0.52 and 0.36 s for Loma Prieta, Northridge and Kobe ... acceleration and bearing displacement for five-story base-isolated structure under Loma Prieta, 1989.
Influence of isolator characteristics on the response of base-isolated structures Vasant A. Matsagar, R.S. Jangid Department of Civil Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India
Abstract The influence of isolator characteristics on the seismic response of multi-story base-isolated structure is investigated. The isolated building is modeled as a shear type structure with lateral degree-of-freedom at each floor. The isolators are modeled by using two different mathematical models depicted by bi-linear hysteretic and equivalent linear elastic–viscous behaviors. The coupled differential equations of motion for the isolated system are derived and solved in the incremental form using Newmark’s step-by-step method of integration. The variation of top floor absolute acceleration and bearing displacement for various bi-linear systems under different earthquakes is computed to study the effects of the shape of the isolator hysteresis loop. The influence of the shape of isolator force-deformation loop on the response of isolated structure is studied under the variation of important system parameters such as isolator yield displacement, superstructure flexibility, isolation time period and number of story of the base-isolated structure. It is observed that the code specified equivalent linear elastic–viscous damping model of a bi-linear hysteretic system overestimates the design bearing displacement and underestimates the superstructure acceleration. The response of base-isolated structure is significantly influenced by the shape of hysteresis loop of isolator. The low value of yield displacement of isolator (i.e. sliding type isolation systems) tends to increase the superstructure accelerations associated with high frequencies. Further, the superstructure acceleration also increases with the increase of the superstructure flexibility. Keywords: Base isolation; Earthquake; Elastomeric bearing; Sliding system; Bearing displacement; Superstructure acceleration; Bi-linear hysteresis; Equivalent linear
1. Introduction Seismic isolation, which is now recognized as a mature and efficient technology, can be adopted to improve the seismic performance of strategically important buildings such as schools, hospitals, industrial structures etc., in addition to the places where sensitive equipments are intended to protect from hazardous effects during earthquake [1–3]. Based on the extent of control to be achieved over the seismic response, the choice of the isolation system varies and thereupon its design is done to suit the requirements of use of the structure. In seismically base-isolated systems, the superstructure is decoupled from the
earthquake ground motion by introducing a flexible interface between the foundation and the base of structure. Thereby, the isolation system shifts the fundamental time period of the structure to a large value and/or dissipates the energy in damping, limiting the amount of force that can be transferred to the superstructure such that inter-story drift and floor accelerations are reduced drastically. The matching of fundamental frequencies of base-isolated structures and the predominant frequency contents of earthquakes is also consequently avoided, leading to a flexible structural system more suitable from earthquake resistance viewpoint. The two most common types of base isolation systems adopted in practice utilize either rubber bearings or sliding systems between the foundation and superstructure for the purpose of isolation from ground motions in the buildings as well as bridges.
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It is very essential to understand the different parameters affecting the response of base-isolated structure when used for seismic protection of the structures. Especially in case of the base-isolated structures, that houses sensitive equipments, determination of acceleration imparted and associated peak displacement are the key issues for the design engineer [4]. Moreover, the pounding and structural impacts in case of baseisolated structures made upon the adjacent structures, when separation gap distances are inadequate, become a major concern because these phenomena may lead to catastrophic failures leading to immense isolator damage. Such failures and damages can be avoided by properly estimating the peak isolator displacement and recommendation of appropriate isolation gap distances. In order to predict peak displacement and determine accurate separation gap distance requirement for a base-isolated structure, it is mandatory to know, in prior, the different parameters that affect the bearing displacement and its consequent effect on the superstructure acceleration. The failures due to such impacts can be avoided by reducing the peak bearing displacement by compromising with increase in superstructure acceleration to an acceptable level i.e. tolerable reduction in effectiveness of isolation. Selection of different parameters characterizing an isolation system is important in view of keeping a control over response quantities especially the excessive bearing displacement at isolator level. The behavior of isolation systems and the baseisolated structures is now well established and codes are developed for designing the base-isolated structures [5–9]. For non-linear isolation systems, the codes allow to use the equivalent linear model to permit the use of response spectrum method for designing the isolated structures. The equivalent linear models are based on the effective stiffness at the design displacement and the equivalent viscous damping is evaluated from the area of the hysteresis loop. The comparison of equivalent linear and actual non-linear model for the response of isolated bridge structures had been demonstrated in the past [10–13] and shown that the equivalent linear model can be used for predicting the actual non-linear response of the system. However, the above studies were restricted to the bridge idealized as a rigid body and the non-linear behavior of the isolator was limited to the lead–rubber bearings idealized by bi-linear characteristics. The equivalent linear model may give different response of isolated structures in comparison to the actual non-linear model for flexible superstructures and the type of non-linear hysteresis loop of the isolator associated with sliding type isolation systems. Therefore, it will be interesting to study the comparison of the two models for different hysteretic behavior of the isolator and the system parameters.
