Journal of Natural Disaster Science, Volume 24, Number 2, 2002, pp91-100

Phase Spectrum Modeling to Simulate Design Earthquake Motion

1)

Tadanobu SATO 1), Yoshitaka MURONO 2) and Akihiko NISHIMURA2)
Disaster Prevention Research Institute, Kyoto University, Kyoto, Japan2)
Railway Technical Research Institute, Tokyo, Japan(Received for 9. Oct., 2002)

ABSTRACT
Modeling the phase of earthquake ground motion is required in order to synthesize design ground motion that iscompatible with the given response spectrum. A simple method is presented for modeling the phase characteristics of earthquake motion using the concept of group delay time and wavelet analysis. Using existing data sets ofobserved earthquake motions, we calculated the group delay time of each earthquake motion on each compact
support of the mother wavelet. Because of the fluctuating nature of the group delay time on each support, themean of the group delay times and standard deviation were calculated. Attenuation relations of the mean delay
time and standard deviation on each support were obtained by regression analysis as functions of the earthquake’s
magnitude and epicentral distance. On the assumption that the group delay time has a normal distribution, a sample group delay time can be generated on each support, and by integrating it with respect to frequency the phase
spectrum is derived. Simulated sample phase spectra were used to simulate earthquake motions compatible with adesign response spectrum. Their characteristics were investigated using ductility demand spectra.

1. INTRODUCTION
Design earthquake motions often are defined by response
spectra. For example, in the seismic design standards for the Japan
Railway facilities (1999) there are two types of design spectra forlevel II earthquake motion: Spectrum I is defined by taking into
account earthquake activities in the inter-plate region; Spectrum II
is defined by considering the near source earthquake motion characteristics caused by intra-plate earthquakes. To assess the aseismic capacity of designed structures, dynamic analyses of structures
must be made using earthquake motion compatible with the design
response spectrum. The phase characteristics of earthquake
motion as well as the amplitude therefore are important issues.
A primitive method developed in early stage studies of thesimulation of design earthquake motions was to generate a nonstationary time history by multiplying an envelope function(Jennings,1969) to a stationary time history simulated using random phase criteria (Shinozuka and Jan,1972). Another method
was to use the phase spectrum of a certain observed earthquake
motion (Arakawa,1984). The phase characteristics of earthquake
motions were not, however, clearly characterized by these methods.
Several pioneer works clarified the nonstationary characteristics of earthquake motion through analyses of its phase characteristics. Osaki (1978) showed similarity between the distribution
width of the phase difference and the duration of earthquake
motion. Izumi and Katsukura (1983) using the concept of groupdelay time, showed that the average arrival time of earthquake
energy and duration of earthquake motion can be evaluated by themean of group delay time and the standard deviation. Kimura(1986) presented a way to simulate earthquake motion by control-

KEY WORDS:

ling the group delay time, but it was not based on observed earthquake motion data.
We developed a method to simulate the group delay time fornear source earthquake motion by assuming that the rupture
process of the earthquake fault can be expressed by a train of
impulses and that the minimum phase concept is effective for evaluating the phase shift due to traveling pass (Sato, Murono andNishimura, 1999). In that method only direct S wave motion wasconsidered, the effect of surface wave motion on the group delay
time was not taken into account. Moreover, to use the method
detailed information on the fault rupture process and geologicalconditions of the transmitting path had to be given apriori.
The parameters that control the group delay time of earthquake motion therefore are modeled as a function of the epicentral
distance, Δ, and earthquake magnitude, M, (i.e., the attenuation
relationship) using observed earthquake motion data sets.
Several studies have developed the attenuation relationshiprelated to the phase characteristics of earthquake motion. Sawada(1986) analyzed the phase difference of earthquake motions andderived an attenuation relationship for duration. Ishii (1987)
derived attenuation relationships of the mean group delay time andits standard deviation for entire time history of earthquake motion.
Satoh (1996) derived attenuation relationships of the mean groupdelay time and its standard deviation for fairly long period earthquake motion (period range 1-10s), using a wave form processed
through a narrow band passed filter with the frequency of 0.02Hz.
The purpose of our study was to develop a model for simulating realistic group delay time of earthquake motion to be used for
dynamic analyses of civil structural systems. The targeted frequency range was 0.2 to 10Hz. The general group delay characteristic is assumed to be modeled by the mean value and its standard

