Proceedings of the International Symposium on Engineering Lessons Learned fromthe 2011 Great East Japan Earthquake, March 1-4, 2012, Tokyo, JapanA STUDY ON MAXIMUM RESPONSES ON
ELASTO-PLASTIC STRUCTURE EXCITED BYSYNTHESIZED GROUND MOTIONS
Ichiro ICHIHASHI1, Akira SONE2, Arata MASUDA3 and Tatsuki NODA4
1

Division of Design Engineering, Graduate School of Science and Technology,
Kyoto Institute of Technology,
Kyoto, Japan, d8821001@edu.kit.ac.jp
2
Professor, Department of Mechanical and System Engineering,
Graduate School of Science and Technology, Kyoto Institute of Technology,
Kyoto, Japan, sone@kit. ac.jp
3
Associate Professor, Department of Mechanical and System Engineering,
Graduate School of Science and Technology, Kyoto Institute of Technology,
Kyoto, Japan, masuda@kit.ac.jp
4
Division of Mechanical and System Engineering, Graduate School of Science and Technology,
Kyoto Institute of Technology,
Kyoto, Japan, m1623034@edu.kit.ac.jp

ABSTRACT: A number of artificial earthquake ground motions compatible with
time-frequency characteristics of recorded actual earthquake ground motions as well as
the given target response spectrum are generated using wavelet transform. The
coefficient of variation (C. O. V.) of maximum displacement on elasto-plastic SDOF
systems excited by these artificial ground motions are numerically evaluated.
Key Words: Seismic motion, earthquake resistant, nonlinear vibration, wavelet, variationof responses

INTRODUCTION
For a design basis earthquake ground motion like a Level-1 or a Level-2 design earthquake groundmotion, an artificial earthquake ground motion is usually defined to be compatible with a given design
response spectrum. It has been known that the maximum values of elasto-plastic responses of astructure subjected to a group of artificial earthquake ground motions generated to be compatible withthe same design response spectrum change in no small way (Kitahara 2001).
For the measure of earthquake ground motion closely related to the maximum value of elasto-plastic
response of a structure, total input energy, input energy rate of its time differential and instant input
energy of its increment value per second, etc. based on the energy theory are recently taken up andrelationship between those factors and maximum ductility factor is investigated (Kato 1975, Ohi 1991,
Kuwamura 1997, Nakamura 1998). The input energy to a structure from an earthquake ground motion

1560shows nonstationary time history due to its nonstationary characteristics and nonlinearity of structure,and it is known that the structural damage rate depends on this nonstationarity. In order to set morerational design basis earthquake ground motion, it is important that the nonstationary characteristics ofa real earthquake ground motion which could hit a target structure in the future must be reflected in thedesign basis earthquake ground motion and its influence on the nonlinear response of the structure istaken account of in the design of the structure.
The method using Fourier phase characteristic (Kimura 1986) and the method using wavelet transform(Maeda 2002) are proposed as for synthesis of an artificial earthquake ground motion considering
nonstationary characteristics of it. Masuda et al. proposed the method generating the artificial
earthquake ground motion compatible with the given response spectrum and having nonstationary
time-frequency characteristics by using wavelet transform of velocity response function (Masuda
2002a, Masuda 2002b).
In this paper, from the viewpoint of generating a rational design basis earthquake ground motion, theinfluence of frequency nonstationary characteristic of earthquake ground motion on the elasto-plastic
response of structure is examined using a group of artificial earthquake ground motions compatible
with the same response spectrum and having the time-frequency characteristics of the recorded
earthquake’s ground motions by Masuda's method, and the examples of the variance of maximum
displacement responses of elasto-plastic structure are shown.
GENERATION OF ARTIFICIAL EARTHQUAKW GROUND MOTION
The relative velocity response x (t ) of a single degree of freedom (SDOF) structure with a natural
period T and damping ratio  subjected to earthquake ground acceleration f (t ) is given by Eq.(1)using the velocity impulse response function.
x (t )   f (t ) gVT , (t   )d

(1)



where, gVT , (t ) is the velocity impulse response function as  2 t2 1   2
2 1   2

T

e
(cos
t

sin
t)

gVT , (t )  
T
T
1 2

0

t0

(2)

t0

The velocity response spectrum SV (T ,  ) is defined as the maximum amplitude of velocity response
in terms of T and  , and SV (T ,  ) is given by

SV (T ,  )  max x (t )  max  f (t ) gVT , (t   )d
t

t



(3)

Introducing a function  (t ) defined as

 (t )  gV1, (t )(4)Using this function (t ) , the impulse velocity response gVT , (t ) of a structure with natural period T
is given by

gVT , (t )   (

t
)
T

(5)

Substituting this equation into Eq.(3), the velocity response spectrum SV (T ,  ) is described asEq.(6) using a function (t ) .

