EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS
Earthquake Engng Struct. Dyn. 2009; 38:1381–1399
Published online 23 February 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.907

A substructure shaking table test for reproduction of earthquake
responses of high-rise buildings
Xiaodong Ji1, ∗, † , Kouichi Kajiwara2 , Takuya Nagae2 , Ryuta Enokida1
and Masayoshi Nakashima1
1 Disaster Prevention Research Institute, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan
2 E-Defense, National Research Institute for Earth Science and Disaster Prevention, Shinjimicho, Miki,

Hyogo 673-0515, JapanSUMMARY
When subjected to long-period ground motions, high-rise buildings’ upper floors undergo large responses.
Furniture and nonstructural components are susceptible to significant damage in such events. This paper
proposes a full-scale substructure shaking table test to reproduce large floor responses of high-risebuildings. The response at the top floor of a virtual 30-story building model subjected to a synthesized
long-period ground motion is taken as a target wave for reproduction. Since a shaking table has difficulties
in directly reproducing such large responses due to various capacity limitations, a rubber-and-mass system
is proposed to amplify the table motion. To achieve an accurate reproduction of the floor responses, acontrol algorithm called the open-loop inverse dynamics compensation via simulation (IDCS) algorithm isused to generate a special input wave for the shaking table. To implement the IDCS algorithm, the model
matching method and the H∞ method are adopted to construct the controller. A numerical example ispresented to illustrate the open-loop IDCS algorithm and compare the performance of different methodsof controller design. A series of full-scale substructure shaking table tests are conducted in E-Defense
to verify the effectiveness of the proposed method and examine the seismic behavior of furniture. The
test results demonstrate that the rubber-and-mass system is capable of amplifying the table motion by afactor of about 3.5 for the maximum velocity and displacement, and the substructure shaking table test
can reproduce the large floor responses for a few minutes. Copyright q 2009 John Wiley & Sons, Ltd.
Received 18 August 2008; Revised 17 January 2009; Accepted 20 January 2009

KEY WORDS:

shaking table test; substructure; IDCS algorithm; model matching method; H∞ method;high-rise building∗ Correspondence to: Xiaodong Ji, Disaster Prevention Research Institute, Kyoto University, Gokasho, Uji, Kyoto†

611-0011, Japan.
E-mail: xiaodong.ji@ks5.ecs.kyoto-u.ac.jp

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1. INTRODUCTION
A long-period, long-duration ground motion induced by a subduction earthquake in the Pacificocean-ridge zone is expected to attack Japan in the next few decades with a very high probability[1]. High-rise buildings are prone to sustain significant responses characterized by many cyclesof vibrations with large velocities and displacements. Two types of damages are likely to occur
in such events. Beam-to-column connections are susceptible to serious low-cycle fatigue failures
resulting from many cycles of plastic deformations [2]. Furniture and nonstructural components
in top stories are likely to be shaken significantly, resulting in sliding, falling, overturning, andcollisions of these elements [3].
To examine the seismic behavior of furniture under large floor responses, large-scale tests area must, because it is very difficult to make reduced-size furniture without the loss of similitude.
Apparently, a full-scale test of a high-rise building as a whole is impracticable. One alternative isthe application of the online hybrid test, which is capable of handling a large-scale structure by
the introduction of substructuring techniques [4–8]. The test, however, adopts quasi-static loading,which is not feasible in the simulation of sliding and overturning behavior of furniture. Recently,
the real-time hybrid test has been explored extensively [9–13], but the development is yet in its
infancy, and the application is limited to simple structures loaded by smaller actuator devices.
Another alternative is a substructure shaking table test using large-scale shaking table facilities.
The simple way is to extract the top portion of a high-rise building as a test specimen and use thefloor response of that portion as the table input motion. Unfortunately, this is also unfeasible inmost cases due to the limitations of various capacities associated with the shaking table, e.g. thecapacities of the maximum displacement and velocity, and the amount of oil supply.
To overcome these difficulties, this paper proposes a new full-scale substructure shaking table
test. Three inventions are included in this effort. One is the substructuring, and only a portionof the structure instead of the entire structure is physically tested on the table. Another is theintroduction of a rubber-and-mass system to amplify the motion of the shaking table, and the lastis the development of a control algorithm to generate a special input wave that can reproduce thefloor response on the test specimen. A schematic view of the substructure shaking test is shownin Figure 1, in which the top floor is physically tested, while the lower portions are replaced by
two layers of rubber-and-mass structures. With substructuring, the tested structure can maintain a

