semi-active control of seismic response of tall with pedium

7/31/2019 Semi-Active Control of Seismic Response of Tall With Pedium

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SEMI-ACTIVE CONTROL OF SEISMIC RESPONSE OF TALL

BUILDINGS WITH PODIUM STRUCTURE USING ER/MR

DAMPERS

W. L. QU1 AND Y. L. XU2

1College of Civil Engineering and Architecture, Wuhan University of Technology, Wuhan 430070, Peoples Republic of

China2Department of Civil and Structural Engineering, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

SUMMARY

A tall building with a large podium structure under earthquake excitation may suffer from a whipping effect due tothe sudden change of building lateral stiffness at the top of the podium structure. This paper thus explores thepossibility of using electrorheological (ER) dampers or magnetorheological (MR) dampers to connect the podiumstructure to the tower structure to prevent this whipping effect and to reduce the seismic response of bothstructures. A set of governing equations of motion for the towerdamperpodium system is first derived, in whichthe stiffness of the member connecting the ER/MR damper to the structures is taken into consideration. Based onthe principle of instantaneous sub-optimal active control, a semi-active sub-optimal displacement controlalgorithm is then proposed. To demonstrate the effectiveness of semi-active control of the system underconsideration, a 20-storey tower structure with a 5-storey podium structure subjected to earthquake excitation isfinally selected as a numerical example. The results from the numerical example imply that, as a kind of intelligentcontrol device, ER/MR dampers can significantly mitigate the seismic whipping effect on the tower structure andreduce the seismic responses of both the tower structure and the podium structure. Copyright 2001 John Wiley& Sons, Ltd.

1. INTRODUCTION

Owing to increasing population, shortage of supply in land, and centralized service requirements,

modern cities often need many tall buildings. Some tall buildings are built as a tower structure with a

large podium to achieve a large open space for parking, shops, restaurants and a hotel lobby at ground

level. In most cases, the tower structure and the podium structure are built together on either a common

box foundation or a common raft foundation. There are no settlement joints or anti-earthquake joints

between the tower structure and the podium structure. The presence of the podium structure, whose

lateral stiffness may be larger than that of the tower structure, leads to a suddenly large lateral stiffness

change of the building at the top of the podium structure under seismic excitation Consequently, the

seismic response of the upper part of the tower structure will be significantly amplified, leading to the

so-called whipping effect. Such a problem cannot be easily solved using conventional structural

modifications.

This paper thus explores the possibility of using ER/MR dampers to connect the podium structureto the tower structure to prevent the whipping effect. Semi-active control of the seismic responses

THE STRUCTURAL DESIGN OF TALL BUILDINGSStruct. Design Tall Build. 10, 179192 (2001)DOI:10.1002/tal.177

Copyright 2001 John Wiley & Sons, Ltd. Received January 2001

Accepted January 2001

* Correspondence to: Y. L. Yu, Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, HungHom, Kowloon, Hong Kong. E-mail: [email protected]


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of both the tower structure and the podium structure can then be achieved by actively controlling

the parameters of the ER/MR dampers according to feedback from the system response. Semi-active

control needs little external power, but structures with semi-active control can adapt themselves to

harsh environments with uncertainties. Though the application of vibration control technology in this

study is similar to that for controlling two neighbouring buildings in principle, only a few investigations

of using semi-active control technology can be found in the literature compared with investigations ofpassive control technology (Gurley et al., 1994; Zhang and Xu, 1999) and active control technology

(Seto, 1994). Klein and Healy (1985) used cables to couple two adjacent buildings in which the cables

could be released and tightened to provide simple semi-active control. Christenson et al. (1999)

recently investigated the semi-active control of adjacent buildings using modern smart dampers with a

clipped-optimal control strategy. They concluded after extensive parametric studies that a smart

damping strategy could achieve nearly the same performance as the active control strategy.

