Active control of irregular buildings considering soil–structure interaction effects

ARTICLE IN PRESS Soil Dynamics and Earthquake Engineering 30 (2010) 98–109

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Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn

Active control of irregular buildings considering soil–structure interaction effects Chi-Chang Lin , Chang-Ching Chang, Jer-Fu Wang Department of Civil Engineering, National Chung Hsing University, 250 Kuo Kuang Road, Taichung, Taiwan 40227, ROC

a r t i c l e in f o

a b s t r a c t

Article history: Received 15 June 2008 Received in revised form 27 February 2009 Accepted 24 September 2009

This paper analyzes the soil–structure interaction (SSI) effect on vibration control effectiveness of active tendon systems for an irregular building, modeled as a torsionally coupled (TC) structure, subjected to base excitations such as those induced by earthquakes. An HN direct output feedback control algorithm through minimizing the entropy, a performance index measuring the trade-off between HN optimality and H2 optimality, is implemented to reduce the seismic responses of TC structures. The control forces are calculated directly from the multiplication of the output measurements by a pre-calculated frequency-independent and time-invariant feedback gain matrix, which is obtained based on a fixedbase model. Numerical simulation results show that the required numbers of sensors, controllers and their installation locations depend highly on the degree of floor eccentricity. For a large two-way eccentric building, a one-way active tendon system placed in one of two frames farthest away from the center of resistance (C.R.) can reduce both translational and torsional responses. The SSI effect is governed by the slenderness ratio of superstructure and by the stiffness ratio of soil to superstructure. When the SSI effect is significant, the proposed control system can still reduce the structural responses, however, with less effectiveness than that of the assumed fixed-base model. Therefore, the TC and SSI effects should be considered in the design of active control devices, especially for high-rise buildings located on soft site. & 2009 Elsevier Ltd. All rights reserved.

Keywords: Active control HN control Direct output feedback Torsionally coupled effect Soil–structure interaction effect

1. Introduction Since 1970, remarkable progress has been made in the field of active control of civil engineering structures subjected to nature forces such as winds and earthquakes [1]. In research studies and practical applications, various control algorithms have been investigated in designing controllers, such as LQ [2,3], LQR [4–6], LQG [7,8], fuzzy logic control [9], sliding mode and PID control [10], fuzzy sliding mode control [11,12], and HN control [13,14]. Among these, the HN control theory has generated many interests because it considers the worst external excitation case in the controller’s design and shows great potential for applications in earthquake engineering. The results from both numerical simulations and experimental tests indicate that HN control is effective. In fact, this theory has already been applied in the design of the control forces of a pair of hybrid mass damper systems to reduce the earthquake-induced bending–torsion motions of an actual building in Tokyo, Japan [15].

 Corresponding author. Tel.: + 886 4 22840438x225; fax: +886 4 22851992.

E-mail addresses: [email protected] (C.-C. Lin), [email protected] (C.-C. Chang), [email protected] (J.-F. Wang). 0267-7261/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.soildyn.2009.09.005

In previous studies, many researchers assumed that a controlled structure is a planar structure built on a fixed base. However, it is generally recognized that a real building is actually asymmetric to some degree even with a nominally symmetric plan. The asymmetric characteristic of the building creates a simultaneous lateral and torsional vibration known as torsion coupling (TC) when under translational excitations. Furthermore, many buildings are constructed on soft medium where the soil– structure interaction (SSI) effect could be significant. The SSI would significantly change the dynamic characteristics of a structure such as natural frequencies, damping ratios, and mode shapes [16]. These changes challenge the design of a structural control system as the dynamic characteristics of a structure are the main inputs in the control algorithm no matter what control strategy is employed. The importance of including the SSI effect on the effectiveness of control system has already been emphasized in the work by Wang and Lin [17], who analyzed the SSI effect on the control performance of multiple tuned mass dampers (MTMDs). In this paper, an optimal, practical, and cost-effective design procedure for an active tendon system is proposed for the vibration control of irregular buildings subjected to earthquake excitations. To understand the SSI effect on control efficacy of HN output feedback, a building structural model considering TC and

ARTICLE IN PRESS C.-C. Lin et al. / Soil Dynamics and Earthquake Engineering 30 (2010) 98–109

SSI effects is investigated. An HN direct output feedback control algorithm is applied to a two-way eccentric building with active tendon systems in outer frames. Through numerical simulations, the control performances of various active force implementations for different TC buildings are first examined to determine the optimal number as well as the location of actuators. The SSI effect is then introduced with ground motions recorded from hard site (1940 El Centro earthquake) and soft site (1985 Mexico City earthquake) conditions to investigate the seismic performance of the proposed active tendon systems. In the previous study, Wu et al. [16] developed an efficient method to evaluate the dynamic responses of buildings with one-way eccentricity considering SSI effect. This paper extends the methodology to evaluate the dynamic responses of a two-way eccentric building–soil system with and without control in frequency and time domain.

2. HN control for irregular building systems Without loss of generality, a single-story TC shear building equipped with active tendon control devices at each outer frame and subjected to bi-lateral ground acceleration, x€ g and y€ g , is shown in Fig. 1. In order to introduce a certain degree of asymmetry into the model, two-way floor eccentricities between the center of mass (C.M.) and the center of resistance (C.R.) along the x and y directions (ex and ey) are considered. The corresponding active control forces are represented as ux,1, ux,2, uy,1 and uy,2. The undamped equations of motion for the fixed-base system model can be expressed as mðx€ þ x€ g Þ þ kx ðx  ey yÞ ¼ ux;1 þux;2

ð1bÞ

Jy€ þ ky y  kx ey ðx  ey yÞ þky ex ðy þex yÞ ¼  By ðux;1  ux;2 Þ þ Bx ðuy;1  uy;2 Þ

