Earthquake Spectra, 2013, DOI: 10.1193/112912EQS339M, final draft; submitted for publication on 11/29/2012; accepted for publication on 7/22/2013.
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 β ( z) 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, f1,app and f2,app , 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 manmade 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]
1
Fig. 1 Los Angeles 54-story office building (CSMIP 24629):photo, vertical cross-section, and typical floor layouts (redrawn from www.strongmotioncenter.org)
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
2
. 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
06/28/1992 04:57:31 PDT 06/28/1992 Big Bear BB 08:05:31 PDT 01/17/1994 Northridge NO 04:30:00 PST Hector 10/16/1999 HM Mine 02:46:45 PDT 07/29/2008 Chino Hills CH 11:42:15 PDT Whittier 03/16/2010 WN Narrows* 04:04:00 PDT 04/04/2010 Calexico CA 15:40:39 PDT * Data not available; H=focal depth Landers
LA
Epicenter
H [km]
ML
Epic/Fault distance [km]
Rec. length [s]
Gnd amax [g]
Struc. amax [g]
1
7.3
170/158
87
0.040
0.130
10
6.5
133
87
0.030
0.067
19
6.4
32/28
180
0.140
0.190
6
7.1
193
106
0.019
0.082
14
5.4
47
83
0.063
0.086
18
4.4
18
-
0.020
0.022
32
7.2
355/286
87
0.009
0.038
34.22N 116.43W 34.20N 116.83W 34.21N 118.54W 34.60N 116.27W 33.95N 117.77W 34.00N 118.07W 32.26N 115.29W
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. 3
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 4
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 5
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 soilstructure 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). 6
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
Fig. 3 The model.
hi , mass density ρi , and shear modulus μi , i = 1,… , n , where n is number of layers, which implies shear wave velocities β i = μi / ρi . consecutive layer interfaces are u1 , u2 ,
The displacements at the roof and at the
…un+1 .
Amplitude attenuation due to material
friction is introduced via the quality factor, Q , and the damping ratio is ζ = 1 / ( 2Q ) . A band-limited impulse response function (IRF) at some level for virtual source at roof, h(z, 0, ωmax ; t) , is obtained by inverse Fourier transform of the corresponding transfer-function
(TF), hˆ ( z , 0; ω ) 7
1 ωmax ˆ −iωt h( z, 0, ωmax ; t ) = ∫ h( z, 0;ω ) e dω 2π −ωmax
(1)
where ωmax is the cut-off frequency. Regularized TFs are practical to use
uˆ( z, ω )uˆ (0,ω ) hˆ( z, 0;ω ) = 2 uˆ(0,ω ) + ε
(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 1/ fmax = 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, fmax , which controls the spatial resolution and the effects of dispersion. A higher value of fmax 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 fmax = 1.7 Hz for the NS and EW responses, and fmax = 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). 8
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 9
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. 10
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 βeq , quality factor Q , and fixed-base frequency 1 / ( 4τ ) of the fitted equivalent uniform model. While βeq 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 ζ = 1 / ( 2Q ) . The fixed-base frequency of the fitted uniform model, 1 / ( 4τ ) , 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).
11
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.
12
Table 2 also shows the apparent frequencies for the first two modes, f1,app and f2,app , identified from the transfer functions, and γ w = 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, γ w , was computed as the weighted average of the peak layer drifts, with weights proportional to the layer heights. We use γ w 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 γ w 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 β i , i = 1, … , 4 estimated by the waveform inversion algorithm and the corresponding normalized standard deviation σ β / β , and the peak layer drifts γi . 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 ρ = 300 kg/m 3 , for all the models. The layer widths are h1 = 27.9 m, h2 = 39.8 m, h3 = 63.6 m and h4 = 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 βeq , f1,app and f2,app during these six earthquakes, with s / μ not exceeding 3.6%, and also small
variability of β i in the layers, not exceeding 4.4%. The range of the peak drifts are also specified in the last column, where γ ≡ γ w for the global parameters, and γ ≡ γ i for the local parameters.
