A parametric study of linear and non-linear passively damped seismic isolation systems for buildings

Transcription

1 Engineering Structures 26 (2004) A parametric study of linear and non-linear passively damped seismic isolation systems for buildings Cenk Alhan, Henri Gavin Department of Civil and Environmental Engineering, Duke University, Durham, NC , USA Received 18 April 2003; received in revised form 21 October 2003; accepted 10 November 2003 Abstract The effects of near-field ground motions with large velocity pulses have motivated passive damping requirements for the protection of seismically isolated structures. Structures in which the first mode damping exceeds 20% or 30% typically do not exhibit classical modes, and simulation via a simple superposition of uncoupled second order equations is not possible. When the damping is produced by viscous or linear visco-elastic devices, we can, however, gain insight into the dynamic behavior of these structures using a convenient first-order formulation and frequency domain methods. When the damping effects are created by nonlinear mechanisms such as yielding or friction, the behavior of the structure is amplitude dependent and analyses are commonly carried out in the time domain. In this paper, frequency domain analysis and earthquake time history analysis are applied to study the influence of isolation damping on higher-mode effects and inter-story drift ratios. Because higher mode effects, plan irregularities, and bi-directional ground motions are all important attributes of the dynamic behavior of these structural systems, a simple comparison of isolation damping mechanisms can not be carried out via simple single or two degree of freedom realizations. In order to incorporate these important details in the study of the dynamic behavior of these structures, a set of 8-story proto-type building models with L-shaped floor plans, different isolation periods, isolation damping characteristics, and levels of isolation stiffnesses are examined. # 2004 Elsevier Ltd. All rights reserved. Keywords: Seismic base isolation; Viscous damping; Hysteretic damping; Vibration control; Earthquake engineering 1. Introduction Seismic isolation systems for building structures are designed to preserve structural integrity and to prevent injury to the occupants and damage to the contents by reducing the earthquake-induced forces and deformations in the super-structure [6]. The compliant elastomeric bearings and frictional sliding mechanisms installed in the foundations of seismically isolated structures protect these structures from strong earthquakes through a reduction of stiffness and an increase in damping [9]. The reduction of stiffness is intended to detune the structure s fundamental period from the characteristic period of earthquake ground motions, Corresponding author. Tel.: ; fax: addresses: hpgavin@duke.edu (H. Gavin), (H. Gavin). and isolated structures have long fundamental periods of vibration as compared to the periods of most commonly encountered earthquake ground motions. Base isolation systems are being implemented in an increasing number of projects in highly seismic areas throughout the world. In full-scale implementation and modelscale investigation, seismic isolation methods performed as desired [1,14]. In addition, during the 1994 Northridge Earthquake, base isolated structures behaved as designed [17]. Ground motion records from recent earthquakes, such as the January 17, 1995 Kobe Earthquake, the August 17, 1999 Kocaeli Earthquake, and the September 21, 1999 JiJi Earthquake show that near-field ground motions may contain strong, long period velocity pulses. This type of ground motion may be very damaging to base-isolated structures. Protection of seismically isolated structures from strong, long-period velocity pulses can be challenging [4,21]. A study by /$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi: /j.engstruct

2 486 C. Alhan, H. Gavin / Engineering Structures 26 (2004) Heaton et al. [4] showed that a base isolated building subjected to a near-field M w 7.0 blind thrust earthquake could result in large isolator drifts (in excess of 50 cm) even when damped at 25% critical damping. Large displacements at the isolation interface during a strong earthquake can lead to buckling or rupture of the isolation bearings [11,16,21]. Koh and Balendra [11] examined the seismic response of base isolated buildings including the P D effects of the isolation bearings and showed that due to large displacements of the bearings under earthquake excitation, buckling may become an important issue especially when the buckling safety factor is low. Kelly [10] discussed the need for high levels of damping in order to limit isolator drifts within