structural health monitoring of a 54-story steel frame building
TRANSCRIPT
1
Structural Health Monitoring of a 54-story Steel Frame Building Using a Wave Method and Earthquake Records
Mohammadtaghi Rahmania) M.EERI and Maria I. Todorovskab) M.EERI
The variations of identified wave velocities of vertically propagating waves
through the structure are investigated for a 54-story steel-frame building in downtown
Los Angeles, California, over a period of 19 years since construction (1992-2010),
using records of six earthquakes. The set includes all significant earthquakes that
shook this building, which produced maximum transient drift ~0.3%, and caused no
reported damage. Wave velocity profiles ( )zβ are identified for the NS, EW and
torsional responses by fitting layered shear beam/torsional shaft models in the recorded
responses, by waveform inversion of pulses in impulse response functions. The results
suggest variations larger than the estimation error, with coefficient of variation about
2-4.4%. About 10% permanent reduction of the building stiffness is detected, caused
mainly by the Landers and Big Bear earthquake sequence of June 28, 1992, and the
Northridge earthquake of January 17, 1994. Permanent changes of comparable
magnitude were identified also in the first two apparent modal frequencies, 1,appf and
2,appf , which were identified from the peaks of the transfer-function amplitudes.
INTRODUCTION
Structural Health Monitoring (SHM) can be a powerful tool to facilitate decision making
on evacuation of an unsafe structure after a strong earthquake (or some other natural or man-
made disaster), to avoid loss of life and injuries from a potential collapse of the weakened
structure from shaking from aftershocks (Todorovska and Trifunac, 2008c). Likewise, it can
confirm a structure to be safe for its occupants, and even serve as a shelter in the aftermath of
a devastating earthquake, when commute is disrupted and overcrowded streets obstruct
emergency response (Hisada et al. 2012). To be effective, SHM methods must work with
a) Ph.D. Candidate, University of Southern California, Dept. of Civil Eng., Los Angeles, CA 90089-2531, Email: [email protected] b) Research Professor, University of Southern California, Dept. of Civil Eng., Los Angeles, CA 90089-2531, Email:[email protected]
Earthquake Spectra, 2013, DOI: 10.1193/112912EQS339M, final draft; submitted for publication on 11/29/2012; accepted for publication on 7/22/2013.
2
real buildings and larger amplitude response, and be reliable, sensitive to damage and
accurate. They should neither miss significant damage nor cause false alarms and needless
evacuation. Ideally, they should also be able to detect localized damage, which is
challenging, and smaller changes due to structural degradation with time, which are difficult
to separate from identified changes due to other factors, such as identification error and
changes in the operating and environmental conditions (Doebling et al., 1996; Chang et al.,
2003; Clinton et al., 2006; Boroschek et al., 2008; Todorovska and Trifunac, 2008b; Herak
and Herak, 2010; Mikael et al., 2013). While the rare records in damaged full-scale buildings
remain invaluable for relating changes in the damage sensitive parameters to levels of
damage of concern for safety (e.g., Todorovska and Trifunac, 2008a,b), the much more
frequent records of smaller and distant earthquakes are also very valuable. (1) Analyses of
multiple earthquake records in full-scale buildings, over longer periods of time, can provide
knowledge about the variability of the damage sensitive parameters due to factors other than
damage and permanent changes due to structural degradation (for a particular structure or
type of structures) in the most realistic conditions. Knowledge of this variability is useful for
making inferences about the state of damage from detected changes. (2) Such analyses also
provide opportunities to test the capabilities of SHM methods being developed. This paper
presents such an analysis for a 54-story steel-frame building in downtown Los Angeles (Fig.
1) and a wave method for SHM. The data consists of records of six earthquakes, over a
Fig. 1 Los Angeles 54-story office building (CSMIP 24629):photo, vertical cross-section, and typical floor layouts (redrawn from www.strongmotioncenter.org)
3
period of 19 years since construction, none of which caused reported damage (Figs 1 and 2,
Table 1). The analysis aims to assess the general variability of the identified wave velocities
in this building, and detect possible permanent changes in the structure by the wave method.
A recently proposed waveform inversion algorithm for the identification of the wave
velocities is applied, which is much more accurate than the ones used previously. The
detected changes are compared with those of the first two apparent frequencies of vibration,
which are also identified. This is the first such analysis with the waveform inversion
algorithm, which examines its capability to detect permanent changes from the scatter. Also,
to the knowledge of the authors, this is the first analysis of the variability of damage sensitive
parameters for this building using any method.
.
Fig. 2 Google map of the epicenters of the earthquakes recorded in the building.
Table 1 List of earthquakes recorded in the building (CSMIP Station 24629; 34.048 N, 118.26 W)
Event name Code Date/ Time Epicenter H
[km] LM Epic/Fault distance
[km]
Rec. length
[s]
Gnd maxa [g]
Struc. maxa [g]
Landers LA 06/28/1992 04:57:31 PDT
34.22N 116.43W 1 7.3 170/158 87 0.040 0.130
Big Bear BB 06/28/1992 08:05:31 PDT
34.20N 116.83W 10 6.5 133 87 0.030 0.067
Northridge NO 01/17/1994 04:30:00 PST
34.21N 118.54W 19 6.4 32/28 180 0.140 0.190
Hector Mine HM 10/16/1999
02:46:45 PDT 34.60N
116.27W 6 7.1 193 106 0.019 0.082
Chino Hills CH 07/29/2008 11:42:15 PDT
33.95N 117.77W 14 5.4 47 83 0.063 0.086
Whittier Narrows* WN 03/16/2010
04:04:00 PDT 34.00N
118.07W 18 4.4 18 - 0.020 0.022
Calexico CA 04/04/2010 15:40:39 PDT
32.26N 115.29W 32 7.2 355/286 87 0.009 0.038
* Data not available; H=focal depth
4
The wave SHM method is based on detecting changes in the velocities of waves
propagating vertically through the structure, which are directly related to the structural
stiffness (Şafak, 1998; 1999; Oyunchimeg and Kawakami, 2003; Todorovska and Trifunac,
2008ab). This study is part of our systematic and in-depth investigation of the wave method,
addressing for the first time important issues such as the accuracy and the spatial resolution
of the identification, and the effects of foundation rocking, wave dispersion and wave
scattering on the estimation (Todorovska, 2009a; Todorovska and Rahmani, 2012; 2013;
Rahmani and Todorovska, 2013). In this study, the building velocity profiles are identified
using the identification algorithm proposed by Rahmani and Todorovska (2013), which
involves fitting a layered shear beam/torsional shaft in recorded earthquake response, in a
carefully chosen low-pass frequency band, by waveform inversion of pulses in impulse
response functions. That is accomplished by nonlinear least squares fit of the amplitudes of
the transmitted pulses, as functions of time, over time intervals approximately equal to the
width of the pulses). The first application of the waveform inversion algorithm, to Millikan
library (9-story RC structure), was concerned with the accuracy of the identification, and
demonstrated that it is much more accurate than the previously used picking of the time of
arrival of the pulses and computing the velocities from the pulse time shifts and the distances
travelled (Rahmani and Todorovska, 2013). The same identification algorithm was later
applied to the 54-story steel building analyzed in this paper, in a study aiming to demonstrate
the validity of this algorithm (which ignores wave dispersion due to bending deformation) for
very tall steel-frame buildings (Todorovska and Rahmani, 2012). That study showed that,
contrary to the common belief, the wave propagation in very tall steel-frame buildings is little
dispersed in the lower frequency range, and that a layered shear beam is an appropriate model
in a band that contains as many as 5-6 of its modes of vibration. The study in this paper
presents the first attempt to detect by this algorithm, and by the wave SHM method, in
general, small changes due to stiffness degradation in a steel frame building. This study also
provides an insight into and a measure for the variability of the vertical wave velocities in
steel-frame buildings, from one earthquake to another, none of which has caused observed
damage. For comparison, the first two apparent frequencies are also analyzed.
