torsion in base-isolated structures with elastomeric ...nagaraja/j4.pdf · tistory base-isolated...

T O R S I O N IN B A S E - I S O L A T E D S T R U C T U R E S W I T H

E L A S T O M E R I C I S O L A T I O N S Y S T E M S

By Satish Nagarajaiah, 1 Associate Member, ASCE, Andrei M. Reinhorn, 2 Member, ASCE, and Michalakis C. Constantinou, 3

Associate Member, ASCE

ABSTRACT" Torsion in base-isolated structures with inelastic elastomeric isolation systems due to bidirectional lateral ground motion is studied. In a companion paper by the writers, torsional coupling in sliding base-isolated structures was investigated. In this paper, which is the second part of the sequence, torsional coupling in elastomeric base-isolated structures is investigated. Various multistoried structural systems with elastomeric isolation systems are investigated, with the objective of studying the influence of: (1) The flexibility of the superstructure; (2) the ratio of uncoupled torsional to lateral frequencies; (3) stiffness eccentricity in the superstructure; (4) eccentricity in the isolation system; (5) higher mode effects; and (6) number of bearings in the isolation system. Response to different ground motions is also studied. The results are used to explain: (1) The behavior of actual buildings; and (2) some inconsistencies in the conclusions of previous studies. It is shown that, although the total superstructure response is reduced significantly due to the effects of elastomeric base isolation, torsional amplification can be significant depending on the isolation and superstructure eccentricity and the lateral and torsional flexibility.

INTRODUCTION

The essential features of an elastomeric isolation system are its lateral flexibility and energy-dissipat ion capacity. The flexibility of the isolation system increases the fundamenta l per iod of the structure, shifting it out of the region of dominant ea r thquake energy. The energy-dissipat ion capacity increases damping, which reduces excessive displacements due to the laterally flexible isolation system. The components of an elastomeric isolation system include: (1) Lamina ted - rubber bearings, re inforced with steel plates, which support the vertical load and provide the lateral flexibility; and (2) dampers that provide energy-dissipat ion capacity. The inherent damping capacity of the rubber in h igh-damping rubber bearings (Kelly 1986) or the damping capacity of the lead-core in lead-rubber bearings (Buckle and Mayes 1990), can also be rel ied upon for energy dissipation.

Coupled lateral- torsional motions occur in base- isolated structures with elastomeric isolation systems, when subjec ted to la teral ground motion, if an eccentricity exists be tween the center of stiffness (CS) and the center of mass (CM) of the superst ructure , as in nonisolated structures (Har t and DiJulio 1975; Kan and Chopra 1981; Reinhorn et al. 1977; Sadek and Tso 1988; Shepherd and Dona ld 1967). Torsional coupling is also possible if such an eccentricity is present in the isolation system (Eisenberger and Rutenberg 1986; Lee 1980; Pan and Kelly 1983).

Several earl ier investigations have addressed torsional coupling in base- isolated structures with e las tomeric isolat ion systems. The results of these

1Asst. Prof., Dept. of Civ. Engrg., Univ. of Missouri, Columbia, Missouri 65211. 2prof., Dept. of Cir. Engrg., State Univ. of New York, Buffalo, NY 14620. 3Assoc. Prof., Dept. of Civ. Engrg., State Univ. of New York, Buffalo, NY. Note. Discussion open until March 1, 1994. To extend the closing date one month,

a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on August 7, 1992. This paper is part of the Journal of Structural Engineering, Vol. 119, No. 10, October, 1993. �9 ISSN 0733-9445/93/0010-2932/$1.00 + $.15 per page. Paper No. 4567.

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earlier investigations are significant for understanding torsional coupling. However, their conclusions may be restricted to the particular systems an- alyzed, and may not be valid in general, which is evident from the following review of previous studies.

Lee (1980) concluded that the dynamic torque amplification (i.e., ratio of dynamic torque to static torque at the CS of the superstructure) is small or negligible, and that the additional corner displacements from the base rotation remain small (within 30% of the lateral displacement at the center of the base), provided the eccentricity of the isolation system is small ( -0 .2L) - -even if the superstructure eccentricity is large (-<0.4L). These basic conclusions were based on the study of single-story structures, having four corner columns and elastomeric isolators, with masses concentrated at the same corners. The model used had excessively large radius of gyration of mass (0.707L; L = longest plan dimension), and the isolators consisted of two laterally independent bilinear hysteretic springs, thus neglecting the nonlinear biaxial interaction between the two directions.

Pan and Kelly (1983) concluded that the effect of torsional coupling on the seismic response of base-isolated structures with small eccentricities is insignificant, due to the combined effect of the time lag between the maximum lateral and torsional response and the influence of the high damping in the isolation system, by studying a rigid-superstructure model that had linear isolators with equivalent isolation damping of 8-10%.

Eisenberger and Rutenberg (1986) concluded that when the eccentricity in the isolation system is small or zero, the torsional response is virtually eliminated, even for large superstructure eccentricities (0.16L). These conclusions were arrived at by reexamining the validity of the findings in both the aforementioned studies, using an equivalent 3D-analysis of multistoried structures with bilinear isolators without biaxial interaction.

Nakamura et al. (1988) concluded that amplification of torsional motions is small or negligible in elastomeric base-isolated structures, for moderate superstructure and isolation system eccentricities (0.1L). The conclusion was based on an experimental study of a single-story torsionally stiff base- isolated structure (ratio of uncoupled torsional to lateral frequency = 1.7) with elastomeric isolation system. However, the preceding conclusion is restricted to torsionally stiff base-isolated structures only.

The torsional behavior of Foothill Communities Law and Justice Center (FCLJC) in California, the first base-isolated building built in the U.S., was studied by Tarics et al. (1984) using 3D nonlinear dynamic analysis. The supe r s t ruc tu re eccen t r i c i ty in the s t ruc tu re is m o d e r a t e at 0 .07L (Papageorgiou and Lin 1989), and the uncoupled torsional frequency is nearly equal to the uncoupled lateral frequency. The results of the analysis (Tarics et al. 1984) indicated that in some cases the additional corner displacment at the base due to rotation is almost equal to the lateral displacement at the center of the base, i.e., corner displacement was nearly twice that at the center of the base. The preceding results contradict the conclusions of all the previous studies. Furthermore, Papageorgiou and Lin (1989) confirmed through the observed response of FCLJC during an earthquake that torsional coupling is a true feature of the structure.

