An e ective approach to structural damage localization in ...scientiairanica.sharif.edu/article_20019_f0e3edc3d677e11325375fac… · 3126 H. Amini Tehrani et al./Scientia Iranica,

Scientia Iranica A (2019) 26(6), 3125{3139

Sharif University of TechnologyScientia Iranica

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

An e�ective approach to structural damage localizationin exural members based on generalized S-transform

H. Amini Tehrania, A. Bakhshia;�, and M. Akhavatb

a. Department of Civil Engineering, Sharif University of Technology, Tehran, P.O. Box 11155-9313 Tehran, Iran.b. School of Civil Engineering, Iran University of Science & Technology, Tehran, Iran.

Received 7 February 2017; received in revised form 22 October 2017; accepted 23 December 2017

KEYWORDSDamage localization;Signal processing;GeneralizedS-transform;Time-frequencyrepresentation;Non-stationarysignals.

Abstract. This paper presents a method for structural damage localization based onsignal processing using generalized S-transform (SGS). The S-transform is a combinationof the properties of the Short-Time Fourier Transform (STFT) and Wavelet Transform(WT) that has been developed over the last few years in an attempt to overcome inherentlimitations of the wavelet and short-time Fourier transform in time-frequency representationof non-stationary signals. The generalized type of this transform is the SGS-transformthat is characterized by an adjustable Gaussian window width in the time-frequencyrepresentation of signals. In this research, the SGS-transform was employed due to itsfavorable performance in the detection of the structural damages. The performance ofthe proposed method was veri�ed by means of three numerical examples and, also, theexperimental data obtained from the vibration test of 8-DOFs mass-sti�ness system. Bymeans of a comparison between damage location obtained by the proposed method and thesimulation model, it was concluded that the method was sensitive to the damage existenceand clearly demonstrated the damage location.

© 2019 Sharif University of Technology. All rights reserved.

1. Introduction

Structural health monitoring has become an evolvingarea during the recent years with an increasing needto ensure the safety and functionality of structures. Itcan be stated that the importance of structural healthmonitoring lies in the serviceability and safety of struc-tures during their service life; of course, appropriateand timely actions are required for their maintenanceand retro�tting. In the past few decades, using signalprocessing tools in structural health monitoring has

*. Corresponding author. Tel.: +98 21 66164243;E-mail addresses: aminitehrani [email protected] (H. AminiTehrani); [email protected] (A. Bakhshi);[email protected] (M. Akhavat).

doi: 10.24200/sci.2017.20019

increased considerably due to the recent advances inthe �eld of sensors and other electronic technologies [1].These advances provide a wide range of responsesignals such as velocity, acceleration, and displace-ment caused by low- to high-intensity earthquakes andenvironmental loads. Doubling et al. [2] provided acomprehensive history of structural health monitoringmethods together with e�ciencies and disadvantages ofeach method. Sohn et al. [3] presented comprehensiveliterature reviews of vibration-based damage detectionand health monitoring methods for structural andmechanical systems.

Signal-based methods investigate changes in thefeatures derived from the series or their correspondingspectra using proper signal processing tools. One of themost commonly used signal processing techniques is theFast Fourier Transform (FFT), which transmits signalsfrom the time domain into the frequency domain. In

3126 H. Amini Tehrani et al./Scientia Iranica, Transactions A: Civil Engineering 26 (2019) 3125{3139

other words, the evaluation of the FFT of a signalresults in a graph in the frequency domain by takinginto account the average size of each spectral amplitudeassessed on the entire length of the analyzed signal.It is evident that the transient phase of the signal,which appears to be a fraction of the whole signal,is entirely distorted [4]. Therefore, the FFT is nota proper tool for processing non-stationary signals.The non-stationarity of the recorded signals is notmerely due to the damage to structures; even thenon-stationarity of the input and the possible inter-action with the ground or adjacent structures canshow the ine�ectiveness of classic techniques [5]. Alarge number of researchers have developed methodsfor overcoming the above-mentioned problems of theFFT in processing non-stationary signals. The short-time Fourier transform [6,7], the complex demodulationtechnique [8], and the Generalized Time-FrequencyDistribution (GTFD) as proposed by Cohen [9] areexamples of these methods.

In recent years, extensive research has been car-ried out in the �eld of structural health monitoringby utilizing the decomposing features of wavelet andwavelet packet transforms. Ovanesova and Suarez [10]used the wavelet transform to detect cracks in aone-story plane frame. They investigated the e�ec-tiveness of wavelet analysis in detecting cracks byusing numerical examples. Ren and Sun [11] used acombination of wavelet transform and Shannon entropyto detect structural damage from vibrational signals.They used wavelet entropy, relative wavelet entropy,and time-wavelet entropy as damage-sensitive featuresfor the detection and determination of the damagelocation. Several recent studies investigating wavelet-based methods can be found in [12-15].