The most remarkable feature of this wave SHM method is its insensitivity to the effects of
soil-structure interaction, even in the more general case when foundation rocking is present
and coupling of the horizontal and rocking responses. Snieder and Şafak (2006) showed, on
an analytical model that does not allow for foundation rocking, that both the transfer-function
5
between roof and base horizontal responses and the corresponding impulse response
functions are not affected by soil-structure interaction, and that the building fixed-base
frequencies and damping can be estimated. However, that is not true for the more realistic
case, when rocking is present, as it is well known from soil-structure interaction studies (e.g.,
Luco et al., 1988). Nevertheless, as demonstrated by Todorovska (2009a) on simulated
response by a soil-structure interaction model with rocking, the pulse time shifts in impulse
response functions, and estimated from them vertical wave velocities, are not affected by
soil-structure interaction. This is supported by analyses of data in structures known to have
been or not to have been damaged (Todorovska and Trifunac, 2008b; Michel et al., 2011).
The pulse amplitudes, however, are affected, and, therefore, the structural damping cannot be
estimated from transfer-functions or impulse response functions of horizontal motions.
Because of this fact, in this paper, we do not attempt to estimate the structural damping and
use it for SHM. We do estimate the apparent quality factor, but only as a byproduct of the
analysis. The insensitivity to the effects of soil-structure interaction is a major advantage of
this SHM method over the methods based on detecting changes in the observed (apparent)
fundamental frequency of vibration, because it eliminates changes in the soil-foundation
system as a possible cause for observed changes (Trifunac et al, 2001ab). An application of
this method to Millikan library (Todorovska, 2009b) helped explain to what degree the
observed wondering of its fundamental frequencies (Clinton et al., 2006) has been due to
changes in the structure as opposed to changes in the soil.
The wave SHM method is based on the view of the building seismic response as wave
propagation, the structure being characterized by its wave velocities, rather than by its
frequencies of vibration as in the traditional vibrational approach (Kanai and Yoshizawa,
1963; Kanai, 1965; Todorovska and Trifunac, 1989; Todorovska and Lee, 1989; Şafak, 1998;
Todorovska et al., 2001; Kawakami and Oyunchimeg, 2004; Snieder and Şafak, 2006; Gičev
and Trifunac, 2007, 2009ab, 2012; Kohler et al., 2007; Trifunac et al., 2010). The local
nature of the wave approach and its advantages to detect localized damage were
demonstrated by Şafak (1999) on an analytical model, assuming that the wave velocities can
be estimated exactly. Wave velocities in buildings have been inferred from time lag of
motion measured by cross-correlation (Ivanović et al., 2001), normalized input-output
minimization (Oyunchimeg and Kawakami, 2003) and pulses in impulse response functions
(Todorovska and Trifunac, 2008ab). For tall buildings, the time lag may also be able to
measure directly from recorded accelerations by following a characteristic peak in the time
6
histories (Şafak, 1999). Measuring time lag from pulses in impulse response functions is
superior to correlation because the characteristics of the excitation, which may mask the
system function, are removed (Snieder and Şafak, 2006). The identification method used in
this study fits a model in observed response by matching, in the least squares sense, impulse
responses for virtual source at roof. Our recent developments of this method, and how they
relate to this study, were described earlier in this section. All of the aforementioned studies
used earthquake response data, in which case the physical source of the excitation is at the
base. It has been demonstrated, for the Factor building, that similar impulse response
functions can be obtained from ambient noise recordings, over 14 or more days of continuous
recording (Preito et al., 2010). This presents an interesting opportunity to estimate the wave
velocity without having to wait for an earthquake. However, in view of the high accuracy
required for SHM, the practical usefulness of the wave method on ambient data, and its
advantages over the modal methods have yet to be demonstrated (Michel and Gueguen, 2010;
Mikael et al., 2013).
Comprehensive reviews of SHM methods, majority of which are vibrational, can be found
in review articles published periodically, e.g. Doebling et al. (1996) and Chang et al. (2003).
Many methods found in SHM literature, other than those that estimate the frequencies of
vibration, turn out not to be robust when applied to actual large amplitude data, and are tested
only on numerically simulated response or on simple lab models. Another category of
methods, found in earthquake engineering literature, which are robust, are the performance
based methods (Ghobarah et al., 1999; Naeim et al. 2006). These methods estimate if some
response characteristic (e.g. the interstory drift) exceeded certain level, rather than if some
structural parameter changed. These methods are also sensitive to the effects of soil-
structure interaction, because they use the total recorded response, which includes foundation
rocking, or the response of fixed-base models calibrated to match the soil-structure system
frequencies. The performance based methods cannot be used to monitor structural
degradation as the methods based on structural parameter identification.
Observed fundamental frequency of vibration of steel buildings excited by multiple
earthquakes have been reported, e.g. by Çelebi et al. (1993), Li and Mau (1997), Rodgers and
Çelebi (2006), and Liu and Tsai (2010). To the knowledge of the authors, only the
Northridge, 1994 earthquake data in this building has been analyzed (e.g. Naeim, 1997;
Todorovska and Rahmani, 2012).
7
Fig. 3 The model.
This paper is organized as follows. The methodology section summarizes briefly the
method, which follows closely Rahmani and Todorovska (2013). In the results section, the
building and data are presented, and the results of the identification for the six earthquakes
and the sample statistics are summarized, followed by exploratory analysis of trends as
function of interstory drift. Finally, the conclusions drawn are presented.
METHODOLOGY
The building is modeled as an elastic, layered shear beam, supported by a half-space, and
excited by vertically incident plane shear waves (Fig. 3). The layers may correspond to
individual floors, or to group of floors. In this paper, the layer boundaries are along the
instrumented floors. Within each layer, the medium is assumed to be homogeneous and
isotropic, and that perfect bond exists between the layers. The building is assumed to move
only horizontally. The layers, numbered from top to bottom, are characterized by thickness
,ih mass density iρ , and shear modulus iμ , , ,i n= 1… , where n is number of layers, which
implies shear wave velocities /i i iβ μ ρ= . The displacements at the roof and at the
consecutive layer interfaces are u1 , u2 , … nu +1. Amplitude attenuation due to material
friction is introduced via the quality factor, Q , and the damping ratio is / ( )Qζ = 1 2 .