In a recent study by the writers (Nagarajaiah et al. 1993) on torsional coupling of sliding-isolated structures, it was concluded that the superstructure eccentricity has significant influence on torsional amplification, and it was also concluded that dynamic torque amplification could be as high as 4.0 for superstructure eccentricities -0 .1 L.

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In the preceding review, the differences in results indicate that the conclusions of aforementioned studies are not applicable in general, but are restricted to the particular system considered and the underlying modeling assumptions. Hence, there is a need for a more comprehensive and system- atic investigation.

In a companion paper (Nagarajaiah et al. 1993), torsional coupling in sliding-isolated structures was investigated. In this paper, which is the seconff part of the sequence, the influence of various system parameters on the lateral torsional response of base-isolated structures with elastomeric systems is investigated systematically. An attempt is made to explain: (1) The behavior of actual buildings such as FCLJC; and (2) some inconsistencies in the conclusions of previous studies. The lateral torsional response of elastic multistoried structural systems with inelastic elastomeric isolators is studied to determine the influence of: (1) The flexibility of the superstructure; (2) the ratio of uncoupled torsional to lateral frequencies; (3) the stiffness eccentricity in the superstructure; (4) the eccentricity in the isolation system; (5) the higher mode effects; and (6) the effect of number of bearings in the isolation system. Response to various types of ground motions is studied. The important effect of biaxial interaction is considered in the models of elastomeric isolators.

INELASTIC DYNAMIC ANALYSIS

The dynamic analysis for the study reported in this paper was carried out using a new computer program 3D-BASIS (Nagarajaiah et al. 1991) de- veloped specifically for nonlinear dynamic analysis of asymmetric plan multistory base-isolated structures with elastomeric or sliding isolation systems. The analytical models and solution algorithm in 3D-BASIS were verified by comparison with experimental results (Nagarajaiah et al. 1991). Fig. 1 shows a typical base-isolated structure and its asymmetric plan. Fig. l(b) shows the cross section of a typical laminated-rubber bearing and a steel damper. The analytical model for elastomeric bearings and steel dampers used in 3D-BASIS is presented in the following section.

Biaxial Model for Eiastomeric Bearings/Dampers in 3D-BASIS Forces mobilized in an elastomeric bearing/damper with biaxial interac-

tion are

F Y Fx = o t - ~ U * + (1 - oOFYZx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

F Y Fy = o~ - f Uy + (1 - oOFrZy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

in which e~ = postyielding to preyielding stiffness ratio; F r = yield force; Y = yield displacement; U* and U~ = displacements at the bearing in the X and Y direction; and Zx and Zy = hysteretic dimensionless quantities that account for the direction and the biaxial interaction of hysteretic forces. The hysteretic dimensionless quantities Zs and Zy are governed by the following coupled differential equations (Park et al. 1986):

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Y

STEEL ROD DAMPER

U~ t ~ _ _ _ _ - - . ~ S U P E R ~

X

BASE

LAMINATED " i RUBBER BEARING

i

FOUNDATION

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(o)

(b)

i ~Center of Geometric - , i kfffness (C c

- - J . _ _

o m ~ T . . . . Center

of mass (C~,) I . r b -i

~.~-..-~ X (c)

FIG. 1. Structural System Considered: (a) Base-Isolated Multistory Structure; (b) Section of Elastomeric Bearing and Steel Damper; (c) Floor Plan of Multistory Structure

Z~2[~ Sign(/)'*Z~) + 13] ZxZy['y Sign(OyZy)+ 1311// O~ (3) -[ZxZy['y Sign(6"~*Zx) + 13] Zy2[~t Sign(UyZy) + 13] ~ ~xUy'T~) . . . .

in which A, 13, 7 = dimensionless quantities. The values of A = 1, t3 = 0.1, and ",/ = 0.9 are used to simulate the behavior of elastomeric bearings and steel dampers in this paper. Solution of (3) along with (1) and (2)

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L• Ste~L rod damper

7 ~ P :7~~Defo rmat ion (a)

E IJ

OJ

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~23

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Disptacement (cm) ~

- - Disptacement ( c m ) -

FIG. 2. Steel Damper Cosidered: (a) Section of Steel Damper; (b) Measured Biaxial Force-Displacement Response of Steel Damper; (c) Simulated Biaxial Force-Dis- placement Response of Steel Damper

enables determination of the forces in each bearing/damper. The validity of the biaxial model presented is verified by comparison with the experimental results of tests on steel-rod dampers under bidirectional motion (Yasaka et al. 1988a). The tested cantilever steel-rod damper, shown in Fig. 2(a), is a scaled model of the steel dampers used in the isolation system of an actual building of Kajima Corp., Japan (Yasaka et al. 1988b). The isolation system of the Kajima building consisted of laminated-rubber bearings and steel-rod dampers. The laminated-rubber bearings were also tested and were found to be elastic. The combined behavior, i.e., the elastic behavior of laminated-rubber bearings and the hysteretic behavior of steel dampers used in Kajima building (Yasaka et al. 1988b) can be represented by the smooth bilinear model presented. The experimental and simulated results from Nagarajaiah et al. (1991) shown in Figs. 2(b) and 2(c) indicate good agreement. Note the significant interaction between forces in the X and Y directions. If biaxial interaction was neglected, the shape of both X and Y direction force-displacement loops would be similar to the Y direction loops shown in Figs. 2(b) and 2(c) or similar to uuiaxial loops.

MULTISTORIED ASYMMETRIC STRUCTURAL SYSTEMS

A series of multistoried 3D-shear buildings are considered in this study. The superstructure is modeled to be elastic at all times while the isolation system exhibits inelastic behavior. The elastomeric isolation system is considered to be located between the base slab and the foundation. Three

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master degrees of freedom (two translational and one torsional degree of freedom) attached to the CM of the floors and the base are considered [see Fig. l(a)]. The floors and the base are assumed to be rigid. The CM of all the floors and the base are assumed to be on the same vertical axis, which is also the reference axis [see Fig. l(a)]. A typical asymmetric floor plan is shown in Fig. 1(c), and is assumed identical for all the floors. A variety of building models can be simulated by changing the mass and stiffness distribution. Torsional coupling is obtained when the distribution of stiffness or mass is eccentric. When a mass offset is considered, the entire vertical axis of the CM of the floors and the base is assumed to be offset with respect to the geometric center (GC) of the building [see Fig. l(c)]. Henceforth, such an offset is referred to as "mass offset" denoted by Ore. T h e mass offset Om represents asymmetric mass distribution with respect to the GC of the structure and the isolation system. The stiffness eccentricity in the superstructure, es, represents the distance measured from the vertical axis of the CM to the vertical axis of the CS along the X or Y axis.

