System Identification Study of a 7-Story Full-Scale BuildingSlice Tested on the UCSD-NEES Shake Table

Babak Moaveni, A.M.ASCE1; Xianfei He2; Joel P. Conte, M.ASCE3;José I. Restrepo, M.ASCE4; and Marios Panagiotou, M.ASCE5

Abstract: A full-scale 7-story reinforced concrete building slice was tested on the unidirectional University of California–San Diego Net-work for Earthquake Engineering Simulation (UCSD-NEES) shake table during the period from October 2005 to January 2006. A rectangularwall acted as the main lateral force resisting system of the building slice. The shake table tests were designed to damage the building pro-gressively through four historical earthquake records. The objective of the seismic tests was to validate a new displacement-based designmethodology for reinforced concrete shear wall building structures. At several levels of damage, ambient vibration tests and low-amplitudewhite noise base excitation tests were applied to the building, which responded as a quasi-linear system with dynamic parameters evolving asa function of structural damage. Six different state-of-the-art system identification algorithms, including three output-only and three input-output methods were used to estimate the modal parameters (natural frequencies, damping ratios, and mode shapes) at different damage levelsbased on the response of the building to ambient as well as white noise base excitations, measured using DC-coupled accelerometers. Themodal parameters estimated at various damage levels using different system identification methods are compared to (1) validate/cross-check the modal identification results and study the performance of each of these system identification methods, and to (2) investigatethe sensitivity of the identified modal parameters to actual structural damage. For a given damage level, the modal parameters identified usingdifferent methods are found to be in good agreement, indicating that these estimated modal parameters are likely to be close to the actualmodal parameters of the building specimen. DOI: 10.1061/(ASCE)ST.1943-541X.0000300. © 2011 American Society of Civil Engineers.

CE Database subject headings: Buildings; Experimentation; Modal analysis; Parameters; Shake table tests.

Author keywords: Building structure; Experimental modal analysis; System identification; Modal parameters; Shake table tests.

Introduction

In recent years, structural health monitoring has been attracting in-creasing attention and is of growing importance in the civil engi-neering research community as a potential tool to develop methodsthrough which structural damage can be identified at the earliestpossible stage, and the remaining useful life of structures can beevaluated (damage prognosis). Civil structures may be suddenlydamaged due to natural and human-made hazards such as earth-quakes, hurricanes, and explosions. Also, material and structuraldegradation resulting from aging and environmental influences

cause progressive structural damage over time. Experimental andoperational modal analyses have been developed as technologiesfor identifying structural dynamic characteristics from recordedstructural vibration data. Changes in dynamic characteristics ofstructures are being used extensively for the purpose of conditionassessment and damage identification of structures. Extensive lit-erature reviews on damage identification based on changes in vi-bration characteristics were provided by Doebling et al. (1996,1998) and Sohn et al. (2003). The success of damage identificationbased on experimental modal analysis depends strongly on theaccuracy and completeness of the identified structural dynamicproperties.

A portion of a reinforced concrete midrise residential building,referred to in this paper as a building slice, was built at full-scaleand tested on the University of California–San Diego Network forEarthquake Engineering Simulation (UCSD-NEES) shake tableduring the period from October 2005 to January 2006. The proto-type building represents a type of low-rise cost-effective construc-tion for residential buildings proposed in substitution of woodframe construction in densely populated areas of southern Califor-nia. The prototype building plan is shown in Fig. 1. The lateralforce resisting system in the short (transversal) direction of thebuilding consists primarily of 3.6-m-long reinforced concreteload-bearing walls. The building was designed using a displace-ment-based method considering two performance levels: immedi-ate occupancy and life safety, each anchored at a specified seismichazard level (Panagiotou and Restrepo 2011). The considered dis-placement-based design methodology resulted in significantlysmaller earthquake equivalent lateral forces and therefore signifi-cantly less longitudinal steel reinforcement in the shear walls thanrequired by current force-based seismic design provisions in the

1Assistant Professor, Dept. of Civil and Environmental Engineering,Tufts Univ., 200 College Ave., Medford, MA 02155. E-mail: babak.moaveni@tufts.edu

2Associate Bridge Engineer, AECOM Transportation, 999 Town &Country Road, Orange, CA 92868. E-mail: Daniel.He@aecom.com

3Professor, Dept. of Structural Engineering, Univ. of California at SanDiego, 9500 Gilman Dr., La Jolla, CA 92093-0085 (corresponding author).E-mail: jpconte@ucsd.edu

4Professor, Dept. of Structural Engineering, Univ. of California at SanDiego, 9500 Gilman Dr., La Jolla, CA 92093-0085. E-mail: jrestrepo@ucsd.edu

5Assistant Professor, Dept. of Civil and Environmental Engineering,Univ. of California at Berkeley, 747 Davis Hall, Berkeley, CA 94720-1710. E-mail: panagiotou@ce.berkeley.edu

Note. This manuscript was submitted on July 28, 2009; approved onAugust 2, 2010; published online on August 25, 2010. Discussion periodopen until November 1, 2011; separate discussions must be submitted forindividual papers. This paper is part of the Journal of Structural Engineer-ing, Vol. 137, No. 6, June 1, 2011. ©ASCE, ISSN 0733-9445/2011/6-705–717/$25.00.

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United States. A portion of the building consisting of one shear wallwith the surrounding slabs was selected as a test specimen. In thisportion of the building, unless supported, the slabs are cantileveredfrom the walls. Thus, to improve the boundary conditions, it wasdecided to support the slab ends on gravity columns. These seismictests provided the unique opportunity to observe the dynamic re-sponse of a building slice built at full-scale using actual construc-tion practices (e.g., tunnel form construction, joints and lap-splices)at various damage states.

Using the measured response of the tested portion of the build-ing to seismic base excitation together with a properly calibratednonlinear finite-element (FE) model, and then extending this cali-brated FE model to the whole prototype building structure, the seis-mic response of the latter can be predicted reliably. The shake tabletesting of the selected building portion reproduces the actual pre-dominant failure modes (spalling of concrete, lap-splice failures,shear-sliding along construction joints, large curvatures, and com-bined shear-flexural damage patterns) of the prototype building.The building slice was subjected to four historical California earth-quake records, including the strong intensity near-fault Sylmarrecord from the 1994 Northridge earthquake, which induced sig-nificant inelastic response and resulting damage to the structure.This project stemmed from the interest in the subject and strongpartnership of a construction industry group comprising primarilycontractors, construction product suppliers, and practicing struc-tural engineers, mostly from southern California. The buildingspecimen responded very satisfactorily to the earthquake groundmotions reproduced by the shake table and met all performanceobjectives. The effects on the experimental response of the inter-action between the walls, the slabs, and the gravity system as wellas the effect of higher modes were found to be very significant(Panagiotou and Restrepo 2011; Panagiotou et al. 2011).