ð1cÞ

where m =floor mass; kx, ky, and ky = story stiffnesses in x, y, and y, directions, respectively; J= mr2 =floor mass polar moment of inertia about z-axis; r =radius of gyration of the floor; Bx and By are half of floor width along x and y directions, respectively. In this study, the assumption of proportional viscous damping is considered so that the superstructure possesses classical normal modes. Theh control force vector ofiT active tendon system uðtÞ ¼ ux;1 ðtÞ ux;2 ðtÞ uy;1 ðtÞ uy;2 ðtÞ is expressed by 3 2 0 0 4kc cos ytx 0 7 6 0 4kc cos ytx 0 7 60 7UðtÞ ð2Þ uðtÞ ¼ 6 7 60 0 4kc cosyty 0 5 4 0 0 0 4kc cosyty h iT where UðtÞ ¼ Ux;1 ðtÞ Ux;2 ðtÞ Uy;1 ðtÞ Uy;2 ðtÞ is the tendon displacement vector, kc, ytx, and yty are stiffness and inclination angles of the tendon, respectively. Defining lex = ex/r, ley = ey/r, lBx =Bx/r, lBy = By/r and considering the structural damping, Eqs. (1a)–(1c) can be rearranged in matrix form as € þ C xðtÞ _ þ KxðtÞ ¼ B1 UðtÞ þE1 wðtÞ M xðtÞ 2

where 2

c11

6 C ¼ 4 c21 c31 2 2

y

3 x 6 7 xðtÞ ¼ 4 y 5; ry 3 c12 c13 c22 c23 7 5; c32 c33

ox

θ

C.R.

h

tx Active Tendons

&x&g

ð3Þ

"

# x€ g wðtÞ ¼ € ; yg

1 6 M ¼40 0

2 1 7 7; E 1 ¼ 6 0 4 5 0 o2y þ o2x l2ey þ o2y l2ex

o2y

o2y lex

o2y lex

2

4kc cos ytx 6 m 6 6 6 B1 ¼ 6 0 6 6 4 4kc cos ytx  lBy m

3

0

0

1

7 0 5;

0

1

3 0 7 1 5; 0

3

4kc cos ytx m

0

0 4kc cos ytx lBy m

0

4kc cos yty m 4kc cos yty lBx m

7 7 7 4kc cos yty 7 7 7 m 7 5 4kc cos yty  lBx m

In control theory, Eq. (3) can be conveniently rewritten in state-space form as X_ c ðtÞ ¼ AX c ðtÞ þ BUðtÞ þ EwðtÞ

Actuator

2

3

o2x ley

0

60 K ¼6 4 o2x ley

x

θ ty

&y&g

mðy€ þ y€ g Þ þ ky ðy þ ex yÞ ¼ uy;1 þ uy;2

ð1aÞ

z

99

ð4Þ

where "

y ux,1 By

uy,2

X c ðtÞ ¼ "

ex

B¼

C.R.



ey uy,1

C.M.

x

Bx

ux,2 Fig. 1. TC building with active tendon control devices: (a) system model and (b) plan view of floor.

0

# xðtÞ ; _ xðtÞ #

A¼

;

E¼

M 1 B1



0

"

M 1 K M 1 C # 0

I

 ;

M 1 E1

are the 6  1 state vector, the 6  6 system matrix, the 6  4 controller location matrix, and the 6  2 external excitation location matrix, respectively. Let us now define a 6  1 control output vector Z(t) and an s  1 output measurement vector Y(t) as ZðtÞ ¼ C 1 X c ðtÞ þ DUðtÞ

ð5Þ

YðtÞ ¼ C 2 X c ðtÞ

ð6Þ

where C1, D and C2 are the 6  6, 6  4 and s  6 matrices that relate to the control output vector and the measurement vector to

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C.-C. Lin et al. / Soil Dynamics and Earthquake Engineering 30 (2010) 98–109

state vector Xc and to the tendon displacement vector U. s is the number of measurement sensors. In this study, the control output vector satisfying DTD =I and DTC1 =0 is defined as " # " # 024 I 22 024 ZðtÞ ¼ a Xc ðtÞ þ UðtÞ ð7Þ 042 044 I 44 to denote the combination of floor lateral displacement responses and tendon displacements. The control weighting factor, a, determines the relative importance between response reduction and control force requirement. The larger the value of a, the greater the reduction of responses. a = 0 represents uncontrolled case. Furthermore, C2 determines the location and number of   measurements. For instance, C2 =I6  6 and C 2 ¼ 033 I 33 represent full-state measurement and only velocity measurement, respectively. For the direct output feedback control, the tendon displacement vector is expressed by UðtÞ ¼ GYðtÞ

ð8Þ

where G is a 4  s time-invariant feedback gain matrix. According to the HN control algorithm, the HN norm of transfer function matrix Tzw(jo) of control output with respect to external excitation takes the form :ZðjoÞ:2 og ð9Þ :T Zw ðjoÞ:1 ¼ sup :wðjoÞ:2 pffiffiffiffiffiffiffi where j ¼ 1, and sup is defined as the supremum over all w(t). Control parameter, g, is a positive attenuation constant that denotes a measurement of control performance. Adopting a smaller value of g means a more stringent control system is required. ||*|| 2 denotes the H 2 norm. It has been proved [18] that an optimal H N control system is asymptotically stable if a matrix P Z 0 exists and satisfies the following Riccati equation: ðA þBGC 2 ÞT P þ PðA þ BGC 2 Þ þ

1

g2

PEET P þ ðC 1 þDGC 2 ÞT ðC 1 þ DGC 2 Þ ¼ 0

where k is a 6  6 Lagrangian multiplier matrix. For simplicity and without loss of generality, let DTC1 = 0 and DTD= I, the necessary and sufficient conditions for minimization of L(G, P, k) are derived and expressed by @L 1 ¼ ðA þ BGC 2 ÞT P þ PðA þ BGC 2 Þ þ 2 PEET P þ C T1 C 1 þ C T2 GT GC 2 ¼ 0 @k g ð14Þ @L 1 1 ¼ ðA þ BGC 2 þ 2 EET PÞk þ kðA þBGC 2 þ 2 EET PÞT þ EET ¼ 0 @P g g ð15Þ @L ð16Þ ¼ BT P kC T2 þ GC 2 kC T2 ¼ 0 @G Thus, the procedures to obtain the HN direct output feedback gain matrix G are: (i) to decide a control output vector Z(t) of Eq. (5), (ii) to select a disturbance attenuation constant g, and (iii) to solve Eqs. (14)–(16) to obtain P, k and G by any iterative scheme. Once the values of the gain elements in the G matrix have been determined, the state-space formulation of the equations of motion can be expressed as X_ c ðtÞ ¼ ðA þBGC 2 ÞX c ðtÞ þ EwðtÞ

ð17Þ

The controlled system poles or eigenvalues are derived from solving the following sets of homogeneous algebraic equations: jl  I  ðA þ BGC 2 Þj ¼ 0