The strength distribution in the isolation system is represented by the center of strength (CYS) (Eisenberger and Rutenberg 1986; Goel and Cho- pra 1990; Sadek and Tso 1988). The CYS is defined as the location of the resultant of yield forces of the bearings. A strength eccentricity or isolation- system eccentricity is defined as the distance between the CM of the base and the CYS of the isolation system, and is denoted by eb. For the sake of simplicity, mass offset and strength eccentricity are considered only in the Y direction [see Fig. l(c)] in this study. The strength eccentricity eb due to the mass offset in the Y direction is defined as follows:

1 eb = i f ; ~ , y j f y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

J

where FY = resultant of yield strength of all bearings in the X direction; f~ = yield strength o f j t h bearing in the X direction; and yj = distance of jth bearing from the CM of base in the Y direction.

The uncoupled fundamental lateral periods of the superstructure in fixed- base condition (i.e., preisolation periods) in the X and Y directions are considered to be identical throughout the study and are designated by Ts and the corresponding circular frequencies by cos. The uncoupled fundamental torsional frequency is designated by 0%,. The lateral and the torsional fequencies to s and COos for a single-story structure (one of the cases studied in this paper) are defined as follows:

, o , = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( . 5 )

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (61 tOos = ~ m r 2

where K, = lateral stiffness in the X or Y direction; K0s = torsional stiffness about the CM (i.e., at the CS of an equivalent uncoupled system with es = 0) of the fixed-base superstructure; m = mass of the superstructure; and r = mass radius of gyration of the floor about the CM. The ratio of torsional- to-lateral frequencies is defined as follows:

Oo, coo, (7) to s

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In the present study, an elastomeric isolation system consisting of laminated-rubber bearings and steel dampers is considered. Such an elastomeric isolation system has been used in an existing building of Kajima Corp., Japan (Yasaka et al. 1988b). This system is representative of most elastomeric systems used in practice (Buckle et al. 1990; Kelly 1986; Roeder et al. 1983; Yasaka et al. 1988b). It is assumed that the laminated-rubber bearing and the steel damper are placed side-by-side, i.e., virtually concentric. The dynamic characteristics of the isolation system are defined by assuming a rigid superstructure above the isolators. The uncoupled lateral periods of the isolation system, calculated considering only the postyielding stiffness of the isolation system, are considered to be identical in both X and Y directions throughout the study. These periods will be designated by Tb and the corresponding frequencies by O~b. The uncoupled torsional frequency of the base is designated by tOob. The lateral frequency ~Ob, the torsional frequency O00b, and the ratio of torsional-to-lateral frequencies of the base Oob are defined in an analogous manner as %, ~o0s, and f~0~. Var- iations in torsional frequency CO0b are obtained by varying the torsional stiffness of the isolation system, assuming that the mass distribution of the structure remains constant. At the same time, for a given number of isolators having a fixed resultant postyielding stiffness Kb and lateral base period Tb, the torsional stiffness gob varies as a function of the distance of the bearings from the CM.

The inelastic lateral-torsional response of a base-isolated structure with elastomeric isolation system--with arbitrary location, number, and stiffness of bearings--depends mainly on the uncoupled period Ts, Tb, on the torsional to lateral frequency ratios 120s, ~Ob, on the normalized superstructure stiffness eccentricity e Jr, and on the normalized strength eccentricity eb/r of the isolation system. The important parameters just mentioned charac- terize best the inelastic lateral-torsional response and allow for a meaningful evaluation of the influence of system parameters on the torsional coupling. Hence the range of these parameters was carefully selected to obtain a better understanding of the influence of the system parameters on torsional coupling. The selection covers a wide spectrum of base-isolated structures with elastomeric isolation systems.

STRUCTURAL SYSTEMS FOR PARAMETRIC STUDY

Multistory structural systems, consisting of several parallel frames with and without eccentric stiffness distribution are considered [see Fig. 1(c)]. The CM, the CS, and the GC of the floors are coincident for the symmetric systems considered, and are not coincident for the asymmetric systems considered. Laminated-rubber bearings and steel dampers [shown in Fig. l(b)] are used in the elastomeric isolation system. Five structural configurations with two isolation variations are considered. The five structural configurations include 10, eight-, five-, two-, and one-story 3D-shear building systems. The two elastomeric isolation systems considered are: (1) A system with 36 bearings with dampers; and (2) a system with four bearings with dampers.

The design of the bilinear elastomeric isolation system is based on pro- visions of the Uniform Building Code (1991) and on ground motions with characteristics of the ATC 0.4 g $2 spectrum. The laminated-rubber bearings and steel-rod dampers were designed such that the cumulative effect in each lateral direction has smooth bilinear hysteretic properties with: (1) Elastic s t i f f n e s s K 1 = 5.84W/m; (2) postyielding stiffness K2 = 0.9W/m; and (3)

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yield force Fe = 0.055W, where W -- total weight of the structure. The rigid-body base period rib = 2.12 S was chosen since it is a typical value of the base period used in base-isolated structures, such as FCLJC (Tarics et al. 1984), and because it is the value used by previous investigators (Nakamura et al. 1988; Pan et al. 1983).

The total weight W of the structure is assumed to be the same in all cases. A variable mass offset is created by shifting a portion of the mass in the Y direction. Hence, both mass moment of inertia and the radius of gyration r vary. The mass is maintained symmetric about the Y axis. Three positions of the eccentric mass are selected leading to mass offsets [see Fig. 1(c)] o,~/L = 0.0, 0.083, and 0.21. The corresponding mass radii of gyration at all floors and at the base are r/L = 0.408, 0.4, and 0.422. The strength and stiffness distribution of the isolation system are chosen such that the CYS and the CS of the isolation system are located, concentrically along the Y axis, between the CM and the GC. Concentric CYS and CS were chosen to verify the conclusion of Eisenberger et al. (1986) who stated that this combination with eb/L = 0 virtually eliminates torque. The strength eccentricities chosen are eb/L = 0, 0.065, and 0.13 corresponding to the mass offset om/L --- 0, 0.083, and 0.21. The aforementioned values of strength eccentricity were chosen as they best reflect the strength and stiffness characteristics of elastomeric isolation systems. Identical superstructure eccentricities es are employed in both X and Y directions [see Fig. l(c)] unless mentioned. In the following parametric study, the superstructure eccentricity is varied from es/r = 0 (no eccentricity) to es/r = 0.5 (which is 21% of the length L of the floor plan).

