Online nonlinear structural damage detection using Hilbert...

Scientia Iranica A (2019) 26(3), 1266{1279

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringhttp://scientiairanica.sharif.edu

Research Note

Online nonlinear structural damage detection usingHilbert Huang transform and arti�cial neural networks

S.M. Vazirizadea;�, A. Bakhshia, and O. Baharb

a. Department of Civil Engineering, Sharif University of Technology, Tehran, Iran.b. International Institute of Earthquake Engineering and Seismology, Tehran, Iran.

Received 3 April 2018; received in revised form 12 December 2018; accepted 3 March 2019

KEYWORDSOnline damagedetection;Structural healthmonitoring;Ensemble empiricalmode decomposition;Moment-resisting steelframe;Arti�cial neuralnetwork;Hilberttransformation.

Abstract. In order to implement a damage detection strategy and assess the conditionof a structure, Structural Health Monitoring (SHM) as a process plays a key role instructural reliability. This paper aims to present a methodology for online detection ofdamages that may occur during a strong ground excitation. In this regard, Empirical ModeDecomposition (EMD) is superseded by Ensemble Empirical Mode Decomposition (EEMD)in the Hilbert Huang Transformation (HHT). Although analogous with EMD, EEMD bringsabout more appropriate Intrinsic Mode Functions (IMFs). IMFs are employed to assessthe �rst-mode frequency and mode shape. Afterwards, Arti�cial Neural Network (ANN)is applied to predict story acceleration based on previously measured values. BecauseANN functions precisely, any congruency between predicted and measured accelerationsindicates the onset of damage. Then, another ANN method is applied to estimate thesti�ness matrix. Though the �rst-mode shape and frequency are calculated in advance, theprocess essentially requires an inverse problem to be solved in order to �nd sti�ness matrix,which is done by ANN. This algorithm is implemented on moment-resisting steel frames,and the results show the reliability of the proposed methodology for online prediction ofstructural damage.© 2019 Sharif University of Technology. All rights reserved.

1. Introduction

Civil structures in general and those subject to seismicexcitation in particular are vulnerable to damage anddeterioration during their service life. Inasmuch asdamages to civil systems have led to unwanted majorloss and casualty, they have gained the attention ofthe scienti�c community. The process of assessing thecondition of a structure in order to detect any imperfec-

*. Corresponding author. Tel.: +15208095049E-mail addresses: [email protected] (S.M.Vazirizade); [email protected] (A. Bakhshi);[email protected] (O. Bahar)

doi: 10.24200/sci.2019.50657.1808

tion is done by visual inspection traditionally and someconventional methods. These antediluvian methods aresusceptible to becoming obsolete. Thus, new methods,that is, Non-Destructive Evaluation techniques (NDE),are designed [1]. Although such new methods asacoustic signals, electromagnetic, radiography, �beroptics, and so forth are not only more e�ective andconvenient but also more economical, these damagedetection methods are not global, but local. Therefore,they are e�ective only for small structures or struc-tural members [2]. Global vibration-based techniqueshave been released recently in order to overcome thischallenge. These methods contain Fourier transform,power spectrum, and spectrum analysis, to name afew [3]. As sensors and data acquisition systems havebecome more a�ordable during the preceding decades,

S.M. Vazirizade et al./Scientia Iranica, Transactions A: Civil Engineering 26 (2019) 1266{1279 1267