A band-limited impulse response function (IRF) at some level for virtual source at roof,
max( , , ; )h z tω0 , is obtained by inverse Fourier transform of the corresponding transfer-function
(TF), ˆ( , ; )h z ω0
8
max
max
maxˆ( , , ; ) ( , ; ) i th z t h z e d
ωω
ωω ω ω
π−
−
10 = 0∫
2 (1)
where maxω is the cut-off frequency. Regularized TFs are practical to use
ˆ ˆ( , ) ( , )ˆ( , ; )ˆ( , )
u z uh zu
ω ωωω ε2
00 =
0 + (2)
where ε is regularization parameter (Snieder and Şafak, 2006) and the bar indicates complex
conjugate. We used ε = 0.1% of the mean square value of the acceleration at the top. Such
small values, used consistently, do not affect the SHM analysis, which is concerned to detect
changes in the identified velocities rather than their exact value.
For the model, both analytical TFs and band-limited IRFs are available derived from the
propagator of the medium (Gilbert and Backus, 1966; Trampert et al. 1993; Todorovska and
Rahmani, 2013). The waveform inversion algorithm is used for the fit, which matches, in the
least squares sense, the IRFs over selected time windows simultaneously at all observation
points (Rahmani and Todorovska, 2013). The width of each time windows is such that it
encloses the corresponding transmitted pulse, which is approximately max1/ f = width of the
source pulse. Both causal and acausal pulses are fitted. For the least squares fit, in this
study, we used the Levenberg-Marquardt option. The Levenberg-Marquardt method for
nonlinear least squares estimation is a fixed regressor, small residual algorithm, which
requires initial values that are close to the true values to insure convergence (Levenberg,
1944; Marquardt, 1963). For that purpose, we used the estimates obtained by the direct
algorithm (Todorovska and Rahmani, 2013). For the data in this study, which did not involve
damage, there was no need to use the more robust but slower simulated annealing option.
The key parameter in the estimation is the choice of cut-off frequency, maxf , which
controls the spatial resolution and the effects of dispersion. A higher value of maxf enables
higher resolution, but too high value leads to distortion of the pulses caused by dispersion.
The optimal value chosen carefully for this study was found to be maxf = 1.7 Hz for the NS
and EW responses, and maxf = 3.5 Hz for the torsional response, which encloses the first 5-6
modes of vibration. Up to this frequency, the building behaves close to a shear
beam/torsional shaft, as shown in Todorovska and Rahmani (2012).
9
RESULTS AND ANALYSIS
Building Description and Strong Motion Data
Los Angeles 54-story office building (Fig. 1) is a steel-frame building in downtown Los
Angeles, California, instrumented by the California Strong Motion Instrumentation Program
(CSMIP) of the California Geological Survey (station No. 24629). As reported by the agency
(www.strongmotioncenter.org), the building has 54 stories (210.2 m) above and 4 stories (14
m) below ground level. It has rectangular base with two rounded sides, 59.7 m × 36.9 m up to
the 36th floor, decreasing in the EW direction to 47.5 m × 36.9 m. Fig. 1 shows photo of the
building, its vertical cross section (EW elevation), and plans of the instrumented levels. The
building was designed in 1988 by the 1985 Los Angeles City Code and Title 24 of the
California Administrative Code, and completed in 1991. The lateral force resisting system is
moment resisting perimeter steel frame (framed tube) with 3 m column spacing. It has
Virendeel trusses and 1.22 m deep transfer girders at the 36th and 46th floors where vertical
setbacks occur. The vertical load carrying system consists of 2.5 inch (6.35 cm) concrete
slabs on 3 inch (7.6 cm) steel decks with welded metal studs, supported by steel frames. The
building is supported by a concrete mat foundation, 2.1 m and 2.9 m thick, and 15 cm
concrete slab on grade. The site geology is alluvium over sedimentary rocks.
The building was instrumented in 1991 with a 20-channel digital accelerometer array
distributed on 6 levels: basement (P4), ground, 20th, 36th, 46th, and Penthouse (54th floor)
(Fig. 1). The instruments are 12-bit resolution SSA-1 recorders with FBA-11 accelerometers.
Fig. 2 shows a map of the building site and the epicenters of the seven earthquakes, reported
to have been recorded in this building, over a period of 19 years (1992 to 2010). Six of
them, for which data are available, are analyzed. No damage has been reported from any of
these earthquakes. Table 1 shows the earthquake name, a two-letter code assigned in this
study, date and time, epicentral coordinates and depth, magnitude, record length, and peak
ground and structural accelerations. Three of these earthquakes were distant but large
(Landers, 1992, Hector Mine, 1999, and Calexico, 2010), one was moderate but near
(Northridge, 1994), one was moderate and distant (Big Bear, 1992), and one was small but
relatively close (Chino Hills, 2008). The processed data were made available equally spaced
at 0.01 s. For the computation of impulse response functions, we interpolated the data to
0.005 s. The Northridge data have been band-pass filtered by Ormsby filter with ramps at
0.06 - 0.12 Hz and 46 - 50 Hz (Lee and Trifunac, 1990). The other data have been band-pass
10
filtered with Butterworth filter, with 3 dB pts at 0.08 Hz and 40 Hz for Landers, and with 3
dB pts at 0.1 Hz and 40 Hz for the other events.
Figs 4 and 5 illustrate the variety of the base excitations and building responses they
produced. Fig. 4 shows pairs of P-4 level (basement) acceleration and penthouse
displacement, and Fig. 5 shows the transient drift, computed from the difference in
Fig. 4 Penthouse displacements and base accelerations observed during the six earthquakes.
Fig. 5 Average drifts observed during the six earthquakes.
11
displacements between at penthouse and P-4 level. Fig. 4 shows that the building response is
poorly correlated with the ground acceleration, and is sensitive to the frequency content of
the excitation. While the Northridge earthquake produced the largest base acceleration, the
more distant Landers and Hector Mine earthquakes produced the largest response (roof
displacement ~55 cm and 50 cm, and average drift of ~0.2%, for EW motions; see Fig. 5).
The Chino Hills earthquake produced the second largest base acceleration, but very small
response.
Identified Parameters and Sample Statistics
Figs 6 and 7 show the observed TF amplitudes and IRFs for the six events, for the NS,
EW and torsional responses. NS or NS average response indicate the average of the NS
responses at the East and West sides of the building. The torsion was computed from the
difference of these motions. The TFs were computed from the ratio of the complex Fourier
transforms of the motions at penthouse and P4 levels. The IRFs were computed for virtual
source at penthouse level. It can be seen that the TFs are very similar, except that, for the
Chino Hills, 2008 earthquake, the peaks corresponding to the fundamental modes are small
or lost. While the high-pass filter might have affected the amplitudes of the first peaks for all
earthquakes, the very small peak amplitude for the Chino Hills earthquake is likely due to the
small signal to noise ratio at low frequencies for this earthquake, which did not excite much
the fundamental mode. The impulse response functions are also very close.
