Stochastic optimal control of wind-excited tall buildings using semi-active MR-TLCDs

Probabilistic Engineering Mechanics 19 (2004) 269–277 www.elsevier.com/locate/probengmech

Stochastic optimal control of wind-excited tall buildings using semi-active MR-TLCDs Y.Q. Nia,*, Z.G. Yingb, J.Y. Wanga, J.M. Koa, B.F. Spencer Jr.c a

Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China b Department of Mechanics, Zhejiang University, Hangzhou 310027, China c Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA Received 15 November 2003; revised 15 December 2003; accepted 2 February 2004

Abstract A new semi-active control device, magneto-rheological tuned liquid column damper (MR-TLCD), has been devised recently by the authors for mitigation of wind-induced vibration response of tall building structures. The developed device combines the benefits of magneto-rheological smart materials and tuned liquid column dampers. In this paper, real-time semi-active vibration control of tall building structures incorporating nonlinear MR-TLCDs under random wind excitation is studied by means of the statistical linearization method and the optimal linear quadratic (LQ) control strategy. The equations of motion of a tall building structure subjected to random wind loading and controlled by using MR-TLCDs at the top floor are first derived and represented in modal coordinate. After linearizing the uncontrollable part of MR-TLCD damping force and incorporating it with structural components, the classical linear quadratic (LQ) control strategy is applied to the linearized structural system to determine optimal control force of the MR-TLCDs. Clipping treatment is performed to ensure the commanded control force implementable by the MR-TLCDs. Wind-excited response of the semi-actively controlled structural system is evaluated by using the frequency-response function and then compared with that of the passively controlled structure to determine the control efficacy. A case study of a 50-story building structure is conducted to illustrate excellent control efficacy of the proposed semi-active MRTLCD control system. q 2004 Elsevier Ltd. All rights reserved. Keywords: Tall building structure; Random wind excitation; Semi-active magneto-rheological tuned liquid column damper; Stochastic optimal control

1. Introduction Mitigating wind-induced vibration response of building structures by using passive tuned liquid column dampers (TLCDs) has been studied extensively [1 – 8]. The TLCD consists of a U-tube container with an orifice in the middle. It dissipates the energy of structural vibration by a combined action of inertia force induced by the movement of the liquid, the restoring force due to gravity on the liquid, and the damping effect caused by an orifice. The TLCD has attracted interest for engineers due to its cost-effectiveness, simplicity in installation, and low maintenance costs. A recent application of the TLCD is its implementation to the 46-story One Wall Centre in Vancouver [9]. Two TLCDs, each consisting of a four-story high, 50,000-gal water tank, * Corresponding author. Tel.: þ 852-2766-6004; fax: þ 852-2334-6389. E-mail address: [email protected] (Y.Q. Ni). 0266-8920/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.probengmech.2004.02.010

were placed at the top of this combined hotel and residential building for mitigating wind-excited vibration. Each TLCD contains two water columns connected by a sluice gate (orifice) to regulate water flow. The damping system is tuned to the natural frequency of the building by regulating water flow through the gates and also monitoring the water levels in the tanks. For a TLCD system, energy dissipation in the water column is due to the passage of the liquid through an orifice with inherent head-loss characteristics. However, the damping force induced by the orifice is nonlinear. The equivalent linearized damping coefficient is responsedependent, so that optimum damping condition cannot be maintained for a wide range of disturbances. Thus, it is highly desirable to develop damping-variable or parameteradjustable TLCDs to achieve optimal control performance for a wide range of loading conditions and to be tolerant of probable structural uncertainty. Some research efforts have