code-mandated limits and used a two degree of freedom model to demonstrate the potential for dynamic amplification in isolated systems with high isolation damping. Although, large levels of damping reduce isolator displacements in the fundamental mode, they impart forces into the structure which can increase structural accelerations and deformations in higher modes, and can increase inter-story displacements [10]. The potential for increased damping to increase high-mode response depends on the flexibility or rigidity of the structure. Feng and Shinozuka [3] and others have shown that increased damping causes increased accelerations in flexible isolated bridge structures. Sadek and Mohraz [19] analyzed a set of 6-dof structures equipped with a damper and reported that for flexible structures, increased damping decreases the base displacement but increases the floor accelerations. However, they only considered extremely low (f¼ 0:05) and extremely high (f¼ 0:40) damping cases, and did not take the intermediate levels of damping into account. Studies discussed above did not examine a broad range of isolation damping and stiffness. Isolation damping and stiffness can also be provided by non-linear isolation mechanisms and therefore the effects of non-linear isolation parameters, i.e. yield displacement, yield force, and the post-yield to pre-yield stiffness ratio, on the structural responses may be investigated. In this study, we examine the potential drawbacks of heavy isolation damping in the context of the response of a realistic eight story base isolated building to nearfield ground motions. The objectives of this paper are to investigate performance limits of passive linear and non-linear isolation systems, to identify levels of heavy isolation damping which increase structural accelerations and inter-story drifts, and to identify appropriate combinations of isolation parameters which reduce base displacements without significantly increasing floor accelerations and inter-story drifts. Section 2 describes the structural model, Sections 3 and 4 focus on base isolation systems with linear viscous damping and non-linear hysteretic damping, respectively, and conclusions are given in Section Structural model To investigate the effect of isolation system parameters on the dynamic response of buildings, the behaviors of a set of prototypical 8-story isolated structures with 46 different isolation systems, modeled with linear stiffness and damping elements and with bi-axial hysteretic elements, are examined via complex frequency response analysis and transient time history response analysis. The responses of these 16 linearly isolated structures are compared to those of 30 non-linearly isolated structures using three historical earthquake records. In these analyses, floor accelerations, drift ratios, and isolation deformation are evaluated. The following simplifying assumptions are made: dynamic soil-structure interaction, P D effects, in-plane shear deformation of the floors, and non-uniform ground motion are neglected; linear material behavior is assumed for the super-structure and the linear isolation system; and the response of the non-linear isolation bearings is simulated with bi-axial hysteretic models. The combination of isolation system damping and superstructure damping results in complex modes; the center of stiffness and center of mass are not coincident; and bi-directional horizontal ground motion excitation is considered in time history analyses. The prototypical structure used in this study has an L-shaped, asymmetric plan, as shown in Fig. 1. The weight of the structure is 33.4 MN and the inter-story clear height is 3 m. In structures such as this, where the center of mass does not coincide with the center of rigidity, the modes exhibit lateral-torsional coupling and are not aligned with the framing layout of the structure. Periods of this structure are associated with modes in which the motion is primarily (but not entirely) in the NS 0 and EW 0 directions, as shown in Fig. 1. The EW 0 axis passes through the center of gravity and the center of stiffness. The NW 0 axis is perpendicular to the EW 0 axis. Each floor has three degrees of freedom as shown in Fig. 1: two lateral degrees of freedom in the x and y directions and a rotational degree of freedom about the vertical axis are placed at the center of the floor mass which is also the geometric center and which is the same at every floor. The asymmetric plan results in modes with both lateral and torsional deformation [20]. Although this structural model has an asymmetric floor plan, the mass and rigidity centers are relatively close to one another, as shown in Fig. 1, and the lateral-torsional coupling effects are not prominent.