Table 2 summarizes the identified global parameters: wave travel time τ over the height
of the building (ground floor to penthouse for the NS and EW, and P4 level to penthouse for
the torsional response), and the wave velocity eqβ , quality factor Q , and fixed-base
frequency / ( )τ1 4 of the fitted equivalent uniform model. While eqβ was identified by the
waveform inversion algorithm, Q was identified from the pulse amplitudes by the direct
algorithm, and represents the apparent damping, which depends on the structural damping
and rocking radiation damping (Todorovska, 2009a; Todorovska and Rahmani, 2013). The
corresponding apparent damping ratio is / ( )Qζ = 1 2 . The fixed-base frequency of the fitted
uniform model, / ( )τ1 4 , in general differs from the actual fixed-base frequency, which
depends on the distribution of stiffness and mass along the height, but can be used as a proxy
of the actual fixed-base frequency to follow its changes (Trifunac and Todorovska, 2008a,b).
12
Fig. 6 Transfer functions of observed NS, EW and torsional responses during six earthquakes..
Fig. 7 Impulse response functions of observed NS and EW responses during six earthquakes.
13
Table 2 also shows the apparent frequencies for the first two modes, ,appf1 and ,appf2 ,
identified from the transfer functions, and wγ = weighted peak transient drift. The apparent
frequencies were estimated manually, based on visual analysis of the shape of the
corresponding peak in the TF. While more elaborate automatic algorithms could have been
used (e.g., as in Carreño and Boroschek, 2011), we believe that the conclusions of this paper
would not have changed. The weighted peak drift, wγ , was computed as the weighted
average of the peak layer drifts, with weights proportional to the layer heights. We use wγ
instead of the peak drift between roof and base, because the drift varied differently along the
height for different earthquakes, and the latter represented poorly the overall deformations of
the building for some of the events. It can be seen that, for the NS response, the largest wγ
occurred during Calexico, 2010 and Landers, 1992 earthquakes (0.1008 cm/m and 0.0987
cm/m), while, for the EW and torsional responses, it occurred during the Landers, 1992
earthquake (0.265 cm/m and 0.00627 mrad/m).
Table 3 shows the identified local parameters, i.e. layer velocities iβ , , ,i = 1 4…
estimated by the waveform inversion algorithm and the corresponding normalized standard
deviation /βσ β , and the peak layer drifts iγ . The largest NS drift occurred in the top layer
during Northridge, and, in the bottom two layers - during Landers and Calexico earthquakes.
The largest EW drifts occurred during Landers and Hector Mine earthquakes in all layers.
The largest torsional drift occurred, in the bottom layer - during Hector Mine, while, in the
other three layers - during the Landers earthquake. The mass density was assumed to be
uniform throughout the building, with 3300 kg/mρ = , for all the models. The layer widths
are 1h = 27.9 m, 2h = 39.8 m, 3h = 63.6 m and 4h = 78.9 m (NS and EW) and 92.9 m (torsion).
Finally, Table 4 summarizes the sample statistics: sample mean μ, sample standard
deviation s and sample coefficient of variation /s μ . They suggest small variability of eqβ ,
,appf1 and ,appf2 during these six earthquakes, with /s μ not exceeding 3.6%, and also small
variability of iβ in the layers, not exceeding 4.4%. The range of the peak drifts are also
specified in the last column, where wγ γ≡ for the global parameters, and iγ γ≡ for the local
parameters.
14
Table 2 Identification results for equivalent uniform model during six earthquakes.
NS at west wall, 0-1.7 Hz; h=210.2 m
Event τ [s]
eqβ
[m/s] βσ
β%
1/ 4τ [Hz]
Q 12Q
ζ = 1,appf [Hz]
2,appf [Hz]
wγ [cm/m]
LA 1.4175 148.3 0.42 0.176 17.5 2.9 0.17 0.53 0.0987
BB 1.4600 144.0 0.46 0.171 20 2.5 0.17 0.52 0.0439
NO 1.5025 139.9 0.48 0.166 25 2 0.165 0.502 0.0964
HM 1.4925 140.8 0.49 0.168 16.7 3 0.16 0.498 0.0739
CH 1.5125 139.0 0.44 0.165 18.5 2.7 --* 0.50 0.0277
CA 1.4725 142.8 0.49 0.170 30.7 1.63 0.165 0.498 0.1008
EW at north wall, 0-1.7 Hz; h=210.2 m
Event τ [s]
eqβ
[m/s] βσ
β%
1/ 4τ [Hz]
Q 12Q
ζ = 1,appf [Hz]
2,appf [Hz]
wγ [cm/m]
LA 1.3875 151.5 0.5 0.180 12.7 3.9 0.2 0.56 0.2653
BB 1.4250 147.5 0.53 0.175 19.2 2.6 0.19 0.56 0.0371
NO 1.490 141.1 0.55 0.168 13.9 3.6 0.185 0.53 0.1188
HM 1.4825 141.8 0.65 0.169 17.2 2.9 0.18 0.528 0.2351
CH 1.4525 144.7 0.49 0.172 14.7 3.4 0.185 0.54 0.0148
CA 1.4350 146.5 0.53 0.174 16.1 3.1 0.19 0.53 0.0839
Torsion, 0-3.5 Hz; h=224.2 m
Event τ [s]
eqβ
[m/s] βσ
β%
1/ 4τ [Hz]
Q 12Q
ζ = 1,appf [Hz]
2,appf [Hz]
610wγ −×[rad/m]
LA 0.810 276.8 1.35 0.309 100 0.5 0.37 1 6.27
BB 0.833 269.1 1.20 0.300 20 2.5 0.37 0.98 1.18
NO 0.865 259.2 1.50 0.289 25 2 0.36 0.935 3.24
HM 0.864 259.5 1.22 0.289 - 0 0.35 0.935 6.03
CH 0.857 261.6 1.17 0.292 33 1.5 0.355 0.965 0.73
CA 0.852 263.1 1.21 0.293 - 0 0.35 0.93 2.17
*The first mode is not readable
15
Table 3 Identification results for equivalent 4-layer model during six earthquakes.
