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Use of Wavelet-Based Damage-Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data Hae Young Noh, S.M.ASCE 1 ; K. Krishnan Nair 2 ; Dimitrios G. Lignos, A.M.ASCE 3 ; and Anne S. Kiremidjian, M.ASCE 4 Abstract: This paper introduces three wavelet-based damage-sensitive features (DSFs) extracted from structural responses recorded during earthquakes to diagnose structural damage. Because earthquake excitations are nonstationary, the wavelet transform, which represents data as a weighted sum of time-localized waves, is used to model the structural responses. These DSFs are defined as functions of wavelet energies at particular frequencies and specific times. The first DSF (DSF 1 ) indicates how the wavelet energy at the original natural frequency of the structure changes as the damage progresses. The second DSF (DSF 2 ) indicates how much the wavelet energy is spread out in time. The third DSF (DSF 3 ) reflects how slowly the wavelet energy decays with time. The performance of these DSFs is validated using two sets of shake- table test data. The results show that as the damage extent increases, the DSF 1 value decreases and the DSF 2 and DSF 3 values increase. Thus, these DSFs can be used to diagnose structural damage. The robustness of these DSFs to different input ground motions is also investigated using a set of simulated data. DOI: 10.1061/(ASCE)ST.1943-541X.0000385. © 2011 American Society of Civil Engineers. CE Database subject headings: Assessment; Monitoring; Stationary processes; Earthquake engineering; Diagnosis; Structural failures; Frequency analysis; Natural frequency. Author keywords: Assessment; Monitoring; Nonstationary processes; Earthquake engineering; Diagnosis; Structural failures; Frequency analysis; Natural frequency. Introduction After an extreme event, such as an earthquake or a hurricane, im- mediate diagnosis of structural damage is of great importance to facilitate emergency responses and to prevent further losses and injuries. Similarly, decisions on repair and rehabilitation after such events can be greatly facilitated with information on the type, lo- cation, and extent of damage. Currently, damage diagnosis is often achieved by visual inspection by professionals. Human inspection, however, can be costly in time and money, perilous in certain situations, inconsistent, limited to only surface inspections, and in- appropriate for immediate assessment of structures over large areas. Therefore, it is necessary to develop an automated system that efficiently and reliably diagnoses structural damage. Consequently, extensive research has been conducted on structural health moni- toring (SHM) in the structural engineering community. Recent developments in sensor technologies and wireless communication systems along with advances in damage diagnosis algorithms have brought us closer to the realization of structural damage assessment without conducting visual inspections. To assess structural damage immediately after an earthquake using the structural responses recorded during the strong motion, it is necessary to develop an algorithm that directly utilizes these recordings. Currently, there are no methods that enable us to diag- nose structural damage using the earthquake responses. Previous work on wireless damage detection algorithms has focused on the use of ambient vibrations obtained before and immediately after the occurrence of an extreme event such as an earthquake (Sohn and Farrar 2001; Nair et al. 2006; Farrar and Worden 2007). Because these previous damage diagnosis algorithms use ambient vibration measurements that are stationary, they cannot be used with earthquake motions that are nonstationary. For this purpose, we have developed a new method that uses wavelet energies of input ground motions and structural acceleration responses to detect damage in the structure from strong earthquake motion. The main advantage of this approach is that it captures the nonstationary characteristics of both earthquake ground motions and structural responses by using the wavelet transform. While the diagnosis algorithm proposed in this paper may be particularly suitable for embedding in wireless monitoring systems, it is not restricted to implementation for such systems and can easily be applied to wired systems. Early work using wavelet analysis for SHM has been carried out from several different perspectives. From a system identi- fication perspective, Basu and Gupta (1997) applied wavelet analysis to obtain the spectral moments and peak structural responses of multi-degree-of-freedom (MDOF) systems subjected to nonstationary seismic excitations. Ghanem and Romeo (2000) also represented the equation of motion in terms of a wavelet basis and solved the inverse problem to estimate time-varying system parameters. Similarly, Kijewski and Kareem (2003) used wavelet analysis for system identification, and Basu (2005) and 1 Postdoctoral Scholar, Dept. of Civil and Environmental Engineering, Stanford Univ., Stanford, CA 94305 (corresponding author). E-mail: [email protected] 2 Senior Financial Modeler, Risk Management Solutions, 7015 Gateway Blvd., Newark, CA 94560-1011. 3 Assistant Professor, Dept. of Civil Engineering and Applied Mechanics, McGill Univ., Montreal, QC H3A2K6, Canada. 4 Professor, Dept. of Civil and Environmental Engineering, Stanford Univ., Stanford, CA 94305. Note. This manuscript was submitted on August 12, 2009; approved on May 13, 2011; published online on January 15, 2011. Discussion period open until March 1, 2012; separate discussions must be submitted for in- dividual papers. This paper is part of the Journal of Structural Engineer- ing, Vol. 137, No. 10, October 1, 2011. ©ASCE, ISSN 0733-9445/2011/ 10-12151228/$25.00. JOURNAL OF STRUCTURAL ENGINEERING © ASCE / OCTOBER 2011 / 1215 J. Struct. Eng. 2011.137:1215-1228. Downloaded from ascelibrary.org by California Institute of Technology on 08/30/14. Copyright ASCE. For personal use only; all rights reserved.

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Page 1: Use of Wavelet-Based Damage-Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data

Use of Wavelet-Based Damage-Sensitive Features forStructural Damage Diagnosis Using Strong Motion Data

Hae Young Noh, S.M.ASCE1; K. Krishnan Nair2;Dimitrios G. Lignos, A.M.ASCE3; and Anne S. Kiremidjian, M.ASCE4

Abstract: This paper introduces three wavelet-based damage-sensitive features (DSFs) extracted from structural responses recorded duringearthquakes to diagnose structural damage. Because earthquake excitations are nonstationary, the wavelet transform, which represents data asa weighted sum of time-localized waves, is used to model the structural responses. These DSFs are defined as functions of wavelet energies atparticular frequencies and specific times. The first DSF (DSF1) indicates how the wavelet energy at the original natural frequency of thestructure changes as the damage progresses. The second DSF (DSF2) indicates how much the wavelet energy is spread out in time. The thirdDSF (DSF3) reflects how slowly the wavelet energy decays with time. The performance of these DSFs is validated using two sets of shake-table test data. The results show that as the damage extent increases, the DSF1 value decreases and the DSF2 and DSF3 values increase. Thus,these DSFs can be used to diagnose structural damage. The robustness of these DSFs to different input ground motions is also investigatedusing a set of simulated data. DOI: 10.1061/(ASCE)ST.1943-541X.0000385. © 2011 American Society of Civil Engineers.

CE Database subject headings: Assessment; Monitoring; Stationary processes; Earthquake engineering; Diagnosis; Structural failures;Frequency analysis; Natural frequency.

Author keywords: Assessment; Monitoring; Nonstationary processes; Earthquake engineering; Diagnosis; Structural failures; Frequencyanalysis; Natural frequency.

Introduction

After an extreme event, such as an earthquake or a hurricane, im-mediate diagnosis of structural damage is of great importance tofacilitate emergency responses and to prevent further losses andinjuries. Similarly, decisions on repair and rehabilitation after suchevents can be greatly facilitated with information on the type, lo-cation, and extent of damage. Currently, damage diagnosis is oftenachieved by visual inspection by professionals. Human inspection,however, can be costly in time and money, perilous in certainsituations, inconsistent, limited to only surface inspections, and in-appropriate for immediate assessment of structures over large areas.Therefore, it is necessary to develop an automated system thatefficiently and reliably diagnoses structural damage. Consequently,extensive research has been conducted on structural health moni-toring (SHM) in the structural engineering community. Recentdevelopments in sensor technologies and wireless communicationsystems along with advances in damage diagnosis algorithms have

brought us closer to the realization of structural damage assessmentwithout conducting visual inspections.

