UNIVERSITY OF SOUTHERN CALIFORNIA
Department of Civil Engineering
BUILDING PERIODS FOR USE IN
EARTHQUAKE RESISTANT DESIGN CODES –
EARTHQUAKE RESPONSE DATA COMPILATION AND
ANALYSIS OF TIME AND AMPLITUDE VARIATIONS
by
M. I. Todorovska, T-Y. Hao, M. D. Trifunac
Report CE 04-02
October 2004
Los Angeles, California
www.usc.edu/dept/civil_eng/Earthquake_eng/
i
ABSTRACT
Most seismic building codes estimate the design forces in structures based on the seismic
“coefficient” C (T ) , where T is the “fundamental vibration period of the building.” For
structures on flexible soil, the relative response is the largest at the first period of the soil-structure
system, which should be substituted in the code equation. This period depends not only
on the structure itself, but also on the properties of the foundation system, of the surrounding
soil, and on the contact conditions between the foundation and the soil. Studies for selected
buildings have shown that this period can vary significantly during earthquake shaking as
function of the level of shaking, reflecting changes in stiffness of the structure and of the soil
(permanent or temporary), and changes in the bonding between the foundation and the soil, and
can be very different from the estimates using ambient vibration data. For further refinement of
the existing and development of new design code procedures, it is important to understand these
changes and estimate their range during strong earthquake shaking, which is done best by
analysis of actual earthquake response data for a large number of buildings.
This report summarizes the results of a one year project, which involved compilation of new
and gathering and analysis of existing processed strong motion data of building responses in the
Los Angeles area, and in particular of buildings that recorded the 1994 Northridge earthquake
and aftershocks, with the objective to estimate the variation of the first system frequency as a
function of the level of shaking and time. Results are shown for seven buildings for which
strong motion data has been archived by U.S. Geological Survey (USGS), and which have been
instrumented by USGS or by the building owner. The “instantaneous” system frequency and
amplitude of response were estimated by two methods—zero crossing analysis, and from the
ridges and skeletons of the Gabor transform. In general, the trend indicated by these data is
decrease during the earthquakes that caused the largest levels of response (1994 Northridge main
event, and the 1971 San Fernando earthquake), and “recovery” during the shaking from the
aftershocks. For one of the buildings, a significant change that occurred during the San
Fernando earthquake (30% reduction) appears to have been permanent. For most buildings, the
frequency changed up to 20%, but for two buildings, the change was about 30%. A permanent
reduction of the frequency is consistent with permanent loss of stiffness, while a “recovery” to
the initial or higher value is consistent with the interpretation that the change was mainly due to
changes in the soil (rather than in the structure itself), or changes in the bond between the soil
and the foundation. Other causes of the temporary changes include contribution of the
nonstructural elements to the total stiffness resisting the seismic forces, and opening of existing
cracks in the concrete structures. The degree to which each of these causes contributed to the
temporary changes cannot be determined from the current instrumentation, but fortunately, what
matters for the building codes is the overall effect.
ii
ACKNOWLEDGEMENTS
This work was supported by the U.S. Geological Survey External Research Program (Grant
No. 03HQGR0013). All views presented in this report are solely those of the authors and do not
necessarily represent the official views of the U.S. Government. The authors are also grateful to
Chris Stephens who kindly supplied the film records of the Northridge earthquake of January 17,
1994, and its aftershocks, that were digitized and processed for this project, from the archives of
National Strong Motion Program maintained by the of the U.S. Geological Survey.
iii
TABLE OF CONTENTS
ABSTRACT.............................................................................................................................................i
ACKNOWLEDGEMENTS....................................................................................................................ii
1. INTRODUCTION ..............................................................................................................................1
2. STRONG MOTION DATA OF 1994 NORTHRIDGE EARTHQUAKE AND AFTERSHOCKS IN
SELECTED BUILDINGS IN THE LOS ANGELES AREA.............................................................3
3. ESTIMATION OF INSTANTANEOUS FREQUENCY.................................................................14
3.1 Methodology ................................................................................................................... 14
3.2 Illustrations .................................................................................................................... 17
4. TIME AND AMPITUDE VARIATIONS OF THE INSTANTANEOUS BUILDING-SOIL
SYSTEM FREQUENCY FOR SEVEN BUILDINGS.....................................................................23
5. SUMMARY AND CONCLUSIONS ...............................................................................................53
6. REFERENCES.................................................................................................................................54
APPENDIX A CATALOG OF PROCESSED DATA FOR 7 BUILDINGS
A.0466 Los Angeles, 15250 Ventura Blvd., Roof (13th floor) (pp. 7)
A.5108 Canoga Park, Santa Susana, ETEC Bldg 462 (6th and 1st Floors) (pp. 46)
A.5450 Burbank, 3601 West Olive Ave., Roof (9th floor) (pp. 11)
A.5451 Los Angeles, 6301 Owensmouth Ave., Roof (12th level) (pp. 8)
A.5453 Los Angeles, 5805 Sepulveda Blvd., Roof (9th floor) (pp. 19)
A.5455 Los Angeles, 16000 Ventura Blvd., Roof (13th floor) (pp. 13)
A.5457 Los Angeles, 8436 West 3rd ST., Roof (10th floor) (pp. 19)
APPENDIX B ISNTANTANEOUS FREQUENCY VERSUS TIME FOR 7 BUILDINGS
B.0466 Los Angeles, 15250 Ventura Blvd., Roof (13th floor) (pp. 7)
B.5108 Canoga Park, Santa Susana, ETEC Bldg 462 (6th and 1st Floors) (pp. 19)
B.5450 Burbank, 3601 West Olive Ave., Roof (9th floor) (pp. 11)
B.5451 Los Angeles, 6301 Owensmouth Ave., Roof (12th level) (pp. 7)
B.5453 Los Angeles, 5805 Sepulveda Blvd., Roof (9th floor) (pp. 13)
B.5455 Los Angeles, 16000 Ventura Blvd., Roof (13th floor) (pp. 11)
B.5457 Los Angeles, 8436 West 3rd ST., Roof (10th floor) (pp. 16)
iv
1
1. INTRODUCTION
The Earthquake Resistant Design Codes have evolved based on principles and procedures
derived from the Response Spectrum Method (Biot, 1942). In most codes, the design shear
forces are quantified using the seismic “coefficient” C (T ) , where T is the “fundamental
vibration period of the building,” and various scaling factors that depend on the seismic zone,
type of structure, soil site conditions, importance of structure etc. Most codes provide simplified
empirical equations for estimation of the building period, T, based on past experience and on
data of response of existing buildings, which is extremely limited (both in quantity and in
quality). Significant new improvements in the code procedures, or improvement of the accuracy
of the existing code equations can be made only if the processed data on recorded earthquake
response is significantly expanded.
For most buildings, the design base shear and the lateral design forces are functions of its
fundamental period, T, via the seismic coefficient function C (T ) . As T cannot be measured
before the structure is completed, the building codes provide simplified empirical equations to
estimate it, based on measured response of existing buildings. Numerous papers on this subject
have approached this problem theoretically (Biot, 1942), using small amplitude ambient and
forced vibration tests of structures (Carder, 1936), and actual earthquake response (Li and Mau,
1979). However, the number of well-documented buildings with one or several earthquake
recordings is typically less than 100. When the recorded data is grouped by the structural
systems (moment resistant frame, shear wall etc.) and building materials (reinforced concrete,
steal, etc.), the number of records in a particular group becomes too small to control the accuracy
of regression analyses, or to separate the “good” from the “bad” empirical models (Goel and
Chopra, 1997; Stewart et al., 1999). This problem is further complicated by the fact that the
foundation soil responds in nonlinear manner, even for very small strains (Hudson, 1970; Luco et
al., 1987). During strong earthquake shaking, the apparent period, T􀀄􀐬 , of the soil-foundation-structure
system can lengthen significantly (Udwadia and Trifunac, 1974), and it may or may not
return to its original pre earthquake value. This period lengthening can reach and exceed a factor
of two, and it adds to the scatter in empirical estimation of the building periods, and to the
ambiguity in choosing a representative period T for evaluation of the seismic coefficient function
C (T ) (Trifunac, 1999; 2000). There are many complex aspects of this problem (e.g. how soil-structure
interaction changes the estimates of T􀀄􀐬 , and how valid and useful are the models
developed so far), which must be addressed by future earthquake hazard reduction research, but
analysis and resolution of those is well beyond the scope of this project. This project addressed
the obvious and essential first step, which is to increase the available data on apparent building
periods T􀀄􀐠 during actual earthquake excitation. Without a major increase of the dataset of
2
structural response to earthquakes, little progress can be made in future developments of the
building codes and of new procedures for earthquake resistant design.
Especially valuable data for estimation of amplitude dependent lengthening and recovery of
T􀀄􀐠 come from buildings with multiple recordings. These multiple recordings can be used to
estimate empirically the dependence of system period T􀀄􀐠 on the overall response amplitudes and
on the strain levels in the soil. Unfortunately, due to limited funding and to the fact that most
records in buildings are on film, only larger amplitude records in buildings are usually processed.
This has left the dataset of structural records deficient in multiple recordings in buildings.