Substructure
Rubber-and-masssystemShaking table
High-rise builiding

Figure 1. Rubber-and-mass system and substructure.
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reasonable scale, and the rubber-and-mass system serves as an amplifier. Because of the nonlinear
nature of the prototype high-rise building, the input wave of the shaking table has to be chosencarefully even if the rubber layers are linearly elastic.
The paper consists of four parts. First, a virtual 30-story steel building is introduced, and the
response of the top floor of this building when subjected to a long-period, long-duration groundmotion is presented. This response becomes a target response to be reproduced on the test specimen.
Second, a control algorithm, i.e. the open-loop inverse dynamics compensation via simulation(IDCS) algorithm, is presented for the identification of the table motion that can reproduce thetarget response. For design of the controller, two methods, the model matching method and the
H∞ method, are adopted. Third, the effectiveness of the control algorithm is calibrated against a
numerical example for a two Degrees of freedom (DOFs) system. Fourth, physical shaking table
tests are implemented for the validation of the proposed full-scale substructure shaking table test,
and behavior of and damage to furniture are discussed.

2. NONLINEAR TIME HISTORY ANALYSIS OF A HIGH-RISE BUILDING MODEL2.1. Numerical model of a high-rise building and ground motion
A virtual 30-story building was considered in this study. The heights of the first story and theother stories were 4.2 and 3.4 m, respectively, and the total height of the building was 102.8 m.
A lumped mass and shear spring model was adopted to simulate earthquake responses of thebuilding, in which the lumped mass represented the story mass, and the shear spring represented
the story stiffness. The parameters of the model were determined according to standard practicesof seismic design in Japan. All story masses were assigned with an identical weight of 1000 tons,
and the distribution of the initial story stiffness along the height was taken to follow the design
story shear distribution. Nonlinearity of the story stiffness was described by a tri-linear kinematic
hardening model with a skeleton curve as shown in Figure 2, in which F, , and k1 denoted
the story shear force, story drift, and initial stiffness, respectively. The shear forces at the firstand second yielding points were assumed as 1.2 and 2.0 times the designed yield strength (Q d ),and the second and third stiffness after yielding was assumed to be 0.7 and 0.1 times the initialstiffness, respectively. The fundamental natural period of the building was estimated as 2.8 s andthe damping ratio was assigned to be 2% for the first mode. The building was subjected to groundmotions in two horizontal directions, and the same numerical model was adopted for both thedirections.
F
2.0·Qd

0.1K1
2nd yielding point
0.7K1

1.2·Qd

1st yielding point
K1
Δ1

Δ2

Δ

Figure 2. Skeleton curve of story restoring force behavior.
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3

6

1.5

Sv(m/s)Acceleration (m/s2)

1384

0
- 1.5
-3

0

50

200

3.7 s
0

1

0

1

(b)

2
3
Period (s)

4

5

4

5

6

3
1.5
0
- 1.5
-3

2
0

250

Sv(m/s)Acceleration (m/s2)

(a)

100
150
Time (s)

4

4
2
0

0

(c)

50

100
150
Time (s)

200

250
(d)

2.8 s
2
3
Period (s)Figure 3. HigashiYuuenchi ground motion: (a) time history in EW direction; (b) velocity response spectrum
in EW direction; (c) time history in NS direction; and (d) velocity response spectrum in NS direction.