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been made to this end. Haroun et al. [10] proposed a concept of hybrid liquid column damper by actively controlling the orifice to produce variable orifice opening ratio; Yalla et al. [11] proposed to convert passive TLCDs into semi-active TLCDs by introducing a controllable valve to adjust the orifice opening. However, no physical devices of TLCDs with variable or controllable damping have been developed. The authors have recently developed a practical semiactive TLCD by using the smart magneto-rheological (MR) fluids [12]. An essential characteristic of MR fluids is their ability to reversibly change from a free flowing, linear viscous liquid to a semi-solid having a controllable yield strength in milliseconds when exposed to a magnetic field [13]. They are thus used as damping fluids to devise semiactive magneto-rheological tuned liquid column dampers (MR-TLCDs) with alterable fluid viscosity. The sharply alterable fluid viscosity results in adjustable and controllable damping force in the MR-TLCD for structural vibration control under a wide range of loading conditions. As a dissipative damper, the MR-TLCD generates the controllable damping force by utilizing the relative motion between the liquid and container and thus, does not have the potential to destabilize the system. Two MR-TLCD prototypes using different types of MR fluids have been fabricated [14,15] and laboratory experiments showed that the devised MR-TLCDs offered much better damping performance than passive TLCDs even in open-loop control mode. In the present paper, after briefing the MR-TLCD design, a semi-active control strategy for tall building structures incorporating MR-TLCDs is developed and its effectiveness for mitigating wind-induced vibration response of a real 50-story building is examined.

2. Design and modeling of MR-TLCDs As shown in Figs. 1 and 2, an MR-TLCD consists of a Utube container filled with the MR fluid and having an orifice opening in the middle of the bottom tube. Magnetic field offered to the fluid is generated by an electromagnet of

Fig. 2. Schematic of MR-TLCD.

length Lp around the bottom tube (usually Lp p B where B is the horizontal length of the liquid column). The MRTLCD device is rigidly connected to the primary structure and capable of dissipating energy through oscillation of the liquid column. With properly designed parameters of the damper, the out-of-phase response of liquid mass exerts an inertia force to the primary structure to counteract external excitations. The damping force results from viscous interaction between the liquid in motion and the rigid container as well as from the hydrodynamic head-loss. Because applying a magnetic field can alter the yield stress of the MR fluid and thus the head-loss, the damping force of an MR-TLCD is adjustable and controllable with the applied magnetic field in a semi-active manner. By modeling the motion of the MR fluid between fixed poles using the parallel-plate theory [16], the governing equation of motion for an MR-TLCD can be represented as [12]

rAD LD y€ þ

¼ 2rAD BD y€ 0

ð1Þ

where y is the displacement of the liquid relative to the container; y0 is the horizontal displacement of the container; AD ; LD and BD denote the cross-sectional area, the total central length, and the horizontal length of the liquid column, respectively; Lp and h are the length and depth of the flow between the fixed poles; r is the liquid mass density; g is the gravitational acceleration; c is a coefficient relying on the flow velocity and has a value ranging from 2.07 to 3.07 [16,17]; ty is the yield stress developed by the applied magnetic field. The overall head-loss coefficient d is determined by

d¼

Fig. 1. Prototype of MR-TLCD.

cty AD Lp y_ 1 þ 2rAD gy rA dl_yl_y þ 2 D h l_yl

48 LD X þ zj
2 H H j Re 1 þ w

ð2Þ

where w is the width of the flow between the fixed poles; H is the height of the bottom tube; zj is the coefficient of minor head-losses, including those of elbows and orifices.

Y.Q. Ni et al. / Probabilistic Engineering Mechanics 19 (2004) 269–277

The Reynold number Re is defined by Re ¼

2rVðw þ hÞ hwh

represented as ð3Þ

where V is the fluid velocity; h is the field-independent viscosity. The damping terms in Eq. (1) are nonlinear due to the presence of orifice and applied magnetic field. Re-writing Eq. (1) as mD y€ þ uð_yÞ þ kD y ¼ 2lmD y€ 0

ð4Þ

where mD ¼ rAD LD ; kD ¼ 2rgAD ; l ¼ BD =LD : The damping force uð_yÞ can be separated into a passive component and a semi-active component and is represented by uð_yÞ ¼ up ð_yÞ þ us ð_yÞ

ð5Þ

with up ð_yÞ ¼

1 rdAD y_ 2 sgnð_yÞ; 2

us ð_yÞ ¼ ty ðcAD Lp =hÞsgnð_yÞ

ð6aÞ ð6bÞ

where up ð_yÞ is the uncontrollable part of the damping force, which is regarded as a passive damping force and will be incorporated into the system equation of motion; us ð_yÞ is the controllable part of the damping force, which is semiactive control force and can be determined according to an optimal control strategy. By adjusting the applied external voltage, the yield stress ty of the MR fluid can alter accordingly and thus the MRTLCD produces a controllable damping force us ð_yÞ in an active manner. As seen in Eq. (6b), the controllable damping force us ð_yÞ is always in the opposite direction of the relative motion, implying that MR-TLCDs can only exert dissipative control force. When the required optimal control force does not meet such a condition, the clipping treatment [18] needs to be performed to ensure the commanded damping force implementable by MR-TLCDs.