Here-in, the seismic response of multi-story structure supported on non-linear base isolation systems is investigated. The specific objectives of the study are: (i) to compare the seismic response of base-isolated flexible building obtained from various bi-linear hysteretic model and its equivalent linear model; (ii) to study the influence of shape of the isolator hysteresis loop and its parameters (i.e. yield displacement and force) on the effectiveness of the isolation system and (iii) to investigate the effects of superstructure flexibility on the response of base-isolated structures.
2. Structural model of base-isolated building Fig. 1(a) shows the idealized mathematical model of the N-story base-isolated building considered for the present study. The base-isolated building is modeled as a shear type structure mounted on isolation systems with one lateral degree-of-freedom at each floor. Following assumptions are made for the structural system under consideration: 1. The superstructure is considered to remain within the elastic limit during the earthquake excitation. This is a reasonable assumption as the isolation attempts to reduce the earthquake response in such a way that the structure remains within the elastic range. 2. The floors are assumed rigid in its own plane and the mass is supposed to be lumped at each floor level. 3. The columns are inextensible and weightless providing the lateral stiffness. 4. The system is subjected to single horizontal component of the earthquake ground motion. 5. The effects of soil–structure interaction are not taken into consideration. For the system under consideration, the governing equations of motion are obtained by considering the equilibrium of forces at the location of each degreesof-freedom. The equations of motion for the superstructure under earthquake ground acceleration are expressed in the matrix form as €g Þ xs g þ ½Cs fx_ s g þ ½Ks fxs g ¼ ½Ms frgð€ xb þ x ½Ms f€
ð1Þ
where [Ms], [Cs] and [Ks] are the mass, damping and stiffness matrices of the superstructure, respectively; xs gare the unknown fxs g ¼ fx1 ;x2 ; ...xN gT , fx_ s g and f€ relative floor displacement, velocity and acceleration €g are the relative acceler€b and x vectors, respectively; x ation of base mass and earthquake ground acceleration, respectively; and {r} is the vector of influence coefficients.
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Fig. 1.
(a) Mathematical model of N-story base-isolated structure, (b) bi-linear hysteretic model, and (c) equivalent linear model of the isolator.
The corresponding equation of motion for the base mass under earthquake ground acceleration is expressed by €b þ Fb k1 x1 c1 x_ 1 ¼ mb x €g mb x
ð2Þ
where mb and Fb are the base mass and restoring force developed in the isolation system, respectively; k1 is the story stiffness of first floor; and c1 is the first story damping. The restoring force developed in the isolation system, Fb depends upon the type of isolation system considered and approximate numerical models shall be used.