Recording strong motion in structures (and in general) is a slow process (as strong
earthquake are rare events) and requires significant financial investment (in instrumentation and
in long term maintenance) and long waiting time to record. Fortunately, the set of processed
building response records can be significantly expanded without having to wait for tens of years
to record, as there is a large amount of such data already recorded (by stations of the National
and of the State of California Strong Motion Programs, and in many tall buildings instrumented
by their owners). Records not digitized by these agencies include those of aftershocks of major
earthquakes, of many smaller earthquakes, and of large but distant earthquakes. For example, in
the Los Angeles metropolitan area, this includes aftershocks of the 1971 San Fernando (M=6.4),
1987 Whittier-Narrows (M=5.9), 1994 Northridge (M=6.7), many “small” earthquakes, such as
1988 Pasadena (M=4.9), 1990 Upland (M=5.2), 1991 Sierra Madre (M=5.8), 1989 Montebello
(M=4.4 and M=4.1), and 2001 West Hollywood (M=4.2), and of large distant earthquakes like
1992 Landers (M=7.5), and 1999 Hector Mine (M=7.1).
The objective of this project was to initiate the augmentation of the pool of processed
structural response data, in particular with multiple recordings corresponding to different levels
of shaking, by digitizing and processing records in buildings of aftershocks of the 1994
Northridge earthquake (as well as of records of the main event not yet processed or processed
inadequately), that have been archived by the U.S. Geological Survey, and to demonstrate the
usefulness of such recordings for estimation of the building periods and improvement of the
building design codes. This report summarizes the processed data and presents results for the
estimated building frequencies from the processed data and their variation as a function of the
level of response and time, from one event to another and during a particular event. It is
organized as follows. Chapter 2 presents a summary of the processed data, Chapter 3—the
methodology used for estimation of instantaneous frequency and illustrations, Chapter 4—results
for the variation of the building frequencies as function of the amplitude of response, Chapter
5—the conclusions, Appendix A—catalog of the data released including tables and plots of time
series of acceleration, velocity and displacement and Fourier spectra of acceleration, and
Appendix B—plots of instantaneous frequency versus time separately for each record considered
in the analysis.
3
2. STRONG MOTION DATA OF 1994 NORTHRIDGE EARTHQUAKE AND
AFTERSHOCKS IN SELECTED BUILDINGS IN THE LOS ANGELES AREA
Figure 2.1 shows a map of the Los Angeles metropolitan area and locations of instrumented
buildings at the time of the 1994 Northridge earthquake, that have been instrumented either by
the U.S. Geological Survey (USGS) and partner organizations, or by the building owners (as
required by the Los Angeles and state Building Codes), and for which the data is archived by
USGS. The latter are often referred to as “code” buildings. All of these building will be referred
to as “USGS instrumented buildings” (as opposed to other buildings in the area that have been
instrumented by the California Division of Mines and Geology, CDMG), and are identified by
the USGS station numbers.
The sensors in these buildings are either three-component SMA-1 or multi-channel CR-1
accelerographs, both recording on film. Many of the “code” buildings (~30 buildings total) have
only one instrument, at the roof. This is due to a change in the original ordinance for Los
Angeles, such that at present only one instrument at the roof is required. Consequently, some
building owners did not continue to maintain or repair the instrument at the base or at the
intermediate floors. This unfortunate fact limits considerably the use of these records, especially
for analyses of soil-structure interaction. Fortunately, the roof records (per se) can be used to
determine approximately the period of the building-foundation-soil system, and the changes of
this period with the amplitudes of the building response. This is due to the fact that near the
system frequencies, the relative roof motion (with respect to the base) is much larger than the
(absolute) motion at the base, which implies that the roof relative motion can be approximated by
its absolute motion.
Since the Northridge earthquake, the analog instrumentation in few of the USGS
instrumented buildings has been replaced by digital ones, and few additional buildings have been
instrumented. For some of these buildings, data of smaller local earthquakes and distant larger
earthquakes has been recorded and released. The recorded level of response for these events,
however, is much smaller than that for the Northridge earthquake.
Figure 2.1 also shows the epicenters of earthquakes that have been recorded in these
buildings. The Northridge main event was followed by a large number of aftershocks (9 of these
had M > 5, and 55 had M > 4). Many of these larger aftershocks, as well as smaller magnitude
but closer aftershocks, were recorded in the instrumented buildings. The aftershock of March 20,
1994 (M = 5.2; “aftershock 392”) was recorded by the largest number of (ground motion)
stations (Todorovska et al., 1999). The Northridge sequence was recorded on multiple films,
archived separately. The largest number of recorded aftershocks known to the investigators of
this project is 86—at station USGS #5455, and about 60 at several other stations. However, it
turned out that the number of aftershock records useable for estimation of the first building-soil
4
Los Angeles
Santa Monica
San Pedro
San Fernando Valley
San Gabriel Mountains
Pacific Ocean
5108
5451
5455
466
5457
5260
793
5277 5284
5450
5259 742
872
572
5106
5465
5462
5258
229
804
5239
482
5293
892
437 5459
5233
5292
982
5082
5456
663
5263
5453
5264
530
5460
Malibu, 1989 Montebello, 1989
Long Beach
1933
S. California
1989
Sierra Madre, 1991
Pasadena, 1988
Northridge
and Aftershocks
1994 Whittier Narrows
and Aftershocks
1987
San Fernando
1972
West Hollywood
2001
34.25
34.00
118.75 118.50 118.25 118.00 117.75
34.50
33.75
33.50
USGS instrumeneted building sites
analysis completed and reported
analysis to be completed
after additional data is processed
other USGS instrumeneted buildings
0 10 20
km
Figure 2.1 Locations of instrumented buildings in the Los Angeles metropolitan area at the time
of the 1994 Northridge earthquake, for which the data is archived by USGS The building sites
are identified by their USGS station number.
frequency was very small—up to 11, and depended not only on the amplitude of recorded
motions, but also on the building itself, i.e. on its first frequency, and on the shape of the Fourier
spectrum of the record relative to that of the noise. In principle, a record is useable for
5
estimation of the building first frequency if the first frequency is within the band where the
signal-to-noise ratio is grater than one, and below the Nyquist frequency. Many records turned
out not to be usable because of too high low frequency cut-off, which is the frequency below
which the recording and processing “noise” becomes larger than the recorded signal. This
frequency is determined automatically by the LeBach data processing software (Lee and
Trifunac, 1990). If the first building-soil system frequency was too close to, or suspected to be
below the cut-off frequency, or could not be determined reliably for some other reason, the
record was discarded. An aftershock record was more likely to be useful for the buildings with
smaller number of stories than for the very high buildings, which have lower first system
frequency. Also, a record from a large but distant earthquake (like 1992 Landers) is more likely
to be useable than a record from a small close by event with similar peak acceleration, because
the ground motion from the former has more energy in the smaller frequencies of the spectrum
and will excite more the first mode, leading to larger amplitudes of the building response in the
smaller frequency range of the spectrum, and hence—larger signal to noise ratio at low
frequencies, and lower long period cut-off frequency for the record. Therefore, it is important to
digitize all “good” Landers earthquake records digitized and added to this analysis (photo copies
of many of these records have been reported in USGS reports), and all “good” Hector Mine
earthquake records.
This report shows results for 7 buildings for which there were three or more adequate
records (of the Northridge sequence or of the 1971 San Fernando earthquake) to estimate the first
building-soil system frequency, and for which at present there are no other “good” records to add
to the analysis, so that the analysis is considered completed. These stations are marked as full
(red) dots in Figure 2.1. For the 15 stations marked by light (yellow) circles, there are some
adequate records of the Northridge sequence, as well as other “good” records that have not yet
been digitized (e.g. of the Landers and/or of the Whittier-Narrows earthquake). For the other
buildings, marked in Fig. 2.1 by solid rectangles, at this time, only one or maybe two adequate
records for such analysis are known to exist to the authors of this report.
Table 2.1 shows a list of earthquakes recorded in “USGS” instrumented buildings. For the
Northridge sequence, only the aftershocks are shown for which there is an adequate record that
has been used in the analysis presented in this report. For most of the buildings, the contributing
aftershocks have not been identified, but are assigned negative aftershock number, the absolute
value of which increases chronologically, and is related to the order of the record on the film
(e.g., aftershock –3 means that this was the third aftershock record on the film following the
main event, and aftershock –105 means that this was the fifth record on the second role of film,
which did not contain the main event). This table also lists the 2001 West Hollywood earthquake
(M=4.2), which occurred close to many of the instrumented buildings (see Figure 2.1), and
which should have been recorded by these buildings.
6
Table 2.1 Earthquakes recorded by USGS instrumented buildings.