The shake table tests were designed to damage the buildingprogressively through four historical earthquake ground motions.At various levels of damage, ambient vibration tests were per-formed and low-amplitude white noise base excitations were ap-plied to the building through the shake table. During theseambient vibration and white noise base excitation tests, the buildingresponded as a quasi-linear system with dynamic parameters evolv-ing as a function of structural damage. Six different state-of-the-artsystem identification methods, consisting of three input-outputand three output-only methods, were applied to the dynamic re-sponse of the building measured using DC-coupled accelerometersto estimate modal parameters (natural frequencies, damping ratios,and mode shapes) of the building in its undamaged (baseline)and various damage states. The system identification methods usedare: (1) multiple-reference natural excitation technique combinedwith eigensystem realization algorithm (MNExT-ERA) (Jameset al. 1993); (2) data-driven stochastic subspace identification

(SSI-DATA) (Van Overschee and de Moor 1996); (3) enhancedfrequency domain decomposition (EFDD) (Brinker et al. 2001);(4) deterministic-stochastic subspace identification (DSI) (VanOverschee and de Moor 1996); (5) observer/Kalman filter identi-fication combined with ERA (OKID-ERA) (Phan et al. 1992); and(6) general realization algorithm (GRA) (de Callafon et al. 2008).The first three algorithms use output-only data (from ambient vi-bration and white noise base excitation tests), whereas the latterthree require measured input and output data (from the white noisebase excitation tests). The three output-only system identificationmethods used for this study were successfully applied by the au-thors to dynamic field test data from the Alfred Zampa MemorialBridge (He et al. 2009).

Different system identification methods provide modal param-eter estimators with different intramethod and intermethod statisti-cal properties (bias, variance, covariance), which depend on theamplitude and frequency content of the input excitation, the degreeof violation of the hypotheses on which these methods were devel-oped (e.g., linear system, broadband excitation, response stationar-ity), and other factors. In a previous study, the authors haveinvestigated the effects of such factors on the performance ofthe three output-only system identification methods used in thispaper based on the dynamic response of the 7-story reinforced con-crete building slice simulated using a three-dimensional nonlinearfinite-element model (Moaveni et al. 2007). It was found that for allthree methods, the estimation bias and variability for the naturalfrequencies and mode shapes are very small, and the estimationuncertainty of the damping ratios is significantly higher than thatof the natural frequencies and mode shapes.

Test Specimen, Test Setup and DynamicExperiments

7-story Reinforced Concrete Building Slice

The test structure, which represents a slice of a full-scale reinforcedconcrete building, consists of a main wall (web wall), a back wall(flange wall) perpendicular to the main wall for transversal stability,a concrete slab at each floor level, an auxiliary posttensioned col-umn to provide torsional stability, and four gravity columns totransfer the weight of the slabs to the shake table. Slotted slab con-nections located between the web and flange walls at floor levelminimize the moment transfer between the two walls whileallowing the transfer of the in-plane diaphragm forces. Fig. 2 showsthe test structure mounted on the shake table. More details about thetest structure can be found in Panagiotou and Restrepo (2011) andPanagiotou et al. (2011).

Instrumentation Layout

The test structure was instrumented with a dense array ofDC-coupled accelerometers, strain gages, potentiometers, and lin-ear variable displacement transducers (LVDTs), all sampling datasimultaneously using a nine-node distributed data acquisition sys-tem. The accelerometer array consisted of 14 channels of acceler-ation measurements on the foundation/pedestal of the test structure,106 on the floor slabs and the web wall, 17 channels on the reactionblock of the shake table, and one triaxial accelerometer on the sur-rounding ground surface (free field), resulting in a total of 140 ac-celeration channels. A set of 54 LVDTs were installed along bothedges (east and west) of the web wall, and eight potentiometerswere installed over the first two stories of the web wall. A totalof 231 strain gages were distributed on the test specimen as follows:143 on the steel reinforcement of the web wall, 64 on the steel

Fig. 1. Plan view of prototype building with cross section of teststructure

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reinforcement of the flange wall, 16 on the gravity columns, and8 on the steel braces connecting the slabs to the posttensionedcolumn. In addition, the displacement responses of six selectedpoints on the structure (three on the top floor slab, two on the flangewall and one on the shake table platen) were measured in threedimensions using six global positioning system (GPS) sensors.The technical characteristics of the accelerometers are: MEMS-Piezoresistive MSI model 3140, amplitude range: �5 g, frequencyrange (min): 0–300 Hz, voltage sensitivity: 400 mV=g. The dataacquisition system used consisted of nine distributed 16-bit reso-lution National Instruments PXI chassis. Each chassis had eightSCXI 1520 modules that were individually configured to handleeight channels of strain gauges, accelerometers, relative displace-ment, and/or pressure transducers.

In this study, measured response data from 28 longitudinal ac-celeration channels (three on each floor slab and one on the webwall at midheight of each story) were used to identify the modalparameters of the test structure. The output-only methods usedin this study were also applied to the strain measurement data ob-tained from the LVDTs located along both edges of the web wall ofthe structure, and the corresponding system identification resultsare reported in Moaveni (2007). Fig. 3 shows the Fourier amplitudespectra (FAS) of two filtered acceleration time histories (between0.5 Hz and 25 Hz using a finite impulse response filter of order1024) recorded at floor levels 1 and 7 during a 0.03 g root-mean-square (RMS) acceleration white noise base excitation test(left column) and an ambient vibration test (right column) per-formed on the structure in its undamaged state. From this figure,it is observed that (1) the FAS plots are very jagged/noisy, whichmay be attributable to rattling of loose connections, especially fromthe small (2 mm maximum) slack between the threaded rod and thenut in the gravity columns when they go from tension to compres-sion or vice versa as well as the slack at both ends of the steel bracesconnecting the slabs to the posttensioned column; (2) the first lon-gitudinal vibration mode has a predominant contribution to the totalresponse, especially at the higher floors, which renders the identi-fication of higher (than the first longitudinal) vibration modes moredifficult; and (3) the FAS of the acceleration response histories at

the first floor appear to have a drop in amplitude at around 11.5 Hz,which is attributable to the application of a notch filter in the controlloop of the table to reduce the effects of the oil column resonance.Mechanical characteristics of the shake table are available inOzcelik (2008).

All the target base excitation signals (white noise base acceler-ation and earthquake ground acceleration records) reproduced onthe shake table were distorted due to the limitations of the shaketable controller and the table-structure interaction. However, the re-produced base excitations were measured on top of the shake tableplaten and used in the input-output system identification methods.The input signal distortion is shown to have little effect on the out-put-only system identification results, as they are found in verygood agreement with those obtained from the input-output meth-ods. In the case of the output-only methods, the effects of table-structure interaction were reduced by using horizontal accelerationsrelative to the base of the specimen. Another effect of table-struc-ture interaction is the rocking of the structure at its base, which ismade of two components: (1) bending of the shake table platen andvertical deformations of the hydraulic bearings, the foundation, andthe surrounding soil; and (2) localized rotation manifested at thebase of the web wall caused by strain penetration of the longitudinalreinforcing bars into the foundation in which they are anchored andpartial debonding (bond-slip) of these bars. Although the secondcomponent of the rocking of the wall contributes up to 7% ofthe peak measured roof relative horizontal displacement for earth-quake tests EQ2 and EQ3 (Panagiotou et al. 20011), it has a muchlower (negligible) relative contribution for lower amplitude ambi-ent and white noise base excitations. The small (i.e., few percentrelative contribution) first component of the rocking of the wallcould not be removed from the measured data before performingsystem identification because of the low signal-to-noise ratio of thevertical accelerometers measuring the rocking of the shake tableplaten at the base of the specimen.

Dynamic Tests Performed

A sequence of dynamic tests (68 tests in total) was applied to thetest structure during the period from October 2005 to January 2006,including ambient vibration tests, free vibration tests, and forcedvibration tests (white noise and seismic base excitations) usingthe UCSD-NEES shake table. The structure was damaged progres-sively through four historical earthquake ground motions, and themodal parameters of the structure were identified at various damage

Fig. 2. Test structure

White Noise Base Excitation Ambient Vibration 1,000

Fig. 3. Fourier amplitude spectra of acceleration response measure-ments at the first and seventh (roof) floors attributable to white noisebase excitation (left) and ambient excitation (right)

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states using different state-of-the-art system identification methodsand different dynamic test data.