Wavelet and wavelet packet transforms have someimperfections such as unavailability of the width oftime-frequency window for all frequencies, which is in-evitable for having windows with adjustable width [16].These transformations present more advantages thanthe FFT and the STFT, yet do not allow for a fairassessment of the local spectrum. In other words,these instruments are incapable to perform a correctevaluation of the spectral characteristics, consideringthe instantaneous variations [4].

Stockwell et al. [17] introduced a new trans-form (S-transform) that combines the characteris-tics of Short-Time Fourier Transform (STFT) andwavelet transform. Unlike the wavelet transform, theS-transform preserves phase-referenced information.This transform has been used by some researchers invarious �elds such as seismology [18,19]. In structuralhealth monitoring, for the �rst time, Pakrashi andGhosh [20] used the S-transform to detect cracks inbeams. Ditommaso et al. [21,22] introduced a �lterbased on S-transform to process the response signals,

considering the evolution of dynamic characteristics ofstructures with time. Ditommaso et al. [23] recentlydeveloped a methodology for damage assessment andlocalization of framed structures subjected to strong-motion earthquakes. In this method, the informationrelated to the evolution of the equivalent viscousdamping factor and mode shape is extracted from therecorded signal using a band-variable �lter and Stock-well Transform. Then, based on the fundamental modeshapes evaluated before, during, and after the seismicevent, the modal curvature variation is quanti�ed, anddamage location is identi�ed.

With all the advantages over all other transforms,such as wavelet and short-time Fourier transform,the S-transform also has some disadvantages one ofwhich is an invariable window that has no parameterfor allowing its width to be adjusted in the time orfrequency domain. To overcome this problem, thegeneralized S-transform is introduced with adjustabletime and frequency resolution.

In this paper, the generalized S-transform is pre-sented to detect the presence and location of damagein structures. Herein, the �rst wavelet transform andS-transform are presented; then, their di�erences inthe time-frequency representation signal are expressedin the form of a simple example. Afterward, thegeneralized S-transform is introduced, and the damagedetection procedure is explained by taking advantageof this transform. The e�ciency of the proposedmethod for damage detection is examined with respectto three di�erent numerical examples. Furthermore,the experimental data from the vibration test of amass-sti�ness system are used to verify the proposedapproach. The results reveal that the proposed methodis robust and e�cient for the detection and localizationof damage.

2. Signal processing tools

2.1. Wavelet transformThe wavelet transform is a method for converting asignal into another form, which either makes certainfeatures of the original signal more amenable to studyor enables the original dataset to be described moresuccinctly. There are, in fact, a large number ofwavelets to choose from for data analysis. The best onefor a particular application depends on both the natureof the signal and what we require from the analysis [24].The wavelet transform of a continuous signal, f(t), canbe expressed as follows [25]:

W (a; b) =1pjaj Z +1

�1f(t) �

�t� ba

�dt; (1)

where a and b are the scale and the transformationparameter, respectively. �a;b is the complex conjugate

H. Amini Tehrani et al./Scientia Iranica, Transactions A: Civil Engineering 26 (2019) 3125{3139 3127

of a;b. Function a;b(t) is obtained from Function (t)(mother wavelet) by scale a and transformation in thetime domain using b.

2.2. S-TransformThe S-transform produces a time-frequency representa-tion of a time series. It uniquely combines a frequency-dependent resolution with simultaneous localization ofthe real and imaginary spectra. This transform wasintroduced by Stockwell et al. [17].

The continuous S-transform of function h(t) withGaussian window is de�ned as follows:

S(�; f) =Z 1�1

h(t)fw(� � t; f)e�i2� ftgdt; (2)

where the Gaussian window (w) is de�ned as follows:

w(� � t; f) =jf jp2�

exp

"�f2(� � t)2

2

#: (3)

In Eq. (2), a voice S(�; f0) is de�ned as a one-dimensional function, of time for a constant frequencyf0, which shows how the amplitude and phase for thisexact frequency change over time. A `local spectrum'of S(�0; f) is a one-dimensional function of frequencyfor a �xed time, t0, [16].

The Gaussian window is chosen due to the follow-ing reasons [16]:

1. It uniquely minimizes the quadratic time-frequencymoment about a time-frequency point;

2. It is symmetric in time and frequency. The Fouriertransform of a Gaussian function is Gaussian;

3. By using it, none of the local maxima in theabsolute value of the S-transform is an artifact.

The width of the Gaussian window changes with1=jf j. This change creates distributions with di�erenttime-frequency resolutions. However, the value of 1=jf jis constant, and the width of the window cannot beadaptively adjusted to the frequency distribution of thesignal.

The S-transform uses a window with variablewidths so that the time window can be wider at thelow-frequency band to gain higher frequency resolution.On the contrary, the time window is narrower at thehigh-frequency band to obtain higher time resolution.In this respect, � is the center of the window and deter-mines the location of the Gaussian window at time axis.