3 C. Alhan, H. Gavin / Engineering Structures 26 (2004) Fig. 1. Plan and elevation of the 8-story base isolated structure. Circles indicate isolator locations. The z-axis passes through the center of mass of each floor. The center of stiffness is at coordinate (1.03,0.55 m). The equations of motion for a damped structure subjected to bi-directional base excitation are: M qðtþþc _qðtþþkqðtþ ¼ MH zðtþ ð1þ where M is a positive definite structural mass matrix, C is a non-negative definite structural damping matrix, K is a positive definite structural stiffness matrix, zðtþ ¼ ½ z x ; z y Š T contains the two orthogonal components of horizontal ground acceleration, H is a two-column boolean earthquake input matrix, and q is a displacement vector which contains the displacement of the centers of masses of the stories. For this isolated structure, the displacement vector q ¼½x 1 ; y 1 ; h 1 ; :::; x 9 ; y 9 ; h 9 Š T contains the positions of the centers of masses at the 9 levels of the building including the isolation level with respect to the ground as shown in Fig. 1. The matrices M iso, K iso, and C iso represent the structural mass, structural stiffness, and structural damping matrices of the isolated structure and M fix, C fix, and K fix model the fixed-base structure which has three fewer degrees of freedom. The lateral inter-story stiffness of a story of the fixed base structure in the x, y, andh directions is given by 2 P i k xi 0 P i k 3 P xiy i K S ¼ 0 i k P 4 yi i k yix i P 5 i k P xiy i i k P yix i i k xi Yi 2 þ k yi Xi 2 ð2þ where k xi and k yi represent the lateral stiffness of the i th column in the x and y directions, and X i and Y i denote the x and y coordinates of the i th column with respect to the center of the coordinate system. The torsional stiffness of each individual resisting element is considered negligible [8]. Formulation of the element stiffness matrix corresponding to the isolation level follows the same procedure. Since the locations of isolation bearings in this model are the same as the locations of the columns, the resulting stiffness matrix of the isolation level, K I, will have the same form as Eq. (2) with the lateral stiffnesses of columns replaced by the lateral stiffnesses of isolation bearings. The block-tri-diagonal symmetric structural stiffness matrices of the fixed base structure, K fix, and the isolated structure, K iso, are then assembled according to the conventions of a shear building model [15]. The mass center and the origin of the coordinate system are coincident for each floor in this model, and the mass matrix of each floor is diagonal. The story mass matrix is: 2 ð 3 dm 0 0 A ð M S ¼ 0 dm 0 ð3þ 6 A ð r 2 dm A

4 488 C. Alhan, H. Gavin / Engineering Structures 26 (2004) where A is the floor area, dm is a differential mass of the floor, and r is the distance from the origin of the coordinate system. The diagonal mass matrices M iso and M fix are assembled by stringing the story mass matrices M S along the diagonal. In this study the damping matrix of the fixed base structure, C fix, is determined from the eigenvector matrix of the general eigenvalue problem K fix U fix ¼ M fix U fix K fix ð4þ where K fix is a diagonal matrix with its i th diagonal element being the i th eigenvalue corresponding to the i th mode and U fix is the mass normalized eigenvector matrix. For the fixed base structure, the damping matrix is calculated from U fix TC fix U fix ¼ C ð5þ where C is a diagonal matrix with its i th diagonal element being equal to 2f i x ni, where f i is the modal damping ratio of the i th mode. From Eq. (5), C fix can be obtained as C fix ¼ U fix TC U fix 1: ð6þ Note that in general, C fix is not mass and stiffnessproportional, and is typically diagonally-dominant and fully populated. The fixed-base periods, T fix, of the superstructure for the first three modes are 0.90, 0.81, and 0.60 s in NS 0, EW 0 and torsional directions. The fixed-base damping ratios, f fix, of the superstructure corresponding to the first three modes are 2%, 3%, and 2%. The floor plan dimensions, natural periods and damping ratios for the fixed-based and nominal-isolated structure are similar to the USC Hospital Building. 