The four historical earthquake records applied to the test struc-ture were: (1) longitudinal component of the 1971 San Fernandoearthquake (Mw ¼ 6:6) recorded at the Van Nuys station (EQ1),(2) transversal component of the 1971 San Fernando earthquakerecorded at the Van Nuys station (EQ2), (3) longitudinal compo-nent of the 1994 Northridge earthquake (Mw ¼ 6:7) recorded at theOxnard Boulevard station in Woodland Hill (EQ3), and (4) 360°component of the 1994 Northridge earthquake recorded at theSylmar station (EQ4). The input white noise base excitation con-sisted of two 8-min-long realizations of a banded white noise(0.25–25 Hz) process with RMS amplitudes of 0.03 and 0.05 g,respectively. Table 1 reports the dynamic tests used in the presentstudy on system identification of this 7-story building structure atvarious damage states.

The use of four earthquake input motions with distinct featuresand intensities allowed monitoring of the development of differentdamage states in the building specimen. Overall, the response wasslightly nonlinear for EQ1, moderately nonlinear for the “medium”intensity input motions EQ2 and EQ3, and highly nonlinear for in-put motion EQ4. During test EQ1, limited yielding occurred in thelongitudinal reinforcing steel of the web wall. After this test, crack-ing of thewebwall was widespread and visible up to the fourth floor.During tests EQ2 and EQ3, moderate yielding occurred in the webwall longitudinal reinforcement. Finally, during test EQ4, localizedplasticity developed at the first floor of the web wall. The roof driftratio, defined as the ratio between the maximum lateral displace-ment at the roof level of the building and the height of the roof rel-ative to the base of the building, was measured as 0.28, 0.75, 0.83,and 2.1% in tests EQ1 to EQ4, respectively. The maximum tensilestrain in the longitudinal reinforcing steel was measured close to thebase of the wall as 0.61, 1.73, 1.78, and 2.85% in tests EQ1 to EQ4,respectively (yield strain of reinforcing steel ¼ 0:22%). In-depthinformation about the behavior of this building specimen duringthe seismic tests EQ1 through EQ4 can be found elsewhere(Panagiotou et al. 2011).

Review of System Identification Methods Used

Six state-of-the-art linear system identification methods were ap-plied to estimate the modal parameters of the test structure at

various damage states. These methods, defined previously, are:(1) MNExT-ERA, (2) SSI-DATA, (3) EFDD, (4) DSI, (5) OKID-ERA, and (6) GRA. These methods are all developed based on lin-ear system theory. If applied to nonlinear response measurements(i.e., data recorded from nonlinear systems), the identified modalparameters are to be interpreted as effective/equivalent modalparameters or modal parameters of an equivalent linear system.In the linear dynamic models identified using these methods, linearviscous damping is the only source of energy dissipation. There-fore, the effects of other actual sources of energy dissipation, suchas friction, yielding, and impact, are lumped into the equivalentviscous damping. An actual reinforced concrete structure is non-linear from the onset of loading, i.e., even when subjected tolow-amplitude excitation, and is also characterized by a numberof energy dissipation mechanisms. It then follows that the modalidentification results obtained using these methods suffer modelingerrors in addition to estimation errors.

The six system identification methods used in this study arebriefly reviewed in this section. The measured acceleration re-sponses were sampled at 240 Hz, resulting in a Nyquist frequencyof 120 Hz, which is much higher than the modal frequencies ofinterest in this study (< 25 Hz). Before applying the system iden-tification methods to the measured data, all the absolute accelera-tion time histories were band-pass filtered between 0.5 and 25 Hzusing a high order (1024) finite impulse response (FIR) filter. Fur-thermore, the absolute horizontal acceleration measurements fromthe white noise base excitation tests were converted to relative ac-celeration by subtracting the base/table horizontal acceleration.

Output-Only System Identification Methods

The first three algorithms use output-only data (i.e., no informationabout the input excitation is used) and are applied to the measuredvibration data from both the ambient vibration and white noise baseexcitation tests. These methods assume a broadband (ideally whitenoise) input excitation. The more this assumption is violated, thelarger the estimation errors of the identified modal parameters are.The three output-only system identification methods used in thisstudy are briefly described next.

Multiple-Reference Natural Excitation Technique Combinedwith Eigensystem Realization AlgorithmThe basic principle behind the natural excitation technique (NExT)is that the theoretical cross-correlation function between two re-sponse measurements made along two degrees of freedom collectedfrom an ambient (broadband) excited structure has the same ana-lytical form as the free vibration response of the structure (Jameset al. 1993). Once an estimation of the response cross-correlationvector is obtained for a given reference channel, the ERA method(Juang and Pappa 1985) can be used to extract the modal param-eters. A key issue in the application of NExT is to select the refer-ence channel so as to avoid missing modes in the identificationprocess because of the proximity of the reference channel to a mo-dal node. In the MNExT, instead of using a single reference re-sponse channel as in the NExT, a vector of reference channels(three reference channels in this study) is used to obtain an outputcross-correlation matrix (He et al. 2009). The response cross-correlation functions were estimated through inverse Fouriertransformation of the corresponding cross-spectral density (CSD)functions. CSD function was estimated based on Welch-Bartlett’s method using Hanning windows of length 116 s (27,840samples) for white noise base excitation and 15 s (3,600 samples)for ambient vibration, with 50% of window overlap. The estima-ted cross-correlation functions were then downsampled to 80 Hzto improve the computational efficiency of the identification

Table 1. Dynamic Tests Used in This Study

Testnumber Date Test descriptiona

Damagestate

39 11/21/2005 8 min WN (0.03 g) + 3 min AV S0

40 11/21/2005 EQ1

41 11/21/2005 8 min WN (0.03 g) + 3 min AV S1

42 11/21/2005 8 min WN (0.05 g) S1

43 11/21/2005 EQ2

46 11/22/2005 8 min WN (0.03 g) + 3 min AV S2

47 11/22/2005 8 min WN (0.05 g) S2

48 11/22/2005 EQ3

49 11/22/2005 8 min WN (0.03 g) + 3 min AV S3.1

50 11/22/2005 8 min WN (0.05 g) S3.1

61 1/14/2006 8 min WN (0.03 g) + 3 min AV S3.2

62 1/14/2006 EQ4

64 1/14/2006 8 min WN (0.03 g) + 3 min AV S4

65 1/14/2006 8 min WN (0.05 g) S4aWN = white noise base excitation test; AV = ambient vibration test.

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algorithm and, finally, they were used to form Hankel matrices ofsize ð28 × 150Þ × 150 (28 output channels, 150 block rows, and150 columns) for applying ERA in the second stage of the modalidentification. After downsampling, the Nyquist frequency (40 Hz)remains higher than the natural frequencies of the vibration modesof interest (25 Hz, see Fig. 3). The identified natural frequenciesand damping ratios of the vibration modes contributing signifi-cantly to the response (longitudinal, torsional, and coupled longi-tudinal-torsional modes) are reported in Table 2 for the ambientvibration data and in Tables 3 and 4 for the white noise base ex-citation response (output) data.