ð18Þ

where l represents complex eigenvalues of the controlled system. The corresponding controlled frequency and damping ratio are given as

oc ¼ jl j;

xc ¼  Reðl Þ=oc

ð19Þ

3. Torsion coupling effect

ð10Þ The controllers that satisfy Eq. (10) are not unique and may have an unbounded closed-loop H2 norm. Thus, one way to design the optimal HN output feedback gain is to solve Eq. (9) with the constraint of H2 norm. For example, the so-called combined H2/HN control problem is that the upper bound of H2 norm is minimized while the bound of HN norm is applied. Another example is to minimize the entropy of transfer function TZw(jo), which takes the form Z g2 1 lnjdet½I  g2 T Zw ðjoÞT Zw ðjoÞjdo ð11Þ En ðT Zw ; gÞ   2p 1 where T Zw ðjoÞ is the conjugate transform of TZw(jo). It has been shown that the entropy of a complex function is an upper bound of its H2 norm. Therefore, the minimization of entropy is equivalent to limit the magnitude of H2 norm. In addition, the entropy of a function is a useful measurement of how its singular values are close to the upper bound, g. By minimizing the entropy, we push all singular values of TZw(jo) away from g. From the results of Stoorvogel [19], the entropy is also expressed by T

En ðT Zw ; gÞ ¼ trfE PEg

Table 1 System parameters of the example building. 2.9235  103

Mass, m (kg)

ð12Þ

where tr{  } denotes the trace of a square matrix. Then, the optimization problem is converted to minimize the entropy in Eq. (12) subject to the constraint of Eq. (10). Incorporated with the constraint, the Lagrangian L can be introduced as n LðG; P; kÞ  tr ET PE þ k½ðA þ BGC 2 ÞT P þ PðA þ BGC 2 Þ  1 ð13Þ þ 2 PEET P þðC 1 þDGC 2 ÞT ðC 1 þDGC 2 Þ

g

A single-story TC shear building whose system parameters are listed in Table 1 is used to demonstrate the TC effect on control effectiveness of the proposed control algorithm. Both transfer function and time-history of floor responses of the irregular building with different number of sensors and controllers under bi-lateral real earthquake excitations are compared to illustrate the optimal number and location of sensors and actuators. All the different control cases are shown in Table 2. The results demonstrate a positive correlation between actuators and performance; the more the actuators, the better the control performance. Based on practical and economic considerations, only one or two control forces are applied in all cases. According to a previous study [13], only direct velocity feedback (DVF) is investigated in this paper since its control performance is as good as that of full-state feedback.

2

ox

3

6 7 Uncoupled natural frequency, 4 oy 5 ðHzÞ 2

oy

3

xx 6 7 Damping ratio, 4 xy 5 ð%Þ xy Tendon stiffness, kc (N/m) Tendon inclination angle,

"

ytx yty

# ð3 Þ

2

3 3:47 6 3:81 7 4 5 4:16 2 3 1:24 6 7 4 1:24 5 1:24 3.721  105   36 36

ARTICLE IN PRESS C.-C. Lin et al. / Soil Dynamics and Earthquake Engineering 30 (2010) 98–109

Table 2 Excitation, measurement, and control force of different control cases. Control case

Excitation Measurement Control force

Case 1

½ x€ g ½ x€ g

 

½ x_



½ x€ g ½ x€ g



½ x_



y€ g 

½ x_

y_



½ x_



Case 2 Case 3 Case 4 Case 5

101

Table 3 Controlled frequencies and damping ratios of different control cases for the oneway eccentric example building.

lBy = lBx

ley lex

Control case w/o control Case 1 ½ x_

Case 6

½ x€ g ½ x€ g

y€ g 

½ x_

Case 7

½ x€ g

y€ g 

½ x_



ry_  ry_ 

½ ux;1

ux;2

  [0.3

0.0]

½ ux;1







[0.3

0.0]

ry_  r y_ 

½

ux;2





[0.3

0.0]

½

ux;2





uy;2  [0.3

0.0]

  [0.3

0.3]

y_

ry_  r y_ 

½

ux;2



uy;2  [0.3

0.3]

y_

r y_ 

½

ux;2





[0.3

0.3]

ux;2

½ ux;1



pffiffiffiffiffiffiffiffi 3=2 pffiffiffiffiffiffiffiffi 3=2 pffiffiffiffiffiffiffiffi 3=2 pffiffiffiffiffiffiffiffi 3=2 pffiffiffiffiffiffiffiffi 3=2 pffiffiffiffiffiffiffiffi 3=2 pffiffiffiffiffiffiffiffi 3=2

a = 0.5 2

2

3

ocx

6o 7 4 cy 5 ðHzÞ 2

ocy xcx

3

6x 7 4 cy 5 ð%Þ

xc y

0 Displacement transfer function

-4 -6 -8

2

a = 0.2 3

3:23 6 3:82 7 4 5 4:48 2 3 34:29 6 1:24 7 4 5 6:58

2

3

3:23 6 3:82 7 4 5 4:48 2 3 13:34 6 1:24 7 4 5 3:29

Case 3

a = 0.2

a = 0.2

2

3 2

3:23 6 3:82 7 4 5 4:46 2 3 4:27 6 1:24 7 4 5 4:75

x-dir.

Case 4

a =0.2 3

3:23 6 3:82 7 4 5 4:47 2 3 12:37 6 1:24 7 4 5 1:76

2

3 3:23 6 3:82 7 4 5 4:47 2 3 11:8 6 4:45 7 4 5 2:75

w/o control Case 1 Case 2 Case 3

10-2

10-3 α=0.2, γ =0.01 10-4 10-1

ux,1 (Case 1)

Displacement transfer function

Gain in translation (10-3)

-2

-10 6

Gain in torsion (10-3)

3:23 6 3:82 7 4 5 4:48 2 3 1:24 6 1:24 7 4 5 1:24

10-1

α=0.2, γ=0.01

ux,2 (Case 1) ux,1 (Case 2) ux,2 (Case 3)

3

0

-3

-6

3

Case 2

10-2

10-3

10-4 0

0.1

0.2

0.3

0.4

0.5

λey Fig. 2. DVF gains of Cases 1–3 with various ley.