Both S-transform and wavelet transform have thecharacteristic of multi-resolution. The S-transformis similar to the wavelet transform in terms of aprogressive resolution; however, unlike the wavelettransform, the S-transform retains the referenced phaseinformation absolutely and has a frequency invariantamplitude response. Moreover, the role of phase inwavelet analysis is not as well understood as it isfor the Fourier transform [16]. In addition, waveletreconstruction leads to information leakage, while theS-transform is nondestructive and reversible.

The synthetic signal shown in Figure 1 is com-posed of two sinusoidal waves with frequencies 10 and50 Hz, respectively. The signal length is 512 ms withthe sampling interval of 2 ms. Figure 1 on the left sideshows the time-frequency distribution of the wavelettransform with DB6 mother wavelet function. TheS-transform of the proposed signal is also shown onthe right side of Figure 1. It is clear that the S-transform re ects the time-frequency representation ofthe synthetic signal more precisely.

2.3. Generalized S-transformThe generalized S-transform is de�ned as follows [26]:

Figure 1. Wavelet Transform and S-transform of the sinusoidal signal.


SSG(�; f; p) =Z 1�1

h(t)fwG(� � t; f; p)

� exp(�2�ift)gdt; (4)

where wG is a generalized Gaussian window. Byapplying appropriate p factor to Eq. (3), wG is givenby:

wG(� � t; f; p) =jf jp2�

exp

"�p� f2(� � t)2

2

#: (5)

Positive parameter p is the width factor in cen-tralizing the Gaussian window that controls the widthof the Gaussian window. A sinusoidal function, as anexample, with a period of 10 sec includes a Gaussianwindow with p � 10 width. In this particular case,Eq. (2) is obtained if p = 1.

If w is an S-transform window, it must satisfy fourconditions [27,28]. These conditions are [28]:Z 1�1<fw(�; f; p)gd� = 1; (6)Z 1

�1=fw(�; f; p)gd� = 0; (7)

w(� � t; f; p) = [w(� � t;�f; p)]�; (8)

@@t

� (� � t; f; p)j t=� = 0: (9)

The �rst two conditions ensure that, when integrationis carried out over all � , the S-transform converges tothe Fourier transform:Z 1�1

S(�; f; p)d� = H(f): (10)

The third condition can ensure the propertyof symmetry between the shapes of the S-transformanalyzing function at positive and negative frequencies.The above-listed conditions provide an appropriate re-lationship between S-transform and Fourier transform.

In Eq. (5), if p < 1, then the Gaussian window inthe time domain expands; thus, the temporal resolu-tion decreases and the frequency resolution increases;however, when p > 1, the reverse of the action occurs.Figure 2 shows the Gaussian window at di�erent valuesof p and f = 3 Hz. As can be seen, by increasing theamount of p, the width of the Gaussian window in thetime domain decreases.

3. Generalized S-transform methodology fordamage detection

The main idea of the generalized S-transform methodis based on the assumption that the e�ect of incoming

Figure 2. Generalized Gaussian window in the timedomain.

load damages produces discontinuities in the structuraldeformation curve such that most of these discontinu-ities are not detectable through the observation of therecorded response signals. As will be explained, the sig-nal used here is the deformation signal of the structureas a function of the node number. Discontinuous pointsof the signal can be detected with high accuracy bythe generalized S-transform with increasing temporalresolution and decreasing frequency resolution. Thisfeature can be used in discontinuity detection of thedeformation signal. Herein, the time axis is replacedby the number of nodes.

To detect the presence and location of damageto structures by using SGS-transform, the followingprocedure is introduced:

1. The structural response signals are measured bythe desired degrees of freedom in both healthy anddamaged structures under applied loads. In beamsthe vertical, and in frames the vertical and lateraldisplacements are usually considered;

2. The time of the occurrence of maximum displace-ment in a healthy structure (tDmax) is obtained. Itcan be used as a basis of the action for damagedetection. In multi-span beams, the maximumvertical displacement at the middle of spans and inframes' columns and beams, the maximum lateraland vertical displacements are employed respec-tively for detection and localization of damage;

3. The total structural deformation signal is calculatedin both undamaged and damaged states. In thecontinuous beams, deformation signal is achievedby using vertical displacement response of all nodesin tDmax . In addition, in the frame, this signal forboth lateral and vertical displacements is achievedby using calculated responses in all frame nodes intDmax;

4. The structural deformation signals are processedin both undamaged and damaged states, and SGS-transform coe�cients (SGS-transform matrix) areachieved by using generalized S-transform. In thispaper, p = 0:5 and p = 0:25 are proposed fordamage detection in frame and beams, respectively.This matrix includes N rows and M columns such


that M is equal to the number of nodal points,and the number of each column represents the nodenumber in the original structure;

5. The structural damage matrix is calculated throughthe following equation:

Di;j =Sdi;j � Shi;j

Shi;j; (11)

where Di;j is the element of the ith row andjth column of the structural damage matrix. Inaddition, Sdi;j and Shi;j are the absolute values ofthe elements of the ith row and jth column of theSGS-transform matrices as calculated in step 4 forboth undamaged and damaged states;

6. Damage matrix elements have high sensitivity toabrupt changes and discontinuities in the defor-mation signal resulting from sti�ness changes inadjacent elements at high frequencies. Hence, thedamage location can be determined by plottingthe lower half of the damage matrix spectrumagainst the node numbers. These rows of thedamage matrix are related to high frequencies. The owchart of the proposed algorithm is shown inFigure 3.

