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1 Puerto Rico Strong Motion
Program
Department of Civil Engineering and Surveying
University of Puerto
Rico at Mayaguez
Mayaguez, Puerto Rico
00681
jclinton{at}uprm.edu
jclinton{at}ecf.caltech.edu
(J.F.C.)
2 Department of Civil
Engineering
Division of Engineering and Applied Science
California
Institute of Technology
Pasadena, California
91125
case{at}caltech.edu
heaton_t{at}caltech.edu
(S.C.B,
T.H.H.)
3 Lockheed Martin Aeronautics
Company
Palmdale, California
93599
javier{at}gps.caltech.edu
(J.F.)
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Abstract |
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 Top Abstract IntroductionHistorical Evidence for Natural...The Current System of...Analysis of the Continuous...M 5.4 22 February...A Linear Transfer Function...Conclusions and CommentAcknowledgmentsReferences |
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Introduction |
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TopAbstractIntroductionHistorical Evidence for Natural...The Current System of...Analysis of the Continuous...M 5.4 22 February...A Linear Transfer Function...Conclusions and CommentAcknowledgmentsReferences |
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This article attempts to document the amount of frequency shifting in the instrumented large structures at the Caltech campus, the Robert A. Millikan Library, and the Broad Center. The current state of instrumentation on campus is described, and the observed changes in fundamental frequencies are correlated with weather, earthquake history, and building usage.
The temporal wandering of the natural frequencies of the Millikan Library has been documented previously (Kuroiwa, 1967; Trifunac, 1972; Udwadia and Trifunac, 1974; Foutch, 1976; Luco et al., 1987; Chopra, 1995). Since its construction in 1967, a decrease in these resonant frequencies may be observed from yearly forced vibration experiments and from strong-motion records. This frequency drop has been interpreted to be due to a corresponding softening in system stiffness. Recent ambient and forced vibration tests indicate that the fundamental natural frequency of the structure is now approximately 22% lower in the east–west direction and 12% lower in the north–south direction than was determined shortly after construction (Kuroiwa, 1967). Strong-motion records indicate that the natural frequencies drop even further during moderately large events. The M 6.1 Whittier Narrows Earthquake, with a epicentral location 19 km from the library, is one example. By comparing the forced vibration measurements prior to the event (Levine et al., 1988) with the coda of the strong-motion records, the building’s east–west and north–south natural frequencies are shown to decrease by 17% and 25%, respectively. The structure recovers stiffness somewhat after a moderately large shaking event, but because of the lack of data in the immediate aftermath of these mainshocks, the recovery time frame cannot be constrained. Further, Kuroiwa (1967) and others noted that the resonant frequencies drop measurably when the applied force during forced vibrations is increased—during construction in 1966 the fundamental east–west frequency dropped 3% when applied force was increased by a factor of 8.
Recent improvements in the quality and quantity of instrumentation in the building and at other sites on the Caltech campus have led to renewed investigation of the structure. Analysis of structural response to previously unrecorded ambient and small-intensity ground motions is now possible.
Data are presented which indicate that not only do the natural frequencies change significantly during strong shaking, as evidenced by analog recordings of large earthquakes in the recent past, but there are also measurable changes in the resonant frequencies of the buildings due to:
These last two factors also affect the recently constructed Broad Center on the Caltech campus (the Broad Center has not yet been shaken or subjected to strong earthquake motions).
The lowering of the natural frequencies during transient events in the Millikan Library is likely due to a combination of two mechanisms, a nonlinear softening of the superstructure itself and an interaction of the structure with the surrounding soil. Changes in occupancy usage are also responsible for natural frequency changes. Note that the construction of partition walls for office space in three entire levels during the spring of 2003 coincides with a significant and permanent raising of the natural frequencies (though the magnitude of the change in frequency is difficult to explain considering the relatively small increase in stiffness expected by the addition of the partition walls).
The natural frequency of a structure, as measured by accelerometers on the structure, is a combination of the fundamental fixed-base period of the structure, as well as the rocking and horizontal translation frequencies of the same structure if it moved as a rigid body on the flexible soil (Trifunac, 1999; Trifunac and Ivanovic, 2003). Throughout this article, all references to the natural frequencies of the structure refer to these combined-system frequencies. Note also that, in addition to the differences in amplitude, forced vibration tests differ from ambient and earthquake motions in how energy is imparted to the system. For forced vibration tests, the energy source located on the roof (the shaker) emits a continuous single frequency, and energy flows down the building and out into the half-space. In ambient and earthquake vibrations, scattered energy with highly variable frequency content enters the building from the bottom, travels to the roof, reflects back down, and eventually dissipates into the half-space.