3.1. One-way eccentricity Cases 1–3 represent a one-way eccentric square building, i.e. ex =0, ey a0, under uni-directional ground acceleration x€ g . The torsional motion (ry) is linked to the translation (x). The building has no response in y direction. The greater the ey, the more significant the TC effect. The direct velocity feedback gains of Cases 1–3 with a =0.2 and g =0.01 to make xcx achieve more than 10% for various ley are calculated from Eqs. (14)–(16) and are shown in Fig. 2. The lines with symbol (J) represent gains for ux,1 and ux,2 in Case 1. The solid and dash lines denote the gains corresponding to translation and torsion measurements for ux,1 (Case 2) and ux,2 (Case 3), respectively. According to Eqs. (4), (6) and (8), negative gains indicate addition of dampings. It is observed from Fig. 2 that in Case 1, the magnitudes of gains corresponding to translation and torsion for ux,2 are greater than those for ux,1. ux,1 decreases the magnitudes of gains as ey increases, and thus degrades the control performance. Similar results are found by comparing Cases 2 and 3 where it is shown that ux,2 has better control efficiency than ux,1. This is a reasonable conclusion since larger values of ey cause larger torsional motion and ux,2, being located at the

rθ-dir.

0

2

4 Frequency (Hz)

6

8

Fig. 3. Transfer functions of floor responses for the one-way eccentric example building.

opposite side of C.R., provides a larger torsional force to reduce torsional as well as translational (x) responses. Table 3 shows the frequencies and damping ratios for the controlled and uncontrolled structure with different values of the control constraint. Looking at the modal damping ratios, the first (x) and third (y) modal damping ratios increase as a increases, while the second (y) one remains unchanged as expected, as shown in Table 3. The comparison among Cases 1–3 indicates that two controllers (Case 1) produce more active damping than the other controller (Cases 2 and 3). This result shows Case 3 generates larger first modal damping (xcx) than Case 2, thus confirming that the control forces ux,2 applied in the x direction do reduce both x and y responses. In Case 4, uy,2 is added to reduce the y responses due to y€ g excitation and this implies that the control damping ratio in the y direction (xcy) increases. Because ex = 0, xcy is not increased as much as xcx. xcx and xcy are slightly changed by including the second control force, uy,2, due to lBx a0. The effectiveness of the control system is also strengthened by the transfer functions of x and y responses with respect to ground

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acceleration x€ g for Cases 1–3 as shown in Fig. 3. It is apparent that Case 3 is as effective as Case 1 in suppressing the first modal response in the x direction. The time-history of floor displacement responses of Case 3 and the response without control under 1940 El Centro earthquake are illustrated in Fig. 4. The values of the peak responses and control forces are listed in Table 4. These results have clearly shown that the torsional response becomes significant when ley =0.3. In Case 3, both translational and torsional responses are suppressed up to 50% by the proposed one-way eccentric active

3

x-dir.

w/o control Case 3

Displacement (cm)

2 1 0 -1 -2

α=0.2, γ=0.01

-3 3

tendon system with peak control force equal to only 7% of the building weight. Larger control force for ux,1 in Case 2 is required to achieve the same control effectiveness as for ux,2 in Case 3. 3.2. Two-way eccentricity Cases 5–7 in Table 2 represent an irregular building with twoway eccentricity under x€ g and/or y€ g excitations. The controlled frequencies and damping ratios with a = 0.2 and g =0.01 are listed in Table 5. The transfer functions of the floor displacements with respect to x€ g are shown in Fig. 5. The comparison between Cases 5 and 1 indicates that two-way torsional coupling increases the controlled system’s dampings in x, y and y directions. Two control forces along the same directions can work together to achieve optimal control performance not only in its translational direction but also in torsion. Similar results are found when comparing Case 4 (Table 3) with Case 6. The expected results show the damping associated with the torsional mode does not increase significantly as there is only one control force applied in each lateral direction. As previously shown in Case 3 for the case of one-way eccentricity, Case 6 shows that one controller placed at each opposite side of C.R. will perform well in reducing all of the structural responses. Since the first modal damping ratio increases significantly, the peak floor

rθ-dir.

Displacement (cm)

2

10-1

-1 10-3

0

4

8 Time (sec.)

12

16

Fig. 4. Time-history floor responses of the one-way eccentric example building with and without control under 1940 El Centro earthquake.

Table 4 Peak displacement responses and control forces for the one-way eccentric example building under 1940 El Centro earthquake. Control case

w/o control

Case 1

Case 2

Case 3

"









" "

#

ðux;1 Þmax ðux;2 Þmax

2:45 1:22   

ðcmÞ # ðNÞ

ðux;1 Þmax =mg ðux;2 Þmax =mg





#

 ð%Þ

 





1:01 0:59



756:74

1926:37   2:64 6:72





1:44 1:01



2446:67



 

8:53 





1:06 0:69

α=0.2, γ=0.01

10-4

 



2042:43    7:12

10-1 Displacement transfer function

-3

ðxÞmax

w/o control Case 5 Case 6 Case 7

10-2

0

-2

ðr yÞmax

x-dir.

1

y-dir. 10-2

10-3

10-4 10-1 rθ-dir. 10-2

Table 5 Controlled frequencies and damping ratios of different control cases for the twoway eccentric example building (a = 0.2, g = 0.01). Control case

w/o control

Case 5

Case 6

Case 7

2

2

2

2

2

ocx

3

6o 7 4 cy 5 ðHzÞ 2

ocy

3

xcx 6x 7 4 cy 5 ð%Þ xcy

3 3:13 6 7 4 3:66 5 4:81 2 3 1:24 6 7 4 1:24 5 1:24

3 3:17 6 7 4 3:65 5 4:77 2 3 14:77 6 7 4 4:78 5 10:76

3 3:15 6 7 4 3:65 5 4:79 2 3 14:33 6 7 4 8:16 5 2:37

3 3:15 6 7 4 3:64 5 4:80 2 3 7:68 6 7 4 2:54 5 2:48

10-3

10-4 0

2

4 Frequency (Hz)

6

8

Fig. 5. Transfer functions of floor responses of the two-way eccentric example building.