4. Numerical study

The capability of the S-transform in structural damageidenti�cation and localization is demonstrated throughthree numerical examples. Numerical examples includea 3-span continuous beam with two �xed ends sub-jected to harmonic load with a resonant frequency, a5-span continuous beam with two �xed ends subjectedto impact loading, and a 4-story 1-bay steel moment-resisting frame under seismic loading. The purposeof using di�erent loading types is to demonstrate thecapability and accuracy of the proposed algorithm toprocess response signals caused by applying variousloading types.

In the proposed examples, since the structuralperformance is exural, all damage scenarios are con-sidered to experience a reduction in the bending sti�-ness (EI) in percentage. Structural damping has beenconsidered based on Rayleigh damping with � = 0:05.In this section, the damage patterns in structuralelements, such as beams and columns, have beenselected so as to simulate the damage occurring as aresult of high-cycle fatigue caused by the presence ofpermanent loads on the bridges (Examples 1 and 2) orlow-cycle fatigue caused by earthquakes (Example 3).

Figure 3. Flowchart of the proposed method based on generalized S-transform.


4.1. Three-span continuous beam4.1.1. Model descriptionA 3-span continuous beam with �xed supports at bothends (A and D supports) is considered as Example 1(Figure 4). Geometric properties of the beam are givenbelow. The total length of the beam is 18 m with threeequal spans of 6 m. The area moments of inertia andcross-section are the same regarding the entire length ofthe beam and are 57680 cm4 and 198 cm2, respectively.Material properties, such as modulus of elasticity andPoisson ratio, are E = 200 GPa and � = 0:2,respectively. The distributed mass with the intensityof 3260 kg/m is also considered. The performance ofthe 3-span beam in load bearing is very similar tothat of multi-span bridge with continuous deck [29].In this example, a harmonic load with a resonantfrequency is applied to excite the beam. Accordingto the studies conducted by Alwash et al. [29], theharmonic excitation at a resonant frequency of a 3-spanbeam with two �xed ends presents credible informationfor damage detection in multi-span bridges. Analysis ofthe beam under incoming loads is done using the �niteelement method in Matlab software. In this regard, thepresented beam shown in Figure 4 is subdivided into54 �nite elements with equal lengths, such that each

Figure 4. Three-span continuous beam model andrelated nodal points.

of the end nodes of the elements has two vertical androtational degrees of freedom.

4.1.2. Harmonic loadingTo apply a harmonic load, the natural frequency ofthe �rst mode of vibration should be calculated. Theestimated value of the �rst-mode natural frequency is10.6 Hz, which is calculated via eigenvalue analysis.Therefore, the harmonic excitation characterized bythe 1000 N amplitude and the frequency equal to the�rst mode natural frequency is applied in the middle ofthe second span of the beam. The time history of theharmonic excitation is shown in Figure 5. In addition,the considered damage scenarios are listed in Table 1.

Herein, undamaged and damaged structures areanalyzed under the mentioned loading condition. Fig-ure 6 represents the beam de ection curve in tDmaxboth in undamaged and damaged states for DamageScenarios (4). The damage matrix of the beam, whichis obtained from Eq. (11), includes 55 columns (thenumber of nodes) and 27 rows.

In Figure 7, the spectrum of rows 15 to 27 of thedamage matrix is shown on the right side, and thediagrams related to the entries of rows 23, 25, and

Figure 5. Time history of the harmonic excitation at theresonant frequency.

Table 1. Considered damage scenarios in the 3-span and 5-span continuous beams under incoming loads.

Damagedscenario

Example 1 (harmonic excitation) Example 2 (impulse excitation)

Damaged elements Damagevalue (%)

Damaged elements Damagevalue (%)

Scenario (1)Span 1 between nodes 12 and 13 20 Span 1 between nodes 15 and 16 20Span 2 between nodes 26 and 27 35 Span 4 between nodes 51 and 52 25Span 3 between nodes 40 and 41 25 Span 5 between nodes 68 and 69 10



Scenario (4)

Span 1 between nodes 10 and 11 20 Span 1 between nodes 8 and 9 20Span 2 between nodes 23 and 24 15 Span 2 between nodes 23 and 24 25Span 2 between nodes 36 and 37 20 Span 3 between nodes 42 and 43 10Span 3 between nodes 45 and 46 30 Span 5 between nodes 64 and 65 20


27 are demonstrated on the left side for all damagepatterns. In Figure 7, all the horizontal axes representthe node numbers. According to Figure 7, it is observedthat the location of all damaged elements has beenidenti�ed with high accuracy. For example, in Damage

Figure 6. Deformation signal of the 3-span beam attDmax in Damage Scenario (4).