1561



SV (T ,  )  max   (


t

t 
) f (t )dt
T(6)

where the integral in the right hand side forms a continuous wavelet transform, in which acts as ananalyzing wavelet  (t ) with the scale parameter T and the sift parameter  . velocity response
spectrum can be described in terms of wavelets, i.e., it can be interpreted as the maximum amplitudeof the wavelet transform of the earthquake ground motion at each scale:

SV (T ,  )  max W { f }(T , )(7)twhere W { f }(T , ) denotes the wavelet transform of the earthquake ground acceleration f (t ) .

Maximum velocity ( m/s )Therefore, using this relation, we can compose the ground acceleration compatible to the given design
response spectrum and the given time-frequency characteristic through inverse wavelet transform.
In this study, the Osaki’s spectrum (Osaki 1995) for the magnitude of 7 and the epicenter distance of
20 km of earthquake, shown in Fig.1, is adopted as the design spectrum.
1

earthquake's magnitude: 7
epicenter distanse: 20 km

0.1

0.01 -2
10

-1

0

10
10
Period ( s )1

10Fig. 1 Osaki’s spectrum
5 time-frequency functions derived actual recorded ground motions which recently caused extensive
damage for structures or maximum ground acceleration more than 1 m/s2 are adopted as the target
time-frequency characteristics to synthesize the artificial ground motions. The list of selected actual
ground motions is Table 1 and the wave forms of selected ground motions are shown in Fig. 2. The
durations of these ground motions are taken shorter time than original record.
Table 1 List of selected actual earthquake’s ground motions

No.

Earthquake name

Origin time

1
2
3
4
5

The 2011 off the Pacificcoast of Tohoku Earthquake

2011/03/11/14:46

The Niigataken Chuetsu-oki
Earthquake in 2007

2007/0716/10:13

Location ofobservatory
Tsukidate(MYG004)
Shiogama(MYG012)
Hitachi(IBR003)
Hokota(IBR013)
Kashiwazaki(NIG018)

Direction
NS
EW
NS
NS
EW

Maximum
acceleration
27.0 m/s2
19.7
16.0
13.5
5.14

The 30 artificial ground motions are synthesized for each time-frequency characteristic. The examplesof synthesized ground motions and the examples of velocity response spectra of synthesized groundmotions are shown in Fig. 3 and Fig. 4 respectively. From Fig. 2 and Fig. 3, No. 1 and No.2
earthquake ground motions and synthesized ground motions have dual shocks and others have single
shock. The synthesized ground motions are fairly similar to original earthquake ground motions. From
Fig. 4 we can see that the velocity response spectra of synthesized ground motions compatible with thetarget spectrum. The mean value of the coefficients of variation (COV) of the velocity response
spectra of the synthesized ground motions are 0.048 ~0.057.

1562

No 1 earthquake
2000
0
-2000
0

10

20

30
40
No 2 earthuake

2000

50

60

70

0
0

Acceleration (cm/s 2 )

-2000

10

20

30

40
50
60
No 3 earthquake

70

80

10

20

30
40
No 4 earthquake

50

60

10

20
30
40
No 5 earthquake

50

60

2000

90

100

0
-2000

0

2000
0
-2000

0

1000
0
-1000

0

10
20
Time (s)30

Fig. 2 Selected actual earthquake’s ground motions
Synthesized ground motion by no 1 earthquake's t-f chracteristic
400
200
0
-200

0

10

20

30

40

50

60

70

Synthesized ground motion by no 2 earthquake's t-f characteristic
200
0Acceleration (cm/s 2 )

-200
0

10

20

30

40

50

60

70

80

90

Synthesized ground motion by no 3 earthquake's t-f characteristic
200
0
-200
0
10
20
30
40
50
60
Synthesized ground motion by no 4 earthquake's t-f characteristic
200
0
-200
0

10

20

30

40

50

60

Synthesized ground motion by no 5 earthquake's t-f characteristic
200
0
-200
0

10
20
Time (s)30

Fig. 3 Examples of synthesized ground motions

1563

100

Maximum velocity response (cm/s)

10

2

10

1

ground motion by no 1earthq.
ground motion by no 2 earthq.
ground motion by no 3 earthq.
ground motion by no 4 earthq.
ground motion by no 5 earthq.
target spectrum
0

10 -2
10

10

-1

10

Period (s)

0

10

1

Fig. 4 Samples of velocity response spectra of synthesized
ground motions

Max velocity response (m/s)

Max. vel. response (m/s)

The time-frequency characteristics derived from 5 actual earthquake ground motions (Table 1) are
shown in Fig. 5 ~ Fig. 9.