Event Date Time ML Latitude Longitude Depth
(km)
San Fernando 02/09/1971 06:00 6.6 34 24 42N 118 24 00W --
Whittier-Narrows 10/01/1987 14:42 5.9 34 03 10N 118 04 34W 14.5
Whittier-Narrows, 12th Aft. 10/04/1987 10:59 5.3 34 04 01N 118 06 19W 13.0
Whittier-Narrows, 13th Aft. 02/03/1988 15:25 4.7 34 05 13N 118 02 52W 16.7
Pasadena 12/03/1988 11:38 4.9 34 08 56N 118 08 05W 13.3
Malibu 01/19/1989 06:53 5.0 33 55 07N 118 37 38W 11.8
Montebello 06/12/1989 16:57 4.4 34 01 39N 118 10 47W 15.6
Upland 02/28/1990 23:43 5.2 34 08 17N 117 42 10W 5.3
Sierra Madre 06/28/1991 14:43 5.8 34 15 45N 117 59 52W 12.0
Landers 06/28/1992 11:57 7.5 34 12 06N 116 26 06W 5.0
Big Bear 06/28/1992 15:05 6.5 34 12 06N 116 49 36W 5.0
Northridge 01/17/1994 12:30 6.7 34 12 48N 118 32 13W 18.4
Northridge, Aft. #1 01/17/1994 12:31 5.9 34 16 45N 118 28 25W 0.0
Northridge, Aft. #7 01/17/1994 12:39 4.9 34 15 39N 118 32 01W 14.8
Northridge, Aft. #9 01/17/1994 12:40 5.2 34 20 29N 118 36 05W 0.0
Northridge, Aft. #100 01/17/1994 17:56 4.6 34 13 39N 118 34 20W 19.2
Northridge, Aft. #129 01/17/1994 20:46 4.9 34 18 04N 118 33 55W 9.5
Northridge, Aft. #142 01/17/1994 23:33 5.6 34 19 34N 118 41 54W 9.8
Northridge, Aft. #151 01/18/1994 00:43 5.2 34 22 35N 118 41 53W 11.3
Northridge, Aft. #253 01/19/1994 21:09 5.1 34 22 43N 118 42 42W 14.4
Northridge, Aft. #254 01/19/1994 21:11 5.1 34 22 40N 118 37 10W 11.4
Northridge, Aft. #336 01/29/1994 11:20 5.1 34 18 21N 118 34 43W 1.1
Northridge, Aft. #392 03/20/1994 21:20 5.2 34 13 52N 118 28 30W 13.1
Hector Mine 10/16/1999 09:46 7.1 34 36 00N 116 16 12W 3.0
West Hollywood 09/09/2001 23:59 4.2 34 04 30N 118 22 44W 3.7
Table 2.2 shows a list of buildings included in the analysis in this report, and Tables 2.3
through 2.10 show a summary of the records of the Northridge sequence that were processed and
found useable for this project. Each table corresponds to a pair of a station and an instrument.
For all but one building (USGS 5108), there was only one instrument on the top floor or on the
roof. Each table shows in the header row the USGS station number, the instrument type and
serial number, station address and location of the instrument within the building, and the station
geographical coordinates. The following rows show the file name containing the processed data
(v1x____.dat has the uncorrected acceleration time series, v2x____.dat has the corrected
acceleration, velocity and displacement time histories, and v3x____.dat has the Fourier and
response spectra), the record reference and log numbers (which have no special meaning for the
7
users), the name of the event (negative aftershock number means unidentified aftershock, as
described in the previous paragraph), the epicentral distance (for the unidentified aftershocks, the
horizontal distance to a central point on the fault plane is shown (as an indicator of the order of
magnitude of the distance), the duration of the digitized record (all recorded length was
digitized), component orientation, and uncorrected peak acceleration. Plots of the corrected
acceleration, velocity and displacement time series, and of Fourier spectra of acceleration (three
components per page) are shown in the appendices, named as A.xxxx where xxxx is the USGS
station number. The processed data (uncorrected acceleration, corrected acceleration, velocity
and displacement, and Fourier and response spectra).is available free of charge from the USC
Strong Motion Research Group web site at
www.usc.edu/dept/civil_eng/earthquake_eng/
Table 2.2 USGS instrumented buildings included in the analysis.
USGS: 0466, SMA-1 185
LOS ANGELES, 15250 VENTURA BLVD., ROOF
(13th floor)
34.157°N,
117.476°W
USGS: 5108 SMA 1276,
SMA 1277
CANOGA PARK, SANTA SUSANA, ETEC Bldg 462
(6th Floor, and 1st Floor)
34.230°N,
118.712°W
USGS: 5450, SMA-1 6146
BURBANK, 3601 WEST OLIVE AVE., ROOF (9th
floor)
34.152°N,
118.337°W
USGS: 5451, SMA-1 4048
LOS ANGELES, 6301 OWENSMOUTH AVE., ROOF
(12th level)
34.185°N,
118.584°W
USGS: 5453, SMA-1 7073
LOS ANGELES, 5805 SEPULVEDA BLVD., ROOF
(9th floor)
34.175°N,
118.465°W
USGS: 5455, SMA-1 4270
LOS ANGELES, 16000 VENTURA BLVD., ROOF
(13th floor)
34.156°N,
118.480°W
USGS: 5457, SMA 5491
LOS ANGELES, 8436 WEST 3rd ST., Roof (10th
floor)
34.072°N,
118.375°W
Table 2.3 List of processed records at station USGS 0466
USGS: 0466
SMA-1 185
LOS ANGELES, 15250 VENTURA BLVD., ROOF (13th floor) 34.157°N
117.476°W
File Name Ref
No.
Log No. Earthquake Distance
[km]
Duration
[s]
Comp Peak Acc.
[g]
v1x0000.dat IAA000 94.000.0 9.0 59.7 N00E 0.550
NORTHRIDGE EARTHQUAKE
UP 0.394
W00N 0.257
v1x0001.dat IAA001 94.000.1 15.9 33.2 N00E 0.151
NORTHRIDGE EARTHQUAKE
(aft. -1) UP 0.097
W00N 0.054
8
Table 2.4 List of processed records at station USGS 5108 – 6th floor
USGS: 5108
SMA 1276
CANOGA PARK, SANTA SUSANA, ETEC Bldg 462 (6th Floor) 34.230°N
118.712°W
File Name Ref
No.
Log No. Earthquake Distance
[km]
Duration
[s]
Comp Peak Acc.
[g]
V1X8900.dat IAA001 94.890.0 16.3 59.8 E00S 0.392
NORTHRIDGE EARTHQUAKE
UP 0.398
N00E 0.595
V1X8901.dat IAA002 94.890.1 16.8 34.2 E00S 0.019
NORTHRIDGE EARTHQUAKE
(aft. 7) UP 0.018
N00E 0.019
V1X8902.dat IAA003 94.890.2 16.1 43.1 E00S 0.032
NORTHRIDGE EARTHQUAKE
(aft. 9) UP 0.040
N00E 0.045
V1X8904.dat IAA005 94.890.4 12.9 37.6 E00S 0.039
NORTHRIDGE EARTHQUAKE
(aft. 100) UP 0.027
N00E 0.025
V1X8905.dat IAA006 94.890.5 15.7 42.8 E00S 0.165
NORTHRIDGE EARTHQUAKE
(aft. 129) UP 0.106
N00E 0.127
V1X8906.dat IAA007 94.890.6 10.9 46.0 E00S 0.129
NORTHRIDGE EARTHQUAKE
(aft. 142) UP 0.071
N00E 0.075
V1X8907.dat IAA008 94.890.7 16.4 34.9 E00S 0.021
NORTHRIDGE EARTHQUAKE
(aft. 151) UP 0.017
N00E 0.035
V1X8908.dat IAA009 94.890.8 16.5 43.1 E00S 0.089
NORTHRIDGE EARTHQUAKE
(aft. 253) UP 0.051
N00E 0.046
V1X8909.dat IAA010 94.890.9 18.5 43.9 E00S 0.034
NORTHRIDGE EARTHQUAKE
(aft. 254) UP 0.031
N00E 0.025
V1X8910.dat IAA011 94.891.0 14.9 42.7 E00S 0.089
NORTHRIDGE EARTHQUAKE
(aft. 336) UP 0.074
N00E 0.070
V1X8911.dat IAA012 94.891.1 16.5 41.8 E00S 0.043
NORTHRIDGE EARTHQUAKE
(aft. 392) UP 0.037
N00E 0.038
9
Table 2.5 List of processed records at station USGS 5108 – 1st floor
USGS: 5108
SMA 1277
CANOGA PARK, SANTA SUSANA, ETEC Bldg 462 (1st Floor) 34.230°N
118.712°W
File Name Ref
No.
Log No. Earthquake Distance
[km]
Duration
[s]
Comp Peak Acc.
[g]
V1X0300.dat IAA013 94.030.0 16.3 60.1 E00S 0.236
NORTHRIDGE EARTHQUAKE
UP 0.226
N00E 0.335
V1X0301.dat IAA014 94.030.1 16.8 34.2 E00S 0.023
NORTHRIDGE EARTHQUAKE
(aft. 7) UP 0.010
N00E 0.022
V1X0302.dat IAA015 94.030.2 16.1 43.1 E00S 0.037
NORTHRIDGE EARTHQUAKE
(aft. 9) UP 0.019
N00E 0.032
V1X0304.dat IAA018 94.030.4 12.9 37.6 E00S 0.030
NORTHRIDGE EARTHQUAKE
(aft. 100) UP 0.015
N00E 0.023
V1X0305.dat IAA019 94.030.5 15.7 42.8 E00S 0.151
NORTHRIDGE EARTHQUAKE
(aft. 129) UP 0.045
N00E 0.181
V1X0306.dat IAA020 94.030.6 10.9 46.5 E00S 0.043
NORTHRIDGE EARTHQUAKE
(aft. 142) UP 0.047
N00E 0.061
V1X0307.dat IAA021 94.030.7 16.4 31.8 E00S 0.016
NORTHRIDGE EARTHQUAKE
(aft. 151) UP 0.010
N00E 0.024
V1X0308.dat IAA022 94.030.8 16.5 43.1 E00S 0.036
NORTHRIDGE EARTHQUAKE
(aft. 253) UP 0.024
N00E 0.039
V1X0309.dat IAA023 94.030.9 18.5 43.9 E00S 0.027
NORTHRIDGE EARTHQUAKE
(aft. 254)
UP 0.016
N00E 0.021
V1X0310.dat IAA024 94.031.0 14.9 42.7 E00S 0.096
NORTHRIDGE EARTHQUAKE
(aft. 336) UP 0.036
N00E 0.090
V1X0311.dat IAA025 94.031.1 16.5 41.8 E00S 0.047
NORTHRIDGE EARTHQUAKE
(aft. 392) UP 0.021
N00E 0.047
10
Table 2.6 List of processed records at station USGS 5450
USGS: 5450
SMA-1 6146
BURBANK, 3601 WEST OLIVE AVE., ROOF (9th floor) 34.152°N
118.337°W
File Name Ref
No.