ARTICLE IN PRESS C.-C. Lin et al. / Soil Dynamics and Earthquake Engineering 30 (2010) 98–109

lateral and torsional responses are dramatically reduced when the structure is subjected to the 1940 El Centro bi-directional earthquake excitations, as shown in Table 6. Fig. 6 also illustrates that one optimally placed controller (Case 7) is sufficient in reducing both lateral and torsional responses of two-way eccentric

Table 6 Peak displacement responses and control forces under 1940 El Centro bidirectional earthquake for the two-way eccentric example building (a =0.2, g =0.01). Control case

w/o control

Case 5

Case 6

Case 7

2

2

2

2

2

3

ðxÞmax 6 ðyÞ 7 4 max 5 ðcmÞ ðryÞmax 2 3 ðux;1 Þmax 6 ðu Þ 7 4 x;2 max 5 ðNÞ ðuy;2 Þmax 2 3 ðux;1 Þmax =mg 6 7 4 ðux;2 Þmax =mg 5 ð%Þ ðuy;2 Þmax =mg

3

2:75 6 7 4 2:06 5 1:58 2 3  6 7 45  2 3  6 7 45 

x-dir. 1.23

2

3

1:01 6 7 4 0:45 5 0:30 2 3 1355:33 6 2066:82 7 4 5  2 3 4:73 6 7 4 7:21 5 

1:19 6 7 4 0:73 5 0:65 2 3  6 2379:10 7 4 5 1058:55 2 3  6 8:30 7 4 5 3:69

3 1:23 6 7 4 1:12 5 0:85 2 3  6 2743:43 7 4 5  2 3  6 7 4 9:57 5 

4. Soil–structure interaction effect A two-way eccentric building (ex a0 and ey a0) supported on soil and subjected to free field ground motion in two orthogonal directions, x€ g and y€ g , as illustrated in Fig. 7, is investigated. In addition to the parameters describing the TC building superstructure in Eqs. (1a)–(1c), a rigid foundation of mass, mb, and axially inextensible columns of height, h, are introduced. The soil is characterized by its mass density, r, shear wave velocity, vs, and Poisson’s ratio, n. The dynamic behavior of the entire system can be completely described by the following eight degrees of freedom: x, y and y, horizontal translations and angle of twist of the floor with respect to the foundation; xb, yb and yb, horizontal translations and angle of twist of the foundation with respect to a fixed coordinate system; and fx and fy, representing the rocking motion about the x-axis and y-axis of the entire building. Applying the conventional substructure method adopted from the SSI analysis, the response of the building subsystem can be solved by

w/o control

2.75

Case 7

hφy

1

x

φx

ex

0 Floor

ey

-1

φy

-2

y¨b

φ¨x

x¨ g

2

θ¨b

1.12

x¨b

φy

Foundation

1

φy

φ¨y

Free Field

y-dir.

θb φx

α=0.2, γ=0.01

-3 3

Displacement (cm)

3

buildings under bi-directional earthquake excitations with peak control force less than 10% of the building floor weight.

θ

Displacement (cm)

3

103

y¨ g

0 -1 z -2 -2.06 -3 3

Soil

y x y

rθ-dir.

yb

Displacement (cm)

2 1.58 1

ex

By

0

θ

ey

C.M.

-1

T Mx

x

.. xg

Vx

-2

C.M.

My

Bx

-0.85

-3

C.R.

xb

Vy m b , I b , Jb .. yg

0

4

8 Time (sec.)

12

16

Fig. 6. Time-history floor responses of the two-way eccentric example building with and without control under 1940 El Centro earthquake.

Floor

Foundation

Fig. 7. Two-way eccentric building–soil interaction system under earthquake excitation: (a) system model, (b) plan view of floor, and (c) plan view of foundation.

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using the interaction forces developed at the foundation–soil interface to replace the soil subsystem. These interaction forces include the horizontal shear, Vx and Vy, the overturning moment, Mx and My, and the torque, T. The undamped equations of motion for the system model can be expressed as

8 9 Xb ðoÞ  Xg ðoÞ > > > > > > > > > Y ðoÞ  Yg ðoÞ > > > < b =  Tby ðoÞ > > > > > > Xf ðoÞ > > > > > > : Y ð oÞ ;

ð22Þ

f

€ Þ þ kx ðx  ey yÞ ¼ 0 mðx€ þ x€ b þ hf y

ð20aÞ

€ Þ þ ky ðyþ ex yÞ ¼ 0 mðy€ þ y€ b þhf x

ð20bÞ

Jðy€ þ y€ b Þ þ ky y  kx ey ðx  ey yÞ þky ex ðy þex yÞ ¼ 0

ð20cÞ

€ Þ þ m x€ þ Vx ¼ 0 mðx€ þ x€ b þ hf b b y

ð20dÞ

€ Þ þ m y€ þ Vy ¼ 0 mðy€ þ y€ b þhf b b x

ð20eÞ

Jðy€ þ y€ b Þ þ Jb y€ b þ T ¼ 0

ð20fÞ

€ þmhðx€ þ x€ þ hf € Þ þ My ¼ 0 ðIy þ Iby Þf b y y

ð20gÞ

€ þ mhðy€ þ y€ þ hf € Þ þ Mx ¼ 0 ðIx þ Ibx Þf b x x

ð20hÞ

where Jb =mass polar moment of inertia about the z-axis of the foundation and Ix, Iy = mass moment of inertia about the x-axis, and y-axis of the floor, Ibx, Iby = mass moment of inertia about the x-axis, and y-axis of the foundation, respectively. Defining xf = hfx, yf = hfy, ty = ry, tby = rbyb where rb is the radius of gyration of the foundation and applying structural damping into Eqs. (20a)–(20c), the equations of motion can be rewritten 8 as 9 > x€ b ðtÞ > > > > 8 9 > 8 9 8 9 > > > _ > > € y€ b ðtÞ > xðtÞ xðtÞ xðtÞ > > > > > > > > < = < < = < = = € _ € t ðtÞ by M yðtÞ þ C yðtÞ þ K yðtÞ ¼ ð21Þ > > > > > > > : t ðtÞ ; > : t_ ðtÞ ; : t€ ðtÞ ; > x€ ðtÞ > > > f > > y y y > > > > y€ ðtÞ > > : ; f

Since the foundation impedance functions of soil subsystem are dependent on the frequency of excitation, o, it is most convenient to relate the interaction shear forces, overturning moments and torque to the resulting foundation translations, rockings, and twist in the frequency domain as follows: 3 2 2 3 KVVx ðoÞ 0 0 0 KVxMy ðoÞ V x ðoÞ 7 6 6 7 6 V y ðoÞ 7 6 0 0 KVyMx ðoÞ 0 KVVy ðoÞ 7 7 6 6 7 7 60 6 7 ð o Þ 0 0 0 K TT 6 T ðoÞ 7 ¼ 6 7 7 6 6 7 6 M x ð oÞ 7 4 0 KMMx ðoÞ 0 KMxVy ðoÞ 0 5 5 4 ð o Þ 0 0 0 K ð o Þ K M y ðoÞ MyVx MMy 2