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T. SATO, Y. MURONO AND A. NISHIMURA

deviation. We calculated these values for the wavelet-transformed
earthquake motions on each compact support of the Meyer wavelet(Yamaguchi, 1990) and used regression analysis to derive theattenuation relationships of these values as functions of the epicentral distance, Δ, and earthquake magnitude, M. As an application
we resimulated the earthquake motions of the Off Sanriku event(1995) and compared them with the observed earthquake motions.
We also show the nonlinear response characteristics of a single
degree of freedom structure to earthquake motions compatible withthe design response spectrum defined in the Japan Railway design
standards.
2. DATA BASE FOR THE ANALYSES
Earthquake motions were selected mainly from data sets
recorded by the 87 type accelerometers of the Japan
Meteorological Agency. Data sets distributed by the Kansai
Earthquake Observation and Research Committee and the K-net of
the Science and Technology Agency also were used. Earthquake
motions that satisfy the following conditions were selected:
1) an earthquake magnitude larger than 6,2) many observed data collectable from one event,3) a hypocenter depth of less than 50km,
4) a maximum acceleration of more than 30 gal,5) a sufficiently long duration.
Condition 1) was selected because our concern was to define
the phase spectrum of level II earthquake motion. Condition 2) isneeded to ensure homogeneous distribution of the earthquake
observation data in the space of the earthquake’s magnitude andepicentral distance. Condition 3) was needed for the selection ofearthquakes that generate sufficient surface wave motion becausesuch motion is not generated if the depth of the hypocenter is morethan 50km. Condition 4) was required because if the maximum
acceleration is less than 30 gal the S/N ratio will not be large
enough to analyze the group delay time. Condition 5) was necessary to obtain observed earthquake motions over time until thearrival of the concerned wave with a targeted group velocity andfrequency band.
Five recent earthquake data sets that satisfy the above conditions are given in Table 1. Figure 1 shows their distribution characteristics in the space of magnitude M and epicentral distance Δ.
Earthquakes with long epicentral distances are those with large
magnitudes, whereas those with short epicentral distances are only
the 1995 Hyogoken Nambu (M7.2) and 1997 North-West
Kagoshima Prefecture (M6.3) earthquakes.
Ground motions of the Off Southwest Hokkaido earthquake(1993, M7.8), the Off East Hokkaido earthquake (1994, M8.1) andthe Off Sanriku earthquake (1994 M7.5) were recorded by the 87
type accelerometers of the Japan Meteorological Agency. The
earthquake motions of the Hyogoken Nambu earthquake include
records from both the Meteorological Agency and Kansai
Earthquake Observation and Research Committee. All records for
the Northwest Kagoshima event are from the K-net of the Scienceand Technology Agency.
3. ANALYTICAL METHOD
Data used for the analyses are the horizontal components of
the observed earthquake motions.

Fig. 1

Relationship between magnitude and epicentral Distanceof the earthquake records used

3.1 Definition of Group Delay TimeGroup delay time is defined by the derivative of the Fourierphase spectrumφ(ω) with respect to circular frequency ω(Papoulis, 1962);
tgr (ω) =

dφ(ω)
dω

(1)

The average value of the group delay time within a certain frequency band with the central frequency ω expresses the arrival
time of a wave component with frequency ω. The distribution
width of the group delay time is related to the duration of the time
history of the wave component. Because of these characteristics ofgroup delay time, its modeling is much easier than direct modelingof the phase spectrum.
The calculated Fourier phase spectrumφ(ω) must be
unlapped to obtain the group delay time based on equation (1).
Izumi and Katsukura (1983) proposed a method to satisfy the condition tgr≧0, but the distribution of the group delay time is concentrated around the end of the time history of earthquake motion
because of the rink effect. To eliminate this defect, an artificial
zero time history was added (Satoh, 1996). Because the calculatedgroup delay time depends on the number of data points in the time
history, we applied the method of Sawada, Morikawa et al (1998),
who showed that the calculated group delay time depends less on
artificially added numbers of zeros to a time history of earthquake
motion.
3.2 Average Group Delay Time and Its Standard Deviation
To show the nonstationary characteristic of earthquake motionin the time and frequency domains, wavelet analysis often is usedbecause resolution in the time and frequency domains is guaranteed through the uncertainty criterion.
The discrete wavelet transformation of the function x (t) and
its inverse transformation were defined by (Yamaguchi, 1990)∞ ＊a j,k =∫-∞
ψj,k・x (t) dt(2)x (t) =Σx j (t) =ΣΣa j,k・ψj,k (t)(3)ψj,k (t) = 2 j/2・ψ(2 jt - k) j, k : integer