Log No. Earthquake Distance
[km]
Duration
[s]
Comp Peak Acc.
[g]
v1x0000.dat IAA000 94.000.0 19.6 47.4 N00E 0.644
NORTHRIDGE EARTHQUAKE
UP 1.060
W00N 0.508
v1x0001.dat IAA001 94.000.1 21.4 21.9 N00E 0.069
NORTHRIDGE EARTHQUAKE
(aft. -1) UP 0.065
W00N 0.085
v1x0005.dat IAA005 94.000.5 21.4 16.3 N00E 0.024
NORTHRIDGE EARTHQUAKE
(aft. -5) UP 0.031
W00N 0.027
v1x0013.dat IAA013 94.001.3 21.4 21.8 N00E 0.036
NORTHRIDGE EARTHQUAKE
(aft. -13) UP 0.041
W00N 0.031
v1x0014.dat IAA014 94.001.4 21.4 11.6 N00E 0.020
NORTHRIDGE EARTHQUAKE
(aft. -14) UP 0.014
W00N 0.011
Table 2.7 List of processed records at station USGS 5451
USGS: 5451
SMA-1 4048
LOS ANGELES, 6301 OWENSMOUTH AVE., ROOF (12th level) 34.185°N
118.584°W
File Name Ref
No.
Log No. Earthquake Distance
[km]
Duration
[s]
Comp Peak Acc.
[g]
v1x0000.dat IAA000 94.000.0 5.4 99.2 N00E 0.552
NORTHRIDGE EARTHQUAKE
UP 0.477
W00N 0.373
v1x0026.dat IAA026 94.002.6 15.55 41.5 N00E 0.077
NORTHRIDGE EARTHQUAKE
(aft. -26) UP 0.068
W00N 0.073
v1x0115.dat IAA115 94.011.5 15.55 38.3 N00E 0.034
NORTHRIDGE EARTHQUAKE
(aft. -115)
UP 0.090
W00N 0.045
11
Table 2.8 List of processed records at station USGS 5453
USGS: 5453
SMA-1 7073
LOS ANGELES, 5805 SEPULVEDA BLVD., ROOF (9th floor) 34.175°N
118.465°W
File Name Ref
No.
Log No. Earthquake Distance
[km]
Duration
[s]
Comp Peak Acc.
[g]
v1X0000.DAT IAA034 94.000.0 7.9 60.9 N00E 0.735
NORTHRIDGE EARTHQUAKE
UP 0.488
W00N 0.663
v1X0001.DAT IAA034 94.000.1 14.0 30.6 N00E 0.231
NORTHRIDGE EARTHQUAKE
(aft. -1) UP 0.104
W00N 0.130
v1X0007.DAT IAA034 94.000.7 14.0 18.1 N00E 0.026
NORTHRIDGE EARTHQUAKE
(aft. -7) UP 0.015
W00N 0.023
v1X0024.DAT IAA034 94.002.4 14.0 21.1 N00E 0.040
NORTHRIDGE EARTHQUAKE
(aft. -24) UP 0.021
W00N 0.053
v1X0026.DAT IAA034 94.002.6 14.0 23.2 N00E 0.057
NORTHRIDGE EARTHQUAKE
(aft. -26) UP 0.047
W00N 0.086
v1X0029.DAT IAA034 94.002.9 14.0 23.2 N00E 0.041
NORTHRIDGE EARTHQUAKE
(aft. -29) UP 0.017
W00N 0.044
v1X0103.DAT IAA103 94.010.3 14.0 24.5 N00E 0.045
NORTHRIDGE EARTHQUAKE
(aft. -103) UP 0.017
W00N 0.037
v1X0104.DAT IAA104 94.010.4 14.0 19.1 N00E 0.017
NORTHRIDGE EARTHQUAKE
(aft. -104) UP 0.015
W00N 0.031
v1X0115.DAT IAA115 94.011.5 14.0 26.9 N00E 0.055
NORTHRIDGE EARTHQUAKE
(aft. -115) UP 0.019
W00N 0.065
12
Table 2.9 List of processed records at station USGS 5455
USGS: 5455
SMA-1 4270
LOS ANGELES, 16000 VENTURA BLVD., ROOF (13th floor)
34.156°N
118.480°W
File Name Ref
No.
Log No. Earthquake Distance
[km]
Duration
[s]
Comp Peak Acc.
[g]
v1x0000.dat IAA000 94.000.0 8.2 57.6 E30S 0.358
NORTHRIDGE EARTHQUAKE
UP 0.337
N30E 0.394
v1x0001.dat IAA001 94.000.1 15.9 28.2 E30S 0.070
NORTHRIDGE EARTHQUAKE
(aft. -1) UP 0.087
N30E 0.104
v1x0007.dat IAA007 94.000.7 15.9 23.1 E30S 0.018
NORTHRIDGE EARTHQUAKE
(aft. -7) UP 0.033
N30E 0.016
v1x0022.dat IAA022 94.002.2 15.9 20.9 E30S 0.033
NORTHRIDGE EARTHQUAKE
(aft. -22) UP 0.072
N30E 0.029
v1x0025.dat IAA025 94.002.5 15.9 34.7 E30S 0.043
NORTHRIDGE EARTHQUAKE
(aft. -25) UP 0.056
N30E 0.057
v1x0046.dat IAA046 94.004.6 15.9 19.2 E30S 0.029
NORTHRIDGE EARTHQUAKE
(aft. -46) UP 0.045
N30E 0.041
13
Table 2.10 List of processed records at station USGS 5457
USGS: 5457
SMA 5491
LOS ANGELES, 8436 WEST 3rd ST., Roof (10th floor) 34.072°N
118.375°W
File Name Ref
No.
Log No. Earthquake Distance
[km]
Duration
[s]
Comp Peak Acc.
[g]
v1x0000.dat IAA030 94.000.0 21.7 53.2 N00E 0.658
NORTHRIDGE EARTHQUAKE
UP 0.253
S90W 0.547
v1x0001.dat IAA030 94.000.1 27.4 28.1 N00E 0.102
NORTHRIDGE EARTHQUAKE
(aft. -1) UP 0.093
S90W 0.262
v1x0004.dat IAA030 94.000.4 27.4 20.3 N00E 0.014
NORTHRIDGE EARTHQUAKE
(aft. -4) UP 0.022
S90W 0.017
v1x0008.dat IAA030 94.000.8 27.4 22.5 N00E 0.031
NORTHRIDGE EARTHQUAKE
(aft. -8) UP 0.023
S90W 0.057
v1x0009.dat IAA030 94.000.9 27.4 27.5 N00E 0.053
NORTHRIDGE EARTHQUAKE
(aft. -9) UP 0.032
S90W 0.053
v1x0010.dat IAA030 94.001.0 27.4 18.7 N00E 0.018
NORTHRIDGE EARTHQUAKE
(aft. -10) UP 0.015
S90W 0.014
v1x0013.dat IAA030 94.001.3 27.4 21.6 N00E 0.027
NORTHRIDGE EARTHQUAKE
(aft. -13)
UP 0.014
S90W 0.032
v1x0017.dat IAA030 94.001.7 27.4 20.5 N00E 0.018
NORTHRIDGE EARTHQUAKE
(aft. -17) UP 0.014
S90W 0.021
v1x0019.dat IAA030 94.001.9 27.4 30.9 N00E 0.100
NORTHRIDGE EARTHQUAKE
(aft. -19) UP 0.124
S90W 0.185
14
3. ESTIMATION OF INSTANTANEOUS FREQUENCY
3.1 Methodology
The instantaneous frequency was estimated by two methods: (a) zero-crossing analysis, and
(b) from the ridge of the Gabor transform, both applied to the relative roof displacement—when
there was a record at the base, or to the absolute displacement—when only the roof response was
recorded—considered as an approximation of the relative displacement in the neighborhood of
the first system frequency. Both methods were applied to the filtered displacement, such that it
contained only motion in the neighborhood of the first system frequency, hence resembling a
chirp signal. The zero-crossing analysis consists of measuring the time between consecutive
zero crossings of the displacement, and assuming this time interval to be a half of the system
period (see Trifunac et al., 2001a, 2001b, 2001c). The Gabor transform method is summarized
in the remaining part of this section, and a more detailed description can be found in Todorovska
(2001).