To assess structural damage immediately after an earthquakeusing the structural responses recorded during the strong motion,it is necessary to develop an algorithm that directly utilizes theserecordings. Currently, there are no methods that enable us to diag-nose structural damage using the earthquake responses. Previouswork on wireless damage detection algorithms has focused onthe use of ambient vibrations obtained before and immediately afterthe occurrence of an extreme event such as an earthquake (Sohnand Farrar 2001; Nair et al. 2006; Farrar and Worden 2007).Because these previous damage diagnosis algorithms use ambientvibration measurements that are stationary, they cannot be usedwith earthquake motions that are nonstationary. For this purpose,we have developed a new method that uses wavelet energies ofinput ground motions and structural acceleration responses todetect damage in the structure from strong earthquake motion. Themain advantage of this approach is that it captures the nonstationarycharacteristics of both earthquake ground motions and structuralresponses by using the wavelet transform. While the diagnosisalgorithm proposed in this paper may be particularly suitable forembedding in wireless monitoring systems, it is not restricted toimplementation for such systems and can easily be applied to wiredsystems.

Early work using wavelet analysis for SHM has been carriedout from several different perspectives. From a system identi-fication perspective, Basu and Gupta (1997) applied waveletanalysis to obtain the spectral moments and peak structuralresponses of multi-degree-of-freedom (MDOF) systems subjectedto nonstationary seismic excitations. Ghanem and Romeo (2000)also represented the equation of motion in terms of a waveletbasis and solved the inverse problem to estimate time-varyingsystem parameters. Similarly, Kijewski and Kareem (2003) usedwavelet analysis for system identification, and Basu (2005) and

1Postdoctoral Scholar, Dept. of Civil and Environmental Engineering,Stanford Univ., Stanford, CA 94305 (corresponding author). E-mail:[email protected]

2Senior Financial Modeler, Risk Management Solutions, 7015 GatewayBlvd., Newark, CA 94560-1011.

3Assistant Professor, Dept. of Civil Engineering and AppliedMechanics, McGill Univ., Montreal, QC H3A2K6, Canada.

4Professor, Dept. of Civil and Environmental Engineering, StanfordUniv., Stanford, CA 94305.

Note. This manuscript was submitted on August 12, 2009; approved onMay 13, 2011; published online on January 15, 2011. Discussion periodopen until March 1, 2012; separate discussions must be submitted for in-dividual papers. This paper is part of the Journal of Structural Engineer-ing, Vol. 137, No. 10, October 1, 2011. ©ASCE, ISSN 0733-9445/2011/10-1215–1228/$25.00.

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Page 2: Use of Wavelet-Based Damage-Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data

Joseph and Minh-Nghi (2005) used wavelet analysis to identifystiffness degradation and damping, respectively, from the equationof motion. A number of papers discuss using modified Littlewood-Paley wavelet packets to identify modal parameters of MDOFsystems, such as natural frequencies, mode shapes, and associatedmodal damping ratios, using ambient vibration responses (Basuand Gupta 2000; Chakraborty et al. 2006; Basu et al. 2008). Froma signal processing perspective, Staszewski (1998) used waveletanalysis combined with various methods such as thresholdingand quantization for data compression and feature selection forfault detection. Hou et al. (2000) and Hera and Hou (2004) detectedsudden changes in acceleration time histories using a discrete wave-let transform; however, their application is limited to the ambientvibration data obtained from the ASCE benchmark structure.Goggins et al. (2007) investigated the degree of correlation betweenwavelet coefficients from ground excitation and building floorresponses and reported that the correlation decreases as the struc-tural behavior changes from linear to nonlinear. Although thisobservation is valuable, Goggins et al. did not offer a classificationscheme to relate the correlation value to a damage state. Later,Curadelli et al. (2007) extracted the instantaneous frequency andthe damping coefficient from free vibration responses and success-fully showed that damage can be detected with these parameters.Spanos et al. (2007) also extracted instantaneous frequency frominelastic seismic structural responses and showed that waveletanalysis can capture the evolution of frequency contents throughthe instantaneous frequencies and can potentially detect globaldamage.

In this paper, we introduce a new damage diagnosis model thatuses the Morlet wavelet to characterize the response motion of astructure subjected to earthquake ground motion. The method isbased on monitoring the damage-sensitive features (DSFs) thatare extracted from acceleration responses of the structure andreflect changes in the structure due to progressive damage. Themain objective of this study is to define several DSFs and test theirperformance to determine their ability to characterize damage.These DSFs are robust to the variations in the input groundmotions, and this framework can be applied to different types ofstructures. For this purpose, we have introduced three DSFs asfunctions of wavelet energies at a particular scale [EscaleðaÞ] andat a particular time [EshiftðbÞ]. The wavelet energy at a particularscale was first introduced by Nair and Kiremidjian (2007) for dam-age detection using ambient vibration data, and wavelet energy at aparticular time was developed by Noh and Kiremidjian (2010).

The three DSFs have been tested to determine their ability andsensitivity to diagnose damage using acceleration response datacollected from two shake-table experiments. In the first experiment,a 30% scale model of a reinforced concrete bridge column was sub-jected to different levels of ground-motion intensity at the GeorgeE. Brown, Jr. Network for Earthquake Engineering Simulation(NEES) facility at the University of Nevada, Reno (Choi et al.2007). The second experiment was conducted with a 1∶8 scalemodel of a four-story steel moment-resisting frame at the NEESfacility at the State University of New York at Buffalo (Lignos et al.2008; Lignos and Krawinkler 2009). Then, a numerical model ofthe second experiment structure is used to simulate accelerationresponses of the structure subjected to 381 different input groundmotions to test the robustness and the sensitivity of the DSFs withvarying input ground motions.

The results of the experimental verification show that the valuesof these DSFs migrate as the extent of damage increases. Thus,these DSFs can be used as an indicator of damage in the structure.The DSFs are also robust to different input ground motions. Furtherdevelopment of classification schemes to map these DSFs to

damage states is necessary to complete the damage diagnosis algo-rithm, and this work is presented in Noh et al. (2011).

This paper is organized as follows: the first part of the paperintroduces three DSFs and shows the relationship between theseDSFs and physical parameters of the structure. Then, the paperpresents the validation of their performance using two sets ofexperimental data and a set of simulated data. Finally, conclusionsare given in the last section.

Damage Sensitive Features

Overview

For this analysis, acceleration responses of a structure and theground acceleration are collected during an earthquake from eachsensor location, for example, from each floor. We first standardizeeach acceleration time history by subtracting its mean. Then, thewavelet transform of the acceleration responses and the corre-sponding wavelet energies are computed. These wavelet energiesindicate how the vibration energy of the acceleration response isdistributed in time and frequency. On the basis of these waveletenergies, we can define three DSFs to indicate structural damage.The first DSF (DSF1) indicates damage by quantifying how muchenergy is lost in the acceleration response at the proximity of thenatural frequency of the original undamaged structure. The secondand third DSFs indicate damage by quantifying how slowly theenergy of the acceleration response decays. The two sections thatfollow describe the procedure to compute these DSFs. More impor-tantly, they investigate the relationship between the DSFs andphysical parameters of the structural system to justify behaviorof the DSFs.