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Historical Evidence for Natural Frequency Wandering—Millikan Library |
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TopAbstractIntroductionHistorical Evidence for Natural...The Current System of...Analysis of the Continuous...M 5.4 22 February...A Linear Transfer Function...Conclusions and CommentAcknowledgmentsReferences |
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After the 1971 San Fernando Earthquake, cracking and spalling of the concrete slabs located on the ground floor entry plaza were noted (Foutch and Jennings, 1978). Further, horizontal cracks along the pour line in the core shear walls between both the basement and first floor, and first and second floors, have been observed in the emergency staircase in the north–south direction. Access to the east–west sides of the core shear wall is not possible. The steam tunnels that connect various buildings on campus also suffered some minor cracking close to the Library. No further structural damage has been observed in the building.
In 1968, the building was instrumented with two permanent triaxial Teledyne-Geotech RFT-250 accelerometers, located on the roof and basement. A 10-channel Kinemetrics CR-1 strong-motion array was also installed in 1979, with channels on the basement, sixth floor, and roof. These systems have since been superseded by a 24-bit continuously recorded digital triaxial accelerometer, the SCSN station MIK (installed in 2001), on the ninth floor, and a 36-channel 19- bit triggered-accelerometer array run by the U.S. Geological Survey (USGS) (installed in 1998), with a minimum of three channels on each floor. A synchronized shaker was permanently installed on the roof of the building in the early 1970s and is still used for forced vibration testing (Hudson, 1962).
Yearly modal analysis of the structure (using temporary deployments of Kinemetrics Ranger SS-1 seismometers) during civil engineering classes at Caltech, as well as the triggered event data from the RFT-250 and CR-1 arrays, have provided us with a relatively detailed history of the evolution of the dynamic properties of the building. A summary of the fundamental natural frequencies observed during strong shaking and selected forced vibration tests is presented in Table 1 (a more complete list is in Clinton, 2004). Figures 2 and 3 present graphical interpretations of Table 1. In Figure 2, the natural frequencies are plotted against the date of the observation. There is a clear trend toward lower natural frequencies as time increases, with major steps occurring during large earthquakes. Figure 3 plots frequency versus the coincident roof acceleration amplitude, on logarithmic axes. There is a clear pattern of frequency dropping with increasing excitation amplitude. The best-fitting line is a good fit to the data, though a very large variance still exists.
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Kuroiwa (1967) first observed variation in the natural frequencies, measuring a decrease in natural frequencies proportional to the applied force imparted by the shaker. This has been consistently observed since then. For example, in the tests carried out in July 2002 by Bradford et al. (2004), during shaking with full weights, an east–west natural frequency of 1.11 Hz was measured, and during shaking with only four side weights, the natural frequency was 1.14 Hz— a difference of 0.03 Hz or 2.5%. This change in weights corresponds to a factor of nearly 2 difference in the amplitude of the rooftop sinusoidal acceleration, and a factor of 2.23 change in the applied force. Similar changes were observed in the north–south fundamental frequency. Thus, with weight configurations variable in some of the forced tests (and unknown in some cases), this level of variability in reporting of results should be noted. It is assumed that the forced vibration test results in Table 1 are taken with the shaker loaded with half-full weights. The shaded area in Figure 2 is a ±0.03 Hz error band for the natural frequencies, which reflects both this loading uncertainty and temporal fluctuations caused by the changing weather conditions, as discussed in this article.
The natural frequencies from strong shaking are determined from the resonance of the structure (measured at the roof channels) in the immediate aftermath of a large event. This is illustrated in Figure 4, which shows the response of the east–west channels of the CR-1 array at Millikan to the 1987 M 6.1 Whittier Narrows Mainshock (Levine et al., 1988).
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Table 1 shows the initial natural frequencies of the building at 1.45 Hz in the east–west direction and at 1.90 Hz in the north–south direction. For this initial test, and many subsequent tests, the higher order modes, including the first torsional frequency, are not clearly and unambiguously identified because of the poor signal-to-noise ratio for the recording systems of the time. (SMA-1s, CR-1s, and Ranger SS-1s all have a dynamic range in the order of 3 orders of magnitude (60 dB), compared with the 144-dB resolution of the 24-bit instruments.)
The recorded history includes four moderately large shaking events, all with roof accelerations of at least 340 cm/sec2 (over 34%g). Several smaller events, including the 1970 Lytle Creek Earthquake, and some more recent events recorded on the digital instruments, with accelerations below 50 cm/sec2, are also included on Table 1 for comparison. During strong motion, the natural frequencies temporarily fall by about 20%. Surprisingly, after each strong-motion event, the structural system stiffens and natural frequencies return to near pre-earthquake levels, usually with a permanent drop in frequency of less than 2.5%. Some events have led to a larger permanent decrease of all subsequent forced vibration resonant frequencies. In the most extreme case, the east–west fundamental frequency dropped permanently by 16.6% in tests subsequent to the 1971 San Fernando event.