the use of vibration data to �nd an e�ective strategyfor the purpose of quantifying structural damage inengineering structures has gained more attention; eventhere are some studies that use this huge amountof data more e�ciently [4]. A review of vibration-based health monitoring methods can be found in [5-10]. The vibration-based methods, considering simplyfrequency, are worthwhile. Nonetheless, these methodsdiscard time information. Additionally, most of thesignals in the real world are not only nonlinear but alsonon-stationary concurrently, which require followingnew methods rather than conventional ones. In orderto solve these problems, time-frequency methods areimplemented, which present both frequency and timesimultaneously. Furthermore, they can be used for non-stationary signals [9]. Hilbert Transform (HT) is a de-vice in time-frequency domain for signal processing andis used for structural damage detection[11-13]. WaveletTransform (WT) and Short Fast Furrier Transform(SFFT) are other tools for time-frequency analyses,and the former also provides variable-sized regionsfor windowing [14]. In doing so, higher frequencyresolutions are brought, and a uniform resolution forall scales is provided [15]. However, WT and SFFThave limitations in Time-Frequency representation ofnon-stationary signals [16]. Huang et al. [17,18] �rstlydeveloped a new method{Hilbert Huang Transforma-tion (HHT){in order to analyze both nonlinear andnon-stationary signals. This aforementioned methodis composed of two portions, that is, Empirical ModeDecomposition (EMD) and HT. In fact, the �rst step,so-called EMD, is a procedure that provides somesignals that stem from the main one. In other words,the mother signal is decomposed into Intrinsic ModeFunctions (IMF). Since the EMD is based on thelocal characteristic time scale of the original data, thisdecomposition method is adaptive and highly e�cient.The HHT method at a galloping rate has been em-ployed in many scienti�c and engineering disciplines ingeneral and structural health monitoring in particularto give new insights into the non-stationary and nonlin-ear signals [19]. Huang et al. [20] asserted that HHT isnot only a more precise de�nition of particular eventsin time-frequency space than wavelet analysis, butalso more physically meaningful interpretation of theunderlying dynamic processes, too. Vincent et al. [21]compared the EMD method with wavelet analysis forstructural damage detection. The structure, whichis a simple 3-DOF system, was monitored during 20seconds. The sti�ness of the �rst story decreased, andthe instantaneous frequencies were calculated from the�rst �ve IMFs. It was concluded that both the EMDand wavelet methods were e�ective in detecting thedamage; however, the EMD method seems to be morepromising for quantifying the damage level. Manyassorted methods have been proposed to improve HHT

in decomposition portion, EMD, and HT portion. Inthis regard, by utilizing the bene�ts of the propertiesof white noise to distribute components with moreproper scales, Ensemble EMD (EEMD) has been pre-sented due to resolving the mode mixing problem inEMD [22,23]. Aied et al. [24] used EEMD to detectsudden sti�ness changes in a bridge model. Further-more, they argued that the application of EEMD seemsto be more adaptive to nonlinear signal than thatof wavelets. Moreover, they asserted that by usingEEMD rather than EMD, the mode mixing problemdiminishes signi�cantly. It should be kept in mind thatenvironmental conditions, such as temperature, mighthave signi�cant e�ects; however, sudden changes insti�ness are always of importance [25]. Nagarajaiahand Basu [26] conducted a very comprehensive re-search on the most common time-frequency techniques;�rst, they developed output-only modal identi�cationas well as evaluated frequencies, damping ratios, andmode shapes of MDOF LTI and LTV systems. Then,besides using HT, they applied SFFT, EMD, andwavelet and discussed the performance of each one.However, they did not employ EEMD. Today, improve-ments in computers' processing power pave the way formore elaborate computations. Arti�cial intelligence,therefore, has become the focus of attention in recentdecades. Arti�cial Neural Networks (ANNs) in termsof machine learning and cognitive science are learningmodels inspired by biological neural networks [27].Suresh et al. [28] considered the exural vibration ina cantilever beam. A neural network was trainedby modal frequency parameters-calculated for variouslocations and crack depths. Furthermore, they inves-tigated two widely used neural networks, namely themulti-layer perceptron network and the radial basisfunction network, and �gured out that the latter wasbetter than the former by virtue of performance andcomputational time. The e�ects of three di�erentlearning rate algorithms-the Dynamic Steepest Descent(DSD) algorithm, the Fuzzy Steepest Descent (FSD)algorithm, and the Tunable Steepest Descent (TSD)algorithm-on the neural network training were stud-ied. Fang et al. [29] studied the Back-PropagationNeural Network (BPNN) and used Frequency ResponseFunctions (FRFs) as its input data in order to assessdamage conditions of a cantilever beam. Eventually,they asserted that this new approach is highly accuratein predicting damage location and severity. Xu etal. [30] presented a method for the direct identi�cationof structural parameters based on neural networks.They used two back-propagation neural networks the�rst of which, emulator, was employed to predict itsdynamic response with su�cient accuracy using time-domain dynamic responses. Then, the di�erence be-tween structure response and neural network predictionwas assessed with Root Mean Square (RMS). The