Data-Driven Stochastic Subspace IdentificationThe SSI-DATA method determines the system model in state-space based on the output-only measurements directly (VanOverschee and de Moor 1996). One advantage of this method com-pared with two-stage time-domain system identification methodssuch as covariance-driven stochastic subspace identification andNExT-ERA is that it does not require any preprocessing of the datato calculate auto/cross-correlation functions or spectra of outputmeasurements. This method involves numerical techniques suchas QR factorization (also known as orthogonal matrix triangulari-zation), singular value decomposition (SVD) and least squares.In the implementation of SSI-DATA, the filtered acceleration datawere downsampled to 80 Hz to increase the computational effi-ciency of the identification algorithm. For each dynamic test, anoutput Hankel matrix was formed, which included 35 block rowswith 28 rows per block (28 longitudinal acceleration channels) forboth white noise and ambient vibration tests, with 24,931 columnsfor white noise tests and 14,331 columns for ambient vibrationtests. The durations of the ambient vibration tests (3 min) and whitenoise base excitation tests (8 min) are different (see Table 1). There-fore, different number of columns were used in forming the corre-sponding Hankel matrices. The natural frequencies and dampingratios of the vibration modes contributing significantly to the re-sponse and identified using SSI-DATA are reported in Table 2

for the ambient vibration data and in Tables 3 and 4 for the whitenoise base excitation response (output) data.

Enhanced Frequency Domain DecompositionThe frequency domain decomposition (FDD) method, a nonpara-metric frequency domain approach, is an extension of the basicfrequency domain approach, also referred to as peak picking tech-nique. According to the FDD technique, the modal parameters areestimated through SVD of the CSD matrix performed at all discretefrequencies. Through this SVD, CSD functions are decomposedinto single-degree-of-freedom (SDOF) CSD functions, each corre-sponding to a single vibration mode of the dynamic system. Con-sidering a lightly damped system, the contribution of differentvibration modes at a particular frequency is limited to a small num-ber (usually one or two). In the EFDD method (Brincker et al.2001), the natural frequency and damping ratio of a vibration modeare identified from the SDOF CSD function corresponding to thatmode. In doing so, the SDOF CSD function is taken back to thetime domain by inverse Fourier transformation, and the frequencyand damping ratio of the mode considered are estimated from thezero-crossing times and the logarithmic decrement, respectively, ofthe corresponding SDOF autocorrelation function. In applying theEFDD method, the measured acceleration data were downsampledto 80 Hz and then the CSD functions were estimated based onWelch-Bartlett’s method using Hanning window with 50% overlap.After estimating the CSD functions, the (28 × 28) response CSDmatrix was singular-value decomposed at each discrete frequency.The modal parameters identified using this method are given inTable 2 based on the ambient vibration data and in Tables 3 and4 based on the white noise base excitation response (output) data.

Input-Output System Identification Methods

The three input-output system identification methods are appliedonly to the 0.03 g RMS white noise base excitation test data, whichincludes the measurement of the input base excitation. These meth-ods are briefly described next.

Table 2. Natural Frequencies and Damping Ratios Identified Based on Ambient Vibration Test Data

State/testnumber System ID method

Natural frequency [Hz] Damping ratio [%]

1st-Lmode

1st-Tmode

1st-L-Tmode

2nd-Lmode

3rd-Lmode

1st-Lmode

1st-Tmode

1st-L-Tmode

2nd-Lmode

3rd-Lmode

S0/test 39 MNExT-ERA 1.92 — 7.05 10.49 24.79 3.0 — 5.0 2.5 0.9

SSI 1.89 — 7.07 10.53 24.58 1.9 — 4.1 2.4 0.4

EFDD 1.90 — 7.22 10.93 24.76 1.3 — 0.2 0.2 0.2

S1/test 41 MNExT-ERA 1.86 — 6.62 10.27 22.98 1.2 — 3.5 3.4 2.4

SSI 1.86 — 6.81 10.24 24.30 2.0 — 2.9 2.6 0.6

EFDD 1.88 — 6.62 10.33 23.45 0.5 — 0.3 0.6 0.0

S2/test 46 MNExT-ERA 1.67 2.04 7.58 10.34 22.78 2.1 1.0 3.4 1.6 1.4

SSI 1.67 2.05 7.58 10.16 22.60 1.3 0.9 3.0 1.4 0.9

EFDD 1.66 2.07 7.54 10.14 22.80 2.5 1.4 0.2 0.2 0.2

S3.1/test 49 MNExT-ERA 1.46 — 7.21 10.06 21.89 3.4 — 2.7 1.4 1.7

SSI 1.46 1.91 7.18 9.28 21.60 2.9 1.4 2.1 1.1 1.6

EFDD 1.44 — 7.03 9.28 21.81 0.8 — 0.1 0.2 0.1

S3.2/test 61 MNExT-ERA 1.58 1.95 — 8.39 22.97 1.5 0.5 — 2.3 1.2

SSI 1.58 1.95 — 8.52 22.83 1.4 1.1 — 2.8 1.0

EFDD 1.58 1.95 — 8.44 22.87 2.0 1.1 — 0.3 0.1

S4/test 64 MNExT-ERA 1.02 — — 5.68 15.04 1.2 — — 2.4 0.8

SSI 1.02 — — 5.69 15.05 1.0 — — 1.8 1.3

EFDD 1.00 — — 5.74 15.10 1.7 — — 0.2 0.1

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Deterministic-Stochastic Subspace IdentificationThe deterministic-stochastic state-space model for linear time-invariant systems can be written as

�xðk þ 1Þ ¼ AxðkÞ þ BuðkÞ þ wðkÞyðkÞ ¼ CxðkÞ þ DuðkÞ þ vðkÞ ð1Þ

where A, B, C, and D = state-space matrices; uðkÞ and yðkÞ = inputand output vectors, respectively; and xðkÞ = state vector. In thedeterministic-stochastic model, the process noise wðkÞ representsdisturbances (small unmeasured excitations) and modeling inaccur-acies, whereas the measurement noise vðkÞ models the sensor in-accuracies. However, in the SSI-DATA, both noise terms (w and v)also include implicitly the input excitation, since it is impossible todistinguish the input information from the noise terms. Consideringthe assumptions that (1) the deterministic input uðkÞ is uncorrelated

with both the process noise wðkÞ and the measurement noise vðkÞ,and (2) both noise terms are not identically zero, a robust identi-fication algorithm was developed by Van Overschee and de Moor(1996) to identify the state-space matrices in the combined deter-ministic-stochastic system. Similar to SSI-DATA, numerical tech-niques such as QR factorization, SVD, and least squares are used inthis method. In the application of this method, the measured input(shake table acceleration) and the relative horizontal accelerationresponse data were downsampled to 80 Hz. For each dynamic test,an input-output Hankel matrix was formed, which included 35block rows with 29 rows each (1 input and 28 output channels)and 24,931 columns using the downsampled data. The naturalfrequencies and damping ratios of the first five significant vibrationmodes (first three longitudinal, first torsional, and first coupled lon-gitudinal-torsional) identified from the 0.03 g RMS white noisebase excitation test data are reported in Table 3.

