indirect structure damage identi cation with the

Scientia Iranica A (2021) 28(4), 2101{2118

Sharif University of TechnologyScientia Iranica

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

Indirect structure damage identi�cation with theinformation of the vertical and rotational mode shapes

M. Ramezani and O. Bahar�

International Institute of Earthquake Engineering and Seismology (IIEES), Tehran, P.O. Box 19395-3913, Iran.

Received 21 September 2020; received in revised form 7 December 2020; accepted 8 May 2021

KEYWORDSIndirect damageidenti�cation;Rotational modeshapes;Improved geneticalgorithm;Model updating;Modal strain energy.

Abstract. The present study aims to propose a robust method to detect the damageseverity and location of the structural elements, focusing on the data type and acquisitionmethod and promoting the model updating tools. The novelty of this method lies in itsrotational mode shape acquisition that provides valuable information on the damage. Inthis method, the damaged elements were indirectly identi�ed by detecting the healthy ele-ments and eliminating them from the search space. Moreover, this method could minimizethe modal strain energy di�erence between the damaged reference model and the numericalmodel using an optimization algorithm. An improved genetic algorithm was then employedto perform the optimization task. In this study, four numerical and two experimentaldamage scenarios were applied to a simply supported beam to examine the performance ofthe proposed method. Data acquisition systems were implemented using vision-based andaccelerometer-based methods. The results indicated that this method could accuratelyidentify the location and severity of damage using only the �rst mode shape since therotational mode shapes were more sensitive to damage than the vertical mode shapes.© 2021 Sharif University of Technology. All rights reserved.

1. Introduction

Signi�cant damages to civil infrastructures caused byrecent natural disasters [1,2] have highlighted the sig-ni�cance of timely damage identi�cation in structuralsystems. Any undesirable change in the materialproperties or geometry of the structure is regarded as adamage. The age of structures, element deterioration,natural disasters, wrong use, and lack of proper main-tenance are the main causes of damages. In case thedamages are not detected, the structure may collapse.Visual inspections and traditional non-destructive testsfail to detect the damages hidden in the nonstructuralelements. Therefore, a global method is required todetect damages at any inaccessible location. Structural

*. Corresponding author. Tel.: +98 21 22831116E-mail address: [email protected] (O. Bahar)

doi: 10.24200/sci.2021.56842.4939

damages reduce the sti�ness of one or more elements.The damage in elements would change the structuraldynamic properties such as the modal frequenciesand mode shapes. The dependency of the dynamicproperties of structures (i.e., natural frequencies andmode shapes) on the physical properties (i.e., massand sti�ness) has led to development of some methodsfor health monitoring of the structures. Contaminatedmeasured data and low sensitivity to damages havehindered the development of basic/primitive meth-ods [3]. Use of convenient tools for measuring the dataor the presence of signi�cant damages can enhancethe performance of these methods [4]. With regardto the limited performance of the basic/primitivemethods, researchers have developed methods such asmode shape curvature, Modal Strain Energy (MSE),model updating, exibility-based methods, etc. thatcould provide more accurate information about thelocation [5{9].

A number of methods that use output-only data

2102 M. Ramezani and O. Bahar/Scientia Iranica, Transactions A: Civil Engineering 28 (2021) 2101{2118

for damage detection require much information anda large number of vibration modes, and their perfor-mances may be signi�cantly attenuated in case of anyreduction in the amount of available information [10].Based on what was mentioned, Pandey and Biswas[7] determined the location of damage using a limitednumber of modes. Yang and Liu [11] proposed analgorithm based on a change in the exibility ofmatrices that could determine the location and severityof damages. Li et al. [12] reduced the e�ect of higher-mode elimination by adding the generalized exibilitymatrix instead of the global exibility matrix.

In recent years, MSE has been the main focus of anumber of studies on the structural health monitoring.This parameter, composed of the element sti�nessand mode shapes of structures, is highly sensitiveto the presence of damage in the structure. Shi etal. [13] proposed a method based on MSE variationsin damaged and healthy elements. In their proposedmethod, elements with a signi�cant change in MSEwere regarded as potentially damaged elements. Thelocation and severity of damages were also determinedthrough a quantifying step. Later, Moradipour etal. [14] improved this method by reducing numericalerrors.

Homaei et al. [15] proposed a direct damage de-tection method based on the criterion of mode shapes.Their method enjoyed several advantages namely sim-plicity, applicability to a large number of elements,accuracy of the location of damage, sensitivity to lowdamage severity, estimation of the number of requiredmode shapes, and low sensitivity to noise-contaminateddata. They investigated the viability of their proposedmethod using three beam examples over di�erent timespans.

Yan and Ren [16] proposed a method basedon the sensitivity of the modal exibility that usedmultiple modes for damage detection. The merit oftheir method was its ability to eliminate the adversee�ect of truncating higher-order modes. Their proposedmethod was evaluated using two beams in the presenceof noise in mode shapes. Yazdanpanah and Seyedpoor[17] introduced a new damage indicator based on themode shape data. The mode shapes as well as theirslope and curvature before and after damage were takeninto account in damage detection. Their method wasveri�ed by two beams and multiple mode data. Shahriand Ghorbani-Tanha [18] considered the ratio of modalkinetic energy change in their study with low sensitivityto common noise. Their method was veri�ed using abeam-like structure considering several required modesand a sensitivity matrix was calculated by the closed-form sensitivity of eigenvalues of the structure.