The Gabor transform is a complex-valued time-frequency distribution, which represents a
projection of the signal onto a series of Gabor wavelets. For a signal f (t)∈L2 (R) , the Gabor
transform is
Gf (b,ω) f (t)g(b,ω) (t)dt, b , ω 0
∞
−∞
= ∫ ∞ < < ∞ > (3.1.1)
where b is time, and ω is circular frequency. The Gabor wavelet, g(b,ω) (t) is a complex
exponential modulated by a Gaussian envelope with a constant shape
( ) ( )
2
( , ) 2 ( ) exp exp
2 b
t b
g ω t iω t b
σ
 − 
= −   − 
 
(3.1.2)
and has Fourier Transform
[ ]
[ ] ( )
( , ) (0, )
2 2
ˆ ( ) exp ˆ ( )
exp 2 exp 1
2
g b i b g
i b
ω ξ ξ ω ξ
ξ πσ ξ ω σ
= −
= − − −   
(3.1.3)
15
Figure 3.1.1 shows the real and imaginary parts of a Gabor wavelet (left) and its Fourier
transform (right) for b = 0 and ω = 2π. Up to a phase shift, the Gabor transform is identical to a
moving window analysis with a Gaussian time window.
-4 -3 -2 -1 0 1 2 3 4
-1.0
-0.5
0.0
0.5
1.0
0.6 0.8 1.0 1.2 1.4
0
1
2
3
2σ t 2σ f
Gabor wavelet, ω=2π, σ=1.5
g(b,ω)(t)
t − s ξ/(2π) − Hz
g(b,ω)(ξ)
Figure 3.1.1 A Gabor wavelet for b = 0 and ω = 2π, in the time domain (left) and in the frequency
domain (right).
The shape of the Gabor wavelet is determined by parameter σ, which is a measure of the
spread of the Gaussian window and hence determines its effective length. This parameter is
preset for the entire transform, which distinguishes the Gabor transform from the continuous
complex wavelet transform with a Morlet wavelet, which is a projection of the function onto a
family of wavelets of variable length. As it can be seen from Fig. 3.1.1, the Gabor wavelet is
localized both in time and in frequency. The smaller the spread of the Gaussian envelope—the
better the localization in time, and the smaller the spread of gˆ(b,ω) (ξ )—the better the localization
in frequency. As it can be seen from eqns (3.1.2) and (3.1.3), the better the localization in time,
the poorer the localization in frequency, which is formally described by the Heisenberg
uncertainty principle, stating that a signal cannot be arbitrarily well localized both in time and in
frequency, i.e. the product of the spread in time, t σ
and the spread in frequency, σω is bounded
1
t 2 σ ⋅σω = C ≤ (3.1.4)
The constant C is the smallest for a Gaussian window, which provides best possible localization
both in time and in frequency. In eqn (3.1.4), the spreads in time and in frequency, t σ
and σω ,
are such that 2
t σ
and 2ω
σ are the variances of the Gabor wavelet and of its Fourier transform,
and in terms of parameter σ
16
2 1 2 0.71
t 2 t σ = σ ⇔ σ = σ
(3.1.5)
2
2
1 1 0.71 1
2 σω σω
σ σ
= ⇔ = (3.1.6)
which gives
1
t ω 2 σ σ⋅ = (3.1.7)
The instantaneous frequency was determined from the ridge of the magnitude of the Gabor
transform—a surface on the time-frequency plane, and the corresponding amplitude was
estimated from the skeleton of the transform. The ridge is the collocation of points (t,ω) where
the magnitude of the transform (i.e. the energy of the signal) is maximum, and the skeleton is the
value of the transform along the ridge. This method is the most robust and efficient from all
methods based on time-frequency distributions for signals that are “noisy” but the amplitude of
the noise is relatively small compared to the signal (Todorovska, 2001). The complex wavelet
transform with the Morlet wavelet was initially considered, which is essentially a Gabor
transform with a window that varies depending on the frequency so that it always contains same
number of wavelengths. The results by both methods were found to be very similar, and no
advantage was seen in using a variable window, as the observed changes of the building
frequency are relatively small—well within an order of magnitude. Using constant window was
convenient in the estimation of the resolution of the method. The Gabor transform was used
with spread σ = 1.5 .
It should be noted here that the spreads in time and frequency, t σ
and σω , of the Gabor
wavelet used define the resolution of the method, namely, the method cannot resolve frequencies
closer than σω , and the estimate of “instantaneous” frequency at a time t is the weighted average
of the frequency within the time widow of the Gabor wavelet, rather than exactly at time t.
Another important fact in interpretation of the results by both the zero-crossing analysis and
Gabor analysis is that they are both based on the premise that the signal—relative building
response in our case—can be represented (uniquely) as the product
f (t) = A(t) cosΦ(t) (3.1.8)
where the amplitude A(t) varies very slowly with time compared to the oscillations of cosΦ(t) ,
in which case the instantaneous frequency can be obtained by differentiating Φ(t)
17
(t) d
dt
ω
Φ
= (3.1.9)
Such signals are called asymptotic. Hence, these methods are most accurate when the
asymptoticity condition is satisfied and produce artifacts when it is violated. Longer window of
the Gabor wavelet (larger σ ) would smooth variations due to such artifacts.
3.2 Illustrations
The methodology for estimation of the instantaneous frequency of building-soil systems is
illustrated in Figs 2.2.1 through 2.2.5 for five records. Figure 3.2.1 shows results for component
N11E of the record of the 1971 San Fernando earthquake at station USGS 466 (Los Angeles,
15250 Ventura Blvd.), Fig. 3.2.2—for component N00E of the record of the 1994 Northridge
earthquake at station USGS 5450 (Burbank, 3601 West Olive Ave.), Fig. 3.2.3—for component
W00N of the record of the 1994 Northridge earthquake at station USGS 5451 (Los Angeles,
6301 Owensmouth Ave.), Fig. 3.2.4—for component N00E of the record of the 1994 Northridge
earthquake at station USGS 5453 (Los Angeles, 5805 Sepulveda Blvd.), and Fig. 3.2.5—for
component E30S of the record of the 1994 Northridge earthquake at station USGS 5455 (Los
Angeles, 1600 Ventura Blvd.). For each record, the plot on the left hand side shows the Fourier
spectrum of the relative roof displacement (solid line), or its approximation by the absolute
displacement when only a roof record was available, the Fourier spectrum of acceleration at the
ground floor (dashed line)—if available, and a smooth approximation of the relative (or absolute)
roof displacement spectrum by the marginal Gabor transform distribution (the smooth line).
There are three plots in the right hand side, as follows. The plot on the top shows the time
history of the roof relative (or absolute) displacement (solid line), and of the ground floor
displacement (dashed line) if available, for the “broad-band” data, which is the output of the
standard data processing. The plot in the middle shows the same time histories but for the
“narrow-band” data, which is the broad-band data filtered so that it contains only the frequencies
in the neighborhood of the first building-soil system frequency. The cut-off and role-off
frequencies, in Hz, of the Ormsby filters used are shown in the upper right corner of these plots.
The plot in the bottom shows the instantaneous frequency versus time estimated by the zero-crossing
analysis (open circles), and Gabor analysis (with σ = 1.5 for all the records). The
shaded rectangle in this plot has width 2 t σ
and height 2σω and is a measure of the resolution of
the Gabor analysis. The Gabor transform at a point (t, f ) in the time frequency plane is the
weighed average of the components of the function (effectively) within such a rectangle centered
at that point. The method cannot resolve frequencies that are closer than σω , and estimates in
time that are closer than t σ
. The resolution in frequency can be increased only if the resolution
in time is decreased (by increasing the time window of the Gabor wavelet, and consequently—
18
Figure 3.2.1 Estimation of the instantaneous frequency for component N11E of the record of the 1971
San Fernando earthquake at station USGS 466 (Los Angeles, 15250 Ventura Blvd.). Left: Fourier
spectrum of the relative roof displacement (solid line), of acceleration at the ground floor (dashed line),
and a smooth approximation of the relative roof displacement spectrum by the marginal Gabor transform
distribution (the red line). Time history of the roof relative displacement (solid line), and of the ground
floor displacement (dashed line), for the “broad-band” data (right-top), and for the filtered data (right-middle).
Right-bottom: instantaneous frequency versus time estimated by the zero-crossing method (open
circles), and from the ridge of the Gabor transform (with 1.5 σ = ). The shaded rectangle has width 2 t σ
and height 2σω , and is a measure of the resolution of the Gabor transform method.
19
Figure 3.2.2 Same as Fig. 3.2.1 but for component N00E of the record of the 1994 Northridge earthquake
at station USGS 5450 (Burbank, 3601 West Olive Ave.).
t σ
), and vice versa. For all the other records considered in this report, such results are presented
in Appendix B
The results in Figures 3.2.1 through 3.2.5 show that the estimates by the zero-crossing and
from the ridge of the Gabor transform are consistent. The estimates by the latter method are
smoother, as the Gabor transform is a smoothing operator. The zero-crossing method is not
accurate when the oscillations of the signal depart too much from a “pure” harmonic, and these
estimates are not shown. Both methods are most accurate when the amplitude of the signal is
20
Figure 3.2.3 Same as Fig. 3.2.1 but for component W00N of the record of the 1994 Northridge earthquake
at station USGS 5451 (Los Angeles, 6301 Owensmouth Ave.).
large and does not vary significantly during one cycle of oscillation, least accurate when the
amplitude is small and varies significantly during one cycle, and are arbitrary when the
amplitude is practically zero. Figures 3.2.1 and 3.2.4 show a significant change (decrease) in the
system frequency for these buildings during a single earthquake, of about 30%, Figure 3.2.2
indicates a change of about 17%, while Fig. 3.2.5 shows no significant change. It is interesting
to see whether the changes of frequency are permanent (indicating possible damage), or
temporary (possibly due to changes in the soil, changes in the bonding condition along the
contact surface between the foundation and the soil, e.g. formation of gaps between the
21
Figure 3.2.4 Same as Fig. 3.2.1 but for component N00E of the record of the 1994 Northridge earthquake
at station USGS 5453 (Los Angeles, 5805 Sepulveda Blvd.).
foundation and the soil, foundation partial uplift, etc.). The answer to this important question
can be explored through records from consecutive earthquakes, the results of which are shown in
Chapter 4.