13
Table 2 Identification results for equivalent uniform model during six earthquakes. NS at west wall, 0-1.7 Hz; h =210.2 m
f1,app
f2,app
γw
[Hz]
[Hz]
[cm/m]
2.9
0.17
0.53
0.0987
20
2.5
0.17
0.52
0.0439
0.166
25
2
0.165
0.502
0.0964
0.49
0.168
16.7
3
0.16
0.498
0.0739
139.0
0.44
0.165
18.5
2.7
--*
0.50
0.0277
142.8
0.49
0.170
30.7
1.63
0.165
0.498
0.1008
f1,app
f2,app
[Hz]
[Hz]
γw [cm/m]
βeq
1/ 4τ
[m/s]
σβ % β
1.4175
148.3
0.42
0.176
17.5
BB
1.4600
144.0
0.46
0.171
NO
1.5025
139.9
0.48
HM
1.4925
140.8
CH
1.5125
CA
1.4725
Event
τ [s]
LA
[Hz]
Q
ζ =
1 2Q
EW at north wall, 0-1.7 Hz; h =210.2 m
βeq
[s]
[m/s]
σβ % β
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
f1,app
f2,app
γ w × 10 −6
[Hz]
[Hz]
[rad/m]
Event
τ
1/ 4τ [Hz]
Q
ζ =
1 2Q
Torsion, 0-3.5 Hz; h=224.2 m
βeq
1/ 4τ
[m/s]
σβ % β
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
Event
τ [s]
LA
[Hz]
Q
*The first mode is not readable
14
ζ =
1 2Q
Table 3 Identification results for equivalent 4-layer model during six earthquakes. NS at west wall, 0-1.7 Hz
γ
β
[m/s]
σβ % β
[cm/m]
LA
98.8
0.4
BB
95.5
NO
Layer 1 Layer 2 Layer 3
Torsion, 0-3.5 Hz
γ
β
[m/s]
σβ % β
[cm/m]
0.087
80.2
0.8
0.7
0.085
79.4
92.5
0.75
0.156
HM
96.4
0.7
CH
88.2
CA
γ × 10−6
[m/s]
σβ % β
0.286
169.1
1.5
13.0
0.7
0.072
165
1.3
3.56
78.2
0.77
0.22
166.4
1.5
6.70
0.11
78
0.8
0.2
162.7
1.2
11.67
0.8
0.073
80.1
1.1
0.041
159.8
1.4
2.16
94.2
0.7
0.075
77.9
0.5
0.084
163
1.2
5.66
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
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
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
Event
Layer 4
EW at north wall, 0-1.7 Hz
β
15
[rad/m]
4-layer model Equivalent uniform model 4-layer model Equivalent uniform model 4-layer model Equivalent uniform model
Torsion, 0-3.5 Hz
EW at north wall, 0-1.7 Hz
NS at west wall, 0-1.7 Hz
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 (%)
γ [cm/m]
β1
94.3
3.65
3.9
0.073-0.156
β2
157.4
3.58
2.3
0.035-0.104
β3
145.3
3.36
2.3
0.019-0.110
β4
169.2
3.88
2.3
0.015-0.106
βeq
142.5
3.40
2.4
1/ 4τ
0.169
0.004
2.4
f1,app
0.166
0.0042
2.5
f2,app
0.51
0.014
2.7
μ
s
s/μ (%)
γ [cm/m]
β1
79
1.1
1.3
0.041-0.286
β2
144.3
6.3
4.4
0.015-0.259
β3
169.1
3.3
1.9
0.013-0.304
β4
187.8
7.1
3.8
0.007-0.230
βeq
145.5
3.9
2.7
1/ 4τ
0.173
0.0044
2.5
f1,app
0.188
0.0068
3.6
f2,app
0.54
0.015
2.8
μ
s
s/μ (%)
γ × 10−6
0.0277-0.1008
0.0148-0.265
[rad/m]
β1
164.3
3.2
2
2.2-13
β2
254.6
4.53
1.8
0.9 -11.4
β3
253.4
5.9
2.3
0.6-4.9
β4
368.4
9.1
2.5
0.3-3.5
βeq
264.9
6.9
2.6
1/ 4τ
0.295
0.0078
2.7
f1,app
0.36
0.0092
2.55
f2,app
0.958
0.0288
3
16
0.7-6.3
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 β 2γ w (~ peak stress) vs. γ w 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 βeq , f1,app and f2,app vs. peak drift γ w . The horizontal bands show the μ ± s interval for the sample, while the bars show the β ± σ interval for the individual fits. The corresponding coefficients of variation ( cvar ≡ 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 ( β eq / β eq , 0 ) 2 , ( f1, app / f1, app , 0 ) 2 and ( f 2, app / f 2, app ,0 ) 2 , where the reference values are those for the Landers earthquake. The changes in βeq suggest overall change in stiffness of ~12%. In the following, we examine the degree and possible causes for the detected variations in βeq , and compare them with the variations of f1,app and f2,app . The changes in βeq 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 βeq 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 βeq over the released 87 s of response 17
Fig. 8 Identified wave velocities in the layers for the six events.
2 Fig. 9 Peak stress (~ β eq γ ) vs. peak strain ( γ ) relations for the six events.
Fig. 10 Roof displacement during the six earthquakes.
18
Fig. 11 Identified global parameters during the six earthquakes vs. weighted peak drift, γ w .
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 βeq 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 f1,app and f2,app 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 f1,app than for βeq . Such difference in behavior is consistent with greater 19
sensitivity of f1,app to nonlinearities in the soil behavior for the RC structures, which are stiffer than steel frame structures (Todorovska, 2009b).
Fig. 12 Reduction of global stiffness, measured by the reduction of βeq , f1,app and f2,app .
Next, we look for such changes in the layer velocities. Figs 11 and 12 show scatter plots for βi and ( β i / β i ,0 ) 2 , 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 hmin that can be resolved is roughly hmin = λmin / 4 , where λmin = β / f max is the shortest wavelength in the data. For this building, and for the choice of f max 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., hmin is about 6 stories. For given f max , the estimation error is larger for thinner and stiffer layers, and therefore is larger for the identified β i than for the identified βeq (Tdorovska and Rahmani, 2013). This is evident in the more noisy appearance of the scatter plots for β i than for βeq , 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 20
Fig. 13 Same as Fig. 9, but for the layer velocities vs. layer peak layer drift, γ i .
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) 21
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).
Fig. 14 Reduction of local (layer) stiffness, measured by the reduction of β i , i = 1,..., 4 , vs.
peak layer drift, γ i .
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. 22
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 23
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 (CMMI0800399). 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 24
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