Table 3. Natural Frequencies and Damping Ratios Identified Based on 0.03 g RMS White Noise Base Excitation Test Data

State/test number System ID method

Natural frequency [Hz] Damping ratio [%]

1st-Lmode

1st-Tmode

1st-L-Tmode

2nd-Lmode

3rd-Lmode

1st-Lmode

1st-Tmode

1st-L-Tmode

2nd-Lmode

3rd-Lmode

S0/test 39 MNExT-ERA 1.71 2.35 8.64 11.49 24.67 3.1 �1:1 5.4 4.8 1.3

SSI 1.66 — 8.67 11.65 24.61 2.2 — 3.6 4.4 0.2

EFDD 1.72 — 8.52 11.88 24.64 2.6 — 1.3 0.5 0.4

DSI 1.72 1.80 9.03 10.70 24.55 2.2 3.5 4.2 4.5 0.5

OKID-ERA 1.70 — 8.95 10.78 25.06 1.7 — 0.1 1.5 1.4

GRA 1.71 — 9.05 11.05 24.31 2.1 — 1.8 1.8 0.5

S1/test 41 MNExT-ERA 1.54 2.34 8.51 11.35 24.65 4.3 1.1 4.7 7.7 0.9

SSI 1.51 1.73 8.37 11.25 24.57 2.4 4.0 4.9 3.3 0.2

EFDD 1.49 — 8.17 11.33 24.56 3.3 — 1.0 0.4 0.2

DSI 1.57 1.78 8.76 10.67 24.42 3.1 2.9 3.9 3.2 0.3

OKID-ERA 1.60 — — 10.75 24.57 3.8 — — 0.3 1.9

GRA 1.54 — 8.65 10.98 24.28 2.0 — 1.1 1.7 0.2

S2/test 46 MNExT-ERA 1.22 — 7.42 10.88 21.26 3.4 — 6.0 0.3 3.9

SSI 1.25 — 7.26 11.1 22.82 4.5 — 4.2 4.9 2.0

EFDD 1.29 — 7.43 10.75 21.08 5.2 — 0.9 0.3 0.2

DSI 1.27 2.02 7.67 10.23 21.56 3.5 11.6 6.7 4.8 2.7

OKID-ERA 1.24 — 7.86 10.93 21.08 2.8 — �0:3 1.7 2.9

GRA 1.24 — 7.60 11.11 21.59 3.0 — 4.0 2.9 0.5

S3.1/test 49 MNExT-ERA 1.11 — 7.01 10.24 19.77 3.5 — 8.6 7.0 5.6

SSI 1.13 2.13 7.14 9.87 20.40 3.9 10.7 3.7 1.9 2.3

EFDD 1.13 — 7.07 10.06 19.83 4.0 — 0.3 0.2 0.1

DSI 1.14 2.20 7.20 — 20.42 3.3 13.3 4.9 — 2.7

OKID-ERA 1.17 — 7.15 10.47 20.40 5.6 — 0.2 1.8 1.1

GRA 1.14 — 7.32 9.77 19.68 3.9 — 4.3 1.6 0.5

S3.2/test 61 MNExT-ERA 1.18 2.37 — 11.00 21.32 5.0 0 — 6.0 5.7

SSI 1.20 2.62 — 10.89 21.04 3.5 4.3 — 5.4 2.0

EFDD 1.23 — — 10.41 20.68 4.6 — — 0.4 0.1

DSI 1.22 — — — 21.47 3.6 — — — 2.7

OKID-ERA 1.20 — — — 20.47 2.8 — — — 2.8

GRA 1.20 — — 10.45 21.11 3.5 — — 2.0 0.3

S4/test 64 MNExT-ERA 0.83 — — 4.68 13.24 3.3 — — 7.6 0.4

SSI 0.85 — — 4.68 14.02 5.6 — — 5.5 2.9

EFDD 0.86 — — 4.71 13.81 3.8 — — 0.4 0.2

DSI 0.85 — — 4.72 13.31 3.6 — — 6.1 4.8

OKID-ERA 0.87 — — 4.79 14.69 2.9 — — 3.6 0.0

GRA 0.88 — — 4.81 13.29 5.5 — — 3.8 0.9

710 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / JUNE 2011

Observer/Kalman Filter Identification Combined with ERAIn this method, developed by Phan et al. (1992), the system input-output relationship is expressed in terms of an observer, which ismade asymptotically stable by an embedded eigenvalue assignmentprocedure. The prescribed eigenvalues for the observer may be real,complex, mixed real and complex, or zero (i.e., deadbeat observer).In this formulation, the Markov parameters of the observer areidentified from input-output data. The Markov parameters of theactual system are then calculated from those of the observer andthen used to obtain a state-space model of the system by ERA.The method performs quite well for finite-dimensional systemsunder the following conditions: (1) the input-output data time his-tories are sufficiently long, (2) the noise is white and zero-mean,and (3) the noise-to-signal ratio is small (Lus et al. 2002). This ap-proach is practical, especially for single input systems, as in thecase of structures subjected to single component base excitation.In the application of OKID-ERA, the filtered and then detrended(by subtracting the mean value) relative acceleration responsesfrom the white noise base excitation tests were downsampled to60 Hz. An output matrix Y ¼ ½yð0Þ…yðpÞ…yðl� 1Þ� of size(28 × 1;000) (i.e., l ¼ 1;000) and an input-output matrix V of size(7;251 × 1;000) (i.e., p ¼ 250) are formed based on inputs uðkÞ,outputs yðkÞ, and input-output vectors

νðkÞ ¼ uðkÞyðkÞ

� �

as

V ¼uð0Þ uð1Þ … uðpÞ … uðl� 1Þ0 νð0Þ … νðp� 1Þ … νðl� 2Þ... ..

. . .. ..

. . .. ..

.

0 0 … νð0Þ … νðl� p� 1Þ

26664

37775 ð2Þ

The Markov parameters of the observer systemM are estimatedas M ¼ YV†, where V† = pseudoinverse of V. Once the true sys-tem’s Markov parameters are retrieved from the observer’s Markovparameters, the ERA is used for estimation of the modal parame-ters, which are reported in Table 3.

General Realization AlgorithmGRA (de Callafon et al. 2008) is a system realization algorithm toidentify modal parameters of linear dynamic systems based on gen-eral input-output data. This algorithm is an extension of ERA.Whereas ERA uses SVD of a Hankel matrix constructed from im-pulse response or free vibration response data, GRA uses SVD of a

weighted Hankel matrix, in which the weighting is defined by theloading. Using GRA, the state-space matrices are estimated in atwo-step process consisting of a state reconstruction, followedby a least squares optimization, yielding a minimum prediction er-ror for the response. In the application of GRA, a weighted Hankelmatrix of size ð28 × 50Þ × 7;000 is formed based on the input-output data. After performing an SVD of this weighted Hankel ma-trix, a state-space realization of the system is estimated from whichthe modal parameters are extracted. The natural frequencies anddamping ratios identified using GRA are reported in Table 3.

Modal Identification Results

Modal parameters of the test structure were identified using the sys-tem identification methods previously outlined based on output-only (for the first three methods) and input-output (for the last threemethods) data measured from low-amplitude dynamic tests (i.e.,ambient vibration tests and 0.03 g and 0.05 g RMS white noisebase excitation tests) performed at various damage states (S0,S1, S2, S3.1, S3.2, and S4). Damage state S0 is defined as theundamaged (baseline) state of the structure before its exposureto the first seismic excitation (EQ1), whereas damage states S1,S2, S3, and S4 correspond to the state of the structure after expo-sure to the first (EQ1), second (EQ2), third (EQ3), and fourth(EQ4) seismic excitations, respectively (see Table 1). Damage stateS0 does not correspond to the uncracked state of the structure, sincethe structure had already been subjected to low-amplitude whitenoise base excitations (0.02–0.03 g RMS) for the purposes ofchecking the instrumentation and data acquisition system and tun-ing the shake table controller. During damage state S3, the bracingsystem between the slabs of the test specimen and the posttensionedcolumn was modified (strengthened and stiffened). Therefore, dam-age state S3 is subdivided into state S3.1 (before modification ofthe braces) and state S3.2 (after modification of the braces). Theidentified modal parameters are presented and discussed in the fol-lowing three subsections based on the type of excitation (ambient,0.03 g, and 0.05 g RMS white noise base excitations).