1 þ dm  clh dr dm kVVx 6 60 6 60 6 6 6 60 6 6 6 6 4 1  cdm

where V x , V y , M x , M y , T , Xb, Yb, Xg, Yg, Tby, Xf and Yf are the Fourier transforms of Vx, Vy, Mx, My, T, xb, yb, xg, yg, tby, xf and yf, respectively. KVVx, KVVy, KMMx, and KMMy and KTT are the translational, rocking and torsional impedances of foundation; KVxMy = KMyVx and KVyMx = KMxVy are the coupling impedance of foundation between translational and rocking motions. The foundation impedances can be further expressed in terms of dimensionless parameters as KVVx ðoÞ ¼ Gs rb kVVx ða0 ; nÞ

ð23aÞ

KVVy ðoÞ ¼ Gs rb kVVy ða0 ; nÞ

ð23bÞ

KMMx ðoÞ ¼ Gs rb3 kMMx ða0 ; nÞ

ð23cÞ

KMMy ðoÞ ¼ Gs rb3 kMMy ða0 ; nÞ

ð23dÞ

KTT ðoÞ ¼ Gs rb3 kTT ða0 ; nÞ

ð23eÞ

KVxMy ðoÞ ¼ Gs rb2 kVxMy ða0 ; nÞ ¼ Gs rb2 kMyVx ða0 ; nÞ ¼ KMyVx ðoÞ

ð23fÞ

KVyMx ðoÞ ¼ Gs rb2 kVyMx ða0 ; nÞ ¼ Gs rb2 kMxVy ða0 ; nÞ ¼ KMxVy ðoÞ

ð23gÞ

v2s

where Gs ¼ r is the shear modulus of soil. The dimensionless frequency parameter is defined as a0 = orb/vs and kVVx, kVVy, kVxMy =kMyVx, kVyMx =kMxVy, kMMx, kMMy and kTT are the dimensionless foundation impedances depending on a0 and v. Two other dimensional parameters have also been identified for conveniently characterizing the SSI system: s =vs/hox, a measure of the relative stiffness of soil with respect to the structure, and g ¼ m=rr2 h, a measure of the relative mass density of structure with respect to the soil. Define dm =mb/m, dr =rb/r, lo = oy/ox, lh =h/r, lsx =Ix/J, lsy = Iy/J, lsxb = Ibx/Jb and lsyb = Iby/Jb. Then, Eqs. (21)– (23) can be rearranged in the frequency domain via Fourier transform to yield 8 9 Xb ðoÞ > > > > 8 9 > > > > > Y ðoÞ > > > > < XðoÞ > = < b = 2 2 ðo M þ joC þKÞ YðoÞ ¼ o 1 Tby ðoÞ > > > : T ðoÞ > ; > > > X ð oÞ > > > y > f > > : Y ðoÞ > ; f

0

0

0

1  clh kVxMy dm

1þ dm  clh dm kVVy

0

1  ckVyMx dm

0

0

dm þ dr  clh dm kTT

0

0

1  ckVyMx

0

l d l cdm kMMx dh þ sx2 þ m2 sxb  lh dr lh lh d2r

0

0

0

0

dh þ

8 8 9 9 clh dr kVVx > 0 > > > > > > > 9 8 > > > > > > > > > > > > 0 Xð o Þ l d k c > > > > > > r VVy h = < < < = = T YðoÞ ¼  0 þ1 Xg ðoÞ  0 Yg ðoÞ > > > > > > : T ð oÞ ; > > > > 2 > > > > 0 d k c > > > > y VyMx r > > > > > > > > : ck : ; ; 0 VxMy

lsy l2h

ð24aÞ

3

78 9 7> Xb ðoÞ > 7> > > > 7> > Y b ð oÞ > > > 7> < = 7 7 Tby ðoÞ 7> > > 7> > X f ð oÞ > > 7> > > > 7> : lsyb dm cdm kMMy 7 Yf ðoÞ ; 5 þ 

l2h

lh dr

ð24bÞ

ARTICLE IN PRESS C.-C. Lin et al. / Soil Dynamics and Earthquake Engineering 30 (2010) 98–109

0

0

1

0

3

6 7 2 where 1 ¼ 4 0 1 0 0 1 5, and c ¼ s2 =g b where b = o/ox 0 0 1 0 0 denotes the normalized frequency of excitation, and X, Y, Ty are the Fourier transforms of x, y and ty, respectively. From Eqs. (24a) and (24b), performing modal decomposition using the modal parameters of the system (i.e., modal frequency oi, modal damping ratio xi, and mode shape matrix U, where i=1, 2 or 3), the floor response in frequency domain due to bi-directional free field earthquake excitations can be easily calculated through modification of its corresponding fixed-base structural response by including the modal SSI transfer matrices Sx(o) and Sy(o) as 8 9 > < XðoÞ > = YðoÞ ¼ UHðoÞS x ðoÞCx X€ g ðoÞ þ UHðoÞS y ðoÞCy Y€ g ðoÞ ð25Þ > : T ðoÞ > ; y where H(o) represents the transfer function matrix of modal displacement of fixed-base building. Cx and Cy indicate the influence vectors of fixed-base irregular building subjected to base excitations, x€ g and y€ g , respectively. The modal SSI transfer matrices S x ðoÞ ¼ diagðSi;xg Þ

ð26bÞ

are diagonal matrices whose generic ith terms, Si;xg and Si;yg are functions of superstructure and soil parameters and expressed as

Gyi G ¼ Rxb xg ðoÞ þ R ðoÞ þ yi Ryb xg ðoÞ Gxi yb xg Gxi Cyi Cxi þ R ðoÞ þ R ð oÞ Gxi yrx xg Gxi yry xg

Si;yg ¼

ð27aÞ

Gxi G R ðoÞ þ Ryb yg ðoÞ þ yi Ryb yg ðoÞ Gyi xb yg Gyi Cyi Cxi R ð oÞ þ R ðoÞ þ Gyi yrx yg Gyi yry yg

ð27bÞ

4.1. Equivalent fixed-base system considering SSI effect From Eq. (25), it is shown that the modal displacement transfer function of an SSI system is obtained by multiplying the fixedbase modal transfer function by the SSI transfer matrix. This subsystem method does not require computing the modal parameters of the entire SSI system. Actually, the entire system would not be linear because the soil impedance is frequencyTable 7 Equivalent system of Case 6 with SSI effect (s = 0.5, lh = 1). Modal parameter

Fixed-base system

Equivalent fixed-base system

Frequency (Hz)

2

2

x-dir. 3

3:13 6 3:66 7 4 5 4:81 2 3 1:24 6 1:24 7 4 5 1:24

y-dir. 3

2:85 6 3:24 7 4 5 4:57 2 3 14:7 6 19:3 7 4 5 15:7

3

True Equivalent

2

y-dir.