NS at west wall, 0-1.7 Hz EW at north wall, 0-1.7 Hz Torsion, 0-3.5 Hz
Event β
[m/s] βσ
β%
γ [cm/m]
β [m/s]
βσβ
% γ
[cm/m] β
[m/s] βσ
β%
610γ −×[rad/m]
Lay
er 1
LA 98.8 0.4 0.087 80.2 0.8 0.286 169.1 1.5 13.0
BB 95.5 0.7 0.085 79.4 0.7 0.072 165 1.3 3.56
NO 92.5 0.75 0.156 78.2 0.77 0.22 166.4 1.5 6.70
HM 96.4 0.7 0.11 78 0.8 0.2 162.7 1.2 11.67
CH 88.2 0.8 0.073 80.1 1.1 0.041 159.8 1.4 2.16
CA 94.2 0.7 0.075 77.9 0.5 0.084 163 1.2 5.66
Lay
er 2
LA 163.2 0.7 0.104 153.1 1.2 0.259 257.6 2.4 11.4
BB 158.3 1.1 0.063 143 1 0.037 262.2 2.0 1.71
NO 153 1.8 0.1 140.7 1.2 0.15 251.3 2.5 4.81
HM 154.5 1.1 0.07 148 1.3 0.23 254 2.0 10.16
CH 156.7 1.1 0.035 134.8 1.8 0.015 250 2.2 0.91
CA 158.5 1.1 0.094 146.4 0.8 0.081 252.7 1.9 4.38
Lay
er 3
LA 150.4 0.5 0.11 174.5 1.2 0.304 263.1 1.5 4.9
BB 148 0.8 0.03 169.3 1 0.037 258.4 1.2 0.77
NO 141.6 1.2 0.097 166.5 1 0.098 250 1.4 2.58
HM 144 0.8 0.079 166.2 1.1 0.26 250 1.1 4.62
CH 145.4 0.8 0.019 171.1 1.6 0.013 249.2 1.3 0.58
CA 142.5 0.7 0.11 167 0.7 0.093 249.6 1.1 0.97
Lay
er 4
LA 174.6 0.5 0.091 197.3 1.1 0.23 377.2 1.4 3.00
BB 170 0.8 0.031 192.7 0.8 0.025 368.3 1.1 0.51
NO 167.7 1.3 0.073 178.4 0.82 0.084 352.1 1.3 1.98
HM 166.6 0.7 0.059 181.8 0.9 0.23 369 1.1 3.52
CH 164 0.7 0.015 185.4 1.2 0.007 367 1.3 0.33
CA 172.3 0.8 0.106 191 0.6 0.078 376.7 1.1 1.00
16
Table 4 Sample statistics for the six earthquakes ( μ =sample mean, s =sample standard deviation, /s μ =sample coefficient of variation).
μ s /s μ (%)
[cm/m]γ
NS
at w
est w
all,
0-1.
7 H
z
4-la
yer m
odel
1β 94.3 3.65 3.9 0.073-0.156
2β 157.4 3.58 2.3 0.035-0.104
3β 145.3 3.36 2.3 0.019-0.110
4β 169.2 3.88 2.3 0.015-0.106
Equi
vale
nt
unifo
rm m
odel
eqβ 142.5 3.40 2.4
0.0277-0.1008 1/4τ 0.169 0.004 2.4
1,appf 0.166 0.0042 2.5
2,appf 0.51 0.014 2.7
μ s /s μ (%)
[cm/m]γ
EW a
t nor
th w
all,
0-1.
7 H
z
4-la
yer m
odel
1β 79 1.1 1.3 0.041-0.286
2β 144.3 6.3 4.4 0.015-0.259
3β 169.1 3.3 1.9 0.013-0.304
4β 187.8 7.1 3.8 0.007-0.230
Equi
vale
nt
unifo
rm m
odel
eqβ 145.5 3.9 2.7
0.0148-0.265 1/4τ 0.173 0.0044 2.5
1,appf 0.188 0.0068 3.6
2,appf 0.54 0.015 2.8
μ s /s μ (%)
610γ −×[rad/m]
Tors
ion,
0-3
.5 H
z
4-la
yer m
odel
1β 164.3 3.2 2 2.2-13
2β 254.6 4.53 1.8 0.9 -11.4
3β 253.4 5.9 2.3 0.6-4.9
4β 368.4 9.1 2.5 0.3-3.5
Equi
vale
nt
unifo
rm m
odel
eqβ 264.9 6.9 2.6
0.7-6.3 1/4τ 0.295 0.0078 2.7
1,appf 0.36 0.0092 2.55
2,appf 0.958 0.0288 3
17
Trends and Permanent Changes
Fig. 8 shows graphically the layer velocities along the building height, the bars being
ordered (top to bottom) in chronological order of the earthquake. The variations in the layer
velocities seem erratic at first sight, possibly due to the fact that the largest drifts along the
height were not necessarily caused by the same event, which we explore later.
Fig. 9 shows wβ γ2 (~ peak stress) vs. wγ for the equivalent uniform model for the six
earthquakes, which suggests essentially linear behavior for the (transient) drift levels this
building experienced (< 0.0265%). The earthquakes are identified by their chronological
order number, as in the remaining figures. Fig. 10 shows plots of the roof displacement for
all events, which was within 60 cm. In the following discussion, we explore possible trends
in the small variations of the parameters, as function of peak drift.
Fig. 11 shows scatter plots of eqβ , ,appf1 and ,appf2 vs. peak drift wγ . The horizontal
bands show the sμ ± interval for the sample, while the bars show the β σ± interval for the
individual fits. The corresponding coefficients of variation ( var /c s μ≡ ) are shown. It can
be seen that the variations among the earthquakes are greater than the estimation error for
each earthquake, and physical causes are likely, which we examine further. The
corresponding reduction in stiffness can be read directly from Fig. 12, as inferred from the
ratios ,( / )eq eqβ β 20 , , , ,( / )app appf f 2
1 1 0 and , , ,( / )app appf f 22 2 0 , where the reference values
are those for the Landers earthquake. The changes in eqβ suggest overall change in stiffness
of ~12%. In the following, we examine the degree and possible causes for the detected
variations in eqβ , and compare them with the variations of ,appf1 and ,appf2 .
The changes in eqβ seen in Figs 11 and 12 suggest that permanent reduction of stiffness,
of about 5%, occurred after the Landers- Big Bear sequence. The two earthquakes occurred
within 3 hours from each other, in the morning of July 28, 1992. Between the two
earthquakes, any significant changes in temperature, mass or human-caused alterations of
structural stiffness are unlikely to have occurred, and the identified changes in eqβ were
likely to have been caused by the earthquakes. Interestingly, the Landers earthquake caused
much larger response but the reduction of stiffness is detected during the subsequent Big
Bear earthquake. A moving window estimation of eqβ over the released 87 s of response
18
Fig. 8 Identified wave velocities in the layers for the six events.
Fig. 9 Peak stress (~ 2
eqβ γ ) vs. peak strain ( γ ) relations for the six events.
Fig. 10 Roof displacement during the six earthquakes.
19
showed that the change did not occur during the released portion of the recorded motion. As
it can be seen from Figs 4 and 5, the released length of the Landers records was too short to
capture the significant response of this building. It is possible that the change in stiffness
occurred during the unreleased portion of the Landers record. It is also possible that the
detected permanent change occurred gradually and was a cumulative effect of the many
cycles of response the building experienced during both earthquakes (Nastar et al., 2010).