Many metaheuristic algorithms developed to solveoptimization problems can be used for damage de-tection such as Genetic Algorithm (GA) [19], Par-

ticle Swarm Optimization (PSO) [20], Arti�cial BeeColony (ABC) algorithm [21], tug of war optimiza-tion algorithm [22], Di�erential Evolution (DE) al-gorithm [23], Enhanced Vibrating Particles System(EVPS) algorithm [24], and Water Evaporation Opti-mization (WEO) algorithm [25], etc. These algorithmscan search the space for possible answers withoutsolving equations. The convergence speed, �ndingglobal optimum, accuracy of answers, and type ofobjective function determine the most appropriatetool for damage detection. Gomez and Silvia [19]investigated two damage detection methods includinga frequency sensitivity-based method and GA. Theycompared these methods and achieved similar results.Some �rst frequencies were required in their method toevaluate the damage location and severity.

Nobahari and Seyedpoor [26] proposed a GAmodi�ed by two new operators which could increasethe speed of convergence. The objective functionof their proposed algorithm was based on correlationwhich required several �rst frequencies. Seyedpoor [20]proposed a two-stage method to identify the locationand severity of damages. At the �rst stage, thelocation of damaged elements was identi�ed by theMSE-based index. Then, the severity of damages wasdetermined by PSO at the second stage. Xu et al. [21]presented a method based on a chaotic ABC. Theirproposed method had some advantages such as ease ofimplementation and good robustness. To overcome theshortcoming of this method, i.e., the slow convergencerate, they utilized the tournament selection mechanisminstead of the roulette mechanism. Furthermore,they introduced a chaotic search mechanism. Theobjective function of this method used the residuals ofnatural frequencies and modal assurance criteria, whichrequired multiple-mode data. Seyedpoor et al. [23] usedan improved DE algorithm as a tool for detecting loca-tion and severity of damage. The objective functionof their method was de�ned in terms of changes innatural frequency. Their proposed model was veri�edby an experimental model and two numerical exampleswith the consideration of measurement noise. Toloueet al. [27] used the damage load vectors computedby variations in intact and faulty exibility matricesfor damage detection of a frame. In this method,acceleration responses were taken into account to derivethe exibility matrix.

Kaveh et al. [24] employed the EVPS optimiza-tion algorithm to detect damages in truss structuresand compared the results of this method with thoseobtained from the vibrating particles system algo-rithm. According to this algorithm, each responseis assumed to be a single-degree-of-freedom systemwith viscous damping approaching its equilibriumpoint. The authors employed the modal shape andfrequency information to de�ne the objective function

M. Ramezani and O. Bahar/Scientia Iranica, Transactions A: Civil Engineering 28 (2021) 2101{2118 2103

and evaluated this method for four truss structuresbased on the information of the �rst few modes. Inorder to reduce the computational e�orts during theoptimization procedure, Kaveh et al. [25] proposed atwo-phased method. In the �rst phase, the structuralfrequencies are calculated; in case the frequencies areequal, the algorithm moves on to the second phase anddetermines the modal shapes. As a result, the numberof cases under study would decrease. They applied thisidea to the accelerated WEO algorithm and identi�edthe damages to the beam, frame, and three-dimensionalstructures.

To establish an accurate �nite element model,Liang et al. [28] proposed a model updating methodby combining the cross-model and cross-mode. Theyemployed natural frequencies to update the modelowing to their ease of measurement, high precision,and insensitivity to noise. They veri�ed the proposedalgorithm using a numerical model and two actualshear structures. Khatir et al. [29] proposed a two-stage approach to damage detection. At the �rststage, the potential damaged elements were identi�edby the local frequency change ratio indicator. Todetect damage severities, the teaching-learning-basedoptimization algorithm was utilized at the secondstage. They compared their method with PSO andbat algorithm. Mishra et al. [30] used an ant lionalgorithm, a population-based search algorithm, withfewer parameters than other meta-heuristic algorithmsfor structural damage detection. The objective func-tion of their method used multiple vibration data suchas some �rst mode shapes and natural frequencies. Kr-ishnanunni et al. [31] studied how natural frequenciesand static displacements could change in case of theoccurrence of the structural damage. They employedthe cuckoo search algorithm that used vibration dataand static displacement measurements for damagedetection. Their method was evaluated using cantileverand �xed both ends of the beams, which required somemode data to meet its objective function.

Inclinometers, or rotation sensors, which arewidely used in industries such as automotive,aerospace, and electronics, are designed to measureangular rotation. These sensors are designed basedon the pendulum behavior caused by gravity. Thependulums used in inclinometers are categorized intothree types, i.e., solid masses, liquids, and gases, whichcan be measured in a variety of ways. Over the pastyears, the performance and accuracy of inclinometershave considerably improved, making them suitable fordamage detection applications. The sampling rate ofthe inclinometers decreases with an increase in themeasurement accuracy. However, the application of theproposed method to collect rotational data is part ofthe novel methodology presented in this study whichexamines whether it can be used for damage iden-

ti�cation. Although the application of inclinometersor rotational responses to understanding the behaviorof the structure has proved to be e�ective, limitedstudies have been conducted on the direct measurementof rotation to identify the damage [32{35], mainlydue to the limitations of sensor technology such aslack of commercially accurate sensors and samplinglimitations.