This paper aims to use ER/MR dampers with a semi-active sub-optimal displacement control

algorithm to control the whipping effect on the tower structure and to reduce the seismic responses of

both structures. The equations of motion for the towerdamperpodium system are derived first. Based

on the principle of instantaneous sub-optimal active control, a semi-active sub-optimal displacement

control algorithm is then proposed. A 20-storey tower structure with a 5-storey podium structure

subjected to earthquake excitation is finally used to demonstrate the effectiveness of semi-activecontrol of the system under consideration.

2. EQUATIONS OF MOTION

Consider a two-dimensional system consisting of a tower structure and a podium structure subjected to

earthquake excitation (Figure 1(a)). Three cases are investigated: (1) the podium structure rigidly

connected to the tower structure; (2) the podium structure totally separated from the tower structure;

and (3) the podium structure connected to the tower structure by ER/MR dampers (smart dampers)

with supporting members. The equations of motion of the system are, however, presented in this paper

for case 3 only.

For a steady and fully developed flow, the shear resistance of MR fluids or ER fluids may be

modelled as having a friction component augmented by a Newtonian viscosity component, i.e. the so-called Bingham model. Based on a fifth-degree polynomial derived by Phillips in 1969 to depict the

Poiseuille flow of Bingham material in a rectangular duct, Gavin et al. (1996) found the approximate

solutions of the fifth-degree polynomial for either flow-type smart dampers or mixed-type smart

dampers. As a result, the relation between the damper force Pd (t) and the velocity e(t) of the damper

piston relative to the damper cylinder can be expressed as (Xu et al., 2000):

Pdt Cd e FdESgne 1

in which

Cd

C1

12oLAp

bh3A

pY F

dE

C

2E

LyE

hA

p P

y 2

For a flow-type damper,

C1 10Y C2E 207 10

10 04TEY TE

bh2yE

12Apo et3

180 W. L. QU AND X. L. XU

Copyright 2001 John Wiley & Sons, Ltd. Struct. Design Tall Build. 10, 179192 (2001)


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C1 10 bh

2ApY C2E 207

10

10 04TE

15V2

10 04T2EY V

bh

2Ap4

where 0 is the Newtonian viscosity, independent of applied electric/magnetic field; ty (E) is the

yielding shear stress controlled by the applied electric/magnetic field; L is the effective axial pole

length; Ap is the effective cross-sectional area of the piston; Py is the mechanical friction force of thedamper; h is the gap between the two electric/magnetic poles; and b is the circumference of the inner

electric/magnetic pole. Clearly, Fd is a function of the yielding shear stress and can be controlled

through a change in the applied field intensity, but Cd is independent of the applied field.

In consideration of real implementation, ER/MR dampers are used to connect the podium structure

to the tower structure through some structural members of certain axial stiffness. Thus, the

mathematical model for the ER/MR damperstructural member system can be seen as a damper and a

spring connected in series, as shown in Figure 1(b). When considering the smart damper and the

structural member to be connected in series, the spring force in the structural member is equal to the

force on the piston of the damper. Equation (1) should thus be correspondingly changed to

Cd

Kd e e

Fd

Kd Sgn e ut 5

where u(t) is the relative horizontal displacement between the tower structure and the podium structure

at the location where the ER/MR damper is installed.

Assume that the podium structure has N storeys, the tower structure has N m storeys, and that n(n < N) smart dampers are used to connect the podium structure to the tower structure. If the sequence

of floor numbers of the tower and podium structures is coded as indicated in Figure 1(a), the equations

Figure 1. Schematic diagram of tower structuresmart damperpodium structure system

For a mixed-type damper,

SEISMIC RESPONSE CONTROL OF TOWER-PODIUM STRUCTURE 181



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of motion of the towerdamperpodium system are derived as follows:

