Received: 28 July 2017
Revised: 2 March 2018
Accepted: 16 April 2018
DOI: 10.1002/eqe.3056
RESEARCH ARTICLE
Damage spectra of global crustal seismic sequences
considering scaling issues of aftershock ground motions
Weiping Wen1,2
| Changhai Zhai1,2
| Duofa Ji1,2
1
Key Laboratory of Structures Dynamic
Behavior and Control of the Ministry of
Education, Harbin Institute of
Technology, Harbin, China
2
Key Laboratory of Smart Prevention and
Mitigation of Civil Engineering Disaster of
the Ministry of Industry and Information
Technology, Harbin Institute of
Technology, Harbin, China
Correspondence
Changhai Zhai, School of Civil
Engineering, Harbin Institute of
Technology, Harbin 150090, China.
Email: zch‐hit@hit.edu.cn
Funding information
National Key R&D Program of China,
Grant/Award Number: 2016YFC0701108;
National Natural Science Foundation of
China, Grant/Award Numbers: 51708161
and 51322801; Scientific Research Fund of
Institute of Engineering Mechanics, China
Earthquake Administration, Grant/Award
Number: 2017D09; China Postdoctoral
Science Foundation, Grant/Award Number: 2016 M601430; Postdoctoral Foundation of Heilongjiang Province, Grant/
Award Number: LBH–Z16085; Fundamental Research Funds for the Central
Universities, Grant/Award Number: HIT.
NSRIF. 201861; Open Foundation by key
Laboratory in Harbin Institute of Technology, Grant/Award Number:
HITCE201607; Outstanding Talents Jump
Promotion Plan of Basic Research of Harbin Institute of Technology
Summary
Recent studies and earthquakes have shown that the mainshock‐aftershock
(MSAS) sequences would induce significant cumulative damage, and it is
urgent to incorporate the MSAS sequences into performance‐based seismic
design and resilience assessment. For this purpose, one of the key problems
is how to accurately and quickly estimate the cumulative damage induced by
MSAS sequences. This manuscript investigates the damage spectra of global
crustal MSAS sequences and proposes a prediction equation of damage spectra
as a tool to quantify the cumulative damage caused by MSAS sequences. The
recorded MSAS ground motions from global crustal seismic sequences are
collected. The structures are modeled by 4 different kinds of single degree of
freedom systems with/without degradation and pinching behavior. The scaling
indicator and scaling factor SF of aftershock ground motions are studied from
the point of view of efficiency and accuracy. The results indicate that scaling
aftershock ground motions with spectra acceleration Sa present the best efficiency. Applying the condition of SF smaller than or equal to 5.0 (ie, SF ≤ 5.0)
can clearly reduce the bias induced by over‐scaling and meanwhile has no
change on the dispersion of results. Strong aftershocks (eg, relative Sa greater
than 0.8) can increase the damage spectra by the minimum level of 20% for
non‐degrading systems, and quantitative effects of aftershocks strongly depend
on the period of vibration, strength reduction factor, hysteretic behavior, and
intensity of the aftershock ground motions. The proposed prediction equation
shows high accuracy on the estimation of the cumulative damage induced by
MSAS sequences.
KEYWORDS
cumulative damage, damage spectra, mainshock‐aftershock (MSAS) sequences, resilience, scaling
1 | INTRODUCTION
It is well known that the mainshock of an earthquake is typically followed by many aftershocks which have smaller magnitude and occur in a limited area around the epicenter of the corresponding mainshock. Due to the short time between the
mainshock and its aftershocks, the mainshock‐damaged structures cannot be repaired prior to the following aftershocks.
The cascading effects of the mainshock and its aftershocks would induce significant cumulative damage on engineering
Earthquake Engng Struct Dyn. 2018;1–18.
wileyonlinelibrary.com/journal/eqe
Copyright © 2018 John Wiley & Sons, Ltd.
1
2
WEN ET AL.
structures located in the area affected by mainshock‐aftershock (MSAS) sequences. However, all current seismic codes in
the world do not consider explicitly the cumulative damage of structure induced by MSAS sequences.
During the last decade, many researchers have paid their attentions to the effects of seismic sequence or multiple
events on the structural performance. Lots of works1-32 have been conducted to study the engineering problems related
to the MSAS sequences, including ground motion characteristics of MSAS sequences,1 response spectra of MSAS ground
motions,2-10 and effects of aftershocks on the damage of multiple degrees of freedoms (MDOF) structures.11-33
How to accurately and quickly estimate the cumulative damage induced by MSAS sequences is the key step in the
process of incorporating the aftershocks into seismic design and resilience assessment. During the previous works focusing on seismic performance of structures under the MSAS sequences, the peak‐deformation‐based damage measure
(DM), (eg, maximum inter‐story drift ratio, maximum displacement of top story) is most widely used to quantify the
damage induced by MSAS ground motions.3-5,11-20,25-27 However, peak‐deformation‐based DM is not a reasonable measure to quantify the cumulative damage caused by MSAS sequences, because aftershocks tend to cause many inelastic
deformation reversals having smaller peak values with respect to the results of corresponding mainshock, while these
lower‐amplitude inelastic deformation reversals still indicate that the aftershocks make the mainshock‐damaged structure vibrate in the nonlinear range (ie, the damage of structure is increased by aftershocks). Zhai7 comprehensively
studied the effects of aftershocks on different kinds of DMs and found that energy‐based DM (ie, normalized hysteretic
energy and modified Park‐Ang damage index in this reference) can more effectively reflect the additional damage
induced by aftershocks. Several works also used the energy‐based DM to quantify the influences of MSAS sequences
on the seismic performance of structure.28,32,33 Besides, residual deformation was used in some works22,23 to study
the seismic performance of structure under the MSAS sequences. The problem related to the residual deformation is
that the aftershock may increase or decrease the residual deformation due to its polarity characteristics22 and thus sometimes residual deformation may present opposite trend on the cumulation damage.
The urban resilience assessment needs to account for the cascading effects of the mainshock and its aftershocks, and
the key step in this process is the assessment of cumulative damage induced by MSAS sequences. Damage spectra, presenting the variation of a damage index (generally use Park‐Ang damage index34 or its modified forms) versus periods
for a series of single degree of freedom (SDOF) systems, provides a simple tool for the assessment of cumulative damage.
Actually, Bozorgnia and Bertero35 have studied the damage spectra in 2003, and their results confirmed that damage spectra are promising for the assessment of performance‐based seismic vulnerability of existing structures. But they did not
focus on the damage spectra of seismic sequence, although they pointed out that the effects of seismic sequence on damage spectra are obvious. Results in Zhai7 confirmed that damage spectra are promising for the assessment of the cumulative damage caused by MSAS sequences. Zhai et al6 studied the damage spectra with recorded and simulated MSAS
sequences. Although they firstly proposed the prediction equation of damage spectra for MSAS sequences, several issues
have limited the application of their results. They just selected the MSAS sequences from the 1999 ChiChi earthquake, and
their results can not reflect the damage potential of MSAS sequences in the whole world. Moreover, the intensity of the
aftershock ground motions, which has significant effects on the damage potential of MSAS sequences, was not specially
considered in their study. Besides, the structure was modeled with a bilinear model while degradation and pinching
behaviors were ignored. Recently, Ghosh et al28 proposed the damage prediction equation for a single‐column box girder
bridge, but their prediction equation is applicable to the specific bridge that they considered only. Moreover, peak ground
acceleration (PGA) was used in their study to scale the ground motions, while reference 9 has noted that spectral acceleration (Sa) is a better scale indicator than PGA if one wants to reduce the response dispersion for MSAS sequences.