East–West Fundamental Frequency
In the east–west direction, the lateral forces are primarily resisted
by the elevator core and the concrete moment frame (the architectural facade of
stiff window frames provide additional stiffness).
Table 1 shows that the very
first significant earthquake motion (from the 1970 M 5.3 Lytle Creek
earthquake, 
= 57 km), with comparatively small rooftop
accelerations of 49 cm/sec2, resulted in a decrease of 10.3% in the
natural frequency as measured in the strong- motion record. A further softening
occurred during the larger magnitude, closer 1971 M 6.6 San Fernando
event ( = 31 km, peak roof accelerations = 306
cm/sec2, with the fundamental frequency measured at about 1.0 Hz
during the strong shaking. Subsequent forced vibration tests indicate the
frequency dropped permanently by 16.6%, to 1.21 Hz, from these two events. No
earthquake recorded since has generated east–west motions that exceeded
the velocities and accelerations of the San Fernando event. Correspondingly,
subsequent natural frequencies from strong-motion and forced vibration do not
show any significant loss of stiffness of the structural system. The most recent
east–west natural frequency recorded from forced vibrations is 1.14 Hz
(Bradford et al., 2004).
The general mode shape has remained constant throughout the history
(Foutch, 1976; Bradford et al., 2004),
though the component of rocking in the east–west mode is not well
characterized by recent studies, e.g.,
Bradford et al. (2004),
which do not account for rocking at the base of the elevator core, because
sensors are located only at the edges of the building.
Foutch et al. (1975), using a dense temporary deployment of sensors, shows that the rocking
contribution at the base is significant.
North–South Fundamental Frequency
For the north–south direction, with the lateral resistance provided by
the massive shear walls, a different pattern emerges. Very little frequency loss
occurs during the Lytle Creek event—even the strong-motion record shows a
decrease of only 1.1%. Instead, it is the San Fernando event, with rooftop
accelerations of 341 cm/sec2, that causes the first major frequency
drop; natural frequencies from forced vibrations fell from 1.9 Hz to 1.77 Hz
after the event. Mode shapes before and after the San Fernando event show major
differences. This can be illustrated by considering the relative contributions
to the displacements at the roof; before the earthquake, less than 3% of the
peak roof displacement is attributed to basement rocking, yet after, and in
subsequent tests, approximately 30% of the roof motion is due to basement
rocking
(Jennings and Kuroiwa, 1968; Foutch, 1976;
Bradford et al., 2004).
Another major decrease occurs during the 1987 M 6.1 Whittier Narrows
event ( = 19 km), where the highest rooftop accelerations (534
cm/sec2) were recorded during the shaking.
Figure 2 shows that this event
caused the largest intraevent frequency drop (nearly 25%), with a 4% permanent
decrease in forced frequencies. Subsequent natural frequency measurements from
forced vibration tests are relatively constant, and no further softening beyond
the 1.33 Hz recorded in Whittier Narrows occurs during strong motions (including
the Northridge Earthquake). The most recent forced north–south natural
frequency is 1.67 Hz
(Bradford et al., 2004).
East–West Second Mode Frequency
At construction, the second east–west mode frequency was determined as
6.2 Hz (Kuroiwa, 1967) (though
it is not clear whether the building had been fully completed at this time; the
heavy cladding may not have been added). During the San Fernando event, the
frequency dropped to 
4.95 Hz
(McVerry, 1980;
Beck and Chan, 1995). Investigations subsequent to this earthquake have indicated the second mode
varies from 4.17 Hz
(Beck and Chan, 1995) to 5.35 Hz
(Teledyne-Geotech-West, 1972).
The most recent measured forced east–west second modal frequency is 4.93
Hz
(Bradford et al., 2004).
Response to Small Earthquakes
Table 1 also contains
fundamental frequencies determined from shaking due to the small M 4.2
Beverly Hills event ( = 26 km) in September 2001. Even though
measured accelerations from the event are about double the accelerations from
the sinusoidally excited forced vibration tests, the measured fundamental
frequencies are higher than those from forced testing. This may be attributed to
changes in the ambient pre-earthquake natural frequency due to climatic changes,
and will be discussed in detail later.
The February 2003 M 5.4 Big Bear event ( = 119 km)
produced accelerations almost double those from the M 4.2 Beverly
Hills event, and yet it had a more significant effect on the fundamental
frequencies—the drop in frequency from Big Bear is much greater than
double the drop observed in Beverly Hills. This suggests that the relationship
between fundamental frequency and acceleration is nonlinear. Further, it is
observed that the fundamental frequency during the Big Bear event drops by 6.1%
in the east–west direction, and only 3.6% in the north–south
direction, even though the north–south accelerations are larger. This
indicates that the east–west direction is more susceptible to softening
under small excitations and that larger motions are required to start
significant softening in the north–south direction (the ambient data will
corroborate this observation). The response of the library to this earthquake
will be studied in more detail.