3. Linear isolation system This section primarily addresses the linear viscous damping in the isolation system. Approximating the energy dissipation mechanisms with a linear viscous damping model results in a simplified linear model which allows a convenient frequency domain analysis Modeling of the linear isolation system Formulation of the damping matrix of the isolated structure requires a methodology which makes use of C fix and the damping of the isolation level, C I. Let C Ixx ; C Iyy, and C Ihh be the total damping of the isolation system in the x, y and h directions, respectively, and let C I and C F be defined as 2 3 C Ixx C I ¼ 4 0 C Iyy C Ihh 2 Pi;j Cfix ij 0 0 C F ¼ Pk;l Cfix kl Pm;n Cfix mn ð7þ where i;j ¼ 1;4;7;:::;22; k;l ¼ 2;5;8;:::;23; and m;n ¼ 3;6;9;:::;24. Here, C F is the viscous damping matrix of the first story, neglecting the lateral-torsional coupling of the damping terms. Then, the damping matrix for the isolated model with linear viscous damping, C iso is 2 3 C I þ C F C F C iso C ¼ 6 F ð8þ C fix In order to study the effect of different levels of isolation damping on structures with different levels of isolation stiffness, two dimensionless parameters, f k and f c, are selected to represent the stiffness and damping levels. The stiffness, K I, and damping, C I, of the isolation system are calculated to give the natural isolation periods, T iso, of the nominal structure to be 2.25, 2.12, and 1.94 s with f iso equal to 0.15 for the first three modes. Recall that the fixed base periods, T fix,of this superstructure are 0.90, 0.81, and 0.60 s for the first three modes, as described in Section 2. The superstructure remains unchanged for all of the isolation systems examined in this analysis. The fundamental isolation period of this proto-type structure falls into the 2 3 s range where most isolation periods of baseisolated buildings lie, as reported by Makris [13]. Some examples of fixed base periods, T fix, and isolated base periods, T iso, of existing buildings include the 5-story New Zealand Parliament Building with T fix ¼ 0:5 and T iso ¼ 3:5 s, the 6-story West Japan Computer Center with T fix ¼ 0:7 andt iso ¼ 3:0 s, and the 8-story USC University Hospital Building with T fix ¼ 0:7 and T iso ¼ 2:25 s [2,12,17], which are close to the fixed and nominal isolation periods of our prototype building. The isolation period and damping ratio of the San Bernardino County Medical Center are 3.0 s and 0.50 [10]. In this study, structures with different isolation damping and stiffness levels are obtained by multiplying K I and C I with scaling factors f k and f c. Note that f k ¼ 1 and f c ¼ 1 correspond to the nominal case. The fundamental mode period, T iso, the fundamental mode damping ratio, f iso, and the corresponding average total lateral stiffness ~ k ¼ f k ðk Ixx þ K Iyy Þ=2 and the average total damping ~c ¼ f c ðc Ixx þ C Iyy Þ=2 of the isolation level for 16 different combinations of f k and f c are given in

5 C. Alhan, H. Gavin / Engineering Structures 26 (2004) Table 1 Fundamental mode periods, T iso, damping ratios, f iso, isolation stiffness, ~ k, and isolation damping rates, ~c, for all cases Case (a) ~c ¼ 19;286 kn=m=s f c ¼ 0:6 (b) ~c ¼ 32;144 kn=m=s f c ¼ 1:0 (c) ~c ¼ 64;288 kn=m=s f c ¼ 2:0 (d) ~c ¼ 96;432 kn=m=s f c ¼ 3:0 T iso f iso T iso f iso T iso f iso T iso f iso (1) k ~ ¼ 144;305 kn=m f k ¼ 0: s s s s 0.72 (2) k ~ ¼ 288;610 kn=m f k ¼ 1: s s s s 0.46 (3) k ~ ¼ 577;220 kn=m f k ¼ 2: s s s s 0.26 (4) k ~ ¼ 865;830 kn=m f k ¼ 3: s s s s 0.18 Table 1. The damping rates are the same in each case, 1 4, and the isolation stiffness is the same in each case, a d. This is demonstrated by the values ~ k and ~c in Table 1. As a result, the damping ratios are larger in the low-stiffness/long-period cases, even though the damping rates are the same for all stiffness cases. For purposes of the parametric study, a broad range of values are considered, including long periods and high damping ratios. The largest values are comparable to those of the San Bernardino County Medical Center Complex frequency response analysis The linear isolation system is considered first for the convenience of complex frequency response analyses including non-proportional damping effects. Transmissibility ratios for roof drift, floor acceleration, base drift, roof rotation drift, and base rotation are calculated for all cases with the excitation in x and y directions, separately. The transmissibilities are observed to be small when the excitation direction is perpendicular to the response direction. In this section, the frequency response plots for x-direction excitation is presented and the trends are similar for the y-direction excitation. Five transmissibility performance indices are defined as follows: T 1 ¼ ðx 9 x 1 Þ=h ð9þ z x T 2 ¼ X9 i¼2 ð x i þ z x Þ=8 z x ð10þ T 3 ¼ x 1 z x ð11þ T 4 ¼ ðh 9 h 1 Þ z x ð12þ T 5 ¼ h 1 ð13þ z x Here, z x and z y are the two orthogonal components of horizontal ground acceleration in x and y directions, and h is the inter-story height. Fig. 2 shows the