In light of the above discussions, this manuscript aims to develop the damage spectra with recorded MSAS
sequences from seismic sequences of global crustal earthquakes. The structure is modeled with/without degradation
or pinching behavior. The influences of scaling characteristics of aftershock ground motions on the damage of SDOF
systems are firstly studied to choose the reasonable scaling indicator and scaling factor for aftershock ground motions.
The effects of vibration period, strength reduction factor, hysteretic models, and aftershock intensity levels are statistically studied. Finally, the prediction equation of damage spectra is proposed for the global crustal MSAS sequences.
2 | GR OUND MOTIONS
This manuscript studies the damage potential of global crustal MSAS sequences. Four famous strong motion dataset
centers, which are Pacific Earthquake Engineering Research Center (PEER) Next Generation Attenuation (NGA) relationships database (http://ngawest2.berkeley.edu/), the European Strong‐Motion dataset (http://www.isesd.hi.is/ESD_
WEN ET AL.
3
Local/frameset.htm), Center for Engineering Strong Motion Data (http://www.strongmotioncenter.org/), and China
Earthquake Network Center (http://www.csndmc.ac.cn/newweb/data.htm#), are visited to select and download the
recorded MSAS ground motions.
All the selected MSAS ground motions satisfy the following criteria: (1) the ground motions are recorded during the
crustal earthquakes; (2) for a given station, the geometric mean of PGAs of 2 horizontal components is greater than
0.1 g; (3) the aftershock ground motion with the largest PGA is selected for the case that several aftershocks have been
recorded at the same station; (4) the ground motions are not recorded on soft soil (ie, average shear‐wave velocity over
the top 30 m of the site Vs30 is smaller than 180 m/s). Two horizontal components of the site that satisfy the above
criteria are used. The total of 274 MSAS ground motions is determined based on the above criteria. Table 1 summarizes
the origin time, magnitude, and number of sites for the selected MSAS sequences.
The number of records used in the statistical analysis is firstly computed based on the usable spectral frequency of
each record, which is greater than 1.25 times the high‐pass corner frequency used in the record processing, as defined in
the NGA database.36 Figure 1 presents the number of MSAS ground motions used in the statistical analysis after
considering the usable spectral frequency of each record. The number of MSAS ground motions used in the statistical
analysis decreases gradually with the increase of periods of systems, because some records need larger high‐pass corner
frequency to remove the effects of noise.
Figure 2 shows the comparison between the characteristics (response spectra and duration are shown here) of
mainshock ground motions and characteristics of aftershock ground motions. The normalized Sa, defined as the ratio
of Sa to PGA (ie, Sa/PGA), is used here to eliminate the influence of PGA. The results of Sa/PGA are shown in
Figure 2A for different regions. The values of normalized Sa of aftershock ground motions are smaller than those of
mainshock ground motions even they have equal PGAs, and this trend becomes more obvious in the moderate‐long
period region, indicating that the response spectra at moderate to long periods decrease more rapidly for aftershock
ground motions than for mainshock ground motions.
The results of significant duration with 70% energy are shown in Figure 2B for different seismic regions. The MSAS
ground motions in Asia and New Zealand generally have larger duration than those in the USA and Europe; this can be
explained by the larger magnitudes (see the Table 1) associated with the MSAS sequences in Asia and New Zealand. The
MSAS ground motions in Europe generally have lower duration due to the smaller magnitudes of MSAS sequences in
this region. The MSAS ground motions in the USA generally have moderate duration due to the moderate magnitudes
of MSAS sequences in this region.
Figure 3 shows a time history for the ChiChi MSAS ground motion recorded at station CHY080 in east‐west
direction. As shown in this figure, the time gap of 30 seconds is added between 2 ground motions to simulate the time
between the mainshock and aftershock.
3 | COMPUTATION OF DAMAGE SPECTRA
The inelastic SDOF systems with a set of 30 periods of vibration between 0.1 and 3.0 seconds with an interval of
0.1 seconds are considered to compute the damage spectra. The viscous damping ratio is assumed to be 5%. The relative
lateral strength of a system is measured by the strength reduction factor R, which is defined as:
R¼
Fe
Fy
(1)
where Fe is the strength demand on an elastic SDOF system during a mainshock ground motion, Fy is the yield
strength of the corresponding inelastic SDOF system with the same mass and initial stiffness. Five values of R are
considered (ie, R = 2, 3, 4, 5, and 6) to study the influences of different yield strength.
The modified Park‐Ang damage index37 is used here to quantify the damage of structure under the MSAS ground
motions. It is defined as
DI ¼
x m ‐x y
EH
μ‐1
EH
þβ
þβ
¼
x u ‐x y
x u F y μu ‐1
F y μu x y
(2)
where μ is the ductility demand; xu is the ultimate deformation capacity of structure under monotonic loading; μu is the
corresponding ultimate ductility capacity of structure under monotonic loading; β is a positive dimensionless parameter
to scale the effect of hysteretic energy dissipation on the final damage of structure. The median value of β (ie, β = 0.15)38
is used as the reference value.
4
WEN ET AL.