3. Mathematical modeling of isolators For the present study, the force-deformation behavior of the isolator is modeled as (i) non-linear hysteretic represented by the bi-linear model and (ii) the code specified equivalent linear elastic–viscous damping
model for the non-linear systems. A comparison of the response of the isolated structure by using the above two models will be useful in establishing the validity of the code specified equivalent linear model. 3.1. Bi-linear hysteretic model of isolators The non-linear force-deformation behavior of the isolation system is modeled through the bi-linear hysteresis loop characterized by three parameters namely: (i) characteristic strength, Q (ii) post-yield stiffness, kb and (iii) yield displacement, q (refer Fig. 1(b)). The bi-linear behavior is selected because this model can be used for all isolation systems used in practice. The characteristic strength, Q is related to the yield strength of the lead core in the elastomeric bearings and friction coefficient of the sliding type isolation systems. The post-yield stiffness of the isolation system, kb is generally designed in such a way to provide the specific
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value of the isolation period, Tb expressed as rffiffiffiffiffiffi M ð3Þ Tb ¼ 2p kb P where M ¼ mb þ N is the total mass of the j¼1 mj base-isolated structure; and mj is the mass of jth floor of the superstructure. Thus, the bi-linear hysteretic model of the base isolation system can be characterized by specifying the three parameters namely Tb, Q and q. The characteristic strength, Q is normalized by the weight of the building, W ¼ Mg (where g is the gravitational acceleration). 3.2. Equivalent linear elastic–viscous damping model of isolators As per Uniform Building Code [8] and International Building Code [9], the non-linear force-deformation characteristic of the isolator can be replaced by an equivalent linear model through effective elastic stiffness and effective viscous damping. The linear force developed in the isolation system can be expressed as Fb ¼ keff xb þ ceff x_ b
ð4Þ
where keff is the effective stiffness; ceff ¼ 2beff Mxeff is the effective viscous damping constant; beff is the effective viscous damping ratio; xeff ¼ 2p=Teff is the effecpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tive isolation frequency; and Teff ¼ 2p M=keff is the effective isolation period. The equivalent linear elastic stiffness for each cycle of loading is calculated from experimentally obtained force-deformation curve of the isolator and expressed mathematically as keff ¼
Fþ F Dþ D
ð5Þ
where F+ and F are the positive and negative forces at test displacements D+ and D–, respectively. Thus, the keff is the slope of the peak-to-peak values of the hysteresis loop as shown in Fig. 1(c). The effective viscous damping of the isolator unit calculated for each cycle of loading is specified as " # Eloop 2 beff ¼ ð6Þ p keff ðjDþ j þ jD jÞ2 where Eloop is the energy dissipation per cycle of loading. At a specified design isolation displacement, D, the effective stiffness and damping ratio for a bi-linear system are expressed as Q D
ð7Þ
4QðD qÞ 2p keff D2
ð8Þ
keff ¼ kb þ beff ¼
4. Solution of equations of motion Classical modal superposition technique cannot be employed in the solution of equations of motion here because (i) the system is non-classically damped because of the difference in the damping in isolation system compared to the damping in the superstructure and (ii) the force-deformation behavior for the isolation systems considered is non-linear. Therefore, the equations of motion are solved numerically using Newmark’s method of step-by-step integration; adopting linear variation of acceleration over a small time interval of Dt. The time interval for solving the equations of motion is taken as 0.02/200 s (i.e. Dt ¼ 0:0001 s).
5. Numerical study Seismic response of a multi-story base-isolated building is investigated under various real earthquake ground motions for bi-linear and equivalent linear isolator characteristics. The earthquake motions selected for the study are N00E component of 1989 Loma Prieta earthquake recorded at Los Gatos Presentation Center, N90S component of 1994 Northridge earthquake recorded at Sylmar Station and N00S component of 1995 Kobe earthquake recorded at JMA. The peak ground acceleration (PGA) of Loma Prieta, Northridge and Kobe earthquake motions are 0.57, 0.60 and 0.86 g, respectively. The displacement and acceleration spectra of the above ground motions for 2% of the critical damping are shown in Fig. 2. The maximum ordinates of the pseudo-acceleration are 3.559, 1.969 and 3.606 g occurring at period of 0.64, 0.52 and 0.36 s for Loma Prieta, Northridge and Kobe earthquakes, respectively. This implies that the selected ground motions are recorded on stations having firm soil or rocky terrain. The response quantities of interest are the top floor absolute acceleration and relative bearing displacement. The above response quantities are of importance because floor accelerations developed in the superstructure are proportional to the forces exerted because of earthquake ground motion. On the other hand, the bearing displacements are crucial in the design of isolation systems. For the present study, the mass matrix of the superstructure [Ms] is a diagonal matrix and characterized by the mass of each floor, which is kept constant (i.e. mj ¼ m for j ¼ 1; 2 . . .N). Further, the base raft of the isolated structure is considered such that the mass ratio, mb =m ¼ 1. The damping matrix of the superstructure, [Cs], is not known explicitly. It is constructed by assuming the modal damping ratio in each mode of vibration for superstructure, which is kept constant. The damping
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Fig. 2.