GROUND MOTIONS FOR PARAMETRIC STUDY

Several typical earthquakes with various frequency contents were used in this study: however, results of only two types of earthquakes are presented. The two earthquakes selected are 1952 Taft earthquake and 1940 E1 Centro earthquake. The Taft earthquake was chosen because it was used as one of the design earthquakes for an actual building (Tarics et al. 1984). The E1 Centro earthquake was chosen because it has been used widely in previous investigations of torsional response of base-isolated structures (Ei- senberger et al. 1986; Lee 1980; Nakamura et al. 1988; Pan et al. 1983).

EFFECT OF NUMBER OF BEARINGS

The importance of the isolation system composition is determined first. The two configurations previously described of 36 and four bearings with dampers supporting a 3D-shear building structural system, with identical system parameters, provide a lower and an upper limit for a realistic as- sessment. For the system with four bearings, the total mass is scaled by a factor of nine so that bearings and dampers would be identical to the ones used in the 36 bearing system. The 3D-shear building structural systems considered are 10, eight, five, and two stories, with a damping ratio of 2% in the first three modes, and 5% in all the higher modes. Identical floor masses are chosen such that the total mass of the floors is four times the mass of the base. In addition, a series of single-story 3D-shear buildings are selected with periods identical to the first three modes of the aforementioned multistory configurations, and with a floor mass equal to the total floor mass of the superstructure they represent. The system configurations considered also allow for a simultaneous investigation of the contribution of

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higher modes of vibration to the response of base-isolated structures with elastomeric isolation systems.

Three combinations of structure-isolation systems are considered in the parametric study: (1) Multistory structures with isolation system of 36 bearings with dampers (36 BMS); (2) single-story structures with isolation system of 36 bearings with dampers (36 BSS); (3) single-story structures with isolation system of four bearings with dampers (4 BSS). The values of parameters Tb, e,/r, eb/r , ~ Q o s , and fl0b are considered to be the same in all the systems, with ratio L/b = 1. The mass of the structure is considered to be eccentric, hence the isolation eccentricity eb/L = 0.065. The fundamental periods T, of the selected structures (i.e., preisolation periods) are 0.3 s, 0.6 s, 0.9 s, and 1.2 s for the two-, five-, eight-, and 10-storied structures, respectively. The torsional-to-lateral frequency ratio of the superstructure 120,, is selected as 1.0 to increase the chance for strong torsional coupling. The same ratio for the base l)0b is selected to be 1.7. This is obtained in the case with 36 bearings, by placing the bearings farther from the CM along the X or Y axis when compared to the case with four bearings. Influences of several ground motions were studied; the 1940 El Centro earthquake excitation produced the maximum differences in the response of different systems previously described.

The peak response values at the CM, for varying supserstructure period Ts and eccentricity e,/r = 0.25, are shown in Fig. 3. The peak response quantities presented are: (1) The structural shears Fxs, Fy,, and the torque T, at the first-story level above the base; (2) the displacements Ux~, Uys and the rotation U0, of the top floor, relative to the base; and (3) the displacements Uxb, Uyb and the rotation Uob of the base, relative to the ground. The preceding peak values are normalized with respect to either the peak ground displacement Ugo (4.29 in.-108.96 ram; for E1 Centro S00E component) or the total weight W and length L. The analysis results (see Fig. 3) show that the single-story structure with either four or 36 isolators (i.e., 4BSS or 36BSS) responds almost identically in both translation and rotation. The same effect as previously described is observed in Fig. 4 for the case of 4BSS and 36BSS, wherein the superstructure eccentricity is varied from a symmetric case to asymmetric case of es/r = 0.5, with a constant superstructure period Ts = 1.2 s and eb/L = 0. It can therefore be concluded that the number of bearings do not affect the torsional and lateral response for a given set of structural parameters T~, Tb, flo~, ~ob, es/r, eb/r, r/L, and L/b.

EFFECT OF HIGHER MODES

Same analysis as previously described is used to emphasize the difference between the multistory structure analysis and single-story structure analysis. Since the analysis is done by direct integration of the equations of motions, the response obtained in Figs. 3 and 4 for multistoried structures (case 36 BMS) includes implicitly the influence of higher modes of vibrations. Com- paring their response with those of single-story structures (case 36BSS), it can be observed that differences appear (see Fig. 3 and 4). Since the differences are not significant, further studies of significant parameters will be done on equivalent single-story systems.

EFFECT OF ISOLATION-SYSTEM ECCENTRICITY

The isolation-system eccentricity eb/r is one of the most important parameters that causes torsional coupling and torsional motions (Eisenberger

2940

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gs

et al. 1986; Lee 1980; Pan et al. 1983). The effect of eccentricity %/r in the presence of a variety of system parameters is examined in this section.

The range of system parameters chosen for study in this section are such that they include the set for FCLJC building (Tarics et al. 1984) also. The parameters are the superstructure frequency ratio ~0s = 0.8, 1.0, 1.7; the base frequency ratio l)0b = 0.8, 1.0, 1.7; the uniaxial superstructure eccentricities e,y/L = 0.072; e J L = 0 in the Y and X directions, respectively; the superstructure period Ts = 0.5 s; the base period Tb = 2.12 s; and the ratio L/b = 4. The system parameters of FCLJC, from Tarics et al. (1984) and Papageorgiou et al. (1989), are: Ts = 0.51 s; Tb = 2 s; [I0s ~ 1.2; f~0a < 1: eJL = 0.072 (only in the direction of longer plan dimension); and L/b = 4.

The values of uniaxial isolation eccentricity chosen are e j L = 0, 0.065, and 0.13; the corresponding maximum corner distance from the CM along the X or Y axis a/r = 1.23, 1.46, and 1.69. The results for the case of Taft N21E excitation in the X direction, with the peak ground acceleration (PGA) scaled to 300% of original PGA, are presented in Fig. 5. The results shown in Fig. 5 are: the ratio of peak corner-base displacement to the peak base displacement at the CM (Ucb); the ratio of the peak corner-structure displacement (relative to base) to the peak structure displacement at the CM (relative to base) (Ucs) and the ratio of the dynamic torque to the static torque at the CS of the superstructure (Tamp). The static torque used for the computation of the factor T~mp is Fx,e,y, where Fx, is the peak structural shear force obtained in the dynamic analysis. The magnitude of factor T, rnp is a good indication of the extent of torsional amplification.