TABLE 1
Seismic
Regions
Asia
Origin time, magnitude, and number of sites of the seismic sequences used in this manuscript
Time (yyyy/mm/dd, hh:mm)
Magnitude
Earthquake
ms
as
ms
as
Number
of Sites
ChiChi, Taiwan
1999/09/20,‐‐
1999/09/20,18:03
7.6
6.2
10
1999/09/20,21:46
6.2
3
1999/09/22,00:14
6.2
14
1999/09/25,23:52
6.3
8
5.8
6.3
1
2
2008/05/16,13:25
5.9
1
2008/05/12.16:21
5.5
1
2005/05/12,17:42
5.3
1
2008/05/12,14:43
6.3
1
2008/05/12,18:04
4.1
1
Wenchuan, China mainland
2008/05/12,14:28
2008/05/13,04:08
2008/05/12,19:11
7.9
NZ
Darfield, New Zealand
2010/09/03,16:35
2011/02/21,23:51
7
6.2
17
USA
Imperial Valley
1979/10/15,23:16
1979/10/15,23:19
6.5
5.0
8
Northridge
1994/01/17,12:31
1994/03/20,21:20
6.7
5.3
14
Chalfant
1986/07/21,14:42
1986/07/31,07:22
6.2
5.4
1
5.7
1
5.3
20
1986/07/21,14:51
Whitter
Europe
1987/10/01,14:42
1987/10/04,10:59
6.0
Petrolia
1992/04/25,11:06
1992/04/26,12:41
7.1
6.5
4
Irpinia, Italy
1980/11/23,19:34
1980/11/23,19:35
6.9
6.2
1
Umbria Marche, Italy
1997/09/26,09:40
1997/10/06,23:24
6.0
5.5
2
4.3
1
1997/10/11,03:20
L'Aquila, Italy
2009/04/06,01:33
2009/04/07,17:47
6.3
5.6
3
Kefallinia island, Greece
1972/09/17,14:07
1972/10/30,14:32
5.6
5.0
1
Friuli, Italy
1976/05/06,20:00
1976/05/07,00:23
6.5
5.2
1
Volvi, Greece
1978/06/20,20:03
1978/07/04,22:23
6.2
5.1
1
Montenegro
1979/04/15,06:19
1979/05/24,17:23
6.9
6.2
1
Alkion, Greece
1981/02/24,20:53
1981/02/25,02:35
6.6
6.3
1
Kalamata, Greece
1986/09/13,17:24
1986/09/15,11:41
5.9
4.9
2
Spitak, Armenia
1988/12/07,07:41
1988/12/07,07:45
6.7
5.8
1
Racha, Georgia
1991/05/03,20:19
1991/05/03,23:41
5.6
4.4
1
Kefallinia island, Greece
1992/01/23,04:24
1992/06/23,04:24
5.6
4.9
1
Mt. Hengill Area, Iceland
1998/06/04,21:36
1998/11/14,14:24
5.4
4.4
1
Izmit, Turkey
1999/08/17,00:01
1999/08/29,10:15
7.6
4.2
1
5.8
1
Duzce, Turkey
1999/11/12,16:57
5.2
2
1999/11/12,17:16
4.7
1
1999/11/19,19:59
4.8
2
6.4
4
1999/09/13,11:55
South Iceland
2000/06/17,15:40
1999/11/12,17:17
2000/06/21,00:51
7.2
6.5
The hysteretic behaviors of structures under the ground motions are simulated with 4 different hysteretic models: (1)
elastic‐perfectly‐plastic (EPP) model; (2) modified clough (MC) model; (3) pinching (PH) model39; and (iv) stiffness
strength degradation (SSD).37
WEN ET AL.
FIGURE 1
5
The number of MSAS ground motions used in the statistical analysis after considering the usable spectral frequency of each
record
FIGURE 2
Comparison between the characteristics of mainshock ground motions and characteristics of aftershock ground motions: A,
normalized spectra acceleration; B, significant duration with 70% energy [Colour figure can be viewed at wileyonlinelibrary.com]
FIGURE 3
Time history for the ChiChi MSAS ground motion recorded at station CHY080 in east‐west direction
The EPP model is extensively used in the field of earthquake engineering due to its simplicity, and it is used here to
simulate the dynamic behavior of non‐degrading structures. The MC model, proposed initially for concrete structures, is
used to consider the stiffness degradation at reloading path and to simulate the flexural behavior of reinforcement concrete (RC) structures. The PH model proposed by Rahnama and Krawinkler39 is used here to simulate the pinching
behavior of non‐ductile RC structure. The default value (ie, 0.4) in Rahnama and Krawinkler39 on the ratio of the
pinching target strength to the yield strength is used. The SSD model based on the 3‐parameter model37 is used to
represent the global behavior of systems exhibiting stiffness degradation and strength deterioration during reloading
branches. It should be noted that the strength degradation rule in the 3‐parameter model is used here, and the
degradation parameter β is the same as that in the modified Park‐Ang damage index. The values of β for various
combinations are shown in Sections 5 and 6. No unloading stiffness degradation and pinching behavior is presented
in the SSD model.
6
WEN ET AL.
Figure 4 presents the hysteretic response of a SDOF system (T = 1.0 s, R = 3, unit mass) under the MSAS ground
motion shown in Figure 3. The results corresponding to the mainshock and aftershock are shown, respectively. Whether
aftershock increases the ductility demand or not depends on the hysteretic behavior of this SDOF system, but it is clear
that the aftershock increases the hysteretic energy for all hysteretic models. The results in Figure 4 also present the
general hysteretic rules of 4 models.
4 | SC ALING INDICATOR AND SC ALING FAC TOR
It is significant to study the effects of aftershock ground motions with different intensity levels on the damage of
structure, due to the fact that the aftershocks with various intensities would happen after the mainshock and the
intensity of aftershock plays an important role in the cumulative damage induced by MSAS ground motions. However,
there are no sufficient recorded MSAS ground motions to statistically study the damage spectra of MSAS sequences in
which aftershocks have identical intensities. Thus, scaling the aftershock ground motions to a given intensity level
provides us a way to simulate the MSAS sequences in which the aftershock has given intensity level. During the scaling
process, the selection of scaling indicator and scaling factor would be very important for the results, because unreasonable scaling indicator or larger scaling factor would make clear bias on the final results. In this section, scaling indicator
and scaling factor for the aftershock ground motions would be studied, respectively, from the point of view of efficiency
and accuracy.
In the probabilistic seismic demand analysis, the efficiency of intensity measure means that lower uncertainty is
included in the probabilistic seismic demand model. This criterion is applied during the selection of scaling indicator,
ie, the scaling indicator induced lower dispersion of damage spectra of MSAS sequences would be used.
The relative intensity measure of the aftershock ground motion is firstly defined as
∇IM ¼
IM as
IM ms
(3)
where IMas and IMms are intensity measures of the aftershock and mainshock ground motion, respectively.
Three commonly used intensity measures, which are peak ground acceleration (PGA), peak ground velocity (PGV),
and spectral acceleration (Sa), are selected as the alternative scaling indicators for aftershock ground motions. For each
FIGURE 4 Hysteretic response of a SDOF system (T = 1.0 s, R = 3) under the MSAS ground motion shown in Figure 3: A, EPP model; B,
MC model; C, PH model; D, SSD model [Colour figure can be viewed at wileyonlinelibrary.com]
WEN ET AL.
7
MSAS ground motion, the aftershock ground motion is scaled to have 3 relative intensity levels (ie, ∇IM=0.5, 0 .8, 1.0).
Due to lower magnitude of aftershock, the intensity of the aftershock ground motion is generally smaller than that of
the mainshock ground motion. Thus, maximum value of ∇IM is assumed as 1.0. This manuscript focuses on the damage
spectra of MSAS sequences with strong aftershock ground motions, and thus minimum value of ∇IM is assumed as 0.5.
In this section, the damage spectra of the EPP systems corresponding to β = 0.15, μu = 10 are used to study the scaling
indicator and scaling factor.
The ratio of coefficient of variation (COV) of damage spectra for MSAS ground motions to that of damage spectra for
mainshock ground motions (ie, COVseq/COVms) is used to study the efficiency of different scaling indicators. The value
of COVms is computed as the ratio of the standard deviation of damage spectra for the mainshock to the mean of damage
spectra for the mainshock. Similarly, the value of COVseq is computed as the ratio of the standard deviation of damage
spectra for the MSAS sequence to the mean of damage spectra for the MSAS sequence. Figure 5 presents the COVseq/
COVms of systems with R = 2 when aftershock ground motions are scaled to ∇IM = 0.8. Figure 6 presents the
COVseq/COVms of systems with R = 5 when aftershock ground motions are scaled to ∇IM = 0.8. The value of
COVseq/COVms greater than 1.0 indicates that the aftershocks increase the dispersion of damage spectra, while value
of COVseq/COVms smaller than 1.0 indicates that the aftershocks decrease the dispersion of damage spectra. The results
in Figures 5 and 6 indicate that aftershock ground motions would decrease the dispersion of damage spectra when it is
scaled with∇Sa, while other scaling indicators would make the aftershocks increase the dispersion of damage spectra in
many cases. Thus, ∇Sais used to scale the aftershock ground motions.