Scenario (4), the number of damaged elements is 4; asshown by the result, the location of damaged elementsis precisely detectable.

4.2. Five-span continuous beam4.2.1. Model descriptionIn Example 2, a 5-span continuous beam with �xedsupports at both ends (A and F supports) is investi-gated (Figure 8). Geometric and material properties ofthe beam are the same as those of the 3-span beam, asexplained in Example 1. The distributed mass with theintensity of 3260 kg/m is also considered in Example 2.To excite the beam, an impact load with a half-sinepulse is applied [29]. Analysis of the beam underincoming loads is conducted using the �nite element

Figure 7. 23rd, 25th, and 27th elements (L-1 to L-4) of the damage matrix and damage matrix spectrum (R-1 to R-4) fordamage scenarios in Example 1.

Figure 8. Five-span continuous beam model and related nodal points.


method. For this purpose, the beam shown in Figure 8is subdivided into 75 �nite elements with an equallength such that each node at two ends of the elementhas two (vertical and rotational) degrees of freedom.

4.2.2. Impact loadingThe time history of the impact load is shown inFigure 9. The impact load amplitude is 100 N that isapplied to the structure as a half-sine pulse within 0.05sec. The point of action is in the middle of the thirdspan at a distance of 3 meters from support C. Theconsidered damage scenarios of the impact excitationare demonstrated in Table 1. The damaged and un-damaged structures are analyzed under the considereddamage scenarios, and deformation curves at tDmax areobtained for each of the patterns. Figure 10 representsthe beam de ection curve in tDmax both in undamagedand damaged states for Damage Scenarios (2). Similarto Example 1, the damage matrix is obtained throughEq. (11) for each damage pattern. The resultingdamage matrix includes 76 columns and 38 rows foreach of the patterns. In Figure 11, the spectrum ofrows 25 to 37 of the damage matrix is shown on theright side, and the diagrams related to the entries of

Figure 9. Time history of the impact excitation.

Figure 10. Deformation signal of 5-span beam at tDmax

in Damage Scenario (2).

rows 33, 35, and 37 are shown on the left side for alldamage scenario. In Figure 11, all the horizontal axesrepresent the node number. In Example 2, accordingto the damage scenarios listed in Table 1, it is observedthat the location of all damaged elements has beenidenti�ed with high accuracy. For example, in DamageScenario (3), the number of damaged elements is 3.According to Figure 11, the locations of damagedelements have been accurately detected.

4.3. Plane frame4.3.1. Model descriptionIn Example 3, a 4-story 1-bay moment-resisting frame(Figure 12) is selected to evaluate the e�ectiveness ofthe proposed method. The span length is 6 m, and theheight of all stories is the same and equal to 3 m. Thegeometric properties of the members are represented inTable 2. The modulus of elasticity is E = 200 GPa, andthe distributed mass with the intensity of 3260 kg/mis considered for all story levels. This intermediatesteel moment-resisting frame is designed based on theIranian Seismic Code (Standard 2800) -3rd edition andvolume 10 of the Iranian National Building Codes forsteel structure design. The earthquake lateral forcee�ective in this frame is calculated by the equivalentstatic method. It is assumed that the model is locatedin a high relatively hazard area with 0.3 g base designacceleration and soil type III (according to Standard2800). In addition, the occupancy importance factorof the structure is considered as moderate. In theextraction of the spectral parameters, the dampingratio is presumed to be 5%.

After the completion of the design process, theframe is analyzed in both intact and damaged statesunder the applied seismic load using �nite-elementmethod. In this regard, the frame shown in Figure 12 issubdivided into 120 �nite elements in such a way thateach element in beam and column is 0.5 m and 0.333m, respectively. Each node at both ends of the elementhas vertical and rotational degrees of freedom.

4.3.2. Seismic loading and considered damagescenarios

After the occurrence of an earthquake, detecting thepresence, location, and severity of damage is a majorstep in the assessment of structural condition. After

Table 2. The geometric properties of the members in the moment frame.

Story

Beams Columns

Cross section(m2m2m2)

Area moment of inertia(m4m4m4)

Cross section(m2m2m2)

Area moment of inertia(m4m4m4)

1st , 2nd 6.791E-3 1.507E-4 0.0216 2.639E-4

3rd , 4th 5.808E-3 1.082E-4 0.0134 1.049E-4


Figure 11. 33rd, 35th, and 37th elements (L-1 to L-4) of the damage matrix and damage matrix spectrum (R-1 to R-4)for damage scenarios in Example 2.

a major earthquake in a region, numerous aftershockswith low magnitude (PGA < 0:1 g) occur, whichcan also be said that structures at the low-intensityaftershocks will not su�er any damage. Thus, it ispossible to use aftershocks as an incoming excitationin the detection of the occurred damage in majorearthquakes.

