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Structural Health Monitoring

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DOI: 10.1177/1475921718820770

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Recovery of the resonance frequencyof buildings following strong seismicdeformation as a proxy for structuralhealth

Ariana Lucia Astorga1 , Philippe Gueguen1, Jacques Riviere1,2,Toshihide Kashima3 and Paul Allan Johnson4

AbstractElastic properties of civil engineering structures change when subjected to a dynamic excitation. The modal frequenciesshow a rapid decrease followed by a relaxation, or slow recovery, that is dependent on the level of damage. In this arti-cle, we analyze the slow recovery process applying three relaxation models to fit real earthquake data recorded in aJapanese building that shows variant structural state over 20 years. Despite the differences in conditions, the differentscales and the complexity of a real-scale problem, the models originally developed for laboratory experiments are welladapted to real building data. The relaxation parameters (i.e. frequency variation, recovery slope, characteristic timesand their amplitudes, and range of relaxation times) are able to characterize the structural state, given their clear con-nection to the degree of fracturing and mechanical damage to the building. The recovery process following strong seis-mic deformation, could, therefore, be a suitable proxy to monitor structural health.

KeywordsSlow dynamics, earthquakes, nonlinear elasticity, recovery, resonance frequency, cracks, structural health monitoring

Introduction

Granular consolidated materials such as rock and con-crete display unusual nonlinear behavior when sub-jected to dynamic loadings. For instance, a transientdrop in elastic modulus—assessed through changes inresonance frequency or wavespeed—can be observedeven at very low (dynamic) strain, order 1026. After theloading, the modulus recovers to its original or newequilibrium state at rates that can last from a few sec-onds to several days, many months, and years.1–3 Thisrecovery effect is called slow dynamics,4 a phenomenonassociated with slow relaxation of the elastic propertiesonce the dynamic disturbance terminates. This non-linear elastic response is considered a universal beha-vior, given that slow dynamics effects have beenobserved in a large variety of rocks and geomaterials,and in scales ranging from laboratory tests1,5–10 to seis-mological observations at the surface and crust of theEarth,2,3,11–15 where the recovery process can beobserved for years. It is also independent of the mate-rial or method used.10

Analogously, atypical nonlinear response was alsoobserved in civil engineering structures. Gueguenet al.16 observed a rapid decrease in resonance fre-quency of the UCLA Factor building (California)under seismic and environmental loadings, followed bya slow recovery to the initial elastic properties. Astorgaet al.17 analyzed nonlinear elasticity in the Annex(ANX) building (Japan) throughout variations of itsfundamental frequency during a long sequence of

1Universite Grenoble Alpes, Universite Savoie Mont Blanc, CNRS, IRD,

IFSTTAR, ISTerre, Grenoble, France2Department of Engineering Science and Mechanics, Pennsylvania State

University, University Park, PA, USA3International Institute of Seismology and Earthquake Engineering (IISEE),

Building Research Institute (BRI), Tsukuba, Japan4Los Alamos National Laboratory, Los Alamos, NM, USA

Corresponding author:

Ariana Lucia Astorga, Universite Grenoble Alpes, Universite Savoie Mont

Blanc, CNRS, IRD, IFSTTAR, ISTerre, CS40700 - 38058 Grenoble Cedex

9, France.

Email: ariana-lucia.astorga-nino@univ-grenoble-alpes.fr

earthquakes. They reported recovery effects at short-and long-term monitoring and found that the recoveryrate is linked to the structural health. Although theunderlying physical mechanisms of the slow dynamicsare not fully understood, previous results suggest a sin-gle origin based on internal strains related to the mate-rial damage.6 What Guyer and Johnson1 first called thebond system (i.e. structure of defects, inter-grain con-tacts, dislocations, cracks at different scales, etc.) isbelieved to play a fundamental role in the recoveryeffect.7,9,13,16–19

Logarithmic time dependence of the recovery wasobserved in several laboratory studies.6,19,20 This beha-vior was also observed in the relaxation of fracturedfault zone materials after earthquakes.2,3,11,12 Shokouhiet al.10 describe the recovery of slow dynamics with arelaxation spectrum that quantifies different relaxationdependencies occurring during the recovery process—meaning that at early time, the recovery is not logarith-mic. Snieder et al.21 propose a relaxation model thatpredicts non-logarithmic behavior at early and latetimes, which allows us to estimate the initial and theend times of the process, with a log-linear tendency. Inour study, we applied the relaxation function21 and therelaxation spectrum10 to real earthquake data collectedin a Japanese building over a 20-year period. Apartfrom testing the models with in situ data, we comparethe relaxation parameters and the level of loading andstructural state, intending to find the evidence for anexus between the extension of the bond system andthe relaxation process after earthquakes that could beused to infer the structural health of buildings.

In the following, we describe the building and dataset, the methodology, and applied models. The recoveryof the slow dynamics is analyzed throughout the evolu-tion of the fundamental frequency with time, that is, aproxy of the elastic properties. Relaxation parametersare then studied as a function of engineering demandparameters (i.e. peak top acceleration (PTA) and struc-tural drift). We present and discuss the results in sec-tions ‘‘Results’’ and ‘‘Discussion’’ respectively, followedby some conclusions.

The ANX building

In this study, we analyzed earthquake data recorded atthe top of the ANX building (Tsukuba, Japan) betweenJune 1998 and May 2018. The building is a steel-framedreinforced concrete structure built in 1998 and moni-tored since then by a dense network composed of three-component accelerometers oriented along the main hor-izontal and vertical directions. The ANX building hoststhe Japanese Urban Disaster Prevention ResearchCenter. It is an eight-story structure with one basement

floor.22,23 The building is supported by a spread foun-dation (;8 m deep). The soil column is layered withclayey sand and sandy-clay materials to a depth of atleast 40 m. The data consist of triggered time historiesof accelerations, sampled at 100 Hz. Earthquake mag-nitudes range from 2.6 to 9.1 (Mw) and epicentral dis-tances from 1 to 1726 km. These earthquakes haveproduced a total structural drift (DMax)—calculated asthe relative displacement between top and base of thebuilding, divided by the height (34 m)—that rangesfrom 1027 to 1023(cm/cm). In this study, we considerDMax as the engineering demand parameter (i.e. theparameter characterizing the response to building load-ing) that is representative of the strain in the system.Figure 1 shows the geographic location and plan andsection views of the ANX building, as well as the strongcorrelation found between the PTA and DMax values.PTA corresponds to the maximum absolute accelera-tion recorded at the top of the building during eachearthquake.

We considered data from a pair of sensors (bottom-top) located at the east corner of the building, orientedalong both X and Y horizontal directions (N270� andN180�, respectively). The elastic frequency of the build-ing is 1.85 Hz (60.04 Hz) in the X-270 direction and1.63 Hz (60.08 Hz) in Y-180 direction.17 These valueswere estimated from the first 10 events recorded at thebeginning of the measurements in 1998. A total of 1630earthquakes were analyzed per direction, including sev-eral great earthquakes that are listed in Table 1.