A synthesized long-period, long-duration ground motion named HigashiYuuenchi ground motion
was adopted as the input excitation, as shown in Figure 3 [14]. The ground motion was estimatedbased on a hypothetical Nankai earthquake (M8.4), with a rupture length of 180 km and a focus
depth of 10 km. The ground motion has a duration of 270 s, and predominant periods of 3.7 and2.8 s in the EW and NS direction, respectively. The peak ground acceleration (PGA) values are
184 and 249 m/s2 , and the peak ground velocity (PGV) values are 0.42 and 0.73 m/s in the twohorizontal directions.
2.2. Earthquake response of the high-rise building
The Newmark- method ( = 0.25 and  = 0.5) was used for the time integration to achieve thenecessary responses. In the HigashiYuuenchi ground motion, the maximum story drift angles occur
in the third story, reaching 0.83 and 1.68% in the EW and NS direction, with the associated story
ductility of 2.0 and 4.1. The peak values of the acceleration, velocity, and displacement of the
top floor are 3.82 m/s2 , 1.54 m/s and 0.93 m in the EW direction, and 4.20 m/s2 , 2.15 m/s and1.24 m in the NS direction. The time histories of the acceleration responses at the top floor andthe corresponding Fourier amplitude spectra are shown in Figure 4. The predominant frequenciesof the floor responses are 0.35 and 0.34 Hz in the EW and NS direction, respectively, exhibitinga notable long-period motion. Owing to the long-period, long-duration vibration, the cumulative
displacement at the top floor reaches about 180 m.
To investigate the furniture behavior, a simple way is to extract the top story as a test substructureand directly use the top floor’s response as the table input. The associated test demands for suchdirect reproduction are shown in Table I. Table I also shows the capacities of the Hyogo Earthquake
Engineering Research Center (E-Defense) facility, featuring a table of 20 m by 15 m in the plan
[15, 16]. The table is actuated in 3D and can accommodate a specimen up to a weight of 12 MN.
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5

Amplitude (m/s2)

Acceleration (m/s2)

SUBSTRUCTURE SHAKING TABLE TEST

2.5
0
-2.5
-5

0

50

100
150
Time (s)

200

250
(b)

5

Amplitude (m/s2)

Acceleration (m/s2)

(a)

2.5
0
-2.5
-5

0

50

(c)

100
150
Time (s)

200

250

0.8
0.35 Hz
0.6
0.4
0.2
0
0.1

1
Frequency (Hz)

10

0.8
0.6

0.34 Hz

0.4
0.2
0
0.1

(d)

1
Frequency (Hz)10

Figure 4. Acceleration responses of top floor: (a) time history in EW direction; (b) Fourier
amplitude spectrum in EW direction; (c) time history in NS direction; and (d) Fourier
amplitude spectrum in NS direction.
Table I. Demands for direct reproduction and capacities of E-Defense’s shaking table.

Acceleration (m/s2 )Velocity (m/s)
Displacement (m)Oil amount (kl)DemandCapacity

4.20
2.15
±1.24
306

9.0
2.0
±1.0
20.0

The specification limits of the maximum acceleration, velocity, and displacement of the table are
9.0 m/s2 , 2.0 m/s, and ±1.0 m. The amount of oil required is related to the cumulative displacementof the shaking table. The oil supply is normally capped because of the accumulator used for thetable facility.
As evidenced in Table I, the expected floor response cannot be used as the direct input, because
its maximum velocity and displacement exceed the capacity limits of the E-Defense table by 7.5
and 24.0%, and the amount of oil required is about 15 times the oil supply limit.
To overcome these limitations, a rubber-and-mass system is proposed, which is set beneath thetest substructure model to amplify the motion of the shaking table. The rubber-and-mass system
consists of multiple rubber layers and lumped masses. Rubber bearings are adopted because of theirstable elastic behavior. The number of rubber layers is determined according to the displacement
demand and the deformation capacity of each rubber layer. By a suitable design of the stiffness ofrubber layers and the weight of masses, the entire test specimen that comprises the substructure
model and rubber-and-mass system is able to achieve linear dynamic properties similar to thoseof the prototype building. However, nonlinearity of the prototype cannot be reflected by the test
specimen, because the rubber-and-mass system works nearly elastically. To achieve an accurate
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reproduction of the top floor’s response, it becomes imperative to generate a special input wave
for the shaking table, rather than to directly use the original ground motion itself.