transmissibility performances in x- direction for roof drift ratio, T 1, floor acceleration, T 2, and base drift, T 3, transmissibilities. Fig. 3 shows the transmissibility performances for roof rotation drift, T 4 and base rotation, T 5. These plots correspond to the extreme cases of levels of stiffness: case 1, where f k ¼ 0:5 (lowest stiffness case), and case 4, where f k ¼ 4 (highest stiffness case), and all values of damping are presented. It can be seen from these figures that increasing damping decreases the first mode responses. However, this trend is reversed in frequency regions between higher mode resonances. Increasing damping increases the response significantly in-between the first and second modes for roof drift and floor accelerations, especially for the low stiffness case (Fig. 2(i) and (iii)). A similar trend can be observed for the roof rotation. However, base drift transmissibility does not increase with high damping for both low and high stiffness cases (Fig. 2(v) and (vi)). For base rotation, one observes some negative effect of high damping, especially for the low stiffness case, case 1, (Fig. 3). These trends vary monotonically through cases 2 and 3. Note that the logarithmic scale indicates that the increase in response in the higher modes due to larger levels of isolation damping are at most only 10% of the reduction in the fundamental mode. The significance of these relatively minor higher mode effects are examined further via time history analyses Time history analysis Time history analyses of the structures are carried out for bi-directional ground motions, for 10 different historical earthquake records: the 1940 Imperial Valley El Centro record, the 1979 Imperial Valley Meloland record, the 1994 Northridge Rinaldi record, the 1994 Northridge Sylmar record, the 1989 Loma Prieta Los Gatos record, the 1995 Kobe JMA record, the 2000 Kocaeli YPT record, the 1999 JiJi Shikhkang record, the 1999 JiJi TCU078 record, and the 1999 JiJi TCU129 record. Representative results obtained for three earthquake records, namely 1940 El Centro, 1995 Kobe JMA, and 1999 JiJi TCU129, which account for low, moderate and large level ground motions, respectively, are presented here. The bi-directional acceleration and displacement response spectra of the three earthquake records for 5% and 10% damping are shown in Fig. 4. In time history analyses, the HHT-a method [7] is used

6 490 C. Alhan, H. Gavin / Engineering Structures 26 (2004) Fig. 2. Transmissibility performances for excitation in the x-direction, (i), (ii), roof drift ratio, T 1, (iii), (iv) floor acceleration, T 2, (v), (vi), base drift, T 3. to compute the responses. The HHT-a method is well suited for numerical integration of large second order systems which may have very high frequency modes which do not contribute significantly to the response, but which may require very short time steps for numerical stability. The HHT-a method enables accu-

7 C. Alhan, H. Gavin / Engineering Structures 26 (2004) Fig. 3. Transmissibility performances for excitation in the x-direction, (i), (ii), roof rotation drift, T 4, (iii), (iv) base rotation, T 5. rate simulation of these systems with a larger time step. In the software developed, the algorithm constants are Fig. 4. Acceleration and displacement bi-directional response spectra for Imperial Valley-El Centro, JiJi-129 and Kobe-JMA earthquake records. a ¼ 0:05, b ¼ 0:276, and c ¼ 0:55, which ensures stability [7]. The time stepused is s. Three criteria, base drift, J 1, floor acceleration, J 2, and story drift ratio, J 3 are considered for the evaluation of performance of the analyzed structures are as follows: J 1 ¼ max t ðd 1 Þ ð14þ J 2 ¼ max t;i ½ðd i d i 1 Þ=hŠ ð15þ "!, #1 X 2 J 3 ¼ max i a 2 xi þ a2 yi t ð16þ t where t is time, h is the inter-story height, i is the floor qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi number, d i ¼ x 2 i þ y 2 i is the displacement of the ith floor, x i is the displacement in the x-direction, y i is the displacement in the y-direction, as shown in Fig. 1, a xi ¼ x i þ z x is the total acceleration in the x-direction, and a yi ¼ y i þ z y is the total acceleration in the y-direction. Recall that z x and z y are the two orthogonal components of horizontal ground acceleration in x and y