3. Description of controlled system Consider a high-rise building structure with n stories and a semi-active MR-TLCD installed at the top floor under random wind loading. For a linear elastic shear-type structure, the equation of motion of the controlled system can be expressed as M X€ þ CX_ þ KX ¼ FW ðtÞ þ FD

271

ð7Þ

where X denotes the n-dimensional horizontal displacement vector of the structure; M; C and K are the n-dimensional mass, damping and stiffness matrices of the structure, respectively; FW ðtÞ is the n-dimensional wind loading vector. FD denotes the n-dimensional control force vector produced by inertial motion of the semi-active MR-TLCD driven under an optimal control strategy and can be

FD ¼ E1 fD ; fD ¼

ð8aÞ

1 ½ð1 2 l2 ÞmD y€ þ uð_yÞ þ kD y l

ð8bÞ

where E1 ¼ {1; 0; …; 0}T is an n-dimensional constant vector, and fD is the interactive force between the MRTLCD and the top floor of the structure. The variable y denotes the relative displacement of the liquid to the container and is governed by Eq. (4). For the random wind excitation, the cross power spectral density of along-wind force is represented by the Davenport spectrum [19] as
 hi hj a S ðvÞ 2 SWij ðvÞ ¼ r2a CD2 V10 ð9Þ Ai Aj cohðhi ; hj ; vÞ 0 2p 100 where SWij denotes the cross spectrum of the wind forces at the ith and jth floors; S0 is the power spectral density of the wind velocity; ra is air mass density; CD is the drag coefficient; V10 is the mean wind velocity at 10 m height; Ai and Aj are the equivalent projection areas about the ith and jth floors, respectively; hi and hj are heights of the ith and jth floors; and a is a constant exponent. The coherence function cohðhi ; hj ; vÞ is described by ( ) Ch lvllhi 2 hj l ð10Þ cohðhi ; hj ; vÞ ¼ exp 2 2pV10 where Ch is an exponential decay constant. The power spectral density of the wind velocity S0 is of the form 2 S0 ðvÞ ¼ 8pKD V10

h0 ¼

h20 ; lvlð1 þ h20 Þ4=3

600v pV10

ð11aÞ ð11bÞ

where KD is the ground coarse coefficient. Suppose that the structural response be expressed by first mð# nÞ modes of the corresponding undamped system in terms of an n £ m-dimensional mode matrix F normalized with respect to the structural mass matrix M: Then the structural displacement response can be represented by X ¼ FY where Y ¼ {y1 ; y2 ; …; ym }T is the m-dimensional modal coordinate vector. The governing equation of the system then becomes Y€ þ JY þ VY ¼ FT FW ðtÞ þ FT E1 fD

ð12Þ

where the m-dimensional diagonal matrices V ¼ FT K F ¼ ½v2i  and J ¼ FT C F ¼ ½2zi vi  in which vi and zi are the structural natural frequency and damping ratio of the ith mode, respectively. For the semi-active MR-TLCD installed at the top floor, the motion of the container is same as the top floor, i.e. x1 ¼ E1T X; and Eq. (4) can be rewritten as y€ þ

1 k uð_yÞ þ D y ¼ 2lE1T FY€ mD mD

ð13Þ

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By combining Eqs. (12) and (13), the following augmented matrix equation for the structure incorporating semi-active MR-TLCD is obtained as  ¼ FAW ðtÞ þ FAs MA Y€ þ CA Y_ þ KA Y þ FAp ðYÞ