3. CONTROL ALGORITHMS
3.1. IDCS algorithm
Identification of an input wave from a target response belongs to an inverse dynamics problem.
Real-time adaptive control approaches, for example, the minimal control synthesis (MCS) method[17], have a strong potential of solving the inverse problem and allowing for the specimen’s
nonlinearity. To date, however, real-time adaptive control is still in the stage of development, andapplications have been limited to relatively simple structures shaken by rather small shaking tables.
On reflecting in the lack of maturity of the concerned technologies, the writers felt that the risk
would be too high yet for the large and complex shaking table used in this study (weighing close
to 20 MN (2000 ton) and controlled in 6DOFs) to adopt a real-time adaptive control strategy. For
this reason, a more robust, stable, and safer offline control approach, named the IDCS algorithm,
is adopted to generate the input wave of the shaking table. The basic idea of the open-loop IDCS
algorithm is to generate a special wave from a numerical feedback loop and to control the behaviorof a physical specimen by this wave [18, 19].
A block diagram description of the open-loop IDCS algorithm is given in Figure 5, in which Be
represents the test specimen; B represents the numerical model of Be ; W denotes the controller;r represents the target wave, i.e. the response of the top floor; u represents the input wave of
the shake table; ye denotes the specimen’s response; and y denotes the calculated response by
the numerical model. The wave u can be obtained from the numerical loop. It is expected that
if the numerical model B accurately represents the test specimen Be , the real response ye of thetest specimen under the excitation of the generated input wave u should be sufficiently close to
the response y of the numerical model. That is, a good controller W , designed to ensure that y
tracks r well, promises a reproduction. To implement the IDCS algorithm, the model matching
method and the H∞ method are adopted, since they show good performance in controller designfor linear time-invariant systems [20, 21].
3.2. Model matching method
The model matching method is a classical method of the controller design for linear systems. The socalled model matching means to determine a controller so that the transfer function of the feedback
system coincides exactly with the expected function. The frequency domain model matching
method is used in this paper, because it is simpler and more convenient in the computational workthan the time domain model matching method [20].

r

W

u

B

Be

y
ye

Figure 5. Principle of open-loop IDCS algorithm.
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SUBSTRUCTURE SHAKING TABLE TEST

r

+
_

e

W

u

B

1387

y

Figure 6. Feedback control system.

Consider a general block diagram for the feedback control system given in Figure 6, in which
the model B is also called the plant, and e represents the error between the system input r and thesystem output y. Let the transfer functions of the plant B and controller W , TB (s) and TW (s), be
expressed by
TB (s) =N B (s)M B (s)(1)

TW (s) =N W (s)
MW (s)(2)in which N B (s) and N W (s) denote the numerator polynomials of the plant B and controller W ,and M B (s) and MW (s) denote their corresponding denominator polynomials. The transfer functionof the feedback system, i.e. the transfer function from r to y can be obtained as follows:
Tr y (s) =N W (s)N B (s)TW (s)TB (s)=1+ TW (s)TB (s) MW (s)M B (s)+ N W (s)N B (s)(3)

To ensure that y accurately tracks r , the transfer function Tr y (s) should equal unity in theconcerned frequency range. The polynomials N B (s) and M B (s) can be calculated from state-space
matrices of the plant systematically. Herein the degrees of N B (s) and M B (s) are denoted as n B
and m B , respectively. The polynomials of the controller, N W (s) and MW (s), are designated to
be of degrees of n W and m W . n W and m W should be larger than n B and m B , respectively, andthe difference between n W and m W should be equal to the difference between n B and m B . By
reasonably assigning poles of the denominator polynomial of the transfer function Tr y (s), themagnitude of transfer function Tr y (s) approaches unity. Then, the coefficients of polynomials,
N W (s) and MW (s), can be determined by the Bott–Duffin inverse operation [22].
Although the model matching method is comprehensive in theory and simple in computation,some drawbacks exist in application, e.g. the assignment of suitable poles has to rely on theexperience of the controller designer, and trials and errors are required to obtain a good solution.
3.3. H∞ method
The H∞ method receives much attention due to its distinguished capability of designing an optimalrobust controller [21, 23]. In this paper, the H∞ mixed sensitivity method is used to construct a
controller. To represent the sensitivity of a feedback control system with respect to the external
disturbance and the plant’s uncertainty, the sensitivity function and the compensation sensitivity
function are defined, respectively, as follows:

Copyright q

S(s) = Tr e (s) =1
1+ TW (s)TB (s)(4)

T (s) = Tr y (s) =TW (s)TB (s)1+ TW (s)TB (s)(5)

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in which S(s) is the sensitivity function, i.e. the transfer function from r to e, and T (s) isthe compensation sensitivity function, i.e. the transfer function from r to y. To attenuate theadverse influence of external disturbance and plant’s uncertainty, functions S(s) and T (s) should
be minimized. However, minimization of S(s) and T (s) cannot be achieved simultaneously for thesame frequency, since the sum of them is unity at any frequency value. Two weighting functions
W S (s) and WT (s) are thus introduced to construct a mixed-sensitivity optimization problem. A
mixed-sensitivity cost function that penalizes both S(s) and T (s) is minimized as follows [24]: Ws (s)· S(s) 