Wavelet Transform and Wavelet Energies

The continuous wavelet transform (CWT) of a functionf ðtÞ ∈ L2ðℜÞ, in which L2ðℜÞ= the set of square integrable func-tions, represents the function or the time history f ðtÞ as a sum ofdilated (by a scale parameter a) and time-shifted (by a shift param-eter b) wavelets. Because wavelets are localized waves that span afinite time duration, CWT can represent time-varying characteris-tics of f ðtÞ. It is mathematically defined (Mallat 1999) as

Wf ða; bÞ ¼Z ∞�∞

f ðtÞ 1ffiffiffia

p ψ��t � ba

�dt ð1Þ

in which ψðtÞ ∈ L2ðℜÞ is called the mother wavelet and � repre-sents a complex conjugate. The mother wavelet ψðtÞ is dilated byvarious scale parameters a and translated by shift parameters b tocreate a set of basis functions called daughter wavelets. Here, thescale is inversely related to the frequency of the wavelet. Thesebasis functions are convoluted with f ðtÞ to compute the waveletcoefficients Wf ða; bÞ. The time-history measurement f ðtÞ is sam-pled at discrete points in the time domain with a constant interval ofΔts, and the shift parameter b is taken at those discrete points. Forsimplicity, those discrete points 1 ×Δts; 2 ×Δts; :::;K ×Δts arereferred to as b ¼ 1; 2; 3;…;K, in which K= the number of datapoints in the measurement. For this analysis, the Morlet waveletis used as the mother wavelet because its shape resembles earth-quake pulses (Nair and Kiremidjian 2007). The Morlet waveletwas originally introduced to analyze seismic recordings (Morletet al. 1982; Goupillaud et al. 1984). Since then, it has been usedin various applications including mechanical fault diagnosis (Linand Qu 2000; Lin and Zuo 2003; Vass and Cristalli 2005) and sys-tem identification (Lardies et al. 2004; Kijewski and Kareem 2003)because of its pulselike shape and the mathematical properties that

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Page 3: Use of Wavelet-Based Damage-Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data

make it suitable for localized harmonic analysis. The Morletwavelet is a special case of the Gabor wavelet, which has the besttime-frequency resolution; in other words, it has the smallestHeisenberg box (Mallat 1999; Hong and Kim 2004). Thus, theMorlet wavelet also has the smallest Heisenberg box and conse-quently is best suited for this study. The analytical expressionfor the Morlet wavelet is

ψðtÞ ¼ ejω0te�t22 ð2Þ

in which ω0 reflects the trade-off between time and frequency res-olutions. The coefficient ω0 ≥ 5 is chosen to satisfy the admissibil-ity condition, and only the real part of the Morlet wavelet is used forcomputational simplicity. Fig. 1 illustrates the Morlet wavelet.

In Eq. (2), ω0 determines the center frequency of the Morletwavelet. The Fourier transform of the mother wavelet, ψðtÞ, hasmaximum amplitude at the center frequency, ω0. For this study,we use the daughter wavelet whose pseudofrequency matches thenatural frequency of the structure. The pseudofrequency of thedaughter wavelet is the frequency at which its Fourier transformhas the maximum amplitude and is defined as the center frequencyω0 divided by its scale a. Because we choose the scale in such a waythat the pseudofrequency of the daughter wavelet matches the natu-ral frequency of the structure, we do not have to worry about thecenter frequency of the original mother wavelet. However, ω0 alsodetermines the time and frequency resolutions of the daughterwavelet with a particular pseudofrequency, which can affect theresults of the analysis. A larger value of ω0 results in a smallerfrequency resolution, while a smaller value of ω0 results in asmaller time resolution for a daughter wavelet with a particularpseudofrequency. Applying the notion of root mean square dura-tion, the time (Δt) and frequency (Δf ) resolutions of the waveletfunction at scale a are as follows (Chui 1992):

Δt ¼ affiffiffi2

p ð3Þ

Δf ¼ 1

2πaffiffiffi2

p ð4Þ

Eq. (5) represents the relationship between the center frequencyof the Morlet wavelet (f 0 ¼ ω0=2π) and the scale a, which isthe scale of the daughter wavelet whose pseudofrequency corre-sponds to the natural frequency of the original undamaged structure(f n). Using this equation, the time and frequency resolutions of

the daughter wavelet with scale a can be represented as functionsof f 0 as follows:

a ¼ f 0f n

ð5Þ

Δt ¼ f 0f n

ffiffiffi2

p ð6Þ

Δf ¼ f n2πf 0

ffiffiffi2

p ð7Þ

The effective window sizes or bandwidths of a wavelet in thetime and frequency domain are 2Δt and 2Δf . More details canbe found in Kijewski and Kareem (2003).

To find the optimal wavelet basis, the Shannon entropy of thescalogram is often used (Coifman and Wickerhauser 1992; Zhuangand Baras 1994; Rosso et al. 2001; Lardies et al. 2004; Hong andKim 2004). In information theory, Shannon entropy represents theamount of information, or the length of the code necessary to con-vey the information. It is also often used as a measure of energyconcentration. A low value of entropy corresponds to a high con-centration of energy; thus, the basis with lower entropy implies thatthe shape of this basis matches the shape of the measurement betterthan other bases with higher entropy. We can define the three wave-let entropies of time (WEt), scale (WEs), and time and scale (WEts)as follows:

WEt ¼ �XKb0¼1

pt;b0 × log2ðpt;b0 Þ ð8Þ

WEs ¼ �Xa∈Sa

ps;a0 × log2ðps;a0 Þ ð9Þ

WEts ¼ �XKb0¼1

Xa0∈Sa

pts;b0;a0 × log2ðpts;b0;a0 Þ ð10Þ

in which Sa= a set of scales used for the entropy analysis,pt;b0 ¼ ½ΣajWf ða; b0 ×ΔtsÞj2�=½ΣbΣajWf ða; b ×ΔtsÞj2�, ps;a0 ¼½ΣbjWf ða0; b × ΔtsÞj2�=½ΣaΣbjWf ða; b × ΔtsÞj2�, pts;b0;a0 ¼½jWf ða0; b0 ×ΔtsÞj2�=½ΣbΣajWf ða; b ×ΔtsÞj2�, and j · j = the abso-lute value of the quantity. For each set of data, the optimal value ofω0 is determined such that the wavelet entropies for the accelerationresponses of the undamaged structure are minimized.

To develop the DSFs, we first examine the pattern of waveletcoefficients from different earthquake responses. Figs. 2 and 3show the variations of the wavelet coefficients computed fromthe structural responses for various damage patterns (DPs) obtainedfrom the two experiments mentioned in the introduction, the detailsof which are explained in the next section. The DPs are numberedin increasing order of the damage extent. These figures show that asthe intensity of the input motion increases, the peaks or ridges ofthe wavelet coefficients shift both in time and in scale. Thesechanges in the pattern of wavelet coefficients correlate well withthe damage status of the structure.