Alten et al. [32] employed acceleration data todetect the damage with emphasis on frequency changes.They also investigated the e�ect of damage on therotation increase. In another study, they examinedthe sensitivity of the rotation to the damage in bridgethrough numerical and experimental analyses. Theyproposed a damage detection method based on thedeformation area di�erence, which could detect damagethrough an area di�erence in the rotation diagrams ofhealthy and damaged structures under static loading.Huseynov et al. [35] evaluated the behavior of a simplysupported beam numerically and experimentally usingthe direct measurement of rotation in healthy and dam-aged structures (di�erence in rotation in uence lines).They employed the di�erence in rotation in uence linesas a criterion for damage identi�cation. In their study,the e�ect of rotation sensitivity to damage as wellas the position of sensors to sensor sensitivity werestudied. The mentioned authors used a high-gradeuniaxial accelerometer to measure the rotation. Theyshowed that the use of modal data alone for damagedetection in their model was not entirely successful, andthat use of strain gauge sensors could partially detectthe damage in case the strain gauges were installed nearthe damage location. However, the results of measuringthe rotation showed the damage in all studied scenarios.In a similar study, Hester et al. [34] directly measuredthe rotation in a simple beam. First, they examined thesensitivity of the rotation as a damage index through anumerical study. This method was then experimentallystudied on a simple beam. Finally, their proposedmethod was evaluated on a bridge with a simplysupported span of 20 meters. Acceptable results wereobtained from damage assessment.

Several studies have contributed to the introduc-tion of damage detection methods. Among them, themethod that can precisely determine the damage sever-ity of elements with a minimum scope of informationis the best one. The main objective of this study is toaccurately localize the damaged elements and quantifytheir severities using only the �rst vibration modeshape. Despite the bene�ts of rotation measurementin terms of damage identi�cation, limited studies havebeen conducted on the direct measurement of thisparameter. In the present study, modal informationwas extracted through the displacement and rotationsignals acquired by the vision-based method and ac-celerometers. Behind this method lies some novel


ideas such as extraction of rotational mode shapesand indirect damage detection that �nally lead to theestablishment of a robust damage detection methodcapable of accurately �nding damage severities usingonly the �rst vibration mode shape without noise andobtaining acceptable results using noise-contaminateddata. At the updating stage of GA, the experiencesconcerning the destruction of actual structures such asdamage occurrence in the limited number of elementswere included to increase the convergence speed andidenti�cation accuracy. In general, this method found asolution to decreasing the number of required vibrationmodes.

2. Theoretical background

Modal characteristics of the undamped linear systemsare described by the following equation (the eigenvalueequations):

[K] f�ig = �i [M ] f�ig ; (1)

where [K] and [M ] are the ND�ND global sti�ness andmass matrices, respectively; ND indicates the numberof Degrees Of Freedom (DOFs) of the system; �i andf�ig are the natural frequency and the ith vibrationmode shape of the system, respectively. While eachnode in the two-dimensional model has three DOFsincluding two transitional and one rotational DOFs,each node in the three-dimensional model has six DOFsincluding three transitional and three rotational DOFs.Mode shape or natural vibration shape indicates howthe points de�ned in the structure or the DOFs ofthe structure would change when vibrating at thenatural frequency of the structure. In fact, the naturalfrequencies and mode shapes show how a structurevibrates under dynamic loads. Although the structuralresponse is a function of the input excitation, it isessentially composed of a set of mode shapes withdi�erent participation factors. This is the reason whyextracting the mode shape of the structure is indepen-dent of the type and intensity of the excitation. Incase of damage occurrence in the structures caused byoverloading, fatigue, etc., the dynamic characteristicswould consequently change due to their dependence onphysical characteristics, as shown in Eq. (1). On thecontrary, structural damages often reduce the sti�nessof one or more elements throughout the system andrarely decrease the mass of the structure. Therefore,mass changes were ignored in this study.

The vibration equation of the damaged system isde�ned as follows:

[K]d�

�di

= �di [M ]�

�di; (2)

where superscript \d" denotes the damaged state.

The global sti�ness of the system is obtained fromthe total elements of the system, as shown in thefollowing:

[K] =NEXj=1

[K]j ; (3)

where [K]j is the ND �ND sti�ness matrix of the jthelement in the global sti�ness, and NE is the totalnumber of elements used in forming global structurematrices. Therefore, the e�ect of damaged elementsin the global sti�ness matrix is de�ned through thefollowing equation:

[K]d = [K] +NEXj=1

� [K]j = [K] +NEXj=1

�j [K]j ; (4)

where � [K]j is the di�erence in the sti�ness in bothhealthy and damaged states and �j is the sti�nessreduction coe�cient of the jth element. Since theelement sti�ness in the damage state is lower thanthe healthy state, the sti�ness reduction coe�cient isconsidered as �1 < �j � 0.

2.1. Calculating MSE of elementsMSE, measured by mode shapes and sti�ness, is anentirely sensible parameter for damage detection. TheMSE values for the ith mode of vibration and thejth element in healthy and damaged states are de�nedthrough the following equations [9]:

MSEij =12f�igT [K]j f�ig ; (5)

MSEdij =12�

�diT

[K]dj�

�di: (6)

Due to the unknown damage severity at �rst, the sti�-ness estimated by the proposed algorithm is consideredfor calculating the MSE of the damaged state.