1268 S.M. Vazirizade et al./Scientia Iranica, Transactions A: Civil Engineering 26 (2019) 1266{1279

second neural network, called parametric evaluationneural network, was proposed to identify the structuralparameters. Saadat et al. [31] proposed a methodcalled the Intelligent Parameter Varying (IPV) thatuses radial basis function networks to estimate theconstitutive characteristics of inelastic and hystereticrestoring forces and showed the e�ectiveness of thismethod. Bandara et al. [32] utilized a neural networkfor the actual damage localization and quanti�cationbased on FRF. Their network inputs were damagepatterns of di�erent damage cases associated withFRF, and outputs were either the damage locationsor severities. By having a su�cient amount of data, adeep learning approach can be applied to both globaland local health condition assessment of structures [33].Entezami et al. [34] used an unsupervised learningmethod to extract features required for structuralhealth monitoring. A quality literature review aboutdi�erent methods for feature extraction can be foundin [35]. In this study, a methodology was proposedto detect damage in nonlinear moment-resisting steelframes. In fact, this method discerns the changesin structure sti�ness as well as the onset of damage,predicts the location of damage, and measures theseverity of damage by implementing EEMD, HT, andANNs. Primarily, the �rst-mode frequency and modeshape are calculated by using EEMD and HT process.Afterward, by using two ANNs, the location andseverity of damages are determined.

2. Signal processing procedures

2.1. Hilbert Transform (HT)HT has proved useful to compute instantaneous fre-quency. By doing so, complex conjugate y(t) of anyreal-valued function can be calculated. HT of a signalx(t) is de�ned by:

H [x (t)] = y (t) =1�PV

+1Z�1

x (t)t� � d�; (1)

where t is the time variable, and PV indicates theprincipal value of the singular integral. With the HT,the analytic signal, z(t), is obtained as follows:

z (t) = x (t) + iy(t) = a (t) ei�(t); (2)

a (t) =px2 (t) + y2 (t); (3)

� (t) = arctan�yx

�; (4)

where a(t) and �(t) are the instantaneous amplitudeand phase function, respectively. Instantaneous fre-quency is derived from the derivative of the phasefunction [36]:

f(t) =1

2�d�(t)

dt: (5)

In spite of the fact that HT has proved useful, one en-counters di�culties for achieving physically meaningfulinstantaneous frequencies by applying HT. There aresome conditions and theories that express these short-comings, e.g., Bedrosian and Nuttall theorems, to namea few. In this regard, many assorted methods, suchas Normalized Amplitude Hilbert Transform (NAHT)method and enhanced Hilbert Huang transform, havebeen proposed [37,38]. Reducing the signal into itsIMFs has improved the chance of getting a meaningfulinstantaneous frequency. Thus, EMD is a procedurethat decomposes a signal into its IMFs.

2.2. Empirical Mode Decomposition (EMD)In 1996, EMD was proposed by Huang et al. [18]for nonlinear and non-stationary data. As mentionedearlier, EMD is a procedure that decomposes a signalinto its IMFs. An IMF is de�ned as a function to satisfytwo conditions:

1. The number of extrema and the number of zerocrossings must either equal or di�er at most by onein the whole data;

2. At any given point, the mean value of the upperenvelope, de�ned by the local maxima, and thelower envelope, de�ned by the local minima, is zero.

To this end, amplitude and frequency of IMFs canbe non-constant and changeable. In other words, am-plitude and frequency are variable as a function of time;in doing so, HT becomes a helpful transformation.