Besides the reasonable scaling indicator, it is also necessary to apply the smaller scaling factor (SF). SF = 1.0 indicates that the un‐scaled ground motions are used. Applying large SF may induce unacceptable bias on the estimation of
seismic demand and thus contaminate the accuracy of statistical results from the scaled ground motions. Based on the
suggestion from previous studies,40-42 the maximum SF during the scaling is limited, and the condition of SF smaller
than or equal to 5.0 (ie, SF ≤ 5.0) is applied here. Figure 7 shows that the number of MSAS ground motions used in
the statistical analysis when SF ≤ 5.0 is applied. The number of MSAS ground motions decreases with the increase of
periods, due to the bigger differences of Sa between the aftershock and mainshock ground motions in the long period
region (as shown in Figure 2A). At least 130 MSAS ground motions can be used in the statistical analysis even for
the long period region and high relative intensity level of aftershock (eg, T = 3.0 s and ∇IM = 1.0).
In order to study the scaling factor from the point of view of accuracy, Figure 8 shows the influences of the reduction
in number of MSAS ground motions on the mean and dispersion of damage spectra. Figure 8A presents the ratio of
FIGURE 5 The influences of aftershocks on the response dispersion of systems with R = 2 when aftershock ground motions are scaled to
∇IM = 0.8: A, EPP model; B, MC model; C, PH model; D, SSD model [Colour figure can be viewed at wileyonlinelibrary.com]
8
WEN ET AL.
FIGURE 6 The influences of aftershocks on the response dispersion of systems with R = 5 when aftershock ground motions are scaled to
∇IM = 0.8: A, EPP model; B, MC model; C, PH model; D, SSD model [Colour figure can be viewed at wileyonlinelibrary.com]
FIGURE 7 The number of MSAS ground motions used in the statistical analysis when usable spectra frequency and SF ≤ 5.0 are both
considered [Colour figure can be viewed at wileyonlinelibrary.com]
FIGURE 8 The influences of number reduction of MSAS ground motions with ∇Sa = 0.8 on the mean and dispersion of damage spectra:
A, mean; B, dispersion [Colour figure can be viewed at wileyonlinelibrary.com]
damage spectra of MSAS ground motions to those of mainshock ground motions (ie, DIseq/DIms) for the cases with/
without the condition of SF ≤ 5.0. After applying the condition of SF ≤ 5.0, the effects of aftershocks on the damage
spectra are clearly reduced. For example, without the condition of SF ≤ 5.0, aftershocks with ∇Sa = 0.8 can increase
WEN ET AL.
9
the damage spectra by nearly 40%, while this magnitude decreases to 30% after applying the condition of SF ≤ 5.0. This
phenomenon can be explained by the fact that over‐scaling of aftershock ground motions would cause great bias on
the seismic demand estimation. However, applying the condition of SF ≤ 5.0 almost has no influence on the
dispersion (in terms of COV). The lines for the dispersion with/without condition of SF ≤ 5.0 are generally overlapped
(Figure 8B). Based on these results, it can be concluded that applying the condition of SF ≤ 5.0 can clearly reduce the
bias induced by over‐scaling and at the same time has no change on the dispersion of results. The value of 5.0 for the
maximum scaling factor is a compromise on the demand of smaller scaling factor and larger number of MSAS ground
motions with a given ∇Sa in the statistical analysis. With the number of recorded MSAS sequences grows, the value of
the maximum scaling factor can gradually decrease.
5 | DAMAGE SPECTRA C HARACTERISTICS
In this section, the influences of aftershocks, hysteretic behavior, μu and β on the damage spectra of MSAS ground
motions are statistically analyzed. In the Sections 5.1 and 5.2, the damage spectra for β = 0.15 and μu = 10 are used
to study the effects of aftershocks and hysteretic behavior.
5.1 | Effects of aftershocks
The effects of aftershocks on the damage spectra are studied by the ratio of damage spectra of MSAS ground motions to
those of mainshock ground motions (ie, DIseq/DIms). Figure 9 presents the results of DIseq/DIms for the EPP systems
under the MSAS ground motions. For the case of∇Sa = 0.5 (Figure 9A), the values of DIseq/DIms increase with the
increase of R for all periods, because smaller aftershocks tend to make no additional damage especially for stronger
structure (ie, smaller R). For the case of ∇Sa = 0.8 (Figure 9B), no clear trend can be observed in the period region
of T < 2.0 seconds, while the same trend as the case of∇Sa = 0.5 appears in the period region of T > 2.0 seconds.
For the case of ∇Sa = 1.0 (Figure 9C), the opposite trend to the case of ∇Sa = 0.5 appears in the whole period region.
This phenomenon can be explained by the fact that the strong aftershocks tend to make the clear additional damage,
and this additional damage is more significant in the relative sense for the stronger structures (ie, smaller R) which
experience smaller damage after mainshock.
FIGURE 9 The results of mean DIseq/DIms for EPP systems under the MSAS ground motions: A, ∇Sa = 0.5; B, ∇Sa = 0.8; C, ∇Sa = 1.0
[Colour figure can be viewed at wileyonlinelibrary.com]
10
WEN ET AL.
The results in Figure 9 actually provide the quantitative effects of aftershocks on damage spectra in the relative
sense. The values of DIseq/DIms for ∇Sa = 0.5 are generally smaller than 1.2, indicating that the aftershocks with
∇Sa = 0.5 can increase the damage spectra by the maximum level of 20%. The values of DIseq/DIms for ∇Sa = 0.8 are
generally greater than 1.2, meaning that the aftershocks with ∇Sa = 0.8 can increase the damage spectra by the
minimum level of 20%. Following this kind of analysis, it can be concluded that the aftershocks with ∇Sa = 1.0 can
increase the damage spectra by the minimum level of 30%. The damage accumulation induced by the MSAS ground
motions is strongly dependent on the strength reduction factor R (related to the damage induced by mainshock) and
the intensity of aftershock ground motions.
It is also necessary to study the effects of aftershocks on the damage spectra of the degrading systems. Figure 10
presents the results of DIseq/DIms for the degrading systems with R = 2 under the MSAS ground motions. For the
convenience of comparison, the results of DIseq/DIms for the EPP systems with R = 2 are also presented in Figure 10.
The values of DIseq/DIms in Figure 10A are smaller than 1.1, indicating that the effects of aftershocks with ∇Sa = 0.5
on the damage spectra of systems with R = 2 are within 10%. The values of DIseq/DIms in the short period region for
the aftershocks with∇Sa = 0.8 and ∇Sa = 1.0 (as shown in Figure 10B and C) can reach 1.3 and 1.6 respectively,
indicating the strong aftershocks with ∇Sa ≥ 0.8 can increase the damage spectra of degrading systems with short
periods and R = 2 in a level of 30% to 60%.