 
 WT (s)· T (s)(6)∞

in which,  is a small positive value and  ∞ denotes the infinity norm of matrix [25]. Solving
Equation (6) by using the improved loop-shifting two-Riccati formulas [26], the transfer functionof the controller W can be obtained.
Although the theory and computation appear rather complicated, the H∞ method is convenient
for application, because the mixed-sensitivity problem can be solved systematically by the robust
control toolbox of the software Matlab [27]. Since the performance of the constructed controller isrelated to weighting functions, it is noted that care must be taken to select these weighting functions.
Selection of weighting functions depends on the experience of the controller designer, and thedesign involves trials and errors until the feedback system meets its design specifications [24].

4. NUMERICAL ILLUSTRATION
4.1. Numerical model
A numerical example is presented to illustrate the feasibility of the open-loop IDCS algorithmand compare the performance of the model matching method with the H∞ method. Since the test
specimen described in the following section is simplified into a two DOFs model, the numericalexample also chooses a model with two DOFs as shown in Figure 7. Consistent with the test
specimen, the masses of m 1 and m 2 were assigned as 3.60×105 and 3.95×105 kg, and thestiffnesses of k1 and k2 were assigned as 5.548 and 3.511 MN/m, respectively. The damping matrix
was assumed to be proportional to the stiffness matrix, and the first modal damping ratio wasdesignated as 3%.

y(t )
m2
k2
m1
k1
u(t )
Shaking table

Figure 7. Numerical simulation model.
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SUBSTRUCTURE SHAKING TABLE TEST1389

If the excitation acceleration of the shaking table and response accelerations of the model aretaken as the input and output, the associated state-space equation can be written asẋ(t) = Ax(t)+Bu(t)(7)y(t) = Cx(t)in which the state vector x(t) = [d(t) ḋ(t)]T consists of the relative displacement vector d(t) andthe relative velocity vector ḋ(t); y(t) denotes the output, i.e. the acceleration vector of the model;u(t) denotes the input, i.e. the excitation acceleration of the shaking table; and the matrices A, B,
and C represent the system matrix, control matrix and observation matrix, which can be constructedfrom the mass, damping, and stiffness matrices [28].
Transforming Equation (7) by the Laplace transform and then rearranging, the transfer function
matrix of the numerical model can be obtained as
T B (s) = C(sI−A)−1 B(8)where s denotes the Laplace variable. Substituting the parameters associated with the numericalmodel, the transfer function from the acceleration of the table to the acceleration response at the
top floor is expressed as:
0.106s 2 +7.614s +136.983
TB (s) = 4
s +0.946s 3 +34.158s 2 +7.614s +136.983(9)

The Bode diagram of the transfer function is shown in Figure 8. The first two modal frequenciesof the numerical model, f 1 and f 2 , were 0.34 and 0.86 Hz.
4.2. Controller designFirst, the model matching method was used to construct the controller. The denominator andnumerator of the controller’s transfer function TW (s) were assumed to be the polynomials of
6 and 4 degrees, respectively. Equation (3) indicates that the denominator of the transfer functionof the feedback system was a polynomial of 10 degrees. Through trial and error, the poles of thedenominator polynomial were assigned as [0.5, 0.5, 0.7, 0.7, 5, 8, 30, 30, 40, 40]T . Then coefficients
30
Magnitude (dB)

0
-30
-60
-90
-120
-150
0.1

f1 f2 110Frequency (Hz)100

Figure 8. Transfer function of numerical model.
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Model matching
methodModelmatching
methodH methodHinfinity method

10
Magnitude (dB)

0
-10
-20
-30
-40
-50

0.1

f1

f2 1

10

100

Frequency (Hz)Figure 9. Transfer function of feedback system.

of the numerator and dominator polynomials of the controller’s transfer function were calculated
from the Bott–Duffin inverse operation. The Bode diagram of the transfer function of the feedback
system is shown in Figure 9. It is notable that the magnitude of the transfer function is very close
to unity at the natural frequencies of the plant.
To compare with the model matching method, the H∞ method was also employed. To makethe feedback system insensitive at the resonant frequencies of the plant, a notch function [28] wasconstructed as shown in Equation (10) to serve as the weighting function Ws (s). On the other hand,to roll-off the control action at high frequencies where model uncertainties cannot be ignored,
the weighting function WT (s) should be chosen to have a high-pass filter’s property. Herein, theweighting functions are given by