The MSE of the entire structure obtained fromthe MSE of the whole elements for the ith mode ofvibration is de�ned as follows [20]:

MSEi =NEXj=1

MSEij : (7)

3. Optimization algorithm

The GA that contains some processes such as reproduc-tion, crossover, and mutation is nature driven. Thisalgorithm is such a powerful optimization tool thatsearches the problem space randomly and purposefully.After every step, GA �nds even better answers. In casethe populations obtained from the reproduction processproduce inappropriate answers related to the previous


step, the former populations will be used for the nextstep. Therefore, this algorithm always moves towardsthe target [36]. One of the most signi�cant advantagesof the GA is the parallel search, which empowers it tosolve huge and nonlinear problems even if the problemshave a local optimum. Therefore, for damage detec-tion problems that comprise numerous variables withnonlinear behavior, the GA is an appropriate choice.Although probabilistic selection rules are used in theGA, the possibility of importing probable answers stillexists, thus increasing the convergence speed.

3.1. Determination of damage severity byImproved Genetic Algorithm (IGA)

The IGA was used for damage detection purposes inthis study. Since the sti�ness reduction coe�cient ofelements has already been introduced already, �j isde�ned as an output parameter of IGA. Upon achievingthe �j coe�cients, the damaged elastic modulus ofelements is determined through the following equation:

Edj = (1 + �j)Ej ; (8)

where Edj and Ej are the elastic moduli of the jthelement in the healthy and damaged states. In orderto improve GA, it must be coded using MATLAB orother programming software. Based on the type ofcoding in this algorithm and the range of variablespresented in Table 1, the chromosomes are generatedby 0 and 1. The number of chromosomes is equal tothe number of population in each generation. Eachchromosome contains the elastic modulus informationof all elements, which should be transferred to base 10to obtain real values that are the same as the sti�nessreduction coe�cients. These coe�cients are convertedto the elastic modulus of the updated model throughEq. (8). The obtained elastic modulus is assignedto the elements to calculate the sti�ness of elements.It is necessary to calculate the MSE of the updatedmodel through Eq. (9) in order to achieve the objectivefunction.

MSEpdij =12

n�pdi

oT[K]j

n�pdi

o; (9)

where superscript \pd " denotes the potential damage.After calculating the MSE of the updated model viaIGA, MSE changes of the jth element in each mode(any desired mode) are calculated using the followingequation:

�MSEij = MSEdij �MSEpdij ; (10)

where subscript i is the number of desired modes.In case the sti�ness reduction coe�cients are

accurately estimated by the IGA, the value of MSE dif-ference will be zero. Therefore, the objective of the IGAis to minimize the sum of MSE changes of all elementsin any desired mode. Since the �rst mode of structurescan usually be obtained more easily than others, thismethod employed it at all steps. The chromosomes thatreduce the objective function are ranked as the morecompatible ones with a greater chance for reproduction.The mutation operator changes binary numbers; hence,the IGA is not trapped in a local optimum.

After entering the best chromosome of the lastgeneration to those obtained from reproduction andmutation steps based on the elitism concept, the new-generation chromosomes are used for �tness determi-nation.

According to Figure 1, GA is the optimizationalgorithm that minimizes the de�ned objective functionto estimate the damage coe�cients. However, inorder to reduce the optimization time and increase theaccuracy of results, several changes have been made inthe usual process of this algorithm; this is the reasonwhy this algorithm is called IGA. These changes are asfollows:

1. Although, in reality, the exact number of dam-aged elements in structures is unknown, it can beclaimed based on the experience that the numberof damaged elements is smaller than the totalnumber of structural elements. This assump-tion can signi�cantly improve the performance ofoptimization algorithms in damage identi�cationproblems. Furthermore, since identi�cation of thedamage coe�cients of structural elements begins

Table 1. The Improved Genetic Algorithm (IGA) parameters.

The population of generations 10Maximum number of generations 100The number of variables 15 for Ex 1

16 for Ex 2, 4 for Ex 3The range of variables [�1; 0]Mutation rate 0.02Coding BinarySelection Roulette wheelCrossover Uniform crossover binary


Figure 1. Determining the severity of damages in the structural elements using Improved Genetic Algorithm (IGA).

randomly, it is likely that the estimated damagecoe�cients at the �rst step are very similar to/verydi�erent from the actual values. This can a�ect thetime required for achieving the accurate damagecoe�cients. Based on these remarks, one of thechromosomes at the �rst step is created when allthe damage coe�cients are equal to zero. Inother words, one of the members of the producedpopulation creates a structure whose elements areall healthy. With this change, healthy elements areidenti�ed quickly and the a�ected elements in thesearch process become the focus;

2. One of the challenges of optimization algorithms isthe large number of variables. With the changesmade at the previous step, the algorithm detectssome of the healthy elements (according to expe-riences, not only the healthy elements but alsothe elements with no damaged element in theirvicinity are quickly identi�ed as healthy elements).By identifying these elements with no improvementmade to the results in a few steps, the programremoves healthy elements from the search space;

3. To start and resume the search task, one of thechromosomes that enters the initial population ischosen as the best chromosome identi�ed from the

previous search process with a larger number ofvariables;

4. In this algorithm, the threshold for determininghealthy and damaged elements is determined as1%. This issue can be discussed from two di�er-ent perspectives: (I) damage detection and (II)improvement of the optimization algorithm per-formance. According to the �rst view, all dam-age detection methods and techniques have falsealarms, which can have di�erent values dependingon the accuracy of the method. False alarm or falsepositive refers to identifying a healthy element asa damaged element. This occurs for the elementsthat are in the vicinity of damaged elements. Theideal objective of damage detection techniques isto detect the location and severity of damagesat the early stages of its formation. However,the already available damage detection methodshave not achieved this objective yet due to somelimitations like noises, measurement issues, sensorlimitations, and physical behavior of the structuredepending on di�erent parameters. In this method,this error is kept at 1% in the analytical model.This view inspires the second view. From thispoint of view, the identi�ed damages below 1%


are not reliable. In fact, this value can be calledthe response reliability limit. Therefore, insteadof applying the optimization algorithm to theseelements, they are identi�ed as healthy and thus,are omitted from the search process.