0.4
0.3
0.2
0.1
0
2
1
Period (s)

0

0

60

40

20

80

0.2
0.1
0
2

100

1
Period (s)Time (s)5000Time (s)Fig. 6 Time-frequency envelope by No. 2
actual earthquake

Fig. 5 Time-frequency envelope by No. 1
actual earthquake

0.4
Max velocity response (m/s)

Max. vel. response (m/s)

0.3

0.3
0.2
0.1
0
2
1
Period (s)

0

0

20

40

60

0.2
0.1
0
2
1
Period (s)Time (s)

0

0

20

40

60

Time (s)Fig. 8 Time-frequency envelope by No. 4
actual earthquake

Fig. 7 Time-frequency envelope by No.3
actual earthquake

1564

Max. vel. response (m/s)

0.4
0.3
0.2
0.1
0
2
1
0

Period (s)

10

0

20

30

Time (s)Fig. 9 Time-frequency envelope by No. 5
actual earthquake

VARIANCE OF MAXIMUM DISPLACEMENT RESPONSES ON BILINEAR SYSTEM
The single degree of freedom system (SDOFS ) which is made of a mass, bi-linear hysteretic springelement and viscous damping element is assumed. The equation of motion for this SDOFS isexpressed as Eq. (8).
mx  cx  F ( x, x)  mxg

Where, m, c, F ( x, x) , x and xg(8)denote mass of SDOFS, damping coefficient, bi-linear restoring

force, relative displacement of mass and acceleration applied to SDOFS respectively. The equation of
motion is numerically solved by linear acceleration method. On the bi-linear restoring force model, theratio of the post-yield stiffness kp to the initial stiffness ke before yield is assumed 1/100. And yieldforce levels Fy are assumed as relative input intensities  in Eq. (9) are 1.0 ~ 2.0.
E (max xg )mFy(9)

Where E (∙)denotes the mean value. The natural period T0 in linear range shown as Eq. (10)T0  2

m
ke

(10)

and the damping ratio  of the viscous damping expressed as Eq. (11)


c
2 mke

(11)

are taken as T0=0.05 ~ 2.0 s and  =0.02 respectively.
The spectra of maximum displacement responses max x of the mass for the natural period T0 of 0.05
s ~2.0 s to the synthesized ground motions with relative intensity α of 2.0 and each type oftime-frequency characteristic are shown in Fig. 10 ~ Fig. 14 . In these figure, the COV of the
maximum displacement responses are also shown. From these figures, the COV vary with the period
and the type of time-frequency characteristic.

1565

0

10

0

-1

10

Max. displacement response (m)COV (dimensionless value)

Max. displacement response (m)COV (dimensionless value)10

no 1 earthq t-falpha=2.0-210

max. displ. resp.
mean value
standard deviation
COV

-3

10

-4

10

10

-2

-1

10

0

1

10

Period (s)10Fig. 10 Max. displacement response by No. 1
synthesized ground motion-2

10

max. displacement
mean value
standard deviation
COV

-3

10

-4

10 -2
10

-1

10

0

Period (s)

10

10

1

0

-1

Max. displacement response (m)COV (dimensionless value)

Max. displacement response (m)COV (dimensionless value)

no 2 earthq t-falpha=2.0

Fig. 11 Max. displacement response by No. 2
synthesized ground motion010

no 3 earthq t-falpha=2.0

10

-2

10

max. displ. resp.
mean value
standard deviation
COV

-3

10

-4

10

-1

10

-2

10

10

-1

0

Period (s)

10

10

1

10

-1

no 4 earthq. t-falpha=2.0

10

-2

10

max. displ. resp.
mean value
standard deviation
COV

-3

10

-4

10

-2

10

Fig. 12 Max. displacement response by No. 3
synthesized ground motion

-1

10

0

Period (s)

10

1

10

Fig. 13 max. displacement response by No. 4
Synthesized ground motion

0

Max. displacement resonse (m)COV (dimensionless value)

10

-1

10

no 5 earthq. t-falpha=2.0-210

max. displ. resp.
mean value
standard deviation
COV

-3

10

-4

10

10

-2

-1

10

0

Period (s)