Modal Parameters Identified Based on AmbientVibration Test Data

The modal parameters identified based on acceleration data fromambient vibration tests are discussed in this section. Fig. 4 shows inpolar plots the complex-valued mode shapes of the five most sig-nificantly excited modes of the test structure identified using SSI-DATA using data from test 46 (damage state S2). The real parts of

Table 4. Natural Frequencies and Damping Ratios Identified Based on 0.05 g RMS White Noise Base Excitation Test Data

State/testnumber

System IDmethod

Natural frequency [Hz] Damping ratio [%]

1st-L mode 2nd-L mode 3rd-L mode 1st-L mode 2nd-L mode 3rd-L mode

S1/test 41 MNExT-ERA 1.39 10.79 24.26 4.1 7.6 2.2

SSI 1.40 11.38 24.29 3.6 3.5 1.4

EFDD 1.40 10.97 24.39 2.8 3.3 1.1

S2/test 46 MNExT-ERA 1.12 10.41 20.29 5.3 7.7 4.3

SSI 1.14 10.24 22.46 3.5 2.4 1.1

EFDD 1.12 10.09 20.09 4.3 3.6 1.6

S3.1/test 49 MNExT-ERA 1.05 10.22 19.12 4.4 6.8 7.9

SSI 1.06 10.23 18.98 3.7 3.3 4.7

EFDD 1.04 10.10 18.88 4.4 6.3 1.1

S4/test 64 MNExT-ERA 0.80 4.62 13.03 3.4 9.7 13.3

SSI 0.81 4.52 13.67 4.6 6.1 8.4

EFDD 0.82 4.62 13.29 3.1 3.1 2.3

JOURNAL OF STRUCTURAL ENGINEERING © ASCE / JUNE 2011 / 711

these mode shapes are displayed in Fig. 5. The five most significantvibration modes identified at this damage state consist of the firstthree longitudinal (1st-L, 2nd-L, and 3rd-L), the first torsional(1st-T), and the first coupled longitudinal-torsional (1st-L-T)modes. The polar plot representation of a mode shape provides in-formation on the degree of nonclassical (or nonproportional)damping (Veletsos and Ventura 1986) characteristics of that mode.If all components of a mode shape (each component being repre-sented by a vector in a polar plot) are collinear, that vibrationmode is classically damped. The more scattered the mode shapecomponents are in the complex plane, the more the system is non-classically (nonproportionally) damped in that mode. However,measurement noise (low signal-to-noise ratio), estimation errors,and modeling errors can also cause a truly classically damped

vibration mode to be identified as nonclassically damped. FromFig. 4, it is observed that the first longitudinal and first torsionalmodes at damage state S2 are identified as perfectly classicallydamped. Some degree of nonproportional damping is identifiedfor the other modes (2nd-L, 3rd-L, and 1st-L-T).

The natural frequencies and damping ratios of the five most sig-nificantly excited modes identified based on ambient vibration ac-celeration data are given in Table 2 for all damage states consideredand the three output-only system identification methods used(MNExT-ERA, SSI-DATA, EFDD). From Table 2, the followingare observed:1. The natural frequencies identified using different methods are

in good agreement at each damage state, whereas the identifieddamping ratios display larger variability across the differentsystem identification methods.

2. The identified natural frequencies of the longitudinal modes(1st-L, 2nd-L, and 3rd-L) decrease consistently with increasinglevel of damage, except from damage states S3.1 to S3.2, whenthe steel braces were modified (stiffened), whereas the identi-fied damping ratios do not exhibit a clear trend as a function ofstructural damage

3. The first torsional vibration mode (1st-T) could only be iden-tified at damage states S2, S3.1, and S3.2.

4. The identified damping ratios are in a reasonable range(0–5%).

5. The modal damping ratios identified using EFDD are system-atically lower than their counterparts identified using the othertwo methods. This can be attributable to the fact that in theapplication of EFDD, a high modal assurance criterion(MAC) value is used to decompose the noisy CSD functionsinto narrow peak (low-damped) SDOF CSD functions.Modal assurance criterion values (Allemang and Brown 1982)

were computed to compare the corresponding complex-valuedmode shapes identified using the three output-only system identi-fication methods considered, and they are reported in Table 5. Thehigh MAC values obtained in most cases (each case being definedby a vibration mode, a pair of system identification methods, and adamage state) indicate a good agreement in general between themode shapes identified using different methods. From Table 5,

Fig. 5. Vibration mode shapes of the building at damage state S2 ob-tained using SSI-DATA based on ambient vibration data

Table 5. MAC Values between Corresponding Mode Shapes Identified Using Different Methods Based on Ambient Vibration Test Data

State/test number System ID method 1st-L mode 1st-T mode 1st-L-T mode 2nd-L mode 3rd-L mode

S0/test 39 MNExT and SSIa 0.99 — 0.26 0.86 0.70

MNExT and EFDD 0.99 — 0.46 0.49 0.74

SSI and EFDD 1.00 — 0.82 0.77 0.97

S1/test 41 MNExT and SSI 0.91 — 0.78 0.89 0.79

MNExT and EFDD 0.97 — 0.84 0.89 0.88

SSI and EFDD 0.79 — 0.84 0.98 0.89

S2/test 46 MNExT and SSI 1.00 0.98 0.98 0.56 0.95

MNExT and EFDD 1.00 0.98 0.96 0.43 0.96

SSI and EFDD 1.00 1.00 0.98 0.97 0.95

S3.1/test 49 MNExT and SSI 0.99 — 0.92 0.39 0.93

MNExT and EFDD 1.00 — 0.78 0.49 0.96

SSI and EFDD 0.99 — 0.92 0.96 0.98

S3.2/test 61 MNExT and SSI 1.00 0.97 — 1.00 0.97

MNExT and EFDD 1.00 0.98 — 1.00 0.98

SSI and EFDD 1.00 0.98 — 1.00 0.99

S4/test 64 MNExT and SSI 1.00 — — 0.99 0.96

MNExT and EFDD 1.00 — — 0.99 0.98

SSI and EFDD 1.00 — — 1.00 0.98aSSI denotes SSI-DATA and MNEXT denotes MNEXT-ERA.

1st-L 1st-T 1st-L-T 2nd-L 3rd-L

Fig. 4. Polar plot representation of complex-valued mode shapes of thebuilding at damage state S2 obtained using SSI-DATA based on am-bient vibration data

712 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / JUNE 2011

it is also observed that the MAC values between correspondingmode shapes identified using SSI-DATA and EFDD are, in general,the highest among the three combinations of system identificationmethods (i.e., MNExT-ERA and SSI-DATA, MNExT-ERA andEFDD, and SSI-DATA and EFDD). The low MAC values betweenmode shapes identified using MNExT-ERA and SSI-DATA andMNExT-ERA amd EFDD for the 1st-L-T mode at damage stateS0 and 2nd-L mode at damage states S2 and S3.1 imply that thesemode shapes identified using MNExT-ERA are the least accurateamong the three output-only methods considered. This could bebecause, in the application of MNExT-ERA, if the selected refer-ence channels are close to modal nodes of a vibration mode, theestimation uncertainty of that mode is relatively large (Heet al. 2009).