To quantify the shift of peaks in scale, we can use the waveletenergy at scale a (EscaleðaÞ) defined by Nair and Kiremidjian (2007):

EscaleðaÞ ¼XKb¼1

jWf ða; b ×ΔtsÞj2 ð11ÞFig. 1. Morlet wavelet basis function

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Page 4: Use of Wavelet-Based Damage-Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data

The square of the norm of the wavelet coefficients is called the sca-logram, referred to as wavelet energy at scale a and time-shift b.Thus, EscaleðaÞ is the sum of all the wavelet energies over time atscale a. As mentioned above, the scale chosen is denoted as a,which corresponds to the natural frequency of the undamagedstructure. It is assumed that a is known prior to the analysis fromwhite noise tests or structural design specifications. Figs. 2 and 3show that EscaleðaÞ is at its maximum at the natural frequency of thestructure for low levels of damage. As the damage extent increases,the peaks of the wavelet coefficients increase in scale. Hence, thechanges in the structure appear to be manifested as a decrease inEscaleðaÞ computed at the scale a. This decrease can be explained bythe fact that as the damage progresses, the structural vibrationloses high-frequency components due to loss of stiffness. Nair andKiremidjian (2007) proved that the EscaleðaÞ of the accelerationresponses at higher scales depends on structural parameters suchas mode shapes, stiffness and damping coefficients, and seismicmasses. Thus, EscaleðaÞ is well correlated with the damage extentof the structure, assuming that the damaged structure is an equiv-alent linear system with reduced stiffness.

The shift of the peaks of wavelet coefficients in time shown inFigs. 2 and 3 is then quantified by the time history of the wavelet

energy at time-shift b (EshiftðbÞ) (Noh and Kiremidjian 2010).EshiftðbÞ is defined as the sum of the scalogram over the scale attime-shift b and is given by

EshiftðbÞ ¼Xa∈S

jWf ða; b ×ΔtsÞj2 ð12Þ

For this algorithm, S is defined as

S ¼ fa; 2ag ð13Þ

in which a is the same scale chosen to compute the EscaleðaÞ. Thereason for choosing S as defined in Eq. (13) is that most of thewavelet energies are concentrated at these scales and the energiesat other scales are possible sources of noise. The distribution ofvibration energy over time is represented by EshiftðbÞ. Figs. 2 and 3show that as the intensity of damage increases, the waveletcoefficients decay more slowly with time. Hence, the time historyof EshiftðbÞ also decays more slowly for more severely damagedcases. Based on EshiftðbÞ, DSF2 and DSF3 are formulated in the nextsection.

Fig. 2. Wavelet coefficients of the acceleration response at the top of the bridge column for different DPs: (a) DP 1, (b) DP 3, (c) DP 5, (d) DP 7,(e) DP 9, (f) DP 11, (g) DP 13

Fig. 3. Wavelet coefficients of the acceleration response at the roof of the four-story steel moment-resisting frame for different DPs: (a) DP 1,(b) DP 2, (c) DP 3, (d) DP 4

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Page 5: Use of Wavelet-Based Damage-Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data

Definition of Damage-Sensitive Features

This section defines three damage-sensitive features by usingEscaleðaÞ and EshiftðbÞ as indicators of structural damage and inves-tigates the sensitivity of these features to structural damage.

DSF1DSF1, which is a function of EscaleðaÞ, is defined as

DSF1 ¼EscaleðaÞEtot

ð14Þ

in which Etot = the total wavelet energy of the acceleration re-sponse. We chose the normalization method such that when twonondamaging ground motions with different amplitudes are appliedto a structure, the DSF1 values from the structural responses areidentical. In this way, DSF1 can be robustly applied to differentamplitudes of ground-motion responses and the values are compa-rable. Because of the normalization, the value of DSF1 variesbetween 0 and 1. We tested two different methods to computeEtot—the sum of the EscaleðaÞ values at all dyadic scales, and thesum of the EscaleðaÞ values at the natural frequency (equivalent to a)and at half of the natural frequency (equivalent to 2a). The sum ofenergies at dyadic scales is equivalent to the signal energy(Mallat 1999), but we observed that using the sum of energiesat only two scales near the natural frequency gives a more accurateresult. A possible explanation is that adding all the energies at alldyadic scales can also accumulate the noise in the measurement,while picking only a few scales is equivalent to applying an effi-cient noise filter.

DSF2To quantify the shift of the wavelet coefficient peaks in time, wedefined the effective time of vibration (ETV) for each accelerationtime history. We then computed the percentage of the sum ofthe EshiftðbÞ outside this ETV as DSF2. The ETV is defined as thetime between t05 and t95, where t05 and t95 are the times when thecumulative sums of EshiftðbÞ for the input ground motion are 5% and95% of the total sum of EshiftðbÞ, respectively. In other words, theETV is the time when strong ground motion occurs. This ideaof the ETV is equivalent to the definition of the 90% cumulativeduration of strong ground motion, which is the interval between thetimes at which 5% and 95% of the total energy has been reached(Trifunac and Brady 1975). The energy refers to the integral ofthe squared acceleration recordings, which is similar to the Ariasintegral (Arias 1970). Kramer (1996) summarized different ap-proaches to define the duration of strong motion using an acceler-ation recording. We calculated the ETV from each input groundmotion, and then computed DSF2 for the corresponding structuralresponses.

Fig. 4 shows an example of the time histories of EshiftðbÞ foran increasing extent of damage and the locations of t05 and t95(indicated by dashed lines) from the bridge column experiment.Because the input ground motions are scaled versions of one earth-quake record, the locations of t05 and t95 are the same for all thetime histories in Fig. 4. DSF2 represents how much the EshiftðbÞ ofthe acceleration response is spread out in comparison to the EshiftðbÞof the input ground motion. It also shows what portion of the wave-let energy occurred outside the strong motion. If the time history ofthe EshiftðbÞ for an acceleration response decays more slowly thanthat of the input ground motion, the DSF2 value will be higher than0.1. This percentage will increase as the damage progresses in thestructure because the time history of the EshiftðbÞ will decay moreslowly. For actual earthquake ground motions, the duration ofthe strong motion should also increase with larger amplitude. Asa result, the response is likely to be even more widely spreadout than that shown in Fig. 4. Thus, we can expect that DSF2 willhave similar increasing trends as the duration of the ground motionincreases. The downside of utilizing DSF2 is that we need theinformation of the input ground motion to compute the ETV. Thus,the communication between sensor units or between sensor unitsand a server computer should be available in both directions for usto compute DSF2. It should be noted that the necessary informationfrom the input ground motion is only two values, t05 and t95, andnot the entire time-history measurement.

DSF3To define DSF3, we first defined the center of EshiftðbÞ (CE) asthe first moment (or the centroid) of the time history of EshiftðbÞ aftert95, and it is given as

CE ¼P

Kb¼t95=Δts

EshiftðbÞ × ðb ×ΔtsÞPKb¼t95=Δts

EshiftðbÞ� t95 ð15Þ

in which K = the number of data points in the measurement. TheCE is a measure of how slowly the wavelet energy decays intime after the strong ground motion. As the damage progresses,the time history of EshiftðbÞ decays more slowly, as shown in Fig. 4,which results in a larger value of the CE. To standardize the value ofthe CE, we normalized the CE of each structural response by theCE of its input ground motion. DSF3 is defined as this normalizedCE. Similar to the computation of DSF2, that of DSF3 also requiresthe input ground-motion recording for the normalization.