(4)

j

j

k

PHASE SPECTRUM MODELING TO SIMULATE DESIGN EARTHQUAKE MOTION

in which the value with the sign * expresses the complex conjugateof the original value, ψ(t) the wavelet function, j the scale factor,
and k the factor expressing the time shift. As seen from equation(4), if scale factor j becomes large the treating frequency range
goes to the high frequency region. If the inverse wavelet transformation exists, ψ(t) is called the analyzing wavelet. We used themethod of Meyer (Meyer 1989) to compose ψ(t) although there
are several ways to define this function. The Fourier transformation of ψ(t) has a compact support for each scale factor j (named
the j th compact support) defined by｛2 j / 3Td ＜ f ＜ 2 j+2/3Td｝(5)

in which Td (=NΔt, N: number of data, Δt: sampling interval) is
the duration of earthquake motion.
For all oberved earthquake earthquake motions with a sampling interval of 0.01(sec) we added zero data until the total number of sampling data for each earthquake motion, N, became
131071 (=217). Using the wavelet transformation of each earthquake motion x(t) we decomposed each time history of earthquake
motion to a component time history of each scale factor j (j=1〜17). We called this the jth component time history, x(j)(t). The
group delay time of this time history, tgr(j)(ω), was calculated. Itsmean, μ(j)tgr, and standard deviation,σ– (j) tgr, on the jth compact
support were then obtained from equations (6) and (7):
(j)

N (j)

(j)

tgr (ωi)
N (j)
i=1

(6)

1 N (j)(t (j) (ω)-μ(j) )2
gritgr
N (j) Σi=1

(7)μtgr =Σ
(j)σtgr =

√

(j)

N (j)

93

(j)

tgr (ωi)N (j)i=1μtgr =Σ

(8)

A sample calculation of these values is shown in Figure2.
4. NUMERICAL EXAMPLE
We used the proposed method to resimulate the earthquake
motion records from the Morioka observation station for the 1993
Far-off Sanriku earthquake.
An example of earthquake motion decomposed into eachcomponent time history x (j)(t) is shown in Figure 3. As scale factor
j becomes smaller, the duration lengthen and the central arrival
time of the component time history is delayed. The Fourier amplitude spectrum on each compact support, A ( j )(ω), is shown in
Figure 4. The Fourier amplitude on the jth compact support overlaps the Fourier amplitudes on both the (j-1)th and (j+1)th compact
supports because the definition of the jth support is given by equation (5). The main frequency range in which the Fourier amplitude
on the jth compact support predominates, is given by｛2 j-1 / Td ＜ f ＜ 2 j /Td｝

(9)

The vertical broken line in Figure 4 shows the frequency range
defined by this equation.
Values ofμ(j)tgr and σ(j)tgr for the jth component time history(j)x (t) based on equations (6) and (7) are shown in Figure 5. The
group delay time tgr (ω) calculated from the observed time historyof the earthquake motion also is shown. The values μ(j)tgr and σ(j)tgr well indicate the group delay time of the observed earthquake
motion. Comparison of Figures 3 and 5 shows that the central

in which j=1,2…,17, N (j) is the number for the time history data at
the jth scale factor defined by N (j)=2 j, and tgr(j)(ωi) the group delay
time of the jth component time history at a circular frequency of ωias defined by equation (8);Fig. 2Concept for calculating average group delay time and its
standard deviation in each conpact support