The dynamic torque amplification T~mp of the elastomeric isolated structures, shown in Fig. 5(a) increases with increasing isolation eccentricity eJL, and has significant values - 5 for large isolation eccentricities. This contradicts the conclusions of Lee (1980), who pointed out that the torque- amplification factor Tamp in elastomeric base-isolated structures is close to unity, unlike nonisolated structures. The order of magnitude of the amplification factor Tamp - 5, indicates that in some cases elastomeric isolated structures amplify the "static torque" no less than the nonisolated structures. However, the structure shear and torque generated in an isolated structure with elastomeric bearings is less than that of the fixed-base structure. It is evident from Fig. 5(a) that an "accidental or nominal isolation eccentricity," eJL = 0.05, can result in significant torque amplification Tamp - 3.0. Fur- thermore, the base frequency ratio f~0b has a significant influence on the torque amplification T, mp, as evident in Fig. 5(b).

The translational motion at the corner of the superstructure is larger than the one at the CM, since the torsional coupling is involved. The corner- displacement magnification Ucs is a function of the isolation eccentricity eb/L and the torsional-to-lateral frequency ratio f~0~, as shown in Fig. 5(c). The corner-displacement magnification Uc~ increases with increasing eJL [Fig. 5(c)], and increases with decreasing 110, [Fig. 5(d)].

The torsional motions at the isolation level are influenced by the isolation eccentricity eJL and the base frequency ratio O0b- The corner-base-displacement magnification O~b, increases with an increase in the isolation eccentricity eJL, and decreases with increasing frequency ratio Oob, as evident from Fig. 5(e) (the results of ratio f~0b = 1 are omitted for clarity of the figure) and Fig. 5(f).

The results of the study by Tarics et al. (1984) for FCLJC building are examined in light of the aforementioned and conclusions. The corner-dis-

2943

5.0

4.5

4.0 3.5

o . .

E 3.0

2.5

2.0 1.5

(0) 1.0

4.0

TAFT 300S EIASTOMERIC ISOLATION SYSTEM - - 5.0

r !

, / / ~ E 3.5 2 , ' / / - , / / / - - - - r ~.o ~ " - - - - L % s = 0 .8

- , / / / . . . . Qes = 1.0 2.5 - - - Qes = 1.7

' I i I L I 2.0 0.00 0.05 0.10 0.15

eb/L

/ " -- e~.= 0.13 2

_ _ ~ A h = 0 . 8 " " - .

~e~1.7 i I t I

0.5 1.0 1,5 ~es

3.5

3.0 t n I ~ 2.5

2.0

1.5

(el 1.0

4.0

~eb:l. 0 / 3.5

3.0

2.0

r . . . . . . . '~

I t 1 I I 1.0 i.oo 0.05 o.lo 0.15

eb/L

', eb/~_O.]3

- - ~ o b : O . 8 "% 9eb:~ o "X-.

I l i I 0.5 1.0 1.5

~es

i (b) 2.0

, i ( d ) 2.0

~.8 - ~ Qes=08 . . . . . Qe# 1.0 /

1.8 - - - Q e ~ l T z , ~

~eb=CL8 i # TM

1.2 : ~

(e} 1.o ~ 1 , I 0.00 0.05 0.10

el7 L I

0.15

1.6

r

[S_._.__~ 1.4

1.2

1,0

X eb/~0"13

- - Qe~0.8 Qe{ 1.0 Qe{ 1.7

, I , j , ~ ( f ] 0,5 1,0 1.5 9.0

Qeb

FIG. 5. C o r n e r - D i s p l a c e m e n t M a g n i f i c a t i o n a n d D y n a m i c T o r q u e A m p l i f i c a t i o n as Function of Isolation Eccentricity eJL and the F r e q u e n c y Rat ios ~os a n d ~ob (wi th Constant : T, = 0.5 s; e,y/L = 0.072; e=/L = O)

placement magnification, 0~ and Ucb, reported by Tarics et al. (1984), are 1.4 - 2.0 and 1.5 - 2.0, respectively, for Taft N21E 300% excitation. The large magnification of corner displacements U , and 0 ~ is due to the low frequency ratios Oo, and ~q0b, respectively. From Figs. 5(d) and 5 ( f ) , we

2944

find that the value of magnificat ion rat io /2 , is - 2 . 0 for a f requency ra t io ll0, = 1.2 [Fig. 5(d)], and also that the value of magnification rat io Ucb is --1.5 for a f requency rat io f~0b = 1.0 [Fig. 5( f ) ] . These results are in good agreement with results of Tarics et al. (1984).

Therefore, increasing isolation eccentrici ty eb/L and decreasing frequency ratios f~0, and ~0b general ly leads to increased corner-displacement magnification U~ and U~b. It is impor tan t to note that a nominal isolation eccentricity eb/L = 0.05 may result in significant torque amplif icat ion Tamp.

EFFECT OF S U P E R S T R U C T U R E E C C E N T R I C I T Y

The study by previous investigators (Eisenberger et al. 1986; Lee 1980) concluded that if the isolation eccentrici ty is zero or small ( e d L <_0.2) , then the corner-displacment magnificat ion Ucb would remain small (Ucb < 1.3), even if supers t ructure eccentrici ty is large ( edL < 0.4), i .e . , the superstructure eccentricity has little effect. Fur the rmore , it was concluded by Lee (1980) that dynamic torque amplif icat ion is small or negligible. How- ever, in a companion paper by the writers (Nagara ja iah et al. 1993) on torsional coupling of s l iding-base-isolated structures, it was concluded that the superstructure eccentrici ty has a significant influence on torsional amplification; a conclusion contrary to ear l ier conclusions. To investigate the reasons underlying these apparen t contradict ions, the effect of superstructure eccentricity on torsional coupling in e las tomeric isolated structures is studied in this section.