For such a signal as x(t), the EMD procedure canbe summarized in six steps as follows:

1. Determining all of the local extrema, that is, max-ima and minima;

2. Considering cubic spline passing all the local max-ima as the upper envelope, emax(t), and an analo-gous procedure for the local minima to achieve thelower envelope, emin(t);

3. Computing the mean value of upper envelope andlower one:

mi;j = [emin(t) + emax(t)]=2;

where i and j indices indicate the number ofassociated IMFs and iterations, respectively. Forthe �rst IMF, i equals 1;

4. Subtracting mi;j from the initial signal to �nd the�rst component, hi;j(t):

if j = 1 : hi;j = x�mi;j ;

if j 6= 1 : hi;j = hi;j�1 �mi;j :


5. Repeating steps 1 to 4 on hi;j(t) k times to obtainthe next IMF, Ci(t):

Ci = hi;k:

6. Calculating r and repeating steps 1 to 5 to extractthe remaining IMFs, ri+1 = x � Ci, and a residue,rn+1(t).

Finally, by using these steps, x(t) is decomposedto n-separate IMFs and a residue, rn+1(t). The originalsignal can be reconstructed by the sum of intrinsicmodes and the residue:

x =nXi=1

Ci+rn+1:

Among the IMFs, high-frequency content is removedgradually from C1(t) to Cn(t), resulting in the lattersignals with lower frequency content.

2.3. Ensemble Empirical Mode Decomposition(EEMD)

Wu and Huang presented EEMD [22,23] to solve themode mixing problem by adding a white noise to theoriginal signal. Mode mixing, which is the consequenceof signal intermittency, causes the existence of dis-parate scales in an IMF or the presence of a similarscale in di�erent IMFs. Consequently, this drawbackmakes IMFs physically meaningless, and it is crucialto get rid of it. Therefore, EEMD is implemented fornot only surmounting mode mixing problem but alsoreducing the sensitivity of a signal to noise pollution,as shown in this study. The statistical characteristicsof white noise with an ensemble of trials bring aboutimprovement in the scale separation problem. Toput it di�erently, the addition of noise to EMD anditeration leads to a better sifting process. It shouldbe mentioned that though the IMFs are polluted bynoise in each trial, these e�ects are canceled out byutilizing ensemble mean. Wang et al. [39] comparedthe applications of EMD and EEMD on time-frequencyanalysis of seismic signals. They found that this newapproach was remarkably capable of solving the modemixing problem. They demonstrated this assertion byapplying an example and revealed the ability of EEMDto decompose the signal into di�erent IMFs and analyzethe time-frequency distribution of the seismic signal.

EEMD is summarized within the following steps:

1. Adding white noise w to the initial signal, x(t), forthe �rst try k = 1:

x1 = x+ wl:

2. Applying steps 1 to 5 of the EMD procedure todecompose signal x1(t) to its IMF.

3. Repeating steps (1) to (2) with di�erent whitenoises for each try.

4. Averaging corresponding IMF calculated by eachtrial to obtain C:

Ci = limn!1

1n

nXl=1

Ci;l;

where n is the number of ensembles.5. Subtracting Ci(t) from the initial signal and repeat-

ing these steps for �nding the remaining IMFs.Finally, the main signal can be reproduced as

in the following equation:

x =nXi=1

Ci+rn+1:

The main advantage of EEMD over EMD is thatthe former is less sensitive to noise than to EMD. Inother words, the IMFs extracted by EEMD are moreanalogous with each other than with those of EMDwhile the uniformity among the decomposed signalsis of considerable importance for extracting the modeshapes.

3. Damage detection procedure

In this section, the rewarding process of �nding severityand location of damage, which is a combination ofsignal processing and arti�cial intelligence, is proposed.Figure 1 plots the schematic process of the proposedmethod for a typical 3-story moment-resisting frame.In the following steps, the procedure is closely exam-ined.