In Figure 13, Tabas ground motion (Sade Station)and Fourier spectrum related to this excitation aredemonstrated. The considered damage scenario forseismic loading is listed in Table 3. Damaged and un-damaged structures are analyzed under the considereddamage scenarios, and the deformation signals in bothhorizontal and vertical degrees of freedom in tDmax arecalculated.

Vertical displacement in beams and horizontaldisplacement in columns represent the behavior ofthe corresponding loading. Thus, vertical and lateraldeformations in tDmax are used to detect damagein beams and columns, respectively. Vertical andlateral deformation signals in tDmax are demonstratedin Figure 14 for Damage Scenario (2). Then, accordingto the algorithm described above by using Eq. (11),the damage matrix is obtained for each of the damage

scenarios of the frame in both directions. Since thenumber of nodal points in this structure is 118, thedamage matrix dimension for each of the patternsincludes 118 columns and 59 rows. In Figure 15, thedamage matrix spectra for detecting damages to beamsare demonstrated on the right side, and the elementsrelated to rows 55, 57, and 59 of the damage matrix aredemonstrated on the left side for all damage scenarios.In Figure 15, all the horizontal axes represent thenode number. According to the damage scenariosin Table 3 and Figure 15 (R-2, R-4, R-6, R-8, R-10), it is observed that all the damaging elements inbeams have been accurately detected, while there isno sign of damage in columns. Due to the high axialsti�ness of the columns, recorded deformation signalsin the vertical direction are not appropriate for damagedetection in columns. In a similar way, R-1, R-3, R-5, R-7, and R-9 in Figure 15 are the damage matrixspectra related to the damage in frame columns. Itis apparent that all the damage elements in columnshave been detected precisely, while there is no sign ofdamage in beams, which is due to the low sensitivityof the generalized S-transform to lateral displacementin beams.


Figure 12. Plane frame model and related nodal points.

Figure 13. Time history of the seismic excitation and itsFourier transform.

5. Experimental study

In the previous section, the damage detection methodwas demonstrated through some numerical simulationstudies. However, it is useful to examine the experi-mental performance of the proposed method using mea-sured data from an experimental study. To evaluate the

Figure 14. Vertical and lateral deformation signals attDmax in Damage Scenarios (2).

e�ciency of the suggested approach, the time historyresult of a mass-spring system with 8-DOFs subjectedto impact load is employed.

The 8-DOFs mass-spring system, tested by Du�eyet al. [30], is formed with eight translating massesconnected by springs. The analytical and experi-mental models of the 8-DOFs mass-spring system arepresented in Figures 16 and 17, respectively. Theundamaged con�guration of the system is the state inwhich all springs are identical and have a linear springconstant. The nominal values of the system parametersare as follows: m1 = 559:6 g (this mass is located atthe end and is greater than others because the hardwarerequires attaching the shaker), m2 through m8 = 419:5g, and spring constants = 56.7 kN/m. Damage issimulated by replacing spring no. 5 with another springthat has a spring constant of about 86% of its originalcounterpart.

The time history of acceleration responses, asregistered by installed sensors in each degree of freedomand supports, is obtained in this experiment. Struc-tural displacement response signals are obtained byusing numerical integration of the acceleration signals.To detect the location of damage from displacementsignals in tDmax, �rst, signal resolution is enhanced byusing a spline interpolation technique. For this pur-pose, the signal is interpolated with a �ner incrementof 0.1. Then, according to the proposed algorithm, thedamage matrix of the 8-DOFs mass-spring system iscalculated through Eq. (11).

The damage matrix spectrum of the experimental8-DOFs system and the diagram related to the entriesof rows 31, 33, and 35 are shown in Figure 18. Theobtained results indicated that the proposed algorithmcould be used as a robust and viable method for damagedetection of real structures.


Table 3. Considered damage scenarios for plane frame under seismic loading.

Damage scenario Damaged elements Damagevalue (%)

Scenario (1)

Column element in the 1st story and between nodes 5 and 6 20

Beam element in the 1st story and between nodes 78 and 79 20

Beam element in the 2nd story and between nodes 90 and 91 25

Beam element in the 3rd story and between nodes 105 and 106 15

Beam element in the 4th story and between nodes 115 and 116 25

Scenario (2)

Column element in the 3rd story and between nodes 25 and 26 25

Column element in the 2nd story and between nodes 49 and 50 20



Scenario (3)





Scenario (4)

Column element in the 4th story and between nodes 30 and 31 25

Column element in the 1st story and between nodes 45 and 46 20


Scenario (5)


Column element in the 3rd story and between nodes 59 and 60 20



6. Conclusions

In this study, an e�ective method based on structuraldeformation signal processing was introduced. Inthe proposed approach, the generalized S-transformwas employed for identi�cation and localization ofstructural damage. Due to the following reasons, theSGS-transform is a tool more powerful than wavelettransform in time-frequency representation and signalprocessing:

1. The SGS-transform uses the frequency parameterinstead of the scale parameter;

2. Wavelet reconstruction leads to information leak-

age, while the SGS-transform is non-destructive andreversible in signal reconstruction;

3. The major di�erence between these two transformsis that the SGS-transform comprises the phasefactor. In other words, the SGS-transform retainsreferenced phase information absolutely;

4. The SGS-transform exhibits a frequency-invariantamplitude response in contrast to the wavelet trans-form.

To validate the e�ciency and applicability of theproposed method, a numerical study of damage local-ization was performed using three examples. Examplesinclude a 3-span beam under the harmonic load, a 5-


Figure 15. 55th, 57th, and 59th elements (L-1 to L-10) of the damage matrix in the beams and columns, and damagematrix spectra (R-1 to R-10) in beams and columns for all damage scenarios in Example 3.