ð14Þ

in which the ðm þ 1Þ-dimensional generalized displacement  the ðm þ 1Þ-dimensional generalized mass vector Y; matrix MA ; damping matrix CA and stiffness matrix KA ; the ðm þ 1Þ-dimensional generalized external force vector FAW ; the passive control force vector FAp ; and the semiactive control force vector FAs are given by ( ) Y Y ¼ ; ð15aÞ y 2 3 1 2 l2 T I 2 m F E D 17 6 m l ð15bÞ MA ¼ 4 5; "

lE1T F

1 #

" AL ¼ ( FN ¼

F¼

Imþ1

2MA21 KA

2MA21 CA

0mþ1

)

2MA21 Bp ( ) 0mþ1 (

U ¼

0mþ1

MA21 FAW 0mþ1 2MA21 Bp

# ;

up ð_yÞ; ;

us ð_yÞ

Z_ ¼ AZ þ FðtÞ þ U

ð15fÞ

FAs ¼ 2Bp us ð_yÞ; ( ) 2FT E1 =l Bp ¼ 1=mD

ð15gÞ

U ¼ 2Bus ð_yÞ

KA ¼

FAW

0m

0m

;

ð15cÞ

V 2kD FT E1 =l

0 kD =mD ( T ) F FW ; ¼ 0

# ;

ð15dÞ

ð15eÞ

ð15hÞ

where Im is the m-dimensional identity matrix. Since the determinant of the generalized mass matrix lMA l ¼ 1 þ ð1 2 l2 ÞmD

m X

f2i1 . 0

ð16Þ

ð18dÞ

ð18eÞ

Eq. (17) is a set of nonlinear equations and can be linearized to cater for the application of LQ control strategy. Applying the stochastic linearization procedure [20,21] to Eq. (17) yields

FAp ¼ Bp up ð_yÞ;

"

0m

ð18cÞ

)

where the coefficient matrix A ¼ AL þ AN and 2 3 0 0mþ1 0mþ1;m 5; AN ¼ 4 0mþ1 0mþ1;m 2MA21 Bp ceq sffiffiffiffiffiffiffiffiffiffi 2E½_y2  ceq ¼ rdAD ; p ( ) 0mþ1 ; B¼ MA21 Bp

CA ¼

J

ð18bÞ

ð19Þ

ð20aÞ

ð20bÞ ð20cÞ ð20dÞ

in which E½· denotes the expectation operator; ceq is the equivalent damping efficient of the passive damping force component up and related to the mean square velocity of the liquid. The linearized state-space equation (19) will be used for optimal control design.

4. Optimal control law

i¼1

where fi1 is the first element of the ith mode vector in mode matrix F; then the generalized mass matrix MA is nonsingular and its inverse matrix MA21 exists. Premultiplying Eq. (14) by MA21 and rewriting it in the state space yield  Z_ ¼ AL Z þ FN ðZÞ þ FðtÞ þ U

ð17Þ

in which the ð2m þ 2Þ-dimensional generalized state vector Z; the ð2m þ 2Þ-dimensional coefficient matrix AL ; the ð2m þ 2Þ-dimensional nonlinear force vector FN ; the external force vector F; and the control force vector U are as follows ( ) Y Z¼ ; ð18aÞ Y_

Perfect and complete observation is assumed in this study and then the optimal control design of system (19) can be performed directly based on the dynamical programming principle. Optimal control law commanding the semi-active damping force component of the MR-TLCD is determined by minimization of the following performance index J in an infinite time interval [22] 1 ðT J ¼ lim LðZðtÞ; us ðtÞÞdt ð21Þ T!1 T 0 where L is a Lagrangian of continuous differential convex functional. The corresponding dynamical programming equation is obtained as [22]   ›V min L þ ðAZ þ UÞT ¼0 ð22Þ us ›Z

Y.Q. Ni et al. / Probabilistic Engineering Mechanics 19 (2004) 269–277

where V is called the value function. Both the Lagrangian functional L in the performance index (21) and the control force U defined by Eq. (20d) are related to the semi-active damping force us to be determined. When the optimal linear quadratic (LQ) control strategy is implemented, the Lagrangian L and the value function V are represented by L ¼ Z T SZ þ Ru2s ;