The �rst step is recording the response of thestructure using installed sensors. It should be men-tioned that because of the earthquake recorded as anexcitation at the base level of the structure as well asthe damping e�ect on the structure, the correspondingresponse is non-stationary. Another point deservingattention is that the procedure is output-only, andin order to underline this fact, there are not anyinstalled sensors on the base although it could be. Asmentioned earlier, the �rst exquisite step is recordingthe structural response in each story. These signals areconsidered as input data for EEMD, and the extractedresults are IMFs. Then, HT is applied to IMFs to �ndinstantaneous frequencies and amplitudes. It should bementioned that each story signal is satis�ed in order toevaluate the �rst-mode frequency, and the result canbe double-checked by the others. Additionally, in mostcases, the structural response is more obvious in higherelevations; thus, the last story response is consideredas the main signal for �nding the dominant frequency.It is further shown that the results are not signi�cantlysensitive to the �rst-mode dominant frequency. In thenext step, thanks to more uniform IMFs by virtueof EEMD, each story' IMFs containing the �rst-mode


Figure 1. Schematic view of the proposed method.

frequency is normalized in relation to the those of thelast story. In doing so, the �rst-mode shape is at hand.

Then, a Radial Basis Function Network (RBFN)is employed in order to predict the response of eachstory. The task of this network is to forecast the accel-eration of each story using its acceleration in previousmoments and other stories in the current and previousmoments. To put it di�erently, by using early secondresponses, as long as the excitation is not destructive,or the former excitation is not strong enough to damagethe structure, this network is trained and ready topredict the behavior of the structure. As mentionedearlier, any di�erences between the predicted andmeasured values signify changes in structural elementson the grounds that RBFNs are remarkably powerfulto predict acceleration response precisely. This stepis extremely worthwhile to locate the damage andfacilitate the procedure for the next step.

The last step is applying another network tomeasure the severity of damage. Although the afore-mentioned steps are a non-model based method, theintact and undamaged structure is required to becompared with the damaged one in this step. Fur-thermore, this network needs some data to train. Inother words, if the relationship between the �rst-modeshape and frequency and its corresponding parametersof the associated structure is available, the structuralparameters can be identi�ed based on the �rst-modeshape and frequency; however, a formidable challengeis to establish a mathematical model for mappingfrom the mode shape and frequency to the structuralparameters. Although the Eigen values and Eigenvectors are simply at hand with known sti�ness matrixand, by doing so, mode shapes and frequencies arecalculated, it is a formidable task to reach a general

solution to assess the sti�ness matrix by the �rst-mode shape and frequency. Therefore, the applicationof optimization algorithms can be helpful in solvingthis inverse problem. These optimization algorithmsinclude Genetic Algorithm (GA), Swarm Intelligence(SI), Particle Swarm Optimization (PSO), Multiparti-cle Swarm Coevolution Optimization (MPSCO), andImproved Multiparticle Swarm Coevolution Optimiza-tion (IMPSCO) employed by Jiang et al. [40] to localizeand quantify the structural damage in comparisonwith GA, etc. Friswell et al. [41] applied the geneticalgorithm and vibration data to the problem of damagedetection. In this study, an ANN is applied on thegrounds that the neural network has the ability toapproximate arbitrary continuous function and map-ping, too. Therefore, it stands to reason that themathematical model is supplanted by a network thatcan be considered as an optimization problem with theseverity of the damages as its variables.

4. Modeling and analysis

The �rst structure is a three-story steel moment-resisting frame with story height and spans of 3 m and4 m, respectively, and its speci�cations are providedin Table 1. The open-source �nite element program,OpenSees, is used for nonlinear dynamic analysis. Theutilized material is Steel01, which is a bilinear model.By using 0% strain hardening, it is transformed intoelastic-perfectly plastic model, in which all of theseparameters are summarized in Table 2, where Fy, E,and �y represent yielding stress, modules of elasticity,and yielding strain. This structure has Rayleigh'sproportional damping with both the �rst and the thirdmodal damping ratios of 0.05. The scaled Northridge


Table 1. Speci�cations of the three-story frame.

Story Mass (kg) Sti�ness of beamsection (MPa)

Sti�ness of columnsection (MPa)

All 8000 8.4164062 17.45226

Table 2. The material properties used for modeling the structure.