This process is repeated until the objective func-tion is reduced to a speci�c value or a certain numberof iterations. After removing the healthy elements, theconvergence speed and detection accuracy increase inthe next iteration of IGA as a result of reducing IGAvariables. In fact, following the convergence of theIGA and removal of the healthy elements, the IGAsearches the new search space in the next iteration.These iterations continue until the objective functionreaches zero. The characteristics and parameter valuesof the IGA are presented in Table 1.

4. Numerical simulations

A numerical simulation was employed in this studyto evaluate the performance of the proposed method.This example includes a simply supported beam [18]with 15 similar elements, 16 nodes, and 30 DOFs,as shown in Figure 2. Increasing the number ofdivided elements facilitates accurate detection of dam-ages; however, more sensors are still required for thederivation of mode shapes.

The simulated beam is considered as a concretebeam where E = 28 GPA and � = 2500 kg/m3.

4.1. Investigation of the proposed methodwithout noise

In this study, damages were simulated by reducing theelasticity modulus. As shown in Table 2, di�erentdamage scenarios and damage severities were employedto examine the proposed method. A parametric studywas carried out in order to investigate the convergenceprocess for DS2 pattern in the simply supported beam.Di�erent possible states of �4 and �6 parameters werestudied with the accuracy of 1%. As shown in Figure 3,the IGA converged into the exact values only afterfew generations, and in case the elements suspectedof damages were known, 900 analyses were required to�nd accurate answers.

The convergence process of the objective functionof the numerical example is shown in Figure 4 thatcontains two curves: \maximum population �tness"and \mean population �tness".

The \maximum population �tness" curve indi-cates the performance of the best chromosome in each

Figure 2. A simply supported beam with 15 elements.

Table 2. Damage scenarios of the simply supportedbeam.

Damage scenario Damagedelement(s)

Damageseverity

DS1 Single damage No. 6 10%

DS2 Double damage No. 4 10%No. 6 15%

DS3 Triple damage No. 4 10%No. 6 15%No. 11 20%

DS4 Multiple damage No. 4, No. 6 10%No. 8, No. 11 15%No. 12 20%

Figure 3. Parametric study of the sti�ness reductioncoe�cient in DS2 pattern.

generation. As shown in these �gures, in case thechildren chromosomes do not have better compatibilitythan the parent chromosomes, the curve becomes hor-izontal. This curve continues horizontally up until theimprovement of one children chromosome comparedwith parent chromosomes. The best chromosome ofeach generation is directly transmitted to the nextgeneration based on the elitism concept used in thisalgorithm. This idea prevents the distraction ofimproper children from the optimization path. The\mean population �tness" curve indicates the averagecompatibility of each generation determined based onthe objective function. The elite chromosome trans-mitted from the prior generation is more likely to be


Figure 4. Convergence process of the simply supported beam: (a) DS1, (b) DS2, (c) DS3, and (d) DS4.

selected for reproduction than other chromosomes, andit can create children chromosomes with a high sharedsimilarity with the parent chromosomes. The �tnessvalues shown in this graph follow a descending trend.

According to Figure 4, all elements are added tothe IGA for damage detection as inputs. Following atemporary convergence, the algorithm recognizes thatthe objective function is enhanced upon reducing thedamage coe�cients of some elements. Therefore, zerovalues are assigned to these elements and are removedfrom the problem space. This process decreases thenumber of variables and increases the accuracy ofdamage estimations. This cycle is repeated as longas the objective function reaches zero. Finally, theIGA determines the damage severity of elements withthe highest accuracy. Since the process of damagedetection starts randomly, it can challenge the timeof achieving accurate damage coe�cients; in otherwords, there is a possibility for the estimated damagecoe�cients at the �rst step to be similar to/di�erentfrom the actual values. As shown in Figure 4, DS1 andDS2 patterns converge to optimal values after threeiterations of IGA and elimination of healthy elementswhen DS3 and DS4 minimize the objective functionsthrough 2 and 4 iterations, respectively.

However, it should be noted that the complexityof the examined structure or sophisticated damagescenario can a�ect the number of IGA iterations in�nding a de�nite answer; therefore, the IGA removesfewer elements from the search space in each iterationand the search space enlarges at a lower speed; yet, itdoes not have any in uence on the accuracy of damagedetections.

Element damage severities of the simply sup-ported beam are shown in Figure 5. According tothis �gure, in case there is no noise in the mode shapedata, the damage severity is accurately calculated byonly one mode (the �rst mode) with no errors after afew steps. In addition, some healthy elements in this�gure are detected as slightly damaged elements at the�rst step of IGA implementation. Furthermore, theestimated damage coe�cients of the damaged elementsare subject to slight error. Through elimination of thenumber of healthy elements detected at the previousstep, the damage coe�cients of the remaining elementsare identi�ed with higher accuracy. As shown inFigure 5, some elements, slightly damaged with smalldamage coe�cients at the early steps, are identi�ed asthe healthy elements and eliminated from the problemspace at the next steps. Although some elements are


Figure 5. Damage severity of the simply supported beam in (a) DS1, (b) DS2, (c) DS3, and (d) DS4.

detected to be damaged at the early steps, their exactvalues can be reached.