Figure 11 presents the results of DIseq/DIms for the degrading systems with R = 5 under the MSAS ground motions.
The values of DIseq/DIms in Figure 11 for the EPP systems are all greater than those for the degrading systems. Based on
the results in Figures 10 and 11, it can be concluded that the aftershocks have larger effects on the damage spectra of the
EPP systems except for the case of R = 2 and ∇Sa = 0.5. This is because the aftershocks tend to make low‐amplitude
inelastic deformation reversals especially for the degrading systems, which can also be seen in Figure 4. Therefore,
aftershocks tend to make the smaller additional damage in the relative sense for degrading systems.
In Figure 11, the values of DIseq/DIms for the PH systems are generally smaller than those for the MC systems and
SSD systems, while values of DIseq/DIms for the MC systems and SSD systems are generally close to each other. This phenomenon indicates that for the degrading systems, aftershocks generally have smallest effects on the damage spectra of
the PH systems and have very similar effects on the damage spectra for the MC systems and SSD systems. For example,
the values of DIms at period of 2.0 seconds in the case of∇Sa = 0.8 and R = 5 (Figure 11B) are 0.90, 0.77, 0.71, and 0.75
for the EPP, MC, PH, and SSD models, respectively, while the corresponding values of DIseq are 1.13, 0.88, 0.76, and
0.84, respectively. Therefore, the values of DIseq/DIms at period of 2.0 seconds show the trend as shown in Figure 11B.
FIGURE 10 The results of DIseq/DIms for systems with R = 2 under the MSAS ground motions: A, ∇Sa = 0.5; B, ∇Sa = 0.8; C, ∇Sa = 1.0
[Colour figure can be viewed at wileyonlinelibrary.com]
WEN ET AL.
11
FIGURE 11 The results of DIseq/DIms for systems with R = 5 under the MSAS ground motions: A, ∇Sa = 0.5; B, ∇Sa = 0.8; C, ∇Sa = 1.0
[Colour figure can be viewed at wileyonlinelibrary.com]
In the moderate‐long period region, the effects of aftershocks on the damage spectra of the degrading systems are
generally within 10%, 20%, and 40% for the relative intensities of ∇Sa = 0.5, 0.8, and 1.0, respectively. But in the short
period region, the effects of aftershocks can exceed 20% and 40% for ∇Sabeing 0.8 and 1.0, respectively.
The results in Figures 9, 10, and 11 indicate that the effects of aftershocks depend on the period of vibration, strength
reduction factor, hysteretic behavior of systems, and intensity of the aftershock ground motions. Moreover, this kind of
effect is generally interdependent for these 4 factors in most cases.
5.2 | Effects of hysteretic behavior
The results in the previous section reveal that the effects of aftershocks on the damage spectra depend on the hysteretic
behavior of systems. This section aims to quantitatively study the effects of hysteretic behavior of systems on the damage
spectra of MSAS ground motions.
The ratio of damage spectra for the degrading systems to those for the EPP systems (ie, DIdegrading/DIEPP) is
computed, and the corresponding results are shown in Figure 12 for the MSAS ground motions with ∇Sa = 0.8. The
values of DIdegrading/DIEPP in the short period region are generally greater than 1.0 except for a few values for R = 6. This
phenomenon indicates that the degrading behavior tends to increase the damage of systems in the short period region,
but the very small lateral strength (ie, large R) may change this trend, and thus effects of hysteretic behavior and R on
the damage spectra are also interdependent. The values of DIdegrading/DIEPP in the long period region are generally lower
than 1.0, indicating that the degrading systems would experience lower seismic damage when they possess the same
ultimate ductility capacity μu as the EPP systems. This may due to the fact that degrading systems tend to experience
more hysteretic cycles with smaller amplitude (as can be seen in Figure 4), and this phenomenon would be more
obvious when the period of vibration becomes longer.
The limiting period that divides the period region where DIdegrading/DIEPP > 1.0 from the period region where
DIdegrading/DIEPP < 1.0 depends on R and decreases with the increase of R. This phenomenon means that smaller lateral
strength (larger R) makes more degrading systems have lower damage than the EPP systems.
Figure 13 presents the results of DIdegrading/DIEPP for the SSD systems under the MSAS ground motions. The similar
trend of DIdegrading/DIEPP versus period (ie, DIdegrading/DIEPP > 1.0 in the short period region and DIdegrading/DIEPP < 1.0
in the long period region) can be observed in Figures 12 and 13. The limiting period that divides the period region where
DIdegrading/DIEPP > 1.0 from the period region where DIdegrading/DIEPP < 1.0 decreases with the increase of ∇Sa.
12
WEN ET AL.
FIGURE 12 The effects of hysteretic behavior on the damage spectra of MSAS ground motions with ∇Sa = 0.8: A, MC model; B, PH
model; C, SSD model [Colour figure can be viewed at wileyonlinelibrary.com]
FIGURE 13 The results of DISSD/DIEPP for the SSD systems under the MSAS ground motions: A, R = 2; B, R = 5 [Colour figure can be
viewed at wileyonlinelibrary.com]
Combining the results in Figures 12 and 13, it can be concluded that larger R or ∇Sawould make more degrading
systems in the moderate‐long period region experience lower damage than the EPP systems. In the long period region,
the effects of degrading behavior are more significant for larger R and∇Sa, while in the short period region, the effects of
degrading behavior are more significant for smaller R and ∇Sa.
5.3 | Effects of β and μu
The value of β = 0.15 used in the above sections is just the median value of this parameter.38 Park and Ang34 pointed out
that the value of β may be different for different kinds of structures. Negro43 provided the suggestions on β for reinforced
concrete (RC) frame structures. This manuscript aims to provide a general tool to estimate the damage of systems under
the MSAS ground motions; thus, different values of β (ie, β = 0.05, 0.10, 0.15, 0.20, and 0.25)35,37,38,43-47 are used here to
study the effects of β on the damage spectra of MSAS ground motions.
The ratio between the damage spectra of β = 0.05 (or β = 0.25) and those of β = 0.15 (ie, DIβ = 0.05/DIβ = 0.15 or
DIβ = 0.25/DIβ = 0.15) are computed, and corresponding results are shown in Figure 14 for the EPP systems with μu = 10
under the MSAS ground motions with∇Sa = 0.8. The values of DIβ = 0.05/DIβ = 0.15 in Figure 14A vary within the range
WEN ET AL.