We proved the relationship between the CE and the extent ofdamage for the single degree of freedom (SDOF) system as shownbelow. Nair and Kiremidjian (2007) showed the relationship be-tween the scalogram and the structural parameters to be as follows:

jWf ða; bÞj2 ≈ πaω2n exp½�a2ω2

nð1� 2ξ2Þ � ω20�fexp½�2bξωn�½jGðpÞj2d1 þ jGðqÞj2

d1� þ 2jGðpÞGðq�Þjg

2ð1� ξ2Þ ð16Þ

in which ωn= the natural frequency, ξ= the damping ratio, ω0 = thecenter frequency of the Morlet wavelet, GðsÞ= the Fourier trans-

form of the input ground motion p, q ¼ jξωn � ωn

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� ξ2

p,

d1 ¼ expð2aω0ωdÞ, and ωd ¼ ωn

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� ξ2

pis the damped natural

frequency. Because the amplitude of the input ground motion issmall after t95, the response motion is less affected by the nonsta-tionary large amplitude input motion. Hence, the structuralresponse will stay within the linear region, which is the main

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assumption for obtaining the relationship shown in Eq. (16) (Nairand Kiremidjian 2007). To derive an expression for the CE in termsof the parameters of the structural system, we defined

A ¼ πaω2n exp½�a2ω2

nð1� 2ξ2Þ � ω20�

2ð1� ξ2Þ ; α ¼ 2jGðpÞGðq�Þj;

β ¼ ½jGðpÞj2d1 þjGðqÞj2d1

�; and X ¼ expð�2ξωnÞ

Then Eq. (16) can be rewritten as

jWf ða; bÞj2 ≈ Aðαþ βXbÞ ð17Þ

Using Eqs. (15) and (17), the CE for an acceleration response ofan SDOF system can be derived as

CE ¼P

Kb¼t95=Δts

EshiftðbÞ × b ×ΔtsPKb¼t95=Δts

EshiftðbÞ� t95

¼LðLþ1Þ

2 αþ β0 PLc¼1 Y

cc

Lαþ β0 PLc¼1 Y

c � 1

¼ Lþ 12

þ β0Y ½ðL�12 ÞYLþ1 � ðLþ1

2 ÞYL þ ðLþ12 ÞY � ðL�1

2 Þ�αLð1� YÞ2 þ β0Yð1� YLÞð1� YÞ � 1

≅ L� 12

þβ0Y2 ½ðLþ 1ÞY � ðL� 1Þ�

ð1� YÞ½αL� ðαL� β0ÞY � ð18Þ

in which L ¼ K � t95=Δts þ 1, Y ¼ XΔts , and β0 ¼ β ×exp½�2ξωnðt95=Δts � 1Þ�. We made the approximation YL ≈ 0because Y < 1 and L ≫ 1. As the extent of damage increases,the natural frequency decreases, and as a result, Y increases. InEq. (18), an increase in Y results in an increase of the numeratorand a decrease of the denominator, which in turn causes the CE toincrease. The sensitivity of the CE with respect to Y is calculated as

∂CE∂Y ≅ β0

2×ð2β0 � 3αL� αL2ÞY2 þ 2αLðLþ 1ÞY þ αLð1� LÞ

ð1� YÞ2½αL� ðαL� β0ÞY �2ð19Þ

It is notable that this sensitivity has the order of 1=Y2. BecauseY < 1, the sensitivity of the CE with respect to Y is sufficientlylarge.

Application of Wavelet-Based DSFs to ExperimentalData

To validate the performance of the algorithm, we applied it totwo sets of experimental data presented in the following sections.The first experiment was conducted at the University of Nevada,Reno, and involved a reinforced concrete bridge column subjectedto the scaled 1994 Northridge earthquake ground motions ofincreasing intensity. For the second experiment, which was con-ducted at the State University of New York at Buffalo, a four-storysteel moment-resisting frame was subjected to the scaled 1994Northridge earthquake ground motions of increasing intensity.Acceleration responses of the structures are collected and thecorresponding damage states of the structures are observed duringthe experiments. Then, the DSFs are extracted from those data tovalidate their sensitivity to structural damage. The robustness ofthese DSFs to different input ground motions is also shown usinga set of simulated data obtained from a numerical model of a four-story steel moment-resisting frame subjected to 40 ground motionsscaled to various intensities.

Reinforced Concrete Bridge Column Experiment

Description of ExperimentThe reinforced concrete bridge column experiment, which wasdesigned according to the 2004 Caltrans Seismic Design Criteriaversion 1.3, was performed at the NEES facility at the Universityof Nevada, Reno (Choi et al. 2007). The height of the columnwas 3,009.9 mm (108.5 in.), and the diameter of the specimenwas 35.56 mm (14 in.). Fig. 5 shows the bridge column andthe experiment setup. We used the acceleration measurementsat the top and the bottom of the column obtained during theexperiment.

For this experiment, the specimen was centered on the shaketable, and the footing was assumed to be fixed at the base. Two121:92 × 121:92 × 243:84 cm (4 × 4 × 8 ft) concrete blocks, eachweighing 88.964 kN (20 kips) were used as the inertia mass. Theseblocks were connected to the top of the specimen to simulate inertiaforces. A steel spreader beam was bolted to the top of the columnhead to provide an axial load of 275.790 kN (62 kips) to thecolumn. The column was subjected to a series of scaled groundmotions with increasing intensity. The ground motion was the fault-normal component of the 1994 Northridge earthquake recorded atthe Rinaldi Station. The amplitude of the input ground accelerationwas scaled by increasing factors of 0.05, 0.10,…,1.65, as shownin Table 1. These tests are referred to as DP 1, 2,…,13 hereafter.Table 1 also shows the RMS values of each of the input groundmotions and the description of damage for each DP. Most of thedamage was concentrated at the bottom of the column. The inputmotion was highly asymmetric in the two loading directions due tothe asymmetric velocity pulse, which is common for near-faultground motions. Thus, the responses of the column were alsoasymmetric. Because the pulse contains most of the energy ofthe earthquake motion, it causes large residual displacement to thecolumn in one direction. According to Choi et al. (2007), the col-umn behaved elastically for DP 1 through DP 4. Most of flexuralcracks formed after DP 4. The first rebar yielding occurred at DP 5,and concrete spalling occurred at the column base during DP 6. AtDP 9, the direction of the residual displacement changed becausethe column period elongated due to damage and became close tothe period of the return pulse in the input record. At DP 11, spiralexposure occurred and the residual displacement became visible(52 mm). Longitudinal bar exposure was observed at DP 12 withresidual displacement of 145 mm (5.69 in.). At the final DP 13,extensive spalling occurred with more exposure of rebar, the

Fig. 4. EshiftðbÞ for the bridge column experiment for different DPs:(a) DP 1, (b) DP 3, (c) DP 5, (d) DP 7, (e) DP 9, (f) DP 11,(g) DP 13

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residual displacement was 339 mm (13.36 in.), and the drift ratiowas 15%.

Results and Discussion

To find the optimal ω0 value, we investigated wavelet entropies forω0 between 5 and 30 rad=s. The ω0 value is limited to 30 rad=sbecause of the time resolution (the time span of the daughterwavelet whose pseudofrequency matches the natural frequencyof the structure becomes longer than the strong motion duration).For this set of data, WEt is minimized at ω0 ¼ 6 (2Δt ¼ 0:88 s,2Δf ¼ 0:36 Hz), and WEs is minimized at ω0 ¼ 17 rad=s(2Δt ¼ 2:53 s, 2Δf ¼ 0:13 Hz). Considering both time andfrequency, WEts is minimum at ω0 ¼ 9 rad=s (2Δt ¼ 1:35 s,2Δf ¼ 0:23 Hz). The results are shown in Fig. 6. Note that thevariations of WEt and WEts are relatively small for ω0 between5 and 17 rad=s. Although ω0 ¼ 5 rad=s is not the optimal solutionfor all three entropies, the difference between the minimum value ofeach entropy and the entropy at ω0 ¼ 5 rad=s is small (lessthan 0.5).