Mfxg Cfxg K KDfxg Kefeg Mf1gxgt

CdjKdj

ej ej FdjKdj

Sgn ej xk2 xk1 j 1; F F F ; n

)6

where [M], [C] and [K] are the (2N m) (2N m) mass, damping and stiffness matrices of the twostructures, respectively; [KD] is the additional (2N m) (2N m) stiffness matrix due to ER/MRdampers, corresponding to the relative floor displacements of the two structures; [Ke] is the additional

(2N m) n stiffness matrix due to the ER/MR dampers, corresponding to the slip displacements ofdampers; {x}, {x} and {x} are the (2N m) floor displacement vector, velocity vector and accelerationvector of the two structures, respectively; {e} is the slip displacement vector of the n smart dampers;

xk2 and xk1 are the displacements ofkth floor of the podium and tower structures, respectively, where

the jth damper is installed. For the sake of clarification, only the matrices [KD] and [Ke] are given as

follows:

KD

HKdHT 0 HKdH

T

0 0 0HKdH

T 0 HKdHT

26664 37775 7

Ke

HKd

0

HKd

2664

3775 8

Kd

Kdl 0 0

0 F FF

0

0 0 Kdn

26664

37775

nn

9

in which [Kd] is the n n damper stiffness matrix; [H] is the N n matrix converting the damperstiffness matrix into the global co-ordinate system; and the superscript T means the transposition of a

matrix.

If the state vector of the system is assumed to be {Z}T

= [{x}T

,{x}T

], equation (6) can be then

rewritten in the state-space form

f Zg AfZg Bfeg fDgxgt

CdiKdj

ej ej FdjKdj

Sgnej xk2 xk1 j 1; F F F ; l

9=; 10

where

A 0 I

M1K Kd M1C

" #Y B

0

M1Ke

" #Y fDg

f0g

f1g

" #

3. SEMI-ACTIVE CONTROL STRATEGY

As the slip displacements of ER/MR dampers can be actively adjusted by changing the yielding shear




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stress of smart materials, optimal control theory can be utilized to obtain the maximum seismic

response reduction of the tower and podium structures. In consideration of the fact that the tower

podium system often has very many dynamic degrees of freedom and the external excitation is

earthquake excitation, instantaneous optimal closed-loop control (Yang et al., 1987) is selected with

some modification. The instantaneous time-dependent performance index is defined by

J fZgTQfZg fegTRfeg 11

where [Q] is the weighting matrix for the structures instantaneous response vector and is a positive

semi-definite matrix; [R] is the weighting matrix for the dampers instantaneous displacement vector

and is a positive definite matrix. For an instantaneous closed-loop control configuration, the

instantaneous optimal damper displacement vector can be obtained by minimizing equation (11)

subject to the constraint of equation (10):

eTg FfZg 12

F t

2R1BTQ 13

where Dt is the time step.

From a practical point of view, the state response vector {Z} of the towerpodium system can rarely

be measured as a whole. It is thus often necessary to replace the state response vector {Z} by an

incomplete state measurement vector fZg, leading to so-called suboptimal control (Kosut, 1970).Letting the matrix [C] denote the measurement matrix, the relation between the original state vector

{Z} and the measurement vector fZg can be expressed as

fZg CfZg 14

Using the minimum error excitation method in the suboptimal control theory, one may have the

instantaneous suboptimal damper displacement vector.

feTg FVCTCVCT1fZg 15

where the matrix [V] is the solution of the following Lyapunov matrix equation:

A BFV VA BFT I 0 16

in which [I] is the unit matrix. The suboptimal damper displacement control, determined by equation

(15), guarantees the tendency toward the minimization of the performance index J in equation (11).

However, the desirable damper displacement vector feTg cannot always be produced by the smartdamper because the motion of the smart damper is dependent on the motion of the structures and only the

damper friction force vector {Fd} due to yielding shear stress in fluids can be controlled through a change

of the applied electric or magnetic field intensity. A reasonable approach is thus to control {Fd} such that

the actual displacement vector {e} traces the suboptimal displacement vector feTg as closely as possible.According to this principle, a suboptimal displacement control strategy is presented, as follows.