Figure 15. 55th, 57th, and 59th elements (L-1 to L-10) of the damage matrix in the beams and columns, and damagematrix spectra (R-1 to R-10) in beams and columns for all damage scenarios in Example 3 (continued).

Figure 16. Schematic of analytical 8-DOFs system [30].

Figure 17. The Experimental 8-DOFs system with theexcitation shaker attached, reproduced with kindpermission from the ASME [30].

span beam under the impact load, and a 4-story 1-bay moment frame under the seismic loading. Fur-thermore, the experimental data from the vibrationtest of a mass-sti�ness system were used to verify theproposed approach. The obtained results indicatedthe robustness of the applied method for achieving themain objectives of damage detection in both numericaland experimental studies.

Acknowledgments

This research was supported by the Civil EngineeringDepartment and the Research and Technology A�airs

Figure 18. (a) Damage matrix spectrum. (b) 31st, 33rd,and 35th rows of the damage matrix in the 8-DOFssystem.

of Sharif University of Technology, Tehran, Iran. Theauthors are grateful for this support.

References

1. Amezquita-Sanchez, J.P. and Adeli, H. \Signal pro-cessing techniques for vibration-based health monitor-ing of smart structures", Archives of ComputationalMethods in Engineering, 23(1), pp. 1-15 (2016).

2. Doebling, S.W., Farrar, C.R., Prime, M.B., and She-vitz, D.W., Damage Identi�cation and Health Mon-itoring of Structural and Mechanical Systems fromChanges in Their Vibration Characteristics: A Liter-


ature Review, Los Alamos National Lab., NM, UnitedStates (1996).

3. Sohn, H., Farrar, C.R., Hemez, F.M., Shunk, D.D.,Stinemates, D.W., Nadler, B.R., and Czarnecki, J.J.,A Review of Structural Health Monitoring Literature:1996-2001, Los Alamos National Laboratory, LosAlamos, New Mexico (2002).

4. Ditommaso, R. Mucciarelli, M., and Ponzo, F.C.\S-Transform based �lter applied to the analysis ofnon-linear dynamic behaviour of soil and buildings",Conference Proceeding, Republic of Macedonia (2010).

5. Ditommaso, R., Parolai, S., Mucciarelli, M., Eggert,S., Sobiesiak, M., and Zschau, J. \Monitoring theresponse and the back-radiated energy of a buildingsubjected to ambient vibration and impulsive action:The falkenhof tower (Potsdam, Germany) ", Bulletinof Earthquake Engineering, 8(3), pp. 705-722 (2010).

6. Gabor, D. \Theory of communication. Part 1: Theanalysis of information", Journal of the Institution ofElectrical Engineers, 93(26), pp. 429-441 (1946).

7. Portno�, M. \Time-frequency representation of digitalsignals and systems based on short-time Fourier anal-ysis", IEEE Transactions on Acoustics, Speech, andSignal Processing, 28(1), pp. 55-69 (1980).

8. Bloom�eld, P., Fourier Analysis of Time Series: AnIntroduction, John Wiley & Sons (2004).

9. Cohen, L. \Time-frequency distributions-a review",Proceedings of the IEEE, 77(7), pp. 941-981 (1989).

10. Ovanesova, A. and Suarez, L. \Applications of wavelettransforms to damage detection in frame structures",Engineering Structures, 26(1), pp. 39-49 (2004).

11. Ren, W.X. and Sun, Z.S. \Structural damage identi�-cation by using wavelet entropy", Engineering Struc-tures, 30(10), pp. 2840-2849 (2008).

12. Cao, M., Xu, W., Ostachowicz, W., and Su, Z.\Damage identi�cation for beams in noisy conditionsbased on Teager energy operator-wavelet transformmodal curvature", Journal of Sound and Vibration,333(6), pp. 1543-1553 (2014).

13. Kaloop, M.R. and Hu, J.W. \Damage identi�cationand performance assessment of regular and irregularbuildings using wavelet transform energy", Advancesin Materials Science and Engineering, 2016, pp. 1-11,Article ID: 6027812, Hindawi (2016).

14. Janeliukstis, R., Rucevskis, S., Akishin, P., and Chate,A. \Wavelet transform based damage detection in aplate structure", Procedia Engineering, 161, pp. 127-132 (2016).