Another permanent reduction of stiffness appears to have occurred in 1994 during the
Northridge earthquake, as suggested by eqβ for the EW and torsional responses, as well as
additional recoverable reduction. The torsional response reveals ~5% permanent reduction
and ~2% recoverable reduction, while, in the EW response it is the opposite. This difference
may be due to higher sensitivity of the torsional response to the permanent changes at smaller
drift levels. The variations of ,appf1 and ,appf2 also suggest permanent and recoverable
changes of stiffness with comparable magnitudes. This differs from what has been found by
similar analyses for RC buildings (Todorovska, 2009b), for which the fluctuations were
greater for ,appf1 than for eqβ . Such difference in behavior is consistent with greater
Fig. 11 Identified global parameters during the six earthquakes vs. weighted peak drift, wγ .
20
Fig. 12 Reduction of global stiffness, measured by the reduction of eqβ , 1,appf and 2,appf .
sensitivity of ,appf1 to nonlinearities in the soil behavior for the RC structures, which are
stiffer than steel frame structures (Todorovska, 2009b).
Next, we look for such changes in the layer velocities. Figs 11 and 12 show scatter plots
for iβ and ,( / )i iβ β 20 , , ,i =1 4… vs. the peak layer drift, similar to those in Figs 9 and 10.
The resolution of the method and error are important issues in fitting layered models. As
shown in Rahmani and Todorovska (2013), the minimum layer width minh that can be
resolved is roughly min min /h λ= 4 , where min max/ fλ β= is the shortest wavelength in the
data. For this building, and for the choice of maxf in this study (1.7 Hz for NS and EW
motions and 3.5 Hz for torsion), the layer thicknesses are larger than the (theoretical)
resolution by a factor of 2-3. For the middle two layers, e.g., minh is about 6 stories. For
given maxf , the estimation error is larger for thinner and stiffer layers, and therefore is larger
for the identified iβ than for the identified eqβ (Tdorovska and Rahmani, 2013). This is
evident in the more noisy appearance of the scatter plots for iβ than for eqβ , which,
nevertheless, clearly show permanent reduction of stiffness. The points for the Chino Hills
earthquake (No. 5) in Layer 1 (NS and torsion), Layer 2 (EW) and Layer 1 (NS) appear to be
outliers. Outliers excluded, Fig. 12 suggests overall permanent change in stiffness typically
between 5% and 10%. The changes are larger in Layers 2 and 4 for EW motions, and Layer
21
4 for torsion, but do not exceed 15%. The largest reduction in these cases, though not all
permanent, occurred during Northridge earthquake (No. 3). An open question remains why
the Chino Hills earthquake estimates show greater deviation from the observed trends. This
was a small local earthquake, which occurred around noon in midsummer, and practically did
not excite the first mode. The temperature at the time of the earthquake was about 80˚F
(weathersource.com), and was likely higher than during the other earthquakes, judging by the
season and time of the day. In depth investigation of the degree to which the temperature,
the nature of the excitation and other factors (environmental and operating conditions)
Fig. 13 Same as Fig. 9, but for the layer velocities vs. layer peak layer drift, iγ .
22
contributed to the more “noisy” estimate for this earthquake is out of the scope of this paper
(Clinton et al., 2006; Boroschek et al., 2008; Herak and Herak, 2010; Mikael et al., 2013).
DISCUSSION AND CONCLUSIONS
System identification and health monitoring analysis of a 54-story steel-frame building in
downtown Los Angeles was presented using recorded accelerations during six earthquakes
over period of 19 years (1992-2010). The set included all significant earthquakes that shook
this building since its construction in 1991. The transient apparent drift, determined from
displacements obtained by double integration of the recorded accelerations, did not exceed
~0.3%, which is less than half of the maximum transient drift for immediate occupancy
(0.7%), and is much smaller than the transient drift of concern for structural safety (2.5%) for
steel moment-frame buildings, as specified in ASCE guidelines (ASCE 2000; ASCE/SEI,
2007). The largest drift occurred during the distant Landers, 1992 and Hector Mine, 1999
earthquakes, while the local Northridge, 1994 earthquake, caused the largest damage in the
area (Table 1 and Fig. 2). No damage was reported from any of these earthquakes.
Fig. 14 Reduction of local (layer) stiffness, measured by the reduction of , 1,..., 4i iβ = , vs. peak layer drift, iγ .
23
The identified wave velocities suggest that the response of the structure was essentially
linear. Nevertheless, they suggest that permanent change in the overall structural stiffness of
~10-12% occurred, mainly caused by the Landers-Big Bear sequence and the Northridge
earthquake. These changes were widespread throughout the structure. The method used in
this study cannot determine the mechanism of the changes. The permanent changes in wave
velocity are comparable with those of the first two apparent frequencies of vibration, which is
consistent with smaller effects of the soil on the variations of the apparent frequency for more
flexible structures, as compared to the stiffer RC structures. While this study, of small
amplitude response of a steel-frame building, did not demonstrate obvious advantages of the
wave method over monitoring changes in the frequencies of vibration, as was the case for the
stiffer RC structures (Todorovska and Trifunac, 2008b; Todorovska, 2009b), it does not
exclude possible advantages in the case of stronger shaking, softer soil, and damage. Also,
the agreement of results by different SHM methods, in general, is useful because of increased
confidence in the results.
Statistical analysis of the estimates for the six earthquakes gave average vertical velocity
of 142.5 m/s for the NS, 145.5 m/s for the EW and 265 m/s for the torsional responses. The
identified variation along the height was larger for the EW response, consistent with the
narrowing of the building with height. The average observed apparent frequency for the
fundamental mode was 0.166 Hz for NS, 0.188 Hz for the EW and 0.36 Hz for torsional
responses, and of the second mode was 0.51 Hz for the NS, 0.54 Hz for the EW and 0.96 Hz
for the torsional responses. The coefficient of variation was small, typically less than 2.5%
and at most ~4.5%, but larger than the estimation error.
The detected variability of the properties of this building can be compared with similar
studies for other steel buildings only in terms of the variations of the apparent frequency of
vibration. For example, Rodgers and Celebi (2006) analyzed the variability of the apparent
frequencies of a 13-story steel building in Alhambra, ~15 km North-East of the 54-story
building, over 16 earthquake in 32 years (four of which were also recorded by the 54-story
building), none of which caused reported damage. Based on their results, we obtained sample
standard deviation of 5-5.6% over 32 years, which is about twice larger, over twice longer
period, from what we obtained for the 54-story building (2.5-3.6% over 19 years). Rodgers
and Celebi (2006), who estimated the frequencies from the Fourier spectra of the recorded
response, found large variations especially at low amplitudes (total variation of ~20%), but
24
no clear trends in the variations both with time and with peak base acceleration. We believe
that estimation of the frequencies from transfer-functions rather than Fourier spectra, and
correlation with peak drift rather than peak base acceleration would have reduced the scatter
and may have revealed some trends in their analysis. (Recall that, in this study, the
Northridge earthquake produced the largest peak ground acceleration, but the third largest
response, and the Chino Hills earthquake produced the second largest peak acceleration but
the smallest response; see Fig. 4.) Analysis of changes in the wave velocities, which are not
sensitive to the effects of soil-structure interaction, may have further reduced the scatter and
revealed permanent changes, like those we found for the 54-story building, and earlier for
Millikan Library (Todorovska, 2009b).