True Equivalent

2

1 1st mode 0 3 2 1 2nd mode 0 3 2 1

1 1st mode 0 3 2 1 2nd mode 0 3 2 1

3rd mode

3rd mode

0

0 0

1 2 Frequency (Hz)

3

0

1 2 Frequency (Hz)

3

Fig. 8. Modal displacement transfer functions of Case 6 with SSI effect.

dependent according to the previous study. In order to understand how the system characteristics change due to SSI effect, let us consider an equivalent single-degree-of-freedom (SDOF) system to represent a single mode of the SSI system. In other words, assume that the following equation is satisfied:

Gi ðo2  o2e Þ þ ið2xe oe oÞ

In Eq. (27a), Rxb xg ðoÞ, Ryb xg ðoÞ, Ryb xg ðoÞ, Ryrx xg ðoÞ and Ryry xg ðoÞ represent the displacement transfer functions of Xb, Yb, Tb, Xf, Yf with respect to the free field displacement, Xg; Gxi, Gyi, Gyi, Cxi, Cyi are the ith modal participation factors.

Damping ratio (%)

x-dir.

ð26aÞ

S y ðoÞ ¼ diagðSi;yg Þ

Si;xg

3

Amplitude of Modal Displacement Transfer Function

1

Amplitude of Modal Displacement Transfer Function

2

105



Gi Si ðoÞ ðo2  o2i Þ þið2xi oi oÞ

ð28Þ

where oe and xe represent the equivalent natural frequency and damping ratio, respectively. To show the feasibility of this ‘‘equivalent’’ SDOF approach, we take Case 6 with SSI effect in which s = 0.5, lh = 1 and the soil impedance functions from the work by Wu et al. [16]. According to Eq. (28), the SSI transfer matrix is calculated then, the optimization solutions for oe and xe of the SSI system with respect to xand y-direction ground accelerations, respectively, are obtained. Table 7 presents the equivalent frequencies and damping ratios in x and y directions. It can be seen that the SSI system has smaller vibration frequencies and larger damping ratios than those corresponding to the fixed-base model because of the flexibility of soil underneath the foundation of the building. Fig. 8 shows a comparison of the true and the equivalent modal displacement transfer functions. From the analysis of such transfer functions, it is clear that the equivalent SDOF system is acceptable to represent the true system. The SSI transfer functions cause different modification in x and y directions due to different lateral stiffnesses in the building.

2

3 2:85 6 3:23 7 4 5 4:58 2 3 14:5 6 19:2 7 4 5 15:9

4.2. Controlled system transfer matrix considering SSI effect When an active control system is installed, its effects will appear as a modification of the original damping and stiffness matrices of the superstructure. Hence, the new damping matrix

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C.-C. Lin et al. / Soil Dynamics and Earthquake Engineering 30 (2010) 98–109

Cc, and stiffness matrix Kc, for the controlled system can be expressed as

Displacement transfer function

10-1

σ= ∞ (w/o) σ= 0.5 (w/o) σ= ∞ σ= 0.5

x-dir.

10-2

ð29Þ 1b BGC 2 1Tb

and K a ¼  are added dampwhere C a ¼  ing and stiffness matrices provided by the control system, and     1a ¼ 033 I 33 , 1b ¼ I 33 033 .

5. Numerical verification of SSI effect

λh = 5

0

2

4 Frequency (Hz)

10-1

6

8

σ= ∞ (w/o) σ= 0.5 (w/o) σ= ∞ σ= 0.5

x-dir. 10-2

10-3 λ h = 0.2

10-4

Kc ¼ K þKa 1a BGC 2 1Ta

10-3

10-4

Displacement transfer function

Cc ¼ C þ Ca;

0

2

4 Frequency (Hz)

6

8

Displacement transfer function

Fig. 9. Transfer functions of (a) slender and (b) squatty buildings with and without control for different site conditions (Case 3).

100

x-dir.

It has been well recognized that the parameters representing the relative stiffness of the soil to the structure, s, and the heightto-base ratio of structure, lh, are the two most important factors in representing the SSI effects. To effectively investigate the influence of these two crucial parameters for the control performance of an active tendon system, which is designed on the fixed-base superstructure, the numerical results in the following discussions are obtained by selecting massless square foundations (dm = 0 and lsxb = lsyb = 0.5) and assuming that the structural floor is of the same size in plane as the foundation (dr = 1 and lsx = lsy = 0.5). As for the two soil parameters, typical values of g ¼ 0:5 and n = 1/3 are chosen. The 1940 El Centro and 1985 Mexico City earthquakes are used as the ground accelerations for hard site and soft site conditions, respectively. The transfer functions of floor lateral displacement to ground acceleration for the TC building of Case 3 in the previous section with two different height-to-base ratios (lh = 5 (slender building) and 0.2 (squatty building)), two different soil conditions (s =0.5 (soft site), and N (hard site) [20]) are presented in Fig. 9 for the controlled and uncontrolled cases. Please note that when the SSI effect is accounted for, the proposed active control can still reduce the structural responses, although its efficiency depends on the

σ=0.5, λh=5

y-dir.

α=0.2, γ=0.01

σ=0.5, λh=0.2

10-1

σ=∞ , λh=5 σ=∞ , λh=0.2

10-2 10-3 10-4 10-5

Case 6 0

0.4

0.8

1.2

1.6

2 0

0.4

0.8

Displacement transfer function

ω/ω1 100

1.2

1.6

2

ω/ω1

x-dir.

σ=0.5, λh=5

y-dir.

α=0.2, γ=0.01

σ=0.5, λh=0.2

10-1

σ=∞ , λh=5 σ=∞ , λh=0.2

10-2 10-3 10-4 10-5

Case 7 0

0.4

0.8

1.2 ω/ω1

1.6

2 0

0.4

0.8

1.2

1.6

2

ω/ω1

Fig. 10. Comparison for controlled transfer functions of buildings with respect to x€ g on different site conditions for (a) Case 6 and (b) Case 7.