10

1

10

Fig. 14 Max. displacement response by No. 5
synthesized ground motion
The COV at α of 0.1, 1.0 and 2.0 and at T0 of 0.1 s, 0.2 s, 0.5 s and 1.0 s for each type oftime-frequency characteristic of the synthesized ground motions are shown in Table 1. The response at
α of 0.1 means linear response. And the spectra of COV at α of 1.0 and 2.0 for each type oftime-frequency characteristic 0f the synthesized ground motions are shown in Fig. 15. It is recognizedthat COV becomes large as the value of α increase and varies with T0 and the type of time-frequency
characteristic from these Table 1 and Fig. 15.

1566

Relative
input
intensityα

Table 1 Coefficient of variation of the max. displacement response
Period Type of synthesized ground motions
Time-frequency
Time-frequency
Time-frequency
Time-frequency
T0(s)

0.2
0.5
1.0
1.0
0.2
0.5
1.0
2.0
0.2
0.5
1.0
(∗) linear response
0.1 (∗)of No. 1 earthq.

of No. 2 earthq.

of No. 3 earthq.

of No. 4 earthq.

Time-frequency
of No. 5 earthq.

0.09880.11110.17770.23070.15810.17850.31920.28240.2091

0.09630.08320.10550.13280.13550.10550.29620.22930.16160.12860.10600.18310.30190.17850.1766
0.3666
0.25730.22390.11490.13560.15080.24340.22410.19250.2526
0.34240.28340.10460.10510.14330.34950.23600.15390.25290.31010.26550

COV of max. displacement10

no 1 earthq alpha=1.0
no 1

2.0

no 2 earthq alpha=1.0
no 2

2.0

no 3 earthq alpha=1.0
no 3

2.0

no 4 earthq alpha=1.0
no 4

-1

10

2.0

no 5 earthq alpha=1.0
no 5
-2

10

10

-1

Priod (s)

10

0

2.0

1

10

Fig. 15 Coefficient of variation of maximum displacement responses by No.1 ~ No.5
synthesized ground motions
The significant differences between the variances of maximum displacement due to the synthesized
ground motions with different type of time-frequency characteristics are studied. In Fig. 16, the ratioof the unbiased variance of the maximum displacement responses due to the ground motions with No.
1 earthquake’s time-frequency characteristic to that with No. 2 earthquake’s time-frequency
characteristic, the ratio of that with No. 1 earthquake’s time-frequency characteristic to that with No.3
earthquake’s characteristic and the ratio of that with No.1 earthquake’s time-frequency characteristic
to that with No. 5 earthquake’s time-frequency characteristic are shown. The values of F-distribution
corresponding to significance level of 5 % are 2.10 and 0.47 drawn by dotted lines. These ratios in thisFig. 16 are inside of the dotted lines at most of the periods, that is to say, there is not significant
difference between the numerator and the denominator of the ratio of the variance at most of theperiods. The ratio of the unbiased variance of the maximum displacement responses due to the groundmotions with No. 3 earthquake’s time-frequency characteristic to that with No. 4 earthquake’stime-frequency characteristic, the ratio of that with No. 2 earthquake’s time-frequency characteristic tothat with No. 4 earthquake’s time-frequency characteristic and the ratio of that with No. 3 earthquake’stime-frequency characteristic to that with No. 5 earthquake’s time-frequency characteristic are shownin Fig. 17. The former two ratios are not inside of the dotted lines at most of the periods. The
remainder, the ratio of that with No.3 earthquake to that with No. 5earthquake, are inside of the dotted

1567

Variance ratio of max. displacemen

Variance ratio of max.diplacement

lines at most of the periods, that is to say, there are significant differences between variance of the
maximum displacement responses due to the ground motions with No. 3 earthquake’s time-frequency
characteristic and that with No.4 earthquake’s one and between variance of that with No.2
earthquake’s one to that with No.4 earthquake’s one at most of the periods. However, there is not
significant difference between that with No.3 earthquake’s one and that with No.5 earthquake’s one at
most of the periods.
110var. by no. 1 earthq /var. by no. 2 earthq
var. by no. 1 earthq /var. by no. 3 earthq
var. by no. 1 earthq /var. by no. 5 earthq
F distribution valuealpha = 2.0

0

10

-2

10

-1

10

0

Period (s)

10

1

10

Fig. 16 Variance ratio of max. displacement
responses(No. 1 earthq / No. 2, No. 1 / No. 3, No. 1 / No. 5)