ð23aÞ

V ¼ Z T PZ

ð23bÞ

and the LQ control damping force is then obtained from Eq. (22) as us ¼ R21 BT PZ

ð24Þ

where S is a positive semi-definite symmetric constant matrix; R is a positive constant; and P is a symmetric constant matrix which is determined from the following algebraic Riccati equation S þ AT P þ PT A 2 R21 PT BBT P ¼ 0

ð25Þ

In general, the LQ control damping force given by Eq. (24) is related to the state vector Z; and does not necessarily meet Eq. (6b). The clipping treatment is therefore performed to define the commanded control damping force ups which is implementable by the MR-TLCD: ( p F sgnð_yÞ F p $ 0 p us ¼ ð26Þ 0 Fp # 0 with F p ¼ R21 BT PZ sgnð_yÞ

ð27Þ

which satisfies the dynamical programming equation (22) under the dissipative damping force constraint (6b) according to the variational principle [23]. The commended optimal damping force defined in Eq. (26) is implementable by the MR-TLCD. Note that the dissipative control force usually mitigates structural vibration response effectively through the dissipation of structural energy. The optimal control force defined by Eq. (26) can be further represented in the form of dissipative force. By partitioning the matrices " # S1 S2 S¼ ; ð28aÞ ST2 S3 " # P1 P2 P¼ ; ð28bÞ PT2 P3 " # A1 A2 A¼ ð28cÞ A3 A4 and letting S1 ¼ 0; the optimal control force can be represented as F p ¼ R21 BTp QT3 Y_ sgnð_yÞ

ð29Þ

273

where Q3 ¼ P3 MA21 which can be determined directly from the following reduced-order Riccati equation S3 2 C TA QT3 2 Q3 C A 2 R21 Q3 Bp BTp QT3 ¼ 0 C0A

ð30Þ

C 0A

and ¼ ½0mþ1;m Bp ceq : Making in which C A ¼ CA þ use of Eq. (26) with Eq. (27) or Eq. (29), voltage (or current) input to the MR fluid and therefore the fluid yield stress ty can be adjusted optimally according to the clipped LQ control strategy for the linearized system.

5. Response evaluation To evaluate the random response of the controlled structural system with semi-active MR-TLCD under wind loading, the clipped optimal control damping force given in Eq. (26) is first linearized statistically as [20,21] T _ ups ¼ Cseq Y

ð31Þ

where the equivalent coefficient vector is    1 › T T _ Q B þE lBp Q3 Ylsgnð_yÞ Cseq ¼ 2R 3 p ›Y_

ð32Þ

Then the augmented matrix equation for the controlled structural system with semi-active MR-TLCD becomes MA Y€ þ C~ A Y_ þ KA Y ¼ FAW ðtÞ C 00A

C 00A

ð33Þ T Bp Cseq :

where C~ A ¼ C A þ and ¼ Eq. (33) expresses a linearized vibration system subjected to external excitation. The random response of a linearized system can be calculated by using the frequency-response function [21]. For the system (33), the frequency-response function matrix is HðjvÞ ¼ ðKA 2 v2 MA þ jvC~ A Þ21 ð34Þ pffiffiffiffi where j ¼ 21: With Eq. (34), the cross power spectrum matrix of the random response is obtained as SY Y ðvÞ ¼ HðjvÞSF ðvÞH T ð2jvÞ

ð35Þ

in which the power spectrum matrix of the random excitation due to wind loading is expressed as " T # F SW ðvÞF 0m£1 SF ðvÞ ¼ ð36Þ 01£m 0 where SW ðvÞ is given in Eq. (9). With Eq. (35), the mean square (MS) response of the controlled system (33) can be evaluated by ðþ1 E½Y 2i  ¼ SY i Y i ðvÞdv; ð37aÞ 21

E½Y_ 2i  ¼

ðþ1 21

v2 SY i Y i ðvÞdv

ð37bÞ

For the semi-actively controlled structural system, the cross power spectrum matrix of the displacement response is then