ARTICLE IN PRESS C.-C. Lin et al. / Soil Dynamics and Earthquake Engineering 30 (2010) 98–109

values of s and lh. In the case of s = N, the control action provides substantial reduction of the peak amplitude of all translational modes, for both squatty and slender buildings. The active control maintains its efficiency when the SSI effect is insignificant. In the case of a squatty building founded on hard site, its transfer function is identical to that shown in Fig. 4. However, when it is built on a soft soil (s =0.5), the control system still provides a reduction in the amplitude of the translational responses but not as substantially as for the case of building on stiff soil as shown in Fig. 9.

107

Fig. 10 shows the transfer functions of the floor lateral displacements for the controlled buildings of Cases 6 and 7 supported on different soil conditions. Here it is interesting to see the SSI effect on the case of a controlled slender building on a relatively soft soil (s = 0.5, lh =5). The natural frequency of the first mode is dramatically reduced with respect to the corresponding fixed-base mode, o1, and this has an effect on the control performance. These results are also confirmed in Table 8 and Figs. 11 and 12 where the peak displacement response and control performance for irregular buildings of Cases 6 and 7 subjected to

Table 8 Peak displacement responses and control forces under 1940 El Centro earthquake (hard site, s = N) and 1985 Mexico earthquake (soft site, s = 0.5) for Cases 6 and 7 with SSI effect. Peak displacement (cm)

s=N

s = 0.5

lh = 0.2 With control

w/o control

With control

w/o control

With control

w/o control

With control

2

2

2

2

2

2

2

3

2

3

1:13 6 7 4 0:63 5 0:65 2 3 1:51 6 7 4 0:92 5 0:80

2.79 1.52

x-dir.

3

2:79 6 7 4 1:82 5 1:60 2 3 2:79 6 7 4 2:01 5 1:60

0 -1 -2

1:13 6 7 4 0:64 5 0:65 2 3 1:52 6 7 4 0:94 5 0:83

σ=∞, λh=5

Displacement (cm)

0.94

1 0 -1 -2

-2.01

-3 3

σ=∞, λh=5

-0.83

1.60

1 0 -1 -2

σ=∞, λh=5

-3 0

4

8 Time (sec.)

12

16

x-dir.

2.75 1.51

3

0:37 6 7 4 0:20 5 0:11 2 3 0:39 6 7 4 0:20 5 0:11

3

0:91 6 7 4 0:36 5 0:22 2 3 0:91 6 7 4 0:36 5 0:22

w/o control Case 7

1 0 -1 -2

σ=∞, λh=0.2 y-dir.

2

0.92

1 0 -1 -2

-1.74

-3 3

rθ-dir.

2

3

0:40 6 7 4 0:20 5 0:11 2 3 0:40 6 7 4 0:20 5 0:11

2

-3 3

y-dir.

2

3

3 w/o control Case 7

1

-3 3

lh = 5

w/o control

Displacement (cm)

Displacement (cm)

3

lh = 0.2

2

2:75 6 7 4 1:74 5 1:54 2 3 2:75 6 7 4 1:74 5 1:54

ðxÞmax 6 ðyÞ 7 4 max 5 ðr yÞmax 2 3 ðxÞmax 6 ðyÞ 7 4 max 5 ðr yÞmax

Displacement (cm)

Case 7

3

Displacement (cm)

Case 6

2

lh = 5

Displacement (cm)

Control case

σ=∞, λh=0.2

rθ-dir.

2 1.54

1 0 -1

-0.80

-2

σ=∞, λh=0.2

-3 0

4

8 Time (sec.)

12

16

Fig. 11. Time-history floor responses of (a) slender and (b) squatty buildings with and without control under 1940 El Centro earthquake.

3 0:64 6 7 4 0:30 5 0:17 2 3 0:74 6 0:33 7 4 5 0:18

ARTICLE IN PRESS C.-C. Lin et al. / Soil Dynamics and Earthquake Engineering 30 (2010) 98–109

w/o control Case 7

0.5 0 -0.5 -0.91

-1 1 Displacement (cm)

1

x-dir.

-0.74 σ=0.5, λh=5

y-dir.

0.5

0.36

0 -0.33

-0.5

x-dir.

w/o control Case 7

0.5 0 -0.40

-0.5

y-dir.

0.5

-0.5

Displacement (cm)

Displacement (cm)

0.5 0 -0.22

-0.18

-0.5 σ=0.5, λh=5

-1 0

σ=0.5, λh=0.2

-1 1

rθ-dir.

25

50 Time (sec.)

75

100

0.20

0.20 0

σ=0.5, λh=5 -1 1

-0.39 σ=0.5, λh=0.2

-1 1 Displacement (cm)

Displacement (cm)

1

Displacement (cm)

108

rθ-dir.

0.5 0.11

0.11

0 -0.5

σ=0.5, λh=0.2

-1 0

25

50 Time (sec.)

75

100

Fig. 12. Time-history floor responses of (a) slender and (b) squatty buildings with and without control under 1985 Mexico earthquake.

the 1940 El Centro and 1985 Mexico City earthquakes are presented. The results show that for a building founded on hard site, no matter if it is slender or squatty, the proposed control system has good control performance. However, for the case of soft site condition, since the SSI effect is significant, the control system is less effective, in particular for the building with smaller lh , which induces larger equivalent damping. Therefore, the SSI effect should be considered in the design of active control devices, especially, for a high-rise building founded on soft site.

6. Conclusions This paper deals with HN direct output feedback control of irregular buildings under earthquake excitations considering the effects of translation-torsion coupling of floor and soil–structure interaction. For a large two-way eccentric building, one-side active tendon located at opposite side of the center of resistance can reduce both two-way floor translational and torsional responses. For a squatty or slender building constructed on a hard site, the SSI effect is significant. The control action based on fixed-base feedback gains can reduce the structural responses, but it is less effective than that obtained from the fixed-base model. On the other hand, for soft site condition, the SSI effect is significant for both squatty and slender buildings. Consequently, the TC and SSI effect should be taken into consideration when designing active control devices in order to prevent response amplification for high-rise buildings.

Acknowledgments This research was partly supported by the Ministry of Education and the National Science Council of the Republic of China in Taiwan under the ATU plan and the Grant no. NSC 932211-E-005-025. These supports are greatly appreciated.

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