1

10var. by no 3 earthq /var. by no 4 earthq
var. by no 2 earthq /var. by no 4 earthq
var. by no 3 earthq /var. by no 5 earthq
F distribution valuealpha = 2.0

0

10

-1

10

-2

10

-1

10

0

Perio (s)

10

1

10

Fig. 17 Variance ratio of max. displacement
responses(No. 3 earthq / No. 4, No. 2 / No. 4, No. 3 / No. 5)

From above descriptions, it is recognized that COV of maximum displacement responses varies with
T0 and the type of the time-frequency characteristic and COV becomes large as α increases. There isnot significant difference between COV of that with the time-frequency characteristics of the recorded
ground motion at KASHIWAZAKI in The Niigataken Chuetsu-oki Earthquake in 2007 and COV ofthat with one at TSUKIDATE and HITACHI in The 2011 off the Pacific coast of Tohoku Earthquake.
CONCLUSIONS
Using the time-frequency characteristics of the recorded strong earthquake ground motions in the 2011
off the Pacific coast of Tohoku Earthquake and Masuda’s method, several groups of synthesized
ground motions are generated. The coefficient of variation (COV) of the maximum displacement
responses on the bi-linear single degree of freedom system are shown. It is recognized that COV varies
with natural period in elastic domain of the system and the type of the time-frequency characteristic of
the synthesized ground motions and COV becomes large as input intensity increases. In this study, we
can not find out clear difference between COV of the maximum displacement responses due to thesynthesized ground motions with the time-frequency characteristic of the recorded ground motion at
KASHIWAZAKI in The Niigataken Chuetsu-oki Earthquake in 2007 and COV of that with one at
TSUKIDATE and HITACHI in The 2011 off the Pacific coast of Tohoku Earthquake.
ACKNOWLEDGMENTS
In this study, we used the strong motion records by K-NET of National Research Institute for Earth
Science and Disaster Prevention (NIED). Authors are grateful for NIED.

REFERENCES
Kato, T. and Akiyama, H. (1975). “Energy Input and Damages in Structures Subjected to Sevier
Earthquake.” (in Japanese), Journal of Structure and Construction Engineering, No.235, 9-18.

1568

Kimura, M. (1986). “On Control of the Characteristics of the Wave in Generating Simulated
Earthquake Ground Motions.” (in Japanese), Journal of Structure and Construction Engineering,
No.367, 30-35.
Kitahara, T. and Itoh, Y. (2001). “Study on Variance of Elasto-plastic Response to a Group of Artificial
Earthquake Ground Motion Compatible with Same Design Response Spectrum.” (in Japanese),
Proceedings of the Japan Concrete Institute, Vol.23, No.3, 1237-1242.
Kuwamura, H., Takeda, T. and Sato, Y. (1997). “Energy Input Rate in Earthquake
Destructiveness-Comparison Between Epicentral and Oceanic Earthquakes.” (in Japanese), Journalof Structure and Construction Engineering, No.491, 29-36.
Maeda, T., Sasaki, F. and Yamamoto, Y. (2002). “Artificial Ground Motion with Non-stationarity
Generated by the Sinc Wavelet.” Journal of Structure and Construction Engineering, No.553,
33-40.
Masuda, A. and Sone, A. (2002a). “Wavelet-based Synthesis of Artificial Earthquake Motions
Consistent with Response Spectra and Nonstationarity of Real Earthquake Records.” (in Japanese),
Proceedings of 11th Japan Earthquake Engineering Symposium, 527-532.
Masuda, A. and Sone, A. (2002b). “Generation of Spectrum-Compatible Earthquake Motion Using
Wavelet Transform.” Proceedings of ASME Pressure Vessels and Piping Conference, SeismicEngineering-2002,PVP, Vol.445-1, 175-180.
Nakamura, T., Hori, N. and Inoue, N. (1998). “Evaluation of Damaging Properties of Ground Motions
and Estimation of Maximum Displacement Based on Momentary Input Energy.” (in Japanese),
Journal of Structure and Construction Engineering, No.513, 65-72.
Ohi, K., Takanashi, K. and Honma, Y. (1991). “Energy Input Rate Spectra of Earthquake Ground
Motions.” (in Japanese), Journal of Structure and Construction Engineering, No.420, 1-7.
Osaki, Y. (1995). “New Introduction to Spectrum Analysis of Earthquake Ground Motion.” (inJapanese), Kajimashuppan, Tokyo, Japan.

1569