Response spectrum for Loma Prieta, 1989, Northridge, 1994 and Kobe, 1995 earthquakes.
ratio of the superstructure, ns, is taken as 0.02 and kept constant for all modes of vibration. The inter-story stiffness of the superstructure is adjusted such that a specified fundamental time period of the superstructure, Ts is achieved. The number of story in the superstructure is considered as 1 and 5. For the five-story building, the inter-story stiffness k1, k2, k3, k4 and k5 are taken in proportion of 1, 1.5, 2, 2.5 and 3, respectively. 5.1. Comparison of response for bi-linear and equivalent linear model In this section, a comparison of earthquake response of base-isolated structure is made for bi-linear and equivalent linear model of isolation systems. The bilinear behavior is selected in a way to represent the force-deformation behavior of the commonly used
isolation systems such as elastomeric (i.e. lead–rubber bearings) and sliding systems (i.e. friction pendulum system). The equivalent linear behavior is considered by selecting the appropriate values of the effective isolation time period, Teff and the effective viscous damping, beff. The design displacement, D, is considered as the maximum isolator displacement of a rigid superstructure isolated by the linear isolation system having the parameters Teff and beff. For the assumed value of the yield displacement, q, depending upon the type of isolation system, the parameters of the bi-linear hysteresis loop are derived such that it has an effective time period as Teff and damping ratio beff from Eqs. (7) and (8), respectively, at the design displacement D. The values of design displacement, D, used for such transformation are 53.61, 34.06 and 32.58 cm under Loma Prieta, Northridge and Kobe earthquake ground
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Fig. 3. Time variation of top floor acceleration and bearing displacement for five-story base-isolated structure under Loma Prieta, 1989 earthquake (Teff ¼ 2 s and beff ¼ 0:1).
Fig. 4. Time variation of top floor acceleration and bearing displacement for five-story base-isolated structure under Northridge, 1994 earthquake (Teff ¼ 2 s and beff ¼ 0:1).
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Fig. 5. Time variation of top floor acceleration and bearing displacement for five-story base-isolated structure under Kobe, 1995 earthquake (Teff ¼ 2 s and beff ¼ 0:1).
Fig. 6.
Comparison of force-deformation behavior for equivalent linear and bi-linear hysteretic model (Teff ¼ 2 s and beff ¼ 0:1).
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motions, respectively, obtained from equivalent linear model with Teff ¼ 2 s and beff ¼ 0:1. In Fig. 3, time variation of top floor absolute acceleration and bearing displacement of a five-story building is plotted for linear and bi-linear isolator models under Loma Prieta, 1989 earthquake motion. The parameters of the equivalent linear system considered are: Teff ¼ 2 s and beff ¼ 0:1. For the bi-linear system, two values of yield displacement i.e. 104 cm and 2.5 cm are considered which corresponds to friction pendulum system and lead–rubber bearing isolators, respectively. The peak superstructure acceleration obtained by bi-linear hysteretic model are 0.665 and 0.701 g for the yield displacement of 2.5 and 104 cm, respectively. The corresponding peak superstructure acceleration obtained from the equivalent linear model is 0.582 g. This implies that the top floor acceleration in a base-isolated structure is underestimated by the equivalent linear model as compared to that obtained from the actual bi-linear hysteretic model. On the other hand, the peak bearing displacement obtained by the bi-linear hysteretic model for the same system
is 42.52 and 40.17 cm, for the yield displacements of 2.5 and 104 cm, respectively, whereas, that obtained from its equivalent linear model is 53.06 cm. This indicates that the bearing displacement in a baseisolated structure is overestimated by the equivalent linear model as compared to that obtained from the bi-linear hysteretic model. Similar trend of comparison between linear and bi-linear models is also observed for Northridge, 1994 and Kobe, 1995 earthquakes motions as shown in Figs. 4 and 5, respectively. Thus, it can be concluded that the equivalent linear model under-predicts the peak superstructure acceleration and over-predicts the bearing displacement as compared to the actual bi-linear hysteretic model. The corresponding force-deformation diagrams are shown in Fig. 6 for equivalent linear and bi-linear models of the isolation system. Fig. 7 shows the comparison of corresponding FFT amplitude spectra (for both equivalent linear and bilinear hysteretic models) of the top floor acceleration for five-story non-isolated and isolated structures under different earthquake motions (refer Figs. 3–5 for the
Fig. 7. FFT spectra of top floor acceleration for five-story base-isolated structure for equivalent linear and bi-linear hysteretic models (Teff ¼ 2 s and beff ¼ 0:1).