22
Figure 3.2.5 Same as Fig. 3.2.1 but for component E30S of the record of the 1994 Northridge earthquake
at station USGS 5455 (Los Angeles, 1600 Ventura Blvd.).
23
4. TIME AND AMPITUDE VARIATIONS OF THE INSTANTANEOUS BUILDING-SOIL
SYSTEM FREQUENCY FOR SEVEN BUILDINGS
This chapter presents, in a compact form, results for the variation of the building-soil system
frequency in time as function of the amplitude of response for seven buildings. The results are
shown in Figures 4.1 through 4.28, which represent seven groups of four figures, each group
corresponding to a particular building. The four figures in a group present the estimates of the
instantaneous system frequency determined by zero-crossing and Gabor transform methods, for
each of the two horizontal components of motion. In each plot, the horizontal axis corresponds
to the instantaneous frequency, the vertical axis corresponds to the amplitude of response (of the
filtered signal) expressed as a rocking angle in radians, and each point corresponds to a particular
instant in time. The rocking angle was computed by dividing the amplitude of the relative (roof
minus base) response, if motion at the base was recorded, or otherwise the absolute horizontal
response of the roof or top floor by an estimate of the building height. The floor height varies
within a building, and the average floor height varies from one building to another. In this
report, an average floor height of 12.5 feet was assumed (1 foot = 30.48 cm). It is noted here that
this rocking angle includes the rigid body rocking, which could not be separated because of
insufficient number of instruments at the base, in addition to motion resulting from deflection of
the structure.
In Figures 4.1 through 4.28, the points corresponding to consecutive instants of time are
connected by a line, each line corresponding to a particular earthquake. In the plots showing
results for the zero-crossing method, the first and last point shown for a particular earthquake are
marked respectively by an open and a closed circle. The backbone curve, drawn by hand,
indicates roughly the trend of the variation of the system frequency as function of the amplitude
of response. From these figures, it can be seen that for the largest motions (during the 1971 San
Fernando and 1994 Northridge earthquakes), the system frequency generally decreased during
the shaking. For all but one building, this change seems to have been temporary, as the system
frequency increased during the shaking by the aftershocks. For one building, permanent change
appears to have occurred during the 1971 San Fernando earthquake (USGS 466). Detailed
interpretation of the causes of these changes is beyond the scope of this project.
The maximum and minimum frequencies determined from the backbone curves in Figs 4.1
through 4.28, and the corresponding maximum and minimum levels of response, are summarized
in Table 4.1, and the percentage change for all the seven buildings is shown in Fig. 4.29. It is
seen that, for the recorded levels of response, the change for most of the buildings is not more
than 20%, but it reaches 30% for two of the buildings.
A complete set of results for the instantaneous frequency versus time for each event, in form
as in Figs. 3.2.1 through 3.2.5, is presented in Appendix B.
24
San Fernando
(1971)
Northridge
(1994)
Northridge
Aft. -1
USGS 0466
N-S (Longitudinal)
Rocking angle (rad)
10-4
10-2
10-3
Apparent frequency, fp (Hz)
0.2 0.4 0.6 0.8
Los Angeles, 15250 Ventura Blvd.
θmax
fmax fmin
θ min
Zero-crossing method
Figure 4.1 Instantaneous frequency versus amplitude of motion for station USGS 466, for N-S
vibrations, determined by the zero-crossing method, for several earthquakes.
25
San Fernando
(1971)
Northridge
(1994)
Northridge
Aft. -1
USGS 0466
N-S (Longitudinal)
Rocking angle (rad)
10-4
10-2
10-3
Apparent frequency, fp (Hz)
0.2 0.4 0.6 0.8
Los Angeles, 15250 Ventura Blvd.
2σf
Gabor transform method
Figure 4.2 Instantaneous frequency versus amplitude of motion for station USGS 466, for N-S
vibrations, determined by the Gabor method, for several earthquakes.
26
San Fernando
(1971)
Northridge
(1994)
Northridge
Aft. -1
USGS 0466
E-W (Transverse)
Apparent frequency, fp (Hz)
0.1 0.2 0.3 0.4 0.5
Rocking angle (rad)
10-4
10-2
10-3
θ max
fmin fmax
θ min
Los Angeles, 15250 Ventura Blvd.
Zero-crossing method
Figure 4.3 Instantaneous frequency versus amplitude of motion for station USGS 466, for E-W
vibrations, determined by the zero-crossing method, for several earthquakes.
27
San Fernando
Northridge (1971)
(1994)
Northridge
Aft. #1
USGS 0466
E-W (Transverse)
Apparent frequency, fp (Hz)
0.1 0.2 0.3 0.4 0.5
Rocking angle (rad)
10-4
10-2
10-3
Los Angeles, 15250 Ventura Blvd.
2σf
Gabor transform method
Figure 4.4 Instantaneous frequency versus amplitude of motion for station USGS 466, for E-W
vibrations, determined by the Gabor method, for several earthquakes.
28
Northridge
(1994)
Aft. #9 Aft. #7
Aft. #100
Aft. #129
Aft. #142
Aft. #253
Aft. #254
Aft. #336
Aft. #392
Rocking angle (rad)
10-5
10-6
10-3
10-4
Apparent frequency, fp (Hz)
1.0 1.5 2.0 2.5 3.0
USGS 5108
θ E-W Comp. max
fmin fmax
θ min
Canoga Park, Santa Susana, ETEC, Bldg. #462
Zero-crossing method
Figure 4.5 Instantaneous frequency versus amplitude of motion for station USGS 5108, for EW
vibrations, determined by the zero-crossing method, for several earthquakes.
29
Northridge
(1994)
Aft. #7
Aft. #9
Aft. #100
Aft. #129
Aft. #142
Aft. #253
Aft. #254
Aft. #336
Aft. #392
USGS 5108
E-W Comp.
Rocking angle (rad)
10-5
10-6
10-3
10-4
Apparent frequency, fp (Hz)
1.0 1.5 2.0 2.5 3.0
Canoga Park, Santa Susana, ETEC, Bldg. #462
2σf
Figure 4.6 Instantaneous frequency versus amplitude of motion for station USGS 5108, for E-W
vibrations, determined by the Gabor method, for several earthquakes.
30
Northridge
(1994)
Aft. #9
Aft. #129
Aft. #142
Aft. #151
Aft. #253
Aft. #336
Aft. #392
Rocking angle (rad)
10-5
10-6
10-3
10-4
Apparent frequency, fp (Hz)
1.0 1.5 2.0 2.5
USGS 5108
θ N-S Comp. max
θ min
fmin fmax
Canoga Park, Santa Susana, ETEC, Bldg. #462
Zero-crossing method
Figure 4.7 Instantaneous frequency versus amplitude of motion for station USGS 5108, for N-S
vibrations, determined by the zero-crossing method, for several earthquakes.
31
Northridge
(1994)
Aft. #9
Aft. #129
Aft. #142
Aft. #151
Aft. #253
Aft. #336 Aft. #392
USGS 5108
N-S Comp.
Rocking angle (rad)
10-5
10-6
10-3
10-4
Apparent frequency, fp (Hz)
1.0 1.5 2.0 2.5
Canoga Park, Santa Susana, ETEC, Bldg. #462
2σf
Gabor transform method
Figure 4.8 Instantaneous frequency versus amplitude of motion for station USGS 5108, for N-S
vibrations, determined by the Gabor method, for several earthquakes.
32
Northridge
(1994)
Aft. -1
Aft. -5
Aft. -13
Aft. -14
USGS 5450
N00E
Apparent frequency, fp (Hz)
0.4 0.6 0.8 1.0
Burbank, 3601 West Olive Ave.
Rocking angle (rad)
10-3
10-5
10-4
10-2
θ max
fmin fmax
θ min
Zero-crossing method
Figure 4.9 Instantaneous frequency versus amplitude of motion for station USGS 5450, for
N00E vibrations, determined by the zero-crossing method, for several earthquakes.
33
Northridge
(1994)
Aft. -1
Aft. -5
Aft. -13
Aft. -14
USGS 5450
N00E
Apparent frequency, fp (Hz)
0.4 0.6 0.8 1.0
Rocking angle (rad)
10-3
10-5
10-4
10-2
Burbank, 3601 West Olive Ave.
2σf
Gabor transform method
Figure 4.10 Instantaneous frequency versus amplitude of motion for station USGS 5450, for N00E
vibrations, determined by the Gabor method, for several earthquakes.
34
Northridge
(1994)
Aft. -1
Aft. -5
Aft. -13
Aft. -14
USGS 5450
W00N
Rocking angle (rad)
10-3
10-5
10-4
10-2
Apparent frequency, fp (Hz)
0.4 0.6 0.8 1.0
θ max
fmin fmax
θ min
Burbank, 3601 West Olive Ave.