Data processing

Time-histories of accelerations were treated followingBoore’s recommendations.24 Mean and trends wereremoved from the signals and a second-orderButterworth filter was used between 0.1 and 50 Hz, thelatter corresponding to the Nyquist frequency. Time-histories of displacements were obtained by doublenumerical integration of the acceleration data. Withthe displacements computed at the bottom and top sen-sors, we determined the maximum structural driftDMax. Moreover, the nonlinear elastic response of thebuilding is analyzed by monitoring the co-seismic varia-tion of its resonance frequency. To achieve that, weused an energy distribution of the Cohen’s class25

known as Wigner–Ville (WV) distribution. This is atime–frequency distribution that allows us to trackinstantaneous small frequency variations over time inthe presence of external perturbation.26

The Wigner–Ville time–frequency distribution

The following procedure is similar to that used byAstorga et al.17 to monitor frequency variations. Here we

2 Structural Health Monitoring 00(0)

present a summary of the methodology. Considering thatthe energy of a signal x(t) can be deduced from thesquared modulus of either the signal or its Fourier trans-form, that is

Ex =

ð+‘

�‘

x tð Þj j2dt =

ð+‘

�‘

X vð Þj j2dv ð1Þ

|x(t)|2 and|X(v)|2 can be interpreted as energy densitiesin time t and frequency v, respectively. One propertyof the energy distributions is that we can integrate thetime–frequency energy density along one variable andobtain the energy density corresponding to the othervariable.25 In the case of civil structures, where energyis carried by the resonance modes, we used the Cohen’sclass distribution to track the time variation of energyat the considered modal frequency. One of the simplestmethods to obtain the time–frequency distribution ofthe energy of a signal is throughout the Wigner–Ville

distribution designed for the analysis of non-stationarysignals, defined as

WVx t,vð Þ=

ð+‘

�‘

x t +t

2

� �x� t � t

2

� �e�j2pvtdt ð2Þ

where x* is the complex conjugate of x(t). For discretesignals, we use a windowed version of equation (2)called the pseudo WV distribution, PWVx, and given by

PWV x t,vð Þ=

ð+‘

�‘

h tð Þx t +t

2

� �x� t � t

2

� �e�j2pvtdt ð3Þ

where h(t) is a classical windowing function. To avoidthe trade-off between time and frequency resolutions, aseparable smoothing function q(t) is applied to the dis-tribution and the smoothed-pseudo WV distribution,sPWVx, is obtained, as follows

Table 1. Strong earthquakes recorded in the ANX building and included in this study. DMax: maximal total structural drift; PTA:peak top acceleration. Both horizontal directions X-270 and Y-180 are reported.

Earthquake Date Magnitude (Mw) PTA (cm/s2) Amax (cm/cm)

X-270 Y-180 X-270 Y-180

Niigata-Che�utsu 23 October 2004 6.9 79.30 67.75 3.01 3 1024 2.11 3 1024

Miyagi 16 August 2005 7.2 112.55 74.59 4.48 3 1024 2.78 3 1024

Igate-Miyagi 14 June 2008 6.9 94.73 90.20 4.03 3 1024 3.64 3 1024

Izu Islands 09 August 2009 6.6 64.09 32.73 2.57 3 1024 1.10 3 1024

Tohoku 11 March 2011 9.1 505.18 596.84 0.0027 0.0028

Figure 1. (a) Geographical position of the ANX building (green square) and epicenters of the earthquakes recorded from 1998 to2018 (circles). Events of Table 1 are highlighted by colored triangles. (b) General scheme of the plan and section views of the ANXbuilding, indicating the sensors whose data were used in this study. A picture of the structure is displayed at the top-left of (b). (c)Correlation between maximum values of acceleration PTA and structural drift DMax found in our data.

Astorga et al. 3

sPWV x t,v; q, hð Þ=

ð+‘

�‘

h tð Þð+‘

�‘

q u� tð Þx u +t

2

� �x� u� t

2

� �du e�j2pvtdt

ð4Þ

The drawback of the equation (4) is that the jointtime–frequency resolution adds new terms to the distri-bution, producing interferences. Moreover, the energyband of the distribution is generally broad, making itdifficult to distinguish slight variations of the frequencycarrying the maximum energy. To remove these diffi-culties, a reassignment method was proposed.27,28 Thereassigned distribution, rsPWVx, whose value at anypoint (t#, v#) is the sum of all the distribution valuesreassigned to each point, can be written as follows

rsPWVx t0,v0; q, hð Þ=

ð+‘

�‘

sPWVx t,v; q, hð Þd t0 � t x; t,vð Þð Þd v0 � v x; t,vð Þð Þdtdv

ð5Þ

where d denotes the Dirac function. The efficiency ofthe rsPWVx for application in building data was provedby Michel and Gueguen29 and applied by Astorgaet al.17 In this study, rsPWVx was applied automaticallyto the whole data set of earthquakes recorded at thetop of the building. We used a fourth order of decima-tion in frequency and 2048 frequency points (N). Timeh and frequency q smoothing windows are Hammingwindows, with N/10 and N/4 points, respectively. Inaddition, instantaneous maximum energy value varia-tions were tracked using a third-order Savitzky–Golayfilter (polynomial smoothing window) with a size equalto 15% of the time window corresponding to the dura-tion of the recording.

An example of the rsPWVx distribution, Figure 2(c)is shown with the corresponding Savitzky–Golaysmoothing function. Characteristic values of the funda-mental frequency observed during each event arepicked from this curve. For example, the apparent ini-tial frequency, fiapp, corresponds to the elastic fre-quency of the structure computed on the ambientvibration pre-event window, that is, between t = 0 sand the time of the first arrival. To obtain the time ofthe first arrival, we applied the short-term average tolong-term average (STA/LTA) trigger algorithm, withSTA/LTA = 2. The co-seismic frequency (fmin) is theminimum value observed. The recovery process is evi-dently starting from this point. Laboratory experimentshave reported analogous behavior30 and have demon-strated the influence of loading on the value of fmin.