Figure 3 indicates there is, in general, a linear relationship between the logarithm of the acceleration amplitude and logarithm of the frequency, though the scatter of the data is large, and, at least for the small amplitudes, may be due to the ambient variations in natural frequencies.
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The Current System of Instrumentation at Caltech |
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TopAbstractIntroductionHistorical Evidence for Natural...The Current System of...Analysis of the Continuous...M 5.4 22 February...A Linear Transfer Function...Conclusions and CommentAcknowledgmentsReferences |
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The Caltech Civil Engineering Department operated an older network of analog film-recording SMA-1s at a number of sites on and around the campus, as well as a 12-channel CR-1 at Millikan Library, which had been operational on campus since the 1970s. However, this network has not been maintained since the mid-1990s and is currently not operational.
Millikan Library (MIK, USGS-Caltech Array)
In January 1998 the USGS and Caltech Civil Engineering Department
installed a 36-channel dense network of FBA-11 accelerometers recording
triggered event data on two 19-bit Mt. Whitney dataloggers with dial-up data
retrieval. A triaxial EpiSensor accelerometer was also installed on the ninth
floor of the structure and has been continuously transmitting 24-bit data since
February 2001, to the Southern California Earthquake Data Center
(SCEDC), as station MIK in the California Integrated
Seismic Network (SCSN).
This improved sensor configuration prompted a detailed forced dynamic analysis (using the existing shaker located on the roof), which was performed in the summer of 2002 (Bradford et al., 2004). The results of this study are summarized in Table 2. At the time of the tests, the approximate first mode frequencies during forced vibration (1/2 weights) are 1.14 Hz for the east–west direction, 1.67 Hz for the north–south direction, and 2.38 Hz for the torsional mode.
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Broad Center (CBC)
This is a three story structure with an irregular floor plan and two deep
basements (see Fig. 5). It was
completed in the summer of 2002 and has been instrumented since February 2003.
The basements are enclosed by stiff shear walls, and the steel superstructure is
braced with stiff unbonded braces in both the north–south and
east–west directions.
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The building houses a 24-bit SCSN station, recording eight channels of EpiSensor accelerometer data. Three triaxial instruments are installed, all on the plan of the unbonded braced frame-line that is the structural core of the building. Two are located near the northwest intersection of the frame line, one on the first floor, with the other on the roof. The final accelerometer, which only has its horizontal channels logged (the datalogger supports only eight channels of data), is near the southeast intersection of the frame line. All eight channels comprise SCSN station CBC. The instrument layout is illustrated by the schematic in Figure 5c.
In the absence of a forced vibration modal analysis for the building, the natural frequencies were investigated by using the CBC ambient data alone. Clinton (2004) describes this method in more detail. Table 2 presents the resonant frequencies determined by this analysis. Because the translational fundamental and first overtone frequencies are very close, which is unusual for typical structures, it is possible some dynamic feature of the building is not appreciated in this analysis. Ideally, further instrumentation and a forced vibration test are needed to confirm the dynamic properties of the Broad Center listed in the table.
525 S. Wilson Avenue, USGS Office (GSA)
GSA is a 24-bit SCSN station with a triaxial EpiSensor
accelerometer located in the basement of the two-story wood-frame house (used as
USGS Pasadena offices). GSA data are often used as a
reference station for data from the Millikan Library and Broad Center. The
station, operating since July 2000, is approximately 150 m due west of the
Millikan Library, and about 200 m south-southwest of the Broad Center.
Robinson Building (CRP)
CRP is a 24-bit SCSN station, located about 18 m below
grade, in the unused Solar Telescope pit of the Robinson Building. It houses a
high-gain broadband (Guralp CMG-1) and a strong-motion (Tokyo-Sokushin velocity
sensor VSE- 355G3) instrument. It is the only station on campus with a high-gain
digital instrument permanently deployed. It has been operational since March
2003 and is about 75 m southwest of Millikan Library.
The Athenaeum (CAC)
SCSN station CAC is a 19-bit K-2 datalogger with a
triaxial accelerometer deployed at the Athenaeum. Located in the basement,
CAC occupies the same site as the old analog Athenaeum/Caltech
station that has recorded earthquakes since the 1960s. Data are continuously
telemetered to the SCSN/SCEDC, but because the information
is only 19- bit, only events that trigger on the network are permanently stored
in the SCEDC.
Continuous and triggered data from these (and all other) SCSN stations may be obtained from the SCEDC at www.data.scec.org. Triggered data from some of the events recorded by the USGS-Caltech array on Millikan Library is available through the National Strong Motion Program at nsmp.wr.usgs.gov.