1 (s 2 +2 h 1 1 s +21 ) 2 (s 2 +2 h 2 2 s +22 )Ws (s) =·s 2 +2h 1 1 s +21
s 2 +2h 2 2 s +22

(10)

WT (s) = s 2

(11)

in which 1 and 2 are the first two circular frequencies of the plant; h 1 and h 2 are the firsttwo modal damping ratios of the plant; and the other parameters are determined by trial and error
as follows: 1 = 2 = 1.25, h 1 = h 2 = 0.5,  = 0.005. Then the transfer function of the controller
was obtained by solving Equation (6) using the improved loop-shifting two-Riccati formulas [26].
Figure 9 shows the Bode diagram of the transfer function of the feedback system, indicating thatthe magnitude of the transfer function at natural frequencies of the plant is close to unity.
4.3. Numerical simulation results
The top floor’s response of the 30-story building is considered as the target wave r . The transfer
function from the target wave r to the shaking table’s input wave u is given by:
Tr u (s) =Copyright q

N W (s)M B (s)TW (s)=
1+ TW (s)TB (s) MW (s)M B (s)+ N W (s)N B (s)2009 John Wiley & Sons, Ltd.

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Acceleration (m/s2)

5
2.5
0
-2.5
-5
0

50

100

150

200

250

200

250

Time (s)(a)

Acceleration (m/s2)

5
2.5
0
-2.5
-5
0

50

(b)

100

150
Time (s)Figure 10. Responses of model in y direction: (a) response with model matching
method and (b) response with H∞ method.

By the convolution integration operation of the transfer function Tr u (s) and the target wave r , theshaking table’s input wave u can be obtained. Furthermore, the wave u is filtered by a lowpass
digital Chebyshev filter to cut highfrequency components. Under the excitation of the generated
waves, the responses of the simulation model were calculated by the time history analysis. The
responses in the y direction are shown in Figure 10. Note that the compass directions associatedwith the axes are EW for the x direction and NS for the y direction. Comparing Figure 10 with
Figure 4(c), it is notable that the model responses when using both the model matching method
and the H∞ method are in good agreement with the target wave. Based on the Fourier amplitude
spectra, the error of reproduction is defined as follows:(Star − Sres )2
Er = 
×100%(13)(Star )2
in which, Star and Sres represent the Fourier amplitude spectrum of the target wave and that of
the model response, respectively. When using the model matching method, the reproduction errors
were 2.9 and 4.8% in the x and y direction, respectively; while when using the H∞ method, thereproduction errors were 4.6 and 6.9%. The results of the numerical example verify that the IDCS
algorithm is effective, and both the model matching method and the H∞ method are similar in theaccuracy of the controller design. In the following shaking table tests, only the model matching
method was used to construct the controller.
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5. EXPERIMENTAL VALIDATION
5.1. Test specimen
To validate the techniques developed for the substructure shaking table test and to investigate theseismic behavior of furniture, a series of full-scale tests were conducted in E-Defense. As shownin Figure 11, a five-story steel frame was placed on a rubber-and-mass system. The steel frame had
a plan dimension of 9 m by 12 m and a total height of 18.5 m. The story stiffness of the frame wasvery large (more than 100 times the stiffness of each rubber layer) so that the frame would behave