Material no. Fy(MPa)

E elastic(MPa)

E plastic(MPa)

�y

1 200 2� 105 0 0.001

Figure 2. Scaled Northridge ground motion accelerationrecord.

earthquake record (which is multiplied by two in orderto push the structure into plastic range) is appliedto the structure and the ground motion, as shown inFigure 2.

Apart from the mentioned points, in order tosimulate damage in columns, each return to the elasticrange after undergoing plastic behavior reduces columnsti�ness to 91% of its previous sti�ness, which is an as-sumed value that indicates those changes less than 15%.This reduction is imposed on modulus of elasticity, andin order to simulate sudden damages and cracks inthe elements, it appears after the occurrence of plasticregion. In this regard, it should be mentioned thatalthough both beams and columns tolerate nonlinearbehavior during a seismic excitation, only columns aresusceptible to degradation. These changes in structuralelement sti�ness are demonstrated in Figure 3 wherethe second and third story columns are not shown,because they remain in the elastic range.

In this study, two salient features of EEMD areconsidered: the noise level and ensemble number. Asthe names imply, the amplitude of noise is determinedaccording to the standard deviation of original data ineach step and the number of iterations, respectively.In spite of the fact that a few hundred ensemblesbring about appropriate results, there is no de�nitevalue for the noise level [23]. Figure 4 shows IMFsusing EMD of the base and the �rst to the thirdstory from left to right. In fact, if the noise levelis 0 and ensemble number is 1, EEMD changes intoEMD, and Figure 5 displays its corresponding HT.Even though IMFs are improved by increasing theensemble number, 5000 is enough in number because no

Figure 3. Changes in beams and �rst story columnssti�ness.

signi�cant di�erence is observed after this number; thevalues more than this number would merely increasethe computational cost. Furthermore, the noise level,which is less than 0.5, is not a completely acceptableremedy. Therefore, it is better to use values more than0.5. Figure 6 shows HT of the seventh and eighth IMFswith the noise level of 1 and ensemble number of 5000.There is a histogram close to every HT, which is anew method. The dominant frequency is assessed bythese histograms rather than the conventional marginaldistribution. Instantaneous frequency of the structurechanges due to the susceptibility of the structure toplastic deformation and instant changes in sti�nessmatrix. Thus, it is probable for HT to provide largevalues that a�ect the marginal distribution. On thecontrary, constructing a histogram is based on thenumber of values that fall into each interval; therefore,


Figure 4. The ground motion acceleration and structure acceleration response IMFs with noise level of 0 and ensemblenumber of 1.

Table 3. Comparsion of FEM results and proposed method results.

Frequency 1stmode (Hz)

Mode shapes

1st story 2nd story 3rd story

Calculated by FEM model 1.0696 0.2964 0.7037 1

Calculated by EEMD and Hilbert 1.07 0.2978 0.7055 1

Percentage of error 0.5� 0.5 0.3 0

�Center of [1.065 1.075) is 1.07. Therefore, maximum error is (1.075-1.0696)/1.0696=0.005.

e�ects of inordinate amplitude in estimating dominantfrequency are squandered. In Figure 6, apart fromthe �rst column relating to ground motion, the sev-enth and eighth IMFs exhibit the lowest meaningfuldominant frequency that emanates from the �rst-modevibration. Furthermore, Figure 7 presents normalizedtotal amplitude of the seventh and eighth IMFs foreach story relative to those of the third story fordi�erent values of the noise level and ensemble number.According to this �gure, as mentioned earlier, the noiselevel at or below 0.5 does not produce decent results.Meanwhile, when the noise level is 1 or 2, the resultsclosely resemble each other. Moreover, Figure 7 depictsthe mentioned normalized amplitude according to theensemble number. In this regard, if the ensemble

number is 5000 or 10000, normalized amplitudes ineach column are the same.

According to Figure 7, the longest duration inwhich variance of values in normalized amplitudesreaches a stable and minimum level is considered as thenormalized mode shape; therefore, for example, whenthe ensemble number is 5000 and noise level is 1, thistimespan spans from 6 to 36 seconds. Table 3 comparesthe �rst-mode shape and frequency calculated by FiniteElement Method (FEM) with the proposed methodbrie y.