When the jth damper displacement ej is approaching the desired suboptimal damper displacement

eTj, the friction force in the damper Fdj should be set to its minimum value Fmin so as to let the jth

damper reach its suboptimal displacement as fast as possible. When the jth damper moves in the

opposite direction from the suboptimal damper displacement, the friction force Fdj in the damper




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should be set to a smaller value of the two quantities: Fmax or the actual damper force Kdj(xk2 xk1 ej) minus a small quantity Fo. In this way, the damper is always in motion to dissipatevibrational energy. This strategy can be stated mathematically as

Fdj

Fmin when ejeTj ej > 0

minfabsKdjxk2 xk1 ej F0; Fmaxg when ejeTj ej0j 1; F F F ; n

8>: 17where Fo, a control force adjustment, can be obtained through a parametric study of the given problem.

4. APPLICATION

For application, a 20-storey shear-type tower structure and a 5-storey shear-type podium structure are

considered. The mass, shear stiffness, and damping coefficient of the tower structure are uniform for

all storeys with a mass of 129 106 kg, a shear stiffness of 40 109 N m1, and a dampingcoefficient of 10 105 Ns m1. For the podium structure, the mass, shear stiffness, and dampingcoefficient are also uniform for all storeys with a mass of 1 29 106 kg, a shear stiffness of 80 109

N m1

, and a damping coefficient of 20 105 Ns m1.For the case of using smart dampers to connect the podium structure to the tower structure, five

smart dampers are used at each floor and a total of 25 identical smart dampers is required at the first

five floors for both the tower and podium structures. The basic parameters of the smart dampers and

material are listed in Table I. The axial stiffness Kd of the member connecting the damper to the

structures is taken as 80000 kN m1

. The weighting matrices [Q] and [R] used in the instantaneous

sub-optimal closed-loop displacement control are selected as follows, after a parametric study:

Q

104 0

F FF

104

103

F FF

103

103

F FF

103

102

F FF

102

2

666666666666666666666666664

3

777777777777777777777777775

g n m

g n

g n m

g n

18

R

106 0

F FF

F FF

0 106

2666664

3777775

55

19

)))

)

n + m

n

n + m

n




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A parametric study for selecting the weighting matrices indicates that the semi-active control

performance is not sensitive to the weighting matrix [R]. The ground motion of both tower and podium

structures is taken as the NS 1940 El-Centro ground excitation. The maximum peak acceleration of

the ground motion is scaled to 01 g for moderate seismic regions. The time steps used in thecomputation for semi-active control force adjustment and for structural response are taken as 001 and00002 s, respectively.

4.1 Semi-active control performance

To evaluate the semi-active control performance on mitigation of the whipping effect and the seismicresponses of both tower and podium structures, three cases are investigated. The first case is when the

podium structure is rigidly connected to the tower structure (Case 1). The second case is when the

podium structure is totally separated from the tower structure (Case 2). The last case is when the

podium structure is connected to the tower structure by smart dampers as specified above (Case 3).

Figure 2(a) depicts the variation of the maximum storey shear force of the tower structure with

height for the three cases while Figure 2(b) illustrates the variations of the maximum storey shear force

of the podium structure with height. The maximum storey shear force of the tower structure in Case 1

jumps from 11 000 kN at the 5th storey to 26 500 kN at the 6th storey. The maximum storey shear

forces of the tower structure above the 6th floor in Case 1 are all much larger than those of the tower

structure in Case 2. Though the maximum storey shear forces of the tower structure in the first five

storeys are reduced to some extent in Case 1, the maximum storey shear forces of the podium

structures are increased in Case 1 compared with Case 2. This is because of the sudden change of

lateral stiffness of the towerpodium system in Case 1, resulting in the so-called whipping effect.

Clearly, the whipping effect is quite unfavourable in the earthquake-resistant design of building

structures.