8 492 C. Alhan, H. Gavin / Engineering Structures 26 (2004) Fig. 5. Performance of structures with different combinations of levels of stiffness and damping, (i), (iv), (vii) base drift, J 1 (ii), (v), (viii) story drift ratio, J 2 (iii), (vi), (ix) floor acceleration, J 3. directions. A threshold is set to discard the values of acceleration lower than 0.2 m/s 2, to avoid including the portions of the records in the average where insignificant or zero ground accelerations are recorded for a long time period. Plots including these performance criteria versus different combinations of f k and f c are shown in Fig. 5. It can be seen from these figures that the base drift is suppressed as the damping is increased for all stiffness levels with the level of suppression being much higher for lower levels of stiffness. However, increasing damping does not always decrease the story drift ratio. On the contrary, increasing damping increases the storydrift ratio for certain combinations of damping and stiffness. This undesirable effect is more pronounced especially for cases with long period isolation systems ðt iso > 2:5 sþ and for pulse-like earthquakes (Kobe and JiJi). This effect is not significant for the El Centro earthquake record. A similar trend is observed for floor accelerations. Floor accelerations increase with increased damping for long period isolation systems in the pulse-like earthquake records. This behavior is, in fact, predicted by the frequency domain analysis. 4. Non-linear hysteretic damping In this section, the structural system is considered to be isolated by elements which exhibit hysteretic material behavior. When damping effects are created by non-linear mechanisms such as yielding or friction, the behavior of the structure is amplitude dependent and analyses are commonly carried out in the time domain.

9 C. Alhan, H. Gavin / Engineering Structures 26 (2004) Modeling of yielding hysteretic isolation system A damped structural system subjected to bi-directional base excitation and possibly experiencing hysteretic material behavior is described by: M qðtþþc _qðtþþkqðtþþrðqðtþ; _qðtþ Þ ¼ MH z ð17þ The term RqðtÞ; ð _qðtþ Þ accounts for the non-linear restoring forces. Typical hysteretic material behavior is shown in Fig. 6 where f y is the yield force, D y is the yield displacement, and a is the post-yield to pre-yield stiffness ratio. When the damping effects are due to material yielding, the restoring forces RqðtÞ; ð _qðtþ Þ may be described by the equations for biaxial hysteresis [18]: F x ¼ ak hi x 1 þð1 aþf y Z x ð18þ F y ¼ ak hi y 1 þð1 aþf y Z y ð19þ where F x and F y are the hysteretic forces in x and y- directions, respectively. Here, Z x and Z y account for the bi-direction and biaxial interaction of hysteretic forces. The differential equation used for biaxial hysteretic behavior is given by: _Z x D y ¼ A _x 1 Zx 2 csignð _x ð 1 Z x ÞþbÞ _x1 Z x Z y csignð_y 1 Z y Þþb _y1 ð20þ h _Z y D y ¼ A_y 1 Zy 2 csignð_y i 1 Z y Þþb _y 1 Z x Z y ðcsignð _x 1 Z x ÞþbÞ _x1 ð21þ in which, A, c, and b are dimensionless quantities to be selected such that A ¼ b þ c for Z x and Z y to be bounded between +1 and 1 when yielding occurs [15]. Here, _x 1 and _y 1 are the relative velocities at the bearing locations in x and y-directions, respectively. In this study, the values of A, c, and b are selected as 1, 0.9, and 0.1, respectively Time history analysis In order to solve for the non-linear restoring forces, Eqs. (20) and (21) are solved via fourth order Runge- Kutta method to obtain the hysteretic quantities Z x and Z y at each time step. These are then inserted in Eqs. (18) and (19) to find the non-linear restoring forces in x and y directions, respectively. The hysteretic behavior, which in turn creates damping, depends primarily on three parameters: yield force, f y, yield displacement, D y, and post-yield to pre-yield stiffness ratio, a. Combinations of different values of these three parameters are used to study the effect of different isolation characteristics. The nominal structure with hysteretic yielding isolation system is defined to have a yielding force of f y ¼ 490 kn, a yielding displacement of D y ¼ 0:007 m and a post-yield to pre-yield stiffness ratio of a ¼ 0:15. With these values, the response of the structure is very close to the nominal structure with the linear isolation system, which corresponds to the case with f k ¼ 1:0 and f c ¼ 1:0 intable 1. Now, to study the effects of f y and D y, two sets of time history analyses, set A and set B, are carried out. For each set of analyses, the post-yield stiffness, ak hi is kept constant. Thus, a values are changed such that all isolation systems would have the same post-yield stiffness of the nominal structure which is K lo ¼ 0:15 490=0:007 ¼ 10; 496 kn=m for analysis set A, and K lo ¼ 0:20 490=0:007 ¼ 14; 004 kn=m for set B. The a values for other isolation systems are changed such that the post-yield stiffness is 10,496 kn/m for set