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Analysis of the Continuous Data Streams |
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TopAbstractIntroductionHistorical Evidence for Natural...The Current System of...Analysis of the Continuous...M 5.4 22 February...A Linear Transfer Function...Conclusions and CommentAcknowledgmentsReferences |
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The JPL weather station logs data every second; the channels used for comparison are rainfall (cumulatively measured per day, millimeters), wind gusts (meters per second), and temperature (degrees Celsius). Subsequent plots present only total rainfall, maximum wind gusts, and the maximum and minimum temperatures.
Entire Station Duration, MIK and CBC
Figure 6 is a spectrogram
plot for the entire history of the station MIK, alongside JPL weather
data. The three individual spectrogram subplots are centered about each of the
east–west, north–south, and torsional fundamental frequencies.
Figure 7 is a similar
spectrogram for the history of station CBC. Because the natural frequencies at
CBC are not well determined, individual spectrogram subplots are presented for
each of the east–west and north–south channels, over a wide range of
frequencies that encompass the observed spectral peaks.
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Each spectrogram is made by dividing the acceleration time series into lengths of time (a slice) and taking the fast Fourier transform (FFT) of this time window. The magnitude of the FFT is then represented by a color contour along the y axis at the time on the x axis to which the FFT corresponds (the midpoint slice time). Plotting this for each slice leads to the composite spectrogram. In Figures 6 and 7, the FFT length is 1 hr long, and there is no time overlap between slices. Each FFT has also been first smoothed over a frequency of 0.002 Hz and then decimated to a sampling frequency of about 0.001 Hz. In these figures, the acceleration amplitude scaling is linear for both MIK and CBC, with upper and lower bounds arbitrarily set to prevent unusual highs, such as the 22 February 2003 Big Bear earthquake, from swamping the color bar.
Because the three fundamental frequencies of Millikan Library are well separated and of large magnitude, the hourly peak in the FFT can be traced over time, as seen in Figure 8. Here the average of all the peaks is determined, and the deviation from this average is plotted. The daily average of the FFT peak is plotted as a thick line, with the hourly FFT peak plotted as a thin line. The timing of small earthquake excitations and forced vibration testing of the structure are highlighted by vertical bars. These are the source of the obvious large deviations from the mean.
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Figure 8 shows considerable variation in natural frequencies over 2 years. In particular, note the sensitivity of the fundamental east–west and torsional modes to rainfall, as evidenced by the large shifts during the winter months, when storms with several days of rainfall are a regular occurrence in southern California. These rain events are infrequent during the summer months. The north–south mode is not as sensitive to the rainfall and, in general, has smaller short-term deviations than the east–west and torsional modes. There is also a steady and unusual rise in the three fundamental frequencies during the spring of 2003, from April to July 2003. This occurs at the same time as a change in usage of three midlevel floors of the library (third, fourth, and fifth), from housing library volumes to providing office space. The books were removed during the summer of 2002, with little apparent change in the natural frequencies. However, the construction of partition walls for the new offices in the spring and summer of 2004 coincides with the gradual rise of about 4% in east–west natural frequency. The rise is less pronounced in the north–south and torsional modes. The natural frequency of a building is proportional to the square root of its stiffness, so an increase of 4% in natural frequency is equivalent to an increase of 8% in modal stiffness. It is surprising that such a large increase in stiffness can be attributed alone to the installation of partition walls in three floors, which only rise up to the false ceiling.
Discounting this recent lengthening trend, the following maximum variation for the daily average over the past 2 years are observed:
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The two lowest north–south and east–west frequencies, near 2.43
Hz and 2.67 Hz, respectively, are not near any machine noise and do seem to
exhibit temporal changes similar to those seen in Millikan Library.
Clinton (2004) presents
additional spectrogram plots that focus on these lower frequencies for all the
horizontal channels. From
Figure 7, these frequencies
exhibit large variations in signal strength over the period, with significant
variation in the natural frequencies themselves. For the first 6 months, there
is periodic variation, with a period of the order of 2 weeks. After this, the
predominant period of variation is much longer, on the order of months. The
lowest east–west frequency seems to vary from 2.55 Hz in mid-October to
2.72 Hz in early February, a variation of 6.5%. The lowest north-south
frequency is of smaller amplitude than the lowest east–west mode and thus
is harder to observe above the ambient noise and is only clearly visible on the
northwest roof channel, BL5. It varies from 2.40 Hz in mid-October and
2.52 Hz in early May, a variation of 4.9%. This north–south mode
appears to disappear in late August and in mid-October drops to 2.4 Hz for
several weeks, coinciding with a large drop in the east–west lowest mode.
Rainstorms, such as in early February and early November, clearly raise these
resonant frequencies in a manner similar to what is observed in Millikan
Library. The spectrograms also have a strong daily and weekly cycle, as much of
the machinery is turned off during the nights and weekends.