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obtained by using the transformation X ¼ FY as SXX ðvÞ ¼ FSYY ðvÞFT

ð38Þ

where the cross power spectrum matrix SYY ðvÞ of the modal displacement response is a sub-matrix of SY Y ðvÞ given in Eq. (35). Then the MS displacement and acceleration responses of the semi-actively controlled structural system can be evaluated by ðþ1 SXi Xi ðvÞdv; ð39aÞ E½Xi2  ¼ 21

E½X€ 2i  ¼

ðþ1 21

v4 SXi Xi ðvÞdv

ð39bÞ

The MS optimal control damping force is T _ _ T E½up2 s  ¼ Cseq E½YY Cseq

ð40Þ

by which the MS yield stress and the MS external voltage (or current) input can be obtained. The MS responses of the corresponding passively controlled structure using MRTLCD as a passive damper can be evaluated similarly by eliminating terms involving the optimal control damping force. The control efficacy of the proposed optimal semi-active control strategy can be evaluated in terms of performance criteria [23]. The performance criteria used here are the percentage reduction of root-mean-square (RMS) response Kresponse and percentage RMS optimal damping force Kus relative to the structural total weight: Kresponse ¼

RMSðresponseÞp 2 RMSðresponseÞs £ 100% RMSðresponseÞp

pffiffiffiffiffiffiffiffi E½up2 s  Kus ¼ £ 100% trðMÞg

ð41Þ ð42Þ

where RMS represents the root-mean-square value; subscripts p and s denote the passively and semi-actively controlled responses of the structure, respectively; and trð·Þ

Fig. 3. Mass-normalized mode shapes of building.

Fig. 4. Wind power spectral density (PSD) at different heights.

is the trace operator of square matrix. Higher values of the percentage response reduction Kresponse and the percentage relative optimal damping force Kus indicate more efficient control capability.

6. Case study A 50-story residential building is now under construction in Hong Kong. The initial design of this building was found to not satisfy the wind-resistant requirement prescribed in the Hong Kong design code, and therefore use of various supplemental damping devices in this building has been studied. The height of the building is 161.65 m and the total mass trðMÞ ¼ 2:774 £ 107 kg. Modal properties of the building have been obtained from a precise three-dimensional finite element model. The natural frequencies of the first five modes in along-wind direction are 0.216, 0.940, 2.278, 3.941 and 5.932 Hz, respectively. Fig. 3 illustrates the mass-normalized mode shapes. With the obtained natural frequencies and mode shapes, a lumped-mass model of 51 DOFs (representing 51 floors) was formulated. The modal damping ratio is assumed to be 3.0% for all

Fig. 5. Wind cross power spectral density (CPSD) between two heights.

Y.Q. Ni et al. / Probabilistic Engineering Mechanics 19 (2004) 269–277

275

Fig. 6. RMS displacement response versus building height.

Fig. 8. Percentage reduction K (%) in RMS responses.

the modes. Since the first mode is dominant in the building vibration, the MR-TLCD system is tuned to the first natural frequency. The parameters of the wind loading spectrum are taken as [19]: ra ¼ 1:28 kg/m3, CD ¼ 1:2; V10 ¼ 45:3 m/s, a ¼ 0:19; KD ¼ 0:02; Ch ¼ 10 and Ai ¼ 1 (for unit equivalent projection area of the wind loading). The parameters of the MR-TLCD are taken as mD ¼ 2:774 £ 105 kg, pffiffiffiffiffiffiffiffi vD ¼ kD =mD ¼ 1:2195 rad/s (near the first natural frequency of the structure), l ¼ 0:8 and d ¼ 30 unless otherwise mentioned. Figs. 4 and 5 show the wind power spectral density at different heights and the wind cross power spectral density between two heights, respectively. The weight parameters in the performance index are taken as R ¼ 106 =trðMÞ2 ; S3 ¼ Diag½0:7; 0:8; 1:2; 1:5; 1:5; 0:3: Fig. 6 shows the RMS displacement response of the semi-actively controlled and passively controlled structure using the MR-TLCD and Fig. 7 shows the RMS acceleration response of the semi-actively controlled and passively controlled structure. It is seen that the structural response reduction by means of the semi-active real-time control is much more than that using the same MR-TLCD as a passive