With the installation of smart dampers, the whipping effect is totally eliminated. There is no sudden

large change of the maximum storey shear force in the tower structure at the 6th storey. The maximum

storey shear forces of the tower structure with smart dampers at the 6th storey above are much smaller

than those of the tower structure to which the podium structure is rigidly connected (Case 1) and are

almost the same as those when the tower structure is separated from the podium structure (Case 2). The

maximum storey shear forces of the tower structure with smart dampers in the first five storeys are

much smaller than those of the tower structure separated from the podium structure (Case 2).

Furthermore, the maximum storey shear forces of the podium structure with smart dampers are the

smallest among the three cases.

The above observations with respect to the maximum storey shear force can also be applied to the

maximum floor displacement response. Figure 3(a) shows the variations of the maximum floor

displacement response of the tower structure with height for the three cases, while Figure 3(b)

describes the variations of the maximum floor displacement response of the podium structure with

height. All these results imply that the semi-active control technology may be a good solution for the

problem under consideration.

Table I. Basic parameters of smart damper and material

Parameters of smart damper Parameters of smart material

L (m) h (m) b (m) Ap (m2) Py (kN) (kPa s) tymin (kPa) tymax (kPa)

05 0002 075 004 005 00002 005 140




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4.2 Two extreme cases of semi-active control performance

As described in Sections 2 and 3, only the friction force Fdj due to yielding shear stress in fluids of the

jth smart damper can be controlled through the change in the applied electric or magnetic field. Thus, if

the electric or magnetic field is set to zero, the smart damper actually becomes a passive viscous

damper, which is called the passive-off mode. The passive-off mode is a kind of minimum guarantee

of the damper performance if the electric or magnetic field loses its function during an earthquake. If

the electric or magnetic field is set to its maximum, the smart damper actually becomes a hybrid

Figure 2. Comparison of maximum storey shear force for different types of connections




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passive damper consisting of both viscous and friction dampers. The later case is often called the

passive-on mode.

Figures 4(a) and 4(b) demonstrate the variations of the maximum storey shear force of the tower and

podium structures, respectively, with the height for the passive-off mode, passive-on mode, semi-

active mode as well as the rigid connection of the two structures. It can be seen from these figures that

the damper control performances in all the three modes are superior to those obtained from the rigid

connected structure. Among the three modes, the semi-active mode has the best performance while the

Figure 3. Comparison of maximum floor displacement for different types of connections




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passive-on mode has the least control capability. The passive-on and passive-off modes considerably

reduce the seismic responses of the 6th floor and above, but at the same time they enlarge the seismic

responses of the first five storeys. For the semi-active mode, the seismic responses of both structures at

all floors are significantly reduced. The maximum control forces of semi-active dampers for achieving

such a seismic response reduction are also computed. The maximum control force of the single smart

damper is 37, 53, 247, 366 and 407 kN at the 1st, 2nd, 3rd, 4th and 5th floor, respectively. These

damper forces are quite small compared with the storey shear forces.

Figure 4. Comparison of maximum storey shear force for different types of control modes




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4.3 Parametric study

To facilitate the design of smart dampers, the effects of several key damper parameters on the smart

damper performance are investigated in this section. These parameters include the maximum yield

shear stress of smart materials, the fluid viscosity of smart materials, the axial stiffness of the member

connecting the smart damper to the structures, and the control force adjustment.

