS set). Analyses were performed by further scaling the amplitude of the motions so that the one-second average 5% damped spectral acceleration values were in the range of 0.2 to 1.6g with increments of 0.2g. This allowed analysis of structures with triple friction pendulum isolators that operated in all possible regimes of displacement as shown in Figure 2 and provided a more comprehensive set of data for evaluation. It should be noted that the scaled SR and SS sets have the same target spectral shapes in the long period range for the same value of spectral acceleration at 1 s. However, the two sets of scaled motions differ Copyright q

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

Station

Northridge (1994)

Sylmar

Loma Prieta (1989)

Gilroy 3 Hollister and Pine

Landers (1992)

Barstow-Vineyard Amboy Hot Springs

Southern Alaska (1979)

Yakutat

Component

Scale factor

90 360 90 0 90 0 90 0 90 0 90 0 09 279

0.99 0.59 1.74 2.20 2.46 1.21 3.03 3.46 4.26 3.79 4.01 3.65 3.20 2.45

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

4. DESCRIPTION OF ISOLATION SYSTEM The isolation system considered in this study had three different geometrical configurations and three different sets of frictional properties. The three geometrical configurations are presented in Table III. The three configurations primarily differ in the displacement capacities, which respectively are 0.48, 1.05 and 1.52 m, and the effective radii of curvature. The effective radius is defined as the difference between the radius of curvature of the relevant spherical surface and the corresponding portion of the height of the slider, Reffi = Ri −h i [2]. Three different sets of friction coefficients are assumed for each of the three geometrical configurations. The first and second sets are characterized by unequal friction values on the two main 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 two main sliding interfaces (referred to as EF). All three sets have the same friction coefficient for the contact surfaces between the rigid slider and the internal plates. Table IV presents the frictional properties 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 the EF configurations, the five regimes collapse to three). Table V presents values of displacement and lateral force (normalized by weight) at each of the transition points between regimes (I to II, II to III, III to IV and IV to V). The nomenclature used for forces and displacements at the transition points and the calculation of their values was based on Fenz and Constantinou [2]. It should be noted that the loops in Figure 4 are shown to extend into displacement regime V although triple friction pendulum isolators are not typically designed to operate in that regime. Copyright q

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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 Designation GC1 GC2 GC3

Copyright q

Effective radii

d2 = d3 (m)

d1 = d4 (m)

dTOT (m)

Reff2 = Reff3 (m)

Reff1 = Reff4 (m)

0.064 0.125 0.178

0.178 0.400 0.584

0.484 1.050 1.524

0.356 0.300 0.902

1.499 2.085 3.823

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Table IV. Friction coefficient values. Designation LF HF EF

Surface 1

Surface 2

Surface 3

Surface 4

0.04 0.07 0.07

0.02 0.02 0.02

0.02 0.02 0.02

0.10 0.14 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 the triple 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 series together with gap elements. The model is capable of capturing the behavior of triple friction Copyright q

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Table V. Displacements and lateral forces at transition points between regimes. Transition displacement (m) Configuration GC1-LF GC1-HF GC1-EF GC2-LF GC2-HF GC2-EF GC3-LF GC3-HF GC3-EF