damper. In particular, the semi-active MR-TLCD in conjunction with the proposed control strategy reduces significantly the acceleration response at all stories of the building. Fig. 8 illustrates the RMS displacement percentage reduction ðKd Þ and the RMS acceleration percentage reduction ðKa Þ by using the proposed control strategy with the RMS optimal damping force being 0.0088% of the total structural weight. About 46% displacement reduction and 72% acceleration reduction are achieved at the top floor. It is found that the percentage reduction in RMS displacement varies slightly with the building height in comparison with the percentage reduction in RMS acceleration. The percentage reduction of RMS acceleration at the higher stories is much larger than that at the lower stories. The RMS interstory drift percentage reduction ðKid Þ is also provided in Fig. 8. It is almost the same as the RMS displacement percentage reduction. The effect of the wind spectrum parameters on the semiactive control efficacy is then studied. Figs. 9 and 10 show the RMS displacement percentage reduction ðKd Þ and the RMS acceleration percentage reduction ðKa Þ under different mean wind velocity V10 but identical weight parameters in

Fig. 7. RMS acceleration response versus building height.

Fig. 9. RMS displacement percentage reduction Kd (%) under different V10 .

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Y.Q. Ni et al. / Probabilistic Engineering Mechanics 19 (2004) 269–277

and acceleration responses of the semi-actively controlled structure using MR-TLCD.

7. Conclusions

Fig. 10. RMS acceleration percentage reduction Ka (%) under different V10 :

the performance index. It is observed that the RMS displacement percentage reduction is enhanced with the increase of the mean wind velocity while the RMS acceleration percentage reduction is slightly affected by the mean wind velocity. Fig. 11 illustrates the percentage reduction of both RMS displacement and acceleration responses with different RMS optimal damping force of the MR-TLCD (response-1: percentage RMS optimal damping force Kus ¼ 0:0088% under weight parameter S3 ¼ Diag½0:7; 0:8; 1:2; 1:5; 1:5; 0:3; response-2: percentage RMS optimal damping force Kus ¼ 0:015% under weight parameter S3 ¼ Diag½1:7; 2:5; 3:5; 5:0; 5:0; 0:5). It is seen that both the displacement and acceleration response reduction capabilities can be enhanced by increasing the percentage RMS optimal damping force Kus ; but the enhancement is not linearly proportional to the increase of the percentage RMS optimal damping force. For allowed maximum RMS optimal damping force, optimal values of the weight parameters in the performance index can be determined by minimizing the RMS displacement

An optimal control method for tall building structures using semi-active MR-TLCDs has been developed based on the dynamical programming principle and the statistical linearization method. The MR-TLCDs combine the benefits of controllable smart materials and tuned liquid column dampers. The dissipative optimal damping force produced by MR-TLCDs through relative motion between the liquid and the container does not have the potential to destabilize the structural system. The proposed real-time control strategy is independent of the external excitation and applicable to tall buildings with arbitrary stories. The cases study of a 50-story tall building demonstrates that a semi-active MR-TLCD installed at the top floor driven by the proposed optimal control strategy can achieve significant response reduction in terms of displacement, interstory drift and acceleration, in comparison with that obtained by using a passive TLCD. It is also shown that the response reduction capability by using the semi-active MR-TLCD in conjunction with the proposed control method increases with the increase of the wind loading intensity.

Acknowledgements The work presented in this paper was supported by a grant from The Hong Kong Polytechnic University through the Area of Strategic Development Programme (Research Centre for Urban Hazards Mitigation) and by a grant from the Zhejiang Provincial Natural Science Foundation, China (Grant No. 101046). These supports are gratefully acknowledged.

References

Fig. 11. Response percentage reduction K (%) with different optimal damping force. (response-1: Kus ¼ 0:0088% and S3 ¼ Diag½0:7; 0:8; 1:2; 1:5; 1:5; 0:3; response-2: Kus ¼ 0:015% and S3 ¼ Diag½1:7; 2:5; 3:5; 5:0; 5:0; 0:5).

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