13
FIGURE 14 The ratio of DI for EPP systems with μu = 10 under the MSAS ground motions with∇Sa = 0.8: A, β = 0.05; B, β = 0.25 [Colour
figure can be viewed at wileyonlinelibrary.com]
of [0.6 0.8], and the values of DIβ = 0.25/DIβ = 0.15 in Figure 14B vary within the range of [1.2 1.4]. The effects of β on the
damage spectra are in the range of 20% to 40%. Smaller β indicates smaller contribution of hysteretic energy dissipation
to the final damage of structure, and thus the values of DIβ = 0.05/DIβ = 0.15 in Figure 14A are smaller than 1.0, while the
values of DIβ = 0.25/DIβ = 0.15 in Figure 14B are greater than 1.0. The values of DIβ = 0.05/DIβ = 0.15 in Figure 14B
decrease with the increase of R, while the values of DIβ = 0.25/DIβ = 0.15 in Figure 14B increase with the increase of R,
indicating that the variation of β has greater influences on the damage spectra of systems with smaller lateral strength
(ie, larger R). This can be explained by the fact that hysteretic energy dissipation corresponds to larger R would
contribute more to the final damage of structure in comparison with smaller R, and β is a parameter to scale the
contribution of hysteretic energy dissipation to the final damage of structure.
The value of modified Park‐Ang damage index is very sensitive to the ultimate ductility capacity μu of structure, and
the actual structures may have different values of μu based on its design details and design targets. In this section,
different values of μu (ie, μu = 6, 8, 10, 12, and14)35,44,45 are used here to study the effects of μu on the damage spectra
of MSAS ground motions.
The ratio between the damage spectra of μu = 6 (or μu = 14) and those of μu = 10 (ie, DI μu ¼6 =DI μu ¼10 or
DI μu ¼14 =DI μu ¼10 ) is computed, and corresponding results are shown in Figure 15 for the EPP systems with β = 0.15
under the MSAS ground motions with ∇Sa = 0.8. The values of DI μu ¼6 =DI μu ¼10 in Figure 15A vary within the range
of [1.72 1.77], and the values of DI μu ¼14 =DI μu ¼10 in Figure 15B vary within the range of [0.69 0.71]. This phenomenon
means that the effects of μu on the damage spectra can exceed 70%, and thus it is necessary to incorporate μu into the
prediction equation of damage spectra. Smaller μu indicates smaller capacity of structure to resist the damage, and thus
the values of DI μu ¼6 =DI μu ¼10 in Figure 15A are greater than 1.0, while the values of DI μu ¼14 =DI μu ¼10 in Figure 15B are
smaller than 1.0. The values of DI μu ¼6 =DI μu ¼10 in Figure 15A decrease with the increase of R, while the values of
DI μu ¼14 =DI μu ¼10 in Figure 15B increase with the increase of R, indicating that the variation of μu has greater influences
on the damage spectra of systems with larger lateral strength (ie, smaller R). This can be explained by the fact that ductility demand corresponding to smaller R would contribute more to the final damage of structure in comparison with
larger R, and variation of μu has more significant effects on the ductility demand item (ie, first item in Equation 2).
It should be noted that the results corresponding to other values of β (ie, β = 0.10 and 0.20) and μu (ie, μu = 8 and 12)
show the same trends as those in Figures 14 and 15, respectively, except for smaller corresponding ratios. Due to the
space limitation, those results are not shown in this section.
FIGURE 15 The ratio of DI for EPP systems with β = 0.15 under the MSAS ground motions with∇Sa = 0.8: A, μu = 6; B, μu = 14 [Colour
figure can be viewed at wileyonlinelibrary.com]
14
WEN ET AL.
6 | PREDICTION EQUATION
The prediction equation of damage spectra is an appealing tool to estimate the cumulative damage induced by MSAS
ground motioms. Based on the statistical analysis in the above sections, it is found that period of vibration, strength
reduction factor, intensity of the aftershock ground motions, hysteretic behavior of systems, β, and μu in modified
Park‐Ang damage index have significant effects on the damage spectra. Moreover, the effects of these factors are
generally interdependent, and this feature makes it impossible to propose a very simplified model to predict the damage
spectra of MSAS ground motions for various cases. However, a very complicated prediction equation is not convenient
for the engineering application in the current status. The prediction equation proposed here for the damage spectra of
MSAS ground motions is a compromise of the accuracy and simplification.
Two‐step regression analysis is used to propose the prediction equation. In the first regression step, the prediction
equation is formulated as a function of T and R. The damage of a system should be zero when this system vibrates
in the elastic range (ie, R = 1); thus, the proposed prediction equation should satisfy the following boundary condition:
DI ðT; R ¼ 1Þ ¼ 0:
(4)
When the period of vibration tends to zero, the small reduction in elastic strength demand Fe (ie, R is greater than
1.0 in a very small level) would make this nonlinear system experience infinite damage.6 Besides, when the period of
vibration tends to infinity and the system becomes exremely flexible, the damage of this system would be independent
of period and just be a function of R due to the equal displacment rule. Thus, the proposed prediction equation should
satisfy the following boundary conditions:
DI ðT→0; R>1Þ ¼ ∞
(5)
DI ðT→∞; R>1Þ ¼ f ðRÞ:
(6)
Based on the above boundary conditions and statistical resutls, the prediction equation is proposed as
DI ¼
aþ
b⋅ðR−1Þc
Td
2
⋅ðR−1Þe
(7)
where T is period of vibration, R is strength reduction factor, and a, b, c, d, and e are regression parameters.
Nonlinear least‐square regression analysis is performed for Equation 7 and the mean damage spectra over MSAS
ground motions. It is found that the values of regression parameters c, d, and e are generally indpendent of ∇Sa, β,
and μu, while just related to the hysteretic models. Thus, the values of regression parameters c, d, and e are suggested
as the results in Table 2.
The regression parameters a and b are dependent on the values of ∇Sa, β, and μu; thus, in the second regression step,
the following models are further regressed to predict the regression parameters a and b. The regression results for the
corresponding parameters are summarized in Table 3.
lna ¼ a1 þ a2 ⋅ lnμu þ a3 ⋅β þ a4 ⋅ expða5 ⋅∇Sa Þ
(8)
lnb ¼ b1 þ b2 ⋅ lnμu þ b3 ⋅β þ b4 ⋅ expðb5 ⋅∇Sa Þ
(9)
With the Equations 7, 8, and 9, one can predict the damage spectra of MSAS ground motions for various cases. The
coefficient of determination for the prediction equation proposed here is greater than 99%. Figure 16 shows the
comparisons between statistical resutls and prediction results for the case of β = 0.15 and μu = 10. The statistical results
(ie, mean damage spectra) which has been used to calibrate the prediction equation during the regression process are
used here for the comparisons. The results in Figure 16 confirm that the prediction equation can provide good
estimation for the damage spectra of MSAS ground motions.
WEN ET AL.