1,31

To consider the loading effect, we used the concept ofstrong motion duration (DSM), corresponding to the

maximal loading energy in a time, much shorter thanthe total duration of the event. Trifunac and Brady32

propose the DSM to be the time between 5% and 95%of the total accumulated energy, computed by theArias Intensity function, IA

33 given by

IA =p

2g

ðt10

a2 tð Þdt ð6Þ

where g is acceleration due to gravity, a(t) the time-history of accelerations, and t1 is the total duration ofthe earthquake. Figure 2(b) shows an example of thecumulated energy curve for one earthquake.Frequencies that correspond to the end of the DSM(f95) and the final frequency at the time of apparenttotal energy, 99.9% (ffapp) were included in the analysisof the slow dynamics recovery. It is noticeable that the

Figure 2. (a) Time-history of accelerations of an earthquakeoccurred in March 2011 and recorded at the top sensor in theX direction of the ANX building. (b) Accumulated energy givenby the Arias intensity distribution, with reported characteristicvalues of energy and the location of the DSM (between the 5%and 95% of energy). (c) Example of the reassigned smoothed-pseudo Wigner–Ville distribution (rsPWVx). This figure shows thevariation of the fundamental frequency with time during theearthquake shown in (a). The solid line is the Savitzky–Golayfunction applied to the maximum energy values (color scale atthe right). The first vertical line (from left to right) correspondsto the time of the first arrival wave (i.e. fiapp). Dashed verticallines correspond to the limits of the DSM (i.e. strongest motionduration). Recovery is divided in two segments: segment 1,between fmin (minimum value of fundamental frequency) and f95

(fundamental frequency estimated at 95% of the earthquakeenergy) and segment 2, from f95 to ffapp (i.e. apparent finalfundamental frequency, at 99.9% of the earthquake energy).

4 Structural Health Monitoring 00(0)

first part of the recovery process falls within the DSMregion (i.e. ‘‘Segment 1’’ in Figure 2(c)), and is thereforelikely affected by conditioning/loading effects. On theother hand, we make the hypothesis that the recoveryafter f95 is not affected by on-going loading but ratheris related to the state of the structure. Therefore, theslow dynamics is studied by analyzing the ‘‘Segment 2’’(i.e. from f95 to ffapp). Note that because of the trig-gered nature of the data, we call ‘‘apparent’’ values theinitial and final frequencies (fiapp and ffapp, respectively)to indicate initial and final values observed in the trig-gered earthquake window.

Relaxation function

Snieder et al.21 consider that the macroscopic relaxa-tion of materials is a combination of different relaxa-tion mechanisms that take place on different temporaland spatial scales. They proposed a relaxation functionto describe this multi-scale phenomenon, assuming thatthe total relaxation is a superposition of decaying expo-nentials. The function depends on two parameters, theminimum and the maximum relaxation times (i.e. tmin

and tmax, respectively) between which the relaxationmechanisms are distributed and display log-time beha-vior (i.e. linear slope in log-time). The authors definethe relaxation function, R(t), as a perturbation to aphysical observable O(t) of a system

O tð Þ= O0 1 + SR tð Þð Þ ð7Þ

where O0 is the equilibrium value of O, and S a scalefactor. The observable might be, for example, the seis-mic velocity, the material density, the elastic modulus(such as the fundamental frequency, in this study), andso on. The total relaxation is given by the followingsuperposition of relaxation processes

R tð Þ=

ðtmax

tmin

1

te�tt dt ð8Þ

The expression (8) must be scaled with equation (7)to describe time-dependent material properties. Theweight factor 1/t is explained by the Arrhenius’ law:assuming a relaxation process with activation energy E(i.e. the minimum energy required to start a chemicalreaction), the corresponding relaxation time is given by

t = AexpE

kBT

� �ð9Þ

where A is a constant, T is the absolute temperature,and kB is the Boltzmann constant.21 Suppose the acti-vation energy has a density of states N(E), meaningthat the number of activation mechanisms between E

and E + dE is equal to N(E)dE. The density of statesP(t) for the relaxation times satisfies P(t) = N(E)dE/dt. Using the expression (9) dt/dE = (A/kBT)exp(E/kBT) = t/kBT, hence

P tð Þ=kBT

tN Eð Þ ð10Þ

N(E) is constant when the distribution of the activa-tion energy is uniform between a minimum Emin and amaximum Emax, and according to expression (10) thedensity of states for a relaxation process with relaxa-tion time, t, is for a fixed temperature, T, proportionalto 1/t.21 Following the expression (9), the minimumand maximum relaxation times are related to the mini-mum and maximum activation energies, by

tmin = Aexp

EminkBT

� �, tmax = Aexp

EmaxkBT

� �ð11Þ

Thus, tmin indicates the time at which the relaxationprocess begins, controlled by the relaxation mechan-isms on the smallest spatial scale that require lessenergy. On the other hand, tmax is associated to thetime at which the relaxation stops (i.e. when an equili-brium between external and internal stresses is reached;beyond this time no relaxation mechanism contributes).tmax depends on the perturbation that triggers therelaxation and ambient conditions.21 Consequently,analyzing the dependence of tmax as a function of thesefactors might be potentially useful to diagnose healingmechanisms.

The relaxation function can be obtained by numeri-cally evaluating the integral of expression (8). The mini-mum relaxation time tmin can be estimated from arelaxation curve, given that the transition to the loga-rithmic dependence (i.e. beginning of relaxation) occursat t ’ tmin. However, previous laboratory experi-ments34 have noted that the log-time recovery can startat very early times, to the point that in some cases, it isnot possible to obtain reliable values of tmin

21 (i.e. amuch greater time resolution would be needed). On theother hand, the maximum relaxation time max can beinferred by estimating the point where the curve flat-tens off. But again, the final transition to a non log-time behavior often occurs after several minutes orhours,35 and might not be captured by non-continuousmonitored systems.

To test the relaxation function (equation (8)),Snieder et al.21 presented a model of pillars describingthe closing of a fracture. Although this model does notsatisfy the Arrhenius’ law, it follows the function prettywell, so that the relaxation function might be applicableto different models. Similarly, no thermally activatedprocesses are included in this study. However, Gueguen

Astorga et al. 5

et al.36 showed that the relaxation function (equation(8)) describes building data reasonably well too.

Figure 3 shows an example of the relaxation functionapplied to the data (i.e. time–frequency variation in the

segment 2 of Figure 2) of one earthquake recorded atthe top of the ANX building. Values of tmin and tmax

are 1.70 and 144.6 s, respectively, representing thebeginning and end of the relaxation process for thisexample. At intermediate times (i.e. between tmin andtmax), the relaxation function follows a log-time ten-dency, where the estimated slope of recovery p = 0.03,as shown in Figure 3(b). This slope was computed byfitting a first-degree polynomial in the data betweentmin and tmax. Y-axis in Figure 3 represents the fre-quency recovery normalized with respect to the finalapparent value, ffapp. X-axis corresponds to time, with alogarithmic scale. Note that it is challenging in Figure 3to precisely determine when the curve flattens off at latetimes and therefore estimate tmax. This is again due tothe use of triggered windows that do not allow us tomonitor the behavior of the relaxation continuously intime and prevent us from determining tmax unequivo-cally in some cases.