Figure 11. Overview and drawings of test specimen: (a) specimen view; (b) elevation in x
direction; and (c) elevation in y direction.
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as a rigid body during the vibration. Five stories were used only to expand the floor areas available
for furniture installation and make it possible to conduct many furniture tests simultaneously. Afterthe installation of furniture, the total mass of the frame model weighted 3.95×105 kg.
Because the maximum displacement demand was larger than the drift (horizontal displacement)
limit accepted by normal rubber bearings, two layers were adopted for the mass-and-rubber system.
To achieve an approximately equal drift between the two layers, the stiffness ratio and the mass
ratio between the first and second rubber layers were designed to be 1.5:1 and 1:1. Four rubber
bearings were placed at the corners of each layer. All rubber bearings were 1000 mm in diameterand 285 mm in height. In reference to the weight of the test structure placed on top of the massand-rubber system, a mass of 3.60×105 kg was chosen for the lumped mass, which was made of
a 0.8 m thick concrete slab of 12 by 15 m in plan. The horizontal stiffness was given as 1.37 and
0.878 MN/m for each rubber bearing in the first and second layers, which was achieved by the
adjustment of the shear modulus of the rubber material.
5.2. Modal parameter identification
To understand the dynamic properties of the test specimen and validate the numerical model,
system identification on the specimen was conducted before the seismic tests. A Gaussian white
noise with a pass-band of 0.12 ∼ 10 Hz, a root mean square (RMS) amplitude of 73 gal, and a
duration of 250 s, was used as the input wave of the shaking table. The accelerations of boththe excitation and response were recorded by servo accelerometers at an interval of 0.005 s. The
transfer function was obtained as a quotient of the auto-spectral density of the output signal
over the cross-spectral density of the input and output signals. In the signal processing, a digital
filter was used to eliminate the measurement noise, a Hanning window was added to reduce
the effects of leakage, and an averaging technique was adopted to reduce the relative standard
deviation of the power spectra [29]. The transfer function from the base to the second floor ofsteel frame is shown in Figure 12. A frequency domain curve-fitting algorithm was employed to
extract the modal parameters from the transfer function [29]. The identified modal frequencies
were 0.35 and 0.98 Hz, close to the frequencies of the numerical model. The identified first modal
damping ratio was 2.9%, which was fortuitously close to the assumed value in the numericalmodel.
5.3. Input wave of the shaking table
The input waves of the shaking table were generated from the numerical loop of the IDCS
algorithm, based on the target waves and the numerical model of the test specimen. To save oil,
the waves generated by the IDCS algorithm were filtered by a low-pass digital Chebyshev filter,
in which the cutoff frequency was set at the second modal frequency. To ensure the safety andfeasibility of the actual tests, the input waves had to be inspected carefully to satisfy the following
restrictions: (1) the displacement, velocity, and acceleration capacity of the shaking table; (2) theoil supply limit of the shaking table; (3) the drift limit of the rubber bearings; (4) the tension limitof the rubber bearings. The displacement and velocity of the shaking table can be calculated byintegrating the acceleration wave, and the demand for oil is related to the cumulative displacementof table. The procedure to calculate the oil demand can be found in Reference [14]. The drifts of
the rubber bearings can be estimated from the time history analysis of the numerical model, andthe tension forces extended to the rubber bearings can be obtained from the estimated overturning
moment.
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20

Measurement
Measured
Identification

Fitting

Magnitude

16
12
8
4
0

0

2

4
6
Frequency (Hz)8

10Figure 12. Transfer function curve of test specimen.

Table II. Inspection of shaking table’s input waves.
Drift (m)Rubber tension (kN)

Displacement (m) Velocity (m/s) Oil supply (kl) Layer 1 Layer 2 Layer 1
Limit
100% case
80% case

1.0
0.40
0.32

2.0
0.75
0.60

20.0
26.2
17.9

0.75
0.46
0.37

0.75
0.49
0.40

784
−659
−897

Layer 2784
9−187

For 100 and 80% of the response at the top floor of the 30-story building, the demands andcapacities are summarized in Table II. The table indicates that for the 100% case, the requirementfor oil is not satisfactory, while for the 80% case, all restrictions are removed. Thus, 80% of the
top floor’s response was chosen as the target response of the test specimen. The associated input
wave for the shaking table test is shown in Figure 13.
5.4. Test results
During shaking, the accelerations of the test specimen’s floors were recorded by servo accelerometers at an interval of 0.005 s. The recorded data revealed that the floor accelerations were nearly
identical for all the five floors, with the differences not greater than 1%. Figure 14 shows theacceleration response at the second floor and the target wave in the y direction. The corresponding
Fourier amplitude spectra obtained from the fast Fourier transform are shown in Figure 15. Fromthese figures, good correlation between the specimen’s response and the target wave is notable. A
similar observation was also obtained from the acceleration data in the x direction. The reproduction errors estimated by Equation (13) were 11.1 and 16.2% in the x and y direction, respectively.
Comparison of the floor response with the table input indicates that the rubber-and-mass system
is capable of amplifying the table motion significantly. The maximum velocity and displacement
were amplified to 3.4 and 3.6 times, respectively.
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Acceleration (m/s2)