Zero-crossing method
Figure 4.11 Instantaneous frequency versus amplitude of motion for station USGS 5450, for W00N
vibrations, determined by the zero-crossing method, for several earthquakes.
35
Northridge
(1994)
Aft.- 1
Aft. -5
Aft.- 13
Aft.- 14
USGS 5450
W00N
Rocking angle (rad)
10-3
10-5
10-4
10-2
Apparent frequency, fp (Hz)
0.4 0.6 0.8 1.0
Burbank, 3601 West Olive Ave.
2σf
Gabor transform method
Figure 4.12 Instantaneous frequency versus amplitude of motion for station USGS 5450, for W00N
vibrations, determined by the Gabor method, for several earthquakes.
36
Northridge
(1994)
Aft. -26
Aft. -115
USGS 5451
N00E (Transverse)
Rocking angle (rad)
10-4
10-3
10-2
Apparent frequency, fp (Hz)
0.2 0.3 0.4 0.5
Los Angeles, 6301 Owensmouth Ave.
θ max
fmin fmax
θ min
Zero-crossing method
Figure 4.13 Instantaneous frequency versus amplitude of motion for station USGS 5451, for N00E
vibrations, determined by the zero-crossing method, for several earthquakes.
37
Northridge
(1994)
Aft. -26
Aft. -115
USGS 5451
N00E (Transverse)
Rocking angle (rad)
10-4
10-3
10-2
Apparent frequency, fp (Hz)
0.2 0.3 0.4 0.5
Los Angeles, 6301 Owensmouth Ave.
2σf
Gabor transform method
Figure 4.14 Instantaneous frequency versus amplitude of motion for station USGS 5451, for N00E
vibrations, determined by the Gabor method, for several earthquakes.
38
Northridge
(1994)
Aft. -26
Aft. -115
USGS 5451
W00N (Longitudinal)
Apparent frequency, fp (Hz)
0.3 0.4 0.5
Rocking angle (rad)
10-3
10-4
10-2 θ max
fmin fmax
θ min
Los Angeles, 6301 Owensmouth Ave.
Zero-crossing method
Figure 4.15 Instantaneous frequency versus amplitude of motion for station USGS 5451, for W00N
vibrations, determined by the zero-crossing method, for several earthquakes.
39
Northridge
(1994)
Aft. -26
Aft. -115
USGS 5451
W00N (Longitudinal)
Apparent frequency, fp (Hz)
0.3 0.4 0.5
Rocking angle (rad)
10-3
10-4
10-2
Loa Angeles, 6301 Owensmouth Ave.
2σf
Gabor transform method
Figure 4.16 Instantaneous frequency versus amplitude of motion for station USGS 5451, for W00N
vibrations, determined by the Gabor method, for several earthquakes.
40
Northridge
(1994)
Aft. -1
Aft. -7
Aft. -29
USGS 5453
N00E
Rocking angle (rad)
10-4
10-2
10-3
Apparent frequency, fp (Hz)
0.3 0.4 0.5 0.6 0.7 0.8 0.9
Van Nuys, 5805 Sepulveda Blvd.
θ max
fmin fmax
θ min
Zero-crossing method
Figure 4.17 Instantaneous frequency versus amplitude of motion for station USGS 5453, for N00E
vibrations, determined by the zero-crossing method, for several earthquakes.
41
Northridge
(1994)
Aft. -1
Aft. -7
Aft. -29
USGS 5453
N00E
Rocking angle (rad)
10-4
10-2
10-3
Apparent frequency, fp (Hz)
0.3 0.4 0.5 0.6 0.7 0.8 0.9
Van Nuys, 5805 Sepulveda Blvd.
2σf
Figure 4.18 Instantaneous frequency versus amplitude of motion for station USGS 5453, for N00E
vibrations, determined by the Gabor method, for several earthquakes.
42
Northridge
(1994)
Aft. -7
Aft. -24
Aft. -26
Aft. -29
Aft. -103
Aft. -104
Aft. -115
USGS 5453
W00N
Rocking angle (rad)
10-4
10-2
10-3
Apparent frequency, fp (Hz)
0.5 0.6 0.7 0.8 0.9 1.0
θ max
fmin fmax
θ min
Van Nuys, 5805 Sepulveda Blvd.
Zero-crossing method
Figure 4.19 Instantaneous frequency versus amplitude of motion for station USGS 5453, for W00N
vibrations, determined by the zero-crossing method, for several earthquakes.
43
Northridge
(1994)
Aft. -7
Aft. 24
Aft. -26
Aft. -29
Aft. -103
Aft. -104
Aft. -115
USGS 5453
W00N
Rocking angle (rad)
10-4
10-2
10-3
Apparent frequency, fp (Hz)
0.5 0.6 0.7 0.8 0.9 1.0
Van Nuys, 5805 Sepulveda Blvd.
2σf
Gabor transform method
Figure 4.20 Instantaneous frequency versus amplitude of motion for station USGS 5453, for W00N
vibrations, determined by the Gabor method, for several earthquakes.
44
Northridge
(1994)
Aft. -1
Aft. -7
Aft. -46 Aft. -25
USGS 5455
E30S
Apparent frequency, fp (Hz)
0.3 0.4 0.5 0.6 0.7
Los Angeles, 16000 Ventura Blvd.
Rocking angle (rad)
10-3
10-4
10-2
θ max
fmin fmax
θ min
Zero-crossing method
Figure 4.21 Instantaneous frequency versus amplitude of motion for station USGS 5455, for E30S
vibrations, determined by the zero-crossing method, for several earthquakes.
45
Northridge
(1994)
Aft. -1
Aft. -7
Aft. -25
Aft. -46
USGS 5455
E30S
Apparent frequency, fp (Hz)
0.3 0.4 0.5 0.6 0.7
Rocking angle (rad)
10-3
10-4
10-2
Los Angeles, 16000 Ventura Blvd.
2σf
Gabor transform method
Figure 4.22 Instantaneous frequency versus amplitude of motion for station USGS 5455, for E30S
vibrations, determined by the Gabor method, for several earthquakes.
46
Northridge
(1994)
Aft. -1
Aft. -22
Aft. -25
Aft. -46
USGS 5455
N30E
Rocking angle (rad)
10-3
10-4
10-2
Apparent frequency, fp (Hz)
0.2 0.3 0.4 0.5 0.6
θ max
fmin fmax
θ min
Los Angeles, 16000 Ventura Blvd.
Zero-crossing method
Figure 4.23 Instantaneous frequency versus amplitude of motion for station USGS 5455, for N30E
vibrations, determined by the zero-crossing method, for several earthquakes.
47
Northridge
(1994)
Aft. -1
Aft. -22
Aft. -25
Aft. -46
USGS 5455
N30E
Rocking angle (rad)
10-3
10-4
10-2
Apparent frequency, fp (Hz)
0.2 0.3 0.4 0.5 0.6
Los Angeles, 16000 Ventura Blvd.
2σf
Gabor transform method
Figure 4.24 Instantaneous frequency versus amplitude of motion for station USGS 5455, for N30E
vibrations, determined by the Gabor method, for several earthquakes.
48
Northridge
(1994)
Aft. 1
Aft. 4 Aft. 8
Aft. 9
Aft. 10
Aft. 13
Aft. 19
USGS 5457
N00E
Rocking angle (rad)
10-4
10-3
10-2
10-5
Apparent frequency, fp (Hz)
0.4 0.6 0.8 1.0
Los Angeles, 8436 West 3rd St.
θ max
fmin fmax
θ min
Zero-crossing method
Figure 4.25 Instantaneous frequency versus amplitude of motion for station USGS 5457, for N00E
vibrations, determined by the zero-crossing method, for several earthquakes.
49
Northridge
(1994)
Aft. 1
Aft. 4
Aft. 8
Aft. 9
Aft. 10
Aft. 13
Aft. 19
USGS 5457
N00E
Rocking angle (rad)
10-4
10-3
10-2
10-5
Apparent frequency, fp (Hz)
0.4 0.6 0.8 1.0
Los Angeles, 8436 West 3rd St.
2σf
Gabor transform method
Figure 4.26 Instantaneous frequency versus amplitude of motion for station USGS 5457, for N00E
vibrations, determined by the Gabor method, for several earthquakes.
50
Northridge
(1994)
Aft. 4
Aft. 8
Aft. 9
Aft. 10
Aft. 17
Aft. 19
USGS 5457
S90W
Apparent frequency, fp (Hz)
0.5 0.75 1.0 1.25
Rocking angle (rad)
10-4
10-5
10-2
10-3
θ max
fmin fmax
θ min
Los Angeles, 8436 West 3rd St.
Zero-crossing method
Figure 4.27 Instantaneous frequency versus amplitude of motion for station USGS 5457, for W90N
vibrations, determined by the zero-crossing method, for several earthquakes.
51
Northridge
(1994)
Aft. 4
Aft. 8
Aft. 9
Aft. 10
Aft. 17
Aft. 19
USGS 5457
S90W
Apparent frequency, fp (Hz)
0.5 0.75 1.0 1.25
Rocking angle (rad)
10-4
10-5
10-2
10-3
Los Angeles, 8436 West 3rd St.
2σf
Gabor transform method
Figure 4.28 Instantaneous frequency versus amplitude of motion for station USGS 5457, for W90N
vibrations, determined by the Gabor method, for several earthquakes.
52
Table 4.1 Maximum and minimum system frequencies and maximum and minimum rocking angles for
seven instrumented buildings.