Modal Parameters Identified Based on 0.03 g RMSWhite Noise Base Excitation Test Data

The modal parameters identified based on acceleration data from0.03 g RMS white noise base excitation tests are presented anddiscussed in this section. Fig. 6 shows in polar plots the complex-valued mode shapes of the five most significant modes of the build-ing identified using MNExT-ERA based on data from test 39(damage state S0). The five most significant vibration modes iden-tified in this case consist of the first three longitudinal (1st-L,2nd-L, and 3rd-L), the first torsional (1st-T), and the first coupledlongitudinal-torsional (1st-L-T) modes. From Fig. 6, it is observedthat the first longitudinal (1st-L) vibration mode is identified asnearly perfectly classically damped, whereas the other identifiedmodes (1st-T, 1st-L-T, 2nd-L, and 3rd-L) display some nonclassicaldamping characteristics.

Table 3 reports the natural frequencies and damping ratios of thefive most significantly excited vibration modes identified using thesix system identification methods based on acceleration data fromthe 0.03 g RMS white noise base excitation tests at all damagestates considered. From Table 3, the following are observed:1. The natural frequencies identified using different methods are

reasonably consistent at each damage state, whereas the iden-tified damping ratios exhibit much larger variability. It appearsthat when a mode is not significantly excited, its modal damp-ing identified using EFDD is consistently very low comparedwith other methods.

2. Some vibration modes (especially the first torsional mode)could not be identified by each of the system identificationmethods.

3. The natural frequencies of all identified vibration modes de-crease consistently with increasing level of structural damage,except from damage state S3.1 to S3.2, when the steel braceswere modified (stiffened), whereas the identified damping ra-tios do not exhibit a clear trend as a function of damage.

4. At each damage state, the identified modal parameters of thefirst longitudinal mode (1st-L) appear to be the least sensitive

to the identification method used, which could be because ofthe predominant contribution of this mode to the total re-sponse.Table 6 reports the MAC values calculated between corres-

ponding complex-valued mode shapes identified using the threeoutput-only system identification methods (i.e., MNExT-ERA andSSI-DATA, MNExT-ERA and EFDD, and SSI-DATA and EFDD),the three input-output system identification methods (i.e., DSI andOKID-ERA, DSI and GRA, and OKID-ERA and GRA), and alsobetween corresponding mode shapes identified using an output-only and an input-output system identification method (SSI-DATAand GRA). From the results in Table 6, the following are observed:1. The MAC values calculated for the 1st-L mode shape identi-

fied using different methods are very close to unity for alldamage states considered (i.e., low level of estimation uncer-tainty), implying that these identified mode shapes are prob-ably close to the actual 1st-L mode shape of the structure.

2. Lower MAC values obtained for the 1st-L-T mode shapeindicate its larger estimation uncertainty compared with othermodes.

3. In general, the MAC values calculated across output-onlymethods are higher than those calculated across input-outputmethods. The output-only methods perform very well (are veryaccurate) when the hypothesis on which they are developed aresatisfied (i.e., when the input excitation is a white noisesequence).

4. The MAC values between corresponding mode shapes identi-fied using SSI-DATA (output-only method) and GRA (input-output method) are, in general, high, indicating that thecorresponding mode shapes are in good agreement, especiallyfor the three longitudinal mode shapes (i.e., 1st-L, 2nd-L, and3rd-L).

Modal Parameters Identified Based on 0.05 g RMSWhite Noise Base Excitation Test Data

To investigate the effect of excitation amplitude (and therefore thelevel of response nonlinearity) on the identified effective modalparameters, the modal parameters of the first three longitudinal vi-bration modes are also identified based on acceleration data from0.05 g RMS white noise base excitation tests and are presented anddiscussed in this section. Table 4 reports the natural frequencies anddamping ratios of the three longitudinal vibration modes identifiedbased on acceleration data from 0.05 g RMS white noise base ex-citation tests at damage states S1, S2, S3.1, and S4. From Table 4,the following are observed:1. The natural frequencies identified using different methods are

reasonably consistent at each damage state, whereas the iden-tified damping ratios exhibit much larger variability.

2. The identified natural frequencies of all identified vibrationmodes decrease with increasing level of structural damage,whereas the identified damping ratios do not exhibit a cleartrend as a function of damage.Figs. 7 and 8 show the natural frequencies and damping ratios of

the three longitudinal modes, respectively, identified using the threeoutput-only methods based on ambient vibration, 0.03 g, and 0.05 gRMS white noise base excitation test data. By comparing the modalparameters identified at different levels of excitation (i.e., ambient,0.03 g, and 0.05 g RMS white noise base excitation), it is observedthat the (effective) natural frequencies of the three longitudinalvibration modes identified based on higher amplitude response dataare, in general, lower than their counterparts identified based onlower amplitude response data at all damage states considered,except between ambient vibration and 0.03g white noise for thesecond mode, and except at damage state S1 for the third mode.

1st-L 1st-T 1st-L-T 2nd-L 3rd-L

Fig. 6. Polar plot representation of complex-valued mode shapes of thebuilding at damage state S0 obtained using MNExT-ERA based onwhite noise base excitation test data

JOURNAL OF STRUCTURAL ENGINEERING © ASCE / JUNE 2011 / 713

The second longitudinal mode has a higher estimation error due tothe proximity of its natural frequency to the oil column frequencyof the shake table, combined with the fact that the output-only sys-tem identification methods used assume broadband excitation. Thedifference is much more significant for the first longitudinal vibra-tion mode (an average of 20% reduction in the identified first modalfrequencies from ambient vibration data to 0.03 g white noise baseexcitation data and 26% reduction from ambient vibration data to0.05 g white noise base excitation data). This is most likely becausethe test structure is nonlinear (even at the relatively low levels ofexcitation considered in this system identification study), with ef-fective modal parameters depending strongly on the intensity of theexcitation and therefore of the structural response, reflecting the

fact that concrete cracks open wider with increasing intensity ofthe excitation. The first natural frequency of the uncracked struc-ture identified based on ambient vibration data (2.23 Hz obtainedusing peak picking method) is significantly higher than its counter-part for the cracked structure at the undamaged state S0 (average of1.90 Hz over the three output-only methods). It is also observedthat, in general, higher modal damping ratios are identified forthe three longitudinal vibration modes during the higher amplitude(0.05 g RMS) base excitation tests. There is an average relativeincrease of 110% in damping ratios of the three longitudinal modesfrom ambient vibration data to 0.03 g white noise base excitationdata and 230% from ambient vibration data to 0.05 g white noisebase excitation data. This reflects the additional hysteretic damping

Table 6. MAC Values between Corresponding Mode Shapes Identified Using Different Methods Based on 0.03 g RMS White Noise Base Excitation TestData

State/test number System ID method 1st-L mode 1st-T mode 1st-L-T mode 2nd-L mode 3rd-L mode