The next step is employing two networks, dis-cussed before. The schematic architecture of the �rstnetwork is illustrated in Figure 8. In fact, this networkneeds a vector with m� n+ n� 1 members, where m


Figure 6. Hilbert transform of the IMFs 7th and 8th with noise level of 1 and ensemble number of 5000.

Figure 7. Normalized total amplitude of the IMFs 7th and 8th for each story relative to story 3.

Figure 8. Overview of emulator arti�cial neural network.

is the required previous moment, and n is the numberof stories. Consequently, m � n and n � 1 de�ne thetotal number of input data based on previous momentsand current moment, respectively. In this study, n is 3due to the structure stories and m is 4, although many

other values have been tested. This network is trainedfor each story (for this 3-story frame, 3 networks arerequired and trained. Their structures are similar, yetare di�erent only in input and output data), and thepredicted acceleration responses are shown in Figure 9.Another subtle point deserving to be mentioned is thatin order to predict the acceleration response in the kthstory, previous moment acceleration responses for the(k� 1)th, kth, and (k+ 1)th stories, as well as currentmoment acceleration responses for the (k � 1)th and(k + 1)th stories, are su�cient. In other words, theresponse data of lower and upper stories are enough;however, the alluded measures have been taken due tothe lack of a sensor at the base level of this frame.Therefore, using a sensor at the base level not onlyimproves the performance of the network but alsodiminishes the input data.


Figure 9. The predicted acceleration response of thestructure.

Figure 10. The di�erence between structure responseand neural network prediction.

Figure 10 presents the di�erence between struc-ture responses and neural network predictions, thatis, absolute di�erence, cumulative di�erence (showinga gentle slope owing to the noise added to inputand output data), and modi�ed cumulative di�erence(eliminating the gentle slope) in rows one to three,respectively. The big jumps in the �rst story errorreveal changes in structure behavior, and the vicinityof two other stories' errors to those big jumps shows thepresence of anomaly in the predicted data, stemmingfrom one common source, i.e., the �rst story.

After locating damages and determining the �rst-mode parameters, the �nal step is the second network,as shown in Figure 11. This network is trainedby 100 sets of training patterns, associated with theperformance of the previous network. The summarizedresults are presented in Table 4. In addition, Figure 12

Figure 11. Overview of the second arti�cial neuralnetwork.

Figure 12. Sensitivity of the proposed algorithm to the�rst-mode dominant frequency.

shows the sensitivity of the proposed algorithm to the�rst-mode dominant frequency. Indeed, because ofusing histogram rather than marginal distribution, the�rst-mode frequency is the interval center; neverthe-less, this �gure sheds light on the fact that the lengthof bins is not the determining factor.

Another example is a 6-story frame subjectedto Fruili earthquake record at the base level. Theprocedure is the same as the previous structure; how-ever, there are two damaged stories{the �rst and sixthstories{the sti�ness of which has actually decreased.Figure 13 shows the amplitude of IMF number 7 foreach story relative to the sixth story and elaboratesthat EEMD is a useful device to assess the �rst-modeshape.

Furthermore, as mentioned earlier, any changesin sti�ness can be detected by the di�erence betweenstructure responses and neural network predictions.

Figure 14 depicts absolute di�erence, cumulativedi�erence, and modi�ed cumulative di�erence aftereliminating the gentle slope. The �rst and sixth storiesrepresent relatively more signi�cant error than those ofthe other stories.

Table 4. Comparsion of FEM results and the proposed method results.

Damaged story Evaluated sti�ness(MPa)

Calculated by FEM(MPa)

Percentage of error

#1 12.70 13.15 3.4


Table 5. Comparsion of FEM results and proposed method results.