63.84 56.53 53.43 54.47
66.59 68.28 62.75 65.90
105.31 96.15 101.64 101.22 –
Teff ¼ 2:5 s Equivalent linear Bi-linear q ¼ 104 cm q ¼ 2:5 cm q ¼ 5 cm
Teff ¼ 3 s Equivalent linear Bi-linear q ¼ 104 cm q ¼ 2:5 cm q ¼ 5 cm Fixed-base 82.01 56.49 53.46 60.05
57.26 46.18 43.13 49.65
53.71 36.58 41.90 41.65
0.475 0.429 0.452 0.450 1.522
0.433 0.440 0.406 0.425
0.649 0.580 0.542 0.554
0.382 0.287 0.267 0.293
0.377 0.407 0.303 0.338
0.559 0.407 0.455 0.454
beff ¼ 0:05 beff ¼ 0:1
beff ¼ 0:05
58.77 51.41 55.05 57.53 –
59.89 52.23 56.96 58.81
44.72 47.89 41.98 40.72
50.12 35.14 43.36 46.37
47.08 31.70 38.37 42.64
34.10 30.68 30.21 29.22
beff ¼ 0:05 beff ¼ 0:1
Bearing displacement (cm)
Top floor acceleration (g)
Bearing displacement (cm) beff ¼ 0:1
Northridge, 1994
Loma Prieta, 1989
Teff ¼ 2 s Equivalent linear Bi-linear q ¼ 104 cm q ¼ 2:5 cm q ¼ 5 cm
Force–displacement model
Table 1 Peak response of a single-story base-isolated structure for linear and bi-linear model (Ts ¼ 0:1 s)
0.267 0.233 0.247 0.257 1.072
0.388 0.340 0.365 0.376
0.452 0.476 0.419 0.408
0.235 0.173 0.204 0.215
0.308 0.236 0.264 0.287
0.349 0.348 0.320 0.313
beff ¼ 0:05 beff ¼ 0:1
Top floor acceleration (g)
37.30 35.75 36.95 38.14 –
36.14 34.75 36.03 36.35
37.86 35.06 40.02 40.80
33.24 31.10 32.83 35.22
32.04 27.87 33.30 33.12
32.54 26.83 32.58 36.54
beff ¼ 0:05 beff ¼ 0:1
Bearing displacement (cm)
Kobe, 1995
0.171 0.162 0.166 0.171 1.048
0.236 0.226 0.233 0.235
0.387 0.361 0.403 0.409
beff ¼ 0:05
0.160 0.146 0.151 0.160
0.214 0.199 0.219 0.217
0.346 0.297 0.336 0.371
beff ¼ 0:1
Top floor acceleration (g)
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62.62 55.92 54.55 55.94
64.23 73.62 69.34 67.46
105.18 100.51 104.78 104.05 –
Teff ¼ 2:5 s Equivalent linear Bi-linear q¼ 104 cm q¼ 2:5 cm q¼ 5 cm
Teff ¼ 3 s Equivalent linear Bi-linear q¼ 104 cm q¼ 2:5 cm q¼ 5 cm Fixed-base 82.37 64.17 61.37 65.53
57.07 47.73 45.24 49.42
53.06 40.17 42.52 41.19
0.526 0.549 0.564 0.539 2.762
0.444 0.574 0.501 0.482
0.673 0.671 0.669 0.661
beff ¼ 0:05
beff ¼ 0:05 beff ¼ 0:1
0.446 0.583 0.414 0.374
0.412 0.554 0.433 0.430
0.582 0.701 0.665 0.612
57.55 52.63 55.03 57.12 –
58.64 53.66 57.06 58.75
45.42 48.74 45.34 44.24
49.35 38.78 44.82 47.41
46.51 35.11 41.85 44.42
34.88 32.60 32.08 30.74
beff ¼ 0:05 beff ¼ 0:1
Bearing displacement (cm)
Bearing displacement (cm) beff ¼ 0:1
Northridge, 1994 Top floor acceleration (g)
Loma Prieta, 1989
Teff ¼ 2 s Equivalent linear Bi-linear q¼ 104 cm q¼ 2:5 cm q¼ 5 cm
Force–displacement model
Table 2 Peak response of a five-story base-isolated structure for linear and bi-linear model (Ts ¼ 0:5 s)
0.304 0.272 0.299 0.312 2.003
0.415 0.477 0.467 0.466
0.508 0.571 0.567 0.550
0.284 0.376 0.269 0.292
0.353 0.471 0.427 0.417
0.405 0.563 0.497 0.449
beff ¼ 0:05 beff ¼ 0:1
Top floor acceleration (g)
36.31 35.13 36.18 37.13 –
35.47 35.13 35.48 35.86
35.19 33.51 37.23 37.44
beff ¼ 0:05
32.49 31.00 32.46 34.37
31.54 29.55 32.95 32.72
30.62 26.91 31.97 34.29
beff ¼ 0:1
Bearing displacement (cm)
Kobe, 1995
0.202 0.255 0.175 0.178 3.627
0.275 0.350 0.254 0.255
0.473 0.486 0.474 0.499
beff ¼ 0:05
0.203 0.352 0.232 0.180
0.262 0.478 0.339 0.303