Horizontal force/Weight

u∗

u ∗∗

u dr 1

u dr 4

F ∗ /W

F ∗∗ /W

Fdr 1 /W

Fdr 4 /W

0.014 0.036 0.036 0.012 0.030 0.030 0.036 0.090 0.090

0.123 0.166 0.036 0.155 0.197 0.030 0.320 0.421 0.090

0.302 0.312 0.392 0.705 0.705 0.830 1.029 1.054 1.258

0.413 0.441 0.392 0.848 0.872 0.830 1.312 1.385 1.258

0.04 0.07 0.07 0.04 0.07 0.07 0.04 0.07 0.07

0.10 0.14 0.07 0.10 0.14 0.07 0.10 0.14 0.07

0.16 0.19 0.19 0.23 0.26 0.26 0.19 0.22 0.22

0.22 0.26 0.19 0.29 0.33 0.26 0.25 0.29 0.22

pendulum isolators in all five regimes of displacement. The analysis included only horizontal components of seismic excitation because experimental studies with vertical excitation reported by Morgan [1] and Fenz and Constantinou [23] concluded that the vertical excitation does not have effects that warrant consideration in the prediction of the isolator displacement response of structures with triple friction pendulum isolators. However, the vertical excitation may have effects on the isolation system forces that warrant consideration. Figures 5–7 present a comparison of peak isolator displacement as calculated by simplified analysis and by nonlinear response history analysis (the latter as average of the peak displacement calculated for each of the motions). For the simplified analysis, the average spectra of the scaled motions were utilized so that direct comparisons to results of response history analysis could be made. Each graph in Figures 5–7 corresponds to one of the nine configurations analyzed, the force–displacement characteristics of which are described in Figure 4 and Table V. Also, each graph identifies the transition displacements (see Table V) so that the reader can easily identify the regime of displacement in which the isolators operated. The results are presented with the average of nonlinear response history analysis plotted on the vertical axis and the results of the simplified analysis plotted on the horizontal axis. Each graph includes the results of the analysis for the SR and SS sets of motions as scaled to represent a range of values of spectral acceleration at a period of 1 s. It should be noted that (a) the results of the response history analysis differ between the two cases of motions due to differences in the details of the scaled motions, and (b) the results of the simplified analysis differ between the two cases of motions due to differences in the average response spectra (whereas the target spectra are identical for the two cases and for the same spectral acceleration at 1 second). In order to discuss the results of analysis, Table VI was prepared. It presents 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 to the various levels of seismic excitation considered. The first value in each range corresponds to the 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 cases identified in the table correspond to operation of the isolators well within the displacement regime V , where significant stiffening occurred. The results of Figure 5–7 demonstrate a systematically good and often conservative estimation of displacement demand by the simplified method of analysis. The most conservative estimation of Copyright q

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

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Figure 6. Comparison of response history (RHA) and simplified analysis (SA) results on displacements for system GC2.

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Figure 7. Comparison of response history (RHA) and simplified analysis (SA) results on displacements for system GC3.

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Table VI. Effective period and effective damping calculated in simplified analysis. Effective period (s) Configuration GC1-LF GC1-HF GC1-EF GC2-LF GC2-HF GC2-EF GC3-LF GC3-HF GC3-EF

Effective damping (%)

Motions SR

Motions SS

Motions SR

Motions SS

1.87–2.76 2.05–2.62 1.67–2.84 1.84–3.45 2.47–3.21 2.75–3.54 3.55–4.59 3.06–4.09 3.21–5.54

1.86–2.70 2.07–2.53 1.60–2.69 2.97–2.80∗ 2.50–3.27 2.90–2.79∗ 3.70–4.60 3.27–4.35 3.49–4.71

30–17 23–18 27–20 23–14 25–21 32–16 21–17 25–22 31–20

31–14 23–15 25–15 23–7 25–18 29–7 21–15 23–20 31–17

∗ Cases of operation in displacement regime V.

displacement demand by the simplified method occurred in cases in which the isolators operated well 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 the simplified 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 that in 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 to the average results of response history analysis. Evidently, the simplified method underpredicts the peak velocity. Among the 90 cases analyzed, the greatest under-estimation of velocity occurred for system GC3 and the cases of strongest excitation where the simplified method under-predicted the peak velocity by as much as 35%. In general, the percentage amount of under-estimation of the peak velocity by the simplified method increased with increasing effective period of the isolation system. In the case of the aforementioned systems GC3 with greatest error in the prediction of velocity, the effective period exceeded 4 s. It should be noted that codes and standards such as ASCE [5] would not allow the use of simplified methods of analysis under the conditions in which either overestimation of displacement occurred (significant stiffening of isolators, very strong excitation) or when underestimation of velocity occurred (effective period exceeding 3 s). Rather, nonlinear response history analysis would be required but subject to limits on the calculated response that are related to the results obtained in simplified analysis. Therefore, the results of simplified analysis are important even when response history analysis is used and the aforementioned limits on response should be revisited in the light of 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 structures with triple friction pendulum seismic isolators, a system that exhibits complex behavior with amplitude dependent stiffness and strength. The simplified analysis method investigated is the one typically described in codes and specifications. The method is based on modeling the isolated structure as a single-degree-of-freedom system with linear stiffness and linear viscous damping Copyright q

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

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Figure 9. Comparison of response history (RHA) and simplified analysis (SA) results on velocities for system GC2.

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Figure 10. Comparison of response history (RHA) and simplified analysis (SA) results on velocities for system GC3.

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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 of simplified analysis using two sets of ground motions that were scaled to match design spectra in the long period range and then scaled to represent earthquakes of different intensities. Nine different configurations were analyzed, each one of which operated in all possible displacement regimes of the triple friction pendulum isolator. The study led to the conclusion that the simplified method of 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 of underestimation of the peak velocity increased with increasing effective period of the isolation system. The greatest overestimation of displacement by the simplified method occurred in cases of operation of the isolators in regime V, where significant stiffening occurs. When the simplified method significantly overestimated (by about 35%) the displacement or significantly underestimated (by about 35%) the velocity, the conditions were such that the simplified methods of analysis would 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.

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