A and 14,004 kn/m for set B. The different values of the parameters, f y ; D y ; a used for these two sets of analyses, A and B, are shown in Table 2. The yield forces and the yield displacements considered in this study are in the typical range that is reported for elastomeric bearing characteristics [5]. Plots of the performance criteria J 1, J 2, and J 3, versus different combinations of f y and D y are given in Figs. 7 and 8. Fig. 7(i), (iv) and (vii) show that for all Table 2 Post-yield to pre-yield stiffness ratios for set A (a A ) and for set B (a B ) Case (a) D y ¼ 0:0035 m (b) D y ¼ 0:0070 m (c) D y ¼ 0:0140 m Fig. 6. A typical hysteretic material behavior. a A a B a A a B a A a B (1) f y ¼ 368 kn (2) f y ¼ 490 kn (3) f y ¼ 613 kn (4) f y ¼ 735 kn (5) f y ¼ 980 kn

10 494 C. Alhan, H. Gavin / Engineering Structures 26 (2004) Fig. 7. Performance of structures (set A) with a post-yield stiffness of 10,496 kn/m (i), (iv), (vii) base drift, J 1 (ii), (v), (viii) story drift ratio, J 2 (iii), (vi), (ix) floor acceleration, J 3. types of earthquake excitations, there is a clear monotonic decrease in base drift as the yield force is increased (cases 1 to 5). As the yield displacement is decreased (cases a to c), there is a decrease in base drift. However, it is important to note that the rate of decrease in base drift diminishes with smaller yield displacements. This effect is particularly clear for the Kobe earthquake where there is almost no difference between case b ðd y ¼ 0:007 mþ and case c ðd y ¼ 0:014 mþ. On the other hand, story drift ratios do not decrease monotonically as the yield force of the isolation system increases (cases 1 to 5). For the lower levels of yield force, the story drift ratio decreases with increasing yield force; for higher levels of yield force, the story drift ratio increases with increasing yield force, depending on the characteristics of the ground motion, also, increasing the yield displacement may increase or decrease the story drift ratio depending upon the level of the yield force. Therefore, we see that there is an appropriate combination of yield force and yield displacement required to obtain a minimum level of story drift. This isolation system corresponds approximately to case 3b. The floor acceleration response follows the same trends as the story drift ratios. Again, for lower values of yield force, increasing the yield force decreases the floor accelerations. For higher values of yield force, increasing f y increases floor accelerations. This effect is clearly evident for the El Centro and Kobe earthquakes. For the JiJi earthquake, the floor acceleration seem to monotonically increase with increasing yield

11 C. Alhan, H. Gavin / Engineering Structures 26 (2004) Fig. 8. Performance of structures (set B) with a post-yield stiffness of 14,004 kn/m (i), (iv), (vii) base drift, J 1 (ii), (v), (viii) story drift ratio, J 2 (iii), (vi), (ix) floor acceleration, J 3. force. Increasing yield displacement increases floor accelerations for low levels of yield force and vice-versa for high levels of yield force. Thus, there is an appropriate combination of yield force and yield displacement required to obtain a minimum level of floor accelerations, and these values appear to depend upon the details of the ground motion record. Fig. 8 shows the results of the analyses of set B with a post-yield stiffness of 14,004 kn/m (as compared to the post-yield stiffness of 10,496 kn/m in set A). It is seen here that increasing post-yield stiffness decreases the base drift while it both increases the story drifts and floor accelerations. This is particularly noted for the Kobe-JMA record, which is a high-level, pulse-like earthquake. The changes are minor for the low-level, moderate earthquake record, El Centro. For the JiJi record, we see a significant increase in inter-story drift ratios, a small decrease in base-drift and a small increase in floor accelerations. On the other hand, the general trend of the curves are very similar to the general trend of curves obtained for set A. Thus, the discussions above for set A are qualitatively the same for the set B simulations. This study also enables a comparison of the two different energy dissipation mechanisms, namely, the linear isolation system and the non-linear hysteretic isolation system, in terms of the three performance criteria that we have defined. Fig. 5(i), (iv), (vii) and Fig. 7(i), (iv), (vii) show that the base drift values obtained for the linear and non-linear isolation systems