0.459 0.526 0.525 0.506
beff ¼ 0:1
Top floor acceleration (g)
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Fig. 8. Variation of top floor acceleration and bearing displacement for a five-story isolated structure against yield displacement of the isolator under Loma Prieta, 1989 earthquake.
time history of top floor acceleration). There is a significant difference between the FFT spectra of the top floor acceleration obtained from the equivalent linear and bi-linear models. The equivalent linear model shows the peak of Fourier amplitude in the vicinity of 0.5 Hz (i.e. this corresponds to the isolation frequency) and insignificant contribution from the other frequencies. On the other hand, for the bi-linear systems, there is contribution in the superstructure acceleration for wide ranging frequencies especially from the higher frequencies. These effects are found to be more pronounced for the bi-linear system with low isolator yield displacement (i.e. q ¼ 104 cm representing sliding type isolation system). These higher frequency contributions in the superstructure acceleration can be detrimental to the sensitive equipments with high frequency placed within the base-isolated structures. Thus, the base isolation systems with very low yield displacement transmit more acceleration in the superstructure associated with high frequencies and this phenomenon is not predicted by the equivalent linear models.
The comparison of the peak response of the isolated structure for equivalent linear and bi-linear models is shown in Tables 1 and 2 for single and five-story structure, respectively. The response is compared for different values of effective isolation time period (i.e. Teff ¼ 2, 2.5, 3 s), effective isolation damping (i.e. beff ¼ 0:05, 0.1) and isolator yield displacement (i.e. q ¼ 104 , 2.5, 5 cm) under three earthquake motions. As observed earlier, the peak top floor acceleration for all earthquake ground motions is higher for bi-linear models in comparison to the equivalent linear for all combinations of system parameters. This confirms that the superstructure acceleration will be under estimated if the bi-linear force-deformation characteristic of the isolator is modeled by an equivalent linear model. On the other hand, the peak bearing displacements predicted by the equivalent linear model is higher than the corresponding bi-linear hysteretic model. In some cases under Kobe, 1995 earthquake motion, the peak bearing displacements estimated by the equivalent linear model are less than the bi-linear
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Fig. 9. Variation of top floor acceleration and bearing displacement for a five-story isolated structure against yield displacement of the isolator under Northridge, 1994 earthquake.
model for q ¼ 2:5 and 5 cm. This is attributed due to the typical variation of the spectral displacement of this earthquake motion, in which the peak displacement decreases with the increase of time period in the range from 1.5 to 3 s (refer Fig. 2). Thus, the equivalent linear model of hysteretic isolator system over-predicts the peak bearing displacements.