Fig. 3

Wavelet decomposed time histories for each scale factor j

94

T. SATO, Y. MURONO AND A. NISHIMURA

arrival time of the jth component time history coincides with μ(j)tgr
and that the duration is well evaluated by σ(j)tgr.
The distribution of group delay time on each compact support
is shown in Figure 6. The examples for j=9 and 11 have the character of a normal distribution. When the value of j become small,the distribution character is distorted and approaches a log normal
distribution. This tendency is confirmed by the fact that the envelop functions for cases of j=8,9 close to the log normal distributionfunction (Figure 3). We assume, however, that the values forgroup delay time have the normal distribution characteristic for all
j values.
Earthquake motion resimulated using the average value of thegroup delay time,μ(j)tgr, and its standard deviation, σ(j)tgr , is shownin Figure 7. We compared the resimulated motion with theobserved one. The group delay time tgr(j)(ω) for the jth component
time history on the compact support {2 j/3Td ≦f≦2 j+2/ 3Td} is generated by assuming the normal distribution of N(μ (j)tgr,σ (j)tgr).
Integration gives the phase spectrum for the jth component time
history, x(j)(t). The Fourier amplitude spectrum, A(j)(ω), on the jthcompact support is assumed to coincide with the Fourier amplitudeof the observed earthquake motion, A(ω), in the frequency range{2 j-1/Td ≦ f ≦2 j/Td } and to be zero outside of this region; i.e., in thefrequency regions of {2 j/3Td ≦f<2 j-1/Td} and {2 j/Td< f ≦2 j+2/3Td}.
We used this kind of simplicity because our purpose was not
to model the Fourier amplitude on the compact support. Using the
calculated φ(j)(ω) and modeled A(j)(ω) for j=1〜17, we resimulated
each component time history and obtained the earthquake motion
by summing up all the jth component time histories. A comparison of the resimulated motion with the observed one is shown in
Figure 7, as is the simulated group delay time. The overall trend of
the simulated group delay time agrees well with that of theobserved earthquake motion. Group delay time calculated fromthe observed motion has a much wider distribution than the simulated one in the period range of 3-10sec because a normal distribution was assumed for all the jth component group delay times. The
shapes of the simulated time histories of acceleration and displacement, however, agree well with those of the observed ones.
5. MODELING OF GROUP DELAY TIME
The mean group delay time, μ(j)tgr, and its standard deviation,
σ(j)tgr, were calculated for all the observed earthquake motions listed in Table 1. Because the trigger time of earthquake motion isrecorded in all the observation data, the time origin of earthquake
motion was defined by adding the theoretically calculated travel
time of the direct S wave generated at the initial rupture point on
the earthquake fault to the trigger time. The concerned period
range in standard aseismic design procedures is 0.1-5 sec. The

Fig. 4

Fourier amplitude A(j) (ω) for each compact support j

Fig. 5

Average group delay time μtgr and its standard deviation
σtgr in each compact support

Fig. 6

Distribution characteristics of group delay time tgr(ω) on
compact support j = 9, 11

PHASE SPECTRUM MODELING TO SIMULATE DESIGN EARTHQUAKE MOTION

95

Table 1. Earthquake data sets used(a) time history of acceleration (Upper: Observed, Lower: Simulated)

Table 2. Regression analyses results(b) time history of displacement (Upper: Observed, Lower: Simulated)(c) Group delay time
Fig. 7Comparison with observed and simulated motions

Fig. 8attenuation relations μ(j)tgr and σ(j)tgr for j=7-14 therefore weredeveloped as functions of earthquake magnitude M and epicentral
distance Δ (km).
(j)

(j)

(j)

(j)

(j)
μtgr
=α1(j)×10β1 M×Δγ1(j)μtgr=α1(j)×10β1 M×Δγ1

(10)
(11)

in which α(j), β(j) and γ(j) are regression coefficients on the jthcompact support. Regression analysis results are shown in Table 2
and Figures 8 and 9. Theβvalues for j=7-8 are forced to be zero
because in this range β has small negative values which meansthat the arrival time of the jth component time history x (j)(t)becomes earlier and the duration longer for the smaller the earthquake magnitude. These characteristics have no physical meaning,rather they are due to the scattering of data. The effective numberof data on the jth compact support is given by

N (j) = 2 j

Period dependence of average group delay times(12)

As the number of data on the compact support for j=7-8 was insufficient, large scattering occurred. The respective values of the correlation coefficients μtgr and σtgr are 0.9 and 0.71〜0.87 (less than
0.5 for j=7).
The value of μ(j)tgr becomes larger as the epicentral distanceincreases. The regression coefficient γ1 has a constant range of
0.68-0.8 for all orders of j. This means that the arrival times of all
the component time histories (x(j)(t), j=7-14) are delayed as thetransmitting path lengthens. Because the value of β1 is less than0.1, the effect of epicentral distance on the average group delay
time is relatively greater than that of earthquake magnitude. The
value of μ(j)tgr depends on the value of j; i.e., the average arrival
time of the component time history is frequency dependent (Figure