The biaxial superstructure eccentrici ty is var ied be tween the symmetr ic case e,/r = 0 and asymmetr ic case es/r = 0.5 (es/L = 0.204) in a structure with superstructure per iod Ts = 1.2 s, in the presence of supers t ructure

TABLE 1. Torsional Ampl i f icat ion- -Ef fect of Superstructure Eccentricity

fla, = 1.7 D.o,. = 1.0 f~o, = 0.8

(1) (2) (3) (71 (8) (9) ( (11 )

G/r = 0.00

G/r = 0.25

e,/r = 0.50

e, lr = 0.00 e,/r = 0.25 e j r = 0.50

1.7 1.0 0.8 1.7 1.0 0.8 1.7 1.0 0.8

1,7 1.7 1.7

1 1 1 1.02 1.13 1.17 1.05 1.32 1.40

(4) (5) (6) (a) Taft 200%

1 0 1 1 1 0 1 1 1 0 1 1 1.09 1.31 1.03 1.27 1.21 1.68 1.48 1.69 1.22 1.67 1.52 1.79 1.19 0.84 1.10 1.86 1.46 0.87 2.00 2.07 1.48 1.67 2.10 2.50

(b) El Centro 100%

0 1 1 1.32 1.05 1.48 1.21 1.03 2,52

1 1.03 1.04

1 1.13 1.26

0 0 0 1.18 1.84 2.08 0.68 1.23 1.21

0 1.32 2.10

1 1 0 1 1 0 1 1 0 1.15 1 .36 0.85 1.68 1 . 9 2 1.43 1.76 2.20 1.78 1.15 2.00 0.73 1.74 2.89 0.77 2.10 2 . 8 1 0.63

1 1 0 1.02 1.89 t .94 1.21 2.97 1.03

"Ucb = Peak corner base displacement/peak base displacement at the center of mass. b~, = Peak corner structure displacement/peak structure displacement at the center

of mass. cT, mp = Dynamic torque/static torque, at the center of stiffness of superstructure.

2945

TAFT 200~ ELASTOMERIC ISOLATION SYSTEi 15 - _ l,~eb= O, 8 i o . . . . ~ob: 1.0

. - - - - - ~eb=l .7 .~- 2.5 ~ ~ .

2 , 0

(a) ~.o 0.00 0.25 0.50

3.0

2.5

i.~u~2.0

1.5

1.0

, Qeb = 0.8 - - - Q e b = 1 0

',\XxT, x - Qeb = 1.7

- e~ I I " I �9

0.5 1,0 1.5 ~es

i(b) 2.0

2,2

2.0

1.8

i ~ 1 . 6

1.4

1.2

(C} 1.0 0.00

,~o~ 0.8, 2.2 -

. . . . . Qeb = 1.0 ~ 2.0 -

Q8 . . . . 1,8

1.4

1.2

z ~ l I 1.0 0,25 0.50 0.5

~eb=O.6 - - =1.o

~ 2el:) = 1,7

I i 1.5

i ( 1,0

f 2 e s

~(d) 2.0

2.6

ca. 2.2 E

1.8

1.4.

(e) 1.0 0.00

Qeb=l.7 3.0 es,~rO.O

Q 8 S - - 0 " 8 ~ c~'2"2E 2.6

f~e s: 1:0,---"'" ~1.8

"Qes=l.7"X_ ....... 1.4

1.0

~eb--17

~,,' " ,,e ~ : 0.5 _ ,\~ \ , , /

- e s ~ O . O ~

esx~0.25" t,r

0,25 0.50 0.5 1.0 1.5 2..0 esx/r f2os

FIG. 6. Corner-Displacement Magnification and Dynamic Torque Amplif ication as Function of Superstructure Eccentricity es/r and Frequency Ratios ,Q~s and ~ b (with Constant: T, = 1.2 s; e~,/r = 0.0)

frequency ratio ~10s = 0.8, 1.0, 1,7; base frequency ratio 12~ = 0.8, 1.0, 1.7; isolation eccentricity eJr = 0; ratio Lib = 1; and air = 1.23. The influence of the preceding parameters on the response of the structure to 200% 1952 Taft earthquake (N21E component in X direction and $69E

2946

component in Y direction) and 100% El Centro earthquake (S00E component in X direction and S90W component in Y direction) are presented in Table 1. In Table 1, constant: Ts = 1.2 s; Tb = 2.12 s; eJr = 0; L/b = 1; a/r = 1.23. Further results are presented in Fig. 6 for Taft earthquake, and in Fig. 4 for constant frequency ratios 120~ = 1.0 and (lob = 1.7 for E1 Centro excitation. It is evident from Fig. 4 that the torsional response increases with increasing eccentricity es/r and, unlike elastic systems, the lateral displacement also increases.

As evident from Figs. 6(a)-6(d) the corner-displacement magnification (Ucs and Ucb) increases with increasing eccentricity es/r and decreasing superstructure and base frequency ratios (10s and (10b. Also, in Fig. 6(c) the ratio Ucb is as high as 2.1 for superstructure eccentricities e j r = 0.5 or eJL = 0.204. These observations, along with similar trends noted in Fig. 5 regarding ratio U= and O~b and magnitude of dynamic torque amplification Tamp, indicate that increasing superstructure eccentricity e Jr and decreasing superstructure frequency ratio fl0, leads to amplification of the torsional response. This behavior is in contradiction to the conclusions of Eisenberger et al. (1986) and Lee (1980), who concluded that when the isolation eccentricity is zero or small (eJL < 0.2) the torsional response is small (O~b < 1.3), even if the superstructure eccentricities are large (e jL = 0.4), thus, implying that the superstructure parameters are not as important as isolation parameters.

Another important conclusion arrived at by Lee (1980) was that the torque-amplification factor Ta~,p, for base-isolated structures, is small or negligible, even for large superstructure eccentricities, i.e., eJL = 0.4. However, the observations about the dynamic torque amplification, Tamp, being high in the previous section [Fig. 5(a) and 5(b] and in a companion paper by the writer (Nagarajaiah et al. 1993) arc contradictory to the preceding conclusion. To clarify these contradictory conclusions, the effect of superstructure eccentricity on the torque amplification is examined next.

The torque-amplification factor Tam p represents the ratio of the dynamic torque to the static torque at the CS of the superstructure. The static torque used for the computation of the factor Tamp is calculated based on Lee's (1980) approach, i.e, the square root sum of squares (SRSS) of the torques F~e~ and Fy~es, where F~, and Fy~ are the peak structural shear forces obtained in the dynamic analysis. The value of the factor T, mp is close to unity in most cases, as shown in Table 1. However, the factors T~p shown in Table 1 do not accurately represent dynamic torque amplification because of the well-known fact that the SRSS method can cause significant errors in the estimation of forces and torques when the lateral and torsional frequencies are close (Wilson et al. 1981). To avoid using SRSS rule uniaxiat e~/r = O, 0.25, 0.5 is considered, with e~y/r = 0 and (10b = 1. All other parameters are kept identical. The results in the form of T, mp are shown in Figs. 6(e) and 6(f). It is clear from these results that the factor Tamp may indeed he as high as 2-2.7 in certain cases. Hence the conclusion the torque-amplification factor Tam p in base-isolated structures, is close to unity, may not be applicable in general.