Station
no. Comp.
fmax, fmin
(Hz)
Δf /
fmax
(%)
θmax, θmin
(×10-3 rad)
Comp.
fmax, fmin
(Hz)
Δf /
fmax
(%)
θmax, θmin
(×10-3 rad)
5108 E00S 2.130, 1.648 22.64 0.49607, 0.00251 N00E 1.899, 1.525 19.68 1.05640, 0.00395
0466 N00E 0.377, 0.312 17.23 4.74628, 0.12339 W00N 0.295, 0.215 27.23 4.66436, 0.31591
5450 N00E 0.691, 0.614 11.16 3.08807, 0.03879 W00N 0.666, 0.576 13.52 5.16651, 0.03820
5451 N00E 0.329, 0.273 17.16 7.38386, 0.18001 W00N 0.434, 0.373 14.14 9.57349, 0.13386
5453 N00E 0.613, 0.434 29.20 7.87023, 0.06008 W00N 0.744, 0.712 5.69 4.88160, 0.02566
5455 E30S 0.434, 0.408 3.86 5.39350, 0.05552 N30E 0.425, 0.363 14.59 5.36059, 0.09933
5457 N00E 0.675, 0.569 15.76 6.34016, 0.02940 S00W 0.866, 0.704 18.62 3.62546, 0.01286
50
0
10
30
20
40
Δf fmax (%)
USGS 5108
USGS 0466
USGS 5450
USGS 5451
USGS 5453
USGS 5457
USGS 5455
Figure 4.29 A summary of the changes of the building-soil system frequencies of the seven buildings
analyzed in this report, determined from the observed trends during multiple earthquake excitations (see
Figures 4.1 through 4.28). For each building, two values are shown, corresponding to the two horizontal
components of motion. The change is expressed as a percentage of the maximum frequency.
53
5. SUMMARY AND CONCLUSIONS
This report presents a summary of new data on the response of seven buildings in the Los
Angeles area to the 1994 Northridge earthquake and its aftershocks, which was digitized and
processed for this project, and an analysis of the building-soil system frequency determined from
these data. Although the number of recorded aftershocks in many of these buildings was large
(up to about 80), only a small number of records were found to be useable for this analysis,
because of the small signal to noise ratio at long periods which lead to high lower cut-off
frequency, higher or too close to the system frequency.
The system frequency was estimated by two methods—zero crossing analysis, and from the
ridge of the Gabor transform. The results by both methods are consistent. The general observed
trend of the variation of the system frequency is decrease during the 1994 Northridge main event,
and the 1971 San Fernando earthquake (one existing record of this earthquake was included in
the analysis), which caused the largest amplitude response. However, for all but one building,
the frequency was again larger during the aftershocks, indicating system recovery. For most
buildings, the frequency changed up to 20%, and for two buildings, the change was about 30%.
A permanent reduction of the frequency is consistent with permanent loss of stiffness, while a
“recovery” to the initial higher value is consistent with the interpretation that the change was
mainly due to changes in the soil (rather than in the structure itself), or changes in the bond
between the soil and the foundation. Other possible causes of the temporary changes are:
contribution of the nonstructural elements to the total stiffness resisting the seismic forces, and
opening of existing cracks in the concrete structures during larger amplitude response. The
degree to which each of these causes contributed to the temporary changes cannot be determined
from the current instrumentation and is beyond the scope of this project. What matters for the
design codes, however, is the overall effect, which can be estimated from the accelerograms
recorded so far.
54
6. REFERENCES
1. Biot, M.A. (1942). “Analytical and experimented methods in engineering seismology,”
Trans., ASCE, 68, 365-409.
2. Carder, D.S. (1936). “Vibration observations,” Chapter 5, in Earthquake Investigations in
California 1934-1935, U.S. Dept. of Commerce, Coast and Geological Survey, Special
Publication No. 201, Washington D.C.
3. Goel, R.K. and A.K. Chopra (1997). “Period formulas for concrete shear wall buildings,” J.
of Structural Eng., ASCE, 124(4), 426-433.
4. Hudson, D.E. (1970). “Dynamic tests of full scale structures,” Chapter 7, 127-149, in
Earthquake Engineering, Edited by R.L. Wiegel, Prentice Hall, N.J.
5. Lee, V.W. and M.D. Trifunac (1990). “Automatic digitization and processing of
accelerograms using PC,” Report No. 90-03, Dept. of Civil Engrg, U. of So. California, Los
Angeles, CA.
6. Li, Y. and S.T. Mau (1979). “Learning from recorded earthquake motions in buildings,” J. of
Structural Engrg, ASCE, 123(1), 62-69.
7. Luco, J.E., M.D. Trifunac and H.L. Wong (1987). “On the apparent changes in dynamic
behavior of a nine story reinforced concrete building,” Bull. Seism. Soc. Amer., 77(6), 1961-
1983.
8. Stewart, J.P. R.B. Seed and G. L. Fenves (1999). “Seismic soil-structure interaction in
buildings II: Empirical findings,” J. of Geotechnical and Geoenvironmental Engrg, ASCE,
125(1), 38-48.
9. Todorovska, M.I. (1992). “Effect of the depth of the embedment on the system response
during building-soil interaction,” Soil Dynamics & Earthquake Engrg, 11 (2), 111-123.
10. Todorovska, M.I. (1993a). “In-plane foundation-soil interaction for embedded circular
foundations,” Soil Dynamics & Earthquake Engrg, 12 (5), 283-297.
11. Todorovska, M.I. (1993b). “Effects of the wave passage and the embedment depth during
building-soil interaction,” Soil Dynamics & Earthquake Engrg, 12 (6), 343-355.
12. Todorovska, M.I. (1995). “A note on distribution of amplitudes of peaks in structural
response including uncertainties of the exciting ground motion and of the structural model,”
Soil Dynamics & Earthquake Engrg, 14 (3), 211-217.
13. Todorovska, M.I. (2001). “Estimation of instantaneous frequency of signals using the
continuous wavelet transform,” Report CE 01-07, Dept. of Civil Engrg., Univ. of Southern
California, Los Angeles, California.
14. Todorovska, M.I., & M.D. Trifunac (1992). “The system damping, the system frequency and
the system response peak amplitudes during in-plane building-soil interaction,” Earthquake
Engrg & Struct. Dynamics, 21 (2), 127-144.
15. Todorovska, M.I., & M.D. Trifunac (1993). “The effects of the wave passage on the response
of base-isolated buildings on rigid embedded foundations,” Rep. No. CE 93-10, Dept. of
Civil Engrg, Univ. of Southern California, Los Angeles, California, pp. 231.
55
16. Todorovska, M.I., A. Hayir & M.D. Trifunac (2001). “Flexible versus rigid foundation
models of soil-structure interaction: incident SH-waves,” Proc. 2nd U.S.-Japan Workshop on
Soil-Structure Interaction, March 6-8, 2001, Tsukuba City, Japan, pp. 19.
17. Todorovska, M.I., M.D. Trifunac, V.W. Lee, C.D. Stephens, K.A. Fogleman, C. Davis and
R. Tognazzini (1999). “The ML = 6.4 Northridge, California, Earthquake and Five M > 5
Aftershocks Between 17 January and 20 March 1994 - Summary of Processed Strong Motion
Data,” Report CE 99-01, Dept. of Civil Engrg., Univ. of Southern California, Los Angeles,
California.
18. Trifunac, M.D. (1999). Comments on “Period formulas for concrete shear wall buildings,”
J. Struct. Eng., ASCE, 125(7), 797-798.
19. Trifunac, M.D. (2000). Comments on “Seismic soil-structure interaction in buildings I:
analytical models, and II: empirical findings,” J. geotech. And Geoenv. Eng., ASCE, 126(7),
668-670.
20. Trifunac, M.D., V.W. Lee and M.I. Todorovska (1999). “Common problems in automatic
digitization of accelerograms,” Soil Dynamics and Earthquake Engineering, Soil Dynamics
and Earthquake Engrg, 18(7), 519-530.
21. Trifunac, M.D., & M.I. Todorovska (1998). “Relative flexibility of a building foundation,”
Proc. US-Japan Workshop on Soil-Structure Interaction, Menlo Park, California, 20-23 Sept.,
1998, pp. 20.
22. Trifunac, M.D. and M.I. Todorovska (1998). “Nonlinear soil response as a natural passive
isolation mechanism–the 1994 Northridge, California, earthquake,” Soil Dynamics and
Earthquake Engrg, 17(1), 41-51.
23. Trifunac, M.D., T.Y. Hao & M.I. Todorovska (2001a). “On energy flow in earthquake
response,” Report CE 01-03, Dept. of Civil Engrg., Univ. of Southern California, Los
Angeles, California.
24. Trifunac, M.D., T.Y. Hao & M.I. Todorovska (2001b). “Energy of earthquake response as a
design tool,” Proc. 13th Mexican National Conf. on Earthquake Engineering, Guadalajara,
Mexico.
25. Trifunac, M.D., M.I. Todorovska and T.Y. Hao (2001c). “Full-scale experimental studies of
soil-structure interaction - a review,” Proc. 2nd U.S.-Japan Workshop on Soil-Structure
Interaction, March 6-8, 2001, Tsukuba City, Japan, pp. 52.
26. Udwadia, F.E. and M.D. Trifunac (1974). “Time and Amplitude Dependent Response of
Structures,” Earthquake Engrg and Structural Dynamics, 2, 359-378.
56