Previous results suggest that the slope of the log-timesegment actually changes under different circumstancesthat can be related to the type of material, environmen-tal conditions, level of loading, or damage.6–8,10,17,36 Inaddition to p, tmin, and tmax, the maximum transitoryvariation of fundamental frequency, Df, was computedwith respect to the final apparent frequency, ffapp. It hasbeen observed that the variation of elastic parameters isstrongly linked to the loading amplitude.10 Likewise, wenotice a solid correlation in our data between Df andPTA and DMax values (Figure 4). To remove the effectsof loading on the variation of frequency, we normalizedDf to the maximum drift, DMax. In this way, the com-putation of p is not conditioned to loading amplitudes.

Figure 3. (a) Relaxation function following one earthquakeregistered in the ANX building and corresponding to thesegment 2 in Figure 2. The solid line is the fit of relaxation. Theestimated minimum and maximum relaxation times are indicatedwith vertical dashed lines (tmin = 1.70 s and tmax = 144.6 s). (b)Log-linear fitting of the data between tmin and tmax, with slopep = 0.03 and Df = 0.07. Y-axis is the fundamental frequencynormalized to the final value (ffapp). Note the logarithmic scaleon the X-axis.

Figure 4. Correlation between the fundamental frequency variation Df and maximum values drift DMax and acceleration, PTA.

6 Structural Health Monitoring 00(0)

Relaxation time spectrum

Shokouhi, et al.10 introduced the concept of relaxationtime spectrum to quantify the recovery process in con-solidated granular systems, measuring speed. Theauthors assumed the recovery to be represented as asum of discrete exponential decays each having anamplitude An and time constant tn. In this way, com-plementary to the model of Snieder et al.,21 the relaxa-tion time spectrum characterizes the recovery processover several orders of magnitude in time. It is given by

z tð Þ=Xm3N

n = 0

Ane�ttn , An ø 0 ð12Þ

where t is measured from t0 (t0 is the time correspond-ing to the end of the external strain, that is, the timecorresponding to f95). N is the number of exponentialsused to fit the function (i.e. N = 10 in this study). Thetime constants tn are assigned a priori and the ampli-tudes An are determined by minimizing the least squaresobjective function

e =

ðtmax

t0

Dc tð Þc0

� z tð Þ� �2

dt ð13Þ

where Dc(t)=c0 = (c(t)� c0)=c0 is the relative shift ofwavespeed (equivalent to frequency changes in ourstudy), where c0 is the wavespeed before the beginningof loading, and c(t) is the wavespeed at time t. tn is cho-sen such that there are m logarithmically spaced timeconstants in each decade,10 as follows

tn = DTbn, b = 101m, n = 0, . . . ,m3N ð14Þ

The result given by equation (12) is a spectrum ofvalues A (i.e. set of values A0, A1, A2,., An). Shokouhi,et al.10 found the relaxation time spectrum to be inde-pendent of the amplitude of loading and of the environ-mental conditions. Hence, the relaxation spectrum canbe considered as the signature of the slow dynamicsrecovery process. Even when the understanding of theconnection between the shape of the relaxation spectraand the internal characteristics of the material is miss-ing, this approach has the potential to unravel theunderlying physical mechanisms of the slow dynamics.10

The main scheme to obtain the relaxation time spec-trum is shown in Figure 5 for one earthquake recorded in2015 at the top of the ANX building. The behavior of thenormalized fundamental frequency (Df/ffapp) as a func-tion of time is shown in Figure 5(a). The set of exponen-tial decays that compose the total relaxation spectrum isshown in Figure 5(b), and their corresponding amplitudesAi are displayed in Figure 5(c), composing the relaxationtime spectrum. The fitting of all the exponential decays is

shown in Figure 5(a). It is noticeable that mechanismshaving relaxation times between t ; 2 s and t ; 100 sare present in this example, where the maximum ampli-tude, Amax, corresponds to relaxation times around 13 s(i.e. green data set in Figure 5(c)).

In this study, we extract the following three para-meters from the relaxation time spectrum:

� Amax corresponds to the maximum amplitudeobserved in the relaxation spectrum.

� The characteristic time, tc, corresponds to the cen-troid, in the time axis, of the spectrum. It was deter-mined using tc = (

PAi3ti)=

PAi, where ti and Ai

are the relaxation times, and their correspondingamplitudes, respectively, for the different exponen-tials used in the fitting (Figure 5(c)).

� The bandwidth, bw, assesses the range of dominantrelaxation times, as commonly used in signal pro-cessing (in the frequency domain typically), to esti-mate the damping ratio and quality factor ofstructures for instance.37 Here, the bandwidth is

Figure 5. General scheme used to obtain the relaxation timespectrum. Example for an earthquake occurring in 2015 andrecorded at the top of the ANX building. (a) Normalizedfundamental frequency (Df/ffapp) as a function of time during therecovery segment, after the strongest loading is finished.Df = fi–ffapp; where fi is the value of fundamental frequency attime ti during the recovery. The circles represent theearthquake data evenly spaced in log-time, and the solid thickline is the fit corresponding to the sum of exponential decays(equation (12)). (b) Exponential decay terms that compose thefit shown in (a). (c) Amplitudes Ai of the different exponentialdecays observed in (b). These amplitudes compose therelaxation spectrum. Maximum amplitude Amax, characteristictime tc, and bandwidth bw are also represented in this example.

Astorga et al. 7

defined using 1=ffiffiffi2p

of the maximum amplitude(Figure 5(c)).

Once applied the relaxation models to fit our data,we are able to obtain the following relaxation para-meters, listed herein for summary:

� tmin and tmax: beginning and end of the relaxation,respectively, they represent energy-related para-meters linked to smallest and largest scale mechan-isms (i.e. smallest and largest cracks, respectively).They are obtained by iterative process using theequation (8).

� p is the recovery slope. It is related to the rapiditywith which the particles in the material rearrangeuntil reaching an equilibrium state. It is estimatedusing a first-degree polynomial between tmin andtmax.

� tc is the characteristic time. Typical relaxation timerelated to the typical size of crack manifested for anearthquake.

� Amax is the maximum amplitude. It corresponds tothe extent of cracks with a certain size.

� bw is the bandwidth. It corresponds to the extensionof the bond system/cracks-size variety.

� Ratio tmax/tmin can be interpreted as the develop-ment of new type/size of cracks, that is, represent-ing the cracks-size variety.

Results

Long-term monitoring of the fundamental frequency inboth directions of the ANX building is shown in Figure6. Each dot corresponds to the minimum value of fre-quency fmin observed during one earthquake. For thesame building, Astorga et al.17 defined four time peri-ods P1, P2, P3, and P4 according to the behavior of thefundamental frequency in time: P1: 1998–2005, P2:2005 to March 2011, P3: March 2011–September 2011,and P4: after October 2011). Astorga et al.17 explainedthe response of the structure along these periods andconcluded that the origin of such a behavior is mostlyrelated to the evolution of the bond system, that is,extension of cracks causing structural softening, as evi-denced by permanent changes in the elastic properties.Compared to Astorga et al.,17 the data set is extendedto 2018 in this study.