Winter Storms—MIK
The temporal variations at MIK
(Fig. 8) can be explained in
part by correlation with the rainfall data from the nearby JPL Weather Station.
This is most clear from the winter of 2002/2003. During this season, heavy winds
and rains were recorded. Each rainfall event prompts an immediate rise in the
natural frequencies, which is largest in the east–west and torsional
modes. After a storm, in the absence of other unusual excitation (such as
windstorm, forced vibration test, or earthquake), a gradual recovery (on the
order of about a week) of the frequencies to near the prerainfall levels is
observed.
Figure 9 is a close-up of the period, including the most severe storm of the period investigated, when over 100 mm of rain fell during 2 days in early February. This event caused a rapid and immediate rise of about 3% for the east– west and Torsional natural frequencies, with a slow decay toward prerainfall levels over a 10-day period.
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Figure 10 shows a
spectrogram of the east–west fundamental and first overtone frequencies
over the same period, February 2003. An obvious rise in frequency of 3% is
observed in the second mode, with a similar return toward prerain frequencies as
shown by the fundamental mode.
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Santa Ana Winds—MIK
Strong winds can also influence the building, though they are usually
accompanied by rainfall, which dominates the library’s response. Between
October and February, dry easterly Santa Ana winds can affect the Pasadena area.
In Figure 11a, an example of
such an intense Santa Ana windstorm (with no rainfall) is presented. The Library
shows a sudden, significant drop in all the fundamental frequencies, in
particular, the east–west mode, which drops by about 3%. Amplitudes of the
fundamental modes increase by about an order of magnitude during the windstorm,
and the torsional mode increases about half as much. Immediately after the
event, the stiffness returns to near pre-event levels. This observation is
consistent with the drops in natural frequency associated with increasing the
weight in forced vibration tests, which also increases the amplitude of
excitation.
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Diurnal Variation and High Temperatures—MIK
Figure 11b shows a typical
example of the building response during hot weather, when there is significant
daily variation of at least 1% for all three fundamental frequencies. This is
likely due to both the changing weather conditions and the daily building usage
cycle. The air conditioning in the building is turned off between the hours of
12 a.m. and 4 a.m. when the library is closed, so the elevators are
also not in use. During the evening, the natural frequencies drop, and, during
the day, they increase again. On particularly hot days (such as 1 and 2
September, 2002, in Fig. 11b),
where temperatures reach 40°C), there are higher frequency peaks. On cool,
overcast and rainy days (such as in
Fig. 9), this diurnal variation
is not as extreme. It is also clear that the torsional mode is more sensitive to
temperature than the translational modes; this mode shows large increases in
frequency above about 30°C, whereas the other fundamental frequencies only
show frequency increases at temperatures nearing 40°C. Maximum daily
variations can be as large as 3%.
These frequency increases during the day seem to be associated with thermal expansion of the concrete, but as the amplitudes of the ambient motion also increase by an order of magnitude during the day, this is inconsistent with the general observation that natural frequencies drop as amplitude increases. For these amplitude levels, the thermal effects are larger than the excitation amplitude effects. Figure 11b includes hourly amplitudes of the FFT peaks, which show this increased noise level in the building associated with the working-hours usage of the building. In the evenings and weekends the building has less noise as the air conditioning and other machinery, such as the elevator, are not in constant operation.
There does not appear to be any long-term trend associated with seasonal trends in temperature, as evidenced from Figure 6.
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M 5.4 22 February 2003 Big Bear Sequence |
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TopAbstractIntroductionHistorical Evidence for Natural...The Current System of...Analysis of the Continuous...M 5.4 22 February...A Linear Transfer Function...Conclusions and CommentAcknowledgmentsReferences |
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Millikan Library
USGS Triggered Data.
Figure 12 shows the time
series for all the east–west channels in the Millikan Library triggered
array. The FFTs of these records centered about the east– west
modes are also shown; the east–west fundamental frequency is at 1.06 Hz
and the east–west first overtone is at 4.55 Hz. These values are
approximately 7% lower than the corresponding natural frequencies determined
during forced vibrations (see
Table 2). Further, the
fundamental frequency is 11% below the ambient natural frequency of 1.19 Hz
(Fig. 13a) just before the
earthquake. and the second mode is 6.2% below the pre-earthquake ambient
frequency of 4.85 Hz
(Fig. 10).
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Note that the amplitude of the peak at 4.55 Hz, which is identified as the second mode, is small at the seventh, eighth, and ninth floors; the time series between 8 and 12 sec is rich in this frequency in the other floors. The mode shape for this frequency (see Fig. 12) shows these floors are near a nodal point, and this explains why it is difficult to identify the second east–west mode using the nineth floor MIK data alone. This is seen in the small magnitude broad second mode peak in Figure 10.