Because there are three different values of optimal ω0, the DSFsare computed using all those ω0 values, including 5 rad=s, in orderto investigate the sensitivity of the DSF values as ω0 changes. We

found that the instantaneous frequency is sensitive to the choiceof ω0. The instantaneous frequency at time t is defined as the pseu-dofrequency of the daughter wavelet that results in the maximumvalue of the scalogram among all the scales at that particular instantof time t. Thus, the instantaneous frequency is the dominant fre-quency of the time-history measurement at time t. We averagedthe instantaneous frequencies between t05 and t95 and computedthe average instantaneous frequency of DP 1 for ω0 between 5and 30 rad=s. The result is shown in Fig. 7(a). The average instan-taneous frequency approaches 1.5 Hz as ω0 increases, and the valueis stable for ω0 > 17 rad=s. This result shows that ω0 for the opti-mum value of WEs results in a reliable estimation of the instanta-neous frequency with the smallest time resolution. We computedthe instantaneous frequencies of all DPs using ω0 ¼ 17 rad=s,and this result is shown in Fig. 7(b). The average instantaneousfrequency of DP 1 is 1.5 Hz. An inspection of the Fourier spectraof this acceleration response shows that the dominant frequencywith maximum spectra is close to 1.2 Hz.

Figs. 8(a)–8(c) show the results for the DSFs using differentvalues of ω0. As we can expect from the small variation ofWEs, the values of DSF1 are similar for different values of ω0.For DSF2 and DSF3, the results are sensitive to the value of ω0.As ω0 increases, the DSF values lose the increasing trend withthe increasing damage. Thus, using ω0 ¼ 5 rad=s is an appropriatechoice for the analysis of this set of data.

On the basis of the Morlet wavelet analysis with ω0 ¼ 5 rad=s,the scale a ¼ 0:54 corresponds to the natural frequency of theundamaged column. We also investigated EscaleðaÞ at all scalesfor the acceleration response at DP 1 and found that EscaleðaÞ isthe largest at scale 0.54. As the damage extent increases, however,the scale where the largest EscaleðaÞ value occurs shifts from scale0.54 to scale 1.08 (lower frequency). This indicates that the dom-inant scale of the acceleration response increases as the damageprogresses, and the change of dominant scale is reflected as thechange in the DSF1 value. The dominant scale (or frequency)can also be obtained from ambient vibration data if available.

Fig. 8(a) shows the results of DSF1. DSF1 is normalized by thesum of twoEscaleðaÞ values at scales 0.54 and 1.08. Thus, it representsthe proportion of wavelet energy at scale 0.54 and how the energyshifts to the higher scale as the damage progresses. Fig. 8(a) showsthatwhen the column is not damaged, theDSF1 value is over 0.9, and

Fig. 5. Shake-table test setup for the bridge column experiment (modified from Choi et al. 2007)

Table 1. Scaling Factor, Input RMS Value, and Description of Damage forEach DP of the Bridge Column Experiment

DP Scaling factor Input RMS (g) Description of damage

1 0.05 0.0053 No damage

2 0.10 0.0105 Small cracks on south side

3 0.20 0.0210 More cracks on both sides

4 0.30 0.0313 More cracks

5 0.45 0.0448 Not available

6 0.60 0.0576 Spalling on south side

7 0.75 0.0696 More spalling, cracks open wider

8 0.90 0.0827 More spalling, cracks open wider

9 1.05 0.0963 Spalling on north side

10 1.20 0.1094 5 spirals exposure

11 1.35 0.1224 More spirals exposure

12 1.50 0.1352 Longitudinal bar exposure

13 1.65 0.1478 Bar exposure on both sides

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as the damage progresses, the DSF1 value decreases, which impliesthat the proportion of the EscaleðaÞ at scale 0.54 (energy at higher fre-quency) decreases while it increases at scale 1.08 (energy at lowerfrequency). According to Choi et al. (2007), most flexural crackswere formed and opened up wider after DP 4. Spalling of concretestarted at DP 5, and the strain exceeded its yield strain at DP 4.This is well reflected in the results of DSF1 that the DSF1 valueis close to 1 up to DP 4 and starts decreasing significantly afterward.

Fig. 8(b) shows the results of DSF2. The DSF2 value is close to10–20% for lower DPs, and it increases as the damage increases.These results are similar to the results of DSF1; the DSF2 value issmall up to DP 4, and the value starts increasing for damage largerthan DP 5. The result implies that as the damage progresses, theresponse motion decays more slowly, which can be correlated toperiod elongation due to stiffness degradation.

Fig. 8(c) shows the DSF3 values, which increase as the damageprogresses, indicating that the EshiftðbÞ decays more slowly withincreased column damage. The DSF3 value starts increasing afterDP 1 and remains constant after DP 4. Based on the bridge columnexperimental data, DSF3 is sensitive to small damage such ascracking, while DSF1 and DSF2 are more sensitive to more severedamage such as spalling.

Four-Story Steel Moment-Resisting Frame Experiment

Description of ExperimentThe second experiment was a series of shake-table tests of a 1∶8scale model for a four-story steel moment-resisting frame withreduced beam moment connections designed according to currentseismic provisions (International Code Council 2003; AISC 2005;SAC Joint Venture 2000). The experiment was conducted at theNEES facility at the State University of New York at Buffalo(Lignos et al. 2008; Lignos and Krawinkler 2009). The structureis shown in Fig. 9 after completion of the erection process onthe shake table. The primary interest is on the model frame shownon the left.

The structure was subjected to the 1994 Northridge earthquakeground motion recorded at Canoga Park Station. The testing se-quence that was executed for the structure included a service levelearthquake (SLE, 40% of the unscaled record), a design level earth-quake (DLE, 100% of the unscaled record), a maximum consideredearthquake (MCE, 150% of the unscaled record), and a collapselevel earthquake (CLE, 190% of the unscaled record). These scaledintensities of the ground motion are referred to as DP 1, 2, 3, and 4in this paper. According to elastic modal identification from whitenoise tests, the structure had a predominant period T1 of 0.45 s in

Fig. 6. Variation of wavelet entropies for the bridge column experiment for different values of ω0: (a) WEt , (b) WEs, (c) WEts

Fig. 7. Instantaneous frequency for the bridge column experiment (a) for different values of ω0 at DP 1, (b) for different DPs at ω0 ¼ 17 rad=s

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the undamaged state. For each DP, accelerations at each floor, in-cluding the ground motion and the roof response, were measured.

Fig. 10 shows the story drift ratio (SDR) of the structure at vari-ous intensities from elastic behavior up to collapse. During theSLE, the structure remained elastic. During the DLE, the structurereached a maximum SDR of about 1.6% with the inelastic actionobserved at the column base and first floor beams. Localized dam-age, such as local buckling of the flange plates, was not noticeable.During the MCE, the frame reached a maximum SDR of about 5%with plastic deformation evident from local buckling of the plates

that represented plastic hinge elements of the frame (Lignos andKrawinkler 2009). During the CLE test, the frame experienced amaximum SDR of about 13% with a full collapse mechanism ofthe first three stories.