First, Figure 6 shows a strong anti-correlation betweenthe loading parameter, considered here as PTA, and thevariation of fundamental frequency over 20 years. Thefundamental frequency progressively decreases in bothdirections during the first period, P1. This reflects theimmediate post-built co-seismic and slow softening of thestructure resulting from the increasing number of cracksopening and closing during each earthquake. The drop offrequency is more prominent during the first years in theX-270 direction, suggesting a preferential distribution ofcracks along elements in this direction. During P1, the

Figure 6. Long-term variation of the minimum value of the fundamental frequency in the ANX building: (top) Y-180 direction and(bottom) X-270 direction. The color bar represents the peak acceleration at the top of the building (note the log-scale). Error barsindicate the average (white circles) and standard deviation values (vertical black lines) computed by bins of 20 consecutive events.Shaded areas indicated as P1, P2, P3, and P4 refer to the structural periods observed in Astorga et al.,17 according to the behavior ofthe frequency variation. Mean and standard deviation values per period are indicated in the text boxes at the right of each plot.

8 Structural Health Monitoring 00(0)

frequency shifted from 1.85 Hz (60.04 Hz) to 1.39 Hz(60.06 Hz) in X-270, representing a mean drop ofapproximately 25%. In Y-180, the drop is about 16.5%(with respect to the first frequency values measured in thebuilding in 1998), that is, from 1.63 Hz (60.08 Hz) to1.36 Hz (60.06 Hz). At the end of P1, a sharp decreaseof frequency is observed, followed by a rapid recovery upto the stable period P2, where the mean frequency is1.33 Hz in both directions, 60.06 Hz in Y-180 and60.14 Hz in X-270. This sharp decrease is coincidentwith the Niigata-Ch�uetsu earthquake that occurred inOctober 2004 (Table 1), which confirms the correlationbetween the co-seismic variation of resonance frequencyand the amount of cracks in structural elements. Similarbehavior was observed after the great Tohoku earth-quake in March 2011, that is, period P3, where the recov-ery took longer to reach the stable period P4 because ofthe long conditioning sequence produced by aftershocks.

Seven years after the 2011 event, despite partialrecovery immediately after the earthquake, the funda-mental frequency has not reached the value measuredprior to the earthquake. Currently (2018), the funda-mental frequency of ANX building is around 1.06 Hz(60.08 Hz) and 1.04 Hz (60.07 Hz) in the Y-180 andX-270 directions, respectively, which represents a 35%

and 43% drop relative to the initial elastic frequencies.The residual frequency variation indicates a permanentchange in the elastic properties and suggests permanentstructural damage has occurred.

Data dispersion represented by standard deviationin periods P2 and P4 are smaller than in P1 and P3,respectively. Astorga et al.17 found that the more thestructure is damaged, the more stable its response toloading over time, characterized by a smaller dispersion(Figure 6). This corresponds to a gradual damagingmechanism in which the number of cracks increases(which reduces the stiffness) and the energy required toopen new cracks lessens.

A correlation between the relaxation parameters andthe structural state is also observed: the mean tendencyof each parameter changes from one period to theother. Results for tmin, tmax, p, tc, bw, and Amax indi-cate a sudden increase from P1 to P2 in both directions(Figure 7 and Table 2). Significant variations are alsoobserved from P2 to P3 and from P3 to P4 (except fortmin). The mean minimum relaxation time tmin is 5.6 sduring the first period in the Y-180 direction. At thebeginning of 2005 (i.e. the transition from P1 to P2),the value shifts to 9.9 s and does not show significantvariation until the end of P3. During P4, the mean

Figure 7. Evolution of the different relaxation parameters as a function of time for the Y-180 (circles) and X-270 (squares)directions of the ANX building: (a) minimum relaxation time tmin, (b) maximum relaxation time tmax, (c) ratio tmax/tmin, (d) log-timeslope p of the interval tmin–tmax, (e) characteristic relaxation time tc, (f) bandwidth bw, and (g) maximum spectral amplitude Amax.Each dot represents the averaged value of 20 consecutive events. Vertical dashed lines correspond to the occurrence of greatearthquakes listed in Table 1. Shaded areas divide the structural periods observed in Figure 6 (i.e. P1, P2, P3, and P4).

Astorga et al. 9

value of tmin increases to 11.9 s. This increase in tmin

represents a delay in the beginning of the relaxationprocess. Results indicate that the more cracked thestructure, the longer it takes for the relaxation to start.The variations of tmin from P1 to P2 and from P3 to P4(Figure 7(a)) are undoubtedly related to the transitionfrom unstable periods of constant softening (i.e. P1 andP3) to stable periods (i.e. P2 and P4), where the systemis cracked due to the expansion of the bond system dur-ing the period immediately before. Same tendency isobserved in the X-270 direction but with larger meanvalues of tmin for all the periods. This is also consistentwith the stronger degradation in the X-270 directionwith respect to Y-180 observed since the beginning ofthe measurements in 1998 (Figure 6).

Values of tmax are likely linked to the maximum sizeof cracks open during an earthquake,36 inasmuch as itrepresents the relaxation mechanisms operating at thelargest scales. The constant increase of the mean valueof tmax from P1 to P2 and from P2 to P3 (Figure 7(b))indicates the emergence of increasingly larger cracksand/or increasing crack density. The decrease of themean tmax from P3 to P4 might be related to some retro-fitting carried out in the building after the 2011 Tohokuearthquake.23 The mean behavior of the maximumrelaxation time tmax is analogous for both directions ofthe building, and once again the direction X-270 showsgreater values, signifying larger cracks in this direction.

Since tmin and tmax represent the initial and finalrelaxation times, respectively, the ratio between thesevalues might be directly linked to the emergence of newrelaxation mechanisms operating at different relaxationtimes. This ratio could then be interpreted as the devel-opment of new type/size of cracks. Values of tmax/tmin

are very dispersed (Figures 7 and 8): on one hand, thereare earthquakes for which tmin and tmax are very differ-ent, that is, related to the creation of new size of cracks.On the other hand, there are earthquakes that did notcreate new types of cracks, and therefore tmax/tmin = 1.Gueguen et al.36 evaluated the recovery of the funda-mental frequency of a building in Ecuador during theMw 7.8 earthquake that occurred in April 2016. Resultsshowed that, during the recovery after the mainshock,

the value of tmax/tmin .1, suggesting the expansion ofthe bond system with new size of cracks. However,tmax/tmin was equal to 1 during the relaxation process(when considering a foreshock and an aftershock withmoderate shaking), implying that new sizes of crackswere created only during the mainshock.