4.2. Investigation of the proposed method withnoise

In the estimation process of the information relatedto the damaged structures, complete noise eliminationis impossible. In fact, the presence of noise in theobtained information from sensors causes a deviationfrom the exact values of damage severity. In thissection, the sensitivity of the proposed method to thepresence of noise in the data is investigated. In order tosimulate the e�ect of noise, the mode shape data withdi�erent noise levels of 0.05%, 1%, 2%, 3%, 4%, and5% are contaminated. The noise-contaminated modeshapes are obtained through the following equation:

��ir = �ir(1 + �r �

� j�max;ij); (11)

where ��ir and �ir are the ith mode shape componentsof rth DOF with and without noise, respectively; and �r is a Gaussian random number with the mean and

variance of zero and one, respectively; further, �� isthe noise level and j�max;ij is the ith largest modeshape component without noise. The Coe�cient ofVariance (COV) is used here as a tool for investigating

the damage estimation accuracy through the followingequation:

COV =��; (12)

where �� and �� are the standard deviation and averageof the damaged coe�cients, respectively, estimated byIGA through 1000 tests. In Figure 6, the averagedamage coe�cients in di�erent scenarios for the simplysupported beam are shown in the presence of 1% noisein mode shapes. As shown in this �gure, damages inhealthy elements are approximately 1%. Since the IGAwhich is based on mode shapes and is polluted by noisesminimizes the objective function, these detections arenot indicative of its poor performance. In other words,the IGA takes into account the mode shape variationsbecause of the similarity between the noises to thedamages.

Figure 7 shows the COV at di�erent noise levelsand in damaged scenarios. The results indicated thatIGA could appropriately identify damage severities upto 1% noise level.

Given that the objective function of this algo-rithm is calculated using Eq. (13), which is directlya�ected by vibration mode shapes, the performance of


Figure 6. Damage averages of simply supported beam elements in (a) DS1, (b) DS2, (c) DS3, and (d) DS4 (with 1%noise).

the IGA will decrease upon increasing noise levels.

�MSEij =12�

�diT

[K]j�

�di� 1

2

n�pdi

oT[K]j

n�pdi

o: (13)

In case of employing pure data, the proposed methodcan exactly identify damage severities. Therefore, byreplacing the objective functions that are not a�ectedby the noise, the performance of IGA can be improvedin the absence/presence of noise and only through the�rst mode.

5. Experimental validation

An experimental setup was studied to investigatewhether the application of rational DOFs could a�ect

the proposed method and to help identify the locationand severity of damages through only one mode. Thespeci�c objective of this example was to identify thedamage location and severity using only one mode.In order to validate the performance of the proposedmethod, a simply supported steel beam including 16segments of 7.5 cm was investigated. The mechanicaland geometrical properties of the beam are listed inTable 3. Since the mass of each acceleration sensor,the wire connected to it, and Plexiglass piece on whichthe sensor was placed were about 30 grams, movingthe sensors did not change the mass distribution andthus, equivalent added masses were used in other placeswith no such sensors. As shown in Table 3, the densityof the beam was considered to be higher than that ofthe current steel elements mainly because the elementmarkers and accelerometers used for data acquisition

Table 3. Properties of the simply supported beam.

Young's modulus(N/m2)

Density(kg/m3)

Length(mm)

Width(mm)

Thickness(mm)

2� 1011 11266 1200 83 3


Figure 7. Coe�cients of variance for the simply supported beam in (a) DS1, (b) DS2, (c) DS3, and (d) DS4.

Figure 8. Experimental modeling of supports: (a) The joint support and (b) the roller support.

as well as the added masses were more signi�cant thanthe beam. Therefore, the mass of these attachmentswas distributed across the total length of the beam.The hinge and roller supports were constructed similarto theoretical examples, as shown in Figure 8.

The overall view of the beam and measuringinstruments including accelerometers, markers, andcomputer system equipped with data logger and cam-era are shown in Figure 9. For this model, two di�erentdamage scenarios were considered. The �rst scenarioincludes a 30% sti�ness reduction in the 6th element.

In the second scenario, in addition to the elementnumber 6, a 20% sti�ness reduction was also consideredfor the 13th element. The transverse cuts were exertedto the desired locations while the measurable damagescontinued throughout the distance between the twomarkers. Given that one of the assumptions of thismethod, before and after the damage, was that themass was constant, the cut pieces were again stuck tothe beam so that the mass would remain unchangedwithout increasing the sti�ness. Figure 10 shows howthe damages are incurred against the predetermined


Figure 9. (a) The overall view of the beam and division of the �lming range. (b) Computer system with data logger. (c)Video camera.

