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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: mrahmani@usc.edu b) Research Professor, University of Southern California, Dept. of Civil Eng., Los Angeles, CA 90089-2531, Email:mtodorov@usc.edu

Earthquake Spectra, 2013, DOI: 10.1193/112912EQS339M, final draft; submitted for publication on 11/29/2012; accepted for publication on 7/22/2013.

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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)

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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

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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

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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

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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).

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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

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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).

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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

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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.

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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).

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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.

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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.

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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

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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.

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