Little excitation of other higher order modes was observed in the earthquake data, including the fundamental torsional mode, so no examination of their wander is attempted. Also note that no significant energy is present at the resonant frequencies in the reference site GSA record.
MIK Continuous Data.
The continuously recording strong- motion channels from the ninth floor of
Millikan provide an excellent opportunity to look at the wander in the
Library’s natural frequencies before, during, and after the mainshock, and
to observe how the structure regains its stiffness after minor shaking.
Figure 13a shows a spectrogram for the three fundamental frequencies and the second east–west mode frequency for a 60-min period around the mainshock. FFTs are taken over a 30-sec period with 15-sec overlap. Although the frequency resolution is poor because of the short FFT length, during the earthquake the previously relatively stable frequencies all decrease by about 10%. Recovery from the earthquake to pre-earthquake levels appears to be almost instantaneous once the shaking has diminished. Aftershocks also shorten the frequencies, but by a lesser amount, commensurate with the smaller amplitudes of motion—for example, in Figure 13b, a M 4.1 aftershock occurs at 170 min, with a measurable decrease in frequency.
Figure 13b presents a spectrogram similar to Figure 13a for a 3-hr period around the same event, with FFTs taken over a 5-min period with a 4-min overlap. This provides improved resolution in the frequency domain, although resolution in the time domain is significantly diminished. The increased length also provides evidence that, though the natural frequencies are shortened a small amount in the immediate aftermath of an event, after about an hour, there is no perceptible difference between the pre- and postevent natural frequencies.
For this level of excitation, the structure regains its stiffness within minutes. This is in contrast to evidence (Udwadia and Trifunac, 1974) that suggests Millikan Library takes weeks or months to return to near pre-earthquake levels after undergoing strong motion. However, this observation is based on a minor shaking, whereas their observations were from San Fernando earthquake motions, which caused rooftop accelerations over 10 times greater than the Big Bear event. Thus, extrapolation of the Big Bear result to the largest shaking that the building has been subjected to may not be valid. Note though that the work by Udwadia and Trifunac was based on two tests performed a few weeks and 22 months after the San Fernando earthquake. In these tests the observed decrease in the east–west natural frequency from the pre-earthquake measurement was 18% and 12%, respectively. Subsequent data (Clinton, 2004) suggests the permanent change from the original natural frequency of 1.45 Hz was about 12%. Udwadia and Trifunac (1974) show FFT segments from the tail of the strong-motion record (about 80 sec after the initial triggering) that indicate the building is already returning to the pre-earthquake state. The differences in measurements may be explained by the many changing variables such as weather and test parameters and may not be due to long-term "healing" of the structure.
Broad Center
The continuous data from the Broad Center (CBC) provide another dataset for
analyzing the effect of a small earthquake on the natural frequencies of a
structure.
Figure 14 is a 3-hr spectrogram of the three horizontal east–west channels of CBC, with FFTs with a length of 5 min. These spectrograms have each FFT scaled to a maximum value of 1, so during the earthquake, the torsional mode is not visible as the translational modes are predominantly excited. For each channel, a wide frequency band of acceleration spectra is plotted; a narrow band of displacement spectra centered on the fundamental frequency is underneath. Displacement spectra accentuate the energy in this frequency relative to the higher frequencies, and because these two fundamental frequencies are not driven by electrical or mechanical noise, fluctuations in frequency can be more easily observed.
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During the earthquake, the natural frequencies all change considerably—it is not clear exactly which frequencies correspond to the values observed during the shaking, because all three frequencies are so close together. It is clear that immediately after the earthquake, the frequencies return to their pre-earthquake levels. Similar changes are seen in the north–south channels (Clinton, 2004).
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A Linear Transfer Function Solution? |
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TopAbstractIntroductionHistorical Evidence for Natural...The Current System of...Analysis of the Continuous...M 5.4 22 February...A Linear Transfer Function...Conclusions and CommentAcknowledgmentsReferences |
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This example attempts to model the known east–west displacements at MIK using a convolution of the east–west acceleration ground motion at GSA with an impulse response of a single degree of freedom (SDOF) model representing the east–west fundamental frequency of Millikan Library. Amplitude amplification is determined by using the participation factor of the first mode, assuming a mass matrix of equal floor mass, and modes shapes as determined from the forced vibration tests. This convolution gives the relative displacement of MIK to GSA, so the ground displacement at GSA is added. A constant damping ratio of 1.63% is employed, which is the value determined from the forced vibration tests (Bradford et al., 2004).