96

T. SATO, Y. MURONO AND A. NISHIMURA(a) Average group delay time μtgr(b) Standard deviation of group delay time σtgr
Fig. 9Average group delay time, its standard deviation in earthquake records, and regression analysis results8) .
The duration of the jth component time history could be evaluated from the value of σ(j)tgr . As the order of j increases the valueof γ2 becomes smaller and β2 larger. The surface wave predominates in the low frequency range of less than 1Hz, whereas the
body wave has its main components in the high frequency range of
1-several Hz. Characteristics of the regression coefficients there-

fore are interpreted as follows: In the frequency region dominatedby the body wave (j≧12), the duration of the component time history is determined mainly by earthquake magnitude, whereas in
the surface wave region (j≦11) the effect of epicentral distance on
its duration becomes larger than that of earthquake magnitude.
Two controversial studies related to the effect of epicentral
distance on the duration of earthquake motion have been reported.

PHASE SPECTRUM MODELING TO SIMULATE DESIGN EARTHQUAKE MOTION

One (Trifunac, 1975) supports a strong effect, the other is against
it (Kamiyama, 1984).
Sato and Uetake et al. (1996) concluded that the effect ofearthquake magnitude on the mean group delay time and its stan-

Table 1. Earthquake data sets used97

dard deviation is very small. Our findings for the mean groupdelay time support their conclusion because the value of β1 issmaller than 0.1. The difference in standard deviations between
our and their results is because they analyzed only long period
motion. The value ofβ2 approaches zero in our research when thevalue of j becomes smaller than 10 (period lager than 1.04sec),
which is consistent with their conclusion.
The attenuation relation of each component time history,
neglecting the effect of earthquake magnitude, also was obtainedby assuming equations (13) and (14). Regression analysis results
are given in Table 3 and shown by bold lines in Figure 9.
γ (j)

(j)
μtgr
=α1(j)×Δ 1

γ (j)

(j)
=α2(j)×Δ 2
σtgr

(13)
(14)

6. VERIFICATION OF ATTENUATION RELATION
To verify the proposed modeling method of the phase spec-(a) Hachinohe (Δ = 186km)(b) Morioka (Δ = 236km)(c) Ishinomaki (Δ = 307km)(d) Onahama (Δ = 459km)
Fig. 10Comparison of observed and simulated motions using the proposed model (Upper: Observed, Lower: Simulated)98

T. SATO, Y. MURONO AND A. NISHIMURA

trum we simulated earthquake motions recorded during the 1994
Far-off Sanriku earthquake at the Hachinohe (Δ=186km),
Morioka (Δ=236km), Ishinomaki (Δ=307km) and Onahama (Δ=459km) observation stations.
We targeted the EW components. Fourier amplitudes on thejth compact support A(j)(ω) were defined as in section 4 using theobserved Fourier amplitudes, A(ω). From the given earthquake
magnitude and epicentral distance we calculated μ(j)tgr and σ(j)tgr.
The group delay time on jth compact support tgr(j)(ω) was simulated
by generating random values based on the normal distribution, N(μ(j)tgr,σ(j)tgr). The phase spectrumφ(j)(ω) was obtained by integrating tgr(j)(ω) with respect to ω. The acceleration time histories composed from φ(j)(ω) and A(j)(ω) and the velocity time histories areshown in Figure 10. The central arrival time and duration of earthquake motions agree well with the observed motions. The non-stationary arrival characteristic of the wave group also agrees wellwith observed earthquake motions.
7.

EARTHQUAKE MOTIONS COMPATIBLE
WITH THE DESIGN RESPONSE SPECTRUM

Earthquake motions compatible with the design response
spectrum defined in the earthquake-resistant design standard of theJapan Railway facilities were simulated (Figure 11). The values of
the average group delay time μ(j)tgr and standard deviation σ(j)tgr on
the jth compact support were obtained for an earthquake of magnitude 8.0 and several epicentral distances (10, 100, 300 km) using
the attenuation relationship developed in section 4. The phase
spectrum on the jth compact support φ(j)(ω) was simulated using
these two values. To simulate an earthquake motion compatible
with the response spectrum, an initial Fourier amplitude spectrum
Aint(ω) is needed. This is assumed to be given by the velocity
response spectrum with zero damping coefficient converted fromthe design acceleration response spectrum as shown in equation(15).
Aint (ω) =β +γ)ωS (ω, 0.05)( 1+αh(15)

in which S(ω, 0.05) is the targeted design spectrum with 5% damping, h the damping constant, and α,β and γgiven parameters(Kwashima and Aizawa, 1986)