5.2. Effects of isolator yield displacement In order to understand the influence of the shape of the bi-linear hysteresis loop of the isolator, the variation of peak top floor acceleration and bearing displacement of a five-story structure is plotted against yield displacement, q in Figs. 8–10 under Loma Prieta, 1989, Northridge, 1994 and Kobe, 1995 earthquakes, respectively. The responses are shown for three isolator characteristic strengths (i.e. Q=W ¼ 0:05, 0.075 and 0.1) and three values of isolation time periods based on
the post-yield stiffness (i.e. Tb ¼ 2, 2.5 and 3 s). It is observed that with the increase in the isolator yield displacement the top floor acceleration decreases substantially. On the other hand, the bearing displacement shows marginal increasing trend with the increase in the isolator yield displacement. This implies that the yield displacement (or the shape of the hysteresis loop) has significant effects on the response of the baseisolated structure. This significant influence of yield displacement on the response of the base-isolated structure is not captured by an equivalent linear viscous model as the q had no effect on the effective stiffness and a very little effect on the effective damping for large design displacement (refer Eqs. (7) and (8)). Further, it is also observed from Figs. 8–10 that with the increase in characteristic strength, Q, the top floor acceleration increases and the bearing displacement decreases. This is expected because for higher isolator characteristic strengths, the isolation system remains much more time in the elastic state which produces less
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Fig. 10. Variation of top floor acceleration and bearing displacement for a five-story isolated structure against yield displacement of the isolator under Kobe, 1995 earthquake.
flexibility in the structural system and thereby less energy dissipation. As a result, the superstructure acceleration increases and bearing displacement decreases with the increase of the isolator characteristic strength. Thus, it can be concluded that the response of base-isolated structure is significantly influenced by the shape and parameters of the bi-linear hysteresis loop of the isolator. 5.3. Effects of superstructure flexibility The flexibility in the base-isolated structure is mainly concentrated at the isolation level, as a result, the response of base-isolated structure can be investigated by modeling the superstructure as rigid [14–16]. However, it will be interesting to compare the seismic response of a base-isolated structure with superstructure modeled as rigid and flexible to study the influence of the superstructure flexibility. Fig. 11 shows the variation of top floor acceleration and bearing displacement of a five-story base-isolated structure
against the superstructure fundamental time period, Ts. The isolation system parameters considered are isolation period, Tb ¼ 2 s, normalized characteristics strength, Q=W ¼ 0:05 and different isolator yield displacement values such as q ¼ 104 , 2.4 and 5 cm. It is observed that there is significant difference in the top floor acceleration obtained when superstructure is rigid (i.e. Ts ¼ 0 s) and flexible (i.e. Ts > 0). There is substantial increase in the top floor acceleration as the fundamental time period of superstructure increases. This implies that the superstructure accelerations will be under-estimated if the superstructure flexibility is ignored and it is modeled as a rigid body. The increase in the superstructure accelerations is found to be more pronounced for the isolation system with low value of yield displacement (i.e. sliding type systems). On the other hand, the bearing displacement is not much influenced with the increase in superstructure flexibility. Similar effects of the superstructure flexibility are depicted in Fig. 12 where the corresponding responses are shown for Tb ¼ 3 s under different
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Fig. 11. Effects of superstructure flexibility on the top floor absolute acceleration and bearing displacement of a five-story structure (Tb ¼ 2 s and Q=W ¼ 0:05).
earthquake motions. Thus, the flexibility of superstructure increases the superstructure acceleration but it does not have significant influence on the bearing displacements.
2.
6. Conclusions Influence of isolator characteristic parameters on the seismic response of multi-story base-isolated structures is investigated. A comparison of the response of the isolated structure for equivalent linear and bi-linear force-deformation behavior of the isolator is made. In addition, the effects of the shape of isolator loop and superstructure flexibility on the seismic response of the base-isolated structure are also investigated. From the trends of the results of the present study, following conclusions are drawn 1. The code specified equivalent linear elastic–viscous damping model for a bi-linear hysteretic model of
3.
4.
5.
the isolator under-predicts the superstructure acceleration and over-predicts the bearing displacement. There is a significant difference in the frequency content of superstructure acceleration of base-isolated structure predicted by the equivalent linear and bi-linear isolator models. The response of base-isolated structure is significantly influenced by the shape and parameters of the bi-linear hysteresis loop of the isolator. The base isolation system with very low yield displacement (i.e. sliding type isolation systems) transmits more earthquake accelerations into the superstructure associated with high frequencies. This phenomenon is not predicted by the equivalent linear models of analysis. The flexibility of superstructure increases the superstructure acceleration. However, the bearing displacements are not much influenced by the superstructure flexibility.
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Fig. 12. Effects of superstructure flexibility on the top floor absolute acceleration and bearing displacement of a five-story structure (Tb ¼ 3 s and Q=W ¼ 0:05).
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