We observe ratios tmax/tmin = 1 or tmax/tmin .1over the periods P1, P2, P3, and P4, with no apparentrelationship with loading parameters (Figure 8) orstructural states (Figure 7). Nevertheless, in Figure 8,we show the sequence of events between 2005 and 2010and results indicate that a series of earthquakes withtmax/tmin = 1 occurs after one or a few consecutiveevents with tmax/tmin .1. This may be equivalent tothat observed by Gueguen et al.36 In Figure 8, the colorscale indicates the maximum drift, DMax. For tmax/tmin

.1, the size of the markers is related to the number ofconsecutive earthquakes, where tmax/tmin .1 (i.e. big-ger markers correspond to a greater amount of succes-sive earthquakes generating new types of cracks). Forexample, in August–September 2005, we observe threeconsecutive earthquakes with tmax/tmin .1 and mean

Figure 8. Ratio tmax/tmin for the events recorded between2005 and 2010 in the Y-180 direction of the ANX building. Fortmax/tmin = 1 (note the log-scale on the Y-axis) each dotrepresents one event. For tmax/tmin .1, the size of the markersis related to the number of consecutive events where tmax/tmin

.1 (shown in the inset legend). The color scale represents themaximum drift. The dashed rectangle encloses threeconsecutive earthquakes creating new types of cracks, followedby some events where no new types of cracks appeared.

Table 2. Mean values of the relaxation parameters for each structural period and for both directions of the building. Graphicrepresentation of these values is shown in Figure 7.

t min (s) t max (s) t max/t min p (%) tc (s) bw (s) Amax (%)

Y-180 X-270 Y-180 X-270 Y-180 X-270 Y-180 X-270 Y-180 X-270 Y-180 X-270 Y-180 X-270

P1 5.6 8.2 13.1 17.3 12.8 9.2 1.1 1.2 9.9 12.6 12.5 15.0 3.3 4.6P2 9.9 17.0 29.2 38.4 12.9 5.4 1.8 1.8 17.2 24.7 23.5 28.4 4.2 6.6P3 9.9 17.6 56.7 69.7 15.1 16.1 1.9 2.5 16.5 31.1 21.9 37.0 5.8 6.2P4 11.9 19.7 34.8 62.8 9.8 14.0 3.1 2.3 19.4 31.6 20.5 34.1 5.9 5.6

10 Structural Health Monitoring 00(0)

DMax = 1024 (yellow dot in the dashed rectangle,Figure 8). Thus, new sizes of cracks were created duringthese events. These are followed by several earthquakes,where no new sizes of cracks emerged, tmax/tmin = 1.However, we observe that earthquakes with big DMaxmight not produce new types of cracks (i.e. yellow dotswith tmax/tmin = 1, Figure 8), and events with lowDMax could open new types of cracks (i.e. blue dotswith tmax/tmin .1, Figure 8), Hence, there is no obviousrelationship between the creation of new sizes of cracksand the maximum drift, DMax. Much more effort isconsequently needed to explain the fact that someearthquakes create new type of cracks and some othersdo not. However, we observe a decrease of the ratiotmax/tmin with time (Figure 8), which characterizes thetendency toward uniformity of the crack features, gen-erating fewer new cracks when successive earthquakesoccur.

TenCate et al.6 reported different recovery slopes fordifferent materials, including intact and damaged con-crete. The authors observed an increase of the recoveryslope for damaged concrete with respect to intact con-crete, and they interpreted this increase by the fact thatthe rate of recovery in time is related to the rapiditywith which the particles in the material rearrange until

reaching an equilibrium state, that corresponds to theend of the recovery. Thus, materials whose bond systemis more ‘‘damaged’’ might be characterized by higherslopes because of the open spaces between the solid par-ticles to be filled in the rearrangement carried out dur-ing the relaxation. We observe a slight but constantincrease in the recovery slope with time (Figure 7(d)),which is consistent with the building becoming progres-sively damaged over the years. The evolution of tc, bw,and Amax is more or less analogous inasmuch as theyshow clear variations in each period (Figure 7(e)–(g)).

A clearer picture of the evolution of these para-meters with respect to the structural state is shown inFigures 9 and 10, for the Y-180 and the X-270 direc-tions, respectively. In these figures, the mean relaxationspectrum is only computed for the stable periods P2and P4, that is, before and after the 2011 Tohoku earth-quake. In these figures, the data were grouped in fiveincreasing ranges of drift amplitudes (i.e. D1 = 1 3

1026–5 3 1026, D2 = 5 3 1026–1 3 1025, D3 =1 3 1025–5 3 1025, D4 = 5 3 1025–1 3 1024, D5 =1 3 1024–5 3 1024), in order to separate loadingamplitude effects from structural state. Before Tohoku(period P2), the variation of frequency Df and log-time recovery slopes increase with strain amplitude

Figure 9. (Top) Recovery of the fundamental frequency with time for the Y-180 direction of the ANX building and different strainamplitudes, during (a) period P2, before the Tohoku earthquake and (b) period P4, after the Tohoku earthquake. Curves representthe mean frequency recovery Df, normalized to the final frequency ffapp. Slopes corresponding to the time-logarithmic segment arerepresented in the insets. (Bottom) Mean relaxation spectra for different strain amplitudes during (left) P2 and (right) P4. Relaxationspectra were obtained from the mean exponential fit shown in the top plots. Different colors correspond to five drift ranges(D1 = 1 3 1026–5 3 1026, D2 = 5 3 1026–1 3 1025, D3 = 1 3 1025–5 3 1025, D4 = 5 3 1025–1 3 1024, D5 = 1 3 1024–5 3 1024). Mean values of tc and bw per period and per strain level are indicated in the legend. Note that time (X-axis) is displayedin log-scale.

Astorga et al. 11

(Figure 9(a) top). Therefore, the corresponding ampli-tudes Amax of the relaxation spectra also increase withthe strain level (Figure 9(a) bottom). Analogous resultsare observed in laboratory experiments when analyzingthe recovery of sandstone at different strain ampli-tudes.10 Besides, Shokouhi et al.10 obtained more or lessequivalent spectrum shapes at different strain ampli-tudes. In our case, a smooth transition of the character-istic relaxation time tc to higher values is observed asthe drift increases (Figure 9 bottom). However, maxi-mum strain amplitudes in laboratory are in the order of1025, which would be comparable to our first tworanges of drift (i.e. yellow and green curves), where theshapes of spectra are rather similar.

For the same ranges of drift, there is an evidentincrease of the frequency drop after the Tohoku earth-quake, that is, in P4, with respect to P2 (Figure 9(b)top). The slope values also increase after the earth-quake, which is coherent with the aforementionedresults on the relationship between the recovery slopeand the state of the bond system. Characteristic relaxa-tion times tc also shift to higher values after Tohokufor all drift categories. This indicates that the typicalsize of cracks activated during P4 is larger than thecharacteristic size of cracks opened during P2 for equiv-alent loading strains (Figure 9(b) bottom).