12 496 C. Alhan, H. Gavin / Engineering Structures 26 (2004) are in the same range, which serve as a basis for comparison. This is particularly true for case 2b which is defined as the nominal system as discussed before. When we consider the two other performance criteria, for the El Centro record, story-drift ratios and floor accelerations are also about the same for the linear and non-linear isolation systems as can be seen from Fig. 5(i), (ii), (iii) and Fig. 7(i), (ii), (iii). However, when a higher level earthquake, i.e. the Kobe JMA record, is considered, we see that there is a significant decrease, about 100% to 200% change in story-drift ratio and floor acceleration ranges when we change the energy dissipation mechanism from linear to non-linear, especially for high stiffness systems. This can be seen from Fig. 5(v), (vi) and Fig. 7(v), (vi). A similar trend is observed in story-drift ratio, between linear and non-linear systems for the JiJi record. 5. Conclusions An eight story, L-shape planned, structural model is used to study the influence of isolation damping and isolation stiffness on higher-mode effects, inter-story drifts and floor accelerations. Two types of isolations systems, namely, linear viscously damped and non-linear yielding hysteretic type isolation systems are studied. The linear systems are examined with both time and frequency domain analyses. The non-linear systems are examined only in the time domain. For the linear isolation systems, a complex frequency response analysis shows that increasing damping decreases the first mode response but increases the response at frequencies between the higher mode resonances, especially in terms of the roof drift and floor accelerations. This effect is more pronounced for structures with lower isolation stiffnesses (longer periods). Time history analyses show that increasing damping in the isolation system increases story drift ratios and floor accelerations for low stiffness isolation systems (T iso > 2:5 s). These effects are more pronounced in pulse-like ground motions, such as Kobe-JMA. Base drift, however, always decreases with increased isolation damping and increased isolation stiffness for all earthquake records. It is important to note that the fast and convenient complex frequency response analysis is predictive of the effects of damping on base drift and floor acceleration response in damped base-isolated buildings. This is confirmed via detailed, computationally intensive time history analyses. For non-linear systems, time history analyses show that increasing yield force and decreasing the yield displacement always decreases the base drift. However, inter-story drifts and floor accelerations are not always reduced by increasing the yield force and decreasing the yield displacement. Increasing the yield force, decreases inter-story drifts upto a point. Further increases in the yield force cause an increase in inter-story drifts. Thus, there exists an appropriate value for the yield force to obtain a minimum interstory drift. Similarly, decreasing the yield displacement can increase or decrease inter-story drifts, depending on the value of the yield force. Therefore, in general, for minimum inter-story drift, an appropriate combination of yield force and yield displacement may be determined. Similar behavior is observed in terms of the floor accelerations. Increasing the post-yield stiffness has a similar effect to increasing the pre-yield stiffness: increasing the post-yield stiffness results in a decrease in base displacements but increases interstory drifts and floor accelerations. The appropriate combination of yield displacement and yield force is dependent upon the characteristics of the earthquake ground motion. For low-level ground motions, dissipation mechanisms do not significantly affect the performance in terms of base drift, story-drift ratio, or floor acceleration. However, for high-level earthquakes, structures with isolation systems with yielding hysteretic mechanisms perform significantly better than structures with linear isolation systems, particularly in terms of storydrift ratios and floor accelerations. These results show that increasing damping decreases the isolator displacements. For the structures analyzed in this study, increasing damping eventually causes an increase in inter-story drifts and floor accelerations. On the other hand, appropriate levels of isolation stiffness and isolation damping can be combined to limit the base drift without significantly increasing floor accelerations and inter-story drift ratios, however, the appropriate values of stiffness and damping is dependent on the type of earthquake record. 6. 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