Figure 5 shows the variations of maximum shear force in the 1st and 6th storeys of the tower

structure and the 1st storey of the podium structure with the maximum yield shear stress of controllable

fluids in the smart dampers. The fluid viscosity of the smart material is fixed at 00002 kPa s. It is seen

that at the beginning, the maximum shear forces in the 1st storey of both the tower and podium

Figure 5. Relationship between maximum storey shear force of structures and maximum yielding shear stress ofsmart material

Figure 6. Relationship between maximum storey shear force of structures and viscous coefficient of smart material




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structures decrease with increasing maximum yield shear stress of the smart material. After the

maximum yield shear stress is increased to a certain level, the maximum storey shear forces remain

almost constant. The maximum shear force in the 6th storey of the tower structure, on the other hand,

tends to increase with increasing maximum yield shear stress at the first stage. It then converges to a

constant value, less affected by the maximum yield shear stress of smart material. Clearly, the

maximum yield shear stress of the smart material should not be so small as to hamper the semi-active

control performance. The maximum yield shear stress of smart material, on the other hand, is not

expected to be too large from a practical point of view. The rational value of the maximum yield shear

stress should thus be selected once the maximum shear forces of both the tower and podium structures

reach a stable level. For the building investigated in this study, (ty)max is taken as 14 kPa.

Figure 6 depicts the variations of maximum shear forces in the 1st and 6th storeys of the tower

structure and the 1st storey of the podium structure with the fluid viscosity of the smart material. The

minimum and maximum yield shear stresses of the smart material are taken as 005 and 140 kPa,respectively. It is seen that the maximum storey shear forces vary remarkably with fluid viscosity. The

maximum storey shear forces become relatively large when the fluid viscosity is either very small or

very large. There is an optimal value of fluid viscosity at which the maximum storey shear force at a

given storey reaches the smallest. However, the optimal value for the 1st storey of the tower structure

is different for those for the 1st storey of the podium structure and the 6th storey of the tower structure.

A compromise must thus be made accordingly. In this study, the fluid viscosity is taken as 00002 kPa s.Figure 7 demonstrates the variations of the maximum storey shear forces in the 1st and 6th storeys of

the tower structure and the 1st storey of the podium structure with the axial stiffness of connecting

member Kd of the smart dampers. The maximum storey shear forces in the three storeys all decrease

with increasing axial stiffness of members at the first stage and then they all converge and remain

almost constant in a wide range of axial stiffness. In this study, the rational axial stiffness is selected as

80000 kN m1

.

The effect of control force adjustment Fo on the performance of semi-active control can be seen

from Figure 8, in which the maximum base shear force of the tower structure is plotted against the

control force adjustment. The control force adjustment is set to be the same for all smart dampers. It is

Figure 7. Relationship between maximum storey shear force of structures and axial stiffness of connectingmembers




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seen from Figure 8 that there is an optimal value of control force adjustment at which the base shear

force of the tower structure is the smallest. A large control force adjustment degrades the control

performance as expected. The optimal control force adjustment for the studied building system is

about 28 kN, which is about 36% of the maximum control force of a single smart damper installed atthe top storey of the podium structure.

5. CONCLUSIONS

The use of ER/MR dampers to connect the podium structure to the tower structure to prevent the tower

structure from the whipping effect when they are subjected to earthquake excitation has been explored

in this study. A set of governing equations of motion for the towerdamperpodium system and asemi-active sub-optimal displacement control algorithm were correspondingly developed. It turned

out that if the key parameters of smart dampers were properly selected, the smart dampers could not

only prevent the tower structure from the whipping effect but also reduce the seismic responses of both

the tower and podium structures at the same time. It was also found that if the electric or magnetic field

lost its function during an earthquake, the smart dampers were still workable as passive viscous

dampers. The selection of beneficial key parameters of smart dampers was also demonstrated through

a numerical example. These key parameters included the maximum yield shear stress of smart

materials, the fluid viscosity of smart materials, the axial stiffness of the member connecting the smart

damper to the structures, and the control force adjustment in the control algorithm.

ACKNOWLEDGEMENTS

The writers are grateful for financial support from the National Natural Science Foundation of China

under Grant NNSF-50038010 and Hong Kong Polytechnic University through its Area of Strategic

Development Programme in Structural Vibration Control.

REFERENCES

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Figure 8. Maximum base shear force of tower structure versus control force adjustment




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