Not only the size of the typical crack enlarged in P4,but also the number of cracks generated with respectto P2, especially for lower strain earthquakes(DMax \ 5x1025). This is revealed by the higher valuesof maximum amplitude Amax observed in Figure 9(b)(bottom) for the green, yellow, and black spectra. In

addition, the bandwidth may be a proxy of the differ-ent types/sizes of cracks created under certain level ofloading. After the Tohoku earthquake, an increase ofthe bandwidth values occurred, being more evident athigher strains (Figure 9(b) bottom). This is related tothe susceptibility of the structure to create a widerrange of crack sizes at a given loading value, in com-parison to those opened during P2. This bandwidthvalue finally characterizes the changes in the bond sys-tem, or in other words, the structural health.

In X-270 direction (Figure 10), no significant varia-tion between P2 and P4 is observed: frequency dropsand slope values are approximately equivalent (Figure10 top). The spectra shapes at different strain ampli-tudes are similar and range of relaxation times bw, max-imum amplitudes Amax, and characteristic times tc arealso rather equivalent (Figure 10 bottom), specially forP4. The data dispersion is also very similar between P2and P4 but also for the different range of strain values.This behavior might be linked to the fact that the struc-ture was already significantly cracked along the X-270direction prior to the 2011 event, as a result of the larg-est variation of frequency observed in Figure 6.

Several post-earthquake observations confirm therelation between the evolution of the relaxation para-meters and the pattern of cracks in the ANX building.For example, Kashima23 reported damage in thebuilding after the Tohoku earthquake in 2011. Visualpost-earthquake structural surveys detected damagearound the expansion joints, splits in the plasterboardof partitioning walls, and several cracks in the con-crete walls.

Figure 10. Similar to Figure 9 but for the X-270 direction of the building.

12 Structural Health Monitoring 00(0)

Discussion

The long-term monitoring of the ANX building (Figure6) shows clear variations of the fundamental frequencyin both horizontal directions, with different trends infour periods, as already defined by Astorga et al.17 Wealso observe an increase of softening (i.e. a decrease ofthe frequency due to a decrease in the modulus) forincreasing values of PTA. The origin of the observedvariations in the elastic properties is mainly due to tran-sient and permanent changes of the structural stiffness,which is controlled by the bond system,1 which in thiscase is the system of cracks and other heterogeneitiescreating weakness in the medium. The structural stateevolution reported for the ANX building manifestsitself in the slow dynamic recovery process followingearthquakes. Different parameters describing theserelaxation effects show distinctive behaviors that areconcomitant with the structural state:

� tmin and tmax (energy-related parameters) valuesincrease with structural softening (i.e. Figure 7(a)and (b)). A stiff medium (i.e. ANX at the beginningof the measurements) is very sensitive to energy:opening and closing of cracks is a fast process mani-festing at any loading amplitude. As the systembecomes softer due to increased mechanical dam-age, the energy required to activate cracks is lessimportant. Hence, a cracked medium takes longerto react and to activate the relaxation process; oncethis process begins, it also takes longer for theenergy to be totally released. This explains the delayin the beginning of the relaxation (i.e. the increaseof tmin) and the elongation of the maximum relaxa-tion time (i.e. the increase of tmax) with the increaseof cracking (i.e. from P1 to P2 in Figure 7(a) and(b)). Mean values of tmin and tmax also show apunctual rise coincident with the occurrence of largeearthquakes within the stable structural period P2(i.e. Figure 7(a) and (b)), which is related to a signif-icant transitory decrease of stiffness during theseevents. Snieder et al.21 determined that loading isone parameter controlling tmax.

� The bandwidth bw and the maximum amplitudeAmax of the relaxation spectrum model are directlyconnected to the extension of the bond system.Whereas the former reflects the variety in cracksizes, the latter is an indicator of the number ofcracks of the same type/size. The increase of crack-ing in the ANX building due to the Tohoku earth-quake and subsequent aftershocks is evident inFigure 9 (bottom), where a wider range of cracktypes is expected even at low-strain amplitudes, andthe number of smaller cracks considerably increasedin comparison to the period preceding the 2011

event. The strong shaking of this great earthquakeopened new structural cracks and enlarged pre-existing ones, which in turn reduced fundamentalfrequencies (and therefore stiffness) permanently(Figure 6). Analogous conclusions were drawn byGueguen et al.36 using lab experiments and realdata in buildings; as well as laboratory experimentsshowing the effects of progressively increasing dam-age by Van Den Abeele et al.;38 or by Rubinsteinand Beroza11 when analyzing seismic velocityreductions in rock after strong motions.

� Initial softening observed in the building (Figure 6)seems to have affected mostly the X-270. The funda-mental frequency in 2005 is alike in both directions,even when the initial elastic frequency was higher inthe X-270 (i.e. 1.85 Hz, with respect to 1.63 Hz inthe Y-180 direction). Although the reasons for thishave not been analyzed, it might be related to a pre-ferential distribution of heterogeneities along thisdirection due to differences in the structural design,connections, cracking, and so on. The X-270 direc-tion is softer than the Y-180 and this is also mani-fested in the recovery process by the behavior of therelaxation parameters, that is, a similar response forDf, bandwidth, maximum amplitude, before andafter Tohoku was observed in Figure 10.

� Just as bandwidth bw, the ratio tmax/tmin is relatedto the extension of the bond system with respect tothe crack sizes. Stable periods (i.e. P2 and P4) showlower mean values of tmax/tmin with respect to theprevious period P1 and P3, respectively, suggestingthat new type of cracks were more prone to emergeduring constant-softening periods (Table 2). Thisresult is coherent with the observation made byAstorga et al.,17 who concluded that new crackswere created during P1 and P3. Moreover, and simi-larly to the results reported by Gueguen et al.,36 it isobserved that some earthquakes are able to createnew sizes/types of cracks, while some others are not(i.e. Figure 8). This might be due to differences inloading conditions. However, no clear correlationwas found between the maximum strain (DMax)and the events originating new types of cracks. Weobserved that the occurrence of one or few eventscreating new types of cracks is followed by severalconsecutive events where no new types of cracksemerge. Note that the latter case does not implythat the bond system is not expanded. For example,the creation of new cracks with the same size of theexistent ones is manifested by the increase of Amax,which is clearly observed in Figure 9 (bottom) afterthe Tohoku earthquake. More analyses are neededto determine the possible link between intensitymeasurements and/or engineering demand para-meters and the creation of new sizes/types of cracks.