The general conclusions of this study, about the capabilities of the wave method for
SHM, is that, with the waveform inversion algorithm, it was able to detect permanent
changes in stiffness in the 54-story steel building, although no damage was observed and the
overall variation of wave velocities was small. Therefore, it is a promising method for SHM
of buildings. The method can be further improved by extending the analysis to higher
frequencies, which would improve its accuracy and spatial resolution, and to two and three
dimensions, which would enable analysis of less regular structures and coupled lateral and
torsional responses. We leave such tasks for the future.
It is also concluded that the records of multiple earthquake excitation, even though small,
were very useful, both for the development of the wave SHM method, providing an
opportunity to test its capabilities, as well as for providing new information about the
changes in stiffness of this building. Such records exist for many buildings in California,
instrumented by the owner or by the federal and state strong motion instrumentation
programs, and can be used for SHM. Although many records in buildings have been released
by the federal and state government programs, and can be conveniently accessed on the web,
the sets for a particular building are incomplete, often missing significant records, and,
therefore, not useful for SHM research to their full potential.
ACKNOWLEDGEMENTS
This work was supported by a grant from the U.S. National Science Foundation (CMMI-
0800399). The strong motion data used was provided by the California Strong Motion
Instrumentation Program (CSMIP) via the Engineering Center for Strong Motion Data
(www.strongmotioncenter.org/). We thank Hamid Haddadi of CSMIP for making available
25
the Landers, 1992, earthquake records on time to include in the revision of this paper. We
also thank Misha Trifunac, Firdaus Udwadia and Gregg Brandow for the insightful
discussions on the wave SHM method and on design and seismic behavior of tall steel
buildings. Finally, we thank the anonymous reviewers for their detailed comments, which
contributed to the clarity of this paper.
REFERENCES
ASCE (2000). Prestandard and commentary for the seismic rehabilitation of buildings, Report FEMA 356, Federal Emergency Management Agency, Washington, DC.
ASCE/SEI (2007). Seismic rehabilitation of existing buildings, ASCE/SEI 41-06, American Society of Civil Engineers, Reston, VA.
Boroschek RL, Lazcano P (2008). Non-damage modal parameter variations on a 22 story reinforced concrete building, Proc. 26th Int. Modal Analysis Conf., IMAC-XXVI-2008, Orlando FL.
Carreño RP, Boroschek RL (2011). Modal parameter variations due to earthquakes of different intensities, in T. Proulx (ed.), Civil Engineering Topics, Volume 4, Proc. 29th Int. Modal Analysis Conf., IMAC-XXIX, DOI 10.1007/978-1-4419-9316-8_30, The Society for Experimental Mechanics, Inc., Springer, New York, NY.
Chang PC, Flatau A, Liu SC (2003). Review paper: health monitoring of civil infrastructure, Struct. Health Monit., 2(3):257-267.
Celebi M, Phan T, Marshal RD (1993). Dynamic characteristics of five tall buildings during strong and low-amplitude motions, The Structural design of Tall Buildings, 2, 1-15.
Clinton JF, Bradford SK, Heaton TH, Favela J (2006). The observed wander of the natural frequencies in a structure. Bull. Seism. Soc. Amer., 96(1):237-257.
Doebling SW, Farrar CR, Prime MB, Shevitz DW (1996). Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review, Report LA-13070-MS, Los Alamos National Laboratory, Los Alamos, NM.
Ghobarah A, Abou-Elfath H, Biddah A (1999). Response-based damage assessment of structures, Earthquake Engng. Struct. Dyn. 28, 79-104.
Gičev V, Trifunac MD (2007). Permanent deformations and strains in a shear building excited by a strong motion pulse, Soil Dyn. Earthq. Eng., 27(8):774-792.
Gičev V, Trifunac MD (2009a). Transient and permanent shear strains in a building excited by strong earthquake pulses, Soil Dyn. Earthq. Eng.,29(10):1358-1366.
Gičev V, Trifunac MD (2009b). Rotations in a shear-beam model of a seven-story building caused by nonlinear waves during earthquake excitation, Struct. Control Health Monit., 16(4): 460-482.
Gičev V, Trifunac MD (2012). Predetermined earthquake damage scenarios (PEDS) for structural health monitoring, Struct. Control Health Monit., 19(8):746-757; DOI: 10.1002/stc.470.
Gilbert F, Backus GE. Propagator matrices in elastic wave and vibration problems, Geophysics 1966; XXXI(2):326-332.
26
Herak M, Herak D (2010). Continuous monitoring of dynamic parameters of the DGFSM building (Zagreb, Croatia), Bull. Earthquake Engineering, 8(3), 657-669.
Hisada Y, Yamashita T, Murakami M, Kubo T, Arata T, Shindo J, Aizawa K (2012). Seismic response and damage of high-rise buildings in Tokyo, Japan, during the 2011 Tohoku earthquake, The 15th World Conf. on Earthqu. Eng. (15WCEE), Sept. 24-28, 2012, Lisbon, Portugal.
Ivanović SS, Trifunac MD, Todorovska MI (2001). On identification of damage in structures via wave travel times, in Proc. NATO Advanced Research Workshop on Strong-Motion Instrumentation for Civil Engineering Structures, M. Erdik, M. Celebi, V. Mihailov, and N. Apaydin (Eds.), June 2-5, 1999, Istanbul, Turkey, Kluwer Academic Publishers; 447-468.
Kanai, K., Yoshizawa, S. (1963). Some new problems of seismic vibrations of a structure. Part 1. Bull. Earthq. Res. Inst., 41:825-833.
Kanai K (1965). Some new problems of seismic vibrations of a structure, Proc. Third World Conf. Earthq. Eng., Auckland and Wellington, New Zealand, January 22-February 1, pp. II-260 to II-275.
Kawakami H, Oyunchimeg M (2004). Wave propagation modeling analysis of earthquake records for buildings. J. of Asian Architecture and Building Engineering, 3(1):33-40.
Kohler MD, Heaton T, Bradford SC (2007). Propagating Waves in the Steel, Moment-Frame Factor Building Recorded during Earthquakes, Bull. Seism. Soc. Am., 97(4):1334-1345.
Lee VW, Trifunac MD (1880). Automatic digitization and processing of accelerograms using PC, Dept. of Civil Eng. Report 90-03, Univ. Southern California, Los Angeles, California.