2
1
0
-1
-2
0

50

100

(a)

150

200

250

200

250

Time (s)Acceleration (m/s2)

2
1
0
-1
-2
0

50

(b)

100

150
Time (s)Figure 13. Time history of shaking table’s input wave: (a) x direction and (b) y direction.

The reproduction errors in the test were larger than those in the numerical example. It occurredbecause of the shaking table’s control and the modeling error for the test specimen, in addition to
the measurement noise. The errors of the Fourier amplitude spectra between the designed input
waves and the measured accelerations of the shaking table were very small, being 2.0 and 1.6% in
the x and y direction. The modeling error for the test specimen, e.g. the neglect of the nonlinearityof rubbers, was speculated to be the main source of reproduction errors.
5.5. Seismic behaviors of furniture
Various types of furnitures were installed on the test specimen’s floors, which simulated rooms
with different functions, e.g. an office set on the second floor, a bed room set on the third floor,
and a living room and kitchen set on the fourth floor. In the simulated offices, furniture set on theleft span was placed naturally on the floor, while the furniture set on the right span was clamped
on the floor or to the partition wall by small channel steels and bolts.
The damage to the furniture is briefly summarized as follows. Most of the furniture not clamped
to the floor or wall overturned under large floor responses. Some scenes after shaking are shownin Figure 16. Compared with that furniture, the set of furniture clamped on the floor or wall
exhibited noticeably better performance, not causing overturning as shown in Figure 16(a). This
demonstrated the effectiveness of clamping. Note that in this test a floor response even larger thanthat described in the previous subsection was adopted. The detail of the furniture damage together
with the adopted motion will be the subject of the companion paper.
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Acceleration (m/s2)

4
2
0
-2
-4
0

50

100

150

200

250

200

250

Time (s)(a)

Acceleration (m/s2)

4
2
0
-2
-4
0

50

100

150
Time (s)(b)

Amplitude (m/s2)

Figure 14. Time history of target wave and specimen’s response in y direction:(a) target wave and (b) response of specimen.

0.6
Response
Response
TargetTa rget

0.4
0.2
0
0.1

1
Frequency (Hz)10

Figure 15. Fourier amplitude spectra of target wave and specimen’s response in y direction.

6. CONCLUSIONSIn this study, a full-scale substructure shaking table test is developed to reproduce the responseof the top floor of a high-rise building when it is subjected to long-period, long-duration groundmotion. Here, the top floor is extracted as the test specimen. To overcome the capacity limitationsof a shaking table, a rubber-and-mass system is proposed to amplify the motion of the shaking
table. An open-loop IDCS algorithm is employed to generate the input wave of the shaking tableto achieve an accurate reproduction of the floor response. A numerical example and shaking table
tests were performed to illustrate and validate the developed techniques.
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Figure 16. Damage to furniture after shaking: (a) office where furniture is clamped; (b) office wherefurniture is not fixed; (c) living room; and (d) kitchen.

The following conclusions are drawn from this study. (1) The designed rubber-and-mass system
is capable of amplifying the input waves of the shaking table effectively. (2) The open-loop IDCS
algorithm can generate suitable input waves of the shaking table by building an accurate numericalmodel and constructing a good controller. Both the model matching method and the H∞ method
show good performance for the controller design. (3) Using the input waves generated by the
IDCS algorithm, the responses at the top floor of a high-rise building were reproduced reasonably
by the substructure shaking table test.
Comparisons of the results between the physical test and the numerical simulation show that the
reproduction errors mainly result from the modeling errors of the rubber bearings, especially theneglect of their nonlinearity. It should be pointed out that the control algorithm used in this study
is offline and does not consider the nonlinearity of the test specimen. Room for improvement does
exist in the development of the substructure shaking table test, including (1) development of acontrol algorithm that can handle the nonlinearity of the test specimen; and (2) development of areal-time feedback control that would allow elimination of the adverse influence of modeling errors.

ACKNOWLEDGEMENTS

The Committee on Disaster Mitigation Research using E-Defense (chaired by Prof. Tsuneo Okada)provided valuable suggestions for the work described in this paper. The authors would like to extend their
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sincere gratitude to the committee. The authors would also like to thank the anonymous reviewers fortheir valuable comments and suggestions.
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DOI: 10.1002/eqe