Figure 10. Details of the damaged elements: (a) Element 6 and (b) element 13.

elements based on Scenarios 1 and 2. Of note, thedamage severity of elements was equal to the ratioof the moment of inertia of the reduced and intactsections. Since the constructed beam has no o�-plane motion, it can be referred to as a 2D model.On the contrary, because the horizontal sti�ness ofthe beam was higher than the vertical and rotationalsti�ness, these DOFs were ignored in the numericalmodel updating. The data required for this methodcontain the �rst mode shape and frequency, and thereare di�erent methods for extracting this informationfrom the structural response. For this reason, a staticload was applied to the middle of the beam and then, itwas suddenly removed to create free vibrations. In thisstudy, the displacement and acceleration responses in

both vertical and rotational directions were obtainedat de�ned points under free vibration using a vision-based method and uniaxial accelerometer sensors, re-spectively. These responses were then used to obtainthe vertical and rotational mode shapes via Fourieranalysis. The output of these two methods would beboth �rst-mode frequency and �rst-mode shape vectorcontaining the mode shapes of both rotational andvertical DOFs.

5.1. Vision-based method for data acquisitionThe mechanism of measuring the movement in thevision-based method involves installing cameras on atripod, setting tracking targets, using template match-ing algorithm, and converting the coordinates. For


Figure 11. Detecting the center of the circles.

Figure 12. Vertical pixel displacement of point 9.

Figure 13. Vertical displacement of point 9.

this purpose, a camera capable of recording video in240 frames per second was used. To increase theresolution of the images, �lming was done at fourstages from di�erent parts of the beam, as shown inFigure 9(a). Black circles on a white background wereused as the targets. These targets were stuck on arigid and negligible mass element called marker withPlexiglass material. By detecting the center of thecircles in each frame, the movement signal was obtainedbased on the position of the image pixels. In order toachieve physical and real displacement, a relationshipwas established between pixel coordinates and realcoordinates. The steps are illustrated in Figures 11to 13. As shown in Figure 13, the recorded noise isapproximately 2:2�10�5 m, which can be decreased orincreased by changing the distance between the cameraand the target.

Figure 14. Vibration of the beam and the targetsinstalled on it.

As discussed in the previous section, the verticaland rotational mode shapes were obtained to detectdamages using only one mode. The most signi�cantfactor that ensures the success of this process is theinformation used as the inputs such as rotational modeshapes. Figure 14 shows how the targets move duringthe vibration. If a rigid connection is made betweenthe marker and the beam, no relative rotation willoccur and the rotation of the marker will be equal tothat of the beam at that location. Based on Eq. (14),by tracking the relative horizontal displacement inmarkers, the rotation of the beam can be obtainedwherever the marker is located.

�t = ArcSin�

�x1t ��x2

tL

�'�

�x1t ��x2

tL

�; (14)

where �t is the time history of rotation; �x1t and �x2

tare the time histories of horizontal displacement of thetargets; and L is the distance from the center of twocircles.

A comparison of the mode shapes obtained fromthe experimental signals and numerical modeling indi-cated some slight discrepancies caused by scaling eachpart of the mode shapes or local imperfections in thebeam. To reduce the e�ect of the scaling errors, themode shapes were obtained using the same methodand reference scale in all scenarios. These di�erenceswere then added to the experimental mode shapes inall scenarios as bias values.

The �rst mode shape of the damaged beamacquired by the experimental data is presented inFigures 15 and 16. A comparison of the rotationalmode shapes of both healthy and damaged beamsin Figure 15(b) revealed that the rotational modeshape of the damaged beam passed exactly betweenpoints 6 and 7, represented as element 6, through therotational mode shape of the healthy beam. Accordingto Figure 16(b), the rotational mode shape of thedamaged beam passed through that of the healthybeam in elements 6 and 13. In this respect, it can


Figure 15. First mode shape (Scenario 1): (a) Vertical mode shape and (b) rotational mode shape.

Figure 16. First mode shape (Scenario 2): (a) Vertical mode shape and (b) rotational mode shape.

Figure 17. First rotational mode shape: (a) Scenario 1 and (b) Scenario 2.

be concluded that the intersection of two lines in therotational mode shape pointed to the location of thedamage. According to Figure 17, a comparison of therotational mode shapes obtained from the numericalresults con�rmed this �nding.

Comparison of the vertical mode shapes in both

scenarios failed to provide any speci�c informationabout the damaged elements. This is the reason whymany researchers prefer using higher modes up untilidenti�cation of the damaged elements.

Some researchers have employed the �rst deriva-tive of the transition mode to identify damages [37]. As


Figure 18. The �rst derivative of the vertical modelshape based on experimental results.

shown in Figure 18, the derivative of the experimentalvertical mode shape of the healthy and damaged struc-tures (Scenario 1) was employed to prove that the rota-tional mode shape obtained using the proposed methodcould provide more complete and accurate informationabout the damage. In this �gure, the rotational modeshapes intersect at the 5th element, indicating that the5th element is mistakenly identi�ed as the damagedelement. This is the reason why direct methods shouldbe utilized to extract rotational mode shapes instead ofindirect methods such as calculating the derivative ofthe translational mode shapes. Figure 18 illustrates thederivation of the curve �t to the vertical mode shape.

In Figure 19, the damage severities of elementsare shown in di�erent steps. In Scenario 1, after threesteps, respectively 9, 1, and 1 element(s) was/wereidenti�ed as healthy; however, the damage can beobserved for all the elements in Scenario 2. Compared

to the numerical example in the previous section, thepresence of noise in the data could signi�cantly reducethe quality of the obtained modal information, thus de-creasing the number of the identi�ed healthy elements.However, as already mentioned in the comparisonresults of rotational mode shapes, despite the reducedquality of modal information, the location and severityof damages can be obtained with acceptable accuracyfor both scenarios. In order to increase the reliabilityof the results of the identi�ed damage, the averagedamage severities in both scenarios were determined,depicted by dashed line in Figure 19. In this �gure,the damage severities below this line can be consideredas unreliable results. It can be concluded that thismethod failed to determine the initial small damages.