TABLE 2
15
The values of regression parameters c, d, and e used in Equation (7)
Hysteretic Model
c
d
e
EPP
0.48
1.68
1.28
MC
0.15
1.52
1.22
PH
0.02
1.36
1.14
SSD
0.00
1.43
1.20
TABLE 3
The values of regression parameters ai and bi (i = 1–5) used in Equations (8) and (9)
Hysteretic
Model
Regression
Parameter
EPP
ai
bi
MC
PH
SSD
Regression Results (i = 1–5)
1
2
3
4
5
0.080
−0.56
1.13
0.0155
2.67
−2.916
−0.55
1.53
0.0030
3.76
ai
0.021
−0.55
1.27
0.0090
2.76
bi
−2.075
−0.54
1.89
0.0153
2.28
ai
0.091
−0.56
0.94
0.0031
3.23
bi
−1.432
−0.55
1.37
0.0133
2.30
ai
0.016
−0.56
1.25
0.0085
2.67
bi
−1.631
−0.55
1.76
0.0131
2.31
FIGURE 16 The comparisons between statistical results and prediction results for the case of β = 0.15 and μu = 10: A, EPP systems,
∇Sa = 0.8; B, EPP systems, R = 4; C, degrading systems, ∇Sa = 0.8
7 | CONCLUSIONS
This manuscript aims to investigate the damage spectra of global crustal MSAS sequence and proposes the prediction
equation to quantify the cumulative damage caused by MSAS sequences. The recorded MSAS ground motions are
16
WEN ET AL.
collected from the global crustal seismic sequence. The scaling indicator and scaling factor of aftershock ground motions
are studied from the point of view of efficiency and accuracy. The influences of aftershocks, hysteretic behavior, μu, and β
on the damage spectra of MSAS ground motions are statistically analyzed. The prediction equation of damage spectra for
MSAS ground motions is proposed with 2‐step regression analysis. The key findings of this manuscript are as folllows:
1. In the 3 different alternative scaling indicators (ie, PGA, PGV, and Sa) used in this manuscript, scaling aftershock
ground motions with Sa present the best efficiency, ie, induce lowest dispersion in terms of coefficient of variation
(COV). Moreover, aftershock ground motions would reduce the dispersion of damge spectra with respect to
mainshock when it is scaled with Sa. The maximum scaling factor of 5.0 used here can clearly reduce the bias
induced by over‐scaling and at the same time has no change on the dispersion of results.
2. Strong aftershocks (eg, ∇Sa being greater than 0.8) can increase the damage spectra by the minimum level of 20%
for non‐degrading systems and maximum influences of aftershocks can exceed 40%. The quantitative effects of aftershocks strongly depend on the period of vibration, strength reduction factor, hysteretic behavior, and intensity of
aftershock ground motions. The aftershocks generally have greater effects on the damage spectra of non‐degrading
systems. For the degrading systems, aftershocks have smallest effects on the damage spectra of the PH systems and
have very similar effects on the damage spectra of the MC systems and SSD systems. In the short period region, the
effects of aftershocks on the damage spectra of degrading systems can exceed 20% and 40% for ∇Sa = 0.8 and 1.0,
respectively.
3. The degrading behavior tends to increase the damage of systems in the short period region, but the very small lateral strength (ie, large R) may change this trend. The degrading systems in the long period region tend to experience
lower seismic damage when they possess the same ultimate ductility capacity μu as the EPP systems, which may be
due to the fact that degrading systems tend to experience more hysteretic cycles with smaller amplitude and this
phenomenon would be more obvious when the period of vibration becomes longer.
4. The effects of β on the damage spectra are in the range of 20% to 40%, and the variation of β has greater influences
on the damage spectra of systems with smaller lateral strength (ie, larger R). The effects of μu on the damage spectra
can exceed 70%.
5. The prediction equation is proposed based on the boundary conditions and statistical resutls. The regression
parameter in the propsoed model is regressed with 2‐step analysis in which nonlinear least‐square regression
method is used. The comparison between statistical results and prediction results indicates that the proposed prediction equation possesses high accuracy on the estimation of the cumulative damage induced by MSAS ground
motions.
Although the comprehensive statistical analysis is performed to propose a general tool for the cumulative damage
assessment, the following 2 issues still need further attentions for the extensive application of the damage spectra proposed here: (1) just 1 aftershock is included in the MSAS sequence, and it may underestimate the cumulative damage
potential of the sequence including more than 1 strong aftershocks; (2) the damage spectra proposed here still need the
further modifications to account for the characteristics of MDOF systems including over‐strength and P‐Delta effects.
A C K N O WL E D G E M E N T
This investigation is supported by the National Key R&D Program of China (2016YFC0701108), the National Natural
Science Foundation of China (No. 51322801, 51708161), Scientific Research Fund of Institute of Engineering Mechanics,
China Earthquake Administration (2017D09), China Postdoctoral Science Foundation (No. 2016 M601430), Postdoctoral Foundation of Heilongjiang Province (No. LBH‐Z16085), the Fundamental Research Funds for the Central Universities (HIT. NSRIF. 201861), Open Foundation by key Laboratory in Harbin Institute of Technology (HITCE201607),
and the Outstanding Talents Jump Promotion Plan of Basic Research of Harbin Institute of Technology. These supports
are greatly appreciated.
ORCID
Weiping Wen http://orcid.org/0000-0001-9354-344X
Changhai Zhai http://orcid.org/0000-0003-3888-6794
Duofa Ji http://orcid.org/0000-0002-5537-5016
WEN ET AL.
17
R EF E RE N C E S
1. Ruiz‐García J. Mainshock‐aftershock ground motion features and their influence in building's seismic response. J Earthq Eng.
2012;16(5):719‐737.
2. Amadio C, Fragiacomo M, Rajgelj S. The effects of repeated earthquake ground motions on the non‐linear response of SDOF systems.
Earthq Eng Struct Dyn. 2003;32(2):291‐308.
3. Hatzigeorgiou GD, Beskos DE. Inelastic displacement ratios for SDOF structures subjected to repeated earthquakes. Eng Struct.
2009;31(11):2744‐2755.
4. Hatzigeorgiou GD. Ductility demand spectra for multiple near‐ and far‐fault earthquakes. Soil Dyn Earthq Eng. 2010;30(4):170‐183.
5. Goda K, Taylor CA. Effects of aftershocks on peak ductility demand due to strong ground motion records from shallow crustal
earthquakes. Earthq Eng Struct Dyn. 2012;41(15):2311‐2330.
6. Zhai CH, Wen WP, Chen Z, Li S, Xie LL. Damage spectra for the mainshock‐aftershock sequence‐type ground motions. Soil Dyn Earthq
Eng. 2013;45:1‐12.
7. Zhai CH, Wen WP, Li S, Chen Z, Chang Z, Xie LL. The damage investigation of inelastic SDOF structure under the mainshock‐aftershock
sequence‐type ground motions. Soil Dyn Earthq Eng. 2014;59:30‐41.
8. Zhai CH, Wen WP, Li S, Xie LL. The ductility‐based strength reduction factor for the mainshock‐aftershock sequence‐type ground
motions. Bull Earthq Eng. 2015;13(10):2893‐2914.
9. Zhai C, Wen W, Ji D, Li S. The influences of aftershocks on the constant damage inelastic displacement ratio. Soil Dyn Earthq Eng.
2015;79:186‐189.
10. Zhang Y, Chen J, Sun C. Damage‐based strength reduction factor for nonlinear structures subjected to sequence‐type ground motions.
Soil Dyn Earthq Eng. 2017;92:298‐311.
11. Hatzigeorgiou GD, Liolios AA. Nonlinear behaviour of RC frames under repeated strong ground motions. Soil Dyn Earthq Eng.
2010;30(10):1010‐1025.
12. Ruiz‐García J, Negrete‐Manriquez JC. Evaluation of drift demands in existing steel frames under as‐recorded far‐field and near‐fault
mainshock‐aftershock seismic sequences. Eng Struct. 2011;33(2):621‐634.