Astorga et al. 13

Conclusion

We have analyzed 1630 real earthquakes of differentmagnitudes, recorded between 1998 and 2018 in theANX building (Japan). The time–frequency distribu-tion based on the WV function was used to monitorthe variation of the fundamental frequency in both hor-izontal directions of the building over time. The maingoals of this study were (1) to corroborate the manifes-tation of nonlinear elastic signatures at the building’sscale, especially the slow dynamics behavior, (2) to testexistent slow dynamics relaxation models developed atthe laboratory scale to real earthquake data and realbuildings and (3) to investigate the behavior of severalrelaxation parameters with respect to different loadingamplitudes and structural states.

Astorga et al.17 detected clear signatures of non-linear elastic behavior in the ANX building, similar towhat is seen in laboratory and seismological scales.Fundamental frequency fluctuations at short (inter-events) and long term (intra-events) show evident bothtransitory and permanent variations of stiffness, con-trolled primarily by the bond system (i.e. cracks andother heterogeneities). The creation and growth ofcracks along with the resultant constant frequencydecrease entail significant energy expended in dama-ging the material, causing rearrangement of the internalstructure, and resulting in variations of detectable phys-ical properties. Two periods of relatively stabledynamic response, where permanent frequency varia-tions are detected in the ANX building are observed (1)between 2005 and March 2011, that is, after a constantsoftening occurred during the first 7 years of the struc-ture and (2) between October 2011 and May 2018, thatis, after the Tohoku earthquake in March 2011 and itsimmediate aftershocks. These permanent stiffness var-iations suggest an extension of the cracks system andare therefore linked to damage.

We focused our study in this recovery process, atime-dependent relaxation mechanism in which the fun-damental frequency shifts back to higher values afterthe loading is finished. We observe such elastic responseat any given loading amplitude. Even after very strongearthquakes, the fundamental frequency recovers: thatis, during the earthquake of 23 October 2004 (Table 1),the frequency dropped significantly. However, therecovery was practically immediate (Figure 6). Afterthe Mw 9 Tohoku earthquake in 2011, the frequencyrecovery is observed during approximately 6 months,although in this case, it is only partial. Seven years afterthis great event, the fundamental frequency seems to bestable around 1.05 Hz in both horizontal directions,after having dropped to 0.80 and 0.65 Hz in the Y- andX-directions, respectively. Although the current

fundamental frequencies represent the 65% and 57% ofthe initial elastic frequencies obtained in 1998 in the Y-180 and X-270 directions, the building is still operative,and no damage is apparently seen.

Parameters linked to the slow dynamics recoveryalso show evidence of the variable structural response.The level of heterogeneity in the material controls thebehavior of relaxation times, recovery rates, and relaxa-tion mechanisms amplitudes. This is manifested byclear variations of these parameters between periods(Figures 7, 9, and 10 and Table 2). Some of the relaxa-tion parameters also seem to be sensitive to loading:that is, within a stable response period (P2), spikes inparameters tmin, tmax, and Amax are more or less coinci-dent with the occurrence of large events (Figure 7).Moreover, variations of frequency Df, characteristicrelaxation times and proxies of the extension of thebond system (i.e. bandwidth and Amax) are well corre-lated to the strain amplitude (Figure 9). This depen-dence is consistent with laboratory observations thatindicate that the nonlinear response is proportional tothe level of dynamic strain.10,39,40

Laboratory experiments are performed under con-trolled conditions and strain amplitudes that do notdamage the material. In practice, real buildings facingreal earthquakes represent a much more complex prob-lem. Multiple uncertainties coming from several sources(i.e. loading, environment, soil, instrumentation, mate-rial, connections, construction process, etc.), the possi-ble mix of modal responses and the interaction betweenmanifold elements, are all factors that make the struc-tural response complex and unique. Despite all that, themodels of the relaxation function and the relaxationspectrum proposed by Snieder et al.21 and Shokouhiet al.,10 respectively, and originally developed forlaboratory experiments, are well adapted to the data ofthe ANX building. Our results are analogous to obser-vations of nonlinear elastic behavior at small (i.e.laboratory) and large (i.e. Earth’s crust) scales; and theyare satisfactory as a first attempt to apply the models toreal data at intermediate-scale (i.e. buildings).

By applying seismic interferometry to boreholeaccelerometric data located right next to the building,Astorga et al.17 concluded that the contribution of thesoil (i.e. soil-structure interaction) on the variation ofthe total response (i.e. soil-structure system) is lessimportant in comparison to the contribution of theresponse of the structure itself. That previous studytogether with the results of this study allows us to con-firm the structural origin of the slow dynamics, clearlyseen in Figures 9 and 10, in relation to the degree offracturing of the structural elements. However, a care-ful analysis of the slow dynamics recovery might allowus to understand the emergent unrevealed behavior

14 Structural Health Monitoring 00(0)

related to cracks growth, friction, contact rates, heal-ing, and so on. Understanding nonlinear elastic beha-vior might be helpful to improve our knowledge indynamic response, allowing us to develop and calibratemodels that are fundamental for predicting real struc-tural behavior.

The procedure followed to obtain the results pre-sented in Figures 9 and 10 could be an easy way todetect changes in the structural response (i.e. damage).For example, the comparison of the structural responsein terms of relaxation parameters before and after aspecific event, for a same level of deformation, can pro-vide us with important information about the extensionand density of heterogeneities, that is, cracks. The auto-matized computation of relaxation parameters appliedto continuous and real-time instrumented buildings,would allow us to detect permanent variations in theresponse (increase of bandwidth, Amax, etc. for equiva-lent loading), which is fundamental for making promptand accurate decisions about structural health.

Acknowledgements

This work is part of the URBASIS program led by P.G. atISTerre/Universite de Grenoble Alpes. The strong motiondata were obtained from the BRI Strong Motion Observation(http://smo.kenken.go.jp/). For the data of the 2011 TohokuEarthquake, we referred to Toshihide Kashima, ShinKoyama, and Izuru Okawa: strong motion records in build-ings from the 2011 off the Pacific coast of TohokuEarthquake, Building Research Data No.135, BuildingResearch Institute, March 2012. (http://www.kenken.go.jp/japanese/contents/publications/data/135/index.html).

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest withrespect to the research, authorship, and/or publication of thisarticle.

Funding

A.L.A. acknowledges IFSTTAR for PhD funding. P.G.acknowledges LabEx OSUG@2020 (Investissements d’ave-nir—ANR10LABX56). P.A.J. acknowledges support by theU.S. Department of Energy, Office of Science, Engineeringand Geoscience. J.R. was supported by a Marie-CurieFellowship (award no.: 655833) from the EuropeanResearch Council. Part of this work was supported by theSeismology and Earthquake Engineering ResearchInfrastructure Alliance for Europe (SERA) project fundedby the EU Horizon 2020 program under the GrantAgreement Number 730900.

ORCID iD

Ariana Lucia Astorga https://orcid.org/0000-0002-7015-4166

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