15. Balafas, K., Kiremidjian, A.S., and Rajagopal, R.\The wavelet transform as a Gaussian process fordamage detection", Structural Control and HealthMonitoring, 25(2), pp. 1-20 (2017).

16. Stockwell, R. \Why use the S-transform. AMS pseudo-di�erential operators: Partial di�erential equationsand time-frequency analysis", Fields Institute Commu-nications, 52, pp. 279-309 (2007).

17. Stockwell, R., Mansinha, L., and Lowe, R. \Localiza-tion of the complex spectrum: the S transform", IEEETransactions on Signal Processing, 44(4), pp. 998-1001(1996).

18. Bindi, D., Parolai, S., Cara, F., Di Giulio, G., Ferretti,G., Luzi, L., Monachesi, G., Pacor, F., and Rovelli, A.\Site ampli�cations observed in the Gubbio basin, cen-tral Italy: hints for lateral propagation e�ects", Bul-letin of the Seismological Society of America, 99(2A),pp. 741-760 (2009).

19. Parolai, S. \Denoising of seismograms using the S-transform", Bulletin of the Seismological Society ofAmerica, 99(1), pp. 226-234 (2009).

20. Pakrashi, V. and Ghosh, B. \Application of S-transform in structural health monitoring", 7th In-ternational Symposium on Nondestructive Testing inCivil Engineering (NDTCE), Nantes, France (2009).

21. Ditommaso, R., Mucciarelli, M., Parolai, S., and Pi-cozzi, M. \Monitoring the structural dynamic responseof a masonry tower: Comparing classical and time-frequency analyses", Bulletin of Earthquake Engineer-ing, 10(4), pp. 1221-1235 (2012).

22. Ditommaso, R., Mucciarelli, M., and Ponzo, F.C.\Analysis of non-stationary structural systems byusing a band-variable �lter", Bulletin of EarthquakeEngineering, 10(3), pp. 895-911 (2012).

23. Ditommaso, R., Ponzo, F., and Auletta, G. \Dam-age detection on framed structures: modal curvatureevaluation using Stockwell transform under seismicexcitation", Earthquake Engineering and EngineeringVibration, 14(2), pp. 265-274 (2015).

24. Addison, P.S., The Illustrated Wavelet TransformHandbook: Introductory Theory and Applications inScience, Engineering, Medicine and Finance, CRCPress (2017).

25. Mallat, S., A Wavelet Tour of Signal Processing, SanDiego, Academic Press (1999).

26. Stockwell, R. \A basis for e�cient representation ofthe S-transform", Digital Signal Processing, 17(1), pp.371-393 (2007).

27. Pinnegar, C. and Mansinha, L. \Time-local Fourieranalysis with a scalable, phase-modulated analyzingfunction: The S-transform with a complex window",Signal Processing, 84(7), pp. 1167-1176 (2004).

28. Wang, Y., The Tutorial: S-Transform, Graduate Insti-tute of Communication Engineering, National TaiwanUniversity, Taipei (2010).

29. Alwash, M., Sparling, B.F., and Wegner, L.D. \In u-ence of excitation on dynamic system identi�cation fora multi-span reinforced concrete bridge", Advances inCivil Engineering, 2009, pp. 1-18, Article ID: 859217,Hindawi (2009).


30. Du�ey, T.A., Doebling, S.W., Farrar, C.R., Baker,W.E., and Rhee, W.H. \Vibration-based damage iden-ti�cation in structures exhibiting axial and torsionalresponse", Journal of Vibration and Acoustics, 123(1),pp. 84-91 (2001).

Biographies

Hamed Amini Tehrani received his BSc and MScdegrees in Civil Engineering from Iran University ofScience and Technology. He is currently a PhD candi-date at the Sharif University of Technology, Iran. Hisresearch interests include structural health monitoring,structural system identi�cation, signal processing, andrandom vibrations in earthquake engineering.

Ali Bakhshi earned his BSc and MSc degrees in Civiland Structural Engineering from Sharif University ofTechnology (SUT) and Shiraz University in 1988 and

1990, respectively. In 1998, he received his PhD fromHiroshima University, Japan, where he joined as afaculty member in the same year. In 2000, he moved toSUT, where he is currently an Associate Professor. Heis actively involved in design and installation of shakingtable facility; since then, he contributed in manyexperimental testing types of research. Dr. Bakhshi'smain interests include experimental and analyticalresearch in structural control, seismic performance ofadobe structures, seismic hazard and risk analyses, andrandom vibrations. He has published many papers ininternational journals and conferences.

Moustafa Akhavat received a BSc degree in CivilEngineering from Azad University of Arak, Iran, andan MSc degree in Earthquake Engineering from IranUniversity of Science and Technology. His researchinterests include soil-structure interaction, earthquakeengineering, and structural health monitoring.

Documents

An e ective approach to structural damage localization in ...scientiairanica.sharif.edu/article_20019_f0e3edc3d677e11325375fac… · 3126 H. Amini Tehrani et al./Scientia Iranica,