Frequency1st mode

(Hz)Mode shapes

1st story 2nd story 3rd story 4th story 5th story 6th story

Calculated by FEM model 0.8326 0.1302 0.3412 0.5525 0.7503 0.8968 1Calculated by EEMD and Hilbert 0.830 0.1303 0.3366 0.5447 0.7493 0.9057 1

Percentage of error 0.9* 0.1 1.3 1.4 0.1 1.0 0�Center of [0.825 0.835) is 0.83. Therefore, the maximum error is (0.8326-0.825)/0.8326=0.00).

Table 6. Comparsion of FEM results and the proposed method results.

Damaged story Evaluated sti�ness(MPa)

Calculated by FEM(MPa)

Percentage of error

#1 13.96 13.96 � 0#6 5.17 5.11 1.2

Figure 13. Normalized amplitude of IMFs 7 for eachstory relative to story 6.

Consequently, the results obtained after usingEEMD and Hilbert transform as well as the resultsobtained after employing ANNs are listed in Tables 5and 6, respectively.

Table 5 shows that there is not any signi�canterror between the �rst-mode frequency calculated byFEM model and EEMD and Hilbert. In addition,Table 6 indicates that the proposed method detectsthe location of damages correctly and estimates thesti�ness changes, too.

Figure 14. The di�erence between structure responseand neural network prediction.

Figure 15 emphasizes the aforementioned fact,meaning that it is not necessary to precisely evaluatethe �rst-mode frequency. In other words, this �guredepicts that changing the �rst-mode frequency in theinput how changes the �nal results in sti�ness of the�rst and sixth stories.

5. Conclusion remarks

In this study, EMD is supplanted by EEMD in theprocess of HT. Even though these methods closelyresemble each other, EEMD brings about more ap-propriate IMFs employed to assess the �rst-modefrequency and mode shape. To put it di�erently,using EEMD rather than EMD in HHT signi�cantlyimproves its performance. By doing so, IMFs are moreuniform, paving the way for assessing the �rst-modeshape. Afterward, an ANN was applied to predict storyacceleration based on the acceleration response of thestructure during the previous moments. ANNs imitatethe behavior of the frame in the elastic range precisely;therefore, any congruency between the predicted and


Figure 15. Sensitivity of the proposed algorithm to the �rst mode dominant frequency.

measured accelerations provides the onset of damage.Then, another ANN method was applied to estimatesti�ness matrix. Though the �rst-mode shape andfrequency are calculated in advance, it is essentiallyrequired for an inverse problem to be solved in orderto �nd sti�ness matrix, and this task is done by thesecond ANN. In other words, these two ANN methodswere implemented to predict the location and measurethe severity of damage, respectively. This algorithmwas implemented on two moment-resisting steel frames,and the results were found acceptable. In fact, anovel technique based on the combination of a time-series method and ANNs was applied, which not onlyovercomes the limitations of time-frequency methodsfor damage detection of nonlinear structures, but alsoprovides an online monitoring method, which is essen-tial in taking preventive measures. Furthermore, thisapproach is not sensitive to noise in input or outputdata, and even 10% error in assessing the �rst-modefrequency is negligible.

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Biographies

Sayyed Mohsen Vazirizade is a PhD candidate atUniversity of Arizona and graduated from Sharif Uni-versity in 2015. His focus of research includes reliabilityengineering and machine learning and application ofuncertainty and AI in engineering structures.

Ali Bakhshi is an Associate Professor at Civil Engi-neering Department at Sharif University of Technology.He obtained his PhD degree from Hiroshima Universityof Technology, Hiroshima, Japan. Furthermore, he


is the Director of Earthquake Simulation NationalLab and the Head of Earthquake Engineering Re-search Center at Sharif University. His main researchinterests include earthquake modeling and randomvibration.

Omid Bahar is an Assistant Professor at InternationalInstitute of Earthquake Engineering and Seismology

(IIEES), Tehran, Iran. He earned his PhD degree inStructural Engineering from Shiraz University, Shiraz,Iran. He spent a short while as a visiting scholarin Professor Kitagawa's Lab. in Hiroshima Universityduring his PhD research program, he worked on theperformance of active structural control during strongground motions. His main research interests includesignal processing and structural health monitoring.

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