5.2. Accelerometer installation method fordata acquisition

Due to the limited number of accelerometers, eight sen-sors have been used including three and �ve sensors forvertical and rotational data acquisition, respectively.To measure the vertical DOFs, the sensors must beattached to the beam such that no relative accelerationis created between the sensors and the beam. Therotational DOFs were also obtained by recording thehorizontal acceleration at a certain level of the beam.To this end, acceleration sensors were placed on Plex-iglass pieces and then, �rmly connected to the beam.However, these accelerometers record the accelerationin line with their horizontal axis, which is not exactlythe same as the horizontal acceleration above thepieces because the horizontal axis of the accelerometersrotates when the beam vibrates. Given that therotation is quite small, the recorded acceleration can beconsidered almost equal to the horizontal accelerationof the top of pieces. In addition, the weight of thesensors can create an inertial force which can causeerrors in the rotational acceleration. Another issue thatcauses errors in data acquisition with accelerometers

Figure 19. Identi�cation of damage severities on di�erent steps: (a) Scenario 1 and (b) Scenario 2.


Figure 20. First rotational mode shape: (a) Scenario 1 and (b) Scenario 2.

Figure 21. Identi�cation of damage severities in di�erent steps: (a) Scenario 1 and (b) Scenario 2.

is the reduced sensitivity of the sensors over time.Each accelerometer has a speci�c capacity that whenaccelerated more than its capacity, damages wouldoccur. For this reason, the manual calibration methodwas employed because the calibration coe�cients in thesensor catalog were not as sensitive as before. Therotational acceleration of each point of the beam canbe calculated through Eq. (15):

��t = ArcSin�

��xtL

�'�

��xtL

�; (15)

where ��xt denotes the time history of horizontalacceleration axis of the accelerometer and L is theheight of Plexiglass piece.

The �rst rotational mode shapes of the damagedbeam are shown in Figure 20. Here, similar to theresults obtained from the vision-based method, therotational mode shapes of the healthy and damagedbeams passed over each other in the location of the cutelement.

Since only a part of the distance between thesetwo sensors was cut, it would be impossible to deter-mine the exact damage in each element. As a rough

approximation, the moment of inertia for the damagedelement can be calculated by the weighted arithmeticmean of the moment of inertia. This approximation isonly valid until high-severity damage does not occur ata limited length. In case this assumption is accepted,the reduced sti�ness rates in elements 2 and 4 are7.5% and 5%. The estimated damage severities areshown in Figure 21. As shown in this �gure, theextent of the estimated damage is close to that of thecalculated values; hence, the accuracy of the results canbe ensured.

6. Conclusion

In this study, a new approach was proposed in whichthe least amount of information on vibration modeshapes, i.e., only the �rst mode, was required fordamage detection. To this end, rotational mode shapescontaining remarkable information about the locationand severity of the damage were employed. Thisinformation was obtained using simple data-acquisitionmethods such as placing accelerometers and vision-based methods as well as geometrical formulations.


After extracting the �rst mode using the proposedmethod, damage coe�cients were obtained by elimi-nating the healthy elements indirectly. This type ofattitude in obtaining the damage coe�cients wouldform a robust algorithm.

Numerical and experimental studies on a simplysupported beam were carried out. The feasibility andreliability of this method were studied using the �rstmode shape. The results of the numerical exampleshowed that the proposed algorithm could exactly de-tect damage severity without noise, only using the �rstmode information. Furthermore, several simulationswere performed to investigate the performance of theproposed method in the presence of noise. In order tosimulate the e�ect of noise, the mode shape data werecontaminated at di�erent noise levels (0.5%, 1%, 2%,3%, 4%, and 5%). The results demonstrated that thismethod had acceptable performance in determiningdamage severities for the noise levels up to 1%. Experi-mental results demonstrated that damage location andseverity estimation could be successfully detected usingonly the �rst mode of vibration in an acceptable rangeand despite the presence/absence of modal informationat both ends of the cut beam. In summary, the resultsshowed that the rotational Degrees Of Freedom (DOFs)played a key role in detecting the location and severityof damages.

Acknowledgement

The present study is a part of the �rst author's PhDthesis at the International Institute of Earthquake En-gineering and Seismology, whose support is gratefullyappreciated.

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Biographies

Meysam Ramezani is currently a PhD candidate atthe International Institute of Earthquake Engineeringand Seismology, Tehran, Iran. He is an experiencedgraduate research assistant working in the academiccontext. His current research interests include struc-tural engineering, vibration control, performance-baseddesign, structural health monitoring, arti�cial intelli-gence, and dynamic simulation using OpenSees. Hewas a strong researcher professional and held MScdegree in Earthquake Engineering from University ofTehran.

Omid Bahar is an Assistant Professor at InternationalInstitute of Earthquake Engineering and Seismology(IIEES), Tehran, Iran. He received his PhD degree inStructural Engineering from Shiraz University, Shiraz,Iran. He spent a short time as a visiting scholarat Professor Kitagawa's Lab. at Hiroshima Universityduring his PhD research program. He worked onthe performance of active structural control duringstrong ground motions. His main research interestsinclude performance-based design, structural control,and structural health monitoring.

indirect structure damage identi cation with the

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