Fig. 12The initial Fourier amplitude of the jth compact support A(j)(ω)
is assumed to coincide with the initial Fourier amplitude Aint (ω) in
the frequency range {2 j-1/Td≦f≦2 j/Td} and to be zero outside ofthis region; i.e., in the frequency regions {2 j/3Td ≦ f < 2 j-1/Td} and{2 j/Td < f ≦2 j+2/3Td}. A jth component time history can be simulated using φ(j)(ω) and A(j)(ω). Earthquake motion of the first step
is obtained by summing up all the component time histories.
Because the calculated response spectrum based on first step earthquake motion does not coincide with the design response spectrum, we modified the amplitude of the initial Fourier amplitude,
Aint (ω), at the circular frequency,ω, by multiplying the ratio r(ω)
between the design response spectrum and the calculated response
spectrum. Assuming the calculated r(ω) Aint (ω) to be the new initial Fourier amplitude of earthquake motion and using the original
phase spectrum of the jth component φ(j)(ω), we could simulate
the earthquake motion of the second step. By repeating thisprocess until ratio r(ω) nealy approached 1.0, we finally could
simulate earthquake motion compatible with the design response
spectrum.
The simulation results are shown in Figure 12. The design
response spectrum shown in Figure 11 was defined by taking into
account only the activity of inter-plate earthquakes (Wang, 1999).
The epicentral distance for these earthquakes in Japan therefore ismore than 30-40 km, so we added the case of a 10 km epicentral
distance to compare the general characteristics of the simulated

Fig. 11The design response acceleration spectrum defined in
the earthquake resistance design standards of Japan
Railway Companies

Earthquake motions compatible with the design response spectrum in figure 11.

PHASE SPECTRUM MODELING TO SIMULATE DESIGN EARTHQUAKE MOTION

Fig. 1399

Ductility demand spectrum for earthquake motions compatible with the response spectrum in figure 11

earthquake motions. In this case, the simulated earthquake motion
shows such typical near-source motion characteristics as a shortduration and pulse-like wave form. As the epicentral distanceincreases, the duration lengthens and longer period motion predominates in the latter part of the earthquake motion.
We calculated the ductility demand response spectra using thesimulated earthquake motions shown in Figure 13. The ductility
demand spectrum was obtained through nonlinear dynamic
response analyses of a single degree of freedom system. The yieldseismic coefficient of the SDF is controlled until the SDF response
reaches the targeted ductility. The chart that expresses the relationship between the yield seismic coefficient and elastic natural
period is called the ductility demand spectrum, in which the targeted ductility is a parameter. The hysteresis model used for the
analyses is a bi-linear type of model proposed by Clough (1966).
The second stiffness is selected as 5% of the first, and a 5% damping coefficient assigned. Although we used the same elastic
response spectrum to simulate the earthquake motions, the strong
effect of epicentral distance can be seen on the elasto-plastic structural response characteristics. A shorter epicentral distancedemands a large ductility whereas the demand decreases as the epicentral distance increases.

ACKNOWLEDGEMENT
We gratefully acknowledge the computer program for wavelet
analysis provided by Professor M. Yamada from University ofTokyo.
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2)

3)

4)

5)

8. CONCLUSION
A method for modeling the phase spectrum of earthquake
motion is proposed that is based on the concept of group delay
time. The findings of this study are1) By using data sets of observed earthquake motions and thewavelet algorithm, the average group delay time and standard
deviation on the compact support of Mayer wavelet can be
expressed as functions of earthquake magnitude and epicentral
distance.
2) By use of derived attenuation relationship, earthquake motions
at sites where earthquake motions were recorded during the1994 Far-off Sanriku earthquake could be simulated. The simulated motions agreed well with the characteristics of the recorded motions.
3) A method to simulate earthquake motion compatible with thedesign response spectrum was developed. Its suitability fordesign purposes was investigated using ductility demand spectra.

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