Fig. 8. Variation of DSF values for the bridge column experiment for different ω0s: (a) DSF1, (b) DSF2, (c) DSF3

Fig. 9. Four-story steel moment-resisting frame after the completion oferection on the shake table (reprinted with permission from Lignos et al.2009)

Fig. 10. Story-drift ratio histories at various levels of input ground-motion intensity for the four-story steel moment-resisting frame experi-ment: (a) first story, (b) second story (reprinted with permission fromLignos 2008)

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Results and DiscussionWe computed the wavelet entropies for ω0 to be between 5 and30 rad=s, and the results are shown in Fig. 11. WEt is minimizedat ω0 ¼ 5 (2Δt ¼ 0:57 s, 2Δf ¼ 0:55 Hz), and WEs is minimizedat ω0 ¼ 30 rad=s (2Δt ¼ 3:40 s, 2Δf ¼ 0:094 Hz). Consideringboth time and frequency, WEts is minimum at ω0 ¼ 8 rad=s(2Δt ¼ 0:88 s, 2Δf ¼ 0:36 Hz). The variations of WEt and WEts

are relatively small for ω0, at between 5 and 30 rad=s, as shown inFig. 11. The DSFs are computed for three ω0 values, and similar tothe results in the first experiment, the DSF1 values are not sensitiveto the value of ω0 while DSF2 and DSF3 perform best withω0 ¼ 5 rad=s. Thus, ω0 ¼ 5 rad=s is used for this set of data.

Fig. 12 illustrates DSF1 normalized with respect to the sum oftwo EscaleðaÞ values at scale 0.406 and 0.812. The results are pre-sented for each DP at each floor. Scale 0.406 corresponds to thepseudofrequency of 2.0 Hz, which is the first natural frequencyof the undamaged frame. Because the natural frequency of the testframe is known based on the white noise tests (Lignos et al. 2008),we can use this scale to compute the DSFs. We observed thatEscaleðaÞ at scale 0.406 has the largest wavelet energy at DP 1 amongall the scales. At the ground level, the DSF1 values do not varymuch as the damage progresses because there is no damage atthe “ground” level. The same figure illustrates that for the upperfloors, DSF1 decreases as the intensity of the input ground motion

increases. With the increasing level of damage, the wavelet energyreduces at scale 0.406, which corresponds to the first natural fre-quency of the undamaged frame, and increases at scale 0.812,which corresponds to a frequency lower than the original naturalfrequency. This change in energy at scale 0.812 can be measuredby 1� DSF1. In Fig. 12, the DSF1 at the second floor does notchange much between DP 3 (MCE) and DP 4 (CLE). This happensbecause the first story is damaged during DP 1 through DP 3, reach-ing a plastic rotation of about 5%. After this plastic deformationlimit, all the beams and the base of the first story columns startdeteriorating in strength. This is reflected in DSF1 as the smallchange of values between DP 3 and DP 4 at the second floor.

The time history of EshiftðbÞ at the roof for each DP is illustratedin Fig. 13. It is shown in the figure that the EshiftðbÞ time historiesspread out further as the intensity of the input ground motion in-creases and damage progresses. Unlike Fig. 4, where the EshiftðbÞtime histories spread out both to the left of t05 and to the rightof t95, the time histories obtained from the steel frame spreadout only to the right of t95. This happens primarily because the con-crete bridge column develops cracks even at the early stages of astrong vibration where the amplitude is still small, i.e., we haveperiod elongation because of cracking. For the steel frame, thisis not the case because its behavior is elastic when the amplitudeof the vibration is small. Moreover, based on white noise tests

Fig. 11. Variation of wavelet entropies for the four-story steel moment-resisting frame experiment for different values of ω0: (a) WEt , (b) WEs,(c) WEts

Fig. 12. DSF1 for the four-story steel moment-resisting frameexperiment

Fig. 13. EshiftðbÞ for the four-story steel moment-resisting frame fordifferent DPs: (a) DP 1, (b) DP 2, (c) DP 3, (d) DP 4

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conducted after each damage level, no stiffness degradation wasobserved in the test frame prior to collapse (Lignos et al. 2009).

Fig. 14 shows DSF2, which quantifies how much the EshiftðbÞtime history is spread out in time. DSF2 is computed using thescales 0.406 and 0.812. At the ground floor, the DSF2 valuesare 10% at all DPs because the ETV is calculated based on theground motion. At all the other floors, the DSF2 value increasesas the damage in the frame progresses. The increase of the valuesat the second floor between DP 3 (MCE) and DP 4 (CLE) is small

for the same reason we mentioned earlier for DSF1. The DSF2 valueat the second floor is the largest among all the floors for each DP.This might be related to the fact that the first story was damagedfirst and had the largest damage extent.

Fig. 15 shows the results of DSF3. The damage in the framecauses the EshiftðbÞ time history to decay more slowly, which resultsin larger values of DSF3. It is similar to the results of DSF2 in thatthe DSF3 values at the ground floor are 1 for all DPs, and the DSF3values at all the other floors increase as the damage progresses.Also, the DSF3 value at the second floor is the largest amongall the floors for each DP, but the differences between the DSF3values at different floors are smaller than those observed forDSF2. The relationships between the values of the DSFs at differentfloors have to be studied carefully because the measured acceler-ations are global in nature, resulting in complex interactionsbetween structural parameters at each floor.

Numerical Model of Four-Story Steel Frame Analysis

A numerical model of the four-story steel moment-resisting framedescribed above is utilized to provide acceleration responses of theframe to examine the sensitivity of the proposed DSFs with respectto different input ground motions. Details about this numericalmodel, which account for component deterioration, can be foundin Lignos and Krawinkler (2009). The sensitivities of all the DSFshave been examined for 381 input ground motions, which arescaled to various intensities from 40 original ground-motionrecordings to investigate the effect of varying excitation profilesand amplitudes of input ground motions on the ability of the DSFsto predict damage. Two aspects of the effectiveness of the DSFs areinvestigated. The first is the consistency of values of the DSFs inthe absence of damage, and the second is the change in values of theDSFs with respect to varying degrees of structural damage. The 40ground motions are selected from large-magnitude earthquakes(6:5 < M < 7:0) recorded at sites that are 13 to 40 km from therupture zone (Medina and Krawinkler 2003). Each ground motionis scaled several times to capture several damage states up to andincluding collapse. Historically, 40 records seem to be adequate forstatistical analysis of structural responses (Vamvatsikos and Cornell2002). These ground motions are used as inputs to the numericalmodel, and the DSFs are computed from the acceleration responsesof the roof. The damage states were estimated from the maximumSDR of the structure. For this study, five damage states are definedas follows: no damage (within the elastic limit) (0% ≤ SDR < 1%),slight damage (1% ≤ SDR < 2%), moderate damage (2% ≤SDR < 3%), severe damage (3% ≤ SDR < 6%), and collapse(SDR ≥ 6%). The threshold SDR values are chosen to representdifferent damage states based on current practice [Applied Technol-ogy Council (ATC) 2005; ASCE 2000].

Fig. 14. DSF2 for the four-story steel moment-resisting frameexperiment

Fig. 15. DSF3 for the four-story steel moment-resisting frameexperiment

Fig. 16. DSF1 for two different input ground motions: (a) the 1989 Loma Prieta earthquake; (b) the 1994 Northridge earthquake

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Figs. 16(a) and 16(b) show how the DSF1 values change forscaled versions of two illustrative ground motions, the 1989 LomaPrieta earthquake recorded at the Capitola Station and the 1994Northridge earthquake recorded at the Northridge-17645 SaticoyStreet Station. For no-damage-state cases (DP 1 and DP 2), theDSF1 value is between 0.9 and 1, and as the damage increases, thisvalue decreases to below 0.4. To further illustrate the consistency ofDSF1 for the no-damage state, the distribution of the DSF1 valuesfor the no-damage state are first computed as shown in Table 2.Seventy-nine scaled ground motions of the 40 recordings resultedin the no-damage state. Table 2 shows that the DSF1 values staybetween 0.87 and 1 for 94% of the cases. The change of theDSF1 values with respect to structural damage is shown in Fig. 17,which contains the scatter plot of the DSF1 values and the corre-sponding maximum SDR in semilog scale and the linear fit forthe data for various degrees of structural damage. We can observethat the DSF1 values decrease exponentially as the damage extentof the structure increases. The correlation coefficient of DSF1 andthe log of the maximum SDR is 0.87. Therefore, DSF1 is stronglycorrelated with the SDR and can be used to estimate the damagestate of the structure accurately. According to this analysis using thenumerical model, DSF1 is robust to the variations in the input

ground motions. Similar results are observed with DSF2 and DSF3,but are not discussed in this manuscript due to space limitation.