S0/test 39 MNExT and SSI 1.00 — 0.90 0.96 0.99

MNExT and EFDD 0.99 — 0.82 0.94 0.97

SSI and EFDD 1.00 — 0.82 0.97 0.99

DSI and OKID 1.00 — 0.58 0.65 0.92

DSI and GRA 1.00 — 0.66 0.24 0.80

OKID and GRA 1.00 — 0.76 0.71 0.94

SSI and GRA 1.00 — 0.70 0.95 0.98

S1/test 41 MNExT and SSI 1.00 0.96 0.51 0.99 0.99

MNExT and EFDD 1.00 — 0.75 0.96 0.99

SSI and EFDD 1.00 — 0.49 0.99 1.00

DSI and OKID 1.00 — — 0.54 0.83

DSI and GRA 1.00 — 0.49 0.39 0.88

OKID and GRA 1.00 — — 0.90 0.56

SSI and GRA 1.00 — 0.30 0.99 0.99

S2/test 46 MNExT and SSI 1.00 — 0.76 0.97 0.67

MNExT and EFDD 1.00 — 0.82 0.98 0.86

SSI and EFDD 1.00 — 0.81 0.99 0.87

DSI and OKID 1.00 — 0.24 0.20 0.91

DSI and GRA 1.00 — 0.59 0.22 0.64

OKID and GRA 1.00 — 0.44 0.89 0.72

SSI and GRA 1.00 — 0.73 0.99 0.98

S3.1/test 49 MNExT and SSI 1.00 — 0.54 0.97 0.73

MNExT and EFDD 1.00 — 0.42 0.98 0.76

SSI and EFDD 1.00 — 0.80 0.99 0.75

DSI and OKID 1.00 — 0.48 — 0.88

DSI and GRA 1.00 — 0.65 — 0.86

OKID and GRA 1.00 — 0.84 0.78 0.94

SSI and GRA 1.00 — 0.74 0.95 0.90

S3.2/test 61 MNExT and SSI 1.00 0.70 — 0.94 0.99

MNExT and EFDD 1.00 — — 0.85 0.89

SSI and EFDD 1.00 — — 0.84 0.90

DSI and OKID 1.00 — — — 0.91

DSI and GRA 1.00 — — — 0.90

OKID and GRA 1.00 — — — 0.97

SSI and GRA 1.00 — — 0.72 0.98

S4/test 64 MNExT and SSI 1.00 — — 0.99 0.51

MNExT and EFDD 1.00 — — 0.96 0.58

SSI and EFDD 1.00 — — 0.96 0.94

DSI and OKID 1.00 — — 0.97 0.68

DSI and GRA 1.00 — — 0.94 0.91

OKID and GRA 1.00 — — 0.93 0.77

SSI and GRA 1.00 — — 0.97 0.73

714 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / JUNE 2011

Nat

ural

Fre

quen

cies

[H

z]

1st-L

Mod

e

2nd-L

Mod

e

3rd -

L M

ode

Ambient Vibration 0.03g White Noise 0.05g White Noise

Damage State

Fig. 7. Natural frequencies of the first three longitudinal modes identified based on ambient and white noise base excitation (0.03 g and 0.05 g RMS)test data using different methods

Dam

ping

Rat

ios

[%]

1st-L

Mod

e

2nd-L

Mod

e

3rd -

L M

ode

Ambient Vibration 0.03g White Noise 0.05g White Noise

Damage State

Fig. 8. Damping ratios of the first three longitudinal modes identified based on ambient and white noise base excitation (0.03 g and 0.05 g RMS) testdata using different methods

JOURNAL OF STRUCTURAL ENGINEERING © ASCE / JUNE 2011 / 715

at a higher level of response nonlinearity, which is identified asequivalent linear viscous damping by the linear system identifica-tion methods used in this study.

Summary and Conclusions

A full-scale 7-story reinforced concrete building slice was tested onthe UCSD-NEES shake table during the period from October 2005to January 2006. The shake table tests were designed to damage thebuilding progressively through several historical earthquake groundmotions reproduced on the shake table. At several levels of damage,ambient vibration tests and low-amplitude white noise base exci-tation tests were applied to the building, which responded as aquasi-linear system with dynamic parameters evolving as a func-tion of structural damage. Six different state-of-the-art system iden-tification methods, including three output-only and three input-output methods, were used to estimate the modal parameters (natu-ral frequencies, damping ratios, and mode shapes) of the building inits undamaged (baseline) and various damage states based on theresponse of the building to ambient as well as white noise baseexcitations measured using 28 DC-coupled accelerometers.

From the results of this system identification study, the follow-ing are observed:1. The natural frequencies identified using different methods are

reasonably consistent at each damage state, whereas the iden-tified damping ratios exhibit much larger variability across sys-tem identification methods.

2. The identified natural frequencies of the three longitudinal vi-bration modes decrease with increasing level of damage exceptfrom damage state S3.1 to S3.2, when the steel braces werestiffened. The identified modal damping ratios do not followa clear trend as a function of structural damage.

3. At each damage state, the identified modal parameters of thefirst longitudinal mode appear to be the least sensitive to theidentification method used, which is most likely due to the pre-dominant contribution of this mode to the total response.

4. The (effective) natural frequencies of the three longitudinal vi-bration modes identified based on higher amplitude responsedata are, in general, lower than their counterparts identifiedbased on lower amplitude response data at all damage statesconsidered, especially for the first longitudinal vibration mode.This is most likely because the test structure is nonlinear (evenat the relatively low levels of excitation considered in this sys-tem identification study), with effective modal parameters de-pending strongly on the amplitude of the excitation andtherefore of the structural response.

5. In general, higher modal damping ratios are identified for thethree longitudinal vibration modes during the higher amplitudebase excitation tests. This is because the additional hystereticdamping at higher level of response nonlinearity is identified asequivalent viscous damping by the linear system identificationmethods used in this paper. It should be emphasized that these“inflated” identified equivalent viscous damping ratios are notto be used to represent the viscous damping component in anonlinear FE model of a structure that explicitly accountsfor the nonlinear material behavior. Panagiotou (2008) showedthat the viscous damping ratios required to best “reproduce”computationally, using nonlinear dynamic time history analy-sis, the experimentally measured response are significantlylower than those identified in this study.

Also in this study, it was found that the modal parameters (naturalfrequencies and mode shapes) computed from a linear elasticfinite-element model of the building structure developed based

on gross section properties are in good agreement with their exper-imental counterparts identified based on ambient vibration data re-corded from the uncracked structure (Moaveni 2007). The modalparameters identified at various damage states in this study providethe input for damage identification of this 7-story building using asensitivity-based finite-element model updating strategy (Moaveniet al. 2010).

This linear system identification study performed on a denselyinstrumented full-scale reinforced concrete building specimenraises the need to apply to this type of testbed structures (1) prob-abilistic system identification methods (e.g., Bayesian system iden-tification) to consistently treat the quantification and propagation ofthe various sources of aleatory and epistemic (modeling) uncer-tainty affecting the system identification results; and (2) nonlinearsystem identification methods (e.g., nonlinear finite-element modelupdating) to capture the various sources (e.g., material, geometric,contact) of nonlinearity affecting the structural response and to usethe dynamic output and possibly input data of the damaging event(e.g., earthquake) in the identification process.

Acknowledgments

Partial support for this research from Lawrence Livermore NationalLaboratory with Dr. David McCallen as program leader and fromEnglekirk Center Board of Advisors are gratefully acknowledged.The authors would also like to thank Dr. Ozgur Ozcelik and thetechnical staff at the Englekirk Structural Engineering Center fortheir help in collecting the test data. Any opinions, findings, andconclusions or recommendations expressed in this material arethose of the authors and do not necessarily reflect those of thesponsors.

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