Therefore, increasing superstructure eccentricity e Jr with decreasing superstructure and base frequency ratio fl0~ and ~0b leads to increased corner- displacement magnifications U~s and O~b. Also significant torque amplification Ta/mp may occur depending on the value of the eccentricity e Jr and frequency ratios (10~ and (10b.

2947

EFFECT OF SUPERSTRUCTURE FLEXIBILITY

T h e s u p e r s t r u c t u r e p e r i o d T, ( f ixed -base p e r i o d ) a long wi th t he base - f r equency ra t io f~0b was f o u n d to h a v e a s ign i f ican t i n f l uence o n t he c o r n e r - d i s p l a c e m e n t m a g n i f i c a t i o n , (Job in s l id ing i so l a t ed s t r u c t u r e s ( N a g a r a j a i a h et al. 1993). H o w e v e r , ea r l i e r i nves t i ga t i ons of e l a s t o m e r i c i so l a t ed s t ruc- tures ( E i s e n b e r g e r et al. 1986; L e e 1980) c o n c l u d e d t h a t t he f lexibi l i ty of the s u p e r s t r u c t u r e Ts has v e r y l i t t le i n f l u e n c e o n t he i so l a t ed s t r u c t u r e response. H e n c e , t h e e f fec t of t h e p e r i o d Ts is s t u d i e d in this sec t ion , T h e resul ts for s t r u c t u r e s w i th a r a n g e o f p e r i o d s T, a n d o t h e r s y s t e m p a r a m e t e r s in i so la ted a n d f i xed -base c o n d i t i o n a re s h o w n in T a b l e 2 for 2 0 0 % 1952

TABLE 2. Torsional Amplification--Effect of Superstructure Flexibility

Period ,Go, = 1.7 ,q~, = 1.0 ,Qo, = 0.8

(1) (2) (3) (4) (7) (8) (9) (1 O) (11 )

0.3 0.3 0.3

0.6 0.6 0.6

0.9 0.9 0.9

12 1.2. 1.2

0.3 0.3

0.6 0.6

0.9 0.9

1.2 1.2

1.7 1.01 1.0 1.01

Fixed - - base 1.7 1.01 1.12 1.0 1.02 1.15

Fixed - - 1.19 base 1.7 1.02 1.15 1.0 1.06 1.14

Fixed - - 1.19 base 1.7 1.06 1.23 1.0 t 1.13 1.21

Fixed [ - - 1.31 [ base

1.7 1.10 1.25 Fixed - - 1.21 base 1.7 1.10 1.16

Fixed - - 1.28 base 1.7 1.10 1.15

Fixed - - 1.27 base 1.7 1.07 1.24

Fixed - - 2.25 base

(6)

(a) Taft 200%

1.16 1.31 1.01 1.16 1.76 1.01 1.24 1 .90 - -

1.05 1.05 1.13 1.02 1.83 - -

1 .52 1.09

1.47 1.21 2.23 - -

1.59 1.19 1.68 1.48 2.43 - -

1.45

1.75

2.04

1.46 1.73 1.89

1.78 1.88 1.81

1.69 1.69 1.70

(b) El Centro 100%

1.50 1.10 1.73 1.52 - - 1.72

1.59 1.12 1.85 1.86 - - 2.48

1.27 1.19 1.71 2.12 - - 2.33

1.77 1.10 1.65 2.15 - - 1.45

1.55 1.66 3.07

1.25 1.73 2.24

1.59 2.06 2.69

2.02 1.84 2.58

1.01 1.77 1.20 1.02 1.89 1.29 - - 2.23 !2 .04

1.04 2.18 2.18 1.08 1.85 1.42

- - 1 .86 1.51

1.15 2.13 1.66 1.37 2.27 1.84 - - 1.95 1.36

1.31 2.58 2.07 1.68 1.92 1.43 - - 1.95 2.04

1.70 1.99

1.66

3.54

1.77

2.42

1.66

1.89

1.09

1.15

1.07

1.10

2.39 2.51

2.15 2.23

1.94

1.61

2.27 1.63

1.83 1.96

1.41 1.75

1.51 1.92

2.20 1.19

"/)c~ = Peak corner base displacement/peak base displacement at the center of mass. b/)c, = Peak corner structure displacement/peak structure displacement at the center

of mass. cT~mp = Dynamic torque/static torque, at the center of stiffness of superstructure.

2948

Taft earthquake (N21E component in X direction and $69E component in Y direction) and in Fig. 3 and in Table 2 for 100% 1940 E1Centro earthquake (S00E component in X direction and $69E component in Y direction). In Table 2, constant: Tb = 2.12 S; es/r = 0.25; eb/L = 0.065; L/b = 1; and a/r = 1.46.

From Fig. 3 it is found that the peak structural shears and torque are virtually unaffected by the period T=, which is consistent with the observations of Eisenberger et al. (19_86) and Lee (1980). However, from Table 2 is is seen that: (1) The ratio U~b increases with increasing period T= and decreasing ~Ob, in general; (2) the corner-displacement magnification ~= and U~b increase with decreasing frequency ratios ~0= and ~0b; and (3) the torque-amplification factor T~mp, in isolated and fixed-base condition; is of comparable magnitude in many cases.

The most important effect of increasing superstructure period T= along with decreasing frequency ratio ~ob in the presence of eccentricities is to increase the corner-displacement magnification U~b.

VERIFICATION USING EXPERIMENTAL STUDY

Nakamura et al. (1988) tested a torsionally stiff structural model, with four corner columns supporting a rigid concrete floor slab. The columns were fixed to a rigid concrete base slab, which was supported by four corner laminated-rubber bearings, with elastic properties, and two cantilever steel dampers, with smooth bilinear hysteretic properties. The system parameters were: T= = 0.8 s; Tb = 2.2 s; ~0= = 1.4; ~0b = 1.7; e=y/L = 0.1; and o,,/L = 0.1 (only in the Y direction). Only 1940 E1 Centro NS component was applied in the X direction. The tests produced a corner-displacement magnification /Job - _1.2 and _0= - 1.2. The low values of corner-displacement magnification Ucb and Uc= can be explained by results in Table 2: for large frequency ratios, ~o= - 1.7 and ~ob = 1.7, the magnification is calculated to be Ucb = 1.1 and 0~ = 1.15 for the period T, = 0.9 s. This is in accord with the conclusion that torsional response is negligible in torsionally stiff structures.