13. Efraimiadou S, Hatzigeorgiou GD, Beskos DE. Structural pounding between adjacent buildings subjected to strong ground motions. Part
II: the effect of multiple earthquakes. Earthq Eng Struct Dyn. 2013;42(10):1529‐1545.
14. Faisal A, Majid TA, Hatzigeorgiou GD. Investigation of story ductility demands of inelastic concrete frames subjected to repeated
earthquakes. Soil Dyn Earthq Eng. 2013;44:42‐53.
15. Ruiz‐García J, Marín MV, Terán‐Gilmore A. Effect of seismic sequences in reinforced concrete frame buildings located in soft‐soil sites.
Soil Dyn Earthq Eng. 2014;63:56‐68.
16. Shin J, Kim J, Lee K. Seismic assessment of damaged piloti‐type RC building subjected to successive earthquakes. Earthq Eng Struct Dyn.
2014;43(11):1603‐1619.
17. Han R, Li Y, van de Lindt J. Seismic risk of base isolated non‐ductile reinforced concrete buildings considering uncertainties and
mainshock‐aftershock sequences. Struct Saf. 2014;50:39‐56.
18. Goda K, Salami MR. Inelastic seismic demand estimation of wood‐frame houses subjected to mainshock‐aftershock sequences. Bull
Earthq Eng. 2014;12(2):855‐874.
19. Li Y, Song R, Van De Lindt JW. Collapse fragility of steel structures subjected to earthquake mainshock‐aftershock sequences. J Struct
Eng. 2014;140(12):04014095.
20. Hatzivassiliou M, Hatzigeorgiou GD. Seismic sequence effects on three‐dimensional reinforced concrete buildings. Soil Dyn Earthq Eng.
2015;72:77‐88.
21. Zhai CH, Zheng Z, Li S, Xie LL. Seismic analyses of a RCC building under mainshock–aftershock seismic sequences. Soil Dyn Earthq Eng.
2015;74:46‐55.
22. Ruiz‐García J, Aguilar JD. Aftershock seismic assessment taking into account postmainshock residual drifts. Earthq Eng Struct Dyn.
2015;44(9):1391‐1407.
23. Ebrahimian H, Jalayer F, Asprone D, et al. A performance‐based framework for adaptive seismic aftershock risk assessment. Earthq Eng
Struct Dyn. 2014;43(14):2179‐2197.
24. Nazari N, van de Lindt JW, Li Y. Quantifying changes in structural design needed to account for aftershock hazard. J Struct Eng.
2015;141(11):04014036.
25. Raghunandan M, Liel AB, Luco N. Aftershock collapse vulnerability assessment of reinforced concrete frame structures. Earthq Eng
Struct Dyn. 2015;44(3):419‐439.
26. Goda K. Record selection for aftershock incremental dynamic analysis. Earthq Eng Struct Dyn. 2015;44(7):1157‐1162.
18
WEN ET AL.
27. Jeon JS, DesRoches R, Lowes LN, Brilakis I. Framework of aftershock fragility assessment—case studies: older California reinforced
concrete building frames. Earthq Eng Struct Dyn. 2015;44(15):2617‐2636.
28. Ghosh J, Padgett JE, Sánchez‐Silva M. Seismic damage accumulation in highway bridges in earthquake‐prone regions. Earthq Spectra.
2015;31(1):115‐135.
29. Jeon JS, DesRoches R, Lee DH. Post‐repair effect of column jackets on aftershock fragilities of damaged RC bridges subjected to successive
earthquakes. Earthq Eng Struct Dyn. 2016;45(7):1149‐1168.
30. Jalayer F, Ebrahimian H. Seismic risk assessment considering cumulative damage due to aftershocks[J]. Earthq Eng Struct Dyn.
2017;46(3):369‐389.
31. Hosseinpour F, Abdelnaby AE. Effect of different aspects of multiple earthquakes on the nonlinear behavior of RC structures. Soil Dyn
Earthq Eng. 2017;92:706‐725.
32. Wang G, Wang Y, Lu W, Yan P, Zhou W, Chen M. Damage demand assessment of mainshock‐damaged concrete gravity dams subjected
to aftershocks. Soil Dyn Earthq Eng. 2017;98:141‐154.
33. Wen W, Zhai C, Ji D, Li S, Xie L. Framework for the vulnerability assessment of structure under mainshock‐aftershock sequences. Soil
Dyn Earthq Eng. 2017;101:41‐52.
34. Park YJ, Ang AHS. Mechanistic seismic damage model for reinforced concrete. J Struct Eng. 1985;111(4):722‐739.
35. Bozorgnia Y, Bertero VV. Damage spectra: characteristics and applications to seismic risk reduction. J Struct Eng. 2003;129(10):1330‐1340.
36. Chiou B, Darragh R, Gregor N, Walter S. NGA project strong‐motion database. Earthq Spectra. 2008;24(1):23‐44.
37. Kunnath SK, Reinhorn AM, Lobo RF. IDARC Version 3.0: a program for the inelastic damage analysis of reinforced concrete structures.
Technical report no. NCEER‐92‐0022. National Center for Earthquake Engineering Research, State University of New York at Buffalo;
1992.
38. Cosenza E, Manfredi G, Ramasco R. The use of damage functionals in earthquake engineering: a comparison between different methods.
Earthq Eng Struct Dyn. 1993;22(10):855‐868.
39. Rahnama M, Krawinkler H. Effects of Soft Soil and Hysteresis Model on Seismic Demands. Standford: John A. Blume Earthquake Engineering Center; 1993.
40. Luco N, Cornell CA. Structure‐specific scalar intensity measures for near‐source and ordinary earthquake ground motions. Earthq
Spectra. 2007;23(2):357‐392.
41. Hancock J, Bommer J, Stafford P. Numbers of scaled and matched accelerograms required for inelastic dynamic analyses. Earthq Eng
Struct Dyn. 2008;37(14):1585‐1607.
42. Roy R, Thakur P, Chakroborty S. Scaling of ground motions and its implications to plan‐asymmetric structures. Soil Dyn Earthq Eng.
2014;57:46‐67.
43. Negro P. Experimental assessment of the global cyclic damage of framed r/c structures. J Earthq Eng. 1997;1(03):543‐562.
44. Warnitchai P, Panyakapo P. Constant‐damage design spectra. J Earthq Eng. 1999;3(3):329‐347.
45. Tiwaris AK, Gupta VK. Scaling of ductility and damage‐based strength reduction factors for horizontal motions. Earthq Eng Struct Dyn.
2000;29(7):969‐987.
46. Panyakapo P. Evaluation of site‐dependent constant‐damage design spectra for reinforced concrete structures. Earthq Eng Struct Dyn.
2004;33(12):1211‐1231.
47. Decanini LD, Bruno S, Mollaioli F. Role of damage functions in evaluation of response modification factors. J Struct Eng.
2004;130(9):1298‐1308.
How to cite this article: Wen W, Zhai C, Ji D. Damage spectra of global crustal seismic sequences considering
scaling issues of aftershock ground motions. Earthquake Engng Struct Dyn. 2018;1–18. https://doi.org/10.1002/
eqe.3056