Conclusions

Three DSFs using the continuous wavelet transform of earthquakeresponses are developed and applied to two sets of experimentaldata and a set of simulated data. The experimental data sets areobtained from recent shake-table experiments of a 30% scaledmodel of a reinforced concrete bridge column and a 1∶8 scale modelof a four-story steel moment-resisting frame. The simulated dataset is obtained from a numerical model of the four-story steelmoment-resisting frame that is used for the second experiment. Thecontinuous Morlet wavelet transform is applied to the accelerationresponse of the structure during the strong ground motion, and thewavelet energies at a particular scale and at a particular time aredefined based on the wavelet coefficients. Then three DSFsare developed as functions of the wavelet energies for structuraldamage diagnosis. DSF1 measures how the wavelet energy at thenatural frequency of the undamaged structure changes as the dam-age progresses in the structure. According to the results fromthe experimental data, the DSF1 value decreases as the damageextent increases. This is because the wavelet energy reduces at ascale corresponding to the first natural frequency of the undamagedstructure with the increasing levels of damage. DSF2 measures howmuch the wavelet energy spread out in time and DSF3 measureshow slowly it decays. The DSF2 and DSF3 values both increaseas the damage extent increases.

For the computation of the three DSFs, different levels ofinformation are required. DSF1 can be calculated using only thestructural response, while the computation of DSF2 and DSF3requires both the structural response and the information fromthe input ground motion. Thus, the communication between thesensor units at different locations has to be set up and synchronizedto compute DSF2 and DSF3 for damage diagnosis. It should benoted that this communication between the sensor units does nothave a large power demand because the information that needsto be transferred is only a few numbers such as t05, t95, and the CEextracted from the input ground motions, and not the entire timehistory measurements.

The three DSFs have different sensitivities to various levels ofdamage according to the results of the applications. The DSF3 valuechanges more for lower levels of damage than for more severelevels of damage. On the other hand, the DSF1 and the DSF2 valuesshow more changes for larger damage. Thus, DSF3 is more sensi-tive to smaller levels of damage and DSF1 and DSF2 are moresensitive to larger levels of damage. Therefore, a combination ofthese DSFs may be required for a robust damage diagnosis.

In addition to the experimental results, a numerical model of thefour-story steel frame is developed, and the DSFs are tested forsensitivity and robustness with structural response data obtainedfrom scaled versions of 40 earthquake ground motions coveringa wide range of magnitudes and distances. The results of these sen-sitivity analyses showed again that the DSFs are directly correlatedto damage states defined through story-drift ratio limits, and theDSF values are robust to the input ground motions.

To apply the method to other types of structures or various otherground motions, additional testing will be necessary. Moreover, itwould be desirable to test the algorithm with data collected fromfield experiments where noise and other environmental conditionsmay show to be a factor. A key advantage of using the DSFs todiagnose damage, however, is that these DSFs can be computeddirectly from the acceleration recording at each sensor location

Fig. 17. Scatter plot of DSF1 and maximum story-drift ratio for thenumerical model of the four-story steel moment-resisting frame

Table 2. Distribution of DSF1 for No-Damage-State Cases

Range Count

0.37–0.43 1

0.43–0.49 2

0.49–0.55 0

0.55–0.62 0

0.62–0.68 0

0.68–0.74 0

0.74–0.80 2

0.80–0.87 0

0.87–0.93 26

0.92–1.00 48

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on the structure and do not rely on the computation of the storydrifts, which is always a challenging process.

In practical applications, it is unlikely to have reference valuesof these DSFs corresponding to each damage state. Thus, a prede-fined system needs to map the values of these DSFs to differentdamage states of the structure when an earthquake occurs. Forthis purpose, the framework to build fragility functions thatdefine the probabilistic relationship between these DSFs and thedamage state of the structure is developed by Noh et al. (2011)to complete the damage diagnosis algorithm.

In summary, we developed DSFs using wavelet energies andtheoretically derived the relationship between these wavelet ener-gies and structural parameters that are important for damage char-acterization. The performance of these DSFs was tested using twosets of experimental data. Further sensitivities and robustness of theDSFs were evaluated using the numerical model of the four-storysteel moment-resisting frame subjected to 381 scaled ground mo-tions. Both the numerical and experimental results systematicallyshowed excellent correlation between the damage estimated bythe DSFs and the observed damage. Although more testing needsto be conducted to verify the performance of these DSFs for ac-tually constructed structures under varying loading conditionsand environmental effects, the results presented in this paper areencouraging and represent a good initial step for automated damagediagnosis following large earthquakes.

Acknowledgments

The research presented in this paper is partially supported by theNational Science Foundation CMMI Research Grant No. 0800932and the Samsung Scholarship. Their support is greatly appreciated.The authors would also like to acknowledge Dr. Hoon Choi (URSCorp.), Professor M.“Saiid” Saiidi (University of Nevada, Reno),and Dr. Paul Somerville (URS Corp.) for providing the experimen-tal data for the first application. Any opinions, findings, and con-clusions or recommendations expressed in this paper are those ofthe authors and do not necessarily reflect the views of the NSF.

Notation

The following symbols are used in this paper:a = scale parameter of wavelet;a = scale corresponding to natural frequency of

undamaged structure;b = time-shift parameter of wavelet;

EscaleðaÞ = wavelet energy at particular scale;EshiftðbÞ = wavelet energy at particular time;

Etot = total wavelet energy of signal;f 0 = center frequency of wavelet (Hz);f n = natural frequency of structure (Hz);

GðsÞ = Fourier transform of input ground motion;K = number of data points in time-history measurement;N = number of scales used in wavelet entropy analysis;p = q ¼ jξωn � ωn

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� ξ2

p;

Sa = set of scales;t05 = time when cumulative sum of EshiftðbÞ from input

ground motion is 5% of total sum of EshiftðbÞ;t95 = time when cumulative sum of EshiftðbÞ from input

ground motion is 95% of total sum of EshiftðbÞ;WEs = wavelet entropy of scale;WEt = wavelet entropy of time;WEts = wavelet entropy of time and scale;

Wf ða; bÞ = wavelet coefficient;

Δf = frequency resolution of wavelet function;Δt = time resolution of wavelet function;Δts = sampling period;ξ = damping ratio;

ψðtÞ = mother wavelet;ω0 = coefficient of Morlet wavelet that determines center

frequency of Morlet wavelet and reflects trade-offbetween time and frequency resolutions (ω0 ≥ 5 ischosen to satisfy the admissibility condition);

ωd ¼ ωn

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� ξ2

p= damped natural frequency;

ωn = natural frequency;� = complex conjugate; and

j · j = absolute value of quantity.

Acronyms

CE = center of EshiftðbÞ after effective time of vibration, t95;CLE = collapse level earthquake;DLE = design level earthquake;DP = damage pattern;

DSF = damage sensitive feature;DSF1 = first DSF;DSF2 = second DSF;DSF3 = third DSF;ETV = effective time of vibration—the time between t05

and t95;MCE = maximum considered earthquake;SDR = story-drift ratio; andSLE = service level earthquake.

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