Levenberg K (1944). A Method for the Solution of Certain Non-Linear Problems in Least Squares, The Quarterly of Applied Mathematics, 2: 164-168 (1944).
Li Y, Mau ST (1997). Learning from recorded earthquake motions in buildings, J. of Structural Engrg, ASCE, 123(1):62-69.
Liu K-S, Tsai Y-B (2010). Observed natural frequencies, damping ratios, and mode shapes of vibration of a 30-story building excited by a major earthquake and typhoon, Earthquake Spectra, 26(2):371-397.
Luco JE, Trifunac MD, Wong HL (1988). Isolation of soil-structure interaction effects by full-scale forced vibration tests, Earthq. Eng. Struct. Dyn. 1988; 16: 1-21.
Marquardt DW (1963). An algorithm for least-squares estimation of nonlinear parameters, Journal of the Society for Industrial and Applied Mathematics, 11(2):431-441, 1963.
Michel C, Philippe Gueguen P (2010). Time–frequency analysis of small frequency variations in civil engineering structures under weak and strong motions using a reassignment method, Structural Health Monitoring, 9(2):159-179.
Michel C, Gueguen P, Lestuzi P (2011). Observed non-linear soil-structure interaction from low amplitude earthquakes and forced-vibration recordings, Proc. 8th International Conf. on Structural Dynamics, EURODYN 2011, Leuven, Belgium, 4-6 July 2011, ISBN 978-90-760-1931-4.
27
Mikael A, Gueguen P, Bard P-Y, Roux P, Langlais M (2013). The analysis of long term frequency and damping wondering in buildings using the random decrement technique, Bull. Seism. Soc. Am., 103(1): 236-246.
Naeim F (1997). Performance of extensively instrumented buildings during the January 17, 1994 Northridge earthquake, JAMA Report No. 97-7530.68, John A Martin and Assoc. Inc., Los Angeles, CA
Naeim F, Lee H, Hagie S, Bhatia H, Alimoradi A, Miranda E (2006). Three-dimensional analysis, real-time visualization, and automated post-earthquake damage assessment of buildings, Struct. Design Tall Spec. Build., 15: 105–138.
Nastar N, Anderson JC, Brandow GE, Nigbor RL (2010). Effects of low-cycle fatigue on a 10-storey steel building, Struct. Design Tall Spec. Build. 19:95–113.
Rahmani M, Todorovska MI (2013). 1D system identification of buildings from earthquake response by seismic interferometry with waveform inversion of impulse responses – method and application to Millikan Library, Soil Dyn. Earthq. Eng., Jose Roësset Special Issue, E. Kausel and J. E. Luco, Guest Editors, 47:157-174, DOI: 10.1016/j.soildyn.2012.09.014.
Oyunchimeg M, Kawakami H (2003). A new method for propagation analysis of earthquake waves in damaged buildings: Evolutionary Normalized Input-Output Minimization (NIOM), J. of Asian Architecture and Building Engineering, 2(1):9-16.
Prieto GA, Lawrence JF, Chung AI, Kohler MD (2010). Impulse response of civil structures from ambient noise Analysis, Bull. Seism. Soc. Am., 100: 2322-2328.
Rodgers JE, Çelebi M (2006). Seismic response and damage detection analyses of an instrumented steel moment-framed building, J. Struct. Eng., 132(1), 1543-1552.
Şafak E (1998). Detection of seismic damage in multi-story buildings by using wave propagation analysis. Proc. Sixth U.S. National Conf. Earthq. Eng. 1998; EERI, Oakland, CA, Paper No. 171, pp. 12.
Şafak E (1999). Wave propagation formulation of seismic response of multi-story buildings. J. Struct. Eng. (ASCE) 1999; 125(4):426-437.
Snieder R, Şafak E (2006). Extracting the building response using interferometry: theory and applications to the Millikan Library in Pasadena, California, Bull. Seism. Soc. Am., 96(2):586-598.
Todorovska MI (2009a). Seismic interferometry of a soil-structure interaction model with coupled horizontal and rocking response, Bull. Seism. Soc. Am., 99(2A):611-625.
Todorovska MI (2009b). Soil-structure system identification of Millikan Library North-South response during four earthquakes (1970-2002): what caused the observed wandering of the system frequencies?, Bull. Seism. Soc. Am., 99(2A):626-635.
Todorovska MI, Rahmani M (2012). Recent advances in wave travel time based methodology for structural health monitoring and early earthquake damage detection in buildings, The 15th World Conf. on Earthqu.Eng. (15WCEE), Sept. 24-28, 2012, Lisbon, Portugal, pp. 10.
28
Todorovska MI, Rahmani M (2013). System identification of buildings by wave travel time analysis and layered shear beam models - spatial resolution and accuracy, Struct. Control Health Monit., 20(5):686–702, DOI: 10.1002/stc.1484.
Todorovska MI, Trifunac MD (2008a). Earthquake damage detection in the Imperial County Services Building III: analysis of wave travel times via impulse response functions, Soil Dyn. Earthq. Eng., 28(5):387–404.
Todorovska MI, Trifunac MD (2008b). Impulse response analysis of the Van Nuys 7-story hotel during 11 earthquakes and earthquake damage detection, Struct. Control Health Monit.; 15(1):90-116.
Todorovska MI, Trifunac MD (2008c). Earthquake damage detection in structures and early warning, Proc. 14th World Conference on Earthquake Engineering, October 12-17, 2008, Beijing, China, Paper S05-03-010, pp. 10.
Todorovska MI, Trifunac MD (1989). Antiplane earthquake waves in long structures, J. Engrg Mech. (ASCE); 115(12):2687-2708.
Todorovska MI, Lee VW (1989). Seismic waves in buildings with shear walls or central core, J. Engrg Mech. (ASCE); 115(12):2669-2686.
Todorovska MI, Ivanović SS, Trifunac MD (2001). Wave propagation in a seven-story reinforced concrete building, Part II: observed wavenumbers, Soil Dyn. Earthq. Eng.:21(3), 224-236.
Todorovska MI, (1988). Investigation of earthquake response of long buildings, PhD dissertation, Civil Eng. Dept, University of Southern California.
Trampert J, Cara M, Frogneux M (1993). SH propagator matrix and SQ estimates from borehole- and
surface-recorded earthquake data, Geophys. J. Int.; 112:290-299. Trifunac MD, Ivanović SS, Todorovska MI (2001a). Apparent periods of a building I: Fourier
analysis, J. of Struct. Engrg, ASCE, 127(5):517-526). Trifunac MD, Ivanović SS, Todorovska MI (2001b). Apparent periods of a building II: time-
frequency analysis, J. of Struct. Engrg, ASCE, 127(5):527-537. Trifunac MD, Todorovska MI, Manić MI, Bulajić BĐ (2010). Variability of the fixed-base and soil-
structure system frequencies of a building – the case of Borik-2 building, Struct. Control Health Monit.; 17(2):120–151.