Figure 15 presents examples of the results for two recent earthquakes. Figure 15a presents data from the 18 June 2003 M 2.0 Pasadena earthquake (a foreshock of a M 2.6 event 20 min later), 5 km from Caltech, which had a maximum acceleration of 1.2 cm/sec2 at MIK. Figure 15b presents data from the 22 February 2003 M 5.4 Big Bear earthquake, 119 km from Caltech, which had a maximum acceleration of 22.6 cm/sec2 at MIK. Figure 15a shows that the model is improved if the ambient SDOF, derived from ambient data immediately before the event, is used instead of the default forced vibration value. For this event, a default SDOF representing the second east–west mode is also included to help model the high-frequency response. In Figure 15b, where the behavior of Millikan during the stronger motion is modeled, both the ambient and forced SDOF give a poor fit. In this case, the natural frequency has moved far from the ambient levels (as seen in Fig. 12) and is best modeled using a natural frequency as determined from the strong-motion records (1.06 Hz). Though the character of the strongest motions is well modeled by this last SDOF, the low-amplitude P-wave and coda motions are not well modeled, suggesting the natural frequency seems to be evolving rapidly, as the excitation amplitude changes. These motions are small, and are certainly within the expected the linear response of the building, yet a single natural frequency cannot predict the motions of the building.
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Current engineering practice has a single natural frequency assigned to the building, usually not even derived from experimental testing, but from formulae in the relevant design code. This natural frequency is assumed constant for a wide range of ground motions. The results presented in this article, and elsewhere in the literature (Trifunac, 1999), suggest that natural frequencies are not constant, even for weak motions.
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Conclusions and Comment |
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In the case of Millikan Library, there is significant wander in the translational and torsional natural frequencies of the structure. It is evident that, in addition to this wander, the method used to determine the natural frequency at a point in time also introduces some variability, because the amplitude of excitation of the building is different for each method. Ambient motion, forced vibrations, and earthquake data will have very different excitation amplitudes.
To the first order though, there has been a significant permanent reduction in natural frequencies for the Millikan Library in the 36 years of its life, 22% east–west and 12% north–south. The natural frequencies of a structure are proportional to the square root of the mass and inversely proportional to the square root of the stiffness. The system mass has not changed significantly during the life span, so this change in natural frequency corresponds to a major decrease in system stiffness.
The large permanent frequency drops appear to be caused almost entirely by strong motions from moderate nearby LA Basin earthquakes. In particular, the 1971 San Fernando M 6.6 earthquake had the largest effect, with permanent drops of 17% east–west and 7% north–south. For the east–west direction, the accelerations of this earthquake were not subsequently exceeded, and there were no more large permanent frequency drops. However, in the north- south direction, the San Fernando roof-top acceleration of 341 cm/sec2 was exceeded with a recording of 534 cm/sec2 during the 1987 M 6.1 Whittier Narrows event. This caused a further permanent drop of 4%.
During the strong shaking, these moderate local events are shown to almost instantaneously shorten the frequencies by more than 20% without apparent structural damage. Because only triggered data are available for large motions, there is no clear evidence of the time frame for stiffness recovery. Continuous data from small earthquake motions, such as from the 2003 M 5.4 Big Bear earthquake, reduce the ambient natural frequencies by up to 10% during the motion, with a full recovery within minutes. Similar observations are documented for the fundamental translational modes of the Broad Center.
Smaller transient excitations also reduce the natural frequencies, such as forced shaking tests (by up to 7% with full shaker weights) and small earthquakes. Heavy rain increases the east–west fundamental and first overtone and the torsional fundamental frequencies by up to 3% in a matter of hours, with little effect to the north–south fundamental frequency. This increase in frequencies occurs even though the amplitude of building motion is increased during storms, increases in displacement amplitudes on the roof typically lead to a decrease in the natural frequencies. When strong winds occur in the absence of rainfall, all the natural frequencies can drop by up to 3%. The diurnal variation is of the order of 1–2%. On a day with a high temperatures near 40°C, rising temperatures raise all the natural frequencies by a further 1–2%.
For transient changes due to extreme weather, the recovery time can vary significantly. Any changes arising from wind and temperature only seem to last as long as the excitation, but for rain events the Library and the Broad Center slowly return to the pre-event fundamental frequencies over about 1 week.
On a longer timescale, changing building usage can be responsible for large changes in the frequencies. Construction of partition walls to provide office space in three midlevel floors of the library over 4 months is shown to coincide with an increase in frequencies during this same period— the east–west fundamental frequency is particularly affected (4% increase over this period). Note that a 4% increase in natural frequency corresponds to an 8% increase in stiffness for the whole structure. The construction of the partition walls was preceded by the removal of the entire book collection from these three floors, which did not have an observable effect on the natural frequencies.
The sources of all these observed wanders are not well understood. The different observed effects may be due to soil–structure interaction, super-structure non-linearities, or a combination of both.
For the particular case of the Millikan Library, Chopra (1995) suggests that the changes in the dynamic behavior of the building during strong motion and weak forced shaking are associated with "cracking and other types of degradation of the so-called non-structural elements." This mechanism does not account for the observed changes due to weather events.
The incr