CONCLUSIONS

The objective of this study was to identify important system parameters and to investigate the influence of various system parameters on the lateral torsional response of base-isolated structures with elastomeric isolation system. The important system parameters identified and the trends of structural response presented for the two earthquakes apply in general.

The main source of torsional motions in etastomeric isolated structures is the isolation system eccentricity eb/L. Increasing isolation eccentricity eb/L leads to increased torque amplification Tamp. An "accidental" isolation eccentricity eb/L = 0.05 may result in significant torque amplification T, mp. Also, increasing isolation eccentricity eJL with decreasing superstructure and base frequency ratios 1)0= an_d ~0b generally leads to increased corner-displacement magnification Uc= and U<o.

Asymmetry and dynamic characteristics of the superstructure are as important as that of the isolation system. Significant torque amplification Tamp may occur depending on the value to the eccentricity e=/r and frequency ratios l"~e= and ~0b. Increasing superstructure eccentricity e=/r with decreasing superstructure and base frequency ratios ~0_= and 1)0, leads to increased corner-displacement magnifications /Q~ and Ucb.

2949

The most important effect of increasing superstructure period Ts, along with decreasing base frequency ratio ft0b, in the presence of eccentricities, is to increase the corner-displacement magnification Ucb.

The number of bearings in an elastomeric isolated structure has very little influence on the response, given a set of parameters, Ts, Tb, ftos, gob, es/r, e Jr, r/L, and L/b.

In conclusion, it can be stated that, although the magnitude of shear and torque generated in an elastomeric isolated structure is less than that of the fixed-base structure, the torsional amplifications may not be negligible, and may lead to torques that cannot be ignored.

ACKNOWLEDGMENTS

The financial support by the National Center for Ear thquake Engineering Research grant no. 90-2102B for this work is acknowledged.

APPENDIX. REFERENCES

Buckle, I. G., and Mayes, R. L. (1990). "Seismic isolation: history, application, and performance--A world overview." Earthquake Spectra, 6(2), 161-202.

Eisenberger, M., and Rutenberg, A. (1986). "Seismic base isolation of asymmetric shear buildings." Engrg. Struct., 8(1), 2-8.

Goel, R. K., and Chopra, A. K. (1990). "Inelastic response of one-storey asymmetric plan systems: effects of stiffness and strength distribution." Earthquake Engrg. Struct. Dyn., 19(3), 949-970.

Hart, G. C., DiJulio, R. M., and Lew, M. (1975). "Torsional response of high-rise buildings." J. Struct. Div., ASCE, 101(2), 397-416.

Kan, C. L., and Chopra, A. K. (1981). "Torsional coupling and earthquake response of simple elastic and inelastic systems." J. Struct. Div., ASCE, 107, 1569-1588.

Kelly, J. M. (1986). "Aseismic base isolation: review and bibliography." Soil Dyn. and Earthquake Engrg., 5(4), 202-217.

Lee, D. M. (1980). "Base isolation for torsion reduction in asymmetric structures under earthquake loading." Earthquake Engrg. Struct. Dyn., 8(3), 349-359.

Nagarajaiah, S., Reinhorn, A. M., and Constantinou, M. C. (1991). "Nonlinear dynamic analysis of 3D-base isolated structures." J. Struct. Div., ASCE, 117(7), 2035-2054.

Nagarajaiah, S., Reinhorn, A. M., and Constantinou, M. C. (1993). "Torsional coupling in sliding isolated structures." J. Struct. Div., ASCE, 119(1), 130-149.

Nakamura, T., Suzuki, T., Okada, H., and Takeda, T. (1988). "Study on base isolation for torsional response in asymmetric structures under earthquake motion." Proc. Ninth World Conf. on Earthquake Engrg., Int. Assoc. of Earthquake Engrg., Tokyo, Japan, V, 675-680.

Pan, T. C., and Kelly, J. M. (1983). "Seismic response of torsionally coupled base isolated structures." Earthquake Engrg. Struct. Dyn., 11(6), 749-770.

Papageorgiou, A. S., and Lin, B. C. (1989). "Study of the earthquake response of the base-isolated law and justice center in Rancho Cucamonga." Earthquake Engrg. Struct. Dyn., 18(4), 1189-1200.

Park, Y. J., Wen, Y. K., and Ang, A. H. S. (1986). "Random vibration of hysteretic systems under bidirectional ground motions." Earthquake Engrg. Struct. Dyn., 14(4), 543-557.

Reinhorn, A. M., Rutenberg, A., and Gluek, J. (1977). "Dynamic torsional coupling in asymmetric building structures." Build. and Envir., 12(4), 251-261.

Roeder, C. W., and Stanton, J. F. (1983). "Elastomeric bearings: a state of the art." J. Struct. Div., ASCE, 109(12), 2853-2871.

Sadek, A. W., and Tso, W. K. (1988). "Strength eccentricity concept for inelastic analysis of asymmetric structures." Proc. Ninth World Conf. on Earthquake Engrg., Tokyo, Japan, V, 91-96.

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Shepherd, R., and Donald, R. A. H. (1967). "Seismic response of torsionally un- balanced buildings." J. Sound and Vibration, 6(1), 20-37.

Tarics, A. G., Way, D., and Kelly, J. M. (1984). "The implementation of base isolation for the foothill communities law and justice center." Rep. to the National Science Foundation.

Uniform building code, earthquake regulations for seismic isolated structures. (1991). Int. Conf. of Build. Officials, Whittier, Calif.

Wilson, E. L., Kiureghian, A. D., and Bayo, E. P. (1981). "A replacement for the SRSS method in seismic analysis." Earthquake Engrg. Struct. Dyn., 9(1), 187- 194.

Yasaka, A., Mizukoshi, K., Iizuka, M., Takenaka, Y., Masada, S., and Fujimoto, N. (1988a). "Biaxial hysteresis model for base isolation devices." Summaries of techniealpapers of annual meeting, Architectural Institute of Japan, Tokyo, Japan, 1,395-400.

Yasaka, A., Koshida, H., and Iizuka, M. (1988b). "Base isolation system for earthquake protection and vibration isolation of structures." Proc. Ninth World Conf. on Earthquake Engrg., Int. Assoc. of Earthquake Engrg., Tokyo, Japan, V, 699- 704.

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